damage mechanics constitutive model for pb/sn solder joints incorporating nonlinear kinematic

Mechanics of Materials 38 (2006) 585–598

www.elsevier.com/locate/mechmat

Damage mechanics constitutive model for Pb/Sn solderjoints incorporating nonlinear kinematic hardeningand 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 con-ducted on thin layer eutectic Pb/Sn solder joints. As compared to previous implementations of the model here we correctthe difficulties introduced by the slow convergency rate of the Owen and Hinton (Owen, D.R.J., Hinton, E., 1980. FiniteElement in Plasticity. Pineridge Press Limited, Swansea, UK) integration scheme. To this end, we time-integrated themodel with a classical return mapping algorithm where rate dependency, nonlinear kinematic hardening of theArmstrong–Frederick type and damage effects are simultaneously coupled. The model is implemented into the commercialfinite element code ABAQUS via its user material subroutine capability and validated against experimental results. Wesimulated 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 ofPb/Sn solder alloys. As a direct advantage of the new model implementation this can be used for axisymmetric and 3Dsimulations as opposed to the plain strain-only capability in the Owen and Hinton (Owen, D.R.J., Hinton, E., 1980. FiniteElement in Plasticity. Pineridge Press Limited, Swansea, UK) integration scheme.� 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 microelec-tronic packaging solder alloys is introduced by low

0167-6636/$ - see front matter � 2005 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2005.11.008

* Corresponding author. Tel.: +1 7168360107; fax: +17166453733.

E-mail address: [email protected] (J. Gomez).

cycle fatigue generated by temperature changes andthe coefficient of thermal expansion mismatchbetween the soldered parts. When the assemblyundergoes a temperature variation the interconnec-tions are stressed mainly in cyclic shear. The stressesimpart elastic and inelastic strains, which are alsocyclic in nature leading to thermomechanical fati-gue. The changes in temperature are due to switch

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586 J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) 585–598

on/off operations or changes in the ambient operat-ing conditions. On the other hand, eutectic solderalloys are routinely used at high homologous tem-peratures. The melting point of the eutectic Pb/Snsolder alloy is 183 �C, and it is at 0.65Tm at roomtemperature, where Tm is the material melting point.Therefore, solder joints exhibit time, temperatureand stress dependent deformation behavior andsuch coupling makes constitutive modeling a diffi-cult task. Material models ranging from purely elas-tic to elasto-plastic using various stress–strainrelations have been proposed for Pb/Sn solderalloys, 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 simpleviscoplastic model without hardening. Wilcoxet al. (1989) proposed a rheological model to repre-sent the inelastic behavior of the material, howeverit is applicable to a limited range of strain rates.The purely phenomenological models proposed byKnocht and Fox (1990), Darveaux and Banerji(1992) and Hong and Burrell (1995) decoupled thecreep and plasticity effects artificially. This decou-pling does not have any physical basis and is justmotivated by mathematical convenience. Classicalforms of decoupled plasticity and creep theorieshave been shown to be quite inferior for modelingcyclic plasticity and creep interaction effects (McDo-well et al., 1994). An extensive literature survey onPb/Sn constitutive models is available in Basaranet al. (1998). Recently Zhao (2000), followed byBasaran and Tang (2002) and Tang (2002) haveextended a creep law originally proposed by Kash-yap and Murty (1981) for eutectic solder alloys intoa thermodynamics damage mechanics based frame-work for low cycle fatigue predictions. In that workdamage is coupled into the model using the strainequivalence principle and the effective stress con-cept. Previous implementations of the model, Tang(2002) and Basaran et al. (accepted for publication),have used the Owen and Hinton (1980) time integra-tion algorithm. This algorithm has several limita-tions. It is developed strictly for plane strainidealizations and its extension to 3D, axisymmetricor plane stress problems is not obvious. Further-more, the algorithm was originally developed withina complete finite element formulation and is not effi-cient when independently implemented into existingcommercial codes. For instance ABAQUS demands

for an integration algorithm that updates the stresstensor, the material Jacobian relating stress tostrains and all the defined state variables at the inte-gration point level. The way to adapt the Owen andHinton (1980) finite element framework to this localscheme is not clear.

