.. Définition-des-indicateurs-de-fatigue-À-l-'échelle-microscopique, 117 4.3.1 Evaluation de la fatigue à l'échelle microscopique, p.117

S. Belhenini, A. Tougui, A. Bouchou, and F. Dosseul, 3D finite element modeling of 3D C2W (chip to wafer) drop test reliability: Optimization of internal architecture and materials, Microelectronics Reliability, vol.54, issue.1, pp.13-21, 2014.
DOI : 10.1016/j.microrel.2013.07.135

Z. Cellier, Une étude sur la fiabilité et le comportement thermomécanique de composant microélectronique type flip-chip, 2013.

H. Berg and W. E. , Advanced IGBT modules for railway traction applications: Reliability testing, Microelectronics Reliability, vol.38, issue.6-8, pp.1319-1342, 1998.
DOI : 10.1016/S0026-2714(98)00150-4

Y. Yamada, Y. Takaku, Y. Yagi, I. Nakagawa, T. Atsumi et al., Reliability of wire-bonding and solder joint for high temperature operation of power semiconductor device, Microelectronics Reliability, vol.47, issue.12, pp.2147-51, 2007.
DOI : 10.1016/j.microrel.2007.07.102

J. Goehre, M. Schneider-ramelow, U. Geißler, and K. Lang, Interface Degradation of Al Heavy Wire Bonds on Power Semiconductors during Active Power Cycling measure by the Shear Test, Integr. Power Electron. Syst. CIPS, pp.1-6, 2010.

M. Ciappa, Selected failure mechanisms of modern power modules, Microelectronics Reliability, vol.42, issue.4-5, pp.653-67, 2002.
DOI : 10.1016/S0026-2714(02)00042-2

M. Ciappa, Some Reliability Aspects of IGBT Modules for High-POwer Applications, pp.10-3929, 1961.

K. Pedersen, P. Kristensen, V. Popok, and K. Pedersen, Micro-sectioning approach for quality and reliability assessment of wire bonding interfaces in IGBT modules, Microelectronics Reliability, vol.53, issue.9-11, 2013.
DOI : 10.1016/j.microrel.2013.07.010

M. Bouarroudj, Z. Khatir, J. Ousten, F. Badel, L. Dupont et al., Degradation behavior of 600V???200A IGBT modules under power cycling and high temperature environment conditions, Microelectronics Reliability, vol.47, issue.9-11, pp.1719-1743, 2007.
DOI : 10.1016/j.microrel.2007.07.027
URL : https://hal.archives-ouvertes.fr/hal-00869421

J. Schultz-harder and K. Exel, Recent developments of direct bonded copper (DBC) substrates for power modules, Int. Conf. Electron. Packag. Technol, vol.11, pp.491-497, 2003.

L. Dupont, Contribution a l'Etude De La Duree De Vie Des Assemblages De Puissance Dans Des Environnements Haute Temperature, ENS de Cachan, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00983098

M. Berthou, P. Retailleau, H. Fremont, A. Guedon-gracia, and C. Jephos-davennel, Microstructure evolution observation for SAC solder joint: Comparison between thermal cycling and thermal storage, Microelectronics Reliability, vol.49, issue.9-11, pp.1267-72, 2009.
DOI : 10.1016/j.microrel.2009.07.040
URL : https://hal.archives-ouvertes.fr/hal-00477613

T. Mattila, H. Xu, O. Ratia, and M. Paulasto-kröckel, Effects of thermal cycling parameters on lifetimes and failure mechanism of solder interconnections, 2010 Proceedings 60th Electronic Components and Technology Conference (ECTC), pp.581-90
DOI : 10.1109/ECTC.2010.5490910

M. Mustafa, J. Suhling, and P. Lall, Experimental determination of fatigue behavior of lead free solder joints in microelectronic packaging subjected to isothermal aging, Microelectronics Reliability, vol.56, pp.136-183, 2015.
DOI : 10.1016/j.microrel.2015.10.021

L. Ladani and A. Dasgupta, Effect of Voids on Thermomechanical Durability of Pb-Free BGA Solder Joints: Modeling and Simulation, Journal of Electronic Packaging, vol.24, issue.3, 2007.
DOI : 10.1115/1.2098812

V. Le, L. Benabou, V. Etgens, and Q. Tao, Finite element analysis of the effect of process-induced voids on the fatigue lifetime of a lead-free solder joint under thermal cycling, Microelectronics Reliability, vol.65, pp.243-54, 2016.
DOI : 10.1016/j.microrel.2016.07.098

Y. Chan and D. Xie, Experimental studies of pore formation in surface mount solder joints, Materials Science and Engineering: B, vol.38, issue.1-2, pp.53-61, 1996.
DOI : 10.1016/0921-5107(95)01317-2

. Louhichi-m, Contribution à la conception thermo-mécanique optimisée d'assemblages sans plomb

D. Yu and L. Wang, The growth and roughness evolution of intermetallic compounds of Sn???Ag???Cu/Cu interface during soldering reaction, Journal of Alloys and Compounds, vol.458, issue.1-2, pp.542-549, 2008.
DOI : 10.1016/j.jallcom.2007.04.047

F. Che, J. Luan, and X. Baraton, Effect of Silver Content and Nickel Dopant on Mechanical Properties of Sn-Ag-Based Solders. 58th Electron, Components Technol. Conf, pp.485-90, 2008.

