A mechanism based approach void growth and coalescence in polymeric adhesive joints

A MECHANISM-BASED APPROACH — VOID GROWTH AND COALESCENCE IN POLYMERIC ADHESIVE JOINTS CHEW HUCK BENG (M.Eng, B.Eng (Hons), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 ii To my parents, and my wife, who truly are the wind beneath my wings iii LIST OF PUBLICATIONS Journal Papers [1] Chew, H B., Guo, T F and Cheng, L., Vapor pressure and residual stress effects on the toughness of polymeric adhesive joints Engineering Fracture Mechanics, 71 (2004), 2435-2448 [2] Chew, H B., Guo, T F and Cheng, L., Vapor pressure and residual stress effects on the failure of an adhesive film International Journal of Solids and Structures, 42 (2005), 4795-4810 [3] Chew, H B., Guo, T F and Cheng, L., Vapor pressure and residual stress effects on mixed mode toughness of an adhesive film International Journal of Fracture, 134 (2005), 349-368 [4] Chew, H B., Guo, T F and Cheng, L., Vapor pressure and voiding effects on thin film damage Thin Solid Films, 504 (2006), 325-330 [5] Chew, H B., Guo, T F and Cheng, L., Effects of pressure-sensitivity and plastic dilatancy on void growth and interaction International Journal of Solids and Structures, 43 (2006), 6380-6397 [6] Chew, H B., Guo, T F and Cheng, L., Pressure-sensitive ductile layers − I Modeling the growth of extensive damage International Journal of Solids and Structures, 44 (2007), 2553-2570 [7] Chew, H B., Guo, T F and Cheng, L., Pressure-sensitive ductile layers − II 3D models of extensive damage International Journal of Solids and Structures, 44 (2007), 5349-5368 [8] Chew, H B., Guo, T F and Cheng, L., Influence of non-uniform initial porosity distribution on adhesive failure in electronic packages IEEE Transactions on Components and Packaging Technologies, (2007), in Press Conference Papers [1] Chew, H B., Guo, T F and Cheng, L., Modeling interface delamination in plastic IC packages Proceedings of APACK 2001 Conference on Advances in Packaging (ISBN 981-04-4638-1), 5-7 Dec 2001, Singapore, pp 381- 388 [2] Chew, H B., Guo, T F and Cheng, L., A mechanism-based approach for interface toughness of ductile layer joining elastic solids JSME/ASME Proceedings of International Conference on Materials and Processing, 15-18 Oct 2002, Hawaii, Vol 1, pp 570-575 [3] Chew, H B., Guo, T F and Cheng, L., Computational study of vapor pressure and residual stress effects on adhesive failure Proceedings of International Conference on Scientific & Engineering Computation, 30 June - 02 July 2004, Singapore iv [4] Chew, H B., Guo, T F and Cheng, L., Computational study of compressive failure of metallic foam Proceedings of International Conference on Computational Methods (ISBN-10 1-4020-3952-2), 15-17 Dec 2004, Singapore, Vol 1, pp 563-568 [5] Chew, H B., Guo, T F and Cheng, L., Vapor pressure and voiding effects on thin film damage Presented in International Conference on Materials for Advanced Technologies, 3-8 July 2005, Singapore [6] Chew, H B., Guo, T F and Cheng, L., Influence of non-uniform initial porosity distribution on adhesive failure in electronic packages Proceedings of 7th Electronics Packaging Technology Conference (ISBN 0-7803-9578-6), 7-9 Dec 2005, Singapore, Vol 2, pp 6-11 [7] Chew H B., Guo, T F and Cheng, L., Void growth and damage ahead of a crack in pressure-sensitive dilatant polymers Proceedings of International Conference on High Performance Structures and Materials (ISSN 1743-3509), 3-5 May 2006, Ostend, Belgium, Vol 85, pp 501-510 [8] Chew, H B., Guo, T F and Cheng, L., Modeling adhesive failure in electronic packages Proceedings of 8th Electronics Packaging Technology Conference (ISBN 1-4244-0664-1), 6-8 Dec 2006, Singapore, Vol 2, pp 787-792 v ACKNOWLEDGEMENTS I wish to acknowledge and thank those people who contributed to this thesis: A/Prof Cheng Li, thesis advisor, for her unwavering guidance and support throughout the course of my studies at NUS She never fails to pepper our conversations with words of encouragement, or to slow down in her hectic schedule and provide a listening ear whenever I needed one The patience and care she demonstrated have not only made my research journey fun and interesting, but most of all have enriched my life with ever greater rewards Dr Guo Tian Fu, research fellow and mentor, for his invaluable guidance and insightful knowledge in continuum mechanics His passion and