Conference Proceedings of the Society for Experimental Mechanics Series
For other titles published in this series, go to www.springer.com/series/8922
Tom Proulx Editor
Experimental and Applied Mechanics, Volume 6 Proceedings of the 2010 Annual Conference on Experimental and Applied Mechanics
Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 068011405 USA [emailprotected]
ISSN 21915644 eISSN 21915652 ISBN 9781441994974 eISBN 9781441997920 DOI 10.1007/9781441997920 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928691 © The Society for Experimental Mechanics, Inc. 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acidfree paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Experimental and Applied Mechanics represents the largest of six tracks of technical papers presented at the Society for Experimental Mechanics Annual Conference & Exposition on Experimental and Applied Mechanics, held at Indianapolis, Indiana, June 710, 2010. The full proceedings also include volumes on Dynamic Behavior of Materials, Role of Experimental Mechanics on Emerging Energy Systems and Materials, Application of Imaging Techniques to Mechanics of Materials and Structures, along with the 11th International Symposium on MEMS and Nanotechnology, and the Symposium on Time Dependent Constitutive Behavior and Failure/Fracture Processes. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. The current volume on the Experimental and Applied Mechanics includes studies on NanoEngineering, MicroNano Mechanics, Measurements and Modeling, Applied Photoelasticity Residual Stress Measurement Techniques, Thermal Methods, BioComposites Cell Mechanics, Interfacial Fracture Phenomena, Mechanics and Mechanobiology of Soft Tissues and Hydrogels, Fatigue, Fracture, Structural Dynamics/Acoustics Residual Stress and Fatigue in Fracture, Recent Progress in DIC Methodology Design and Processing of Composites, Inverse Problems, and Characterization of Materials Experimental and Applied Mechanics covers the wide variety of subjects that are related to the broad field of experimental or applied mechanics. It is SEM’s mission to disseminate information on a good selection of subjects. To this end, research and application papers relate to the broad field of experimental mechanics. The organizers would like to thank the authors, presenters, session organizers and session chairs for their participation in this track The Society would like to thank the organizers of the track, Ryszard J. Pryputniewicz, Worcester Polytechnic Institute; John Lambros, University of Illinois at UrbanaChampaign; Hugh A. Bruck, University of Maryland, College Park for their efforts. Bethel, Connecticut
Dr. Thomas Proulx Society for Experimental Mechanics, Inc
Contents
1 High Accuracy Optical Measurements of Surface Topography C.A. Sciammarella, L. Lamberti, F.M. Sciammarella 2 Industrial Finishes of Ceramic Surfaces at the Microlevel and its Influence on Strength F.M. Sciammarella, C.A. Sciammarella, L. Lamberti, B. Burra 3 Elastic Properties of Living Cells M.C. Frassanito, L. Lamberti, P. Pappalettere
1
9 17
4 Standards for Validating Stress Analyses by Integrating Simulation and Experimentation E. Hack, G. Lampeas, J. Mottershead, E.A. Patterson, T. Siebert, M. Whelan
23
5 Avalanche Behavior of Minute Deformation Around Yield Point of Polycrystalline Pure Ti G. Murasawa, T. Morimoto, S. Yoneyama, A. Nishioka, K. Miyata, T. Koda
31
6 The Effect of Noise on Capacitive Measurements of MEMS Geometries J.L. Chee, J.V. Clark 7 Effects of Clearance on Thick, SingleLap Bolted Joints Using Throughthethickness Measuring Techniques J. Woodruff, G. Marannano, G. Restivo 8 Deformation and Performance Measurements of MAV Flapping Wings P. Wu 9 Dynamic Constitutive Behavior of Aluminum Alloys: Experimental & Numerical Studies S. Abotula 10 Estimating Surface Coverage of Gold Nanoparticles Deposited on MEMS N. Ansari, K.M. Hurst, W.R. Ashurst
43
53 63
65 67
viii
11 Functionally Graded Metallic Structure for Bone Replacement S. Bender
69
12 Dissipative Energy as an Indicator of Material Microstructural Evolution N. Connesson, F. Maq uin, F. Pierron
71
13 Temporal Phase Stepping Photoelasticity by Load or Wavelength M.J. Huang, H.L. An
73
14 Determination of the Isoclinic Map for Complex Photoelastic Fringe Patterns P. Siegmann, C. Colombo, F. DíazGarrido, E. Patterson
79
15 A Study on the Behaviors and Stresses of Oring Under Uniform Squeeze Rates and Internal Pressure by Transparent Type Photoelastic Experiment J.S. Hawong, D.C. Shin, J.H. Nam 16 Strength Physics at Nanoscale and Application of Optical Interferometry S. Yoshida 17 Light Generation at the Nano Scale, Key to Interferometry at the Nano Scale C.A. Sciammarella, L. Lamberti, F.M. Sciammarella 18 Mechanical Characterization of Nanowires Using a Customized Atomic Force Microscope E. Celik, I. Guven, E. Madenci
87 95 103
117
19 Blast Performance of Sandwich Composites With Functionally Graded Core N. Gardne, A. Shukla
127
20 Thermal Softening of an UFG Aluminum Alloy at High Rates E.L. Huskins, K.T. Ramesh
129
21 Blast Loading Response of Glass Panels P. Kumar, A. Shukla
131
22 Novel Approach to 3D Imaging Based on Fringe Projection Technique D.A. Nguyen
133
23 Measuring Shear Stress in Microfluidics Using Traction Force Microscopy B. Mueller
135
24 Efficiency Enhancement of Dyesensitized Solar Cel C.H. Chien, M.L. Tsai, T.H. Su, C.C. Hsieh, Y.H. Li, L.C. Chen, H.D. Gau
137
25 Integration of Solar Cell With TNLC Cell for Enhancing Power Characteristics C.Y. Chen, Y.L. Lo
145
ix
26 Fullfield Measuring System for the Surface of Solar Cell S.A. Tsai, Y.L. Lo
151
27 Photographic Diagnosis of Crystalline Silicon Solar Cells by Electroluminescence T. Fuyuki, A. Tani
159
28 Novel Interfacial Adhesion Experiments With Individual Carbon Nanofibers T. Ozkan, I. Chasiotis
163
29 Dynamic Constitutive Behavior of Reinforced Hydrogels Inside Liquid Environment S. Padamati
165
30 Osteon Size Effect on the Dynamic Fracture Toughness of Bone M. Raetz
167
31 Radial Inertia in Noncylindrical Specimens in a Kolsky Bar O. Sen
169
32 Core Deformation of Sandwich Composites Under Blast Loading E. Wang, A. Shukla
171
33 Influence of Frictionstirwelding Parameters on Texture and Mechanical Behavior Z. Yu, H. Choo, W. Zhang, Z. Feng, S. Vogel
173
34 Measurement of Stresses in Pipelines Using Hole Drilling Rosettes G.R. Delgadillo, F. Fiorentini, L.D. Rodrigues, J.L.F. Freire, R.D. Vieira
175
35 Incremental Computation Technique for Residual Stress Calculations Using the Integral Method G.S. Schajer, T.J. Rickert
185
36 Experimental Investigation of Residual Stresses in Water and Air Quenched Aluminum Alloy Castings B. Xiao, Y. Rong, K. Li
193
37 Residual Stress on AISI 300 Sintered Materials C. Casavola, C. Pappalettere, F. Tursi
201
38 Practical Experiences in Hole Drilling Measurements of Residual Stresses P.S. Whitehead
209
39 Destructive Methods for Measuring Residual Stresses: Techniques and Opportunities G.S. Schajer
221
40 The Contour Method Cutting Assumption: Error Minimization and Correction M.B. Prime, A.L. Kastengren
233
x
41 Measurements of Residual Stress in Fracture Mechanics Coupons M.R. Hill, J.E. VanDalen, M.B. Prime
251
42 Analysis of Large Scale Composite Components Using TSA at Low Cyclic Frequencies J.M. DulieuBarton, D.A. Crump
259
43 Determining Stresses Thermoelastically Around Neighboring Holes Whose Associated Stresses Interact A.A. Khaja, R.E. Rowlands
267
44 Crack Tip Stress Fields Under Biaxial Loads Using TSA R.A. Tomlinson
275
45 Extending TSA With a Polar Stress Function to Noncircular Cutouts A.A. Khaja, R.E. Rowlands
279
46 Novel Synthetic Material Mimicking Mechanisms From Natural Nacre A. Juster, F. Latourte, H.D. Espinosa
289
47 Mechanical Characterization of Synthetic Vascular Materials A.R. Hamilton, C. Fourastie, A.C. Karony, S.C. Olugebefola, S.R. White, N.R. Sottos
291
48 MgO Nanoparticles Affect on the Osteoblast Cell Function and Adhesion Strength of Engineered Tissue Constructs M. Khandaker, K. Duggan, M. Perram
295
49 Mechanical Interactions of Mouse Mammary Gland Cells With a Threedimensional Matrix Construct M.d.C. LopezGarcia, D.J. Beebe, W.C. Crone
301
50 Tracking Nanoparticles Optically to Study Their Interaction With Cells J.M. Gineste, P. Macko, E. Patterson, M. Whelan 51 Coherent Gradient Sensing Microscopy: Microinterferometric Technique for Quantitative Cell Detection M. Budyansky, C. Madormo, G. Lykotrafitis
307
311
52 Rate Effects in the Failure Strength of Extraterrestrial Materials J. Kimberley, K.T. Ramesh, O.S. Barnouin
317
53 Determination of Dynamic Tensile Properties for Low Strength Brittle Solids R. Chen, F. Dai, L. Lu, F. Lu, K. Xia
321
54 The Mechanical Response of Aluminum Nitride at Very High Strain Rates G. Hu, K.T. Ramesh, J.W. McCauley
327
xi
55 Dynamic and Quasistatic Measurements of C4 and Primasheet P1000 Explosives G.W. Brown, D.G. Thompson, R. DeLuca, P.J. Rae, C.M. Cady, S.N. Todd 56 Dynamic Characterization of Mock Explosive Material Using Reverse Taylor Impact Experiments L. Ferranti, Jr., F.J. Gagliardi, B.J. Cunningham, K.S. Vandersall
329
337
57 Mechanical Behavior of Hierarchicallystructured Polymer Composites A.L. Gershon, H.A. Bruck
347
58 Composite Design Through Biomimetic Inspirations S.A. Tekalur, M. Raetz, A. Dutta
355
59 Nanocomposite Sensors for Wide Range Measurement of Ligament Strain T. Hyatt, D. Fullwood, R. Bradshaw, A. Bowden, O. Johnson
359
60 Advanced Biologicallyinspired Flapping Wing Structure Development L. Xie, P. Wu, P. Ifju
365
61 Characterization of Electrodeelectrolyte Interface Strengths in SOFCs S. Akanda, M.E. Walter
373
62 Die Separation Strength for Deep Reactive Ion Etched Wafers D.A. Porter, T.A. Berfield
383
63 Temperature Moisture and Mode Mixity Dependent EMC Copper (Oxide) Interfacial Toughness A. Xiao, G. Schlottig, H. Pape, B. Wunderle, K.M.B Jansen, L.J. Ernst
393
64 An Integrated Experimental and Numerical Analysis on Notch and Interface Interaction in Samematerials A. Krishnan, L.R. Xu
405
65 Differentiation of Human Embryonic Stem Cells Encapsulated in Hydrogel Matrix Materials M. Salick, R.A. Boyer, C.H. Koonce, T.J. Kamp, S.P. Palecek, K.S. Masters, W.C. Crone
415
66 Nonlinear Viscoelasticity of Native and Engineered Ligament and Tendon J. Ma, E.M. Arruda
423
67 Spinal Ligaments: Anisotropic Characterization Using Very Small Samples R.J. Bradshaw, A.C. Russell, A.E. Bowden
429
68 InFlight Wingmembrane Strain Measurements on Bats R. Albertani, T. Hubel, S.M. Swartz, K.S. Breuer, J. Evers
437
xii
69 The Mechanical Properties of Musa Textilis Petiole N.S. Liou, S.F. Chen, G.W. Ruan
447
70 Analysis of Strain Energy Behavior Throughout a Fatigue Process O. ScottEmuakpor, T. George, C. Cross, M.H.H. Shen
451
71 Relating Fatigue Strain Accumulation to Microstructure Using Digital Image Correlation J. Carroll, W. Abuzaid, M. Casperson, J. Lambros, H. Sehitoglu, M. Spottswood, R. Chona
459
72 High Cycle Fatigue of Structural Components Using Critical Distance Methods S. Chattopadhyay
463
73 Crack Propagation Analysis of New Galata Bridge K. Ozakgul, O. Caglayan, O. Tezer, E. Uzgider
471
74 Stressdependent Elastic Behaviour of a Titanium Alloy at Elevated Temperatures T.K. Heckel, A.G. Tovar, H.J. Christ
479
75 Numerical and Experimental Modal Analysis Applied to the Membrane of Micro Air Vehicles Pliant Wings U.K. Chakravarty, R. Albertani
487
76 Objective Determination of Acoustic Quality in a Multipurpose Auditorium B. Hayes, C. Braden, R. Averbach, V. Ranatunga
501
77 Identification and Enhancement of Onstage Acoustics in a Multipurpose Auditorium C. Braden, B. Hayes, R. Averbach, V. Ranatunga
517
78 Fractional Calculus of Hydraulic Drag in the Free Falling Process Y. Wan, R.M. French
529
79 Fatigue Cracks In Fibre Metal Laminates in the Presence of Rivets and Cold Expanded Holes D. Backman, E.A. Patterson
541
80 Effect of Nonlinear Parametric Model Accuracy in Crack Prediction and Detection T.A. Doughty, N.S. Higgins
549
81 Optical Based Residual Strain Measurements J. Burnside, W. Ranson, D. Snelling
557
82 Correlating Fatigue Life With Elastic and Plastic Strain Data S.M. Grendahl, D.J. Snoha, B.S. Matlock
561
xiii
83 Monitoring Crack Tip Plastic Zone Size During Fatigue Loading Y. Du, A. Patki, E. Patterson
569
84 Studying Thermomechanical Fatigue of Hastelloy X Using Digital Image Correlation M. Casperson, J. Carroll, W. Abuzaid, J. Lambros, H. Sehitoglu, M. Spottswood, R. Chona
575
85 Understanding Mechanisms of Cyclic Plastic Strain Accumulation Under High Temperature Loading Conditions W. Abuzaid, H. Sehitoglu, J. Lambros, J. Carroll, M. Casperson, R. Chona
579
86 Simulated Corrosionfatigue via Ocean Waves on 2024aluminum E. Okoro, M.N. Cavalli
583
87 The Effect of Processing Conditions on the Properties of Thermoplastic Composites F.P. Cook, S.W. Case
591
88 Calculation of Shells and Plates Constructed From Composite Materials R. Tskvedadze, G. Kipiani, D. Tabatadze
599
89 Reducing Build Variation in Arched Guitar Plates E. Efendy, M. French
607
90 Rotation Angle Measurement Using an Electrooptic Heterodyne Interferometer J.F. Lin, C.J. Weng, K.L. Lee, Y.L. Lo
621
91 Digital Shadow Moiré Measurement of OutofPlane Hygrothermal Displacement of TFTLCD Backlight Modules W.C. Wang, Y.H. Chang
627
92 Theory and Applications of Universal Phaseshifting Algorithm T. Hoang, Z. Wang, D. Nguyen
641
93 Evaluation of Crash Energy Absorption Capacity of a Tearing Tube Y. Ko, K. Ahn, H. Huh, W. Choi, H. Jung, T. Kwon
647
94 Fracture Studies Combining Photoelasticity and Coherent Gradient Sensing for Stress Determination S. Kramer
655
95 Strength and Fracture Behavior of Diffusion Bonded Joints A.H.M.E. Rahman, M.N. Cavalli
677
96 Fabrication and Characterization of Novel Graded Bone Implant Material S. Bender, S.D. El Wakil, V.B. Chalivendra, N. Rahbar, S. Bhowmick
683
xiv
97 A Dynamic Design Model for Teaching T.K. Kundra
689
98 Video Demonstrations to Enhance Learning of Mechanics of Materials Inside and Outside the Classroom M. Dietzler, W.C. Crone
697
99 Experimentation and Productmaking Workshop Simplified for Easy Execution in Classroom T. Nakazawa, M. Matsubara, S. Mita, K. Saitou
703
100 Illustrating Essentials of Experimental Stress Analysis Using a UShaped Beam M.E. Tuttle
711
101 Demonstration of Rodwave Velocity in a Lecture Class D. Goldar
721
102 Influence of Frictionstirwelding Parameters on Texture and Tensile Behavior Z. Yu, H. Choo, W. Zhang, Z. Feng, S. Vogel
725
103 Comparing Two Different Approaches to the Identification of the Plastic Parameters of Metals in Postnecking Regime A. Baldi, A. Medda, F. Bertolino
727
104 Determination of Hardening Behaviour and Contact Friction of Sheet Metal in a Multilayered Upsetting Test S. Coppieters, P. Lava, H. Sol, P. Van Houtte, D. Debruyne
733
105 Measuring the Elastic Modulus of Soft Thin Films on Substrates M.J. Wald, J.M. Considine, K.T. Turner 106 Characterization in Birefringence/Diattenuation of an Optical Fiber in a Fibertype Polarimetry T.T.H. Pham, P.C. Chen, Y.L. Lo 107 Predictive Fault Detection for Missile Defense Mission Equipment and Structures J.S. Yalowitz, R.K. Youree, A. Corder, T.K. Ooi 108 Portable Maintenance Support Tool Enhancing Battle Readiness of MDA Structures and Vehicles T. Niblock, J. O'Day, D. Darr, B.C. Laskowski, H. Baid, A. Mal, T.K. Ooi, A. Corder 109 A Compact System for Measurement of Absorbance of Light A.J. Masi, M. Sesselmann, D.L. Rodrigues 110 Fracture Mechanics Analysis in Frost Breakage of Reservoir Revetment on Cold Regions X. Liu, L. Peng
741
749 757
765 773
781
xv
111 Combined Experimental/Numerical Assessment of Compression After Impact of Sandwich Composite Structures M.W. Czabaj, A.T. Zehnder, B.D. Davidson, A.K. Singh, D.P. Eisenberg 112 Mechanical Behavior of Cocontinuous Polymer Composites L. Wang, J. Lau, N.V. Soane, M.J. Rosario, M.C. Boyce 113 Laboratory Evaluation of a Silicone Foam Sealant for Field Application on Bridge Expansion Joints R.B. Malla, B.J. Swanson, M.T. Shaw
793 801
805
114 Effect of Prior Cold Work on the Mechanical Properties of Weldments M. Acar, S. Gungor, P.J. Bouchard, M.E. Fitzpatrick
817
115 Use of Viscoplastic Models for Prediction of Deformation of Polymer Parts N.G. Ohlson
827
116 Mechanical Characterization of SLM Specimens With Speckle Interferometry and Numerical Optimization C. Barile, C. Casavola, G. Pappalettera, C. Pappalettere 117 Torsion/compression Testing of Grey Cast Iron for a Plasticity Model T.A. Doughty, M. LeBlanc, L. Glascoe, J. Bernier
837 845
118 Nucleation and Propagation of PortevinLe Châtelier Bands in Austenitic Steel With Twinning Induced Plasticity L.G. Hector, Jr., P.D. Zavattieri
855
119 Identification and Enhancement of Onstage Acoustics in a Multipurpose Auditorium C. Braden, B. Hayes, R. Averbach, V. Ranatunga
865
120 Determination of Objective Architectural Acoustic Quality of a Multipurpose Auditorium B. Hayes, C. Braden, R. Averbach, V. Ranatunga
867
121 Notchinterface Experiments to Determine the Crack Initiation Loads A. Krishnan 122 Mechanical Characterization of Alternating Magnetic Field Responsive Hydrogels at Microscale S. Meyer, L. Nickelson, R. Shelby, J. McGuirt, J. Peng, S. Ghosh
869
871
123 Investigation of a Force Hardening Spring System E. Jones, D. Prisco
873
124 Resonance Behavior of Magnetostrictive Sensor in Biological Agent Detection M. Ramasamy, B.C. Prorok
875
xvi
125 Detection of Damage Initiation and Growth of Carbon Nanotube Reinforced Epoxy Composites V.K. Vadlamani
877
126 Nanomechanical Characterization of Polypropylene Fibers Exposed to Ultraviolet and Thermal Degradation N.D. Wanasekara
879
127 Detachment Dynamics of Cancer Cells C.C. Wong, J. Reboud, J. Soon, P. Neuzil, K. Liao
881
128 Compression Testing of Biomimetic Bones With 3D Deformation Measurements H. Yao, W. Tong
883
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
HIGH ACCURACY TOPOGRAPHY
OPTICAL
MEASUREMENTS
OF
SURFACE
C.A. Sciammarella*, L. Lamberti**, F.M. Sciammarella* *
Northern Illinois University, Department of Mechanical Engineering, 590 Garden Road, DeKalb, IL 60115, USA ** Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Viale Japigia 182, Bari, 70126, ITALY Email: [emailprotected], [emailprotected], [emailprotected] Abstract Surface characterization is a very important aspect of industrial manufacturing. All engineering parts are strongly affected in their performance by properties depending on surface topography. For this reason methodologies able to provide a functional representation of surface topography are of paramount importance. Among the many techniques available to get surface topography optical techniques play a fundamental role. A digital moiré contouring technique recently proposed by the authors provides a new approach to the study of surface topography. This paper presents further developments in surface topography analysis particularly with respect to the resolution that can be currently obtained. The validity of the proposed approach is checked by analyzing existing standards for surface roughness determination. Optical results are compared with NIST certified standard specimen. 1. INTRODUCTION Surface characterization is a very important aspect of industrial manufacturing because structural behavior of all engineering parts is strictly related with surface topography. For example, fatigue strength decreases dramatically as surface roughness increases. Another example of the deep relationships between mechanical behavior and surface properties occurs in contact problems where distribution of strains and stresses can be determined at different scales each of which corresponds to a specified level of roughness. Surface topography can be reconstructed by means of mechanical probes or non contact optical techniques. The latter approach is certainly preferable as optical methods provide fullfield information at high vertical and lateral resolution. Ref. [1] illustrated the successful application of a new measurement technique based on the use of non conventional illumination to the analysis of the contact between rough surfaces. The methodology proposed in [1] is based on the emission of coherent light from the same surface that is under analysis through the phenomenon of the generation of plasmons. This paper presents further developments in surface topography analysis particularly with respect to the resolution that can be currently obtained. The validity of the proposed approach is checked by analyzing existing standards for surface roughness determination. Optical results are compared with NIST certified standard specimen. The paper is structured as follows. After the introduction section, Section 2 describes the methodology prescribed by standards for roughness determination. Section 3 outlines the theoretical basis of the measurement technique. The experimental setup is described in Section 4. Results are presented and discussed in Section 5. Finally, the main findings of the study are summarized in Section 6. 2. MEASUREMENTS OF SURFACE ROUGHNESS ACCORDING TO ANSI B46.1 STANDARDS Figure 1a shows the HQC226 precision standard (Holts, Dambury, CT) analyzed in this paper. The specimen geometry is sketched in Figure 1b: the standard is shaped as a saw tooth pattern. The specimen is realized by molding. The nominal distance from peak to peak is 100 Pm. The nominal height of the peak is 6 Pm. Accordingly to the ANSI B46.1 standard [2], the value of roughness Ra is defined as the area of a triangle divided by width (also mean of height, area in blue in Figure 1b). In the NIST procedure, the profile of the standard is measured by a stylus with radius of curvature of 2 Pm. A region of 4 mm is assessed. The stylus follows the precision standard surface and records the waviness of the surface. The point coordinates describing the profile are filtered. Since the nominal wavelength of asperities is 100 Pm, anything larger than that is removed by the filter. The wavelength corresponding to the lowest frequency filter to be used in the analysis of experimental data is defined as the sampling length. Most standards recommend the measurement length be at least 7 times longer than the sampling length. The Nyquist–Shannon’s sampling theorem states that the measurement length should be at least ten times longer than the wavelength of T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_1, © The Society for Experimental Mechanics, Inc. 2011
1
2
interesting features. The assessment length or evaluation length is the length containing the data used for the analysis. One sampling length is usually discarded from each end of measurement length. Figure 1c shows the five locations on the standard where traces were run. The measured values for Ra are also indicated in the figure: the mean value of Ra is 3.03276 microns (i.e. 119.4 microinches).
a)
b)
c) Figure 1. a) HQC226 standard; b) Schematic of the standard; c) Roughness values measured by NIST in the target area
3. DESCRIPTION OF THE CONTOURING MODEL The theoretical foundation of the measurement method was explained in [1]. The experimental set up includes a double interface creating a cavity. This cavity then acts as a passive optical resonator so that resonances can be observed from the emitted light. In order to utilize this information three aspects should be considered: (i) direction of the emerging wave vectors; (ii) changes of the gap depth; (iii) polarization of the light. [1] In the experimental setup of [1] the cavity was represented as a FabryPerot resonator. The illumination system consisted of a beam going through a glass plate and impinging in the surface of the glass plate at an angle with respect to the normal to the plate larger than the limit angle. The system of fringes thus generated is equivalent to the first diffraction order of a sinusoidal grating of pitch [1]:
p
O 2 sin T i
(1)
3
where Ti is the angle that the beam forms with the surface. The illumination process of the surface was modified in this research by introducing an actual grating on top of the surface to be measured. The actual process of fringe formation is very complex because the illumination of the surface is obtained through evanescent fields. In the limited context of this article it is not possible to present all the derivations but results can be summarized as follows. For single illumination, it holds:
h
p 2 n sin T o
(2)
where p is the pitch of the utilized grating, n is the diffraction order observed and To is the angle of illumination with respect to the normal to the surface of the grating. The value of To is computed with the following equation:
To
§1 O· S arc sin ¨¨ ¸¸ 2 ©2 p¹
(3)
That is, the inclination of the beam with respect to the normal to the surface is larger than that of the limit angle of total reflection. Consequently, the illumination of the surface is done through the evanescent field. In the applications presented in this paper the utilized values are: p = 2.5 Pm; O = 0.635 Pm. With the above values, for n=1, it follows:
To
§ 1 0 . 635 · 90 o arcsin ¨ ¸ 2 .5 ¹ ©2
82 . 704
o
(4)
The resulting sensitivity is:
2.5 sin (82.704q)
S
2.520 Pm
(5)
In the case of double illumination, the corresponding equation is:
h
p 2 2 n sin T o
(6)
while sensitivity is:
S
2. 5 2 2 sin (82.704q)
1.260 Pm
(7)
From Eq. (2), we obtain:
h ( x , y)
S
I( x , y ) 2S
(8)
where h is the height of the surface with respect to the reference plane that must be defined: in this case, the glass surface is utilized as the reference plane; S is the sensitivity computed using Eq. (7) with n=1; the factor
I( x , y) is the order n. I( x , y) is obtained by the usual procedure applied in the moiré method when the 2S
carrier is recorded. 4. EXPERIMENTAL SET UP The experimental setup is very similar to that described in [1] when a NIST traceable standard was used to successfully validate the method. Figure 2 shows the experimental setup. Details on the laser source and the
4
viewing system are presented in Table 1. By using double illumination it is possible to increase the sensitivity of the measurements, which was calculated in Section 3 as 1.27 Pm.
Figure 2. View of experimental setup
Table 1. Details of the experimental setup used in profile measurements
Illumination
Laser
Mitutoyo – Measuring Microscope – 176847A Mitutoyo; Magnification: 10X, NA: 0.3; WD: 16 mm; Depthoffield: 8.5 Pm Stocker Yale Lasiris; HeNe (O=660 nm), 50 mW
CCD camera
Bassler A640f; 1624×1236 pixels
Microscope Objective
Imaging
Figure 3a shows the view of 119.5 Pin rough side of the HQC226 sample illuminated with white light. The bright lines correspond to the top of each tooth. Figure 3b shows the same specimen with the 2.5 Pm pitch grating superimposed on it. The field of view represented in the image covered 16 peaks/valleys for a nominal length of 1.5 mm.
a)
b)
Figure 3. (a) View of the HQC226 standard illuminated with white light; b) View of the standard with superimposed grating
5
Figure 4 shows the image of the standard specimen with the superimposed grating obtained by illuminating the specimen itself with coherent light. Besides the grating lines there is a system of fringes resulting from the complex interaction between the diffraction pattern coming from the grating and the wavefront coming from the standard surface. These orders become modulated by the depth of the surface. From the region limited by the rectangular mask marked in red on the image, a square area was extracted and then repixelated to 2048u2048 in order to precisely reconstruct the profile of one tooth.
Figure 4. View of the standard with the superimposed grating when coherent illumination is used
All image patterns recorded were processed with the HoloMoiré Strain AnalyzerTM (HMSA) Version 2.0 [3] fringe analysis software package developed by Sciammarella and his collaborators and supplied by General Stress Optics Inc. (Chicago, IL USA). The HMSA package includes a very detailed library of stateoftheart fringe processing tools based on Fourier analysis (Fast Fourier Transform, filtering, carrier modulation, fringe extension, edge detection and masking operations, removal of discontinuities, etc.). The software has been continuously developed over the years and can deal practically with any kind of interferometric pattern. All experimental results are presented and discussed in the next section. 5. RESULTS AND DISCUSSION By simply counting moiré fringes it is possible to reconstruct the profile of the specimen surface. The moiré pattern results from subtracting the frequency of the reference grating from the modulated frequency. The roughness can then be measured by multiplying the total depth thus obtained by one half of the tooth width. Although very simple, this approach allowed surface roughness to be estimated at a good level of accuracy. For example, the computed value for Ra was 2.99 Pm vs. 3.03276Pm measured by NIST (see Figure 1c). A more detailed analysis of the specimen profile is shown in Figure 5a. The size of the region of interest shown in the figure is 700 Pm. The average measured pitch is 101.24 Pm with a standard deviation of r0.322 Pm. The average measured depth is 6.078 Pm. Consequently, the average value of Ra is 3.039 Pm which is within the range of NIST’s measurements. Figure 5b shows the detailed view of the tooth included in the 2048u2048 repixelated region. It can be seen that the local height of the tooth profile is 6.0645 Pm, practically the same as the nominal height of 6 Pm, while the local length of the tooth is 101.3 Pm that is very close to the nominal length of 100 Pm. The 3D map of the reconstructed surface is shown in Figure 6. The average Ra of the standard is 3.05054 Pm and oscillates between 3.0175 Pm and 3.07848 Pm. By extracting different profiles it was possible to make an estimate of the average surface finish of the standard. The average depth thus determined is 6.035±0.1367 Pm. Therefore, the finish of surface standard can be estimated as 0.1367/0.635, that is about O/5.
6
a)
b) Figure 5. a) Profile of the HQC226 standard reconstructed with the contouring technique; b) Detail of one tooth
Figure 6. 3D MATLAB representation of the reconstructed surface of half of one tooth
6. CONCLUSION This paper presented an advanced optical method for digital moiré contouring. The method was tested on HQC226 precision standards. Experimental results were within the range of measurement of the standard. It appears that the optical method presented here is a very powerful tool that can be used for the determination of surface roughness values for any kind of material and to analyze relationships between surface properties and mechanical behavior (see for example the study on flexural strength of ceramic materials presented in [4]) The fact that this technique can be extended to a traditional far field microscope opens the possibilities to many other kinds of analysis. 7. REFERENCES [1] Sciammarella C.A., Lamberti L., Sciammarella F.M., Demelio G.P., Dicuonzo A. and Boccaccio A. “Application of plasmons to the determination of surface profile and contact strain distribution”. Strain, 2010. In Press. [2] ASME Standard B46.1 (2002) Surface Texture, Surface Roughness, Waviness and Lay, American Society of Mechanical Engineers, New York, NY.
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[3] General Stress Optics Inc. (2008) HoloMoiré Strain Analyzer Software HoloStrain, Version 2.0. General Stress Optics Inc., Chicago, IL (USA), http://www.stressoptics.com [4] Sciammarella F.M., Sciammarella C.A., Lamberti L. and Burra V. “Industrial finishes of ceramic surfaces at the microlevel and its influence on strength”. SEM Annual Conference & Exposition on Experimental & Applied Mechanics, June 710, 2010, Indianapolis, Indiana.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
INDUSTRIAL FINISHES OF CERAMIC SURFACES AT THE MICROLEVEL AND ITS INFLUENCE ON STRENGTH F.M. Sciammarella*, C.A. Sciammarella*, L. Lamberti**, V. Burra* *
Northern Illinois University, Department of Mechanical Engineering, 590 Garden Road, DeKalb, IL 60115, USA ** Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Viale Japigia 182, Bari, 70126, ITALY Email: [emailprotected], [emailprotected], [emailprotected] Abstract The mechanical properties of ceramic materials are influenced by surface finishing procedures. This paper presents a brief introduction to an advanced methodology of digital moiré contouring utilized to get surface information in the microrange that is a generalization of a method developed for metallic surfaces [1]. Five types of surface treatments are considered. Two of the finishes utilize diamond grinding with a rough grit (100) and a smoother grit (800). The third type is laser assisted machining of the ceramic. The fourth type is simply applying the laser without machining. The final type is the as received ceramic resulting from the process of fabrication. A total of 63 specimens, nine of each kind were tested in fourpointbending. The strength of the specimens was statistically analyzed using the Gaussian and the Weibull distributions. The statistical strength values are correlated with the statistical distributions of surface properties obtained using this advanced digital moiré contouring method. The paper illustrates the practical application of this method that was developed to analyze surfaces at micro level and beyond. 1. INTRODUCTION Ceramics more specifically Si3Ni4 are known to have very high coefficient of friction, excellent compression strength and resistance to corrosion especially at elevated temperatures. These qualities make them great candidates for a variety of engine components as well as for bearings [2,3]. There are two major factors however that currently limits the use of ceramics for these types of applications. The first has to do with control of surface defects, due to their brittle nature a reduction or elimination of flaws that can prove critical in the failure of ceramics is required. The second factor has to do with cost. The manufacturing costs to diamond grind can be as much as 70% to 90% of the total component cost for complex components [4]. The wide application of advanced ceramics has been restrained largely because of the difficulty and high cost associated with shaping these hard and brittle materials into products. Laser assisted machining (LAM) of ceramics is proving to be a viable alternative to conventional manufacturing methods in terms of cost. Machining time can be cut down drastically and with proper arrangement other more complex machining operations may be possible. Information on the development of a commercially viable system for LAM of ceramics is presented in [5]. One of the main issues now is ensuring that the surface finish of these ceramics be good enough or better than the conventional diamond grind process. Therefore, in order to carry out measurements in the micro range and beyond one must have a very robust yet accurate non contact method. This paper describes the experimental procedures and measurements made on the ceramic samples that were utilized for the bending tests. With this advanced digital moiré contouring method a spatial resolution on the order of a micron was achieved. This level of accuracy enables the micro/nano contouring of the surfaces which can then be represented by a variety of surface roughness measurements such as Ra. These values are then compared to the fourpointbending tests where it does show an experimental relationship between the surface roughness and its bending strength. 2. ADVANCED DIGITAL MOIRÉ CONTOURING METHOD Most of the information on how this method was developed and why it works were explained in [1]. While this work was described for metallic surfaces we can only assume at this point that there must be similar conditions with the ceramic material that enabled us to obtain the experimental results presented in this paper. In other words we must assume that there is some photonic absorption by the ceramic that enables a similar effect to take place. That is why the focus in this section lies on extracting the necessary information from the experimental measurements of the ceramic test pieces. It is important to remember that the experimental set up should have a double interface creating a cavity. This cavity then acts as a passive optical resonator so that resonances can be observed from the emitted light. For T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_2, © The Society for Experimental Mechanics, Inc. 2011
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this information to be used successfully there are three factors that must be defined. The first relates to the direction of the emerging wave vectors, second is the effect of the changes of the gap depth and the third deals with polarization of the light. Again we can refer to [1] to see the detailed solutions and equations. This approach will provide the necessary equations to reproduce the topography of the ceramic surface at the micron and submicron level. The derivation of these equations given in [1] assimilates the cavity that was represented to a FabryPerot resonator. For this contouring method the illumination system consisted of a beam going through a glass plate and impinging to the surface of the glass plate at an angle with respect to the normal to the plate larger than the limit angle. In this way a system of fringes was generated. This system of fringes was equivalent to the first diffraction order of a sinusoidal grating of pitch [1]:
p
O 2 sin T i
(1)
where Ti is the angle that the beam forms with the surface. In the present case, the illumination process of the surface was modified by introducing an actual grating on top of the surface to be measured. The actual process of fringe formation is very complex because the illumination of the surface is obtained through evanescent fields. In the limited context of this paper it is not possible to present all the derivations but the result is the following for single illumination:
h
p 2 n sin T o
(2)
where p is the pitch of the utilized grating, n is the diffraction order observed and To is the angle of illumination with respect to the normal to the surface of the grating. The value of To is computed with the following equation:
To
§1 O· S arc sin ¨¨ ¸¸ 2 ©2 p¹
(3)
That is, the inclination of the beam with respect to the normal to the surface is larger than that of the limit angle of total reflection. Consequently, the illumination of the surface is done through the evanescent field. In the applications presented in this paper the utilized values are: p 2.5 Pm , O .653 Pm . With the above values, for n=1, it follows:
To
§ 1 0 . 635 · 90 o arcsin ¨ ¸ 2 .5 ¹ ©2
82 . 704
o
(4)
The resulting sensitivity is:
2.5 sin (82.704q)
S
2.520 Pm
(5)
In the case of double illumination, the corresponding equation is:
h
p 2 2 n sin T o
(6)
while sensitivity is:
S
2. 5 2 2 sin (82.704q)
1.260 Pm
(7)
From Eq. (2), we obtain:
h ( x , y)
S
I( x , y ) 2S
(8)
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where h is the height of the surface with respect to the reference plane that must be defined: in this case, the glass surface is utilized as the reference plane; S is the sensitivity computed using Eq. (7) with n=1; the factor
I( x , y) is the order n. I( x , y) is obtained by the usual procedure applied in the moiré method when the 2S
carrier is recorded. 3. EXPERIMENTAL SET UP There were two different providers of ceramic rods (length = 6”, diameter =1”) with a total of five different surface conditions. For ceramic rod K the treatments were: As Received, Diamond Grind (100 grit), and LAMC. For ceramic rod B the treatments were: As Received, Diamond Grind (800 grit), Laser Glazed, and LAMC. Once the samples were made each rod was sectioned into three arc segments with a smaller size (19 mm thick, 19 mm wide, and 50 mm long (Figure 1a). These arc segment specimens were tested under flexure (Figure 1b) with the machined surface in tension at loads less than 4000 N (880 pounds); this geometry gave three test bars from each 25mm diameter silicon nitride rod. This arc segment specimen geometry was used for other ceramics [6]. The complete testing procedure (specimens, fixturing, calculations) is described in detail by Quinn [7].
(a)
(b)
Figure 1. (a) Schematic of arc samples used for 4pointbending tests; (b) view of experimental setup.
Prior to having samples subject to the fourpointbending test, samples were analyzed using the advanced digital moiré contouring method with the set up shown in Figure 2a. This set up is very similar to that described in [1] when a NIST traceable standard was used to successfully validate the method. For these measurements a special fixture (Figure 2b) was designed to hold these sliced specimens along with the grating, ensuring proper contact. It was important to maintain the total internal reflection at the interface of the grating and the ceramic; for this purpose an angle of illumination computed with equation (4) was utilized. By using double illumination it is possible to increase the sensitivity of the measurements, which was calculated in Section 2 as 1.27 microns.
(a)
(b)
Figure 2. (a) View of entire set up for advanced digital moiré contouring method; (b) close up view of fixture.
For the first set of data taken from the K samples the field of view was 450u450 microns. There was one image that was recorded per sample with a total of 9 samples per condition: there were provided 27 separate images to be analyzed. For the second set of data taken from the B samples the field of view was 325u325
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microns. This time three images were recorded per sample giving a total of 27 images to analyze per condition. Two procedures were utilized depending on the actual roughness of the surface. If the roughness of the surface is larger than the sensitivity of the set up the standard processing with fringe unwrapping is applied. If the roughness is less than the grating pitch the two images taken (left and right) are subtracted from each other. Since there is less then a fringe between the two patterns, we are dealing with fractional orders. Consequently, there is no need to perform any fringe unwrapping and the results obtained from the subtraction give us the surface profile of the ceramic directly. All image patterns recorded were processed with the HoloMoiré Strain AnalyzerTM (HMSA) Version 2.0 [8] fringe analysis software package developed by Sciammarella and his collaborators and supplied by General Stress Optics Inc. (Chicago, IL USA). The HMSA package includes a very detailed library of stateoftheart fringe processing tools based on Fourier analysis (Fast Fourier Transform, filtering, carrier modulation, fringe extension, edge detection and masking operations, removal of discontinuities, etc.). The software has been continuously developed over the years and can deal practically with any kind of interferometric pattern. Figure 3a shows the ceramic surface for the diamond grind condition (800 grit). Figure 3b shows the final result TM obtained after subtraction and processing using HMSA software. All the results are provided in the next section.
(a)
(b)
Figure 3. (a) View of diamond grind surface from microscope; (b) view of final result after processing.
4. EXPERIMENTAL RESULTS There were two types of measurements performed on the ceramic samples: 1) Mechanical and 2) Optical. Prior to the mechanical testing, the samples were placed into the advanced digital moiré contouring system to obtain full field images of the ceramic surface so that surface roughness (Ra) measurement could be made. After the images were taken, the samples were put into a fourpointbending test to determine the bend strength. This section provides the results obtained from testing. 4.1 Mechanical Testing Tables I & II show the results obtained from the fourpointbending test. Specimens were tested at room temperature in an Instron testing machine, using an articulated fixture (40 mm20 mm spans) with rolling tool steel bearings (Figure 1b). The cross head rate was 0.125 mm/min. Each specimen was tested with the curved face in tension (down). Table I. Experimental results from fourpointbending tests for ceramic K
Diamond Grind Bar ID Strength (MPa) 1A 433.2 1B 419.8 1C 464.2 2A 408.3 2B 452.6 2C 496.2 3A 530.9 3B 505.3 3C 510.9
LaserAssisted Machine Bar ID Strength (MPa) 4A 447.2 4B 300.2 4C 422.6 5A 622.6 5B 606.6 5C 615.6 6A 600.3 6B 609.1 6C 540.9
Bar ID 7A 7B 7C 8A 8B 8C 9A 9B 9C
AsReceived Strength (MPa) 475.7 604.0 544.2 588.3 552.9 540.8 528.1 468.6 414.8
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Table II. Experimental results from fourpointbending tests for ceramic B
AsReceived Bar # Stress (MPa) 1B1 410 1B2 401 1B3 388 2B1 476 2B2 445 2B3 499 3B1 429 3B2 441 3B3 437
800 Grit Dia. Grind Bar Stress (MPa) 1A1 456 1A2 434 1A3 625 2A1 559 2A2 502 2A3 457 3A1 474 3A2 544 3A3 430
Laser Glaze Bar # Stress(MPa) 15B1 524 15B2 541 15B3 494 15A1 570 15A2 511 15A3 488 16A1 580 16A2 459 16A3 588
LAM Turned Bar # Stress (MPa) 18A1 624 18A2 618 18A3 523 16B1 543 16B2 469 16B3 571 18B1 605 18B2 561 18B3 531
Because ceramics fail in a brittle fashion statistical analysis for determining bend strength must be implemented. In this particular case, a two parameter Weibull Analysis was performed on bend stress results. Despite the low number of measurements this was done to give some statistical meaning to the experimental data. The scale parameter determines the most probable location where (in this case) the bend strength would be and the shape parameter (m) indicates the distribution. A high m value means that there is almost no spread in the data and that most values should fall within the most probable range (95%), a low m value means there is a lot of spread in the data. In addition to obtaining the 95% confidence value, the upper and lower bounds are also determined. All data were obtained directly through MATLAB which include canned functions to perform these calculations. Tables III & IV provide the results obtained using the Weibull Analysis from MATLAB. Table III. Results from Weibull Analysis of Bend Strength for ceramic K
LAM 4 Diamond (100) As Received LAM 5&6
Flexural Strength (MPa) Lower Bound 95% 360.69 416.35 462.91 487.99 516.50 549.00 596.79 609.77
m values Upper Bound 480.60 514.42 583.54 623.04
Lower Bound 3.01 7.78 6.66 18.81
95% 8.24 13.09 11.27 38.37
Upper Bound 22.55 22.02 19.08 78.26
Table IV. Results from Weibull Analysis of Bend Strength for ceramic B
As Received Diamond (800) LASER Only LAM
Flexural Strength (MPa) Lower Bound 95% 429.27 452.10 482.55 526.53 522.05 547.88 554.10 581.97
m values Upper Bound 476.14 574.51 574.99 611.25
Lower Bound 4.94 8.26 8.52 8.33
95% 7.97 13.39 14.31 14.05
Upper Bound 12.85 21.72 24.01 23.67
4.2 Image Analysis The images recorded using the HMSATM software provided the full field view of the ceramic surface. Figure 4a shows the 3D view of the diamond grind surface (100 grit). Figure 4b shows the 3D view of the LAM surface.
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(a)
(b)
Figure 4. (a) 3D view of diamond grind (100 grit) surface (b) 3D view of LAM surface.
A dedicated MATLAB program was created to take the output file from the HMSATM software which contains (1024 u 1024) the actual surface information to calculate the Ra value. Then for each sample this value was stored and then placed into the Weibull Analysis program that was used for the bend strength. Table V & VI shows the data in terms of Ra (microns) and its corresponding m value. Table V. Results from Weibull Analysis of Ra for ceramic K
LAM 4 Diamond (100) As Received LAM 5&6
Ra (microns) Lower Bound 95% 0.951 1.376 1.156 1.336 0.981 1.260 0.856 0.956
m values Upper Bound 1.991 1.544 1.619 1.067
Lower Bound 1.11 1.93 1.15 2.88
95% 1.63 2.44 1.47 4.00
Upper Bound 2.40 3.08 1.88 5.57
Table VI. Results from Weibull Analysis of Ra for ceramic B
As Received Diamond (800) LASER Only LAM
Ra (microns) Lower Bound 95% 0.968 1.088 1.170 1.332 0.966 1.063 0.846 0.912
m values Upper Bound 1.223 1.517 1.171 0.983
Lower Bound 5.61 4.29 5.22 6.35
95% 7.55 5.37 6.53 8.23
Upper Bound 10.15 6.71 8.16 10.65
5. DISCUSSION AND CONCLUSIONS Looking through Tables IIIVI it is clear that the LAM process provides the best results in terms of bending strength and surface roughness. For each table the LAM holds among the highest m value meaning that it creates a high confidence in terms of repeatability. One other interesting outcome from the first round was that sample #4 from the LAM trial experienced much lower bend strength values than #s 5 and 6. At first it was not clear why but after analyzing the Ra we can see that #4 values are much higher than those of #s 5 and 6. In this first run the LAM parameters were not controlled and therefore it is possible that some other factor may have influenced the higher Ra value (i.e. cutting tool, force of tool, etc.). During the second trials all possible factors were controlled and observed. This is evidenced by the data from Table II being more consistent. Figure 5 is a plot of all the data obtained from the Weibull Analysis that shows Bend Strength (MPa) vs. Ra (microns). We can clearly see that in fact there is a correlation between the values measured for bend strength and those measured for surface roughness. Further investigation will be carried out to see if a determination can be made into why this relationship exists.
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Figure 5. Relationship between roughness and bend strength for different specimens
The correlation is an experimental fact that needs to be explained. The correlation indicates that as the Ra values increase the strength decreases. It can also be seen that variations of Ra of few hundreds nanometers have a sizable effect on the strength. Figure 5 shows that from Ra=900 nm to Ra=1400 a considerable change of the strength occurs. Further investigations will be carried out to see the factors that can explain the reason for the observed correlation. It appears that the optical method presented here is a very powerful tool that can be used for the determination of surface roughness values for any kind of material. The fact that this technique can be extended to a traditional far field microscope opens the possibilities to many other kinds of analysis. 6. ACKNOWLEDGEMENTS This work would not have been possible without the financial support of NIU’s College of Engineering and Engineering Technology R&D outreach program – Rapid Optimization for Commercial Knowledge (ROCK) Director Dr. Richard Johnson, Assistant Director Alan Swiglo, Senior Project Manager Dr. Joe Santner. The authors would also like to the folks at Reliance Tool & Manufacturing, Dick Roberts, Jeff Staes, and Ricardo Deleon for all of their expertise in Manufacturing. Recognition must also go to Dr. Stephen Gonczy of Gateway Materials Technology Inc. for his support and expertise in ceramics. Finally thanks to Mike Matusky current graduate student in the ME Dept. at NIU for all of his work in the LAM project. 7. REFERENCES [1] Sciammarella C.A., Lamberti L., Sciammarella F.M., Demelio G.P., Dicuonzo A. and Boccaccio A. “Application of plasmons to the determination of surface profile and contact strain distribution”. Strain, 2010. In Press. [2] Takebayashi H., Johns T.M, Rukkaku K., and Tanimoto K., "Performance of Ceramic Bearings in HighSpeed Turbine Application" Society of Automotive Engineers, Paper No. 901629, Milwaukee, WI (1990). [3] Chudecki J.F., "Ceramic Bearirgs  Applications and Performance Advantages in Industrial Applications," SAE Technical Paper Series 891904 (1989) [4] Marinescu I.D., Handbook of Advanced Ceramic Machining, CRC Press, (2007). [5] Sciammarella F.M., Santner J., Staes J., Roberts R., Pfefferkorn F. and Gonczy S., “Production Focused
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Laser Assisted Machining of Silicon Nitride”, Proceedings of the 34th International Conference on Advanced Ceramics & Composites (ICACC), January 2010. [6] Swab J.J., Wereszczak A.A., Tice J., Caspe R., Kraft R.H. and Adams J.W., “Mechanical and Thermal Properties of Advanced Ceramics for Gun Barrel Applications”. Army Research Laboratory Report ARLTR3417, February (2005). [7] Quinn G.D., “The Segmented Cylinder Flexure Strength Test”, Ceramic Eng. and Sci. Proc., 27 [3], 295– 305 (2006). [8] General Stress Optics Inc. (2008) HoloMoiré Strain Analyzer Software HoloStrain, Version 2.0. General Stress Optics Inc., Chicago, IL (USA), http://www.stressoptics.com
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
ELASTIC PROPERTIES OF LIVING CELLS M.C. Frassanito, L. Lamberti and C. Pappalettere Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale Viale Japigia 182, Bari, 70126, ITALY Email: [emailprotected], [emailprotected], [emailprotected] Abstract Constitutive behavior of living cells is in deep relationship with their biological properties. For that reason gathering the most detailed information on cell mechanical behavior as it is feasible becomes extremely useful in assessing therapeutic protocols. However, investigation of elastic properties of living cells is a fairly complicated process that requires the use of advanced sensing devices. For example, Atomic Force Microscopy (AFM) is a research tool largely used by biomedical engineers and biophysicists for studying cell mechanics. Experimental data can be given in input to finite element models to predict cell behavior and to simulate the tissue response to biophysical, chemical or pharmacological stimuli of different nature. The paper analyzes different aspects involved in the numerical simulation of AFM measurements on living cells focusing in particular on the effect of constitutive behavior. 1. INTRODUCTION Nanoindentation is a widely used technique to measure the mechanical properties of films with thickness ranging from nanometers to micrometers. In case of biological samples, a convenient tool to probe living cells at the nanometer scale is the Atomic Force Microscope (AFM) because it enables measurements of samples in a physiologic aqueous cell culture environment [15]. The AFM was designed primarily to provide high resolution images of nonconductive samples and, among the plethora of application subsequently developed, the instrument can also be operated as a nanoindenter to gather information about the mechanical properties of the sample. AFM is based on the following principle: a nanometer sized sharp tip placed at the free end of a cantilever is put into contact with the surface of the sample [6]. The building block of the instrument is shown in Figure 1.
Figure 1. Schematic diagram of Atomic Force Microscopy (AFM). Deflection of the AFM cantilever probe is sensed from the reflection of a laser onto a fourquadrant photodetector, and the position of the probe is controlled by a piezoelectric ceramic actuator (PZT).
When the tip scans, indents or otherwise interacts with the sample, it determines a deflection of the microscopic sized cantilever probe which is tracked by a laser based optical method. The cantilever is rectangular or “V”shaped, 100 to 300 micron long and about half a micron thick, microfabricated of silicon or siliconnitride. The tip actually comes in contact with the sample while the cantilever act as a soft spring to measure the contact force. The spring constant K of the cantilever is determined by its physical and geometric properties and the value of k typically ranges from 0.01N/m to 1N/m for biomechanics application. Then, deflection h of the cantilever is converted into a contact force with the standard equation of the spring: T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_3, © The Society for Experimental Mechanics, Inc. 2011
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F=Kh. The tip dimension determines the spatial resolution of the instrument: sharpened pyramids, etched silicon cones, carbon nanotubes and other high aspect ratio tip have been developed for scanning samples with ultra high resolution. [6] When AFM is used as a nanoindenter, elasticity measurements are performed by pushing a tip onto the sample of interest and monitoring force versus distance curves. This results in a deformation which is the sum of the deformation of the tip and the deformation of the investigated sample (indentation) under the tip. The relationship between force and indentation depends upon the tip geometry and the mechanical properties of the specimen. In indentation experiments, usage of ultrasharp tip is avoided because such tips have been shown to penetrate the cell membrane and cause damage to living cells. In this case pyramid shaped or conical tip are preferred. In order to determine reliably the mechanical properties of the samples with AFM technology, it is important to identify accurately the influence on the indentation problem of the methodological issues related with the shape and the size of the probe tip. In particular, when indenting a soft material, the contact area of the probe increases with indentation. It follows that the resulting force depth relationship is non linear and it is difficult to distinguish the contribution of the intrinsic properties of the sample and of the tip geometry. In literature, classical infinitesimal strain theory is mostly applied to extract a Young’s modulus for the material as nonlinearity of the indentation response is usually attributed solely to the tip geometry. This approach is appealing due to the rather simple form of the theoretical equation. However, when studying AFM indentation of biological samples, the application of the Hertz theory [7,8] is questionable [4]. In fact, the key assumptions of the Hertz theory are that the sample is a hom*ogeneous, isotropic, linear elastic half space subject to infinitesimally small strains while most biological materials are heterogeneous, anisotropic and exhibit non linear constitutive behavior. Besides this, AFM indentation of soft material is typically 20500nm which cannot be considered infinitesimal compared to the thickness of the sample (often < 10ȝm) or to the size of the indenter tip (~1050nm radius of curvature). Analyses of AFM indentation based on infinitesimal strain theory may be inappropriate and a much better understanding of contact mechanics between the AFM tip and the soft biological material is obtained by finite element modeling. The aim of this study is to analyze AFM indentation data in case of soft specimens. Results of finite element analyses carried out on a 5Pm thick membrane indented by a silicon nitride sharp tip are presented in the paper. The indentation process is simulated for two different constitutive behaviors of the membrane: linear elasticity and hyperelasticity. 2. FINITE ELEMENT ANALYSIS Finite element modelling and analysis were carried out with the ANSYS® Version 11.0 general purpose FEM software developed by ANSYS Inc., Canonsburg (PA), USA. The FE model of the biological membrane is shown in Figure 2. The mesh of the conical tip included 3040 PLANE42 4node elements and 3300 nodes. The membrane was modelled as a linearly elastic material or a twoparameter Mooney Rivlin hyperelastic material: in the former case the mesh included 7500 PLANE42 4node elements and 7500 nodes while in the latter case the mesh included 7500 PLANE182 4node elements (the PLANE182 element supports hyperelasticity) and the same number of nodes. The surface of the membrane is covered by a layer including 1001 target elements TARGE169 while the indenter tip is covered by a layer including 509 contact elements CONTACT175.
Figure 2. Finite element model of the nanoindentation experiment
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The tip considered in the study was shaped as a blunt cone. This geometry is commonly used in AFM experiments on biological samples. Figure 3 summarizes the main geometric parameters of the tip.
Figure 3. Schematic of the geometry of the blunt conical tip
Two different constitutive models were hypothesized for the biological membrane: (i) linearly elastic; (ii) hyperelastic, following the twoparameter MooneyRivlin (MR) law [911]. The former model is described by the Hooke’s law V = EH. The strain energy density function for the MooneyRivlin model is:
W
a 10 I1 3 a 01 I 2 3
(1)
where a10 and a01 are the MooneyRivlin constants given in input to ANSYS as material properties. Strain invariants are defined, respectively, as I1 =tr[C] and I 2 ={tr2[C] tr2[C]2} where [C] is the CauchyGreen strain tensor. In most of the AFM studies presented in literature, the indentation problem is analyzed on the basis of infinitesimal strain theory. In order to assess the accuracy in the estimation of the material properties when applying the so called “Hertz model” [7,8] to extract the Young’s modulus of the material, finite element results were compared with the theoretical predictions provided by the Hertzian model. When a rigid axisymmetric probe indents an hom*ogeneous, semiinfinite elastic material, infinitesimal theory predicts the following relationship between force F and indentation depth G:
F
2 S E f ( G)
(2)
The generalized elastic modulus E is equivalent to:
E for linear elastic material; 2(1  Q 2 ) 4(a 10 a 01 ) for an hyperlastic material;
and f(G) is a function depending on the indenter geometry which determines the relationship between the depth and the indentation response. For a blunt cone indenter with tip angle D, contact radius a and transition radius b into spherical tip of radius R (see nomenclature in Figure 3), the f(G) function can be expressed as [12]:
f ( G)
2 2 °ª a2 § S § b ·· a a 2 b2 ¨¨ arcsin¨ ¸ ¸¸ ®«aG S °¯¬ 2tg (D) © 2 © a ¹ ¹ 3R
1/ 2
§ b a 2 b2 ¨¨ 3R © 2tg (D)
·¸º½°¾ ¸» ¹¼ °¿
(3)
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In the finite element model developed in this paper the nanoindentation process was simulated by increasing progressively the force F applied to the indenter. This is actually the axial component of the reaction force exercised on the indenter tip. The axial displacement of the indenter computed by ANSYS as the tip surface elements come into contact with the membrane target layer is the indentation depth G. We were interested in the relationship between F and G as those are the output parameters measured by AFM. In order to reproduce experimental conditions usually encountered in AFM measurements on biological samples, membrane displacements were not constrained in the direction orthogonal to the tip movement. The augmented Lagrangian model available as default in ANSYS was utilized in the computation. Finite element analyses accounted for the large deformations experienced by the hyperelastic membrane: the geometric nonlinearity option was activated by switching on the NLGEOM command in ANSYS. Convergence analysis was carried out in order to obtain mesh independent solutions. All finite element analyses were run on a standard personal computer. 4. RESULTS AND DISCUSSION The following parameters were given in input to the FE program:  tip angle D equal to 20°;  blunt cone tip radius of curvature equal to 10nm;  membrane thickness equal to 5Pm;  Young’s modulus of the tip (silicon nitride) equal to 150 GPa;  Young’s modulus of the membrane modeled as linearly elastic equal to 50 kPa;  MooneyRivlin constants a10 and a01 of the membrane modeled as hyperelastic equal to 10 kPa and 2.5 kPa, respectively: these value yield the same equivalent Young’s modulus of 50 KPa. Figure 4 shows the FG curve predicted by ANSYS for the indentation of a blunt conical tip onto a linearly elastic membrane. As expected in view of the Hertzian model depicted by Eq. (4), numerical values were fitted very precisely by a linear regression.
Figure 4. Forceindentation curve (blue dots) computed by ANSYS for a blunt cone tip and linearly elastic membrane compared with the Hertzian theoretical model (dashed red line)
From the slope of the linear fitting, the value of the equivalent Young’s modulus can be determined by combining equations (2) and (4). One can obtain the value 69.9 kPa which is fairly close to the value of 50 kPa given in input to the model. The obtained results confirm that in the case of linear elasticity and in presence of very small indentations the Hertzian model describes well the contact problem.
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In the case of hyperlastic membrane the FG curve was again well fitted by a linear regression but the slope resulted much smaller than in the previous case of linear elasticity. The Young’s modulus derived from data fitting is 0.835 kPa, about 85 times lower than the value included in the FE model. This indicates that the Hertzian model should not be applied to hyperelastic membranes to derive reliably the value of equivalent Young’s modulus and confirms that the indentation response results from the combination of geometric and material nonlinearities. 5. CONCLUDING REMARKS This paper analyzed the suitability of Hertzian model in nanoindentation of soft materials. It was found that the classical theory does not work well. However, a deeper analysis should be carried out in order to understand better this complicated phenomenon. The main limitation of this study is the fact that the stiffness value was derived from a closed form relationship. The best approach to material characterization is to formulate the identification problem in fashion of an optimization problem where the unknown material properties are included as design variables. The cost function to be minimized is the error functional : ҏdefined by summing over the differences between displacements measured experimentally and those predicted numerically by a finite element model. Each time the optimizer perturbs material properties, the finite element model is updated and a new analysis is executed until the process converges. The identification problem for a material with NMP unknown properties can be stated as follows:
ª ° « 1 °Min «:(X1 , X 2 ,..., X NMP ) N CNT ° « ° ¬ °G (X , X ,..., X NMP ) t 0 ° p 1 2 °° ® L U °X 1 d X 1 d X 1 ° L U °X 2 d X 2 d X 2 °... ° °X NMP 1 L d X NMP 1 d X NMP 1 U ° °¯X NMP L d X NMP d X NMP U
NCNT
¦ j 1
2 º §G j Gj · » ¨ FEM ¸ ¨ u j ¸ » FEM © ¹ » ¼
(4)
where the design vector X (X1,X2,…,XNMP) includes the NMP unknown material parameters to be determined in the identification process. The constraint functions Gp(X) may be introduced in the optimization in order to ensure that the hypothesized constitutive behavior is physically reliable and may satisfy constraints on numerical stability especially in case of high nonlinearity. In the above equation, G and G denote, respectively, the values of indentation at the jth loading step predicted by the finite element model and their counterpart measured experimentally. Experimental values are taken as target in the identification process since AFM measurements do not require values of material properties to be known a priori. Conversely, correct values for material properties must be given in input to the finite element program in order to calculate the value of indentation at a given load. The above mentioned approach is currently being utilized in order to reproduce experimental tests involving AFM measurements carried out on biological specimens having the same material properties considered in this study. j
j
REFERENCES [1] Costa K.D. “Single cell elastography: probing for disease with the atomic force microscope” Disease Markers, 19, 139154, 2004. [2] Suresh S. “Biomechanics and biophysics of cancer cells”, Acta Biomaterialia, 3, 413438, 2007. [3] Vinckier A., sem*nza G. “Measuring elasticity of biological materials by atomic force microscopy”, FEBS Letters, 430, 1216, 2008.
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[4] Costa K.D., Sim A. J.F., Yin C.P. “NonHertzian approach to analyzing mechanical properties of endothelial cells probed by atomic force microscopy”, Journal of Biomechanical Engineering, 128, 176184, 2006. [5] Charles R., Sekatski S., Dietler G., Catsicas S., Lafont F., Kasas S. “Stiffness tomography by atomic force microscopy”, Biophysical Journal, 97, 674677, 2009. [6] Bhushan B. “Handbook of Nanotechnology”, Springer, 2007. [7] Hertz H. “On the contact of elastic solids”, J. Reine Angew. Mathematik, 92, 156171, 1881. [8] Johnson K.L. “Contact Mechanics”, Cambridge University Press, New York, 1985. [9] Mooney M. A theory of large elastic deformation. Journal of Applied Physics, 11, 582592, 1940. [10] Rivlin R.S. “Large elastic deformations of isotropic materials I. Fundamental concepts”. Philosophical Transactions of the Royal Society of London, A240, 459490, 1948. [11] Rivlin R.S., “Large elastic deformations of isotropic materials IV. Further developments of the general theory”. Philosophical Transactions of the Royal Society of London, A241, 379397, 1948. [12] Briscoe B.J., Sebastian K.S., Adams M.J., “The effect of indenter geometry on the elastic response to indentation”. Journal of Physics D: Applied Physics, 27, 11561162, 1994.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Standards for Validating Stress Analyses by Integrating Simulation and Experimentation Erwin Hack1, George Lampeas2, John Mottershead3, Eann Patterson4, Thorsten Siebert5 and Maurice Whelan6 1
Laboratory of Electronics/Metrology/Reliability, EMPA, Duebendorf, Switzerland Department of Mechanical Engineering and Aeronautics, University of Patras, Greece 3 Department of Mechanical Engineering, University of Liverpool, UK 4 Composite Vehicle Research Center, Michigan State University, East Lansing, MI 48824, USA. [emailprotected] 5 Dantec Dynamics GmbH, Ulm, Germany 6 Institute for Health and Consumer Protection, European Commission DG Joint Research Centre, Italy. 2
ABSTRACT A reference material and a series of standardized tests have already been developed for respectively calibrating and evaluating optical systems employed for measuring inplane static strain (for draft standard see: www.twa26.org). New work has commenced on the design of a reference material (RM) for use with instruments or systems capable of measuring threedimensional displacements and strains during dynamic events. The rational decisionmaking process is being utilized and the initial stages have been completed, i.e. the identification and weighting of attributes for the design, brainstorming candidate designs and evaluation of candidate designs against the attributes. Twentyfive attributes have been identified and seven selected as being essential in any successful design, namely: the boundary conditions must be reproducible; a range of inplane and outofplane displacement values must be present inside the field of view; the RM must be robust and portable; there is a means of verifying the performance in situ; and for cyclic loading it must be possible to extract data throughout the cycle. More than thirty candidate designs were generated and have been reduced to nine viable designs for further evaluation. In parallel with this effort to design a reference material, work is also in progress to optimize methodologies for conducting analyses via both simulations and experiments. Image decomposition methods are being explored as a means to making quantitative comparisons fullfield data maps from simulations and experiments in order to provide a comprehensive validation procedure. 1. INTRODUCTION 1.1 Engineering context The Olympic motto is "Citius, Altius, Fortius" which translates as "Faster, Higher, Stronger"; for the modern design engineer this could be modified only slightly to “Faster, Lighter, Stronger”. In an era where global competition dominates industry, everyone wants everything to be faster so that you can gain edge on your competitors and generally speaking it is easier to achieve swiftness with lighter designs while the dangers inherent in great speed imply the need for greater strength. Lighter and stronger also provide advantages in terms of environmental footprint since lighter usually implies more energy efficient both in terms of service and manufacturing resources, and stronger offers the potential for a longer life cycle. In the quest for “Faster, Lighter, Stronger” structural analysis plays a crucial part in optimizing the performance of devices, machines and vehicles of all types and ensuring that safety requirements are achieved. In most design processes, computational modeling is the dominant form of structural analysis and so the reliability of the model becomes a critical issue for engineers involved in making design decisions. Schwer [1] has described in outline a ‘Guide for verification and validation in T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_4, © The Society for Experimental Mechanics, Inc. 2011
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computational solid mechanics’ [2]. In this context verification is defined as being two processes: first, identifying and eliminating errors of logic and programming from the code used for modeling; and second, quantifying the errors arising from the code as a consequence of discretizations required for the modeling. So verification can be largely performed without reference to the realworld whereas validation is concerned with establishing how well the model represents the realworld, at least in the context of the anticipated use of the model for solid mechanics or more particularly, design. It is recommended that validation should be achieved by reference to experiments conducted specifically for this purpose but the guide provides no insight or guidelines for the conduct of such experiments. This is not surprising since the computational mechanics community was responsible for the guide’s preparation; however, the experimental mechanics community has not been idle in this regard and made a first step with a draft proposed standard for the calibration and evaluation of optical system for inplane, static strain measurement in 2007 [3]. An outline description of this proposed standard is provided below and then work in progress to extend it to include threedimensional measurements in dynamic loading cases is reported. 1.2 Calibration and evaluation of measurement systems for inplane, static strain Technical Working Area 26: Fullfield Optical Stress and Strain Measurement [4] of VAMAS [5] was formed in 1999 with the aim of bringing together those concerned with the use of optical techniques for full field measurements of stress and strain in order to develop internationally accepted standards. In 2002 a consortium of European organisations embracing universities, research laboratories, instrument manufacturers, end users and national laboratories was formed to pursue SPOTS (Standardisation Project for Optical Techniques for Strain measurement) [6]. SPOTS was an EC shared cost RTD contract (no. G6RDCT200200856 (SPOTS)) which lasted for three years and in 2006 issued a proposed standard for calibration and assessment of optical strain measurement systems [4, 6]. This document has been endorsed by VAMAS TWA26 following independent international review and is currently awaiting recommendation by VAMAS to ISO. The SPOTS standard relates to optical systems designed for making measurements of static strain over a field of view which can be approximated to a plane. A reference material, shown in figure 1, and accompanying experimental protocol are provided for the calibration of such instruments at any scale. Indeed the standard recommends that the calibration should be conducted at the same scale as the planned experiment and for the same range of strain values.
Figure 1: Threedimensional view of the physical reference material (EU Community Design Registration 000213467) which is scalable to any size and can be manufactured in any material. (©SPOTS consortium)
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Briefly, the reference material consists of a monolithic frame surrounding a beam subject to fourpoint bending. The gauge section of the reference material is the central portion of the beam. The beam is connected to the frame by a series of whiffletrees which are designed to minimize the constraints applied to the beam. The frame ensures that the boundary and loading conditions are easily reproducible which earlier roundrobins had demonstrated were a limiting issue in making comparisons between datasets [7]. The load can be applied in compression by placing the reference material on a platen and loading the nipple on the top surface, or in tension using the two circular holes along its centerline. The load is monitored as a relative displacement of the upper and lower portions of the frame which can be measured via a calibrated displacement transducer at the top left and top right corners. The use of calibrated displacement transducers provides the first link in the chain of traceability to the length standard. The importance of traceability in this context has been discussed by Hack et al [8]. The flowchart in figure 2 illustrates the experimental protocol for the use of the reference material and an exemplar of its use is provided by Whelan et al [9]. The values to be reported from the calibration process are highlighted in a box on the right of the flow chart and include the uncertainties in the calibration which are the minimum uncertainties that would be obtained in any subsequent experiment using the calibrated instrument. These uncertainties allow confidence limits to be defined for measured strain data which significantly improves the quality of data provided for the validation process. Select material for RM & estimate uncertainty in X
Manufacture RM
u(X)
Measure dimensions of RM & estimate their uncertainty (see figure 5) Select calibrated instrument to measure displacement
Apply a displacement load & measure strain in RM with instrument to be calibrated u(W) u(a) u(c)
u(vk)
Values to be reported
Assess differences (eq. 5) between measured & predicted strain values Perform linear leastsquares fit to field of deviations & evaluate D & E and their uncertainties (eq. 7 & 9)
Adjust instrument & repeat for acceptable calibration
field of deviations, dk(i,j)
Dk & Ek and u(Dk) & u(Ek)
Calculate RM uncertainty, uRM (eq. 10) Assess acceptability of calibration (figure 6) Repeat for 3 increments of load Assess acceptability of calibration (figure 6) & calculate calibration uncertainty, ucal (eq. 12)
ucal(H)
Figure 2: Flowchart for performing a calibration of an optical system for fullfield strain measurement using the SPOTS reference material where u is uncertainty, W, a and c are characteristic dimensions, X is Poisson ratio and D and E are fit parameters relating to the correlation of the experimental and analytical strain distribution for the gauge section of the beam. References to figures and equations relate to the SPOTS standard found in [6] (© SPOTS consortium)
The SPOTS consortium also provided the design and protocol for a standardised test material [6, 10] for use in evaluating an optical system against its design specification or other instruments. The strain distribution in the gauge section of the standardised test material is significantly more complicated than the simple linear distribution in the reference material and is designed to be a challenge to the capabilities of the most sophisticated instrument. As shown in figure 3, the gauge section consists of a disc in contact with a semiinfinite halfplane and is surrounded by a monolithic frame as in the reference material. The standardised test material can only be loaded in compression and contains the same feature for protecting the gauge section from overload and for monitoring the displacement load although in this case traceability to the length standard is not required. The rigid motion implicit in the contact loading presents difficulties for many strain measurement techniques and can be reduced substantially by loading the test material upside down as shown in figure 3. However, it should be noted that many realworld scenarios will involve significant rigid body motion.
y displacement
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Figure 3: A physical standardised test material (EU Community Design Registration 000299094) manufactured from aluminium having a disc diameter of 50mm being tested using an ESPI system (Dantec Dynamics Q300) with typical results shown inset. (© SPOTS consortium)
2. VALIDATION OF DYNAMIC ANALYSES 2.1 The ADVISE Project The SPOTS project was completed in early 2006 and a new consortium was formed during 2008 under the title of ADVISE (Advanced Dynamic Validations using Integrated Simulation and Experimentation) [11] with the purpose of extending the work into three dimensions and dynamic loading. ADVISE is a threeyear program partly funded by the EU 7th Framework Programme. The members were selected to provide continuity from the SPOTS project, to create an international collaboration and to span the innovation process from concept through product development and manufacture to enduse. The partners in ADVISE are: Airbus (UK), Centro Ricerche Fiat (Italy), Dantec Dynamics GmbH (Germany), EC Joint Research Centre, EMPA Swiss Federal Laboratories for Materials Testing and Research (Switzerland), High Performance Space Structure Systems GmbH (Germany), Michigan State University (USA), University of Liverpool (UK) and the University of Patras (Greece). The goals of the ADVISE project are: to develop a reference material and associated protocol for the calibration of optical systems capable of measuring displacements in a wide range of dynamic applications so that a system can be certified and the measurement uncertainties quantified; and to develop a methodology for making quantitative comparisons between fullfield data sets from such systems and computational models. Composite structures used in the transport industry are being used as a vehicle for testing these capabilities. 2.2 Design of a reference material The rational decision making model [12] is being employed to guide the design of the new reference material following its successful use in the SPOTS project and because it has the considerable advantage of providing clear opportunities for input by the wider experimental mechanics community. In outline, this model involves the definition and subsequent weighting, in terms of importance, of attributes that the design either must (essential attributes) or should (desirable attributes) possess. Candidate designs are then developed, often through brainstorming, and then evaluated on the extent to which they possess the attributes. Designs that do not or could not be modified to possess all of the essential attributes are discarded and the remainder are ranked based on the weighted sum of the degree of possession of each attribute. The top ranked designs are taken forward for detailed embodiment and further evaluation.