Z. Ma, F. Chalon, R. Leroy, N. Ranganathan, and B. Beake, Local and Global Properties of a Lead-Free Solder, Journal of Electronic Materials, vol.4, issue.9, pp.658-70, 2013.
DOI : 10.1038/nmat1429

A. El-daly, A. El-taher, and S. Gouda, Development of new multicomponent Sn???Ag???Cu???Bi lead-free solders for low-cost commercial electronic assembly, Journal of Alloys and Compounds, vol.627, pp.268-75, 2015.
DOI : 10.1016/j.jallcom.2014.12.034

G. Limaye, High temperature vibration fatigue life prediction and high strain rate material characterization of lead-free solders, 2013.

Q. Tao, L. Benabou, V. Le, H. Hwang, and D. Luu, Viscoplastic characterization and post-rupture microanalysis of a novel lead-free solder with small additions of Bi, Sb and Ni, Journal of Alloys and Compounds, vol.694, pp.892-904, 2017.
DOI : 10.1016/j.jallcom.2016.10.025

M. A. Motalab, T. Bieler, H. Jiang, L. Lehman, T. Kirkpatrick et al., Influence of Sn grain size and orientation on the thermomechanical response and reliability of Pb-free solder joints, IEEE Trans Components Packag Technol, vol.2731, pp.370-81, 2008.

M. Grieu, A. Lalonde, D. Emelander, J. J. Larson, C. Rietz et al., Etude de la fatigue des joints brases de composants electroniques soumis a des sollicitations thermomecaniques, vibratoires et combinees Ecole nationale supérieure des mines de Paris Quantitative Metallography of ? -Sn Dendrites in Sn-3.8Ag-0.7Cu Ball Grid Array Solder Balls via Electron Backscatter Diffraction and Polarized Light Microscopy, J Electron Mater, vol.33, pp.1545-1554, 2004.

L. Yin, L. Wentlent, L. Yang, B. Arfaei, A. Oasaimeh et al., Recrystallization and Precipitate Coarsening in Pb-Free Solder Joints During Thermomechanical Fatigue, Journal of Electronic Materials, vol.15, issue.411, pp.241-52, 2012.
DOI : 10.1007/978-1-4615-3910-0

B. Zhou, Characterizatation of Tin crystal orientation evolution during thermal cycling in lead-free solder joints, 2012.

P. Darbandi, T. Lee, T. Bieler, and F. Pourboghrat, Crystal plasticity finite element study of deformation behavior in commonly observed microstructures in lead free solder joints, Computational Materials Science, vol.85, pp.236-279, 2014.
DOI : 10.1016/j.commatsci.2014.01.002

P. Darbandi, D. Henderson, J. Woods, T. Gosselin, and J. Bartelo, Crystal plasticity finite element of deformation behaviour in SAC305 solder joint The microstructure of Sn in near-eutectic Sn-Ag-Cu alloy solder joints and its role in thermomechanical fatigue, J Mater Res Then Warrendale, vol.3419, pp.1608-1620, 2004.

A. Telang, T. Bieler, S. Choi, and K. Subramanian, Orientation imaging studies of Sn-based electronic solder joints, Journal of Materials Research, vol.15, issue.09, pp.2294-306, 2002.
DOI : 10.1557/S0883769400062448

S. Terashima, T. Kohno, A. Mizusawa, K. Arai, and O. Okada, Improvement of Thermal Fatigue Properties of Sn-Ag-Cu Lead-Free Solder Interconnects on Casio???s Wafer-Level Packages Based on Morphology and Grain Boundary Character, Journal of Electronic Materials, vol.55, issue.411, pp.33-41, 2009.
DOI : 10.1016/j.msea.2005.08.083

M. Matin, E. Coenen, W. Vellinga, and M. Geers, Correlation between thermal fatigue and thermal anisotropy in a Pb-free solder alloy, Scripta Materialia, vol.53, issue.8, pp.927-959, 2005.
DOI : 10.1016/j.scriptamat.2005.06.034

T. Mattila and J. Kivilahti, The Failure Mechanism of Recrystallization-Assisted Cracking of Solder Interconnections, 2012.
DOI : 10.1016/j.commatsci.2010.09.035

