UB Electronic Packaging Laboratory, Department of Civil Engineering, 102 Ketter Hall, University at Buffalo, Buffalo, NY 14260, USA

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1 Mechanics of Materials 38 (2006) Damage mechanics constitutive model for Pb/Sn solder joints incorporating nonlinear kinematic hardening and rate dependent effects using a return mapping integration algorithm Juan Gomez *, Cemal Basaran UB Electronic Packaging Laboratory, Department of Civil Engineering, 102 Ketter Hall, University at Buffalo, Buffalo, NY 14260, USA Received 17 December 2004 Abstract A thermodynamics-based damage mechanics rate dependent constitutive model is used to simulate experiments conducted on thin layer eutectic Pb/Sn solder joints. As compared to previous implementations of the model here we correct the difficulties introduced by the slow convergency rate of the Owen and Hinton (Owen, D.R.J., Hinton, E., Finite Element in Plasticity. Pineridge Press Limited, Swansea, UK) integration scheme. To this end, we time-integrated the model with a classical return mapping algorithm where rate dependency, nonlinear kinematic hardening of the Armstrong Frederick type and damage effects are simultaneously coupled. The model is implemented into the commercial finite element code ABAQUS via its user material subroutine capability and validated against experimental results. We simulated monotonic shear, cyclic shear and fatigue shear experiments performed on homemade thin layer solder joints. The simulation results are in good agreement with the experiments and the model accurately describes the true behavior of Pb/Sn solder alloys. As a direct advantage of the new model implementation this can be used for axisymmetric and 3D simulations as opposed to the plain strain-only capability in the Owen and Hinton (Owen, D.R.J., Hinton, E., Finite Element in Plasticity. Pineridge Press Limited, Swansea, UK) integration scheme. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Constitutive modeling; Damage mechanics; Rate dependence; Integration scheme; Solder joints; Microelectronics packaging 1. Introduction The most common cause of failure in microelectronic packaging solder alloys is introduced by low * Corresponding author. Tel.: ; fax: address: jdg8@buffalo.edu (J. Gomez). cycle fatigue generated by temperature changes and the coefficient of thermal expansion mismatch between the soldered parts. When the assembly undergoes a temperature variation the interconnections are stressed mainly in cyclic shear. The stresses impart elastic and inelastic strains, which are also cyclic in nature leading to thermomechanical fatigue. The changes in temperature are due to switch /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.mechmat

2 586 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) on/off operations or changes in the ambient operating conditions. On the other hand, eutectic solder alloys are routinely used at high homologous temperatures. The melting point of the eutectic Pb/Sn solder alloy is 183 C, and it is at 0.65T m at room temperature, where T m is the material melting point. Therefore, solder joints exhibit time, temperature and stress dependent deformation behavior and such coupling makes constitutive modeling a difficult task. Material models ranging from purely elastic to elasto-plastic using various stress strain relations have been proposed for Pb/Sn solder alloys, such as in Adams (1986), Kitano et al. (1988), Wilcox et al. (1989), Lau and Rice (1990), Knocht and Fox (1990), Darveaux and Banerji (1992), Hong and Burrell (1995), Basaran et al. (1998, accepted for publication) and many others. For instance, Adams (1986) proposed a simple viscoplastic model without hardening. Wilcox et al. (1989) proposed a rheological model to represent the inelastic behavior of the material, however it is applicable to a limited range of strain rates. The purely phenomenological models proposed by Knocht and Fox (1990), Darveaux and Banerji (1992) and Hong and Burrell (1995) decoupled the creep and plasticity effects artificially. This decoupling does not have any physical basis and is just motivated by mathematical convenience. Classical forms of decoupled