River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Time Dependent Constitutive Behavior and Fracture/Failure Processes, Volume 3 Tom Proulx Proceedings of the 2010 Annual Conference on Experimental and Applied Mechanics River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series
River Publishers Tom Proulx Editor Time Dependent Constitutive Behavior Proceedings of the 2010 Annual Conference on Experimental and Applied Mechanics and Fracture/Failure Processes, Volume 3
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-843-9 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2011 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Preface Symposium on Time Dependent Constitutive Behavior and Failure/Fracture Processes represents one 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, Experimental and Applied Mechanics, along with the 11th International Symposium on MEMS and Nanotechnology. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. The current volume on the Symposium on Time Dependent Constitutive Behavior and Failure/Fracture Processes includes studies on: Characterization and modeling of behavior at multiple scales; viscoelasticity, viscoplasticity; transport, chemically and electronically active processes; multiphase and biomaterial systems; thermodynamics; shape memory; mechanics of testing; dynamic rate-dependent behaviors; large deformations; residual stresses; time (rate)-dependent damage and failure; time (rate)-dependent polycrystalline, single crystal and nanocrystalline behaviors; multifunctional materials; mechanics of processing; design methods; environmental interactions; experimental methods and techniques; linear and non-linear time-dependent behavior; time (rate)-dependent composite materials of all types; numerical analysis; physical aging; rheological properties; temperature, pressure, and moisture effects on time dependence; damping. The papers in the Symposium address constitutive, time (rate)-dependent constitutive and fracture/failure behavior of a broad range of materials systems, including prominent researchers in both applied and experimental mechanics. Solicited papers involve non-negligible time-dependent mechanical response in cases incorporating non-mechanical fields. Papers address modeling and experimental aspects of the subject areas. The organizers thank the presenters, authors and session chairs for their participation in this symposium.
vi The Society would like to thank the organizers of the track, H. Jerry Qi, University of Colorado; Richard B. Hall, Air Force Research Laboratory; Hongbing Lu, University of North Texas; Gyaneshwar P. Tandon, University of Dayton Research Institute; Bonnie R. Antoun, Sandia National Laboratories; Y. Charles Lu, University of Kentucky for their efforts. Bethel, Connecticut Dr. Thomas Proulx Society for Experimental Mechanics, Inc
Contents 1 Thermal and Mechanical Characterization of a Healable Polymer 1 C. Nielsen, H. Weizman, S. Nemat-Nasser 2 Fracture Behavior of Polymeric Foams Under Mixed-Mode Loading 5 E.E. Gdoutos 3 Coupled Experimental and Computational Analysis of Fracture Path Selection in PMMA Blocks 13 C.L. Tsai, Y.L. Guan, R.C. Batra, D.C. Ohanehi, J.G. Dillard, E. Nicoli, D.A. Dillard 4 Procedures for Mixed Mode Fracture Testing of Bonded Beams in a Dual Actuator Load Frame 25 E. Nicoli, D.A. Dillard 33 6 Experimental Study of Voids in High Strength Aluminum Alloys 39 H. Jin, W.-Y. Lu, J. Korellis, S. McFadden 7 Local Strain Accommodation in Polycrystalline Ni-Base Superalloys 41 J. Walley, R. Wheeler, M.D. Uchic, M.J. Mills 8 Coupled Thermal-mechanical Experiments for Validation of Pressurized, High Temperature Systems 49 B.R. Antoun, J.F. Dempsey, G.W. Wellman, W.M. Scherzinger, K. Connelly, V.J. Romero 9 Predictive Simulation of a Validation Forging Using a Recrystallization Model 51 D.J. Bammann, C.S. Marchi, N.Y.C. Yang 10 Characterization of Liquefied Natural Gas Tanker Steel From Cryogenic to Fire Temperatures 57 B.R. Antoun, K. Connelly, G.W. Wellman, J.F. Dempsey, R.J. Kalan 11 Nonlocal Microdamage Constitutive Model for High Energy Impacts 59 R.K. Abu Al-Rub, A.N. Palazotto 5 Linear Viscoelastic Behavior of Poly(Ethylene Therephtalate) Above Amorphous F. Bédoui Visco-elastic Property Crystallinity: Experimental and Micromechanical Modeling Tg VS A.A. Brown, B.R. Antoun, M.L. Chiesa, S.B. Margolis, D. , J.M. Simmons, O’Connor
viii 12 Impact Testing and Dynamic Behavior of Materials 67 L.W. Meyer, N. Herzig, F. Pursche, S. Abdel-Malek 13 Development of an Internal State Variable Model to Describe the Mechanical Behavior of Amorphous Polymer and its Application to Impact Testing 75 J.L. Bouvard, D. Ward, E.B. Marin, D. Bammann, M.F. Horstemeyer 14 Examination of Validity for Viscoelastic Split Hopkinson Pressure Bar Method T. Tamaogi, Y. Sogabe 85 16 93 F. Wang, Y. Wang, B. Fu, H. Lu 17 Theoretical and Computational Modelling of Instrumented Indentation of Viscoelastic Composites 101 Y.-P. Cao, K.-L. Chen 18 Obtaining Viscoelastic Properties From Instrumented Indentation 119 Y.-T. Cheng 19 Mechanical Properties Measurement of Sand Grains by Nanoindentation 121 20 Design and Implementation of Coupled Thermomechanical Failure Experiments 131 B.R. Antoun, J.F. Dempsey, G.W. Wellman, W.M. Scherzinger, K. Connelly 21 The Role of Interface and Reinforcement in the Finite Deformation Response of Polyurethane-Montmorillonite Nanocomposites 133 A.K. Kaushik, M. Yang, P. Podsiadlo, A.M. Waas, N.A. Kotov, E.M. Arruda 22 Time-Temperature Superposition and High Rate Response of Thermoplastic Composites and Constituents 139 P.D. Umberger, S.W. Case, F.P. Cook 23 Measuring Time Dependent Diffusion in Polymer Matrix Composites 147 S.P. Pilli, V. Shutthanandan, L.V. Smith 24 New Strain Rate Dependent Material Model for Fiber Reinforced Composites 149 L.W. Meyer, M. Mayer 25 Effect of Crystallinity and Fiber Volume Fraction on Creep Behavior of Glass Fiber Reinforced Polyamide 159 T. Sakai, Y. Hirai, S. Somiya 26 Hybrid Metal-Ceramic Thermo-oxidation Protection Layers for Polymer Matrix Composites 165 K.V. Pochiraju, G.P. Tandon 27 Degradation Phenomena Under Water Environment of Cotton Yarn Reinforced Polylactic-acid 175 S. Somiya, T. Ooike 77 F. Wang, B. Fu, R.A. Mirshams, W. Cooper, R. Komanduri, H. Lu 15 Weldability and Toughness Evaluation of the Ceramic Reinforced Steel Matrix Composites E. Bayraktar, F. Ayari, D. Katundi, J.-P. Chevalier, F. Bonnet (TIB2-RSMC) Nonlinear Viscoelastic Nanoindentation of PVAc
