Experimental and Applied Mechanics, Volume 6: Proceedings of the 2010 Annual Conference on Experimental and Applied Mechanics

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 06801-1405 USA [emailprotected]

ISSN 2191-5644 e-ISSN 2191-5652 ISBN 978-1-4419-9497-4 e-ISBN 978-1-4419-9792-0 DOI 10.1007/978-1-4419-9792-0 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 acid-free 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 7-10, 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 Nano-Engineering, Micro-Nano Mechanics, Measurements and Modeling, Applied Photoelasticity Residual Stress Measurement Techniques, Thermal Methods, Bio-Composites 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 Urbana-Champaign; 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 Micro-level 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

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

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

6 The Effect of Noise on Capacitive Measurements of MEMS Geometries J.L. Chee, J.V. Clark 7 Effects of Clearance on Thick, Single-Lap Bolted Joints Using Through-the-thickness 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

53 63

65 67

viii

11 Functionally Graded Metallic Structure for Bone Replacement S. Bender

12 Dissipative Energy as an Indicator of Material Microstructural Evolution N. Connesson, F. Maq uin, F. Pierron

13 Temporal Phase Stepping Photoelasticity by Load or Wavelength M.J. Huang, H.L. An

14 Determination of the Isoclinic Map for Complex Photoelastic Fringe Patterns P. Siegmann, C. Colombo, F. Díaz-Garrido, E. Patterson

15 A Study on the Behaviors and Stresses of O-ring 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 Nano-scale 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 Dye-sensitized 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 TN-LC Cell for Enhancing Power Characteristics C.-Y. Chen, Y.-L. Lo

145

26 Full-field 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 Non-cylindrical 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 Friction-stir-welding 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

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. Dulieu-Barton, 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 Non-circular 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 Three-dimensional Matrix Construct M.d.C. Lopez-Garcia, 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

55 Dynamic and Quasi-static Measurements of C-4 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 Hierarchically-structured Polymer Composites A.L. Gershon, H.A. Bruck

347

58 Composite Design Through Biomimetic Inspirations S.A. Tekalur, M. Raetz, A. Dutta

355

59 Nano-composite Sensors for Wide Range Measurement of Ligament Strain T. Hyatt, D. Fullwood, R. Bradshaw, A. Bowden, O. Johnson

359

60 Advanced Biologically-inspired Flapping Wing Structure Development L. Xie, P. Wu, P. Ifju

365

61 Characterization of Electrode-electrolyte 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 Same-materials 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 In-Flight Wing-membrane 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. Scott-Emuakpor, 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 Stress-dependent 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 On-stage 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 Corrosion-fatigue via Ocean Waves on 2024-aluminum 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 Electro-optic Heterodyne Interferometer J.-F. Lin, C.-J. Weng, K.-L. Lee, Y.-L. Lo

621

91 Digital Shadow Moiré Measurement of Out-of-Plane Hygrothermal Displacement of TFT-LCD Backlight Modules W.-C. Wang, Y.-H. Chang

627

92 Theory and Applications of Universal Phase-shifting 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 Product-making 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 U-Shaped Beam M.E. Tuttle

711

101 Demonstration of Rod-wave Velocity in a Lecture Class D. Goldar

721

102 Influence of Friction-stir-welding 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 Post-necking Regime A. Baldi, A. Medda, F. Bertolino

727

104 Determination of Hardening Behaviour and Contact Friction of Sheet Metal in a Multi-layered 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 Fiber-type 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

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 Co-continuous 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 Portevin-Le Châtelier Bands in Austenitic Steel With Twinning Induced Plasticity L.G. Hector, Jr., P.D. Zavattieri

855

119 Identification and Enhancement of On-stage Acoustics in a Multipurpose Auditorium C. Braden, B. Hayes, R. Averbach, V. Ranatunga

865

120 Determination of Objective Architectural Acoustic Quality of a Multi-purpose Auditorium B. Hayes, C. Braden, R. Averbach, V. Ranatunga

867

121 Notch-interface Experiments to Determine the Crack Initiation Loads A. Krishnan 122 Mechanical Characterization of Alternating Magnetic Field Responsive Hydrogels at Micro-scale 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 Nano-mechanical 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

HIGH ACCURACY TOPOGRAPHY

OPTICAL

MEASUREMENTS

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 E-mail: [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 full-field 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 10.1007/978-1-4419-9792-0_1, © The Society for Experimental Mechanics, Inc. 2011

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).

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 Fabry-Perot 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]:

O 2 sin T i

(1)

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:

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:

§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:

§ 1 0 . 635 · 90 o arcsin ¨ ¸ 2 .5 ¹ ©2

82 . 704

(4)

The resulting sensitivity is:

2.5 sin (82.704q)

2.520 Pm

(5)

In the case of double illumination, the corresponding equation is:

p 2 2 n sin T o

(6)

while sensitivity is:

2. 5 2 2 sin (82.704q)

1.260 Pm

(7)

From Eq. (2), we obtain:

h ( x , y)

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

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 – 176-847A Mitutoyo; Magnification: 10X, NA: 0.3; WD: 16 mm; Depth-of-field: 8.5 Pm Stocker Yale Lasiris; He-Ne (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.

Figure 3. (a) View of the HQC226 standard illuminated with white light; b) View of the standard with superimposed grating

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 re-pixelated 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 Holo-Moiré 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 state-of-the-art 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 re-pixelated 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.

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.

[3] General Stress Optics Inc. (2008) Holo-Moiré 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 micro-level and its influence on strength”. SEM Annual Conference & Exposition on Experimental & Applied Mechanics, June 710, 2010, Indianapolis, Indiana.

INDUSTRIAL FINISHES OF CERAMIC SURFACES AT THE MICRO-LEVEL 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 E-mail: [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 micro-range 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 four-point-bending. 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 four-point-bending 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 10.1007/978-1-4419-9792-0_2, © The Society for Experimental Mechanics, Inc. 2011

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 sub-micron level. The derivation of these equations given in [1] assimilates the cavity that was represented to a Fabry-Perot 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]:

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:

p 2 n sin T o

(2)

§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:

§ 1 0 . 635 · 90 o arcsin ¨ ¸ 2 .5 ¹ ©2

82 . 704

(4)

The resulting sensitivity is:

2.5 sin (82.704q)

2.520 Pm

(5)

In the case of double illumination, the corresponding equation is:

p 2 2 n sin T o

(6)

while sensitivity is:

2. 5 2 2 sin (82.704q)

1.260 Pm

(7)

From Eq. (2), we obtain:

h ( x , y)

I( x , y ) 2S

(8)

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 25-mm 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 4-point-bending tests; (b) view of experimental setup.

Prior to having samples subject to the four-point-bending 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

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 Holo-Moiré 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 state-of-the-art 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 four-point-bending 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 four-point-bending test. Specimens were tested at room temperature in an Instron testing machine, using an articulated fixture (40 mm-20 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 four-point-bending 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

Laser-Assisted 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

As-Received Strength (MPa) 475.7 604.0 544.2 588.3 552.9 540.8 528.1 468.6 414.8

Table II. Experimental results from four-point-bending tests for ceramic B

As-Received 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 3-D view of the diamond grind surface (100 grit). Figure 4b shows the 3-D view of the LAM surface.

(a)

(b)

Figure 4. (a) 3-D view of diamond grind (100 grit) surface (b) 3-D 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 III-VI 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.

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

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 ARL-TR3417, 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) Holo-Moiré Strain Analyzer Software HoloStrain, Version 2.0. General Stress Optics Inc., Chicago, IL (USA), http://www.stressoptics.com

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 E-mail: [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 [1-5]. The AFM was designed primarily to provide high resolution images of non-conductive 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 four-quadrant 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 silicon-nitride. 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 10.1007/978-1-4419-9792-0_3, © The Society for Experimental Mechanics, Inc. 2011

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 ultra-sharp 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 non-linearity 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 (~10-50nm 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 hyper-elasticity. 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 4-node elements and 3300 nodes. The membrane was modelled as a linearly elastic material or a two-parameter Mooney Rivlin hyperelastic material: in the former case the mesh included 7500 PLANE42 4-node elements and 7500 nodes while in the latter case the mesh included 7500 PLANE182 4-node 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

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 two-parameter Mooney-Rivlin (MR) law [9-11]. The former model is described by the Hooke’s law V = EH. The strain energy density function for the Mooney-Rivlin model is:

a 10 I1 3 a 01 I 2 3

(1)

where a10 and a01 are the Mooney-Rivlin 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 Cauchy-Green 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, semi-infinite elastic material, infinitesimal theory predicts the following relationship between force F and indentation depth G:

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)

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 hyper-elastic membrane: the geometric non-linearity 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; - Mooney-Rivlin 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 F-G 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. Force-indentation 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.

In the case of hyperlastic membrane the F-G 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 nano-indentation 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 non-linearity. 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

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, 413-438, 2007. [3] Vinckier A., sem*nza G. “Measuring elasticity of biological materials by atomic force microscopy”, FEBS Letters, 430, 1216, 2008.

[4] Costa K.D., Sim A. J.F., Yin C.P. “Non-Hertzian 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.

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 in-plane 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 three-dimensional displacements and strains during dynamic events. The rational decision-making process is being utilized and the initial stages have been completed, i.e. the identification and weighting of attributes for the design, brain-storming candidate designs and evaluation of candidate designs against the attributes. Twenty-five attributes have been identified and seven selected as being essential in any successful design, namely: the boundary conditions must be reproducible; a range of in-plane and out-of-plane 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 full-field 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 10.1007/978-1-4419-9792-0_4, © The Society for Experimental Mechanics, Inc. 2011

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 real-world whereas validation is concerned with establishing how well the model represents the real-world, 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 in-plane, 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 three-dimensional measurements in dynamic loading cases is reported. 1.2 Calibration and evaluation of measurement systems for in-plane, static strain Technical Working Area 26: Full-field 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. G6RD-CT-2002-00856 (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: Three-dimensional 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)

Briefly, the reference material consists of a monolithic frame surrounding a beam subject to four-point 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 whiffle-trees 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 round-robins 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 center-line. 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 least-squares 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: Flow-chart for performing a calibration of an optical system for full-field 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 semi-infinite half-plane 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 real-world scenarios will involve significant rigid body motion.

y displacement

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 three-year 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 end-use. 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 full-field 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.

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 in-plane and out-of-plane 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

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)

The ADVISE consortium met in Fall 2009 to brainstorm candidate designs for the reference material that would be appropriate for cyclic loading, single high-speed loading events that result in deformations that vary linearly with load and single high-speed loading events that result in deformations that vary non-linearly 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 full-field strain measurement [3, 4, 6]. The latter is applicable only to in-plane, 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 three-dimensional 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 non-linear 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 round-robin 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

2 2

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)

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, 23:245-252 (2007) 2. ASME V&V 10-2006, 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 full-field strain measurement’, Optics and Lasers in Engineering, 45(5):550-564, 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, McGraw-Hill, Milano, pp.695-6, 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.3-4, pp.391-396, 2005. 9. M.P. Whelan, D. Albrecht, E. Hack, E.A. Patterson, ‘Calibration of a speckle interferometry full-field strain measurement system’, Strain, 44(2):180-190, 2008. 10. E.A. Patterson, P. Brailly, R.L. Burguete, E. Hack, T. Siebert, M.P.Whelan, ‘A challenge for high performance full-field strain measurement systems’, Strain, 43(3):167-180, 2007. 11. www.dynamicvalidation.org 12. N. Cross, Engineering Design Methods (John Wiley & Sons, London, 1989)

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, 4-3-16, Jonan, Yonezawa, Yamagata 992-8510, Japan, Japan. [emailprotected] **Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, 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 stress-strain curve, local strain distribution and minute deformation arising in specimens under tensile loading. In this time, the in-house 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 in-house system on the basis of Digital Image Correlation (DIC). Also, minute deformation behavior is measured by in-house 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 stress-strain 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 “Self-similar 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 well-known 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 micro-order specimen of pure Ni by focused ion beam process. Then, they conducted uniaxial compression test for the micro-order 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 10.1007/978-1-4419-9792-0_5, © The Society for Experimental Mechanics, Inc. 2011

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 stress-strain curve, local strain distribution and minute deformation arising in specimens under tensile loading. In this time, the in-house 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 in-house system on the basis of Digital Image Correlation (DIC). Also, minute deformation behavior is measured by in-house 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 close-packed 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

2.2 Measurement Method of Local Deformation Behavior (LDB) Digital Image Correlation (DIC) software is in-house 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 in-house software suited on a LabVIEW platform. We can semi-automatically 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 in-house software suited on a LabVIEW platform. 2.3.2 Analysis of AE-count data Also, we can obtain the relationship between cumulative AE counts and time from detected AE signals. Furthermore, we can calculate the AE-count speed-time relation from the cumulative AE counts-time curves. AE-count speed is the slope of cumulative AE counts-time curves at each time. Avalanche behavior can be seen from these curves. Calculation method of AE-count speed is shown in Fig. 3. Firstly, an arbitrary point on the cumulative AE counts-time 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 AE-count speed at the point x0 . This calculation is conducted at all points on cumulative AE counts-time curve. Obtained AE-count speed-time relation is shown in Fig. 3(c). Calculation is automatically conducted for all data by in-house data analysis software suited on a LabVIEW platform.