plasticity and creep theories have been shown to be quite inferior for modeling cyclic plasticity and creep interaction effects (McDowell et al., 1994). An extensive literature survey on Pb/Sn constitutive models is available in Basaran et al. (1998). Recently Zhao (2000), followed by Basaran and Tang (2002) and Tang (2002) have extended a creep law originally proposed by Kashyap and Murty (1981) for eutectic solder alloys into a thermodynamics damage mechanics based framework for low cycle fatigue predictions. In that work damage is coupled into the model using the strain equivalence principle and the effective stress concept. Previous implementations of the model, Tang (2002) and Basaran et al. (accepted for publication), have used the Owen and Hinton (1980) time integration algorithm. This algorithm has several limitations. It is developed strictly for plane strain idealizations and its extension to 3D, axisymmetric or plane stress problems is not obvious. Furthermore, the algorithm was originally developed within a complete finite element formulation and is not efficient when independently implemented into existing commercial codes. For instance ABAQUS demands for an integration algorithm that updates the stress tensor, the material Jacobian relating stress to strains and all the defined state variables at the integration point level. The way to adapt the Owen and Hinton (1980) finite element framework to this local scheme is not clear. In this paper the constitutive model equations are integrated using a classical return mapping algorithm (Simo and Hughes, 1998). The algorithm is developed to update the stresses and the material Jacobian matrix at each time step and at a given Gauss point, and therefore can be implemented into available software that allows user defined subroutines. Three points are of interest in the present implementation and not explicitly described in the original work by Simo and Hughes (1998). First, is the coupling into the model of a nonlinear kinematic hardening rule of the Armstrong Frederick type (Armstrong and Frederick, 1966). We have followed Lubarda and Benson (2002) to incorporate the nonlinear kinematic hardening effects. Second is the implementation of the rate dependent effects in the form of a viscous overstress law of the Perzyna type. We have followed Alfano et al. (2001) to incorporate the rate dependent effects. Third is the coupling of damage into the constitutive model equations and the integration scheme. We have made use of the effective stress concept and the strain equivalence principle to couple damage into the model. This work is organized as follows. After defining notation, the first part of the paper describes the constitutive model equations. Attention is called upon the kinematic hardening rule which is different from the one previously implemented by Basaran and Tang (2002) which was in fact nonlinear but not exactly of the Armstrong Frederick type. The second part describes the integration algorithm where we present the algorithmic versions for the tangent stiffness matrix. Subsequently, we test the model and integration scheme simulating several experiments performed on thin layer eutectic Pb/Sn solder joints Notation Tensorial (indicial) and matrix notation are used throughout this article. In matrix notation second order tensors will be mapped into column vectors and fourth order tensors will be mapped into matrices. In indicial notation repeated indices are assumed to follow the summation convention unless explicitly stated otherwise and a free index will take

3 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) on the values 1, 2, 3. An index after a comma will represent a derivative with respect to a Cartesian coordinate. The inner (dot) product between two tensors will be denoted by the symbol : and the inner product between two first order tensors will be denoted by the symbol Æ. The tensor product will be denoted by the symbol. In the article the norm of a tensor v will be defined like [v : v] 1/2 if v is a second order tensor or like [v Æ v] 1/2 if v is a first order tensor and denoted by the symbol k k in both cases. The unit second order tensor d ij will be mapped into the column vector ^I and the fourth order unit tensor will be mapped into the matrix P. In the incremental equations of the constitutive model a time derivative will be denoted by a superimposed dot. In the algorithmic equations the subscript n will refer to quantities at the beginning of the increment and subscripts n + 1 will refer to quantities updated at the end of the increment. Finally, the superscript D will refer to a quantity computed considering the effects of damage. 