Effect of Loading Condition and Stress State on DamageEvolution of Silicon Particles in an Al-Si-Mg-Base CastAlloy

M.D. DIGHE, A.M. GOKHALE, and M.F. HORSTEMEYER

Damage evolution of Si particles in a Sr modified cast A356(T6) Al alloy is quantitatively characterizedas a function of strain under tension, compression, and torsion. The fraction of damaged Si particles,their size distributions, and orientation distribution of particle cracks are measured by image analysisand stereological techniques. Silicon particle cracking and debonding are the predominant damagemodes. Particle debonding is observed only under externally applied tensile loads, whereas particlecracking is observed under all loading conditions. The relative contributions of Si particle debondingand fracture to the total damage strongly depend on stress state and temperature. For all loadingconditions and stress states studied, the average size of damaged Si particles is considerably largerthan the bulk average size. The rate of damage accumulation is different for different loading conditions.At a given strain level, Si particle damage is lowest under compression and highest under torsion.The anisotropy of the damage is highly dependent on the deformation path and stress state. Underuniaxial tension, the cracks in the broken Si particles are mostly perpendicular to the loading direction,whereas in the compression test specimens they are parallel to the loading direction. The Si particlecracks in the torsion and notch-tension test specimens do not exhibit preferred orientations. Thequantitative microstructural data are used to test damage evolution models.

I. INTRODUCTION verify the existing theoretical models for the damage evolu-tion and fracture,[8,10,12,15,16] and they are required for devel-

THE Al-Si-Mg-base cast alloys are widely used for auto- opment and validation of internal state variable (ISV)–basedmotive and aerospace structural applications. The cast micro- damage mechanics models for damage evolution and frac-structure of these alloys typically contains aluminum rich ture[17,18] of cast Al-alloy components.dendrite cells, silicon particles in the interdendritic regions, Earlier studies on the damage evolution of silicon particlesmicroporosity, and numerous minor phases. Mechanical in Al-Si-Mg alloys only considered damage evolution under

uniaxial tensile stress, and they are mostly of qualitativeproperties of cast Al-Si-Mg-base alloys depend on thenature. Few investigations report quantitative data on theamount and size of macrodefects such as internal oxideevolution of silicon particle damage as a function of tensilelayers and shrinkage macroporosity,[1,2,3] microporosity,[4,5]

strain in Al-Si alloys. Yeh and Liu[8] reported damage evolu-dendrite cell size,[6,7] and size and shape of Si particles[8]

tion in A357 alloy, Gurland and Plateau[9] studied damagepresent in the interdendritic regions. Deformation and frac-evolution as a function of tensile strain in Al-13 pct Si alloy,ture of these alloys is usually associated with gradual micro-and Doglione et al.[11] investigated damage evolution of Sistructural damage accumulation.[8–12] In good qualityparticles in A356 alloy. Recently, Caceres and Griffiths[10]castings having low porosity and macrodefects, the dominantstudied the cracking of silicon particles in a cast Al-7Si-damage micromechanisms under overload or LCF condi-0.4Mg alloy during plastic deformation under uniaxial ten-tions are fracture[8–12] and debonding[12] of the Si particles.sion. Effects of temperature[20] and strain rate[21] on the dam-As the Si particles are clustered in the interdendritic regions,age of Si particles in A356 alloy have been reported by thetheir fracture and debonding facilitates crack initiation in theauthors, but to the best of the authors’ knowledge, damageinterdendritic regions. The interlinkage of such microcracksevolution related to Si particles in Al-Si-Mg base (or anyusually causes the final failure (Figure 1). Therefore, toother Al alloys) has not been studied as a function of com-understand and model the deformation and fracture behaviorpression, torsion, or various triaxial stress states to under-of cast Al-Si-Mg alloys, it is necessary to quantify the evolu-stand the effect of loading condition and stress state on thetion of microstructural damage processes such as fracturedamage evolution processes.and debonding of silicon particles as a function of strain

The objective of this contribution is to report detailedunder various loading conditions. Such data are useful toquantitative microstructural analysis of the damage evolu-tion of Si particles in a cast A356 alloy (which is a typicalAl-Si-Mg-base alloy) as a function of strain under tension,compression, and torsion. The Si particle damage is alsoM.D. DIGHE, Advanced Technology Manager, is with the Hi TecMetal