enthusiasm for research work was contagious and had been, and most surely will continue to be, a strong inspiration to me Fellow postgraduates, Chong Chee Wei, and Tang Shan, for their friendships and the moral support they had lent when I most needed it Directors and staff of EFE Engineering Pte Ltd, who were very supportive of my undertaking throughout the first two years of my research journey on a part-time basis Eunice See, my other half, for having absolute confidence in me, and for giving me the space and understanding I needed to work on my research at an excruciatingly slow pace Finally, I am forever indebted to my parents and brother for their most precious gift to me — love I can never thank them enough for the love, understanding, patience and encouragement that they had unselfishly given when I most needed them Not forgetting my furry companion Jo-Jo, who even after 12 wonderful years, never fails to teach me the simplicity of love vi TABLE OF CONTENTS DEDICATION ii LIST OF PUBLICATIONS iii ACKNOWLEDGEMENTS v LIST OF TABLES x xi LIST OF FIGURES LIST OF SYMBOLS xvii SUMMARY xix INTRODUCTION BACKGROUND THEORY AND MODELING 2.1 Micromechanics of ductile fracture 2.2 Mathematical models for void growth 2.3 Mechanism-based models 2.3.1 Traction-separation relation 2.3.2 Cell element approach 2.4 Pressure-dependent yielding 11 2.5 Crazing 14 2.5.1 Craze formation and growth 15 2.5.2 Micromechanical modeling 17 2.5.3 Pressure-sensitivity effects 18 MECHANISMS OF FAILURE FOR ADHESIVE LAYER WITH CENTERLINE CRACK 20 3.1 Introduction 20 3.2 Modeling aspects 22 3.2.1 Adhesive properties 22 3.2.2 Material model 24 3.2.3 Boundary value problem 25 Uniform initial porosity distribution 27 3.3.1 Failures of low and high porosity adhesives 28 3.3.2 Temperature/moisture effects on failures of low porosity adhesives 31 3.3.3 Temperature/moisture effects on failures of high porosity adhesives 37 3.3 vii 3.4 Failures of low and high porosity adhesives 37 3.4.2 37 3.4.1 3.5 Non-uniform initial porosity distribution Vapor pressure induced adhesive failures 39 Concluding remarks 42 PARALLEL DELAMINATION ALONG INTERFACES OF DUCTILE ADHESIVE JOINTS 44 4.1 Introduction 44 4.2 Problem formulation 45 4.3 Results and discussion 47 4.3.1 Film-substrate CTE mismatch 48 4.3.2 Residual stress in film 50 4.3.3 Vapor pressure at film-substrate interface 52 4.3.4 Porosity of film-substrate interface 53 4.3.5 Strain hardening of film 53 4.3.6 Thickness of film 54 Concluding remarks 57 4.4 INTERFACIAL TOUGHNESS OF DUCTILE ADHESIVE JOINTS UNDER MIXED MODE LOADING 59 5.1 Introduction 59 5.2 Modeling aspects 60 5.2.1 Material model 60 5.2.2 Small-scale yielding 62 Crack growth procedure and validation 65 5.3.1 Parametric dependence 65 5.3.2 Model validation 65 Steady-state toughness 68 5.4.1 Vapor pressure effects 68 5.4.2 Residual stress and vapor pressure effects 70 5.4.3 Layer thickness effects 73 Concluding remarks 74 5.3 5.4 5.5 PRESSURE-SENSITIVITY AND PLASTIC DILATANCY EFFECTS ON VOID GROWTH AND INTERACTION 76 6.1 Introduction 76 6.2 Material model 78 viii 6.3 Numerical modeling 79 6.3.1 The axisymmetric cell 79 6.3.2 Modeling aspects 80 Single void results 81 6.4.1 Initially spherical voids 83 6.4.2 Initially ellipsoidal voids 85 6.4.3 Implications to IC package failure 87 6.5 Multiple size-scale void interaction 92 6.6 Concluding remarks 96 6.4 PRESSURE-SENSITIVE DUCTILE LAYERS: MODELING THE GROWTH OF EXTENSIVE DAMAGE 98 7.1 Introduction 98 7.2 Problem modeling 99 7.2.1 Discrete void implementation 100 7.2.2 Internal pressure 102 7.2.3 Model parameters 103 Unit-cell behavior 103 7.3.1 Equibiaxial straining 104 7.3.2 Uniaxial straining 105 7.4 Failure mechanisms in pressure-insensitive adhesives 106 7.5 Damage evolution in pressure-sensitive adhesives 108 7.5.1 Associated normality flow, β = α 108 7.5.2 Non-associated flow, β < α 111 7.5.3 Relative cell size 113 7.6 Void coalescence and fracture toughness trend 115 7.7 Vapor pressure effects on adhesive damage 118 7.8 Concluding remarks 120 7.3 PRESSURE-SENSITIVE DUCTILE LAYERS: 3D MODELS OF EXTENSIVE DAMAGE 122 8.1 Introduction 122 8.2 Problem formulation 123 8.3 Model comparison 126 8.3.1 Three-dimensional versus two-dimensional discrete voids 127 8.3.2 Discrete voids versus computational cell elements 128 ix 8.4 Shape evolution and intervoid ligament reduction 130 8.4.1 Pressure-sensitivity effects 130 8.4.2 Relative