S. Terashima, K. Takahama, M. Nozaki, and M. Tanaka, Recrystallization of Sn Grains due to Thermal Strain in Sn-1.2Ag-0.5Cu-0.05Ni Solder, MATERIALS TRANSACTIONS, vol.45, issue.4, pp.1383-90, 2004.
DOI : 10.2320/matertrans.45.1383

D. Benoit, Fiabilité mécanique des assemblages électroniques utilisant des alliages du type SnAgCu, 2011.

H. Ma and J. Suhling, A review of mechanical properties of lead-free solders for electronic packaging, Journal of Materials Science, vol.34, issue.2, pp.1141-58, 2009.
DOI : 10.1016/j.matchar.2004.11.013

T. Bieler and A. Telang, Analysis of Slip Behavior in a Single Shear Lap Lead-Free Solder Joint During Simple Shear at 25??C and 0.1/s, Journal of Electronic Materials, vol.50, issue.310, pp.2694-701, 2009.
DOI : 10.1016/S0921-5093(00)01762-7

J. Gong, Mesomechanical modelling of SnAgCu solder joints in flip chip, Computational Materials Science, vol.43, issue.1, pp.199-211, 2008.
DOI : 10.1016/j.commatsci.2007.07.039

J. Harper, L. Shepard, and J. Dorn, Creep of aluminum under extremely small stresses, Acta Metallurgica, vol.6, issue.7, pp.509-527, 1958.
DOI : 10.1016/0001-6160(58)90114-7

L. Zhang and J. He, Reliability behavior of lead-free solder joints in electronic components, Journal of Materials Science: Materials in Electronics, vol.124, issue.3, pp.172-90
DOI : 10.1115/1.1462625

L. Anand, Constitutive equations for hot-working of metals, International Journal of Plasticity, vol.1, issue.3, pp.213-244, 1985.
DOI : 10.1016/0749-6419(85)90004-X

J. Chaboche, A review of some plasticity and viscoplasticity constitutive theories, International Journal of Plasticity, vol.24, issue.10, pp.1642-93, 2008.
DOI : 10.1016/j.ijplas.2008.03.009

E. Busso and M. Kitano, A Visco-Plastic Constitutive Model for 60/40 Tin-Lead Solder Used in IC Package Joints, Journal of Engineering Materials and Technology, vol.114, issue.3, pp.331-338, 1992.
DOI : 10.1115/1.2904181

D. Mcdowell, A nonlinear kinematic hardening theory for cyclic thermoplasticity and thermoviscoplasticity, International Journal of Plasticity, vol.8, issue.6, pp.695-728, 1992.
DOI : 10.1016/0749-6419(92)90024-7

J. Lemaitre and J. Chaboche, Mécanique des matériaux solides, 1996.

M. Amagai, M. Watanabe, M. Omiya, K. Kishimoto, and T. Shibuya, Mechanical characterization of Sn???Ag-based lead-free solders, Microelectronics Reliability, vol.42, issue.6, pp.951-6610, 2002.
DOI : 10.1016/S0026-2714(02)00017-3

I. Shohji, T. Yoshida, T. Takahashi, and S. Hioki, Tensile properties of Sn???Ag based lead-free solders and strain rate sensitivity, Materials Science and Engineering: A, vol.366, issue.1, 2004.
DOI : 10.1016/j.msea.2003.09.057

T. Korhonen, P. Turpeinen, L. Lehman, B. Bowman, G. Thiel et al., Mechanical properties of near-eutectic Sn-Ag-Cu alloy over a wide range of temperatures and strain rates, Journal of Electronic Materials, vol.123, issue.12, pp.1581-1589, 2004.
DOI : 10.1007/s11664-004-0101-2

S. Wiese and K. Wolter, Microstructure and creep behaviour of eutectic SnAg and SnAgCu solders, Microelectronics Reliability, vol.44, issue.12, pp.1923-1954, 2004.
DOI : 10.1016/j.microrel.2004.04.016

M. Pei and J. Qu, Constitutive modeling of lead-free solders, Int. Symp. Adv. Packag. Mater. Process. Prop. Interfaces, pp.45-54, 2005.

Q. Wang, L. Liang, and Y. Liu, Experimental Determination of Mechanical Properties for Lead-Free Material SnAgCu, 7th Int. Conf. Electron. Packag. Technol. ICEPT06, pp.1-4, 2006.

R. Darveaux and C. Reichman, Mechanical Properties of Lead-Free Solders. 57th Electron, Components Technol. Conf, pp.695-706, 2007.

