Computational Materials Science 85 (2014) 236–243

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Computational Materials Science

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Crystal plasticity finite element study of deformation behaviorin commonly observed microstructures in lead free solder joints

0927-0256/$ - see front matter � 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2014.01.002

⇑ Corresponding author. Tel.: +1 5178991820.E-mail address: payamdarbandi@yahoo.com (P. Darbandi).

Payam Darbandi a,⇑, Tae-kyu Lee c, Thomas R. Bieler b, Farhang Pourboghrat a

a Mechanical Engineering, Michigan State University, United Statesb Chemical Engineering and Materials Science, Michigan State University, United Statesc Cisco Systems, Inc., San Jose, CA, United States

a r t i c l e i n f o

Article history:Received 10 August 2013Received in revised form 2 January 2014Accepted 3 January 2014Available online 29 January 2014

Keywords:Finite element analysisLead-Free solderCrystal plasticityElectron backscattering diffraction (EBSD)Plastic shear deformation

a b s t r a c t

The anisotropy of the tin phase in a Pb-free tin based solder joint has a significant effect on heterogeneousdeformation and therefore, the reliability of solder joints. In this study the ability of crystal plasticityfinite element (CPFE) modeling to account for elastic and plastic anisotropy in tin based solder jointswas examined using shear deformation applied on a simplified representation of a real microstructureof four specific SAC305 solder balls. Commonly observed microstructures in lead free solders are eithersingle crystals or a particular microstructure with solidification twin relationship with about 55–65�rotations about a common [100] axis (known as beach-ball microstructure [6]). In this study two differ-ent single crystals and two different beach-ball microstructures were investigated using CPFE modeling.

Simulation results show the ability of CPFE to predict the heterogeneous deformation due to the aniso-tropic elastic and plastic properties of tin in lead free solder joints.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Due to the requirement to eliminate lead in electronic assem-bly, the tin–silver–copper (SAC) family of alloys has been heavilyused in electronic assembly in recent years. A significant challengeresulting from introducing these alloys is that the fracture proba-bility differs from Sn–Pb solders because of the anisotropy of tin[1–3]. The location in the package that is most prone to failure can-not be predicted as easily as in Sn–Pb alloys, because the singlecrystal or large grain microstructure and the crystal orientationin lead-free solder alloys greatly affect damage initiation andevolution. Miniaturization is a very important issue in solder jointsbecause the dimension of a joint is similar to the size of the Sngrains. The design of solder joints will increasingly require consid-ering the grain scale anisotropy. Studies of the microstructure andSn grain morphologies in lead-free solder joints show peculiarmicrostructures, such as the beach-ball morphology, in which cyc-lic twins form (rotations of 60� about a common [100] axis) duringsolidification [6], which are illustrated in several of the figures inthis paper. The strongly anisotropic elastic modulus and coefficientof thermal expansion (CTE) play an important role in thermo-mechanical loading of solder joints, leading to complex heteroge-neous stress states acting on joints [4]. Due to this anisotropy,crystal features such as grain size and grain morphology have

significant effects on the mechanical properties and reliability ofsub-mm scale joints. There are very few prior studies about mod-eling the real microstructure of lead free solders [5–11]. Park et al.[10] used an anisotropic linear elastic constitutive model in an FEMsimulation to simulate three-dimensional elastic strains measuredexperimentally in lead-free solder balls. Although they could pre-dict the location of damage near the grain boundary, the observa-tion of plastic deformation by Bieler et al. [6] in the samples thatexperience thermo-mechanical loading suggests that applicationof more sophisticated plastic constitutive models are required forprediction of damage evolution in lead free solders that fail by duc-tile fracture caused by plastic deformation.

More recently, Maleki et al. [11] used a J2 plasticity constitutivemodel to investigate the effect of aging conditions on deformationbehavior of the eutectic micro-constituent in SnAgCu lead-free sol-der. Although their study accurately modeled the geometry of Sngrains and intermetallic, the isotropic plasticity associated withJ2 model is unrealistic and oversimplified.