Figure 3. Method for calculating the AE-count speed

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 in-house 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 in-house software suited on a LabVIEW platform. Tests are performed at strain rate, 0.08%/min. A CCD camera (HC-HR70, Sony) is set in front of the specimen as shown in Fig. 5(b). A 50mm lens (VCL-50Y-M, 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 300kHz-2MHz (AE-900M, NF corporation) are mounted on another surface of the specimen like Fig. 5(c). The output signals are amplified 40 dB by the pre-amplifier (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 GPIB-USB-HS, National Instrument) to control tensile machine, a monochrome PCI frame grabber (mvTITAN-G1, Matrix Vision) to acquire 1024×768 8-bit grayscale digital image from the CCD camera and two 12-bit AD converters (PCI-3163, Interface) to acquire AE signals from the AE sensors. Then, the in-house 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

3. RESULTS 3.1 Macroscopic Stress-strain Curves of Polycrystalline Pure Ti Figure 6 shows macroscopic stress-strain (and stress-time) 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 stress-strain curve than that of 0° and 30° specimens, and 90° specimen shows more larger one than others, around yield point.

Figure 6. Macroscopic stress-strain (-time) curves for polycrystalline pure Ti 3.2 Local Deformation Behavior Around Yield Point of Polycrystalline Pure Ti Figures 7(a)~(d) demonstrate stress-time 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

(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 stress-strain 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 stress-time curve, cumulative AE counts-time curve and AE-count speed-time 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 AE-count speed-time 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

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. AE-count 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 Multi-scale 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 fast-moving 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 multi-scale 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 __

In general, it is well known that (1012) or (1011) twin boundary in deformed polycrystalline pure Ti can be observed by high-resolution 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 in-plane 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 close-packed 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 initiation-related 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.

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

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 stress-strain 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 in-house software suited on a LabVIEW platform. Obtained results are as followings. (1) From the results of macroscopic stress-strain curves, the results of 0° and 30° specimens show the same behavior. On the other hand, 45° specimen shows a little larger stress-strain 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 stress-strain 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. AE-count 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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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 on-chip measurements in-the-field. 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 off-chip 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 off-chip 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 commonly-used 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 well-suited for batch-fabricated industrial scale measurements, or for post-packaged measurements in the inthe-field. Such abilities may be achieved with electronically-probed 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 10.1007/978-1-4419-9792-0_6, © The Society for Experimental Mechanics, Inc. 2011

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

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 inner-most 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 .

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)

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)

a ,layout

a ,layout + Δw )

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

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 square-bracketed 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 square-bracketed 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 off-the-self 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.

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 low-noise features: a Faraday cage to reduce noise interference from external electro-magnetic fields; an air-damped table to reduce the noise due to building vibrations; low-reflective interior cage walls to reduce the noise due to light; and low-parasitic 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 experimentally-fitted 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.

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 peak-to-peak. 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 thermally-induced vibration due to small stiffness values. Since stiffness scales by a factor of length-scale 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.

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 thermally-induced vibrations, we have

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 thermally-induced 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 off-the-shelf 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 thermally-induced 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

[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. 129-142, 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. 1743-1761, 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. 173-181, 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, “Sub-zeptofarad sensitivity scanning capacitance microscopy,” Canadian Conference on Electrical and Computer Engineering, v. 1, pp. 455-459, 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. 380-389, (2000). J. Green, et. al., “New iMEMS Angular Rate-Sensing Gyroscope,” Analog Devices, Dialogue, pp. 37-03, (2003). Hutter and Bechhoefer, “Calibration of Atomic-Force Microscope Tips,” Review of Scientific Instruments, No. 64 (7) pp. 1868-1873, (1993).

Effects of Clearance on Thick, Single-Lap Bolted Joints Using Through-theThickness 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, single-lap, 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 single-lap, 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 single-lap 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 10.1007/978-1-4419-9792-0_7, © The Society for Experimental Mechanics, Inc. 2011

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 single-lap, 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 single-lap 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 through-the-thickness 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 non-invasively 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 single-lap joint at regular intervals through-the-thickness above the hole. Numerical analyses were performed using ANSYS as preprocessor and LS-DYNA 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 lap-joint 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 S-Glass material and an epoxy resin. The S-Glass was chosen due to having superior tensile strength than the E-glass. 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).

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).

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).

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

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.

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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.

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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 hand-rolling 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, lap-bolted 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 bolt-hole 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 S-glass in an Epoxy resin resulting in a 12.7mm (0.5 inch) thick composite panel. Point measurements

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). “Three-dimensional finite element analysis of single-bolt, single-lap composite bolted joints: part II-effects of bolt-hole clearance” Composite Structures 71: 159-175 [2] McCarthy, M.A., Lawlor, V.P., et al. (2002) “Bolt-hole clearance effects and strength criteria in single-bolt, single-lap, composite bolted joints.” Composites Science and Technology 62: 1415-1431. [3] Chen, Wen-Hwa, Lee, Shyh-Shiaw, et al. (1995) “Three-dimensional contact stress analysis of a composite laminate with bolted joint.” Composite Structures 30: 287-297 [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] Lopez-Anido, 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). “Three-Dimensional Strain Analysis of Single-Lap Bolted Joints in Thick Composites using Fiber-Optic Strain gages and Finite Element Method”, Journal of Strain Analysis for Engineering Design, Accepted for publication.

Deformation and Performance Measurements of MAV Flapping Wings Wu, Pin. University of Florida, MAE-A #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 hummingbird-size 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 four-camera 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 10.1007/978-1-4419-9792-0_8, © The Society for Experimental Mechanics, Inc. 2011

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. 1094-1097. 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., “Three-dimensional Kinematics of Hummingbird Flight,” J. of Exp. Bio., Vol. 210, 2007, pp. 2368-2382.

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 7075-T4, 2024-T3, 6061-T6 and 5182-O were considered for investigation. Copper-110 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 LS-DYNA 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 strength-to-weight 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 quasi-static case. Maiden and Green [2] conducted experiments using Hopkinson pressure bar setup to and found out that Aluminum 6061-T651 and 7075-T6 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 high-strain rate constitutive behavior of aluminum alloys. Hence, for the first time, this paper discuss about both experimental and numerical investigation of high-strain rate behavior of four different aerospace aluminum alloys 7075-T4, 2024-T3, 6061-T6 and 5182-O under above said range. EXPERIMENTAL DETAILS Quasi-static Characterization The quasi-static compression tests were performed using Instron materials testing system-5585 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 5182-O 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 AA5182-O 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 axi-symmetric split Hopkinson pressure bar setup was modeled using the commercially available LS-DYNA 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 LS-DYNA 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 10.1007/978-1-4419-9792-0_9, © The Society for Experimental Mechanics, Inc. 2011

RESULTS AND DISCUSSION

Fig. 1. Typical real time strain pulses obtained from SHPB

Fig. 3. Force equilibrium conditions for Al 7075-T4

Fig. 2. True stress strain curve for Al 7075-T4

Fig. 4. Comparison of numerical dynamic true stressstrain behavior with those of experiments for Al 7075-T4

Dynamic Constitutive Response The typical real-time strain pulses obtained for Al 6061-T6 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 quasi-static true stress-strain curves for Al 7075-T4 in Fig. 2. It can be noticed from the figure that Al 7075-T4 shows no significant rate sensitivity and there is no change in the yield strength as the strain rate increases. Force equilibrium conditions of Al 7075-T4 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, 113-123, 1953. 2. Maiden, C. J., Green, S. J.: Compressive Strain-Rate Tests on Six Selected Materials at Strainrates from −3 10 to 10° in./in./sec. Journal of Applied Mechanics, 33, 496-504 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, 676-700, 1949.

Estimating surface coverage of gold nanoparticles deposited on MEMS

N. Ansaria,*, K. M. Hursta and W. R. Ashursta Department of Chemical Engineering, Auburn University, Auburn, Al-36849, 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 10.1007/978-1-4419-9792-0_10, © The Society for Experimental Mechanics, Inc. 2011

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

( wL )

[2]

4ʌ M3eff2 ǻ f R

[3]

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 gas-expanded 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 gas-expanded 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 gas-expanded liquid nanoparticle deposition technique for reducing the adhesion of silicon microstructures”, Nanotechnology, 20(18), 185303(9pp), 2009.

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 micro-indentation. 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 Gradzka-Dahlke 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 ball-point 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 10.1007/978-1-4419-9792-0_11, © The Society for Experimental Mechanics, Inc. 2011

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. Quasi-static 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, 901-904, 2005 [3] Gradzka-Dahlke M. et al., Modification of Mechanical Properties of Sintered Implant Materials…, Journal of Materials Processing Technologies, 204, 199-205, 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, 21-27, 1983

Dissipative energy as an indicator of material microstructural evolution N. Connesson1,a 1

F. Maquin2,a

F. Pierron3,a 3

[emailprotected], [emailprotected], [emailprotected]

Laboratoire de mécanique et procédés de fabrication (LMPF), Arts et Métiers ParisTech, rue St Dominique, BP 508, 51006 Châlons-en-Champagne 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 [4-6]. 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.

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 10.1007/978-1-4419-9792-0_12, © The Society for Experimental Mechanics, Inc. 2011

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 Kelvin-Voigt 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) 1141-1144 [2] G. Fargione, A. Geraci, G. La Rosa, A. Risitano, International Journal of Fatigue 24, (2002) 11-19 [3] D. Caillard, J.L. Martin, Thermally activated mechanisms in crystal plasticity (Pergamon, Amsterdam London 2003) 85-123 [4] A. Granato, K. Lücke, Journal of Applied Physics, 27:6, (1956) 583-593 [5] T. Tanaka, S. Hattori, Bulletin of the JSME, (21:161) (1978) 1557-1564 [6] H. Mughrabi, F. Ackermann, K. Herz, ASTM Special Technical Publication, 675, (1979) 69-105 [7] A. Chrysochoos, H. Louche, International Journal of Engineering Science 16, (2000) 1759-1788 [8] F. Maquin, F. Pierron, Mechanics of Materials 41, (2009) 928-942

Temporal Phase Stepping Photoelasticity by Load or Wavelength

M.J. Huang and H.L. An Mechanical Engineering, National Chung Hsing University, 250, Kuo-Kuang Road, Taichung, Taiwan 40227, R.O.C.

ABSTRACT Phase-stepping techniques in photoelasticity own the isoclinic-isochromatic 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 electro-optic 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 [1-4]. However, the interaction between the isoclinic parameter and relative retardation makes the phase retrieval work very difficult. Many papers [5-19] 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 isoclinic-isochromatic 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 10.1007/978-1-4419-9792-0_13, © The Society for Experimental Mechanics, Inc. 2011

n I 4s I0 n I2s I0 1 1 ° tan ® s s 4 ¯° n I3 I0 n I1 I0

½° ¾ °¿

(1)

sin I ½ 1 tan1 ® ¾, 4 ¯ cos I ¿

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 ^

1 4

I sj

1 3

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

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

Fig. 2. The optical arrangement of circular polariscope

Table 1 Polariscope arrangement and the intensity results for phase stepping algorithm

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

Ib 21 Ia (1 cos G )

3S / 4

S /4

Ib 21 Ia (1 cos G )

3S / 4

0 0

Ib 21 Ia (1 sin 2I sin G )

3S / 4

S /4

Ib 21 Ia (1 cos 2I sin G )

S /4

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

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 newly-developed 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

(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 NSC98-2221-E-005-009.

References [1] A.V. Sarma, S.A. Pillai, G. Subramanian and T.K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Experimental Mechanics, pp. 24-29, 1992. [2] K. Ramesh and V. Ganapathy, “Phase-shifting methodologies in photoelastic analysis – the application of Jones calculus,” Journal of Strain Analysis, vol. 31, pp. 423-432. [3] K. Ramesh and S.K. Mangal, “Automation of data acquisition in reflection photoelasticity by phaseshifting methodology” Strain, pp. 95-100, 1997. [4] S. Yoneyama, Y. Morimoto and R. Matsui, “Photoelastic fringe pattern analysis by real-time phase-shifting method,” Optics and Lasers in Engineering, vol. 39, pp. 1-13, 2003. [5] Z.F. Wang and E.A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Optics and Lasers in Engineering, vol. 22, pp. 91-104, 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. 387-400, 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. 421-436, 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. 2891-2896, 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. 612-617, 2007 [10] T.Y. Chen, “A simple method for the determination of photoelastic fringe order,” Experimental Mechanics, vol. 40, pp. 256-260, 2000. [11] T.Y. Chen, “Digital determination of photoelastic birefringence using two wavelengths,” Experimental Mechanics, vol. 37, pp. 232-236, 1997. [12] M.J. Huang and Po-Chi Sung, “Regional phase unwrapping algorithm for photoelastic phase map,” Optics Express, vol. 18, pp. 1419-1429, 2010.