N. Bai, X. Chen, and H. Gao, Simulation of uniaxial tensile properties for lead-free solders with modified Anand model, Materials & Design, vol.30, issue.1, pp.122-130, 2009.
DOI : 10.1016/j.matdes.2008.04.032

D. Herkommer, J. Punch, and M. Reid, A reliability model for SAC solder covering isothermal mechanical cycling and thermal cycling conditions, Microelectronics Reliability, vol.50, issue.1, pp.116-142, 2010.
DOI : 10.1016/j.microrel.2009.08.008

M. Kuna and S. Wippler, A cyclic viscoplastic and creep damage model for lead free solder alloys, Engineering Fracture Mechanics, vol.77, issue.18, pp.3635-3682, 2010.
DOI : 10.1016/j.engfracmech.2010.03.015

T. Chiu, J. Lin, H. Yang, and V. Gupta, Reliability model for bridging failure of Pb-free ball grid array solder joints under compressive load, Microelectronics Reliability, vol.50, issue.12, pp.2037-50, 2010.
DOI : 10.1016/j.microrel.2010.06.012

G. Limaye, High Temperature Vibration Fatigue Life Prediction and High Strain Rate Material Characterization of Lead-Free Solders, 2013.

L. Zhang, J. Han, Y. Guo, and C. He, Anand model and FEM analysis of SnAgCuZn lead-free solder joints in wafer level chip scale packaging devices, Microelectronics Reliability, vol.54, issue.1, pp.281-287, 2014.
DOI : 10.1016/j.microrel.2013.07.100

R. Asaro and J. Rice, Strain localization in ductile single crystals, Journal of the Mechanics and Physics of Solids, vol.25, issue.5, pp.309-347, 1977.
DOI : 10.1016/0022-5096(77)90001-1

F. Roters, P. Eisenlohr, L. Hantcherli, D. Tjahjanto, T. Bieler et al., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Materialia, vol.58, issue.4, pp.1152-211, 2010.
DOI : 10.1016/j.actamat.2009.10.058

R. Lebensohn and C. Tomé, A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys, Acta Metallurgica et Materialia, vol.41, issue.9, pp.2611-2635, 1993.
DOI : 10.1016/0956-7151(93)90130-K

W. Lee, L. Nguyen, and G. Selvaduray, Solder joint fatigue models: review and applicability to chip scale packages, Microelectronics Reliability, vol.40, issue.2, pp.231-275, 2000.
DOI : 10.1016/S0026-2714(99)00061-X

H. Solomon, Strain-Life Behavior in 60Sn40Pb Solder, J Electron Packag, vol.111, 1989.
DOI : 10.1115/1.3226525

S. Knecht and L. Fox, Constitutive relation and creep-fatigue life model for eutectic tin-lead solder, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol.13, issue.2, pp.424-457, 1990.
DOI : 10.1109/33.56179

J. Bannantine, Fundamentals of metal fatigue analysis, p.273, 1990.

M. Miner, Cumulative Damage in Fatigue, J Appl Mech, vol.7467, pp.159-69, 1945.

J. Morrow, Fatigue properties of metals. Fatigue Design Handbook, 1968.

J. Lee and H. Jeong, Fatigue life prediction of solder joints with consideration of frequency, temperature and cracking energy density, International Journal of Fatigue, vol.61, 2014.
DOI : 10.1016/j.ijfatigue.2013.10.021

J. Pang, B. Xiong, and T. Low, Low cycle fatigue models for lead-free solders, Thin Solid Films, vol.462, issue.463, pp.462-463408, 2004.
DOI : 10.1016/j.tsf.2004.05.037

C. Andersson, Comparison of isothermal mechanical fatigue properties of lead-free solder joints and bulk solders, Materials Science and Engineering: A, vol.394, issue.1-2, pp.20-27, 2005.
DOI : 10.1016/j.msea.2004.10.043

D. Dugdale, Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids, vol.8, issue.2, pp.100-104, 1960.
DOI : 10.1016/0022-5096(60)90013-2

K. Park and G. Paulino, Cohesive Zone Models: A Critical Review of Traction-Separation Relationships Across Fracture Surfaces, Applied Mechanics Reviews, vol.3, issue.6, 2013.
DOI : 10.1016/j.engfracmech.2012.02.007

J. Planas, M. Elices, and G. J. , The cohesive zone model: advantages, limitations and challenges, Eng Fract Mech, vol.69, issue.01, pp.137-163, 2002.

E. Glaessgen, E. Saether, D. Phillips, and V. Yamakov, Multiscale Modeling of Grain-Boundary Fracture: Cohesive Zone Models Parameterized from Atomistic Simulations, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 14th AIAA/ASME/AHS Adaptive Structures Conference<BR> 7th, pp.1-13, 2006.
DOI : 10.1103/PhysRevLett.95.015502

E. Iesulauro, Decohesion of grain boundaries in three-dimensional statistical representations of aluminum polycrystals, 2006.

S. Wen, L. Benabou, Z. Sun, and P. Dahoo, Identification expérimentale de modèles de zones cohésives à partir de techniques d'imagerie thermomécanique Univeristé Monpellier II A thermo-mechanical cohesive zone model for solder joint lifetime prediction, Int J Fatigue, vol.8749, pp.18-30, 2012.