Group, Cleveland, OH 44092. A.M. GOKHALE, FASM, Professor, is with quantified in notch tension test specimens to understand thethe Department of Materials Science and Engineering, Georgia Institute of effect of triaxial stress state on the damage development.Technology, Atlanta, GA 30332-0245. M.F. HORSTEMEYER, Manager, It is shown that the damage evolution strongly depends onis with the Center for Materials and Engineering Science, Sandia National

the loading condition and stress state. The experimentalLaboratories, Livermore, CA 94551-0969.Manuscript submitted February 28, 2001. data have been utilized to compute the size dependence of

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of the dislocation pileup. The model predicts that the effec-tive shear stress required for particle cracking may increasewith an increase in the particle size (D) if the length ofpileup (L) is much larger than D.

Yeh and Liu[8] reported damage evolution of Si particlesas a function of strain in a cast A357 alloy under uniaxialtensile stress. They observed that silicon particle fractureinitiates at the onset of plastic deformation, and the extentof this damage increases with an increase in plastic strain.Further, the fraction of fractured particles increases morerapidly with plastic strain in the alloys having higher yieldstress. At all plastic strain levels, most of the cracks in theparticles tend to be perpendicular to the tensile axis. Yehand Liu developed a model for particle cracking that is basedFig. 1—Fracture caused by interlinkage of microcracks through the alumi-

num dendrites. on a dislocation pileup mechanism. Their model gives thefollowing relation between the fraction of cracked particles,f, plastic strain «, and the applied tensile stress, s.

f 5 [ADm2L2M 2s 5(« 2 «0)]/[5n 1 1] [3]the probability of Si particle fracture under tension, com-pression, and torsion, to test the damage models, and to

In this equation, n is the exponent of the assumed powercompute parameters of an ISV-based damage evolutionlaw stress-strain relationship, «0 is the total strain at whichmodel that incorporates damage nucleation, growth, andmicroyielding occurs, A is a constant, L is the dislocationcoalescence.[17,18]

pileup length, M is the two-dimensional Schmidt factor, Dis the particle size, and m is the average number of pileupsin a slip band. If one assumes that « À «0, and the powerII. BACKGROUNDlaw hardening equation given by s 5 K ? «n, then Eq. [3]

Numerous particle-cracking models have been proposed takes the following form.for damage evolution of particulate phases in metals and

f 5 {[AK 5 Dm2L2M 2]/[5n 1 1]} ? « (5n11) [4]alloys.[8–15] These models mainly differ in the failure crite-rion, and on this basis, they can be classified into two groups: Equations [3] and [4] predict that at a given value of plastic(1) models that utilize stress or strain based failure crite- strain, the fraction of broken particles, f, is directly propor-rion[12,13,24,25] and (2) energy based models.[9] Further, some tional to the particle size D.models are based on continuum mechanics,[1,5,9,12,18,19,23]

Doglione et al.[11] have reported an experimental studywhereas others are based on dislocation theory.[8,13,24] Parti- on the Si particle damage in A356-T6 alloy. They performedcle cracking theories[9,26,27] predict that (1) the stress required experiments on a chill cast alloy having average Si particlefor particle cracking decreases with an increase in particle size of 2.6 mm (and aspect ratio 5 1.3) and on a sand castsize, and therefore, larger particles are expected to crack alloy having average size of 3.4 mm (and aspect ratio 5before the smaller ones; and (2) higher aspect ratio particles 1.4). They measured the fraction of broken particles as acrack more easily than equiaxed or spherical particles. All function of tensile strain for both the alloys. Interestingly,the models pertain to damage evolution under tension. Very they did not observe any significant difference in the varia-few models have been experimentally verified, perhaps due tion of the fraction of broken Si particles with plastic strainto lack of detailed quantitative microstructural data on the in the sand cast and chill cast alloy specimens, although thedamage evolution. average Si particle size in the two groups differed by about

Gurland and Plateau[9] have given the following equation 30 pct.for critical tensile stress s to develop a crack in a particle Recently, Caceres and Griffiths[10] studied the cracking ofof size D. silicon particles in a cast Al-7Si-0.4Mg alloy during plastic

deformation. Different microstructures were produced bys 5 [Eg /(Dq2]1/2 [1]varying the solidification rate and time for solution treat-