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1
2
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a.rangeofdisplacementinsideFOV b.outͲofͲplane&inͲplanedisplacements c.RMcanbeoperateduptokHz d.operationalatanyfrequency e.displacementfieldpredictable f.insituverifyingtheperformance g.forcyclicloading:nohysteresis h.cyclicloading:dataextractedthro'cycle i.definedstart&endconditions j.NDTforlargedisplacements
Figure 4: Attributes and their weightings for the displacement field in the dynamic reference material. The weightings (1  unimportant, 2  preferred, 3  important, 4  highly desirable, or 5 – essential) assigned by individual partners in the project were averaged (green) and the same exercise repeated for the wider community (blue). Some attributes were suggested by the community during the weighting process (white) and so were not weighted by a significant number of participants. (© ADVISE consortium)
The ADVISE consortium developed a set of attributes in early 2009 which were presented to the community at a VAMAS TWA26 workshop held at the Society for Experimental Mechanics Conference in Albuquerque, NM in June 2009. Participants at the workshop and members of the Society were invited by email to suggest additional attributes and to weight them. The results are presented in figures 4 and 5 together with the weightings based on input from the ADVISE partners. The essential attributes were chosen as those for which the sum of the mean and a standard deviation was greater than 4.0 when the data from the ADVISE consortium was merged with that from the community. The essential attributes are listed below: x there is a range of displacement values inside the field of view x inplane and outofplane displacements available x there is a means of verifying the performance in situ x for cyclic loading: data can be extracted throughout the cycle x the boundary conditions are reproducible x it is portable x it is robust 0
1
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k.surfacetextureisdefined l.completelyselfͲcontained m.incorporatesasourceofexcitation n.designisscaleable o.manufacturingiseasy p.fabricationfromdifferentmaterials q.lowmaterial&fabricationcosts r.easytosetup s.easytooperate t.boundaryconditionsarereproducible u.operationalatdifferenttemperature v.portable w.robust x.disp.sensitivityadjustableatdifferentscales y.excitationofviscoelastic,elastic,plastic
Figure 5: Attributes and their weightings for the physical embodiment of the dynamic reference material. The weightings (1 unimportant, 2  preferred, 3  important, 4  highly desirable, or 5 – essential) assigned by individual partners in the project were averaged (green) and the same exercise repeated for the wider community (blue). Some attributes were suggested by the community during the weighting process (white) and so were not weighted by a significant number of participants. (© ADVISE consortium)
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The ADVISE consortium met in Fall 2009 to brainstorm candidate designs for the reference material that would be appropriate for cyclic loading, single highspeed loading events that result in deformations that vary linearly with load and single highspeed loading events that result in deformations that vary nonlinearly with load. More than thirty candidate designs were generated but after assessment for the possession of the essential attributes only nine designs have been retained for further evaluation. At the time of writing these designs are undergoing evaluation involving both experimentation and computational modeling in order to gain a full understanding of their potential, prior to selecting a final design for detailed development and preparation of an accompanying protocol for its use. 3. DISCUSSION AND CONCLUSIONS A guide for the verification and validation of computational solid mechanics was published in 2006 [1, 2] and has been complemented by a proposed standard for the calibration and evaluation of optical systems for fullfield strain measurement [3, 4, 6]. The latter is applicable only to inplane, static strain distributions but nevertheless represented a substantial effort that included one of the first attempts to quantify the uncertainties involved in measuring strain over a wide field of view. A new project was launched in late 2008 to extend the proposed standard to include threedimensional strain analysis in dynamic cases. A single reference material is being sought that would allow calibration of optical systems capable of measuring displacement and, or strain in three dimensional components subject to dynamic loading which might or might not be cyclical and that may or may not induce nonlinear responses. This is wide and challenging design brief which is being tackled with the aid of the rational decision making model and with input from the community. The consortium undertaking the development has an international membership with representation from across the innovation process and appears to be broadly representative of the wider experimental mechanics community as indicated by the excellent correlation of the weighting of design attributes shown in figure 6. The new design of reference material is due for completion in late 2011 and following a period of testing within the ADVISE consortium will be available via a roundrobin for more comprehensive tests within the wider community. A protocol for its use and the reporting of results will also be produced. In parallel, research is underway to establish viable methods for making quantitative comparisons of large data fields generated from computational and experimental approaches to structural analysis. Ultimately it is expected that this new work will be incorporated into a revised draft standard by VAMAS TWA26. 5
Consortium scores
y = 0.9664x + 0.1291 R2 = 0.4465
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2 2
3
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Community scores
Figure 6: Correlation of the weightings by the ADVISE consortium partners and the members of the scientific community of the attributes required or desired in the dynamic reference material. (© ADVISE consortium)
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ACKNOWLEDGEMENTS The authors cheerfully acknowledge the inputs of the following persons in the ADVISE project: Richard Burguete, Mara Feligiotti, Alexander Ihle, George Lampeas, John Mottershead, Andrea Pipino, Hans Reinhard Schubach, Thorsten Siebert, and Victor Wang. The ADVISE project is a Seventh Framework Programme Collaborative Project within Theme 7: Transportation including Aeronautics (Grant no. 218595) funding of which is gratefully acknowledged. REFERENCES 1. L.E. Schwer, An overview of the PTC 60/V&V 10: guide for verification and validation in computation solid mechanics’, Engineering with Computers, 23:245252 (2007) 2. ASME V&V 102006, Guide for verification and validation in computational solid mechanics, American Society of Mechanical Engineers, New York, 2006. 3. E.A. Patterson, E. Hack, P. Brailly, R.L. Burguete, Q. Saleem, T. Siebert, R.A. Tomlinson, M.P. Whelan, ‘Calibration and evaluation of optical systems for fullfield strain measurement’, Optics and Lasers in Engineering, 45(5):550564, 2007. 4. www.twa26.org 5. www.vamas.org 6. www.opticalstrain.org 7. D. –A. Mendels, E. Hack, P. Siegmann, E.A. Patterson, L. Salbut, M. Kujawinska, H.R. Schubach, M. Dugand, L. Kehoe, C. Stochmil, P. Brailly, M.P. Whelan, ‘Round robin exercise for optical strain measurement’, Proc. 12th Int. Conf. Exptl. Mechanics, Advances in Experimental Mechanics edited by C. Pappalettere, McGrawHill, Milano, pp.6956, 2004. 8. E. Hack, R.L. Burguete, E.A. Patterson, ‘Traceability of optical techniques for strain measurement’, Proc. BSSM Int. Conf. on Advanced Experimental Mechanics, Southampton, UK, published as Applied Mechanics & Materials, vols.34, pp.391396, 2005. 9. M.P. Whelan, D. Albrecht, E. Hack, E.A. Patterson, ‘Calibration of a speckle interferometry fullfield strain measurement system’, Strain, 44(2):180190, 2008. 10. E.A. Patterson, P. Brailly, R.L. Burguete, E. Hack, T. Siebert, M.P.Whelan, ‘A challenge for high performance fullfield strain measurement systems’, Strain, 43(3):167180, 2007. 11. www.dynamicvalidation.org 12. N. Cross, Engineering Design Methods (John Wiley & Sons, London, 1989)
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Avalanche Behavior of Minute Deformation Around Yield Point of Polycrystalline Pure Ti G. Murasawa*, T. Morimoto*, S. Yoneyama**, A. Nishioka*, K. Miyata* and T. Koda* *Department of Mechanical Engineering, Yamagata University, 4316, Jonan, Yonezawa, Yamagata 9928510, Japan, Japan. [emailprotected] **Department of Mechanical Engineering, Aoyama Gakuin University, 5101 Fuchinobe, Sagamihara, Kanagawa 2298558, Japan. ABSTRACT The aim of present study is to investigate the avalanche behavior of minute deformation around yield point of polycrystalline pure Ti. Firstly, we prepare commercial polycrystalline pure Ti (99.5%) thin plate, and investigate the pole figures and inverse pole figure distribution for rolling direction on the surface of specimen, which is obtained from Electron Backscatter Diffraction Patterns (EBSD). Secondarily, tensile specimens are cut out from 0°, 30°, 45° and 90° relative to rolling direction of thin plate. Then, we attempt to measure macroscopic stressstrain curve, local strain distribution and minute deformation arising in specimens under tensile loading. In this time, the inhouse measurement system integrated with tensile machine control system, local strain distribution measurement system and minute deformation measurement system is constructed suited on a LabVIEW platform. Local strain distribution is measured by inhouse system on the basis of Digital Image Correlation (DIC). Also, minute deformation behavior is measured by inhouse other one on the basis of acoustic emission (AE). Finally, we discuss about the mechanism of avalanche behavior of minute deformation around yield point of polycrystalline pure Ti on the basis of results in present study. 1. INTRODUCTION Macroscopic deformation behavior, such as stressstrain curve, is subject to the minute deformation behavior like the motion of slip or twinning deformation. Such minute deformation behavior reveals the phenomena of “Intermittent deformation behavior”, “Spatial clustering and avalanche behavior” and “Selfsimilar or scale free”. These phenomena are not continuous deformation, but discontinuous deformation. Although continuum approaches are selectively useful for describing deformation, they completely fail to account for wellknown discontinuous deformation phenomena. Steel and Aluminum alloys usually show local deformation behavior such as the propagation of Luders band, Portevin Le Chatelier effect and necking. Recently, some researchers began to study about the nucleation and propagation of shear band for some metal materials (Kuroda et al., 2007; Tong et al., 2005; Zhang et al., 2004; Cheong et al., 2006; Tang et al., 2005; Hoc et al., 2001; Louche et al., 2001). These studies are that predicting macroscopic deformation is tried by continuum approaches, but it is difficult to describe the detailed behavior based on the mechanism of deformation for solid materials. In recent years, Weiss and Uchic are aggressively studying about the minute deformation behavior for solid materials. Weiss measured the motion of slip under creep deformation for single and poly crystalline ice by using Acoustic Emission (AE) technique. They reported that minute deformation shows avalanche behavior and its behavior is different between single and poly crystalline. Also, Uchic fabricated the microorder specimen of pure Ni by focused ion beam process. Then, they conducted uniaxial compression test for the microorder specimen. They reported that minute deformation shows the phenomenon of scale free. The aim of present study is to investigate the avalanche behavior of minute deformation around yield point of polycrystalline pure Ti. Firstly, we prepare commercial polycrystalline pure Ti (99.5%) thin plate, and investigate T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_5, © The Society for Experimental Mechanics, Inc. 2011
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the pole figures and inverse pole figure distribution for rolling direction on the surface of specimen, which is obtained from Electron Backscatter Diffraction Patterns (EBSD). Secondarily, tensile specimens are cut out from 0°, 30°, 45° and 90° relative to rolling direction of thin plate. Then, we attempt to measure macroscopic stressstrain curve, local strain distribution and minute deformation arising in specimens under tensile loading. In this time, the inhouse measurement system integrated with tensile machine control system, local strain distribution measurement system and minute deformation measurement system is constructed suited on a LabVIEW platform. Local strain distribution is measured by inhouse system on the basis of Digital Image Correlation (DIC). Also, minute deformation behavior is measured by inhouse other one on the basis of acoustic emission (AE). Finally, we discuss about the mechanism of avalanche behavior of minute deformation around yield point of polycrystalline pure Ti on the basis of results in present study. 2. MATERIAL AND METHODS 2.1 Materials Material is commercial polycrystalline pure Ti (99.5%) thin plate (Nilaco co.). Pure Ti shows hexagonal closepacked structure. The deformation mode is mainly twinning deformation. Figure 1(a) shows an optical image of microstructure for present Ti. Also, Fig. 1(b) shows pole figures, (0001), (1012) and (1011) planes, of present Ti plate. Figure 1(c) demonstrates the inverse pole figure distribution for rolling direction on the surface of specimen, which is obtained from Electron Backscatter Diffraction Patterns (EBSD). Tensile specimens are cut out from 0°, 30°, 45° and 90° relative to rolling direction of thin plate as shown in Fig. 2(a). Figure 2(b) shows specimen configuration. The stainless steel tabs are attached to the specimen by epoxy resin. Also, random pattern is created on the surface of specimen by spraying black and white paint for measuring strain distribution by Digital Image Correlation (DIC) method. __
(a) Microstructure (b) pole figures (c) inverse pole figure distribution Figure 1. Microstructure, pole figures and inverse pole figure distribution for polycrystalline pure Ti
Figure 2. Specimen configuration
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2.2 Measurement Method of Local Deformation Behavior (LDB) Digital Image Correlation (DIC) software is inhouse software, and the details of DIC are presented in the references. In these references, Yoneyama et al. proposed the principal method and its algorism of DIC to calculate displacement field from two pictures (i.e., undeformed image and deformed image). Also, they have applied this method to the measurement of stress intensity factor around crack tip and bridge deflection. A test system, local deformation behavior (LDB) measurement system, is also constructed on the basis of DIC as an inhouse software suited on a LabVIEW platform. We can semiautomatically measure strain distribution by using the test system. The test system consists of three parts as follows: (A) Image acquisition, (B) Digital image correlation, (C) Calculation of local strain distribution. Firstly, the images used in DIC are taken at an interval during the deformation of specimen. Secondary, two images (i.e., undeformed image and deformed image) are selected from whole images, and the displacement all over the surface of specimen can be calculated by comparing these two images in DIC. Thirdly, the local strain all over the surface of specimen is calculated from the distribution of displacement. Its details of test system, especially in the calculation method of strain field obtained from displacement field, are presented in the references. 2.3 Measurement Method of Minute Deformation Behavior and Data Analysis Method Acoustic emission (AE) is a powerful technique for studies about minute deformation behavior that arise from the motion of dislocation and twinning deformation. Acoustic waves are generated by dislocation and twinning glide, and many information of deformation can be obtained from detected acoustic waves. 2.3.1 Measurement of AE In present study, the AE signals generated from twinning deformation are monitored by four small AE sensors, which are mounted on the surface of the specimen. Four small AE sensors are aligned on a center straight line of the surface of the specimen as shown in Fig. 2(c). In present experiment, we use No.1 and No.4 sensors as guard sensor, and No.2 and No.3 sensors as effective sensor in order to get AE signals only from effective area. A method to distinguish the AE signals from effective area with that from other area, is as follows; (1) Four AE signals are monitored for a event during deformation, (2) Arrival times (No.1 sensor: t1, No.2 sensor: t2, No.3 sensor: t3, No.4 sensor: t4) are recorded from the AE signals, (3) If t1< t2 or t4< t3, then AE signals are regarded as a signal out of effective area. In other case, AE signals are regarded as a signal from effective area. These processes are automatically conducted during deformation of materials by inhouse software suited on a LabVIEW platform. 2.3.2 Analysis of AEcount data Also, we can obtain the relationship between cumulative AE counts and time from detected AE signals. Furthermore, we can calculate the AEcount speedtime relation from the cumulative AE countstime curves. AEcount speed is the slope of cumulative AE countstime curves at each time. Avalanche behavior can be seen from these curves. Calculation method of AEcount speed is shown in Fig. 3. Firstly, an arbitrary point on the cumulative AE countstime curve is selected, and its before and after points, 50 points, are pointed as shown in Fig. 3(a). Secondarily, these 51 points are approximated by quadratic least squares method. Thirdly, the slope of approximated quadratic curve is obtained at the point x0 as shown in Fig. 3(b). The value of slope is AEcount speed at the point x0 . This calculation is conducted at all points on cumulative AE countstime curve. Obtained AEcount speedtime relation is shown in Fig. 3(c). Calculation is automatically conducted for all data by inhouse data analysis software suited on a LabVIEW platform.
Figure 3. Method for calculating the AEcount speed
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2.3.3 Wave analysis of AE Figure 4 shows a typical AE signal monitored from No.2 and No.3 sensors. For each event, wave analysis allows us to record two parameters, the arrival time t0 (the time at which the signal reaches threshold value Amin) and the maximum amplitude A0. Recording is automatically conducted for all data by inhouse wave analysis software suited on a LabVIEW platform.
Figure 4. Detected AE wave 2.4 Experimental Setup of System Integrated with LDB and AE Measurement The integrated experimental setup is shown in Fig. 5(a). Uniaxial tensile loading is performed for specimens at room temperature (21℃). Autograph (AGS 5KNG, Shimazdu) is used in present tensile loading test, and it is controlled by inhouse software suited on a LabVIEW platform. Tests are performed at strain rate, 0.08%/min. A CCD camera (HCHR70, Sony) is set in front of the specimen as shown in Fig. 5(b). A 50mm lens (VCL50YM, Sony) is attached to the CCD camera. The images of specimen are taken into computer every 120 second during tensile loading. Four small AE sensors with frequency bandwidth of 300kHz2MHz (AE900M, NF corporation) are mounted on another surface of the specimen like Fig. 5(c). The output signals are amplified 40 dB by the preamplifier (9916, NF corporation) and then 20 dB by the discriminator (AE9922, NF corporation), and fed to oscilloscope and computer. We set the proper threshold value (about 1mV) of AE signal. Data are sampled at an interval of 100 ns. We use the computer with a GPIB interface (NI GPIBUSBHS, National Instrument) to control tensile machine, a monochrome PCI frame grabber (mvTITANG1, Matrix Vision) to acquire 1024×768 8bit grayscale digital image from the CCD camera and two 12bit AD converters (PCI3163, Interface) to acquire AE signals from the AE sensors. Then, the inhouse software integrated with tensile machine control system, LDB measurement system and AE measurement system is constructed suited on a LabVIEW platform as shown in Fig. 5(d).
Figure 5. Experimental setup
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3. RESULTS 3.1 Macroscopic Stressstrain Curves of Polycrystalline Pure Ti Figure 6 shows macroscopic stressstrain (and stresstime) curves at room temperature (21°C) for different specimens which are cut out from 0°, 30°, 45° and 90° relative to rolling direction of pure Ti thin plate. From these results, it is seen that the results of 0° and 30° specimens show the same behavior. On the other hand, 45° specimen shows a little larger stressstrain curve than that of 0° and 30° specimens, and 90° specimen shows more larger one than others, around yield point.
Figure 6. Macroscopic stressstrain (time) curves for polycrystalline pure Ti 3.2 Local Deformation Behavior Around Yield Point of Polycrystalline Pure Ti Figures 7(a)~(d) demonstrate stresstime curve and longitudinal local strain distribution for 0°, 30°, 45° and 90° specimens of polycrystalline pure Ti under uniaxial tensile loading at room temperature (21°C). Local strain distribution is measured at the points indicated as (1)~(10) or (12) in Figs. 7(a)~(d). Also, local strain distributions are acquired every 120s, which is corresponding to the nominal strain of 0.16%. The value of strain is shown in right hand scale level in Fig. 7. The longitudinal and transversal axes of the results for local strain distribution in Figs. 7(a)~(d) show the position on picture used in DIC, which are along to x (horizontal axis) direction and y (vertical axis) direction in picture. From even a cursory examination of Fig. 7(a)~(d), it is seen that all specimens reveal
(a) 0° specimen
(b) 30° specimen
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(c) 45° specimen (d) 90° specimen Figure 7. Longitudinal strain distribution under uniaxial tensile loading for polycrystalline pure Ti inhom*ogeneous deformation behavior, and the cluster of local strain gradually increase around yield point of stressstrain curve. Also, the initiation for the cluster of local strain depends on the cut direction of specimen, and it becomes late from 0° to 90°. 3.3 Minute Deformation Behavior Around Yield Point of Polycrystalline Pure Ti Figure 8 gives stresstime curve, cumulative AE countstime curve and AEcount speedtime curve around yield point for all specimens during tensile loading. As shown in these figures, it can be seen that these curves depend on the cut direction of specimen. Also, from the results of AEcount speedtime curves, 0° and 30° specimens
Figure 8. Cumulative AE count – time curves and AE count speed – time curves under uniaxial tensile loading for polycrystalline pure Ti
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suddenly show the phenomenon such as avalanche of minute deformation around the early stage of yield point, and especially 0° specimen demonstrates large avalanche behavior. AEcount speed becomes equilibrium state after showing avalanche behavior during tensile loading. On the other hand, minute deformation does not appear around the early stage of yield point for 45° and 90° specimens. Then, their specimens show slow avalanche behavior at the latter stage of yield point or the beginning stage of plastic flow, as compared with former ones. 4. DISCUSSION 4.1 Multiscale Deformation Structure We could measure the macroscopic inhom*ogeneity of local strain and the avalanche behavior of minute deformation for polycrystalline pure Ti under uniaxial loading by using digital image correlation and acoustic emission techniques. From these results, following deformation structure can be supposed. Figure 9 shows the schematic illustration of macroscopic inhom*ogeneity arising in polycrystalline pure Ti under uniaxial loading. Firstly, the clusters of local strain appear during the elastic deformation for macroscopic stress – strain curve. Secondarily, the regions of cluster drastically increase around yield point. Thirdly, the increase of clusters becomes constant under plastic deformation. Figure 10 illustrates the mechanism of deformation for polycrystalline pure Ti under uniaxial loading. As shown in this figure, a cluster consists of a lot of twinning deformation regions. A twinning deformation region consists of some grains at which are appeared twinning bands (Fig.10(b)). Also, some researchers reported that the amplitudes of the acoustic signals are related to the area swept by the fastmoving dislocations and hence to the energy dissipated during deformation events. In present study, AE count is measured as the number of twinning deformation region. The amplitudes of the acoustic signals are related to the size of twinning deformation region. Also, a twinning band in a grain is illustrated as zigzag shape of crystal structures (Fig.10(c)). It is well known that polycrystalline pure Ti shows anisotropic characteristics in macroscopic stress – strain curve. Then, the key points of mechanism of deformation are “initiation behavior” and “avalanche behavior” of minute deformation.
Figure 9. Increase of macroscopic inhom*ogeneity local strain
Figure 10. Schematic illustration of multiscale deformation for structure for polycrystalline pure Ti
4.2 Mechanism of Initiation for Minute Deformation Initiation of minute deformation is governed by the direction of grains in polycrystalline. Texture data is much important information to obtain the direction of grains for polycrystalline. In present study, texture data of polycrystalline pure Ti is obtained by EBSD equipment. Figure 1(b) shows pole figures, (0001), (1012) and (1011) planes, of present Ti plate. Figure 1(c) demonstrates the inverse pole figure distribution for rolling direction on the surface of specimen __
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__
In general, it is well known that (1012) or (1011) twin boundary in deformed polycrystalline pure Ti can be observed by highresolution electron microscopy. It has been also reported that its structure has essentially a pure mirror symmetry. The structure is characterized by a mirror plane which corresponds to the coalescence of two separate atomic planes (0001) into a single plane (1012) or (1011). The angle of misorientation is close to 87°. From the results of Fig.1, it is seen that (1012) and (1011) planes randomly exist for the inplane direction. On the other hand, the result of (0001) plane strongly shows anisotropy and its direction is parallel to rolling direction of specimen. Macroscopic experimental results show the anisotropic characteristics in macroscopic stress – strain curve and in initiation of the cluster of local strain, and in minute deformation behavior. That is to say, (0001) plane is something to do with the mechanism for the initiation of minute deformation. Twinning deformation occurs in a grain in polycrystalline under shear deformation for metal materials with hexagonal closepacked structure. Figure 11 shows the schematic illustration of the mechanism for the initiation of twinning band. In this time, it is assumed that the twinning plane is parallel to habit plane under applied load. Also, lattice transformation due to applied load occurs around habit plane (Fig.11(a)). Then, twinning deformation instantaneously develops as shown in Fig.11(b). In this time, we can see that the degree of initiation of twinning is strongly caused by the direction of plane which is firstly transformed lattice in a grain region by applied load. In other words, (0001) plane is initiationrelated direction of plane. The closer (0001) plane before deformation is to plane which is firstly transformed lattice around habit plane, the easier occurrence of twinning is. In present study, (0001) plane is parallel to the rolling direction. The twinning deformation becomes hard to occur if the longitudinal direction of tensile specimen leaves from the rolling direction. __
__
Figure 11. Schematic illustration of the mechanism for initiation of twinning deformation due to applied load 4.3 Mechanism of Avalanche Behavior for Minute Deformation In present study, avalanche behavior can be investigated by evaluating AE counts. Furthermore, it is assumed that the amplitudes of the acoustic signals are related to the size of twinning deformation region, and the avalanche behavior can be connected with the size of twinning deformation region. Also, from the results of local strain distribution measurement and such an AE measurement, we can lead the mechanism of avalanche behaviors, such as the avalanche behavior of minute deformation causes the macroscopic inhom*ogeneity of local strain and so on. From the results of Fig.8, 0° and 30° specimens show strong avalanche behavior around yield point. On the other hand, 45° and 90° specimens show slow avalanche behavior at the latter stage of yield point or the beginning stage of plastic flow, as compared with former ones. This can imply that twinning deformation regions actively increase at the timing between the neighborhood of yielding point and the beginning stage of plastic flow. Next, we take the size of twinning deformation region around yield point under uniaxial tensile loading into consideration for polycrystalline pure Ti. Figure 12 displays the cumulative number of event – amplitude relation obtained from AE signals for different specimens which are cut out from 0°, 30°, 45° and 90° relative to rolling direction of pure Ti thin plate. The cumulative number of event – amplitude relation is obtained by following ways. Firstly, 2000 AE signals during measurement are prepared, and maximum voltage is detected for each AE signal.
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Figure 12. The cumulative number of event – amplitude relation obtained from all AE signals of present measurement
Figure 13. The cumulative number of events – amplitude relations for each 250 events during measurement
This maximum voltage is corresponding to the size of twinning deformation region for each event. Secondarily, these maximum voltages are listed in order of amplitude. Thirdly, listed voltages are numbered from highest one to lowest one. The number and voltage are the cumulative number of event and amplitude in Fig. 12. As shown in Fig.12, the cumulative number of events – amplitude relation for 90° specimen is the same as 45° specimen. On the other hand, it seems that the relation tends to shift to high amplitude with decreasing of the cut orientation. This implies that larger size of twinning deformation regions exist with decreasing of the cut orientation during measurement. Therefore, we try arranging the size of twinning deformation regions with the number of counts. Figure 13 demonstrates the cumulative number of events – amplitude relations for each 250 events during measurement, for 0°, 30°, 45° and 90° specimens. From these figures, we can see the dependency of the specimen orientation on the cumulative number of events – amplitude relations. In other words, as considered with results of Figs.7 and 8 during measurement, twinning deformation regions of smaller size nucleate for 90° specimens. On the other hand, those of larger size nucleate for 0° specimens during measurement. Figure 14 shows the schematic illustration of the mechanism for the initiation of twinning deformation region and its
Figure 14. Schematic illustration of the mechanism for avalanche of minute deformation
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cluster arising in polycrystalline pure Ti. As illustrated in this figure, firstly, twinning deformation regions nucleate with a certain space under mechanical deformation. Then, larger twinning deformation regions increase at the space between initiated smaller twinning deformation regions. The size of twinning deformation regions grow up under mechanical deformation, and it becomes the cluster of local strain which we can confirm from the results of local strain behavior measurements. While the size of twinning deformation regions is growing up under mechanical deformation, AE counts show the avalanche behavior. Then, it is much important for us to study “how do the twinning deformation regions behave in polycrystalline pure Ti during mechanical loading”. The study about the information for the initiated location of twinning deformation region, is future work. 5. CONCLUSIONS Firstly, we prepare commercial polycrystalline pure Ti thin plate, and investigate the pole figures and inverse pole figure distribution for rolling direction on the surface of specimen. Secondarily, tensile specimens are cut out from 0°, 30°, 45° and 90° relative to rolling direction of thin plate. Then, we measure macroscopic stressstrain curve, local strain distribution and minute deformation arising in specimens under tensile loading. Local deformation behavior measurement system is constructed on the basis of DIC and minute deformation behavior measurement system is constructed on the basis of AE. Then, we developed both of their inhouse software suited on a LabVIEW platform. Obtained results are as followings. (1) From the results of macroscopic stressstrain curves, the results of 0° and 30° specimens show the same behavior. On the other hand, 45° specimen shows a little larger stressstrain curve than that of 0° and 30° specimens, and 90° specimen shows more larger one than others, around yield point. (2) From the results of local strain distribution, all specimens reveal inhom*ogeneous deformation behavior, and the cluster of local strain gradually increase around yield point of stressstrain curve. Also, the initiation for the cluster of local strain depends on the cut direction of specimen, and it becomes late from 0° to 90°. (3) From the results of minute deformation behavior, 0° and 30° specimens suddenly show the phenomenon such as avalanche of minute deformation around the early stage of yield point, and especially 0° specimen demonstrates large avalanche behavior. AEcount speed becomes equilibrium state after showing avalanche behavior during tensile loading. On the other hand, minute deformation does not appear around the early stage of yield point for 45° and 90° specimens. Then, their specimens show slow avalanche behavior at the latter stage of yield point or the beginning stage of plastic flow, as compared with former ones.
ACKNOWLEDGEMENTS The author would like to express my deep gratitude to Prof. Mitsutoshi Kuroda (Yamagata University, Japan) and Assosiate Prof. Takuya Uehara (Yamagata University, Japan) for fruitfully discussing about present study. Also, The author would like to express my deep gratitude to Dr. Tadaaki Satake (Yamagata University, Japan) for the help of using EBSD.
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Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
The Effect of Noise on Capacitive Measurements of MEMS Geometries
Joo Lien Chee1 and Jason V. Clark1,2 School of Electrical and Computer Engineering 2 School of Mechanical Engineering Purdue University Birck Nanotechnology Center, 1205 W. State Street, West Lafayette, IN 47907 [emailprotected], [emailprotected] 1
ABSTRACT Small variations in the geometry of Micro Electro Mechanical Systems (MEMS) can yield very large variations in performance. Variations in geometry are a consequence of the MEMS fabrication process, and are unavoidable with current fabrication technology. To achieve quick, accurate, and precise measurements of MEMS geometry, we have previously reported on our pioneering use of capacitance to measure MEMS geometry. The ability to capacitively probe MEMS geometries has the potential to more precisely obtain geometric uncertainties, and to realize autonomous onchip measurements inthefield. The precision of measurement method depends on the precision of the capacitance meter, which is subject to various sources of noise. In this paper, we examine the effect of this noise using our offchip capacitive measurement method. In our present approach, we consider four sources of noise and analyze how they individually contribute to the uncertainty in the extraction of MEMS geometry. The four sources of noise are: noise from the voltage source, internal noise of the capacitance meter, noise from external electromagnetic fields, and thermal noise. We verify our analytical results with simulation and validate our results with experiment. With offchip capacitive probing, we find that the uncertainty in geometric extraction is most strongly affected by external electromagnetic fields, moderately affected by noise from the measurement equipment and thermal noise, and least affect by the applied voltage. We measure the uncertainty in geometry due to shielded and unshielded conditions, and we predict the uncertainty in geometry due to the voltage source and thermal noises. 1 INTRODUCTION Although Micro Electro Mechanical Systems (MEMS) have been used in a wide variety of application areas, intrinsic variations in the structural dimensions of each fabricated device impedes technological advancement. These structural variations are primarily caused by the totality of natural variations within a fabrication process [13]. It is because of these variations that the predicted geometry from layout does not match fabrication. In addition to variations in geometry, there are variations in material properties for each MEMS device on a chip. The change in geometry overcut in going from layout to fabricated device is often small, on the order of a tenth of a micron. However, small changes in geometry can translate to large changes in performance. For instance, Clark [4] showed that a 0.25 micron (10%) variation in overcut of the width of a commonlyused folded flexure results in a change in spring stiffness of 100%. And the change in stiffness is exacerbated if the uncertainty of Young’s modulus (typically 10%) is taken into account. An overview of several conventional tools used to characterize MEMS geometry is provided by Novak [5]. The tools include: optical microscopy, contact stylus profilometry, electron microscopy, interferometry microscopy, scanning confocal microscopy, atomic force microscopy, atomic force microscopy, laser Doppler vibrometers, and digital holography. Although these tools are adequately suited for individualized laboratory investigations, they are not wellsuited for batchfabricated industrial scale measurements, or for postpackaged measurements in the inthefield. Such abilities may be achieved with electronicallyprobed measurements of MEMS geometries. Work in this area has been pioneered by Gupta [11] and Clark [4]. In Gupta’s method, electrical measurements of static deflections and resonance frequencies are fed into a 3D computer modeling and simulation tool to determine geometry. In Gupta’s method, it is assumed that material properties of the process are known. In Clark’s method, T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_6, © The Society for Experimental Mechanics, Inc. 2011
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electrical measurements of static deflections are fed into analytical equations to determine geometry, independent of material properties. In one version Clark’s method, MEMS fabricated within close proximity of each other are assumed to share the same unknown variations. The latter metrology method is called electro micro metrology (EMM). We chose to use EMM in this work to avoid having to measure material properties, which would create additional uncertainties in our analyses. We developed an EMM test bed to capacitively measure the planar geometry of our MEMS structure in [6]. Our test bed has a geometric resolution of ~60 nanometers (nm) which corresponds to our measured capacitance uncertainty of ~1 femtofarads (fF). We expect that this resolution can be improved by lowering the magnitude of the uncertainty in capacitance to the zeptofarad (zF) regime [7,12] or by improving the sensitivity through design optimization [4]. In this paper, we use our test bed to study how noise affects the extracted uncertainty in geometry due to the uncertainty in measured capacitance. We consider four sources of noises: Noise from our voltage source, form our capacitance meter, from external electromagnetic fields, and from thermal noise. We measure how each source of noise affects geometric uncertainty using an EMM model. Following the introduction, the rest of this paper is organized as follows. In Section 2 we describe how EMM theory is used to extract geometric uncertainty in terms of noise. In Section 3 we describe our EMM test bed which we use to obtain measurement parameters for the theoretical equations discussed in the previous section. In Section 4, we describe our noise models and determine the effect of such noise or our EMM measurement of geometry. Last, in Section 5 we summarize what we learn from this work. 2 THEORY 2.1 ELECTRO MICRO METROLOGY (EMM) EMM exploits the strong and sensitive electrical to mechanical coupling at the microscale to expresses mechanical properties as functions of electrical measurands. We express planar geometry as a function of changes in capacitance as follows. Consider two test structures a and b . One of our structures is shown in Figure 1. The difference between structures a and b is that the layout widths wlayout of the flexures of b are slightly larger than those of a due to a layout parameter n chosen by the MEMS designer; that is, wb ,layout = n wa ,layout . We assume that structures a and b are fabricated within close proximity to each other such they share the same unknown material properties and geometrical errors. The fabricated widths ( wa and
wa
wb ) are related to layout widths ( wa ,layout and wb ,layout ) by wa = wa ,layout + Δw
(1)
and
wb = nwa ,layout + Δw ,
(2)
where Δw is the geometrical difference in going from layout to fabrication, which is to be determined. It is important to understand that although the true width of a MEMS fabricated flexure varies along the length of its flexure due to
Figure 1: Folded flexure comb drive resonator. The four large square anchors are each attached to a flexure. There are 4 flexures of equal length and width. The 2 innermost beams of both folded flexures attach to a backbone with comb drives. The device is composed of nickel for our related RF investigations. The flexure lengths are 2000μm, flexure widths of a are 7μm and of b are 8 μm, thickness is 20μm, distance between flexures are 500μm, comb drive gaps are 8μm, there are 42 fingers on each drive, and the layout parameter (see Equation 2) is n = 8 / 7 .
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the coarse sidewalls, the expressions in (1) and (2) correspond to the effective width that a smooth sidewalled flexure (i.e. that of an analytical model) needs to be such that its performance matches that of a true flexure with coarse sidewalls. In the electrical domain, the electrostatic force that is generated on a comb drive is F = 12 V ∂C ∂y . Applying a 2
voltage V across a comb drive produces an electrostatic force F and corresponding deflection y , which results in a change in capacitance ΔC . So for each of our structures a and b , we have
1 ΔCa Fa = V 2 2 Δya
(3)
and
ΔCb 1 Fb = V 2 , 2 Δyb
(4)
where partial derivative symbols have been replaced with difference symbols due to the linearity of comb drives; that is ∂C ∂y → ΔC Δy . Equations (3) and (4) account for fringing field effects, asymmetry in comb fingers, and planar residual strain gradients as long as the deflection is only in the y direction. Since the comb drives in a and b are assumed to share the same geometric errors, then a voltage V applied across the comb drives generates the same electrostatic forces. That is,
Fa (V ) = Fb (V ) .
(5)
Hence, equating (3) and (4) yields a ratio between unknown displacements and measured capacitances,
Δya ΔCa = . Δyb ΔCb
(6)
And from the mechanical domain, using Hooke’s law for small deflections, we have
Fa = ka Δya
(7)
and
Fb = kb Δyb ,
(8)
Due to (5) we are able to equate (7) and (8), which yields a ratio between unknown displacements and unknown stiffnesses,
Δya kb = . Δyb k a
(9)
Since (6) and (9) correspond to the same displacement ratios Δya Δyb , equating (6) and (9) yields a ratio in stiffenss in terms of capacitance,
kb ΔCa = . ka ΔCb
(10)
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3
3
Now the stiffnesses ka and kb are proportional to unknown flexure widths cubed ( wa and wb ), Young’s modulus
E , layer thickness h , and inversely proportional to beam length cubed L3 . Since we are assuming that the two structures share the same unknown geometric and material properties, then the unknown quantities E , h , L cancel each other out in the ratio. And (10) reduces to
ΔCa wb3 = ΔCb wa3
(n w = (w
+ Δw )
(11)
3
a ,layout
a ,layout + Δw )
3
,
where we have substituted (1) and (2) for the fabricated widths on the second line. Equation (11) has one unknown, Δw , which the difference between layout and fabrication. Solving (11) for Δw we find
n ( ΔCb ΔCa ) − 1 13
Δw = wa ,layout
( ΔCb
ΔCa ) − 1 13
.
(12)
In essence, by using EMM we are able to express in (12) the change in geometry in going from layout to fabrication ( Δw ) in terms of precisely measured changes in capacitance ( ΔCa and ΔCb ), and exactly known layout parameters ( wa ,layout and n ). 2.2 GEOMETRIC UNCERTAINTY IN TERMS OF CAPACITIVE UNCERTAINTY In Section 2.1 we determined the change in geometry as a function of change in capacitance, Equation (12). According to our EMM modeling assumptions, uncertainty in geometry is attributed to uncertainty in capacitance. A measurement of capacitance is has the form
C ±δC
(13)
where various sources of noise is lumped into capacitance uncertainty δ C . Depending on analysis, δ C may be taken as the last flickering digit on a capacitance meter of a single measurement, or δ C may be taken as the standard deviation resulting from a multitude of repeated measurements. Substituting (13) for the capacitance quantities in (12) we have 13
Δw ± δ w = wa,layout
⎛ ΔC ± δ C ⎞ n⎜ b ⎟ −1 ⎝ ΔCa ± δ C ⎠ , 13 ⎛ ΔCb ± δ C ⎞ ⎜ ⎟ −1 ⎝ Δ Ca ± δ C ⎠
(14)
where we have assumed that each measurement of capacitance contains the same order of uncertainty. Expanding (14) in a Taylor series about δ C yields to first order
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Δw ±δ w = wa,layout
⎡ 13 13 ⎤ ⎛ ΔCb ⎞ ⎛ ΔCb ⎞ ⎥ ⎢ n⎜ ⎟ −1 ⎢ 2 wa,layout ( n − 1) ( ΔCa  ΔCb ) ⎜ ΔC ⎟ ⎥ ⎝ ΔC a ⎠ ⎝ a ⎠ ⎥, ±δ C ⎢ 13 2 1 3 ⎢ 3 ⎥ ⎛ ΔCb ⎞ ⎛ ⎛ ΔC ⎞ ⎞ b ⎢ ⎥ ⎜ ⎟ −1 ⎜ ⎟ ΔCa ΔCb ⎜ −1 ⎜ ⎝ ΔCa ⎟⎠ ⎟ ⎝ ΔCa ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦
(15)
where the second term on the right hand side is the uncertainty in geometry δ w , and the squarebracketed quantity in this term is the sensitivity in geometry Δw as a function of change in capacitance δ C . That is
⎡ 13 ⎤ ⎛ ΔCb ⎞ ⎥ ⎢ ⎢ 2 wa,layout ( n − 1) ( ΔCa  ΔCb ) ⎜ ΔC ⎟ ⎥ ⎡ ∂Δw ⎤ ⎝ a ⎠ ⎥. δ w= δC ⎢ = δC ⎢ 2 ⎥ ⎢ 3 ⎥ ⎣ ∂δ C ⎦ ⎛ ⎛ ΔC ⎞1 3 ⎞ b ⎢ ⎥ ΔCa ΔCb ⎜ ⎜ − 1⎟ ⎜ ⎝ ΔCa ⎟⎠ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦
(16)
( ) . It is 8
In practice, the order of the squarebracketed expression (the sensitivity) can be quite large, say, O 10 therefore necessary that the uncertainty in capacitance δ C be much smaller than the sensitivity, such that the product
δ C [ ∂Δw ∂δ C ] achieves the
desired uncertainty in geometry. Techniques to reduce the uncertainty in geometry include: reducing the uncertainty in capacitance δ C by using a capacitance meter with better precision; reducing δ C by reducing the magnitude of noise contributions from various sources (see below); or reducing the sensitivity
[∂Δw ∂δ C ]
by modifying the geometric
configuration a priori. That is, geometry affects stiffness and or capacitance, which affects the change in capacitance due to a given applied voltage. For instance, in Figure 2 we show the dependence of sensitivity of our test structure (shown in Figure 1) as a function the number of comb drive fingers and comb drive gap. We use such analysis to improve EMM measurement for a given capacitance meter’s uncertainty. 3 EMM TEST BED A salient attribute of EMM is its ease of use for MEMS metrology by electrical probing of capacitance. In our present test bed we use an offtheself capacitance meter for $15. Typically, capacitance meters are available in most laboratories. This attribute of EMM is in stark contrast to most other metrology methods that required expensive equipment not readily available in most labs, and that require specialized training due to the intricacy in using such equipment.
Figure 2: Sensitivity as a function of geometry. (Top): Sensitivity ∂Δw ∂δ C as a function of the number of comb fingers N . The sensitivity at N = 45 is 5.81E8 m F . (Bottom): Sensitivity ∂Δw ∂δ C as a function of comb finger gap g . The sensitivity at
g = 3μ m is 5.5E 6 m F . The circle on the curve indicates the sensitivity with respect to Figure 1.
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We have developed a test bed especially designed to characterize EMM. Our test bed is shown in Figure 3. The test bed comprises the following lownoise features: a Faraday cage to reduce noise interference from external electromagnetic fields; an airdamped table to reduce the noise due to building vibrations; lowreflective interior cage walls to reduce the noise due to light; and lowparasitic shielded capacitance probes. Other components of the test bed include a MEMS wafer is held in place by a vacuum chuck; micromanipulators for positioning the electrical probes; a small 200x digital microscope camera mounted onto a boom stand; shielded probe leads; and an inexpensive capacitance meter chip with 4 attofarad resolution, by Analog Devices AD7746 [8]. However, our current test bed configuration shown in Figure 3 yields a capacitance noise floor of 1.2 femtofarads, leaving much room for further improvement. Measurement using our present test bed yields a difference between layout to fabrication of 2.5 μm with an uncertainty of ±0.062μm. We validated our measurement against scanning electron microscope in [6]. Analysis for various sources of noise contributions to the uncertainty in our measurement follows. 4 NOISE MODELS In our noise model we assume to have four source of noise: noise from the power supply voltage source; noise from internal electronics; noise from external electromagnetic radiation; and thermal noise. In Equation (13) we lump the totality of noise into the uncertainty in capacitance δ C . In Figure 4 we provide our computational noise model, which we used to better understand our experimental results. Our computational model comprises an experimentallyfitted finite element model of our device modeled in COMSOL [10], coupled to a various sources of noise inputs to the model by using SIMULINK [9]. Our computer model is displayed graphically in Figure 5.
Figure 3: EMM test bed. The MEMS structure under test and the testing apparatus are enclosed within a Faraday cage to reduce the noise due to external electromagnetic interference. Major components of the test bed are: (i) a 7.6 cm wafer comprising a several folded flexure comb drive structures; (ii) a vacuum chuck to hold the wafer in place; (iii) micromanipulators to position electronic probes; (iv) a digital microscope camera mounted on a boom stand; and (v) the Faraday cage.
4.1 NOISE DUE TO VOLTAGE SUPPLY As discussed in Section 1, we obtain a change in capacitance by applying a voltage across the comb drive. The voltage produces an electrostatic force which deflects the comb drive to yield a change in capacitance. However, if the applied voltage is perturbed, then so too is the corresponding comb drive force. This disturbance in force changes the comb drive deflection, which affects the magnitude of measured capacitance. The voltage supply used in our present
Figure 4: SIMULINK schematic our noise model. Using SIMULINK, we explore various types and combinations of noise sources to our finite element model of our MEMS (see Figure 1). Sources of noise contributions include the power supply, system electronics, external electromagnetic fields, and thermal noise. Although the output of our model is displacement, we show in Section 5 that deflection is directly related to EMM geometry. See Figure 5 for COMSOL model.
49
analysis is the Agilent 6645A High Voltage Power Supply. We selected this voltage source due to its wide voltage range, which is necessary for our related RF investigations. The 170V range is able to actuate our MEMS o structure 200μm. At the nominal 25 C operating condition, the power supply has a DC noise voltage of 0.06% of programmed voltage plus an additional 51mV, and an AC noise voltage of 7mV peaktopeak. With the voltage supply acting alone in our computational model, we apply the maximal voltage plus and then minus the uncertainty in voltage. The model yields an uncertainty in deflection that corresponds to an uncertainty in geometry of 4nm. That is, the change in geometry ( Δw ) from layout to fabrication that is due to error in the applied voltage supply alone is Δw ± 4nm. 4.2 NOISE FROM INTERNAL ELECTRONICS AND FROM EXTERNAL ELECTROMAGNETIC RADIATION Our present test bed yields an uncertainty in capacitance of 1.2fF. We obtain this value by enclosing our MEMS in a Faraday cage, connecting it to our capacitance meter, and determining the standard deviation of our measured capacitance distribution curve. For each measurement, we disconnected the electrical probes from the electrode pads, which create a slightly different contact parasitic capacitance upon each trial. After 90 trials, we compute the uncertainty as the standard deviation of our distribution curve. We apply this 1.2fF uncertainty in capacitance alone to our computational model (Figure 4) as a change in capacitance to determine the equivalent deflection. In doing so, we compute an 800nm displacement, which corresponds to an equivalent geometric uncertainty of 62nm. By opening the Faraday cage to expose our MEMS to radio, light, and other forms of electromagnetic radiation, we measure an uncertainty in capacitance of 5.1fF. Similarly, this value of uncertainty was obtained by taking a multitude of measurements of capacitance and calculating the standard deviation of the distribution data. Upon applying the 5.1fF uncertainty due to external
Figure 5: Finite element model. For computational efficiency, we model half of the symmetric folded flexure of our microdevice and do not show the comb drives. To produce the correct performance, we double its stiffness and we apply a resultant electrostatic comb drive force instead of including electrostatic physics about a multitude of comb drive fingers. Not including the field calculations saves a lot of computational time. We configure the geometry of the models to match the true geometries. The color map corresponds to total deflection. I.e., blue for zero deflection to red for maximum deflection. The length and width of the flexures are 2497.5μm and 9.5μm. The layer thickness and Young’s modulus are 20μm, and 215GPa\+14.59%, where the 14.59% adjustment is used to match simulation to the measured deflection.
electromagnetic (EM) radiation to our computational model (Figure 4) we compute a deflection of 4.9μm, which corresponds to an uncertainty in geometry of 263.5nn. Figure 6 shows the two states of our test bed: shielded external EM radiation versus exposed to external EM radiation. 4.3 THERMAL NOISE MEMS are susceptible to thermallyinduced vibration due to small stiffness values. Since stiffness scales by a factor of lengthscale L , then as the size of the device decreases, then
Figure 5: Test bed states for external radiation. With the Faraday cage open to EM exposure (left), the uncertainty in capacitance is 5.1fF. And with experiment shielded from EM radiation (right) the uncertainty in capacitance is 1.2 fF.
50
so to dose the stiffness. As given in [13] by Hutter and Bechhoefer, the relationship between the expected amplitude of deflection and temperature is
1 1 k y 2 = k BT , 2 2
(17)
where kB is Boltzmann constant and T is the absolute temperature. Solving (17) for the expected amplitude due to thermallyinduced vibrations, we have
y=
k BT . k
(18)
The stiffness k of our computational model is Figure 4: Displacement amplitude versus temperature. 0.267N/m. From (18), a temperature of 300K is The stiffness used for this plot is extracted from our MEMS expected to yield an amplitude of motion of shown in (Figure 5). Using three flexure changes in widths of 0.12nm. This corresponds to an uncertainty 2.4μm, 2.5μm, and 2.6μm, the corresponding stiffnesses are contribution in geometry of 0.014nm. We plot 0.276N/m, 0.267N/m, and 0.259N/m. For temperatures the expected relationship between deflection ranging from 100K to 600K, the thermallyinduced and temperature for our device Figure 7. displacement amplitude ranges from ~7nm to ~18nm. Measuring thermal noise is beyond the resolution of our present device and capacitance meter. Improvements to our experiment are underway. 5. CONCLUSION Since small variations in geometry can yield very large variations in performance in many MEMS, measurement of geometry of each device may be required. Electro Micro Metrology (EMM) offers a quick and practical way to measure geometry by using offtheshelf equipment that is readily available in most laboratories. Depending on the analyst’s desired precision in measurement, an understanding the effect of various sources of noise on EMM can be used to reduce the uncertainty in EMM measurements of geometry. In this paper we examined the effect of four sources of noise on geometric measurement: noise from the voltage source, internal noise of the capacitance meter, noise from external electromagnetic fields, and thermal noise. By applying our experimental measurements of noise to our EMM analytical model and computer model, we determined the effect of noise on capacitive measurements of geometry. With our test bed equipment and MEMS sample, we found the following. By not shielding our experiment from external electromagnetic fields from the environment, we found that this condition resulted in the largest uncertainty in measurement of geometry, δ w = 263nm. By shielding our experiment, the effect on geometric uncertainty due to noise within the capacitance meter (i.e. meter precision) was found to be δ w = 62nm. And predicted geometric uncertainties due to variation in voltage from the voltage source and due to thermallyinduced vibrations had the smallest effect, δ w = 4nm and δ w = 0.12nm respectively. The next step in this effort is to use the results found in this work to further reduce the uncertainty in EMM measurement of geometry for both unshielded and shielded measurements. We expect this can be done by modifying our MEMS design to improve its sensitivity to such measurement, and by improving the resolution of capacitance measurement. We are also examining the case of variation mismatch by relaxing our assumption that both structures share the same variations in geometry. REFERENCES
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
H. I. Smith, “Review of submicron lithography,” Superlattices and Microstructures, v. 2, n. 2, pp. 129142, 1985. B. Wu and A. Kumar, “Extreme ultraviolet lithography: A review,” Journal of Vacuum Science and Technology B: Microelectronics and Nanometer Structures, v. 25, n. 6, pp. 17431761, 2007. nd M. J. Madou, “Fundamentals of Microfabrication (2 ed.),” CRC Press, 2002. J. V. Clark, “Electro Micro Metrology,” Ph.D. Dissertation, University of California Berkeley, Berkeley, California, 2005. E. Novak, “MEMS metrology techniques,” Progress in Biomedical Optics and Imaging – Proceedings of SPIE, v. 5716, pp. 173181, 2005. J. Chee, J. V. Clark, and D. Peroulis, “Measuring the Planar Geometry of MEMS by Measuring Comb Drive Capacitance” (In review). T. Tran, D. R. Oliver, D. J. Thomson, and G. E. Bridges, “Subzeptofarad sensitivity scanning capacitance microscopy,” Canadian Conference on Electrical and Computer Engineering, v. 1, pp. 455459, 2002. Analog Devices AD7746, Analog Devices, Inc., 3 Technology Way, Norwood, MA 02062, United States. Matlab Simulink, The Mathworks, 3 Apple Hill Drive, Natick, MA 01760. COMSOL Multiphysics, COMSOL, 1 New England Executive Park, Suite 350, Burlington, MA 01803. R. K. Gupta, “Electronically Probed Measurements of MEMS Geometries,” Journal of Microelectromechanical Systems, vol. 9, no.3, pp. 380389, (2000). J. Green, et. al., “New iMEMS Angular RateSensing Gyroscope,” Analog Devices, Dialogue, pp. 3703, (2003). Hutter and Bechhoefer, “Calibration of AtomicForce Microscope Tips,” Review of Scientific Instruments, No. 64 (7) pp. 18681873, (1993).
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Effects of Clearance on Thick, SingleLap Bolted Joints Using ThroughtheThickness Measuring Techniques
John Woodruff, Giuseppe Marannano, Gaetano Restivo Composite Vehicle Research Center, Michigan State University 2727 Alliance Drive, Lansing, MI 48910 Abstract Composite materials have increasingly become more common in ground transportation. As this occured thicker panels, as compared to composite panels used in aviation, become necessary in order to withstand high impact loads and day to day degradation. The effectiveness of these panels was often limited by the strength of the joint in which the panel was attached to the frame of the vehicle. Investigating methods of reducing strain concentrations within these joints would increase the effectiveness in using composite materials in ground transportation applications by increasing the load necessary for joint failure to occur. In this study, fiber optic strain gages were embedded in a composite panel along the bearing plane of a thick, singlelap, bolted joint. The gages allow for the strain profile above the hole to be determined experimentally. Several clearance values were then implemented in the bolt to determine their effect on the strain concentrations. Strain increased at every gage, by nearly the same proportion, when clearance was increased from zero to three percent. When clearance was further increased to five percent strain only continued to increase at gages three and four, with one and two remaining similar in value to what was seen at three percent clearance. Ultimately, like in thin composite panels, the zero percent clearance condition was the stiffest. Introduction Large advancements have occurred in the field of composite materials for use in automotive and aerospace application. Advancements have been generated by demands for “greener”, more fuel efficient vehicles that also maintained or increased in overall safety capabilities. In order to maintain the safety aspects of a ground transportation vehicle, thicker panels were developed. Panels were made thicker in order to provide a safe ride in a more hazardous environment. Such an environment would include heavy day to day wear and tear, as well as the prospect of collisions with other objects or vehicles. Other hazards, such as projectiles, have been included in this environment when composite panels were used in military vehicles. Failure of the composite panel by impact with these objects has been a real possibility. The location where the composite panel is fastened to the metallic frame is of great importance. At this location stress concentrations develop upon impact with objects in the course of the vehicle’s journey. And stress concentrations may cause the panel to fail. If a composite panel is to be utilized to its full potential, then the load transfer between the composite panel and the aluminum frame of the vehicle has to be further developed. A singlelap, bolted joint has typically been used to represent the fastening of the composite panel to the aluminum frame vehicle for experimental purposes. The test setup typically consists of a single composite plate bolted to an aluminum plate of equal dimensions. Bolting is chosen as the means of fastening since it provides the fastest method of securing or removing a panel from a vehicle frame, while still providing a strong connection. The singlelap bolted joint is a standard test setup for composite fastening. Clearance between the bolt and the hole of the composite panel has been an important factor in the strength of the joint. When the lap joint is pulled in tension, the bolt tilts, which provides a variation in the contact surface area between the bolt and the hole surface. The larger the contact surface area that is maintained during testing the better the load distribution between the bolt and the hole surface. Initial clearance between the bolt and the
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_7, © The Society for Experimental Mechanics, Inc. 2011
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hole has a significant effect on the ability to maintain the maximum surface area contact and the strength of the joint. In order to properly develop the singlelap, bolted joint, a literature review was conducted. Papers were compiled and reviewed in order to determine what was currently known about clearance in thick composite panels. Studies reviewed included many numerical and experimental techniques. A study was performed with a thin singlelap joint with an 8 mm diameter hole, which was tested for small clearance values [1]. It was seen that as clearance increased, the contact surface area decreased, from 160 – 170 degrees around the hole for a zero percent clearance condition, to 105 – 110 degrees at the largest clearance value of three percent. The same author performed another study which had shown the effects of clearance on stiffness and how bearing strength was affected [2]. In this study, increased clearance showed a decrease in joint stiffness. However ultimate bearing strength was not affected by clearance. Other numerical studies have found similar results when clearance is tested to determine its effects [3]. One such study concluded that clearance decreased the load capacity of the joint and was overall a negative design characteristic. This was a very general statement, but in line with other research groups findings. Interest in further analyzing clearance effects on the lap joint has led to developments in measuring techniques. Fiber optic strain gages of various types have been considered for use in taking throughthethickness measurements within the composite panel. These gages have the advantage of being very small in size, are immune to electromagnetic interference and they can be embedded noninvasively into a composite panel [5]. The material properties in the region of the gage do not change and point measurements of strain are available. Measurements can then be used to experimentally validate finite element models. One study exists as an attempt at developing an understanding of the strain profile at low loads [6]. In this study, fiber optic strain gages were embedded within a thick 12.7mm (0.5”) composite panel of the singlelap joint at regular intervals throughthethickness above the hole. Numerical analyses were performed using ANSYS as preprocessor and LSDYNA as solver. The overall goal was to evaluate the magnitude of contact strains around the hole and through the thickness of the composite. The values were analyzed and compared with the FEM results: the finite element analysis correlated reasonably well with the experiments. An investigation of error causes was also carried out, in particular to evaluate the influence of incorrect gage positioning and the effect of friction coefficients. The next logical step in the development of the lapjoint was then to use embedded fiber optic strain gage technology to understand the effects of clearance. Creating a specimen similar to [6] provided a more thorough understanding of the actual strain profile when tested through a higher loading range. Also, changes in the strain profile above the hole when clearance is present were experimentally determined. Composite Manufacturing A composite specimen for testing was constructed using a hand layup process with vacuum bagging. The process included attaching the Bragg grating fiber optic strain gages to the plies prior to creating the specimen. Then, once the gages are secured and their location marked, the layup process began. The plies were inserted with the attached gages in the proper order to know their location in the thickness direction of the final specimen. Finally a vacuum bag was used to pull out any unnecessary resin and obtain a higher fiber volume fraction in the specimen. The specimen tested was constructed from a plain weave SGlass material and an epoxy resin. The SGlass was chosen due to having superior tensile strength than the Eglass. The epoxy resin is 635 Epoxy and uses a 3:1 ratio of Epoxy to hardener. The epoxy, hardener and fiber are supplied by US Composites. The actual dimensions of the panel are chosen as ratios of the hole diameter. The hole diameter was known to be 12.7mm (0.5 inches) for this panel. A thickness to hole diameter ratio of 1:1 was used. Also, the ratio of edge distance to hole diameter was 4:1 moving laterally from the hole, and 3:1 from the top of the plate to the center of the hole. The locations of the gages within the composite plate are shown in Figure (1).
55
Figure 1: Dimensions of the FOS gage locations within the composite panel
The two fiber optic strain gages to be used in this experiment are provided by Technica SA and are located at the places designated as 3 and 4 on the above diagram. In order to obtain data for the 1 and 2 locations the specimen was simply reversed. The gages are Bragg grating fiber optic strain gages and have a gage length of 2mm with a maximum strain output of approximately 12,000 microstrain. The 2mm gage length was chosen since it is small, and works accurately for taking strain readings at specified points in the presence of a large strain gradient. Also, a 3mm protective armor cable was used to protect the internal fiber optic cable from shearing off at the ingress/egress point after construction. Further protection was provided at the adapter, where the cable connects to the interrogator. The gages were applied to a ply prior to the hand layup process. This was done by first marking the edges of the specimen and the location of the center of the gage with a thin black cotton string as shown in Figure (2).
56
Figure 2: Location of the gage and edges of specimen String was used so that during the layup process the plies with gages can be aligned via the string. Lastly, the gage was glued in place using the same epoxy and hardener that will be used during the hand layup process and is shown in Figure (3).
Figure 3: Gages are glued in place with epoxy resin
The composite panel was created large enough so that specimens for tensile tests could be cut from the same panel that the test specimen would be made from. Tensile tests were used to determine the material properties of the composite panel for use in a finite element model. By cutting tensile test specimens from the same panel the lap joint specimen was made from there is a high degree of accuracy in determining the material properties of the lap joint test specimen itself. The hand layup process was performed by inserting the plies with the gages at the desired interval. Overall, 60 plies were used including the two with gages attached. Since the gages were inserted at locations 0.1 inches and 0.2 inches in from the front surface, this meant they were located as plies 12 and 24. A vacuum bag system was used to pull extra resin out of the specimen after the layup process was completed. The setup for the panel construction can be viewed in Figure (4).
57
Figure 4: Vacuum bag setup for hand layup process.
Experimental Setup The specimen was tested using a tensile testing machine. Wedge grips were used to hold in place a mounting device created to hold the specimen. A mounting device was used since the lap joint specimen was too thick to fit into the wedge grips of the MTS machine. The device mentioned is pictured in Figure (5).
(A)
(B)
Figure 5: (A) Mechanical mounting device, (B) Specimen loaded into MTS Displacement was set to 1.0 mm per minute and the specimen was loaded from 0 – 10,500 N. This loading range was sufficient to develop a linear trend for stiffness data. Fiber optic strain gage data was compiled by Labview software. Testing was performed to determine the optimum bolt to hole clearance condition. For these tests the hole size would remain constant while the bolt would be varied in diameter to reflect clearance values of 0,1,2,3,4
58
and five percent. As a precaution, testing was first performed on gage locations three and four since lower strains were expected at the gages. After these tests were successfully concluded, tests were carried out for gage locations one and two. Experimental Results As mentioned, tensile tests were first performed to determine material properties for use in FEA. Young’s modulus for the through the thickness direction was specified as that of the matrix material. Table (1) shows the material properties. A completed FEA analysis was not available at the time of submission of the paper but will be shown during the presentation. Embedded Specimen Ex (GPa) 19.56 12 0.12133 Table 1: Material properties of the composite specimen Stiffness testing on the specimen provided results that are similar to what was seen in tests on much thinner panels. The stiffest condition was seen when there was no clearance between the bolt and the bolt hole. All clearance results can be seen in Table (2). The zero clearance condition was seen to be the optimal configuration of the joint, with the five percent clearance the poorest performing configuration.
!"#$ % &' &' ' '' ( ")
&%'
Table 2: Stiffness characteristics of each test configuration. Fiber optic strain gages have provided very interesting data of the strain profile through the thickness of the specimen as shown in Figure (7). Gage two had higher strains than gage one for all tests. Increased clearance from zero to three percent increased the strain seen at all gage by similar proportions to the values at zero percent. However, once clearance was further increased to five percent only the strains at the gages furthest from the interface between the aluminum plate and the composite panel continued to see increased strain.
59
*)#!+,,, "# ./" 0#"#
*)#
"# ") "
(A) *)#!+,,, "# ./" 0#"#
*)#
"# ") "
(B)
60
*)#!,+,,, "# ./" 0#"#
*)#
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(C) Figure 7: (A) 0 percent clearance, (B) 3 percent clearance, (C) 5 percent clearance Discussion During the fabrication of the specimen there was a slight misalignment of the Bragg grating fiber optic strain gages. Although less then a millimeter of horizontal misalignment was determined from visual inspection after construction, there was still error in strain readings associated with the misalignment. The cause of the misalignment was the handrolling process of applying the resin. Experimentation has shown the optimal value of clearance as well as interesting data on the strain profile during loading for the thick, single, lapbolted joint. There is much alignment between the stiffness data created in these tests and what has been seen for similar research projects on joints that were constructed of mainly thinner panels. In the embedded specimen, stiffness was seen to be optimized when clearance does not exist at all. There was an 11 percent decrease in stiffness from the zero percent to the one percent conditions. Such a large decrease in stiffness has shown the importance of maintaining a tight tolerance on the bolthole clearance for use of such a composite plate when used in field applications. The strain profile through the thickness at the two gage locations was very revealing. Past numerical studies have all shown that the highest strain values are at the interface of the composite plate and the aluminum plate and decrease substantially away from the interface of the two plates. Experimental evidence provided by the embedded Bragg grating fiber optic strain gages has revealed that the highest strain values are located a little further in from this interface at the location of gage two. However, gages one and two did both read substantially higher than gages three and four. The difference in what was seen in experimentation and other numerical studies was largely due to the sliding of the gages during the embedding process. Conclusions The focal point of this study was to determine the effects of clearance on the strain profile through the thickness of the specimen and the optimal stiffness value for a thick composite panel. The composite specimen contained 60 layers of Sglass in an Epoxy resin resulting in a 12.7mm (0.5 inch) thick composite panel. Point measurements
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for strain in the bearing plane of the composite specimen were determined experimentally using Bragg grating fiber optic strain gages embedded into the specimen. Analysis of the data provided conclusions regarding the effects of clearance on strain concentrations and on the stiffness of the joint. The concentration was highest at gage two, which was toward the front of the specimen, but further into the thickness than gage 1. Also, strain seen under three percent of clearance was increased from the zero percent clearance conditions at all gages by approximately the same proportion. A further increase in strain to five percent only resulted in an increase in strain at gages three and four, the gages furthest from the interface between the two plates. Stiffness was seen to follow similar trends to thin composite panels. Any increase in clearance beyond the zero clearance condition lead to a decrease in stiffness. Zero clearance was then the optimum condition at 10.02 KN / mm and five percent was the least stiff at 8.7 KN / mm.
REFERENCES [1] McCarthy, M.A., McCarthy, C.T. (2004). “Threedimensional finite element analysis of singlebolt, singlelap composite bolted joints: part IIeffects of bolthole clearance” Composite Structures 71: 159175 [2] McCarthy, M.A., Lawlor, V.P., et al. (2002) “Bolthole clearance effects and strength criteria in singlebolt, singlelap, composite bolted joints.” Composites Science and Technology 62: 14151431. [3] Chen, WenHwa, Lee, ShyhShiaw, et al. (1995) “Threedimensional contact stress analysis of a composite laminate with bolted joint.” Composite Structures 30: 287297 [4] Baldwin, C., Mendex, Alexis. (2005) “Introduction to Fiber Optic sensing with Emphasis on Bragg Grating Sensor Technologies; short course 102” SEM Annual Conference and Exposition on Experimental and Applied Mechanics [5] LopezAnido, R., Fifield, S. (2003). “Experimental Methodology for Embedding Fiber Optic Strain Sensors in Fiber Reinforced Composites Fabricated by the VARTM/SCRIMP Process”. Structural Health Monitoring 247 – 254. [6] Restivo, G., Marannano G., Isaicu, G.A. (2010). “ThreeDimensional Strain Analysis of SingleLap Bolted Joints in Thick Composites using FiberOptic Strain gages and Finite Element Method”, Journal of Strain Analysis for Engineering Design, Accepted for publication.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Deformation and Performance Measurements of MAV Flapping Wings Wu, Pin. University of Florida, MAEA #231, Gainesville, FL, 32611
INTRODUCTION If bumblebees and hummingbirds could speak to us, could they tell us how they fly? Probably not. “How they fly” has been a fascinating question to biologists and aerodynamicists. Recently, attention is directed to micro air vehicle (MAV) research, which is aimed to develop sub 150 mm wingspan aircraft for reconnaissance and surveillance. The hummingbird poses as a perfect emulation target: they can dash like a jet fighter, hover like a helicopter, and they are on the MAV length scale. Warrick et al.1 examined the aerodynamics of hummingbird hovering with digital particle image correlation to capture the airflow structure. The authors found that the hummingbird’s upstroke and downstroke are not symmetrical in producing lift (thrust): the downstroke responsible for about 75% of the body weight and the upstroke about 25%. This is very different from insects, which have a more symmetrical load distribution. The differences are results of wing kinematics and structure. If a robotic hummingbird or insect is to be developed, understanding the causal relationship between kinematics, deformation and aerodynamics is essential. On the other hand, Tobalske et al.2 documented the kinematics of hummingbirds in forward flight at different speeds. The authors used a few parameters to described wing trajectories and angles. However such description may be considered insufficient for reconstructing the same kinematics. Therefore, in order to facilitate the research of flapping wing MAVs, an experimental method that can describe the complete wing kinematics and deformation, and correlate with aerodynamic loads, is called for. This paper presents an experimental technique for studying hummingbirdsize flapping wings in MAV research. A sophisticated experimental setup featuring a customized digital image correlation system is described; several anisotropic flexible membranous wings are tested and post processed results are presented. EXPERIMENTAL SETUP The experimental setup consists of a flapping mechanism, a force and torque sensor, a digital image correlation (DIC) system, a vacuum chamber, composite wings and computer user interface, shown in Figure 1. The flapping mechanism actuates the wings in a frequency range of 0~40 Hz at ±35º. It can be adjusted for different amplitudes. A force and torque sensor (load cell) Nano 17 is mounted underneath the mechanism.
Figure 1. The experimental setup for flapping wing kinematics and deformation measurements. The Nano17 (0.3 gram sensitivity) is used to measure the aerodynamic loads produced by the flapping wings. For measuring wing kinematics and deformation, a fourcamera DIC system with stroboscope is used. The use of two pairs of cameras allows capturing wing surface at large flapping and rotation angles. A vacuum chamber is used to isolate inertial deformation from aerodynamic effects. The user interface allows the computer to control all T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_8, © The Society for Experimental Mechanics, Inc. 2011
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instruments simultaneously. The measurement procedure is shown in Figure 2, including input, operation, data acquisition and analysis. On the right, data correlation results between average thrust and deformation is plotted.
Figure 2. Measurement procedure on the left, correlation results on the right. RESULTS AND CONCLUSIONS Results are shown in Figure 2 and Figure 3. In Figure 2, the average thrust data correlates well with maximal wing tip deflection and twist. In Figure 3, upstroke and downstroke kinematics are shown with color contour describing the out of surface deformation (w/c, normalized to the chord length 25 mm). The grey line indicates the rigid body kinematics without any deformation. Comparison between the results in air and vacuum identifies the deformation caused by aerodynamic effects. In conclusion, an experimental technique is developed to measure the kinematics, deformation and performance of flapping wings for micro air vehicle research.
Figure 3. Wing kinematics and deformation measurement at 25 Hz in both air and vacuum. REFERENCES 1. Warrick, D.R., Tobalske, B.W., and Powers, D.R., “Aerodynamics of the Hovering Hummingbird,” Nature, Vol. 435 No. 23, June, 2005, pp. 10941097. DOI: 10.1038 2. Tobalske, B.W., Warrick, D.R., Clark, C.J., Powers, D.R., Hedrick, T.L., Hyder, G.A., and Biewener, A., “Threedimensional Kinematics of Hummingbird Flight,” J. of Exp. Bio., Vol. 210, 2007, pp. 23682382.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Dynamic Constitutive Behavior of Aluminum Alloys: Experimental & Numerical Studies Sandeep Abotula Department of Mechanical, Industrial and Systems Engineering University of Rhode Island, Kingston, RI 02881, USA ABSTRACT Split Hopkinson pressure bar (SHPB) setup was used to investigate dynamic constitutive behavior of aerospace Aluminum alloys both experimentally and numerically. The study was conducted in the strain rate regime of 500/s 10000/s. Both regular solid and modified hollow transmission bars were employed in realizing this strain rate regime. Four different Aluminum alloys namely 7075T4, 2024T3, 6061T6 and 5182O were considered for investigation. Copper110 alloy pulse shaper was used to obtain better force equilibrium conditions at the barspecimen interfaces. Plastic kinematic model was used to model rate dependent behavior of Aluminum alloys using commercially available LSDYNA software. It was identified from the final results that experimentally determined dynamic constitutive behavior matches very well with that of numerical in the strain rate regime of 2000/s 5000/s. INTRODUCTION The growing requirement for fuel efficient vehicles has made a renewed interest in aluminum alloys as a replacement for other metals in aerospace and automobile bodies due to their high strengthtoweight ratio. When studying the crashworthiness of these vehicles, dynamic behavior of these alloys must be well understood in developing numerical models. For the first time, Campbell [1] reported dynamic constitutive behavior of Aluminum alloy by subjecting long rods to compression impact and showed significant difference in the flow stress when it was compared with quasistatic case. Maiden and Green [2] conducted experiments using Hopkinson pressure bar setup to and found out that Aluminum 6061T651 and 7075T6 alloys showed no strain rate sensitivity in that strain range. Based on above literature search, it was identified that there exists no consistency in the results of highstrain rate constitutive behavior of aluminum alloys. Hence, for the first time, this paper discuss about both experimental and numerical investigation of highstrain rate behavior of four different aerospace aluminum alloys 7075T4, 2024T3, 6061T6 and 5182O under above said range. EXPERIMENTAL DETAILS Quasistatic Characterization The quasistatic compression tests were performed using Instron materials testing system5585 as per ASTM standard E9. Experiments were performed at 10mm/min extension rate and the tests were continued until crosshead extension reaches a value of 2.5mm. Since AA 5182O was received as a sheet material, a dog bone specimen configuration as per ASTM standard E8M was tested using Instron materials testing system 5582. Dynamic Characterization Traditional Split Hopkinson Pressure Bar (SHPB) was used to study the dynamic behavior of aluminum alloys. Incident and transmission bars (Maraging steel) have the diameter of 12.5mm and length up to 1220mm. Copper 110 alloy was used as a pulse shaper to generate force equilibrium conditions. Since the AA5182O has thickness of 1.70mm, a diameter of 3.22mm hollow transmission bar was used. Due to brevity of space, the description and theory of SHPB is not explained here and it can be referred in Kolsky[3] paper. NUMERICAL MODELING DETAILS To study the dynamic behavior of these aluminum alloys, three dimensional axisymmetric split Hopkinson pressure bar setup was modeled using the commercially available LSDYNA software package for strain rates ranging from 1700/s to 7000/s. All components were modeled using 3D Solid 164 element type. CowperSymonds model based on plastic kinematic material model available in LSDYNA was used to model strain rate constitutive material behavior of all four aluminum alloys. T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_9, © The Society for Experimental Mechanics, Inc. 2011
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RESULTS AND DISCUSSION
Fig. 1. Typical real time strain pulses obtained from SHPB
Fig. 3. Force equilibrium conditions for Al 7075T4
Fig. 2. True stress strain curve for Al 7075T4
Fig. 4. Comparison of numerical dynamic true stressstrain behavior with those of experiments for Al 7075T4
Dynamic Constitutive Response The typical realtime strain pulses obtained for Al 6061T6 alloy for an average strain rate of 3025/s is shown in Fig. 1. The dynamic true stress strain curves for different strain rates are plotted against the quasistatic true stressstrain curves for Al 7075T4 in Fig. 2. It can be noticed from the figure that Al 7075T4 shows no significant rate sensitivity and there is no change in the yield strength as the strain rate increases. Force equilibrium conditions of Al 7075T4 is shown in Fig. 3. and it can be seen that good equilibrium can be achieved by using C110 alloy as a pulse shaper. Comparison of experimental and numerical true stress strain behavior is plotted in Fig. 4. Figure shows that the experimental results match very well with the numerical results. ACKNOWLEDGEMENTS I would like to thank Dr.Vijaya Chalivendra for guiding me throughout the project. Also I would like to thank Dr. Arun Shukla for his valuable suggestions on this paper. REFERENCES 1. Campbell, J. D.: An Investigation of the Plastic Behavior of Metal Rods Subjected to Longitudinal Impact. Journal of mechanical Physics Solids, 1, 113123, 1953. 2. Maiden, C. J., Green, S. J.: Compressive StrainRate Tests on Six Selected Materials at Strainrates from −3 10 to 10° in./in./sec. Journal of Applied Mechanics, 33, 496504 1966. 3. Kolsky, H.: An investigation of mechanical properties of materials at very high strain rates of loadings, Proceedings of the Physical Society of London, B62, 676700, 1949.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Estimating surface coverage of gold nanoparticles deposited on MEMS
a
N. Ansaria,*, K. M. Hursta and W. R. Ashursta Department of Chemical Engineering, Auburn University, Auburn, Al36849, USA * Corresponding author: [emailprotected] [N. Ansari]
Introduction Commercialization of a whole spectrum of useful MEMS is still hindered by surface phenomena that dominate at the micron scale. Altering the roughness and surface chemistry of MEMS surfaces by depositing nanoparticles on them is being considered by the MEMS community as a useful strategy to address tribological issues. Although, gold nanoparticle monolayer is reported to reduce adhesion in MEMS, determining its surface coverage still remains a challenge [1]. A technique to determine the surface coverage of deposited gold nanoparticles is needed, so that its effect on the tribology of MEMS surfaces can be studied. Design, Fabrication and Testing Procedure of the Test Structure The design of our test structure (referred to as Resonator) is based on a single mask scheme, thereby making its fabrication facile and inexpensive. The Resonator is fabricated using the standard surface micromachining technology on a SOI wafer. The resonating structure is fabricated in a 1.84 μm thin film of Si(100) and is suspended 2 μm above the substrate, which is 500 μm thick. Fig. 1 is an optical image of a fabricated and released Resonator
Figure 1: Optical image of a released Resonator. The Resonator is actuated electrostatically. Electrical contacts are made by touching the structure with sharp tungsten probe tips. The resonating structure, ground plane and one set of comb electrodes are grounded and an AC voltage with a DC offset is applied to the other set of comb electrodes. The resonator is observed under high magnification as the frequency of the driving signal is gradually increased. Resonance is determined optically using the pattern etched on the resonating structure. Estimation of Surface Coverage The resonance frequency (fR) of the resonator is given by eq.1, where kx is the stiffness of the folded beams supporting the suspended structure and Meff is the effective mass of the resonating structure. The stiffness of the folded beams as given by eq.2 depends on the Young’s modulus (E) and the thickness (t) of the structural film as T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_10, © The Society for Experimental Mechanics, Inc. 2011
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fR
1 kx 2ʌ Meff
[1]
well as the width (w) and length (L) of the folded beams. The deposition of gold nanoparticles on the resonating structure increases the mass of the resonating structure, which in turn decreases its resonance frequency according to eq.1. By comparing the resonance frequency of gold nanoparticle coated and uncoated resonators, the increase in the mass of the coated resonator, which is the mass of deposited gold nanoparticles can be determined using eq.3. This increase in the mass of the coated resonator is then used to determine the coverage of gold nanoparticle monolayer deposited on the resonator. kx
ǻMeff
2Et
3
( wL )
[2]
4ʌ M3eff2 ǻ f R
[3]
kx
Deposition of Gold Nanopaticles After fabrication, the Resonator is released by etching it in conc. (49 wt.%) HF. The etchant is completely displaced by deionized (DI) water, after which the Resonator is placed in hot H2O2 (Temp. = 75ºC) for 10 mins. H2O2 is further displaced by DI water, which in turn is displaced by isopropanol (IPA). After completely rinsing away the IPA with anhydrous hexane, gold nanoparticles are deposited on the Resonator using the CO2 gasexpanded liquid technique reported by Hurst et. al. [1]. Results and Discussion Length (μm) 300 350 400 450 500
Resonance Frequency, fR +/ std.dev. (KHz) Uncoated Coated 8.34 +/ 0.01 8.28 +/ 0.01 6.14 +/ 0.01 6.09 +/ 0.01 4.72 +/ 0.01 4.68 +/ 0.01 3.92 +/ 0.01 3.88 +/ 0.01 3.64 +/ 0.01 3.61 +/ 0.01
¨Meff × 1012 (Kg) 4.0 +/ 0.31 3.7 +/ 0.36 4.1 +/ 0.45 4.5 +/ 0.55 4.2 +/ 0.71
Coverage of Au nanoparticles (%) 61 +/ 4.7 59 +/ 5.6 63 +/ 6.9 68 +/ 8.3 61 +/ 10.1
Table 1: Surface coverage of gold nanoparticles deposited using the CO2 gasexpanded liquid technique on a MEMS chip. Resonators from different regions of the chip coated with gold nanoparticles are tested to determine the uniformity of coating. Table 1 above lists the resonance frequencies of both uncoated and gold nanoparticle coated resonators along with the lengths of their folded suspension beams. The results reported in table 1 indicate that the coating is uniform throughout the chip and it has a coverage of 64 +/ 4 %. Conclusions We have demonstrated a simple technique based on optically determined resonance to determine the coverage and uniformity of a gold nanoparticle monolayer deposited on a MEMS chip using the CO2 gasexpanded liquid technique. This technique can also be used to compare the number densities of gold nanoparticle solutions, since the coverage of gold nanoparticle monolayer depends strongly on the number density of the starting gold nanoparticle solution. References 1. Hurst, K. M., Roberts, C. B., Ashurst, W. R., “A gasexpanded liquid nanoparticle deposition technique for reducing the adhesion of silicon microstructures”, Nanotechnology, 20(18), 185303(9pp), 2009.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Functionally Graded Metallic Structure for Bone Replacement S. Bender University of Massachusetts Dartmouth, 285 Old Westport Rd. North Dartmouth MA, 02747 ABSTRACT Processes for the creation and characterization of functionally graded metallic structures for use as artificial bone tissue were investigated. The metallic structure consists of a solid surface layer, a graded porosity layer and a hom*ogeneous porous core. Porous compacts with varied densities were created using traditional powder metallurgy techniques. The surfaces of the compacts were subjected to a densification process with the use of a specially designed indentation tool. Investigations on the effect of the initial density of the compact and the depth of indentation during the deification process on the densified layer and graded porosity region were studied. Compacts with an initial density of 86 % of the true density were indented to depths of approximately 1.8, 1.25, 1, and 0.65 mm. Compacts with initial densities of 67, 70, and 73% were indented to a depth of 1 mm. Optical microscopy and scanning electron microscopy (SEM) were implemented to characterize the morphology of the porous structure. Results show that deeper indentation during the densification process yielded a larger densified layer. The variation of Young’s modulus along the porosity gradation is investigated using microindentation. The graded structures are also investigated for fracture parameters and crack growth behavior using digital image correlation techniques. INTRODUCTION Bone tissue replication is a rapidly evolving field of study. Many methods and materials are used to create artificial bone structures with porosity distributions. Pelletier et al. [1] tested poly(methyl methacrylate) cured at different temperatures and found at higher temperatures they could obtain a solid structure with a porous core. Tape casting of alumina ceramics has also been used to create a porous region in a solid structure and was show to have enhanced toughness values by Zeming et al [2]. Most of the research involving metallic structures has been focused on purely porous structures, such as the work by GradzkaDahlke et al. [3]. This paper suggests a method for creating a metallic structure with a porous core, a graded porosity layer, and a solid surface layer. EXPERIMENTAL METHODS Iron compacts were created using traditional powder metallurgy techniques. The powder was compacted in a die and later sintered in an argon atmosphere. An initial density of the sintered compact of 86% of the true density of iron was obtained. Since stainless steel powder does not posses the favorable compactability similar to iron, loose sintering techniques, similar to those described by El Wakil [4], were used to produce the porous stainless steel substrates. To achieve a variety of substrate densities the sintering temperature was varied. Sintering temperatures of 1300, 1350, and 1400°C produced initial densities of 67, 70, and 73% respectively. The surface densification of the porous substrate was achieved through the use of a specially designed indentation tool. The tool consists of a ballpoint which is used to penetrate into surface of the substrate. Iron substrates with an initial density of 86% were densified to depths of 1.78, 1.27, 1, and 0.64 mm. The stainless steel substrates of initial densities of 67, 70, and 73 % were all densified to a depth of 1 mm. RESULTS Cut and polished cross sections of the densified surface were imaged using optical microscopy. Image analysis using MATLAB tools, was performed to determine the morphology of the porous structure from the solid surface to the porous core. Figure 1 shows the image analysis for the maximum and minimum densification depths (1.78 and .64 mm). The length of the solid region is observed to increase as the depth of indentation increases. The corresponding solid region lengths are 0.3, 0.65, 0.75 and 0.85 mm for densification depths of 0.64, 1, 1.27, and 1.78 mm respectively. The density of the solid region also appeared to increase about 1% for each of the densification depths. Figure 2 shows the results of the stainless steel substrates of varied initial densities that were densified to a depth of 1 mm. It was observed that the difference in the initial density had a negligible effect on the length of the solid surface. An increase in initial density did show a decrease in slope of the change in T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_11, © The Society for Experimental Mechanics, Inc. 2011
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a) b) Figure 1. Image analysis of density distribution for densified iron samples. a) 1.78 mm indentation depth, b) .64 mm indentation depth
porosity in the graded porous region. SEM images of the 1.78 mm densification depth on an iron substrate were taken at increasing depth below the surface of densification. In the region of the solid surface it was observed that the pores were mostly collapsed. While images toward the core region reveled a decreased ratio of collapsed pores to rounded pores. The SEM images at 0.5 mm intervals below the densified surface are shown in figure 3.
