The Effect of Random Voids in the Modified Gurson Model...material with random initial void volume fraction. The elastic modulus of Sn15,16 is E = 46.9 GPa and its Poisson’s ratio

The Effect of Random Voids in the Modified Gurson Model

HUIYANG FEI,1 KYLE YAZZIE,2 NIKHILESH CHAWLA,1,2,3

and HANQING JIANG1,4

1.—Mechanical and Aerospace Engineering, School for Engineering of Matter, Transport andEnergy, Fulton Schools of Engineering, Arizona State University, Tempe, AZ 85287-8706, USA.2.—Materials Science and Engineering, School for Engineering of Matter, Transport and Energy,Arizona State University, Tempe, AZ 85287-8706, USA. 3.—e-mail: [email protected].—e-mail:[email protected]

The porous plasticity model (usually referred to as the Gurson–Tvergaard–Needleman model or modified Gurson model) has been widely used in thestudy of microvoid-induced ductile fracture. In this paper, we studied theeffects of random voids on the porous plasticity model. Finite-element simu-lations were conducted to study a copper/tin/copper joint bar under uniaxialtension using the commercial finite-element package ABAQUS. A randomlydistributed initial void volume fraction with different types of distribution wasintroduced, and the effects of this randomness on the crack path and macro-scopic stress–strain behavior were studied. It was found that consideration ofthe random voids is able to capture more detailed and localized deformationfeatures, such as different crack paths and different ultimate tensilestrengths, and meanwhile does not change the macroscopic stress–strainbehavior. It seems that the random voids are able to qualitatively explain thescattered observations in experiments while keeping the macroscopic mea-surements consistent.

Key words: Gurson model, porous plasticity, random voids, finite-elementanalysis

INTRODUCTION

Nucleation, growth, and coalescence of microvoidsin metallic materials are important mechanisms inductile failure.1,2 In Pb-free solder alloys, particu-larly Sn-rich solders with Ag and Cu, second-phaseparticles of Ag3Sn and Cu6Sn5 are present. Theseparticles, which can be as large as tens of microm-eters or as small as submicrometer in size, havebeen shown to nucleate voids.3 Many investigatorshave conducted fractographic analyses that show apredominant distribution of voids, with the second-phase particles at the base of these voids.4 In othercases, voids can be nucleated close to Cu6Sn5

intermetallic compound (IMC) formed during reflowof Sn-rich solder and Cu.

Therefore, theoretical descriptions and predictivenumerical simulations of ductile failure due to

incubation and coalescence of microvoids are ofgreat interest. In the 1960s, Rice and Tracey5

investigated ductile enlargement of a single void inan infinite matrix, and proposed that the voidgrowth rate increases due to the hydrostatic stressrkk. In the 1970s, Gurson2 developed a porousplasticity model based on a unit cell with a singlevoid in a finite matrix. Gurson2 established a yieldcriterion for porous ductile metals that depends onnot only the von Mises effective stress re (as inconventional plasticity) but also the hydrostaticstress rkk and void volume fraction f. Later, Tvergaardand Needleman studied ductile crack growth basedon the Gurson model, and further developed andmodified this model to account for plastic workhardening and damage evolution.6–11 This porousplasticity model is referred to as the Gurson–Tvergaard–Needleman (GTN) model or modifiedGurson model, being widely used to model micro-void-induced ductile fracture. The GTN model hasbeen included in the commercial finite-element

(Received March 26, 2011; accepted November 7, 2011;published online December 1, 2011)

Journal of ELECTRONIC MATERIALS, Vol. 41, No. 2, 2012

DOI: 10.1007/s11664-011-1816-5� 2011 TMS

177

package ABAQUS as one of the plastic models tostudy the evolution of damage in metallic materialscontaining microvoids and to analyze ductile frac-ture. Recently, Wen et al.12,13 extended the GTNmodel to account for the effect of void size andobserved that the yield surfaces for micrometer- andsubmicrometer-sized voids are significantly largerthan given by the Gurson model.

Experimentally, the size and location of nucleatedmicrovoids are randomly distributed; for example,Fig. 1 shows a single Cu-Sn-Cu joint tested in ten-sion. The joint was made of 99.99% pure tin (Sn)solder and oxygen-free high-conductivity (OFHC)copper bars. The thickness of the pure Sn joint was0.5 mm, and the diameter of the cylindrical speci-men was 6.35 mm. Figure 1a is an optical micro-graph of the side view of the joint before loading.Figure 1b shows scanning electron microscopy(SEM) images of the fracture surfaces of the twospecimens. The fracture surfaces on the left andright are opposing matching fracture surfaces.Clearly, the voids are distributed randomly. It is,thus, necessary to be able to simulate the behavior

of Pb-free solder alloys with random distribution ofnucleating voids.