In Eq. [1], q is the stress concentration factor at the particle, ment, in order to study the effect of microstructural variablesE is the weighted average of the elastic moduli of the particle on damage evolution. It was observed that the eutectic siliconand matrix, and g is the interfacial energy of the crack. Thus, particles progressively crack with applied plastic strain inthe stress required to crack a particle is inversely proportional tension. In coarser structures, the particle cracking occursto the square root of the particle size, and therefore, very rapidly, while in finer structures, it is more gradual.according to this model, larger particles crack at lower They used dispersion hardening models to calculate thestresses. stresses in the silicon particles. Assuming that the particle

Smith and Barnby[22] developed a model for particle crack- fracture follows Weibull statistics, they suggested the follow-ing on the basis of a dislocation pileup mechanism. They ing equation for the probability of particle cracking.showed that the effective shear stress, t, for cracking aparticle of size D is given as follows:

p 5 1 2 exp F2VV0 1

sp

s02mG [5]

t 5 [D/(L 1 D)]1/2 ? [2 (mg (L(1 2 n ))]1/2 [2]

In Eq. [2], n is the Poisson’s ratio, g is the crack surface In Eq. [5], p represents the probability of cracking of aparticle of volume V 5 (1/6)pD3; sp is the stress in theenergy, m is the matrix shear modulus, and L is the length

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particle; V0 5 (1/6)pD30 and m are constants; and D is the of torsion test specimens) and were polished using standard

metallographic procedures. Details are given elsewhere.[12]size of the silicon particles. These authors concluded thatthe Weibull statistics could represent the fracture behavior The microstructures were observed and quantified in the

unetched condition. In the case of compression samples, theof the silicon particles in uniaxial tension.region near the center of the specimen was selected formeasurements. For the torsion samples, the thinnest regionIII. EXPERIMENTALshowed the maximum damage. The damage was uniform

A. Materials and Heat Treatment within this thin section and hence the thin section wasselected for the measurements. In the case of notched tensileThe experiments were performed on a commercial A356test specimens, the measurements were performed only inalloy of nominal composition Al-7 wt pct Si-0.4 wt pct Mg.regions within 3 mm of the fracture surface, where theRectangular plate castings of dimensions 203.2 3 139.7 3damaged particles were concentrated.25.4 mm were made by using iron chills on the top, bottom,

To quantify the Si particle damage, contiguous fields ofand side opposite to the riser end of the plate to simulate aview (at magnification 500 times) were grabbed and storedpermanent mold casting. A “no-bake” silica sand was usedin the computer memory to create large-area “tiled” micro-for the riser and the down sprue. A ceramic filter was placedstructural regions at high resolution.[30,31,32] Particle countingbetween the riser and the down sprue. The melt was grainon such a large area high-resolution image eliminates therefined, strontium modified, and degassed using a rotarybias in particle counting due to edge effects for all practicaldegasser. The pouring temperature was in the range of 950purposes.[30,31,32] Broken and debonded silicon particles wereto 977 K. The castings were cooled for a period of aboutmeasured interactively over 200 to 300 such contiguous16 hours and then removed from the molds. The cast platesfields. The number density, size distribution, volume frac-were solution treated at 810 K for 16 hours, hot watertion, and nearest neighbor distribution of damagedquenched at 344 K, and then artificially aged at 492 K for(fractured/debonded) Si particles were measured in each4 hours to achieve a T6 temper. Mechanical test specimensspecimen. This was done by classifying the damaged Siwere machined from the low porosity (,0.1 pct) regions ofparticles into two groups, namely, broken (fractured) andthe heat-treated cast plates near the chill end.debonded particles. Any particle that showed a separationat the interface with the Al rich matrix was classified as aB. Mechanical Testsdebonded particle. On the other hand, if a particle showedA series of interrupted tests were carried out at different cracking, then it was counted as a broken particle. Thestrain levels under uniaxial tension, compression, and tor- geometric attributes of these damaged particles such as thesion. All test specimens were drawn from similar locations maximum dimension, minimum dimension, shape factor,in different plate castings, and their microstructures were perimeter, area, and equivalent circle diameter were meas-statistically similar. Tension, compression, and torsion tests ured by means of extensive image analysis algorithms. Inwere performed at room temperature and under displacement addition, data were gathered on the orientations and thecontrol at the strain rate of 1024. Tensile tests were carried centroid coordinates of the damaged particles. To quantifyout according to the ASTM E18 standard. The interrupted the anisotropy of the silicon particle damage with respecttensile tests were conducted at different strain levels ranging to the loading axis, the orientation of the cracks with respectfrom 0.3 to 7.5 pct (failure strain). The uniaxial compression to the loading axis and the orientation of the Si particlestests were performed on specimens of 25.4 mm diameter with respect to the loading axis were measured. The overalland 50.8 mm length. The compression test specimens were bulk microstructure was also quantified in a similar manner.designed to produce homogenous deformation throughout