Figure 2. Image analysis for densification of stainless steel
Figure 3. SEM images of pore morphology at increasing depths below the surface of densification for iron substrate densified to a depth of 1.78 mm
EXPERIMENTS IN PROGRESS Micro indentation experiments using a Vickers indentation tip are in progress to determine the mechanical properties of the solid surface and the graded porous region. Quasistatic fracture and flexure tests are also in progress. Both hom*ogenously porous and densified samples are used to determine the porosity gradient’s effect on fracture toughness and flexure strength. The results of these experiments will be presented at the conference. REFFERENCES [1] Pelletier M. H. et al., Pore Distribution and Mechanical Properties of Bone Cement Cured at Different Temperatures, Acta Biomaterialia, Actbio 995, 2009 [2] Zeming He et al, Dynamic Fracture Behavior of Layer Alumina Ceramics…, Materials Letters, 59, 901904, 2005 [3] GradzkaDahlke M. et al., Modification of Mechanical Properties of Sintered Implant Materials…, Journal of Materials Processing Technologies, 204, 199205, 2008 [4] El Wakil S.D., Khalifa H., Cold Formability of Billets by Loose Powder Sintering, The International Journal of Powder Metallurgy and Powder Technologies, 19, 1, 2127, 1983
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Dissipative energy as an indicator of material microstructural evolution N. Connesson1,a 1
F. Maquin2,a
2
F. Pierron3,a 3
[emailprotected], [emailprotected], [emailprotected]
a
Laboratoire de mécanique et procédés de fabrication (LMPF), Arts et Métiers ParisTech, rue St Dominique, BP 508, 51006 ChâlonsenChampagne cedex, France
1. Introduction Rapid fatigue limit estimation methods are of strong interest to industry. Some authors proposed rapid experimental methods to estimate the fatigue limit based on the material temperature increase under cyclic loading [1,2]. Yet, heat dissipation phenomena need to be more thoroughly studied in order to give better physical grounds to such methods. Usually, the dissipative energy is explained by internal friction phenomena and is mainly attributed to dislocation movements [3]. The dissipative energy should thus depend on the material dislocation characteristics. Yet, these characteristics are usually modified during the loading history [46]. In this work, it was thus chosen to experimentally study the correlation between the dissipative energy and the cold work of a dual phase steel (DP600, Yield limit at 0.02% of 220 MPa), both phenomena being related to the microstructural dislocation structure. 2. Experimental procedure and results A specimen has been machined from a 2 mm thick DP600 steel plate and has been uniaxially loaded. The applied harmonic load has been defined by Rσ=σmin/σmax=0.1 where σmin and σmax are respectively the minimal and maximal stresses. For each loading sequence, as the plastic strain is monotonic with these loading characteristics, the solicitation has been maintained during a high enough number of cycles so that the material reached a steady state (stabilized mean strain). The material cold work has then been characterized after each p test: the residual plastic strain ε has been measured while no loading was applied to the specimen by using a strain gauge. During each loading sequence, the specimen temperature variation has also been measured with an infrared camera. The dissipative energy per cycle E m has then been estimated by solving the heat balance equation [7] d1
and by using the experimental method described in [8]. As this measurement has been performed in steady state condition and with a positive loading ratio, no alternate plastic strain occurred during the thermal acquisition: here, the dissipated energy is only due to internal friction phenomena and has not been influenced by the yield limit variation due to cold work (if the loading ratio Rσ were negative, the alternate plastic strain could have been modified with the cold work). p
Therefore, this experimental setup provides the plastic strain ε measurement while the specimen is not loaded and the dissipative energy Edm measurement under cyclic loading and in steady state behaviour. 1 Four measurements sequences (Phase I to IV) have been developed in this study to monitor the correlation between dissipative energy and plastic strain. These test sequences have been sketched in the upper graph of Figure 1(a). In this representation, each point represents three successive tests: • a cyclic loading at the maximal stress σmax (represented by the point ordinate) applied until a steady behaviour is reached, • then a dissipative energy measurement in the same loading conditions, p • and eventually a residual plastic strain measurement ε after unloading.
p
The cumulated plastic strain ε at the end of each test is presented in the bottom graph of Figure 1(a). The four measurement sequences have been designed to reach different goals: the material initial behaviour is analyzed during Phase I where small plastic stains occurred. The material is then plastically strained during Phase II and its dissipative energy evolution is analyzed. Phase III and IV are then performed to monitor the changes in the material dissipative energy due to the previous two phases. T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_12, © The Society for Experimental Mechanics, Inc. 2011
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The dissipative energy per cycle is presented for each phase versus the alternate stress σa and the maximal stress σmax in Figure 1(b). The dissipative energy per cycle increases with the alternate stress, which can be explained with a simple KelvinVoigt model. The dissipative energies during Phase I and the first steps of Phase II (Tests up to D0) are identical. Thus, the plastic strain occurring during Phase I does not change the dissipative energy behaviour; the dissipative energy measurement is reproducible if the material is not plastically strained (ie, no change in the dislocation density). The dissipative energies of the tests D0 to D10, which have all been performed at the same maximal stress σmax=240 MPa and in steady state behaviour, increase gradually along with the material cumulated plastic strain (sub graph in Figure 1(b)). The material microstructural changes occurring during Phase II have thus progressively increased the dissipated energy. Moreover, the dissipative energy during Phase III is always greater than the dissipative energy during Phase II. These results confirm that the dissipated energy is an indicator of the material microstructural state. Eventually, the measurements achieved during Phase IV point out a good reproducibility of the dissipative energy measurements on the same specimen. Phase III and IV are thus the characterization of the material dissipative energy stabilized behaviour and this curve could be seen as the "signature" of this material dislocation structure.
(a) Maximal stress σmax applied for each test and cumulated plastic strain at the end of each test
(b) Dissipative energy measurement versus alternate stress σa (bottom scale) and maximal stress σmax (top scale) Fig 1. Experimental sequence and results
3. Conclusion The dissipative energy of a dual phase steel has been proved to increase with the material cold work. The dissipative energy is thus an indicator of the material state. As this measurement has been achieved during a few hundreds of cycles at Rσ=0.1, correlation between the material dissipative energy and its microstructure evolution during fatigue tests still need to be studied and will be addressed in future work. References [1] H.F. Moore, J.B. Kommers, Chemical and Metallurgical Engineering 25, (1921) 11411144 [2] G. Fargione, A. Geraci, G. La Rosa, A. Risitano, International Journal of Fatigue 24, (2002) 1119 [3] D. Caillard, J.L. Martin, Thermally activated mechanisms in crystal plasticity (Pergamon, Amsterdam London 2003) 85123 [4] A. Granato, K. Lücke, Journal of Applied Physics, 27:6, (1956) 583593 [5] T. Tanaka, S. Hattori, Bulletin of the JSME, (21:161) (1978) 15571564 [6] H. Mughrabi, F. Ackermann, K. Herz, ASTM Special Technical Publication, 675, (1979) 69105 [7] A. Chrysochoos, H. Louche, International Journal of Engineering Science 16, (2000) 17591788 [8] F. Maquin, F. Pierron, Mechanics of Materials 41, (2009) 928942
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Temporal Phase Stepping Photoelasticity by Load or Wavelength
M.J. Huang and H.L. An Mechanical Engineering, National Chung Hsing University, 250, KuoKuang Road, Taichung, Taiwan 40227, R.O.C.
ABSTRACT Phasestepping techniques in photoelasticity own the isoclinicisochromatic interaction problem, which causes phase ambiguity zone on the photoelastic phase map and bothers engineers of this field very much. Temporal phase unwrapping is an effective method for circumventing the above problem. In this work, the load stepping and wavelength stepping approaches are applied both but individually on photoelastic samples to compare the different characteristics of them including in accuracy of results, ease of applied technologies, and automation of stepping etc. Load stepping approach is not applicable for stress frozen sample while wavelength stepping is. The constant increment of loading percentage is not easy to be controlled under complex loading conditions. The control of wavelength stepping if integrated with electrooptic components is very flexible and versatile. However, the birefringence error of optical wave plate for different wavelength should be carefully calibrated to minimize the errors. Experimental works are studied to practically verify the differences and limitation of these two approaches.
Keywords: Photoelasticity, temporal phase unwrapping, load stepping, Isoclinic, Isochromatic. 1. Introduction Phase unwrapping (PU) consists of retrieval of the true phase field from wrapped format data, which is restricted in a [ʌ, ʌ] for 2ʌ modulo. This problem is encountered in the phase stepping digital photoelasticity [14]. However, the interaction between the isoclinic parameter and relative retardation makes the phase retrieval work very difficult. Many papers [519] had been published to solve the related problems. Wang and Patterson [5] used signal analysis and fuzzy sets theory to investigate these difficulties. The ambiguity existing problem can be also overcome with the development of load stepping. Ramesh and Tamrakar [6] proposed a new methodology for data reduction in load stepping to remove the noise points in the model domain. However, the methodology requires three times the number of images required for normal phase stepping method. Ramesh et. al. [7, 8] proposed a new interactive methodology to remove the ambiguity. Temporal phase unwrapping is an effective way to circumvent this problem. It is usually implemented by the load stepping method [9, 10] but however, this technique can not be applicable for stress frozen sample. An alternative is to change the wavelength of the light source to generate phase stepping. Chen [11] successfully utilized this technique by two different wavelengths and checked the phase changes between them that had successfully solved the isoclinicisochromatic interaction problem. In this work, some comparisons are further investigated. 2. PHASE STEPPING PHOTOELASTICITY A general optical arrangement of a plane polariscope and a circular polariscope are shown in Fig. 1 and Fig. 2, respectively. White light is used as the light source and a general digital camera with RGB filters is used as the fringes recorder. Table 1 summarizes the used phase stepping parameters and their corresponding intensity results. The isoclinic ӽw and isochromatic Ӭw parameters can be obtained as follows.
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_13, © The Society for Experimental Mechanics, Inc. 2011
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n I 4s I0 n I2s I0 1 1 ° tan ® s s 4 ¯° n I3 I0 n I1 I0
Iw
½° ¾ °¿
(1)
sin I ½ 1 tan1 ® ¾, 4 ¯ cos I ¿
I
Gw
W 1 ^Iw `,
(2)
° I I sin2I I8 I10 cos 2I ½° tan1 ® 9 7 ¾ I5 I 6 °¯ °¿ I sin G ½ tan1 ® a ¾, ¯ Ia cos G ¿
(3)
(4)
where
W 1 ^
I0
1 4
I
I sj
1 3
I
`
s 1
I2s I3s I4s ,
j ,R
I j ,G I j ,B for j
(5)
1,...,4,
represents the unwrapping operator, and n
caused by isochromatic parameter. tan
1
means normalization operator to eliminate any effect
^ ` of Eq. (1) and Eq. (3) is arctan2 function with their results ranging
between S / 4 ~ S / 4 and S ~ S , respectively. Since the isochromatic data is dependent on the isoclinic data, the isoclinic data should be correctly unwrapped and ranged in the range of S / 2 ~ S / 2 before substituting into Eq. (3). Provided the substituted isoclinic data are correct, the isochomatic calculation is ambiguity free and that makes the following retrieval work of the isochromatic data extremely easy.
Fig. 1. The optical arrangement of plane polariscope
75
Fig. 2. The optical arrangement of circular polariscope
Table 1 Polariscope arrangement and the intensity results for phase stepping algorithm
Ӷ
ӯ
Ӫ
Intensity equation


I1,i
IB,i 21 I A,i sin2 21 G i 1 cos 4D
S /8
I2,i
IB,i 21 I A,i sin2 21 G i 1 sin 4D
S /4
I3,i
IB,i 21 I A,i sin2 21 G i 1 cos 4D
3S / 8
I4,i
IB,i 21 I A,i sin2 21 G i 1 sin 4D
3S / 4
S /4
S /2
I5
Ib 21 Ia (1 cos G )
3S / 4
S /4
I6
Ib 21 Ia (1 cos G )
3S / 4
0 0
I7
Ib 21 Ia (1 sin 2I sin G )
3S / 4
S /4
S /4
I8
Ib 21 Ia (1 cos 2I sin G )
S /4
I9
Ib 21 Ia (1 sin 2I sin G )
S /4
3S / 4
S /4
I10
Ib 21 Ia (1 cos 2I sin G )
3. LOAD STEPPING AND WAVELENGTH STEPPING In this work, the load stepping work of Ref. 9 is used. Two isochromatic fringes of loading stages with and without load increment are checked. Compare the isochromatic phases of the sample with load increment with that of same sample without load increment. Whenever its isochromatic phases of the load increment one is less than that of the one without load increment, that means its corresponding isoclinic data should be added or subtracted by Ӹ /2 depending on its isoclinic data is negative or positive, respectively. This restored isoclinic data is then further substituted into the isochromatic formulation to eliminate the original (wrapped) isoclinic induced phase ambiguities of the isochromatic fringes. Since the load increment percentage is known, which should be accurately controlled by the photoelastic engineer to ensure the percentage of the load increment, and the isochromatic data differences of the tested sample before and after load increment implementation are also collected experimentally, their division yields the correct isochomatic phase of the tested sample. It is simple and effective. However, this technique is not applicable for the stress frozen sample because the loading condition has been frozen into the sample which can not be loaded any further. It is known that the isochromatic data is proportional to the applying loading but inversely proportional to the wavelength of the light source. Therefore, similarly as load stepping, an alternative way to shift the isochromatic
76
phase is to keep loading fixed but to change the wavelength of the photoelastic experiment. And because it is implemented by different color filtering which is not done mechanically, sometimes it is much easier to be implemented digitally like using color camera filtering technique. The fast growing digital camera market makes the color filtering technique remarkable progress. 4. EXPERIMENT RESULTS A curve beam under tension is tested here. Fig. 3(a) shows the wrapped isoclinic. Fig. 3(b) is the isochomatic fringes under the substitution of Fig. 3(a) into isochromatic formulation. Using load stepping technique, the wrapped isoclinic can be restored into Fig. 3(c) and its corrected isochromatic result is shown as Fig. 3(d). It is clearly shown that the results are with certain degree of noise, which very probably is due to the point detection between two different loading isochromatics. The result is not as good as that from spatial photoelastic unwrapping. The sample is also unwrapped using wavelength stepping technique and the results are shown as Fig. 4(a) and Fig. 4(b). It is also shown that since the temporal (wavelength) unwrapping is done by checking individual isochromatic data in correspondence. Therefore, isoclinic and isochromatic results are very easily influenced by any local noise and thus lead to the results of Fig. 4(a) and Fig. 4(b). It is very clear that these results are not as good as those unwrapped by the newlydeveloped regional phase unwrapping algorithm [12] for photoelastic data.
(a)
(b)
ʳʳ
(c) (d) Fig. 3. The isoclinic and isochromatic of curve beam under tension (a) wrapped isoclinic (b) isochromatic derived from wrapped isoclinic (c) unwrapped isoclinic (d) isochromatic data derived from unwrapped isoclinic
77
(a)
(b)
Fig. 4. Wavelength stepping result (a) isoclinic (b) isochromatic. 5. DISCUSSION AND CONCLUSION Two approaches, load stepping and wavelength stepping, of temporal photoelastic phase unwrapping are discussed in this study. It is found that temporal phase unwrapping is very easy to unwrap its isoclinic and isochromatic data by the comparison of data of any point between the two different temporal conditions. It is also because this kind of data comparison way that unwrapped results are full of noise than those from spatial unwrapping algorithm. In addition, data obtain from wavelength stepping are much better than those from loading stepping technique. It can be give reasons from the difficulty of accurate control of the loading condition and is a difficult task which should be carefully done by the photoelastic engineers. Acknowledgments The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under contract Number NSC982221E005009.
78
References [1] A.V. Sarma, S.A. Pillai, G. Subramanian and T.K. Varadan, “Computerized image processing for wholefield determination of isoclinics and isochromatics,” Experimental Mechanics, pp. 2429, 1992. [2] K. Ramesh and V. Ganapathy, “Phaseshifting methodologies in photoelastic analysis – the application of Jones calculus,” Journal of Strain Analysis, vol. 31, pp. 423432. [3] K. Ramesh and S.K. Mangal, “Automation of data acquisition in reflection photoelasticity by phaseshifting methodology” Strain, pp. 95100, 1997. [4] S. Yoneyama, Y. Morimoto and R. Matsui, “Photoelastic fringe pattern analysis by realtime phaseshifting method,” Optics and Lasers in Engineering, vol. 39, pp. 113, 2003. [5] Z.F. Wang and E.A. Patterson, “Use of phasestepping with demodulation and fuzzy sets for birefringence measurement,” Optics and Lasers in Engineering, vol. 22, pp. 91104, 1995. [6] K. Ramesh and D.K. Temrakar, “Improved determination of retardation in digital photoelasticity by load stepping,” Optics and Lasers in Engineering, vol. 33, pp. 387400, 2000. [7] V.S. Prasad, K.R. Madhu and K. Ramesh, “Towards effective phase unwrapping in digital hotoelasticity,” Optics and Lasers in Engineering, vol. 42, pp. 421436, 2004. [8] K. Ashokan and K. Ramesh, “A novel approach for ambiguity removal in isochromatic phasemap in digital elasticity,” Measurement Science and Technology, vol. 17, pp. 28912896, 2006. [9] A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Optics and Lasers in Engineering, vol. 45, pp. 612617, 2007 [10] T.Y. Chen, “A simple method for the determination of photoelastic fringe order,” Experimental Mechanics, vol. 40, pp. 256260, 2000. [11] T.Y. Chen, “Digital determination of photoelastic birefringence using two wavelengths,” Experimental Mechanics, vol. 37, pp. 232236, 1997. [12] M.J. Huang and PoChi Sung, “Regional phase unwrapping algorithm for photoelastic phase map,” Optics Express, vol. 18, pp. 14191429, 2010.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Determination of the isoclinic map for complex photoelastic fringe patterns
Philip Siegmann1, Chiara Colombo2 , Francisco DíazGarrido3 and Eann Patterson4 1
Departamento de Teoría de la Señal y Comunicaciones, Universidad de Alcalá, 28805 Madrid, SPAIN. 2 Department of Mechanics, Politecnico di Milano, 20156 Milano, ITALY 3 Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, Campus las Lagunillas, Edif. A3, 23071, Jaén, SPAIN. 4 Composite Vehicle Research Center, Michigan State University, East Lansing, MI 48824, USA. ABSTRACT Most of the existing algorithms used for processing phaseshifted photoelastic data attempt to compute the unambiguous or demodulated isoclinic map in order to obtain the unambiguous or continuous isochromatic map. However, in some cases experiments on engineering components yield isoclinic maps that are severely corrupted due to the interaction between isoclinics and isochromatic. The result is that some of these algorithms fail in the direct demodulation of isoclinic maps from phaseshifted photoelastic data. An indirect way to obtain the isoclinic map by computing first the unambiguous isochromatic map is presented. The employed approach is based on a regularisation process that, by minimising a cost function, selects the appropriate value of the relative retardation angle at each pixel. In this way, an unambiguous map can be straightforwardly unwrapped and calibrated to generate an isochromatic map. The unambiguous isoclinic angle map is then calculated using the regularized isochromatic map. The process has been demonstrated to be robust and reasonably quick for crack tip fringe patterns.
Keywords: Photoelasticity, isoclinics, isochromatics, cost function.
1. INTRODUCTION Photoelastic stress analysis is based on the temporary birefringence exhibited by all transparent materials when subjected to stress [1]. In a few incidences, such as in the glass industry, this property is used directly to evaluate stresses however in most cases either a photoelastic model is made or a transparent coating is bonded to an opaque part using a reflective adhesive. Twodimensional models can be viewed directly while subject to load whereas stress must be ‘frozen’ into threedimensional models that can be subsequently sliced physically or optically. Coatings act as strain witnesses and reproduce the surface strains in the component, and are considered to be locally planar. In all cases, a planar section or slice is viewed in polarised light using a polariscope, revealing fringes of constant color known as isochromatics, which are contours of constant principal stress difference, and black fringes known as isoclinics which are loci of points where the directions of the principal stresses are aligned to the polarising axes of the polariscope. Traditionally, photoelastic analysis has involved the tedious manual analysis (counting) of fringes however, in the last twenty years a wide range of digital methods have been developed for photoelasticity and have been reviewed in detail elsewhere [2]. Among these methods phasestepping or phaseshifting has gained in popularity since it was first applied to photoelasticity by amongst others, Hecker and Morche [3]. Phasestepping is usually T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_14, © The Society for Experimental Mechanics, Inc. 2011
79
80
conducted in monochromatic light conditions and involves the stepped or incremental rotations of a polariser and quarter waveplate of the polariscope, usually the output pair of optical elements are employed. Digital images are captured at each increment or step as illustrated in Figure 1; and then the set of images are combined to generate maps of the relative retardation or phase difference, which is related to the isochromatic fringe order, and of the isoclinic angle (shown in Figure 2). The relative retardation is directly proportional to the principal stress difference but when generated with these techniques its sign is ambiguous or undetermined, i.e. it is unclear whether the value relates to the maximum minus the minimum principal stress or the minimum minus the maximum principal stress. In most phasestepping techniques, the calculation of the relative retardation requires the use of the evaluated isoclinic angle which is related to the direction of either the maximum or minimum principal stress but which, i.e. maximum or minimum, is unknown. Indeed, left uncorrected most phasestepping methods will generate a map of interlocking areas in which the relative retardation alternates between the two possible definitions, i.e. proportional to maximum minus minimum principal stress and minimum minus maximum.
I2 )
I3 )
I4 )
I5 ) I1) I6 )
Fig.1 Geometry and dimensions of the 2 mm thick polycarbonate CT specimen with a 7.9 mm long fatigue crack [13] (top left) and the corresponding six phaseshifted fringe patterns [14] (bottom left and right) from obtained with an applied load of 90N. Note that I1 corresponds to a darkfield circular polariscope. Much effort has been expended on developing robust methods of processing phasestepped photoelastic data that eliminate the ambiguity described above. One possible option is to collect images at several wavelengths; however, this significantly increases the required number of images and makes realtime data capture almost impossible. In ambient conditions, using monochromatic light a theoretical minimum of four images are required to solve for the relative retardation and isoclinic angle and can be collected simultaneously [5]. Hence, it is desirable to restrict the number of images required to as close to four as possible. In 1993 Wang and Patterson [4] developed a method of resolving the ambiguity based on six phasestepped images combined with fuzzy set classification. This technique and its derivatives [6, 7] are fast and almost completely automatic in that very little
81
user interaction is required however, they fail when the fringe density is high or the shape of the fringes becomes very complex. Two approaches to creating more robust, automatic techniques for resolving the ambiguity have been taken recently. Ramji and Ramesh [8] have used ten images combined with a pixel by pixel application of the quality map proposed by Siegmann et al [7] which yields a more robust approach at the expense of complexity and time required for data collection and processing. The second approach proposed by Quiroga and GonzálezCano [9] utilises regularised phasetracking (RPT) using five phaseshifted images [10] to obtain the relative retardation and the isoclinic angle [11]. Recently, this second approach has been combined with methods derived from the earlier work of Wang and Patterson [4] to generate a fast, robust and essentially automatic algorithm [12]. The motivation for the study reported here is the solution of problems associated with processing the stress field around a fatigue crack propagating in polycarbonate compact tension specimens (Figure 1). The corrupted isoclinic map (Figure 2) does not allow the use of most of the uptodate existing techniques [4, 6  8] and only those based on regularised phasetracking [9, 12] have been found to be successful with that due to Siegmann et al [12] being much faster.
a)
b)
?
c)
Fig. 2. a) The ambiguous (without demodulation) isoclinic phase map obtained from the data in figure 1 using equation (2) and b) the vertical and c) the horizontal profiles along the lines shown in (a)with changes in the principal stress directions indicated by green arrows and ellipses.
2. PHASESTEPPING ALGORITHM The sixstep photoelastic phasestepping approach is selected here because it is employed in industrial and research laboratories although other phasestepping methods can be used. The six steps are chosen such that the combination of the intensity levels are related to the isoclinic angle, θ and the relative retardation δ as follows:
I 5 − I 3 = 2 I v sin δ sin 2θ , I 4 − I 6 = 2 I v sin δ cos 2θ , I 2 − I1 = 2 I v cos δ ,
(1)
82
where Iν accounts for stray light. A number of combinations of the optical elements of the polariscope will generate appropriate images however in this work those proposed by Patterson and Wang [14] were employed and are illustrated for a fatigue crack in Figure 1. From these intensity patterns, the ambiguous isoclinic angle and the wrapped and ambiguous relative retardation can be obtained using:
θ a = arctan ¨
§ I5 − I3 · ¸, © I4 − I6 ¹
(2)
ª ( I 5 − I 3 ) sin 2θ a + ( I 4 − I 6 ) cos 2θ a º », I1 − I 2 ¬ ¼
(3)
1 2
δ w, a = arctan «
Grey scale images for θ a and δ w, a in the CT specimen are shown in Figures 2(a) and 3(a) and, as can be seen, their values are ambiguous and the modulo of retardation is between 0 and 2π. It should be pointed out that in equation (3), the four quadrant arctan function [15] is used to obtain Figure 3(a) and, as has been highlighted before the form of equation (3) guarantees a high modulation over the field [16] thereby reducing the isoclinicisochromatic interaction [12]. An unambiguous isoclinic angle would allow an unambiguous relative retardation map to be obtained since both are related through equation (3). However, it is not always possible to retrieve an unambiguous or demodulated isoclinic angle. This is the case in the CT specimen, where strong variations of both the isoclinic angle and relative retardation or isochromatic fringe order create large areas with isoclinicisochromatic interactions.
a)
b)
c)
d)
Fig. 3. a) Ambiguous and wrapped isochromatic phase map (in radians) obtained using the data in Fig.2a.; b) Inverse of (a); c) Quality map ; d) Regularized and wrapped isochromatic phase map (in degrees)
3. REGULARIZATION ALGORITHM 3.1 Regularization process The method chosen to process the photoelastic fringe patterns associated with the fatigue crack in the CT specimen is based on the one proposed by Siegmann et al [12]. The ambiguity problem is solved by processing
83
the information directly from δ w, a provided by equation (3). From the experimental data there are for each pixel in the image of the sample two possible values of the wrapped relative retardation, namely δ w+, a = δ w, a and
δ w−, a = −δ w, a , depending on the sign given to the isoclinic angle. The two maps δ w+, a and δ w−, a are shown in Figure 3(a) and (b), and were obtained using the four quadrant arctangent function (arctan2) in equation (3) [15] as mentioned previously. In order to select at each pixel the correct value of δ w, a , a path guided algorithm is followed through all of the pixels of the image and at each new pixel, the selection of the correct value of the relative retardation ( δ w ) is performed by taking the one that minimizes a cost function ( U Γ ):
U Γ (δ w ) = min ª¬U Γ ( δ w+, a ) ,U Γ (δ w−, a ) º¼ ,
(4)
This cost function ensures the continuity of either the value of the relative retardation or the gradient of the value of the relative retardation with respect to the already processed pixels, and is defined as:
U Γ ( i; λ ) = Q ( i ) AΓ ( i ) +
10λ BΓ ( i ) , Q (i )
(5)
where i=(x,y) is the pixel position, Γ is a window around i used to consider the spatial dependence, λ is a regularization parameter, Q is a quality map, and the two terms AΓ and BΓ account for the continuity in the value and the gradient of the selected relative retardation respectively, and are defined as:
AΓ ( i ) = ¦ j∈Γ
1 δ w ( j) − δ w±,a ( i ) s ( j) , 2π
§ ∂ hδ w ( j) ∂ hδ a±, w ( i ) · § ∂ hδ w ( j) ∂ hδ a±, w ( i ) · − − ¨¨ ¸¸ + ¨¨ ¸¸ s ( j) , ∂x ∂y © ∂x ¹ © ∂y ¹ 2
BΓ ( i ) = ¦ j∈Γ
2
(6)
where s is equal to 1 if the pixel has already been processed and 0 otherwise. The partial derivatives are computed over a distance h in the xdirection as follows:
∂ hδ ( x , y ) ∂x
=
δ ( x + h, y ) − δ ( x − h, y ) 2h + 1
,
(7)
and in a corresponding manner for the ydirection. A quality map is employed to perform two fundamental functions: to select the path followed by the process; and to act as the weighting factor between AΓ and BΓ . Thus, the quality map is defined as:
( w, a ( i )) .
Q ( i ) = sin 2 δ
(8)
Figure 3(c) shows the quality map for the data in Figure 1, and Figure 3(d) shows the computed unambiguous relative retardation obtained by following this quality map and using expression (5) as the cost function in the regularization process. The parameters used to process these data were λ=1, Γ=7×7, and h=3. The regularization process described above requires a relative long processing time, which is strongly dependent on the size of the images. For the data in Figure 1, 150 s were required to regularize 177x154 pixels using a Pentium 4 with CPU speed of 3.20 GHz and with the algorithm programmed in Matlab. However, it is much faster than the one proposed in [9], because it eliminates the need to perform a minimization process at each pixel; instead the new cost function has only to be evaluated for the two values of the relative retardation.
84
3.2 Determination of the isoclinic angle The demodulation of the isoclinic angle can be performed at the same time as the regularization process is applied to the relative retardation using the same cost function. As for the retardation, there are two possible values of the isoclinic angle at each pixel:
θ + = θa
and
θ − = θa + π / 2 .
The selection of the unambiguous
isoclinic angle at each pixel is performed in a similar manner: if δ w = δ w+, a then
θ = θ + , else if δ w = δ w−, a
then
θ = θ . Figure 5 shows the resultant demodulated isoclinic map superimposed with a mesh of arrows at equidistant points, aligned with the directions of the corresponding principal stresses. −
Fig. 4. Unwrapped and regularized isochromatic phase map (units: calibrated fringe order).
4. DISCUSSION AND CONCLUSIONS A robust and fast algorithm for processing phasestepped photoelastic data has been demonstrated for a specimen geometry that has proven troublesome in the past as consequence of the large amount of interaction between the isoclinic and isochromatic parameters, and the density and complexity of the fringes in the vicinity of the crack tip. Most existing algorithms employing a small number of phasestepped images failed to produce satisfactory results or required an excessively long time for processing. In the past, such issues tended to be avoided by masking out a large area around the crack tip because it was not relevant to the analysis of fatigue cracks based on linear elastic fracture mechanics; however, in this case, the strains associated with the plastic enclave and wake generated around the crack during fatigue were of primary interest, and hence it was necessary to process the photoelastic fringe patterns as close to the crack as possible. The problem could have been resolved by using a much larger number of phasesteps and corresponding number of images captured at a single monochromatic wavelength or a number of wavelengths; however since a propagating fatigue crack was the subject of interest increasing the complexity and time required for data capture was not an appropriate option. Similarly long processing times were unattractive so combining the robustness of earlier work by Quiroga and GonzálexCano [9] with the speed of algorithms produced by Patterson and his coworkers [4, 7] was a good proposition. The resultant algorithm works with almost no user intervention after defining the area of interest and the selection of the three parameters λ, Γ and h. The algorithm appears to be more robust when there is a large variation of relative retardation and tends to introduce noise when the relative retardation is near constant in magnitude. The algorithm is straightforward to implement and the elimination from the algorithm of the need to perform a minimization of a cost function at each pixel, as in the method of Quiroga and GonzálexCano [9], renders it significantly faster.
85
a)
b)
c)
Fig. 5. a) Demodulated isoclinic phase map; b) and c) shows respectively the vertical and horizontal profiles of the demodulated (solid line) and not demodulated (dashed line) isoclinic angle along the same lines as shown in Fig. 2a 5. REFERENCES [1] [2] [3]
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
rd
J.W. Dally and W.F. Riley. Experimental Stress Analysis, 3 Ed., New York. (McGrawHill Inc, 1991). K. Ramesh. Digital Photoelasticity. Berlin. (Springer, 2000). F. W. Hecker and B. Morche, Computeraided measurement of relative retardations in plane photoelasticity. In Experimental Stress Analysis, ed. H. Weiringa. Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1986, pp. 53242. Z.F. Wang and E.A. Patterson, “Use of Phasestepping with Demodulation and Fuzzy Sets for Birefringence Measurement,” Opt. Laser. Eng. 22, 91–104 (1995). J.R. Lesniak, S.J. Zhang and E.A. Patterson, ‘The design and evaluation of the poleidoscope: a novel digital polariscope,’ Experimental Mechanics, 44(2):128135 (2004). N. Plouzennec, J.C. Depuré and A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech. 23, 3033 (1999). P. Siegmann, D. Backman and E.A. Patterson, “A robust approach to demodulating and unwrapping phasestepped photoelastic data,” Exp. Mech. 45(3), 278289 (2005). M. Ramji and K. Ramesh “Whole field evaluation of stress components in digital photoelasticity – Issues”, implementation and application, Opt. Laser. Eng. 46(3), 257271, (2008) J.A. Quiroga and A. GonzálexCano, “Separation of isoclinics and isochromatics from photoelastic data with a regularized phasetracking technique,” Appl. Opt. 39, 29312940 (2000). M. Servin, J.L. Marroquin and F.J. Cuevas, “Demodulation of a single interferogram by use of a twodimensional regularized phasetracking technique,” Appl. Opt. 36(19), 45404548 (1997). J. Villa, J.A. Quiroga and E. Pascual. “Determination of isoclinics in photoelasticity with a fast regularized estimator,” Opt. Laser Eng. 46, 236242 (2008). P. Siegmann, F. DíazGarrido and E. Patterson, “Robust approach to regularize an isochromatic fringe map”, Appl. Opt. 48(22), E24E34 (2009). M.N. Pacey, M.N.James and E.A. Patterson, ‘A new photoelastic model for studying fatigue crack closure’, Experimental Mechanics, 45(1):4252 (2005). E.A. Patterson and Z.F. Wang. “Towards Fullfield Automated Photoelastic Analysis of Complex Components,” Strain, 27, 49–56 (1991). MATLAB Function Reference at: www.mathworks.com/access/helpdesk/help/techdoc/ref/atan2.html J.A. Quiroga and A. GonzálezCano. “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe pattern,” Appl. Opt. 36(32), 83978402 (1997).
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
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Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Strength Physics at Nanoscale and Application of Optical Interferometry
Sanichiro Yoshida Southeastern Louisiana University, Department of Chemistry and Physics SLU 10878, Hammond, LA 70402, USA, [emailprotected]
ABSTRACT The theoretical basis of the physical mesomechanical approach of strength physics is described. Based on a fundamental principle of physics known as local symmetry, this approach is capable of describing deformation and fracture comprehensively, and applicable to any scale level. It is thus useful to describe strength physics at the nano/microscopic without relying on phenomenology. An optical interferometric technique capable of resolving displacement at subnanometer level is introduced. Being capable of quantitative measurement, this technique is useful for characterization of material strength at the nano/microscopic level. 1. Introduction Much remains unexplained in physics behind the transition from deformation to fracture of solid state materials. Most prevailing theories of plasticity and fracture are empirical and phenomenological. They can describe phenomena observed before and after the appearance of a crack respectively, but not comprehensively. In particular, strength physics of nano/microscopic systems has not yet been systematically formulated. These systems are too small to apply continuum mechanics, and at the same time, too large to apply quantum mechanics. It is widely known that constitutive parameters of nan/microscopic systems can be substantially different form macroscopic systems of the same material. This situation leads to difficulty in designing nano/microscopic systems for required strength. A breakthrough theoretical development is essential. In this regard, a field theoretical approach developed by a recent theoretical system called physical mesomechanics [1, 2] is advantageous. Based on a fundamental physical principle known as local symmetry [3], this approach has intrinsic mechanism to describe different stages of deformation including the fracturing stage, and different scale levels comprehensively. This fills the gap between continuum mechanics and quantum mechanics with respect to discontinuity of materials. Continuum mechanics handles materials as collections of point masses and therefore does not have a builtin mechanism to formulate defects. Quantum mechanics, on the other hand, handles materials as groups of molecules connected by electronic orbits or other intermolecular interactions. Discontinuity could be considered via potentials, but for a large number of molecules there are too many interactions to be taken into account and it is not practical. From this viewpoint, the concept of mesoscopic structural level introduced by physical mesomechanics is important. Essentially, the mesoscopic level is the smallest volume element with finite volume that has rotational degrees of freedom. At one scale level above, different rotations of these mesoscopic elements represent rotational mode of deformation. This is contrastive to the formalism of elastic theory where the rotational part of the strain tensor represents rigid body rotation. In the mesomechanical approach, discontinuities in a bulk material such as defects can be naturally introduced as the space between neighboring mesoscopic volume elements experiencing different rotations. While allowing these volume elements to rotate differently, physical mesomechanical approach requests that the underlying physics is invariant before and after the rotational deformation. This is where the concept of local symmetry is used in the theory. Moreover, this approach characterizes the progress of deformation as development from a smaller scale level to a larger scale level where the deformation at the lower scale level serves as a source of deformation at T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_16, © The Society for Experimental Mechanics, Inc. 2011
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the larger scale level. Therefore, advancement in strength physics at nano/microscopic scale level is important for macroscopic mechanics as well. Quantitative measurement of displacement and other parameters characterizing strength physics is another important issue in nano/microscopic level. Recent advancement in microscopy enables us to view extremely small objects. However, quantitative measurement is not as easy as qualitative observations. Often, the size of the object is beyond the diffraction limit determined by the wavelength of the light. From this standpoint, optical interferometry is quite powerful. It uses the phase of light to quantify the size of object without being limited by the diffraction limit of the light source. The aim of this paper is to describe the theoretical basis of the physical mesomechanical approach of strength physics, and introduce an optical interferometric technique useful for measurement of small displacement. After discussing the deformation theory focusing on the dynamics of defects, the resulting dynamics of deformation at larger scale level, and the irreversibility of plastic deformation, we will consider an optical interferometric configuration suitable for outofplane displacement at the scale level beyond the diffraction limit of typical light sources. Note that the same interferometric principle is applicable to other types of measurement such as inplane displacement and strain with moderate modification of the optical configuration.
2. Formulation 2.1 Deformation transformation and local symmetry
⃗( +
, +
, +
)
K⃗0
[ ( 2,
K⃗1
[ ( 1,
2 , 2 )]
1 , 1 )]
⃗( , , ) Fig. 1 Line element K⃗0 transforms to K⃗1 via coordinate dependent displacement ⃗.
Fig. 2 Different parts of the object transform coordinate dependent transformation matrix
Consider a line element vector transforms from K⃗0 to K⃗1 in Fig. 1. By defining displacement vector ⃗ ( , , ), we can describe this deformation as differential displacement between the tip and tail of the line element vector; ⃗( + , + , + ) − ⃗( , , ). The corresponding distortion matrix [ ] can be put as below.
⎡ ⎢ [ ]=⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎦
(1)
Now consider that the transformation varies from location to location, i.e., [ ] depends on the coordinates. Since components of [ ] contain derivatives, the coordinates dependence of this matrix indicates that the derivative depends on the coordinates, and that therefore it is necessary to replace the usual derivatives with covariant derivatives. In other words, it is necessary to introduce a gauge so that the underlying physics is invariant [1, 4].
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−*
=
(2)
Here is the covariant derivative and is the gauge. In the present context, the gauge represents the interrelationship among different parts of the material experiencing different elastic deformation. Dynamics of plastic deformation can be described as the characteristics of this gauge.
2.2 Defect dynamics Replacing the usual derivatives in the conventional Lagrangian of elastic deformation with covariant derivatives and deriving an equation of motion from the resultant Lagrangian, Egorushkin [5] obtained the following set of equations. ∂
=−
∂x α
F
F
(ln
)
(3.1)
=−
(3.2)
=0
F
1 2
F
(3.3)
=
1 ̃2
Q
+
−
(3.4)
Q
=
−
Q
(3.5)
Here and are the dimensionless flux density and density of line defects (discontinuities in displacement ). Egorushkin [6] interprets these equations as follows. The righthand side of eq. (3.1) is the rate of elastic deformation, F is the LeviCivita density, ̃ 2 is the velocity of the plastic deformation front, is the elastic stress,
is the plastic part of distortion, and
is the sound velocity in the material, Eq. (3.1) is the continuity
equation of a medium with defects. Eq. (3.2) represents the consistency of plastic deformation. Note that it indicates that the defect density changes over time in proportion to the rotation of the defect flow, not divergence. Eq. (3.3) represents the continuity of line defects; there is no charge (source) for the rotational component of plastic deformation field. Eq. (3.4) indicates that either a change in the defect flow rate (the first term on the righthand side) or elastic stress (the second term on the righthand side) produces change in the rotation of defect density (the lefthand side). The second and third terms on the righthand side represent the source of plastic deformation whereas the second term represents the effect that elastic stress yields plasticity and the third term represents the initial plasticity. Eq. (3.5) indicates that the elastic velocity changes over time due to the elastic stress minus plastic stress. Also note that in this case the source is elastic stress’s spatial variation, not the stress itself as is the case of eq. (3.4) (plastic case).
2.3 Continuum mechanics local and global Requesting local symmetry under transformation [ ] and using a different Lagrangian, we can derive another set of field equations [1, 4]. From the physical point of view, this set of field equations represents the conservation of translational momentum [7].
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∇∙ ⃗ = ∇× ⃗ =
(4.1) ⃗
∇× ⃗ =− ∇∙ ⃗ = 0
(4.2) 1 2
⃗
−⃗
(4.3) (4.4) Q
= K K̇ / 2 and ⃗ = K ( Q K ) / 2 are the time and space components of the symmetry charges Q is the metric tensor, is the dimensionless elastic constant and is the associated with the transformation. characteristic length of the local area where distortion matrix [ ] can be considered to be constant. where
By putting 2 = / where and are the shear modulus and density of the material, we can interpret eq. (4.3) to be the equation of motion representing the dynamics of a unit volume [7]. With this interpretation, 0 appearing on the righthand side of eq. (4.1) represents strain concentration. The first term on the righthand side of eq. (4.3) represents the recovery force in plasticity whereas the second term ⃗ represents longitudinal force on the unit volume exerted by the ⃗ field. In the plastic regime, the longitudinal force is dissipative [4, 7]. Eq. (4.4) indicates there is no source for rotational mode of deformation. From eqs. (4.2) and (4.3), we can derive a wave equation, which represents the decaying wave characteristic of the displacement field in the plastic regime [8]. Experiments indicate that when a deformation develops at a certain level, the ⃗ field shows strain concentration represented by 0 [8]. When this happens, the field loses oscillatory character, or the wave decays [7]. Another experiment shows that acoustic emission accompanies the appearance of this type of strain concentration [9]. This observation can be interpreted as the strain concentration is generated by the above defect dynamics as an irreversible process as discussed in the next section. 2.4 Irreversibility in plasticity Irreversibility of plastic deformation must be described in accordance with thermodynamics. According to the second law of thermodynamics, the entropy change of a system undergoing any infinitesimal reversible process is given by / , where is the heat supplied to the system and is the absolute temperature of the system. In a deforming material, part of the material exerts force on the neighboring part causing stress. As an example, consider a part of a material being stretched by neighboring parts in Fig. 3. The increase in volume of this part causes the entropy change of this part to be positive, because the degree to which the probability of the system is spread out over different possible microstates increases. Here a microstate specifies all molecular details about the system including the position and velocity of every molecule. If this deformation process is to be reversible, this positive change in entropy must be compensated by a negative change in entropy by some other part of the material so that the total entropy of the whole material is zero and that the entropy must flow from the other part to this part in the form of heat . Consequently, the temperature of the other part of the material must decrease. This phenomenon is known as thermoelastic effect.
Fig. 3 Part of material being stretched by neighboring parts (left). As a consequence, the volume of this part increases and so does the entropy. The positive change in the entropy can be compensated by a heat flow in from other parts (right). If the deformation is irreversible, the entropy of this part must be increased without exchange of heat or the entropy increase is greater than the heat flow, > / . Egorushkin [6] expresses this situation by the following expression.
99
= −∇ ∙
+
(5)
where is the density of the material, is the entropy flow and is the entropy production. In ref. 6, Egorushkin derives an expression for the entropy product from eqs. (3.2) and (3.4) as
=
F(∇ )2 2
−
2
([
], ∇ ) +
Here F is the thermal conductivity,
ℎ
. is the temperature,
(6) is the density of the material,
is the linear defect
density, is the elastic stress, ℎ = − ̃ where ̃ being the plastic deformation front velocity and the flux of linear defect (the plastic front flows, as the defects flows, in the opposite direction under very high stress associated with the work needed to push the defects to go through the material [10]), and ℎ represents the work associated with the defect flow due to hydrostatic tension. Note that the righthand side of entropy production expression (6) contains elastic stress , showing the mechanism of transition from elastic to plastic deformation. When the temperature gradient is zero, the condition of irreversibility becomes
>0
(7)
The entropy of metals is associated with lattice vibration [11]. An increase in entropy thus is associated with some sort of acoustic phenomenon. This is consistent with the abovementioned observation that when the material shows irreversible deformation causing energy dissipation, acoustic emission is recorded. 3. Supporting experimental results Experimental verification of the above theory at the nano/microscopic level is a future subject. However, there is experimental evidence observed at the macroscopic scale level that supports the gist of the above theory. Fig. 4 showsa plastic deformation wave, a transverse wave solution to the abovementioned wave equation derived from eqs. (4.2) and (4.3), observed in a tensile experiment on an aluminum alloy thin plate at a constant pulling rate [8]. The displacement was monitored at three reference points along the tensile axis. It is seen that the oscillatory characteristic of the displacement at the lowest reference point is advanced indicating that the wave travels from the lowest part toward the highest part of the specimen. Also shown is the stress variation recorded simultaneously. The displacement wave rises when the stress level is about to reach the yield stress, indicating that the wave characteristics are associated with plastic deformation.
Fig. 4 Plastic deformation wave observed in an aluminum alloy specimen under a tensile load. Displacement perpendicular to the tensile load is recorded at three reference points P1 – P3.
Fig. 5 Temperature rise observed in a similar experiment as Fig 4 with a notch at the vertical center of the specimen.
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Fig. 5 shows the temperature rise measured in a similar experiment to Fig. 4, where we applied a tensile load to a brass thin plate with a notch on one side near the vertical center. The upper plots are the temperatures recorded near the notch and at an end of the specimen near the grip of the tensile machine, and the lower plot is the loading curve. Clearly seen is that the temperature constantly decreases from the beginning of the experiment until the applied load reaches the yield point. This effect, known as the thermoelastic effect, indicates that even with the notch, the initial deformation experienced by this specimen is accompanied by heat flow and therefore at least to some extent reversible. However, when the deformation passes the yield point, the temperature rises. This indicates that the stretch causes totally irreversible deformation at this point and on.
4. Optical interferometric techniques for measurement of small displacement 1.2
Laser light
1 R= 0.7
It/Ii
0.8
R= 0.9
0.6 0.4
Output reflector
0.2
Input reflector
9
10
11
12
13
14
15
16
round trip phase (rad)
Fig. 6 Optical resonator consisting of a flat output reflector and curved input reflector
Fig. 7 Transmission of optical power as a function of round trip optical phase
Consider an optical resonator consisting of a pair of mirrors in Fig. 6. A laser light is incident to this resonator from the right. The amount of the optical power coupled into the resonator depends on the resonator’s reflectance and the optical phase corresponding to one round trip in the resonator = 4 /O., where is the resonator length and O is the laser wavelength. The ratio of the transmitted/reflected power to the incident power can be given by the following equations.
= (1−
(1− )2 )2 +4
2(
, /2)
= (1−
4 )2 +4
2(
/2) 2(
/2)
(8)
Here , and are the incident, transmitted and reflected optical intensity. Fig. 7 shows the dependence of ⁄ on for = 0.7 and = 0.9 . The phase at which the curve shows the peak corresponds to the resonant resonator length. Thus by monitoring transmitted power, or alternatively the reflected power, we can measure the mirrors position relative to the resonance. With a servo control, it is possible to increase the sensitivity drastically. Fig. 8 shows the principle schematically. This method is known as the PoundDreverHall scheme [12] and used in ground based gravitational wave detectors of LIGO (Laser Interferometer for Gravitational wave Observatory) project [13]. With the use of a frequency stabilized laser, the designed sensitivity of initial LIGO detector corresponding to the length change on the order of 10−18 m has been achieved [14]. In this scheme, the incident laser is phase modulated to generate a pair of side bands. The carrier frequency is tuned to resonate when the resonator length is at the nominal value, whereas the side band frequency is offtuned and therefore reflected under the resonance condition. The photo detector picks up the reflected light. When the resonator length is at the resonance, the photo detector signal contains only the side band; when the resonator is off resonance, it contains both the carrier and side band, and therefore, the output voltage contains the beat signal as eq. (9) indicates.
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Pickoff mirror
Actuator
Photo detector
Mixer
Laser source
Electrooptical phase modulator
Fig. 8 Optical resonator with servo control .Photo detector picks off reflected power to create error signal and actuator correct the total reflector position to zero the error signal.
( ) = cos(
+ : ) cos(
+I+
)≅
cos(: ) ± sin(: )
(9)
Here I = /2 is an artificially introduced phase, and : is the modulation frequency. The electric mixer demodulates this photo detector output signal at the modulation frequency. The inphase term of the resultant voltage (the first term on the righthand side of the below equation) becomes proportional to [15, 16].
∫ ( )
= ∫{ cos(: ) ± sin(: )}
∝
(10)
The actuator corrects the reflector’s position by zeroing the error signal represented by eq. (10). By monitoring the error signal, we can know the mirror’s location with reference to the resonant location precisely. 5. Summary We have quickly reviewed physical mesomechanical formulation of strength physics. Based on the physical principle known as local symmetry, this formalism is capable of describing deformation and fracture comprehensively, regardless of the scale level. The irreversibility of plasticity is explained as the increase in entropy as a consequence of linear defects’ motion due to local elastic stress exerted by surrounding materials. On the larger scale level, plastic deformation is characterized as a transverse, decaying wave nature of the displacement field. The irreversible motion of the defects at the lower scale level is the source of energy dissipation in the wave dynamics, causing the displacement wave to decay. These formulations are yet to be experimentally verified at the nano/microscopic level. There are a number of experimental observations that support the formulations at the macroscopic level. To conduct experiments at the nano/microscopic level, quantitative measurement of displacement is essential. We have discussed an optical interferometric technique suitable for extremely small displacement. It is expected that with various optical configurations, this or similar interferometric techniques are used in the future for advancement of strength physics at the nano/microscopic scale level. Acknowledgement The financial support by the Southeastern Louisiana University Alumni Association is highly appreciated. References 1. V. E. Panin, Yu. V. Grinyaev, V. E. Egoruskin, I.L. Buchbinder and S. N. Kul’kov, “Spectrum of excited states and the rotational mechanical field in a deformed crystal”, Sov. Phys. J. 30, 2438 (1987)
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2. V. E. Panin (ed.), Physical Mesomechanics of Heterogeneous Media and ComputerAided Design of Materials, vol. 1, Cambridge International Science, Cambridge (1998) 3. J. P. Elliott and P. G. Dawber, Symmetry in physics, Macmillan, London (1979) 4. S. Yoshida, “Field theoretical approach to dynamics of plastic deformation and fracture”, AIP Conference Proceedings, Vol. 1186, pp. 108119 (2009) 5. V.E. Egorushkin, “Dynamics of plastic deformation: waves of localized plastic deformation in solids”, Sov. Phys. J. 33, 2 135149 (1990) 6. V.E. Egorushkin, “Dynamics of plastic deformation: waves of localized plastic deformation in solids”, Rus. Phys. J. 35, 4 316334 (1992) 7. S. Yoshida, “Dynamics of plastic deformation based on restoring and energy dissipative mechanisms in plasticity”, Physical Mesomechanics, 11, 31, 140146 (2008) 8. S. Yoshida et al. "Observation of plastic deformation wave in a tensileloaded aluminumalloy" Phys. Lett. A, 251, 5460 (1999) 9. S. Yoshida, “Optical interferometric study on deformation and fracture based on physical mesomechanics”, J. Phys. Meso. Mech., 2, 4, 5 – 12 (1999) 10. V. E. Egorushkin, private communication (2008) 11. C. Kittel, Introduction to solid state physics, 2nd edition, John Wiley and sons, Inc New York 118156 (1956) 12. R.Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B, 31, 97105 (1983) 13. B. C. Barish and R. Weiss, “LIGO and detection of Gravitational waves,” Physics Today, 52, 10, 4450 (1999) 14. LIGO Scientific Collaboration, “LIGO: The Laser Interferometer GravitationalWave Observatory”, Rep. Prog. Phys. 72, 076901 (pp 25) (2009) 15. P. R. Saulson, Fundamentals of Interferometeric Gravitationalwave Detectors (World Science, 1994) 16. S. Yoshida, “Detection of 1021 strain of spacetime with an optical interferometer”, Proc 2005 SEM Annual conference, June 7–9, 2005 Portland, Oregon, paper No. 303, 7 pages (2005)
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
LIGHT GENERATION AT THE NANO INTERFEROMETRY AT THE NANO SCALE
SCALE,
KEY
TO
C.A. Sciammarella12, L. Lamberti2 and F.M. Sciammarella1 1
Northern Illinois University, Department of Mechanical Engineering, 590 Garden Road, DeKalb, IL 60115, USA 2 Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Viale Japigia 182, Bari, 70126, ITALY Email: [emailprotected], [emailprotected], [emailprotected] Abstract The feasibility of recording optical information at the nanometric level was considered for a long time restricted by the wavelength of light. The concept of wavelength of light in classical optics is a direct consequence of the standard solution of the Maxwell equations for purely harmonic functions. Propagating harmonic light waves in vacuum or air satisfy the required mathematical conditions imposed by the Maxwell equations. Hence, in classical optics, the concept of wavelength of light was associated with that type of waves. Mathematically speaking, one can derive solutions of the Maxwell equations utilizing Fourier integrals and show that light generated in a volume with dimensions much smaller than the wavelength of light will have periods in the subwavelength region. Every oscillator, whether a mass on a spring, a violin string, or a Fabry– Perot cavity, share common properties deriving from the mathematics of vibrating systems and the solutions of the differential equations that govern vibratory motions. In this paper, some common properties of vibrating systems are utilized to analyze the process of light generation in nanodomains. Although simple, the present model illustrates the process of light generation without getting into the very complex subject of the solution of quantum resonators.
1. INTRODUCTION The optical observation of nanosize objects is at the frontier of current optical technology. Many important advances have been achieved in this field and today it is possible to get results that less than one decade ago were thought not possible. For a long time, a classical solution of the Maxwell equations corresponding to a harmonic oscillator yield the basic concept of wavelength leading to the well known limits of optical resolution. However, one can derive solutions of Maxwell equations utilizing Fourier integrals and show that light generated in a volume with dimensions much smaller than the wavelength of light will have periods falling in the subwavelength region. In [1,2], there are presented experimental investigations resulting in the reconstruction of simple geometry objects (prisms and spheres) with accuracies within few nanometers. In more recent work [3, 4], a similar methodology was applied to complex geometries of microscopic surfaces reaching nanometer accuracy. The paper returns to some of the original work presented in [1,2] to illustrate some of the mechanisms through which information in the nanorange can be obtained. These mechanisms are based on the phenomena related with the evanescent optical fields and the properties of optical resonators. Complex optical processes and intricate systems are briefly described and then analyzed in view of resonator models. The goal is to provide an overview of complex optical phenomena that would require quantum optics to be correctly and completely explained. However, a heuristic overview can be given in terms of classical oscillator systems.
2. OPTICAL SYSTEM TO VIEW IMAGES Figure 1 shows the optical system utilized to observe objects at the nanometric range. An optical microscope is used to view the interface between the microscope slide and the saline solution. The observed objects are located on the microscope slide. Following the classical arrangement of T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_17, © The Society for Experimental Mechanics, Inc. 2011
103
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TIR, a HeNe laser beam with nominal wavelength 632.8 nm impinges normally to the face of a prism designed to produce limit angle illumination on the interface between a microscope slide (supported by the prism itself) and a saline solution of sodium chloride contained in a small cell supported by this slide. Consequently, evanescent light is generated inside the saline solution. The resulting effect of this particular optical system is complex. The prism has residual stresses that provide one of the important tools in the process of observation. Figure 2 presents an approximate model of the glass prism boundary.
Figure 1. Optical system utilized to make observations in the nanometric range
Figure 2. Schematic representation of interfaces: 1) prismmicroscopic slide, 2) microscopic slidesaline solution
The presence of residual stresses can be approximately modeled by the presence of a grating that produces different diffraction orders. However, these diffraction orders correspond to evanescent optical waves and not to ordinary waves. The effect of the evanescent diffraction
105
orders is to excite electromagnetic oscillations in the inner space of the microscope slide that acts as a FabryPérot interferometer. The electromagnetic oscillations of the FabryPérot result in the generation of interference fringes that are actually photoelastic fringes that play an important role in the subwavelength observations.
Figure 3. Plot of sinTr vs. fringe order for the ppolarization plane wavefronts generated by the evanescent waves. The order 0 corresponds to the direction of the impinging laser beam.
In Figure 3, the sine of the emerging wavefronts is plotted with respect to fringe orders recorded from the FT of the analyzed image. It can be seen that all the emerging orders except the first one are in the range of complex sine. Since values of sinҏșr are greater than 1, then angles șr are complex numbers, with a real part and an imaginary part. This implies a generalization of the trigonometric function to complex values of the argument. Utilizing the plane wave complex solutions of the Maxwell equations, one arrives to a system of fringes whose variable intensity which will be recorded by the sensor can be expressed as follows:
I( x )
ª 2S n º I0 I1 cos « N 0 sol » x n g »¼ ¬« p 0
(1)
where: po is the pitch of the fringes, No is the fringe order, nsol is the index of refraction of the saline solution, ng is the index of refraction of the microscope slide. The relationship between po and the wavelength of light is given by the equation:
sin I
Oe 2p n sol 0 N0
(2)
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where I is the angle of the diffraction order with respect to the zero order. The effective wavelength O e given by the ratio:
Oe
O N
(3)
In the above equation, Ȝe is the actual wavelength of the light generating the evanescent wave and the integers N=1,2,3 represent the fraction of the wavelength generating interference fringes. In Figure 4, the theoretical results provided by Eq. (1) are plotted in abscissas vs. the experimental values represented in ordinates. The cross correlation indicates an excellent matching between the experimental and the theoretical values (“System 1” and “System 2” refer to ppolarization and spolarization beams, respectively). In the present case, the effective wavelength was obtained by setting N=2: that is, interference fringes correspond to an effective wavelength which is half of the actual wavelength of light. It appears from the figure that the experimental values are in excellent agreement with theoretical computations done with Eq. (1).
Figure 4. Measured fringe spacing vs. theoretical values of pitch computed via Eq. (1)
Figure 5 shows the values of the measured pitches as a function of the fringe order. This relationship can be put in the following analytical form:
pn
p0 N0
(4)
The values of the resulting hyperbolic law written above provide, for each order No, the pitch pn of fringes as a function of the fundamental pitch po of each of the two families of interference fringes generated by the ppolarized and the spolarized waves, respectively. Point and triangle dots in
107
the figure correspond to values measured experimentally. The continuous curves portray the correlation equations shown in Figure 5.
Figure 5. Orders plotted vs. measured fringes pitches
The process of formation of fringes that are multiple interference fringes produced by the interaction between the outer layers of the prism and the microscope glass slide is now analyzed. In Ref. [5], there is thorough discussion on the multiple interference fringes arising from light incident at the limit angle at an interface where there are high gradients of the index of refraction caused by residual stresses in the glass. The region of high gradient behaves as a diffraction grating that generates a number of wavefronts that emerge at different angles (Figure 2) and then enter the microscope.
3. ANALYSIS OF THE BIREFRINGENCE THAT PRODUCES THE OBSERVED FRINGES Figure 6 shows the laser light incident at the interface between the saline solution and the microscope slide. Two beams are entering the saline solution at angles Ip and Is, for the p and spolarized waves, respectively. These two beams have originated in the prism due to the birefringence generated by residual stresses at the boundary between glass slide and prism. It will be shown that these two beams experience a S/2 rotation inside the glass slide and continue their trajectory to the interface between the glass slide and the saline solution as they were originated at the interface between the glass slide and the prism. Since there is not a significant difference between the index of refraction of the prism and the glass slide, the beams will continue approximately with the same direction impinging in the boundary region between the microscope slide and the saline solution.
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Figure 6. Beam illuminating the microscope observations
Figure 7. Angle of incidence of the illuminating beam and actually observed beam.
Figure 7 shows again the wavefronts involved in the formation of the observed images. The illuminating beam is totally reflected at the interface microscope slide saline solution. The illuminating beam generates the evanescent waves that, according to the preservation of the momentum, will produce in the medium in contact with the interface (i.e. the saline solution) propagating wavefronts. As stated earlier, two beams were observed: the ordinary and the extraordinary beams entering the saline solution (Figure 6). The angles of inclination of these beams are respectively Ipp and Isp where the first subscripts “p” and “s” indicate the type of polarization while the second subscript indicates that these wavefronts come from the prism. The two wavefronts are originated by the ppolarized beam and the spolarized beam produced by the birefringence of the prism outer layers. The pwave front order emerges at the angleIpm in the saline solution while the swave front emerges at the angle Ism in the saline solution. Those angles can be determined experimentally by utilizing the following relationships:
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sper °sin Ipm ° ® °sin I sper sm °¯
O p pm
(5)
O psm
where ppm and psm, respectively, correspond to the spatial frequencies of the two families of fringes formed by the p and swaves travelling through the thickness of the glass slide. The pitches ppm and psm were measured experimentally by retrieving the sequence of orders produced by the two beams from the FT of the formed images as shown in Figure 5. The experimental values hence are:
sper °°sin I pm ® °sin I sper sm °¯
0.6328 3.356 0.6328 2.238
°I pm sper ® sper °¯I sm
a sin 0.1886 10.860q a sin 0.2828 16.425q
(6)
4. ROLE OF THE MICROSCOPE SLIDE AS A FABRYPEROT INTERFEROMETER An optical cavity or optical resonator is a system of mirrors that produces standing waves. The electrical field confined in the cavity is reflected multiple times inside the cavity producing standing waves for certain resonance frequencies that depend on the geometry of the cavity and properties of the mirrors. The standing wave patterns thus generated are called modes: each mode is characterized by a frequency fn, where the subscript n is an integer. Optical cavities are characterized by the quality factor, or Q factor. The Q factor is a dimensionless parameter that characterizes the resonator's bandwidth relatively to its center frequency. Low Q’s indicate high losses of energy in the cavity and a wide bandwidth. High values of Q indicate a low rate of energy loss in the cavity with respect to the stored energy of the oscillator and a narrow bandwidth. In general, Q is defined in terms of the ratio of the energy stored in the resonator to the energy being lost in one cycle:
Q
2S
Energy Stored Energy loss per cycle
(7)
For a simple type of resonator with two flat surfaces, the Q factor is given by [6]:
Q
2Sf 0" cT
(8)
where fo is the resonance frequency, " is the length of the cavity, c is the speed of light in the vacuum, and T is the transmittance of the faces of the cavity. The behavior of the resonator is described by the eigenvalues that characterize the system. If separation of variables is possible, the eigenfunction can be defined in terms of a geometrical factor G(r), with r indicating the spatial coordinates, and a temporal factor M(t ) :
< ( t , r ) M( t )G (r ) The M(t ) function is the solution of the harmonic oscilator:
(9)
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d 2M 1 dM Q leak Zo M 2 dt dt
where Zo is the resonant angular frequence and
(10) 1 Q leak is the energy leaked, in the present case,
to the saline solution. In the case analyzed in the paper there will be many wave solutions because the actual phenomena depend on the solution of the forced equation:
d 2M 1 dM Q leak Zo M 2 dt dt
F(t )
(11)
where F(t) represents the different harmonics that enter the oscilator and create resonances at different frequencies. It can be proven that the maximum energy will concentrate in the direction of the normal to the face of the slide. Say D the inclination of the rays with respect to the normal, " the thickness of the interferometer. The transmission reaches unity if " =mO/2cosD and D=0 [6]. Hence there will be a 0, or fundamental resonance mode, with maximum energy. As a consequence of the resonator oscillations, the beams generating the interference fringes emerge as they have experienced a change in direction of S/2, as it was concluded by analysis of the experimental results.
5. FORMATION OF THE INTERFERENCE FRINGES The mechanism of formation of interference fringes can be explained as follows. At the interface between the prism and the glass slide the p and swaves arrive at angles larger than the critical angle, Figure 2. If we neglect birefringence in the glass slide, the following relationships can be written at the boundary between the glass slide and the prism taking into consideration the S/2 rotation of the wavefronts:
° n pp sin D s ® °¯ n sp sin D p
n g sin I ps n g sin I ss
(12)
where: npp and nsp are respectively the indices of refraction for the p and swaves travelling in the prism; ng is the refraction index in the glass slide; Dp and Ds indicate the equivalent angles of incidence of the p and swaves, respectively, due to the S/2 rotation. It can be written:
°sin D p ® °¯sin D s
cos I pp
(13)
cos I sp
where Ipp is the inclination angle of the ray of ppolarization in the prism and Isp is the corresponding inclination angle of the ray of spolarization. Equations (13) can be rewritten as:
sin D p ° ® °¯sin D s
cos I pp
1 sin 2 I pp
cos I sp
1 sin 2 I sp
(14)
111
By substituting equations (12) in (14), we obtain:
°sin D p ° ° ® ° °sin D s °¯
§n · 1 ¨¨ g ¸¸ © n pp ¹ §n · 1 ¨ g ¸ ¨n ¸ © sp ¹
2
1 2 2 n pp n g n pp
(15)
2
1 2 2 n sp n g n sp
The above equations can be rewritten as:
n sin D p ° pp ® ° n sp sin Ds ¯
n pp n g 2
n sp n g 2
2
(16)
2
At the interface between the prism and the glass slide, the law of refraction for the two propagating p and swaves yields:
° n pp sin D p n g sin Ips ® °¯ n sp sin D s n g sin Iss
(17)
At the interface between glass and the saline solution, the law of refraction for the two waves can be written as:
°n g sin I ps ® °¯n g sin I ss
n sol sin I pm n sol sin I sm
(18)
By combining equations (17) and (18), it follows:
° n pp sin D p n sol sin Ipm ® °¯ n sp sin Ds n sol sin Ism
(19)
By combing equations (16) and (19), we obtain:
n 2 n 2 g ° pp ® °¯ n sp 2 n g 2
n sol sin Ipm n sol sin Ism
(20)
Finally, from Eq. (20), the angles of the p and swave fronts emerging in the saline solution can be expressed as:
112
°sin Ipm ° ® ° °sin Ism ¯
n pp n g 2
2
n sol n sp n g 2
(21) 2
n sol
The theoretical values of the angles of inclination of the p and swavefronts can be compared with the corresponding values measured experimentally. There are two unknowns in Eqs. (21): the indices of refraction npp and nsp The index of refraction of the glass slide ng was provided by the manufacturer: 1.5234. The index of refraction of the saline solution was determined from the solution concentration: 1.36. The MaxwellNeumann equations can be written for the p and swave fronts travelling through the stressed region of the prism,
n pp n o
AV1 BV 2
(22)
n sp n o
BV1 AV 2
(23)
where no is the index of refraction corresponding to the prism in the unstressed state while npp and nsp correspond to the indices of refraction of the extraordinary wavefronts in the stressed prism. Equations (22) and (23) contain also the constants A and B and the residual compressive stresses V1 and V2 generated in prism by the manufacturing process. By comparing the expressions of the index of refraction no of the ordinary beam it follows:
no
n pp AV1 BV 2
no
n sp BV1 AV 2
(24)
where the two members of equation (24) must coincide. The optimization problem can be stated in fashion of Eq. (25). The objective function < represents the average difference between the theoretical angles at which the p and swave fronts emerge in the saline solution and their counterpart measured experimentally. The MaxwellNeumann equations are included as constraints in the optimization process: the equality constraint on the ordinary wave, equation (24), must be satisfied within 2% tolerance. There are 6 optimization variables: the two indices of refraction npp and nsp in the error function; the two constants A and B and the residual stresses V1 and V2 in the constraint equation. The last unknown of the optimization problem, the refraction index no corresponding to the unstressed prism, was instead removed from the vector of design variables by combing the MaxwellNeumann relationships into the constraint equation (24). Optimization runs were started from three different initial points: (i) lower bounds of design variables (Run A); (ii) mean values of design variables (Run B); (iii) upper bounds of design variables (Run C). The optimization problem (25) was solved by combining response surface approximation and line search [7]. Sequential Quadratic Programming performed poorly as design variables range over very different scales. Results of different optimization runs are reported in Table 1. Besides the residual error on the optimized value of < and the value of constraint margin (24), the table shows also the resulting value of the index of refraction of the ordinary beam no, the photoelastic constant C=AB and the shear stress W in the prism determined as (V1ҟV2)/2. It can be seen that the residual error on the emerging wave angles is always smaller than 0.4%. The error on the refraction index of the ordinary beam is less than 1.8%. The convergence curves corresponding to the three optimization runs are shown in Figure 8. The optimization process was completed within an average number of 20 iterations. Some
113
oscillations in the error function were seen only in the first iterations but were progressively reduced as the search algorithm approached the optimum design.
° ª ª ° 2 2 « «§¨ n pp n g ° sper « sin I pm «¨ ° n sol 1« °Min «< (n , n , A, B, V , V ) ¨ pp sp 1 2 « sper «¨ ° 2 sin I pm « « ° ¨ « «¨ ° « «¬© ° ¬ ° n pp (AV 1 BV 2 ) ° d 1.02 °0.98 d n sp (BV1 AV 2 ) °° ® ° °1.5235 d n pp d 1.575 ° °1.5235 d n sp d 1.575 ° ° ° 0.8 10 11 d A d 0.3 10 11 ° 11 11 ° 3.5 10 d B d 2.5 10 ° ° ° 180 10 6 d V1 d 120 10 6 ° °¯ 180 10 6 d V 2 d 120 10 6
2
· § n 2 n 2 sp g ¸ ¨ sper sin I sm ¸ ¨ n sol ¸ ¨ sper ¸ ¨ sin I sm ¸ ¨ ¸ ¨ ¹ ©
· ¸ ¸ ¸ ¸ ¸ ¸ ¹
Table 1. Results of optimization runs
Design parameter npp nsp no 2 A (m /N) B (m2/N) C=AҟB (m2/N) V1 (MPa) V2 (MPa) W (MPa) Error < (%) Error on no (%)
Run A 1.5447 1.5716 1.5399 0.516710ҟ11 ҟ11 2.809610 ҟ11 2.292910 135.30 145.01 4.8549 0.3872 1.730
Run B 1.5447 1.5709 1.5399 0.301710ҟ11 ҟ11 2.964510 ҟ11 2.662810 142.50 145.83 1.6657 0.2732 1.673
Run C 1.5468 1.5699 1.5405 0.699410ҟ11 ҟ11 3.139010 ҟ11 2.439610 161.85 164.90 1.5279 0.3394 1.485
6. CONCLUSIONS As stated in the introduction of the paper, light generation at the nanolevel is one key in the process of making optical measurements at the nanorange. In the case analyzed in this paper, the light generation occurs by a complex process involving several fundamental aspects.
2
ºº »» »» »» »» »» »» »¼ »¼
114
1) Creation of excitation or forcing function F(t) by generating diffraction orders via evanescent light fields. 2) Presence of a resonator, in this case a FabryPerot interferometer, that provides the mechanism of linkage between two medias, glass slide and saline solution. 3) Introduction of a medium, in this case the saline solution, capable of generating propagating waves from the excitation generated by the evanescent fields. The presence of a resonator is a fundamental step in the process of creating subwavelength light fields. The resonator produces eigenmodes that are localized in the space. The localization of modes in this example was implemented by a simple mirrorcavitymirror structure. The resonator must be a leaking resonator because its role is to transmit energy from the evanescent fields to a medium capable to convert evanescent field electromagnetic waves to light propagating waves. 85 80
Lower bound
75 70
Mean value
65
Error on PSI (%)
60
Upper bound
55 50 45 40 35 30 25 20 15 10 5 0 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Iterations
Figure 8. Convergence curves for different optimization runs
Through the above presented analysis we have successfully supported the model of conversion of evanescent field energy into propagating light energy. The analysis relates the pitch of the observed fringes and the residual stresses at the boundary between the glass prism and the glass microscope slide. The solution shows two principal stresses that are almost equal as it should be in the case of a rectangular block of glass cooled down from the molten state to room temperature. Away from edges the residual stresses are equal in magnitude and both compressive as was found by optimization. The numerical values of the residual stresses are low and well within acceptable limits. The photoelastic constants of the glass are also of the correct magnitude and very close to the values found in the literature for glass. The optimization process has been a fundamental and powerful tool to verify that the model can predict successfully the observed phenomena.
7. REFERENCES [1] Sciammarella C.A. “Experimental mechanics at the nanometric level”, Strain 44: 319, 2008. [2] Sciammarella C.A., Lamberti L., and Sciammarella F.M. ”The equivalent of Fourier Holography at the nanoscale”, Experimental Mechanics 49: 747773, 2009.
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[3] Sciammarella C.A., Lamberti L., Sciammarella F.M., Demelio G.P., Dicuonzo A. and Boccaccio A. “Application of plasmons to the determination of surface profile and contact strain distribution”. Strain, 2010. In Press. [4] Sciammarella F.M., Sciammarella C.A., Lamberti L. and Burra V. “Industrial finishes of ceramic surfaces at the microlevel and its influence on strength”. SEM Annual Conference & Exposition on Experimental & Applied Mechanics, June 710, 2010, Indianapolis, Indiana. [5] Guillemet C. L’interférométrie à ondes multiples appliquée à détermination de la répartition de l’indice de réfraction dans un milieu stratifié, PhD Dissertation, Faculté de Sciences, University of Paris, Imprimerie Jouve, Paris (France), 1970. [6] Davies C.C. Laser and ElectroOptics: Fundamentals and Engineering. Cambridge University Press, Cambridge (UK), 2002. [7] Vanderplaats GN. Numerical Optimization Techniques for Engineering Design. VR&D Inc., Colorado Springs (USA), 1998.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
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T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_18, © The Society for Experimental Mechanics, Inc. 2011
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Sample @ «0 i » « 2 ¬1 1 ¼ ¬ ¼ «i sin Ȧt ¬« 2
Ȧt º 2 » ªE0 º e iȦ0 t » Ȧt « 0 » cos » ¬ ¼ 2 ¼»
i sin
(3)
where E0 is the amplitude of the incident electric field, P(0°) represents the Jones matrix of the polarizer aligned with xaxis, Q(0°) represents the Jones matrix of the quarterwave plate, whose slow axis is aligned with xaxis, S( < , ' ) represents the Jones matrix of the sample. Furthermore, EO(45°, Ȧt) represents the Jones matrix of the EO modulator driven by a saw tooth voltage waveform with an angular frequency Ȧ and its slow axis is oriented at 45° related to the xaxis, and A(45°) represents the Jones matrix of the analyzer whose transmission axis forms an angle 45° with the xaxis. As a result, the intensity of the detected signal is given by I1
Idc (1+ sin 2Ȍ * cos ǻ * sin Zt cos 2Ȍ * cos Zt )
Idc +R1 sin(Zt ĭ1 )
(4)
where Idc E02 / 4 is the dc component of the output intensity, and E02 is the intensity of the input light. R1 represents the amplitude, and ĭ1 represents the phase. Therefore, as the polarizing elements are arranged as shown in Fig.1, the measured principal axis direction of the sample would be the slow axis. The signal as shown in Eq.(4) will be further proceeded in CCD and introduced in the next section.