2. Material constitutive model and integration scheme 2.1. Elastic constitutive relationship (Hooke s law) For a classical Von Mises rate independent plasticity model with isotropic hardening the elastic constitutive relationship is written using Hooke s law in rate form as _r ¼ C : ð_e _e p _e h Þ ð1þ where _e, _e p and _e h are the rates of total strain, plastic strain and thermal strain respectively and C is the elastic constitutive tensor. In Eq. (1) : represents the inner product between the fourth order tensor C and the elastic strain _e e ¼ _e _e p _e h Yield surface An elasto-plastic domain is defined according to the following yield function: rffiffi 2 F ðr; aþ ¼kS Xk KðaÞ 3 ks Xk RðaÞ ð2þ where F(r, a) is a yield surface separating the elastic from the inelastic domain, r is the second order stress tensor, a is a hardening parameter which specifies the evolution of the radius of the yield surface, X is the deviatoric component of the back stress tensor describing the position of the center of the yield surface in stress space, S is the deviatoric component of the stress tensor given by S ¼ r 1 ^p bi where 3 ^p is the hydrostatic pressure and bi qisffiffi the second 2 order identity tensor and RðaÞ KðaÞ is the 3 radius of the yield surface in stress space Flow rule The evolution of the plastic strain is represented by a general associative flow rule: _e p ¼ c of or c^n ð3þ where ^n ¼ of is the normal to the yield surface in or stress space, _e p has already been defined as the plastic strain rate and c is a nonnegative parameter representing the amount of plastic flow Isotropic hardening Isotropic hardening is described by the evolution of the radius of the yield surface in Eq. (2). The present evolution equation follows Chaboche (1989) and given by: rffiffi 2 KðaÞ ¼ Y 0 þ R 1 ð1 e ca Þ ð4aþ 3 where a is a plastic hardening parameter or plastic strain trajectory evolving according to Eq. (4b), Y 0 is the initial yield stress, R 1 is an isotropic hardening saturation value and c is the isotropic hardening rate: rffiffiffi 2 _a ¼ c ð4bþ 3 From Eqs. (3) and (4b) it can be seen that a ¼ R qffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 1 2 t 0 3 _ep : _e p ds which is precisely the standard definition of equivalent plastic strain The nonlinear kinematic hardening (NLK) rule The NLK rule describing the evolution of the center of the yield surface in stress space is the one from Chaboche (1989) and originally proposed by Armstrong and Frederick (1966). Nonlinearities are introduced as a recall term to the Prager (1956) linear hardening rule given in Eq. (5) and where c 1 and c 2 are material parameters: _X ¼ c 1 _e p c 2 X _a ð5þ

4 588 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) In Eq. (5) the first term represents the linear kinematic hardening rule as defined by Prager (1956). The second term is a recall term, often called a dynamic recovery term, which introduces the nonlinearity between the back stress X and the actual plastic strain. When c 2 = 0 Eq. (5) reduces to the Prager (1956) linear kinematic hardening rule. The NLK equation describes the rapid changes due to the plastic flow during cyclic loadings and plays an important role even under stabilized conditions (after saturation of cyclic hardening). In other words, these equations take into account the transient hardening effects in each stress strain loop. After unloading, dislocation remobilization is implicitly described due to the back stress effect and the larger plastic modulus at the beginning of the reverse plastic flow (Chaboche, 1989). In the original formulation by Tang (2002) the NLK hardening rule is written like _X ¼ 1½ 2 X 3 1_e p X _aš where 1 and X 1 are material parameters. Comparing this expression