Determination of the isoclinic map for complex photoelastic fringe patterns

Philip Siegmann1, Chiara Colombo2 , Francisco Díaz-Garrido3 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 phase-shifted 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 phase-shifted 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. Two-dimensional models can be viewed directly while subject to load whereas stress must be ‘frozen’ into three-dimensional 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 phase-stepping or phase-shifting has gained in popularity since it was first applied to photoelasticity by amongst others, Hecker and Morche [3]. Phase-stepping is usually T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/978-1-4419-9792-0_14, © The Society for Experimental Mechanics, Inc. 2011

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 phase-stepping 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 phase-stepping 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 phase-shifted fringe patterns [14] (bottom left and right) from obtained with an applied load of 90N. Note that I1 corresponds to a dark-field circular polariscope. Much effort has been expended on developing robust methods of processing phase-stepped 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 real-time 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 phase-stepped images combined with fuzzy set classification. This technique and its derivatives [6, 7] are fast and almost completely automatic in that very little

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 phase-tracking (RPT) using five phase-shifted 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 up-to-date existing techniques [4, 6 - 8] and only those based on regularised phase-tracking [9, 12] have been found to be successful with that due to Siegmann et al [12] being much faster.

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. PHASE-STEPPING ALGORITHM The six-step photoelastic phase-stepping approach is selected here because it is employed in industrial and research laboratories although other phase-stepping 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)

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 ¨

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 hole-drilling cannot be calculated directly from theoretical considerations, they are usually obtained by numerical procedures such as finite-element 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 micro-indentations and has the configuration of delta or rectangular rosette [11]. A delta rosette, also called 60-deg ISSR, contains three six-faced 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/Ring-core method is a combination of the ISSR method and ring-core cutting method [12, 13]. The ringcore cutting method is a stress-relief method. The ring-core 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 ring-core center and the ISSR are shown in Figure 1. Similar to strain gauge hole-drilling method, the relieved strains and slopes are used to back-calculate 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/ring-core method. X-RAY DIFFRACTION METHOD Different from strain gauge and ISSR methods, X-ray diffraction method and neutron diffraction method are nondestructive methods. When a monochromatic X-ray 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

195

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 X-rays 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

O (Core Center) Figure 1.

Principal Stress V1 Direction

~100Pm

25 mm 122 mm

Rectangular and polar coordinates in the ISSR/ring-core 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 X-ray 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 X-rays 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

b) Figure 3.

Experimental set-up for: a) air quenching; b) water quenching

196

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) hole-drilling method, ISSR ring-core method, X-ray diffraction method and neutron diffraction method were used. For the casting quenched in air flow, only resistance strain rosette (RSR) hole-drilling 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.

a) RSR hole-drilling and ISSR/ring-core on top of the water-quenched aluminum casting; b) RSR hole-drilling 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 water-quenched 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.

197

RSR R X-ray

N Neutron

ISSR R

RSR on side o

RSR S

Figure 5.

Locations of measurements using different methods

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]. X-ray 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 X-ray 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

198

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 X-ray 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 X-ray 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).

Sample 2L, lines c/d

Longit.

stress, MPa

Transv.

20 0 -20 -40 -60

Inner side Surface -10

-8

-6

Outer side Surface -4

-2

through thickness position, mm

a) Figure 7.

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

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

199

Figure 8 compares the residual stress distributions measured at the thick wall of the air-quenched aluminum casting to those of the water-quenched casting. It is seen that the residual stresses in the air-quenched aluminum casting are less than 10 MPa and pretty uniform, but in the water-quenched aluminum casting, the absolute values of residual stresses vary from 40 MPa to 100 MPa. Small and uniform residual stresses in air-quenched 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 hole-drilling method, Interferometric Strain/Slope Rosette, X-ray 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. 417-423. [2] Rose, A., Kessler, O., Hoffmann, F., 2006, "Quenching Distortion of Aluminum Castings-Improvement by Gas Cooling," Materialwissenschaft Und Werkstofftechnik, 37(1) pp. 116-121. [3] Li, K., Xiao, B., and Wang, Q., 2009, "Residual Stresses in as-Quenched Aluminum Castings," SAE International Journal of Materials & Manufacturing, 1(1) pp. 725-731. [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. 567-574. [5] Lu, J., and James, M., 1996, "Handbook of measurement of residual stresses," Fairmont Press,Inc, Lilburn, GA, USA, pp. 237. [6] ASTM designation: E837-01, 2002, ASTM international, Philadelphia, PA, pp. 703-712. [7] Vishaymicro-Measurements, 2007, "Tech Note TN-503:Measurement of Residual Stresses by the Hole-Drilling Strain Gage Method," 2009pp. 33. [8] Schajer, G. S., 1988, "Measurement of Non-Uniform Residual Stresses using the Hole-Drilling Method. Part IStress Calculation Procedures," J.Eng.Mater.Technol.(Trans.ASME), 110pp. 338-343. [9] Schajer, G. S., 1988, "Measurement of Non-Uniform Residual Stresses using the Hole-Drilling Method. Part IIPractical Application of the Integral Method," J.Eng.Mater.Technol.(Trans.ASME), 110pp. 344-349. [10] Xiao, B., Li, K., and Rong, Y., 2010, "Automatic Determination and Experimental Evaluation of Residual Stress Calibration Coefficients for Hole-Drilling Strain Gage Integral Method," Strain, In Press, Accepted Manuscript. [11] Li, K., 1995, "Interferometric 45 and 60 Strain Rosettes," Applied Optics, 34(28) pp. 6376-6379. [12] Li, K., and Ren, W., 2007, "Application of Minature Ring-Core 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 Ring-Core 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 Gnaeupel-Herold, T., April 22, 2005, "Residual Stresses in GM aluminum Castings," NIST Center for Neutron Research, Gaithersburg, MD. [15] Hornbach, D., May 24,2005, "X-ray Diffraction Determination of the Surface and Subsurface Residual Stresses in Two A356 Aluminum Castings," Lambda Technologies, 227-12260, Cincinnati, Ohio.

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, e-mail: [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 3-dimensional 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 [1-2]: 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 [1-4]. 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 0-65 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 10.1007/978-1-4419-9792-0_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 [9-12] 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 5-7 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 thermo-physical properties of the material (thermal expansion coefficient, thermal conductivity, density, etc). This paper studied the magnitude of residual stresses developed in SLM-fabricated 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. 59-64. 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. 195-198. 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. 159-166. Klingbeil N., Beuth J., Chin R., Amon C., Proc. of Solid Freeform Fabrication Sym., University of Texas, Austin, 1998, pp. 367-374. Mercelis P., Kruth JP., Residual stresses in selective laser sintering and selective laser melting, Rapid Prototype J, 2006, vol. 12, pp. 254-265. 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), 93-104, 2009. ASTM E 837 Standard method for determining residual stresses by the hole-drilling 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 TN-503-6. Schajer G.S., Measurement of non-uniform 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 non-uniform 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. 344-349. 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. 3-11 (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. Euro-uRapid 2002 International User's Conf., A-5, 2002.

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 strain-to-stress 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 PC-controlled miniature 3-axis drilling machines for orbital drilling in 1999. Two machines have been used over a 10-year 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 test-surface. A hole is drilled into the test-surface 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 – Type-1 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 air-turbine (ca. 250,000 rpm). T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/978-1-4419-9792-0_38, © The Society for Experimental Mechanics, Inc. 2011

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Fig. 1 Type-1 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 pre-loaded to create a longitudinal tensile stress. Subsequently, parts of the plate surfaces were masked while other parts were shot-peened. Target gauges (type EA-06-031RE-120) 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 (un-peened) 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 near-surface 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 shot-peening process is excessive; there is no trend for sub-surface stresses to revert to the un-peened state. This clearly demonstrates the severe limitation of the calculation method when applied to non-uniform 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 – Type-2 Drilling Machine and Integral Method A second drilling machine was manufactured. This machine was developed from the Type-1 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 type-2 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 strain-to-stress 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 shot-peened 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 Type-2 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 near-surface shot-peening 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 sub-surface tensile peak in excess of 400 MPa (greater than the un-peened stress) occurs at depth 0.2 mm. Values of sub-surface 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 un-loaded 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 shot-peening 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: high-temperature cyano-acrylate 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 Type-2 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 test-pieces and material samples. At Stresscraft, two machines and operators were used to drill, typically, 1000 to 1200 gauges per year. 1999 – 3-Axis 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 3-axis 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 ball-screw driven by 200-step/rev stepper motor via a toothed belt. The machine is shown in Fig. 6. The drill head holder is fixed to a tee-slotted 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/ ball-screw 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 MS-Excel file for processing.

Fig. 7 PC control panel for the 3-axis hole drilling machine It was clear from the first trials that results obtained using the 3-axis 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 switch-off 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 3-axis 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 sub-increments to extract more stress distribution detail from near-surface 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 near-surface increments for 031-size gauges and 32 Pm near-surface increments for 062-size 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 test-pieces

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 near-surface tensile stresses and stress gradients which are not revealed by the coarser increment set. Fig. 9 gives a summary of residual stresses at 031-size gauges applied to a series of titanium alloy test-pieces and drilled using 16 Pm near-surface increments. The conditions of the material surface for the five gauges are : 1. Wire-EDM cut surface 2. EDM surface; recast layer removed using a fine abrasive 3. Shot-peened 4. Shot peened (as 3) followed by smoothing using a fine abrasive to remove peening indentations 5. Shot-peened 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; High-strength nickel alloy disc forging This example is concerned with the determination of residual stresses in an aircraft engine disc forging following quenching, heat-treatment 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 Rolls-Royce 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 equi-spaced 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

Fig. 10 Section through a compressor disc

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Near-surface 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 sub-surface stresses reported are not affected by the machining process. The gauge type used was the Measurements Group CEA-06-062UL-120 as shown in Fig. 11. Each gauge was drilled using a 3-axis 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 out-of-plane distortions during machining. x The level of circumferential variation of stresses around the disc. Irregular temperature gradients during quenching or heat-treatment lead to circumferential variations in residual stresses and possible ovalisation distortions during machining. Stress distributions in Fig. 12 confirm the maximum depth of machining-affected 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; High-strength 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 Rolls-Royce 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 shot-peened 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 centre-lines and at positions displaced by - 1 mm and +1 mm from the centre-lines. Measurements were to be made at both the OD and ID at 3 equi-spaced circumferential locations. The wall thickness at the weld is ca. 6 mm.

Gauge positions OD centre

OD+1 mm

OD-1 mm

ID-1 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 centre-line; each gauge was installed with a circumferential offset to avoid any detectable interference from previously drilled holes (Fig. 14). The gauge type used was EA-06-031RE-120.

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 near-surface drilling increments of 16 Pm; sub-orbits 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, near-surface 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 run-out of the peening effect (at depth ca. 200 Pm). It has been demonstrated that sub-surface 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 non-uniform 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 PC-controlled 3-axis 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 Rolls-Royce 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 Non-uniform Residual Stresses by the Hole-drilling Method, Proc., SESA XIV (1), pp. 181-194, 1956 [2] RENDLER, N.J. and VIGNESS, I., Hole drilling Strain-gauge Method of Measuring Residual Stresses, Exp. Mech., 6 (12), pp. 577-586, 1966 [3] ASTM E 837-85 to –08e1, Determining Residual Stresses by the Hole drilling Strain-Gauge Method, 1985 to 2008 [4] TECHNICAL NOTES TN-503-2 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 Non-Uniform Residual Stresses Using the Hole Drilling Method, J. Eng. Materials and Technology, 110 (4), Part I: pp.338-343, Part II: pp.3445-349, 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

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 non-destructive methods because of the specimen damage they cause, the non-destructive 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 high-precision 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 “locked-in” 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 well-established specialty. There are two general classes of residual stress measurement methods, destructive and non-destructive. The non-destructive 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

material cutting and deformation measurement technology

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 10.1007/978-1-4419-9792-0_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 stress-free. 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 thin-walled 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 thin-wall 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 shot-peening [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 hole-drilling 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 full-field optical techniques such as Moiré interferometry [19], ESPI [20] and DIC [21]. Figure 4(a) illustrates the process. The hole-drilling 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. Hole-drilling methods: (a) conventional hole-drilling method, (b) ring-core method, (c) deep hole method (from [29]). In the basic implementation of the hole-drilling method, the surface deformations are measured as the hole is drilled to a depth approximately equal to the hole radius. The in-plane 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 hole-drilling 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 ring-core method [25,26] is an “inside-out” variant of the hole-drilling 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 ring-core method is a circular generalization of the two-slots method [2]. As with the hole-drilling method, the ring-core method has a basic implementation to evaluate in-plane stresses [26], and an incremental implementation to determine the stress profile [27]. The ring-core method has the advantage over the hole-drilling 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 ring-core 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 hole-drilling 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 hole-drilling calculations. The slitting method has the advantage over the hole-drilling method that it can evaluate the stress profile over the entire specimen depth, the surface strain gauge providing data for the near-surface 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 hole-drilling method provides all three in-plane 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 full-field residual stress measurements, typically within the cross-section of a prismatic specimen. It consists of cutting through the specimen cross-section 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 2-dimensional map of the residual stress distribution over the entire material cross-section. In comparison, other techniques such as layer removal and hole-drilling give one-dimensional 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) stress-free 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 high-speed 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 high-precision 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 Electro-Discharge Machining (EDM) to residual stress measurements further advanced measurement capabilities to difficult-to-machine materials [38], deep-hole 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 full-field 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)

(b)

(c)

Figure 7. Focused Ion Beam machining. (a) splitting [39], (b) ring-coring [41], (c) slitting [42].