The time varying signal I1 in Eq.(4) can be integrated successively over the four quarters of modulation period T. This integration is performed in a parallel manner using a twodimensional (2D) CCD. The three frames of the image obtained by integrating the timevarying signal over the individual quarters of the modulation period are given by [10].
154
S1 x, y
T 4
³ I x, y, t dt 1
2T 4
S 2 x, y
³ I x, y, t dt 1
T 4
3T 4
S 3 x, y
³ I x, y, t dt 1
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I sin 2< cos ' I dc cos 2< T I dc dc 4 Z Z I sin 2< cos ' I dc cos 2< T I dc dc 4 Z Z
(5)
I sin 2< cos ' I dc cos 2< T I dc dc Z Z 4
From Eq.(5), the 2D distributions of the ellipsometric parameters ʻ < , ' ) can be obtained as:
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stress state caused by the internal (operation) pressure and soil movement
>V @M .
Consequently, if
>V @
>V @p
> @
and by the stress states caused by construction
is measured, if
>V @p is
accounted for by simple analytic
calculation or by the unloading and loading of some of the internal pressure, and if
> @
>V @R
is estimated by some
experimental procedure, the soil interaction stress state V M can be determined. Measuring the stress states for at least three points allows for calculating the moment and axial loads acting on the pipe section. An overly deterministic approach may be used to determine the unknowns and to minimize the error associated with the measurement and calculation procedure. This calculation procedure is conceptually simple and straightforward but it suffers from drawbacks due to serious uncertainties regarding the tube fabrication and pipeline construction induced residual stress distributions, and due to the uncertainties generated in the strain measurements [13]. A similar calculation scheme using the blindhole technique combined with strain measurements made with a suitable, portable optical interferometer proved to be competitive when compared to measurements made with strain gage rosettes [4]. Unfortunately, the scheme’s overall analysis technique suffered from a lack of representativeness once it measured the residual fabrication stresses using a short segment of pipe in the laboratory. The present paper aims to provide laboratory test results that are helpful in considering how residual stresses are distributed along the perimeter of a section and how these distributions change along short longitudinal lengths of a pipe. It also aims to show how measurements made with conventional strain gages compare with strains predicted by simple analytic or numerical techniques, and to show what the uncertainty generated by these measurements and predictions is. The tests carried out in the present investigation utilized a special Ushaped device designed and built for applying combined states of stress, which were generated by a normal force, by a bending moment and by internal pressure. The horizontal leg of the Ushaped device was the test specimen, which consisted of a segment of an American Petroleum Institute API 5L X60 steel pipe that could be loaded by internal pressure. The vertical legs of the Ushaped device were welded to the ends of the pipe in order to cap its ends and to apply the normal forces and bending moments. The resulting applied stress states in the pipe walls were measured using rosettes of electrical resistance strain gages that were suitable for applying the holedrilling technique. Three sets of data points were collected during the tests. The first set consisted of measurements of strains caused by applying simple or complex load combinations. The second set consisted of strain measurements made after the blindholes were drilled in order to determine the residual stresses caused during the fabrication of the steel pipe. The third set of data consisted of measurements of strain variations caused by unloading the device. All the data
177
from the experiment were analyzed and compared with strains calculated by means of analytic (Strength of Materials) and numeric (Finite Elements) methods. EXPERIMENTAL METHODS The special Ushaped device designed and built for applying combined states of stress is presented in Figure 1. The device consisted of a segment of API 5L X60 steel pipe with length, diameter and thickness equal to 910mm, 323.3mm and 9.7mm, respectively. The pipe segment was capped at both ends by welding on two reinforced plates. These plates were long enough to cap the pipes at one end and to support two threaded spindles in the other end. The spindles and the long plate arms made it possible to apply axial forces and bending moments to the pipe segment. The spindles were instrumented with four electrical resistance uniaxial strain gages in a full bridge arrangement to measure the applied forces. The resulting applied stress states in the pipe walls caused by normal force, by bending moment and by internal pressure were measured using rosettes of electrical resistance strain gages (PA06062RE120L from EXCEL Sensors Ind. Com. Exp. Ltda., SP, Brazil) that were suitable for applying the holedrilling technique. All residual stress measurements and calculations followed the standard American Society of Testing Materials ASTM E837 [5]. Drilling the blind holes was done with special 1.56mm squareended drills using low speed (4cps). Final hole diameters were measured to be 2R0 =1.59mm. The median diameter of the center of the strain gages of the rosette was 2R = 5.2mm. Measurements of stress variations were made for each 0.5mm depth step along the drilling process to a maximum hole depth of 2mm. Except for the residual stress measurements taken at the seam weld, all the data of the experiment proved to be very close to the uniform stress states along the 2mm range of measurement depth. The proper procedure recommended by the ASTM E837 standard [5] was used to verify this claim of uniformity.
Example of section along which rosettes were bonded Pressurized water intake
API 5L X60 line pipe segment: external diameter 324mm, thickness 9.57mm, length 910mm Lateral arm that caps the pipe and transmits axial force and bending moment Full Wheatstone strain gage bridges to measure applied force on the spindles Spindle (used two) to apply forces to the lateral arms
Nut to apply forces on the spindle Ball bearing and shaft to support device Figure 1: Ushaped device built to apply combined loading of internal pressure, axial force and bending moment to the API 5L X60 line pipe segment Figure 2 depicts the location of the data points used herein (sections 1 and 2) and in other previous investigation reports  sections 3 and 4 from [1,2] and section 5 from [6]  which used the same piece of pipe test specimen.
178
Circumferential weld Seam weld at 0o angular reference
E=28 mm (3) Section 1 (20 rosettes)
Section 1 (20 data points) Section 2 (8 data points) Section 3 (10 data points at 0o, + 22.5o, + 45o, + 90o, + 135o and 180o) Section 4 (10 data points, same position as section 3) Section 5 (3 data points, + 135o and 180o )
Section 2 (8 rosettes)
E (4) (5)
Figure 2: Data points used in the present and in previous investigations [13,6] using the same piece of pipe segment. The following procedure was used in the tests to load the pipe, and to collect and analyze the strain data from sections 1, 2 and 5. Data from sections 3 and 4 have already been published [2] and will be used in the present article for comparison purposes. 1 – Surface preparation and rosette bonding and cabling to signal conditioner. 2 – Pressure loading (5MPa), strain data collection, and pressure unloading. 3 – Axial force and (coupled) bending moment application of 20kN and 24.7kNm, respectively; strain data collection and unloading. 4 – Combined pressure (5MPa) and axial force  bending moment loading of 20kN and 24.7kNm, respectively; strain data collection. 5 – Hole drilling and measurement of strain data variations caused by the release of residual strains. Calculation of the residual stresses at each analyzed point using ASTM E837. 6 – General unloading of the device and data collection of variations in the straingage readings caused by the unloading. Calculation of the nominal strains for each gage direction based on the strain data collected using Kirsch equations for an infinite plate with a throughthickness circular hole under a biaxial state of stress [7]. RESULTS AND DISCUSSION Analytical and numerical solutions were developed using the conventional Strength of Materials and Finite Element methods to generate elastic stress and strain distributions so that they could be compared with the strain gage results determined for the external loading applied to the device. Although it is beyond the scope of this study to show the detailed solutions and results of both analyses, it should be mentioned that both agreed quite well except for the presence of some ovalization due to pressure loading, which appeared in the FE and which was not expected with the basic Strength of Materials solution. The FE analysis was developed using a welltested commercial program. Two highly detailed grid solutions were developed that yielded similar results, except for points located at or very near the seam weld due to different options involving more detail or less in order to simulate the geometry of this area.
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Figure 3: Circumferential and longitudinal strains measured by the holedrilling rosettes (before drilling) compared with the finite element strain results determined for hydrostatic internal pressure loading of 5MPa, coupled axial force and bending moment loading of 20kN and 24.7kNm, and combined loading. Figure 3 presents a plot that summarizes the comparison among experimental and numerical circumferential and longitudinal strains determined for simple (internal pressure or axial force and coupled bending moment) or combined loading cases. Each point coordinate is given by the calculated strain and the measured strain at the same location and direction for a given loading case. The data analyzed were taken from tests that measured strains for the instrumented points located in sections 1 and 2 shown in Figure 2. The root mean square deviation RMSD was calculated and was equal to 50μİ in the case shown in the Figure, which corresponds to the morerefined FE grid, and 54μİ in the case of the lessrefined FE grid. Data points that contributed more to this o deviation were those determined for rosettes at or near the seam weld (rosettes at positions between 0 and + o 22.5 of sections 1 and 2 – see Figure 2).
Figure 4: Comparison between the expected nominal circumferential shape and the actual measured shape. Existing deviations in both shapes are magnified. The deviations’ approximate scale is given in the figure.
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To better understand the causes of overall deviations in the experimental and numerical results, measurements of the actual profile of both sections and its possible deviation from circularity were taken. Figure 4 presents these deviations for section 1. In the seam weld region, the pipe displays dents and protuberances of about 0.5mm, except at the center weld position, where the deviation is 2mm, caused by the weld reinforcement. It is well known that localized bending stresses in smooth dents or protuberances acting on the outer surface layers of the walls of thin pipes loaded by internal pressure can be significantly higher than the calculated nominal hoop of longitudinal stresses. A simple analysis renders a stress concentration factor of 1 6 u e / t , where e is the circularity deviation (which can be positive if inward and negative if outward) and t is the thickness of the thin pipe. Considering that the pipe’s thickness is 9.7mm, the stress concentration factor for e = 0.5mm can be as high as 1.31. The overall shape of section 1 shows noncircularities that can be credited to the steps taken during the forming process, called UOE. These steps are shown in Figure 5 and include the formation of the U shape, the Oshaped completion of the rounding process, the seam process consisting of the submerged arc welding SAW to close the section gap, and finally, the Eexpansion of the pipe. The “E” expansion is made by an internal tool that travels along the tube and contains shoes that expand in the radial direction. The expansion takes place by means of equal steps that develop along the length of the tube, starting at and running from one end of the tube to the other in order to improve and guarantee circularity.
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Figure 5: Illustration of the fabrication of the UOESAW line pipe It is believed that the UOE fabrication process greatly influences the heterogeneous residual stress distributions that exist for pipes fabricated by this process. Residual stress measurements were measured at several points on the external surface of the present test specimen in order to investigate their variation along different sections of the short pipe segment. These results complement other data acquired using the same test specimen [1,2,6] Figures 6 and 7 present residual stress measurement results for the longitudinal and circumferential directions determined by the hole drilling technique applied to all rosettes positioned in sections 1 to 5 of the segment of steel pipe tested. These results correspond to step 5 in the test and analysis procedure described in the previous section. The plots in Figures 6 and 7 present the residual stress distributions along the external perimeter of all five sections analyzed. The longitudinal and circumferential stress plots presented in Figures 6 and 7 show steep variations in both stresses along the perimeter of each section and from section to section. The stress values range from zero value to positive or negative values that are of the order of the material yield strength. A word of caution is warranted: results where residual stresses are larger than 50% of the material yield strength may be seen as merely qualitative [7]. Another word of caution should be given regarding the fitting curves shown in Figures 6 and 7. These curves are included to connect the data point sequence and to make it easier to visualize. However, they are by no means proposed as shape distributions of the residual stress variations in the investigated cross sections.
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Given the nature of uncertainties generated in the pipe fabrication process, and considering the fact that in the present case the blindhole measurement technique generated measurements with low uncertainties  as will be discussed later using Figure 8  it can be concluded that the resulting measurements for neighboring points on the cross sections that are as close together as 16.6mm (sections 3 and 4) replicate quite well. This conclusion does not apply when sections are located far from each other or when they are close to the ends of the tube. Data on residual stresses for the 48 measuring point locations displayed in the plots covering the perimeter of the five cross sections positioned along the 910mm long segment of the pipe show a large spread that has to be credited to the fabrication procedure. Although at some locations  particularly for the longitudinal stress distributions similar trends may be observed, there are differences of more than 200MPa for the stresses in some angular locations. Thus, it can be concluded that there are variations in these distributions, and that such variations may be credited to heterogeneity distributions of residual fabrication stresses along the short length of the tested segment. These variations may be caused either by different “U” or “O” shapes generated during the forming of the tube or by different cold form straining along sections of the pipe segment caused by the progressive “E” cold expansion steps occurring along the length of the tube. Residual Stresses  Longitudinal Pipe Direction Longitudinal Residual Stress (MPa)
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Figure 7: Circumferential residual stress distributions along the circumferential position of five sections comprising a total length of 636mm (distance between sections 1 and 5)
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Figure 8: Comparison among radial strains measured along the gages belonging to the rectangular and 45 directions of the rosettes before unloading and drilling, with strains calculated along the same directions after drilling and unloading using the Kirsch equations applied to a through hole in an infinite plate under biaxial state of stress. Circlehighlighted data points belong to rosettes placed on the seam weld, where the hypothesis of uniformity of stresses along the measuring depth is not valid. In order to determine the intrinsic accuracy of the hole drilling method using the triaxial strain gage rosettes, the linear strains generated by the unloading of the device were measured. These strain measurements are directly related to the unloading of the primary biaxial strain or stress states existing previously in the neighborhood of each hole. As the primary strains related to the combined loading process were measured before the holes were drilled and recorded on a data sheet, applying the Kirsch equations for a through hole in an infinite plate to the measured strains caused by the unloading process can help to analytically determine strains to be compared with the initial strains [7]. The analytical strains were calculated in the radial direction for the central position of each strain gage belonging to each rosette. The plotting of points on a graph whose coordinates are the primarily measured radial strain for each strain gage and the corresponding strain calculated from the unloading process o and Kirsch equations should fall over a 45 line. Deviations in the plotted points were relatively small and an overall uncertainty was evaluated by calculating the root mean square deviation RMSD, which turned out to be 26μİ. Figure 8 shows a graph illustrating a comparison among the radial strains measured and among the gages o belonging to the rectangular and 45 directions of the rosettes before unloading and drilling, after drilling and unloading, and after using the Kirsch equations. Agreement is shown to be good except for the circle highlighted data points. These points belong to rosettes placed on the seam weld, where the hypothesis of uniformity of stresses along the measuring depth is not valid. Moreover, the high strain values that some of them show may indicate plastic behavior around the hole, rendering the application of the elastic equations used in the problem invalid. The above method of calculating uncertainties was adapted from reference [7], which uses a similar method to calibrate material and geometric factors to be used in the blindhole residual stress technique.
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CONCLUSIONS This article has presented residual stress distributions along five cross sections of one short segment of UOESAW thin walled line pipe classified as API 5L X60. It corroborates the following conclusions, which were also arrived at in a previous article [2]: a) the residual stress distributions are complexshaped along a given cross section; b) these distributions are similar for cross sections very close to each other, but they differ significantly when the cross sections are reasonably spaced in a given segment of pipe; c) similar pipe segments from a given pipeline may have completely different residual stress distributions. A consequence of these conclusions is that the use of a residual stress measurement technique such as the blindhole drilling method to determine general primary and soil movement loading in operating pipeline will furnish inaccurate results even if a reasonable number of measurement points are used to describe the stress states of points along the analyzed cross sections. Furthermore, the present investigation used a device that made it possible to apply combined loading: axial force, bending moment and internal pressure. Stress states at several points were measured using strain gage rosettes, and the results were compared with Strength of Material and Finite Element solutions. The overall uncertainty evaluated from the comparison of strains determined from the experimental and analytic and numerical methods turned out to be around 50μİ. The uncertainty of the hole drilling method was also evaluated by comparing the initially imposed combined strain state and the calculated strains from the measured unloading of the combined loads. Calculations in this case used the analytic Kirsch solution for an infinite plate with a through hole under a biaxial stress loading. Uncertainty evaluated in this way was only 26μİ. ACKNOWLEDGMENTS The authors wish to acknowledge Fluke Engenharia, Macaé, RJ, Brazil, for the construction of the Ushaped device used in the experiments. REFERENCES 1. L.D. Rodrigues, “Measurement of residual stresses in pipes driving the determination of efforts in buried pipelines,” MSc. Dissertation, Department of Mechanical Engineering, Pontifical Catholic University of Rio de Janeiro / PUCRio, Brazil, 2007 (in Portuguese). 2. L.D.Rodrigues, J.L.F.Freire, R.D.Viera, “Measurement of residual stresses in UOESAW line pipes,” Experimental Techniques, 5862, January / February, 2008. 3. G.W.R. Delgadillo, “Residual stress measurements in pipes subjected to combined loads,” MSc. Dissertation, Department of Mechanical Engineering, Pontifical Catholic University of Rio de Janeiro / PUCRio, Brazil, 2009 (in Portuguese). 4. J.C. Freitas, A.A. Gonçalves Jr., M.R. Viotti, “A historical case in the BrazilBolivia natural gas pipeline: 5 years of stress monitoring at the Curriola river slope,” Paper IBP10422009, Proceedings of the Rio Pipeline 2009 th Conference and Exposition, Instituto Brasileiro de Petróleo, Gás e Biocombustíveis, Av. Almirante Barroso 52/26 floor, Rio de Janeiro, Brazil. 5. ASTME83708, “Standard test method for determining residual stress by the hole drilling straingage method,” American Society for Testing Materials, 100 Barr Harbor Drive, West Conshohocken, PA, USA, January, 2008. 6. F. Fiorentini, ¨Measurement of applied stresses in a pipe under combined loadings using the blindhole residual stress technique,¨ Mechanical Engineering Department, ENSAM ChâlonsenChampagne, France, 2008. 7. TechNote; TN5036, “Measurement of residual stresses by the holedrilling strain gage method,” Vishay MicroMeasurements Group, Raleigh, NC, USA, August, 2007.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Incremental Computation Technique for Residual Stress Calculations Using the Integral Method Gary S. Schajer
Theo J. Rickert
Dept. Mechanical Engg, Univ. British Columbia, Vancouver, Canada V6T 1Z4 [emailprotected]
American Stress Technologies Cheswick, PA [emailprotected]
Abstract The Integral Method for determining residual stresses involves making surface deformation measurements after a sequence of small increments of material removal depth. Typically, the associated matrix equation for solving the residual stresses within each depth increment is illconditioned. The resulting error sensitivity of the residual stress evaluation makes it essential that data measurement errors are minimized and that the residual stress solution method be as stable as possible. These two issues are addressed in this paper. The proposed method involves using incremental deformation data instead of the total deformation data that are conventionally used. The technique is illustrated using an example ESPI holedrilling measurement.
Introduction Methods for measuring throughthickness profiles of residual stress in materials typically involve measuring surface deformations at a sequence of steps as stressed material is incrementally removed [1]. Examples of such techniques are holedrilling [2,3], slitting [4,5] and layer removal [6,7]. Evaluation of the throughthickness stress profile from the measured deformations requires the solution of an inverse equation [8]. Such calculations are well known to be illconditioned, causing amplification of modest measurement errors into relatively much larger errors in the evaluated stresses. This error amplification places severe demands on the quality of the measurement technique to ensure that stress evaluation errors remain within an acceptable range. Care must also be taken with the mathematical method used to evaluate the residual stress profile from the measured data. Numerical techniques have been developed to reduce the effects of measurement errors through data averaging [9] and regularization [10]. This paper describes a technique that can improve the quality of the measured data and help stabilize the residual stress calculation method. In present practice, deformation data measured at each step during an incremental material removal process are referenced to the initial uncut condition. The proposed procedure is to alter the computational approach so that the stresses are instead evaluated in terms of the deformation change during each material removal step. When using optical methods such as ESPI [11,12,13], Moiré interferometry [14,15,16] or Digital Image Correlation [17,18], this change reduces the time interval between pairs of image sets, thereby reducing drift, improving optical correlation and enhancing image quality. Mathematically, the change to using incremental data leads to a betterconditioned set of equations to be solved because they become more diagonally dominant [19]. The proposed method is illustrated using an ESPI holedrilling residual stress measurement as an example.
Theory For compactness of discussion, the theoretical procedure will be presented using ESPI hole drilling as a specific example. However, the ideas presented apply equally to other incremental material removal methods for measuring residual stress, such as slitting and layer removal, and also to other deformation measurement methods such as Moiré interferometry. Figure 1 shows the crosssection of a hole drilled in a material containing inplane residual stresses. These stresses vary smoothly with depth from the specimen surface, as shown by the dashed line. When using the Integral method, the stresses are assumed to vary in the stepwise manner shown. The depth steps for the stresses correspond to the hole depth steps used for the incremental holedrilling measurements. The ESPI measurement method involves taking a reference set of phasestepped images of the specimen surface around the hole before the start of hole drilling, and then sets of images after each incremental increase in hole depth T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_35, © The Society for Experimental Mechanics, Inc. 2011
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[20]. In the conventional technique, the images measured after each increment in hole depth are correlated with the initial set, from which the total deformation of the specimen surface around the hole is determined. The relationship between the measured deformation f measured after each step and the stresses ı within each step can be expressed as a matrix equation [21]
Gı
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where G is a matrix and f and ı are vectors. Matrix element Gij represents the total surface deformation measured after hole depth increment i caused by a unit stress within increment j. Figure 2 illustrates this representation. Equation (1) shows how the total deformation measured after a given hole depth increment i is the sum of the effects of the stresses within all the increments 1 < j < i within the hole. There is no sensitivity to stresses below the hole, so matrix G is lower triangular. For strain gauge measurements, the elements of matrix G are single numbers representing the relationship between the measured strain and the interior stresses [21]. For ESPI measurements, each element of the measured deformation vector f represents the n measured pixels around the drilled hole, where n is a large number, typically many thousands or even millions of pixels. This excess of data causes equation (1) to be overdetermined. The lower triangular shape of G is in blocks, each block comprising n rows. In this case, the equation can be solved in a least squares sense using [19,22].
GT G ı
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ı Figure 1. Crosssection of a hole drilled into a material with residual stresses varying with depth.
(2)
The residual stress evaluation involves computing the matrix G, typically using finite element calculations [21], measuring the deformation quantities f, and then solving equation (2) to determine the stresses ı. Equation (3) illustrates the character of matrix G using typical numerical values from strain gauge measurements [21]. Strain gauge measurements were chosen for compactness for this example because each strain gauge gives a single strain number in each matrix position, rather than n rows.
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ª 0.0490 « «  0.0671 «  0.0754 « «  0.0792 «  0.0810 ¬
º » » » (3)  0.0507  0.0242 »  0.0547  0.0305  0.0116 »  0.0563  0.0325  0.0151  0.0037 »¼  0.0399
Figure 2. Physical interpretation of matrix coefficients Gij for the holedrilling method [21].
The negative signs of the matrix elements in Equation (3) occur because the strains are measured while the holedrilling causes the residual stresses to be subtracted rather than added. The smallest elements of matrix G occur along the diagonal, which is a mathematically undesirable feature because it gives a small determinant value, causing the matrix to be illconditioned [14]. The decreasing diagonal elements of G as the matrix grows downward, mirrors the rapidly decreasing sensitivity of the surface deformations to stresses at greater distances from the specimen surface. Thus, the matrix illconditioning progressively gets worse as the hole gets deeper. Equation (1) can be reformulated in terms of the differential deformation d that occurs within each hole depth increment, where di = fi – fi1, for 2 < i < j.
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To accommodate the change on the right side, matrix G needs to be expressed in differential form. Each row of the differential matrix D equals the corresponding row of G minus the preceding row. Exceptionally, the first rows are identical. For the example values of G in Equation (3), the corresponding values in D are:
D
ª 0.0490 « «  0.0181 «  0.0083 « «  0.0038 «  0.0018 ¬
º »  0.0399 » »  0.0108  0.0242 »  0.0040  0.0063  0.0116 »  0.0016  0.0020  0.0035  0.0037 »¼
(5)
Matrix D is diagonally dominant, and so is much better conditioned than matrix G. Thus, it is expected to be much less prone to error amplification. For the strain gauge case where matrices G and D are square, equations (1) and (4) are just linear variants of the same equation, and thus they give identical solutions. However, for overdetermined solutions using leastsquares calculations such as equation (2), significantly different results are achieved. In this case, the data associated with the largest matrix elements tend to be weighted more heavily. A leastsquares calculation using matrix D in equation (6) gives a more stable result than with matrix G in equation (2) because the largest matrix elements are along the diagonal.
DT D ı
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ESPI Measurements Figure 3 schematically shows the apparatus used here for the example ESPI holedrilling measurements (PRISM, American Stress Technologies, Cheswick, PA). Light from a coherent laser source is split into an illumination light and a reference light. The illumination light illuminates the specimen surface, which is imaged by a CCD camera. The reference light passes through a piezo phase stepper and goes via a fiber link directly to the CCD surface. The two light beams interfere on the CCD surface to form a speckle pattern. The local phase of this speckle pattern is determined at each pixel from a set of four phasestepped speckle images [20]. Deformations of the measured surface are evaluated by making subsequent sets of phasestepped speckle images, computing the local phase angles, and subtracting the initial phase angle measurements. For the optical arrangement shown in Figure 3, the phase changes indicate surface displacements in the direction of the bisector of the illumination and object beams, called the sensitivity vector. The specimen used for this study was from a construction steel, StE355, with its surface shotpeened to an intensity of 0.20  0.25. This treatment created equibiaxial compressive residual stresses near the material surface, with tensile stresses extending into the interior to maintain equilibrium. The shotpeened specimen was chosen because it has a steep stress gradient near the surface and thus gives a challenge to the stress evaluation method.
Figure 3. Schematic of equipment used for ESPI measurements (from Steinzig [14]).
A highspeed (50,000rpm) electric drill mounted on a precision traverse (not shown in Figure 3) was used to drill a hole for the example holedrilling residual stress measurements. An initial ESPI phase map was measured on the specimen surface before starting the hole drilling. Further ESPI phase maps were measured after 16 successive hole depth increments of 25μm. Exceptionally, the first hole depth increment was 10μm to 1 provide greater surface detail. The hole diameter was 1.59mm ( /16”).
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Figure 4 shows the fringe pattern for the final (16th) ESPI measurement. This measurement was completed within one hour after the initial surface measurement taken before commencing the hole drilling. Figure 4(a) shows the fringe pattern for the 16th ESPI measurement referenced to the initial (0th) ESPI measurement. The central ellipse indicates the edge of the drilled hole. This elliptical shape appears because the circular hole was imaged nonperpendicularly. The camera was set at an angle of 27° to allow space for operation of the electric drill. The stress calculation takes this angle into account [22]. The two outer ellipses enclose the pixels used for the residual stress calculation. The positions of these ellipses were chosen to exclude the pixels immediately adjacent to the hole edge, which tend to get damaged by the passage of the chips from the hole cutting, and the pixels in the far field, which are too far away from the hole to contain much useful data.
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Figure 4. Fringe patterns for the 16th ESPI measurement: (a) referenced to the initial measurement, (b) referenced to the immediately prior measurement
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Figure 5. Residuals for the 16th ESPI measurement: (a) referenced to the initial measurement, (b) referenced to the immediately prior measurement.
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Figure 4(a) shows a distinctive fringe pattern contained within a somewhat noisy background. This noisy background is typical of ESPI measurements. Figure 4(b) shows the fringe pattern for the same 16th ESPI th measurement, but here referenced to the immediately prior (15 ) ESPI measurement. Since there had been very th th little further surface deformation between the 15 and 16 depth increments, the fringe pattern in Figure 4(b) is barely visible. This small deformation presents no difficulty because the calculation method can successfully respond to the small deformations indicated. However, the muchreduced noise within the data is notable because it improves the quality of the ESPI data and significantly reduces noise in the residual stress evaluations.
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A further indication of the lower noise contained in prior referenced data can be seen in the residual plots for the two types of data referencing. The ”residual” is the quantity remaining after the theoretical data corresponding to the computed stresses (as described below) are subtracted from the measured data. Ideally, the measured and theoretical data should be identical, so a small residual is a desirable feature. Figure 5(a) shows the residual in the 16th initially referenced ESPI measurement. It shows a pattern of noise similar to that seen in Figure 4(a). Figure 5(b) shows the corresponding residual in the 16th prior referenced ESPI measurement. As expected, it shows a similar pattern of reduced noise to that seen in Figure 4(b).
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Figure 6. Residual stress vs. depth profiles: (a) computed using initially referenced ESPI measurements, (b) computed using prior referenced ESPI measurements
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Figure 6(a) shows the stresses calculated using equation (2) with initially referenced ESPI measurements. The curves show the expected peak compression below the surface, with x and y stresses approximately equal, as also observed by Lord et al. [18]. However, measurement noise has caused substantial irregularities of the graphs. Also of concern are the indicated tensile stresses immediately adjacent to the surface and the differences between the x and ystresses. Figure 6(b) shows the stresses calculated using equation (6) with each ESPI measurement referenced to the prior measurement. The calculated stresses are much more realistic; they have a smoother trend, the indicated stresses at the surface are compressive, and the x and ystresses are closely similar. The tensile stresses reported at the surface in Figure 6(a) are an artefact caused by the large offdiagonal elements in the lower left corner of matrix G, such as shown in equation (3). During the leastsquares evaluation in equation (2), these large matrix elements have correspondingly large influence, causing the interior stresses to distort the evaluation of the nearsurface stresses. Matrix D is diagonally dominant, and so does not display this effect when used in equation (6). Figure 7 shows a further important feature of using priorreferenced data. Figure 7(a) shows the results of a progressively growing series of xstress calculations using equation (2). The first calculation of the series computes only the first stress using the first ESPI measurement, the second computes the first two stresses using the first two ESPI measurements, and so on until the 16th measurement. This sequence illustrates the cumulative effects of the interior stresses on the surface stress calculation. Figure 7(a) shows that the surface stress is realistically calculated when only few subsurface stresses are included in the calculation, but that it becomes increasingly tensile as further subsurface stresses are involved. In addition, the effects of measurement noise are readily apparent in Figure 7(a). In contrast, the corresponding results using prior referenced ESPI data and equation (6) show consistently compressive stresses at the surface and much reduced noise elsewhere.
Discussion Figures 47 clearly show the significant improvement achieved in calculations of residual stresses when using prior referenced ESPI data in place of initially referenced data. A practical question arises as to whether a similar advantage could be obtained when doing traditional holedrilling residual stress measurements with strain gauges. While there is a great similarity between strain gauge and ESPI holedrilling, there are also significant differences. An important difference is that strain gauge measurements provide a justsufficient amount of data. There are no excess data, and thus the “least squares” solutions to equations (2) and (6) provide identical “exact” results. In addition, the lack of excess data provides no quality advantage in using prior referenced measurements. Any drift that may occur during the course of a set of strain gauge measurements similarly affects initial and prior referenced data. Thus, there is no particular advantage or disadvantage in using prior referenced data with traditional straingauge holedrilling measurements. In contrast, prior referencing does significantly improve the average quality of the excess data available with optical techniques such as ESPI, and it provides a more balanced “bestfit” to those data.
Conclusions In present practice, optical data measured at each step of an incremental material removal process for measuring residual stresses are referenced to the initial uncut condition. In the procedure proposed here, the measurement and computational approaches are altered so that the stresses are evaluated in terms of the deformation change during each material removal step. This change provides two important benefits, the first that the time between corresponding pairs of optical measurements is minimized, thereby greatly improving image and phase unwrapping quality, and the second that the associated mathematical relationship between measured deformations and residual stresses is much better conditioned and gives more stable results. An example ESPI holedrilling measurement demonstrates the substantial measurement quality and calculation stability improvements. The proposed method is effective particularly when used with optical measurements, where there is a great excess of data available. For measurements using strain gauges, there typically are no excess data, in which case the method then gives neither advantage nor disadvantage.
Acknowledgments Financial support for this work was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), and by American Stress Technologies, Cheswick, PA. Mr. Anthony An kindly assisted with the experimental work.
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Schajer, G. S. and Prime, M. B. “Use of Inverse Solutions for Residual Stress Measurements.” Journal of Engineering Materials and Technology. Vol.128, No.3, pp.375382, 2006. ASTM. “Determining Residual Stresses by the HoleDrilling StrainGage Method.” ASTM Standard Test Method E83708. American Society for Testing and Materials, West Conshohocken, PA. 2008. Lu, J. “Handbook of Measurement of Residual Stresses,” Chapter 2: “HoleDrilling and Ring Core Methods.” Fairmont Press, Lilburn, GA, 1996. Prime, M. B. “Residual Stress Measurement by Successive Extension of a Slot: The Crack Compliance Method.” Applied Mechanics Review, Vol.52, No.2, pp.7596. 1999. Cheng, W., and Finnie, I. “Measurement of Residual Hoop Stress in Cylinders Using the Compliance Method.” Journal of Engineering Materials and Technology, 108, pp.8792, 1986. Sachs, G. and Espey, G. “Measurement of Residual Stresses in Metal.” Iron Age, 148, Sept. 18, pp. 63– 71; Sept. 25, pp. 36–42, 1941. Treuting, R. G. and Read, W. T. “A Mechanical Determination of Biaxial Residual Stress in Sheet Materials,” Journal of Applied Physics, Vol.22, No.2, pp.130134, 1951. Parker, R. L. “Geophysical Inverse Theory.” Princeton University Press, New Jersey, 1994. Schajer, G. S. “Strain Data Averaging for the HoleDrilling Method.” Experimental Techniques, Vol.15, No.2, pp.2528, 1991. Schajer, G. S. “HoleDrilling Residual Stress Profiling with Automated Smoothing.” Journal of Engineering Materials and Technology, Vol.129, No.3, pp.440445, 2007. Nelson, D.V. and McCrickerd, J.T. “Residualstress Determination Through Combined Use of Holographic Interferometry and BlindHole Drilling.” Experimental Mechanics, Vol.26, No.4, pp.371378, 1986. Díaz, F. V., Kaufmann, G. H. and Möller, O. Residual Stress Determination Using Blindhole Drilling and Digital Speckle Pattern Interferometry with Automated Data Processing.” Experimental Mechanics, Vol.41, No. 4, pp.319323, 2001. Steinzig, M. and Ponslet, E. “Residual Stress Measurement Using the Hole Drilling Method and Laser Speckle Interferometry: Part I.” Experimental Techniques, Vol.27, No.3, pp.4346, 2003. McDonach, A., McKelvie, J., MacKenzie, P. and Walker, C. A. “Improved Moiré Interferometry and Applications in Fracture Mechanics, Residual Stress and Damaged Composites.” Experimental Techniques, Vol.7, No.6, pp.2024, 1983. Nicoletto, G. “Moiré Interferometry Determination of Residual Stresses in the Presence of Gradients,” Experimental Mechanics, Vol.31, No.3, pp.252256, 1991. Wu, Z., Lu, J. and Han, B. “Study of Residual Stress Distribution by a Combined Method of Moiré Interferometry and Incremental Hole Drilling.” Journal of Applied Mechanics, Vol.65, No.4 Part I: pp.837843, Part II: pp.844850, 1998. McGinnis, M.J., Pessiki, S. and Turker, H. “Application of Threedimensional Digital Image Correlation to the Coredrilling Method.” Experimental Mechanics, Vol.45, No.4, pp.359367, 2005. Lord J.D., Penn D. and Whitehead, P. “The Application of Digital Image Correlation for Measuring Residual Stress by Incremental Hole Drilling.” Applied Mechanics and Materials, Vols. 1314, pp 6573, 2008. Dahlquist, G., Björck, Å., and Anderson, N., 1974, Numerical Methods, PrenticeHall, Englewood Cliffs, NJ. Sirohi, R. S. “Speckle Metrology.” Marcel Dekker, New York, 1993. Schajer, G. S. “Measurement of NonUniform Residual Stresses Using the HoleDrilling Method.” Journal of Engineering Materials and Technology, Vol.110, No.4, Part I: pp.338343, Part II: pp.344349, 1988. Schajer, G. S. and Steinzig, M. “FullField Calculation of HoleDrilling Residual Stresses from ESPI Data.” Experimental Mechanics, Vol.45, No.6, pp.526532, 2005.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Experimental Investigation of Residual Stresses in Water and Air Quenched Aluminum Alloy Castings
Bowang Xiao1, Yiming Rong1 and Keyu Li2 1. Manufacturing Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA. [emailprotected] 2. Mechanical Engineering Department, Oakland University, Rochester Hills, MI 48309, USA
ABSTRACT Cast aluminum alloys are usually subject to heat treatment including quenching for improved mechanical properties. A significant amount of residual stresses can be developed in aluminum castings during heat treatment. This paper investigates experimentally residual stress differences between water quenched aluminum castings and air quenched ones. The residual stresses in aluminum castings were measured mainly using resistance strain rosettes holedrilling method. Other measured methods such as Interferometric Strain/Slope Rosette (ISSR), Xray diffraction method and neutron diffraction method were also applied in this investigation. In comparison with water quenching, air quenching significantly reduces the residual stresses. KEYWORDS Resistance Strain Gauge, Incremental Holedrilling Method, Residual Stress, Quenching, Aluminum Alloy Casting 1. INTRODUCTION With the increasing demand of reducing weight and improving fuel efficiency, cast aluminum components have been widely used in critical automotive components such as engine blocks, cylinder heads, and suspension parts. In order to increase mechanical properties, cast aluminum alloys are usually subject to a T6 /T7 heat treatment which includes a solution treatment at a relatively high temperature, quenching in a cold medium such as water or forced air flow, and age hardening at an intermediate temperature. A significant amount of residual stresses can be developed in aluminum castings during the heat treatment process [14]. Residual stresses are those remaining in a component after manufacture or heat treating processing. The existence of residual stresses in a structural component can have a significant influence on its performance. Fatigue performance of cast aluminum components can be dramatically affected by the presence of residual stresses, in particular, the tension residual stresses in the surface layer. Therefore, it is of increasing interest to measure and control residual stresses in critical components. Many methods have been developed to measure residual stresses in manufactured parts. Mechanical approaches such as hole drilling, curvature measurements, and crack compliance measure residual stress by changes in component distortion. Diffraction techniques such as Xray diffraction and neutron diffraction measure elastic strains in the components due to residual stresses [5]. In this investigation, the residual stresses in aluminum castings after water quenching and air quenching were measured mainly using RSR (Resistance Strain Rosette) in conjunction with holedrilling method. Other measured methods such as Interferometric Strain/Slope Rosette (ISSR), Xray diffraction and neutron diffraction were also applied in this investigation. The purpose of this investigation is to see the differences of residual stresses in castings between after water quenching and after air quenching. 2. THEORERICAL BACKGROUND OF RESIDUAL STRESS MEASUREMENT METHODS This section reviews the residual stress measurement methods applied in the measurement of residual stresses.
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RESISTANCE STRAIN ROSETTES HOLEDRILLING METHOD The widely used RSR (holedrilling) method is included in the ASTM standard E837 [6]. The detailed procedures are also documented in Measurement Group’s technique note 503 [7]. In the measurement of residual stress using center holedrilling method, a strain gauge rosette is mounted onto the surface of the sample and a small hole is drilled at the center of the three strain gauges and used to measure relieved strains during the holedrilling. The general expression for the relieved radial strains due to a plane biaxial residual stress state is Equation 1 [7]. The measured strains are then used to backcalculate residual stresses, and the integral holedrilling method was developed to calculate residual stresses in multiply layers drilling for nonuniform residual stress fields [8].
where
H1 H2
A(V x V y ) B(V x V y ) cos 2D
H3
A(V x V y ) B(V x V y ) cos 2J
A(V x V y ) B(V x V y ) cos 2E
(1)
H 1, 2,3 =measured strain relieved from strain gauge 1, 2 and3, respectively V x , V y =stress in x and y direction, respectively A, B
D EJ
=calibration coefficients =angle measured counterclockwise from the x direction to the axis of the strain gauge 1, 2 and 3,
respectively Since the coefficients A and B for blind holedrilling cannot be calculated directly from theoretical considerations, they are usually obtained by numerical procedures such as finiteelement analysis [7, 8]. Some tables of the coefficients defined in Equation 3 were published [8, 9]. It is suggested that the coefficients A and B can be interpolated or extrapolated from the published nondimensionless coefficients [8, 9]. However, errors are always introduced in this procedure. More accurate residual stresses can be calculated if the interpolation can be avoided, e.g. determine the calibration coefficients directly using FEA for a specific measurement [10]. INTERFEROMETRIC STRAIN/SLOPE ROSETTE (ISSR) The ISSR is a laser based technique that measures the strain changes based on diffraction and interference of laser light [11]. ISSR consists of three microindentations and has the configuration of delta or rectangular rosette [11]. A delta rosette, also called 60deg ISSR, contains three sixfaced indentations and the three indentations form an equilateral triangle. Under illumination by an incident laser beam, the six facets of each indentation in a delta rosette reflect and diffract the light in six directions. The motion of the fringes is related to the displacements between the indentations and hence to the strains and slopes. The strains and slopes are determined through tracing the shifts of fringe patterns. The ISSR/Ringcore method is a combination of the ISSR method and ringcore cutting method [12, 13]. The ringcore cutting method is a stressrelief method. The ringcore cutting is to remove a ring of the material around the ISSR and the relieved strains and slopes are measured by the ISSR. The relative positions of the ringcore center and the ISSR are shown in Figure 1. Similar to strain gauge holedrilling method, the relieved strains and slopes are used to backcalculate residual stresses with calibration coefficients which are calibrated numerically [12, 13]. It is worthy to mention that the gage length of the ISSR on the core area is normally in the range of 50 μm and 250 μm while the resistance strain rosette (RSR used for the hole drilling method) has a 5mm gage length and the core size with the ISSR is less than 1 mm which is also much smaller than that used in the RSR. Thus, more localized residual stresses, especially in the small areas with the high stress gradients, can be measured by the ISSR/ringcore method. XRAY DIFFRACTION METHOD Different from strain gauge and ISSR methods, Xray diffraction method and neutron diffraction method are nondestructive methods. When a monochromatic Xray beam irradiates a solid material, it is scattered by the atoms composing the material. Because of the regular distribution of atoms (for a perfect crystalline material), the scattered waves lead to interferences similar to visible light diffraction by an optical diffraction pattern. If the material is composed of many grains (crystallites) randomly oriented, there is always a group of them suitably oriented to produce a diffracted beam. If the specimen is stressed, due to elastic deformation, the lattice spacing
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varies. Thus, the crystal lattice is used as a strain gauge which can be read by diffraction experiments. A diffraction peak is the result of Xrays scattering by many atoms in many grains, so a change in the lattice spacing will result in a peak shift only if it is hom*ogeneous over all these atoms and grains. The strain determined from peak shift measurements is representative of a macroscopic elastic strain (residual or applied). Therefore, residual stresses can be calculated from the residual strains measured in several directions and the elasticity constants of the material [5]. Please note that any crystal defects (vacancies, dislocations, stacking faults, etc) lead to a local fluctuation of the lattice spacing which results in a peak broadening and this method can only measured residual stresses on surface [5].
P (ISSR Center) 10 mm
y 25 mm
* r T
M
x
O (Core Center) Figure 1.
Principal Stress V1 Direction
~100Pm
25 mm 122 mm
Rectangular and polar coordinates in the ISSR/ringcore method
Figure 2.
Sketch drawing of the aluminum alloy casting
NEUTRON DIFFRACTION METHOD The physical principle of residual strain measurement by neutron is the same as Xray diffraction. However, the deep penetration of thermal neutrons into engineering materials means that the strain information obtained nondestructively are at depth complements. In other words, the neutron diffraction can determine residual strains (stresses) throughout the thickness of a component, whereas Xrays provide a measurement of the strains average over a few microns near the surface [5]. 3. AIR QUENCHING AND WATER QUENCHING OF ALUMINUM ALLOY CASTING
a)
b) Figure 3.
Experimental setup for: a) air quenching; b) water quenching
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In air quenching tests, the aluminum casting shown in Figure 2 was first heated to a designated solution temperature in the furnace, and then quickly moved out of the furnace (within 10 seconds) and placed onto the fixture which was under the forced air generated by a blower as shown in Figure 3a. The input voltage of the blower was adjusted by a variac so that different air velocities can be obtained. In this investigation, the forced air velocity was calibrated to 18m/s using an anemometer by setting a proper input voltage. A quenching bed shown in Figure 3b was built. The aluminum alloy casting was heated in the furnace and moved out and fixed to the pneumatic lifting system after it had reached a uniform specified temperature. The pneumatic system then lowered to immerse the casting into water at a constant speed. In this experiment, the water was heated up to a high temperature to simulate real production condition for water quenching of cylinder head. For experiments with agitation, the water was pump and circulated. The water flow velocities at the location where the test casting was quenched were calibrated and found to be very uniform at 0.08m/s. After the test casting was cooled down to water temperature, the test casting was taken out by the pneumatic lifting system. 4. RESIDUAL STRESS MEASUREMENT AND COMPARISON Residual stresses on the aluminum alloy castings were measured use different methods. For the casting quenched in water, resistance strain rosette (RSR) holedrilling method, ISSR ringcore method, Xray diffraction method and neutron diffraction method were used. For the casting quenched in air flow, only resistance strain rosette (RSR) holedrilling method was applied. Figure 4a shows the measurement of residual stresses at the top of the thin legs using RSR and ISSR methods on one of castings, and Figure 4b shows the measurements of residual stresses on thin and thick legs using RSR methods.
a) Figure 4.
b)
a) RSR holedrilling and ISSR/ringcore on top of the waterquenched aluminum casting; b) RSR holedrilling measurements on the side of air quenched and water quenched castings
Figure 5 illustrates the locations of all the applied methods. Please note that Figure 5 is for the purpose of illustrating relative locations of these methods and in reality different methods might be used on different castings. All measurements are at or near to surfaces, except the neutron diffraction method, which measures residual stresses inside of the castings. Figure 6 compares the residual stresses on the top surface of the thin legs of waterquenched casting. Figure 6a shows the measured residual stress distributions using ISSR technique. It is seen that the distribution of the residual stresses changes dramatically near the top surface. For the location measured, the residual stress varies from 200 MPa on the surface to about 10 MPa in the area about 0.7 mm below the surface. Figure 6b shows the residual stresses measured using RSR. Similar to the results shown in Figure 6a, a substantial compressive residual stress was developed at the top surface of the thin leg during water quenching process. By comparing the data in both figures, it can be concluded that the results from two methods (RSR and ISSR) are comparable.
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RSR R Xray
N Neutron
ISSR R
RSR on side o
RSR S
Figure 5.
a)
Locations of measurements using different methods
b)
Figure 6. Residual stresses after water quenching: a) Residual stresses measured by ISSR method at the top surface of the thin leg; b) Residual stresses measured by RSR at the top surface of the thin leg Figure 7 compares the residual stresses measured at thick leg by both RSR and neutron diffraction methods [14]. Xray diffraction measurements [15] were also made at the location as shown in Figure 5. The RSR measured the residual stress distribution from the surface to 0.45 mm deep, the Xray diffraction measured the residual stresses from the surface to 1 mm deep by etching and removing the surface material, and neutron diffraction method measured residual stresses across the whole thickness but started from 1.5 mm below the surface. By comparing the data in Figure 7, it is seen that the general trend of residual stress distribution measured at this
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location by RSR and neutron techniques is similar. The residual stresses measured by neutron are negative (the outer side surface), which agree with those measured by RSR. The residual stresses measured by both methods indicate that the stresses increase from surface to inside. The minimum principle residual stress measured by RSR method increases from about 100 MPa to 80 MPa as the depth increases from 0.05 mm to 0.45 mm. It is expected that the minimum principle residual stresses increases to about 40 MPa when the depth increases to 1.5 mm, which is the measurement result by neutron method. The maximum residual stress measured by RSR increases from about 50 MPa to 40 MPa as the depth increases from 0.05 to 0.45mm. It is expected that the maximum residual stresses increases to about 20 MPa when the depth increases to 1.5 mm, which is the measurement result by neutron diffraction method. The minimum residual stress measured by Xray increases from about 130 MPa to 90 MPa as the depth increases from 0.0127 to 0.5 mm, , which agrees with the measurement result by RSR method (Figure 7a). The maximum residual stress measured by Xray increases from about 70 MPa to 40 MPa as the depth increases from 0.0127 to 0.5 mm, which is agreeable with the measurement result by RSR method (Figure 7a).
60
Sample 2L, lines c/d
Longit.
stress, MPa
40
Transv.
20 0 20 40 60
Inner side Surface 10
8
6
Outer side Surface 4
2
2
4
6
8
10
through thickness position, mm
a) Figure 7.
b)
Residual stresses measured on thick leg using (a) RSR; and (b) Neutral diffraction method [3].
Residual stresses at thick wall Residual stresses ( MPa)
0 0
40 60
0.4
0.6
0.8
1
Max Principal after air quenching Min Principal after air quenching Max principal after water quenching
80 100 120
Figure 8.
0.2
20
Depth ( mm )
Comparison of the measured residual stresses at thick wall of the aluminum castings after air and water quenching
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Figure 8 compares the residual stress distributions measured at the thick wall of the airquenched aluminum casting to those of the waterquenched casting. It is seen that the residual stresses in the airquenched aluminum casting are less than 10 MPa and pretty uniform, but in the waterquenched aluminum casting, the absolute values of residual stresses vary from 40 MPa to 100 MPa. Small and uniform residual stresses in airquenched castings are good to the fatigue life and distortion control. 5. CONCLUSIONS The residual stresses in aluminum castings generated in air quenching and water quenching were measured using resistance strain rosettes holedrilling method, Interferometric Strain/Slope Rosette, Xray diffraction method and neutron diffraction method in this investigation. A significant amount of residual stresses can be developed in cast aluminum alloys during water quench after solution treatment. In comparison with water quenching, air quenching significantly reduces the residual stresses. 6. ACKNOWLEDGEMENT The authors would like to thank Dr. Qigui Wang from General Motors Company and Prof. Richard D. Sisson from CHTE (Center for Heat Treating Excellence) at WPI for their valuable help in the experimental work. REFERENCES [1] Elkatatny, I., Morsi, Y., Blicblau, A. S., 2003, "Numerical Analysis and Experimental Validation of High Pressure Gas Quenching," International Journal of Thermal Sciences, 42(4) pp. 417423. [2] Rose, A., Kessler, O., Hoffmann, F., 2006, "Quenching Distortion of Aluminum CastingsImprovement by Gas Cooling," Materialwissenschaft Und Werkstofftechnik, 37(1) pp. 116121. [3] Li, K., Xiao, B., and Wang, Q., 2009, "Residual Stresses in asQuenched Aluminum Castings," SAE International Journal of Materials & Manufacturing, 1(1) pp. 725731. [4] Li, M., and Allison, J. E., 2007, "Determination of Thermal Boundary Conditions for the Casting and Quenching Process with the Optimization Tool OptCast," Metallurgical and Materials Transactions B, 38B(4) pp. 567574. [5] Lu, J., and James, M., 1996, "Handbook of measurement of residual stresses," Fairmont Press,Inc, Lilburn, GA, USA, pp. 237. [6] ASTM designation: E83701, 2002, ASTM international, Philadelphia, PA, pp. 703712. [7] VishaymicroMeasurements, 2007, "Tech Note TN503:Measurement of Residual Stresses by the HoleDrilling Strain Gage Method," 2009pp. 33. [8] Schajer, G. S., 1988, "Measurement of NonUniform Residual Stresses using the HoleDrilling Method. Part IStress Calculation Procedures," J.Eng.Mater.Technol.(Trans.ASME), 110pp. 338343. [9] Schajer, G. S., 1988, "Measurement of NonUniform Residual Stresses using the HoleDrilling Method. Part IIPractical Application of the Integral Method," J.Eng.Mater.Technol.(Trans.ASME), 110pp. 344349. [10] Xiao, B., Li, K., and Rong, Y., 2010, "Automatic Determination and Experimental Evaluation of Residual Stress Calibration Coefficients for HoleDrilling Strain Gage Integral Method," Strain, In Press, Accepted Manuscript. [11] Li, K., 1995, "Interferometric 45 and 60 Strain Rosettes," Applied Optics, 34(28) pp. 63766379. [12] Li, K., and Ren, W., 2007, "Application of Minature RingCore and Interferometric Strain/Slope Rosette to Determine Residual Stress Distribution with Depth—Part I: Theories," Journal of Applied Mechanics, 74pp. 298. [13] Ren, W., and Li, K., 2007, "Application of Miniature RingCore and Interferometric Strain/Slope Rosette to Determine Residual Stress Distribution with Depth—Part II: Experiments," Journal of Applied Mechanics, 74pp. 307. [14] Luzin, V., Prask, H., and GnaeupelHerold, T., April 22, 2005, "Residual Stresses in GM aluminum Castings," NIST Center for Neutron Research, Gaithersburg, MD. [15] Hornbach, D., May 24,2005, "Xray Diffraction Determination of the Surface and Subsurface Residual Stresses in Two A356 Aluminum Castings," Lambda Technologies, 22712260, Cincinnati, Ohio.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
RESIDUAL STRESS ON AISI 300 SINTERED MATERIALS C. Casavola, C. Pappalettere, F. Tursi Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Viale Japigia, 182 – 70126 Bari, email: [emailprotected] ABSTRACT  Selective Laser Melting (SLM) is one of the most interesting technologies in the rapid prototyping processes because it allows to build complex 3D geometries. Moreover, full density can be reached and mechanical properties are comparable to those of bulk materials. However, the most important drawback is related to the thermal transient encountered during solidification which generates highly variable residual thermal stresses. Parameters such as laser scanner strategy, laser velocity and power should be optimized also in order to minimize residual stresses that are strictly dependent on the manufacturing process and cannot be completely avoided. Geometry of parts should be optimized in order to keep residual stresses and distortions low. This paper presents a study on residual stress distribution on SLM rectangular plates built by means of a new scanning strategy, implemented by dividing the fused zone in very small square sectors. Residual stresses measurement on SLM samples are performed by means of the hole drilling technique. Specimens made of AISI Maraging 300 steel are investigated and the residual stress profiles are compared with those related to previous measurements on SLM disks coming from the same process parameters.
INTRODUCTION Selective Laser Melting (SLM), such as Laser Cladding (LC) and Laser Sintering (LS), represents a modern
technology for building complex 3dimensional parts. All these processes allow different layer of material to be combined and are especially used for the deposition of wear and corrosion protective coatings. Besides, powder metallurgy with SLM is capable to yields materials and components with a very wide range of properties. However, SLM is very similar to a welding process and it is necessary to understand the interaction between manufacturing parameters and mechanical and metallurgical properties of parts. High precision in the dimension of SLM parts is strictly related to very high laser energy density. Large temperature gradients associated with the process, however, are the cause of high thermal stresses which may even lead to cracking, delaminating, and large bending distortion. Developing of residual stress depends on thermal phenomenon [12]: when a new layer of powder is deposited onto the existing ones, temperature gradients develop because of the rapid heating of the upper layer induced by the laser. At the same time, heat conduction through previously solidified layers is comparatively slow. The heated top layer expands first and then cools and shrinks. In all cases, the surrounding material, whose yield limit is reduced by the high temperature, constrains these movements and produces plastic strains. In many cases these processes can lead to macroscopic curvature of the product (fig. 1) as each new layer is added into the structure. Some studies demonstrate that the laser strategy used to melt the powder may affect significantly residual stresses and distortions [14]. In addition, product geometry, particularly the length and moment of inertia, affect the magnitude of residual stresses [5]. A base plate is generally used as rigid constraint welded to the part in order to reduce macroscopic distortions. This paper presents experimental measurements of residual stresses in SLM specimens produced from AISI Maraging 300 steel. The strain gage hole drilling method (HDM) is used for measuring residual stress profiles in different position of a rectangular specimen. Non uniform stress field have been found into the thickness of specimens. Experimental strain released have been processed with different method (ASTM, power series, integral method) and discussed. Results have been compared with residual stresses on SLM specimens of circular shape, in order to evaluate the influence of geometry on SLM process.
MATERIALS The SLM equipment used to built specimens is characterized by a Nd:YAG laser source with a wavelength of 1.064 ȝm, a spot diameter (d) of 200 ȝm and a maximum output power of 100 W in the continuous mode. Operation in pulsed mode in the 065 kHz band is also possible. Powder layers is deposited in one direction T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_37, © The Society for Experimental Mechanics, Inc. 2011
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using a knife. The building chamber is filled with nitrogen to prevent oxidation of the parts. The layer thickness is set to 30 ȝm. The powder used consists of spherical particles and has the composition of the AISI 18 Maraging 300 steel. The measured density of SLM parts is 8010 kg/m3. Mechanical properties are as follow: tensile strength 1152 MPa, Yield strength 985 MPa, Young modulus 166 GPa. The laser sintering strategy used in the preparation of specimen is random, in order to distribute the fused zones over the whole surface of the component without localizing in some limited region high thermal gradients which may cause warping and generate residual stresses. Each layer is divided in small square 2 sectors of 5×5 mm [6]. In order to reduce deformations that may lead to failures in the building process, specimens are manufactured onto a 15 mm thick substrate (building plate). Besides, parts are built using supports of 4 mm height in order to facilitate removal from the building platform. It has been observed that supports with a spare dimension of 2 mm cause the disengagement of the specimen from the building plate due to high thermal stress (Fig. 1). A spare dimension of 1 mm is recommended in order to avoid distortions of SLM parts. Specimen geometry is reported in figure 2. The rectangular specimen is oblique on the building plates in order to facilitate the powder deposition by means of a knife. Positions 1, 2 and 3 indicate the locations of residual stress measurements. These locations are chosen in order to have a correspondence between disks studied in [7] and rectangular specimens made with identical material and process parameters. This should point out the influence of geometry on residual stresses generated by SLM.
Supports
Buildingplate
a) Squaresectors5x5mm2
b) Figure1–DistortiononSLMrectangularspecimen
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Knife direction
Figure 2 – Geometry of SLM rectangular specimens (127 mm x 35 mm) with holes location (1,2,3) and disks position (diameter 35 mm)
METHODOLOGY AND EXPERIMENTAL PLAN The hole drilling method is utilized for the residual stresses measurements. It consists in drilling a very small hole into the specimen; consequently, residual stresses relax in the hole and stresses in the surrounding region change causing strains also to change; a strain gage rosette, specifically designed and standardized [8], measures these strains. Residual stresses can be calculated as suggested in [8]. The most recent version of ASTM standard contemplate a methodology for the very frequent case of non uniform stress field within the specimen thickness. Power series and Integral method [912] have also been considered in computing the experimental data. The accuracy of residual stresses calculations from the measured strain values obviously depends also on the level of accuracy at which elastic modulus and Poisson’s ratio are known. For this reason, E and Q values used in this work have been obtained from tensile tests on the same material. Distribution and magnitude of residual stresses generated in SLM components depend on a number of factors. In order to reduce some of these factors, the dimensions of the building plate are much larger than the dimensions of the manufactured component which is rectangular. In addition, the scanning laser strategy has been optimized by trying to have scan vectors always oriented along the normal to the part’s elongation. These facts should allow to maximize the adhesion between layers thus ensuring a nearly full density which limits the magnitude of residual stresses [13]. Specimens of circular geometry (disks of 35 mm diameter) studied in [7] exhibited very small warping effect which are instead very pronounced in specimens where one dimension predominates over the others (Fig. 1). The aim of the present work is to investigate on the influence of geometry (that is rectangular, with one dimension much larger than others, or circular) on residual stress values measured in the same location. As it can be observed in Fig. 2, the SLM rectangular specimens have been built so that SLM disks are comprised into its edge. Three SLM rectangular specimens have been studied. In all cases residual stresses measurements have been executed by HDM at location 1, 2 and 3. Fig. 3 shows the drilling device utilized. Strain values have been measured with System 5000 by Micro Measurements.
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Figure 3. Measuring device for HDM (RS 200 Milling Guide by Micro Measurement)
EXPERIMENTAL RESULTS Released strains have been measured and strain versus hole depth have been obtained for each measurements. Figure 4 shows residual stresses values computed according to ASTM method (case of non uniform stress field), Power series method and Integral method. It can be observed the high variability of the ASTM results with respect to the others. As with any other mathematical calculation, the quality of the calculated residual stresses depends directly on the quality of the input data. In this case, even if it should be observed that strain measurements are very sensitive to the effects of small experimental errors, the same data have been utilized for the 3 different computing procedures but results disagree: each of the 3 method utilize different approaches for stabilizing and smoothing the non uniform residual stress field. Since the strain variations during drill depends on many factors hardly predictable and checkable a priori, it seems that the best calculation method should be evaluated for each particular experiment. In the present work, the Power series method seems to be the most suitable for residual stresses calculations.
Figure 4 – Residual stresses values (specimen 2 – location 1)
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Figures 57 show the trend of the calculated residual stresses in terms of Von Mises equivalent stress plotted versus the perforation depth. Measurements have been executed on 3 rectangular plates of the same geometry and manufacturing process.
Figure 5 – Residual stress values on rectangular plate 1
Figure 6 – Residual stress values on rectangular plate 2
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Figure 7 – Residual stress values on rectangular plate 3
Residual stresses are variable inside the tested components. For all considered specimens and regardless of the hole location on the rectangular shape, residual stress value decrease sharply as we move far from free surface. In all cases, that is for each location, residual stress near the free surface are tensile and rather high (>100 MPa). Anyway, stress magnitude is lower than material yield point (Vy=990 MPa). Residual stress values at a depth hole ranging from 0.1 mm (i.e., in the vicinity of the zones heated by the laser beam in the last stage of the SLM process) to 1÷1.5 mm are considered to be representative. Results obtained at 1 mm hole depth, beyond which residual stress values do not change significantly, indicate that residual stresses are fairly small. This behavior probably occurs because the very large number of laser passes required to complete 7 mm thick parts may be considered to be equivalent to a sort of thermal treatment which relaxes internal stresses. This behavior seems to occur in the same way, both for rectangular plates and for disks: figure 8 summarizes residual stresses on SLM disks [7]. It seems that in any case, both for rectangular specimens and disks, there is a strong reduction of residual stress through the thickness (about 1 mm depth) with respect to the surface. Moreover, residual stresses on location 2 are lower than in other locations. Table 1 summarizes Von Mises  residual stress values measured on rectangular plates. Let us consider the effect of residual stress location on the building plate: the comparison between values of residual stresses measured for specimens located the same on the build plate indicated that residual stresses corresponding to position 2 seems to be lower than other cases. It could be explained remembering that location 2 is in the centre of the building plate, where the scanning strategy makes it possible to line up the laser always orthogonally to the layer surface; location 1 and 3 have slope with respect to laser beam, so the density of energy in different location could be not uniformly distributed.
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Figure 8 – Residual stress values on disks [7]
Rectangular plate 3 2 1 Mean St Dev
Position Hole depth 1,5 [mm] 1 2 3 16 30 66 42 30 35 51 38 24 36 18
33 5
42 22
1 126 61 129
Position Hole depth 0,1 [mm] 2 145 164 154
3 179 186 111
105 38
155 10
159 41
Table 1 – Von Mises residual stress on rectangular plates
CONCLUSIONS Selective laser melting (SLM) is a technological process which utilizes a laser beam to generate the energy for melting the powder and building parts of complicated geometry. However, this procedure may have some inherent drawbacks such as warping, cracking and residual stresses. The last melted layer generally shrinks during cooling while the layer underneath, already solidified constrains it and prevents further shrinking. Since this mechanism occurs for each layer at each step of the SLM process, residual stresses may develop inside the manufactured component. The whole phenomenon is very complicated and depends also on thermophysical properties of the material (thermal expansion coefficient, thermal conductivity, density, etc). This paper studied the magnitude of residual stresses developed in SLMfabricated components of rectangular shape. Residual stresses were measured by means of the hole drilling method in three different locations on building plate (1, 2 and 3) and then compared with residual stresses measured on circular specimens in the same locations on building plate. Experimental results were preliminary processed with ASTM, Power series and Integral method and then Power series method was chosen because of its ability in stabilizing non uniform stress field. Results of experimental tests provided the following indications: (i) residual stresses near free surface (i.e., at 0.1 mm hole depth) are tensile and higher than their counterpart measured at 1.5 mm. Stress magnitude decreased moving towards inner layers. This effect was more observed both for disks [7] and for rectangular specimens. (ii) Residual stress values became practically constant at 1.0 mm hole depth. (iii) The influence
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of positioning on residual stresses: the position of the component on the build plate affects magnitude of residual stress, in fact the central position (i.e., Pos. “2”) corresponds to the lowest stress values.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Nickel A.H., Barnett D.M., Prinz F.B., Thermal stress and deposition patterns in layered manufacturing, Mat. Science and Engineering A317, 2001, pp. 5964. Shiomi M., Osakada K., Nakamura K., Yamash*tal T., Abe F., Residual Stress within Metallic Model Made by Selective Laser Melting Process, CIRP Annals  Manufacturing Technology, 2004, 53 (1), pp. 195198. Jacobs P., Rapid Prototyping and Manufacturing / Fundamental of Sterolithography, Society of Manufacturing Engineers, 1992, Dearbon (MI). McIntosh J., Danforth S., Jamalabad V., Proc. of Solid Freeform Fabrication Sym., University of Texas, Austin, 1997, pp. 159166. Klingbeil N., Beuth J., Chin R., Amon C., Proc. of Solid Freeform Fabrication Sym., University of Texas, Austin, 1998, pp. 367374. Mercelis P., Kruth JP., Residual stresses in selective laser sintering and selective laser melting, Rapid Prototype J, 2006, vol. 12, pp. 254265. C. Casavola, S.L. Campanelli, C. Pappalettere – Preliminary investigation on the residual strain distribution due to the Selective Laser Melting Process, J of Strain Analysis (Professional Engineering Publishing, London, UK), vol.44 (1), 93104, 2009. ASTM E 837 Standard method for determining residual stresses by the holedrilling strain gage method, Annual Book of ASTM Standards, 2008. Vishay Micro Measurements, Measurement of residual stresses by the hole drilling strain gage method, Tech Note TN5036. Schajer G.S., Measurement of nonuniform residual stresses using the hole drilling method. Part I – Stress calculation procedures, J of Engineering Materials and Technology, 1988, vol. 110, pp. 344349. Schajer G.S., Measurement of nonuniform residual stresses using the hole drilling method. Part II Practical Application of the integral method, J of Engineering Materials and Technology, 1988, vol. 110, pp. 344349. Ajovalasit, Rassegna del metodo del foro per tensioni costanti  Studi per una proposta di raccomandazione sull’Analisi Sperimentale delle Tensioni Residue con il Metodo del Foro, Quaderno AIAS n. 3, 1997, pp. 311 (in Italian). Over, C., Meiners, W., Wissenbach, K., Lindemann, M., Hutfless, J., Laser Melting: A New Approach for the Direct Manufacturing of Metal Parts and Tools, Proc. EurouRapid 2002 International User's Conf., A5, 2002.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Practical Experiences in Hole Drilling Measurements of Residual Stresses Philip S. Whitehead, Stresscraft Ltd, St Winefride’s Chapel, Pick Street, Shepshed, Leics., LE12 9BB, UK. [emailprotected] ABSTRACT Strain gauge hole drilling is one of the most widely used destructive methods for measuring residual stresses. This paper describes hole drilling from 1987 to the present day at Stresscraft. Early procedures consisted of simple installations at readily accessible sites. In subsequent years, demands increased for hole drilling on more diverse component shapes and materials. Critical details of the methodology required for credible and reliable measurements are identified and discussed. These include strain measurements, the hole forming process and straintostress calculation procedures. Developments were made to improve the reproducibility and reliability of the method and accessibility at difficult target sites. Significant developments have included implementation of the Integral Method in 1989 (after G. S. Schajer) and the introduction of PCcontrolled miniature 3axis drilling machines for orbital drilling in 1999. Two machines have been used over a 10year period to drill approximately 15,000 gauges. While the fundamental elements of the method remain unchanged, in extreme cases, gauges can now be installed and drilled at sites that can only be viewed using miniature cameras. A number of examples of installations and results are presented and discussed to demonstrate the development of the method. Introduction Hole drilling is a popular method used for the determination of residual stresses in engineering components and structures. In this method, a ‘target’ strain gauge rosette is bonded to the testsurface. A hole is drilled into the testsurface relieving stresses at the hole boundary. Changes in strain output resulting from the drilling process are recorded; residual stresses are then calculated from the recorded strains using suitable coefficients. The equipment required for hole drilling is relatively simple; an operator working alone can complete a number of measurements in a day. Because of this simplicity and speed, hole drilling is an inexpensive method and, despite its destructive nature, this has led to its popularity. In many early instances of hole drilling, single sets of relaxed strains were measured at the end of the drilling process which were processed to generate residual stress values which refer to a single depth. In 1956, Kelsey [1] investigated the stresses which vary with depth using the hole drilling method; Rendler and Vigness [2] developed the method further on which later work was based. During the early 1980s, Stresscraft Ltd was involved exclusively in the production and analysis of photoelastic models. In 1987, requests were received to tender for a number of hole drilling projects linked to photoelastic model work. A survey of current practices was carried out. These included a standard for the hole drilling method established by ASTM [3] and the Measurements Group Technical Note [4] which described the use of target strain gauges, a hole drilling machine and a stress calculation procedure. It was decided that it would be feasible to perform measurements of this type and that hole drilling would be a useful addition to the services we provided. 1987 – Type1 Drilling Machine and Equivalent Uniform Stresses It was observed that the requirements for hole drilling could be divided into three main activities, namely relaxed strain measurement, hole drilling and calculation of residual stresses from the relaxed strains. While strain gauge training was in progress, the company manufactured its own drilling machine so that the required features could be included. The machine is shown in Fig. 1 and includes : x a combination of fixed and spring loaded rolling ball bearings in the drill guide for low friction linear motion without backlash or wear issues, x an optical head for insertion into the drill guide for alignment and diameter measurement, x a drill head (with micrometer feed attachment) for insertion into the drill guide for the drilling process. x two methods of drill rotation – from a remote electric motor via a flexible drive cable drive (ca. 15,000 rpm) or from a dental airturbine (ca. 250,000 rpm). T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_38, © The Society for Experimental Mechanics, Inc. 2011
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Fig. 1 Type1 Drilling Machine
Fig. 2 Results Data Sheet
A BASIC language program was written for the calculation of residual stresses from relaxed strains. This was to be performed using the method described in the Technical Note [4] as required by customers. It was decided that results from each gauge would be presented on a single sheet for incorporation into the report to the customer. The sheet provides tables of depths, relaxed strains and calculated residual stresses, graphical distributions of stresses vs. depth, and a diagram of the gauge indicating the installation directions and principal stress directions. A typical results sheet is shown in Fig. 2; this type of layout remains little changed to this day. The program also provided for the recording and archiving of strain data in electronic format. When hole drilling work was started it was understood that there was a need for close attention to all details of gauge installation, hole drilling and strain measurements. However, the importance of the choice of method used to calculate residual stresses from relaxed strains was not fully appreciated. Subsequently, stress calculation methods became the subject of much investigation within the company. One example of this work was based on a series of 6 mm thick alloy steel plate testpieces preloaded to create a longitudinal tensile stress. Subsequently, parts of the plate surfaces were masked while other parts were shotpeened. Target gauges (type EA06031RE120) were installed on the plates and drilled. Clear guidelines for the detection of stresses which vary with depth and the limitations of uniform stress calculations are given in ASTM E 837 [3]. The Technical Note [4] recommends that the equivalent uniform stress (EUS) is plotted against depth so that trends in distributions can be observed. The EUS is the stress which, if uniformly distributed from the surface to the bottom of the hole would produce the equivalent
Fig. 3 Distributions of stresses in a steel plate (EUS)
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strain at the surface. Results of the EUS calculation are shown in the distributions of stresses vs. depth in Fig. 3. Features of the stress distributions in Fig. 3 include : x Residual stresses at gauge 1 (unpeened) are distributed in a generally uniform manner; tensile longitudinal stresses of ca. 300 MPa are calculated for the depth range 0.1 mm to 0.5 mm; transverse stresses are small. x Intense nearsurface compression was detected at the peened surface. However, at the final depth, residual stresses at gauge 2 are significantly smaller than at gauge 1; no tensile stresses were calculated for gauge 2. The apparent penetration of the shotpeening process is excessive; there is no trend for subsurface stresses to revert to the unpeened state. This clearly demonstrates the severe limitation of the calculation method when applied to nonuniform stresses. In practice, it was not always possible to reconcile the concept of EUS with customers’ requirements for results which could be confidently related to real levels of residual stresses. At this stage in the development of hole drilling at Stresscraft, target gauges could be installed to a satisfactory standard and strain measuring equipment used to obtain reliable relaxed strain measurements. However, there was need for improvement in the hole driller design and, more importantly, the stress calculation method. 1989 – Type2 Drilling Machine and Integral Method A second drilling machine was manufactured. This machine was developed from the Type1 driller but reduced in size for operation in confined situations. The guide (with rolling bearings) was configured so that the drill head could be removed sideways for operation within disc bores of I200 mm. An electrical contact system was incorporated to aid drill / target datum depth detection, but this was not popular with drill operators who preferred to view the drilling cutter and gauge through a x10 magnification eyepiece. The drill collet was extended for operation as close as possible to step and shoulder features. A type2 machine is shown in Fig. 4. With this machine the setting up time for hole drilling was reduced and the scope for hole drilling at positions which could not be accessed previously was significantly extended. The limitations of the existing stress calculation method remained. A search was undertaken to find an alternative straintostress calculation scheme. This revealed the papers by Schajer [5] in which the EUS Method, the Power Series Method, and the Integral Method were compared. Schajer describes in detail how stresses in any depth increment are relaxed as that increment is drilled and how further relaxation occurs during drilling of subsequent increments because of the change in compliance of the structure. The Integral Method appeared to have the properties required for the evaluation of residual stresses with the severe gradients encountered in the shotpeened example given above. It was decided to produce and test a BASIC program version of the Integral Method incorporating the coefficients provided. This program was first used to calculate residual stresses in the previously described steel plates. Fig. 5 shows the distributions of residual stresses calculated using strains from the
Fig. 4 Type2 Drilling Machine
Fig. 5 Distributions of stresses in a steel plate (Integral Method)
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same data set used for EUS results in Fig. 3, but on this occasion using the Integral Method. The gauge 2 results in Fig. 5 show the same intensity of nearsurface shotpeening stresses as determined using the EUS method (Fig. 3). In the case of the Integral Method results, however, the rates of decay from the surface are more severe; a subsurface tensile peak in excess of 400 MPa (greater than the unpeened stress) occurs at depth 0.2 mm. Values of subsurface stresses from both gauges at depth 0.5 mm are very similar. This was seen to be a very encouraging result. Prior to introduction of the Integral Method, it was decided to develop a better understanding of the behaviour of the relaxation of stresses caused by hole drilling. Models were created and processed (using Algor finite element software) to reproduce the Integral Method coefficients [5]. Coefficients were obtained by mapping displacements over the active gauge area. The level of agreement between coefficients from Stresscraft models and those from Schajer [5] was found to be good. In addition, hole drilling measurements were made on cantilever beams gauged and drilled in unloaded and loaded conditions. Results from these tests confirmed the ability of the Integral Method to detect stress gradients, albeit at a reduced gradient from the shotpeening case. Following introduction of the Integral Method program (1989), it became apparent that the residual stresses calculated in this way were more sensitive to any deficiencies in the experimental procedure. A programme of testing was undertaken to enhance the quality of the relaxed strain data which would, in turn, reduce the levels of uncertainties in calculated stress values. This work included testing of : x more rigorous gauge installation procedures: it was determined that swab etching target sites to a matt finish provided the most reliable installations, x gauge adhesives: hightemperature cyanoacrylate types were found to be more reliable, x gauge types: the use of flexible ‘open’ construction gauges produced thinner adhesive layers (and more reliable strain transfer) than encapsulated gauges for installations other than on completely flat surfaces, x sourcing of inverted cone drilling cutters with ‘sharp’ corners (no chamfer) so that hole profiles were identical to those of FE models; x 100% inspection and measurement of all drilling cutters for form, chips, size, etc. Many details resulting from work of this type were incorporated into the NPL Good Practice Guide [6]. At this stage in hole drilling development at Stresscraft, it had been demonstrated that the use of the Integral Method for stress calculations had produced a significant advance in the credibility of residual stress results. In addition, the requirement for meticulous attention to detail in all procedures was made clear. During the period 1989 to 1998, four Type2 drilling machines were constructed. Residual stress measurements were carried out for more than 30 customers on aerospace components, diesel engines, pressure vessels, racing engines, forgings and a wide variety of testpieces and material samples. At Stresscraft, two machines and operators were used to drill, typically, 1000 to 1200 gauges per year. 1999 – 3Axis Drilling Machine For drilling tougher materials, it had sometimes been the practice to use a rotating eccentric ring between two elements in the drilling machines to provide an orbital motion during drilling. This addition was found to be very satisfactory in decreasing rates of drill wear and damage. It was also noted that the use of such a motion reduced the forces involved in the drilling process, the magnitudes of thermally induced strains and the length of the settling period between drilling and strain recording. In order to improve the performance of hole drilling, it was decided to build a new type of machine to achieve this type of motion. A machine design was evolved
Fig. 6 3axis hole drilling machine
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which was built from 3 miniature dovetail slides. Motion for each slide was provided by a 1 mm pitch recirculating ballscrew driven by 200step/rev stepper motor via a toothed belt. The machine is shown in Fig. 6. The drill head holder is fixed to a teeslotted plate; this gives a more versatile arrangement for holding the tool than previous drillers. Step signals for the motors were provided by an input/output (I/O) card in a PC via power supply units. A program was written in MS Visual Basic to create a control panel linking mouse clicks to I/O card outputs. The stepper motor/ ballscrew drives were set to achieve a resolution of 1 Pm per motor step. A screen control panel is shown in Fig. 7; ‘z’ refers to the default drilling direction (vertically downward) and ’x’ and ‘y’ to directions normal to the drill axis (in the horizontal plane). The LHS of the panel covers the selection of gauge size, orbit eccentricity, depth increment pattern and feed rate and provides a list of increment depths. The upper part of the panel centre shows the selected drill vector, the position of the drill head and controls for the movement of x, y and z directions. Beneath this section lie controls for movement along and normal to the selected drilling vector along with drill motor, orbit and increment drilling controls. A ‘Cycle’ control provides for datum depth detection; after the drill cutter has been set ca. 1000 Pm above the gauge upper surface, a single mouse click advances the drill head by 1002 Pm, completes a single orbit and then retracts by 1000 Pm. By advancing 2 Pm and retracting in this way, the operator can view the progress of the cutter through the gauge and adhesive layer and see the first contact with the target surface. The RHS of the panel provides for control of a P3 strain indicator; measured strains are recorded and tabulated for each depth increment. After setting up the drilling parameters and strain indicator, the ‘Drill and Record’ command allows the operator to drill through an entire increment pattern and record each set of strains on a single button click; after the final increment, strain values are written to an MSExcel file for processing.