M. Erinc, P. Schreurs, and M. Geers, Intergranular thermal fatigue damage evolution in SnAgCu lead-free solder, Mechanics of Materials, vol.40, issue.10, pp.780-91, 2008.
DOI : 10.1016/j.mechmat.2008.04.005

Q. Tao, L. Benabou, L. Vivet, V. Le, and F. Ouezdou, Effect of Ni and Sb additions and testing conditions on the mechanical properties and microstructures of lead-free solder joints, Materials Science and Engineering: A, vol.669, pp.403-419, 2016.
DOI : 10.1016/j.msea.2016.05.102

Q. Zhang and Z. Zhang, In situ observations on creep fatigue fracture behavior of Sn???4Ag/Cu solder joints, Acta Materialia, vol.59, issue.15, pp.6017-6045, 2011.
DOI : 10.1016/j.actamat.2011.06.010

K. Mysore, G. Subbarayan, V. Gupta, and R. Zhang, Constitutive and Aging Behavior of Sn3.0Ag0.5Cu Solder Alloy, IEEE Transactions on Electronics Packaging Manufacturing, vol.32, issue.4, pp.221-253, 2009.
DOI : 10.1109/TEPM.2009.2024119

M. Motalab, Z. Cai, J. Suhling, and P. Lall, Determination of Anand constants for SAC solders using stressstrain or creep data

D. Chan, X. Nie, D. Bhate, G. Subbarayan, W. Chen et al., Constitutive Models for Intermediate- and High-Strain Rate Flow Behavior of Sn3.8Ag0.7Cu and Sn1.0Ag0.5Cu Solder Alloys, IEEE Transactions on Components, Packaging and Manufacturing Technology, vol.3, issue.1, pp.133-179, 2013.
DOI : 10.1109/TCPMT.2012.2211022

Z. Ping, H. Bing-ting, Z. Jie-min, and Y. Ying, Parameter fitting of constitutive model and FEM analysis of solder joint thermal cycle reliability for lead-free solder Sn-3.5Ag, pp.10-1007, 2009.

L. Zhang, S. Xue, L. Gao, G. Zeng, Z. Sheng et al., Determination of Anand parameters for SnAgCuCe solder, Modelling and Simulation in Materials Science and Engineering, vol.17, issue.7, pp.1-10, 2009.
DOI : 10.1088/0965-0393/17/7/075014

Y. Zhu, X. Li, R. Gao, and C. Wang, Low-cycle fatigue failure behavior and life evaluation of lead-free solder joint under high temperature, Microelectronics Reliability, vol.54, issue.12, pp.2922-2930, 2014.
DOI : 10.1016/j.microrel.2014.08.016

L. Zhang, S. Xue, L. Gao, and Y. Chen, Effects of trace amount addition of rare earth on properties and microstructure of Sn???Ag???Cu alloys, Journal of Materials Science: Materials in Electronics, vol.44, issue.1, pp.1193-1202, 2009.
DOI : 10.1016/j.microrel.2007.10.005

K. Otiaba, M. Okereke, and R. Bhatti, Numerical assessment of the effect of void morphology on thermo-mechanical performance of solder thermal interface material, Applied Thermal Engineering, vol.64, issue.1-2, pp.51-63, 2014.
DOI : 10.1016/j.applthermaleng.2013.12.006

M. Okereke and A. Akpoyomare, A virtual framework for prediction of full-field elastic response of unidirectional composites, Computational Materials Science, vol.70, pp.82-99, 2013.
DOI : 10.1016/j.commatsci.2012.12.036

A. Melro, P. Camanho, and S. Pinho, Generation of random distribution of fibres in long-fibre reinforced composites, Composites Science and Technology, vol.68, issue.9, pp.2092-102, 2008.
DOI : 10.1016/j.compscitech.2008.03.013

L. Zhang, Etude de fiabilite des modules d'electronique de puissance a base de composant SiC pour applications hautes temperatures, 2014.
URL : https://hal.archives-ouvertes.fr/tel-00988235

L. Dupont, Z. Khatir, S. Parmentier, and S. Bontemps, Electrical characterizations and evaluation of thermo-mechanical stresses of a power module dedicated to high temperature applications, 2005 European Conference on Power Electronics and Applications, pp.1-11, 2005.
DOI : 10.1109/EPE.2005.219580

X. Perpina, L. Navarro, X. Jorda, and M. Vellvehi, Reliability and lifetime prediction for IGBT modules in railway traction chains, Reliab. Saf. Railw, pp.193-222, 2012.

