Multi-inclusion unit cell models for metal matrix composites

Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcementsH.J.B€o hm*,A.Eckschlager,W.Han1Christian Doppler Laboratory for Functionally Oriented Materials Design,Institute of Lightweight Structures and Aerospace Engineering,Vienna University of Technology,Gusshausstrasse27-29,A-1040Vienna,AustriaAbstractA multi-inclusion unit cell approach is employed to study the elastic and elastoplastic behavior of metal matrix composites reinforced by randomly oriented shortfibers.Periodic arrangements of15identicalfibers of spheroidal or cylindrical shape with an aspect ratio of5and a reinforcement volume fraction of15%are generated by a random sequential adsorption algorithm.The overall responses of the resulting unit cells under uniaxial tensile loading and the corresponding microscale stress and strainfields are evaluated via thefinite element method.In addition,a microge-ometry containing15identical spherical particles at the same volume fraction is studied for comparison.Effects of the reinforcement types and shapes in the elastic and elastoplastic ranges are studied and the predicted microfields are discussed in terms of their phase averages and the corresponding standard deviations.Weibull-type fracture probabilities are used to assess the vulnerability of thefibers or particles to brittle fracture.Ó2002Elsevier Science B.V.All rights reserved.PACS:07.05.Tp;62.20.FeKeywords:Discontinuously reinforced metal matrix composites;Randomfiber orientation;Unit cell models;Mechanical properties1.IntroductionShortfiber reinforced metal matrix compos-ites(MMCs)––like other shortfiber reinforced materials––may contain aligned,nonaligned or randomly orientedfibers.Most production and processing routes give rise to microgeometries in which thefibers are neither perfectly aligned nor fully random,but showfiber orientation distribu-tions or textures between these two extremes,see e.g.[1–3].Numerous models have been reported in the literature for describing the thermomechanical and thermophysical behavior of composites containing aligned discontinuous reinforcements.The major-ity of the analytical estimates have been based on Eshelby’s[4]equivalent inclusion approach and on Hashin–Shtrikman formalisms[5],see e.g.[6–9], and most of the numerical work has used unit cell descriptions,see e.g.[10–13].In addition,rigorous bounds for the elastic overall behavior of such materials are available[14,15].Accordingly,mod-eling the thermoelastic and thermoelastoplasticComputational Materials Science25(2002)42–53*Corresponding author.Tel.:+43-1-58801-31712;fax:+43-1-58801-31799.E-mail address:hjb@ilfb.tuwien.ac.at(H.J.B€o hm).URL:http://ilfb.tuwien.ac.at.1Present address:Alcan Mass Transportation Systems,CH-8048Zurich,Switzerland.0927-0256/02/$-see front matterÓ2002Elsevier Science B.V.All rights reserved. PII:S0927-0256(02)00248-3responses of aligned shortfiber reinforced MMCs has reached a fairly high level of development.The situation is not as satisfactory,however, with respect to materials reinforced by nonaligned or randomly orientedfibers.A number of authors proposed to modify Mori–Tanaka schemes to in-corporate ellipsoidal randomly oriented inclusions [16]or nonaligned reinforcements with prescribed fiber orientation distribution functions,see e.g. [3,17–19],and approaches of this type were ex-tended into the elastoplastic range via secant plas-ticity schemes,compare e.g.[20,21].Even though these methods produce reasonable results for many applications(among them elastic two-phase mate-rials with randomly oriented reinforcements),they have the drawback that there are a number of sit-uations in which they generate nonsymmetric elasticity matrices,see e.g.[22,23].The reason for this behavior lies in the construction of Mori–Ta-naka formalisms,which is based on aligned ellip-soidal arrangements of inclusions and is subject to an intrinsic incompatibility with nonaligned rein-forcements.The Hashin–Shtrikman procedure of Ponte Casta~n eda and Willis[24]can resolve these difficulties for many microgeometries,but for strongly nonalignedfiber-like inclusions its un-conditional applicability is typically restricted to reinforcement volume fractions of a few percent. Similar limitations hold for the double inclusion estimates of Hori and Nemat-Nasser[25].Another meanfield description for randomly oriented ellipsoidal inclusions,the Kuster–Toks}o z model [26],is basically