Darbandi et al. [14] used a crystal plasticity model to investigatethe effect of grain orientation and its relationship with morphology(grain position within a simplified geometry of solder ballmicrostructure). Specifically, this study investigated the complexinteraction between the activity of slip systems, location of a par-ticular grain within the solder ball and orientation of differentgrains.

Since the damage evolution highly depends on stress, strain,and the activity of slip systems, providing a model that can predict

P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243 237

the local stress and strain state will be helpful from both ascientific and industrial point of view. Furthermore, the use of amicrostructure scale FE mesh in a crystal plasticity finite element(CPFE) model is also important for interpreting experimental

ðrjþ1; qjþ1Þ ¼Min lðxÞ ¼ ðrtrial

jþ1 �PÞ : C�1 : ðrtrial

jþ1 �PÞ þ ðq� qjÞ : E�1 : ðq� qjÞ

h iSubjected to : f

P; qð Þ 6 0

( )ð2Þ

measurements, because at these scales the experimentallyimposed boundary conditions are very difficult to measure ormonitor.

In this study, a CPFE model is used to simulate deformationmechanisms and anisotropy associated with the slip phenomenain Sn to compare with corresponding experimental characteriza-tion of shear deformation, in order to assess the capability of theCPFE model to predict the kinematics of plastic deformation andevolution of microstructural features in different solder joints.Firstly, two different single crystal orientations were investigatedto evaluate the capability of CPFE to predict the kinematic plasticdeformation. Next, the beach-ball microstructure which is com-monly observed in lead free solder was investigated, in two differ-ent samples to evaluate the reliability of CPFEM model to predictthe plastic deformation in lead free solders. Thus, this work pro-vides a basis for an integrated incremental model developmentstrategy based upon experiments, modeling and comparativeanalysis.

2. Model description

The crystal plasticity model developed by Zamiri et al. [12,13]for FCC metals was used to study tin. Firstly, the existing crystalplasticity model was modified in order to account for the morecomplicated crystal structure of tin. Tin has a body centered tetrag-onal crystal structure with 32 possible slip systems. Since notmuch is known from the literature about the slip activity of tin slipsystems, slip resistance or hardening characteristics, a modifiedcrystal plasticity model was used to simulate the deformation ofsolder balls under shear loading in order to identify the likely ac-tive slip systems and hardening properties that allow comparisonwith experiments. The crystal plasticity model calculates shearrate for each slip system, allowing the user to identify the most ac-tive slip systems for a given increment of plastic deformation. Theincremental hardening of slip systems is also a function of themagnitude of shear rates, and hardening parameters. By comparingthe simulation results with microscopic and macroscopic measure-ments, estimates for the hardening parameters of slip systems fortin were identified [14]. Once these parameters were fitted to a setof experimental dataset, they were no longer modified when thecode was used to simulate deformation of tin solder balls underdifferent loadings. The formulation of crystal plasticity model isbriefly described below showing the relationship between slip sys-tems, shear rates, and hardening parameters with the rate of plas-tic deformation.

A velocity gradient in plastic deformation in the material coor-dinate system can be decomposed into a rate of deformation and aspin tensor:

LP ¼ DP þWP ð1Þ

An elasto-plastic problem is usually defined as a constrainedoptimization problem aimed at finding the optimum stress tensorand internal variables for a given strain increment. In such a prob-lem, the objective function is defined based on the principle of

maximum dissipation. This function has terms describing theincremental release of elastic strain energy and dissipation dueto incremental plastic work, and the constraint is the yield func-tion:

whereP

is the design variable (stress tensor) to be found, q is avector containing internal variables such as strain hardening andkinematic hardening parameters that need to be found, C is thematerial stiffness matrix, f(

P, q) is the yield function, and E is

the so-called matrix of generalized hardening moduli. One of thesolutions to the above problem is the following equation for theplastic rate of deformation:

Dp ¼ k@f ðr; qÞ@r

ð3Þ

In a crystal plasticity problem, deformation is defined by yieldfunctions for each slip system in a crystal. Assuming the Schmidlaw is valid for plastic deformation of a single crystal, then forany slip system a yield function can be defined as:

faðr; qÞ ¼jr : Paj

say� 1 ð4Þ

The constraints of problem (2) can be combined and replaced byan equivalent single constraint defined as:

f ðr; qÞ ¼ 1q

lnXm

i¼1

exp qjr : Paj

say� 1

!" #" #ð5Þ

where i represents the index of summation for slip systems, m is thenumber of active slip systems, sa

y is the critical shear stress on slipplane a, and Pa is the symmetric part of the Schmid tensor, Ia, thatdescribes the orientation of a slip system, defined as:

Pa ¼ 12ðIaÞ þ Ia

� �Th i

¼ 12

ma � na þ ma � nað ÞT� �

ð6Þ

where na is a unit normal to the slip plane, and ma is a unit vectordenoting the slip direction.

The plastic deformation matrix can be expressed as:

Dp ¼Xn

i¼1

_cPa ð7Þ

where _c are slip rates. The spin tensor, which represents the mate-rial rotation due to slip, can be expressed as:

Xp ¼Xn

i¼1

_cwa ð8Þ

where wa matrix is the anti-symmetric part of Ia, defined as:

wa ¼ 12ðIaÞ � ðIaÞTh i

¼ 12

ma � na � ma � nað ÞT� �

ð9Þ

Using Eqs. (3), (5), and (7), it can be shown that during the plas-tic deformation of a single crystal, the slip rate on any slip systemcan be expressed by:

_ca ¼ k

sgnðr:PaÞsa

yexp q

mjr:Pa jsa

y� 1

� �h imPn

b¼1 exp qmjr:Pa jsa

y� 1

� �h i ð10Þ

where b represents the index of summation for slip systems, n is thenumber of active slip systems, m and q are material parameters that

Table 1Elastic Constants (GPa) of Tin used in numerical analysis.

Parameter C11 C22 C33 C44 C55 C66 C12 C13 C23

72.3 72.3 88.4 22.0 22.0 24.0 59.4 35.8 35.8

238 P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243

control the shape of the single-crystal yield surface, which havebeen shown to have a direct relationship with the stacking fault en-ergy (SFE) of the material as following:

q ¼ 5CGb� 10�3 ð11Þ

where C is the SFE of the material, G is the shear modulus, and b isthe magnitude of Burgers vector. For most materials m = 1. Theparameter k is the Lagrange multiplier which has been shown tobe a measure of the rate of plastic work in a crystal [12].

The generalized Schmid factor can be found for each slip systema by finding the scalar product of the normalized stress tensor(using the Frobenius norm) and the Schmid matrix. In order to de-fine the resistance against shear, Hill [16], and Asaro and Needle-man [17] proposed the following formulation:

_sa ¼XN

b¼1

haaj _caj ð12Þ

where j _caj is the plastic slip rate on the active slip system b, and hab

are the components of the hardening matrix, where haa are knownas the self-hardening moduli while hab for a–b are known as the la-tent-hardening moduli. Hutchison [18] proposed the followingmodel for evolution of the components of hardening matrix

hab ¼ hb qþ ð1� qÞdab� �ð13Þ

Here q is the so-called latent-hardening ratio, which is the harden-ing on a secondary slip system caused by slip on a primary slip sys-tem (Kapoor and Nemat-Nasser [19]) and can be measured by theratio of the latent-hardening rate to the self-hardening rate of a slipsystem with typical values in the range of 1 < q < 1.4. The parameterq can be considered as 1 for coplanar slip systems and 1.4 for non-coplanar slip systems. There are different types of hardening mod-els presented by researchers for the evolution of hb (so-called self-hardening). One of the most well-known formulations is:

hb ¼ h0 1�sb

y

ss

����������

a

sgn 1�sb

y

ss

!ð14Þ

where h0, a, and ss are slip system hardening parameters, which areconsidered to be identical for all slip systems. h0 denotes the initialhardening rate, ss the saturation value of the slip resistance, and a isthe exponent describing the shape of the stress–strain yield

Fig. 1. Schematic diagrams of a 3-D visualization of the CPFE mesh located in a the 4 � 4 sand shear test apparatus is also shown.

function. These parameters can be obtained by fitting the modelto experimental data.