ix 28 Advanced Accelerated Testing Methodology for Life Prediction of CFRP Laminates 183 Y. Miyano, M. Nakada 29 Dynamic Properties of Foam With Negative Incremental Bulk Modulus 193 Y.-C. Wang, T. Jaglinski, H.-T. Chen 30 A Note on Automated Time-Temperature and Time-Pressure Shifting 199 M. Gergesova, B. Zupančič, I. Emri 31 Application of Fractional Derivatives Models to Time-dependent Materials 213 M. Sasso, G. Palmieri, D. Amodio 32 Tissue- and Microstructural-level Deformation of Aortic Tissue Under Viscoelastic/Viscoplastic Loading 223 D. Shahmirzadi, A.H. Hsieh 33 Strain Accumulation Process in Periodically Loaded Polymers 229 34 Viscoelastic and Viscoplastic Mechanical Behavior of Polymeric Nanofibers: An Experimental and Theoretical Approach 235 M. Naraghi, I. Chasiotis 35 Effect of Polyacrylate Interlayer Microstructure on the Impact Response of Multi-layered Polymers 241 J.S. Stenzler, N.C. Goulbourne 36 Visco-elastic Properties of Carbon Nanotubes and Their Relation to Damping 259 D. Qian, Z. Zhou 37 Ballistic Missile Defense System (BMDS) Solutions Using Remendable Polymers 267 T. Duenas, J. Schlitter, N. Lacevic, A. Jha, K. Chai, F. Wudl, L. Westcott-Baker, A. Mal, A. Corder, T.K. Ooi 38 Experimental Characterization and Modeling of Shape Memory Material for Downhole Completion Applications 275 C. Feng, G.D. Shyu, S. Gaudette, M. Johnson 39 Mechanics of Persulfonated Polytetrafluorethylene Proton Exchange Membranes 283 M.N. Silberstein, M.C. Boyce 40 The Influence of Pressure on the Large Deformation Shear Response of a Polyurea 287 M. Alkhader, W.G. Knauss, G. Ravichandran 41 Micromechanics Models for Predicting Tensile Properties of Latex Paint Films 297 E.W.S. Hagan, M.N. Charalambides, C.R.T. Young, T.J.S. Learner, S. Hackney 42 Time Dependent Recovery of Shape Memory Polymers 307 F. Castro, K.K. Westbrook, J. Hermiller, D.U. Ahn, Y. Ding, H.J. Qi 43 Structural Relaxation Near the Glass Transition: Observing Kovacs Kinetic Phenomenology by Mechanical Measurements 313 Y. Guo, R.D. Bradshaw B. Zupancic, I. Emri
x 44 Creep Mechanisms in Bone and Dentin Via High-Energy X-ray Diffraction 321 45 High Local Deformation Correlates With Optical Property Change in Cortical Bone 327 X. Sun, J.H. Jeon, S. Fuhs, J. Blendell, O. Akkus 46 Probing Pre-failure Molecular Deformation in Cortical Bone With Fluorescent Dyes 333 X. Sun, J.H. Jeon, J. Blendell, O. Akkus 47 The Influence of MgO Particle Size on Composite Bone Cements 339 M. Khandaker, S. Tarantini 48 Small-scale Mechanical Testing: Applications to Bone Biomechanics and Mechanobiology 345 M.M. Saunders 49 Determination of Fracturing Toughness of Bamboo Culms 353 N.-S. Liou, M.-C. Lu 50 Biomechanical Analysis of Ramming Behavior in Ovis Canadensis 357 P. Maity, S.A. Tekalur 51 Deformation and Failure Mode Transition in Hard Biological Composites 365 R. Rabiei, S. Bekah, F. Barthelat 52 Measurement of Structural Variations in Enamel Nanomechanical Properties Using Quantitative Atomic Force Acoustic Microscopy 373 W. Zhao, C. Cao, and C.S. Korach 53 Influence of Diamond-like Carbon Coatings on the Fatigue Behaviour of Spinal Implant Rod 383 54 Modeling Creep and Fatigue Properties of Bone at Nanoscale Level 391 F. Yuan, A. Singhal, A.C. Deymier-Black, D.C. Dunand, L.C. Brinson 55 High-energy X-ray Diffraction Measurement of Bone Deformation During Fatigue 395 A. Singhal, J.D. Almer, D.R. Haeffner, D.C. Dunand 56 Investigation of Cyclic Impact Fatigue, Grain-to-grain Interaction, and Residual Stress in Zirconia Dental Materials 399 H. Bale, N. Tamura, J.C. Hanan A.C. Deymier-Black, A. Singhal, F. Yuan, J. Almer, D. Dunand Y.C. Pan, J. Don, T.P. Chu, A. Mahajan
Thermal and Mechanical Characterization of a Healable Polymer Christian Nielsen1, Haim Weizman2 and Sia Nemat-Nasser1,* 1 Center of Excellence for Advanced Materials, Dept of Mechanical and Aerospace Engr, 2 Dept of Chemistry and Biochemistry, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0416, USA * sia@ucsd.edu ABSTRACT A cross-linked polymer capable of reforming broken bonds is considered a healable polymer. One such polymer, 2MEP4FS, has previously been shown to regain full toughness under ideal fracture and healing conditions. Here, a more purified 2MEP4FS polymer is characterized using thermal and mechanical techniques and compared with the previous 2MEP4FS polymer. Differential scanning calorimetry (DSC) confirms the presence of the thermally reversible Diels-Alder bonds necessary for healing. Dynamic mechanical analysis (DMA) establishes mechanical properties and the glass transition temperature. Fracture tests are conducted using the double cleavage drilled compression (DCDC) geometry. Compression drives symmetric cracks up and down a rectangular column of material with a central through-thickness hole. Correlating the applied stresses and crack lengths with a finite element model, critical stress intensity factors are estimated. The cracks are healed with a thermal treatment and light pressure, and the sample is retested. Over the course of multiple fracture and healing cycles, changes in the critical stress intensity factor are used to establish healing efficiency. Keywords: polymer, healing, fracture, crack, DSC, DMA, DCDC EXTENDED ABSTRACT A healable polymer, 2MEP4FS, is experimentally studied using thermal and mechanical techniques. The polymer was designed by Wudl et al. as a modified version of a previously reported polymer, 2MEP4F [1]. The product of a Diels-Alder reaction between monomers 2MEP and 4FS, 2MEP4FS contains thermally reversible cross-linking bonds that can be reformed after fracture. This healing ability is not found in traditional cross-linked polymers like epoxy. Testing has previously shown 2MEP4FS capable of complete mechanical recovery after a fracture event [2]. Under ideal conditions, macrocracks were healed and regrown numerous times with no significant changes in the critical stresses required