cell size effects 133 Damage and fracture of pressure-sensitive adhesives 136 8.5.1 Damage evolution ahead of crack 136 8.5.2 Void coalescence and fracture toughness trends 140 8.6 Softening-rehardening yield characteristics 143 8.7 Concluding remarks 146 8.5 SUMMARY OF CONCLUSIONS 149 9.1 Mechanisms of failure in adhesive joints 149 9.2 Interfacial toughness of adhesive joints 150 9.3 Pressure-sensitivity and plastic dilatancy effects 151 9.4 Industrial implications 153 9.5 Recommendations for future work 153 REFERENCES 156 APPENDIX A – THE CRACK DRIVING FORCE 165 APPENDIX B – RADIAL EQUILIBRIUM SOLUTION FOR AXISYMMETRIC VOID GROWTH 167 APPENDIX C – VOID GROWTH OF AN AXISYMMETRIC PLANE STRAIN UNIT-CELL 169 APPENDIX D – STRAIN LOCALIZATION BEHAVIOR OF A UNITCELL 172 x LIST OF TABLES 6.1 Peak axial stress for σ /E = 0.002, ν = 0.3, N = 0.1 6.2 Critical mean stress for several void shapes under ψ = σ /E = 0.002, ν = 88 0.3, N = 0.1, f0 = 0.005 7.1 Critical mean stress and applied loads for cells ahead of the crack in pressure-sensitive adhesives with D = h/2, σ /E = 0.01, N = 0.1 112 Average critical porosity across cell distribution and cell sizes for several pressure-dependent parameters 117 7.2 86 160 [66] Keenan, J.H., Keyes, F.G., Hill, P.G., Moore, J.G., 1978 Steam Tables: Thermodynamic Properties of Water Including Vapor, Liquid, and Solid Phases New York: A Wiley-Interscience Publication, John Wiley & Sons [67] Kim, J., Gao, X., Srivatsan, T.S., 2003 Modeling of crack growth in ductile solids: a three-dimensional analysis International Journal of Solids and Structures 40, 7357-7374 [68] Kinloch, A.J., 1982 The science of adhesion — Part Mechanics and mechanisms of failure Journal of Materials Science 17, 617-651 [69] Koplik, J., Needleman, A., 1988 Void growth and coalescence in porous plastic solids International Journal of Solids and Structures 24, 835-853 [70] Kramer, H.H., 1983 Microscopic and molecular fundamentals of crazing Advances in Polymer Science 52, 1-56 [71] Krishnamachari, S.I., 1993 Applied stress analysis of plastics — a mechanical engineering approach Van Nostrand Reinhold, New York [72] Lee, B.J., Mear, M.E., 1992 Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids Journal of the Mechanics and Physics of Solids 40, 1805-1836 [73] Lee, B.J., Mear, M.E., 1999 Evolution of elliptical voids in power-law viscous solids Mechanics of Materials 31, 9-28 [74] Li, F.Z., Pan, J., 1990a Plane-strain crack-tip fields for pressure-sensitive dilatant materials Journal of Applied Mechanics 57, 40-49 [75] Li, F.Z., Pan, J., 1990b Plane-stress crack-tip fields for pressure-sensitive dilatant materials Engineering Fracture Mechanics 35, 1105-1116 [76] Liu, P., Cheng, L., Zhang, Y.W., 2003 Interface delamination in plastic IC packages induced by thermal loading and vapor pressure — a micromechanics model IEEE Transactions on Advanced Packaging 26(1), 1-9 [77] Liu, S., Mei, Y., 1995 Behavior of delaminated plastic IC packages subjected to encapsulation cooling, moisture absorption, and wave soldering IEEE Transactions on Components, Packaging and Manufacturing Technology Part A 8(3), 634-645 [78] Luo, S., Wong, C.P., 2005 Influence of temperature and humidity on adhesion of underfills for flip chip packaging IEEE Transactions on Components and Packaging Technologies 28, 88-94 [79] Madhusudhana, K.S., Narasimhan, R., 2002 Experimental and numerical investigations of mixed mode crack growth resistance of a ductile adhesive joint Engineering Fracture Mechanics 69(7), 865-883 [80] McClintock, F.A., 1968 A criterion for ductile fracture by growth of holes Journal of Applied Mechanics 35, 363-371 [81] Moran, B., Shih, C.F., 1987 A general treatment of crack tip contour integrals International Journal of Fracture 35, 363-371 [82] Narasimhan, R., 2004 Analysis of indentation of pressure sensitive plastic solids using the expanding cavity model Mechanics of Materials 36, 633-645 161 [83] Needleman, A., 1987 A continuum model for void nucleation by inclusion debonding Journal of Applied Mechanics 54, 525-531 [84] Omi, S., Fujita, K., Tsuda, T., Maeda, T., 1991 Causes of cracks in SMD and type-specific remedies IEEE Transactions on Components Packaging and Manufacturing Technology 14(4), 818-823 [85] O’Brien, T.K., 1987 Generic aspects of delamination in fatigue of composite materials Journal of the American Helicopter Society 32(1), 13-18 [86] Pardoen, T., Hutchinson, J.W., 2000 An extended model for void growth and coalescence Journal of the Mechanics and Physics of Solids 48, 2467-2512 [87] Pardoen, T., Hutchinson, J.W., 2003 Micromechanics-based model for trends in toughness of ductile metals Acta Materialia 51, 133-148 [88] Passaglia, E., 1984 Distribution of stress in a craze of the tip of a uniformly extending crack Polymer 25, 1727-1733 [89] Passaglia, E., 1987 Crazes and fracture in polymers Journal of Physics and Chemistry of Solids 48, 1075-1100 [90] Perrin, G., Leblond, J.