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 8. Full-field 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) 2-D 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 600-1200 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 hole-drilling measurement [45]. Each light or dark line represents a contour line of in-plane surface displacement, in the x-direction in Figure 8(a). For typical optical arrangements, the in-plane displacement increment between fringe lines is about 0.5Pm. The full-field 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 non-uniform 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 full-field 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 hole-drilling measurement. Digital Image Correlation [21,48-50] is a versatile optical technique for measuring surface displacements in two or three dimensions. The 2-D technique, schematically illustrated in Figure 8(e), involves applying a textured pattern on the specimen surface and imaging the region of interest using a high-resolution 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 well-calibrated optical system, displacements of +/- 0.02 pixel can be resolved. Figure 8(f) shows an example vector plot of the measured displacements in a hole-drilling 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. [39-42]. 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)

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

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 hole-drilling 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 hole-drilling method. The solution procedures for the various measurement methods were developed on an ad-hoc 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 E837-08 [7] for hole-drilling, 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 full-field 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

extracting the data from the image with a minimum of human interaction, preferably none

using the available data in a compact and stable computation, preferably a linear one.

Least-squares [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 rigid-body motions, thermal strains, plasticity, material non-isotropy, 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, full-field 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 full-field 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 ultra-high-precision cutting tools, laser interferometers, high-resolution 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.35-40, 1981. 2. Schwaighofer, J. “Determination of Residual Stresses on the Surface of Structural Parts”, Experimental Mechanics, Vol.4, No.2, pp.54-56, 1964. 3. Jullien, D. and Gril, J. “Growth Strain Assessment at the Periphery of Small-diameter Trees using the Twogrooves Method: Influence of Operating Parameters Estimated by Numerical Simulations”, Wood Science and Technology, Vol.42, No.7, pp.551-565, 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.89-98, 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 E1928-07, 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.172-175, 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.157-166, 1987. 11. Tebedge, N., Alpsten, G. and Tall, L. “Residual-stress Measurement by the Sectioning Method”, Experimental Mechanics, Vol.13, No.2, pp. 88-96, 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.130-134, 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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14. Hospers, F. and Vogelesang, L. B. “Determination of Residual Stresses in Aluminum-alloy Sheet Material”, Experimental Mechanics, Vol.15, No.3, pp.107-110, 1975. 15. Östlund, M., Östlund, S., Carlsson, L.A. and Fellers, C. “Experimental Determination of Residual Stresses in Paperboard”, Experimental Mechanics, Vol.45, No.6, pp.493-497, 1985. 16. Lu, J. (ed.) “Handbook of Measurement of Residual Stresses”, Fairmont Press. Lilburn, USA, 1996. 17. Measurements Group. “Measurement of Residual Stresses by Hole-Drilling Strain Gage Method”, Tech Note TN-503-6, Vishay Measurements Group, Raleigh, NC, 2001. 18. Rendler, N. J. and Vigness, I. “Hole-Drilling Strain-gage Method of Measuring Residual Stresses, Experimental Mechanics, Vol.6, No.12, pp.577-586, 1966. 19. 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.837-843, Part II: pp.844-850, 1998. 20. Nelson, D.V. and McCrickerd, J.T. “Residual-Stress Determination Through Combined Use of Holographic Interferometry and Blind-Hole Drilling”, Experimental Mechanics, Vol.26, No.4, pp.371-378, 1986. 21. McGinnis, M. J., Pessiki, S. and Turker, H. “Application of Three-dimensional Digital Image Correlation to the Core-drilling Method.” Experimental Mechanics, Vol.45, No.4, pp.359-367, 2005. 22. Grant, P. V., Lord, J. D. and Whitehead, P. S. “The Measurement of Residual Stresses by the Incremental Hole Drilling Technique”, Measurement Good Practice Guide, No.53. National Physical Laboratory, UK, 2002. 23. ASTM. “Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain-Gage Method”, Standard Test Method E837-08, American Society for Testing and Materials, West Conshohocken, PA, 2008. 24. Schajer, G. S. “Measurement of Non-Uniform Residual Stresses using the Hole-Drilling Method”, Journal of Engineering Materials and Technology, Vol.110, No.4, Part I: pp.338-343, Part II: pp.344-349, 1988. 25. Milbradt, K.P. “Ring-Method Determination of Residual Stresses”, Proc. SESA, Vol.9, No.1, pp.63-74, 1951. 26. Kiel, S. “Experimental Determination of Residual Stresses with the Ring-Core Method and an On-Line Measuring System”, Experimental Techniques, Vol.16, No.5, pp.17-24, 1992. 27. Ajovalasit, A., Petrucci, G. and Zuccarello, B. “Determination of Non-Uniform Residual Stresses using the Ring-Core Method”, Journal of Engineering Materials and Technology, Vol.118, No.2, pp.224-228, 1996. 28. Leggatt, R. H., Smith, D. J., Smith, S.D. and Faure, F. “Development and Experimental Validation of the Deep Hole Method for Residual Stress Measurement”, J. Strain Analysis, Vol.31, No.3, pp.177-186, 1996. 29. DeWald, A. T. and Hill, M. R. “Improved Data Reduction for the Deep-Hole Method of Residual Stress Measurement”, Journal of Strain Analysis, Vol.38, No.1, pp.65-78, 2003. 30. Amadei, B. and Stephansson, O. “Rock Stress and its Measurement”, Chapman and Hall, London, 1997. 31. Prime, M. B. “Residual Stress Measurement by Successive Extension of a Slot: The Crack Compliance Method”, Applied Mechanics Reviews, Vol.52, No.2, pp.75-96, 1999. 32. Germaud M., Cheng, W., Finnie, I., and Prime, M. B. “The Compliance Method for Measurement of Near Surface Residual Stresses - Analytical Background”, Journal of Engineering Materials and Technology, Vol.119, No.4, pp.550-555, 1994. 33. Prime, M. B. “Cross-Sectional Mapping of Residual Stresses by Measuring the Surface Contour After a Cut”, Journal of Engineering Materials and Technology, Vol.123, No.2, 2001. 34. Pagliaro, P., Prime, M.B., Clausen, B., Lovato, M.L., Robinson, J.S., Schajer, G.S., Steinzig, M.L., Swenson, H. and Zuccarello B. “Mapping Multiple Residual Stress Components Using the Contour Method and Superposition”, Intl. Conference on Residual Stresses, Denver, CO, August 6-8, 2008. 35. DeWald, A.T. and Hill, M.R. “Multi-Axial Contour Method for Mapping Residual Stresses in Continuously Processed Bodies”, Experimental Mechanics, Vol.46, No.4, pp.473–490, 2006. 36. Flaman, M. T. “Brief Investigation of Induced Drilling Stresses in the Center-Hole Method of Residual-Stress Measurement”, Experimental Mechanics Vol.22, No.1, pp.26 –30, 1982. 37. Beaney, E. M. “Accurate Measurement of Residual Stress on any Steel Using the Centre Hole Method”, Strain, Vol.12, No.3, pp.99-106, 1976. 38. Lee, H.T., Rehbach, W.P., Hsua, F.C., Tai, T.Y. and Hsua, E. “The study of EDM Hole-Drilling Method for Measuring Residual Stress in SKD11 Tool Steel”, Journal of Materials Processing Technology, Vol.149, No.1-3, pp.88–93, 2004. 39. McCarthy, J., Pei, Z., Becker, M. and Atteridge, D. “FIB Micromachined Submicron Thickness Cantilevers for the Study of Thin Film Properties”, Thin Solid Films, Vol.358, No.1, pp.146-151, 2000.

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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 2-D, cross-sectional residual-stress map. A part is cut in two using a precise and low-stress 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 [2-4], 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 cross-sectional 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 [5-10], friction welding [6,7,11-13] and fusion welding [14-21]. 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 two-dimensional (2-D) stress maps have significant limitations. The neutron diffraction method is nondestructive but sensitive to micro-structural 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 [27-30].

T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/978-1-4419-9792-0_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 stress-free 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 x-direction 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 x-direction, 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.

Anti-Symmetric Errors The issues with both transverse displacements and anti-symmetric 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 anti-symmetric portion caused by shear stress.

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Figure 2. Asymmetric contours can be separated into the symmetric portion and the anti-symmetric portion, which averages away. Shown for the example of shear stresses.

The effect of shear stresses, and indeed all anti-symmetric 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 x-direction, based on the average contour. The shear stresses (Wxy and Wxz) are constrained to zero in the solution. This stress-free constraint is automatically enforced in most implicit, structural, finite-element 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 Poisson-type 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 anti-symmetric 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 cross-sectional dimension unless specific information about the stress variation is known [12]. Figure 3 shows another example of an anti-symmetric error that averages away. If cut path is crooked in space, the resulting surface contours are anti-symmetric.

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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 anti-symmetric 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 re-cut 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 zero-width crack causes only effects that average away, as shown in Figure 4. Considering anti-symmetric 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

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 two-dimensional 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 user-defined 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 stress-free 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/(1-Q ); 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 y-displacement 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

241

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

0.9

(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.

242

1.2

Theoretical

0.8

Including Cutting Error

Stress

0.6

0.4

0.2

-0.2

-0.4

-0.6 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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 zig-zag 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 peak-to-valley 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.

243

0.004 0.003

w/t

0.002

5.0% 4.3%

0.001

3.6% 3.0%

0.000

2.3% 1.6%

-0.001

1.0% contour

-0.002 -0.003 -0.004 0.0

0.2

0.4

0.6

0.8

1.0

a/t

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%

Maximum deviation

12.0% 10.0% 8.0%

Simulation

6.0%

Representative experimental slot width

4.0% 2.0% 0.0% 0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

Slot width relative to beam thickness Figure 10. The peak values of bulge error from Figure 9 are normalized relative to the peak-to-peak 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 finite-element 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

244

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 stress-relieved stainless steel beam, 30.0mm deep. Residual stresses were induced in the beam by loading it into the plastic range using a four-point 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 stress-strain 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 x-ray diffraction, and within the associated uncertainty ranges, the results agreed with those of the stress-strain 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 zinc-coated 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.

246

Bend test prediction Initial results After correction

150

Residual Stress (MPa)

100 50 0 -50 -100 -150 0

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 non-ideal 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

Free side Clamped side Smoothed fits

0.008

Contour, x (mm)

0.006 0.004 0.002 0.000 -0.002 -0.004 -0.006 -0.008 -0.010 0

Position, y (mm) Figure 15. The surface contours measured on the two halves of beam A are very different.

247

300

Independent results Contour Bend test prediction Free Side Slitting Clamped Side neutron Average

250 200

Stress (MPa)

150 100 50 0 -50 -100 -150 -200 0

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 over-predicted the stress magnitude, see Figure 16, the correction was also over-predicted, which led to a non-convergent 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

Initial

200

150

Iter. 1 Iter. 2

150

100

Iter. 3

Bend test

Iter. 5 50

MPa

Last 2 iters.

Iter. 4

100

Iter. 6

-50 -50

-100 -150

-100

-200 mm

-150

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.