for crack propagation. Some tests even indicated the material was tougher after healing. Current work seeks to confirm these results for material made from a new batch of monomers synthesized and purified using a different procedure than [2]. Typically, retesting the material would not be necessary, but the new monomers are believed to be more chemically pure. Current batches of 2MEP are observed to be a brighter white color [3]. Regarding 4FS, the previous purification process was determined to be incomplete and a new process was developed. The purity of the new monomers was confirmed using NMR. Monomers from [2] are not available for conclusive chemical characterization and comparison. Differential scanning calorimetry (DSC) is used to verify the presence of reversible bonds in the current polymer. Several milligrams of cured 2MEP4FS are heated in a sealed pan at a known temperature rate, and the amount of heat energy required is recorded. At the end of the test, when the polymer is at the peak temperature, the sample is removed and quenched in a bath of liquid nitrogen. By rapidly cooling the sample, separated reversible bonds will not be able to reform. Retesting the sample yields an exothermic peak around 80°C, where the bonds have gained sufficient mobility to reconnect. Proceedings of the SEM Annual Conference June 7-10, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc. for Experimental Mechanics Series 15, DOI 10.1007/978-1-4419-9794-4_1, © The Society for Experimental Mechanics, Inc. 2011 1 T. Proulx (ed.), Time Dependent Constitutive Behavior and Fracture/Failure Processes, Volume 3, Conference Proceedings of the Society
Dynamic mechanical analysis (DMA) gives the complex modulus of the material as a function of temperature, from which the glass transition temperature can be extracted. A single cantilever clamp tests a rectangular block of material using fixed-fixed bending. For a 1 Hz prescribed displacement oscillation, the storage modulus and glass transition temperature are observed to be slightly higher than the results presented in [2]. The healing potential of the current 2MEP4FS polymer is evaluated using the double cleavage drilled compression (DCDC) fracture test [2,4,5]. The test applies uniaxial compression to a column of material with rectangular cross-section and central through-thickness hole. This loading and geometry creates regions of tension at the apex and base of the hole, leading to symmetric mode I cracks. As the applied displacement is increased, these cracks propagate in a stable manner up and down the specimen. Once the cracks have grown to a sufficient length, the DCDC test is stopped and the healing process [2] is initiated. Light lateral pressure is applied to bring the crack faces together and the sample is heated to 85°C. After a period of 30 minutes, the pressure is released and the temperature is increased to 95°C for another 30 minutes. The sample is allowed to slowly cool to room temperature overnight before DCDC fracture testing and healing are repeated (Figure 1). By correlating the lengths of the cracks during each DCDC test with the measured forces, the fracture toughness of the material is estimated [6]. The crack length and force data are correlated with a set of finite element models, and internal energy, energy release rate and fracture toughness are calculated. After the sample is healed and retested, the healing efficiency is defined by the ratio of the new fracture toughness to the virgin fracture toughness. The initial series of DCDC tests are conducted using the DMA sample repurposed for DCDC testing. The results are not ideal, with healing efficiencies well below the 100% observed in [2]. The healing pressure, times and temperatures are varied with some improvement. The virgin fracture toughness of the current 2MEP4FS polymer is slightly larger than [2]. The DMA results and estimated DCDC fracture toughness indicate improved monomer purity may increase some material properties. Higher purities should translate to more cross-linking bonds, which would have this effect. More DCDC tests with a larger, dedicated sample are needed to give a better indication of healing potential. b c d a Figure 1. Crack morphology during a typical DCDC test and healing cycle: (a) the sample is pre-cracked and ready for testing; (b) the cracks start to grow; (c) the test is stopped; and (d) the sample is healed. 2
ACKNOWLEDGEMENTS This work was conducted with the support of Air Force Office of Scientific Research grant FA9550-08-1-0314 to UC San Diego. REFERENCES [1] Chen X et al, µNew thermally remendable highly cross-linked polymeric materials¶, Macromolecules 36(6), 1802-1807, 2003. [2] 3ODLVWHG 7 HW DO µ4XDQWLWDWLYH HYDOXDWLRQ RI IUDFWXUH KHDOLQJ DQG UH-healing of a reversibly cross-linked SRO\PHU¶ Acta Materialia 55, 5684-5696, 2007. [3] Plaisted T, Laboratory tour and conversation, San Diego, California USA, October 29, 2009. [4] -DQVVHQ & µ6SHFLPHQ IRU IUDFWXUH PHFKDQLFV VWXGLHV RQ JODVV¶ th International Congress on Glass, Kyoto Japan, Ceramic Society of Japan, 1974. [5] Plaisted T et al, µ&RPSUHVVLRQ-induced axial crack propagation in DCDC polymer samples: experiments and PRGHOLQJ¶ ,QWernational Journal of Fracture 141, 447-457, 2006. [6] 1LHOVHQ & µGeometric Effects in DCDC Fracture Experiments¶ 3URFHHGLQJV RI WKH 6(0 $QQXDO &RQIHUHQFe, Albuquerque, New Mexico USA, 2009. 3