In this paper the constitutive model equations areintegrated using a classical return mapping algo-rithm (Simo and Hughes, 1998). The algorithm isdeveloped to update the stresses and the materialJacobian matrix at each time step and at a givenGauss point, and therefore can be implemented intoavailable software that allows user defined subrou-tines. Three points are of interest in the presentimplementation and not explicitly described in theoriginal work by Simo and Hughes (1998). First,is the coupling into the model of a nonlinear kine-matic hardening rule of the Armstrong–Fredericktype (Armstrong and Frederick, 1966). We have fol-lowed Lubarda and Benson (2002) to incorporatethe nonlinear kinematic hardening effects. Secondis the implementation of the rate dependent effectsin the form of a viscous overstress law of the Per-zyna type. We have followed Alfano et al. (2001)to incorporate the rate dependent effects. Third isthe coupling of damage into the constitutive modelequations and the integration scheme. We havemade use of the effective stress concept and thestrain equivalence principle to couple damage intothe model. This work is organized as follows. Afterdefining notation, the first part of the paperdescribes the constitutive model equations. Atten-tion is called upon the kinematic hardening rulewhich is different from the one previously imple-mented by Basaran and Tang (2002) which was infact nonlinear but not exactly of the Armstrong–Frederick type. The second part describes the inte-gration algorithm where we present the algorithmicversions for the tangent stiffness matrix. Sub-sequently, we test the model and integration schemesimulating several experiments performed on thinlayer eutectic Pb/Sn solder joints.

1.1. Notation

Tensorial (indicial) and matrix notation are usedthroughout this article. In matrix notation secondorder tensors will be mapped into column vectorsand fourth order tensors will be mapped into matri-ces. In indicial notation repeated indices areassumed to follow the summation convention unlessexplicitly stated otherwise and a free index will take

J. Gomez, C. Basaran / Mechanics of Materials 38 (2006) 585–598 587

on the values 1, 2, 3. An index after a comma willrepresent a derivative with respect to a Cartesiancoordinate. The inner (dot) product between twotensors will be denoted by the symbol ‘‘:’’ and theinner product between two first order tensors willbe denoted by the symbol ‘‘Æ’’. The tensor productwill be denoted by the symbol �. In the article thenorm of a tensor v will be defined like [v : v]1/2 if vis a second order tensor or like [v Æ v]1/2 if v is a firstorder tensor and denoted by the symbol ‘‘k k’’ inboth cases. The unit second order tensor dij will bemapped into the column vector I and the fourthorder unit tensor will be mapped into the matrixP. In the incremental equations of the constitutivemodel a time derivative will be denoted by a super-imposed dot. In the algorithmic equations the sub-script n will refer to quantities at the beginning ofthe increment and subscripts n + 1 will refer toquantities updated at the end of the increment.Finally, the superscript D will refer to a quantitycomputed 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 plas-ticity model with isotropic hardening the elasticconstitutive relationship is written using Hooke’slaw in rate form as

_r ¼ C : ð_e� _ep � _ehÞ ð1Þwhere _e, _ep and _eh are the rates of total strain, plasticstrain and thermal strain respectively and C is theelastic constitutive tensor. In Eq. (1) ‘‘:’’ representsthe inner product between the fourth order tensorC and the elastic strain _ee ¼ _e� _ep � _eh.

2.2. Yield surface

An elasto-plastic domain is defined according tothe following yield function:

F ðr; aÞ ¼ kS � Xk �ffiffiffi2

3

rKðaÞ

� kS � Xk � RðaÞ ð2Þwhere F(r,a) is a yield surface separating the elasticfrom the inelastic domain, r is the second orderstress tensor, a is a hardening parameter which spec-ifies the evolution of the radius of the yield surface,X is the deviatoric component of the back stress ten-

sor describing the position of the center of the yieldsurface in stress space, S is the deviatoric compo-nent of the stress tensor given by S ¼ r� 1

3pbI where

p is the hydrostatic pressure and bI is the second

order identity tensor and RðaÞ �ffiffi23

qKðaÞ is the

radius of the yield surface in stress space.