-B., 2000 Accelerated void growth in porous ductile solids containing two populations of cavities International Journal of Plasticity 16, 91120 [91] Pijnenburg, K.G.W., Steenbrink, A.C., Van der Giessen, E., 1999 Shearing of particles during crack growth in polymer blends Polymer 40(21), 5761-5771 [92] Quinson, R., Perez, J., Rink, M., Pavan, A., 1997 Yield criteria for amorphous glassy polymers Journal of Materials Science 32, 1371-1379 [93] Reimanis, I.E., Dalgleish, B.J., Branhy, M., Ruhle, M., Evans, A.G., 1990 Effects of plasticity on the crack propagation resistance of a metal/ceramic interface Acta Metallurgica et Mater 38, 2645-2652 [94] Rice, J.R., Tracey, D.M., 1969 On the ductile enlargement of voids in triaxial stress fields Journal of the Mechanics and Physics of Solids 17, 201-217 [95] Ritter, J.E., Fox, J.R., Hutko, D.I., Lardner, T.J., 1998 Moisture-assisted crack growth at epoxy-glass interfaces Journal of Materials Science 33(18), 4581-4588 [96] Ruggieri, C., Panontin, T.L., Dodds Jr, R.H., 1996 Numerical modeling of ductile crack growth in 3-D using computational cell elements International Journal of Fracture 82(1), 67-95 [97] Seppälä, E.T., Belak, J., Rudd, R.E, 2004 The onset of void coalescence during dynamic fracture of ductile metals Physics Review Letters 93(24), Art No 245503 [98] Sha, Y., Hui, C.Y., Ruina, A., Kramer, E.J., 1995 Continuum and discrete modeling of craze failure at a crack tip in a glassy polymer Macromolecules 28, 24502459 [99] Sha, Y., Hui, C.Y., Ruina, A., Kramer, E.J., 1997 Detailed simulation of craze fibril failure at a crack tip in a glassy polymer Acta Materialia 45, 3555-3563 162 [100] Sha, Y., Hui, C.Y., Kramer, E.J., 1999 Simulation of craze failure in a glassy polymer: rate dependent drawing and rate dependent failure models Journal of Materials Science 34, 3695-3707 [101] Shih, C.F., 1991 Cracks on bimaterial interfaces: elasticity and plasticity aspects Materials Science and Engineering A143, 77-90 [102] Shih, C.F., Asaro, R.J., 1988 Elastic-plastic analysis of cracks on bimaterial interfaces: Part I — small scale yielding Journal of Applied Mechanics 55, 299-316 [103] Smelser, R.E., Becker, R., 1989 Abaqus user subroutines for material modeling In: Abaqus User’s Conference Proceedings, May 24-26, Stresa, Italy Hibbitt, Karlsson & Sorensen, Inc., Providence, RI, p 207-226 [104] Spearing, M., Beaumont, P.W.R., Ashby, M.F., 1991 Fatigue damage mechanics of notched graphite-epoxy laminates ASTM Special Technical Publication, p 617637 [105] Spitzig, W.A., Richmond, O., 1979 Effect of hydrostatic pressure on the deformation behavior of polyethylene and polycarbonate in tension and in compression Polymer Engineering and Science 19, 1129-1139 [106] Spitzig, W.A., Richmond, O., 1984 The effect of pressure on the flow stress of metals Acta Metallurgica 32, 457-463 [107] Sternstein, S.S., Ongchin, L., 1969 Yield criteria for plastic deformation of glassy high polymers in general stress fields Polymer preprints 10(2), 1117-1124 [108] Strohband, S., Dauskardt, R.H., 2003 Interface separation in residually-stressed thin-film structures Interface Science 11, 309-317 [109] Stoughton, T.B., Yoon, J.-W., 2004 A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming International Journal of Plasticity 20, 705-731 [110] Subramanya, H.Y., Viswanath, S., Narasimhan, R., 2006 A three-dimensional numerical study of mode I crack-tip fields in pressure-sensitive plastic solids International Journal of Solids and Structures, doi:10.1016/j.ijsolstr.2006.08.003 [111] Suo, Z., 1990 Failure of brittle adhesive joints Applied Mechanics Review 43(5), S276-279 [112] Swallowe, G.M., 1999 Mechanical properties and testing of polymers — an A-Z reference Kluwer Academic Publishers [113] Tandon, G.P., Weng, G.J., 1988 A theory of particle-reinforced plasticity Journal of Applied