Fracture Behavior of Polymeric Foams Under Mixed-Mode Loading E.E. Gdoutos Office of Theoretical and Applied Mechanics of the Academy of Athens School of Engineering, Democritus University of Thrace GR-671 00 Xanthi, Greece egdoutos@civil.duth.gr ABSTRACT The present work deals with the crack growth behavior in polymeric foams under mixed-mode loading conditions. Polymeric foams are anisotropic materials and cracks generally propagate under mixed-mode conditions. Due to the anisotropy of the material crack kinking occurs even though the applied load is perpendicular to the crack plane. The strain energy density criterion is used for the determination of the critical load of crack initiation and crack growth path under mixed-mode loading. A stress analysis of the plate is performed by a commercial finite element computer program. Results are obtained for the fracture trajectories for various polymeric foams. The study takes place within the frame of linear elastic fracture mechanics of anisotropic media. Introduction Cellular materials have extensively been used in sandwich construction due to their excellent properties, such as high specific modulus and strength, low weight, good thermal insulation and low cost. The mechanical behavior of cellular materials has been studied in [1-4]. It was found that the compressive stress-strain behavior of PVC cellular foams consists of an initial relatively small and stiff elastic regime, an extended stress plateau regime and a final regime in which densification of the material takes place. In the stress plateau regime the cells of the foam collapse, while the average stress remains almost constant during the instability spread through the material. Axial compresVLRQ SURGXFHV OLWWOH ODWHUDO VSUHDGLQJ UHVXOWLQJ WR DOPRVW ]HUR 3RLVVRQ¶V UDWLR :KHQ DOO RI WKH FHOOV FRllapse the material is densified and its stiffness increases. As a consequence of such behavior foams change their volume during plastic compression. This is contrary to dense solids which are incompressible during plastic deformation. On the contrary, the uniaxial stress-strain behavior in tension is nonlinear elastic without any identifiable yield region. The objective of this work is to study the mixed-mode crack growth behavior in a cross-liked polymeric foam under the commercial name Divinycell H250 with a density of 250 Kg/m3. The case of a plate with a crack perpendicular to the applied uniaxial stress is analyzed by finite elements. The results of stress analysis are coupled with the strain energy density theory to obtain crack growth trajectories for various values of the angle of orientation of the axes of anisotropy of the material with respect to the loading direction. Mechanical Characterization of Foam Materials The study will include many types of fully cross-linked PVC closed-cell foams under the commercial name Divinycell H80, H100, H160, H250 with densities of 80, 100, 160 and 250 kg/m3, respectively, and balsa wood. Figure 1 shows the stress-strain curves of Divinycell H250 in tension and compression. Note that the uniaxial stress-strain behavior in tension is nonlinear elastic without any identifiable yield region. In uniaxial compression the material is nearly elastic-perfectly plastic in the initial stage of yielding. Mechanical properties of materials studied are shown in Table 1. All Divinycell H80, H100, H160 and H250 materials exhibit axisymmetric anisotropy with much higher stiffness and strength in the cell (3-direction) than the in-plane direction. The ratio of the stiffness in the cell (eProceedings of the SEM Annual Conference June 7-10, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc. 5 for Experimental Mechanics Series 15, DOI 10.1007/978-1-4419-9794-4_2, © The Society for Experimental Mechanics, Inc. 2011 T. Proulx (ed.), Time Dependent Constitutive Behavior and Fracture/Failure Processes, Volume 3, Conference Proceedings of the Society
direction) to the in-plane direction is of the order of 1.5. The anisotropy of balsa wood is more pronounced with the above ratio equal to 42. All materials display different behavior in tension and compression with tensile strengths much higher than corresponding compressive strengths. Fig. 1: Stress-strain curves of Divinycell H250 PVC foam, in tension and compression. Strain Energy Density Criterion The basic quantity in the strain energy density theory for crack problems is the strain energy density function dW/dV, which can be put in the form [5-7]: r S dV dW (1) where S is the strain energy density factor and r is the distance measured from the crack tip. For plane elastic problems under conditions of plane stress the strain energy density function is given by . dV dW y y xy xy x x V H V H W J 2 1 (2) where ıx, ıy, ljxy are the stress and İx, İy, Ȗxy are the strain components. The strain energy density factor S is given by [5-7]: 2 22 2 12 1 2 2 11 1 2 S A k A k k A k (3) where Tension / Throughthe-thickness Tension / In-plane Compression / In-plane Compressn / Through-the-thickness 6
> @ CE AE C E AC A ǯ ǯ ǯ ǯ ǯ ǯ 26 16 12 2 66 2 22 2 11 11 2 2 2 4 1 D$ D D D D D (4a) > @ CE DE AE BE B CD EF AD BC A ǯ ǯ ǯ ǯ ǯ ǯ D$ D D D D D 26 16 12 66 22 11 12 2 4 1 (4b) > @ DF BF B D F BD A ǯ ǯ ǯ ǯ ǯ ǯ 26 16 12 2 66 2 22 2 11 22 2 2 2 4 1 D D D D D D (4c) with » » ¼ º « « ¬ ª ¸ ¸ ¹ · ¨ ¨ © § » » ¼ º « « ¬ ª ¸¸ ¹ · ¨¨ © § 1 2 1 2 2 2 1 2 1 1 2 2 1 2 1 2 1 z s z s s s , B Re z s z s s s s s A Re (5a) » » ¼ º « « ¬ ª ¸¸ ¹ · ¨¨ © § » » ¼ º « « ¬ ª ¸¸ ¹ · ¨¨ © § 1 1 2 2 1 2 2 1 1 2 1 1 1 1 s s z z , D Re z s z s s s C Re (5b) » » ¼ º « « ¬ ª ¸¸ ¹ · ¨¨ © § » » ¼ º « « ¬ ª ¸¸ ¹ · ¨¨ © § 2 2 2 1 1 2 2 1 2 1 1 2 1 1 1 z s z s s s , F Re s s z z s s E Re (5c) and 1 2 1 2 K K k k . S S (6) ,Q WKH DERYH HTXDWLRQV Įij are the compliance coefficients of the anisotropic material relating stress and strain, K1 and K2 are the stress intensity factors which dictate the stress field in the neighborhood of the crack tip, z1 = x1 + iy1, z2 = x1 - iy1 are complex numbers, and the other coefficients are related to the anisotropic behavior of the material [4-6]. Consider a plate with a through-the-thickness crack of length 2a that is subjected to a uniaxial stress ı perpendicular to the crack plane. The axȚV [ ޗ RI RUWKRWURS\ RI WKH PDWHULDO PDNHV DQ DQJOH ij ZLWK WKH FUDFN D[LV [ )LJ 7KH FRPSOLDQFH FRHIILFLHQWV Įij ޗ UHIHUUHG WR WKH V\VWHP [ \ )LJ DUH UHODWHG WR WKH FRHIILFLHQWV Dij refereed to the system xy by the following equations [5-7] , sin sin cos sin cos sin cos ǯD D M D D M M D M D M D M M2 2 2 26 2 16 4 22 2 2 66 12 4 11 11 , cos sin cos cos cos sin sin ǯD D M D D M M D M D M D M M2 2 2 26 2 16 4 22 2 2 66 12 4 11 22 , cos sin cos sin ǯD D D D D D M M D D M M 2 2 2 1 2 16 26 2 2 66 12 22 11 12 12 , cos sin cos sin ǯD D D D D D M M D D M M 2 2 2 2 4 16 26 2 2 66 12 22 11 66 66 7