248

150

100

MPA

0 0

-50

-100

-150

initial iter. 1 iter. 2 iter. 3 iter. 4 iter. 5 iter. 6 bend

-200

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. Anti-symmetric 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

249

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 stress-free 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 DE-AC5206NA25396. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royalty-free 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 70:2211-2264. [2] Prime MB (2001) Cross-Sectional Mapping of Residual Stresses by Measuring the Surface Contour after a Cut. Journal of Engineering Materials and Technology 123:162-168. [3] Prime MB, Sebring RJ, Edwards JM, Hughes DJ, Webster PJ (2004) Laser Surface-Contouring and Spline Data-Smoothing for Residual Stress Measurement. Experimental Mechanics 44:176-184. [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 126:465-473. [6] Hatamleh O, Lyons J, Forman R (2007) Laser Peening and Shot Peening Effects on Fatigue Life and Surface Roughness of Friction Stir Welded 7075-T7351 Aluminum. Fatigue and Fracture of Engineering Material and Structures 30:115-130. [7] Hatamleh O (2008) Effects of Peening on Mechanical Properties in Friction Stir Welded 2195 Aluminum Alloy Joints. Materials Science and Engineering: A 492:168-176. [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 44:13-27. [9] Liu KK, Hill MR (2009) The Effects of Laser Peening and Shot Peening on Fretting Fatigue in Ti-6Al-4V Coupons. Tribology International 42:1250-1262. [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 209:4822-4829. [11] Woo W, Choo H, Prime MB, Feng Z, Clausen B (2008) Microstructure, Texture and Residual Stress in a Friction-Stir-Processed AZ31B Magnesium Alloy. Acta Mat. 56:1701-1711. [12] Prime MB, Gnaupel-Herold T, Baumann JA, Lederich RJ, Bowden DM, Sebring RJ (2006) Residual Stress Measurements in a Thick, Dissimilar Aluminum Alloy Friction Stir Weld. Acta Mat. 54:4013-4021. [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 25:640-650. [14] Zhang Y, Pratihar S, Fitzpatrick ME, Edwards L (2005) Residual Stress Mapping in Welds Using the Contour Method. Materials Science Forum 490/491:294-299. [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 41-42:391-400.

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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 Multi-Pass Groove Weld Specimens Using Neutron Diffraction and the Contour Method. Materials Science Forum 524/525:671-676. [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 85:118-127. [18] Zhang Y, Ganguly S, Edwards L, Fitzpatrick ME (2004) Cross-Sectional Mapping of Residual Stresses in a VPPA Weld Using the Contour Method. Acta Mat. 52:5225-5232. [19] Thibault D, Bocher P, Thomas M (2009) Residual Stress and Microstructure in Welds of 13%Cr-4%Ni Martensitic Stainless Steel. Journal of Materials Processing Technology 209:2195-2202. [20] Hacini L, Van Lê N, Bocher P (2009) Evaluation of Residual Stresses Induced by Robotized Hammer Peening by the Contour Method. Experimental Mechanics 49:775-783. [21] Turski M, Edwards L (2009) Residual Stress Measurement of a 316L Stainless Steel Bead-on-Plate Specimen Utilising the Contour Method. International Journal of Pressure Vessels and Piping 86:126-131. [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 11:187-193. [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 15:147-154. [24] Martineau RL, Prime MB, Duffey T (2004) Penetration of HSLA-100 Steel with Tungsten Carbide Spheres at Striking Velocities between 0.8 and 2.5 km/s. International Journal of Impact Engineering 30:505-520. [25] Holden TM, Suzuki H, Carr DG, Ripley MI, Clausen B (2006) Stress Measurements in Welds: Problem Areas. Materials Science and Engineering A 437:33-37. [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) Multi-Axial Contour Method for Mapping Residual Stresses in Continuously Processed Bodies. Experimental Mechanics 46:473-490. [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 Residual-Stress Components Using the Contour Method and Multiple Cuts. Experimental Mechanics 50:187-194. [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 Multi-Axial Contour Method. Acta Mat. 56:44174428. [31] Bueckner HF (1973) "Field Singularities and Related Integral Representations." Mechanics of Fracture G. C. Sih, ed., 239-314. [32] Cheng W, Finnie I, Gremaud M, Prime MB (1994) Measurement of near-Surface Residual-Stresses Using Electric-Discharge Wire Machining. Journal of Engineering Materials and Technology-Transactions of the ASME 116:1-7. [33] Shin SH (2005) FEM Analysis of Plasticity-Induced Error on Measurement of Welding Residual Stress by the Contour Method. Journal of Mechanical Science and Technology 19:1885-1890. [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, PVP2008-61490. [35] Abaqus 6.9, ABAQUS, inc., Pawtucket, RI, USA, 2009. [36] Mayville R, Finnie I (1982) Uniaxial Stress-Strain Curves from a Bending Test. Experimental Mechanics 22:197-201. [37] Schajer GS, Prime MB (2006) Use of Inverse Solutions for Residual Stress Measurements. Journal of Engineering Materials and Technology 128:375-382. [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, 1-3 Jun 1998, Society for Experimental Mechanics, Inc., 497-499.

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 opening-direction 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 follow-up to recently reported work that contains details of fracture tests on the same set of coupons [1]. METHODS Material and geometry Aluminum alloy 7075-T6 was selected for this test program due to its prevalent use in a variety of applications. This alloy exhibits low-energy ductile fracture with a rising R-curve. The material was received as clad plate 4.8 mm thick. Handbook [2] mechanical properties of 7075-T6 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 561-98 – “Standard Practice for R-Curve Determination”). The C(T) T. Proulx (ed.), Experimental and Applied Mechanics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series 17, DOI 10.1007/978-1-4419-9792-0_41, © The Society for Experimental Mechanics, Inc. 2011

251

252

coupon is well suited for this work because of its accepted use in fracture toughness testing, simple geometry, and one-dimensional 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 L-T 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

0.5

KIc (MPa m ) 29.0

Table 1 – Mechanical properties of 7075-T6 plate [2]

Fig. 1 – 7075-T6 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 through-thickness compressive residual stress in the treated area. The LSP process uses laser-induced 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 4-18-3) [5,6]. It was further found that 4-18-3 provided significant high-cycle 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 4-18-1, or using three layers, at 4-18-3, where three-layer 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: as-machined (AM); one-layer LSP applied near the front face (LSP-1N); three-layer LSP applied near the front face (LSP-3N); one-layer LSP applied far from the front face (LSP-1F); and, three-layer LSP applied far from the front face (LSP-3F). 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 Two-dimensional 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 area-scanning profilometer was used to measure the resulting out-of-plane 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 three-dimensional, linear elastic finite element model of one-half 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 thickness-average residual stress as a function of position across the crack plane, with the two-dimensional 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 load-line) 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 five-point 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 561-98 to determine the K-R 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 K-R 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 K-R 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 LSP-3N 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 mid-thickness. 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 (LSP-1N and LSP-3N) 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 (LSP-1F and LSP-3F) 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 one-layer and three-layer LSP are quite similar, with 3-layer LSP coupons having about twice as much residual stress. Thickness average residual stress for the LSP-3N and one of the AM coupons is plotted in Fig. 5. AM coupons have a thickness-average stress of nearly 0 MPa at all points across the coupon, which is consistent with plate rolling. Thickness-average stress in the LSP-3N 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 LSP-3N, KRS was also computed using the Green’s function and contour thickness-average 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 far-from-front-face LSP coupons (LSP-1F, LSP-3F) have positive KRS and the near-front-face LSP coupons (LSP-1N, LSP-3N) have negative KRS. KRS for the three-layer LSP coupons (LSP-3F, LSP-3N) are roughly double those for the corresponding one-layer LSP coupons (LSP-1F, LSP-1N). For LSP-3F, 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 LSP-1F and LSP-3F (17.7 and 0.5 35.9 MPa m ) is nearly twice the magnitude of the most negative KRS in coupons LSP-1N and LSP-3N (-10.5 and 0.5 -16.8 MPa m ). Based on the slitting results (Fig. 6), target initial crack lengths were selected for K-R 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 LSP-1F and LSP-3F 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 LSP-induced residual stress. R-curves in terms of Kapp are shown for all coupons in Fig. 8. Results for the two AM coupons exhibit typical 0.5 elastic-plastic 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 LSP-1N and LSP-3N coupons exhibit greater toughness (in terms of Kapp) and the LSP-1F and LSP3F coupons exhibit reduced toughness, with the 3-layer 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 R-curves were prepared using Eqn. (1) and using values of K RS determined from Eqn. (2). The corrected R-curves for the different coupon subsets are similar (Fig. 9). The LSP-1N and LSP-3N results exhibit significantly greater elastic-plastic 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 R-curves of all coupon subsets are in good agreement, though the LSP-3F coupons exhibit a significantly shallower R-curve. 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 LSP-3F coupons to 51.6 MPa m for LSP-3N 0.5 0.5 coupons. Mean KQ,Tot values ranged from 32.1 MPa m (LSP-3F) to 37.4 MPa m (LSP-1N), 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); one-sigma error bars are plotted for LSP-3N, other conditions exhibit similar levels of uncertainty

Fig. 5 – Slitting residual stress for LSP-3N plotted with contour through thickness average for LSP-3N 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 LSP-3N condition computed from derivative of strain data (Eqn. (2)), and measured residual stress from slitting and contour thickness-average stress

Fig. 8 – R-curve resulting from applied stress intensity factor

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Fig. 9 – R-curve 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 high-strength 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 low-energy 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., hand-forgings or welded materials) likely would provide less-consistent results. Third, the coupon geometry had a large width-to-thickness 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 non-straight crack fronts (e.g., as for welded joints [14]). While slitting was the best method for the present coupons, a significantly smaller width-to-thickness ratio might require accounting for through-thickness stress variation (e.g., measured by the contour method) and its affect on KRS along the crack front. Follow-on 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 R-curve 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/AR-MMPDS-01. [3] Fabbro, R., Peyre, P., Berthe, L., and Scherpereel, X., 1998, “Physics and applications of laser-shock processing,” Journal of Laser Applications, 10, pp. 265-279.

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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. 1021-1036. [5] Pistochini, T., 2003, “Fatigue Life Optimization in Laser Peened 7050-T7451 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 7050-T7451 aluminum, 7085T651 aluminum, and Ti-6Al-4V titanium alloy,” M.S. Thesis, Department of Mechanical and Aeronautical Engineering, University of California, Davis. [7] Prime, M.B., 2001, “Cross-sectional mapping of residual stresses by measuring the surface contour after a cut,” Journal of Engineering Materials and Technology, 123, pp. 162-168. [8] Prime, M.B., Sebring, R.J., Edwards, J.M., Hughes, D.J., and Webster, P.J., 2004, “Laser surface-contouring and spline data-smoothing for residual-stress measurement,” Experimental Mechanics, 44, pp. 176-184. [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. 375-382. [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 cut-compliance technique,” Fatigue and Fracture of Engineering Materials and Structures, 30, pp. 214-222. [12] Newman, J.C., Yamada, Y., James, M.A., 2010, “Stress-intensity-factor equations for compact specimen subjected to concentrated forces,” Engineering Fracture Mechanics, 77, pp 1025-1029. [13] Wu, X.-R. and Carlsson, A.J., 1991, Weight function and stress intensity factor solutions, Pergamon Press, pp. 37-38. [14] Towers, O.L., Dawes, M.G., 1985, “Welding Institute research on the fatigue precracking of fracture toughness specimens,” in Elastic-Plastic Fracture Test Methods: The User's Experience, ASTM STP 856, American Society for Testing and Materials, Philadelphia, PA, p. 23-46. [15] Lee, M.J., Hill, M.R., 2007, “Intralaboratory repeatability of residual stress determined by the slitting method,” Experimental Mechanics, 47, pp. 745-752.

Analysis of Large Scale Composite Components Using TSA at Low Cyclic Frequencies

Author: J. M. Dulieu-Barton, School of Engineering Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ, UK, [emailprotected] Co-Author: 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 lay-up 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, non-contacting technique for the evaluation of stresses in engineering components, e.g. [2-5]. The technique uses an infra-red 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 tee-joint [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 10.1007/978-1-4419-9792-0_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 infra-red detection system manufactured by Cedip Infrared Systems was used to obtain the TSA data. The system comprises an infra-red 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 ‘lock-in’ 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 K xV x K y V y

(2)

261

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 914C-TS-5-34% prepreg tape as the face sheet material, one with quasi-isotropic (QI) sheets [0°, 45°, 45°, 90°, 0°, 45°, -45°, 90°, 0°, 45°, -45°, 90°]s about a core of Nomex honeycomb, and one with cross-ply (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 infra-red 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 full-field plot of the temperature change on the surface of the sandwich panel. This temperature field was then calibrated to obtain a full-field 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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'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)

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 lay-up 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 lay-up i.e. one with all plies aligned longitudinally or transversely. The surface ply in the panel faces sheets is orientated with the x-direction 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 servo-hydraulic 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 non-adiabatic 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 (MPa-1 x 10-6)

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 out-of-plane 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 full-field 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 non-adiabatic 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 non-adiabatic conditions prevail.

(a)

(b)

(c)

(d)

Figure 2: Full-field 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 ply-by-ply 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

1. Stanley P, Chan WK. The application of thermoelastic stress analysis to composite materials. J Strain Anal Eng 1988; 23(3): 137-142.