to Eq. (5) it can be seen that c 1 corresponds to the term 1 2 X 3 1 and c 2 corresponds to the term 1 in the original formulation. It is clear that the model originally proposed in Tang (2002) cannot be reduced to the linear kinematic case due to the presence of the pre-multiplicative factor 1 and in fact that model is not strictly of the Armstrong Frederick type Consistency parameter In Eqs. (3) and (4b) c is a nonnegative plasticity (consistency) parameter representing the irreversible character of plastic flow and obeying the following properties: 1. For a rate independent material model c obeys the so-called loading/unloading and consistency condition: c P 0 and F ðr; aþ 6 0 ð6þ c _F ðr; aþ ¼0 ð7þ 2. For a rate dependent material model conditions specified by Eqs. (6) and (7) are replaced by a constitutive equation of the form: huðf Þi c ¼ ð8þ g where g represents a viscosity material parameter, hiare Macauley brackets and u(f) is a specified function defining the character of the viscoplastic flow. When g! 0 the constitutive model approaches the rate independent case (Simo and Hughes, 1998). In the case of a rate independent material F satisfies conditions specified by Eqs. (6) and (7) and additionally stress states such F(r,a) > 0 are ruled out. On the other hand, in the case of a rate dependent material, the magnitude of the viscoplastic flow is proportional to the distance of the stress state to the surface defined by F(r,a) = 0. Using this fact and using Eq. (8) we have that the following relationship can be established: F ¼ HðcgÞ ð9þ where H(cg) =u 1 (cg) Viscoplastic creep law The relation between c and g expressed in Eqs. (8) and (9) is a general constitutive equation and different forms of the constitutive relationship describing the evolution of the viscoplastic strain can be implemented. In this particular model the creep law is the one proposed by Kashyap and Murty (1981) and extended to the multiaxial case by Basaran et al. (1998) and given by: _e vp ¼ AD 0Eb kh n hf i b E d p e Q=^Rh of or ð10þ where A is a dimensionless material parameter which is temperature and rate dependent, D i ¼ D 0 e Q=^Rh is a diffusion coefficient with D 0 representing a frequency factor, Q is the creep activation energy, br is the universal gas constant, h is the absolute temperature in Kelvin, E(h) is a temperature dependent Young s modulus, b is the characteristic length of crystal dislocation (magnitude of Burger s vector), k is Boltzmann s constant, d is the average grain size, p is a grain size exponent, n is a stress exponent for plastic deformation rate, where 1/n indicates strain sensitivity. From (10) we can identify hu(f)i = hfi n and g ¼ kh AD 0 E n 1 b ðd b Þp e Q=^Rh Damage coupled model Making use of the strain equivalence principle, Lemaitre (1990), we can write: _r ¼ð1 DÞC : ð_e _e vp _e h Þ ð11þ

5 rffiffi F ¼kS X D 2 k ð1 DÞ KðaÞ 3 ks X D k ð1 DÞRðaÞ ð12þ where D is a damage metric and the evolution of the backstress after considering damage reads _X D ¼ð1 DÞðc 1 _e vp c 2 X _aþ ð13þ 2.9. Return mapping algorithm Consider the following trial (elastic predictor) state: S tr nþ1 ¼ S n þð1 DÞ2lDe nþ1 ð14þ where l is the shear modulus and De n+1 is the deviatoric strain increment. Using Eq. (13) we can compute the increment of the back stress: dx D nþ1 ¼ð1 DÞfc 1 de vp nþ1 c0 2 Dc½bX D n þð1 bþx D nþ1 Šg ð15þ qffiffi where c 0 2 ¼ 2c 3 2 and we have used a generalized midpoint rule for the recall term with the extreme values b = 0 and b = 1 corresponding to the backwards and forwards Euler rules respectively. Now, from Eqs. (3) and (12) we can obtain the algorithmic counterpart of the viscoplastic strain increment: de vp nþ1 ¼ Dc S nþ1 X D nþ1 ks nþ1 X D nþ1 k substitution of Eq. (16) into Eq. (15) yields dx D nþ1 ¼ a nþ1 Dc S nþ1 X D nþ1 ks nþ1 X D nþ1 k c0 2 c 1 X D n c 1 ð1 DÞ! ð16þ ð17þ with a nþ1 ¼ 1þc 0 ð1 DÞð1 bþ Dc. 2 Using the flow rule specified by Eq. (3), allow us to express Eq. (14) like S nþ1 ¼ S tr nþ1 Dcð1 DÞ2l S nþ1 X D nþ1 ks nþ1 X D nþ1 k ð18þ Introducing the relative stress n D nþ1 ¼ S nþ1 X D nþ1 we have from Eq. (18): n D nþ1 ¼ S nþ1 X D nþ1 S tr nþ1 Dcð1 DÞ2l S nþ1 X D nþ1 ks nþ1 X D nþ1 k X D n dx D nþ1 