a dilute estimate.For recent dis-cussions of the relationships between some of the above models see[27,28].Berryman’s self consis-tent scheme[29]does not suffer from the above restrictions,but is applicable to granular rather than matrix–inclusion microtopologies.At present no rigorous bounds for the overall thermome-chanical behavior of materials with nonaligned reinforcements are available in the general case,but the‘‘standard’’Hashin–Shtrikman bounds[30] hold for elastic composites with randomly oriented fibers,which show isotropic overall thermome-chanical responses.Another group of models for describing non-aligned shortfiber reinforced composites are based on the assumption that the contribution of a given fiber to the overall stiffness and strength depends solely on its orientation with respect to the applied load and on its length,interactions between neighboringfibers being neglected.Among such models are the Fukuda–Kawata probabilistic theory[31,32]and laminate analogy approaches [33,34].In the latter group of methods nonaligned reinforcement arrangements are approximated by a stack of layers each of which handles onefiber orientation and,where appropriate,onefiber length.Neither of these approaches provides full overall elastic tensors.Planar multi-fiber unit cell models[35,36]for nonaligned composites have achieved an impres-sive level of development,but by definition intro-duce strong idealizations in terms of the spatial arrangements and orientations of thefibers.In addition,their predictions are compromised by unrealistic out-of-plane constraints as discussed in [37]for particle reinforced materials.Approaches based on the superposition of submodels,each of which describes different planar normal sections [38],are not suitable for overcoming these re-strictions.Studies using three-dimensional discrete micromodels of nonaligned shortfiber reinforced composites have included work in which thefinite element method was applied to evaluate the creep response of arrangements of alternatingly tilted inclusions[39]as well as multi-inclusion unit cell models and related approaches relying on bound-ary element methods to study the elastic behavior of materials containing nonaligned reinforce-ments,see[40,41].The present state of the art in theoretical de-scriptions of the thermomechanical behavior of composites reinforced by nonaligned shortfibers suggests that the exploration of improved model-ing approaches is of considerable interest.One promising way of doing this consists in extend-ingfinite-element-based multi-inclusion unit cell models recently developed for particle reinforced MMCs[37,42,43]tofiber-like inclusions.Within such a framework,the case of randomfibers is more difficult to handle than otherfiber orien-tation distributions in terms of generating and meshing appropriate microgeometries.Accord-ingly,the present study concentrates on unit cell models for describing the elastoplastic behavior ofH.J.B€o hm et al./Computational Materials Science25(2002)42–5343MMCs reinforced by randomly oriented shortfi-bers.2.Three-dimensional multi-inclusion unit cell mod-elsA number of strategies for constructing matrix–inclusion microgeometries with random inclusion positions have been reported in the literature,viz. methods based on random sequential adsorption (RSA)schemes[37,42],on Monte Carlo methods [44],and on simulated annealing procedures[45]. The latter two approaches may be viewed as iter-atively adjusting appropriately chosen‘‘starting’’phase arrangements until the required phase dis-tribution statistics(which do not have to be uni-form)are achieved.In contrast,RSA schemes in their basic form sequentially add inclusions to a volume by randomly generating candidate inclu-sion positions,which are accepted if a reinforce-ment placed there does not overlap any previously accepted inclusion and are rejected otherwise. Their main drawback is a tendency to show geo-metrical frustration when trying to attain high reinforcement volume fractions,i.e.the process cannot be continued once‘‘free’’positions capa-ble of accommodating further inclusions become scarce or are lacking.All of these approaches can be adapted to handle statistically distributedfiber orientations.The unit cells employed in the present study use arrangements of identical cylindrical,spheroidal or spherical reinforcements that were generated by a sequential adsorption approach(RSA)modified to provide for a user specified