the specimen by machining grooves into the specimenIV. RESULTS AND DISCUSSIONends.[28] Lubrication was placed into the grooves to alleviate

frictional forces to prevent barreling of the specimen.[29]

A. Qualitative Microstructural Observations and ResultsThese quasi-static tests were interrupted at different com-In the present context, fracture and/or debonding of Sipression strain levels ranging from 0.1 to 25 pct.

particles constitutes damage. Figures 2 through 6 illustrateTorsional testing involved holding one end of the speci-the damaged Si particles in tension, compression, torsion,men and rotating or twisting the other end. The Lindholmand notch-tension specimens. Careful examination of micro-specimen geometry was used for the torsion test specimens.structure leads to the following observations.The specimens were notched to a very thin section in the

middle portion to minimize the gradient in the strain from (1) In tensile test specimens, the cracks in the Si particlesone end to the other. The interrupted torsion tests were are mostly perpendicular to the loading direction (Figureperformed at torsional strain levels ranging from 0.1 to 45 2), whereas in the compression test specimens, thepct. cracks are parallel to the loading direction (Figure 3).

The notch Bridgman tension test specimens had a notch On the other hand, Si particle cracks in the torsion testradius of 2.97 mm and specimen diameter of 9.525 mm. specimens (Figure 4) and notch-tension test specimensNotch tensile tests were performed at 222, 294, and 394 K, (Figure 5) do not exhibit any preferred orientations.at a strain rate of 0.01 per second. These tests were continued Therefore, the anisotropy of cracks and damaged Sito final fracture of the specimens. particles strongly depends on the loading condition and

stress state.C. Quantitative Metallography (2) For all loading conditions and stress states studied, the

“average” size of damaged Si particles appears to beThe specimens were cut in the center along vertical planescontaining the applied load direction (torsional axis in case considerably larger than the bulk average size of the Si

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Fig. 5—Optical micrograph depicting damaged Si particle in a notch-ten-Fig. 2—Optical micrograph depicting damaged Si particles in a tensile testsion test specimen tested at 394 K.specimen at 7.5 pct strain level.

Fig. 3—Optical micrograph depicting the damaged Si particle in a compres-sion test specimen at 22.5 pct strain level.

Fig. 6—Debonded Si particles of the tensile test specimen strained to 7.5pct strain level.

cracking of the silicon particles appears to be a phenome-non that is strongly influenced by its local microstruc-tural environment, which is stochastic in nature.

(3) Both fractured (broken/cracked) and debonded Si parti-cles are present in uniaxial tension (Figure 6) and notch-tension test specimens, whereas compression and torsiontest specimens exhibit only fractured particles.

B. Evolution of Silicon Particle Damage with Strain

Figures 7 through 9 show the number fraction of damagedFig. 4—Optical micrograph depicting damaged Si particles in a torsion (broken or debonded) Si particles as a function of straintest specimen at 5 pct strain level.

under uniaxial tension, compression, and torsion. Figure 10shows a plot of number fraction of damaged Si particles asa function of temperature for notch-tension test specimens.particles. However, all large Si particles are not dam- The notch-tension test data have been reported in an earlieraged, and some small Si particles are cracked. Small contribution.[20] Inspection of these plots leads to the follow-fractured Si particles are mostly found in clusters of ing observations and conclusions.many small particles. Also, there are some isolated large

particles that have not cracked at all. Therefore, the (1) For all loading conditions studied, damaged Si particles

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Fig. 7—Number fraction of damaged (broken/debonded) Si particles vs Fig. 9—Number fraction of damaged Si particles vs strain for torsionglobal remote tensile strain under uniaxial tension. test specimens.