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) ð19þ substituting Eq. (17) into Eq. (19) gives S nþ1 X D nþ1 þ Dc½ð1 DÞ2l þ a nþ1š S nþ1 X D nþ1 ks nþ1 X D nþ1 k ¼ B n ð20þ which is obtained after letting B n ¼ S tr nþ1 X D n þ b nþ1 DcX D n and b nþ1 ¼ c0 2 c 1 a nþ1. From Eq. (20) it can be shown that the normal to the yield surface can be expressed in terms of the data at the beginning of the step, therefore n nþ1 S nþ1 X D nþ1 ks nþ1 X D nþ1 k ¼ B n kb n k ð21þ Taking the trace product of Eq. (19) with itself yields: ks nþ1 X D nþ1 kþdc½ð1 DÞ2l þ a nþ1š ¼fkS n X D n k2 þkð1 DÞ2lDe nþ1 þ b nþ1 DcX D n k2 þ 2ðS n X D n Þ : ½ð1 DÞ2lDe nþ1 þ b nþ1 DcX D n Šg1=2 ð22þ Using Eq. (12) for the rate independent case or Eq. (9) for the rate dependent case we have the following nonlinear scalar equation in the consistency parameter which can be solved by a local Newton method: n gðdcþ ¼ ks n X D n k2 þkð1 DÞ2lDe nþ1 þ b nþ1 DcX D n k2 þ 2ðS n X D n Þ o 1=2 : ½ð1 DÞ2lDe nþ1 þ b nþ1 DcX D n Š rffiffiffi rffiffi! DÞ K a n þ Dc 3 3 Dc½ð1 DÞ2l þ a nþ1 Š H Dcg Dt ð23þ Once Eq. (23) is solved for Dc the following updating scheme can be used: rffiffi 2 a nþ1 ¼ a n þ Dc ð24þ 3 e vp nþ1 ¼ evp n þ Dc B n ð25þ kb n k! X D nþ1 ¼ X D n þ a S nþ1 X D nþ1 nþ1 Dc ks nþ1 X D nþ1 k c0 2 X D n c 1 ð26þ

6 590 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) n D nþ1 ¼ð1 DÞKða nþ1þ B n kb n k S nþ1 ¼ n D nþ1 þ X D nþ1 r nþ1 ¼ jð1 DÞtrðe nþ1 ÞbI þ 2lð1 DÞ e nþ1 e vp n c B n nþ1 kb n k eh nþ1 ð27þ ð28þ ð29þ The solution of the scalar nonlinear equation given by Eq. (23) via a local Newton iteration is one of the main distinguishing features of the present algorithm with the previous Owen and Hinton (1980) scheme. The use of this local Newton Raphson approach preserves the quadratic convergence of the global Newton algorithm used by ABAQUS while the Owen and Hinton (1980) scheme does not Linearization (consistent Jacobian) Differentiating Eq. (29) with respect to the total strain at the end of the step yields: dr nþ1 ¼ð1 DÞ C 2l^n nþ1 odc 2lDc o^n nþ1 : de nþ1 ð30þ where o Dc can be found from Eq. (22) such o Dc ¼ n nþ1 ð31þ K 3 where we have used K 3 ¼ K 1 þ K 2 a nþ1 K 1 ¼ 1 þ K0 3l þ 2lð1 DÞ K 2 ¼ a0 nþ1 Dc 2lð1 DÞ þ n nþ1b nþ1 2lð1 DÞ ½b nþ1ð1 hþdc 1Š : X D n 1 oh þ 2lð1 DÞ o Dc and o^n nþ1 can be obtained from Eq. (21) like o^n nþ1 ¼ o^n nþ1 ob n ob n 1 kb nþ1 k ðp ^n ob n nþ1 ^n nþ1 Þ : ð32aþ and ob n ¼ 2lð1 DÞ P 1 3 b I bi þðb 0 nþ1 Dc þ b nþ1þx n odc Letting K 4 ¼ b 0 nþ1 Dc þ b nþ1 and substituting ob n in Eq. (32a) yields o^n nþ1 ¼ o^n nþ1 ob n ob n 2lð1 DÞ kb n k P ^n nþ1 ^n nþ1 1 3 b I bi þ 1 kb n k ð bi ^n nþ1 ^n nþ1 Þ : K 4 ^n nþ1 X D n K 3 ð32bþ Using Eqs. (31) and (32b) into Eq. (30) results in C EVPD nþ1 ¼ð1 DÞjbI bi þ 2lð1 DÞd nþ1 P 1 3 b I bi where 2lð1 DÞ 2lð1 DÞ h nþ1^n nþ1 ^n nþ1 kb n k DcðP ^n nþ1 ^n nþ1 Þ : K 4 K 3 ^n nþ1 X D n Dc2lð1 DÞ d nþ1 ¼ 1 kb n k h nþ1 ¼ 1 Dc2lð1 DÞ K 3 kb n k and Formulation of the damage function ð33þ The damage evolution model is based on Basaran and Yan (1998) where the relation between the disorder and entropy is established using statistical mechanics and the second law of thermodynamics. The thermodynamic framework assumes that damage and the disorder are analogous concepts and the thermodynamic disorder can be used to model the damage evolution. The damage evolution function is given by D ¼ D cr ½1 e ðde D/Þ=ðN 0kh=m sþ Š ð34þ where D cr is a damage threshold, De D/ is the difference between the changes in the internal energy and the Helmholtz free energy with respect to a reference state, N 0 is Avogadro s number, k is Boltzmann s constant and m s is the specific mass of the material. The difference between the changes in the internal energy and the Helmholtz free energy with respect to a reference state, is obtained as follows:

7 the internal energy equation, which is an expression of the first law of thermodynamics, reads q de dt ¼ r : Din þ q^c rq ð35þ where D in is the rate of deformation tensor and $ is the gradient operator. For the particular case of small strains and small displacements D in ¼ dein, ^c dt is the internal heat production rate and q is the rate of heat flux through the surface. The Helmholtz free energy is written in terms of the stress tensor thus q dw dt ¼ r : Din ð36þ combining Eqs. (35) and (36) the difference between the changes in the internal energy and the Helmholtz free energy with respect to a reference state is obtained: De D/ ¼ 1 q Z t2 t 1 Z t2 r : D in dt þ J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) Z t2 t 1 ^cdt t 1 rqdt ð37þ 3. Simulation of monotonic and fatigue shear testing on Pb/Sn thin layer solder joints The proposed constitutive model was validated against results from experiments performed on thin layer solder joints of Pb37/Sn63 prepared at the UB-Electronic Packaging Lab. The solder joints are 460 lm thick, which is a thickness comparable to the diameter of solder joints in actual electronic packaging. Displacement controlled experiments were conducted for monotonic and cycle shear on an MTS 858 material testing system with ATS 7510 box thermal chamber. The thin layer solder joints were made by reflowing Pb37/Sn63 solder wire of mm diameter with flux of rosin core. After reflowing, the solder joints were left aging at room temperature (22 C) for 7 days to allow for metallurgy to develop between the solder alloy and the copper plates. As a fixture for the specimen, MTS 647 hydraulic grips with extension rods were used. The test was performed under displacementcontrolled conditions and the proper corrections on the total displacement to account for the stiffness of the fixture were made. Details about the testing can be found in Tang (2002). Figs. 1 and 2 show the used testing system and specimen attached to copper plates. The monotonic and mechanical shear tests were performed at different temperatures and Cu strain rates under isothermal conditions. Table 1 shows the test conditions Material properties Fig. 1. Testing system. 9.02mm 38.00mm Pb37/Sn63 Solder Joint 135 Cu 84.00mm Fig. 2. Thin layer solder joint attached to copper plates mm For the verification study the material constants were taken from the experimental study reported in Tang (2002) and from experimental results by Adams (1986). According to Adams (1986), Young s modulus varies significantly for the same Pb/Sn composition from one specimen to another. Furthermore, there is a big scatter on the values

8 592 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) Table 1 Testing conditions Strain rate (s 1 ) Temperature ( C) ISR Monotonic shear Case II E Case III E E E E Cyclic shear Case IV E Fatigue shear Case V E ISR: inelastic strain range. of Young s module reported in the literature. Pb/Sn solder alloy is a highly temperature dependent and rate dependent material. In addition, its material properties are also very sensitive to its microstructure. The variations of E, l and r y with temperature h (K) used in this work are as follows: Fig. 3. Monotonic shear testing under strain rate s 1 at 40 C. E ðgpaþ ¼52:10 0:1059h l ðgpaþ ¼19:44 0:0395h Y 0 ðmpaþ ¼60:069 0:140h ð38aþ ð38bþ ð38cþ Table 2 Material parameters used in the constitutive model Material parameters Elastic Young s modulus (GPa) h Shear modulus (GPa) h Isotropic hardening R 00 (MPa) h c r y h Kinematic hardening c c Flow rule A 7.60E+09 D 0 (mm/s 2 ) 48.8 b (mm) 3.18E 07 d (mm) 1.50E 02 n 1.67 p 3.34 Q (mj/mol) 4.47E+07 Fig. 4. Monotonic shear testing under strain rate s 1 at 22 C.

9 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) Fig. 5. Monotonic shear testing under strain rate s 1 at 60 C. Fig. 6. Monotonic shear testing under strain rate s 1 at 100 C. The phase size data corresponding to Adams (1986) is obtained after comparison of Adams results with Kashyap and Murty (1981) creep test data. Following this approach, Chandaroy (1998) found d = 15 lm. The optimum values of the kinematic hardening properties to fit the experimental data were found to be c 1 = 2040 MPa and c 2 = 180. The constant A is dependent on temperature, and the value of A is fitted based on Adams (1986) data. A linear regression was performed to obtain the relationship between A and temperature. The regressed equation yielded the following relationship A ¼ b 1 h b 2 or equivalently loga = logb 1 + b 2 logh where logb 1 = and b 2 = All the Fig. 7. Monotonic shear testing under strain rate s 1 at different temperature.