Y. Gu and T. Nakamura, Interfacial delamination and fatigue life estimation of 3D solder bumps in flip-chip packages, Microelectronics Reliability, vol.44, issue.3, pp.471-83, 2004.
DOI : 10.1016/j.microrel.2003.11.002

K. Otiaba, R. Bhatti, N. Ekere, S. Mallik, and M. Ekpu, Finite element analysis of the effect of silver content for Sn???Ag???Cu alloy compositions on thermal cycling reliability of solder die attach, Engineering Failure Analysis, vol.28, pp.192-207, 2013.
DOI : 10.1016/j.engfailanal.2012.10.008

M. Yunus, K. Srihari, J. Pitarresi, and A. Primavera, Effect of voids on the reliability of BGA/CSP solder joints, Microelectronics Reliability, vol.43, issue.12, pp.2077-8610, 2003.
DOI : 10.1016/S0026-2714(03)00124-0

Q. Yu, T. Shibutani, D. Kim, Y. Kobayashi, J. Yang et al., Effect of process-induced voids on isothermal fatigue resistance of CSP lead-free solder joints, Microelectronics Reliability, vol.48, issue.3, pp.431-438, 2008.
DOI : 10.1016/j.microrel.2007.08.008

M. Dudek, L. Hunter, S. Kranz, J. Williams, S. Lau et al., Three-dimensional (3D) visualization of reflow porosity and modeling of deformation in Pb-free solder joints, Materials Characterization, vol.61, issue.4, pp.433-442, 2010.
DOI : 10.1016/j.matchar.2010.01.011

A. Fleischer, L. Chang, and B. Johnson, The effect of die attach voiding on the thermal resistance of chip level packages, Microelectronics Reliability, vol.46, issue.5-6, pp.794-804, 2006.
DOI : 10.1016/j.microrel.2005.01.019

R. Al-raoush and K. Alshibli, Distribution of local void ratio in porous media systems from 3D X-ray microtomography images, Physica A: Statistical Mechanics and its Applications, vol.361, issue.2, pp.441-56, 2006.
DOI : 10.1016/j.physa.2005.05.043

A. Scott, I. Sinclair, S. Spearing, M. Mavrogordato, and W. Hepples, Influence of voids on damage mechanisms in carbon/epoxy composites determined via high resolution computed tomography, Composites Science and Technology, vol.90, pp.147-53, 2014.
DOI : 10.1016/j.compscitech.2013.11.004

L. Benabou, V. Le, Z. Sun, P. Pougnet, and V. Etgens, Effects of voids on thermal-mechanical reliability of lead-free solder joints, MATEC Web of Conferences, vol.41, 2014.
DOI : 10.1007/s11664-011-1812-9

R. Pyrz, Quantitative description of the microstructure of composites. Part I: Morphology of unidirectional composite systems, Composites Science and Technology, vol.50, issue.2, pp.197-208, 1994.
DOI : 10.1016/0266-3538(94)90141-4

C. Schneider, W. Rasband, and K. Eliceiri, NIH Image to ImageJ: 25 years of image analysis, Nature Methods, vol.42, issue.7, pp.671-675, 2012.
DOI : 10.2144/000112257

J. Kim, Analysis of direct-soldered power module/heat sink thermal interface for electric vehicle applications. Virginia polytechnic institute and State university, 2001.

K. Stinson-bagby, Microstructural evolution in thermally cycled large-area lead and lead-free solder joints, 2002.

Y. Kariya, S. Tajima, and S. Yamada, Influence of Crystallographic Orientation on Fatigue Reliability of β-Sn and β-Sn Micro-Joint, MATERIALS TRANSACTIONS, vol.53, issue.12, pp.2067-71, 2012.
DOI : 10.2320/matertrans.MB201204

G. Taylor, The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.145, issue.855, pp.362-87, 1934.
DOI : 10.1098/rspa.1934.0106

J. Mandel, W. T. Generalisation-de-la-theorie-de-plasticite-de, and . Koiter, Generalisation de la theorie de plasticite de W. T. Koiter, International Journal of Solids and Structures, vol.1, issue.3, pp.273-95, 1965.
DOI : 10.1016/0020-7683(65)90034-X

R. Hill, Generalized constitutive relations for incremental deformation of metal crystals by multislip, Journal of the Mechanics and Physics of Solids, vol.14, issue.2, pp.95-102, 1966.
DOI : 10.1016/0022-5096(66)90040-8

R. Hill and J. Rice, Constitutive analysis of elastic-plastic crystals at arbitrary strain, Journal of the Mechanics and Physics of Solids, vol.20, issue.6, pp.401-414, 1972.
DOI : 10.1016/0022-5096(72)90017-8

R. Asaro and . Plasticity, Crystal Plasticity, Journal of Applied Mechanics, vol.50, issue.4b, pp.921-955, 1983.
DOI : 10.1115/1.3167205

Y. Huang, A User-Material Subroutine Incorporating Single Crystal Plasticity In the ABAQUS Finite Element Program, 1991.

J. Hutchinson, Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.348, issue.1652, 1976.
DOI : 10.1098/rspa.1976.0027

. Hibbett and S. Karlsson, ABAQUS/standard: User's Manual, 1998.