Fig. 10—Number fraction of debonded and broken Si particles vs tempera-ture for notch-tension test specimens.

amount of hydrostatic tensile stress is required to initiateSi particle debonding. In the compression and torsion testspecimens, the local hydrostatic tensile stresses around SiFig. 8—Number fraction of damaged Si particles vs strain under uniax-particles are not sufficiently large to initiate particle debond-ial compression.ing. On the other hand, as the Si particle cracking is observedunder all the loading conditions, initiation of cracks in Siparticles is likely to be governed by principal tensile stressare observed at very low strain levels (0.3 pct). It can be for fracture. This is consistent with the observations that, inconcluded that under all loading conditions, the particle compression, the cracks are perpendicular to the local tensiledamage initiates at the onset of local plastic deformation. axis (parallel to applied compressive loading axis), and in(2) For all loading conditions, the number fraction of the tension the cracks are perpendicular to the loading axis,damaged Si particles increases monotonically with indicating mode-I type fracture. On the other hand, in torsion,strain. However, the rate of damage accumulation varies the principal stress axis rotates as the Si particles and Alfor different loading conditions. rich grains reorient themselves, and therefore, there is no(3) At a given strain level, the Si particle damage is lowest preferred direction for crack orientations.under compression and highest under torsion. Figure 11 shows relative contributions of debonding andparticle fracture to the total Si particle damage as a functionof tensile strain. It is interesting to observe that the rate of

C. Relative Contributions of Particle Debonding and increase of fraction of debonded Si particles with tensileCracking Damage Mechanisms strain is significantly lower than the rate of increase of

fractured Si particles with strain. There is no significantAll the earlier experimental studies on damage evolutionof Si particles in Al-Si-Mg base alloys only considered increase in the fraction of debonded particles at high strain

levels. At the 7.5 pct strain level (failed tensile test speci-particle cracking. In the earlier studies, Si particle debondingis either combined with particle cracking or is completely men), the ratio of debonded Si particles to fractured Si

particles in the bulk microstructure is about 0.3. However,ignored. In the present investigation, both fractured anddebonded Si particles are observed in the uniaxial tension interestingly enough, on the fracture profile of the same

specimen, the ratio of the number of debonded Si particlestest specimens and notch-tension test specimens, whereasonly particle cracking is observed under compressive and to fractured Si particles was observed to be about two. This

may be due to preference of the fracture path for debondedtorsional loading conditions. This implies that a critical

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Fig. 12—Relative contributions of debonding and particle fracture to thetotal Si particle damage as a function of temperature in notch-tensiontest specimens.

damage due to particle fracture is larger than that due todebonding. This transition is related to the relative influenceof temperature on the elastic moduli of silicon, which then

Fig. 11—Relative contributions of debonding and particle fracture to the changes the stress state, for fracture vs the plastic responsetotal Si particle damage as a function of uniaxial tensile strain.

of the aluminum matrix in conjunction with the changingelastic moduli of the silicon for debonding. Clearly, thetemperature influence on the aluminum is greater since weobserve a greater difference in the debonding than siliconparticles rather than fractured particles, and/or profuse

debonding (rather than cracking) of the particles ahead of fracture over the temperature range valuated in this study.The present work leads to the following new observationsthe crack tip in the final stages of the fracture process due

to the presence of a highly triaxial stress state. and conclusions regarding Si particle debonding and crack-ing mechanisms in the Sr modified A356 (T6) alloy.Figure 12 shows a plot of the fraction of debonded and

broken Si particles as a function of temperature for notch (1) As Si particle debonding is observed only in tensile testtension test specimens. These data bring out an important (smooth as well as notch tension) specimens, it may beeffect of state of stress on the Si particle damage. In the said that Si particle debonding requires an externallyuniaxial tension specimens, at any given strain level, the applied tensile stress that gives rise to a critical localfraction of debonded Si particles is lower than the fraction hydrostatic tensile stress. On the other hand, Si particleof broken Si particles (Figure 11). However, in the notch cracking is observed under all loading conditions (ten-tension specimens, the fraction of debonded Si particles is sion, compression, and torsion).higher than the fraction of broken Si particles, except at the (2) The extent of particle damage due to debonding dependshighest temperature. Because the stress triaxiality is higher on the local hydrostatic stress level. A larger triaxialfor the notch tensile test than for the uniaxial tensile test, stress state (predominant in notch tensile test specimens)debonding would be expected to be higher for the notch increases the damage due to particle debonding.tensile test. This verifies that debonding is mainly driven (3) Under an externally applied tensile load, damage dueby the hydrostatic tensile stress but the Si particle fracture to particle cracking increases with tensile strain at ais driven by the local maximum principal stress. significantly higher rate than damage due to particleThe data in Figure 12 reveal that the fraction of debonded debonding.Si particles varies significantly with the temperature, (4) The extent of damage due to particle debonding is morewhereas the fraction of fractured Si particles is not as sensi- sensitive to temperature than particle cracking.tive to the change in the temperature. Fracture of silicon ismainly a function of the maximum principal stress, whichchanges minimally with temperature. However, debonding