In this work, we studied the effect of a randomdistribution of voids using the modified Gursonmodel in the finite-element package ABAQUS.Random sets of initial void volume fractions f withdifferent random distribution, such as rectangularor Gaussian distributions, were assigned to ele-ments. The effects of mean void volume fraction, f,its standard deviation, and the type of distributionwere studied. It was found that capturing the ran-domness of the voids in the model gives moredetailed and local information of void nucleationand may explain some of the scatter in ultimatestrength and crack path observed experimentally.

THE GTN MODEL

The GTN model assumes a ductile metal matrixmaterial either in fully dense form or with a smallinitial distribution of voids. The void volume frac-tion f is defined as the volume of voids divided by thetotal volume of the porous material; f = 0 describes

Fig. 1. (a) Optical image of the side view of a copper/tin/copper joint. (b) Scanning electron microscopy image of the crack surface of a copper/tin/copper joint.

Fei, Yazzie, Chawla, and Jiang178

a fully dense material, whereas f = 1 represents afully voided material. The GTN model proposes ayield surface given by2,6

/ ¼ re

ry

� �2

þ2q1f cosh � 3q2

2ryrkk

� �� 1þ q3f 2� �

¼ 0;

(1)

where ry is the yield stress of the matrix material;the three parameters (q1, q2, and q3) were intro-duced by Tvergaard6 to make the original Gursonmodel agree with numerical studies. Tvergaard6

suggested that q1 is dependent on the strain-hard-ening behavior of the metal; for example, q1 = 1.25for n = 20 and q1 = 1.8 for n = 5, where n is thestrain-hardening exponent. It was also suggestedthat q2 = 1.0 and q3 = q1

2. Setting the void volumefraction f = 0, the GTN collapses to the classical J2

theory, which uses the von Mises yield criterion.Figure 2 shows representative yield surfacesexpressed as the relationship between normalizedvon Mises effective stress and hydrostatic stress forq1 = 1.8, q2 = 1.0, and q3 = 3.24. One can clearly seethat the yield condition depends on both the vonMises effective stress and the hydrostatic stress,and with increasing void volume fraction, f, thematerial tends to yield at smaller von Mises stressand hydrostatic stress.

The evolution of the void volume fraction growthrate, _f , has two contributions, namely the nucle-ation rate and the growth rate,

_f ¼ _fnucl þ _fgr; (2)

where _fnucl changes due to nucleation of new voidsand _fgr changes due to growth of existing voids. Chuand Needleman8 assumed that the nucleation ofnew voids is plastic strain controlled and given by

_fnucl ¼ A_�eplm; (3)

where

A ¼ fN

sN

ffiffiffiffiffiffi2pp exp � 1

2

�eplm � eN

sN

� �� �

is a normal distribution and _�eplm is the rate of

equivalent plastic strain �eplm of the material.

Needleman and Rice14 suggested that void nucle-ation may take place over a narrow range of nucle-ation strain, characterized by mean eN and standarddeviation sN. Usually, one can take eN as a strainvalue before the ultimate tensile strength (UTS) inthe stress–strain curve. fN is the upper limit of thetotal void volume fraction that can be nucleated,which is consistent with the volume fraction ofsecond-phase particles of the metal. The voidgrowth rate is given by

_fgr ¼ ð1� f Þ_eplkk; (4)

where _eplkk is the rate of volumetric plastic strain of

the material. Equation (4) was derived from theplastic incompressibility of the matrix material (notthe material containing incompressible matrix andcompressible voids).

FINITE-ELEMENT ANALYSIS

Compared with classical metal plasticity, theGTN model introduces a new state variable, thevoid volume fraction, f, into the constitutive rela-tions. Therefore, the randomness of the voids can beimplemented via random assignment of the initialvoid volume fraction to each element. The randomlyassigned initial void volume fraction obeys a certaintype of random distribution, such as rectangular orGaussian. To handle a large number of elements, aC++ code was developed to generate and assignrandom initial void volume fractions to each ele-ment, based on the prescribed distribution of initialvoids.