J. Kysar, Addendum to A User-Material Subroutine Incorporating Single Crystal Plasticity in the ABAQUS Finite Element Program, Mech Rep, pp.1-3, 1997.

M. Matin and W. Vellinga, Microstructure evolution in a Pb-free solder alloy during mechanical fatigue, Materials Science and Engineering: A, vol.431, issue.1-2, pp.166-74, 2006.
DOI : 10.1016/j.msea.2006.05.144

M. Fujiwara and T. Hirokawa, The Strength of Main Obstacles to Dislocation Motion in White Tin Crystals, Journal of the Japan Institute of Metals and Materials, vol.51, issue.9, pp.830-838, 1987.
DOI : 10.2320/jinstmet1952.51.9_830

R. Fiedler and I. Vagera, On the Burgers Vectors in ??-Sn Single Crystals, Physica Status Solidi (a), vol.12, issue.2, pp.419-443, 1975.
DOI : 10.1002/pssa.2210320209

R. Masson and A. Zaoui, Self-consistent estimates for the rate-dependentelastoplastic behaviour of polycrystalline materials, Journal of the Mechanics and Physics of Solids, vol.47, issue.7, pp.1543-6810, 1999.
DOI : 10.1016/S0022-5096(98)00106-9

A. Zaoui, An extension of the self-consistent scheme to plasticity -flowing polycrystals, pp.325-369, 1979.

G. Cailletaud and P. Pilvin, Utilisation de mod??les polycristallins pour le calcul par ??l??ments finis, Revue Europ??enne des ??l??ments Finis, vol.12, issue.4, pp.515-556, 1994.
DOI : 10.1016/0020-7683(72)90072-8

L. Benabou and Z. Sun, Homogenization scheme for brittle intergranular decohesion in polycrystalline aggregates, Mechanics Research Communications, vol.55, pp.114-123, 2014.
DOI : 10.1016/j.mechrescom.2013.10.010

L. Benabou, Z. Sun, P. Pougnet, and P. Dahoo, Abstract, Journal of Mechanics, vol.227, issue.05, pp.525-556, 2015.
DOI : 10.1115/1.1371781

L. V. Benabou, L. Etgens, V. Tao, and Q. , Micromechanical model for describing intergranular fatigue cracking in a lead-free solder alloy, Struct Integr Procedia 2016

C. Gerard, Mesures de champs et identification de modeles de plasticite cristalline, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00177914

R. Quey, P. Dawson, and F. Barbe, Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.17-20, pp.1729-1774, 2011.
DOI : 10.1016/j.cma.2011.01.002
URL : https://hal.archives-ouvertes.fr/hal-00858028

M. Kovac and L. Cizelj, Modeling elasto-plastic behavior of polycrystalline grain structure of steels at mesoscopic level, Nuclear Engineering and Design, vol.235, issue.17-19, pp.1939-50, 2005.
DOI : 10.1016/j.nucengdes.2005.05.009

S. Patibandla and R. , Metal fatigue and basic theoretical models: A review, Alloy Steel -Prop. Use, pp.203-239, 2011.

K. Tanaka and T. Mura, A Dislocation Model for Fatigue Crack Initiation, Journal of Applied Mechanics, vol.48, issue.1, pp.97-103, 1981.
DOI : 10.1115/1.3157599

M. Sangid, H. Maier, and H. Sehitoglu, A physically based fatigue model for prediction of crack initiation from persistent slip bands in polycrystals, Acta Materialia, vol.59, issue.1, pp.328-369, 2011.
DOI : 10.1016/j.actamat.2010.09.036

A. Manonukul and F. Dunne, High- and low-cycle fatigue crack initiation using polycrystal plasticity, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.460, issue.2047, pp.1881-903, 2004.
DOI : 10.1098/rspa.2003.1258

H. Mughrabi, Microstructural fatigue mechanisms: Cyclic slip irreversibility, crack initiation, non-linear elastic damage analysis, International Journal of Fatigue, vol.57, pp.2-8, 2013.
DOI : 10.1016/j.ijfatigue.2012.06.007

P. Mu, Study of crack initiation in low-cycle fatigue of an austenitic stainless steel, 2011.
URL : https://hal.archives-ouvertes.fr/tel-00628063

M. Matin, W. Vellinga, and M. Geers, Aspects of coarsening in eutectic Sn???Pb, Acta Materialia, vol.52, issue.12, pp.3475-82, 2004.
DOI : 10.1016/j.actamat.2004.03.045

M. Erinc, P. Schreurs, and M. Geers, Integrated numerical???experimental analysis of interfacial fatigue fracture in SnAgCu solder joints, International Journal of Solids and Structures, vol.44, issue.17, pp.5680-94, 2007.
DOI : 10.1016/j.ijsolstr.2007.01.021