D. Size Dependence of Silicon Particle Damageis a function of the thermoelastic constitutive response ofthe silicon particles and the interface between the particles In the present work, the largest dimension of a particle

(major axis) is regarded as the measure of Si particle size.and the matrix, and the thermoplastic response of the matrix.Since the thermoplastic response of the aluminum matrix is In the bulk microstructure of the present specimens, the

average Si particle size is 5.7 mm. Figure 13 shows thehighly dependent on temperature, the interface response isalso expected to depend on temperature, which gives rise overall Si particle size distribution in the bulk microstructure,

whereas Figure 14 shows the size distribution of damagedto the temperature dependence of the particle debonding.As the rate of silicon particle debonding increases signifi- (broken and/or debonded) Si particles in the tensile test

specimen with 7.5 pct strain. Observe that the sizes of thecantly with decrease in temperature, the contribution of thisfracture micromechanism to crack initiation is expected to damaged Si particles are significantly larger than the overall

bulk average size of 5.7 mm. This trend is observed for allbe higher at lower temperatures.One final note about Figure 12 is that there appears to be loading conditions and stress states studied in this investiga-

tion. To quantify the size dependence of probability of parti-a transition temperature at which dominant damage mecha-nism changes. At cold and ambient temperatures, the fraction cle damage, it is essential to analyze a fraction of particles

in the overall population having a given size that are dam-of debonded Si particles is larger than the fraction of brokenparticles. However, at the highest temperature studied, the aged at a given strain. This fraction P is essentially the

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Fig. 15—A histogram plot of the probability of particle damage P vs Siparticle size D for failed tensile test specimen (7.5 pct strain).

Fig. 13—Overall Si particle size distribution in the bulk microstructure.

Fig. 16—A histogram plot of the probability of particle damage P vs SiFig. 14—Size distribution of damaged Si particles in the tensile test speci-particle size D for a compression test specimen (22.5 pct strain).men with 7.5 pct strain.

probability that a particle of given size will be damaged ata given strain level for a given loading condition, and it isformally defined as follows.

P 5 [Number of damaged Si particles

of given size]/[number of Si particles [6]

of same size in overall bulk population].

Figure 15 shows a histogram plot of the fraction P vs Siparticle size D in a failed tensile test specimen (7.5 pctstrain). Observe that P increases with an increase in theparticle size in a nonlinear manner. Figures 16 and 17 showsimilar plots for compression and torsion test specimens.These data show that a significant fraction of largest particles(.28 mm size range) is damaged, but only a small fractionof particles in the average size range (4 to 6 mm) or belowaverage size are damaged. This clearly shows that larger Fig. 17—A histogram plot of the probability of particle damage P vs Si

particle size D for a torsion test specimen (45 pct strain).particles have significantly higher probability of fracture,

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Fig. 20—Orientation distribution of cracks in Si particles in a tensile testFig. 18—Average size of damaged Si particles vs strain for compressionspecimen (7.5 pct strain).test specimens.

Fig. 19—Average size of damaged Si particles vs strain for torsion testFig. 21—Orientation distribution of cracks in Si particles in a compressionspecimens.test specimen (22.5 pct strain).