ǯ 16 22 11 12 66 16 2 2 2 26 1 [ sin cos 2 cos2 ]sin2 cos cos 3sin 2 sin 3cos sin , D D M D M D D M M D M M M D M M M ǯ 26 22 11 12 66 16 2 2 2 26 1 [ sin sin 2 cos2 ]sin2 sin 3cos sin 2 cos cos 3sin , D D M D M D D M M D M M M D M M M (7) 2Į y' y ij x x' ı Fig.1 A cracked plate with a crack perpendicular to the applied load at an angle with the direction of the axis of material orthotropy of the material According to the strain energy density theory unstable crack growth takes place in the radial direction along which S becomes minimum. This condition is mathematically put in the form: . S , S 0 0 2 2 ! wT w wT w (8) 7KLV HTXDWLRQ LV XVHG IRU WKH GHWHUPLQDWLRQ RI WKH FULWLFDO DQJOH șc of initial crack growth. Unstable crack growth occurs when Smin șc) takes its critical value Sc which is an intrinsic material parameter, that is, S . S c min cT (9) 8
Equations (8) and (9) will be used for the determination of the critical quantities at crack instability for the case of Fig. 1. Results Results were obtained for an orthogonal plate with a crack perpendicular to the applied uniaxial stress. The axis of PDWHULDO V\PPHWU\ PDGH DQ DQJOH ij ZLWK UHVSHFW WR WKH FUDFN D[LV )LJ 7KH VWUHVV DQDlysis of the plate was performed by the ABACUS computer program. Figs 2 and 3 present the finite element idealization of the specimen in the vicinity of the crack tip. Fig. 4 presents the contours of strain energy density function near the crack tip. Note that due to material orthotropy the contours are not symmetrical, but inclined with respect to the crack axis. Fig. 5 presents the variation of strain energy density function dW/dV along the circumference of a circle centered at the crack tip for ij = 0, 300 and 600 7KH YDOXHV RI ș DW ZKLFK G: G9 SUHVHQWV ORFDO PLQLPD DUH WKH FULWLFDO YDlues of the angle of initial crack growth. Fig. 6 presents the variation of șc versus the DQJOH ij RI WKH RULHQWDWLRQ RI the axis of material orthotropic symmetry with respect to the crack axis. Note that the critical angle șc increases YHUVXV ij UHDFKLQJ D PD[LPXP YDOXH DIWHU ZKLFK LW GHFUHDVHV DQG EHFRPHV ]HUR ZKHQ WKH FUDFN LV DORQJ WKH D[LV of material symmetry. Fig. 2 Finite element mesh near the crack tip 9
Fig. 3 Detailed mesh )LJ &RQWRXUV RI VWUDLQ HQHUJ\ GHQVLW\ IXQFWLRQ QHDU WKH FUDFN WLS IRU ij 0 10
-180 -120 -60 0 60 120 180 ș (degrees) 0.0 0.4 0.8 1.2 dW/dV(x10-3kN mm/mm3) ij=0Ƞ ij=30Ƞ ij=60Ƞ șc Fig. 5 Variation of strain energy density function dW/dV versus polar angle ș around the circumference of a circle surrounding the crack WLS IRU ij 0 and 600. Crack grows in the direction of local minimum of strain energy density function 0 30 60 90 ij (degrees) 0 2 4 6 8 10 șc (degrees) Fig. 6 Critical angle of crack growth șc versus angle ij of orientation of axes of material symmetry with respect to the crack axis 11
Conclusions The crack growth in polymeric foams which present mechanical anisotropic behavior was studied. The case of a cracked plate subjected to a uniaxial stress perpendicular to the crack plane with the axes of material anisotropy at an angle with respect to the crack plane is analyzed. From the results of stress analysis in conjunction with the strain energy density theory the mixed-mode crack growth behavior of the plate was obtained. Results for the angle of initial crack growth for various orientations of the axes of anisotropy of the material with respect to the loading direction were reported. References [1] Gibson L.J. and Ashby, M.F., ³Cellular Solids,´ Cambridge University Press (1997). [2] Gdoutos, E.E., Daniel, I.M. and Wang, K.-A. ³0XOWLD[LDO Characterization and Modelling of a PVC Cellular FRDP ´J. Therm. Comp. Mat. 14, 365-373 (2001). [3] Gdoutos, E.E., Daniel, I.M. and Wang, K.-A. ³Failure of Cellular Foams under Multiaxial LRDGLQJ ´ Comp Part A, 33, 163-176 (2002). [4] Gdoutos E.E. and Abot, J.L., ³Indentation of a PVC Cellular FRDP ´ ,Q Recent Advances in Experimental Mechanics - In Honor of Isaac M. Daniel, Kluwer Academic Publishers pp. 55-64 (2002). [5] Gdoutos, E.E., ³3UREOHPV RI 0L[HG-0RGH &UDFN 3URSDJDWLRQ ´ 0DUWLQXV 1LMKRII 3ublishers (1984). [6] Gdoutos, E.E., ³)UDFWXUH 0HFKDQLFV &ULWHULD DQG $SSOLFDWLRQV ´ Kluwer Academic Publishers (1990). [7] Gdoutos, E.E., ³)UDFWXUH 0HFKDQLFV ± $Q ,QWURGXFWLRQ ´ 6HFRQG Edition (2005). 12
Coupled Experimental and Computational Analysis of Fracture Path Selection in PMMA Blocks C. L. Tsai1, Y. L. Guan1, R. C. Batra1, D. C. Ohanehi1, J. G. Dillard2, E. Nicoli1, D. A. Dillard1, 1Department of Engineering Science and Mechanics 2Department of Chemistry Virginia Polytechnic Institute and State University Blacksburg, Virginia Abstract While developing experimental and computational tools for analyzing crack path selection and failure loci in adhesively bonded joints, we have initially applied these tools for studying crack paths in pre-notched monolithic blocks of polymethyl methacrylate (PMMA), a common material for conducting brittle fracture experiments. Specimen configurations similar to the compact tension specimen but of varying length/width ratios were used to explore the effect of the T-stress on destabilizing the crack from growing straight along its original direction. Asymmetric versions of this geometry were also used to determine the effect of imposed mode mixity on crack path selection. These test configurations provided useful data for checking the robustness of the computational software based on a meshless local Petrov-Galerkin formulation of the boundary-value problem. The PMMA was assumed to be linear elastic, homogeneous and isotropic. A crack was assumed to initiate when the maximum principal tensile stress reached a critical value and propagate in the direction of the eigenvector of this stress. Effects of the mode-mixity on the crack propagation have been studied. Introduction Crack trajectories in adhesive bonds and brittle materials under mixed-mode loading are of interest in applications in biomedical implant, microelectronic, transportation, and energy devices and machinery. PMMA (polymethyl methacrylate), a transparent polymer, is the material used in the current study on crack propagation. PMMA has been employed in numerous studies to observe crack trajectories [1-5] in a range of specimen geometries including semi-circular bend (SCB), four-point bend (FPB), Brazilian disc (BD), and diagonally loaded square plate (DLSP). The current work covers a first series of tests to