2.1. Experimental procedures

Four specially prepared samples of a 4 � 4 ball grid array (BGA)were cut from a 15 mm � 15 mm body-size plastic ball grid array(PBGA) package with Sn–3.0Ag–0.5Cu (wt.%) solder joints. The sol-der joints were approximately 0.5 mm in diameter. The sampleswere aged at 100 �C for 500 h, sectioned, and carefully polished(without metallurgical mounting) so that the plane of shear atthe center of the joint could be observed on one set of four balls.A fixture shown in Fig. 1 was used to impose a shear strain, so thatevidence of local strain effects could be detected topographicallyby small variations in out-of-plane shear on the previously pol-ished surface.

Optical microscopy, analysis of polarized light microscopy, andorientation imaging microscopy™ (OIM), were conducted initially,and after the deformation. To permit accurate orientation mea-surements, samples were lightly re-polished after the deformationto improve surface smoothness. The resin in the package and boardwere carefully painted with carbon paint to cover non-conductingsurfaces. Also, copper tape was used to provide a conductive pathto prevent the charging that causes image distortion.

OIM data were obtained using a CamScan 44FE electron micro-scope. The accelerating voltage was 20 kV and the working distancewas 33 mm. A TSL EDAX OIM system with a digiview SEM detectorwas utilized to generate the orientation maps. Maps were cleanedup using the neighbor confidence index (CI) correlation to replacepixels that had low CI with their neighbors having a high CI. Thisstep is required to remove the minority data points coming fromun-indexed or wrongly indexed pixels, typically arising from inter-metallic phases, which were not considered in this work. A user de-fined crystal orientation map was employed to correlate theorientation of the Sn c-axis with respect to solder/package interface.For this orientation map 5 colors (Purple–blue–green–yellow–red)

older joint array sectioned from a PBGA package. The solder ball assembly geometry

Table 2Hardening parameters of tin for different slip systems used in numerical analysis.

Parameters so ss ho a q m q

23 (for all sets) 40 (for all sets) 100 for sets (1,2,5,7,) 150 for sets (3,4,6,8,9,10) 2.0 (for all sets) 1.4 1 60

P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243 239

were used to represent the orientation of the c-axis with respect tothe solder/package interface. When the c-axis is on the interfaceplane, the orientation is expressed as ‘‘red’’, and when the c-axis

Fig. 2. optical micrographs of two different solder balls with corresponding c-axis orientah) 0.65 mm shear displacement. After shear, strains are concentrated in upper right regiorotated about the c-axis (note overlaid unit cell prisms) it during shear. (For interpretativersion of this article.)

(a)

Fig. 3. Comparison of simulation and experimental results for single crystal deformedC3D10 M elements were used in (b).

is perpendicular to the surface the orientation is ‘‘purple’’. Othercolors (blue, green and yellow) fill the rest of the c-axis orientationspace.

tion maps with respect to the substrate normal direction, before (a–d) and after (e–ns. In both joints, the c-axis orientation was retained (no color change) as the crystalon of the references to color in this figure legend, the reader is referred to the web

(b)

solder joints illustrated in Fig. 2 565 C3D10 M elements were used in (a), and 569

Fig. 4. (a) Optical image of joint b in Figs. 2 and 3, showing local unit cellorientations with shaded slip planes and corresponding plane traces, and slipvectors (blue lines) with high Schmid factors (there is some evidence for (121) slipin the highly sheared upper right part of the joint). Red and yellow–green edges ofthe unit cells represent the crystal x and y axes, respectively. (b) Comparison of theaverage activity in the simulation of the four most active slip systems based on all ofthe integration points in the single crystal joint. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of thisarticle.)