which should be accounted for in the models for damageevolution. It is important to point out that in this analysis, stress triaxiality. Figures 20 through 22 show plots of orienta-

tion distribution of cracks in broken Si particles for speci-fractured and debonded Si particles are combined togetherin a single group. The data in Figures 15 through 17 show mens strained under tension, compression, and torsion. In

these data, the crack orientation is specified by the anglethat for all three loading conditions, at a given strain level,the probability of particle damage increases with the size. between the loading direction (torsion axis in the case of

torsion specimens) and the crack. Observe that in the tensileIn the uniaxial tensile test specimens, the average size ofdamaged Si particles does not vary significantly with strain. test specimen (Figure 20) most of the cracks have an orienta-

tion angle in the range of 80 to 100 deg; i.e., they areHowever, in the compression and torsion test specimens,the average size of damaged Si particles decreases with the approximately perpendicular to the tensile axis. However,

in the compression test specimen strained up to 22.5 pctincrease in the strain (Figures 18 and 19). These observationssupport the hypothesis that, at least for compression and compressive strain (Figure 21), most of the cracks are in

the orientation range of 0 and 30 deg (or 150 to 180 deg)torsion loading conditions, larger particles are the first tofracture, and the extent of damage in the smaller particle with respect to the loading axis, i.e., they are mostly parallel

to the loading axis (i.e., direction of compressive load).size range gradually increases with strain.These cracks are oriented perpendicular to the induced ten-sile stress component resulting from the Poisson effect. Fig-

E. Anisotropy of Si Particle Damage ure 22 shows the distribution of the orientations of the crackswith respect to torsion axis for a torsion test specimen. TheseAs mentioned earlier, the morphological anisotropy of the

damage is highly dependent on the deformation path and cracks do not show strong preferred orientations. These data

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Fig. 23—Plots of ln ( f ) vs ln (« ) for damaged (broken or debonded) Siparticles and only broken Si particles in uniaxial tension test.

Fig. 22—Orientation distribution of cracks in Si particles in a torsion testspecimen (45 pct strain).

and Gokhale[17] is used to model the results from the castA356 (T6) aluminum data under compression, tension, and

clearly demonstrate that the anisotropy of the damage is torsion. The integrated form of the void nucleation ratehighly dependent on the loading condition and stress state. equation is given by

h (t) 5 Ccoeff exp 1« (t)d 1/2

KIC f 1/3 HaF 427

2J3

2

J23G

[8]

F. Damage Evolution Models

The experimental observations on the size dependence ofdamage are consistent with the model proposed by Gurland

1 bJ3

J23/2 1 cZZ I1

!J2ZZJ2and Plateau.[9] According to their model (Eq. [1]), the stress

required to crack a particle is inversely proportional to thesquare root of the particle size, and therefore, larger particles where h (t) is the void nucleation density (which is the same

as the number density of damaged Si particles), « (t) is theshould crack at lower stresses. Consequently, the modelpredicts that the probability of particle damage should strain at time t, and Ccoeff is a material constant. The material

parameters a, b, and c relate to the volume fraction of dam-increase with the particle size, as experimentally observedunder all loading conditions in the present work. aged Si particles arising from local microstresses in the

material. These constants are determined from experimen-Yeh and Liu[8] proposed a model for the fracture of siliconparticles under uniaxial tension. Their model assumes a tally measured data on the Si particle damage evolution

under tension, compression, and torsion at different strainstrain controlled particle fracture phenomenon, where thefracture occurs due to the pile-up stresses developed due to levels. The stress state dependence on damage evolution is

captured in Eq. [8] by using the stress invariants denoteddislocation movement during the deformation process (Eqs.[3] and [4]). A power law hardening equation given by s 5 by I1, J2, and J3, respectively. The term I1 is the first invariant

of stress (I1 5 skk). The term J2 is the second invariant ofK ? « n is assumed in Yeh and Liu’s model, and therefore,one can write deviatoric stress (J2 5 1/2 SijSij), where Sij 5 sij 2 1/3dijskk .

The term J3 is the third invariant of deviatoric stress (J3 5f 5 {[AK 5Dm2L2M 2]/[5n 1 1]} ? « (5n11) [7] SijSjkSiki). The parameter f is the volume fraction of siliconparticles, d is the average silicon particle size, and KIC isTherefore, if Yeh and Liu’s model is applicable, a plot of

ln ( f ) vs ln (« ) should be linear and the slope of the linear the bulk fracture toughness of A356 alloy.For the cast A356 aluminum alloy in our study, KIC 5plot should yield the value of the strain hardening coefficient

n. Figures 23 shows these plots for the total fraction of 17.3 MPa!m, d 5 6 mm, and f 5 0.07. The stress stateparameters were determined to be a 5 615.46 GPa, b 5damaged Si particles (i.e., fractured and/or broken), as well