explore crack trajectories under mixed-mode loading of monolithic PMMA specimens. Using angled-cracked plate specimens, Smith et al. studied the mixed-mode fracture response of PMMA blocks (and brittle materials, in general) and explained crack trajectories in terms of the sign of the T-stress [3]. The Tstress, an important nonsingular term in Williams expansion[6] of stresses near a crack tip, is tangent to the crack, depends on the specimen geometry and loading conditions, and governs the stability of the growing crack. Aliha et al., using four specimen geometries, demonstrated the geometry dependence of the measured fracture toughness, and explained the dependence using the sign of the T-stress [1]. The mode mixity was varied from pure mode I to pure mode II through changes in the angle of the initial crack for various specimens, and the corresponding fracture toughness was computed. The crack trajectories for a series of SCB specimens were very instructive, showing a straight crack for pure mode I (initial crack inclination angle = 0o) and ending with a crack kink for mode II (initial crack inclination angle = 50o) [1]. For the current study, experiments were conducted on notched PMMA blocks that were similar to compact tension (CT) specimens, but had varying length/width ratios to explore effects of fracture mode mixity and the Tstress. The findings of these tests can provide an understanding of crack propagation in monolithic and isotropic materials and validate computational models being developed. For the numerical studies, the meshless local Petrov-Garlekin formulation of the boundary-value problem using the symmetric smoothed particle hydrodynamics (SSPH) basis functions for the trial solution was employed. Values of various parameters in the weight (or the test) function were optimized by studying the mode-I fracture problem for which analytical solution is known. Results of the computational studies were used to compute the stress concentration factor (SCF), the stress intensity factor (SIF), and the T-stress for simple tensile deformations of a square plate with a hole at the center. Proceedings of the SEM Annual Conference June 7-10, 2010 Indianapolis, Indiana USA ©2010 Society for Experimental Mechanics Inc. for Experimental Mechanics Series 15, DOI 10.1007/978-1-4419-9794-4_3, © The Society for Experimental Mechanics, Inc. 2011 13 T. Proulx (ed.), Time Dependent Constitutive Behavior and Fracture/Failure Processes, Volume 3, Conference Proceedings of the Society
The same software was used to analyze test configurations, and computed results were compared with the experimental findings. The software was also used to design experimental configurations that will give desired mode-mixity. Experimental Study Using PMMA Fracture tests were conducted on CT and modified CT specimens of 12.7 mm thick PMMA sheets. All tests were conducted on an Instron 5800R machine using a 5kN load cell. With one arm of the CT specimen fixed by the pin and the clevis, the other arm of the specimen was displaced downwards by the loading clevis. CT Specimen Tests A standard CT specimen (recommended in ASTM D 5045), shown in Fig. 1, has the thickness B=12.7mm (1/2’’), w=25.4mm (1’’) and a=12.7mm (0.5’’) [7]. Tests on the CT specimens were conducted using a crosshead displacement rate of 10 mm/min, as recommended in ASTM D5045. To ensure consistency in KIc values for all the specimens, sharp initial cracks were necessary. The sharp initial cracks were produced by driving a wedge into the specimens. The initial cracks were made carefully with a razor blade and a wooden hammer after machining the notches. The blade and two shims were inserted in the notch to center the blade and to keep it parallel to the centroidal axis of the notch. Then, gentle hammer blows were applied to the back of the blade, initiating a tiny crack, only a few millimeters in front of the blade, and avoiding a large initial crack, propagating too far into the specimen. A picture of three specimens, prior to the tests, is shown in Fig. 2. Relationships between the load and the crosshead displacement for three specimens are exhibited in Fig. 3. From these results, the average value of KIc was determined to be 1.056 ± 0.029 MPaξ݉ . As expected, the crack trajectories remained straight. Fig. 1. PMMA CT Specimen with w=25.4mm (1’’) and a=12.7mm (0.5’’); B, specimen thickness = 12.7mm (0.5’’). (All dimensions in mm). 14
Fig. 2. Notches and Initial Cracks before the Tests. Fig. 3. Load-Extension Curves for the three CT specimens. Effect of Notch Asymmetry on Crack Path Trajectory Unlike specimens described above that had a notch placed along the horizontal centroidal axis, specimens with the notch and the starting crack not located along the horizontal centroidal axis(“modified” or “asymmetric” CT 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 CT 1 CT 2 CT 3 Extension (mm) Load (N) 15
specimens) were also tested. The ratio of the specimen height above the notch to that below it is 2 (cf. Fig. 4.), and an unloaded specimen is shown in Figs. 4 and 5. In the asymmetric specimen, the mode mixity initiated as soon as the crack started propagating. Also, the diameter of the holes was reduced to 3.2mm (1/8’’) to make room Fig. 4. PMMA asymmetric CT specimens with w=25.4mm (1’’), a=12.7mm (0.5’’), and the ratio of the specimen height above the notch to that below is 2; B, specimen thickness = 12.7mm (0.5’’). (All dimensions in mm). Fig. 5a. Asymmetric CT specimens with notch and initial crack for the notch in the narrower arm of the asymmetric geometry. Because of the mode mixity, the crack path was inclined in the direction of the smaller arm (on the top in Fig. 5b). However, the crack path changed direction and became straight (almost perpendicular to the right edge in Fig. 5b) when the crack approached the edge. Additional details on this observation will be discussed in the section on the numerical study and comparisons will be made with the computed results. 16