10 594 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) temperature dependent material properties used in this work are reported in Table Simulations vs test results Fig. 9. Monotonic shear testing under strain rate s 1 at 22 C. Figs. 3 7 show a comparison between numerical results and testing under monotonic shear for different temperatures at a strain rate of s 1. Good predictions are generally obtained and the model in fact captures the temperature dependence. Figs present results for the same type of test at room temperature for different strain rates. It can be seen that the model effectively captures the strain rate dependency. Fig present cycling loading simulations at room temperature, strain rate of s 1 and different inelastic strain ranges again good correlation between experimental and numerical simulation results is obtained. Figs present the results for several fatigue cycles under displacement controlled conditions and the corresponding evolution of the damage parameter. The computational simulations are generally in good agreement with the experimental results. The scatter in the results is more likely due to the differences in the geometry between actual specimens and the numerical model. For instance, there are voids in the actual specimens that are not explicitly represented in the numerical model; therefore some differences should be expected. In order to compare the stress strain response with the testing results the proper corrections were made to account for Fig. 8. Monotonic shear testing under strain rate s 1 at 22 C. Fig. 10. Monotonic shear testing under strain rate s 1 at 22 C.

11 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) Fig. 11. Monotonic shear testing under strain rate s 1 at 22 C. Fig. 13. Cyclic shear simulation vs test data at 22 C, strain rate s 1 and ISR = the voids in the solder joint and the stiffness of the used load train. 4. Conclusions In this paper we have implemented a thermodynamics-based damage mechanics coupled constitutive model for Pb/Sn solder alloys within a classical return mapping scheme but considering the combined effects of damage, viscoplasticity and a nonlinear kinematic hardening rule of the Armstrong Frederick type. The model capabilities were verified via comparisons with experimental results from tests performed on homemade thin layer solder joints. The comparisons show that the temperature and rate dependent effects that Fig. 12. Monotonic shear testing at different strain rates at 22 C.

12 596 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) Fig. 14. Cyclic shear simulation vs test data at 22 C, strain rate s 1 and ISR = Fig. 16. Cyclic shear simulation vs test data at 22 C, strain rate s 1 and different inelastic strain range. Fig. 15. Cyclic shear simulation vs test data at 22 C, strain rate s 1 and ISR = Fig. 17. Isothermal fatigue at strain rate s 1 at 22 C with inelastic strain range characterize the behavior of Pb/Sn solder alloys under different loading conditions are effectively captured by the model. Moreover the observed scattering in the results is due to the imperfect nature of the actual solder joints while the finite element model assumes perfect conditions. Although here

13 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) Fig. 18. Damage evolution under fatigue with ISR = Fig. 19. Isothermal fatigue at strain rate s 1 at 22 C with inelastic strain range Fig. 20. Damage evolution under fatigue with ISR = we have used a particular creep law suitable for Pb/ Sn solder alloys, the model formulation is general and can be used for other type of constitutive relationships like in the case of Pb-free solders. Moreover in the limiting case of g! 0 the model describes rate independent response. The constitutive model implemented in the form of a return mapping algorithm can be straightforwardly extended to 3D and axisymmetric idealizations. This is in contrast with previous implementations of the same model using the Owen and Hinton (1980) scheme which is restricted to plane strain problems and exhibits an inferior convergency rate. There are several reasons for such a low convergency rate. First, it is a global finite element iteration developed to solve material nonlinear problems characterized by rate dependent response whereas ABAQUS demands for an integration algorithm at the local level (i.e., one that updates the stress tensor, the material Jacobian and all other defined state variables at the Gauss point level). When the Owen and Hinton scheme is ported into ABAQUS the quadratic rate of convergency exhibited by the Newton scheme is lost. In the case of the return mapping algorithm implemented herein, the nonlinear scalar equation in the consistency parameter (see Eq. (23)) is also solved via a local Newton Raphson scheme thus preserving the convergency rate of the global iterative approach used by ABAQUS. Moreover, the algorithmic version of the material Jacobian resulting in the adapted version of the Owen and Hinton (1980) algorithm does not reduce to its equivalent in the continuum theory for small time steps. That is not the case for the return mapping algorithm where it can be shown that the algorithmic term C EVPD nþ1 in Eq. (33) approaches its continuum counterpart in the limit of vanishing Dt or equivalently of small Dc. Asa