as for the fraction of broken Si particles. Although both 58.64 GPa, c 5 30.011 GPa, and Ccoeff 5 90. Figure 24 showsthe comparison of this damage model with the experimentalplots appear to be linear, the values of the strain hardening

coefficient n calculated from their slopes are negative. There- results for compression, tension, and torsion, respectively.The agreement between the predictions of the model andfore, it can be concluded that Yeh and Liu’s model is not

applicable to the present data. the experimental data is very much expected, as the samedata have been used for calculation of the model parameters.Recently, Horstemeyer and Gokhale[17] have proposed a

continuum model that incorporates void nucleation (i.e., par- An interesting feature of Horstemeyer and Gokhale’smodel[17] is that once the parameters of the model are known,ticle cracking and debonding) as well as void growth and

void coalescence. This model has been implemented into a it is possible to predict the fraction of damaged Si particlesfor any stress state. Therefore, Horstemeyer and Gokhale’sfinite element code.[18] In this model, the total damage evolu-

tion is divided into three equations, one each for nucleation, model can be tested by using the damage data on notchtension specimens, which were not used to compute thegrowth, and coalescence. In this model, the Si particle dam-

age by fracture or debonding implies void nucleation. There- model parameters. At room temperature, the experimentallymeasured value of the fraction of damaged Si particles isfore, data discussed thus far in this article relate to only void

nucleation. The void nucleation equation of Horstemeyer equal to 1.7 pct for the notch tension specimen. For the same

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 33A, MARCH 2002—563

particles have a significantly higher probability ofdamage.

5. At least for compression and torsion loading conditions,larger particles are the first to fracture, and the extentof damage in the smaller particle size range graduallyincreases with strain.

6. For all loading conditions, the number fraction (and vol-ume fraction) of the damaged Si particles increases mono-tonically with strain.

7. The rate of damage accumulation is different for differentloading conditions. Further, for a given strain level, the Siparticle damage is lowest under compression and highestunder torsion.

8. The morphological anisotropy of damage is highlydependent on the deformation path and stress triaxiality.

Fig. 24—Comparison of damage nucleation data with Horstemeyer and Under uniaxial tension, cracks in the broken Si particlesGokhale model illustrating the differences in Si particle damage evolution are mostly perpendicular to the loading direction,under different deformation paths. whereas in the compression test specimens, they are par-

allel to the loading direction. On the other hand, the Siparticle cracks in the torsion test specimens and notch-loading condition, temperature, and specimen geometry, thetension test specimens do not exhibit any preferredfraction of damaged Si particles calculated (Reference 18orientations.provides details of such calculations) from the model is

1.8 pct. Therefore, there is a good agreement between theprediction of the model and the data. However, comparison

ACKNOWLEDGMENTSwith more experimental data involving different stress statesis clearly necessary to confirm the model’s predictive The authors thank Richard Osborne (General Motorscapability. Corp.) and Srinath Viswanathan (Oak Ridge National Labo-

ratory) for numerous useful discussions, G.A. Schulke(Chrysler Corp.) for casting and heat treatment of A356 alloyV. SUMMARY AND CONCLUSIONSplates, and Westmoreland Mechanical Testing and Research

A detailed microstructural analysis has been carried out Laboratories for conducting the mechanical tests. Theto quantify damage evolution of Si particles in a cast A356 research conducted at the Georgia Institute of Technology(T6) alloy, as a function of strain, under tension, compres- (AMG and MDD) was supported by research grants fromsion, and torsion. Analysis of the experimental data leads the United States National Science Foundation (Grant No.to the following conclusions. DMR-9816618) and American Foundry Society. The finan-

cial support is gratefully acknowledged.1. For all loading conditions studied, particle damage initi-ates at the onset of local plastic deformation.

2. Silicon particle debonding requires an externally applied REFERENCEStensile stress that gives rise to a critical local hydrostatic

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26A, pp. 2359-72.and therefore, it is observed under all loading conditions5. E.N. Pan, C.S. Lin, and C.R. Loper: Am. Foundrymen Soc. Trans.,

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8. Jien-Wei Yeh and Wen-Pin Liu: Metall. Mater. Trans. A, 1996, vol.particle fracture.27A, pp. 3558-68.4. For all loading conditions and stress states studied, the

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