Figure 3a illustrates the model problem and theboundary conditions. The model problem consists ofa copper/tin/copper joint which reproduces the jointexperiment shown in Fig. 1. The Sn is 0.5 mm thickand 6.35 mm in diameter. Axisymmetric elementsare used, namely CAX4R (four-node axisymmetricelement with reduced integration) for copper andCAX3 (three-node axisymmetric element) for Sn.The symmetric boundary conditions were applied inthe symmetry axes, and the bottom of the copperhad fixed displacement in the vertical direction. Theprescribed displacement boundary condition wasapplied at the top of the copper in the verticaldirection. Mesh convergence was examined byvarying the element numbers of pure Sn, and finallythe simulations used 8000 CAX3 elements in pureSn. All simulations were quasistatic with applieddisplacement.

Copper was modeled as an elastic material withYoung’s modulus of E = 116.5 GPa and Poisson’sratio of m = 0.34. Sn was modeled as a GTN plasticFig. 2. Schematic of the yield surface of the modified Gurson model.

The Effect of Random Voids in the Modified Gurson Model 179

material with random initial void volume fraction.The elastic modulus of Sn15,16 is E = 46.9 GPa andits Poisson’s ratio is m = 0.36. The experimentallymeasured stress–strain curve of pure tin was usedfor the plastic portion of the curve, as given inFig. 3b. To best fit the stress–strain curve, thestrain-hardening index was obtained as n = 5. Inthe simulation, we only used the hardening part upto the ultimate tensile stress (UTS). The threeparameters in the yield surface (Eq. (1)) were takenas q1 = 1.8, q2 = 1.0, and q3 = 3.24 for n = 5 basedon Tvergaard.6 The mean value of the nucleationstrain was taken as eN = 0.1 (less than the strain atthe UTS of 0.3), and the standard deviation waschosen as sN = 0.02; the upper limit of the total voidvolume was taken as fN = 0.01. The initial voidvolume fraction is on the order of that observedexperimentally. It must be noted that, although theaforementioned parameters are empirical, the sim-ulation results based on these empirical valuesqualitatively captured the failure process of a pureSn joint.

Here our main interest is to characterize thestress–strain response of the Sn segment. The fol-lowing methods were used to calculate the stressand strain in Sn: Before necking, the stress state isuniaxial, so that the nominal normal stress r wascalculated as the reaction force at the loading point(i.e., top of the copper) divided by the original cross-sectional area. The strain in Sn was calculated asthe height change of the Sn part divided by theoriginal height of the Sn.

RESULTS AND DISCUSSION

First, we studied the effect of the initial voidvolume fraction on the fracture path in Sn. Wecompared three situations as shown in Fig. 4, inwhich the contours indicate the void volume frac-tion. Figure 4a shows a case with a uniform initialvoid volume fraction throughout all elements in theSn segment (Fig. 4a), and Fig. 4b, c shows two caseswith randomly distributed initial void volume frac-tions in Sn. The initial void volume fraction wastaken as f = 0.8%, and the range of variation of theinitial void volume fraction for the randomly dis-tributed voids was from 0.3% to 1.3%. A rectangulardistribution was used to generate the randomlydistributed initial voids for Fig. 4b, c. The random-ness of the initial void volume fraction can beobserved in Fig. 4b, c. After deformation, the threecases exhibited different fracture paths as shown inFig. 4d–f. Because of the randomly distributed ini-tial void volume fraction, the localizations of voidnucleation and growth were also random, which ledto different fracture patterns. Figure 5 comparesthe macroscopic stress–strain response of the Snsegment for the three cases in Fig. 4a–c. It wasfound that almost identical macroscopic stress–strain responses (both hardening and softening)were obtained for the three cases (Fig. 4a–c). Thissuggests that, by considering the randomly distrib-uted voids in the GTN model, local microstructuraleffects that control the fracture path can be gleaned,while the macroscopic stress–strain behaviorremains unchanged. It should be noticed that thesoftening segments of the output curves result fromthe increase of the void volume fraction, since the

Fig. 3. (a) Geometry and boundary conditions of the axisymmetricmodel for the simulation. (b) Stress–strain curve measured for pureSn at strain rate of 0.001/s.

Fig. 4. Dependence of crack path on the randomness of the initialvoid volume fraction. Contours are for void volume fraction.

Fei, Yazzie, Chawla, and Jiang180

input has only the hardening segment (as shown inFig. 3b). This observation agrees qualitatively withexperiments in which different specimens preparedunder identical conditions may have differentcracking paths but exhibit very similar macroscopicstress–strain curves.