2.3. Flow rule

The evolution of the plastic strain is representedby a general associative flow rule:

_ep ¼ coFor� cn ð3Þ

where n ¼ oFor is the normal to the yield surface in

stress space, _ep has already been defined as theplastic strain rate and c is a nonnegative parameterrepresenting the amount of plastic flow.

2.4. Isotropic hardening

Isotropic hardening is described by the evolutionof the radius of the yield surface in Eq. (2). Thepresent evolution equation follows Chaboche(1989) and given by:

KðaÞ ¼ffiffiffi2

3

rY 0 þ R1ð1� e�caÞ ð4aÞ

where a is a plastic hardening parameter or plasticstrain trajectory evolving according to Eq. (4b), Y0

is the initial yield stress, R1 is an isotropic harden-ing saturation value and c is the isotropic hardeningrate:

_a ¼ffiffiffi2

3

rc ð4bÞ

From Eqs. (3) and (4b) it can be seen that

a ¼R t1

t0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi23

_ep : _ep

qds which is precisely the standard

definition of equivalent plastic strain.

2.5. The nonlinear kinematic hardening (NLK) rule

The NLK rule describing the evolution of thecenter of the yield surface in stress space is the onefrom Chaboche (1989) and originally proposed byArmstrong and Frederick (1966). Nonlinearitiesare introduced as a recall term to the Prager(1956) linear hardening rule given in Eq. (5) andwhere c1 and c2 are material parameters:

_X ¼ c1 _ep � c2X _a ð5Þ


In Eq. (5) the first term represents the linear kine-matic hardening rule as defined by Prager (1956).The second term is a recall term, often called adynamic recovery term, which introduces the non-linearity between the back stress X and the actualplastic strain. When c2 = 0 Eq. (5) reduces to thePrager (1956) linear kinematic hardening rule. TheNLK equation describes the rapid changes due tothe plastic flow during cyclic loadings and playsan important role even under stabilized conditions(after saturation of cyclic hardening). In otherwords, these equations take into account thetransient hardening effects in each stress–strainloop. After unloading, dislocation remobilizationis implicitly described due to the back stress effectand the larger plastic modulus at the beginning ofthe reverse plastic flow (Chaboche, 1989). In the ori-ginal formulation by Tang (2002) the NLK harden-ing rule is written like _X ¼ 1½2

3X1 _ep � X _a� where 1

and X1 are material parameters. Comparing thisexpression to Eq. (5) it can be seen that c1 corre-sponds to the term 1 2

3X1 and c2 corresponds to

the term 1 in the original formulation. It is clear thatthe model originally proposed in Tang (2002)cannot be reduced to the linear kinematic case dueto the presence of the pre-multiplicative factor 1and in fact that model is not strictly of theArmstrong–Frederick type.

2.6. Consistency parameter

In Eqs. (3) and (4b) c is a nonnegative plasticity(consistency) parameter representing the irreversiblecharacter of plastic flow and obeying the followingproperties:

1. For a rate independent material model c obeysthe so-called loading/unloading and consistencycondition:

c P 0 and F ðr; aÞ 6 0 ð6Þ

c _F ðr; aÞ ¼ 0 ð7Þ

2. For a rate dependent material model conditionsspecified by Eqs. (6) and (7) are replaced by aconstitutive equation of the form:

c ¼ huðF Þig

ð8Þ

where g represents a viscosity material parameter,h i are Macauley brackets and u(F) is a specifiedfunction defining the character of the viscoplastic

flow. When g! 0 the constitutive modelapproaches the rate independent case (Simo andHughes, 1998). In the case of a rate independentmaterial F satisfies conditions specified by Eqs.(6) and (7) and additionally stress states suchF(r,a) > 0 are ruled out. On the other hand, inthe case of a rate dependent material, the magni-tude of the viscoplastic flow is proportional to thedistance of the stress state to the surface definedby F(r,a) = 0. Using this fact and using Eq. (8)we have that the following relationship can beestablished:

F ¼ HðcgÞ ð9Þwhere H(cg) = u�1(cg).