Fig. 7 PC control panel for the 3axis hole drilling machine It was clear from the first trials that results obtained using the 3axis driller were very good and levels of consistency not previously seen were achieved because of the inherent characteristics of the machine : x drilling cutter wear was significantly reduced; cutting edge chipping and damage were eliminated, x settling times (from drill switchoff to strain recording) were reduced indicating a reduction in heat generated during the drilling process; forces generated during drilling were also found to be significantly reduced, x control of the drill feed and orbit eccentricity size gives the operator a high degree of control over the process which can be suited to specific materials and installations. Settings can be recorded and all holes in a series of gauges can be drilled using the same parameters. A second, identical 3axis driller was completed to increase gauge throughput and allow further development work to be carried out. During driller calibration, it was determined that the accuracy of depths achieved was an order
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of magnitude better than for conventional drillers. To exploit this, some test gauges were drilled in which the nearsurface increments were divided into a number of subincrements to extract more stress distribution detail from nearsurface relaxed strains. Because of difficulties in interpolating Integral Method coefficients within the triangular matrix close to the zero depth point, the number of sets of increments at which coefficients were calculated for each hole diameter was increased to 11 (66 finite element models). This provided coefficients without the need for depthwise interpolation. Drill depth increment patterns were adopted which used 16 Pm nearsurface increments for 031size gauges and 32 Pm nearsurface increments for 062size gauges. Results from gauges drilled using finer increments are shown in Figs. 8 and 9
Fig. 8 Residual stresses in a nickel alloy disc; machined surface
Fig. 9 Residual stresses in titanium alloy testpieces
Fig. 8 shows an example of residual stresses obtained from hole drilling in a machined nickel alloy disc. The two sets of data show stresses obtained using both ‘coarse’ (64 Pm) and ‘fine’ (16 Pm) calculation increments. Stress distributions calculated using fine increment show nearsurface tensile stresses and stress gradients which are not revealed by the coarser increment set. Fig. 9 gives a summary of residual stresses at 031size gauges applied to a series of titanium alloy testpieces and drilled using 16 Pm nearsurface increments. The conditions of the material surface for the five gauges are : 1. WireEDM cut surface 2. EDM surface; recast layer removed using a fine abrasive 3. Shotpeened 4. Shot peened (as 3) followed by smoothing using a fine abrasive to remove peening indentations 5. Shotpeened and smoothed (as 4) followed by further fine abrasion to remove ca. 50 Pm. The two following examples describe hole drilling measurements carried out on aerospace components. Example 1; Highstrength nickel alloy disc forging This example is concerned with the determination of residual stresses in an aircraft engine disc forging following quenching, heattreatment and ‘rough’ machining into the condition of supply (COS) shape. Such measurements are made to fulfil part of the aviation authorities requirements for engine certification. A section through a typical high pressure compressor disc forging for a RollsRoyce Trent engine is shown in Fig. 10. Target strain gauges are to be installed and drilled at four positions on the disc; A and D at the hub flanks and B and C at the rim flanks, with four gauges equispaced around the circumference at each position (0°, 90°, 180° and 270°). Hole drilling results will determine the magnitudes of circumferential and radial stresses at each target site.
I170 mm
I550 mm
B A
D
C
Fig. 10 Section through a compressor disc
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Nearsurface stresses are heavily influenced by the machining process and are not directly related to the state of stress within the disc structure. It is a requirement that the measurements demonstrate that the subsurface stresses reported are not affected by the machining process. The gauge type used was the Measurements Group CEA06062UL120 as shown in Fig. 11. Each gauge was drilled using a 3axis machine at 11 x 128 Pm increments to give a final hole depth of 1,4 mm. This depth of penetration leaves an adequate machining allowance for subsequent processing; in this instance, the hole drilling procedure is not destructive. Stress distributions from the sixteen gauges are shown in Fig. 12; the scale of stress is V/Vy (where Vy is the material yield strength). Fig. 11 Installation of a target gauge at the hub flank
Fig. 12 Distributions of residual stresses in a nickel alloy disc forging Residual stress results are examined for features including : x The presence of compressive circumferential stresses at the hub flanks; as the disc is machined, there is a significant redistribution of stresses, but circumferential compression at the hub flanks is usually retained. x The balance of stresses across the hub flanks. The top/bottom balance of the quenching process may give rise to a significant imbalance of front/rear stresses producing outofplane distortions during machining. x The level of circumferential variation of stresses around the disc. Irregular temperature gradients during quenching or heattreatment lead to circumferential variations in residual stresses and possible ovalisation distortions during machining. Stress distributions in Fig. 12 confirm the maximum depth of machiningaffected stresses as 0,4 mm; this depth differs at the four positions. Circumferential stresses at the hub flanks (A and D) are similar with the exception of those at gauge D.270 where the level of compression is less intense. In this case, the disc was subjected to a further assessment of hub flank stresses using additional target gauges near the 270° circumferential position.
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Example 2; Highstrength nickel alloy drum weld Some measurement procedures require special attention because of considerations of the gauge installation, material machining, stress gradients and other features. The RollsRoyce Trent HP compressor drum assembly (Fig. 13) presents an unusual set of challenges. The assembly is fabricated from two discs and a rear cone joined at two welds. The assembly material is a very tough high strength nickel superalloy and has been shotpeened in the measurement areas. In addition to conventional measurements at the disc hubs and diaphragms, etc, it was required to install and drill gauges in the region of the front and rear welds at the weld centrelines and at positions displaced by  1 mm and +1 mm from the centrelines. Measurements were to be made at both the OD and ID at 3 equispaced circumferential locations. The wall thickness at the weld is ca. 6 mm.
Gauge positions OD centre
OD+1 mm
OD1 mm
ID1 mm ID centre
ID+1 mm
front weld
Fig. 13 Schematic view of HP compressor drum assembly and gauge positions Outer diameter (OD) measurements were reasonably straightforward with easy access. Each target area was swab etched to reveal the weld centreline; each gauge was installed with a circumferential offset to avoid any detectable interference from previously drilled holes (Fig. 14). The gauge type used was EA06031RE120.
Fig. 14 Gauge and drilled holes at the OD
Fig. 15 Gauge and drilled holes at the ID
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Gauges were drilled using nearsurface drilling increments of 16 Pm; suborbits used during machining (the feed rate) were set to 2 Pm. The ID sites between the two discs are not visible directly; all etching, installation, soldering and drilling activities must be carried out while viewing using a miniature video camera and fibre light source. The radial distance between the hub bore and weld is ca. 160 mm while the axial distance between the adjacent hub flanks is 40 mm. An extended soldering iron and a number of other special tools were made to install the ID gauges. Fig. 15 shows a gauge installed at the ID and two previously drilled holes.
Fig. 16 Schematic view of the assembly and driller set up to drill at the front weld ID
The assembly and driller are shown in Fig. 16; the driller body (with eccentrically mounted air turbine) is assembled within the space between the two discs, inserted into the holder connected to the driller and then advanced towards the gauge and finally locked in position. Final movements for the alignment of the drill and gauge are made using the driller x and y movement commands while viewing via the video camera. Fig. 17 shows the driller at the front of the assembly during setting up.
Fig. 17 Drilling machine at the front of the compressor assembly
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ca. 6 mm
Fig. 18 Distributions of stresses at the weld OD and ID Typical distributions of circumferential stresses from a group of gauges are given in Fig. 18; the scale of stress is V/Vy (where Vy is the material yield strength). At each gauge, nearsurface compression resulting from the shotpeening process is clearly shown; the peening process appears to be more intense at the ID than at the OD. Transitions from compression to tension occur between depths 100 Pm and 200 Pm. Tensile stress maxima occur immediately after the runout of the peening effect (at depth ca. 200 Pm). It has been demonstrated that subsurface stresses at the weld centre are greater than at those at the 1 mm and +1 mm axial positions. Conclusions Over a period of 23 years, a number of developments have been applied to the hole drilling method which have added significantly to the value of the results produced. Throughout this time, the exemplary performance of commercially available target strain gauges and strain indicators has been a consistent feature. Early in this period, the introduction of the Integral Method transformed the way in which nonuniform stresses could be evaluated from relaxed strain data, while demanding a higher standard of gauge installation and hole drilling practice. It has been found that hole drilling to the required standard is best performed using an orbital drilling motion. At Stresscraft, this has been achieved using PCcontrolled 3axis drilling machines which can be configured to drill holes with a wide range of parameters and to enable gauges to be drilled in difficult locations. Acknowledgements The author wishes to thank the staff members at Stresscraft Ltd for their support and patient work in carrying out hole drilling measurements and manufacturing the drilling machines. Thanks are also due to RollsRoyce plc for permission to publish the two examples included in this paper and to Gavin Baxter and Jim Orr for their assistance in the presentation of this material.
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References [1] KELSEY, R.A., Measuring Nonuniform Residual Stresses by the Holedrilling Method, Proc., SESA XIV (1), pp. 181194, 1956 [2] RENDLER, N.J. and VIGNESS, I., Hole drilling Straingauge Method of Measuring Residual Stresses, Exp. Mech., 6 (12), pp. 577586, 1966 [3] ASTM E 83785 to –08e1, Determining Residual Stresses by the Hole drilling StrainGauge Method, 1985 to 2008 [4] TECHNICAL NOTES TN5032 to 5, Measurement of Residual Stresses by the Hole drilling Strain Gauge Method, Vishay Measurements Group, 1986 to 1993 [5] SCHAJER, G.S., Measurement of NonUniform Residual Stresses Using the Hole Drilling Method, J. Eng. Materials and Technology, 110 (4), Part I: pp.338343, Part II: pp.3445349, 1988 [6] GRANT, P.V., LORD, J.D. and WHITEHEAD, P.S., The Measurement of Residual Stresses by the Incremental Hole Drilling Technique, NPL Good Practice Guide No. 53 Issue 2, February 2006
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Destructive Methods for Measuring Residual Stresses: Techniques and Opportunities Gary S. Schajer Dept. Mechanical Engineering, Univ. British Columbia, Vancouver, Canada V6T 1Z4 [emailprotected]
Abstract Destructive methods are commonly used to evaluate residual stresses in a wide range of engineering components. While seemingly less attractive than nondestructive methods because of the specimen damage they cause, the nondestructive methods are very frequently the preferred choice because of their versatility and reliability. Many different methods and variations of methods have been developed to suit various specimen geometries and measurement objectives. Previously, only specimens with simple geometries could be handled, now the availability of sophisticated computational tools and of highprecision machining and measurement processes have greatly expanded the scope of the destructive methods for residual stress evaluation. This paper reviews several prominent destructive measurement methods, describes recent advances, and indicates some promising directions for future developments.
Introduction Residual stresses greatly influence the service performance of practical engineering components, notably their strength, fatigue life and dimensional stability. Thus, it is important to know the size of these stresses and to account for them during the design process. The “lockedin” character of residual stresses makes them challenging to measure because there are no external loads that can be manipulated. Thus, residual stress measurement methods differ significantly from general stress measurement methods. Over the years, these methods have grown both in range and sophistication, and have collectively become a wellestablished specialty. There are two general classes of residual stress measurement methods, destructive and nondestructive. The nondestructive methods [1] have the obvious advantage of specimen preservation, and are particularly useful for production quality control and for measurement of valuable specimens. However, they typically involve either very costly measurement equipment or they give empirically calibrated or comparative results. The destructive residual stress measurement methods generally have the opposite characteristics. While they either partially damage or fully destroy the specimen, they often require fairly simple equipment and generally give reliable and widely applicable results. Many different destructive methods for measuring residual stresses have been developed for specific applications. They differ in specimen and material removal geometry, the measurement method and the required computations. However, they all share the same general principle, that is, they all determine the residual stresses from the deformations measured as some stressed material is cut or removed from the specimen. This approach mirrors the common method used for measuring stresses due to external loads, where the deformations are measured as the loads are applied or removed. The “load” associated with residual stresses is in the material itself, and is “removed” as the material is cut or removed. The challenge introduced by this process is that the stress is removed from one part of the specimen while the measurements are made in a different part. This feature significantly complicates the calculations required. This paper gives an overview of the various destructive methods for measuring residual stress. discussion focuses on the three main areas of development: x
specimen shapes and material removal geometry
x
material cutting and deformation measurement technology
x
stress calculation techniques
The
Substantial advances have been made in all three areas. It is the purpose here to describe the characteristics of the various techniques, give information on method selection, and to highlight recent developments and trends. T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_39, © The Society for Experimental Mechanics, Inc. 2011
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Specimen Shapes and Material Removal Geometry The various destructive techniques for measuring residual stress have been developed on a mostly adhoc basis for particular applications and specimen shapes. Some of these methods have a wide range of application, and some are very specific. The more commonly used methods are described here. Excision provides a simple quantitative method for measuring residual stresses. It involves attaching one or more strain gauges on the surface of the specimen, and then excising the fragment of material attached to the strain gauge(s). This process releases the residual stresses in the material, and leaves the material fragment stressfree. The strain gauge(s) measure the corresponding strains. Excision is typically used with thin plate specimens, where the cutting of a small material fragment around the strain gauge(s) is straightforward. Application on thicker specimens is also possible. Full excision is possible, but is not often done because the inconvenience of the undercutting process required to excavate the material fragment. Partial excision by cutting deep slots at each end of the strain gauge [2,3] is a more practical procedure. Figure 1(a) illustrates the splitting method [4]. A deep cut is sawn into the specimen and the opening (or possibly the closing) of the adjacent material indicates the sign and the approximate size of the residual stresses present. This method is commonly used as a quick comparative test for quality control during material production. The “prong” test shown in Figure 1(b) is a variant method used for assessing stresses in dried lumber [5].
(a)
(b)
(c)
(d)
Figure 1. The splitting method, (a) for rods (from [4]), (b) for wood (from [5]), (c) for axial stresses in tubes [6], (d) for circumferential stresses in tubes [6]. The splitting method is often used to assess the residual stresses in thinwalled tubes. Figure 1 shows two different cutting arrangements [6], (c) for evaluating longitudinal stresses and (d) for circumferential stresses. The latter arrangement is commonly used for heat exchanger tubes, and is specified by ASTM standard E1928 [7]. The thinwall tube splitting method illustrated in Figures 1(c) and (d) is also an example of Stoney’s Method [8], sometimes called the curvature method. This method involves measuring the deflection or curvature of a thin plate caused by the addition or removal of material containing residual stresses. The method was originally developed for evaluating the stresses in electroplated materials, and is also used for assessing the stresses induced by shotpeening [9]. The sectioning method [10,11] combines several other methods to evaluate residual stresses within a given specimen. By choosing the combination and sequence of methods, a highly specific measurement can be made tailored to particular specimen. The sectioning method typically involves attachment of strain gauges and cutting out parts of the specimen. The strain reliefs measured as the various parts are progressively cut out provide a rich source of data from which both the size and location of the original residual stresses can be determined.
Figure 2. Sectioning method (from [10]).
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Figure 2 shows an example of the sectioning method, where a sequence of cuts was made to evaluate the residual stresses in a welded pipe [10]. The layer removal method is a generalization of Stoney’s method. It involves observing the deformation caused by the removal of a sequence of layers of material. The method is suited to flat plate and cylindrical specimens where the residual stresses are known to vary with depth from the surface, but to be uniform parallel to the surface. Figure 3 illustrates examples of the layer removal method, (a) on a flat plate specimen, and (b) on a cylindrical specimen. The method involves measuring deformations on one surface, for example using strain gauges, as parallel layers of material are removed from the opposite surface [12]. In the case of a hollow cylindrical specimen, deformation measurements can be made on either the outside or inside surface, while annular layers are removed from the opposite surface. When applied to cylindrical specimens, the layer removal method is commonly called “Sachs’ Method” [13]. The method is a general one; it is typically applied to metal specimens, e.g., [14], but can be applied to other materials, e.g., paperboard [15].
(a)
(b)
Figure 3. Layer removal method. (a) flat plate, (b) cylinder. The holedrilling method [16,17] is probably the most widely used destructive method for measuring residual stresses. It involves drilling a small hole in the surface of the specimen and measuring the deformations of the surrounding surface, traditionally using strain gauges [18], and more recently using fullfield optical techniques such as Moiré interferometry [19], ESPI [20] and DIC [21]. Figure 4(a) illustrates the process. The holedrilling method is popular because it gives reliable and rapid results with many specimen types, and creates only localized and often tolerable damage. The measurement procedure has been well developed [22] and is standardized as ASTM E837 [23].
(a)
(b)
(c)
Figure 4. Holedrilling methods: (a) conventional holedrilling method, (b) ringcore method, (c) deep hole method (from [29]). In the basic implementation of the holedrilling method, the surface deformations are measured as the hole is drilled to a depth approximately equal to the hole radius. The inplane stresses within the hole depth can
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be determined from the surface deformations using tabulated calibration coefficients [23]. In a more detailed implementation of the holedrilling method, the surface deformations are measured after each step in a series of small hole depth steps. This incremental procedure allows the stress profile within the hole depth to be determined [24]. The ringcore method [25,26] is an “insideout” variant of the holedrilling method. Whereas the holedrilling method involves drilling a central hole and measuring the resulting deformation of the surrounding surface, the ring core method involves measuring the deformation in a central area caused by the cutting of an annular slot in the surrounding material. Figure 4(b) illustrates the geometry. The ringcore method is a circular generalization of the twoslots method [2]. As with the holedrilling method, the ringcore method has a basic implementation to evaluate inplane stresses [26], and an incremental implementation to determine the stress profile [27]. The ringcore method has the advantage over the holedrilling method that it provides much larger surface strains. However, is less frequently used because it creates much greater specimen damage and is much less convenient to implement in practice. The deep hole method [28,29] is a further variant procedure that combines elements of both the holedrilling and ringcore methods. It involves drilling a hole deep into the specimen, and then measuring the diameter change as the surrounding material is overcored. Figure 4(c) illustrates the geometry. The main feature of the method is that it enables the measurement of deep interior stresses. The specimens can be quite large, for example, steel and aluminum castings weighing several tons. On a yet larger scale, the deep hole method is often used to measure stresses in large rock masses [30]. The slitting method [31,32] is also conceptually similar to the holedrilling method, but using a long slit rather than a hole. Alternative names are the crack compliance method, the sawcut method or the slotting method. Figure 5 illustrates the geometry. Strain gauges are attached either on the front or back surfaces, or both, and the relieved strains are measured as the slit is incrementally increased in depth. The slit can be introduced by a thin saw, milling cutter or wire EDM. The residual stresses perpendicular to the cut can then be determined from the measured strains using finite element calculated calibration constants, in the same Figure 5. Slitting method. way as for holedrilling calculations. The slitting method has the advantage over the holedrilling method that it can evaluate the stress profile over the entire specimen depth, the surface strain gauge providing data for the nearsurface stresses, and the back strain gauge providing data for the deeper stresses. However, the slitting method provides only the residual stresses normal to the cut surface, whereas the holedrilling method provides all three inplane stresses. Additional cuts can be made to find other stress components, in which case the overall procedure resembles the sectioning method. The Contour Method [33] is a newly developed technique for making fullfield residual stress measurements, typically within the crosssection of a prismatic specimen. It consists of cutting through the specimen crosssection using a wire EDM, and measuring the surface height profiles of the cut surfaces using a coordinate measuring machine or a laser profilometer. Figure 6 illustrates the process. The original residual stresses shown in Figure 6(a) are released by the cut and cause the material surface to deform (pull in for tensile stresses, bulge out for compressive stresses), as shown in Figure 6(b). The originally existing residual stresses normal to the cut can be evaluated from finite element calculations by determining the stresses required to return the deformed surface shape to a flat plane. In practice, to avoid any effects of measurement asymmetry, the surfaces on both sides of the cut are measured and the average surface height map is used. The contour method is remarkable because it gives a 2dimensional map of the residual stress distribution over the entire material crosssection. In comparison, other techniques such as layer removal and holedrilling give onedimensional profiles, while excision gives the residual stress only at one point. The contour method provides measurements of the stresses normal to the cut surface. If desired, stresses in other directions can also be determined by making additional cuts, typically perpendicular to the initial cut [34]. This approach effectively combines the contour method with the sectioning method. Taking a different approach, it is also possible to infer other stress components from the plastic deformations that original induced the residual stresses [35].
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(a)
(b)
(d)
(c)
Figure 6. Contour Method (from Prime [33]). (a) original stresses, (b) stressfree surface after cutting, (c) stresses required to restore a flat surface, (d) an example measured stress profile of a railway rail. Material Cutting and Deformation Measurement Technology The majority of the destructive methods for measuring residual stress have histories that trace back over many decades. During their evolution, the various methods have greatly developed in terms of their practical implementations, notably the methods used for cutting the specimen material and for measuring the resulting deformations. Early implementations, e.g., [6], used saws and twist drills to cut material, and mechanical gauges to measure deformations. These were superseded by highspeed cutters and strain gauges, e.g., [17], while the most recent implementations use EDM and electronic optical metrology, e.g., [28,19]. Exceptionally, the Contour Method [33] is a recently developed method. However, it’s successful implementation is also critically dependent on the availability of modern highprecision material cutting and surface measurement techniques. Certainly, the advent of complementary modern technologies has substantially opened up, and will continue to open up, many new opportunities in terms of test materials, specimen shapes and detail/accuracy of the residual stress evaluations. The discussion here focuses on the more recent cutting and measurement technologies. The two main requirements for a material cutting/removal technique are that the process should be accurate and should not induce additional residual stresses (“machining stresses”). The introduction of highspeed cutters [36] and abrasive machining [37] in the 1970s and 1980s, significantly advanced both these objectives. The more recent introduction of ElectroDischarge Machining (EDM) to residual stress measurements further advanced measurement capabilities to difficulttomachine materials [38], deephole drilling [28] and to the Contour Method [33]. In the latter case, EDM can justifiably be described as an “enabling technology” because the Contour Method only became feasible with the advent of the very high accuracy cutting available from wire EDM. A further example of an “enabling technology” is the application of Focused Ion Beam (FIB) instruments to make microscopic cuts [39], holes [40], rings [41], slots [42] etc., with dimensions is the low micron range. This development dramatically extended the dimension scale of the various residual stress measurement techniques into the micro and nano range. Figure 7 illustrates the application of FIB cutting to the splitting, ring core and slitting methods. The measurement techniques used for residual stress measurements have followed the same developmental trend. In particular, optical methods have increasingly been applied [43], notably because of their fullfield measurements and their adaptability to a wide range of machining geometries and size scales. The example applications shown in Figure 7 exemplify this adaptability, with three very different machining geometries applied at a microscopic scale.
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(a)
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Figure 7. Focused Ion Beam machining. (a) splitting [39], (b) ringcoring [41], (c) slitting [42].
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Fig. 8. Fullfield optical techniques for surface deformation measurements. (a) Moiré interferometry (from [19]), (b) Moiré fringe pattern (from [45]), (c) ESPI (from [46]), (d) ESPI fringe pattern, (e) 2D Digital Image Correlation (from [48]), (f) DIC vector plot (from [50]).
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Moiré interferometry [19,44,45] provides a sensitive technique for measuring the small surface displacements that occur during residual stress measurements. Figure 8(a) schematically shows a typical optical arrangement [19]. Light from a single coherent laser source is split into two beams that illuminate the specimen surface with the symmetrical geometry shown in the diagram. A diffraction grating consisting of finely ruled lines, typically 6001200 lines/mm, is replicated or made directly on the specimen surface. Diffraction of the beams creates a “virtual grating”, giving interference fringes consisting of light and dark lines. Figure 8(b) shows an example holedrilling measurement [45]. Each light or dark line represents a contour line of inplane surface displacement, in the xdirection in Figure 8(a). For typical optical arrangements, the inplane displacement increment between fringe lines is about 0.5Pm. The fullfield character of the measurements illustrated in Figure 8(b) provides opportunities not available from single point measurements. The availability of “excess” data provides the possibility to improve stress evaluation accuracy and reliability by data averaging, and to be able to identify errors, outliers or additional features. For example, the different fringe spacings in the upper and lower halves of Figure 8(b) show that the residual stresses are nonuniform within the surface plane. Electronic Speckle Pattern Interferometry (ESPI) [20,46,47] provides a further important method for measuring the surface displacements around a drilled hole. It has several similarities to the Moiré method and also involves measuring the interference pattern that is created when mixing two coherent light beams. Figure 8(c) shows a typical ESPI arrangement [46]. The light from a coherent laser source is divided into two parts, one of which (the illumination/object beam) is used to illuminate the specimen surface so that it can be imaged by a CCD camera. The second (the reference beam) is fed directly to the CCD camera so that it creates an interference pattern on the CCD surface. Evaluation of this interference pattern provides a fullfield map of the deformation of the measured surface, which can then be analyzed in a similar way as with Moiré interferometry. Figure 8(d) shows an example fringe pattern for a holedrilling measurement. Digital Image Correlation [21,4850] is a versatile optical technique for measuring surface displacements in two or three dimensions. The 2D technique, schematically illustrated in Figure 8(e), involves applying a textured pattern on the specimen surface and imaging the region of interest using a highresolution digital camera. The camera, which is set perpendicular to the surface, records images of the textured surface before and after deformation. The local details within the two images are then mathematically correlated, and their relative displacements determined. The algorithms used for doing this have become quite sophisticated, and with a wellcalibrated optical system, displacements of +/ 0.02 pixel can be resolved. Figure 8(f) shows an example vector plot of the measured displacements in a holedrilling measurement. The Moiré and ESPI techniques are based on the interference of coherent laser light, and they can measure surface displacements of fractions of a micron. Thus, they can provide sensitive measurements of the surface strains of structures ~1mm and larger. In contrast, Digital Image Correlation provides deformation measurements that are relative to the area being imaged. Thus, the DIC method can be applied over a wide range of length scales, for example on concrete structures measured in meters, e.g. [21], and on microscopic specimens measured in microns, viewed in a scanning electron microscope, e.g. [3942]. In addition, the DIC method simultaneously provides displacement measurements in both axial directions, whereas separate Moiré and ESPI measurements must be made to resolve each direction. However, the strain sensitivity of DIC is typically about an order of magnitude less than with the Moiré and ESPI techniques, and is thus best reserved for measurements of large strains or small specimens. Stress Calculation Techniques Advances in stress computational methods, numerical techniques and computer hardware have also provided “enabling technologies”, analogous in supportive effect to the advances in material cutting and surface deformation measurement techniques. The computational challenge when doing a destructive measurement of residual stress arises because the measured deformations occur by stressed material removal rather than by load removal. This feature introduces two complicating factors, the geometry of the specimen is changed and the deformations are typically measured at a different location than the original residual stresses. Because of these factors, the relationship between the measured deformations and the corresponding residual stresses does not have the simple proportional stress/strain relationships typical when working with applied loads. Instead, it commonly has the form of an inverse equation, schematically shown in Equation (1) [51].
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d(h)
³
h
G(H, h) V(H) dH
(1)
where d(h) are the data, typically strains or displacements, measured at position h, typically the radius or the distance from the measured surface, ı(H) is the stress at position H, and G(H,h) is the “kernel function” that relates the strains/displacements to the stresses. For simple specimen geometries, the kernel function G(H,h) can be determined analytically. For example, the strains measured on the inside surface of a hollow cylinder, at radius = a, as concentric layers are removed from the outside surface, at radius = r (Sachs’ Method) are [52]: H T (r ) QH a (r)
1 Q2 E
³
r a
2 r 2
r a2
V(R) dR
(2)
Thus, it is seen that strain measured at any stage within the measurement sequence depends on the different stresses originally contained within all the removed material, and not just the stress at the last load removal point. This feature is the reason for the name “inverse equation”. The “forward” solution for the strains given the stresses is straightforward, but the “inverse” solution for the stresses given the strains is more difficult. The need for analytical deformation/stress relationships such as Equation (2) initially limited the types of tests that could be done to specimens with simple geometry, for example the planar and cylindrical layer removal methods illustrated in Figure 3. The advent of finite element calculations greatly widened the scope of possible measurements. For example, finite element calculations enabled stress/strain relationships to be determined for the holedrilling method [37,24] with much greater accuracy than available through the previous experimental calibrations. In addition, the numerical calculations provided stress/strain results for cases where experimental evaluation of the kernel function G(H,h) is not possible, thereby allowing substantial extensions of residual stress measurement capabilities. An example of this is the Integral Method [53,24] for measuring stress vs. depth profiles using the holedrilling method. The solution procedures for the various measurement methods were developed on an adhoc basis, and are as varied as the specimen geometries to which they refer. However, for the majority of cases the deformation/residual stress relationships have the same fundamental mathematical form as Equation (1), and thus can be solved by the same mathematical procedure [54]. This procedure is now specified in the ASTM Standard Test Method E83708 [7] for holedrilling, and is also effective for layer removal and slitting measurements [55]. A characteristic of solutions to inverse equations is that they tend to amplify noise. Thus, small measurement errors can create much larger relative errors in the stress solution. Although reflected in the mathematics, this characteristic has a physical origin. It is caused by the spatial separation of the locations of the measured data and the calculated residual stresses, the greater their separation, the greater the error sensitivity. Mathematical techniques such as regularization [56] can be used to stabilize the residual stress solution and reduce noise sensitivity. The availability of large quantities of measured data from fullfield optical measurements greatly extends the opportunities for residual stress evaluations. Every pixel within an image contains information, thus a typical image can provide thousands or millions of surface deformation data. Initial computation methods using optical data mirrored the previous procedures used with discrete sensors such as strain gauges, e.g., [57], and focused on selected data at a few specific locations. The performance of these methods can be significantly enhanced by including the contributions of the substantial quantity of additional data available beyond the few selected points. The challenge is to use the large amount of available data in an effective and compact way. Some desirable features of a residual stress computation method for use with optical data include: x
taking advantage of the wealth of data available within an optical image
x
extracting the data from the image with a minimum of human interaction, preferably none
x
using the available data in a compact and stable computation, preferably a linear one.
Leastsquares [58,59] and orthogonalization [60] techniques have been developed that meet these criteria. The substantial data averaging that is implicit in the calculations substantially reduces measurement noise and enhances measurement accuracy. Error checking and outlier rejection also become possible [61]. In addition, the data quantity allows the calculations to take into account further behaviors in the measurements such as rigidbody motions, thermal strains, plasticity, material nonisotropy, etc. More complex residual stress fields can be assessed, for example, the slitting method has been extended so that the stress distribution along the length of the slit can be determined in addition to the stress distribution within the slit depth [47].
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Concluding Remarks The destructive methods for measuring residual stresses provide a versatile methodology that can be applied to a wide range of specimen and stress geometries. They are based on the elastic deformations that occur during residual stress release. This is a direct phenomenon, and thus, the methods generally give accurate and reliable results. This feature often makes them the measurement type of choice, even though some specimen damage is caused. The various specific methods have evolved over several decades and their practical applications have greatly benefited from the development of complementary technologies, notably in material cutting, fullfield deformation measurement techniques, numerical methods and computing power. These complementary technologies have stimulated advances not only in measurement accuracy and reliability, but also in range of application; much greater detail in residual stress measurement is now available. A good example of this advance is the Contour Method. This fullfield method relies on highly accurate material cutting and surface measurement technologies and individually evaluated finite element calculations. Future developments in destructive residual stress measurement techniques are likely to follow present trends. The ongoing development of complementary technologies will continue to provide new opportunities for technical progress and sophistication. The research work cited here is but a small fraction of the output of an active and vibrant field; the number of researchers and the quantity of their published work continue to grow. These are signs both of practical interest and of opportunity to make advances. Most destructive methods have origins from the days of dial gauges, but the advent of ultrahighprecision cutting tools, laser interferometers, highresolution cameras, etc. have substantially refreshed and advanced them. Certainly, a future review such as this should in just a few years be able to report many further advances and interesting applications. Acknowledgment This work was financially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). References 1. Ruud, C. “A Review of Nondestructive Methods for Residual Stress Measurements”, Journal of Metals, Vol.33, No.7, pp.3540, 1981. 2. Schwaighofer, J. “Determination of Residual Stresses on the Surface of Structural Parts”, Experimental Mechanics, Vol.4, No.2, pp.5456, 1964. 3. Jullien, D. and Gril, J. “Growth Strain Assessment at the Periphery of Smalldiameter Trees using the Twogrooves Method: Influence of Operating Parameters Estimated by Numerical Simulations”, Wood Science and Technology, Vol.42, No.7, pp.551565, 2008. 4. Walton, H. W. “Deflection Methods to Estimate Residual Stress”, in Handbook of Residual Stress and Deformation of Steel, Totten, G., Howes, M., and Inoue, T. (eds.), ASM International, pp.8998, 2002. 5. Fuller, J. “Conditioning Stress Development and Factors That Influence the Prong Test”, USDA Forest Products Laboratory, Research Paper FPL–RP–537, 6pp, 1995. 6. Baldwin, W. M. “Residual Stresses in Metals”, Proc. American Society for Testing and Materials, Philadelphia, PA, 49pp., 1949. 7. ASTM. “Standard Practice for Estimating the Approximate Residual Circumferential Stress in Straight Thinwalled Tubing”, Standard Test Method E192807, American Society for Testing and Materials, West Conshohocken, PA, 2007. 8. Stoney. G. G. “The Tension of Thin Metallic Films Deposited by Electrolysis”, Proc. Royal Society of London, Series A, Vol.82, pp.172175, 1909. 9. Cao, W. Fathallah, R. Castex, L. “Correlation of Almen Arc Height with Residual Stresses in Shot Peening Process”, Materials Science and Technology, Vol.11, No.9, pp.967–973, 1995. 10. Shadley, J. R., Rybicki, E. F. and Shealy, W. S. “Application Guidelines for the Parting out in a Through Thickness Residual Stress Measurement Procedure”, Strain, Vol.23, pp.157166, 1987. 11. Tebedge, N., Alpsten, G. and Tall, L. “Residualstress Measurement by the Sectioning Method”, Experimental Mechanics, Vol.13, No.2, pp. 8896, 1973. 12. Treuting, R.G. and Read, W.T. “A Mechanical Determination of Biaxial Residual Stress in Sheet Materials”, Journal of Applied Physics, Vol.22, No.2, pp.130134, 1951. 13. Sachs, G. and Espey, G. “The Measurement of Residual Stresses in Metal”, The Iron Age, Sept 18, pp.6371, 1941.
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40. Sabaté, N., Vogel, D., Keller, J., Gollhardt, A., Marcos, J., Gràcia, I., Cané, C. and Michel, B. “FIBBased Technique for Stress Characterization on Thin Films for Reliability Purposes”, Microelectronic Engineering, Vol.84, No.58, pp.17831787, 2007. 41. Korsunsky, A. M., Sebastiani, M., Bemporad, E. “Focused Ion Beam Ring Drilling for Residual Stress Evaluation”, Materials Letters, Vol.63, pp.1961–1963, 2009. 42. Winiarski, B, Langford, R. M., Tian, J., Yokoyama, Y., Liaw, P. K. and Withers, P. J. “Mapping Residual Stress Distributions at the Micron Scale in Amorphous Materials”, Metallurgical and Materials Transactions A, Vol.41, 2010. 43. Nelson, D.V. “Residual Stress Determination by Hole Drilling Combined with Optical Methods”, Experimental Mechanics, Vol.50, No.1, pp.145–158, 2010. 44. McDonach, A., McKelvie, J., MacKenzie, P. and Walker, C. A. “Improved Moiré Interferometry and Applications in Fracture Mechanics, Residual Stress and Damaged Composites.” Experimental Techniques, Vol.7, No.6, pp.2024, 1983. 45. Nicoletto, G. “Moiré Interferometry Determination of Residual Stresses in the Presence of Gradients,” Experimental Mechanics, Vol.31, No.3, pp.252256, 1991. 46. Steinzig, M. and Ponslet, E. “Residual Stress Measurement Using the Hole Drilling Method and Laser Speckle Interferometry: Part I.” Experimental Techniques, Vol.27, No.3, pp.4346, 2003. 47. Montay, G., Sicot, O., Maras, A., Rouhaud, E. and François, M. “Two Dimensions Residual Stresses Analysis Through Incremental Groove Machining Combined with Electronic Speckle Pattern Interferometry”, Experimental Mechanics, Vol.49, pp.459–469, 2009. 48. Sutton, M. A., McNeill, S. R., Helm, J. D. and Chao, Y. J. “Advances in TwoDimensional and ThreeDimensional Computer Vision.” Chapter 10 in “Photomechanics”, ed. P. K. Rastogi, SpringerVerlag, Berlin Heidelberg, 2000. 49. Hung, M.Y.Y., Long, K.W. and Wang, J.Q. “Measurement of Residual Stress by Phase Shift Shearography”, Optics and Lasers in Engineering, Vol.27, No.1, pp.61–73, 1997. 50. Lord, J.D., Penn, D. and Whitehead, P. “The Application of Digital Image Correlation for Measuring Residual Stress by Incremental Hole Drilling”, Applied Mechanics and Materials, Vol.1314, pp.6573, 2008. 51. Parker, R. L. “Geophysical Inverse Theory.” Princeton University Press, New Jersey, 1994. 52. Lambert, J. W. “A Method of Deriving Residual Stress Equations”, Proc. SESA, Vol.12, No.1, pp.91–96, 1954. 53. BijakZochowski, M. “A Semidestructive Method of Measuring Residual Stresses.” VDIBerichte, Vol.313, pp.469476, 1978. 54. Schajer, G. S. and Prime, M. B. “Use of Inverse Solutions for Residual Stress Measurements.” Journal of Engineering Materials and Technology. Vol.128, No.3, pp.375382, 2006. 55. Schajer, G. S. and Prime, M. B. “Residual Stress Solution Extrapolation for the Slitting Method Using Equilibrium Constraints”, Journal of Engineering Materials and Technology. Vol.129, No.2, pp.227232, 2007. 56. Tikhonov, A., Goncharsky, A., Stepanov, V. and Yagola, A. “Numerical Methods for the Solution of IllPosed Problems,” Kluwer, Dordrecht, The Netherlands, 1995. 57. Focht, G. and Schiffner, K. “Determination of Residual Stresses by an Optical Correlative Hole Drilling Method.” Experimental Mechanics, Vol.43, No.1, pp.97104, 2003. 58. Ponslet, E. and Steinzig, M. “Residual Stress Measurement Using the Hole Drilling Method and Laser Speckle Interferometry: Part II.” Experimental Techniques, Vol.27, No.4, pp.1721, 2003. 59. Baldi, A. “A New Analytical Approach for Hole Drilling RS Analysis by Full Field Method”, Journal of Engineering Materials and Technology, Vol.127, No.2, pp. 165169, 2005. 60. Schajer, G. S. and Steinzig, M. “FullField Calculation of HoleDrilling Residual Stresses from ESPI Data.” Experimental Mechanics, Vol.45, No.6, pp.526532, 2005. 61. An, Y. and Schajer, G.S. “Pixel Quality Evaluation and Correction Procedures in ESPI”, Experimental Techniques, (in press), 2010.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
The Contour Method Cutting Assumption: Error Minimization and Correction Michael B. Prime ([emailprotected]), Alan L. Kastengren* Los Alamos National Laboratory, Los Alamos, NM 87545 * Now at the Energy Systems Division, Argonne National Laboratory ABSTRACT The recently developed contour method can measure 2D, crosssectional residualstress map. A part is cut in two using a precise and lowstress cutting technique such as electric discharge machining. The contours of the new surfaces created by the cut, which will not be flat if residual stresses are relaxed by the cutting, are then measured and used to calculate the original residual stresses. The precise nature of the assumption about the cut is presented theoretically and is evaluated experimentally. Simply assuming a flat cut is overly restrictive and misleading. The critical assumption is that the width of the cut, when measured in the original, undeformed configuration of the body is constant. Stresses at the cut tip during cutting cause the material to deform, which causes errors. The effect of such cutting errors on the measured stresses is presented. The important parameters are quantified. Experimental procedures for minimizing these errors are presented. An iterative finite element procedure to correct for the errors is also presented. The correction procedure is demonstrated on experimental data from a steel beam that was plastically bent to put in a known profile of residual stresses.
INTRODUCTION Residual stresses play a significant role in many material failure processes like fatigue, fracture, stress corrosion cracking, buckling and distortion [1]. Residual stresses are the stresses present in a part free from any external load, and they are generated by virtually any manufacturing process. Because of their important contribution to failure and their almost universal presence, knowledge of residual stress is crucial for prediction of the life of any engineering structure. However, the prediction of residual stresses is a very complex problem. In fact, the development of residual stress generally involves nonlinear material behavior, phase transformation, coupled mechanical and thermal problems and also heterogeneous mechanical properties. So, the ability to accurately quantify residual stresses through measurement is an important engineering tool. Recently, a new method for measuring residual stress, the contour method [24], has been introduced. In the contour method, a part is carefully cut in two along a flat plane causing the residual stress normal to the cut plane to relax. The contour of each of the opposing surfaces created by the cut is then measured. The deviation of the surface contours from planarity is assumed to be caused by elastic relaxation of residual stresses and is therefore used to calculate the original residual stresses. One of the unique strengths of this method is that it provides a full crosssectional map of the residual stress component normal to the cross section. The contour method is useful for studying various manufacturing processes such as laser peening [510], friction welding [6,7,1113] and fusion welding [1421]. Some of the applications are quite unique such as mapping stresses in a railroad rail [22], in the region of an individual laser peening pulse [23], and under an impact crater [24]. The only common methods that can measure similar twodimensional (2D) stress maps have significant limitations. The neutron diffraction method is nondestructive but sensitive to microstructural changes [25], time consuming, and limited in maximum specimen size, about 50 mm, and minimum spatial resolution, about 1mm. Sectioning methods [26] are experimentally cumbersome, analytically complex, error prone, and have limited spatial resolution, about 1 cm. A limitation of the original contour method is that only one residual stress component is determined. Recent developments have extended the contour method to the measurement of multiple stress components [2730].
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_40, © The Society for Experimental Mechanics, Inc. 2011
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In an overly simplistic view, one might think that the contour method requires the assumption of a perfectly flat cut. Such as assumption is overly restrictive. In this paper, the actual assumptions about the cut are developed. Several potential error sources are shown to be removed by standard data processing. Other errors are explored in more detail numerically to establish procedure to minimize the errors. An experimental demonstration is given with a correction for errors arising because of the cut.
THEORY This section reviews previously published theory for the contour method in order to allow for later detailed discussion of the assumption about the cut. The contour method [2] shown in Figure 1 is based on a variation of Bueckner’s superposition principle [31].
Figure 1 Superposition principle to calculate residual stresses from surface contour measured after cutting the part in two [28].
In A, the part is in the undisturbed state and contains the residual stress to be determined. In B, the part has been cut in two and has deformed as residual stresses were released by the cut. For clarity, only one of the halves is shown. In C, the free surface created by the cut is forced back to its original flat shape. Assuming elasticity, superimposing the partially relaxed stress state in B with the change in stress from C would give the original residual stress throughout the part:
V ( A)
V ( B ) V (C )
(1)
where V without subscripts refers to the entire stress tensor. This superposition principle assumes elastic relaxation of the material and that the cutting process does not introduce stress that could affect the measured contour. With proper application of this principle it is possible to determine the residual stress over the plane of the cut. Experimentally, the contour of the free surface is measured after the cut and analytically the surface of a stressfree model is forced back to its original flat
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configuration by applying the opposite of the measured contour as boundary conditions. Because the stresses in B are unknown, one cannot obtain the original stress throughout the body. However, the normal and shear stresses on the free surface in B must be zero (Vx, Wxy and Wxz). Therefore, C by itself will give the correct stresses along the plane of the cut:
V x( A)
V x(C )
( A) (C ) W xy W xy ( A) W xz
(2)
(C ) W xz
The measured surface contour has an arbitrary reference plane, resulting in three arbitrary rigid body motions in defining the surface. These three arbitrary motions are uniquely determined by the need for the stress distribution over the cross section to satisfy three global equilibrium conditions: force in the xdirection and moments about the y and z axes. It is not necessary to explicitly enforce these constraints. In a finite element calculation, appropriate boundary conditions are applied to the cut plane, including three extra constrains to prevent rigid body motions. The remainder of the body is unconstrained. In the static equilibrium step used to solve for stress, the free end of the body will automatically translate and rotate such that the equilibrium conditions are fulfilled. A small convenience is taken in the data analysis. Modeling the deformed shape of the part for C in Figure 1 would be tedious. Instead, the surface is flat in the finite element model, and then the part is deformed into the shape opposite of the measured contour. Because the deformations are quite small, the same answer is obtained but with less effort.
CUTTING ASSUMPTION From a theoretical point of view, the relevant assumption for the superposition principle in Figure 1 is that the material points on the cut surface are returned in C to their original configuration. Experimental limitations results in two broad types of departures from that assumption. The first departure is that the surface contour measurement only gives information about the normal (x) displacement. So the material points are not returned to their original configuration in the transverse directions. The second departure is that in order for the measured surface contour to accurately return the material points to their original locations in the xdirection, those material points must have come from a common plane in the original configuration (A). A transformation of this theoretical assumption into more practical assumptions can accommodate some of these issues without sacrificing any accuracy. This naturally leads to considering symmetric errors separate from antisymmetric errors, as explained in the following.
AntiSymmetric Errors The issues with both transverse displacements and antisymmetric errors can be examined by considering the case of shear stresses [2]. The top of Figure 2 shows a beam that was cut in half on a plane that had both normal and shear stresses. The two halves have different contours. The equivalent surface traction components for released stresses are given by Tx V x nx , Ty W xy nx , where n is the surface normal vector. By a Poisson effect, the Ty from the released shear stress has an effect on the contour. As shown in Figure 2 in reference [2], the tractions for releasing Vx are symmetric about the cut plane, but the tractions for releasing Wxy are antisymmetric. Hence, the contours in Figure 2 can be decomposed into the symmetric portion caused by normal stress and the antisymmetric portion caused by shear stress.
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Figure 2. Asymmetric contours can be separated into the symmetric portion and the antisymmetric portion, which averages away. Shown for the example of shear stresses.
The effect of shear stresses, and indeed all antisymmetric errors, are removed by averaging the contours on the two halves. The lack of information on transverse displacements also does not cause errors. In the analysis, the surface is forced back (step C in Figure 1) to the original flat configuration only in the xdirection, based on the average contour. The shear stresses (Wxy and Wxz) are constrained to zero in the solution. This stressfree constraint is automatically enforced in most implicit, structural, finiteelement analyses if the transverse displacements are left unconstrained. Even if residual shear stresses were present on the cut plane, averaging the contours measured on the two halves of part still leads to the correct determination of the normal stress Vx [2]. Even when there are no shear stresses, there are also transverse displacements from a Poissontype effect when normal stresses are relaxed. Because there was no shear stress, there is no traction associated with those displacements. Therefore, the solution that constrains the shear stresses to zero on the cut surface, as described above, is still correct. Averaging the contours on the two halves to remove antisymmetric errors requires another assumption. The portion of the contour caused by the normal stress must be symmetric, which requires that the stiffness be the same on the two sides of the cut. For hom*ogeneous materials, this assumption is certainly satisfied when a symmetric part is cut precisely in half. In practice, the part only needs to be symmetric about the cut within the region where the stress release has a significant effect. The length of this region is about 1.5 times the Saint Venant’s characteristic distance. The characteristic distance is often as the part thickness, but is more conservatively taken as the maximum crosssectional dimension unless specific information about the stress variation is known [12]. Figure 3 shows another example of an antisymmetric error that averages away. If cut path is crooked in space, the resulting surface contours are antisymmetric.
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Figure 3. The effect of a crooked cut goes away when the two surfaces contours are averaged.
Figure 4 shows another antisymmetric effect that in principle averages away, but that should be minimized anyway. This time the cut is path is straight in space, but the part is moving as the stresses relax. In Figure 4, the movement occurs because the part is not clamped symmetrically. An experimental example later in this paper shows that the effect can be quite large. Averaging two very different contours can introduce some uncertainty, so both sides of the cut should be clamped to minimize such movement. It will be shown later that other errors are also reduced by good clamping.
clamp
plane of cut current location of material points that were on cutting plane in undefomred state clamp x y
undeformed shape
deformed shape as cut progresses and stresses relax
Figure 4. Movement of the cut plane as stresses relax during cutting. In this example, this is caused by asymmetric clamping.
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Symmetric Errors There are other errors that cause symmetric effects that do not average way. Before discussing those, realize that some cutting errors depend on how the cut is made. So far, the only method that has been successfully applied for the contour method is wire Electric Discharge Machining (EDM). Thus, this discussion focuses on EDM cutting. Several of the symmetric error sources are relatively straightforward: x Local cutting irregularities, such as wire breakage or overburning at some foreign particle. These are usually small length scale (order of wire diameter) and are removed by the data smoothing process or manually from the raw data x Change in width of cut. This can occur in heterogeneous materials since the EDM cut width varies for different materials. Sometimes a change in the part thickness (wire direction) can also cause this. x Wire vibration causing a “bowed” cut [2]. This can usually be avoided by using good settings on the wire EDM. x Stresses induced by the cutting process can cause errors. Such errors have been studied for wire EDM cutting for use with incremental slitting (crack compliance) [32]. Such effects are generally negligible if “skim cut” or “finish cut” EDM settings are used. Those are low power settings that give higher accuracy and a better surface finish. The symmetric error sources listed above do not depend on stress magnitude, which leads to a straightforward approach to the issue. A test cut in stress free material would have no contour caused by stresses but would show these errors. Any such error could then be corrected for by subtracting the error off of the measured contour. Most good practitioners make such a test cut standard practice. The simplest way to achieve a nearly stress free test cut is to use the actual test part. After surface contours are measured, one can cut a thin layer off of one of the cut surfaces. The relevant stress components are zero on the free surface and very small if the test cut occurs only a small distance from the original cut surface. The surface contour is measured on the new surface of the larger piece. Some of those assumptions, primarily the constant cut width, may be less accurate near the edges of the cut surface. For example, the EDM cut width may flare out a little bit at the top and bottom of the cut. Contour results can therefore be quite uncertain near the edges of the cut and should not be generally be reported in that region. With special care, good results have been achieved very close to the edges [4,23]. The contour cut also cannot recut previously cut surfaces. This is one of the main reasons that EDM is used for the contour method. In principle, EDM might slightly recut the surface when cutting into a compressive stress. In practice, as sketched in Figure 5, constraint at the cut tip minimizes the amount the cut surfaces can pinch in close behind the wire.
Figure 5. Even cutting into compressive stress, constraint limits the amount the cut surfaces can pinch in close behind the wire.
Most of the rest of this paper will concern a particular symmetric error that can cause significant bias in the contour method results if it is not dealt with. This error, called the “bulge” error, is illustrated in Figure 6. The cutting process makes a cut of constant with w in the laboratory reference frame. As the cutting proceeds, stresses relax and the material at the tip of the cut deforms. The material at the cut tip that was originally w wide has stretched. However, the physical cut will still be only w wide, which means that the width of material removed has been reduced when measured relative to the original state of the body. Since the fundamental assumption is that step C of Figure 1 returns the material points to their original configuration, this causes an error that will not
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be averaged away. The effect is obviously dependent on the stress state at the cut tip relative of the original stress state. That change in stress can be minimized by securely clamping the part. Also, this effect is a result of the finite width of the cut; a zerowidth crack causes only effects that average away, as shown in Figure 4. Considering antisymmetric effects that average away as well as the bulge and other asymmetric errors, the fundamental assumption about the cut can be more compactly stated. The cutting process is assumed to remove a constant width of material when measured relative to the state of the body prior to any cutting.
The Bulge Error: undeformed shape deformed shape as slot is cut and stresses relax t
w
x
a y
cutting occurs on plane fixed in laboratory reference frame current location of material points that were on cutting plane in undeformed state normal displacement of material point about to be cut, u(y=a,a)
Figure 6. The “bulge” error. As cutting proceeds, the material at the tip of the cut deforms from stress relief. This changes the width of the cut relative to the original state of the body and causes errors.
Cut tip plasticity can also cause symmetric errors. Such errors violate the assumption that stress relaxation is elastic, rather than assumptions about the cut, but they cause very similar effects. Like the bulge error, this effect also depends on the stress state at the cut tip. Thus, any efforts to minimize the bulge error should also reduce plasticity errors. An FEM study showed that plasticity errors were generally small when clamping during cutting was secure [33]. Another study that looked at experimental results showed some significant errors that were explained, using FEM, by plasticity effects [34]. However, those experimental errors might have been a combination of both bulge and plasticity errors.
Summary of Assumptions x x x x
The cutting process does not recut surfaces that have already been cut. The stiffness of the part is symmetric with respect to the cut within the regions where stresses are relaxed by the cut The cut removes a finite width of material when measured relative to the state of the body prior to any cutting. There is no plasticity at the cut tip
The ideal cut would be zero width, introduce no stresses, and allow no plasticity at the tip of the cut.
FINITE ELEMENT STUDY
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Finite Element Models The bulge error was estimated using twodimensional ABAQUS [35] finite element simulations of sequentially cutting a slot into a beam, similar to Figure 6. A known field of residual stresses was used as an initial condition for the model using a userdefined subroutine. The cutting process was then simulated by sequentially removing elements. After each step, the displacement of appropriate material points originally on the cut plane were recorded to estimate errors in the planar cut assumption. Finally, residual stresses were calculated using the principle of Figure 1 but including the errors, and the difference between the calculated residual stresses and the known residual stresses were analyzed. Several assumptions were used for the simulations. Isotropic, linear elastic material behavior is assumed throughout the analysis. It is also assumed that the cut is perfectly planar in space; hence, the deviations come only from defomation of the material. For meshing convenience, a square slot bottom was used. Since an EDM cut has a round bottom, this assumption may introduce errors. Dimensions of the beam were normalized to a beam thickness, t, of one unit. A total beam length of four units was chosen in order to minimize any effects from the free ends. Plane stress deformation was assumed. Legendre polynomials were chosen for the residual stresses because Legendre polynomials of order two and greater automatically satisfy force and moment equilibrium over the beam cross section. For each simulation, the appropriate Legendre polynomial was used to initialize Vx(y) in the region of the cut. The other two stress components, Vy and Wxy , were initialized to zero in the region of the cut. The stresses elsewhere in the beam were specified such that local equilibrium and the stressfree boundary conditions were satisfied everywhere. The residual stress distributions were all normalized to a peak value of unity. The choice of the elastic modulus E has no effect on the final stress values calculated from the displacements, since it proportionally affects both the error magnitude and the actual contour. Nonetheless, an E of 1000 was used rather in order to give reasonably scaled displacements; the Vmax/E ratio of 0.001 is consistent with typical residual stress magnitudes in metals. Since E has no effect on the final stresses calculated, plane strain can be 2 simulated by replacing E with E/(1Q ); plane stress and plane strain will give the same results if the analysis is consistent. The deviations from a planar cut assumption were estimated by considering the displacement of the material point where cutting is about to occur. Referring to Figure 6, the material point at the bottom corner of the slot is the next to be cut. The material point that was originally on the plane where cutting is occurring has displaced. In the actual cutting process, the wire will cut this point flush with the original cut plane. Therefore, the deviation from the flat cut assumption is given by the displacement of this material point normal to the cut plane, in the x direction. The ydisplacement is not important. A Lagrangian coordinate system is used to track the deformations. The displacements u(x,y,a) refers to the xdirection displacement after the cut is at depth a of the material point that was at the coordinate (x,y) in the original configuration of the system. Because the only displacement that is relevant for this simulation is the displacement of points at the fixed value of x for the cut plane, the displacement is reported just as u(y,a). The final contour of the cut surface is given by the displacements after the cut and is designated by u(y,final). The results of the sequential cutting simulation were used as input to a subsequent analysis to calculate the effect of the cutting deviation on the residual stress measurements of the contour method. The final contour of the cut surface from the simulation was combined with the cutting error to give the displacement boundary conditions used to calculate the residual stress per the third part of Figure 1:
u y u y, final u y
a, a ,
(1)
where the initial negative sign appears because calculating the original residual stresses requires applying the opposite of the measured contour. This stress calculation makes two additional assumptions. The small displacements of the material on the cut plane were not accounted for in the material removed for the simulation. The simulation removed elements with edges that were originally on the cut plane but were no longer on the cut plane at the time of removal. Because the displacements are small, the difference has a negligible effect on the results of the stress calculations. An
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exact simulation of the material removal would require adaptive remeshing or some other process, and such extra effort is not justified by the small changes that would result.
Simulation Results: General Features of the Contour Method and the Errors There are several interesting features in almost all of these simulations. Typical deflection and cutting error profiles are shown in Figure 7 for a simple clamping arrangement: on the top and bottom surfaces of the beam at one thickness away from the cut, symmetric on both sides of the cut. The slot width is 0.01. The quadratic Legendre polynomial stress is used. The bulge error has a shape that is not the same as the stress profile. Rather, the bulge error is approximately proportional to the stress intensity factor, KIrs, at the cut tip from the accumulated effect of releasing residual stress. Combining the bulge error with the theoretical contour gives the contour including the error. The general shape of the contour with cutting error closely parallels that of the deflection without cutting error. The deflection profile has a slightly smaller peak when the cutting error is included. It also shows a phase shift compared to the profile without cutting error, with the peak values shifted closer to the beginning of the cut. Near the end of the cut, the deflection error rises sharply, causing a large difference between the two profiles.
0.0004
0.0003
Contour Including Cutting Error 0.0002
Deflection
Bulge Error 0.0001
Theoretical Contour no Error 0
0.0001
0.0002
0.0003 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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(Position or cut depth)/t
Figure 7. Bulge error for a FEM simulation of a quadratic stress profile.
The stress profiles in Figure 8 show most of the same features as the deflection profiles. The stress error is high near the beginning of the cut and very high near the end of the cut. This matches with the experiences in the actual measurement process, where the edges often show erratic behavior. However, the effect seems to be exaggerated in these simulations. More work is needed to improve the fidelity of the simulations at the beginning and end of the cut. Near the middle of the cut, the stress error is quite reasonable and seems to follows a smooth curve. Again, it does not seem to be simply related to the initial stress distribution, though a great deal of the error seems to be a phase shift between the profiles with and without cutting error. The peak compressive stress is slightly reduced in magnitude.
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1.2
1
Theoretical
0.8
Including Cutting Error
Stress
0.6
0.4
0.2
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0.6 0
0.1
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0.3
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Position/t
Figure 8. Stresses calculated from the contours of Figure 7 showing the stress error caused by the bulge error.
Effect of Clamping A variety of clamping arrangements were simulated by constraining various nodes in the FEM model from moving. Because of the difficulty of achieving experimental clamping similar to the perfect constraint, only qualitative conclusions are presented: o A significant benefit is achieved by going from clamping on one side of the cut to clamping both sides of the cut. o In the simulations, a significant benefit is achieved by clamping the material all along the cut rather than just at the top and bottom surfaces. However, this is difficult to achieve experimentally. o Modest but diminishing benefits are achieved by moving the clamping closer to the cut.
Effect of Slot Width Another simulation examined the effect of slot width on the error. A zigzag stresses profile typical of beam bending was used as the initial stress. The beam was clamped symmetrically on both sides of the cut. The FEM mesh was adjusted to give different slot widths. Figure 9 shows curves of bulge error as a function of cut depth relative to the beam thickness. The final contour (without error) is also plotted to illustrate the scale of the errors. The bulge error gets larger for larger cut width. The peak magnitude of the bulge error was extracted from each curve in Figure 9, normalized by the peaktovalley magnitude of the contour, and plotted versus slot width in Figure 10. The peak error increases with slot width, with the slope being greater for narrow slots. A typical experimental slot width, taken from the beam tests to be discusses in the next section, is indicated in order to place the errors in proper context.
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0.004 0.003
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Figure 9. Bulge error as a function of cut depth for different slot widths relative to the beam thickness. The final contour (without error) is also plotted to illustrate the scale of the errors.
14.0%
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4.0% 2.0% 0.0% 0.0%
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Slot width relative to beam thickness Figure 10. The peak values of bulge error from Figure 9 are normalized relative to the peaktopeak amplitude of the measured contour and plotted versus slot width.
PROCEDURE FOR CORRECTING BULGE ERROR The previous analysis, when applied to realistic slid widths, shows that the errors in the residual stress measurement are very reasonable, especially when compared to the errors in other residual stress measurement techniques. However, a method to correct these errors could improve the measurements by this technique. An iterative finiteelement analysis is proposed to correct the deflection error. The basic principle is to reverse the previous analysis technique. The experimentally measured or recorded profile is used to calculate residual stress. Since the errors in the contour method are relatively small, and the measurable deflection profile closely
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parallels the theoretical profile, the calculated stress profile can be used as an initial guess for the theoretical profile. Then the FEM simulation described above is used to estimate the bulge error. The bulge error is used to correct the measured contour and then to calculate stresses now including an estimated bulge error. This process is then repeated until the stress converges.
EXPERIMENTAL The bulge error and the iterative correction are examined on two bent beam specimens with known residual stress profiles. To prepare the specimen, strain gages were attached to the top and bottom of a stressrelieved stainless steel beam, 30.0mm deep. Residual stresses were induced in the beam by loading it into the plastic range using a fourpoint bend fixture. On removal of the load, the beam unloaded elastically, leaving permanent plastic deformations and substantial axial residual stresses. During this loading and unloading, the total load and corresponding strains were measured. The profile of the residual stresses was calculated from the measured strain data using the stressstrain curve identification method described by Mayville and Finnie [36]. More details on these specific beams are given elsewhere [2,37]. The first beam, beam A, was used as a known stress specimen for measurements using incremental slitting (crack compliance). As is standard with the slitting method [38], the beam was only clamped on one during the cutting as shown in Figure 11. The one sided clamping will make for a larger bulge error and different contours on the two halves of the beam. Before destructive measurement, the residual stresses in the beam were measured by neutron and xray diffraction, and within the associated uncertainty ranges, the results agreed with those of the stressstrain curve calculation [37,39]. A 200 Pm diameter hard brass wire was used for the cutting. The cutting was done in 36 steps, at approximately 0.8mm intervals to a final depth of 29.26 mm, 97.5% of the beam depth. After cutting, the width of the cut was measured as 267 Pm. Unfortunately, because the beam had not been cut all the way through, the remaining ligament was fractured by hand. That left the final 0.74 mm of the surface without a usable contour.
Figure 11. Beam A just prior to EDM cutting for the slitting method. The central section of the beam is 30 mm thick and 10 mm wide.
The second beam, beam B, was measured with the contour method [2]. As shown in Figure 12, it was clamped on both sides of the cut. This time the EDM cut used a 100 Pm diameter zinccoated brass wire and gave a cut width of 140 Pm. The beam was cut all the way in half.
245
Figure 12. Beam B was clamped on both sides of the cut prior to EDM cutting.
Beam B Beam B is examined first because it was cut under more ideal conditions: a narrow slot and clamped on both sides of the cut. The contours on the two halves of the beam were very similar, so the average was used for all subsequent calculations. By symmetry, only half of the beam was modeled (in 2D) with FEM. Displacement boundary conditions were applied on the top and bottom surfaces of the beam where the clamps from Figure 12 were located. Figure 13 shows the contours calculated with the iterative correction procedure. Because the correction is small, the correction has converged by the second iteration. Figure 14 shows the stress results compared with the stresses predicted in the bend test. The correction, while not large, has moved the contour results closer to the bend test prediction. Now the results generally agree within the uncertainty in the bend test prediction.
Figure 13. Iterative correction to measured contour on beam B.
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Bend test prediction Initial results After correction
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Position, y (mm) Figure 14. Contour method results before and after bulge correction compared with the prediction from the bend test.
Beam A Because it was cut in such a nonideal way for the contour method, with a wide slot and one sided clamping, beam A represents a challenging test for the iterative correction. Figure 15 shows the drastic difference between the contours measured on the two halves of the beam. Figure 16 shows results from beam A calculated using the average contour as well as using the contour from each half individually. The results are compared with bend test prediction and slitting and neutron diffraction measurements [39]. In spite of the large difference in the contours, the stresses calculated using the average contour do an impressive job of matching the bend test prediction and independent measurements. 0.010
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Position, y (mm) Figure 15. The surface contours measured on the two halves of beam A are very different.
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Independent results Contour Bend test prediction Free Side Slitting Clamped Side neutron Average
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x (mm) Figure 16. Stress results from beam A calculated using the average contour as well as using the contour from each half individually. Results compared with bend test prediction and slitting and neutron diffraction measurements.
A full 2D (no symmetry) FEM model was used to perform the iterative correction on the beam A results. Displacement boundary conditions were applied on the top and bottom surfaces of the beam where the clamps from Figure 11 were located. The contours had to be extrapolated to cover the final 0.74 mm of the surface that had been fractured off. Because the contour on the free side of beam A overpredicted the stress magnitude, see Figure 16, the correction was also overpredicted, which led to a nonconvergent solution. So for the first iteration only, half of the predicted correction was applied. Figure 17 shows that the correction was then reasonably stable and converged after several iterations. The average of the last two iterations compares quite well with the stresses predicted by the bend test. 250
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Figure 17. Iterative correction applied to the unclamped half of beam A.
Considering the starting point for the clamped half of beam A, a good result from the iterative correction was not expected. The results in Figure 15 and Figure 16 do not even have the correct shape. Figure 18 shows the iterative correction. Even though the results are poor at the ends of the beam, the central results slowly converge towards the expected results from the bend test analysis.
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Figure 18. The iterative correction applied to the clamped half of beam A.
DISCUSSION Both the bulge effect and plasticity errors (only briefly mentioned in this paper) are apparently dependent on the stress state at the cut tip, which can be characterized by the intensity factor, KIrs, at the cut tip from the accumulated effect of releasing residual stress. Similar strategies could be used for both effects. More secure clamping, e.g., [20], would experimentally minimize the errors. Analytical corrections based on fracture mechanics analysis might be successful. Because of the dependence on KIrs, such errors will depend on the direction of the cut. Although the bulge error decreases for decreasing cut width, other errors might increase. Some difficulties getting good cuts with wire under 100 Pm diameter have been observed. The iterative correction procedure using FEM is promising. Some improvements need to be made on handling the beginning and end of the cut. The procedure is conceptually straightforward and was demonstrated in 2D. There is no conceptual difficulty in applying the procedure in 3D, but keep track of displacements at all the correct nodes for each cut depth would be tedious. A scripting procedure could simplify matters. Scripting interfaces are now available for many commercial finite element codes.
CONCLUSION The assumptions about the cut for the contour method are x The cutting process does not recut surfaces that have already been cut. x The stiffness of the part is symmetric with respect to the cut within the regions where stresses are relaxed by the cut x The cut removes a finite width of material when measured relative to the state of the body prior to any cutting. x There is no plasticity at the cut tip Deviations from these assumptions can cause errors: x Cutting errors can be divided into errors that cause perturbations to the measured contours that are antisymmetric with respect to the cut plane, and those that are symmetric. Antisymmetric errors are removed by averaging the contours measured on the two halves of the part. Symmetric errors do not average away. x The bulge error, a symmetric error, tends to cause stress profiles to show both a reduced peak stress magnitude and a spatial shift of the peaks x The bulge error depends on the direction of the cut x The bulge error is reduced the more securely material near the cut is clamped
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x x x
The bulge error decreases for decreasing slot width The bulge error can be modeled with FEM and an iterative correction is possible. Larger errors are expected at the beginning and end of the cut.
The ideal cut would be zero width, introduce no stresses, and allow no plasticity at the tip of the cut. Some conclusions can be made about best practices for the contour method: x The part should be clamped securely on both sides of the cut during cutting x A stressfree test cut should be used as a control check on cutting assumptions. The test cut can be achieved after the contour measurement by cutting a thin slide off of the cut surface. x Contour results are uncertain near the edges of the cut and should not be reported in that region unless special care is used to get better results there.
ACKNOWLEDGEMENTS This work was performed at Los Alamos National Laboratory, operated by the Los Alamos National Security, LLC for the National Nuclear Security Administration of the U.S. Department of Energy under contract DEAC5206NA25396. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royaltyfree license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes.
REFERENCES [1] Withers PJ (2007) Residual Stress and Its Role in Failure. Reports on Progress in Physics 70:22112264. [2] Prime MB (2001) CrossSectional Mapping of Residual Stresses by Measuring the Surface Contour after a Cut. Journal of Engineering Materials and Technology 123:162168. [3] Prime MB, Sebring RJ, Edwards JM, Hughes DJ, Webster PJ (2004) Laser SurfaceContouring and Spline DataSmoothing for Residual Stress Measurement. Experimental Mechanics 44:176184. [4] Johnson G, (2008), "Residual Stress Measurements Using the Contour Method," Ph.D. Dissertation, University of Manchester. [5] DeWald AT, Rankin JE, Hill MR, Lee MJ, Chen HL (2004) Assessment of Tensile Residual Stress Mitigation in Alloy 22 Welds Due to Laser Peening. Journal of Engineering Materials and Technology 126:465473. [6] Hatamleh O, Lyons J, Forman R (2007) Laser Peening and Shot Peening Effects on Fatigue Life and Surface Roughness of Friction Stir Welded 7075T7351 Aluminum. Fatigue and Fracture of Engineering Material and Structures 30:115130. [7] Hatamleh O (2008) Effects of Peening on Mechanical Properties in Friction Stir Welded 2195 Aluminum Alloy Joints. Materials Science and Engineering: A 492:168176. [8] DeWald AT, Hill MR (2009) Eigenstrain Based Model for Prediction of Laser Peening Residual Stresses in Arbitrary 3D Bodies. Part 2: Model Verification. Journal of Strain Analysis for Engineering Design 44:1327. [9] Liu KK, Hill MR (2009) The Effects of Laser Peening and Shot Peening on Fretting Fatigue in Ti6Al4V Coupons. Tribology International 42:12501262. [10] Hatamleh O, DeWald A (2009) An Investigation of the Peening Effects on the Residual Stresses in Friction Stir Welded 2195 and 7075 Aluminum Alloy Joints. Journal of Materials Processing Technology 209:48224829. [11] Woo W, Choo H, Prime MB, Feng Z, Clausen B (2008) Microstructure, Texture and Residual Stress in a FrictionStirProcessed AZ31B Magnesium Alloy. Acta Mat. 56:17011711. [12] Prime MB, GnaupelHerold T, Baumann JA, Lederich RJ, Bowden DM, Sebring RJ (2006) Residual Stress Measurements in a Thick, Dissimilar Aluminum Alloy Friction Stir Weld. Acta Mat. 54:40134021. [13] Frankel P, Preuss M, Steuwer A, Withers PJ, Bray S (2009) Comparison of Residual Stresses in Ti6al4v and Ti6al2sn4zr2mo Linear Friction Welds. Materials Science and Technology 25:640650. [14] Zhang Y, Pratihar S, Fitzpatrick ME, Edwards L (2005) Residual Stress Mapping in Welds Using the Contour Method. Materials Science Forum 490/491:294299. [15] Edwards L, Smith M, Turski M, Fitzpatrick M, Bouchard P (2008) Advances in Residual Stress Modeling and Measurement for the Structural Integrity Assessment of Welded Thermal Power Plant. Advanced Materials Research 4142:391400.
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[16] Kartal M, Turski M, Johnson G, Fitzpatrick ME, Gungor S, Withers PJ, Edwards L (2006) Residual Stress Measurements in Single and MultiPass Groove Weld Specimens Using Neutron Diffraction and the Contour Method. Materials Science Forum 524/525:671676. [17] Withers PJ, Turski M, Edwards L, Bouchard PJ, Buttle DJ (2008) Recent Advances in Residual Stress Measurement. The International Journal of Pressure Vessels and Piping 85:118127. [18] Zhang Y, Ganguly S, Edwards L, Fitzpatrick ME (2004) CrossSectional Mapping of Residual Stresses in a VPPA Weld Using the Contour Method. Acta Mat. 52:52255232. [19] Thibault D, Bocher P, Thomas M (2009) Residual Stress and Microstructure in Welds of 13%Cr4%Ni Martensitic Stainless Steel. Journal of Materials Processing Technology 209:21952202. [20] Hacini L, Van Lê N, Bocher P (2009) Evaluation of Residual Stresses Induced by Robotized Hammer Peening by the Contour Method. Experimental Mechanics 49:775783. [21] Turski M, Edwards L (2009) Residual Stress Measurement of a 316L Stainless Steel BeadonPlate Specimen Utilising the Contour Method. International Journal of Pressure Vessels and Piping 86:126131. [22] Kelleher J, Prime MB, Buttle D, Mummery PM, Webster PJ, Shackleton J, Withers PJ (2003) The Measurement of Residual Stress in Railway Rails by Diffraction and Other Methods. Journal of Neutron Research 11:187193. [23] Evans A, Johnson G, King A, Withers PJ (2007) Characterization of Laser Peening Residual Stresses in Al 7075 by Synchrotron Diffraction and the Contour Method. Journal of Neutron Research 15:147154. [24] Martineau RL, Prime MB, Duffey T (2004) Penetration of HSLA100 Steel with Tungsten Carbide Spheres at Striking Velocities between 0.8 and 2.5 km/s. International Journal of Impact Engineering 30:505520. [25] Holden TM, Suzuki H, Carr DG, Ripley MI, Clausen B (2006) Stress Measurements in Welds: Problem Areas. Materials Science and Engineering A 437:3337. [26] Schajer GS (2001) "Residual Stresses: Measurement by Destructive Testing." Encyclopedia of Materials: Science and Technology, Elsevier, 8152–8158. [27] DeWald AT, Hill MR (2006) MultiAxial Contour Method for Mapping Residual Stresses in Continuously Processed Bodies. Experimental Mechanics 46:473490. [28] Pagliaro P, (2008), "Mapping Multiple Residual Stress Components Using the Contour Method and Superposition," Ph.D. Dissertation, Universitá degli Studi di Palermo, Palermo. [29] Pagliaro P, Prime MB, Swenson H, Zuccarello B (2010) Measuring Multiple ResidualStress Components Using the Contour Method and Multiple Cuts. Experimental Mechanics 50:187194. [30] Kartal ME, Liljedahl CDM, Gungor S, Edwards L, Fitzpatrick ME (2008) Determination of the Profile of the Complete Residual Stress Tensor in a VPPA Weld Using the MultiAxial Contour Method. Acta Mat. 56:44174428. [31] Bueckner HF (1973) "Field Singularities and Related Integral Representations." Mechanics of Fracture G. C. Sih, ed., 239314. [32] Cheng W, Finnie I, Gremaud M, Prime MB (1994) Measurement of nearSurface ResidualStresses Using ElectricDischarge Wire Machining. Journal of Engineering Materials and TechnologyTransactions of the ASME 116:17. [33] Shin SH (2005) FEM Analysis of PlasticityInduced Error on Measurement of Welding Residual Stress by the Contour Method. Journal of Mechanical Science and Technology 19:18851890. [34] Dennis RJ, Bray, D.P., Leggatt, N.A., Turski, M. , (2008), "Assessment of the Influence of Plasticity and Constraint on Measured Residual Stresses Using the Contour Method." 2008 ASME Pressure Vessels and Piping Division Conference, Chicago, IL, USA, PVP200861490. [35] Abaqus 6.9, ABAQUS, inc., Pawtucket, RI, USA, 2009. [36] Mayville R, Finnie I (1982) Uniaxial StressStrain Curves from a Bending Test. Experimental Mechanics 22:197201. [37] Schajer GS, Prime MB (2006) Use of Inverse Solutions for Residual Stress Measurements. Journal of Engineering Materials and Technology 128:375382. [38] Cheng W, Finnie I (2007) Residual Stress Measurement and the Slitting Method, Springer Science+Business Media, LLC, New York, NY, USA. [39] Prime MB, Rangaswamy P, Daymond MR, Abeln TG, 1998, "Several Methods Applied to Measuring Residual Stress in a Known Specimen." Proc. 1998 SEM spring conference on experimental and applied mechanics, 13 Jun 1998, Society for Experimental Mechanics, Inc., 497499.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Measurements of Residual Stress in Fracture Mechanics Coupons
Michael R. Hill1, John E. VanDalen2 and Michael B. Prime3 1 Mechanical and Aerospace Engineering Department, University of California, One Shields Avenue, Davis, CA 95616 USA, [emailprotected] 2 Hill Engineering, LLC, McClellan, CA USA 3 Los Alamos National Laboratory, Los Alamos, NM USA
ABSTRACT This paper describes measurements of residual stress in coupons used for fracture mechanics testing. The primary objective of the measurements is to quantify the distribution of residual stress acting to open (and/or close) the crack across the crack plane. The slitting method and the contour method are two destructive residual stress measurement methods particularly capable of addressing that objective, and these were applied to measure residual stress in a set of identically prepared compact tension (C(T)) coupons. Comparison of the results of the two measurement methods provides some useful observations. Results from fracture mechanics tests of residual stress bearing coupons and fracture analysis, based on linear superposition of applied and residual stresses, show consistent behavior of coupons having various levels of residual stress. INTRODUCTION Fracture mechanics testing typically relies on test coupons being free from residual stress, though limited guidance is provided to ensure that coupons are stress free. For materials that cannot be tested in coupons free from residual stress, it may be that measured residual stress can be used to obtain a test outcome (i.e., fracture mechanics properties) independent of residual stresses. The superposition of stress intensity factors provides a basis for residual stress corrections that might be applied for properties determined under generally linearly elastic, small scale yielding conditions, such as lowenergy fracture (brittle or ductile in nature) or fatigue crack growth in metallic materials. Superposition of stress intensity factors (SIFs) under monotonic or cyclic loading can be expressed simply as
KTot = KApp + KRS
(1)
where KApp is the SIF from applied loads, KRS is the SIF from residual stress, and KTot is the total SIF. Testing standards (e.g., ASTM E399, E647, and so forth) provide expressions for KApp while KRS would be determined using newly established standard procedures. The objective of this work is to provide example measurements of the openingdirection residual stress on the crack plane, and the residual stress intensity factor, in fracture coupons containing various levels of residual stress. This paper is a followup to recently reported work that contains details of fracture tests on the same set of coupons [1]. METHODS Material and geometry Aluminum alloy 7075T6 was selected for this test program due to its prevalent use in a variety of applications. This alloy exhibits lowenergy ductile fracture with a rising Rcurve. The material was received as clad plate 4.8 mm thick. Handbook [2] mechanical properties of 7075T6 are listed in Table 1. Compact tension, C(T), coupons were used in this study, as described in several of the ASTM fracture toughness testing standards (e.g., ASTM E 56198 – “Standard Practice for RCurve Determination”). The C(T) T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_41, © The Society for Experimental Mechanics, Inc. 2011
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coupon is well suited for this work because of its accepted use in fracture toughness testing, simple geometry, and onedimensional crack, characterized by the crack length a. Coupon geometry had thickness B of 3.81 mm and a characteristic width dimension W of 50.8 mm (Fig. 1). Coupons were cut such that crack growth occurred in the LT orientation. To obtain the 3.8 mm coupon thickness from the stock material, material was machined in equal amounts from each side so that the coupons lay in the T/2 plane and the original clad layer was removed. A starter notch with integral knife edges for crack mouth opening displacement (CMOD) was cut into each coupon using a wire electric discharge machine (EDM). The EDM notch was 0.3 mm wide and had various lengths, as described below. Su (MPa) 552
Sy (MPa) 490
E (GPa) 71
0.33
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Table 1 – Mechanical properties of 7075T6 plate [2]
Fig. 1 – 7075T6 C(T) Coupon (dimensions in mm)
Fig. 2 – Location of LSP regions (square regions enclosed by dashed line, having side length of 22.9 mm) (a) near the front face (12.7 mm from the front face) and (b) far from the front face (12.7 mm from the back face) (also shown are initial crack lengths used for fracture tests of the coupon sets)
Residual stress treatment Laser shock peening (LSP) was used to produce residual stress bearing coupons. In thin geometries, like the C(T) coupons used here, LSP can generate throughthickness compressive residual stress in the treated area. The LSP process uses laserinduced shocks to drive plastic deformation into the surface of a part [3,4]. For this work, LSP was applied using industrial facilities at Metal Improvement Company (Livermore, CA). LSP parameters were chosen to achieve high levels of residual stress in the C(T) coupons. Earlier work in high strength aluminum alloys found that deep residual stress was induced in 19 mm thick coupons using an LSP 2 parameter set of 4 GW/cm irradiance per pulse, 18ns pulse duration, and 3 layers of treatment (denoted 4183) [5,6]. It was further found that 4183 provided significant highcycle fatigue life improvements in 19 mm thick bend bars. For the present work, LSP was used to obtain two amounts of residual stress effect and either positive or negative residual stress effect on fracture toughness. To obtain sets of coupons with different amounts of residual stress, LSP was applied using a single layer, at 4181, or using three layers, at 4183, where threelayer LSP induces a greater amount of residual stress. To obtain sets of coupons with either positive or negative residual stress effects on fracture, LSP was applied in two different areas, each a square with side length of 22.9 mm and located either near to or far from the front face of the C(T) coupon (Fig. 2). Applying LSP near the front face results in compressive residual stress on the crack faces, providing a negative contribution to fracture (i.e., a negative residual stress intensity factor), while applying LSP far from the from the front face results in tensile residual stress on the crack faces, providing a positive contribution to fracture. In total, five coupon conditions were used in this study: asmachined (AM); onelayer LSP applied near the front face (LSP1N); threelayer LSP applied near the front face (LSP3N); onelayer LSP applied far from the front face (LSP1F); and, threelayer LSP applied far from the front face (LSP3F). LSP coupons were treated on both sides, alternating sides between each layer application until each side was treated with the desired number of layers.