Secondly, we studied the effect of the range of therectangular distribution of the initial void volumefraction on the macroscopic stress–strain behavior.Figure 6 shows three cases that have the sameaverage initial void volume fraction of 0.8% butdifferent ranges of rectangular distribution of 0% to1.6%, 0.3% to 1.3%, and 0.6% to 1.0%, along with auniformly distributed initial void volume fraction of0.8%. It was found that the different ranges did notaffect the overall stress–strain curve but did affectthe UTS. Specifically, the uniform case had thehighest UTS while the case with the widest range(i.e., 0% to 1.6%) showed the smallest UTS. This isbecause the overall stress–strain behavior is deter-mined by the total void content in the material.Thus, all four cases, regardless of the distribution,exhibited similar behavior. It should be noted that amuch larger range of voids may have elicited dif-ferent results, i.e., have had a more profound effecton the deformation behavior. However, the UTS ofthe material is determined by the localization of thevoids: Larger voids give smaller UTS, whichexplains why the case with the widest range of ini-tial void volume fraction had the smallest UTS.

Thirdly, we studied the effect of the differenttypes of random distribution on the macroscopicstress–strain behavior. Figure 7 shows the stress–strain curve for two types of distributions, namelyrectangular and Gaussian. The average initial voidvolume distribution was taken as 0.8%, the rangefor the rectangular distribution is 0.3% to 1.3%, andthe standard deviation of the Gaussian distributionis 0.02. The results suggest that the macroscopicstress–strain behavior is not sensitive to the distri-

bution types, which is similar to the observation ofFig. 6, i.e., that the macroscopic stress–strainbehavior is mainly determined by the total voids ofthe materials. More explicitly, the macroscopicstress–strain behavior represents the overall load-bearing capability, which is related to the totalfraction of voids. Since the two distributions havecomparable average and maximum void volumefraction, the macroscopic stress–strain behaviorsare similar.

Finally, we want to discuss a common phenome-non relating to the crack path that was found in allthe aforementioned simulations. We found that allcracks initiated at the symmetry of axes and prop-agated towards the free surface of the cylindricalbar, even though the crack paths were different. Asthe tensile load increases, the stress and the plasticstrain in the Sn segment increase, which in turn

Fig. 5. Macroscopic stress–strain behavior of the tin segment withuniform and randomly distributed voids.

Fig. 6. Effect of the range of random distribution on the macroscopicstress–strain behavior.

Fig. 7. Effect of the type of random distribution on the macroscopicstress–strain behavior.

The Effect of Random Voids in the Modified Gurson Model 181

leads to increase of the void volume fraction. In theyield surface given by Eq. (1), for given void volumefraction, the yield condition is determined by twoquantities, namely the von Mises effective stress re

and the hydrostatic stress rkk. Here, compressivehydrostatic stress is considered negative andhydrostatic stress under tension is defined as posi-tive. Figure 8 shows the von Mises effective stressre and the hydrostatic stress rkk in the Sn segmentalong the radial direction. The von Mises stressalong the radial direction does not change signifi-cantly. On the contrary, the hydrostatic stress dropssignificantly in the vicinity of the free surface.Based on the yield surface given by Fig. 2, the ele-ments close to the symmetry axis (with largerhydrostatic stress) will yield first, followed by theelements close to the free surface (with smallerhydrostatic stress). Once the element yields, thevoid volume fraction starts to increase, as charac-terized by Eqs. (2)–(4). Fracture occurs once thevoid volume fraction reaches a critical value.

CONCLUSIONS

In this work, we studied the effects of randomvoids on the porous plasticity model, i.e., the modi-fied GTN model. Finite-element simulations wereconducted using the commercial finite–elementpackage ABAQUS. We introduced randomly dis-tributed initial void volume fraction with differenttypes of distribution and then studied the effects ofthis randomness on the crack path and macroscopicstress–strain behavior. We found that this consid-

eration of the random voids is able to capture moredetailed and localized deformation features, such asdifferent crack paths and different UTS values, andmeanwhile does not change the macroscopic stress–strain behavior. It seems that the random voids areable to qualitatively explain the scattered observa-tions in experiments while keeping the macroscopicmeasurement consistent.

ACKNOWLEDGEMENTS

The authors are grateful for financial support forthis work from the National Science Foundation,Division of Materials Research—Metals Division,DMR-0805144 (Drs. Allan Ardell, Bruce Macdonald,and Harsh Chopra, Program Managers). We alsoappreciate the Fulton High Performance Computingat Arizona State University for enabling us to con-duct our simulations.

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Fig. 8. Variation of hydrostatic stress and von Mises effective stress along the radial direction of the pure Sn segment.

Fei, Yazzie, Chawla, and Jiang182

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