2. Diaz FA, Patterson EA, Tomlinson RA, Yates JR. Measuring stress intensity factors during fatigue crack growth using thermoelasticity. Fatigue Fract Eng M. 2004; 27(7): 571-584. 3. Lin ST, Miles JP, Rowlands RE. Image enhancement and stress separation of thermoelastically measured data under random loading. Exp Mech 1997; 37(3). 4. El-Hajjar R, Haj-Ali R. A quantitative thermoelastic stress analysis method for pultruded composites. Comp Sci Tech 2003; 63(7): 967-978.

5. Emery, T. R., Dulieu-Barton. J.M., Earl, J., and Cunningham, P.R., A generalised approach to the calibration of orthotropic materials for thermoelastic stress analysis. Composites Science and Technology, 2008, 68(3-4), p 743-752 6. Paynter, R.J.H., Dutton, A.G., The use of a second harmonic correlation to detect damage in composite

structures using thermoelastic stress measurements. Strain, 2003. 39(2): p. 73-78. 7. Dulieu-Barton, J.M., Quinn, S., Shenoi, R.A., Read, P.J.C.L., and Moy, S.S.J, Thermoelastic stress

analysis of a GRP tee joint. Applied Composite Materials, 1997. 4(5): p. 283-303. 8. Johannes, M., Dulieu-Barton, J.M, Bozhevolnaya, E., Thomsen, O.T., Characterisation of local effects at

core junctions in sandwich structures using thermoelastic stress analysis. The Journal of Strain Analysis for Engineering Design, 2008. 43(6): p. 469-492.

9. Crump, D.A., Dulieu-Barton, J.M., and Savage, J, The manufacturing procedure for aerospace secondary sandwich structure panels, Journal of Sandwich Structures and Materials, 2009, 0, In Press (DOI:1099636208104531) 10. Crump, D.A., Dulieu-Barton, J.M., and Savage, J. Design and commission of an experimental test rig to apply a full-scale pressure load on composite sandwich panels representative of an aircraft secondary structure, Measurement Science and Technology, 2010, 21, 16pp 11. Cunningham, P.R., Dulieu-Barton, J.M., Shenoi, R.A., Damage location and identification using infra-red

thermography and thermoelastic stress analysis. Proceedings of SPIE, 2002. 4704: p. 93-103. 12. Cunningham, P.R., Dulieu-Barton, J.M., Dutton, A.G., Shenoi, R.A., Thermoelastic characterisation of

damage around a circular hole in GRP component. Key Engineering Materials, 2001. 204-205: p. 453463. 13. Sambasivam, S., Quinn, S and Dulieu-Barton, 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

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]

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 different-size 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 plane-problems 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 traction-free 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 [1-5]. 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 different-size 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 10.1007/978-1-4419-9792-0_43, © The Society for Experimental Mechanics, Inc. 2011

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Fig. 1: Aluminum Plate . 2. Relevant Equations For plane-stress isotropic TSA under cyclic proportional loading condition, the quantitative information of figure 1 can be expressed as [4-6] ∗

∆ =

[∆(

)]

(1)

where S* is the thermoelastically-recorded 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. TSA-wise, 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 X-axis 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 counter-clockwise from this downward X-axis. Based on the symmetry about X-axis, a relevant Airy’s stress function in polar co-ordinates satisfying equilibrium and compatibility can be written as [6]

§ · c' a 0 b0 ln r c0 r 2 ¨¨ a1' r 1 d1' r 3 ¸¸ cos T r © ¹

¦> a r ' n

b r ' n

( n 2)

c r ' 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 counter-clockwise from the longitudinal vertical X-axis (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 traction-free 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

(7)

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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

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

· ¸¸ 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 traction-free 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 Ultra-Flat 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 X-axis (figure 1), the recorded TSA data obtained were averaged about the longitudinal X-axis so that only one half of the plate is considered during thermoelastic stress analyses. Individual analyses were carried-out 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 traction-free 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 traction-free 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 traction-free 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) X-direction from ANSYS and TSA after having imposed stress compatibility between the holes.

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Y/d2 Y

X/d2

0.5

1.5

2.5

0.5

1.5

2.5

3.5

Fig. 7: Common area between small and large circles.

Fig. 8: Contour plots of σXX/σ0 from ANSYS.

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 multiple-perforated 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 traction-free 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, traction-free 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 ‘full-field’ 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, 2877-2888 (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, S-J., Matthys, D.R. and Rowlands, R.E., “Separating Stresses Thermoelastically in a Central Circularly Perforated Plate using an Airy Stress Function” , Strain, 45, 516-526 (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).

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, safety-critical components. Examples of such components are wing skin panels, which, with their array of stiffeners and holes, present a complex mixed-mode 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 mixed-mode 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 crack-tip strain fields . One of the most promising of these techniques is thermoelastic stress analysis (TSA), due to its simple surface preparation and non-contacting, 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 infra-red 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 crack-tip stress fields in mode I, and some work on mixed-mode 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 biaxially-loaded 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 crack-tip 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 Dulieu-Barton’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 crack-tip 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 crack-tip stress intensity factors and crack-tip position by using genetic algorithms to aid direct curve-fitting 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 mixed-mode loads, the method to find the crack-tip 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

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mixed-mode 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 crack-tip 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 non-singular term 13 in Williams’s crack-tip stress field solution. The T-stress 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 T-stress on the crack growth rate, crack-tip constraints, crack closure, and the shape and size of the plastic 16,17 zone ahead of the crack-tip 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 T-stress 18 results were found to be sensitive to crack-tip position. A new technique was proposed to find the crack-tip from thermoelastic images based on the Y or phase image, and this crack-tip location was used in the method to determine K and the T stress. This crack-tip location technique appeared to be more reliable than another similar 6 18 technique proposed by Diaz , especially in T-stress determination. The work by Zanganeh also investigated the use of higher order terms on the accurate determination of both K and T-stress, 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 T-stress accurately and the results for T-stress 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 T-stress accurately. The majority of this work was carried out under Mode I loads and only one mixed-mode 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 mixed-mode 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 mixed-mode 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 crack-tip 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 crack-tip 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 mixed-mode 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 mixed-mode, and how these data relate to the crack-tip location, the crack-tip 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 mixed-mode loading. These experiments will provide additional understanding of the relationship between different load magnitudes, loading modes and the crack-tip behaviour, and further knowledge of the how the TSA data can characterise local crack-tip 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 mixed-mode 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 mixed-mode 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 mixed-mode methodologies for K and crack-tip 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 crack-tip 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: 623-636, 2004. 2 Sanford R. J., Determining fracture parameters with full-field optical methods, Experimental Mechanics, 29(3):241-247, 1989. 3 Tomlinson, R. A, Nurse, A. D. and Patterson, E. A., “On determining stress intensity factors for mixed-mode cracks from thermoelastic data”, Fatigue and Fracture of Engineering Materials and Structures, 20, (2), 217-226, 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), 49-55, 1999. 9 Stanley P and Dulieu-Smith J M 1993 Progress in the thermoelastic evaluation of mixed-mode stress intensity factors Proc. SEM Spring Conf. on Exp. Mech. Dearborn 617–26, 1993 10 Dulieu-Barton, 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, 301-313, 2000 11 Dulieu-Barton, JM and Worden, K, Genetic Identification of crack-tip parameters using thermoelastic isopachics, Meas. Sci. Technol. 14, 176–183, 2003 12 th Hebb, R I, Dulieu-Barton, 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, 283-298, 1998 15 Cotterell, B. Notes on the paths and stability of cracks. Int. J. Fracture, 2(3), 526-533, 1966 16 Roychowdhury, S. and Dodds Jr., R.H. Effect of T-stress on fatigue crack closure in 3-D small-scale yielding. Int. J. Solids Struct., 41, 2581–2606, 2004 17 Smith, D.J., Ayatollahi, M.R. and Pavier, M.J. The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract. Engng. Mater. Struct. 24(2), 137–50, 2001

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Zanganeh, M, Tomlinson, R A, and Yates J.R., T-stress determination using thermoelastic stress analysis, Journal of Strain Analysis for Engineering Design, 43, 529-537, 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

Extending TSA with a Polar Stress Function to Non-Circular Cutouts.

A. A Khaja1 and R. E. Rowlands2 Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706 1

[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 series-form 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, [2-10]. This paper demonstrates the ability to thermoelastically stress analyze finite components containing a non-circular 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 non-circular 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 10.1007/978-1-4419-9792-0_45, © The Society for Experimental Mechanics, Inc. 2011

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2b = 19.05mm (0.75 “)

139.7 mm (5.5““)

y r

139.7mm (5.5 “)

2a = 38.1 mm (1.5”)

76.2 mm (3””)

Fig. 1: Symmetrically-loaded tensile aluminum plate containing an elliptical hole. 2. Relevant Equations Under proportional plane-stress 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 elliptically-perforated vertically-loaded 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 non-symmetry. In addition to the measured temperature data, the traction-free conditions are imposed discretely on the boundary of the elliptical hole. It is convenient to employ elliptical coordinates when satisfying the traction-free 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)

(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 x-axis, 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 co-ordinate system can be transformed to those in the Cartesian coordinates, equations 9 through 11, and in elliptical co-ordinates, 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 traction-free 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 traction-free 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 traction-free 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 6061-T6, 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 Ultra-Flat 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 load-induced 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 traction-free 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 finite-plate 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 traction-free 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 traction-free 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 counter-clockwise from the positive x-axis, 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 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 TSA-based 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 full-field, 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 cut-out in a finite component. Stress functions formulated in terms of complex variables are mathematically more cumbersome. While the traction-free conditions at the hole are applied here discretely rather the analytically, the present approach also enjoys the advantage over the complex-variable 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 cut-outs (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, 257-265 (1995).

[2]

Wang, J.T., Lotts, C.G., and Davis, D.D, JR., “Analysis of Bolt-Loaded Elliptical holes in Laminated Composite Joints” Journal of Reinforced Plastics and Composites, 12(2), 128-138 (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, 695-751 (2007).

[4]

Madenci, E., and Ileri, L., “An Arbitrarily Oriented, Rigid, Elliptic Inclusion in a Finite Anisotropic Medium”, Int’l Journal of Fracture, 62, 341-354 (1993).

[5]

Persson, E., and Madenci, E., “Composite Laminates with Elliptical Pin-Loaded Holes”, Engineering Fracture Mechanics, 61, 279-295 (1998).

[6]

Hwu, C., and Wen, W.J., “Green's Functions of Two-Dimensional 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”, McGraw-Hill Book Company, Inc. (1951).

[8]

Pilkey, W.D., and Pilkey, D.F., “Stress Concentration Factors”, a Wiley-Interscience 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), 58-63 (1991).

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 60208-3111 * 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 [3-7]). 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 brick-and-mortar 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 in-situ 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 scaled-up 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 three-point bending using a pre-notched 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 10.1007/978-1-4419-9792-0_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 mother-of-pearl: A key feature in the material hierarchical structure. Journal of the Mechanics and Physics of Solids, 2007. 55(2): p. 306-337. 2.

Barthelat, F. and H. Espinosa, An experimental investigation of deformation and fracture of nacre–mother of pearl. Experimental Mechanics, 2007. 47(3): p. 311-324.

Sarikaya, M. and I.A. Aksay, eds. Biomimetics, Design and Processing of Materials. Polymers and complex materials, ed. AIP. 1995: Woodbury, NY.

Mayer, G., Rigid Biological Systems as Models for Synthetic Composites Science, 2005. 310(5751): p. 1144-1147.

Buehler, M.J. and T. Ackbarow, Fracture mechanics of protein materials. Materials Today, 2007. 10(9): p. 46-58.

Ashby, M.F., et al., The Mechanical Properties of Natural Materials. I. Material Property Charts. Proceedings: Mathematical and Physical Sciences, 1995. 450: p. 123-140.

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. 5597-5600.

Espinosa, H., et al., Merger of structure and material in nacre and bone-Perspectives on de novo biomimetic materials. Progress in Materials Science, 2009. 54(8): p. 1059-1100.

Launey, M. and R. Ritchie, On the fracture toughness of advanced materials. Advanced Materials, 2009. 21(2103-2110).

10.

Tang, H., F. Barthelat, and H.D. Espinosa, An elasto-viscoplastic interface model for investigating the constitutive behavior of nacre. Journal of the Mechanics and Physics of Solids, 2007. 55(7): p. 1410-1438.

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 Pre-Hardening Stresses. The Journal of Strain Analysis for Engineering Design, 2008. 43(8): p. 705-717.

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 Urbana-Champaign, Urbana, United States 5 Arts et Métiers ParisTech, Paris, France *1304 W. Green St., 61801, Urbana, Illinois, United States of America, 217-333-1041, [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 micron-scale 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 locally-placed 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 layer-by-layer 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.

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Fig. 1 Schematic of vascular system geometries. Table 1 Vascular system dimensions Label

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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

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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 5-30 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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Type I (n=9) Type II (n=4)

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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 Halpin-Tsai 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 Halpin-Tsai 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 Halpin-Tsai prediction, while the Type III sample elastic modulus lays between the two theoretical curves. The deviation of the Type III sample from the Halpin-Tsai 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 three-point 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 D5045-99 [9].