1 Mixed {) and h ] for planes and directions recognize the symmetry of thetragonal crystal structure.

240 P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243

The finite element analysis was performed on four samplesusing the commercial finite element code ABAQUS [15]. The crystalplasticity material model was implemented using a user materialsubroutine in FORTRAN (VUMAT). Details of the model can befound in Darbandi et al. [14]. The joints were modeled as asymmet-rically truncated spheres to correspond with measured cross sec-tions. Element types and number of elements used are describedin captions of relevant figures. The material parameters used areshown in Tables 1 and 2.

A 3-D visualization of the CPFE mesh is presented in Fig. 1,showing the position of one of the joints analyzed with respectto the rest of the package.

Two different single crystal joints (Figs. 2–4) and two differentbeach ball morphology joints (Figs. 5–7) were studied to investi-gate the capability of CPFE to model the kinematics of deformationand microstructural features. The two single crystals were sub-jected to 0.65 mm displacement. One of the beach-ball solder ballswas subjected to 0.1 mm shear displacement, and the second onewas displaced by 0.4 mm.

In order to reconstruct the real microstructure of a joint withbeach ball morphology from OIM and polarized light microscopymeasurements, one can assume that the beach ball microstructureis part of a sphere that is equally divided by three planes into sixparts with the same volume. Due to the fact that the beach ballmicrostructure can be produced by 60� rotations about an hai axis,

the cutting plane can be assumed to be a plane with its normal par-allel to a common h100}hai axis. This modeling method allows themorphology to be correlated appropriately with crystal orienta-tions. (The morphology obtained using the polarized light micros-copy is not sufficient to reconstruct the microstructure because itdoes not give the orientation of the common hai axis required todetermine the geometrical inclination of the grain boundarythrough the depth of the solder balls).

3. Results and discussion

3.1. Modeling of single crystal solder balls using CPFE

Fig. 2(a and b) shows two different undeformed single crystaljoints. The crystal orientations overlaid on the figures in (c andd) show that the c-axis is nearly parallel to the interface accordingto the c-axis color code. While both of these joints have a ‘red’ ori-entation, the c-axis direction differs with respect to the plane ofshear. (The top of the map in both joints is exaggerated in the ver-tical direction due to the topographic effects of the polish and the70� tilt of the sample during the measurement). The optical imagesin Fig. 2(e and f) show that deformation was concentrated in theupper part of the joint, where the cross sectional area was thesmallest. The OIM maps for the samples after deformation inFig. 2(g and h) show that the orientation of the c-axis with respectto the interface did not change much (same color), but rotation isobserved in both samples about the c-axis, especially in the highlydeformed area near the top.

In both joint orientations, the crystal rotated about the c-axis inthe direction of shear about an axis near the [001] direction. InFig. 2(g), the c-axis is inclined to the plane of shear, and the rota-tion axis is close to the c-axis. In Fig. 2(h), the c-axis rotation axisis nearly perpendicular to the plane of shear. In both cases, modestrotations about an axis near (001) imply that slip on the (110)plane is facile and stable.

For the simulation, optical micrographs (Fig. 2(a and b)) wereused to define the geometry for CPFE analysis. The initial orienta-tions were used to establish the basis for the kinematics of plasticdeformation (Fig. 2(c and d)). The simulations also show that largeamounts of deformation took place in the upper right area indicat-ing the localization of plastic deformation in these regions in Fig. 3.The deformation simulated by the CPFE model compared closely tothe deformed shapes observed in the optical microscopy micro-graphs. This shows that the crystal plasticity model can accuratelypredict the kinematics of plastic deformation due to the shearloading.

Fig. 4 shows the surface features on the previously polished sur-face of the joint in Fig. 3(b). The micrograph shows different planetraces on the sample surface in different locations. Using planetrace analysis based upon local orientations measured afterdeformation (details in Zhou et al. [20]), slip on {110)1 planes isdominant. The complimentary shear strains required for shear defor-mation are apparent in the perpendicular {110) slip traces. Thehighest Schmid factors (m, based upon a pure shear stress assump-tion) are for slip in h110] directions, but they are also high for slipin h111i directions. The most active slip systems in the simulationplotted in Fig. 4(b) also indicate that the most active slip systemsare those with high Schmid factors.