2.7. Viscoplastic creep law

The relation between c and g expressed in Eqs.(8) and (9) is a general constitutive equation and dif-ferent forms of the constitutive relationship describ-ing the evolution of the viscoplastic strain can beimplemented. In this particular model the creeplaw is the one proposed by Kashyap and Murty(1981) and extended to the multiaxial case by Basa-ran et al. (1998) and given by:

_evp ¼ AD0Ebkh

hF iE

� �n bd

� �p

e�Q=Rh oFor

ð10Þ

where A is a dimensionless material parameterwhich is temperature and rate dependent, Di ¼D0e�Q=Rh is a diffusion coefficient with D0 represent-ing a frequency factor, Q is the creep activationenergy, bR is the universal gas constant, h is the abso-lute temperature in Kelvin, E(h) is a temperaturedependent Young’s modulus, b is the characteristiclength of crystal dislocation (magnitude of Burger’svector), k is Boltzmann’s constant, d is the averagegrain size, p is a grain size exponent, n is a stressexponent for plastic deformation rate, where 1/nindicates strain sensitivity.

From (10) we can identify hu(F)i = hFin and

g ¼ khAD0En�1b

ðdb ÞpeQ=Rh.

2.8. Damage coupled model

Making use of the strain equivalence principle,Lemaitre (1990), we can write:

_r ¼ ð1� DÞC : ð_e� _evp � _ehÞ ð11Þ


F ¼ kS � X Dk � ð1� DÞffiffiffi2

3

rKðaÞ

� kS � X Dk � ð1� DÞRðaÞ ð12Þ

where D is a damage metric and the evolution of thebackstress after considering damage reads

_X D ¼ ð1� DÞðc1 _evp � c2X _aÞ ð13Þ

2.9. Return mapping algorithm

Consider the following trial (elastic predictor)state:

Strnþ1 ¼ Sn þ ð1� DÞ2lDenþ1 ð14Þ

where l is the shear modulus and Den+1 is thedeviatoric strain increment. Using Eq. (13) we cancompute the increment of the back stress:

dX Dnþ1 ¼ ð1�DÞfc1 devp

nþ1� c02 Dc½bX Dn þð1�bÞX D

nþ1�gð15Þ

where c02 ¼ffiffi23

qc2 and we have used a generalized

midpoint rule for the recall term with the extremevalues b = 0 and b = 1 corresponding to the back-wards and forwards Euler rules respectively. Now,from Eqs. (3) and (12) we can obtain the algorithmiccounterpart of the viscoplastic strain increment:

devpnþ1 ¼ Dc

Snþ1 � X Dnþ1

kSnþ1 � X Dnþ1k

ð16Þ

substitution of Eq. (16) into Eq. (15) yields

dX Dnþ1 ¼ anþ1 Dc

Snþ1 � X Dnþ1

kSnþ1 � X Dnþ1k� c02

c1

X Dn

!ð17Þ

with anþ1 ¼ c1ð1�DÞ1þc0

2ð1�DÞð1�bÞDc.

Using the flow rule specified by Eq. (3), allow usto express Eq. (14) like

Snþ1 ¼ Strnþ1 � Dcð1� DÞ2l

Snþ1 � X Dnþ1

kSnþ1 � X Dnþ1k

ð18Þ

Introducing the relative stress nDnþ1 ¼ Snþ1 � X D

nþ1 wehave from Eq. (18):

nDnþ1 ¼ Snþ1 � X D

nþ1

� Strnþ1 � Dcð1� DÞ2l

Snþ1 � X Dnþ1

kSnþ1 � X Dnþ1k

� X Dn � dX D

nþ1 ð19Þ

substituting Eq. (17) into Eq. (19) gives

Snþ1 � X Dnþ1 þ Dc½ð1� DÞ2lþ anþ1�

� Snþ1 � X Dnþ1

kSnþ1 � X Dnþ1k¼ Bn ð20Þ

which is obtained after letting Bn ¼ Strnþ1 � X D

n þbnþ1 DcX D

n and bnþ1 ¼c0

2

c1anþ1.