Mechanics — Transactions ASME 55, 126-135 [114] Tay, A.A.O., Lin, T.Y., 1999 Influence of temperature, humidity, and defect location on delamination in plastic IC packages IEEE Transactions on Components and Packaging Technologies 22(4), 512 — 518 [115] Teh, L.K., Teo, M., Anto, E., Wong, C.C., Mhaisalkar, S.G., Teo, P.S., Wong, E.H., 2005 Moisture-induced failures of adhesive flip chip interconnects IEEE Transactions on Components and Packaging Technologies 28, 506-516 [116] Tijssens, M.G.A., van der Giessen, E., Sluys, L.J., 2000a Modeling of crazing using a cohesive surface methodology Mechanics of Materials 32, 19-35 163 [117] Tijssens, M.G.A., van der Giessen, E., Sluys, L.J., 2000b Simulation of mode I crack growth in polymers by crazing International Journal of Solids and Structures 37, 7307-7327 [118] Trigg, A., 2003 Applications of infrared microscopy to IC and MEMs packaging IEEE Transactions on Electronics Packaging Manufacturing 26(3), 232-238 [119] Tvergaard, V., Needleman, A., 1984 Analysis of the cup-cone fracture in a round tensile bar Acta Metallurgica 32, 157-169 [120] Tvergaard, V., 1990 Material failure by void growth to coalescence Advances in Applied Mechanics 27, 83-151 [121] Tvergaard, V., Huang, Y., Hutchinson, J.W., 1992 Cavitation instabilities in a power hardening elastic-plastic solid European Journal of Mechanics A/Solids 11, 215-231 [122] Tvergaard, V., Hutchinson, J.W., 1992 The relation between crack growth resistance and fracture process parameters in elastic-plastic solids Journal of the Mechanics and Physics of Solids 40(6), 1377-1397 [123] Tvergaard, V., Hutchinson, J.W., 1994 Toughness of an interface along a thin ductile layer joining elastic solids Philosophical Magazine A 70(4), 641-656 [124] Tvergaard, V., Hutchinson, J.W., 1996 On the toughness of ductile adhesive joints Journal of the Mechanics and Physics of Solids 44(5), 789-800 [125] Tvergaard, V., 1998 Interaction of very small voids with larger voids International Journal of Solids and Structures 35, 3989-4000 [126] Tvergaard, V., 1999 Effect of large elastic strains on cavitation instability predictions for elastic-plastic solids International Journal of Solids and Structures 36, 5453-5466 [127] Tvergaard, V., Hutchinson, J.W., 2002 Two mechanisms of ductile fracture: void by void growth versus multiple void interaction International Journal of Solids and Structures 39, 3581-3597 [128] Utz, M., Peng, Q., Nandagopal, M., 2004 Athermal simulation of plastic deformation in amorphous solids at constant pressure Journal of Polymer Science: Part B: Polymer Physics 42, 2057-2065 [129] Van Melick, H.G.H., Govaert, L.E., Meijer, H.E.H., 2003 Localisation phenomena in glassy polymers: influence of thermal and mechanical history Polymer 44, 35793591 [130] Varias, A.G., Suo, Z., Shih, C.F., 1991 Ductile failure of a constrained metal foil Journal of the Mechanics and Physics of Solids 39(7), 963-986 [131] Wang, J., Lu, M., Zou, D., Liu, S., 1998 Investigation of interfacial fracture behavior of a flip-chip package under a constant concentrated load IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part B 21(1), 79-86 [132] Xia, L., Shih, C.F., 1995a Ductile crack growth — I A numerical study using computational cells with microstructurally-based length scales Journal of the Mechanics and Physics of Solids 43(2), 233-259 164 [133] Xia, L., Shih, C.F., 1995b Ductile crack growth — II Void nucleation and geometry effects on macroscopic fracture behavior Journal of the Mechanics and Physics of Solids 43, 1953-1981 [134] Xia, L., Cheng, L., 1996 Quasi-statically steady crack growth in rate dependent solids — One-parameter field encompassing different constraints International Journal of Fracture 82(2), 97-114 [135] Xia, L., Shih, C.F., Hutchinson, J.W., 1995 A computational approach to ductile crack growth under large scale yielding conditions Journal of the Mechanics and Physics of Solids 43(3), 389-413 [136] Zhang, S., Hsia, K.J., Pearlstein, A.J., 2002 Potential flow model of cavitationinduced interfacial fracture in a confined ductile layer Journal of the Mechanics and Physics of Solids 50, 549-569 165 APPENDIX A THE CRACK DRIVING FORCE For the interfacial crack problem in Chapters and 5, the stress field near the crack-tip is akin to that for an interface crack between dissimilar isotropic bimaterials (Hutchinson and Suo, 1992) The near-tip complex