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Residual stress and KRS measurements Twodimensional residual stress distributions were measured on the prospective crack plane of AM and LSP3N coupon blanks using the contour method [7,8]. Measurements were made on C(T) blanks that had holes, but did not have initial notches. Wire EDM was used to cut the coupons in half and expose the crack plane. After cutting, an areascanning profilometer was used to measure the resulting outofplane deformation of the cut surface, on both halves of the coupon. The measured deformations of the two halves were averaged and smoothed to remove effects of cut path wandering, shear stress, and surface roughness. The negative of the averaged and smoothed deformations were applied as displacement boundary conditions in a threedimensional, linear elastic finite element model of onehalf of the coupon. The stress resulting from the elastic finite element calculation provided the experimental estimate of residual stress in the coupon. Thickness average residual stresses were measured in all coupon conditions using the slitting method. A strain gage with an active grid length of 0.8 mm was applied to the center of the back face of the coupon (opposite the crack mouth). Wire EDM was used to incrementally extend a slit through the coupon, with strain recorded after each increment of slit depth. Residual stress as a function of position across the coupon was determined from strain versus slit depth data using the approach recently described by Schajer and Prime [9] and adapted to the geometry of the C(T) coupon. Measured residual stresses from slitting and contour are compared to one another to determine the degree of consistency among the methods employed. The comparison is made on thicknessaverage residual stress as a function of position across the crack plane, with the twodimensional stresses from the contour measurements averaged at a set of positions across the crack plane. The slitting method was also used to determine the residual intensity factor as a function of crack length, KRS(a), following Schindler’s method for a thin rectangular plate [10]. KRS(a) was computed from the influence function Z(a) provided in [10], the plane stress modulus of elasticity E´ = E (given in Table 1), and the derivative of strain with respect to slit depth
KRS (a) =
E d (a) . Z(a) da
(2)
The influence function Z(a) of [10] does not account for the holes present in the C(T) coupon, which was assumed to be of negligible effect. In addition, care was taken to account for the different definitions of crack size used by ASTM (measured from the loadline) and Schindler (measured from the front edge), which can cause error (or misinterpretation of results, as in [11]). The derivative of strain with respect to crack length was computed using a moving fivepoint quadratic polynomial, with slope evaluated numerically at the middle point. Residual stress intensity factors were computed from measured residual stress using a Green’s function for the C(T) coupon recently published by Newman, et al [12] and numerical integration, paying careful attention to the singularity of the Green’s function [13]. Residual stress intensity factors computed from measured residual stresses are compared against those determined from Eqn. (2). Fracture toughness tests were performed according to ASTM E 56198 to determine the KR crack growth resistance curve for each of the five coupon conditions. Replicate tests were run for each condition. Details can be found in our earlier work [1]. The KR results are presented in terms of applied loading alone, and in terms of total residual stress intensity factor (Eqn. (1)). In addition, initiation toughness was determined in the coupons using data collected during KR testing, but using the data reduction procedures of ASTM E 399 to determine KQ. RESULTS Residual stress fields on the crack plane determined using the contour method are illustrated in Fig. 3 for two AM and one LSP3N coupons. Residual stress in the AM coupons have peak values around ±20 MPa and have a similar through thickness distribution at all points across the coupon width, which is consistent with residual stress from plate rolling. The LSP coupon has a maximum compressive residual stress of 290 MPa that occurs on the surface near the middle of the coupon, and a maximum tensile stress of 350 MPa that occurs at the front face of the coupon. In the peened region (from x = 12.7 to x = 35.6 mm, where x is measured from the front face of the coupon), residual stress is compressive at the surfaces and grows more positive monotonically with position toward the coupon midthickness. Outside the peened region, residual stress is nearly uniform through thickness but varies with position across the width. The residual stress distribution features inside the LSP region are consistent with double sided peening and outside the LSP region are consistent with plate bending and axial stresses that arise in the coupon to achieve residual stress equilibrium (zero net force and moment across the contour plane).
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Slitting residual stress measurements show that the location of the peened area significantly affects the stress distribution across the coupon (Fig. 4). LSP near the front face of the coupon (LSP1N and LSP3N) produces tensile stresses at the front face that give way to compression 12 to 16 mm from the front face. LSP far from the front face (LSP1F and LSP3F) creates tensile stresses at the front face and over a region extending 26 to 30 mm from the front face. For the same peened area, the shape of the stress distributions for onelayer and threelayer LSP are quite similar, with 3layer LSP coupons having about twice as much residual stress. Thickness average residual stress for the LSP3N and one of the AM coupons is plotted in Fig. 5. AM coupons have a thicknessaverage stress of nearly 0 MPa at all points across the coupon, which is consistent with plate rolling. Thicknessaverage stress in the LSP3N coupon has maximum tension at the front face (240 MPa) and maximum compression (150 MPa) near the middle of the LSP area (x = 28 mm). The slitting residual stress measurement has a higher gradient near the right edge of the peened area (x = 36 mm) than shown for the contour residual stress, but the two measurements are in good agreement (Fig. 5). The difference may be due to surface fitting in the contour data reduction, which softens (spatial) stress gradients. Estimates of KRS were determined from strain versus depth data using two different procedures, and these are compared in Fig. 6. The first procedure used Eqn. (2), above, and results are shown as symbols in Fig. 6. The second procedure used the residual stress from slitting, reported in Fig. 4, the Green’s function of Newman, et al [12], and numerical integration, and the resulting KRS estimates are shown as lines in Fig. 6. There is good agreement between the two calculation procedures. For condition LSP3N, KRS was also computed using the Green’s function and contour thicknessaverage residual stresses (Fig. 5). The estimates of KRS show good agreement, except for the negative peak in KRS, which appears softened when using the contour stresses and may be the result of surface smoothing (Fig. 7). At initial crack lengths useful for fracture tests (0.35W = 18 mm ao 0.55W = 28 mm), KRS estimates show that KRS(a) is either negative or positive, depending on the location of the LSP area (Fig. 6). At these crack lengths, the farfromfrontface LSP coupons (LSP1F, LSP3F) have positive KRS and the nearfrontface LSP coupons (LSP1N, LSP3N) have negative KRS. KRS for the threelayer LSP coupons (LSP3F, LSP3N) are roughly double those for the corresponding onelayer LSP coupons (LSP1F, LSP1N). For LSP3F, the positive 0.5 KRS reaches 35.9 MPa m , which is quite large, exceeding the material plane strain fracture toughness of 0.5 29.0 MPa m (Table 1). The magnitude of the most positive KRS in coupons LSP1F and LSP3F (17.7 and 0.5 35.9 MPa m ) is nearly twice the magnitude of the most negative KRS in coupons LSP1N and LSP3N (10.5 and 0.5 16.8 MPa m ). Based on the slitting results (Fig. 6), target initial crack lengths were selected for KR fracture tests. The target crack lengths chosen for each condition are shown (to scale) with the LSP area in Fig. 2. Compressioncompression precracking was performed for the LSP1F and LSP3F coupons due to the high magnitude, positive KRS (for further precracking and other details of fracture testing, refer to our earlier work [1]). Results of fracture testing showed significant and expected effects from LSPinduced residual stress. Rcurves in terms of Kapp are shown for all coupons in Fig. 8. Results for the two AM coupons exhibit typical 0.5 elasticplastic blunting behavior until about 30 MPa m , after which monotonically increasing (nearly linear) crack growth resistance is obtained. Compared with the AM results at a given crack length, the LSP1N and LSP3N coupons exhibit greater toughness (in terms of Kapp) and the LSP1F and LSP3F coupons exhibit reduced toughness, with the 3layer coupons having a larger effect in both cases. Comparing results for the two coupons within each coupon subset shows the data to be repeatable. Residual stress corrected Rcurves were prepared using Eqn. (1) and using values of K RS determined from Eqn. (2). The corrected Rcurves for the different coupon subsets are similar (Fig. 9). The LSP1N and LSP3N results exhibit significantly greater elasticplastic blunting than for the AM coupons, which is consistent with the higher level of applied loading at which they initiate crack extension. After blunting, the Rcurves of all coupon subsets are in good agreement, though the LSP3F coupons exhibit a significantly shallower Rcurve. Values of initiation toughness are very significantly affected by residual stress in terms of applied load (KQ) 0.5 but are nearly invariant in terms of total stress (KQ,Tot) (Fig. 10). The mean KQ for AM coupons was 33.1 MPa m , a value slightly greater than KIc (Table 1), which is expected given the limited coupon thickness. Mean KQ values 0.5 0.5 for residual stress bearing coupons ranged from 9.69 MPa m for LSP3F coupons to 51.6 MPa m for LSP3N 0.5 0.5 coupons. Mean KQ,Tot values ranged from 32.1 MPa m (LSP3F) to 37.4 MPa m (LSP1N), with the latter value being somewhat of an outlier.
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Fig. 3 – Contour measurement results illustrating the residual stress distribution on the plane of the crack for AM and LSP3N coupons (“rep” indicates a replicate (identical) coupon)
Fig. 4 – Slitting residual stress measurement results (AM condition not measured); onesigma error bars are plotted for LSP3N, other conditions exhibit similar levels of uncertainty
Fig. 5 – Slitting residual stress for LSP3N plotted with contour through thickness average for LSP3N and AM conditions
Fig. 6 – Comparison of KRS determined from slitting strain data: symbols are determined from derivative of strain data (Eqn. (2)); lines result from integrating residual stress of Fig. 4 with the Green’s function for the C(T) coupon published by Newman, et al. [12]
Fig. 7 – K RS for LSP3N condition computed from derivative of strain data (Eqn. (2)), and measured residual stress from slitting and contour thicknessaverage stress
Fig. 8 – Rcurve resulting from applied stress intensity factor
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Fig. 9 – Rcurve resulting from superposition of applied and residual stress intensity factors; KRS(a) from symbols in Fig. 6
Fig. 10 – Comparison of KQ and KQ,Tot for the five coupon conditions (bars and values provide mean, error bars indicate range from two coupons) ; KRS(a) from symbols in Fig. 6
SUMMARY The present work describes measurements of residual stress in fracture coupons, the determination of residual stress intensity factors (as a function of crack length) from measured stress and more directly using Schindler’s method [10], and correlation of observed fracture behavior in highstrength aluminum coupons. The results demonstrate consistent fracture property measurements in residual stress bearing coupons, where residual stress was taken into account; however, the consistency here was dependent on specific methodological choices. First, the material employed is one that exhibits lowenergy fracture, and as such, has fracture behavior that is influenced directly (in fact, linearly) by KRS . It remains to be determined whether consistent toughness data could be obtained in residual stress bearing coupons of a significantly more ductile material. Second, fracture toughness and KRS measurements were performed on coupons prepared identically. A test program employing coupons with greater variability (e.g., handforgings or welded materials) likely would provide lessconsistent results. Third, the coupon geometry had a large widthtothickness ratio (W/B 13), which allowed significant residual stress influence while avoiding complications arising from variations of KRS along the crack front and the potential for nonstraight crack fronts (e.g., as for welded joints [14]). While slitting was the best method for the present coupons, a significantly smaller widthtothickness ratio might require accounting for throughthickness stress variation (e.g., measured by the contour method) and its affect on KRS along the crack front. Followon work would need to determine the range of material, coupon geometry, and residual stress fields for which consistent fracture toughness properties can be obtained. Method selection is an important step when making material property measurements, and the inclusion of residual stress in fracture property testing would require additional standardization activity. Here, the slitting method directly provided KRS(a) from Eqn. (2), which was readily combined with test data. The values of KRS(a) from Eqn. (2) were in very good agreement with those determined from residual stress and Green’s function. Slitting was straightforward to implement and the method offers good repeatability (see replicate results for the present coupons reported in [1], as well as repeatability of residual stress reported in [15]). Active standardization of the slitting method, within ASTM Task Group E28.13.02, supports its potential use in fracture toughness testing standards. The good agreement for initiation toughness KQ,Tot and Rcurve behavior shown here among coupons containing significantly different distributions of residual stress suggest a course of further work to extend standard fracture toughness tests so that they include residual stress effects for an appropriate range of material, coupon geometry, and residual stress fields. REFERENCES [1] VanDalen, J. E. and Hill, M. R., “Evaluation of residual stress corrections to fracture toughness values,” Journal of ASTM International, 5(8), Paper ID JAI101713. [2] United States Department of Transportation – Federal Aviation Administration, 2003, Metallic Materials Properties Development and Standardization, Report DOT/FAA/ARMMPDS01. [3] Fabbro, R., Peyre, P., Berthe, L., and Scherpereel, X., 1998, “Physics and applications of lasershock processing,” Journal of Laser Applications, 10, pp. 265279.
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[4] Montross, C.S., Wei, T., Ye, L., Clark, G., and Mai, Y.W., 2002, “Laser shock processing and its effects on microstructure and properties of metal alloys: a review,” International Journal of Fatigue, 24, pp. 10211036. [5] Pistochini, T., 2003, “Fatigue Life Optimization in Laser Peened 7050T7451 and 300M Steel,” M.S. Thesis, Department of Mechanical and Aeronautical Engineering, University of California, Davis. [6] Luong, H., 2006, “Fatigue life extension and the effects of laser peening on 7050T7451 aluminum, 7085T651 aluminum, and Ti6Al4V titanium alloy,” M.S. Thesis, Department of Mechanical and Aeronautical Engineering, University of California, Davis. [7] Prime, M.B., 2001, “Crosssectional mapping of residual stresses by measuring the surface contour after a cut,” Journal of Engineering Materials and Technology, 123, pp. 162168. [8] Prime, M.B., Sebring, R.J., Edwards, J.M., Hughes, D.J., and Webster, P.J., 2004, “Laser surfacecontouring and spline datasmoothing for residualstress measurement,” Experimental Mechanics, 44, pp. 176184. [9] Schajer, G.S., and Prime, M.B., 2006, “Use of Inverse Solutions for Residual Stress Measurements,” Journal of Engineering Materials and Technology, 128, pp. 375382. [10] Schindler, H.J. and P. Bertschinger, 1997, “Some steps towards automation of the crack compliance method to measure residual stress distributions,” in Proceedings of the 5th International Conference on Residual Stress, Linköping. [11] Ghidini, T., Donne, C. D., 2007, “Fatigue crack propagation assessment based on residual stresses obtained through cutcompliance technique,” Fatigue and Fracture of Engineering Materials and Structures, 30, pp. 214222. [12] Newman, J.C., Yamada, Y., James, M.A., 2010, “Stressintensityfactor equations for compact specimen subjected to concentrated forces,” Engineering Fracture Mechanics, 77, pp 10251029. [13] Wu, X.R. and Carlsson, A.J., 1991, Weight function and stress intensity factor solutions, Pergamon Press, pp. 3738. [14] Towers, O.L., Dawes, M.G., 1985, “Welding Institute research on the fatigue precracking of fracture toughness specimens,” in ElasticPlastic Fracture Test Methods: The User's Experience, ASTM STP 856, American Society for Testing and Materials, Philadelphia, PA, p. 2346. [15] Lee, M.J., Hill, M.R., 2007, “Intralaboratory repeatability of residual stress determined by the slitting method,” Experimental Mechanics, 47, pp. 745752.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Analysis of Large Scale Composite Components Using TSA at Low Cyclic Frequencies
Author: J. M. DulieuBarton, School of Engineering Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ, UK, [emailprotected] CoAuthor: D. A. Crump, University of Southampton
ABSTRACT Thermoelastic stress analysis (TSA) has been applied to large scale honeycomb core sandwich structure with carbon fibre face sheets. The sandwich panel was subjected to a pressure load using a custom designed test rig that could only achieve low cycle frequencies of 1 Hz. Two calibration approaches have been discussed and investigated to allow the use of the thermoelastic response as a validation tool for the stress distribution predicted by an FE model. The TSA data was calibrated using thermoelastic constants derived experimentally using tensile strips of the face sheet material. It has been shown that by using constants obtained from the tensile strips manufactured with the same layup as the face sheet of the sandwich panel it was possible to achieve a good correspondence between the predicted stress distribution and the measured TSA response.
1. Introduction Thermoelastic stress analysis (TSA) [1] is a well established, noncontacting technique for the evaluation of stresses in engineering components, e.g. [25]. The technique uses an infrared camera to obtain the small temperature change associated with the thermoelastic effect in a loaded component or structure. It is assumed that the small temperature change occurs isentropically; to eliminate heat transfer TSA is usually performed using a cyclic load. TSA has been successfully applied to realistic composite structures, e.g. large wind turbine blades [6] and to a marine teejoint [7]. The aim of the current paper is to demonstrate that TSA can be used as an accurate and qualitative validation tool for finite element analysis (FEA) of complex sandwich structures with carbon fibre face sheets. In [8] TSA was applied successfully to a validation of FE models of sandwich beams with core junctions. The introduction of the core junction caused a large stress gradient to occur both through the
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_42, © The Society for Experimental Mechanics, Inc. 2011
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thickness and in the plane of the face sheet. In TSA, stress gradients drive the non adiabatic behaviour. In the case where the face sheets were manufactured from relatively low thermal conductivity material it was possible to obtain good results that compared well with strain gauge readings and therefore validated the FEA. For high conductivity aluminium alloy face sheets it was shown that adiabatic conditions could not be achieved. In the present paper, sandwich structure with carbon fibre face sheets are studied, which have a much greater thermal conductivity than the glass fibre face sheets used in [8]. This paper starts with a description of the TSA technique and its application to large carbon fibre sandwich structure representative of a secondary wing structure panel in an aircraft, known as a ‘generic panel’ [9]. The generic panels are subjected to a pressure load, representative of the aerodynamic pressure on the wing of an aircraft, using a specialist test rig [10]. Two different approaches are discussed for calibration of the thermoelastic response from the face sheet material.
To achieve adiabatic
behaviour in composite materials it is usually recommended that a frequency of at least 10 Hz is applied [11, 12]. The requirement for this relatively high loading frequency limits the use of TSA to laboratory experiments, and also the size of components that TSA can be readily applied to. Therefore, the calibration approaches are tested over a range of loading frequencies. However, to achieve a constant cyclic pressure the generic panels can only be loaded at a rate of 1 Hz using the test rig. Hence, this paper discusses the calibration of TSA data recorded using a low cycle frequency and the effect this has on the measured stresses. 2. Thermoelastic stress analysis (TSA) and experimental setup The Cedip Silver 480M infrared detection system manufactured by Cedip Infrared Systems was used to obtain the TSA data. The system comprises an infrared camera with an InSb detector with 320 x 256 pixels at pitch of 30 μm and allows frame rates between 5 and 380 Hz. The images are recorded and processed using AltairLI software. This system is radiometrically calibrated so it is possible to obtain the temperature change directly as well as using the mean temperature field to correct for any changes in the specimen temperature. A ‘lockin’ signal from the test machine is used to synchronise the TSA measurement, and in this way the structural response due to the loading can be averaged over a number of cycles and the accuracy improved. The output from the detector provides the change in surface temperature, ǻT, resulting from the change in the sum of the principal stresses on the surface of the material. For an orthotropic material, such as the composites investigated in this paper, ǻT can be related to the stresses in the material, ıx and ıy, as follows [7]:
ǻT
T Įx ı x Įy ıy ȡCp
(1)
where Įx and Įy, are the coefficients of linear thermal expansion in the longitudinal and transverse material direction, ıx and ıy are the stresses in these directions, ǻT is the change in temperature, T is the ambient temperature, ȡ is the density and Cp is the specific heat at constant pressure. It is possible to combine the material constants in Equation (1), i.e. Įx, Įy, ȡ and Cp into two thermoelastic constants Kx and Ky as follows [10]:
'T
T K xV x K y V y
(2)
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where K x
Dx and K y UC p
Dy . UC p
In previous work [9], a generic sandwich panel was designed to capture some of the features representative of composite sandwich secondary wing structure (see Figure 1). Two generic panels were manufactured using Hexcel’s 914CTS534% prepreg tape as the face sheet material, one with quasiisotropic (QI) sheets [0°, 45°, 45°, 90°, 0°, 45°, 45°, 90°, 0°, 45°, 45°, 90°]s about a core of Nomex honeycomb, and one with crossply (CP) sheets [0°, 90°]6s about the core. A custom designed rig was used to apply a pressure load, representative of aerodynamic loading, to the sandwich panel. The rig used the movement of the actuator in an Instron servohydraulic test machine to pull the sandwich panel over a water filled pressure cushion that is fully constrained by the rig and the panel. The rig has been designed to allow uninterrupted optical access to the top surface of the sandwich panel, thereby enabling the use of optical measurement techniques such as TSA. The panels were loaded cyclically at 1 Hz with a mean pressure of 10 kPa (1.5 psi) and an amplitude of 5 kPa (0.75 psi), thereby imparting a pressure range of 10 kPa (1.5 psi). The panel was attached to the test rig horizontally and therefore to measure the response from the surface it was necessary to mount the TSA detector vertically with the lens directed downwards; this would not have been feasible with the older bulk cooled infrared detector systems. The detector was located at a distance that gave approximately 600 pixels across the 300 mm width of the panel. Therefore, the spatial resolution was approximately 2 pixels/mm, and the system recorded images at a frame rate of 100 Hz. To achieve this spatial resolution for the entire panel it was necessary to record 32 images per panel, and then ‘stitch’ the data together using a Matlab routine. This provided a fullfield plot of the temperature change on the surface of the sandwich panel. This temperature field was then calibrated to obtain a fullfield stress measure.
Figure 1: Design for generic sandwich panel [9]
3. Calibration approach
One of the major challenges in calibrating the thermoelastic response from composite materials is the requirement for accurate thermal and mechanical material properties both longitudinally and transversely. By using Equation (2) it is possible to calibrate the thermoelastic response using measured thermoelastic constants
Kx and Ky. The unaxial stress state in a tensile strip test provides the opportunity to measure the calibration constant, and by using longitudinal and transverse specimens it is possible to calculate Kx and Ky as follows:
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Kx
'T and K y 'V xT
'T 'V yT
(3)
The values obtained from equation (3) inserted into Equation (2) and then by rearrangement the TSA data can be expressed in terms of a stress in such a way that it can be used for FEA validation: 'T TK x , TSA
Ky 'V x 'V y Kx
(4)
FE
The left hand side of Equation (4) represents the TSA data calibration and the right hand side the form into which the FE data must be processed to make it comparable with the TSA data.
In this paper two calibration
approaches are investigated to establish if the stresses in equation (4) should be the global stresses or surface ply stresses in the face sheet. The key feature is to determine if this has an effect on the interpretation of the thermoelastic response. The first, ‘global calibration’, uses constants derived from tensile strips manufactured from material with the same layup as the face sheets of the generic panel, i.e. QI and CP depending on the panel. The second, ‘UD calibration’, uses constants derived from tensile strips manufactured with a unidirectional layup i.e. one with all plies aligned longitudinally or transversely. The surface ply in the panel faces sheets is orientated with the xdirection in all cases so the UD surface ply thermoelastic constants are the same for both the QI and CP generic panels. Tensile strips of 15 mm width were loaded in a servohydraulic test machine at both 1 and 10 Hz to investigate the effect of loading frequency on the derived thermoelastic constant. As a result of the different stiffness and strengths of the materials it was necessary to apply different loads to the longitudinal and transverse specimens than that used for the QI and CP strips. The longitudinal UD specimen was loaded at 10 ± 9.0 kN, the transverse UD specimen at 0.15 ± 0.1 kN whilst all the QI and CP specimens were loaded at 3.5 ± 3.0 kN. The specimens were not coated with paint as the response was sufficient in the uncoated condition. Table 1 provides the thermoelastic constant for each of the tensile strips for both loading frequencies. The longitudinal UD 0° specimen provides a constant that is almost sixty times smaller than the transverse UD 90° specimen. The derived thermoelastic constants show increases in the response between the 1 and 10 Hz loading o o frequency of 11% for the UD 0 specimen and 5% for the UD 90 specimen. As all the plies are aligned in the
same direction there is no through thickness stress gradient and hence little heat transfer. The small changes in response may be caused by the stress gradient between the surface resin rich layer and fibre reinforced substrate. This has been noticed in previous research [13] but is not considered in the present paper. For QI and CP specimen there is a considerable difference between the measured response at 1 Hz and 10 Hz; this is expected as there will be a stress gradient change ply by ply in these specimens.
For the QI specimens at 1 Hz the
longitudinal specimen provides a K value is around 50% less than that for the transverse specimen. At 10 Hz the transverse constant is more than three times greater.
It is clear that at a 1 Hz loading frequency the QI
specimens are not behaving adiabatically because of through thickness heat transfer from the subsurface plies. For the CP specimens at 1 Hz the longitudinal K value is practically the same as the transverse. Here it could be speculated the heat transfer through the thickness produces an almost hom*ogenous material where the orientation of the surface ply is unimportant for the value of the calibration constant. When the loading frequency
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is increased to 10 Hz the transverse K value is almost 3 times greater than the longitudinal in a very similar manner to the QI.
The presence of ±45° plies in the QI specimen may be responsible for reducing the
hom*ogenising effect of heat transfer at low loading frequencies. It is important to note, particularly for the UD material, the data contains more than 50% noise. This is because 'T is of the order 20 mK and very close to the minimum resolvable value of about 4 mK. This is because the Dx value is small for the UD 0o material and the applied stress must be small for the UD 90o material. There is clear nonadiabatic behaviour at 1 Hz for the materials that comprise the panel face sheets. As the loading rig can only operate at this level, then it must be established if applying the thermoelastic constants derived from these loading frequencies will provide a sufficiently accurate stress solution. Table 1: Thermoelastic response and constants from calibration strips
Loading at 1 Hz
Loading at 10 Hz
Specimen
ı (MPa)
ǻT
T (°K)
K (MPa x 10 )
ı (MPa)
ǻT
T (°K)
K (MPa1 x 106)
UD 0°
187.50
0.0175
294.0
0.32 ± 0.19
187.50
0.0197
294.0
0.36 ± 0.22
UD 90°
4.17
0.0235
295.1
19.12 ± 4.71
4.17
0.0247
295.1
20.09 ± 5.29
QI 0°
122.75
0.0576
295.6
1.59 ± 0.83
122.75
0.0351
295.6
0.97 ± 0.53
QI 90°
125.00
0.1161
298.5
3.11 ± 1.19
125.00
0.1388
298.5
3.50 ± 1.75
CP 0°
124.56
0.0710
295.0
1.93 ± 1.08
124.56
0.0643
295.0
1.75 ± 1.05
CP 90°
131.06
0.0783
295.0
2.03 ± 0.78
131.06
0.0991
295.0
2.56 ± 0.88
1
6
4. Analysis of generic panels – TSA vs FE
The generic panel was modelled using ANSYS 11. The relatively simple geometry of the panel is shown in Figure 1. The face sheets were treated as orthotropic blocks of material using element Shell181 with material properties derived experimentally. The core was assumed to be a single anisotropic solid volume modelled using brick element Solid 185. After a convergence study using panel deflection, an element size of 0.01 m was selected [10]. The service constraints were represented in the model simply by imposing zero deflection on three edges of the model; i.e. the two short edges and one of the longer edges. The bolted connections used in the experimental work are not modelled so there will be differences between the FE and experimental results at the boundary. The pressure load of 10 kPa was treated in the FE model as a force perpendicular to each of the 2150 surface nodes, i.e. 0.96 N per node. As the panel is relatively thin in comparison to its length and width, and experiences a relatively large outofplane deformation a geometrically nonlinear solver was used. The TSA data from the panels and stresses from the FE models were manipulated into the form in Equation (4) using both the UD and global values of K. Figure 2 shows the fullfield images for the CP panel as an example, although the images from the QI panel show similar trends. Figure 2 (a and b) show the result of using the UD values on the FE model and TSA data respectively. There is very little correlation between the FE and TSA. This must be attributed to the fact that the surface ply in the TSA is not behaving in an adiabatic manner. However some trends are apparent, even though in general the FE model gives much larger stresses than the TSA, the
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stress concentrations are still discernable. Figure 2 (c and d) show the result of using the global K values. Qualitatively there is obviously much better correlation between the images than for the UD calibration. The stress distribution appears to be in agreement, and the level of stress appears to be similar. To provide a quantitative comparison of the stress sum calibrated in this manner, a line of data is plotted through the stress concentration at the top right corner of the core for each of the panels and calibration approach. Figure 3 (a) plots the line of data for the CP panel calibrated with the UD approach comparing the FE and TSA. The FE model predicts a peak value of approximately 3200 MPa, whilst experimentally the TSA provides 1400 MPa. Figure 3 (b) shows the line for the CP panel calibrated using the global approach.
The FE predicts a peak value of
approximately 130 MPa, whilst experimentally the TSA provides 150 MPa. Using the global calibration approach the FE model under predicts the stress peak by only 15%. This indicates that using the thermoelastic constants derived from the global calibration experiments, at the actual loading frequency, provides a means of obtaining reasonable stress values from nonadiabatic thermoelastic data. Figure 3 (c and d) plots the line of data for the QI panel using the UD and global calibration approaches respectively. These plots support the conclusion from the CP panel that the global calibration can provide a means of deriving useful stress data from the thermoelastic response even if nonadiabatic conditions prevail.
(a)
(b)
(c)
(d)
Figure 2: Fullfield stress sum data for CP panel (a) FE model using UD calibration, (b) TSA using UD calibration, (c) FE model using global calibration and (d) TSA using global calibration
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(a)
(b)
(c)
(d)
Figure 3: Line plot through stress concentration (a) CP panel with UD calibration, (b) CP panel with global calibration, (c) QI panel with UD calibration and (d) QI panel with global calibration
5. Conclusions
A method for validating FE models of large carbon fibre sandwich panels has been established using TSA. To ensure adiabatic conditions in a composite material it is usual to apply a cyclic load of at least 10 Hz. However, by obtaining calibration constants from tensile strips loaded at 1 Hz it has been shown possible to process fullfield TSA data from a large representative carbon fibre sandwich panel loaded at just 1 Hz. The TSA data and stresses from an FE model were processed into a form that was comparable using two calibration approaches. The first used values measured from UD composite tensile specimens, whilst the second used values measured from composite specimens manufactured from the same layup as the face sheets of the generic panel. The processed TSA and FE data did not compare well when the UD calibration approach was applied in terms of both the stress distribution and the stress values. However, by applying the global calibration approach the stress obtained from the TSA and FE compared well in both distribution and value. The results are very encouraging in the sense that the idea of calibrating at the actual loading frequency provides a means of interpreting thermoelastic data from components that are subject to non adiabatic behaviour. However, the work presented in the paper represents only a small part of the overall picture regarding non adiabatic behaviour in laminated composite component. The work provides an initial indication that non adiabatic behaviour can be accounted for and that representative low loading frequencies can be used for TSA.
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In future work the non adiabatic behaviour will be investigated further by using a plybyply FE model and by evaluating the stress in the surface ply and thereby identifying when adiabatic behaviour is occurring. Further more detailed investigations into the effect of the loading frequency on the thermoelastic response of the tensile strips will be undertaken. References
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thermography and thermoelastic stress analysis. Proceedings of SPIE, 2002. 4704: p. 93103. 12. Cunningham, P.R., DulieuBarton, J.M., Dutton, A.G., Shenoi, R.A., Thermoelastic characterisation of
damage around a circular hole in GRP component. Key Engineering Materials, 2001. 204205: p. 453463. 13. Sambasivam, S., Quinn, S and DulieuBarton, J.M., “Identification of the source of the thermoelastic
response from orthotropic laminated composites”, 17th International Conference on Composite Materials (ICCM17), 2009, Edinburgh, 11 pages on CD
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Determining Stresses Thermoelastically around Neighboring Holes whose Associated Stresses Interact
A. A Khaja1 and R. E. Rowlands2 Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706 1
[emailprotected]
2
[emailprotected]
ABSTRACT Since stresses at a geometric discontinuity can be influenced by the neighboring structural compliance, this paper emphasizes determining the individual components of stress in a finite tensile plate containing two differentsize neighboring holes. Stresses associated with the respective holes interact. Relatively little information is available for such cases, purely analytical solutions are extremely difficult for finite geometries, and numerical approaches are challenging if the external loading is not well known. An effective method for determining stresses in perforated finite planeproblems is to synergize measured temperature information with a series representation of an Airy stress function. A separate stress function is employed here with each hole. The coefficients of the stress functions are evaluated from the recorded temperatures, imposing the tractionfree conditions analytically at the edge of the holes, as well as satisfying stress compatibility between the holes. The number of Airy coefficients utilized is substantiated experimentally. TSA results agree with those from FEM and commercial strain gages. The technology has implications relative to reducing stress concentration factors. 1. Introduction Numerous publications have been appeared investigating the effect of single geometric discontinuity in finite/infinite plate members, for example references [15]. Engineering members often contain holes whose tensile stress concentration factor can be reduced by introducing a nearby hole or notch such that their stress fields interact with each other. It is therefore important to investigate the effects on the stress field of an original hole of adding an auxiliary hole. The present study emphasizes determining the individual components of stress in a region containing two differentsize neighboring holes in a finite tensile plate. Individually analyzing the stresses around each hole and due to stress compatibility between the two holes provide meaningful insight into how the stresses associated with each hole interact. These issues are pursued here by means of a finite tensile aluminum plate containing a circular hole plus a smaller neighboring hole, figure 1.
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_43, © The Society for Experimental Mechanics, Inc. 2011
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Fig. 1: Aluminum Plate . 2. Relevant Equations For planestress isotropic TSA under cyclic proportional loading condition, the quantitative information of figure 1 can be expressed as [46] ∗
=
∆ =
[∆(
1
+
2
=
+
=
+
)]
(1)
where S* is the thermoelasticallyrecorded signal, K is the thermomechanical coefficient (usually determined experimentally) and ∆ is the change in S, S is also called the isopachic stress or the first stress invariant. TSAwise, the doubly perforated plate of figure 1 is analyzed by employing individual stress functions associated with the individual coordinate systems, these separate coordinate systems having their origins at the center of the respective holes. The longitudinal Xaxis of figure 1 is positive down in both cases (top small and bottom large hole) and in each case the angle θ is measured positive counterclockwise from this downward Xaxis. Based on the symmetry about Xaxis, a relevant Airy’s stress function in polar coordinates satisfying equilibrium and compatibility can be written as [6]
I
§ · c' a 0 b0 ln r c0 r 2 ¨¨ a1' r 1 d1' r 3 ¸¸ cos T r © ¹
N
¦> a r ' n
n
b r ' n
( n 2)
c r ' n
n
d r ' n
( n 2 )
(2)
@cos(nT )
n 2 , 3...
where r is the radius measured from the center of a hole, angle T is measured counterclockwise from the longitudinal vertical Xaxis (figure 1) and N is the terminating value of the above series (N can be any positive integer greater than one).
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Individual components of stresses can be obtained by differentiating the stress function
V rr
1 wI 1 w 2I r wr r 2 wT 2
(3)
V TT
w 2I wr 2
(4)
V rT
w § 1 wI · ¨ ¸ wr © r wT ¹
(5)
Imposing the tractionfree boundary conditions analytically on the boundary of the hole (Vrθ = 0 and Vrr = 0 at r = R (where R = r1, radius of the small hole or R = r 2, radius of the large hole) and for all values of θ) results in some of the originally independent coefficients becoming dependent functions of other independent coefficients. The individual components of stresses become
V rr
ª 3r § r ·3 º §1 § 2R4 · 3r r3 · ¨¨ 2 3 5 ¸¸b0 «2 ¨ ¸ » c0 ¨¨ 3 2r ¸¸ cos Td1' 2R 2R ¹ R © R ¹ ¼» ©r © r ¹ ¬«
3R 2 1 r 4 R 4 cos(2T )b2' R 2 3r 4 R 2 4r 2 cos(2T )d 2' § 24r 12r 3 12 · § 18r 8r 3 10 · ¨¨ 6 8 5 ¸¸ cos(3T )c3' ¨¨ 4 6 3 ¸¸ cos(3T ) d3' R r ¹ R r ¹ © R ©R
V TT
>
@
° ( n 2 1) r ( n 2) R 2 ( n 1)(n 2) r n ( n 1) r ( n 2) R ( 2 n 2) cos(nT )bn' ½° ® ¦ ( n 2) ( 2 n 2) 2 ( n 2) 2 n '¾ R (1 n )r R (n 1)(n 2)r cos(nT )d n °¿ n 4 , 5,... ° ¯ (n 1)r N
>
@
(6)
§ 1 § § 2R 4 · 3r 5r 3 · 3r 5r 3 · ¨¨ 2 ¸ ¨ ¸ ¨¨ 3 6r ¸¸ cos Td 1' b 2 c 3 5 ¸ 0 3 ¸ 0 ¨ R R ¹ 2R 2R ¹ © r © © r ¹ 2 2 4 6 ' 2 4 2 (3) R 12r 3r R cos(2T )b2 R 3r R cos(2T )d 2'
>
@
§ 24r 60r § 18r 40r 12 · 2· ¨¨ 6 8 5 ¸¸ cos(3T )c 3' ¨¨ 4 6 3 ¸¸ cos(3T )d 3' R r ¹ R r ¹ © R © R 2 ( n 2 ) 2 n N ½ R ( n 1)(n 2)r cos(nT )bn' ° ( n 1) r ° ¦ ® ( n 2) ( 2 n 2) 2 ( n 2) 2 n ' ¾ R (1 n )r R (n 2)(n 1)r cos(nT )d n ° n 4 , 5... ° ¯ (n 1)r ¿ 3
>
>
3
@
@
(7)
270
V rT
ª 3r cot(3T ) 3r 3 cot(3T ) º ª 3r cot(3T ) 3r 3 cot(3T ) º b « » 0 « »c0 R 2R 3 2R 5 R3 ¬ ¼ ¬ ¼ 4 § 2R · ¨¨ 3 2r ¸¸ sin Td 1' 3R 2 6r 2 3r 4 R 6 sin(2T )b2' © r ¹
§ 24r 36r 3 12 · R 2 3r 4 R 2 2r 2 sin(2T )d 2' ¨¨ 6 8 5 ¸¸ sin(3T )c 3' R r ¹ ©R 3 § 18r 24r 6· ¨¨ 4 6 3 ¸¸ sin(3T ) d 3' R r ¹ ©R
>
@
° (n 2 1)r ( n 2 ) R 2 n(n 1)r n (n 1)r ( n 2 ) R ( 2 n 2 ) sin(nT )bn' ½° ¦ ® ¾ ( n 2) ( 2 n 2) R (1 n 2 )r ( n 2 ) R 2 n(n 1)r n sin(nT )d n' °¿ n 4 , 5... ° ¯ ( n 1) r N
>
@
(8)
and the isopachic stress is
S
V rr V TT
§ 2r 3 4r 3 ¨ 4 b 0 ¨ R5 R3 ©
· ¸¸c 0 8r cos Td 1' 12r 2 cos(2T )b2' 4r 2 cos(2T )d 2' ¹
§ 32r 3 48r 3 8 ' ¨¨ 6 3 cos( 3 T ) c 3 8 R r © R
¦ >4(n 1)r N
n
· ¸¸ cos(3T )d 3' ¹
cos(nT )bn' 4(n 1)r n cos(nT )d n'
(9)
@
n 4 , 5...
Unless stated otherwise (and although not needed), an additional twenty tractionfree boundary conditions, VYY = 0 and VXY = 0, were also imposed at h/2 = 10 locations along the right edge of the vertical plate of figure 1. Combining the measured temperature related data from TSA, S = S*/K, and expressions for the isopachic stress at different values of r and θ, the following matrix equation can be written as [A](m+h)xk[c]kx1 = [d](m+h)x1
(10)
where matrix [A] involves the m (m1 for the small hole and m2 for the large hole) Airy isopachic equations of equation 9 and h = 20 expressions for VYY = VXY = 0 at the right edge of the vertical plate. Vector {c} contains the k unknown coefficients. Vector {d} consists of the m measured TSA data values of S=S*/K corresponding to the data points employed in the S to form matrix [A] as well as the h = 20 zeros. Equation 10 was solved using the ‘\’ matrix division operator or ‘pinv’ pseudo inverse operator which uses the algorithm for least squares in MATLAB.
3. Experimental Details and Results The tensile aluminum plate of figure 1 (E = 68.95 GPa and Poisson’s ratio ν = 0.33) was sprayed with Krylon UltraFlat black paint to provide an enhanced and uniform emissivity. Figure 2 is a photograph of the loaded aluminum plate. The plate was sinusoidally loaded between 1334.46 N (300 lb) and 5782.68 N (1300 lb) with a mean value of 3558.57 N (800 lb) at a frequency of 10 Hz. A separate uniaxial tension calibration specimen was used to determine the thermomechanical coefficient, K = 210.3 U/MPa (1.45 U/psi).
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Fig. 2: Specimen loading and recording.
Fig. 3: Actual recorded TSA image, S*, for a load range of 4448.22 N (1000 lb).
Figure 2 is a photograph of the loaded aluminum plate and mechanically cooled Delta Therm camera DT1410 having a sensor array of 256 horizontal x 256 vertical pixels, whereas figure 3 is a recorded TSA image displayed using the commercially provided Delta Vision software. This image was averaged and recorded over a period of two minutes. As the plate is symmetrical about the longitudinal Xaxis (figure 1), the recorded TSA data obtained were averaged about the longitudinal Xaxis so that only one half of the plate is considered during thermoelastic stress analyses. Individual analyses were carriedout for each of the large and the small hole and k = 9 was found to be an appropriate number of Airy coefficients for each of the small and large holes. For the analysis of the large hole of the plate of figure 1 a total of m 2+h = 2,517 input values were used. Of these, 2,497 are measure TSA input values whose source locations as shown in figure 4 and 20 are the tractionfree conditions VYY = VXY = 0 imposed at the ten locations along the right edge of the vertical plate of figure 4. After evaluating all the unknown Airy coefficients (b0, c0, d1c, b2c, d2c, c3c, d3c, b4c and d4c) from the measured S* for each of the small and large holes from equations 9 and 10, the individual components of stress were obtained from equations 6 through 8. These TSA results are compared with those from finite element analysis (ANSYS) and commercial strain gages. Tangential stresses, σθθ, are normalized with respect to the far field stress, σ0, and are plotted around the boundary of the large hole in figure 5. Figure 6 compares the strains (static equivalent specimen load of F = 4,448.22 N = 1,000 lb) along the line CD extending from the large hole (figure 1) obtained from Thermoelastic Stress Analysis with those from finite element analysis (ANSYS) and strain gages.
(a) (b) Fig. 4: TSA source locations (a) m1 = 5,350 and (b) m2 = 2,497 for the small and large hole.
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Fig. 5: Plot of σθθ/σ0 along the boundary of the large hole from TSA and ANSYS
Fig. 6: Strain εXX along CD of figure 1 from TSA, ANSYS and strain gages
4. Satisfying Stress Compatibility between the Holes Although the measured input TSA data around either one of the holes automatically includes the consequences of the other hole, until now there was no attempt here to incorporate explicitly the consequence of the stresses associated with one hole on those of the other hole. This is now done so by imposing compatibility conditions for the analyses of the stresses associated with the large and the small holes, i.e., ensuring the stresses σ XX, σYY, σXY given by the individual analysis for each of the small and the large hole are equal in the common region between the holes, figure 7. For this, the rectangular stress components, σ XX, σYY, σXY, in the common overlapping area of figure 7 associated with each hole were expressed in terms of their Airy coefficients according to equation 10. Additional conditions thus make use of the 406 discrete locations in this common region. Therefore, each stress component, σXX, σYY, σXY, provides 406 additional conditions i.e., c = 406*3 = 1218 more conditions which were then employed to solve a matrix expression similar to equation 10 by which to evaluate the unknown Airy coefficients. The total number of input values used now becomes m 1+h+c = 6,588 for the small hole and m2+h+c = 3,735 for the large hole, where c = 1,218 conditions, h = 20 tractionfree conditions at the longitudinal edge of the plate, m1 = 5,350, and m 2 = 2,497 measured TSA values with k = 9 coefficients for the analysis of each hole. The analysis associated with each of the small and the large hole of figure 8 was again carried out individually, and enforced (in addition to the tractionfree conditions at the respective hole and 10 locations at the edge of the plate) compatibility of the stresses by each of these analyses at the 406 points in the common region between the holes. The individual regions of the respective TSA source locations engulfing these separate holes of figure 7 are those of figure 4. The inner radius for each of these regions is the [radius of hole + (4 x actual pixel size)] and the outer radius associated with the small and large holes are 22.86mm (0.9”) and 19.05mm (0.75”) or X/r 1 = 2.4 and X/r2 = 1.5, respectively. Figures 8 and 9 compare the normalized stress, σXX/σ0 in the (loading) Xdirection from ANSYS and TSA after having imposed stress compatibility between the holes.
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Y/d2 Y
X
X/d2
Y
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 7: Common area between small and large circles.
Y
X
X
Fig. 8: Contour plots of σXX/σ0 from ANSYS.
Y
Y
X
X
Fig. 9: Contour plots of σXX/σ0 from TSA for small hole (right image) m 1+h+c = 6,588 input values, and m 1 = 5,350 TSA values, and for large hole (left image) m 2+h+c = 3735 input values and m 2 = 2,497 TSA values; h = 20, c = 406*3, k = 9 coefficients.
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5. Summary, Discussion and Conclusions The present study was motivated by a relative lack of stress information in multipleperforated finite members when the different size geometric discontinuities are sufficiently close together that their respective stress fields interact. Current TSA results correlate well with those from strain gages and FEM. While the details are omitted here for space reasons, force equilibrium is also satisfied. Imposing the tractionfree conditions (σrr = σrθ = 0) on the edge of the holes analytically enables all of the Airy coefficients to be evaluated from measured TSA data without the need for any supplemental experimental stress analysis. That it is unnecessary to know the external boundary conditions is advantageous. The present method of mathematically enforcing compatibility of the stresses which were associated initially only in the neighborhood of each of the individual holes involved averaging these respective stress components in a common region between the holes. The Airy coefficients are evaluated using least squares where the number of input values/equations (i.e., TSA measured temperature data, tractionfree conditions and additional compatibility conditions) well exceeds the number of these coefficients, k. Under various conditions, k = 9 is found to be an appropriate number of Airy coefficients to use for the analyses of either the small or the large hole. The technique gives the three individual components of ‘fullfield’ stress, including on the edge of the holes, from only the measured temperature information and provides an enhanced understanding of how the stresses at a geometric discontinuity can be influenced by the neighboring structural compliance. This ability to determine the stress consequence of adding auxiliary holes or notches in the neighborhood of initial geometric discontinuities is important in engineering design. One could conceptually utilize photoelastic or measured displacement (moiré, speckle, digital image correlation) methods for such problems. However, unlike those approaches, TSA enjoys the advantages of neither having to differentiate the measured information, nor prepare a model or apply a birefringent coating to the component.
6. References [1]
Rhee, J., Cho, H. K., Marr, R.J. and Rowlands, R. E., “On Local Compliances, Stress Concentrations and Strength in Orthotropic Materials”, Society of Experimental Mechanics, Portland, OR (2005).
[2]
Cho H. K. and Rowlands, R. E., “Reducing Tensile Stress Concentration in Perforated Hybrid Laminate by Genetic Algorithm,” Composite Science and Technology, 67, 28772888 (2007).
[3]
Jindal, U.C., “Reduction of Stress Concentration around a Hole in a Uniaxially Loaded Plate”, Journal of Strain Analysis, 18(2) (1983).
[4]
Lin, SJ., Matthys, D.R. and Rowlands, R.E., “Separating Stresses Thermoelastically in a Central Circularly Perforated Plate using an Airy Stress Function” , Strain, 45, 516526 (2009).
[5]
Foust, B. E., “Individual Stress Determination in Inverse Problems by Combining Experimental Methods and Airy Stress Functions”, Thesis, University of Wisconsin  Madison (2002).
[6]
Joglekar, N., “Separating Stresses using Airy Stress Function and TSA: Effects of Varying the Amount and Source Locations of the Input Measured TSA Data and Number of Airy Coefficients to Use”, Thesis, University of Wisconsin–Madison (2009).
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Crack tip stress fields under biaxial loads using TSA
R. A. Tomlinson *Department of Mechanical Engineering, The University of Sheffield, Sheffield, S1 3JD, ENGLAND; email: [emailprotected]
INTRODUCTION The aerospace industry is striving to design less conservative and hence more efficient structures in order to meet weight reduction targets, and consequently give improvements in fossil fuel use. With this focus comes the need for the development of more accurate techniques for the assessment of the structural integrity of complex, lightweight, safetycritical components. Examples of such components are wing skin panels, which, with their array of stiffeners and holes, present a complex mixedmode loading problem where cracks can change their growth direction. This paper focuses on the exploration of experimental mechanics methods which can be used to understand such mixedmode fatigue failure problems. BACKGROUND  TSA OF CRACKS UNDER BIAXIAL LOADS In recent years considerable progress has been made using optical experimental mechanics techniques for the 1,2 investigation of fatigue cracktip strain fields . One of the most promising of these techniques is thermoelastic stress analysis (TSA), due to its simple surface preparation and noncontacting, full field, data collection capabilities. Thermoelasticity is based on the principle that under adiabatic and reversible conditions, a cyclically loaded structure experiences temperature variations that are proportional to the sum of the principal stresses. These temperature variations may be measured using a sensitive infrared detector and thus the cyclic stress field on the surface of the structure may be obtained. For over a decade, the Experimental Mechanics Research Group at the University of Sheffield has been performing significant work using TSA techniques for the understanding and description of cracktip stress fields in mode I, and some work on mixedmode applications. The initial work was by Tomlinson, who developed a methodology which she had used previously in photoelasticity, for the determination of stress intensity factors KI 3 and KII using the Muskhelishvili approach . Subsequent experiments were performed using this technique on 4 5,6 biaxiallyloaded slots and cracks (by Marsavina ), and mode I fatigue cracks through welds (by Diaz ). The latter also explored the utilisation of the phase signal, which represents the phase shift between the thermoelastic signal recorded by the infrared detector and the reference signal from the test machine. The result of this research was to provide more information about the cracktip stress field and plastic zones, and also the development of a method for tracking the crack tip. This work was particularly useful in the investigation of crack 7 closure. Zanganeh used Diaz’s bespoke software for investigating the interaction of crack tips, and then further developed the software to investigate T stresses. Other methodologies have been developed by other 8 researchers and these are documented elsewhere . One of the most active, however is DulieuBarton’s team at 9,10,11,12 to describe the the University of Southampton which has established a technique using a cardioid method relationship between the stress intensity factor, the cracktip location and stress field surrounding the crack measured using TSA and the theoretical description proposed by Westergaard. The technique has been developed to allow estimations of the cracktip stress intensity factors and cracktip position by using genetic algorithms to aid direct curvefitting of the cardioid (which mathematically describes the stress field) to the measured isopachics from TSA. This method appears to work well for mode I loaded cracks and slots, but for 12 mixedmode loads, the method to find the cracktip from the cardioid patterns gives spurious results which appear to be artefacts of the uniaxial loading regime. Thus it is proposed in the current study to investigate
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_44, © The Society for Experimental Mechanics, Inc. 2011
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mixedmode cracks under controlled biaxial loads, where the crack will not change direction whilst data are being collected, thus preventing spurious results. In addition to characterising the cracktip stress field using the stress intensity factor, in recent years it has been shown that the T stress may be determined using TSA data. T stress is defined as the second nonsingular term 13 in Williams’s cracktip stress field solution. The Tstress is a constant stress parallel to the crack and is a 14 measure of the constraint around the tip of a crack in contained yielding problems and can be used to identify 15 how a crack growth will deviate from the perfect path due to irregularities in the microstructure . The effects of the Tstress on the crack growth rate, cracktip constraints, crack closure, and the shape and size of the plastic 16,17 zone ahead of the cracktip have also been assessed . Following successful use of TSA for stress intensity 18 factor determination, a method was proposed to determine the T stress from such data , however the Tstress 18 results were found to be sensitive to cracktip position. A new technique was proposed to find the cracktip from thermoelastic images based on the Y or phase image, and this cracktip location was used in the method to determine K and the T stress. This cracktip location technique appeared to be more reliable than another similar 6 18 technique proposed by Diaz , especially in Tstress determination. The work by Zanganeh also investigated the use of higher order terms on the accurate determination of both K and Tstress, using methods based on both Williams’s formulation and the Muskhelishvili technique. It was shown that Muskhelishvili and the two term Williams’s solutions both give the same results and using them to determine the stress intensity factors does not affect the accuracy of the results significantly when compared to using three terms of Williams’s solution. However, the two term Williams’s solution was not sufficient to determine the Tstress accurately and the results for Tstress using this model were dissimilar to those predicted by a finite element method. It was shown that using up to three terms of the Williams’s solution makes it possible to determine both the SIF and Tstress accurately. The majority of this work was carried out under Mode I loads and only one mixedmode case was considered, thus there is great potential to continue this work under biaxial loading. 12 It was suggested by Hebb et al that the inclusion of higher order terms in the theoretical description of the stress field could be more important for the accurate determination of the stress intensity factor in mixedmode problems than in mode I. In previous work by Tomlinson which used the Mushkelishvili approach to determine K it was found that only three terms were needed to obtain convergence of the solution, however the values of KII determined still differed from the theoretical solution by up to 16%. It is now considered that the uniaxial loading, the location of the crack tip, and the resolution of the data recorded could have influenced the accuracy of these 4 data. The subsequent work by Marsavina and Tomlinson using the Mushkelishvili approach on biaxially loaded slots and a single example of a fatigue crack concluded that the TSA technique had potential for investigating fatigue cracks under mixedmode loads. Hence the availability at MSU of the TSA equipment together with a biaxial test machine presents the opportunity to explore this research further. In addition to the determination of cracktip parameters, it has been proposed by Patterson’s group at Michigan 19 6 State University that an understanding of the plastic zone size may be obtained from the TSA phase data. Diaz 18 and then Zanganeh had already concluded that the maps of the phase difference between the applied load and the measured temperature contained important information which could be used to identify plasticity in the 19 specimen, and used the signal to identify the cracktip location. Patki has developed this idea further to measure the size of the plastic zone and used the information to document experimentally for the first time the effects of overloads. Again, these experiments have only been performed under Mode I conditions and so there is the potential to use this method for mixedmode loading in order to gain greater understanding of such stress fields in a more practically relevant experimental arrangement. Although the great potential of the TSA method has been demonstrated for Mode I fatigue cracks, it can be concluded that there is still not a full understanding of the meaning of the TSA data obtained from cracks under mixedmode, and how these data relate to the cracktip location, the cracktip plasticity, the mathematical descriptions of the stress field, and subsequently the characterising parameters. Therefore it is proposed to combine the expertise of Tomlinson and Patterson in this area with the facilities at MSU in order to explore these aspects further. PROPOSED EXPERIMENTS The study will start by reviewing the methodology for quantification of the plastic zone size which has been 19 developed by Patterson’s team . Experiments will be conducted in mode I on aluminium alloy 2024, with the application of larger overloads than those carried out by Patki, and then the technique will be applied using the biaxial test machine in mixedmode loading. These experiments will provide additional understanding of the relationship between different load magnitudes, loading modes and the cracktip behaviour, and further knowledge of the how the TSA data can characterise local cracktip phenomena.
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It is then proposed to use the bespoke software developed at the University of Sheffield for the determination of K and T stress in order to explore the TSA response of propagating mixedmode fatigue cracks under biaxial loads. The bespoke software allows the number of terms to be varied easily, so the hypothesis that “the inclusion of higher order terms in the theoretical description of the stress field is more important for the accurate determination of K in mixedmode problems than in mode I” will be explored. Both the Muskhelishvili approach and the Williams approach will be used. It has already been suggested here that mode I crack growth from the slanted cracks under uniaxial loads caused 12 spurious results in Hebb’s experiments , but it is also possible that specimen movement under such loads could distort the TSA data field collected. The mixedmode methodologies for K and cracktip location may be more sensitive to specimen movement under such loads than for mode I applications. The biaxial test machine has four actuators which can be adjusted independently to minimise rigid body movement of a specimen. The ability to minimise such movement of the cracktip will be crucial to quality data collection, an added advantage of biaxial loading. 4 The initial biaxial experiments will be performed using cruciform shaped specimens made from 150M36 steel which are identical to those used by Marsavina. This will allow direct comparison with previous results, but by using more modern machine control and data collection systems than these previous tests, better quality results are anticipated. It is then proposed to use the same cruciform specimen design with other materials such as the 19 aircraft grade aluminium 2024 used by Patki . This will allow the potential of the technique to be explored. REFERENCES 1
Olden E. J., Patterson E. A., Optical analysis of crack tip stress fields: A comparative study, Fatigue and Fracture in Engineering Materials and Structures, 277: 623636, 2004. 2 Sanford R. J., Determining fracture parameters with fullfield optical methods, Experimental Mechanics, 29(3):241247, 1989. 3 Tomlinson, R. A, Nurse, A. D. and Patterson, E. A., “On determining stress intensity factors for mixedmode cracks from thermoelastic data”, Fatigue and Fracture of Engineering Materials and Structures, 20, (2), 217226, 1997 4 Marsavina, L, and Tomlinson, R A, Thermoelastic investigations for fatigue life assessment, Experimental Mechanics,44, 487 – 494, 2004 5 Diaz, F A, Patterson, E A, and Yates, J R, Some improvements in the analysis of fatigue cracks using thermoelasticity, International Journal of Fatigue, 26, 365–376, 2004 6 Diaz, F A, Patterson, E A, Tomlinson, R A, and Yates, J R, Measuring stress intensity factors during fatigue crack growth using thermoelasticity, Fatigue and Fracture in Engineering Materials and Structures, 27, 571 – 583, 2004 7 Yates J.R, Zanganeh, M., Tomlinson, R.A. Brown, M.W., and F.A. Diaz Garrido , Crack paths under mixed mode loading, Engineering Fracture Mechanics, 75, 319–330, 2008 8 Tomlinson, R A, and Olden, E J, Thermoelasticity for the analysis of crack tip stress fields  a review, Strain, 35, (2), 4955, 1999. 9 Stanley P and DulieuSmith J M 1993 Progress in the thermoelastic evaluation of mixedmode stress intensity factors Proc. SEM Spring Conf. on Exp. Mech. Dearborn 617–26, 1993 10 DulieuBarton, J M, Fulton, M C, and Stanley, P, The analysis of thermoelastic isopachic data from crack tip stress fields, Fatigue and Fracture in Engineering Materials and Structures, 23, 301313, 2000 11 DulieuBarton, JM and Worden, K, Genetic Identification of cracktip parameters using thermoelastic isopachics, Meas. Sci. Technol. 14, 176–183, 2003 12 th Hebb, R I, DulieuBarton, J M , Worden, K, and Tatum, P, Curve Fitting of Mixed Mode Isopachics, 7 International Conference on Modern Practice in Stress and Vibration Analysis, Journal of Physics: Conference Series, 181, 2009 13 Williams, M.L. On the stress distribution at the base of a stationary crack. J. App. Mech., 24, 109–114, 1957 14 Ayatollahi, M.R., Pavier, M.J. and Smith, D.J. Determination of T stress from finite element analysis for mode I and mixed mode I/II loading. Int. J. Fracture, 91, 283298, 1998 15 Cotterell, B. Notes on the paths and stability of cracks. Int. J. Fracture, 2(3), 526533, 1966 16 Roychowdhury, S. and Dodds Jr., R.H. Effect of Tstress on fatigue crack closure in 3D smallscale yielding. Int. J. Solids Struct., 41, 2581–2606, 2004 17 Smith, D.J., Ayatollahi, M.R. and Pavier, M.J. The role of Tstress in brittle fracture for linear elastic materials under mixedmode loading. Fatigue Fract. Engng. Mater. Struct. 24(2), 137–50, 2001
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Zanganeh, M, Tomlinson, R A, and Yates J.R., Tstress determination using thermoelastic stress analysis, Journal of Strain Analysis for Engineering Design, 43, 529537, 2008 19 Patki, A S and Patterson, E A , Thermoelastic stress analysis of Fatigue Cracks subject to overloads, Fatigue and Fracture of Engineering Materials and Structures, submitted for special issue on Experimental Fracture Mechanics, 2009
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Extending TSA with a Polar Stress Function to NonCircular Cutouts.
A. A Khaja1 and R. E. Rowlands2 Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706 1
[emailprotected]
2
[emailprotected]
ABSTRACT This paper extends ability to thermoelastically stress analyze components containing other than circular discontinuities. Combining recorded temperatures with a stress function for stress analysis (i.e., TSA) has proven to be effective. Previous publications applied TSA to members having various shaped cutouts by employing a stress function in complex variables [1]. A shortcoming of such an approach is that one can typically evaluate the stresses only incrementally around the entire edge of a hole or notch. Real variables are also easier to use than complex variables and a general seriesform of the Airy stress function exists in (real) polar coordinates which enables the stresses be determined simultaneously around an entire cutout. The TSA approach using polar coordinates is illustrated here for the case of an elliptical hole in a finite tensile plate. Little information is available for such finite situations, purely analytical solutions are extremely difficult for finite geometries, and both theoretical and numerical approaches are challenging if the external loading is not well known. The number of Airy coefficients to retain is evaluated experimentally, and the TSA results are substantiated independently. 1. Introduction Holes of various shapes are frequently employed in engineering structures. Elliptical holes often occur in finite engineering components but such situations are difficult to stress analyze, particularly for finite components whose external boundary conditions are not well known. Relatively little information is available for finite members containing elliptical holes and even less for elliptical notches, [210]. This paper demonstrates the ability to thermoelastically stress analyze finite components containing a noncircular geometric discontinuity using a stress function in real (polar) coordinates. This is important because very few theoretical solutions are available for finite geometries, figure 1 and strictly numerical (finite element or finite difference) methods necessitate knowing the external loading and geometry. The reliability of the present TSA results is substantiated by comparison with those from finite element analysis and commercial strain gages. The major contribution of this paper is the demonstrated ability to thermoelastically stress analyze finite components containing a noncircular geometric discontinuity using a stress function in real (polar) coordinates.
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_45, © The Society for Experimental Mechanics, Inc. 2011
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280
2b = 19.05mm (0.75 “)
139.7 mm (5.5““)
F
y r
θ
A
x
B
139.7mm (5.5 “)
2a = 38.1 mm (1.5”)
76.2 mm (3””)
F
Fig. 1: Symmetricallyloaded tensile aluminum plate containing an elliptical hole. 2. Relevant Equations Under proportional planestress isotropy, the measured TSA temperature signal, S*, is related to the stresses by (1) where S is sometimes called the first stress invariant, the trace of the stress tensor or the isopachic stress value, K is the isotropic thermoelastic coefficient, and are the sum of the stresses in the principal directions, in polar coordinates, in Cartesian coordinates and in elliptical coordinates, respectively, figure 2.
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(edge of the elliptical hole) tangent to at
variable, fixed (hyperbola) r at (
)
(x, y;
; r, θ)
tangent to at θ Fig. 2: Coordinate representations For thermoelastically stress analyzing the ellipticallyperforated verticallyloaded finite plate of figure 1, one can utilize the Airy’s stress function in polar coordinates. In this particular case the coordinate origin is at the center of the hole, figure 1, such that the situation is symmetric about both x and y axes, thereby simplifying the stress function and hence the number of Airy coefficients needed. Since the plate of figure 1, is symmetrical about both horizontal and vertical axes, the raw recorded TSA data were averaged throughout the four quadrants so as to eliminate any possible nonsymmetry. In addition to the measured temperature data, the tractionfree conditions are imposed discretely on the boundary of the elliptical hole. It is convenient to employ elliptical coordinates when satisfying the tractionfree conditions at the edge of the elliptical hole. For the present case of figure 1, a relevant Airy stress function is (2)
Individual components of stresses can be evaluated by differentiating the stress function of equation 2 as shown in equations 3 through 5:
V rr V TT V rT
1 wI 1 w 2I r wr r 2 wT 2 w I wr 2 w § 1 wI · ¨ ¸ wr © r wT ¹
(3)
2
(4)
(5)
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Upon differentiating equation 2 according to equations 3 through 5, the individual polar components of stress can be written as follows: (6)
(7)
(8) Quantity r is the radial coordinate measured from the center of the cavity and angle θ is measured counterclockwise from the horizontal xaxis, figure 1. N is the terminating index value of the series (since in practice one can only handle a finite number of terms) and it can be any positive even integer. Using the transformation matrix, stresses acting in the polar coordinate system can be transformed to those in the Cartesian coordinates, equations 9 through 11, and in elliptical coordinates, equations 12 through 14, as follows:
(9)
(10)
(11)
(12)
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(13)
(14)
where
, figure 2.
Adding stresses in polar coordinates (equations 6 and 7) or in Cartesian coordinates (equations 9 and 10) or in elliptical coordinates (equations 12 and 13) gives the following equation for isopachic stress: S=
=
=
(1)
(15)
Comparing equations 6 through 15 shows that coefficients present in the expression for isopachic stress are also present in the expressions for individual stresses. However, the individual components of stress also contain the Airy coefficients bo, an and cn (equations 6 through 14) which are absent in the isopachic stress, equation 15. Therefore the separate stress components cannot be determined from only thermoelastic data through the expression for the isopachic stress (equation 15). However imposing the tractionfree conditions i.e., zero normal and shear stress around the boundary of the hole, together with measured temperature information, does enable all of the Airy coefficients to be evaluated. Imposing the tractionfree boundary conditions discretely on the boundary of the elliptical hole ( at r = R (where R maps the boundary of the elliptical hole for a = 19.05mm (0.75”) and b = 9.525mm (0.357”))) and for all values of θ), together with measured TSA data, evaluates all the unknown Airy coefficients, such that
(16)
Thus, by incorporating the boundary conditions around the central hole discretely, all the unknown Airy coefficients bo , co , an , bn , cn , dn for n = 2, 4, 6.…N can be evaluated from the measured TSA data. Equation 16 can be written in simplified form as follows:
>A@(mh) xk ^c`kx1 ^d `(m h) x1
(17)
where matrix [A] also includes h/2 = 73 expressions for each of imposed at the boundary of the elliptical hole. Vector {c} of equation 17 contains the k unknown Airy coefficients (i.e., bo , co , an , bn , cn , dn for n = 2, 4,…N). The stress vector {d} of equation 17 therefore contains the tractionfree normal and shear stresses at the boundary of the elliptical hole as well as measured values of S within the plate. Measured TSA data can be
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noisy so it was advantageous to use more measured input values of S plus known boundary conditions than unknowns and to determine the Airy coefficients by least squares using the ‘\’ matrix division operator or ‘pinv’ pseudo inverse operator in the commercial MATLAB software. 3. Experimental Details Figure 1 shows the aluminum specimen (alloy 6061T6, ultimate strength = 275 to 311 MPa (40 to 45 ksi) and yield strength = 241 to 275 MPa (35 to 40 ksi)), its geometry, dimensions, and orientation and location of the coordinate axes. The plate was sinusoidally loaded using a MTS hydraulic testing machine, at a mean value of 3558.57 N (800lb), maximum value of 5782.68 N (1300lb) and a minimum value of 1334.46 N (300lb) at a frequency of 10Hz, figure 3. The aluminum plate was polished with a 400 grid sand paper and then sprayed with Krylon UltraFlat black paint to provide an enhanced and uniform emissivity.
Fig. 3: Specimen in loading frame with Delta Therm DT1410 infrared camera Figures 3 show the specimen loading, and the TSA and strain gage recording equipment. The loadinduced temperature effect was recorded by a TSA Delta Therm model DT1410 camera which has a sensor array of 256 horizontal x 256 vertical pixels, figures 3 and 4. Figure 4 shows an actual TSA image, S* as recorded and displayed by the Delta Vision software which has data acquisition and interpretation tools. From a separately prepared and tested uniaxial tensile specimen (same material and “identical” (thickness, etc.) coating of flat black paint) and tested using the same TSA recording characteristics, the thermoelastic coefficient was found to be K = 265.42 U/MPa (1.83 U/psi).
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Fig.4: Actual recorded TSA image, S*, for a load range of 4448.22 N (1000lb). 4. Analysis and Results Since the plate is symmetrical about the both the x and y axes (figure 1), the recorded TSA data were averaged throughout the four quadrants. However, recognizing the typical unreliability of TSA data near an edge (figure 4), no measured information were utilized originating within at least two data positions (~ 0.96 mm or 0.0378s) of the edge of the hole. The m = 1703 measured TSA values of S* utilized, figure 5, and the h = 146 tractionfree conditions ( ) on the boundary of the hole (i.e. for a total of m+h = 1849 input values) were combined to evaluate the k unknown Airy coefficients, equation 17 for analyzing the elliptically perforated finiteplate of figure 1. In this analysis k = 26, was found to be a realistic number of Airy coefficients.
Y X
Fig.5: TSA source locations (m = 1,703) for 1,849 input values (m+h = 1,849). After evaluating all of the unknown Airy coefficients (bo, co, an, bn, cn, dn for n = 2, 4, 6.…N = 12) from the measured data, S*, at the source locations shown in figure 5 along with the h = 146 tractionfree conditions on the boundary of the hole of figure 1, the individual components of stress can be obtained from equations 6 through 14. TSA results are compared with the strain gages and finite element analysis in figures 6 through 8. Figures 6 contain contour plots of the normalized Cartesian component of stress (σyy/σ0) using the TSA (and imposed tractionfree stresses on the edge of the hole) evaluated coefficients and from ANSYS. The x and y axes of figures 6 are normalized by a = 19.05 mm = 0.75s (half of the major axis of the elliptical hole of figure 1). The /σ0, is plotted on the edge of the hole in figure 7. The stresses normalized hoop stress in elliptical coordinates, are normalized with σ0 = 9.19 MPa (1333.33 psi) which is based on the applied tensile load, F, divided by the gross area. Angle θ of figures 7 is measured counterclockwise from the positive xaxis, and the radial coordinate,
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r, is measured from the center of the elliptical hole (x = y = 0) of figures 1 and 2. Figure 8 compares the Cartesian components of strains εθθ = εyy from ANSYS, TSA and strain gages along line AB of figure 1. The x axis of figure 8 is normalized by a = 19.05 mm = 0.75s (half of the major axis of the elliptical hole), figure 1.
Y
Y X
X
(a) (b) Fig. 6: Contour plots of σyy/σ0 from TSA (a) and ANSYS (b).
Fig. 7: Plot of hoop stress /σ0 on the boundary of the hole (2a= 38.10 mm (1.5”), 2b = 19.05 mm (0.75”)) from ANSYS and TSA (m+h = 1,849 input values, k = 26 coefficients and m = 1,703 TSA values).
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Fig. 8: Strain εyy along AB of figure 1 from TSA, ANSYS and strain gages for m+h = 1,849 input values, m = 1,703 TSA values, k = 26 and F = 4448.2 N (1000 lb) 5. Summary Combining a stress function with the recorded temperature information, plus perhaps some local boundary conditions, makes this general TSA approach amenable to a variety of engineering problems, thereby enhancing TSA’s practical applicability. The demonstrated use of real variables (in polar coordinates) to evaluate the stresses in a finite plate containing an elliptical hole is particularly noteworthy. The TSAbased results agree with those from FEM and commercial strain gages. Summing σyy across twice the area associated with line AB of figure 1, i.e., (where t is the thickness), gives a load of 4409.1 N (991.2 lbs), which agrees with the applied load of 4448 N (1000 lbs). Since recorded TSA data are unreliable at edges, the described scheme employs temperature information only beyond at least two pixel locations from the boundary of the hole. Nevertheless, and in addition to giving individual components of stress fullfield, the current technique provides accurate stress values at the boundary of the elliptical hole. Perhaps the most advantageous feature of the present approach is its ability to use a stress function in real (rather than complex) variables for other than a circular cutout in a finite component. Stress functions formulated in terms of complex variables are mathematically more cumbersome. While the tractionfree conditions at the hole are applied here discretely rather the analytically, the present approach also enjoys the advantage over the complexvariable technique of references [1] and [11] in that the latter method necessitates an iterative application for implementation around an entire hole. Unlike the present TSA approach, strictly numerical and analytical methods require accurately knowing the farfield boundary conditions. Theoretical solutions to finite components are also extremely difficult, and few are available. The present general TSA approach is applicable to elliptical holes in more complicated components, under more complicated loading, as well as more complicated shaped cutouts (holes, notches). In addition to not restricted to cases possessing symmetry, the current concepts could well find application to other areas of experimental mechanics.
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6. References [1]
S. T. Lin and R. E. Rowlands, “Thermoelastic Stress Analysis of Orthotropic Composites,” Experimental Mechanics, 35, 257265 (1995).
[2]
Wang, J.T., Lotts, C.G., and Davis, D.D, JR., “Analysis of BoltLoaded Elliptical holes in Laminated Composite Joints” Journal of Reinforced Plastics and Composites, 12(2), 128138 (1993).
[3]
Oterkus, E., Madenci, E. and Nemeth. M. P., “Stress Analysis of Composite Cylindrical Shells with an Elliptical Cutout", Journal of Mechanics of Materials and Structures,” 2, 695751 (2007).
[4]
Madenci, E., and Ileri, L., “An Arbitrarily Oriented, Rigid, Elliptic Inclusion in a Finite Anisotropic Medium”, Int’l Journal of Fracture, 62, 341354 (1993).
[5]
Persson, E., and Madenci, E., “Composite Laminates with Elliptical PinLoaded Holes”, Engineering Fracture Mechanics, 61, 279295 (1998).
[6]
Hwu, C., and Wen, W.J., “Green's Functions of TwoDimensional Anisotropic Plates Containing an Elliptic Hole”, Int’l Journal of Solids and Structures, 27, 1705–1719 (1991).
[7]
Timoshenko, S., and Goodier, J.N., “Theory of Elasticity”, McGrawHill Book Company, Inc. (1951).