Fig. 4 SENB sample geometry and dimensions. Sharp pre-cracks were initiated by loading the beams in cyclic fatigue until cracks extended from a notch that was scored with a razor blade. Pre-crack 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 (MPa-m ) 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 Halpin-Tsai 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) Self-healing materials with microvascular networks. Nat Mater 6:581-585. [2] Toohey KS, Sottos NR, White, SR (2009) Characterization of microvascular-based self-healing coatings. Exp Mech 49:707-717. [3] Therriault D, White SR, Lewis JA (2003) Chaotic mixing in three-dimensional microvascular networks fabricated by direct-write assembly. Nat Mater 2:265-271. [4] Lewis JA (2006) Direct ink writing of 3D functional materials. Adv Funct Mater 16:2193-2204. [5] Hansen CJ, Wu W, Toohey KS, Sottos NR, White SR, Lewis JA (2009) Self-healing materials with interpenetrating microvascular networks. Adv Mater 21:1-5. [6] Williams HR, Trask RS, Bond IP (2007) Self-healing composite sandwich structures. Smart Mater Struct 16:1198-1207. [7] Williams HR, Trask RS, Bond, IP (2008) Self-healing sandwich panels: Restoration of compressive strength after impact. Compos Sci Technol 68:3171-3177. [8] O’Brien DJ, Sottos NR, White SR (2007) Cure-dependent viscoelastic Poisson’s ratio of epoxy. Exp Mech 47:237-249. [9] ASTM D5045-99. Standard Test Methods for Plane-Strain 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 50:255-263.

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) PCL-hydrogel, (2) PCL-hydrogel with cells, (3) PCL-hydrogel with cells and MgO nanoparticles, and (4) PCL-hydrogel 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 [4-5]. 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 bone-ligament or bone-cartilage reconstructions. The interface is the weakest place in the grafted bone-ligament or bone-cartilage 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 PCL-hydrogel 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 PCL-hydrogel seed with MgO nanoparticles specimen using a custom made tensile test setup to the influence of adhesion strength between PCL-hydrogel 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 10.1007/978-1-4419-9792-0_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. PCL-Hydrogel 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.

(a)

C (b) (c) (d) Figure 2 (a) Hydrogel samples in the sylgard mold, (b-d) 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 PCL-hydrogel 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 PCL-hydrogel 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™ LTA-HL actuator) and motion controller (SMC 100). All instruments were calibrated before testing.

Figure 3 Experimental setup. EXPERIMENT AND DATA ANALYSIS Four different types of PCL-hydrogel tissue constructs were tested: PCL-hydrogel, PCL-hydrogel with cells, PCLhydrogel with cells and MgO nanoparticles, and hydrogel with MgO nanoparticles. The PCL-hydrogel 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 PCL-hydrogel 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 load-displacement curve and A is the cross-sectional 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 hydrogel-based soft tissue grafts with the use of 22 nm MgO particles. On the other hand, increased cell adhesion of mouse osteoblast cells (MT3T3E1) to hydrogel-based 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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24 hrs 4 hrs Cell culture time

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Figure 5 (a) Fluorescent intensity of hydrogel with cells and hydrogel with cells plus 22 nm MgO particles, and (b) confocal microscope image of the hydrogel samples with the cells. Figure 6(a) compares the load-displacement curves among PCL-hydrogel, PCL-hydrogel with cells and PCLhydrogel with cells plus 22 nm MgO. The load-displacement response of all specimens is characterized as initially elastic response, followed by a short inelastic region and then stable descending response. Figure - 5(b) compares the interface fracture strength values. Results shows decreased interface strength between cell-seeded PCL hard tissue graft and hydrogel when the MgO nanoparticles were mixed with hydrogel. The interface strength of PCL-hydrogel with no cells or nanoparticles was lower than PCL-hydrogel with cell samples but higher than PCL-hydrogel with cells and MgO nanoparticles. 0.35 Hydrogel

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Figure 6 (a) Load vs. displacement plot of different kinds of PCL-hydrogel specimens, (b) variation of interface fracture strength for various kinds of PCL-hydrogel specimens. CONCLUSIONS Cells were successfully grown in hydrogel with MgO nanoparticles with no significant difference (p< .01) between samples of hydrogel with cells. The interface tensile strength of PCL-hydrogel with cells had 6% greater tensile

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strength than PCL-hydrogel with cells and MgO nanoparticles. However, PCL-hydrogel with cells and MgO nanoparticles had the weakest tensile strength. This result showed that PCL-hydrogel 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 PCL-hydrogel 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), 2529-2543 (2000). 2. Lee, K.Y., Mooney D.J., "Hydrogels for tissue engineering," Chemical Reviews, 101(7), 1869-1879 (2001). 3. Sun, W., Darling A., Starly B., Nam J., "Computer-aided tissue engineering: Overview, scope and challenges," Biotechnology and Applied Biochemistry, 39(1), 29-47 (2004). 4. Kim, J.Y., "Fabrication of a sff-based three-dimensional 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), 1169-1185 (2002). 6. Shor, L., "Fabrication of three-dimensional polycaprolactone/hydroxyapatite tissue scaffolds and osteoblastscaffold interactions in vitro," Biomaterials, 28(35), 5291-5297 (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), 611-618 (2008). 8. Ricker, A., Liu-Snyder 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), 125-1 (2008). 9. Lee, C.S.D., Gleghorn J.P., Won Choi N., Cabodi M., Stroock A.D., Bonassar L.J., "Integration of layered chondrocyte-seeded alginate hydrogel scaffolds," Biomaterials, 28(Compendex), 2987-2993 (2007). 10. An, Y.H., Draughn R.A., Mechanical testing of bone and the bone-implant interface. CRC (2000).

Mechanical Interactions of Mouse Mammary Gland Cells with a Three-Dimensional Matrix Construct M.d.C. Lopez-Garcia, 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 four-dimensional imaging, tracking X, Y, Z positions over a three to four hour time frame. Three-dimensional 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 [Lopez-Garcia, 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 1ii-iii 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 10.1007/978-1-4419-9792-0_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 (GFP-NMuMG) 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 cell-gel 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 1i-iii 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 [Lopez-Garcia, 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 Pre-doctoral Fellowship (JPFP-NASA); 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 (SDE-GWIS).

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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 cell-matrix adhesions to the third dimension. Science, 2001. 294(5547): p. 17081712. Friedl, P. and E.B. Brocker, The biology of cell locomotion within three-dimensional extracellular matrix. Cellular And Molecular Life Sciences, 2000. 57(1): p. 41-64. Jo, B.H., et al., Three-dimensional micro-channel fabrication in polydimethylsiloxane (PDMS) elastomer. Journal Of Microelectromechanical Systems, 2000. 9(1): p. 76-81. Lopez-Garcia, M.d.C., D.J. Beebe, and W.C. Crone, “Mechanical Interactions of Mouse Mammary Gland Cells with Collagen in a Three-Dimensional 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. 491-499.

Tracking nanoparticles optically to study their interaction with cells Jean-Michel 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 nano-biotechnology. 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 dark-field microscopy. However, fluorescent-based techniques tend to suffer from photo-bleaching and dark-field 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 micro-spheres of diameter 5 to 10um using Mie theory. In 2006, Guerrero-Viramontes et al. [5] demonstrated the approach by tracking glass particles of diameter 20 to 30um using a 70mW He-Ne 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 three-dimensions 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 10.1007/978-1-4419-9792-0_50, © The Society for Experimental Mechanics, Inc. 2011

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Figure 1: Image (top left), z-stacks for the xz- (top right) and yz- (bottom left) planes and a composite three-dimensional composite view (bottom right) for 300nm diameter polystyrene particles obtained by collecting images at 1um steps of the objective over 30um in the z-direction 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 three-dimensional 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 (z-direction) 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 P-725.4CD) with a servo controller (Physik Instrumente Gmbh & Co. KG. model E-665) to an inverted optical microscope (Olympus IX71) with 100W halogen light source (Olympus U-LH100L-3) 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 peak-topeak amplitude in the range 100um to 150um about a mean position that corresponded to the approximate center of the particle of interest. For pseudo-static investigations images were captured at 20 frames/second using a 12bit monochrome CCD camera (Hamamatsu C8484-05G) with a resolution of 1344u1024 pixels and for tracking motion a high-speed, 8-bit monochrome camera (Photron FASTCAM-PCI 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 (L0655-1ML and L05530-1ML respectively Aminemodified polystyrene, fluorescent blue, Signa-Aldrich Inc., St Louis, MO) were used in this investigation and were diluted 1000-fold 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 (Menzil-Gläser, 21u23mm #4). The particles flash in real-time 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 xy-plane and for these pixels the location of the particle in the z-direction, i.e. along the light path was identified by scanning along the z-direction 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.

309

200

Intensity

180 160 140 120 100 80

time [half-period]

Figure 2: Image of 500nm diameter particles captured with the condenser aperture closed to a minimum with the objective stationary (left): and the variation of intensity with time (right) over a three half-periods of oscillation of the microscope objective for the pixel marked, A in the image which is approximately located at the center of a particle.

3. RESULTS A typical image obtained with a static but defocused objective is shown in figure 2 together with the time-varying 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: Grey-scale 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 z-direction a 3u3 patch of data centered on the particle center in the xy-plane was used rather than a single line of pixels in the z-direction. 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.

310

70.0

69.5

Z (Pm)

70.5

69.0 14

(P m )

24 23

11 10

)

Figure 4: A typical trajectory (in black) of a 500nm particle illustrating the long-term, 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 three-dimensions 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 three-dimensional single particle tracking, J Phys. Chem., 111:32-35, 2007 2. Louit, G., Asahi, T., Tanaka, G., Uwada, T., Masuhara, H., Spectral and 3-Dimensional 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 5-nm gold beads in live cells, Biophysical Journal, 91:45984604, 2006. 4. Ovryn, B., Three-dimensional forward scattering particle image velocimetry applied to a microscope field-ofview, Experiments in Fluids [Suppl.], S175-184, 2000. 5. Guerrero-Viramontes, J.A., Moreno-Hernandez, Mendoza-Santoyo, F., Funes-Gallanzi, M., 3D particle positioning from CCD images using the generalised Lorenz-Mie and Huygens-Fresnel theories, Meas.Sci. Technol., 17:2328-2334, 2006. 6. Toprak, E., Balci, H., Blehm, B.H., Selvin, P.R., ‘Three dimensional particle tracking via bifocal imaging’, Nano Letters, 7(7):2043-2045, 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):1703-1706, 2008.

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, Storrs-Mansfield, CT, 06269 1 [emailprotected] ABSTRACT Micro-CGS, an integration of the interferometric optical method Coherent Gradient Sensing (CGS) [1-3] with an inverted microscope, is introduced. Micro-CGS extends the capabilities of Classical CGS into the cellular level. As such, it provides full-field, real-time, micro-interferometric quantitative imaging. Micro-CGS 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 30-micron glass microspheres. KEYWORDS: Micro-interferometer, 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), Dark-Field (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 [4-5]. In this framework, we developed micro-CGS which offers an increased stability and direct measurement of the surface curvature of microstructures. Classical CGS is a well-established large-scale 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 10.1007/978-1-4419-9792-0_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 high-density 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, E-1, etc. For aesthetic purposes only E-1, 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 E-1,1, E-1,0, E-1,-1, …, from E-1, 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, D-1, 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 Micro-CGS is based upon the introduction of a microscopic objective into the Classical CGS experimental setup. A design schematic of the Micro-CGS setup can be seen in Figure 2. A 635-nm He-Ne 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. He-Ne 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: Micro-CGS schematic Upon exiting the microscope objective, the beam is re-collimated, expanded (10X) and directed through the pair of Ronchi diffraction gratings (1000 lines/inch) as in the classical CGS setup. A CCD camera, Casio EX-F1, 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.

313

Fig. 3: Optical setup of Micro-CGS utilized in the experimentation IMAGE PROCESSING The software utilized to post-process 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 Plano-convex 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 phase-angle 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 Plano-convex lens, (b) corresponding frequency power spectrum after converting to Fourier domain

314

The program then converts the image to the Fourier domain by employing Fast-Fourier 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 low-frequency 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 full-field 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 y-axis 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)

Unwrapped Fringe Order

(b)

Fig. 6: a) Full field unwrapped phase, b) Unwrapped fringe order vs. distance on y-axis of Figure 6a in pixels

315

This particular case established linearity in fringe order for the plano-convex 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 [1-3] 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 30-micron glass microspheres (±1%), manufactured by Thermo-Scientific, 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 x2-axis in the form of a reasonably flat plane in the field of the bead as exhibited in Figure 7b.