T. Luther and C. Könke, Polycrystal models for the analysis of intergranular crack growth in metallic materials, Engineering Fracture Mechanics, vol.76, issue.15, pp.2332-2375, 2009.
DOI : 10.1016/j.engfracmech.2009.07.006

I. Benedetti and M. Aliabadi, A three-dimensional grain boundary formulation for microstructural modeling of polycrystalline materials, Computational Materials Science, vol.67, pp.249-60, 2013.
DOI : 10.1016/j.commatsci.2012.08.006

L. Benabou and Z. Sun, Analytical homogenization modeling and computational simulation of intergranular fracture in polycrystals, International Journal of Fracture, vol.2, issue.2, pp.59-75
DOI : 10.1615/IntJMultCompEng.v2.i1.60

T. Slack and F. Sadeghi, Cohesive zone modeling of intergranular fatigue damage in rolling contacts, Tribology International, vol.44, issue.7-8, pp.797-804, 2011.
DOI : 10.1016/j.triboint.2011.02.003

V. Le, L. Benabou, Q. Tao, and V. Etgens, Modeling of intergranular thermal fatigue cracking of a leadfree solder joint in a power electronic module, Int J Solids Struct, vol.106107, 2017.

N. Valoroso and L. Champaney, A damage-mechanics-based approach for modelling decohesion in adhesively bonded assemblies, Engineering Fracture Mechanics, vol.73, issue.18, pp.2774-801, 2006.
DOI : 10.1016/j.engfracmech.2006.04.029
URL : https://hal.archives-ouvertes.fr/hal-00109715

C. Moriconi, G. Henaff, and D. Halm, Influence of hydrogen coverage on the parameters of a cohesive zone model dedicated to fatigue crack propagation, Procedia Engineering, vol.10, pp.2657-62, 2011.
DOI : 10.1016/j.proeng.2011.04.443

C. Davila, P. Camanho, and M. De-moura, Mixed-mode decohesion elements for analyses of progressive 164

I. Simonovski and L. Cizelj, Cohesive element approach to grain level modelling of intergranular cracking, Engineering Fracture Mechanics, vol.110, pp.364-77, 2013.
DOI : 10.1016/j.engfracmech.2013.05.011

Z. Sun, L. Benabou, and P. Dahoo, Prediction of thermo-mechanical fatigue for solder joints in power electronics modules under passive temperature cycling, Engineering Fracture Mechanics, vol.107, 2013.
DOI : 10.1016/j.engfracmech.2013.05.009
URL : https://hal.archives-ouvertes.fr/hal-00830637

J. Chaboche, F. Feyel, and Y. Monerie, Interface debonding models: a viscous regularization with a limited rate dependency, International Journal of Solids and Structures, vol.38, issue.18, pp.3127-6010, 2001.
DOI : 10.1016/S0020-7683(00)00053-6

M. Vu, S. Geniaut, P. Massin, and J. Marigo, Numerical investigation on corner singularities in cracked plates using the G-theta method with an adapted ?? field, Theoretical and Applied Fracture Mechanics, vol.77, pp.59-68, 2015.
DOI : 10.1016/j.tafmec.2015.02.003
URL : https://hal.archives-ouvertes.fr/hal-01246477

L. Vitos, A. V. Ruban, H. Skriver, and J. Kollár, The surface energy of metals, Surface Science, vol.411, issue.1-2, pp.186-202, 1998.
DOI : 10.1016/S0039-6028(98)00363-X

W. Tyson, Surface free energies of solid metals: Estimation from liquid surface tension measurements, Surface Science, vol.62, issue.1, pp.267-76, 1977.
DOI : 10.1016/0039-6028(77)90442-3

M. Kamaya and M. Itakura, Simulation for intergranular stress corrosion cracking based on a three-dimensional polycrystalline model, Engineering Fracture Mechanics, vol.76, issue.3, pp.386-401, 2009.
DOI : 10.1016/j.engfracmech.2008.11.004

S. Shawish, . El, L. Cizelj, and I. Simonovski, Modeling grain boundaries in polycrystals using cohesive elements: Qualitative and quantitative analysis, Nuclear Engineering and Design, vol.261, pp.371-81, 2013.
DOI : 10.1016/j.nucengdes.2013.01.023

N. Dao and M. Vu, Load sequence effects on the fatigue crack growth in a cylinder subjected to combined rotary bending moment and axial force loads, Theoretical and Applied Fracture Mechanics, vol.82, pp.117-141, 2016.
DOI : 10.1016/j.tafmec.2015.12.011

D. Pierce, S. Sheppard, P. Vianco, J. A. Regent, and J. Grazier, Fatigue Life Prediction Methodology for Lead-Free Solder Alloy Interconnects: Development and Validation, J Electron Packag, vol.130, 2008.
DOI : 10.1115/1.2837515