Fig. 5b. Asymmetric CT Specimen with Crack after Test. Effect of Ratio of Specimen Length / Crack Length on Crack Path Trajectories The ratio of the specimen length and the initial crack length (w/a) was expected to be important for crack stability. Accordingly, PMMA specimens with different values of w/a were tested. The crack length (a) was varied from 12.7mm (0.5’’), to 25.4mm (1’’), and 38.1mm (1.5’’) while the value of w=101.6 mm (4’’) was kept constant; thus w/a equaled 8, 4, and 2.67. The other dimensions were the same as those for the CT specimen. The crack paths are shown in Fig. 6. Fig. 6. Trajectories of the crack path of the specimens with different initial crack lengths (From top to bottom: a=12.7mm, 25.4mm, and 38.1mm.) 17
As the ratio, w/a, was varied from 8, to 4, and 2.67, the crack path became curved and deviated towards one side. The curvatures of the three crack trajectories were almost the same. For specimens with a=25.4mm and 38.1mm, (w/a 4), the crack curved to one side quickly as soon as propagation started. However, when the initial crack length, a, was 12.7mm (w/a=8), the crack propagated along a straight line for almost 20mm before the deviation started. This suggests that, for small crack lengths, the T-stress is negative and the crack is stable initially. When the crack length increased to some critical value, the T-stress turned positive, destabilizing the crack, and the crack deviated to one side. Additional details will be shown and discussed when test results are compared with those from numerical simulations. Numerical Studies Both the finite element (FE) and meshless methods were employed in the numerical studies; the former used the commercial software, ABAQUS™, and the latter our in-house developed computer code. Finite Element Analysis The X-FEM (extended finite element method) implemented in ABAQUS™ v.6.9 using 4-node plane strain element, CPE4R, was employed to analyze deformations of the asymmetric CT specimen used in the experimental studies [8, 9]. Assigned boundary conditions simulated as closely as possible those likely to occur in the test configurations. With the loading points shown in Fig. 7a, the following boundary conditions were used in the numerical work. Load point 1: fixed in x and y directions: ux= 0, uy= 0; Load point 2: fixed in the x-direction and y-displacement prescribed: ux= 0, uy= -1 mm. From tensile tests on PMMA at a strain rate of 0.00014/s and room temperature, Elices and Guinea [10] obtained the following average values: Young’s modulus ܧ =3000±30 MPa, yield limit stress ߪ 0.2 =43.9±0.7MPa, rupture stress ܴߪ =74.9±0.2MPa, and Poisson’s ratio ߭ =0.4; these values were used in our simulations. The maximum principal stress of the damage initiation was set equal to ߪ 0.2 =44 MPa. Damage evolution was based on fracture energy ( ܩܿܫ = ܩܿܫܫ = ܭܿܫ 2 (ܧ /(1െ߭ 2)) =312.2 ܬ/݉ 2), linear softening, and mixed mode behavior of power law Į [10]. X-FEM results are show in Figure 7b and will be discussed after meshless methods are introduced. Analysis of the Problem by the SSPH Method Meshless methods were introduced in 1970’s, and include the Smooth Particle Hydrodynamics (SPH) method [11], Element-Free Galerkin method (EFGM) [12-14], Reproducing Kernel Particle Method (RKPM) [15], Meshless Local Petrov-Galerkin (MLPG) [16], Modified Smoothed Particle Hydrodynamics (MSPH) [17, 18] and Symmetric Smoothed Particle Hydrodynamics (SSPH) [19, 20]. Like the FE and the boundary element methods, meshless methods are used to find an approximate solution of an initial-boundary-value problem with the difference that no element connectivity is needed in a meshless method. Various meshless methods differ in the construction of basis functions for the trial solution. The EFGM [12-14] has been used to study linear elastic fracture mechanics (LEFM) problems and uses basis functions found by the moving least squares (MLS) method. However, it employs a background mesh to numerically evaluate various integrals appearing in the weak formulation of the problem and thus is not truly meshless. The enriched basis functions [21] are used to capture the stress singularity near a crack-tip without having a very fine distribution of particles (nodes) there. The computational efficiency can be improved by using an appropriate weight or test function [22] and basis functions that better capture singularity of fields near the crack tip [23]. Advantages of the SSPH method are that spatial derivatives of the trial solution are computed without differentiating the basis functions, and the stiffness matrix is symmetric. Our implementation of the SSPH method does not have the enriching terms to capture the stress singularity. We thus use a fine particle distribution around the crack tip. Additional details of the numerical scheme are provided in [24] where results for the CT specimen are also included. 18
Fig. 7a. Boundary conditions for the asymmetric CT specimen used while analyzing deformations with the commercial software ABAQUS™ Fig. 7b. Comparison of computed and experimental crack trajectories for an asymmetric CT specimen [The curve denoted by “SSPH Method” will be dicussed in the next section.] Analysis of deformations of the Double edge notched (DEN) specimen by the SSPH Method Since the T-stress is expected to play a significant role in the analysis of the crack problem, we first describe results for the mode-I problem obtained by the SSPH method. Deformations of the standard DEN specimen with H=3mm, B=1mm, the crack length ratio (a/B) = 0.2, E = 70GPa and Q 0.3 have been analyzed. The specimen is subjected to a uniform axial traction S as shown in Fig. 8. Because the specimen geometry and the boundary conditions are symmetric about the x2-axis, deformations of only the right-half of the specimen are studied. The 19