From Eq. (20) it can be shown that the normal tothe yield surface can be expressed in terms of thedata at the beginning of the step, therefore

nnþ1 �Snþ1 � X D

nþ1

kSnþ1 � X Dnþ1k¼ Bn

kBnkð21Þ

Taking the trace product of Eq. (19) with itselfyields:

kSnþ1 �X Dnþ1kþDc½ð1�DÞ2lþ anþ1�

¼ fkSn �X Dn k

2 þkð1�DÞ2lDenþ1þ bnþ1 DcX Dn k

2

þ 2ðSn�X Dn Þ : ½ð1�DÞ2lDenþ1þ bnþ1 DcX D

n �g1=2

ð22Þ

Using Eq. (12) for the rate independent case or Eq.(9) for the rate dependent case we have the follow-ing nonlinear scalar equation in the consistencyparameter which can be solved by a local Newtonmethod:

gðDcÞ ¼ kSn � X Dn k

2 þ kð1� DÞ2lDenþ1

nþ bnþ1 DcX D

n k2 þ 2ðSn � X D

n Þ

: ½ð1� DÞ2lDenþ1 þ bnþ1 DcX Dn �o1=2

� 1� DÞffiffiffi2

3

rK an þ

ffiffiffi2

3

rDc

!

� Dc½ð1� DÞ2lþ anþ1� �HDcgDt

� �ð23Þ

Once Eq. (23) is solved for Dc the following updat-ing scheme can be used:

anþ1 ¼ an þffiffiffi2

3

rDc ð24Þ

evpnþ1 ¼ evp

n þ DcBn

kBnkð25Þ

X Dnþ1 ¼ X D

n þ anþ1 DcSnþ1 � X D

nþ1

kSnþ1 � X Dnþ1k� c02

c1

X Dn

!ð26Þ


nDnþ1 ¼ ð1� DÞKðanþ1Þ

Bn

kBnkð27Þ

Snþ1 ¼ nDnþ1 þ X D

nþ1 ð28Þ

rnþ1 ¼ jð1� DÞ trðenþ1ÞbI þ 2lð1� DÞ

� enþ1 � evpn � cnþ1

Bn

kBnk� eh

nþ1

� �ð29Þ

The solution of the scalar nonlinear equation givenby Eq. (23) via a local Newton iteration is one of themain distinguishing features of the present algo-rithm with the previous Owen and Hinton (1980)scheme. The use of this local Newton–Raphson ap-proach preserves the quadratic convergence of theglobal Newton algorithm used by ABAQUS whilethe Owen and Hinton (1980) scheme does not.

2.10. Linearization (consistent Jacobian)

Differentiating Eq. (29) with respect to the totalstrain at the end of the step yields:

drnþ1 ¼ ð1� DÞ C � 2lnnþ1 �oDcoenþ1

��2lDc� onnþ1

oenþ1

�: denþ1 ð30Þ

where o Dcoenþ1

can be found from Eq. (22) such

oDcoenþ1

¼ nnþ1

K3

ð31Þ

where we have used

K3 ¼ K1 þ K2

K1 ¼ 1þ K 0

3lþ anþ1

2lð1� DÞ

K2 ¼a0nþ1 Dc

2lð1� DÞ þnnþ1bnþ1

2lð1� DÞ ½bnþ1ð1� hÞDc� 1� : X Dn

þ 1

2lð1� DÞoHoDc

andonnþ1

oenþ1can be obtained from Eq. (21) like

onnþ1

oenþ1

¼ onnþ1

oBn

oBn

oenþ1

� 1

kBnþ1kðP� nnþ1 � nnþ1Þ :

oBn

oenþ1

ð32aÞ

and

oBn

oenþ1

¼ 2lð1� DÞ P� 1

3bI � bI� �

þ ðb0nþ1 Dcþ bnþ1ÞX n �oDcoenþ1

Letting K4 ¼ b0nþ1 Dcþ bnþ1 and substituting oBnoenþ1

inEq. (32a) yields

onnþ1

oenþ1

¼ onnþ1

oBn

oBn

oenþ1

� 2lð1� DÞkBnk

P� nnþ1 � nnþ1 �1

3bI � bI� �

þ 1

kBnkðbI � nnþ1 � nnþ1Þ :