To assess the most active slip systems from experimental mea-surements, rotation axes can indicate which of the four slip direc-tions on a {110) plane was the most active using the orientationmap data in Fig. 2(d and h). The rotation axes were determined

te

(a)

(c)

(e) (f)

(d)

(b)

Fig. 5. Pole figures obtained from OIM for two solder joints with beach ball microstructure (a and b), FE meshes colored according to the c-axis color scale in the verticaldirection before deformation (c and d) and after deformation, (e and f) showing the shape after indicated displacements were imposed and released. 2550 C3D10M elementswere used in (c and e) and 2600 C3D10M elements were used in (d and f). (a) indicates the orientations of (c and e) before deformation and (b) shows the orientation of (d andf) after deformation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

(d)

X

Y0.1 mm

(c)

(e)

Fig. 6. (a) X direction c-axis orientation maps for beach ball microstructure before deformation and, (b) after 0.1 mm shear, backscattered SEM image tilted �45� about X axisshowing a ledge in the lower left area (c), and a path along which misorientation and topography are traced in the experiment (b). A similar trace in the CPFE simulation (d), isplotted with experimental data in (e). The CPFE model is geometrically simpler and does not include the material indicated by the dotted line in (d), but it is able to semi-quantitatively capture the localized deformation observed experimentally in the lower left corner.

P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243 241

between the initial and final orientations in nine locations evenlydistributed in a 3 � 3 grid (after correcting for the samplealignment in the measurements). In each of the nine locations,the rotation angle was 5.4�, about an average rotation axis of

h�15 ± 4.5–6.5 ± 3.7 100]. This is an average of 5.4 ± 4� from theh�15–15 100] rotation axis that is 90� from the {1–10)h111] slipplane normal and slip directions. As the h�15–15 100] axis is 21�from the [001] axis, the measured rotation axes are much closer

(a) (b)

Fig. 7. Polarized light micrograph (a) of beach ball microstructure that experienced a large shear displacement of 0.4 mm (red arrow), (b) deformed solder ball predicted byCPFE illustrating distribution of shear stress in the 1–2 plane indicating a higher stress due to more shear localization in the upper area. The outline of the simulation is shownin (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

242 P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243

to the expected rotation axes for h111i slip than the [001] rotationaxis for h110] slip (these vectors are illustrated in Fig. 4(b)). Thisindicates that the h111i slip system is much more active thanthe h110] slip direction, and this suggests that its critical resolvedshear stress is smaller. Furthermore this indicates that one of theh111i directions is strongly favored, because equal amounts of slipin the two h111i directions in the same {110) plane will give theequivalent of h110] slip. Fig. 4(b) shows the average amount ofshear on the four most active slip systems obtained from CPFEmodel (which is based on the average initial orientation). This sim-ulation shows that (1–10)[110] slip is dominant, which is not whatthe experiment shows, but this is a consequence of having no sig-nificant difference in the initial slip resistance on the various slipsystems used in the simple material model [14]. This experimentand others like it will lead to future refinements in the model. Nev-ertheless, even this overly simple material model is able to effec-tively capture the kinematics of deformation.

3.2. Modeling of heterogeneous plastic deformation in Beach-ballmicrostructure of solder joints

In order to simulate the behavior of solder balls in differentlocations and at different stages of thermo-cycling in a real ballgrid array package, the behavior of solder balls subjected to bothsmall and large shear displacements have been investigated. Fromexamination of incremental strains resulting from small to largershear displacements, it is apparent that orientations near the cen-ter of the ball do not change very much, so the final orientation inthe central area can be used as a good approximation for the initialcrystal orientation. This allows analytical comparison with priorcharacterization studies where grain orientations were character-ized only after the deformation. Two cases are presented: the first,where the orientation was measured before and after a small sheardisplacement (Figs. 5 and 6), and second, a sample characterized inprior work where solid modeling of the beach ball and sectioningbased upon orientations in the center of the deformed sample wereused to identify a highly probable initial microstructure condition(Figs. 5 and 7).