[8]
Pilkey, W.D., and Pilkey, D.F., “Stress Concentration Factors”, a WileyInterscience Publication, (2008).
[9]
Lekhnitskii, S.G., “Anisotropic Plates”, Gordon and Breach, New York, (1968).
[10]
Savin, G.N., “Stress Concentration around Holes”, Pergamon Press, London, (1961).
[11]
Huang, Y.M., and Rowlands, R.E., “Quantitative Stress Analysis Based on the Measured Trace of the Stress Tensor”, Journal of Strain Analysis, 26(1), 5863 (1991).
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Novel Synthetic Material Mimicking Mechanisms from Natural Nacre
Allison Juster, Felix Latourte, and Horacio D. Espinosa* Dept. of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd., Evanston, IL 602083111 * Corresponding author: [emailprotected] ABSTRACT The biomimetics field has become very popular as Mother Nature creates materials with superior strength and toughness out of relatively weak material constituents. This concept is attractive because current synthetic materials have yet to achieve this level of performance from the same weak material constituents. Nacre, from Red Abalone shells, is among the natural materials exhibiting outstanding toughness, while being comprised of a brick and mortar structure of 95% brittle ceramic tablets and 5% soft organic biopolymer mortar. During loading, that tablets slide relative to each other. This generates progressive interlocking which constitutes nacre’s primary toughening mechanism [1, 2]. We have translated this concept of tablet sliding and interlocking to create a novel composite material. Fabrication of the material will be discussed as well as design parameters. Results from tensile tests will be presented as well as comparison of the synthetic material to natural nacre. Implications to the synthetic materials community will be presented. Introduction Materials from Nature make use of hierarchical structures to yield high performance materials from weak material constituents (nacre, bone, wood [37]). Nacre is one of these natural materials found in Abalone shells that exhibit remarkable strength and toughness despite comprising of 95% brittle aragonite ceramic [3]. Because of its hierarchical structure, nacre is orders of magnitude tougher than the pure aragonite ceramic. At the microscale, nacre resembles a brickandmortar structure where the aragonite ceramic tablets are the bricks and the biopolymer lining of the tablets is the mortar [8, 9]. Under loading it has been shown that the brick tablets slide relative to each other, interlocking progressively as they slide to spread damage across the sample [1, 2, 10]. Because of nacre’s damage spreading capabilities, it dissipates energy over large areas giving it superior toughness. Understanding the hierarchical mechanisms in natural materials, specifically nacre, will aide in the development of synthetic materials with extraordinary mechanical properties. Through insitu atomic force microscope (AFM) fracture experiments and digital image correlation (DIC), we are able to quantify tablet sliding at the nanoscale and elucidate the microscale mechanisms in the tablet geometry that generate progressive interlocking. Using this comprehensive nanoscale investigation of tablet sliding as a guide, we then translate these toughening mechanisms into a scaledup composite material. We integrate the interlocking mechanism through a dovetailed structure that interlocks during loading, and use a soft polymer as the interface material to enable tablet sliding. We conduct a parametric analysis varying the geometry of tablets in the artificial composite material and arrive at an optimal design whose failure mode corresponds directly to that of natural nacre. Using the same DIC process as mentioned above, we see the same normalized degree tablet sliding as we observed in natural nacre. Experiments Natural nacre was tested in threepoint bending using a prenotched fracture sample [2]. To observe tablet sliding, the sample was imaged with an AFM during testing. With these images, we can observe the tablet sliding evolution of natural nacre. To quantify the tablet sliding, post processing was performed using DIC with subnanometer resolution (~1/10 pixel or 0.4nm) [11].
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_46, © The Society for Experimental Mechanics, Inc. 2011
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The artificial composite material was tested using a MTS Syntec 20/G tensile testing machine. Images were taken using a Navitar macro lens and CCD camera. The same DIC technique was used to quantify tablet sliding in the artificial material. Through this testing, we were able to analyze the effect of changing geometric parameters on the artificial material. Results By comparing the post processing from the natural nacre and artificial experiments, we have found that our optimized artificial sample has the same normalized failure mode as natural nacre. Furthermore, our artificial sample gives a 100% increase in energy dissipation, compared to the monolithic tablet material. We also see damage spreading throughout the sample which gives this increase in energy dissipation. With this study we can conclude that tablet sliding and interlocking is a major toughening mechanism in natural nacre. References 1. Barthelat, F., et al., On the mechanics of motherofpearl: A key feature in the material hierarchical structure. Journal of the Mechanics and Physics of Solids, 2007. 55(2): p. 306337. 2.
Barthelat, F. and H. Espinosa, An experimental investigation of deformation and fracture of nacre–mother of pearl. Experimental Mechanics, 2007. 47(3): p. 311324.
3.
Sarikaya, M. and I.A. Aksay, eds. Biomimetics, Design and Processing of Materials. Polymers and complex materials, ed. AIP. 1995: Woodbury, NY.
4.
Mayer, G., Rigid Biological Systems as Models for Synthetic Composites Science, 2005. 310(5751): p. 11441147.
5.
Buehler, M.J. and T. Ackbarow, Fracture mechanics of protein materials. Materials Today, 2007. 10(9): p. 4658.
6.
Ashby, M.F., et al., The Mechanical Properties of Natural Materials. I. Material Property Charts. Proceedings: Mathematical and Physical Sciences, 1995. 450: p. 123140.
7.
Gao, H.J., et al., Materials become insensitive to flaws at nanoscale: Lessons from nature. Proceedings of the National Academy of Sciences of the United States of America, 2003. 100(10): p. 55975600.
8.
Espinosa, H., et al., Merger of structure and material in nacre and bonePerspectives on de novo biomimetic materials. Progress in Materials Science, 2009. 54(8): p. 10591100.
9.
Launey, M. and R. Ritchie, On the fracture toughness of advanced materials. Advanced Materials, 2009. 21(21032110).
10.
Tang, H., F. Barthelat, and H.D. Espinosa, An elastoviscoplastic interface model for investigating the constitutive behavior of nacre. Journal of the Mechanics and Physics of Solids, 2007. 55(7): p. 14101438.
11.
F. Latourte, A.S., A. Chrysochoos, S. Pagano, B. Wattrisse, An Inverse Method Applied to the Determination of Deformation Energy Distributions in the Presence of PreHardening Stresses. The Journal of Strain Analysis for Engineering Design, 2008. 43(8): p. 705717.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Mechanical Characterization of Synthetic Vascular Materials AR Hamilton1,2, C Fourastie2,5, AC Karony1, SC Olugebefola2, SR White2,3, NR Sottos2,4 * 1
Department of Mechanical Science and Engineering, 2 Beckman Institute for Advanced Science and Technology, 3 Department of Aerospace Engineering, 4 Department of Materials Science and Engineering, University of Illinois at UrbanaChampaign, Urbana, United States 5 Arts et Métiers ParisTech, Paris, France *1304 W. Green St., 61801, Urbana, Illinois, United States of America, 2173331041, [emailprotected] ABSTRACT Biological materials in living organisms are furnished with a vascular system for the transportation and supply of necessary biochemical components. Many critical functions are supported by these vascular systems, including the maintenance of homeostasis, growth, autoimmune responses, regeneration, and repair. Synthetic materials with vascular systems have been created via a number of fabrication techniques in order to mimic some of these functionalities. The direct ink writing technique has been used to create polymer matrix, vascularized materials with micronscale channels capable of autonomic repair. Liquid healing agents are delivered via the vascular system to sites of damage, where they polymerize in the crack plane, forming an adhesive bond with the surrounding material and recovering the mechanical integrity of the material. Characterizing the mechanical integrity of these materials is critical to optimizing their strength and toughness. The spacing between vascular conduits and the presence of locallyplaced particle reinforcement have been shown to affect the local strain concentrations measured in these materials. In this work we study the effect of vascular geometry and local microchannel reinforcement on the bulk properties of the vascular material. Dynamic mechanical analysis is conducted to evaluate the stiffness of various vascular designs, and single edge notch beam (SENB) fracture samples are used to quantify the effect on fracture toughness. SAMPLE PREPARATION Epoxy matrix samples containing vascular networks were manufactured using direct ink writing. A fugitive organic ink (60% microcrystalline wax, 40% mineral oil, by weight) was deposited in a layerbylayer manner with depositions made at each location where a microchannel was desired in the final sample. The resulting ink scaffold was then infiltrated with liquid epoxy and allowed to cure. The ink was subsequently removed by moderate heating (80°C) and applying vacuum to the microchannel outlets. The resulting samples contained embedded microvascular networks of 200 μm diameter channels that were fully connected in three dimensions. The samples had a 0–90° stacking sequence between subsequent layers of channels, with connections from one layer to the next at the channel intersections. As detailed in Fig. 1 and Table 1, three different microvascular geometries (denoted Types I, II, III) were created for testing by varying the spacing between channels in the xy plane. Channel spacing in the yz plane was the same among all microvascular sample geometries.
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_47, © The Society for Experimental Mechanics, Inc. 2011
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Fig. 1 Schematic of vascular system geometries. Table 1 Vascular system dimensions Label
Architecture
a (mm)
b (mm)
c (mm)
Type I Type II Type III
Offset Offset Aligned
2.0 1.0 2.0
0.7 0.7 0.7
2.0 2.0 2.0
Plain samples, with no vasculature, were produced using the same matrix material as vascular samples. This epoxy matrix was formed by mixing stoichiometric quantities (100:40) of Epon 828™ (DGEBA) and Epikure 3274™ (aliphatic amine). At a minimum, all samples were cured for the first 24 hours at room temperature, followed by 24 hours at 30°C. DYNAMIC MECHANICAL ANALYSIS Storage and loss moduli were measured by dynamic mechanical analysis over a range of frequencies from 0.0110 Hz. Dynamic tests were performed using a TA Instruments model RSA III at a strain amplitude of 0.1%. The storage modulus of plain epoxy samples was determined as a function of cure time beyond the minimum 48 hour cure cycle given above. The average storage modulus at 0.1 Hz for three samples is plotted in Fig. 2 as a function of cure time.
Average storage modulus (GPa)
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
5
10
15
20
25
30
35
Additional cure time at room temperature (days)
Fig. 2 Storage modulus measured at 0.1 Hz in plain epoxy samples as a function of cure time at room temperature beyond the first 48 hours. Each data point represents an average of three samples tested. The data in Fig. 2 shows no significant change in the elastic modulus of the matrix material after the initial 48 hour cure cycle. Therefore, the moduli of vascular samples tested within a window of 530 days could reasonably be considered independent of cure time. Microvascular samples were subjected to the same dynamic testing regime as plain samples. The storage modulus measured at 0.1 Hz was taken as an estimate of the elastic modulus. The average moduli for each vascular architecture (as defined in Fig. 1) are plotted as a function of microchannel volume fraction in Fig. 3.
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4
Type I (n=9) Type II (n=4)
3.5
Type III (n=4)
Elastic modulus (GPa)
Rule of Mixures HalpinTsai
3
2.5
2
1.5
1 5
10
15
20
25
Microchannel volume fraction (%)
Fig. 3 Storage modulus measured at 0.1 Hz in microvascular samples plotted as a function of microchannel volume fraction. Each data point represents the average of the sample set (indicated for each sample type in the legend). Theoretical predictions, based on the rule of mixtures and the HalpinTsai equation, are plotted as dashed lines. The elastic moduli obtained from dynamic mechanical analysis are compared with two theoretical predictions in Fig. 3. The theoretical curves were generated using either a rule of mixtures approximation or the HalpinTsai equation for particulate inclusions. For these calculations the elastic modulus of the matrix material was taken from the data plotted in Fig. 2. The Poisson’s ratio of the matrix material was estimated as 0.40 for the HalpinTsai prediction, based on previous studies of this epoxy system [8]. All of the vascular sample types fall within the range of elastic moduli bounded by these model predictions. The Type I and Type II samples elastic moduli are better approximated by the HalpinTsai prediction, while the Type III sample elastic modulus lays between the two theoretical curves. The deviation of the Type III sample from the HalpinTsai trend may be an effect of the square lattice arrangement of microchannels, as opposed to the rhombic lattice in the Types I and II samples. SINGLE EDGE NOTCH BEAM FRACTURE Single edge notch beams (SENB) loaded in threepoint bending were used to quantify the fracture toughness of the neat matrix material and of microvascular samples. The beam geometry detailed in Fig. 4 was determined based upon ASTM standard D504599 [9].
Fig. 4 SENB sample geometry and dimensions. Sharp precracks were initiated by loading the beams in cyclic fatigue until cracks extended from a notch that was scored with a razor blade. Precrack lengths were measured optically before monotonic loading to failure at a displacement rate of 10 μm/s. The average fracture toughnesses for the neat matrix material and vascular samples are tabulated in Table 2. Sample Plain Vascular (Type I)
1/2
KIC (MPam ) 1.18 ± 0.09 0.87 ± 0.08
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Crack propagation occurred at lower loads in vascular fracture samples compared with plain samples, resulting in the reduced fracture toughness listed in Table 2. However, the reduced bending stiffness of the vascular samples was not taken into account in the calculation of KIC. Furthermore, the effect of the initial position of the precrack relative to the microchannels has not yet been fully explored. CONCLUSIONS AND FUTURE WORK The elastic moduli of epoxy beams containing different microvascular network geometries were measured using dynamic mechanical analysis. The presence of microchannels reduced the overall flexural modulus of the material. The results are within the range of values predicted using the rule of mixtures and the HalpinTsai equation for particulate inclusions. Preliminary fracture testing indicates a drop in the fracture toughness due to the presence of a microvascular network, however more analysis and testing are required to clarify this effect. In a previous study, local reinforcement around a microchannel was shown to significantly reduce the level of strain surrounding a channel loaded in tension [10]. In the future, we intend to explore the effect of this locallyplaced reinforcement on the bulk mechanical properties of vascular materials. REFERENCES [1] Toohey KS, Sottos NR, Lewis JA, Moore JM, White SR (2007) Selfhealing materials with microvascular networks. Nat Mater 6:581585. [2] Toohey KS, Sottos NR, White, SR (2009) Characterization of microvascularbased selfhealing coatings. Exp Mech 49:707717. [3] Therriault D, White SR, Lewis JA (2003) Chaotic mixing in threedimensional microvascular networks fabricated by directwrite assembly. Nat Mater 2:265271. [4] Lewis JA (2006) Direct ink writing of 3D functional materials. Adv Funct Mater 16:21932204. [5] Hansen CJ, Wu W, Toohey KS, Sottos NR, White SR, Lewis JA (2009) Selfhealing materials with interpenetrating microvascular networks. Adv Mater 21:15. [6] Williams HR, Trask RS, Bond IP (2007) Selfhealing composite sandwich structures. Smart Mater Struct 16:11981207. [7] Williams HR, Trask RS, Bond, IP (2008) Selfhealing sandwich panels: Restoration of compressive strength after impact. Compos Sci Technol 68:31713177. [8] O’Brien DJ, Sottos NR, White SR (2007) Curedependent viscoelastic Poisson’s ratio of epoxy. Exp Mech 47:237249. [9] ASTM D504599. Standard Test Methods for PlaneStrain Fracture Toughness and Strain Energy Release Rate of Plastic Materials. Annual Book of ASTM Standards, Vol. 08.01: Plastics (I) (ASTM International, West Conshohocken, Pennsylvania, USA, 2003). [10] Hamilton AR, Sottos NR, White SR (2010) Local strain concentrations in a microvascular network. Exp Mech 50:255263.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
MgO nanoparticles affect on the osteoblast cell function and adhesion strength of engineered tissue constructs Morshed Khandaker and Kelli Duggan Department of Engineering and Physics, University of Central Oklahoma, Edmond, OK 73034 Melissa Perram Department of Chemical Engineering, Purdue University, West Lafayette, IN 47907
ABSTRACT The objective of this research was to evaluate the influence of magnesium oxide (MgO) on the adhesion strength between hard tissue and soft tissue constructs. The scope of works for this research were: (1) to determine the viability of osteoblast cells in hydrogel and hydrogel with 22 nm MgO particles, (2) to design and construct a test setup for the measurement of adhesion strength of engineered tissue constructs, and (3) to determine if MgO nanoparticles affect on the adhesion strength of the engineered tissue constructs. Mouse osteoblast cells (MT3T3E1) were cultured on polycaprolactone (PCL) scaffold, hydrogel scaffold, as well as hydrogel scaffold with 22 nm MgO particles. The viability of cells was determined in: hydrogel and hydrogel with 22 nm MgO particles. Tensile test were conducted on: (1) PCLhydrogel, (2) PCLhydrogel with cells, (3) PCLhydrogel with cells and MgO nanoparticles, and (4) PCLhydrogel with MgO nanoparticles to measure the adhesion strength between these hard tissue and soft tissue constructs. This research found the increase of osteoblast cells adhesion on hydrogel scaffold containing 22 nm MgO particles and decrease of adhesion strength between PCL and hydrogel, when both PCL and hydrogel were seeded with cells and 22 nm MgO particles. BACKGROUND Each year, there are thousands of new injuries to soft tissue due to high energy impact [1]. Most of these injuries occur where soft tissue meets hard tissue in the body [2]. Several engineered tissue grafts have been developed for the reconstruction of the injured hard and soft tissues [3]. In this research, Polycaprolactone (PCL) based hard tissue and hydrogel based soft tissue grafts have been investigated. PCL scaffold is made using a precision deposition system which allows for the scaffold to have precise pore sizes and porosity, promoting an increase in cell adhesion [45]. The hydrogel resembles the mechanical properties of tissues in the body [6]. It has porous structure which allows the cells to grow [7]. Both of these scaffolds are biodegradable and biocompatible in the human body and can be used for boneligament or bonecartilage reconstructions. The interface is the weakest place in the grafted boneligament or bonecartilage reconstructions. There has not been much research done on improving the interface strength of the engineered tissue constructs. A strong bond of engineered tissue constructs will be beneficial for future tissue engineering. If this interface could be strengthened it would have great medical applications. MgO nanoparticles have been shown to improve osteoblast cell adhesion on bone cement [8]. But the influence of MgO nanoparticles on the interface strength on engineered tissue constructs is still unknown. This study hypothesized that the MgO nanoparticles could improve the osteoblast cell adhesion on hydrogel scaffold and the adhesion strength between PCLhydrogel interfaces could be enhanced by the inclusion of nano sizes MgO particles on PCL and hydrogel tissue grafts. Both qualitative and quantitative cell viability test were conducted to examine the osteoblast cell adhesion on hydrogel scaffold. Mechanical test was conducted on PCLhydrogel seed with MgO nanoparticles specimen using a custom made tensile test setup to the influence of adhesion strength between PCLhydrogel interfaces.Materials and methods MATERIALS AND METHODS Cell viability tests Cells were cultured in cell culture flask, and hydrogel with different percentages of MgO nanoparticles (2%, 1%, 0.5%, and 0.1%) for the cell viability tests (Figure 1). The viability testing of cells was conducted in these mediums both quantitatively and qualitatively. A hemocytometer is used to count cell density and cell viability of cells cultured on the flask. Trypan blue was mixed with the cells. It dyed the dead cells blue. The cells were T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_48, © The Society for Experimental Mechanics, Inc. 2011
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counted using the hemocytometer. The viability was evaluated by dividing the living cells by the total number of cells. A live/dead kit was used for the cell viability of cells on hydrogel samples. The samples were viewed under confocal microscope (Olympus BX61W1) for the qualitative measurement. Two different dyes were used in the kit to detect the live and dead cells. They are calcein AM and ethidium. Calcein AM entered the live cells and under blue light the cells turn to green. Ethidium entered the dead cell’s nuclei and under green light, turns the cells to red. The ratio of live to dead cells was determined using the sliced confocal microscopic images. Fluorescent intensity tests were conducted on hydrogel with cells and hydrogel with nanoparticles using Fluostar Optima Microplate Reader (BMG Labtech) for determining the quantitative measurement the cell viability of cells on hydrogel samples.
Figure 1 Cell viability test on various kinds of hydrogel specimens: A. Control, Hydrogel with no cells. B. Hydrogel with cells C. Hydrogel with 2% nano D. Hydrogel with 1% nano E. Hydrogel with 0.5% nano F. Hydrogel with 0.1% nano. PCLHydrogel specimen preparation Mouse osteoblast cells (MT3T3E1) was obtained from the American Type Culture Collection (ATCC). The 5% sodium alginate was added to 5% CaCl2 to make hydrogel scaffold ((height ~1.5 mm, scaffold diameter~21 mm) independently in another well plate according to Lee et al. [9]. Osteoblast cells were cultured on hydrogel and TM hydrogel with MgO nanoparticles scaffolds according to ATCC instructions (Figure 2(a)). 3D Insert PCL scaffold scaffold (height ~1.5 mm, scaffold diameter~21 mm, fiber diameter ~300μ, fiber spacing~300μ) were purchased from 3D Biotek, LLC. The cells were cultured on the PCL well plate according to 3D Biotek instructions (Figure 2 (b to d)). The hydrogel scaffolds were carefully placed on top of the PCL and 3.9 Pascals of pressure was added on top using sterilized aluminum blocks.The hydrogel and PCL tissue constructs were cultured for 3 weeks.
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C (b) (c) (d) Figure 2 (a) Hydrogel samples in the sylgard mold, (bd) PCL samples at different time periods: 2 days, 1 week and 2 weeks, respectively.
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Design and manufacture of the setup and instrumentation A custom made three point bend tester was used for the tensile experiment of the PCLhydrogel samples. The complete setup is shown in Figure 3. A xyz stage was assembled with the test setup for microscopic viewing purposes of the samples. The PCLhydrogel samples were carefully glued with two holders as shown in figure. A 250 gram load cell (Futek™ LCM300, model number FSH02630) with a sensor (Futek™ IPM500) was fastened to the one end of the holder. The other end of the loadcell was connected with a high precision microactuator ® (Newport™ LTAHL actuator) and motion controller (SMC 100). All instruments were calibrated before testing.
Figure 3 Experimental setup. EXPERIMENT AND DATA ANALYSIS Four different types of PCLhydrogel tissue constructs were tested: PCLhydrogel, PCLhydrogel with cells, PCLhydrogel with cells and MgO nanoparticles, and hydrogel with MgO nanoparticles. The PCLhydrogel specimens were tested under moist condition using DMEM solution. The loading rate was 0.001 mm/s. The load and displacement data were continuously recorded until the failure of the PCLhydrogel joint. The load and GLVSODFHPHQW GDWD ZHUH SURFHVVHG WR GHWHUPLQH WKH FULWLFDO LQWHUIDFH WHQVLOH VWUHQJWK ıf, using the relationship [10]: ıf=Pcr/A, where Pcr is the maximum value of the loaddisplacement curve and A is the crosssectional area of 2 LQWHUIDFH$ ʌG /4). RESULTS Absorbance tests were conducted (Figure 4) to determine an adequate amount of MgO nanoparticles to add to the hydrogel to increase adhesive strength, while not decomposing the hydrogel itself. The 0.5% and 0.1% were most closely compatable with the hydrogel with cells, so it was determined that the 0.5% would be sufficient to use in the hydrogel.
Figure 4 Absorbance test on cell seeded hydrogel with different percentage of 22 nm MgO additives.
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Figure 5(a) compares the fluorescent intensity of hydrogel with cells and hydrogel with cells plus 22 nm MgO particles. Cell viability test shows that 36 μm MgO particles were detrimental to cell viability with 74% decrease in cells. Although increased cell adhesion of mouse osteoblast cells (MT3T3E1) to hydrogelbased soft tissue grafts with the use of 22 nm MgO particles. On the other hand, increased cell adhesion of mouse osteoblast cells (MT3T3E1) to hydrogelbased soft tissue grafts with the use of 22 nm MgO particles. The confocal microscope was used to see many different layers of the hydrogel and determine if the cells are living. Figure 5(b) shows that there are more alive cells (green color) than dead cells (red color). 6.0x10
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strength than PCLhydrogel with cells and MgO nanoparticles. However, PCLhydrogel with cells and MgO nanoparticles had the weakest tensile strength. This result showed that PCLhydrogel tissue constructs had the strongest tensile strength when only cells were present in the tissue constructs and the addition of nanoparticles decreased the adhesion between PCL and hydrogel. This research concludes that appropriate amount of MgO nano additives is required for the proper cell function of the PCLhydrogel tissue constructs. ACKNOWLEDGEMENTS This publication was made possible by 2009 OKEPSCoR summer research opportunity award. REFERENCES 1. Hutmacher, D.W., "Scaffolds in tissue engineering bone and cartilage," Biomaterials, 21(24), 25292543 (2000). 2. Lee, K.Y., Mooney D.J., "Hydrogels for tissue engineering," Chemical Reviews, 101(7), 18691879 (2001). 3. Sun, W., Darling A., Starly B., Nam J., "Computeraided tissue engineering: Overview, scope and challenges," Biotechnology and Applied Biochemistry, 39(1), 2947 (2004). 4. Kim, J.Y., "Fabrication of a sffbased threedimensional scaffold using a precision deposition system in tissue engineering," Journal of Micromechanics and Microengineering, 18(5), (2008). 5. Zein, I., "Fused deposition modeling of novel scaffold architectures for tissue engineering applications," Biomaterials, 23(4), 11691185 (2002). 6. Shor, L., "Fabrication of threedimensional polycaprolactone/hydroxyapatite tissue scaffolds and osteoblastscaffold interactions in vitro," Biomaterials, 28(35), 52915297 (2007). 7. Gleghorn, J.P., "Adhesive properties of laminated alginate gels for tissue engineering of layered structures.," Journal of Biomedical Materials Research  Part A, 85(3), 611618 (2008). 8. Ricker, A., LiuSnyder P., Webster T.J., "The influence of nano mgo and baso4 particle size additives on properties of pmma bone cement," International Journal of Nanomedicine, 3(1), 1251 (2008). 9. Lee, C.S.D., Gleghorn J.P., Won Choi N., Cabodi M., Stroock A.D., Bonassar L.J., "Integration of layered chondrocyteseeded alginate hydrogel scaffolds," Biomaterials, 28(Compendex), 29872993 (2007). 10. An, Y.H., Draughn R.A., Mechanical testing of bone and the boneimplant interface. CRC (2000).
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Mechanical Interactions of Mouse Mammary Gland Cells with a ThreeDimensional Matrix Construct M.d.C. LopezGarcia, Graduate Research Assistant, Material Science Program, Wisconsin Institutes for Medical Research D.J. Beebe, Professor, Department of Biomedical Engineering, Wisconsin Institutes for Medical Research W.C. Crone, Professor, Department of Engineering Physics University of Wisconsin – Madison, 1500 Engineering Dr, Madison, WI 53706 [emailprotected] Abstract One risk factor associated with breast cancer is tissue or mammographic density which is directly correlated with the stiffness of the tissue. We undertook a study of mammary gland cells and their interactions with the extracellular matrix in a microfluidic platform. Mammary gland cells were seeded within collagen gels inside microchannels, using concentrations of 1.3, 2, and 3 mg/mL, along with fluorescent beads to track strains in the gel. The cells and beads were observed via fourdimensional imaging, tracking X, Y, Z positions over a three to four hour time frame. Threedimensional elastic theory for an isotropic material was employed to calculate the stress. The technique presented adds to the field of measuring cell generated stresses by providing the capability of measuring 3D stresses locally around the single cell and using physiologically relevant materials properties for analysis. Introduction There are many risk factors related to breast cancer. One risk factor we would ultimately like to study is tissue or mammographic density. In a study by Boyd et al. it was found that women with a dense tissue are at four to six times greater risk of breast cancer than women with low density tissue [Friedl, et al, 2000]. This risk factor may account for as many as 30% of breast cancer cases. It was found that the increasing density in the mammogram was associated with increasing collagen and decreasing fat concentrations in the breast tissue [Cukierman, et al, 2001]. Increased density of this material correlates with an increase of stiffness in the tissue. Breast tissue is a multiphase material, in this case, mostly fat and collagen, exhibits the properties of both constituent phases. As the collagen concentration increases, the density of the tissue increases. Since the mechanical property of tissue stiffness increases with its collagen concentration [Jo, et al, 2000], we infer that the increase in density of the tissue is also related to the increase of stiffness in the material. This is one indicator of the role that mechanical properties has in the development of breast cancer. In the research presented here we study the influence that a normal murine mammary gland cell has on its surrounding environment when encapsulated in a collagen matrix. Three different stiffnesses of collagen matrix materials were used in the broader study [LopezGarcia, et al, in press]. In the research discussed below we will focus on results obtained in the 2 mg/mL collagen gel. Methods This study was carried out in a microfluidic platform composed of PDMS fabricated via a photolithography technique [Jo et al. 2000; McDonald and Whitesides 2002]. Typically two channels per dish were assembled. Figure 1i shows a diagram of the glass bottom Petri dish with the channels, and Figure 1iiiii shows the top and side views of the channel when seeded with the cells. T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_49, © The Society for Experimental Mechanics, Inc. 2011
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In brief, normal murine mammary gland cells transfected to express green fluorescent protein (GFPNMuMG) were seeded in a collagen matrix within the microfluidic device. All experiments were carried out at passages less than Pn+18. The microfluidic channel walls were coated with a collagen solution 50 Pg/mL in PBS a few hours before the experiment. The cells were then suspended in collagen concentrations of either 1.3, 2.0, or 3 mg/mL. To this mixture, 0.2 PL of red fluorescent 1 Pm carboxylate modified beads was added and mixed well. A 5 PL volume of the cellgel mixture was poured into the channel. The channels were then covered and incubated. A droplet of media was added to the ports to feed the cells. They were incubated overnight, for 17 hours. Figure 1iiii shows a diagram of cells and beads seeded into the channel.
Figure 1 i) Schematic diagram of a glass bottom Petri dish with a PDMS straight channel. ii) Close up of the top view of one channel. Green objects are GFP fluorescing cells and red objects are red fluorescing beads. The channel has an inlet and outlet port on each end. iii) Close up of side view of one channel. iv) Frame of reference is shown with respect to the side view. Bead b has negligible displacement caused by the cell and is the reference bead. Bead a is being displaced by the cell. Bead aƍ is the position of bead a at a later time point. In the direction of the z axis, the strain that the cell has caused can be calculated by the distance between beads a and b at the later time point, ǻzƍ, and dividing by the original distance between beads a and b, ǻz. This analysis is done for all three directions, but is only shown here in z for clarity.
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Results and Discussion Although a number of different measures were made in the full study [LopezGarcia, et al, in press], here we will focus on one particular aspect observation: the fluctuation in the elastic stresses over time which was observed for many bead pairs. To determine if this fluctuation was produced by the cell and not an artifact of the technique the cell shape was studied as well as the path shapes and the behavior of the stress peaks. The area and the perimeter for cells in each concentration were measured and compared to the peaks in the stress graph. It was found that the shape of the cell changes throughout the experiment, but no strong correlation could be found with the stress peaks. On occasion, as shown in Figure 2a, b, and c for a cell in a 2 mg/mL collagen gel, the area and/or perimeter coincide with the peaks, but not consistently. One limitation, however, is that area and perimeter measurements of the cell can only be made in the XY plane. The path of the bead was also compared to the stress peaks. In the example shown in Figure 2d the peaks occur at times when the cell is pulling towards itself. Time between peaks was calculated for various tests as well as the relationship of the time between peaks and distance from cell as well as collagen concentration but no clear correlation was found. (Data not shown.) Figure 2d also demonstrates how complex the displacement behaviors of the bead were throughout the observation period. The bead has backwards and forwards movement due to how the collagen was being displaced by the cell. This behavior was also seen in the Z direction. This information would not have been available if intermediate timepoints had not been obtained. In Figure 3 a cell moving in a 2 mg/mL gel is shown. The images shown are for the focal plane of the cell, the lines shown are paths of the beads that are adsorbed onto the collagen gel. There are 20 minutes between each image. The path between the first two timepoints is blue and change into turquoise, green, and then yellow as time progresses. The paths that are formed between the first and second timepoints tend in the opposite direction of the cell in the rear region; and they tend in the same direction of the cell in the frontal region. At this moment, the cell appears to move in a thrust between 0 – 20 minutes. During this thrust, the paths in the front are generally greater in length than the paths in the back of the cell. After the thrust, the paths in the front and in the back are of a similar size, and in the same direction. Conclusion In this work the strains generated by NMuMG cells in a three dimensional collagen matrix were measured using a microfluidic platform by tracking the displacements of beads that were embedded in the collagen. Fluctuations in the stress behavior over time were observed. Displacement of the material determined from the complex paths of the beads throughout the time of observation were also measured. This powerful tool can give way to future works in understanding 3D mechanical interactions between cells and their surrounding ECM and is an important step in that direction. Acknowledgements The authors would like to thank Patricia Keely’s lab for allowing the use of their microscope, especially Dr. Matt Conklin for his time and assistance and Prof. Keely for helpful conversations. The authors would also like to thank Erich Zeiss for all of his help with the imaging and use of Slidebook Software. This work was possible via funding from the Harriet Jenkins Predoctoral Fellowship (JPFPNASA); Graduate Engineering Research Scholars (GERS) of the College of Engineering, University of Wisconsin – Madison; and the Ruth Dickie Research Scholarship from the University of Wisconsin Beta Chapter of Sigma Delta Epsilon  Graduate Women in Science (SDEGWIS).
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Figure 2 a) Graph shows the stresses generated by a cell in a 2 mg/mL gel. The cell b) area and the c) perimeter were measured at each time point and used to calculate the shape factor. The area, perimeter, and shape factor were used to compare to the stress behavior. On the top right c) the path in X and Y for the bead being studied is shown. The cell is in the direction in which the green arrow is pointing. The pulses in the stress graph all coincide with the times at which the cell is pulling the material marked by that bead towards itself.
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Figure 3. Cell moving in a 2 mg/mL collagen gel. Paths of beads adsorbed onto collagen are shown. The progression of the path is in chronological order by color: blue, turquoise, green, yellow. Each path section represents a 20 min interval. White arrow indicates direction of cell movement. References Cukierman, E., et al., Taking cellmatrix adhesions to the third dimension. Science, 2001. 294(5547): p. 17081712. Friedl, P. and E.B. Brocker, The biology of cell locomotion within threedimensional extracellular matrix. Cellular And Molecular Life Sciences, 2000. 57(1): p. 4164. Jo, B.H., et al., Threedimensional microchannel fabrication in polydimethylsiloxane (PDMS) elastomer. Journal Of Microelectromechanical Systems, 2000. 9(1): p. 7681. LopezGarcia, M.d.C., D.J. Beebe, and W.C. Crone, “Mechanical Interactions of Mouse Mammary Gland Cells with Collagen in a ThreeDimensional Construct,” accepted for publication in Annals of Biomedical Engineering, March 2010. McDonald, J.C. and G.M. Whitesides, Poly(dimethylsiloxane) as a material for fabricating microfluidic devices. Accounts Of Chemical Research, 2002. 35(7): p. 491499.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Tracking nanoparticles optically to study their interaction with cells JeanMichel Gineste1,Peter Macko1, Eann Patterson2 and Maurice Whelan1 1
Institute for Health and Consumer Protection, European Commission DG Joint Research Centre, Italy. 2 Composite Vehicle Research Center, Michigan State University, East Lansing, MI48824, USA. [emailprotected] ABSTRACT Nanoparticles are by definition too small to be visible in an optical microscope and devices such as scanning electron microscopes must be used to resolve them. However electron beams quickly lead to cell death and so it is difficult to study the interaction of nanoparticles with living cells in order to establish whether such interactions could be damaging to the cell. A simple modification to a conventional inverted optical microscope is proposed here which renders the location of nanoparticles readily apparent and permits tracking of them in threedimensions. Particles in the range 100nm to 500nm have been tracked with a temporal resolution of 200ms. The technique, although motivated by the desire to study the interaction of nanoparticles with cells, has a wide range of potential applications in the fields of food processing, pharmaceuticals and nanobiotechnology. 1. INTRODUCTION The Rayleigh limit defines the minimum resolution of an optical microscope based on the wavelength of light being employed so that most nanoparticles of interest are not resolvable in a conventional optical microscope. Nanoparticles are of interest for wide range of applications from colloidal systems in which they control the behavioral characteristics to cell biology where they may be associated with toxic effects. Consequently, significant research effort has been expended on inventing ways to image nanoparticles, identify their location and track their movement in an environment that permits the study of their interaction with living cells. Metallic particles are relevant easy to image since many of them can be designed to exhibit fluorescence and, or to produce large scatter patterns using darkfield microscopy. However, fluorescentbased techniques tend to suffer from photobleaching and darkfield microscopy works best in confocal microscopes where particles of diameter 80 to 180nm can be readily imaged [1, 2]. Metallic particles can be induced to generate both a plasmon resonance and a photothermal signature when excited by laser and then particles as small as 5nm diameter can be imaged using a scanning sensor [3]. An alternative approach is to defocus the microscope which magnifies the diffraction effects that most microscope users go to considerable effort and expensive to eliminate. These diffraction effects are based on forward scattering of the light from the particles which Ovryn [4] in 2000 showed could be modeled for microspheres of diameter 5 to 10um using Mie theory. In 2006, GuerreroViramontes et al. [5] demonstrated the approach by tracking glass particles of diameter 20 to 30um using a 70mW HeNe laser. In 2007 Kvarnstrom and Glasby used defocusing combined with local quadratic kernel estimates to locate particles of diameter 500nm and in the same year Toprak et al. [6] implemented a similar approach in a bifocal microscope and successful tracked particles of diameter 200nm. Then in 2008, Patterson and Whelan [7, 8] defocused a simple optical inverted microscope and demonstrated that the location of both nominally opaque metallic particles and nominally transparent particles as small as 3nm in diameter could be identified. While finding the location of large nanoparticles using defocusing can be relatively straightforward, tracking them in threedimensions is more challenging and the forward scattering from a small nanoparticle can be very faint so that even locating them can be problematic. The technique described here has been developed to overcome these difficulties without adding significantly to the complexity of the optical microscope or interfering with other capabilities which are of interest to those wanting to study cells. T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_50, © The Society for Experimental Mechanics, Inc. 2011
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Figure 1: Image (top left), zstacks for the xz (top right) and yz (bottom left) planes and a composite threedimensional composite view (bottom right) for 300nm diameter polystyrene particles obtained by collecting images at 1um steps of the objective over 30um in the zdirection along the light path. [2008_01 experiments\tuesday22jan]
2. METHODOLOGY The forward scatter pattern observed when small particles are viewed a conventional optical microscope is essentially a threedimensional interference pattern consisting of a series of light and dark fringes as can be seen in the example in figure 1. The amplitude of this pattern is strongest along the light path (zdirection) through the center of a spherical particle. Thus, if the focal plane of the microscope objective is moved along this line then the pixels at the x,y position of the center of the particle would be expected to flash between bright and dark. This was arranged to occur by fitting a piezo actuator (Physik Instrumente Gmbh & Co. KG. model Pifco P725.4CD) with a servo controller (Physik Instrumente Gmbh & Co. KG. model E665) to an inverted optical microscope (Olympus IX71) with 100W halogen light source (Olympus ULH100L3) and narrow bandwidth filter (Olympus 43IF550W45). The microscope was fitted with an adjustable field aperture which was closed to its minimum diameter of 1mm in order to maximum the diffraction effects. The piezo actuator was oscillated with a peaktopeak amplitude in the range 100um to 150um about a mean position that corresponded to the approximate center of the particle of interest. For pseudostatic investigations images were captured at 20 frames/second using a 12bit monochrome CCD camera (Hamamatsu C848405G) with a resolution of 1344u1024 pixels and for tracking motion a highspeed, 8bit monochrome camera (Photron FASTCAMPCI R2) was used with at a frame rate of 500 images/second and resolution of 256u240 pixels. These cameras when combined with a u60 objective and u10 lens on the camera port gave a pixel size of 0.1082um and 0.1272um respectively at the objective plane. Polystryrene particles of nominal diameters 100nm and 500nm (L06551ML and L055301ML respectively Aminemodified polystyrene, fluorescent blue, SignaAldrich Inc., St Louis, MO) were used in this investigation and were diluted 1000fold in phosphate buffered saline solution. Conventional glass microscope slides generate large diffraction patterns from the relatively low quality glass surface when used in a microscope set up in this mode and so the solution was sandwiched between glass slide covers (MenzilGläser, 21u23mm #4). The particles flash in realtime which allows the location of particles to be identify manually with ease. To allow automatic data processing a series of images were collected over several cycles of oscillation of the objective and the mean and standard deviation of the intensity as a function of time were calculated for each pixel in the field of view. The local maxima in the map of standard deviations were taken as the center of particles in the xyplane and for these pixels the location of the particle in the zdirection, i.e. along the light path was identified by scanning along the zdirection to find the area of maximum intensity and the edge of it furthest from the objective was taken as the center of the particle as indicated in the model of Ovryn [4]. This process was repeated every 200ms when tracking a particle so that the x, y, z coordinates could be found as a function of time.
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3. RESULTS A typical image obtained with a static but defocused objective is shown in figure 2 together with the timevarying intensity signal for the center of a particle of nominal diameter 500nm over three half periods of oscillation of the objective lens. A number of particles are apparent in the image and are at different depths in the solution and hence their forward scatter patterns appear to be different sizes because they are being sectioned at different distances from the center of the particles. The maps of standard deviation for solutions of 500nm and of 100nm particles are shown in figure 3 and the centers of the particles are distinctive against the background. In these grey scale maps the minimum standard deviation is close to zero and is shown white while the maximum standard deviation is black and occurs at the center of the particle. The size and intensity of the signature of a particle in these maps is both a function of the particle diameter but also its position relative to the center of oscillation of the objective. A particle whose forward scatter pattern is completely enveloped by the oscillation of the objective will appear much more distinct, such as the 500nm particle in the top left of the image, than one for which only part of the forward scatter pattern lies within the oscillation amplitude of the objective, such as in the bottom center and right of the image from the 500nm sample.
Figure 3: Greyscale maps of the standard deviation of intensity as a function of time over five cycles of the motion the objective for solutions of 500nm (left) and 100nm (right) diameter particles.
The results in figure 4 were obtained using the high speed camera to track the Brownian motion of a 500nm diameter particle using an objective amplitude of 100um. When identifying the particle center in the zdirection a 3u3 patch of data centered on the particle center in the xyplane was used rather than a single line of pixels in the zdirection. This strategy made the algorithm considerable more robust and less susceptible to noise and do not attenuate the resolution of the x, y or z coordinates.
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Figure 4: A typical trajectory (in black) of a 500nm particle illustrating the longterm, high resolution 3d tracking of the nanoparticle (in red and in blue are respectively the projections of the 3d trajectory onto the xy and yz planes).
4. DISCUSSION AND CONCLUSIONS The results demonstrate that nanoparticles of the order of 100nm diameter can be readily tracked in three dimensions with respect to time with a temporal resolution of 200ms using an optical microscope with only a small modification that does not interfere with other functions of the instrument. This is a significant advance on other techniques for tracking in threedimensions which require bifocal arrangements [6] or scanning sensors [1 – 3]. When used without the data processing, i.e. in manual mode but with the objective oscillating, the nanoparticles flash in the field of view rendering them far more apparent than in a simple defocused optical microscope [7, 8]. Consequently the technique has considerable potential in nanobiotechnology, nanoengineering, pharmaceuticals and food processing where introducing fluorescent particles or creating plasmons may not be compatible with the processes under observation. REFERENCES 1. Xu, C.S., Cang, H., Montiel, D., Yang, H., Rapid & quantitative sizing of nanoparticles using threedimensional single particle tracking, J Phys. Chem., 111:3235, 2007 2. Louit, G., Asahi, T., Tanaka, G., Uwada, T., Masuhara, H., Spectral and 3Dimensional tracking of single gold nanoparticles in living cells studied by Rayleigh light scattering microscopy, J Phys. Chem. C.,113:1176611772, 2009. 3. Lasne, D., Blab, G.A., Berciaud, S., Heine, M., Groc, L., Choquet, D., Cognet, L., Lounis, B., Single nanoparticle photothermal tracking (SNaPT) of 5nm gold beads in live cells, Biophysical Journal, 91:45984604, 2006. 4. Ovryn, B., Threedimensional forward scattering particle image velocimetry applied to a microscope fieldofview, Experiments in Fluids [Suppl.], S175184, 2000. 5. GuerreroViramontes, J.A., MorenoHernandez, MendozaSantoyo, F., FunesGallanzi, M., 3D particle positioning from CCD images using the generalised LorenzMie and HuygensFresnel theories, Meas.Sci. Technol., 17:23282334, 2006. 6. Toprak, E., Balci, H., Blehm, B.H., Selvin, P.R., ‘Three dimensional particle tracking via bifocal imaging’, Nano Letters, 7(7):20432045, 2007. 7. Patterson, E.A., Whelan, M.P., Tracking nanoparticles in an optical microscope using caustics, Nanotechnology, 19:105502, 2008. 8. Patterson, E.A., Whelan, M.P., Optical signatures of small nanoparticles in a conventional microscope. Small, 4(10):17031706, 2008.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Coherent Gradient Sensing Microscopy: Microinterferometric Technique for Quantitative Cell Detection
M. Budyansky, C. Madormo, G. Lykotrafitis1 Department of Mechanical Engineering, University of Connecticut 191 Auditorium Road, UNIT 3139, StorrsMansfield, CT, 06269 1 [emailprotected] ABSTRACT MicroCGS, an integration of the interferometric optical method Coherent Gradient Sensing (CGS) [13] with an inverted microscope, is introduced. MicroCGS extends the capabilities of Classical CGS into the cellular level. As such, it provides fullfield, realtime, microinterferometric quantitative imaging. MicroCGS is based on the introduction of a microscope objective into the classical CGS setup. An experimental setup is detailed and resulting interferograms are shown. A digital image processing program is created to compute specimen curvature from the given fringe patterns. The experimental method is validated by the successful measurement of the curvature of 30micron glass microspheres. KEYWORDS: Microinterferometer, Fringe pattern analysis, Optical system, Image processing, Phase INTRODUCTION A majority of the optical techniques currently used in microscopy, such as Differential Interference Contrast (DIC), DarkField (DF), and Phase Contrast (PC) microscopy, are qualitative in nature. Recently, there has been an emergence of quantitative optical techniques that obtain spatial and temporal information from living cells [45]. In this framework, we developed microCGS which offers an increased stability and direct measurement of the surface curvature of microstructures. Classical CGS is a wellestablished largescale quantitative optical interferometric technique which has been successfully used in the field of fracture mechanics as well as the measurement of residual stresses in thin films [1, 3]. CGS can be used in two operating modes, transmission and reflection. The working principle of transmission mode CGS is shown in Figure 1.
Fig. 1: Working principle of CGS
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_51, © The Society for Experimental Mechanics, Inc. 2011
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A collimated laser beam, or a nearly planar wavefront, passes through a specimen. Due to the refractive index of the specimen the planar wavefront is deformed. This deformation is directly correlated to the thickness of the specimen, assuming a constant refractive index throughout the specimen. Two highdensity diffraction gratings, G1 and G2, laterally shear the beam. The principle axis of the diffraction gratings is along the x2 direction. When the incident wavefront, S(x1, x2), encounters the first diffraction grating, G1, it is diffracted into several wavefronts E0, E1, E1, etc. For aesthetic purposes only E1, E0, E1 are shown. The resulting wavefronts are again diffracted at the second diffraction grating, G2. This produces an additional set of wavefronts, E1,1, E1,0 E1,1, …, from E1, E0,1, E0,0, E0,1, …, from E0, and E1,1, E1,0, E1,1, …, from E1, etc. A focusing lens is then used to combine parallel wavefronts at the filter plane. A diffraction pattern is produced consisting of several diffraction spots D1, D0, D1, etc. The single diffraction spot D1 is isolated because it contains information on the curvature of the specimen [1, 3]. This diffraction spot is filtered out and imaged by a CCD camera. EXPERIMENTAL SETUP MicroCGS is based upon the introduction of a microscopic objective into the Classical CGS experimental setup. A design schematic of the MicroCGS setup can be seen in Figure 2. A 635nm HeNe collimated laser is spatially filtered and passed through a specimen. A microscope objective is introduced into the optical path for magnification purposes. The objective used is an Olympus 40x objective. HeNe 635 nm Laser Collimator Spatial Filter Sample Stage
X2  Orientation Focusing G1 & G2 Lens
Objective
D+1 Filter CCD Camera Plane Image Plane
Collimating Lens Mirror Expander Fig. 2: MicroCGS schematic Upon exiting the microscope objective, the beam is recollimated, expanded (10X) and directed through the pair of Ronchi diffraction gratings (1000 lines/inch) as in the classical CGS setup. A CCD camera, Casio EXF1, focused onto the image plane is employed to capture the generated interferograms. The experimental optical setup is shown in Figure 3. The orientation of the diffraction gratings dictates the axis along which the setup will measure curvature. For the current setup the gratings are perpendicular to the x2 direction and thus they provide surface gradient information along the x2 axis. In order to acquire a complete description of the sample curvature, a second optical path with the pair of gratings oriented perpendicular to the x1 direction has to be implemented.
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Fig. 3: Optical setup of MicroCGS utilized in the experimentation IMAGE PROCESSING The software utilized to postprocess the fringes was coded in MATLAB 2009. The algorithm begins by reading in the interferogram captured by the CCD camera [1]. The CCD camera records a pixel by pixel intensity distribution of the interferogram shown in Figure 4a. This interferogram corresponds to the x2 axis surface gradients of a 75mm Planoconvex lens. The intensity distribution of the CGS interferogram can be described by Eq. 1 I x1, x2
a x1, x2 b x1, x2 cos G x1, x2
(1)
where I x1, x2 is the intensity of the fringe pattern at point x1, x2 , a x1, x2 is the background intensity level, b x1, x2 is the fringe visibility, and G x1, x 2 is the phaseangle term [1].
It can also be expressed in a complex form as Eq. 2 I x1, x2
where c x1, x2
a x1, x2 c x1, x2 c * x1, x2
(2)
1 iG x ,x b x1, x2 e 1 2 and c * Z1,Z2 is the complex conjugate of c Z1,Z2 . 2
(a)
(b)
x2 x1 Fig. 4: (a) Interferogram for 75mm diameter Planoconvex lens, (b) corresponding frequency power spectrum after converting to Fourier domain
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The program then converts the image to the Fourier domain by employing FastFourier transform. The resulting frequency domain can be seen in Figure 4b. The intensity distribution in the Fourier domain is shown as Eq. 3 I Z1,Z2
A Z1,Z2 C Z1,Z2 C * Z1, Z2
(3)
where Z1 and Z2 are the spatial frequencies in the Fourier domain, A Z1,Z2 contains lowfrequency information, and C * Z1, Z2 is the complex conjugate of C Z1,Z2 . Bandpass filtering is used in order to eliminate A Z1,Z2 and either C Z1,Z2 or C * Z1, Z2 . The inverse Fourier transform of the remaining term is taken to obtain c Z1,Z2 or c * Z1,Z2 . The phase can then be found by using the following expression
G x1, x2
tan1
Im ª¬c x1, x2 º¼
(4)
Re ª¬c x1, x2 º¼
This is the fullfield wrapped phase as seen in Figure 5a. A single column representation is provided in Figure 5b to display values oscillating between S and S .
(b)
Wrapped Phase
(a)
Fig. 5: (a) Full field wrapped phase of Fig. 4a, (b) Phase in radians vs. distance on yaxis of the Fig. 5a in pixels After obtaining the wrapped phase, a phase unwrapping algorithm is used to unwrap the phase shown in Figure 5a. The resulting full field unwrapped phase is displayed in Figure 6a, accompanied by a single column representation shown in Figure 6b.
(a)
x1
Unwrapped Fringe Order
x2
(b)
Fig. 6: a) Full field unwrapped phase, b) Unwrapped fringe order vs. distance on yaxis of Figure 6a in pixels
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This particular case established linearity in fringe order for the planoconvex lens which has constant curvature. To approximate the curvature N 22 of the specimen, the unwrapped phase is differentiated with respect to x2 direction according to Equation 5 [13] for CGS in transmission.
N 22 
w 2S( x1, x2 ) wx22
U wn
(5)
2' wD
where ȡ is the grating pitch, wn is the change in the number of fringes, ǻ is the distance between the diffraction gratings, and wD is the distance between the fringes. Finally, a thickness profile is generated to obtain the thickness profile along the same direction as the curvature by integrating the unwrapped phase [6] RESULTS A sample of 30micron glass microspheres (±1%), manufactured by ThermoScientific, in aqueous solution were a) imaged with the aforementioned experimental setup. The corresponding fringe interferogram of each bead captured in the CCD camera is shown in Figure 7a. The interferograms display a predicted constant curvature through the equidistant straight parallel fringes obtained about the y axis surface gradients. In addition, adjacent beads exhibit nearly the same fringe pattern and fringe spacing signifying identical curvature detection between these glass beads. For each of the labeled beads a rectangular cross section was isolated from Figure 7a, and analyzed by the currently in development image processing software. The results were averaged and presented in Figure 7b showing a calculated curvature ratio between numerical vs. analytical solutions N 22 / N bead . The average of the four beads has an average standard deviation of approximately 2.0% between the four beads. The numerical curvature values were also produced with an average difference of 1.5% from the analytical value (1/radius of curvature). The curvature was nearly constant along the x2axis in the form of a reasonably flat plane in the field of the bead as exhibited in Figure 7b.
2
30 μm 1 x2
3
(b) Normalized Curvature μm1
(a)
4 x1 Fig. 7: a) Interferogram captured for 30micron glass beads in aqueous solution b) Curvature ratio of calculated vs. analytical bead curvature N 22 / N bead
CONCLUSION A proposed adaptation to the optical method of Coherent Gradient Sensing (CGS), known as microCGS, has been presented. The constructed experimental setup was employed to produce interferograms from microbeads of known curvature. The preliminary curvature results obtained for a set of 30 micron beads produced an average error of 1.5% with respect to the known value. It is believed that with further refinements to the digital image processing program this result will become more accurate. The accuracy of the results signifies that microCGS can be used to successfully obtain curvature of static microspecimens. In addition, the optical system produced repeatable fringe interferograms within the beads, signifying the successful detection capability of microCGS. Moving forward, the intention of the research is to further extend the capabilities of microCGS to cellular specimens. By characterizing the thermally induced surface vibrations of biological cells, their mechanical properties can be then determined.
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ACKNOWLEDGEMENTS We would like to acknowledge Ares J. Rosakis of the California Institute of Technology for very helpful discussions on Coherent Gradient Sensing and providing us with the Ronchi gratings used in the experimental setup. This work was supported by UCRF Large Grant J980. REFERENCES 1. Lee H, Rosakis AJ, & Freund LB, Fullfield optical measurement of curvatures in ultrathinfilmsubstrate systems in the range of geometrically nonlinear deformations. Journal of Applied Physics 89(11):61166129. (2001). 2. Lee YJ, Lambros J, & Rosakis AJ, Analysis of Coherent Gradient Sensing (CGS) by Fourier optics. Optics and Lasers in Engineering 25(1):2553. (1996). 3. Rosakis AJ, Singh RP, Tsuji Y, Kolawa E, & Moore NR, Full field measurements of curvature using coherent gradient sensing: application to thin film characterization. Thin Solid Films 325(12):4254. (1998). 4. Ikeda T, Popescu G, Dasari RR, & Feld MS, Hilbert phase microscopy for investigating fast dynamics in transparent systems. Optics Letters 30(10):11651167. (2005). 5. Lue N, et al., Live cell refractometry using Hilbert phase microscopy and confocal reflectance microscopy. J Phys Chem A 113(47):1332713330. (2009). 6. Lambros J & Rosakis AJ, An experimental study of dynamic delamination of thick fiber reinforced polymeric matrix composites. Experimental Mechanics 37(3):360366. (1997).
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
37,/97:/2+@/+/==>97+>/2/ 7/2+83+6 98C 7/>/9/ >2+> 3= ,/63/@/. >9 ,/ 90 +=>/0/>93=96+>/+8C/00/>=90=:/37/8+83=9>2/=:/37/8 A/9 /8=?2+> >2/ 0+/= A/238 7 +2/ =:/37/8 $:/37/8= A/C90 517 +8.+:9C90 T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_52, © The Society for Experimental Mechanics, Inc. 2011
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869+;371 =2/ ;/5=+ 0 for aluminium. This behaviour can be understood in such a way that the amount of the elastic deformation becomes smaller in respect to the fraction of plastic deformation on total strain, i.e. flow stress and thus the critical resolved shear stress are reduced, which leads to activation of dislocation movement even at low strains. This behaviour is in accordance with the observations made by Paton et al. [8]. They note that the critical resolved shear stress of Dtitanium alloyed with different amounts of aluminium reduces rapidly at temperatures above 600°C. A summary of ED as a function of temperature and stress is given in figure 7. 120
room temperature
110 350°C
ED [GPa]
100
450°C
90 80
550°C
70 650°C
60 50 600
400
200
200
400
600
Stress [MPa]
Figure 7: Calculated dependence of elastic modulus ED on stress and temperature 3.4 Change of Unloading Stiffness during Fatigue Life Figure 3 shows that at 650°C the Young’s modulus decreases steadily. This behaviour also effects the values of + + k and k as will be shown in this section. At low numbers of fatigue cycles, the absolute values of k and k are + about the same. At about half of the fatigue life (i.e. some 600 cycles for the specimen investigated) k becomes smaller, while the absolute value of k rises. The behaviour of the two constants is shown in figure 8. Some selected hysteresis loops are shown in figure 9. At 100 and 600 cycles the shape of the hysteresis loops is similar, however, due to the reduced Young’s modulus the nonlinear effects have to be compensated through different kfactors.
Absolute Values of k [TPa]
5 4 3 2 Reihe2 k+
1
kReihe3 0 0
200
400
600
800
1000
1200
Number of Cycles +
Figure 8: Evolution of the parameters k and k with number of cycles, T = 650°C, 'H/2 = 0.7%
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Figure 9: Measured (a) and kcompensated (b) hysteresis loops of Ti6242 for N = 100, 600 and 1000 cycles, T = 650°C, 'H/2 = 0.7% 4 SUMMARY and CONCLUSIONS The alloy investigated, Ti6242, posses a high yield stress and a relatively low Young’s modulus. Therefore, the deviation from the equilibrium atomic spacing at high elastic strains is relatively large, resulting in a distinct deviation of the linear Hooke’s law. The stiffness behaviour has been evaluated during unloading from compressive and tensile peak stresses. The study has shown that the nonlinear behaviour is different in tension and compression. It also changes with temperature. Up to 350°C one can observe constant values for the Young’s modulus E0 and the nonlinear compensation factor k during fatigue life. However, in particular at 650°C increased thermally activated dislocation glide and grain boundary softening result in a continuous change of the kvalues during fatigue life. The results suggest that the description of nonlinear elastic effects through an extension of Hooke’s law by a quadratic term lead to a reasonable description of the material’s behaviour. Yet, it cannot be expected that this value is independent of (un)loading situation and temperature. REFERENCES [1] Mehmke R: Zum Gesetz der elastischen Dehnungen. Z Math Phys 42, 327338, 1897. [2] Sommer C, Christ HJ, Mughrabi H: NonLinear Elastic Behaviour of the Roller Bearing Steel SAE 52100 during Cyclic Loading. Acta Met Mat 39, 11771187, 1991. [3] Boyer R, Welsch G, Collings EW: Materials Properties Handbook: Titanium Alloys, Materials Park, OH: ASM International. 337375, 1994. [4] Heckel TK, Guerrero Tovar A, Christ HJ: Fatigue of the NearAlpha TiAlloy Ti6242. Exp Mech, in print, doi: 0.1007/s1134000992385, 2010. [5] Rösler J, Harders H, Bäker M: Mechanical Behaviour of Engineering Materials. Berlin: Springer, 2007. [6] Singh N, Gouthama, Singh V: Low Cycle Fatigue Behavior of Ti Alloy Timetal 834 at 873 K, Int J Fatigue 29, 843851, 2007. [7] Hörnqvist M, Karlsson B: Development of the Unloading Stiffness during Cyclic Plastic Deformation of a HighStrength Aluminium Alloy in Different Tempers. Int J Mat Res 98, 11151123, 2007. [8] Paton NE, Williams JC, Rauscher GP: The Deformation of DPhase Titanium. In: Jaffee RI, Burte HM, editors. Titanium ’72 Science and Technoloy, New York, NY: Plenum Press. 10491070, 1973.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Numerical and Experimental Modal Analysis Applied to the Membrane of Micro Air Vehicles Pliant Wings
Uttam Kumar Chakravarty Postdoctoral Fellow, School of Aerospace Engineering, Georgia Institute of Technology Atlanta, GA 30332 and Roberto Albertani Affiliate Research Assistant Professor, Department of Mechanical and Aerospace Engineering University of Florida, Research and Engineering Education Facility, Shalimar, FL 32579 Email: [emailprotected]
ABSTRACT Hyperelastic latex membrane is an integral structure of micro air vehicles and plays an important role in their wings performance. This paper presents finite element analysis (FEA) models for characterizing latex hyperelastic membrane at both static and dynamic loadings, validated by experimental results. The membrane at different pretension levels are attached with a circular steel ring and statically loaded using steel spheres of different sizes placed at the center of the membrane. The deformation of the membrane is measured by visual image correlation (VIC), a noncontact measurement system and strain energy is calculated based on MooneyRivlin material model. It is found that the deflection and strain energy of the membrane computed by experimental and FEA models are correlated well, although discrepancy is expected among experimental and FEA results within reasonable limits due to the variation of the thickness of the membrane. The experimental modal analysis is conducted by imposing a structural excitation to the ring for investigating the membrane vibration characteristics at both atmospheric pressure and reduced pressure in a vacuum chamber. The threedimensional shape of the membranes during a burstchirp excitation at different pretension levels is dynamically measured and recorded and the natural frequencies are computed by performing the fast Fourier transform of the outofplane displacement at several points of the circular membrane. Experimental results show that the natural frequencies increase with mode and pretension of the membrane, but decrease due to increase in ambient pressure. A preliminary FEA model is developed for the natural frequencies of the membrane at different isotropic and nonisotropic pretension levels at vacuum environment. Keywords Micro air vehicle (MAV) · Hyperelastic membrane · Finite element analysis (FEA) · Visual image correlation (VIC) · Natural frequencies · Damping ratio INTRODUCTION The design and operation of micro air vehicles (MAVs) of similar proportions to natural fliers emphasizes the intricate but vital aeroelastic features mastered by biological systems. A particular form of these enhanced flying abilities benefit from the use of flexible lifting surfaces: either fixed or flapping. Birds and bats twist and bend their wings while maneuvering for optimal aerodynamics. Locusts use specialized dome shaped sensory organs (campaniform sensillae) within the structure of their wings [1]. These feedback sensors respond specifically to wing deformation in order to trigger the wing structure to operate at resonance frequency. Furthermore, bats can control their wing characteristics by changing the level of pretension in their wing membrane, thus effectively changing the wing camber and the passive aeroelastic dynamic feedback of the membrane to the aerodynamic loading [2]. Clearly, wing stiffness distribution and flexibility are essential aspects when considering natural fliers. The wellknown class of materials which function in supportive systems through deformation have been classified by T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_75, © The Society for Experimental Mechanics, Inc. 2011
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biologists as pliant materials and include proteins, soft connective tissues and cartilage. Recently there has been significant progress in the understanding of the aerodynamics of low Reynolds number artificial and natural flight [3], though the structural intimate mechanical behavior is still under early numerical modeling efforts. The aerodynamic models and flight control design of fixed [3] and flapping wings [4] must include the wing flexibility and structural dynamics, an area where very little experimental data is available. Many researchers [58] examine added mass effects on structural vibration. Yadykin et al. [5] review the added mass effects on a plane flexible plate oscillating in a fluid. Modal analysis of a taught, triangular membrane is given by Sewall et al. [9]. The membrane is excited with a shaker, and modal parameters are measured with an eddy current probe. Results are given with membrane pretension as a variable for vibration in both air and in a vacuum. Gaspar et al. [10] give similar data for a square membrane, with the eventual goal of progressing research in gossamer spacecraft. Excitation is provided with both an automated impact hammer and a shaker, and measurements are made with a scanning laser Doppler vibrometer (LDV). Modal parameters are reported as a function of membrane prestress and excitation locations. Jenkins and Korde [11] provide a comprehensive review of experimental analysis of membrane vibrations, as well as their own research on a circular membrane excited by a small audio speaker located at the membrane’s center. A laser vibrometer is used to garner data, with good agreement to analytical results for highly localized mode shapes caused by the small speaker. Graves et al. [12] present dynamic deformation measurements of a MAVRICI semispan wing in a wind tunnel. Measurements from select locations over the wing are made with a high speed videogrammetric system operating at 60 Hz during flutter and limit cycle oscillations. Spectral analysis of the data indicates the prevalent wing modal frequencies. Similar work is presented by Burner et al. [13], with an extension of the videogrammetric technique to inflight dynamic measurements. Finite element simulations to explain the structural behavior of natural and artificial fixed and flapping wings is at an early stage of development. The large amplitude of wing deformations, the nonlinear interaction with the flow, and the lack of quantitative experimental results for validation limits the numerical models’ applications. Critical experimental work showed the dependency of modal characteristics of microstructures with ambient pressure [14]. Experiments were conducted in the wind tunnel on a membrane sheet stretched between two rigid posts at a variety of angles of attack and Reynolds numbers [2]. Shape and strain measurements using visual image correlation (VIC) and aerodynamic coefficients evaluations for different configurations of membrane MAV wings in the wind tunnel have been performed in steady conditions [15, 16]. The flight performance of MAVs with flexible wings, shown in Fig. 1, has indicated several desirable properties directly attributable to the elastic nature of the wing: primarily, passive shape adaptation. Such adaptation also introduces a higher level of uncertainties in the structural dynamics. The object of this work is to present results from experimental modal analysis techniques to evaluate the basic parameters of latex membrane of MAV thin flexible wing structures. This is an ongoing research project and as the first step, the latex membrane at different pretension levels is attached with the circular steel ring, as illustrated in Fig. 2, and characterized at static and dynamic loading environments. Different sizes spheres are placed at the middle of the membrane as the static load as shown in Fig. 2(b). The steel spheres are painted for avoiding the light reflection from the surface of the spheres so that deformation of the membrane specimen can be captured properly. The deflection is recorded using noncontact measurement technique and strain energy of the membrane is computed. The membrane is also vibrated for investigating the mode shapes (eigenfunctions), natural frequencies, and modal damping factors. The ambient pressure is varied to examine the added mass effect on the natural frequencies of the membrane. A static finite element analysis model is developed for computing the deflection and strain energy of circular membrane at different pretension levels. A preliminary numerical model for the dynamics of hyperelastic membrane is also proposed and validated with experimental data. EXPERIMENTAL SETUP AND PROCEDURE Test Specimen Rubber latex membrane at different pretension levels is attached with circular steel ring of inner and outer diameters 102.5 and 113.2 mm, respectively, as shown in Fig. 3. The thickness and density of the latex 3 membrane are estimated 0.1016 ± 0.0508 mm and 980 kg/m [17]. The average thickness of the membrane is measured by the authors and 0.15 ± 0.01 mm is a reasonable assumption for the FEA models. MooneyRivling material model [18, 19] is considered for this hyperelastic latex membrane. The MooneyRivling material
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parameters are calculated based on the uniaxial tension experimental data, conducted by Stanford et al. [20], and the parameters are C1 = 18.088E4 Pa and C2 = 18.088E3 Pa, considering C1/C2 = 0.1. Three and two different pretension levels are considered for static and vibration tests, respectively. The average pretension levels of the membrane are shown in the Table 1.
Fig. 1 Micro air vehicle with a flexible membrane wing from the MAV Lab at the University of Florida, Gainesville, Florida
(a)
(b)
Fig. 2 (a) Latex membrane specimen attached with circular ring subject of the experiments (b) The painted steel sphere of weight 0.66 N placed at the center of the membrane for static test Table 1 Average pretension levels of the membrane specimen for static and vibration tests
Strain xx yy
Average pretension levels for static tests Low Medium High 0.0266 0.1009 0.1288 0.0136 0.0140 0.0380
Average pretension levels for vibration tests Low High 0.0541 0.0895 0.0630 0.0305
Membrane Deformation Measurement Because the measurements of the strains along a thin membrane must be performed using a noncontact method, a visual image correlation (VIC) technique is used. The VIC is a noncontacting fullfield measurement technique originally developed by researchers at the University of South Carolina [21]. The underlying principle is to calculate the displacement field of a test specimen by tracking the deformation of a subset of a random
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speckling pattern applied to the surface. Two precalibrated cameras digitally acquire this pattern before and after loading, using stereotriangulation techniques. The VIC system then tries to find a region (in the image of the deformed specimen) that maximizes a normalized crosscorrelation function corresponding to a small subset of the reference image (taken when no load is applied to the structure). Such a technique is known as temporal tracking. Images are captured with two highspeed Phantom v7 CMOS cameras, capable of storing 2,900 frames in an incamera flashmemory buffer. Typical data results obtained from the VIC system consist of geometry of the surface in discrete coordinates (x, y, z) and the corresponding displacements (u, v, w). A postprocessing option involves calculating the inplane strains (xx, yy, xy). This is done by mapping the displacement field onto an unstructured triangular mesh, and conducting the appropriate numerical differentiation (the complete definition of finite strains is used). Static Test Wan and Liao [22] characterize thin circular flexible membrane by applying an external load to the film center via a rigid spherical capped shaft. Circular membrane is also characterized statically under the weight of a spherical ball [20, 23, 24]. Central alignment is ensured as the ball rolls to the membrane centre spontaneously by gravity. For this paper, steel spheres of different sizes are also placed at the center of the membrane attached with the rigid ring for performing the static deformation tests (shown in Fig. 3). The membrane is stretched due to the weight of the spheres. The deformation of the membrane is recorded with the VIC system. Total strain energy stored in the membrane is calculated using the strain information from the VIC system and considering Mooney– Rivlin material model.
Fig. 3 Spherical indentation test setup of latex membrane attached with circular ring
Fig. 4 Latex membrane attached with the circular steel ring mounted on the shaker
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Vibration Test Traditional experimental modal analysis techniques (such as an impact hammer in conjunction with an array of accelerometers) are not suitable for the testing of thin, lightweight membrane structures: noncontact measurement techniques must be applied. Since one of the key factors for modal analysis is generating frequency response functions (FRFs), an external system is needed to generate the input to the structures. The base excitation using external elements is considered the most promising excitation technique for microstructures [16] and is adopted in this work. The choice of excitation method is a very relevant factor affecting the quality of the computed FRFs. The application of white noise or a random sinusoidal sweep to the structure does not achieve the required structural excitation, especially in the membrane wings with low or no pretension (slack). The burst chirp is considered the best choice for all ranges of membrane pretension. Experiments on more rigid structures would most likely be able to make use of the random excitation methods, but the nature of our structures indicate that the complex burst chirp signal is the best method, with the cleanest results. Typical frequency sweeps range from 2 to 500 or 1,000 Hz. The specimen, mounted on the shaker (shown in Fig. 4), is placed in the vacuum chamber (shown in Fig. 5) where ambient pressure can be controlled for investigating the added mass effect on the membrane dynamics. The outofplane displacement (w) is recorded using VIC system [16]. Fast Fourier transform (FFT) is carried out for the natural frequencies of the circular membrane specimen at different pretension levels.
Fig. 5 Membrane specimen mounted on the shaker and positioned inside the vacuum chamber. Partial VIC system is also shown in the figure Damping Ratio Calculation The damping characteristic [2528] of the circular membrane specimen at different pretension levels is investigated at atmospheric pressure. The outofplane displacement (w) of the specimen is recorded using VIC system due to the impact test by hammer. The amplitude of the picks decreases due to the presence of damping. The damping ratio () is computed based on the logarithmic decrement () and from the following equation [25]: = , (1) 2 2 (2 ) + where =
1 w0 ln , w0 and w n (w0 > wn) are the amplitudes of two picks that are n periods away. n wn
PostProcessing of Experimental Data The strain energy density (Wse) of the latex membrane is computed based on Mooney–Rivlin material model [18, 19].
W se = C1 (I 1 3) + C 2 (I 2 3) , where C1, C2 are MooneyRivlin material parameters; and I1, I2 are strain invariants.
(2)
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I1 and I2 can be calculated from the following equations:
I 1 = 12 + 22 + 23 ,
(3)
I 2 = ( 1 2 ) 2 + ( 2 3 ) 2 + ( 3 1 ) 2 ,
(4)
where i ( i = 1, 2, 3) are principal stretches. Total strain energy is estimated from the strain energy density, multiplying by volume of the membrane specimen. COMPUTATIONAL MODEL Computational models are developed for investigating both static and vibration characteristics of the latex membrane at different pretension levels, attached with circular ring. It is found that Stanford et al. [20] present a static numerical model for computing the deflection due to the indentation by sphere. In this paper, an axisymmetric static FEA model of circular membrane with fixed boundary condition and steel spheres of different @ sizes placed at the center is developed using finite element software, Abaqus 6.9 [29]. This problem is modeled considering axisymmetric due to geometrical symmetry and reducing the computational cost. Three different pretension levels (denoted as low, medium, and high in Table 1) and hyperelastic Mooney–Rivlin material model [18, 19] are considered for the membrane. @
Gonçalves et al. [30] develop theoretical and Abaqus 6.5 [29] FEA models for investigating the dynamic behavior of a radially stretched circular hyperelastic membrane without added mass effect. In this paper, a finite element analysis (FEA) model is presented for studying the vibration characteristics of the circular membrane with fixed @ boundary condition and at different pretension levels using finite element software, Abaqus 6.9 . A 3@ dimensional Abaqus FEA model is developed for the membrane attached with circular ring specimen. A slice of 0 0.5 3dimensional circular membrane specimen at two different pretension levels (denoted as low and high in Table 1) is chosen due to symmetry and reducing the computational cost. Mooney–Rivlin material model is also selected for the membrane. The membrane is vibrated inside vacuum environment and added mass effect is not considered for this preliminary FEA model. This is an ongoing project work and FEA model for investigating the effect of the added mass on the vibration characteristics of membrane at different pretension levels will be developed and compared with experimental results as part of the future work.
(a)
(b)
Fig. 6 Deformed shape plots of the low pretension membrane specimen from (a) Experimental VIC data and (b) FEA model due to 0.66 N weight of the sphere RESULTS AND DISCUSSION Static Test The static test is performed using the steel spheres placed at the center of the membrane, as shown in Fig. 3. The deformation and strain states are captured using VIC system. The deformation of low pretension membrane from VIC data due to 0.66 N weight of the sphere is shown in Fig. 6(a). Finite element analysis (FEA) model of the @ latex membrane attached with the circular ring of the same geometry is also developed using Abaqus software
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for the static tests placing the steel spheres at the center of the membrane. CAX4H (4–node bilinear axisymmetric quadrilateral, hybrid, constant pressure) type of elements are considered for the membrane. But CAX4R (4–node bilinear axisymmetric quadrilateral, reduced integration, hourglass control) and CAX3 (3–node linear axisymmetric triangle) types of elements are selected for the steel spheres. The friction coefficient between the sphere and membrane surface is assumed 0.2. The deformation of low pretension membrane from axisymmetric FEA model at 4,880 degrees of freedom (DOF) due to 0.66 N weight of the sphere is depicted in Fig. 6(b). Mooney–Rivlin hyperelastic material model is selected for computing the deformation and total strain energy of the membrane specimen. The maximum deflection is occurred at the center of the membrane due to placing the sphere at that location. The experiment is repeated for several times for each sphere. The average maximum deflection and total strain energy with 95% confidence intervals are shown in Figs. 7 and 8, respectively. The strain energy at zero weight indicates the energy due to the pretension of the membrane in Fig. 8. The deformation and strain energy increase with the weight of the sphere and pretension level of the membrane. FEA and experimental results can vary within reasonable limits due to the variation of the thickness of the membrane [17].
Fig. 7 Variation of maximum deflection of the membrane specimen due to the indentation of different weight spheres
Fig. 8 Variation of total strain energy of the membrane specimen due to the indentation of different weight spheres
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Fig. 9 Outofplane displacement (w) vs. time plots at five different locations for the high pretension membrane specimen from VIC data at atmospheric pressure
Fig. 10 FFT of Outofplane displacement (w) from Fig. 9
(a)
(b)
Fig. 11 Actual and average strains distribution over the low pretension membrane specimen
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(a)
(b)
Fig. 12 Actual and average strains distribution over the high pretension membrane specimen
(a) Undeformed Shape
(c) 2
nd
Mode Shape
st
(b) 1 Mode Shape
rd
(d) 3 Mode Shape
Fig. 13 Mode shapes of the high pretension membrane specimen from VIC data at atmospheric pressure Vibration Test The membrane specimen is mounted on the shaker and placed inside the vacuum chamber, shown in Fig. 5. The shaker is controlled using a complex burst chirp signal at ranges from 2 Hz to 500 Hz. The deformation images are captured using VIC system at the rate of 1,000 frames per second. The outofplane deformation (w) is computed at a specific location of the membrane for the fast Fourier transform (FFT). The variation of w with time at five different locations is shown in Fig. 9. FFT of w is carried out for the frequency plots and shown in Fig. 10.
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The spikes in the figures indicate the natural frequencies. FFT is computed more than one location of the membrane so that all nature frequencies can be identified properly. For example, second modes and corresponding frequencies may not be captured if FFT is carried out at the center of the membrane only.