K4

K3

nnþ1 � X Dn

� �ð32bÞ

Using Eqs. (31) and (32b) into Eq. (30) results in

CEVPDnþ1 ¼ð1�DÞjbI �bI þ2lð1�DÞdnþ1 P�1

3bI �bI� �

�2lð1�DÞ�hnþ1nnþ1� nnþ1�2lð1�DÞkBnk

�DcðP� nnþ1� nnþ1Þ :K4

K3

nnþ1�X Dn

ð33Þ

where

dnþ1 ¼ 1� Dc2lð1� DÞkBnk

and

�hnþ1 ¼1

K3

� Dc2lð1� DÞkBnk

2.11. Formulation of the damage function

The damage evolution model is based on Basaranand Yan (1998) where the relation between the dis-order and entropy is established using statisticalmechanics and the second law of thermodynamics.The thermodynamic framework assumes that dam-age and the disorder are analogous concepts andthe thermodynamic disorder can be used to modelthe damage evolution. The damage evolution func-tion is given by

D ¼ Dcr½1� e�ðDe�D/Þ=ðN0kh=�msÞ� ð34Þwhere Dcr is a damage threshold, De � D/ is the dif-ference between the changes in the internal energyand the Helmholtz free energy with respect to a ref-erence state, N0 is Avogadro’s number, k is Boltz-mann’s constant and �ms is the specific mass of thematerial. The difference between the changes in theinternal energy and the Helmholtz free energy withrespect to a reference state, is obtained as follows:


the internal energy equation, which is an expressionof the first law of thermodynamics, reads

qdedt¼ r : Din þ qc�r � q ð35Þ

where Din is the rate of deformation tensor and $ isthe gradient operator. For the particular case ofsmall strains and small displacements Din ¼ dein

dt , cis the internal heat production rate and q is the rateof heat flux through the surface. The Helmholtz freeenergy is written in terms of the stress tensor thus

qdwdt¼ r : Din ð36Þ

combining Eqs. (35) and (36) the difference betweenthe changes in the internal energy and the Helm-holtz free energy with respect to a reference stateis obtained:

De� D/ ¼ 1

q

Z t2

t1

r : Din dt þZ t2

t1

cdt

�Z t2

t1

r � qdt ð37Þ

Fig. 1. Testing system.

Cu

9.02mm

38.00mm

Pb37/Sn63Solder Joint

135° Cu

84.00mm

18.50mm

Fig. 2. Thin layer solder joint attached to copper plates.

3. Simulation of monotonic and fatigue shear

testing on Pb/Sn thin layer solder joints

The proposed constitutive model was validatedagainst results from experiments performed on thinlayer solder joints of Pb37/Sn63 prepared at theUB-Electronic Packaging Lab. The solder jointsare 460 lm thick, which is a thickness comparableto the diameter of solder joints in actual electronicpackaging. Displacement controlled experimentswere conducted for monotonic and cycle shear onan MTS 858 material testing system with ATS7510 box thermal chamber. The thin layer solderjoints were made by reflowing Pb37/Sn63 solderwire of 0.032 mm diameter with flux of rosin core.After reflowing, the solder joints were left aging atroom temperature (22 �C) for 7 days to allow formetallurgy to develop between the solder alloyand the copper plates. As a fixture for the specimen,MTS 647 hydraulic grips with extension rods wereused. The test was performed under displacement-controlled conditions and the proper correctionson the total displacement to account for the stiffnessof the fixture were made. Details about the testingcan be found in Tang (2002). Figs. 1 and 2 showthe used testing system and specimen attached tocopper plates. The monotonic and mechanical sheartests were performed at different temperatures and

strain rates under isothermal conditions. Table 1shows the test conditions.