minimum distance between neighboring inclusions,for uniformly dis-tributedfiber orientations,and for the periodicity of the volume elements.Whereas the geometri-cal tests necessary for maintaining the specified minimum distance between neighboring inclusions can be implemented in a straightforward way for spherical particles,for nonaligned spheroids(i.e. ellipsoids of rotation)and cylinders checking against violation of this condition can become a fairly complex task[46]that may require consid-erable computational resources.In order to allow a direct assessment offiber shape effects,special phase arrangements were generated which can be used with either spheroidal or cylindricalfibers that occupy the same positions and have the same orientations,aspect ratio and reinforcement volume fraction.Fig.1(a)and(b) show such a matching pair of unit cells,each of which contains15fibers of equal size and aspect ratio a¼5at a total reinforcement volume frac-tion of n¼0:15.In addition,a cell with15iden-tical spherical particles of the same reinforcement volume fraction is displayed in Fig.1(c).In all three cases the minimum distance between neigh-boring inclusions was set at0.0075times the side length of the unit cell(which corresponds toabout Fig.1.Periodic unit cells with randomly positioned reinforcements in the form of(a)15randomly oriented identical spheroidal short fibers(a¼5,arrangement RSFRC/sph),(b)15randomly oriented identical cylindrical shortfibers(a¼5,arrangement RSFRC/cyl) and(c)15identical spherical particles(a¼1,arrangement PRC/sp).The reinforcement volume fraction is n¼0:15in all cases.44H.J.B€o hm et al./Computational Materials Science25(2002)42–535.6%of the radius of the spheres in Fig.1(c)). Fibers intersecting one or more surfaces of the unit cell were split into an appropriate number of parts in accordance with periodicity.The constituents’material data were chosen to correspond to elastic SiCfibers embedded in an elastoplastic matrix of Al2618-T4,which is de-scribed by a J2plasticity model.A modified Lud-wik strain hardening law was used for the matrix, which describes the actualflow stress r y in terms of the initial yield stress r y;0and the accumulated equivalent plastic strain e eqv;p asr y¼r y;0þhðe eqv;pÞn;ð1Þwhere h and n are the hardening coefficient and the hardening exponent,respectively.Initially stress-free constituents and perfect interfacial bonding were assumed throughout the study.The material parameters used in the analyses closely follow those used in[47]and are listed in Table1,where E denotes the Young’s modulus and m the Poisson’s ratio.The unit cells were meshed with10-node tetra-hedra using the preprocessor code PATRAN V.8.0 (MacNeal–Schwendler Corp.,Los Angeles,CA, 1998),element counts approaching100,000.The elastoplastic responses of the unit cells under uni-axial tensile loading were evaluated with thefinite element program ABAQUS/Standard V.5.8(Hib-bitt,Karlsson and Sorensen Inc.,Pawtucket,RI, 1998),load controlled geometrically nonlinear analyses being used.Multi-point displacement constraints were employed to implement the peri-odic boundary conditions[48]and modified tet-rahedral elements(3D10M)were used to avoid volume locking in fully yielded matrix regions.The number of nodes in the unit cells with15randomly positionedfibers exceeded130,000.Although mi-crogeometries with a higher number of reinforce-ments are clearly desirable for describing the complex phase arrangements of actual materials,the above model size approaches the limit of what can be handled at present with workstation-type computers and standard FE packages.Phase averages of the microfields as well as the corresponding standard deviations were evaluated via a feature of ABAQUS that allows volume in-tegrals of functions to be approximated by sum-ming up the appropriate functions values at the integration points weighted by the volumes asso-ciated with the integration points.This algorithm was also used to evaluate Weibull-type fracture probabilities[49]for the reinforcements following the definitionPðjÞfr¼1ÀexpÀ1V0ZVðjÞ:r1>0r1ðxÞr fmd V:ð2ÞHere r1ðxÞstands for the distribution of maximum principal stress within the j th inclusion,VðjÞ: r1>0denotes the part of the inclusion in which r1 is tensile,and V0is a reference volume that was chosen equal to the volume of a single reinforce-ment.The material behavior of thefibers or par-ticles enters via the Weibull modulus m and the characteristic