30 μm 1 x2

(b) Normalized Curvature μm-1

(a)

4 x1 Fig. 7: a) Interferogram captured for 30-micron 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 micro-CGS, 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 micro-CGS can be used to successfully obtain curvature of static micro-specimens. In addition, the optical system produced repeatable fringe interferograms within the beads, signifying the successful detection capability of micro-CGS. Moving forward, the intention of the research is to further extend the capabilities of micro-CGS 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, Full-field optical measurement of curvatures in ultra-thin-film-substrate 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):25-53. (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(1-2):42-54. (1998). 4. Ikeda T, Popescu G, Dasari RR, & Feld MS, Hilbert phase microscopy for investigating fast dynamics in transparent systems. Optics Letters 30(10):1165-1167. (2005). 5. Lue N, et al., Live cell refractometry using Hilbert phase microscopy and confocal reflectance microscopy. J Phys Chem A 113(47):13327-13330. (2009). 6. Lambros J & Rosakis AJ, An experimental study of dynamic delamination of thick fiber reinforced polymeric matrix composites. Experimental Mechanics 37(3):360-366. (1997).

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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 D-titanium 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 non-linear effects have to be compensated through different k-factors.

Absolute Values of k [TPa]

5 4 3 2 Reihe2 k+

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 k-compensated (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 non-linear 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 non-linear 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 k-values during fatigue life. The results suggest that the description of non-linear 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, 327-338, 1897. [2] Sommer C, Christ H-J, Mughrabi H: Non-Linear Elastic Behaviour of the Roller Bearing Steel SAE 52100 during Cyclic Loading. Acta Met Mat 39, 1177-1187, 1991. [3] Boyer R, Welsch G, Collings EW: Materials Properties Handbook: Titanium Alloys, Materials Park, OH: ASM International. 337-375, 1994. [4] Heckel TK, Guerrero Tovar A, Christ H-J: Fatigue of the Near-Alpha Ti-Alloy Ti6242. Exp Mech, in print, doi: 0.1007/s11340-009-9238-5, 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, 843-851, 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, 1115-1123, 2007. [8] Paton NE, Williams JC, Rauscher GP: The Deformation of D-Phase Titanium. In: Jaffee RI, Burte HM, editors. Titanium ’72 Science and Technoloy, New York, NY: Plenum Press. 1049-1070, 1973.

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 E-mail: [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 pre-tension 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 non-contact measurement system and strain energy is calculated based on Mooney-Rivlin 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 three-dimensional shape of the membranes during a burst-chirp excitation at different pre-tension levels is dynamically measured and recorded and the natural frequencies are computed by performing the fast Fourier transform of the out-of-plane displacement at several points of the circular membrane. Experimental results show that the natural frequencies increase with mode and pre-tension 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 pre-tension 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 pre-tension 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 10.1007/978-1-4419-9792-0_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 [5-8] 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 pre-tension 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 pre-stress 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 MAVRIC-I 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 in-flight 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 pre-tension 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 non-contact 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 pre-tension levels. A preliminary numerical model for the dynamics of hyperelastic membrane is also proposed and validated with experimental data. EXPERIMENTAL SET-UP AND PROCEDURE Test Specimen Rubber latex membrane at different pre-tension 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. Mooney-Rivling material model [18, 19] is considered for this hyperelastic latex membrane. The Mooney-Rivling material

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parameters are calculated based on the uni-axial 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 pre-tension levels are considered for static and vibration tests, respectively. The average pre-tension 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 pre-tension levels of the membrane specimen for static and vibration tests

Strain xx yy

Average pre-tension levels for static tests Low Medium High 0.0266 0.1009 0.1288 0.0136 0.0140 0.0380

Average pre-tension 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 non-contact method, a visual image correlation (VIC) technique is used. The VIC is a non-contacting full-field 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 pre-calibrated cameras digitally acquire this pattern before and after loading, using stereo-triangulation techniques. The VIC system then tries to find a region (in the image of the deformed specimen) that maximizes a normalized cross-correlation 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 high-speed Phantom v7 CMOS cameras, capable of storing 2,900 frames in an incamera flash-memory 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 post-processing option involves calculating the in-plane 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: non-contact 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 pre-tension (slack). The burst chirp is considered the best choice for all ranges of membrane pre-tension. 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 out-of-plane 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 pre-tension 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 [25-28] of the circular membrane specimen at different pre-tension levels is investigated at atmospheric pressure. The out-of-plane 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

Post-Processing 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 Mooney-Rivlin 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 pre-tension 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 pre-tension 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 3-dimensional circular membrane specimen at two different pre-tension 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 pre-tension levels will be developed and compared with experimental results as part of the future work.

(a)

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Fig. 6 Deformed shape plots of the low pre-tension 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 pre-tension 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 pre-tension 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 pre-tension of the membrane in Fig. 8. The deformation and strain energy increase with the weight of the sphere and pre-tension 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 Out-of-plane displacement (w) vs. time plots at five different locations for the high pre-tension membrane specimen from VIC data at atmospheric pressure

Fig. 10 FFT of Out-of-plane displacement (w) from Fig. 9

(a)

(b)

Fig. 11 Actual and average strains distribution over the low pre-tension membrane specimen

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(a)

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Fig. 12 Actual and average strains distribution over the high pre-tension membrane specimen

(a) Undeformed Shape

Mode Shape

(b) 1 Mode Shape

(d) 3 Mode Shape

Fig. 13 Mode shapes of the high pre-tension 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 out-of-plane 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

Mode Shape

(b) 1 Mode Shape

(d) 3 Mode Shape

Fig. 14 Mode shapes of the high pre-tension membrane specimen from FEA model at vacuum environment

Fig. 15 Frequency of the first mode vs. degrees of freedom (DOF) plot for the high pre-tension membrane specimen. DOF are calculated based on among one-four 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 pre-tension membrane specimen

Fig. 17 Natural frequencies vs. ambient pressure plots for the high pre-tension membrane specimen The actual and average strains distribution over the low and high pre-tension membrane specimens are shown in Figs. 11 and 12, respectively. The first three mode shapes of high pre-tension 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 pre-tension 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 pre-tension 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 pre-tension and ambient pressure. Natural frequencies increase with the membrane pre-tension 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 non-uniform pre-tension 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 non-uniform pre-tension membrane than that at uniform pre-tension.

Fig. 18 The variation of first natural frequency (from FEA model) with the pre-tension levels of the membrane specimen

Fig. 19 Variation of the out-of-plane displacement (w) of the center of the high pre-tension membrane specimen with time due to impact test by hammer Damping Ratio Calculation The out-of-plane 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 pre-tension 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 pre-tension 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 pre-tension 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 non-contact measurement system, is used for measuring the deformation of the membrane. @ The membrane-steel 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 pre-tension. 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 pre-tension level is placed inside a vacuum chamber, the three-dimensional dynamic shape is measured using the VIC technique and the natural frequencies are found from the FFT of the out-of-plane 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 pre-tension 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 non-uniform pre-tension 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 set-up. 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 out-of-plane 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 out-of-plane 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 pre-tension 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 FA9550-09-1-0072, 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 Wright-Patterson 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., “Laminar-Turbulent 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 Three-Sided 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 US-European 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 Thera-Band® 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, Wiley-VCH, 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 shaft-loaded 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/McGraw-Hill, Boston, MA, 2007. [26] Feldman, M., “Non-linear 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., “Non-linear 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 02909-2499, 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.

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 pre-determined set of locations on the audience seating area. Recommendations for improvements are presented throughout the paper. INTRODUCTION As a sub-discipline of classical mechanics, ‘acoustics’ is the science concerned with the study of sound. As a sub-discipline 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 check-up 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 10.1007/978-1-4419-9792-0_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 – Low-frequency 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 high-frequency 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 2-sample T-tests 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 p-value can be calculated such as to reject or accept the null hypothesis. If the p-value < 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 Hamilton-Fairfield 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 diffuse-field 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).

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 free-field level of an omni-directional 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).

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), energy-like 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 well-accepted 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:

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 15-20 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 Omni-Directional. Source Used: B&K – Type 4292 Dodecahedron. Power Amplifier: B&K – Type 2734-A. Sound Interface: Edirol UA-25EX. Recording Device: HP Notebook computer using WinMLS 2004 acoustic measurement and analysis software. m) The signal used was a sine-sweep 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 omni-directional 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 In-Situ 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] In-situ 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

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 electro-acoustic enhancements. The electro-acoustic

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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 electro-acoustic 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 well-accepted 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 electro-acoustic 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 fine-tune 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 On-stage Acoustics in a Multipurpose Auditorium”. Society for Experimental Mechanics (SEM) conference proceedings paper #178, (2010) [2] ISO 3382. “Acoustics-Measurement 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, 618-628, (1988) [4] Long, M., “Architectural Acoustics”. Elsevier, Chapter 17, 666-682, 654-655, (2006) [5] Egan, D., “Architectural Acoustics”. J.Ross Publishing, 64, 147-150, (2007) [6] Meyer Sound. “Glossary of terms”. http://www.meyersound.com/support/papers/speech/glossary.htm#sti [7] IEC 60268-16. “Sound system equipment – Part 16: Objective rating of sound intelligibility by STI”, International Electrotechnical Commission, (2004) [8] Dammerud, J.J., “In-situ 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

Identification and Enhancement of On-stage 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 on-stage 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, mid-frequency 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 10.1007/978-1-4419-9792-0_77, © The Society for Experimental Mechanics, Inc. 2011

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for Glate in the auditorium is 1 – 3 dB [5]. Glate measured on-stage 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 3382-1:2009 [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 3382-1:2009 [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 on-stage 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 3382-1:2009, “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 2734-A (loudspeaker amplifier), Edirol UA-25EX 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 sine-sweep that began at 125 Hz and ended at 20 kHz. The acoustic shell used was a Wenger Legacy® Acoustical Shell. This is a portable-type 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 Center-stage 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 omni-directional 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 source-receiver 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 source-receiver 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

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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 “in-situ” 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 “reflection-free 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 source-receiver 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 500-1000 Hz octave bands are presented in Table 1. In the case of ST1, the mean spatial average is calculated over the octave bands from 250-2000 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 (a-e).

Table 1. Mean Spatial Averages of Objective Parameters Spatial Average: 500-1000 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: 250-2000 Hz Bands Parameter

Without Shell

With Shell

ST1 (dB)

-15.0

-8.9

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Figure 6 (a-e). 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 1000-2000 Hz bands were lower than the “without shell” values. However, four of the nine measured locations show differences in the mid-frequency C80 (“without shell” vs. “with shell”) which exceed the JND (Just Noticeable Difference) of 1 dB. The measured differences in mid-frequency C80 values are by no means consistent across all stage locations – six of the nine locations showed a decrease in mid-C80, 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 mid-C80 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 mid-frequency 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 late-arriving 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 source-receiver 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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frequency-averaged values. As was mentioned above, the standard deviation of the mid-frequency C80 values on stage was 1.4 dB “without shell” and 0.9 dB “with shell”. The standard deviation of the mid-frequency 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 mid-frequency 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 mid-frequency 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 re-emphasized 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. 249-262, 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. 26-36, 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 3382-1:2009: 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. 532-544, 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. 232-239, 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 7-10, Indianapolis, IN. 10. Dammerud, J.J. “In-situ 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. E-mail. 16. Skalevik, M. “Orchestra Canopy Arrays – Some Significant Features.” Joint Baltic-Nordic Acoustics Meeting. Sweden, 2006. 17. Dammerud, J.J. “Re: Results of Glate.” Message to the author. 11 Mar. 2010. E-mail.

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 10.1007/978-1-4419-9792-0_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] electro-analytical 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 3-D 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 Ex-F1, 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.

531

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 1.5393e-004 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

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

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

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

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 1.56e-004 3 998.2 kg/ 3 Physics parameters of Plug #3 Plug Volume Water Density Buoyancy 1.526048 N 1.56e-004 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

532

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 1.56e-004 3 998.2 kg/ 3 Figure 2 is a sample of the picture acquired during the falling process. 0.16206 kg

1.56e-004

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

+ 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

534

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 *

+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:

(− +

+4∗

∗

−4∗

∗

)

(6)

(− −

+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.

535

(A) Log(Hydraulic Drag) verse P and K

(B) Log(Hydraulic Drag) verse P, fixed K=0.5 8

10 8 6

Log(Hydraulic Drag)

4 4 2 0 -2 -4

-6 -8 3 2.5

-2

0.8

1.5

0.6 1

0.4 0.5

0.2

-4 0

0.5

1.5 P

2.5

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 p-2 in (13) when p>2, and the reason one needs to solve (9) in different ranges of .

536

= , one could have :

Similarly, note, ( )= +

3−

2− ,4−

1 2

(−

2− ,3 (−

2−

2− ,2 (−

)

2−

Experimental and Applied Mechanics, Volume 6: Proceedings of the 2010 Annual Conference on Experimental and Applied Mechanics - PDF Free Download (2024)

References