analytical expressions for the stress field near the crack-tip are [25]: » » » ¼ º « « « ¬ ª » ¼ º « ¬ ª ) 2 3 )sin( 2 ) 1 sin( 2 3 )sin( 2 sin( ) 2 3 )sin( 2 ) sin( 2 3 )sin( 2 1 sin( ) 2 cos( 2 22 12 12 11 T T T T T T T T T S V V V V r KI ( ) 0 0 0 r T » 2 ¼ º « ¬ ª (1) where T ,r are the cylindrical coordinates of a point with the origin located at the crack tip. From Eq. (1), the T- stress along the crack tip, obtained by setting 0 T , is given by 22 11V V T (2) Fig. 8. Double edge notched (DEN) specimen and particles distribution of half model The computed value, -0.501S, of the T-stress agrees well with of the -0.508S obtained by Kfouri [26]. Analysis of deformations of the Double Cantilever Beam (DCB) specimen by the SSPH Method The crack trajectories computed by the SSPH method for the DCB specimen shown in Fig. 9 were compared with those found experimentally. Dimensions and material properties for the DCB specimen are: length, W mm 25.4 , H mm 2 30.5 , a mm 12.7 , 3.10 E GPa and 0.35 X . A plane strain state of deformation was assumed to prevail in the DCB. We employed the crack initiation criterion used by Erdogan [27], i.e., a crack initiates when the maximum principal tensile stress reaches a critical value and it propagates in the direction of the eigenvector of this stress. In cylindrical coordinates with the origin at the crack-tip, the stress fields near the crack-tip are [27] 2B 2H a S B 2H S 20
Fig. 9 Double cantilever beam specimen T T T T T S V 2 2 cos 2 sin 2tan 2 3 2 1 sin 2 cos 2 1 T K K r II I rr » ¼ º « ¬ ª ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § (3) T T T T S VTT 2 2 sin sin 2 3 2 cos 2 cos 2 1 T K K r II I » ¼ º «¬ ª (4) > @ T T T T S VT 2 (3cos 1) sin sin 2 cos 2 1 T K K r II I r (5) where rrV , TT V and T Vr are, respectively, the radial, the circumferential and the shear stresses, I K and II K are the Mode I and the Mode II stress intensity factors, respectively, and T is the non-singular axial stress. The crack propagation angle 0T is found from 0 TT V T w w or equivalently from > @ 0 cos 2 sin 3 16 2 1) (3cos sin 0 0 0 0 T S T T T c II I T r K K (6) Once the crack initiation criterion has been met, the crack is propagated through the distance 0.02a . Thus the crack propagation analysis may be summarized as follows: Step1. The displacement, strain and stress fields are determined by using the SSPH method. Step2. The SIFs are evaluated using the interaction integral based on the results of step1. Step3. The crack propagation angle 0T is determined from Eq. (6). Step4. The crack is advanced by 0.02a in the direction that makes the angle 0T with the horizontal axis. Step5. Repeat steps1 through 4 until the desired load has been applied on the end faces of the specimen. In Fig. 7b, we have compared the computed crack path with that found experimentally; and it is clear that the two crack trajectories agree well with each other. W 2H a 2/3H 4/3H 21
Conclusions The compact tension PMMA specimens were tested in mode I with a displacement rate of 10 mm/min, and the value of KIc was found to be 1.056 ± 0.029 MPaξ݉ . A set of asymmetric DCB (modified CT) specimens, with the notch and the initial crack not located along the specimen center line, were tested with the goal of investigating crack trajectory deviations due to the mode mixity introduced at the beginning of the crack propagation. Referring to Figs. 4 and 6, the crack first propagated in the lengthwise direction of the specimen, then, briefly, in a transverse direction, followed by the lengthwise direction, and finally nearly perpendicular to the long edge of the specimen. Direction changes in the crack trajectory may be attributed sign switches in the T-stress. The effect of the specimen length to the initial crack length ratio, w/a, was also studied by testing the modified CT specimens with w/a = 8, 4 and 2.67. For the modified CT specimen with w/a 4, the crack deviated quickly when propagation started. However, for the specimen with w/a = 8, the crack propagated in a straight line for almost 20 mm before it deviated. These results are in qualitative agreement with those of Aliha and Ayatollahi determined from mixed-mode tests on semi-circular beam specimens of PMMA [1]. Deformations of pre-notched specimens were also analyzed by the X-FEM implemented in the commercial software, ABAQUS™, and the meshless SSPH method implemented in the in-house developed computer software. As should be clear from the results plotted in Fig. 7, the crack path predicted by the SSPH method is in better agreement with that found experimentally than the crack path computed using the X-FEM. However, this needs to be checked for several test configurations before the superiority of the SSPH method over the X-FEM can be ascertained. Acknowledgements The authors are grateful for support from the National Science Foundation (NSF/CMMI Award No. 0826143). YG would also like to thank the China Scholarship Council for partial support during this work. References 1. Aliha, M.R.M., Ayatollahi, M. R., Geometry effects on fracture behaviour of polymethyl methacrylate. Materials Science and Engineering A-Structural Material Properties Microstructure and Processing, 2010 527(3): p. 526-530. 2. Ayatollahi, M.R., Aliha, M. R. M., Hassani, M. M. , Mixed mode brittle fracture in PMMA - An experimental study using SCB specimens Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing, 2006 417(1-2): p. 348-356. 3. Smith, D.J., Ayatollahi, M. R., Pavier, M. J., The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Engng Mater Struct, 2001 24: p. 137–150. 4. Mahajan, R.V., and Ravi-Chandar, K., An experimental investigation of mixed-mode fracture. International Journal of Fracture, 1989 41: p. 235-252. 5. Gomez, F.J., Elices, M., Planas, J., The cohesive crack concept: application to PMMA at -60 oC. Engineering Fracture Mechanics, 2005 72(8): p. 1268-1285 6. Williams, M.L., On stress distribution at base of stationary crack. American Society of Mechanical Engineers -- Transactions -- Journal of Applied Mechanics, 1957 24(1): p. 109-114. 7. ASTM, D5045 - 99(2007)e1 Standard Test Methods for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials. 2007, ASTM International: West Conshohocken, PA. 8. Hibbitt, K., Sorensen, Inc., ABAQUS/standard user’s manual, v.6.9. 2009, Pawtucket, Rhode Island. 9. Giner, E., Sukumar, N., Tarancon, J. E., Fuenmayor, F. J., An ABAQUS implementation of the extended finite element method. Engineering Fracture Mechancs, 2009 76(3): p. 347-368. 10. Elices, M., Guinea, G. V., The cohesive zone model: advantages, limitations, and challenges. Engineering Fracture Mechanics, 2002 69(2): p. 137-163. 11. Lucy, L.B., A numerical approach to the testing of the fission hypothesis. Astron J, 1977 82: p. 1013– 1024. 22
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