(a) Undeformed Shape
(c) 2
nd
Mode Shape
st
(b) 1 Mode Shape
rd
(d) 3 Mode Shape
Fig. 14 Mode shapes of the high pretension membrane specimen from FEA model at vacuum environment
Fig. 15 Frequency of the first mode vs. degrees of freedom (DOF) plot for the high pretension membrane specimen. DOF are calculated based on among onefour layers of elements along the thickness direction of the membrane specimen
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Fig. 16 Natural frequencies vs. ambient pressure plots for the low pretension membrane specimen
Fig. 17 Natural frequencies vs. ambient pressure plots for the high pretension membrane specimen The actual and average strains distribution over the low and high pretension membrane specimens are shown in Figs. 11 and 12, respectively. The first three mode shapes of high pretension membrane from VIC experimental data and FEA model are shown in Figs. 13 and 14, respectively. C3D4H (4–node linear tetrahedron, hybrid, linear pressure) type of elements are selected for the FEA model of the membrane. The variation of the frequency of the first mode with DOF for the high pretension circular membrane specimen is depicted in Fig. 15. The frequencies of the first mode at different DOF levels are computed considering among one to four layers of elements at the thickness direction of the membrane specimen. It is found from Fig. 15 that the frequency converges at higher DOF. Natural frequencies and mode shapes are computed at 242,168 DOF for the FEA model inside vacuum environment (no added mass effect). The first three natural frequencies at two different pretension levels of the membrane are computed and shown in Figs. 16 and 17. The frequencies at zero ambient pressure in Figs. 16 and 17 indicate that the frequencies are computed at vacuum by the FEA model. The experiments in the vacuum chamber show the dependency of the modal frequencies on the membrane pretension and ambient pressure. Natural frequencies increase with the membrane pretension and decrease with ambient pressure. It is also found that the frequencies increase with the mode, as expected. The variation of first natural frequency (from FEA
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model) with the uniform and nonuniform pretension levels of the membrane specimen is shown in Fig. 18. It is found from Fig. 18 that the frequency increases in a higher rate for nonuniform pretension membrane than that at uniform pretension.
Fig. 18 The variation of first natural frequency (from FEA model) with the pretension levels of the membrane specimen
Fig. 19 Variation of the outofplane displacement (w) of the center of the high pretension membrane specimen with time due to impact test by hammer Damping Ratio Calculation The outofplane displacement (w) of the membrane specimen is recorded using VIC system due to the impact test by hammer. The variation of w with time at the center (x=0, y=0) location of the high pretension membrane specimen, caused by the impact test, is shown in Fig. 19. It is found from Fig. 19 that the picks of w decrease with time due to the damping. The exponential underdamped decay curve is also depicted in Fig. 19. It is found that the damping ratio of the high pretension membrane specimen at atmospheric pressure is 0.29%.
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CONCLUSIONS This paper presents the preliminary phase of an effort towards the theoretical and experimental characterization of the dynamics of an elastic latex membrane at different pretension levels. Experiments in static conditions are used for preliminary finite element analysis models validation. The elastic membranes with specific levels of pretension are attached to a rigid steel ring and loaded using steel spheres of different masses as dead weights. The VIC technique, a noncontact measurement system, is used for measuring the deformation of the membrane. @ The membranesteel sphere system FEA model developed using Abaqus exhibits a good correlation with experimental data in terms of the membrane displacement and elastic strain energy in function of the mass of the sphere and membrane pretension. The dynamic experiments are based on the base excitation of the steel ring using a shaker with a burst chirp control signal with frequency sweeps range from 2 to 500 or 1,000 Hz. The membrane specimen at a specific pretension level is placed inside a vacuum chamber, the threedimensional dynamic shape is measured using the VIC technique and the natural frequencies are found from the FFT of the outofplane deformation time histories of selected points on the surface of the membrane. The chamber’s ambient pressure is varied for investigating the added mass effect on the mode shapes and natural frequencies of the membrane. It is found from the experimental results that natural frequencies of the membrane decrease with the increase in ambient pressure of the vacuum chamber. As a result, added mass decreases the natural frequencies. Natural frequencies of the membrane increase with mode and pretension level. @
A preliminary FEA model is developed with Abaqus with the primary objective of estimating the natural frequencies of the membrane at different uniform and nonuniform pretension levels in the vacuum thus without added mass effect. The FEA model at vacuum (without added mass effect) can predict the first natural frequency reasonably well although second and third frequencies are quite high comparing to those calculated by experiments at different ambient pressures. Some sort of variation is expected among the frequencies predicted by experiments and FEA model due to the variation of the ambient pressure and the thickness of the membrane. It is not possible to perform experiments at ambient pressure less that 9.325 kPa with the available vacuum chamber setup. So, the modal characteristics can not be found through experiments at vacuum (without added mass effect) to compare with the FEA results. The FEA model for investigating the effect of added mass on the modal characteristics is under progress and will be validated by experimental results when it should be available in future. The outofplane deflection at the center of the membrane is computed due to the hammer impact test at atmospheric pressure and it is found that the amplitudes of the picks of the outofplane deflection decrease after ten periods away in exponential manner which indicate underdamped vibration behavior. The damping ratio of the membrane at atmospheric pressure is calculated from the logarithmic decrement of the amplitudes of two picks after ten periods away. A computational model for investigating the added mass effect and damping on the vibration characteristics of the elastic membrane at different pretension levels will be developed and validated by experimental results as part of the future work. The added mass effect in relation to the size and mass of the membrane will also be investigated. ACKNOWLEDGMENTS The authors would like to thank the support from the Air Force Office of Scientific Research under contract FA95500910072, with Prof. Victor Giurgiutiu (initiator) and Dr. David Stargel as project monitors. The continuing support for research activities from the Air Force Research Laboratories at Eglin AFB and WrightPatterson AFB is also greatly appreciated. The authors would also like to thank the experimental support from Joshua Martin, Department of Electrical and Computer Engineering, University of West Florida, Shalimar, Florida. REFERENCES [1] Raney, D., Slominski, E., “Mechanization and Control Concepts for Biologically Inspired Micro Air Vehicles,” Journal of Aircraft, Vol. 41, pp. 1257–1265, 2004. [2] Song, A., Breuer, K., “Dynamics of a Compliant Membrane as Related to Mammalian Flight,” AIAA Paper 2007–665, Jan. 2007. [3] Lian, Y., Shyy, W., “LaminarTurbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil,” AIAA Journal, Vol. 45, pp. 1501–1513, 2007. [4] Ho, S., Nassef, H., p*rnsinsirirakb, N., Tai, Y., Ho, C., “Unsteady Aerodynamics and Flow Control for Flapping Wing Flyers,” Progress in Aerospace Sciences, Vol. 39, pp. 635–681, 2003.
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[5] Yadykin, Y., Tenetov, V., Levin, D., “The added mass of a flexible plate oscillating in a fluid,” Journal of Fluids and Structures, Vo. 17, No. 1, pp. 115–123, 2003. [6] Noca, F., “Apparent mass in viscous, vortical flows,” American Physical Society, 54th Annual Meeting of the Division of Fluid Dynamics, San Diego, California, Nov. 2001. [7] Deruntz, J. A., Geers, T. L., “Added mass computation by the boundary integral method,” International Journal for Numerical Methods in Engineering, Vol. 12, pp. 531–550, 1978. [8] Han, R. P. S., Xu, H., “A simple and accurate added mass model for hydrodynamic fluid—Structure interaction analysis,” Journal of the Franklin Institute, Vol. 333, No. 6, pp. 929–945, 1996. [9] Sewall, J., Miserentino, R., Pappa, R., “Vibration Studies of a Lightweight ThreeSided Membrane Suitable for Space Applications,” NASA Technical Paper, TP 2095, 1983. [10] Gaspar, J., Solter, M., Pappa, R., “Membrane Vibration Studies Using a Scanning Laser Vibrometer,” NASA Technical Memorandum, TM 211427, 2002. [11] Jenkins, C., Korde, U., “Membrane Vibration Experiments: An Historical Review and Recent Results,” Journal of Sound and Vibration, Vol. 295, pp. 602–613, 2006. [12] Graves, S., Bruner, A., Edwards, J., Schuster, D., “Dynamic Deformation Measurements of an Aeroelastic Semispan Model,” Journal of Aircraft, Vol. 40, pp. 977–984, 2003. [13] Burner, A., Lokos, W., Barrows, D., “Aeroelastic Deformation: Adaptation of Wind Tunnel Measurement Concepts to Full Sale Vehicle Flight Testing,” NASA Technical Memorandum, TM 213790, 2005. [14] Ozdoganlar,O., Hansche, B., Carne, T., “Experimental Modal Analysis for Microelectromechanical Systems,” Experimental Mechanics, Vol. 45, pp. 498–506, 2006. [15] Albertani, R., Stanford, B., Hubner, J., Ifju, P., “Aerodynamic Coefficients and Deformation Measurements on Flexible Micro Air Vehicle Wings,” Experimental Mechanics, Vol. 47, pp. 625–635, 2007. [16] Albertani, R., Stanford, B., Sytsma, M., Ifju, P., “Unsteady Mechanical Aspects of Flexible Wings: an Experimental Investigation Applied on Biologically Inspired MAVs,” MAV07 3rd USEuropean Competition and Workshop on MAV Systems & European Micro Air Vehicle Conference and Flight Competition 2007, ISAESUPAERO, Toulouse, France, Sept. 2007. [17] Technical data of The Extra Thin TAN TheraBand® Band. The Hygenic Corporation, 1245 Home Avenue, Akron, OH 44310, USA. [18] Mooney, M., “A Theory of Large Elastic Deformation,” J. Appl. Phys., Vol. 11, pp. 582–592, 1940. [19] Macosko, C. W., Rheology: principles, measurement and applications, WileyVCH, NY, 1994. [20] Stanford, B., Albertani, R., Ifju, P. G., “Static Finite Element Validation of a Flexible Micro Air Vehicle, “ SEM Annual Conference on Experimental and Applied Mechanics, Saint Louis, MS, USA, June 2006. [21] Sutton, M., Cheng, M., Peters, W., Chao, Y., McNeill, S., “Application of an Optimized Digital Image Correlation Method to Planar Analysis,” Image and Vision Computing, Vol. 4, pp. 143–151, 1986. [22] Wan, K., Liao, K., “Measuring mechanical properties of thin flexible films by a shaftloaded blister test,” Thin Solid Films, Vol. 352, No. 1–2, pp.167–172,1999. [23] Liu, K. K., Ju, B. F., “A novel technique for mechanical characterization of thin elastomeric membrane,” J. Phys. D: Appl. Phys., Vol. 34, pp. L91–L94, 2001. [24] Yang, W. H., Hsu, K. H., “Indentation of a Circular Membrane,” J. Appl. Mech., Vol. 38, pp. 227–230, 1971. rd [25] Alciatore, D. G., Histand, M. B., Introduction to Mechatronics and Measurement Systems, 3 ed., WCB/McGrawHill, Boston, MA, 2007. [26] Feldman, M., “Nonlinear system vibration analysis using Hilbert transformation–I. Free vibration analysis method ‘FREEVIB’,” Mechanical Systems and Signal Processing, Vol. 8, No. 2, pp. 119–127, 1994. [27] Feldman, M., “Nonlinear system vibration analysis using Hilbert transformation–I. Forced vibration analysis method ‘FORCEDVIB’,” Mechanical Systems and Signal Processing, Vol. 8, No. 3, pp. 309–318, 1994. [28] Wren, G. G. Kinra, V. K., “An Experimental Technique for Determining a Measure of Structural Damping,” Journal of Testing and Evaluation, JTEVA, Vol. 16, No. 1, pp. 77–85, 1988. @ [29] Abaqus FEA, SIMULIA, Rising Sun Mills, 166 Valley Street, Providence, RI 029092499, USA: <www.simulia.com>. [30] Gonçalves, P. B., Soares, R. M., Pamplona, D., “Nonlinear vibrations of a radially stretched circular hyperelastic membrane,” Journal of Sound and Vibration, Vol. 327, pp. 231–248, 2009.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Objective Determination of Acoustic Quality in a Multipurpose Auditorium
B.Hayes, C.Braden, Undergraduate Students, Miami University, Hamilton, OH R. Averbach, Conductor and Associate Professor, Miami University, Oxford, OH V. Ranatunga, Associate Professor, Miami University, Middletown, OH 501 E. High St., Oxford, OH, 45056, [emailprotected]
ABSTRACT Significant differences in listening qualities have been reported by the audience attending wide varieties of functions ranging from orchestral music to stage drama and public announcements at the multipurpose auditorium of the Miami University Hamilton campus. Preliminary investigations have been conducted to examine the difference in sound qualities utilizing impulse responses at various seating sections throughout the auditorium, based on standardized measurement procedures. Detailed comparisons of measured acoustic quality parameters have been made against the available data in literature for statistically determined optimal acoustic conditions. Each measured range of parameter values has been analyzed to determine the distribution characteristics over a predetermined set of locations on the audience seating area. Recommendations for improvements are presented throughout the paper. INTRODUCTION As a subdiscipline of classical mechanics, ‘acoustics’ is the science concerned with the study of sound. As a subdiscipline of acoustics, ‘room acoustics’ deals with the study of sound in enclosed spaces. In all rooms, acoustical conditions can be controlled. Usually this control is established within the architectural design and construction of the space. However, as with the auditorium studied in this research, that is not always the case. Some buildings are modified over time and as a result their design intent is altered. Other buildings may have not been intended for use in large varieties of functions, but have obtained a demand for such versatility over time. In the event that the quality of acoustics in a performance space is questionable, and/or a simple checkup is desired, this research will be particularly useful. Using a multipurpose auditorium on the Hamilton campus of Miami University as an example, the aim of this research is to clearly describe a simple yet effective method of quantifying the quality of acoustics in a performance space for music, drama, and speech events, as it pertains to the audience perception only. (See reference [1] for an analysis of the ‘stage acoustics’ in this same auditorium). This 4500 cubic meter example auditorium is named Parrish Auditorium and will be referred to as such throughout the paper. Subjective and objective measures are employed to gain such an acoustic evaluation. Interviews and surveys are suggested to isolate subjectively perceived problems, while the ISO 3382 standard [2] defines several objective room acoustic parameters used as diagnostics. By studying subjective impressions and comparing objective measures to well accepted optimum parameter values, the acoustic quality of a space can be determined. Upon diagnosing the acoustic quality of a space using this research, a proposal for implementing acoustic improvements will have strong scientific support. This study is an application of research by expert acousticians that can be implemented by those with a basic understanding of Physics, Integral Calculus, and basic Experimentation Techniques. Section 1 deals with acoustic modeling and experimentation, section 2 analyzes measurement results, while section 3 makes conclusions and recommendations for improving the example multipurpose auditorium. T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_76, © The Society for Experimental Mechanics, Inc. 2011
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1. Acoustical Modeling and Experimentation
1.1 Observing Subjective Impressions Many studies have been done on subjective preferences as it relates to room acoustics for music, speech, and drama performances. Several acoustic attributes which are found to help describe such preferences are defined as followed (most definitions are obtained from table 19.1 reference [4]): Intimacy – The sensation that music is being played in a small room. Balance – Equal Loudness among the various orchestra and vocal participants. Reverberation The sound that remains in the room after the source is switched off. Warmth – Lowfrequency reverberation, between 75 and 350 Hz. Liveness – Reverberation above 350 Hz. Direct Sound – Sound which has not encountered any reflections before reaching the listener. Reverberant Sound – Sound that has encountered at least 1 reflection before reaching the listener. Auditorium Sound Projection – The ability of sound to be projected from the source to the audience area. Blend  A harmonious mixture of orchestral sounds. Brilliance – A bright, clear, ringing sound, rich in harmonics, with slowly decaying highfrequency components. Clarity – The degree to which rapidly occurring individual sounds are distinguishable. Speech Intelligibility – How well a speaker can be heard and understood in a room. Envelopment – The impression that the sound is arriving from all directions and surrounding the listener. Presence – The sense that we are close to the source. Spaciousness – The perceived widening of the source beyond its visible limits. Texture – The subjective impression that a listener receives from the sequence of reflections returned by the hall. Timbre – The quality of sound that distinguishes one musical instrument from another. Tonal Color – The subjective perception of a combination of sounds in sequence or played simultaneously. Uniformity – The evenness of the sound distribution. The weightings of acoustical attributes for music performances are reported by Marshall Long in reference [4]. This reported order of importance is given as; 1) Intimacy; 2) Liveness; 3) Warmth; 4) Loudness of direct sound; 5) Loudness of reverberant sound; 6) Balance and blend; 7) Diffusion; 8) Ensemble. As Long describes after reporting this order based on recollection, other parameters such as the Interaural cross correlation coefficient (IACCE3), Early decay time (TE), Surface diffusivity index (SDI), and Bass ratio (BR) act as better controls and allow for better repeatability in testing. If an adequate sample of musicians and listeners and appropriate instrumentation is available, such a study is more reliable than the subjective study described in this paper. If an adequate sample of musicians/listeners is available, an effective method of analyzing subjective impressions is based on 2sample Ttests done on various acoustical attributes. Each studied attribute is rated on a scale (possibly from 0 to 10) by listeners given various acoustic conditions (curtains open, curtains closed, orchestra enclosure in place, orchestra enclosure not in place, etc). In such a model, the null hypothesis (H0) is that there is no change to the attribute as a result of changing the space. The alternate hypothesis (Ha) is that the attribute has increased as a result of implementing the acoustic control. By using the sample scores, a pvalue can be calculated such as to reject or accept the null hypothesis. If the pvalue < 5%, reject the null. Egan [5] uses a similar method, while also providing a survey sheet which can be utilized with the addition of a numeric scale. Examining subjective impressions can frequently lead to the quick detection of acoustic strengths and weaknesses in a space. Results for Parrish Auditorium showed a very strong correlation between subjective impressions and the objective measurements. It was suggested by the Miami University and HamiltonFairfield Symphony orchestra conductors that; 1) the space lacked adequate reverberation; 2) sound projection from the stage to the audience is poor; 3) The musicians have trouble hearing themselves and each other during the performance. However, given this is a multipurpose auditorium, subjective impressions for other performances such as speech, drama, and amplified music were also investigated. For these performances, the acoustic quality was subjectively determined to be very good. Further comparisons and interpretations of these subjective results when compared to objective measurements are made in section 2.
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1.2 Objective Measures The ISO 3382 standard [2] defines and sets forth measurement procedures for objective parameters which closely relate to the subjective attributes mentioned. Each parameter could be measured and analyzed, but that is beyond the scope of this study. The focus here is on parameters which were determined to have strong correlations with the proposed subjective impressions. Using the subjective impressions from musicians, conductors, and staff at Miami University, five applicable parameters were selected for investigation. Table 1.2a below lists the selected parameters along with their related attribute(s). Table 1.2a – Investigated objective parameters along with related subjective attribute(s).
Objective Parameters Reverberation Time – RT60 Sound Strength  G Clarity – C80 Definition – D50 Speech Transmission Index  STI
Related Attribute(s) Reverberation, Liveness, Warmth, Sound Projection, Loudness Music Brilliance, Reverberation Speech Intelligibility Speech Intelligibility, Clarity
Reverberation time (RT60) is the time it takes for the sound level to drop 60 dB after a sound source stops. Preferred values in a multipurpose auditorium are suggested by Egan [5] as 1.6 < RT 60 < 1.8 seconds. Equation 1 is based on classic diffusefield theory for estimation purposes. To calculate RT 60 based on room impulse response measurements, backward integration of the decay curves was employed (ref. [2] section 5.3.3).
3
Where, V = Volume of the space (m ), α = material coefficient of absorption, S = surface area of any material in 2 the space (m ), n = number materials in the space Sound Strength (G) is the sound pressure level at any given location in an auditorium relative to the freefield level of an omnidirectional source measured 10 m away [4]. This measure is dependent on distance from the source and is similar to our perception of loudness. Preferred range of G is 4 < G < 6 dB given by Long[4].
Clarity (C80) is a measure of balance between early and late arriving energy. It is the degree to which rapidly occurring individual sounds are distinguishable. Barron gives the preferred range as 2 < C80 < +2 dB.
Definition (D50) is the parameter which indicates the clarity of speech [10]. From 0 (bad) to 1 (excellent).
2
2
Where p = squared sound pressure measured in a room (Pa), p10 = reference pressure in a free field 10m from the source (Pa), t = time (ms). An explanation of acquiring the G reference pressure is given in section 1.6. Speech Transmission Index (STI). This is a machine measure of speech intelligibility whose value varies from 0 (completely unintelligible) to 1 (perfect intelligibility) [6]. An algorithm for the determination of STI is described in reference [7]. The ISO 3382 standard [2] gives clear definitions in A.2.1 and A.2.3 for equations 2 and 3. Both suggest that integrating the exponentially decaying squared sound pressure over time give quantities of acoustic energy. As a
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confirmation of this statement, Miami University professor of Physics, Dr. Herbert Jaeger, states, “In the context of sound pressure and sound pressure level (dB), energylike quantities are usually given as intensities. Intensity is the amount of energy that crosses a given surface area per unit time. So we are talking about energy/ time surface area or power/surface area. It also turns out that intensity = p^2/Z where Z is the characteristic impedance. In other words, the intensity is proportional to the square of the pressure, as the characteristic impedance is practically constant. So the ratio of p1^2/p2^2 is equal to the ratio of two intensities. But intensity integrated over time becomes energy per surface area. In that sense you can think of p^2 as an energy quantity.” Therefore, the squared pressure multiplied by time is equal to energy per surface area. Hence, ISO 3382 parameters are calculated using measured ratios of acoustic energy which vibrate the microphone membrane surface area as they are received from the source. 1.3 Various Architectural Acoustic Controls In the same way that one might adjust a home theater or stereo equalizer, the previously defined attributes can also be adjusted through the usage of various architectural acoustic controls. The selection of such controls depends upon the nature of necessary improvement(s). Generally speaking, if speech is unintelligible in a space and unamplified music is good, the space requires more absorptive materials. These materials would reduce reverberation and allow the listener to perceive more of the direct sound. Such is the case in churches and other venues where speech intelligibility is desired. The design and selection of effective architectural acoustic controls depends heavily on topics in sound transmission, absorption, and reflection. Figure 1.3a visually depicts these topics. In the case of the multipurpose Parrish Auditorium, having the ability to vary the acoustics is essential. Therefore, given the subjectively determined problems described above, controls were selected which reflect more acoustic energy from the stage to the audience. This is done by adding panels which surround the orchestra, called an orchestra enclosure or acoustic shell (shown in figure 1.3b). Figure 1.3c shows other types of controls.
Figure 1.3a – Reflection, transmission, and absorption
Figure 1.3b – An orchestra enclosure (acoustic shell). Figure was obtained from (http://www.wengercorp.com/Acoustic/Shell_Diva.html)
Figure 1.3c – Other types of acoustic controls. (top figure) is from reference [9 ] Fitzwilliam college auditorium. (bottom left and right) are from (http://www.wengercorp.com/Acoustic/Panels.html)
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1.4 Modeling Objective Parameters In this section, parameters RT 60 and G are modeled using two methods: 1) Spreadsheet formulation; 2) Autodesk Ecotect software. Method 2 requires a 3D CAD model of the auditorium. Parameters C80, D50, and STI are only measured and compared to wellaccepted optimum values in a later section. Although not necessary in the calculation of RT 60 (Eq. 1), a 3D CAD model was constructed using Solid Works prior to using either method. The 3D CAD model was found to be a convenient tool for quickly gathering accurate geometric information such as the volume of the room and surface area of each material. For example, from the architectural drawings it was found that the walls of the audience seating area are plaster. By measuring the area of all the plaster surfaces in the 3D model, one can quickly determine the total surface area associated with each material. The volume of the entire space was determined from the 3D model by extruding a separate body in the auditorium air space. This body was then measured using Solid Works evaluation tools. The volume and surface areas can also be determined by using the drawings. In this case, complex geometry can be simplified. Using equation 1, the auditorium volume, the surface area of each material, and the coefficient of absorption for each material, a spreadsheet can be formulated from which to estimate RT 60. One can also estimate RT60 with and without the acoustic controls in place. Figure 1.4a reports estimated RT60 values of Parrish Auditorium with and without an acoustic shell on the stage. These values are averaged between the 500 Hz and 1000 Hz frequencies with a 0% auditorium occupancy. David Egan (reference [5]) states the optimum range for a multipurpose auditorium to be 1.6 < RT60< 1.8 seconds.
Figure 1.4a – Estimated RT60 with and without an acoustic shell
Clearly, RT60 without an acoustic shell is estimated to be quite low when compared to the suggested optimum range. However, for speech, drama, and amplified music the low reverberation is desired. The objective parameter G is estimated using Barron’s revised theory [3] given by the equation:
3
Where r = source to receiver distance (m), RT 60 = reverberation time (dB), and V = Volume of the space (m ). This relationship given by Barron’s revised theory indicates G as a function of r, RT60, and V. Reference [9] also describes several applications of using equation 5. Hence, with RT60 and V determined, and the source to receiver distance varying, G is estimated in figure 1.4b for the ‘with shell’ and ‘without shell’ acoustic states.
Figure 1.4b – Estimated G for with and without shell acoustic states
Predictions for both RT60 and G suggest a considerable increase as a result of implementing a full acoustic shell.
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The next method of estimating RT60 and G was performed using the Autodesk Ecotect software. For this analysis, the 3D model was converted from its Solid Works file format into a stereolithography (stl) file for which Ecotect could open. Materials were then selected and assigned to their associated surfaces. With the geometry imported and materials defined, RT 60 (averaged over 500 Hz and 1000 Hz frequencies with a 0% auditorium occupancy) without a shell was estimated by the Ecotect software using equation 1. The Ecotect estimated RT 60 is 0.99 seconds. The ‘with shell’ state was not estimated using this software. Although G could not directly in itself be estimated using Ecotect, applying a ‘Ray Tracing’ analysis with and without the shell in place allows for a relevant simulation of how acoustic energy is projected in this space. With a 200 Watt sound source placed in the center of the stage, figure 1.4c and 1.4d display how this acoustic energy would be projected ‘without a shell’ and ‘with a shell’ respectively. These simulated results give clear indications as to how sound strength would be increased. Note the highly increased density of acoustic energy in figure 1.4d.
@25 ms
@55 ms
@85 ms
Figure 1.4c – Energy projection over time without shell
Figure 1.4d – Energy projection over time with shell
1.5 Hypothesis Formulation Based on estimated objective parameters, gaining subjective impressions, and communicating with expert acousticians, it was understood that the suggested lack of sound projection and low reverberation time in the audience seating area of Parrish Auditorium is caused by a large surface area of 2 cm thick curtains and large fly space over the stage. These curtains make up 40% of the stage material composition and have the highest coefficient of absorption (shown in figure 1.5a). Therefore, because the musicians are surrounded by these materials during a performance, the sound waves are being absorbed and are losing much of their acoustic energy before ever leaving the stage. Thus, the hypothesis is that the stage materials are impeding upon the opportunity for quality auditorium and stage [1] acoustics for unamplified music performances.
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Figure 1.5a shows the impact of the curtains as a result of their total surface area and high coefficient of absorption. Also note that the higher percentage of stage sound produced will strike these curtains.
Figure 1.5a – Material composition of the stage area with coefficients of absorption
Thus, by surrounding an orchestra with reflective panels (an acoustic shell), this lost energy will be more efficiently distributed to the musicians and audience. 1.6 Measurement and Testing The ISO 3382 standard [2] gives detailed instructions for measurement conditions, procedures, equipment, evaluating decay curves, determining uncertainty, spatial averaging, and stating results. Therefore, it would be redundant for the author to cover each topic. Section 9.2 of the standard also gives a template for reporting tests. The following list was populated using this suggested template: a) All measurements reported below conform to the testing procedures designated by ISO 3382. b) The tested space was Parrish Auditorium on the Miami University campus in Hamilton, Ohio. c) Sketch Plan of Parrish Auditorium along with selected source and receiver locations (figure 1.6a):
Figure 1.6a – Sketch plan of auditorium along with source and receiver locations
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d) Room Volume [V] = 4500 cubic meters = 158,916 cubic feet e) Total Capacity = 451 seats. Thick upholstered seats. Seats were raised during tests. f) Ceiling material is steel decking. Wall material is plasterboard and is angled in as you move away from the stage. Audience floor is cement with some carpet coverage. Stage floor is wood. g) There were 2 people in the auditorium during each test. h) There were 2 variable acoustic conditions tested: 1.Without an orchestra enclosure (shell) and 2.With an orchestra enclosure. It is a Wenger Legacy shell. Estimated age is 1520 years old. The shell is located at Millet Hall on the Oxford campus and was borrowed for the test. 15 shell towers and 11 ceiling panels were used. The towers completely surrounded the area around an orchestra. The shell ceiling only covered about half of the area over the orchestra. i) All curtains and line batons were raised to their highest positions such as to maximize the acoustics of Parrish in the ‘without shell’ state. The auditorium blower system was switched off during each test. j) Stage furnishing conditions are described in section (h) above. k) The temperature was approximately 21 degrees Celsius during the test. l) Microphone used: Earthworks M30 OmniDirectional. Source Used: B&K – Type 4292 Dodecahedron. Power Amplifier: B&K – Type 2734A. Sound Interface: Edirol UA25EX. Recording Device: HP Notebook computer using WinMLS 2004 acoustic measurement and analysis software. m) The signal used was a sinesweep from 125 Hz to 20 kHz. n) For the audience area, 3 source positions on stage were used (shown as circles in figure 1.6a). 16 microphone locations were selected in the auditorium (4x4 square grid). o) Measurements were taken from 2/15 through 2/19/2010. Miami University engineering students acquired these measurements. The source and receiver sample size and locations were selected such that parameter distributions from auditorium front to back and left to right could be analyzed. Spatial averaging was applied in correspondence to ISO 3382 section 8. Results of this test are analyzed in section 2 of this paper. Prior to obtaining more reliable instrumentation, simplified impulse response tests were performed in this space using similar source and receiver grid locations. In place of the B&K dodecahedron speaker, balloons were employed as an alternative sound source. Figure 1.6b shows a Miami University engineering student preparing to supply the auditorium with a sound impulse from which it will respond to (or in other words, he is going to pop the balloon with a needle). Such tests performed with balloons were found to be less reliable. Because not every balloon is created equally, varying thicknesses in the balloon latex membrane can cause equivalently sized balloons to provide different sound levels. If balloons are ones only means for producing a sound impulse, Miami University Physics professor, Dr. Herbert Jaeger, suggests the following: 1) Use balloons with tight manufacturing tolerances 2) Blow each balloon up to the same geometric size; 3) Apply the needle to the same location for each balloon pop. It was also suggested that by having two microphones recording for each balloon pop, corrections can be applied to compensate for balloon imperfections. One of the recording microphones would remain stationary throughout the entire process in order to gain correction data. Although not covered in this research, further investigations concerning the reliability of balloon impulse response tests would be useful.
Figure 1.6b – Balloon impulse tests
Calibrating of the sound source level for measuring G Calibrating the sound source level for measuring G was done without the use of an anechoic chamber. Normally this free field (without reflection) environment is subjected to identical impulse response tests as were performed on the diffuse space (auditorium). By taking 29 free field measurements around an omnidirectional source (dodecahedron or balloon) and by using the same measurement settings as in the diffuse field, one can determine the reference pressure as an input for the denominator of equation 2. This is done by averaging the
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total acoustic energy in all 29 measurements. This process is described in detail by the ISO 3382 standard. However, as is the case with the Parrish auditorium project, finding an appropriately sized free field environment (such as an anechoic chamber) can be challenging. Therefore, an InSitu method of acquiring this reference pressure level was utilized. This method was suggested by Jens Jørgen Dammerud who is a Phd candidate at the University of Bath in England and is found in reference [8]. The system shown in Figure 1.6c was derived from the Dammerud [5] Insitu method. If this method is employed, the user must be certain to have an analysis system capable of calculating only the energy between the arrival of the direct sound and the first reflection from the floor. It should also be noted that this method is only proved valid at frequencies greater than 250 Hz. Figure 1.6d shows the time between the direct sound and the reflection
Source Receiver
8 ms 3m
2.5m
2m
Figure 1.6c – Setup for calibrating G Measurements
Figure 1.6d – Impulse response from direct sound to the reflection from the floor.
from the floor. The vertical axis is proportional to sound pressure (Pa) while the horizontal axis is time (ms). After squaring pressure, the integral sum of the area under the curve is acoustic energy. In this study WinMLS 2004 software was used for sending the impulse signal to the source and for recording the auditorium response. The strength calibration menu in this software allows for the selection of all 29 measurement files upon which the calibration data is calculated. Before selecting the 29 files, in WinMLS 2004 the window type can be specified as 8 ms from the direct sound. This sets the lower limit of integration in the denominator of equation 2 as the direct sound, and the upper limit as being 8 ms after the direct sound. The point is to use only the energy which is measured between the direct and the first reflection.
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2. Results Section 2 is divided in to two sections. Section 2.1 is concerned with the analysis of measurements pertaining to the quality of acoustics for unamplified music performances (orchestra, choir, etc), while section 2.2 concentrates on the quality of acoustics where speech intelligibility is desired. Parrish Auditorium is used as an example in both sections. All acquired acoustic data were recorded using WinMLS 2004. These data were then exported to MS Excel where data reduction and analysis was performed. 2.1 Acoustics for Unamplified Music Performances The subjectively proposed problems with the acoustics in Parrish Auditorium are that the space lacks adequate reverberation, sound isn’t being projected efficiently from the stage to the audience, and musicians cannot hear themselves and each other well (from section 1.1). Based on the hypothesis formulated in section 1.5, the objective parameters RT60, G, and C80 are measured with and without an acoustic shell in place (Fig 2.1a, 2.1b).
Figure 2.1a – Measurements with shell
Figure 2.1b – Measurements without shell
Analysis of RT60 (Main subjective attribute is reverberation) Figure 2.1c shows results for measured RT60 spatially averaged over frequencies, source, and receiver locations.
Figure 2.1c – Overall RT60 measured results
The % difference between theory and measured RT 60 ‘without shell’ is less than 1%. However, results show the % difference between estimated (figure 1.4a) and measured RT 60 ‘with shell’ is 20%. This difference is speculated to come from the 451 highly absorbent seats throughout the audience area. Such thick seats are very comfortable but provide difficulties when higher reverberation in a space is desired. These measurements are all performed relative to the 0% occupancy state of the auditorium. Therefore, when the auditorium is occupied the reverberation time (as well as other parameter values) will change further. If it is desired to obtain optimal RT 60, 3 options are available: 1) Construct a new auditorium; 2) Do a major renovation to Parrish Auditorium; 3) Use artificial electroacoustic enhancements. The electroacoustic
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enhancements would consist of phase shifted speakers placed throughout the space to provide the added reverberation. Such a system would provide a variable amount of artificial reverberation to the auditorium. Excel generated contour surface plots are used to investigate distribution characteristics of RT 60 (see 2.1d and 2.1e). These plots provide an overview as if the observer were on the ceiling looking down at the floor.
Figure 2.1d – Distribution of RT60 without shell
Figure 2.1e – Distribution of RT60 with shell
Although low, RT60 has good symmetry from left to right. Higher values in the back of the audience seating area are caused by an accumulation of acoustic energy from reflections against the rear wall and corners. Analysis of G (Main subjective attributes are loudness and sound projection) Figure 2.1f shows results for measured Gmid levels vs distance from spatially averaged over source and receiver locations. The preferred range of G suggested by Long [4] is given as 4 < G mid < 6 dB.
Figure 2.1f – Measured G vs distance from source
As predicted in section 1.4, G is increased as a result of the shell. Because the shell blocked off the large curtains and volume over the stage, and because the walls of the shell absorb very little sound, sound projected much more effectively to the audience. However, as can be observed from figure 2.1f, measured strength is above the preferred range of G. This would suggest that the addition of a full acoustic shell would amplify sound levels too much. Figures 2.1g and 2.1h show distribution characteristics for G measured at the 16 microphone locations.
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Figure 2.1g – Distribution of G without shell
Figure 2.1h – Distribution of G with shell
Both figures show excellent symmetric distribution characteristics. Small anomalies in figure 2.1h could be due to details such as instrumentation tolerances. Both acoustic states clearly show a well diffused space, although as depicted in figure 2.1f, the acoustic shell increases strength above the preferred range.
Analysis of C80 (Subjective attributes are musical brilliance and reverberation) Figure 2.1i shows results for measured C80 with and without an acoustic shell. The optimal range displayed
Figure 2.1i – Overall measured C80
is suggested that for unamplified music the optimal range is 2dB < C80 < +2dB. C80 is also inversely related to RT60. Therefore, as the reverberation time increases (as was the case when the shell was implemented), C 80 will decrease. The shell only acts to decrease the intelligibility of speech in this space and would serve no practical purpose being used during these events. However, with the shell in place C80 is shown to increase the quality of acoustics for unamplified music. As with RT 60, an optimal C80 value in Parrish Auditorium could be achieved by incorporating an electroacoustic reverberation enhancement system which was previously described.
2.2 Acoustics for Speech Intelligibility Interviews with various subjects who attend wide varieties of functions at Parrish Auditorium indicate speech intelligibility to be very good in this space. Objective parameters D50 and STI are measured and compared to wellaccepted optimum values in literature.
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Analysis of D50 (Subjective attribute is speech intelligibility) Figure 2.2a shows results for measured D50 ‘without shell’ when averaged and compared to optimal values.
Figure 2.2a – Measured D50 result
Clearly, definition in Parrish is leaning towards accurate subjective impressions of good acoustics. Figure 2.2b displays distribution of D50 over the audience seating area.
Figure 2.2b – Distribution of D50
Overall distribution of D50 is good with the exception of less than 50% between points 6 and 7. By selecting adaptable wall panels similar to the top of figure 1.3c, D50 can be improved.
Analysis of STI (Subjective attributes are clarity and speech intelligibility) Figure 2.2c shows results for measured STI values when averaged and compared to optimal values [6].
Figure 2.2c – Measured STI result
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As with the analysis of D50, incorporating adaptable absorbent wall panels in strategic locations would also place STI into the optimal range (between 0.75 and 1.0). However, the subjective impressions of the acoustics for speech intelligibility are objectively confirmed to be correct. That is, the acoustics are fairly good. Figure 2.2d shows distribution characteristics for STI over the 16 measurement locations based on the ‘without shell’ acoustic state only.
Figure 2.2d – Distribution of STI
The distribution of STI is highly symmetric relative to the source locations and side walls. Measurement locations 6 and 7 are points with the largest distance from any reflective surfaces. The STI distribution reaches its minimum value at a location between points 6 and 7. Therefore, one could postulate that because this surface minimum lies at a location centrally relative to the source, back walls, and side walls, there may be a need for more surfaces closer to this point. These surfaces could potentially be overhead as an extension to the auditorium’s existing acoustic clouds.
3. Conclusions In conclusion, the summation of all the investigated acoustic characteristics of Parrish Auditorium makes the addition of adaptable acoustic equipment a significant enhancement. The main enhancements as a result of the acoustic shell would be improved musicians support on stage and an increase of sound strength (G) in the auditorium. Although the analysis of G in section 2.1 shows the full acoustic shell to increase G above the preferred limits, the author suggests that a partial acoustic shell (several towers and ceiling panels which do not enclose the musicians completely) would produce a significant improvement. This is because during an actual unamplified music performance the audience (which was not present in testing) will absorb acoustic energy. Therefore, when Parrish Auditorium has an audience and there is a partial acoustic shell present, G is expected to decrease to a preferable level. RT60 and C80 could also be greatly enhanced with the addition of an artificial electroacoustic system as was suggested in section 2.1. This system would allow for variable control of the reverberation in Parrish upon demand. Although the shell would make the most impact to the acoustics in this space, one could finetune the acoustics by also adding adaptable or rotatable absorbent/reflective panels to the auditorium walls and ceiling. For this multipurpose auditorium, as well as many others, having the ability to adapt the acoustic environment to a variety of event functions would be a very desirable and potentially profitable characteristic to obtain.
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ACKNOWLEDGMENTS The author would like to thank Jens Jørgen Dammerud, Phd candidate at the University of Bath, and Brad Hoover, physicists and musician, for mentoring the research team in various aspects of acoustics. The author would also like to thank the Miami University administration and faculty from the many departments which helped in various aspects of research. The author would also like thank the Department of Engineering Technology for facility and financial support. REFERENCES [1] Braden, C., Hayes, B., Ranatunga, V., Averbach, R., “Identification and Enhancement of Onstage Acoustics in a Multipurpose Auditorium”. Society for Experimental Mechanics (SEM) conference proceedings paper #178, (2010) [2] ISO 3382. “AcousticsMeasurement of room acoustic parameters”, Part One  Performance Spaces, International Organization for Standardization, (2009) [3] Barron, M., Lee, L.J., “Energy relations in concert auditoria”, Acoustic Society of America, 84, 618628, (1988) [4] Long, M., “Architectural Acoustics”. Elsevier, Chapter 17, 666682, 654655, (2006) [5] Egan, D., “Architectural Acoustics”. J.Ross Publishing, 64, 147150, (2007) [6] Meyer Sound. “Glossary of terms”. http://www.meyersound.com/support/papers/speech/glossary.htm#sti [7] IEC 6026816. “Sound system equipment – Part 16: Objective rating of sound intelligibility by STI”, International Electrotechnical Commission, (2004) [8] Dammerud, J.J., “Insitu calibration of the sound source level for measuring G”. University of Bath Phd candidate paper, http://home.no/jjdamm/stageac/insituGcalib.pdf, (2008) [9] Aretz, M., Orlowski, R., “Sound strength and reverberation time in small concert halls”. Elsevier, Applied Acoustics. 70, (2009) [10] Nittobo Acoustics. “Definition of room acoustic parameters”, http://www.noe.co.jp/en/product/pdt3/ot/ot03.html
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Identification and Enhancement of Onstage Acoustics in a Multipurpose Auditorium C. Braden, B. Hayes, Undergraduate Students, Miami University, Hamilton, OH R. Averbach, Conductor and Associate Professor, Miami University, Oxford, OH V. Ranatunga, Associate Professor, Miami University, Middletown, OH 501 E. High St., Oxford, OH, 45056, [emailprotected] ABSTRACT The objective of this research is to identify the measurable sound quality parameters of an existing multipurpose auditorium in which the acoustics are not adequate for orchestral purposes. The major problems have been identified as the lack of sufficient projection of sound from the stage area, the inability of the musicians to hear themselves and each other, and the poor reverberation in the auditorium. Objective architectural acoustic parameters have been identified and analyzed by conducting acoustical measurements following standardized procedures. One of the measured objective parameters is 'Musician Support', which depends on the amount of early reflections available on stage for the musicians. An acoustic shell, which is a set of large panels that surround an orchestra, will increase the level of early reflections. The goal of this study is to assess the effectiveness of an acoustic shell in improving the onstage acoustics of this particular auditorium. An objective analysis of these parameters is presented with and without the shell in place, in order to quantify the overall enhancement for the musicians provided by the acoustic shell. INTRODUCTION The acoustical analysis and enhancement of auditoriums is a subject which has been investigated by many researchers over the past few decades. The study of platform/stage acoustics for orchestras in particular was greatly affected by the findings of Gade, who introduced the parameter ST1 (now referred to as STearly) for quantifying ‘Musician Support’ [1]. This and other objective acoustical parameters are used in an attempt to assess the quality of acoustics for the musicians on stage. The particular stage in this study was analyzed using the following objective acoustic parameters: Reverberation Time (RT60) – The time required for the sound level in a room to drop 60 decibels after a source ceases to emit sound. A space is said to more “live” as RT60 increases. For multipurpose auditoriums, an optimal RT60 is within the range of 1.6 to 1.8 seconds [2]. An RT60 of closer to 2 seconds is preferred for orchestral performances. Clarity (C80) – The degree to which rapidly occurring individual sounds are distinguishable. For orchestral performances, a lower clarity provides sounds which flow more smoothly into one another. A study of eight concert halls showed that for the most preferred halls, midfrequency C80 values on stage were in the range of approximately 2 to 4 dB [3]. Musician Support (ST1 or STearly) – A measure relating to how well musicians can hear themselves and others around them. This parameter depends on the amount of early reflections available from surrounding stage surfaces. The optimal range for ST1 is between 13 and 11 dB [4]. Late Sound Level (Glate or G80∞) – A parameter calculated from measured G (sound strength – closely related to “loudness”) and C80 values. It is a measure of the level of “reverberant sound”. Glate calculated for the auditorium and the stage indicates the ability of the auditorium to provide an audible level of reverberant sound to the musicians on stage. Based on the recent research of J.J. Dammerud of the University of Bath, the optimal range T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_77, © The Society for Experimental Mechanics, Inc. 2011
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for Glate in the auditorium is 1 – 3 dB [5]. Glate measured onstage was found to be about 2 dB above Glate in the auditorium, for the most preferred halls in his study [3]. RT60 is calculated by the “integrated impulse response method” outlined in section 5.3 of ISO 33821:2009 [6]. In this case, RT60 is calculated based on “T 30”, the decay time over the range of 5 to 35 dB extrapolated to 60 dB. The equations for calculating C80, G, and ST1 are stated in Annexes A and C of ISO 33821:2009 [6] and are also based on the integrated impulse response. Glate, stage and Glate, auditorium values were calculated based on G and C 80 values. The equation for calculating G late is below [7]: Glate =
(1)
BACKGROUND An Engineering Perspective The design of multipurpose auditoriums to cater to a wide variety of events is particularly demanding due to the varying acoustical needs of the venue. Motivation for this research originated from the accounts given by two area conductors on the orchestral acoustics of an existing multipurpose auditorium on the Miami University Hamilton campus in Hamilton, OH. Parrish Auditorium, shown in Figure 1, is relatively small with 451 seats, and is mostly used for theatre, lectures, and small amplified musical groups. The conductors’ subjective impressions were identical, and highlighted three main acoustical deficiencies in Parrish for orchestral purposes: 1) Low reverberation, 2) Lack of sufficient projection of sound from the stage to the audience area, and 3) Musicians struggling to hear themselves and each other. The main objective of this paper is to quantify the improvement of the onstage acoustics for musicians provided by the addition of an acoustic shell. From an engineering standpoint, this requires the identification and analysis of measurable objective architectural acoustic parameters related to musician perception (defined in the above section). Optimal ranges for these objective parameters have been identified based on subjective studies with musicians. The variation between the measured parameter values and the identified optimal ranges can then be compared, with and without the shell in place, in order to show the quantitative improvement of the acoustics provided by the shell.
Figure 1. Seating Area in Parrish Auditorium – View from Stage (Dodecahedron Loudspeaker at Center)
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A Musical Perspective Our observations on the importance of musicians being able to hear effectively while performing in an orchestra are significant in the conception of an acoustic setting. If musicians can better hear each other, and themselves, while playing, they can better adjust their intonation and manipulate tonal colors with ease. As in chamber music where musicians listen and interact with each other, the same holds true for a large ensemble. In both cases, an optimal acoustic setting is desirable. With excellent acoustics, the sound will not only become richer and more resonant, but also more lively and varied due to the phenomenon of resonance. The phenomenon of resonance is a scientific explanation of the Physics of overtones; it is what is sensed by the musicians and the audience when the orchestra plays in tune. The sound becomes richer and multiplies due to many different pitches ringing within one predominant tone. Even after each musician tunes their instrument in relation to the A given by the oboe, they must be constantly aware of their intonation within the texture and adjust accordingly. Being “in tune” is not a solid, but rather having the buoyancy to float within the ringing tones of the overtone series. Dynamic changes like crescendo and diminuendo should be observed within the context, as dictated by the musical score. These changes should be determined by the instruments that carry the main melodic line, and the accompanying instruments should subordinate the dynamic changes to this line in order that the main theme is always clearly heard. For example, if the strings are accompanying one or more woodwinds which carry the melodic line, their crescendos should never cover the solo woodwinds. What is most important in this case is that the strings always give the soloists space to breathe and soar above them. Fugal passages should be performed in a way that the subject can always be followed and identified. This is particularly difficult when the subject is stated in the inner voices. Several solutions can be used, such as dynamic adjustments that are frequently not written in the music, the use of contrasting articulations, accents, etc. All of this can happen only if the musicians are able to clearly hear each other and themselves. The strings are the most hom*ogeneous section of the orchestra and should be thought of as a choir. String instruments can be particularly expressive especially when the hom*ogeneity of sound, articulation and musical thought are achieved. They must blend as in a choir, or the purity of sound is missing. When the players are listening better to each other, it is possible for them to blend more effectively. The trombones and tuba form another relatively hom*ogenous group of instruments, however much smaller than the strings. The same holds true of the French horn section. These instruments form their own inner choirs within the brass section of the orchestra, which can be viewed as a larger choir encompassing all the brass instruments. The main challenge in dealing with the brass section is to achieve a similar hom*ogeneous sound as with the strings. It is particularly challenging because of the many different timbres of the various instruments within the brass section. In addition, it is more difficult for brass players to hear themselves in comparison to string instruments. Since the bells are facing away from the face and ears, these performers are usually hearing a reflected sound of their tone. The French horns are the most difficult instruments to blend with the others because their bells often face the back of the stage, while the bells of all the other instruments face the audience. All brass instruments use a rather similar embouchure (formation of lips at the mouthpiece), something that cannot be said of the woodwind instruments. Woodwind instruments call for a wide variety of reeds, both single and double. In addition, the woodwinds, have at least three different registers that produce much more variance in timbre than in the other instrumental sections. Blending sounds and tone colors, the clarity of attacks, and hom*ogeneity of various types of articulation is a significantly bigger challenge for the woodwind instruments. The tone of each woodwind or brass instrument should positively relate to the others players in their section. Frequently there are dialogues or imitations of themes that go from one instrument to another. The players need to be able to listen to each other and try to find a logical way to phrase and articulate passages, so that the theme can flow artistically from one instrument to the other. Each player should find a unique voice and approach to the music, while yet maintaining a uniform style with the other instruments.
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With the advent of recordings, standards for precision and tone quality in orchestras has increased significantly. Contemporary music presents significant technical challenges, and the treatment to orchestral performance is leaning more and more in the direction of a chamber music approach. Because of this, there exists a constant demand to improve the acoustical environment where musicians perform so that the musicians can better hear each other and perform with the utmost accuracy. No matter how large the orchestra may be, players must be able to feel a sense of intimacy in order to communicate with one another and the audience. Herein, lies the key for an artistically effective performance. EXPERIMENTAL SETUP All measurements were performed in accordance with ISO 33821:2009, “Measurement of room acoustic parameters” [6]. Impulse responses were obtained using an Earthworks M30 Measurement Microphone (omnidirectional microphone), Brüel & Kjær Type 4292 OmniPower Sound Source (dodecahedron loudspeaker), Brüel & Kjær Power Amplifier Type 2734A (loudspeaker amplifier), Edirol UA25EX Audio Capture (USB sound interface), and an HP Compaq 8510p Notebook PC using WinMLS 2004 acoustic measurement and analysis software (for sending output signal and analyzing input signal). The output signal was a sinesweep that began at 125 Hz and ended at 20 kHz. The acoustic shell used was a Wenger Legacy® Acoustical Shell. This is a portabletype shell with flat panels. For the “with shell” measurements, fifteen towers, 1.83 m wide by 4.88 m tall, were set up to enclose an area of approximately 100 2 m on the stage (with the top sections angled downward). Eleven ceiling panels were used  each was 1.22 m by 2.44 m. Six ceiling panels were located near the back of the enclosure, and five were located near the front, as shown in Figure 2. They were at an average height of approximately 5 m above the stage. The ceiling panels were spaced such as to obtain the best ceiling coverage possible with the limited available line batons and rigging weight capacity. The ceiling panels covered about half of the area above the orchestra enclosure.
Figure 2. Acoustic Shell Setup with Loudspeaker Centerstage 3
The volume of Parrish Auditorium is approximately 4500 m . During testing, the HVAC system was switched off in order to eliminate unnecessary ambient noise. In addition, all stage curtains were raised to their highest positions to provide the best possible stage acoustics for the “without shell” state. All access doors were closed, and there were two people present in the auditorium during testing. The temperature was approximately 21 degrees Celsius. The state of the platform was empty (except for the shell during the “with shell” measurements) – there were no chairs, music stands, or instruments present on stage during testing. The omnidirectional microphone receiver was in a vertical position for all measurements.
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Measurement positions for the corresponding objective parameters are shown below in Figures 3 and 4. For RT60, C80, and G, source and receiver locations were chosen in an attempt to represent different instrumental groups. Locations for these measurements were also chosen to be at least 1 m away from reflecting surfaces (shell walls), as depicted by the pink line in Figure 3. The height of the source was 1.5 m, and the height of the receiver was 1.2 m. For these measurements, 18 unique combinations of sourcereceiver locations were used, as follows: Source @ 3, Receiver @ 7, 8, 9 Source @ 4, Receiver @ 1, 7, 8, 9 Source @ 6, Receiver @ 8 Source @ 7, Receiver @ 1, 2, 3, 4, 5 Source @ 8, Receiver @ 2, 3, 4, 5, 6 For the measurement of ST1, a different schematic was created (shown in Figure 3). ST1 requires that the source and receiver are 1 m apart. Gade also suggests keeping both source and receiver at least 4 m from walls to ensure that reflections from the walls are included in the support measure [8]. In this case, positions for the source were chosen to be at least 4 m from the shell walls, as depicted by the pink line in Figure 3. Receiver positions were 1 m upstage from source positions. For ST1 measurements, the height of both source and receiver was 1 m. In total, 8 unique sourcereceiver locations were used. In order to calculate Glate in the auditorium, a measurement system was required for obtaining G and C80 in the auditorium. The grid schematic used for the auditorium measurements, consisting of 16 receiver locations, is shown below in Figure 4. The source remained at “Figure 3  position P6” during these measurements. The source height was 1.5 m and the receiver height was 1.2 m. A more detailed explanation of the acoustic assessment of the auditorium area is given in the paper by Hayes, et al. [9].
Figure 3. Measurement Positions for RT60, C80, and G on Stage
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Figure 4. Measurement Positions for ST1 on Stage
Figure 5. Measurement Positions for C80 & G in the Auditorium
523
Calibrating Sound Strength (G) The calculation of G requires calibration measurements to be taken at a 10 m distance in a “free field” (an environment free of reflections). A true free field environment is hard to come by. One such possibility is the use of an anechoic chamber, which approximates a free field. However, anechoic chambers are also hard to come by. Therefore, an “insitu” method described by Dammerud [10] was used for the purpose of calibrating the sound level without the use of a free field. This method simply requires defining a “reflectionfree zone” around the loudspeaker and microphone, defined in Dammerud’s paper. The configuration used was a source height of 2 m, receiver height of 2.5 m, and a sourcereceiver distance of 2 m. In total, 29 measurements were taken – the source was rotated approximately 12.5° between each measurement. Hayes et al. further discusses the calibration of G [9]. RESULTS The parameters RT60, C80, G, and ST1 were calculated from the measured impulse responses using WinMLS 2004 software. The results of G on stage, G in the auditorium, and C80 in the auditorium are not shown here, but were used to calculate Glate, stage and Glate, auditorium values. This was done using a spreadsheet created by Dammerud [11]. Mean values of the spatial averages over the 5001000 Hz octave bands are presented in Table 1. In the case of ST1, the mean spatial average is calculated over the octave bands from 2502000 Hz. Spatial averages are also plotted in the four octave bands from 250 to 2000 Hz for each of the objective parameters. These results are shown in Figure 6 (ae).
Table 1. Mean Spatial Averages of Objective Parameters Spatial Average: 5001000 Hz Bands Parameter
Without Shell
With Shell
RT60 (s)
1.03
1.29
C80 (dB)
6.8
6.5
Glate, stage (dB)
0.0
4.4
Glate, auditorium (dB)
0.3
2.7
Spatial Average: 2502000 Hz Bands Parameter
Without Shell
With Shell
ST1 (dB)
15.0
8.9
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Figure 6 (ae). Spatial Average Values in Octave Bands
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DISCUSSION Spatial Averaged Results The addition of the shell on stage produced measurable differences in the objective parameters, most notably Glate, stage, Glate, auditorium, and ST1. An acoustic shell is not necessarily meant to increase the reverberation time as much as it is to provide early reflections for the musicians and also project sound from the stage to the audience. However, the addition of a shell increased the level of the reverberant sound (G late) in both the auditorium and stage areas. The shell increased RT 60 on stage by 0.26 s. This is approximately a 25% increase, and brings the reverberation time approximately 50% closer to the optimal range of 1.6 to 1.8 seconds. The standard deviation of the midfrequency average between individual measurement positions was only 0.04 s for “without shell” and “with shell”. According to Egan, RT60 on stage should be approximately equal to that of the auditorium, which was found to be the case, with and without the shell, based on comparisons with the auditorium area study by Hayes et al. [9]. The C80 spatial average value did not change much with the addition of the shell. The “with shell” spatial average values for the 10002000 Hz bands were lower than the “without shell” values. However, four of the nine measured locations show differences in the midfrequency C80 (“without shell” vs. “with shell”) which exceed the JND (Just Noticeable Difference) of 1 dB. The measured differences in midfrequency C80 values are by no means consistent across all stage locations – six of the nine locations showed a decrease in midC80, while the other three locations showed an increase. One interesting thing to note is that the addition of the shell decreased the standard deviation of the midC80 between different positions from 1.4 to 0.9 dB. This is a standard deviation decrease of about 35%. Therefore, it can be said that the shell makes the clarity on stage slightly more uniform, which is desirable. However, even with the addition of the shell, the midfrequency C80 values at all locations were still too high to be within the optimal range of approximately 2 to 4 dB. As briefly described in an earlier section, Glate is simply the latearriving portion of the sound strength (from 80 ms to ∞). The level of Glate measured in the auditorium must be high enough for the musicians on stage to perceive a sense of reverberance in the auditorium and a sense that their sound is being projected from the stage. The level of Glate, auditorium is not the only concern, but also the difference between Glate, auditorium and Glate, stage. Glate measured on the stage will be higher than Glate measured in the auditorium due to decreased sourcereceiver distances, based on Barron’s revised theory (discussed further in Hayes, et al. [9]). An acoustic shell further increases Glate, stage. If Glate measured on the stage is too much higher than that of the auditorium, the stage area may be acting as a separate acoustic space, creating its own reverberant response. This is undesirable, as it makes the sound on stage overwhelming and also prohibits the musicians from hearing the reverberant response from the auditorium. For the small set of popular stages investigated in Dammerud’s recent study, the average value of Glate, auditorium was found to be within 1 – 3 dB [5], and the average value of Glate, stage was found to be approximately 2 dB above the average Glate, auditorium [3]. It should be noted that these suggested “optimal” values are currently more hypotheses than proven facts, as these topics have not been investigated much previously. In the “without shell” state, Glate, auditorium was below the 1 – 3 dB range, and Glate, stage was only 0.3 dB above Glate, auditorium. Comparing to the suggested optimal values, this would contribute to the initial perceived subjective problems of “lack of projection” and “lack of reverberation”. With the addition of the shell, Glate, auditorium was increased to 2.7 dB, which is within the range proposed by Dammerud. Glate, stage was increased to 4.4 dB, which is 1.7 dB above Glate, auditorium – again, within the suggested optimal range. In the “without shell” state, the average ST1 value was 2 dB below the optimal range. The addition of the acoustic shell greatly increased the ST1 value, to the point that it was approximately the same amount above the optimal range as it originally was below. It should be noted that the preferred height of overhead reflectors is approximately 7 m [12]. It is reasonable to assume that raising the overhead ceiling panels to the preferred height would decrease the ST1 value. According to O’Keefe [13], changing the height of overhead reflectors only produces ST1 changes in the range of 0.5 to 1.5 dB. However, assuming the upper end of that range, raising the overhead ceiling panels could bring the ST1 value very close to the optimal range. Spatial Variation of C80 and G The ISO standard states that the parameters C80 and G are normally not spatially averaged, as they “are assumed to describe local acoustical conditions” [6]. Although there is some relevance to spatial variations of these parameters, when quantifying stage acoustics, the typical method is to quote single position and mid
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frequencyaveraged values. As was mentioned above, the standard deviation of the midfrequency C80 values on stage was 1.4 dB “without shell” and 0.9 dB “with shell”. The standard deviation of the midfrequency G values on stage was 1.2 dB “without shell” and 0.5 dB “with shell”. This only resulted in a standard deviation of 0.5 dB for the calculated midfrequency Glate, stage values, “with shell” and “without shell”. The standard deviation of midfrequency C80 in the auditorium was 0.9 dB “without shell” and 1.0 dB “with shell”. For midfrequency G in the auditorium the standard deviations were 0.6 dB “without shell” and 0.9 dB “with shell”. This resulted in a standard deviation for Glate, auditorium of 0.4 dB “with shell” and “without shell”. For a more detailed spatial analysis of C 80 and G in the auditorium, see Hayes et al. [9]. Occupancy It should be reemphasized that these measurements were taken with the stage and auditorium in an unoccupied state. If measured in an occupied state, these parameter values would change. However, the optimal ranges for the studied objective parameters refer to unoccupied states. A scale model investigation by Dammerud showed a decrease in RT60 on stage by about 0.5 s when an orchestra was present on stage [14]. According to a discussion with Dammerud, C80 on stage will change significantly with the presence of an orchestra, but this change is difficult to predict because it is dependent upon the stage enclosure design. However, he states that C 80 can be expected to decrease above 1000 Hz due to attenuation of the direct sound [15]. It is the suggested practice to measure ST1 with chairs, music stands, and instruments on stage, in order to better represent the platform conditions when an orchestra is present. Several researchers have found that ST1 decreases anywhere from 1 to 2.5 dB when chairs, stands, instruments, and even musicians are present (compared to an empty platform) [13, 14, 16]. According to Dammerud [17], Glate, stage will decrease slightly with the addition of an orchestra – more so when an acoustic shell is present, and Glate, auditorium should not decrease significantly with the addition of an audience, given adequate absorption characteristics of the audience seating. CONCLUSIONS The addition of the acoustic shell in this study proved a significant benefit based on observed changes in several objective acoustical parameters. The addition of the shell increased RT 60, Glate, stage and Glate, auditorium, and also ST1. The increase in ST1 was slightly excessive, but by raising the ceiling panels to approximately 7 m above the stage, ST1 is expected to decrease to a level which would be close to optimal. This is expected to enhance the ability of the musicians to hear themselves play, as well as hear the other musicians around them. This, in turn, would have numerous beneficial effects on the ensemble, such as better intonation, resonance, blend, and hom*ogeneity, as well as increased ability to make dynamic adjustments. The average C 80 for the stage did not change by a significant amount. With the addition of the shell, RT 60 was brought to within 0.3 s of the optimal range for multipurpose auditoriums. Taking into consideration the fact that G late, stage and Glate, auditorium were increased to the suggested optimal state, the reverberance perceived by the musicians on the stage should be much improved, providing an enhanced sense of projection and acoustical response from the auditorium. Based on these results, the authors recommended that the Miami University Hamilton campus purchase an acoustical shell, similar to the one in this study, in order to enhance the orchestral acoustics of Parrish Auditorium. A future study with an orchestra present would prove beneficial to provide a subjective analysis of the state of the acoustics with and without the shell. ACKNOWLEDGEMENTS We extend our deepest thanks to Jens Jørgen Dammerud, PhD candidate from the University of Bath, who has so graciously offered his expertise in stage acoustics for the purpose of furthering this project. We would also like to thank Miami University and the Department of Engineering Technology for their financial and facilities support. REFERENCES 1. Gade, A.C. “Investigation of musicians’ room acoustic conditions in concert halls. Part II: Field experiments and synthesis of results.” Acustica 69, pp. 249262, 1989. 2. Egan, M.D. Architectural Acoustics. Ft. Lauderdale, FL: J. Ross Publishing, 2007. 3. Dammerud, J.J., and M. Barron. “Concert hall stage acoustics from the perspective of the performers and physical reality.” Proceedings of the Institute of Acoustics. 30, pp. 2636, 2008. 4. Barron, M. Auditorium Acoustics and Architectural Design. London: E & FN Spon, 1993.
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5. Dammerud, J.J. “EstimatingGl_fromTandV.” Stage acoustics for symphony orchestras in concert halls. Updated 1 Oct. 2009. Accessed 25 Jan. 2010. . 6. ISO 33821:2009: Acoustics – Measurement of room acoustic parameters – Part 1: Performance spaces, International Organization for Standardization, Geneva, Switzerland. 7. Beranek, L. “Concert Hall Acoustics – 2008.” J. Audio Eng. Soc. 56, pp. 532544, 2008. 8. Jeon, J. Y. and M. Barron. “Evaluation of stage acoustics in Seoul Arts Center Concert Hall by measuring stage support.” J. Acoust. Soc. Am. 117, pp. 232239, 2005. 9. Hayes, B., Braden C., Averbach R., Ranatunga, V. “Determination of Objective Architectural Acoustic Quality of a Multipurpose Auditorium.” Paper presented at: SEM Annual Conference & Exposition on Experimental and Applied Mechanics; 2010 Jun 710, Indianapolis, IN. 10. Dammerud, J.J. “Insitu calibration of the sound source level for measuring G.” Stage acoustics for symphony orchestras in concert halls. Updated 1 Oct. 2009. Accessed 2 Feb. 2010. . 11. Dammerud, J.J. “Ge_Gl_fromGandC80.” Stage acoustics for symphony orchestras in concert halls. Updated 1 Oct. 2009. Accessed 26 Jan. 2010. . 12. Barron, M. and J.J. Dammerud. “Stage Acoustics in Concert Halls – Early Investigations.” IOA Proceedings, Copenhagen, Denmark, May 2006. 13. O'Keefe, J. "Acoustical Measurements on Concert and Proscenium Arch Stages.” IOA Proceedings, London UK, Feb. 1995. 14. Dammerud, J.J. “Model investigations – preliminary.” Akutek.info. Updated 3 Oct. 2009. Accessed 8 Mar. 2010. . 15. Dammerud, J.J. “Re: C80.” Message to the author. 2 Mar. 2010. Email. 16. Skalevik, M. “Orchestra Canopy Arrays – Some Significant Features.” Joint BalticNordic Acoustics Meeting. Sweden, 2006. 17. Dammerud, J.J. “Re: Results of Glate.” Message to the author. 11 Mar. 2010. Email.
Proceedings of the SEM Annual Conference June 710, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc.
Fractional Calculus of Hydraulic Drag in the Free Falling Process
Yuequan Wan Department of Mechanical Engineering Technology, Purdue University Email: [emailprotected]
Richard Mark French, Associate Professor, Department of Mechanical Engineering Technology, Purdue University. 138 KNOY, 401 N. Grant St, West Lafayette, IN47906, USA. Email: [emailprotected]
ABSTRACT: A new approach to describe the hydraulic drag received by a falling body has been developed through fractional calculus, and the analytical solution has been given. This new method treats the measurement of the hydraulic drag as the fractional derivative of the falling body’s displacement. The existing methods could be classified into two categories. The first ones assume the drag could be described by a quadratic equitation of the body’s velocity and use the classical Newton law to describe the falling process. The second ones do introduce the fractional calculus to describe the dynamic process but still treat the drag as the quadratic of velocity, which make the physics meaning of parameters are obscure. Compared to existing methods, the new approach introduced in this paper is original from the perspectives of basic hypothesis and modeling. To evaluate the performance of this method, series experiments have been conducted with the help of high speed camera. The data fit the new method successfully, and compared to the existing approaches, the new one has the overall better performance on the accuracy to describe the dynamic process of the falling body and owns intuitive physical explanation of its parameters. KEYWORDS: fractional calculus, hydraulic drag, falling process, NOTATION: Hydraulic drag (Newton) Velocity (m/s)
T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/9781441997920_78, © The Society for Experimental Mechanics, Inc. 2011
529
530
Mass of the plug (kg) ,
Fractional order of derivative Gravity of the plug (Newton) Buoyancy of the plug (Newton)
, ,
Constant coefficients
1. Introduction The fractional calculus, in which people consider the order of integral and derivative as any real number, has a [1] history nearly as long as the ordinary calculus, which contains only integer orders . Recently, the applications of [2][3] [4][5] this technology have been successfully found in many fields, such as in viscoelasticity , control theory and [6][7] electroanalytical chemistry During a free falling process, traditionally people use quadratic function to approximate the hydraulic drag [8] received by the falling body. When Reynolds number is low, the hydraulic drag is always described by a linear term of the falling speed, and when facing high Reynolds number situation, people consider the drag as the second order polynomial function of the falling velocity. This is considered to be the classic method which has existed for many years. Aided by the computation fluid dynamics (CFD), now people could simulate the hydraulic drag received at different velocity and fit the simulation result to facilitate further use such as estimating the falling position of an object at any given time. However, the CFD result depends on computational algorithm, environment parameters’ setup, and model’s quality. Besides, high quality CFD work, especially when dealing with 3D problem, always consumes great amounts of CPU hours. And its result could still be quite different from the experiment, especially when using it to predict the falling position. Both the quadratic function and CFD methods are based on regular calculus method. The application of fractional calculus in studying the hydraulic drag during a falling process could be categorized in the fractional viscoelasticity field. There have been scholars applied fractional calculus in the dynamical [9] [10] , however, this method equation of the falling process, and gained high quality fitness to the historical data still assume the hydraulic drag is quadratic of the falling velocity, and made modification to the Newton second law while involving terms which do not have intuitive physical interpretations. In this paper, a new approach which assumes the hydraulic drag could be described by fractional calculus has been developed. And to support, series experiment under controlled environment have been conducted with the help of high speed camera. The result shows that, the new approach from this paper could fit the experiment data soundly, and the hydraulic drag received during this falling process could also be easily accessed. Compared with the existing methods, the advantages of the new approach based on fractional calculus are immediate evident. 2. Case Study and Experiment In this paper, the free falling process has been studied is quite straightforward. It is a process that a polyurethane plug falling in a tube filled with water. Therefore this experiment could be conducted under fully control, and the results could be generalized into many more complex situations. By filling different materials into the plug, its mass could be modified while keeping the shape and surface unchanged. Different mass of plug lead to different falling process in terms of velocity’s range, and provide more data to evaluate the performance of each method. The high speed camera has been used to acquire the actual positions during the plug’s falling. The camera used is a Casio Exillim ExF1, and the image processor is a Photron 1024. With respect to different falling process, the number of frames taken in one time unit could be changed to provide best accuracy. The case studied in this paper and experiment setup could be described in the figure 1. In figure 1, frame (A) describes the case and frame (B) demonstrates the experiment setting. With repeated experiment, groups of data have been collected with respect to five different mass of the plug. Table 1 shows the physics parameters and the falling process data for the first plug.
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Water Tube
Hydraulic Drag
Scale High Speed Camera
Buoyancy
Gravity
(A)
(B)
Figure 1, Case and Experiment Demonstration
Plug Mass 0.16471 kg
Table 1: Physics Parameters and Falling Process of Plug#1 Physics Parameters Plug Volume Water Density Buoyancy 1.5058 N 1.5393e004 3 998.2 kg/ 3
Plug Gravity 1.6142 N
Falling Position (cm) Against Time (s) Position
Time
Position
Time
Position
Time
Position
Time
Position
Time
7.6956
0.6667
28.2317
1.3333
0.0667
9.3293
0.7333
30.7927
1.4
56.0366
2
88.4756
2.6667
59.1463
2.0667
91.5854
2.7333
0.2439
0.1333
10.9146
0.8
33.2927
1.4667
62.3171
2.1333
95
2.8
0.6404
0.2
12.7877
0.8667
35.9146
1.5333
65.4878
2.2
1.2195
0.2667
14.5795
0.9333
38.7195
1.6
68.5366
2.2667
1.8293
0.3333
16.8981
1
41.5244
1.6667
72.2355
2.3333
2.7439
0.4
18.9024
1.0667
44.3293
1.7333
75.5484
2.4
3.7195
0.4667
20.9431
1.1333
46.8825
1.8
78.3537
2.4667
4.939
0.5333
23.3537
1.2
50.061
1.8667
81.5244
2.5333
6.2846
0.6
25.6707
1.2667
52.8219
1.9333
85
2.6
The physics parameters of the other four plugs are summarized in the table 2.
Plug Mass 0.15716 kg Plug Mass 0.16062 kg
Plug Mass
Table 2, Physics Parameters of Plug #2~#5 Physics Parameters of Plug #2 Plug Volume Water Density Buoyancy 1.526048 N 1.56e004 3 998.2 kg/ 3 Physics parameters of Plug #3 Plug Volume Water Density Buoyancy 1.526048 N 1.56e004 3 998.2 kg/ 3 Table 4, physics parameters and falling process of Plug #4 Physics Parameters of Plug #4 Plug Volume Water Density Buoyancy
Plug Gravity 1.5402 N Plug Gravity 1.5741 N
Plug Gravity
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1.526048 N 998.2 kg/ 3 Physics Parameters of Plug #5 Plug Mass Plug Volume Water Density Buoyancy 0.16544 kg 1.526048 N 1.56e004 3 998.2 kg/ 3 Figure 2 is a sample of the picture acquired during the falling process. 0.16206 kg
1.56e004
3
Figure 2, Sample of Picture Acquired
Figure 3, CFD Model Demonstration
1.5882 N Plug Gravity 1.6213N
533
For comparison purpose, a CFD model has also been built, and varies simulation have been carried out under different flow velocities. The figure 3 demonstrates the CFD model and the table 3 consists of the data collected during the simulation. Based on the simulation result of the drag force at different velocities, one could obtain a fine fit of the hydraulic force into a polynomial equation with no interception as the equation (1), and the fitted result is showed in the figure 4. = 0.7785
2
+ 0.0317
(1)
Table 4, CFD Result of the Hydraulic Force (N) at Different Velocity (m/s) Force
Velocity
Force
Velocity
Force
Velocity
Force
Velocity
Force
Velocity
0.000182
0.01
0.012472
0.11
0.041023
0.21
0.084929
0.31
0.143984
0.41
0.000593
0.02
0.014622
0.12
0.04473
0.22
0.090154
0.32
0.150709
0.42
0.001209
0.03
0.016932
0.13
0.048588
0.23
0.095522
0.33
0.157592
0.43
0.002015
0.04
0.019399
0.14
0.052598
0.24
0.101049
0.34
0.164649
0.44
0.003001
0.05
0.022024
0.15
0.056762
0.25
0.106729
0.35
0.171813
0.45
0.004162
0.06
0.024801
0.16
0.061078
0.26
0.112566
0.36
0.179128
0.46
0.005493
0.07
0.027735
0.17
0.065545
0.27
0.118547
0.37
0.186646
0.47
0.00699
0.08
0.030827
0.18
0.070164
0.28
0.124676
0.38
0.194274
0.48
0.008654
0.09
0.034071
0.19
0.074935
0.29
0.130949
0.39
0.202057
0.49
0.010482
0.1
0.037469
0.2
0.079856
0.3
0.137392
0.4
0.210005
0.5
0.25
Hydraulic Force (N)
0.2
0.15
0.1
0.05
0 0
0.1
0.2
0.3 Velocity (m/s)
0.4
0.5
Figure 4, Fitted Hydraulic Force Received by the Plug at Different Velocities
0.6
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3. Methodology In this paper, the study is focused on the performances of the method which use fractional calculus to describe the hydraulic drag. To make comparisons, traditional method and CFD method are also involved. Begin with the most classic quadratic method, which assumes the hydraulic drag could be described by a quadratic function as showed in equation (2) 2
=a*
+b *
(2)
Apply the Newton second law in the studied case, one could get: ∗
−
=
− (a *
2
+b * )
(3)
And the solutions of (3) are: ( )=
+ ( − ) ∗ (1 − ∗ exp (−
( )=
∗ +
∗ log − +
a m
∗ exp −
∗ (α − β) ∗ t))−1 ∗( − )∗
(4) +( − )∗t−
The ( ) in (4) represents the displacement of the falling plug, and the = =
1 2∗ 1 2∗
m a
∗ log (α − β) and
(5)
in (4) and (5) are:
(− +
2
+4∗
∗
−4∗
∗
)
(6)
(− −
2
+4∗
∗
−4∗
∗
)
(7)
Equations (2) through (5) are the classical method to describe the free falling process. The constant coefficients in [11] (3) could be optimized by searching algorithm, such as the classic genetic algorithm , the one has been used in this study. Based on the CFD result, equation (1), one could obtain the constant coefficient in (3) immediately. However, as mentioned in introduction, the result based on CFD varies depending on the change of several factors. The coefficients find by CFD could be better if one optimized those factors, but tradeoff will include time and CPU hours. For the fractional calculus method focused in this study, the basic assumption is that the hydraulic drag received by any falling body could be approximated through the fractional calculus, as described by (8) ( )
=
(8)
Apply the basic Newton law, one could have 2 ( ) 2
=
−
−
( )
(K>0)
(9)
( )
There are several accepted definitions of the fractional derivative, . And the one involved in this study is [12] [13] defined by Caputo . This definition gives the solid physics meaning of the initial condition , which makes the application of equation (9) is straightforward. The figure 5 shows the trajectory of the hydraulic drag verses different and . For the purpose of better demonstration, the hydraulic drag is showed in its log formation in figure 5. And the calculation is based on the experiment data in table 1. As showed in figure 5 (A), when the fractional order changes from 0 to 3 and the constant coefficient changes from 0 to 1, the hydraulic drag could vary roughly from −6 to 10 Newton, a very large range. This shows the fractional calculus equation (8) could be very capable to describe the hydraulic drag that changing dramatically. One could fix the coefficient K and plot another trajectory as in the figure 5 (B), this figure demonstrates how the hydraulic drag changes with respect to the fractional order, and the plot implies that the variation of hydraulic drag mainly stems from the change of the fractional order.
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(A) Log(Hydraulic Drag) verse P and K
(B) Log(Hydraulic Drag) verse P, fixed K=0.5 8
6
10 8 6
Log(Hydraulic Drag)
Log(Hydraulic Drag)
4 4 2 0 2 4
2
6 8 3 2.5
2
1
2
0.8
1.5
0.6 1
0.4 0.5
0.2
P
4 0
K
0.5
1
1.5 P
2
2.5
3
Figure 5, Hydraulic Drag Trajectory Based on Fractional Calculus Following the Caputo’s definition of fractional derivative, people could solve equation (9) in different ranges of the order p. For 2 0)
−2 )
+
−2,1 (−
−2 )
+ (13)
[15]
defined as in (14). (14)
Notice that, in (14) > 0, > 0, this is the reason we have p2 in (13) when p>2, and the reason one needs to solve (9) in different ranges of .
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= , one could have :
Similarly, note, ( )= +
3−
2− ,4−
1 2
(−
2− ,3 (−
2−
2−
)+
2− ,2 (−
)
2−
)+
(1