3.1. Material properties

For the verification study the material constantswere taken from the experimental study reportedin Tang (2002) and from experimental resultsby Adams (1986). According to Adams (1986),Young’s modulus varies significantly for the samePb/Sn composition from one specimen to another.Furthermore, there is a big scatter on the values

Table 1Testing conditions

Strain rate (s�1) Temperature (�C) ISR

Monotonic shear

Case II1 �402 223 604 1.67E�03 100

Case III1 1.67E�012 1.67E�023 1.67E�034 1.67E�04 22

Cyclic shear

Case IV1 0.0052 0.0123 0.02

1.67E�03 22

Fatigue shear

Case V1 0.0222 1.67E�04 22 0.004

ISR: inelastic strain range.

Fig. 3. Monotonic shear testing under strain rate 1.67 · 10�3 s�1

at �40 �C.


of Young’s module reported in the literature. Pb/Snsolder alloy is a highly temperature dependent andrate dependent material. In addition, its materialproperties are also very sensitive to its microstruc-ture. The variations of E, l and ry with temperatureh (K) used in this work are as follows:

Table 2Material parameters used in the constitutive model

Material parameters

ElasticYoung’s modulus (GPa) 52.10–0.1059hShear modulus (GPa) 19.44–0.0395h

Isotropic hardeningR00 (MPa) 37.47–0.0748hc 383.3ry 60.069-0.140h

Kinematic hardeningc1 2040c2 180

Flow ruleA 7.60E+09D0 (mm/s2) 48.8b (mm) 3.18E�07d (mm) 1.50E�02n 1.67p 3.34Q (mJ/mol) 4.47E+07

E ðGPaÞ ¼ 52:10� 0:1059h ð38aÞl ðGPaÞ ¼ 19:44� 0:0395h ð38bÞY 0 ðMPaÞ ¼ 60:069� 0:140h ð38cÞ


at 22 �C.


at 60 �C.Fig. 6. Monotonic shear testing under strain rate 1.67 · 10�3 s�1

at 100 �C.


The phase size data corresponding to Adams (1986)is obtained after comparison of Adams resultswith Kashyap and Murty (1981) creep test data.Following this approach, Chandaroy (1998) foundd = 15 lm. The optimum values of the kinematichardening properties to fit the experimental datawere found to be c1 = 2040 MPa and c2 = 180.

Fig. 7. Monotonic shear testing under strain ra

The constant A is dependent on temperature, andthe value of A is fitted based on Adams (1986) data.A linear regression was performed to obtain therelationship between A and temperature. The re-gressed equation yielded the following relationshipA ¼ b1 � hb2 or equivalently logA = logb1 + b2 loghwhere logb1 = 41.368 and b2 = �13.692. All the

te 1.67 · 10�3 s�1 at different temperature.


at 22 �C.


temperature dependent material properties used inthis work are reported in Table 2.

3.2. Simulations vs test results

Figs. 3–7 show a comparison between numericalresults and testing under monotonic shear for differ-ent temperatures at a strain rate of 1.67 · 10�3 s�1.Good predictions are generally obtained and themodel in fact captures the temperature dependence.Figs. 8–12 present results for the same type of test atroom temperature for different strain rates. It can beseen that the model effectively captures the strainrate dependency. Fig. 13–17 present cycling loadingsimulations at room temperature, strain rate of1.67 · 10�3 s�1 and different inelastic strain rangesagain good correlation between experimental andnumerical simulation results is obtained. Figs. 18–20 present the results for several fatigue cycles underdisplacement controlled conditions and the corre-sponding evolution of the damage parameter. Thecomputational simulations are generally in goodagreement with the experimental results. The scatterin the results is more likely due to the differences inthe geometry between actual specimens and thenumerical model. For instance, there are voids inthe actual specimens that are not explicitly repre-sented in the numerical model; therefore some dif-


at 22 �C.

ferences should be expected. In order to comparethe stress–strain response with the testing resultsthe proper corrections were made to account for

Fig. 10. Monotonic shear testing under strain rate1.67 · 10�3 s�1 at 22 �C.