stress intensity factor Ktip for elastic media is related to the near-tip energy release rate Gtip by ả 1 − ν − ν s |Ktip |2 + Gtip = μ μs cosh2 π (A.1) where μ and μs are the shear moduli given by μ = E/(2(1+ν)) and μs = Es /(2(1+ν s )), and is the oscillation index = ln ả (3 − 4ν) /μ + 1/μs (3 − 4ν s ) /μs + 1/μ (A.2) The energy dissipated in the adhesive is small, since the layer thickness is relatively small compared to the remote boundary Under small-scale yielding conditions, Gtip can be approximately equated to the applied energy release rate G in (3.6), i.e G = Gtip Substituting (3.6) and (A.1) in (A.3), the following relationship is obtained: s µ ¶ − ν − ν s |Ktip | Es + K= − ν2 μ μs cosh π s (A.3) (A.4) The near-tip and far-field mode mixities are also loosely related (see Shih, 1991) The near-tip stress intensity factor has two sources: Ktip = Kb + Kp (A.5) where Kb is the contribution of the background stress, and Kp results from the tractionvapor pressure exerted on the crack faces To evaluate Kp , consider a bimaterial interface crack of length 2a with uniform pressure pt imposed on the crack faces England (1965) showed that Kp and pt are related by √ Kp (2a)i = pt πa(1 + 2i ) (A.6) 166 See Fig in Shih and Asaro (1988) From (A.4), (A.5) and (A.6), the applied remote K can be interpreted as an effective crack driving force from the contributions of the background stress and the traction-vapor pressure on the crack faces 167 APPENDIX B RADIAL EQUILIBRIUM SOLUTION FOR AXISYMMETRIC VOID GROWTH This appendix provides the homogenized solution for spherically symmetric void growth subjected to internal pressure p in: (a) a fully-plastic pressure-sensitive dilatant nonhardening matrix; (b) an incompressible elastic-plastic matrix α = β = B.1 Pressure-dependent fully-plastic solution Consider a thick-walled spherical shell of inner and outer radii ri and ro , subjected to internal pressure p and remotely applied radial stress Σm Assuming that the matrix of the shell is fully-plastic non-hardening, the yielding condition is then given by (2.4) with σ = σ From the flow rule in (2.6), one obtains ˆ trdp = 3β ˙ p (B.1) which defines the plastic dilatancy factor β Next, the elastic deformation of the matrix is neglected and the radial solution for spherically symmetric void growth is derived With respect to the orthonormal frame {er , eθ , eφ } of a spherical coordinate system, the radial problem has the nonzero stress components, σ rr and σ θθ = σ φφ , and the nonzero velocity, υ r The dilatancy condition (B.1) for void growth can be solved by υ r = Dm ³r ´ o 1+2β r r with ˙ p = 2Dm ³ ro ´ 1+2β + 2β r (B.2) where r is the radial distance from the void center and Dm = trD (≥ 0) is the spherical part of the macroscopic strain rate D The radial equilibrium solution satisfying (2.4) with σ = σ takes the form ˆ σ rr + p σ + 3αp σ θθ + p σ + 3αp = = ∙ ³ r ´ 6α ¸ 2α o 1+2α 1+2α 1−f , 3α r ∙ 6α ¸ 2α 1 − α 1+2α ³ ro ´ 1+2α 1− f 3α + 2α r where f = (ri /ro )3 is the void volume fraction (B.3) (B.4) 168 From the conservation of mass in the unit cell, one can derive the evolution law for the void volume fraction ´ ³ 2β f˙ = f 1+2β − f trD (B.5) by using the radial velocity (B.2) Compared to the incompressible case β = 0, Eq (B.5) shows that nonzero plastic dilatancy suppresses the void growth rate Identifying the radial stress σ rr at r = ro in (B.3) with the macroscopic mean stress Σm yields ´ 2α ³ Σm + p = − f 1+2α σ + 3αp 3α (B.6) From this simple unit-cell analysis, one observes that the major influence of pressuresensitivity is on the macroscopic stresses By contrast, plastic dilatancy mainly controls the void growth rate and has little influence on the overall stresses Equation (B.6) can be considered as the macroscopic yielding condition for hydrostatic stress As α → 0, it reduces to the following form ả Ă Â m + p + f2 = 2f cosh σ0 which is the extended Gurson model (Guo and Cheng, 2002) for pressure-insensitive solids, in the absence of the macroscopic Mises stress Σe B.2 Elastic-plastic solution, α = β = For a spherical cavity in a finitely deformed incompressible elastic-plastic solid subjected to internal vapor pressure p and externally applied radial stress Σm , Guo and Cheng (2001) showed that the radial equilibrium solution takes the form Z ε2 H (ε) dε Σm + p = σ0 − exp (−3ε/2) ε1 (B.7) Here, the uniaxial relationship between true stress