14 598 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) final improvement with respect to the previous implementation of the model, Tang (2002), we have corrected the kinematic hardening rule allowing it to simultaneously treat linear and nonlinear hardening rules. In the current implementation the described damage mechanics constitutive model presented herein constitutes a powerful tool in the evaluation of fatigue life in the case of low cycle fatigue conditions commonly exhibited by Pb/Sn solder joints used in electronic packaging applications. References Adams, P.J., Thermal fatigue of solder joints in microelectronic devices. M.S. thesis. Department of Mechanical Engineering, MIT, Cambridge, MA, Alfano, G., De Angelis, F., Rosati, L., General solution procedures in elasto/viscoplasticity. Comput. Meth. Appl. Mech. Eng. 190, Armstrong, P.J., Frederick, C.O., A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N731. Basaran, C., Tang, H., Implementation of a thermodynamic framework for damage mechanics of solder interconnect in microelectronic packaging. In: Proceedings of IMECE, 2002 ASME International Mechanical Engineering Congress and Exposition, New Orleans, Louisiana. Basaran, C., Yan, C., A thermodynamic framework for damage mechanics of solder joints. J. Electron. Packaging, Trans. ASME 120, Basaran, C., Chandaroy, R., Zhao, Y., Influence of grain size and microstructure on dynamic response of solder joints. 98-WA/EEP-12, ASME Publications. Basaran, C., Zhao, Y., Tang, H., Gomez, J., accepted for publication. A damage mechanics based unified constitutive model for Pb/Sn solder alloys. ASME J. Electron. Packaging. Chaboche, J.L., Constitutive equations for cyclic plasticity and viscoplasticity. Int. J. Plast. 3, Chandaroy, R., Damage Mechanics of Microelectronic Packaging Under Combined Dynamic and Thermal Loading. Ph.D. Dissertation. Civil Structural and Environmental Engineering Department, University at Buffalo, The State University of New York. Darveaux, R., Banerji, K., Constitutive relations for tinbased solder joints. IEEE Trans. Comp. Hybrids Manuf. Technol. 15 (6), Hong, B.Z., Burrell, L.G., Nonlinear finite element simulation of thermoviscoplastic deformation of C4 solder joints in high density packaging under thermal cycling. IEEE Trans. Comp. Packaging Manuf. Technol. Part A 18 (3), Kashyap, B., Murty, G., Experimental constitutive relations for the high temperature deformation of a Pb Sn eutectic alloy. Mater. Sci. Eng. 50, Kitano, B.P., Kawai, S., Shimizu, L., Thermal fatigue strength estimation of solder joints of surface mount IC packages. In: Proceedings of the 8th Annual International Electronic Packaging Conference, IEPS, Dallas, Texas, November 1988, pp Knocht, S., Fox, L.R., Constitutive relation and creepfatigue life model for eutectic tin lead solder. IEEE Trans Comp. Hybrids Manuf. Technol. 13, Lau, J., Rice, J.R., Thermal stress/strain analyses of ceramic quad flat pack packages and interconnects. In: Electronic Components and Technology 40th Conference, vol. 1, Los Vegas, Nevada, May 1990, pp Lemaitre, J., A Course on Damage Mechanics. Springer. Lubarda, V., Benson, D., On the numerical algorithm for isotropic-kinematic hardening with the Armstrong Frederick evolution of the back stress. Comput. Meth. Appl. Mech. Eng. 191, McDowell, D.L., Miller, M.P., Brooks, D.C., A unified creep-plasticity theory for solder alloys. In: Fatigue Testing of Electronic Materials, ASTM STP 1153, 1994, pp Owen, D.R.J., Hinton, E., Finite Element in Plasticity. Pineridge Press Limited, Swansea, UK. Prager, W., A new method of analyzing stresses and strains in work-hardening plastic solids. J. Appl. Mech. 23 (4), Simo, J., Hughes, T., Computational Inelasticity. Interdisciplinary Applied Mathematics. Springer. Tang, H., A thermodynamic damage mechanics theory and experimental verification for thermomechanical fatigue life prediction of microelectronics solder joints. Ph.D. dissertation. University at Buffalo, The State University of New York. Wilcox, J.R., Subrahmanyan, R., Li, C., Thermal stress cycles and inelastic deformation in solder joints. In: Proceedings of the 2nd ASME International Electronic Material and Processing Congress, Philadelphis, PA, April 1989, pp Zhao, Y., Thermomechanical behavior of ball grid array solder joints under thermal and vibration loading; testing and modeling. Ph.D. dissertation. University at Buffalo, The State University of New York.