Fig. 5 shows the measured tricrystal orientations in pole figures,and the microstructure, mesh, and deformed mesh for the twojoint samples studied. The c-axis OIM map for a beach ball samplebefore and after 0.1 mm shear displacement is shown in Fig. 6(aand b). A comparison of the c-axis map before and after deforma-tion reveals the most noteworthy change to take place in the crys-tal orientation due to localized plastic deformation in the lower leftcorner of the c-axis map. In this area, the orientation color changedfrom green to blue, as made evident by overlaid prisms. A corre-sponding backscattered electron micrograph in Fig. 6(c) indicatesthe formation of a ledge in the lower left corner area, where signif-icant local orientation changes were measured with OIM. The im-age of the surface in Fig. 6(c) is tilted so that it makes the ledgeeasier to see. The tilted image was one of a stereo pair from which

it was possible to prove that the material to the right of the ledgewas at a lower elevation than the material to the left.

Fig. 6(e) illustrates the change in surface topography obtainedusing the CPFE analysis. It is apparent that the CPFE model wasable to predict a negative displacement from strain localizationsimilar to what happened in the real experiment. Fig. 6(e) com-pares the orientation and topography along a diagonal path thatcrosses the region where the topographic features were observed(diagonal lines in Fig. 5(d and c). As shown in Fig. 6(e), the crystalorientation changed along this path by about 25� from the outeredge of the sample, and, the surface elevation dropped by 1 lmat the ledge, and then came back to the original height at the cop-per interface. The path in the CPFE simulation also shows a similartrend, though the large size of the elements makes for a rough rep-resentation of the topography. Also, because the CPFE model wasgeometrically simpler, without modeling the shape of the Cu padaround which the solder flowed, the experimental geometry hasmore solder in the perimeter region. However, when the trace cov-ers similar paths with respect to the corner of the copper pad, thesurface topography from the CPFE matches the experimentaltrends reasonably closely (noting that a ledge cannot be predictedby the model).

Fig. 7 shows a simulation of the microstructure of a different de-formed sample for which the initial condition was not measured.With computational deformation, the distribution of strain is quitesimilar to the observed strain in the real microstructure. The differ-ences in shape made evident by overlaying the shape of the CPFEmodel on the experimentally deformed sample may arise fromthe fact that initial solder joints are not always perfect truncatedspheres (that were modeled), so it is not possible to assess the dif-ferences between the model and the experiment quantitatively. Inparticular, the localization of shear strain in the upper grain, whichis a ‘‘red’’ orientation (where the orientation of the c-axis is parallelto the interface) shows consistency with the real microstructurewhere the greatest amount of shear occurs in the ‘‘red’’ orientation.This is an outcome of slip along the c-axis, which is known to be afacile slip system [6].

4. Conclusions

In this study, the capability of CPFE modeling was examinedusing shear deformation applied on real microstructures ofSAC305 solder balls. The simulation results clearly show the capa-bility of crystal plasticity model to predict the heterogeneousstrain that is observed in correlated physical samples. The modelsused estimates of material properties that are considered to be rea-sonable based upon limited prior study of slip behavior in Sn. Inaddition to the kinematics of plastic deformation, more detailscan be predicted using this model, such as accurate prediction ofsome the microstructural features such as formation of ledges aswell as activity of slip systems. Therefore, this model can be usedas a useful tool to predict damage, as the damage phenomena is

P. Darbandi et al. / Computational Materials Science 85 (2014) 236–243 243

controlled by the plastic deformation, and it can be further refinedwith constraints obtained by detailed comparisons with experi-mental data.

Acknowledgements

This work was supported by NSF-GOALI Contract 1006656 andCisco Systems Inc., San Jose, CA.

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