and logarithmic strain of the material is described by σ/σ = H (ε) For an elastic-plastic power law hardening solid, H (ε) = ε/ε0 if |ε| < ε0 ; otherwise H (ε) = (|ε| /ε0 )N sign(ε) where ε0 = σ /E is the reference strain The lower and upper integration limits ε1 and ε2 in (B.7) are the two-end strains of the void, which can be determined solely by the current and initial void volume fractions f and f0 : ε1 = ln µ f0 − f f − f0 ¶ , ε2 = ln µ 1−f − f0 ¶ (B.8) 169 APPENDIX C VOID GROWTH OF AN AXISYMMETRIC PLANE STRAIN UNIT-CELL The homogenized solution for a plane strain hollow cylinder subjected to internal pressure p and remotely applied radial stress Σρ is derived in this appendix The overall axisymmetry reduces the problem to a one-dimensional problem in the radial direction In Appendices C.1 and C.2, attention is directed towards two special cases of plastic flow for void growth in a fully-plastic non-hardening matrix: (i) normality flow β = α, and (ii) incompressible plastic flow β = The solution for void growth in an incompressible elastic-plastic matrix α = β = is provided in Appendix C.3 C.1 Normality flow, β = α When the matrix material obeys the normality flow rule (β = α) , the macroscopic stresses in the radial and axial directions Σρ and Σz and the evolution law for the void volume fraction f can be derived as follows Σρ + p σ + 3αp Σρ − Σz σ + 3αp and = = i ±1 h − f (1g )/2 ả i g h (1−g±1 )/2 f −f 3α g + i h ±1 f˙ = f (1−g )/2 − f trDp where Dp is the macroscopic plastic strain rate and √ √ − α2 − 3α √ g=√ − α2 + 3α (C.1) (C.2) (C.3) (C.4) In the above, the sign ± distinguishes between void growth (+) and compression (−) p For the axisymmetric plane strain cell, Σρ and Dρ are the only effective work- conjugate pair with nonzero work-rate since Dp is proportional to a unit in-plane tensor R As such, the average work-rate of the cell, V −1 V σ : dp dv, has the normalized form i ±1 Σ + pI h : Dp = − f (1−g )/2 trDp σ + 3αp 3α (C.5) 170 This formulation suggests an approximate work-rate equivalence between axisymmetric and non-axisymmetric void growth (or compression), which can be used to estimate the upper-limit stress level for a unit-cell with elastic-plastic matrix The analytical derivation is found to corroborate closely with the numerical results in Fig 7.2 ±1 Observe that the factor f (1−g )/2 − f in the range ≤ α ≤ is identically greater than zero for both void growth and compression Under normality flow conditions, the radial stress Σρ is consistently greater than the axial stress Σz due to the plastic dilatancy of the matrix Thus Σρ − Σz in (C.2) defines the macroscopic effective stress For void growth, the factor f (1−g)/2 − f is not a monotonic decreasing function of f but attains a maximum at f = 1g ả2/(1+g) (C.6) When increases from to , f ∗ correspondingly increases from to , while the factor f (1−g)/2 −f decreases monotonically from to Hence an increase in pressure-sensitivity α under an associated flow significantly reduces the void growth rate (C.3) C.2 Incompressible plastic flow, β = For incompressible plastic flow (β = 0) , the axisymmetric cell is under a state of pure hydrostatic stress ´ 3α Σρ + p ³ √ = − f 3α± σ + 3αp 3α Σz − Σρ = 0, (C.7) and the void growth rate takes the classical form f˙ = (1 − f ) trDp (C.8) From (C.1) and (C.7), the effects of vapor pressure are noted to be less severe for pressure-sensitive polymers, like the die-attach or molding compound in electronic packages In the limiting case of α → 0, both flow rules considered above consistently leads to the pure hydrostatic stress state Σz − Σρ = 0, Σρ + p = ∓ √ ln f σ0 (C.9) and the classical void evolution law (C.8) as well Equation (C.9) can be cast into the 171 macroscopic yield condition ! Ã√ Σγγ + 2p = + f2 2f cosh σ0 (C.10) where Σγγ = Σx + Σy = 2Σρ In the absence of internal pressure, this equation can be identified with the Gurson model (Gurson, 1977) for cylindrical void growth with vanishing macroscopic effective stress C.3 Elastic-plastic solution, α = β = For a cylindrical cavity in a finitely deformed incompressible elastic-plastic solid subjected to externally applied in-plane radial stress Σρ and internal pressure p, the radial equilibrium solution takes the form: Σρ + p = σ0 Z ε2 ε1 H (ε) dε ¡ √ ¢ − exp − 3ε (C.11) Here, the uniaxial relationship between true stress and logarithmic strain of the material is described by