Fig. 11. Monotonic shear testing under strain rate1.67 · 10�4 s�1 at 22 �C.

Fig. 13. Cyclic shear simulation vs test data at 22 �C, strain rate1.67 · 10�3 s�1 and ISR = 0.005.


the voids in the solder joint and the stiffness of theused load train.

4. Conclusions

In this paper we have implemented a thermo-dynamics-based damage mechanics coupled consti-tutive model for Pb/Sn solder alloys within a

Fig. 12. Monotonic shear testing at

classical return mapping scheme but consideringthe combined effects of damage, viscoplasticityand a nonlinear kinematic hardening rule of theArmstrong–Frederick type. The model capabilitieswere verified via comparisons with experimentalresults from tests performed on homemade thinlayer solder joints. The comparisons show thatthe temperature and rate dependent effects that

different strain rates at 22 �C.



Fig. 16. Cyclic shear simulation vs test data at 22 �C, strain rate1.67 · 10�3 s�1 and different inelastic strain range.

Fig. 17. Isothermal fatigue at strain rate 1.67 · 10�4 s�1 at 22 �Cwith inelastic strain range 0.022.


characterize the behavior of Pb/Sn solder alloysunder different loading conditions are effectivelycaptured by the model. Moreover the observed scat-

tering in the results is due to the imperfect nature ofthe actual solder joints while the finite elementmodel assumes perfect conditions. Although here

Fig. 18. Damage evolution under fatigue with ISR = 0.022.Fig. 20. Damage evolution under fatigue with ISR = 0.004.


we have used a particular creep law suitable for Pb/Sn solder alloys, the model formulation is generaland can be used for other type of constitutive rela-tionships like in the case of Pb-free solders. More-over in the limiting case of g! 0 the modeldescribes rate independent response.

The constitutive model implemented in theform of a return mapping algorithm can be straight-

Fig. 19. Isothermal fatigue at strain rate 1.67 · 10�4 s�1 at 22 �Cwith inelastic strain range 0.004.

forwardly extended to 3D and axisymmetric ideal-izations. This is in contrast with previousimplementations of the same model using the Owenand Hinton (1980) scheme which is restricted toplane strain problems and exhibits an inferior con-vergency rate. There are several reasons for such alow convergency rate. First, it is a global finite ele-ment iteration developed to solve material nonlinearproblems characterized by rate dependent responsewhereas ABAQUS demands for an integration algo-rithm at the local level (i.e., one that updates thestress tensor, the material Jacobian and all otherdefined state variables at the Gauss point level).When the Owen and Hinton scheme is ported intoABAQUS the quadratic rate of convergency exhib-ited by the Newton scheme is lost. In the case of thereturn mapping algorithm implemented herein, thenonlinear scalar equation in the consistency param-eter (see Eq. (23)) is also solved via a local Newton–Raphson scheme thus preserving the convergencyrate of the global iterative approach used byABAQUS. Moreover, the algorithmic version ofthe material Jacobian resulting in the adapted ver-sion of the Owen and Hinton (1980) algorithm doesnot reduce to its equivalent in the continuum theoryfor small time steps. That is not the case for thereturn mapping algorithm where it can be shownthat the algorithmic term CEVPD

nþ1 in Eq. (33)approaches its continuum counterpart in the limitof vanishing Dt or equivalently of small Dc. As a


final improvement with respect to the previousimplementation of the model, Tang (2002), we havecorrected the kinematic hardening rule allowing itto simultaneously treat linear and nonlinear harden-ing rules.

In the current implementation the describeddamage mechanics constitutive model presentedherein constitutes a powerful tool in the evaluationof fatigue life in the case of low cycle fatigue condi-tions commonly exhibited by Pb/Sn solder jointsused in electronic packaging applications.

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damage mechanics constitutive model for pb/sn solder joints incorporating nonlinear kinematic

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