σ/σ = H (ε) For an elastic-plastic power law hardening solid, H (ε) = ε/ε0 if |ε| < ε0 ; otherwise H (ε) = (|ε| /ε0 )N sign(ε) where ε0 = σ /E is the reference strain The lower and upper integration limits ε1 and ε2 in (C.11) are the two-end strains of the void, which can be determined solely by the current and initial void volume fractions f0 and f : ε1 = √ ln µ f0 − f f − f0 ¶ , ε2 = ln 1f f0 ả A similar form to the integral in (C.11) was used by Huo et al (1999) (C.12) 172 APPENDIX D STRAIN LOCALIZATION BEHAVIOR OF A UNIT-CELL To shed some light on the distinctive characteristics of softening-rehardening in polymers (Fig 8.13), the strain localization behavior of a representative material volume, consisting of a periodic assemblage of hexagonal unit-cells, is examined Each hexagonal cell can be approximated as a cylindrical cell containing a single spherical void A cylindrical coordinate system, with radial and axial coordinate directions (ρ, z), is adopted in this appendix Roller boundary conditions are applied along the symmetry planes of the quarter geometry of the cell volume to be analyzed The computations are carried out under a prescribed stress triaxiality using the numerical procedure outlined by Smelser and Becker (1989), with the dominant loading in the axial direction The material parameters adopted for the axisymmetric unit-cell study are σ /E = 0.01, ν = 0.4, with f0 = 0.08 The plastic strain contours of the representative unit-cell under stress triaxiality levels of T = 2/3 and are shown in Figs D.1 and D.2 respectively for several levels of effective strain Ee Softening-rehardening parameters of η = 20 and 100 are considered The former represents a polymer with strong rehardening characteristics, while the latter represents one with dominant softening response Under low stress triaxiality (Fig D.1), the yielding characteristics for both η = 20 and 100 are similar Plastic yielding initiates from the equator of the void at low loads, with nucleation of several inclined shear bands from the void surface At higher loads, these shear bands propagate along the diagonal of the cell (near 60◦ inclination), with nucleation of additional shear bands from the void surface The plastic strain localizes within each shear band under subsequent deformation Under high stress triaxiality (Fig D.2), results indicate the absence of the diagonal shear bands previously seen in Fig D.1 Instead, the plastic yielding is mainly confined to the horizontal ligament between neighboring voids for both η = 20 and 100 173 (a) η = 20 Ee = 0.029 Ee = 0.056 Ee = 0.103 (b) η = 100 Ee = 0.036 Ee = 0.074 Ee = 0.124 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Figure D.1: Plastic strain contours for η = 20 and 100 under low stress triaxiality T = 2/3 for elastic-plastic material, i.e α = β = The effects of pressure-sensitivity on the plastic strain contours are next examined in Fig D.3 for η = 20 under moderate triaxiality levels of T = With an increase in the friction angle ψ α (= ψ β ) from 0◦ to 20◦ , which is in the range relevant to polymers, the failure pattern evolves from shear banding at ψ α = 0◦ , to internal necking at ψ α = 20◦ This suggests that internal necking is the likely failure mode for pressure-sensitive constrained adhesives (e.g die-attach in IC packages) 174 Ee = 0.187 (a) η = 20 Ee = 0.018 Ee = 0.066 (b) η = 100 Ee = 0.115 Ee = 0.018 Ee = 0.034 >1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Figure D.2: Plastic strain contours for η = 20 and 100 under high stress triaxiality T = for elastic-plastic material, i.e α = β = 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Figure D.3: Plastic strain contours for η = 20 under stress triaxiality T = at Ee = 0.031 for several pressure-sensitivity levels: ψ α = 0◦ , 10◦ , 20◦ ... failures in detail A primary motivation behind this thesis is to understand how variations in temperature and moisture degrade mechanical properties of polymeric materials and adhesives and activate... 2003) and deformation fields in unvoided polymers and adhesives (e.g Li and Pan, 199 0a, b; Chowdhury and Narasimhan, 200 0a, b; Subramanya et al., 2006) A clearer picture of these distinctive characteristics... activate damage mechanisms which in turn lead to adhesive cracking and interface delamination These aspects of failure lie outside the scope of conventional elastic fracture mechanics based on a