strength r f.The values used for these parameters are listed in Table1.For an in-depth discussion of issues related to using Weibull fracture probabilities for modeling the failure be-havior of brittle inclusions embedded in a ductile matrix,see[50].3.Discussion of resultsIn the following,results pertaining to spherical reinforcements are marked by the prefix PRC and results for composites reinforced by randomly oriented shortfibers of aspect ratio a¼5are marked by the prefix RSFRC.Predictions ob-tained from the matching pair of unit cells shownTable1Material parameters used for the elastoplastic Al2618-T4matrix(modified Ludwik hardening law)and the elastic SiC reinforcements E(GPa)m r y;0(MPa)h(MPa)n m r f(GPa)Al2618-T4matrix700.301847230.49––SiC reinforcement4500.17–––3 1.0H.J.B€o hm et al./Computational Materials Science25(2002)42–5345in Fig.1(a)and(b),which contain15identical randomly oriented spheroidal or cylindricalfibers each,are denoted as RSFRC/sph and RSFRC/cyl, respectively.Data generated with a unit cell con-taining15identical spherical particles,compare Fig.1(c),are designated as PRC/sp and predic-tions obtained from a set of three different unit cells containing15randomly oriented cylindrical fibers each are denoted as RSFRC/3cyl.Unless stated otherwise,results are understood to pertain to ensemble averages over three perpendicular loading directions.3.1.Elastic behaviorIn Table2predictions for the macroscale and, where applicable,microscale elastic responses of statistically isotropic SiC/Al MMCs subjected to uniaxial unit loads are compared.The homoge-nized Young’s moduli are denoted by EÃand the overall Poisson’s ratios by mÃ.To allow an assess-ment of the microfields,the von Mises equivalent stresses r eqv and the maximum principal stresses r1 are given in terms of phase averagesÆstandard deviations for both matrix and reinforcements, which are indicated by superscripts(m)and(r), respectively.In addition to the unit cell predic-tions a number of analytical results are listed.The Hashin–Shtrikman bounds[30]for isotropic com-posites(HSB),which are combined with Zimmer-man’s[51]procedure for bounding mÃ,hold for materials reinforced by particles and by randomly oriented shortfibers.The three-point bounds for materials reinforced by identical noninterpene-trating spheres[52](PRC/3PB)and the generalized self-consistent scheme[53](PRC/GSCS)apply to the particle reinforced unit cell.As specific ap-proaches for materials reinforced by randomly oriented shortfibers,the modified Mori–Tanaka procedure of Benveniste[16](RSFRC/MTM), Berryman’s self-consistent scheme[29](RSFRC/ SCS)and the Kuster–Toks}o z model[26](RSFRC/ KTM)were evaluated.The unit cell predictions for the Young’s moduli and Poisson’s ratios for all configurations studied were found to comply with the Hashin–Shtrikman bounds.The results for both the overall elastic moduli and for the phase averaged microstresses clearly fall into two groups,particle reinforced composites and shortfiber reinforced composites. Within the former group,the unit cell predictions can be seen to comply with the three-point bounds. For the composites reinforced by randomly ori-ented shortfibers,good agreement was found be-tween the unit cell predictions and the analytical results.Whereas no major differences are evident for the microstressfields in the matrix,both the phase averages and widths(described by the standard deviations in Table2)of the distributions of the equivalent and maximum principal stresses in the reinforcements are significantly higher for the nonaligned shortfibers than for the particles. The interpretation of the phase averaged stresses for thefibers,however,is not straightforward be-cause the loads acting on a givenfiber depend markedly on its orientation with respect to theTable2Analytical and numerical predictions for overall and microscale elastic responses of SiC/Al MMCs reinforced by particles or randomly oriented shortfibers(n¼0:15nominal)under a uniaxial tensile load of1MPa(see text for identification of models)EÃ(GPa)mÃrðmÞeqv (MPa)rðmÞ1(MPa)rðrÞeqv(MPa)rðrÞ1(MPa)HSB87.6–106.10.246–0.305––––PRC/3PB87.9–89.20.283–0.287––––PRC/GSCS87.80.2860.890.91 1.64 1.53PRC/sp87.90.2860.90Æ0.160.91Æ0.20 1.69Æ0.20 1.56Æ0.20 RSFRC/MTM89.80.2850.860.88 1.78 1.67 RSFRC/SCS91.20.2840.850.87 1.88 1.76 RSFRC/KTM90.30.2850.860.88 1.81 1.70 RSFRC/sph89.40.2850.89Æ0.160.89Æ0.18 1.86Æ0.46 1.70Æ0.55 RSFRC/cyl90.00.2840.88Æ0.150.89Æ0.18 1.92Æ0.52 1.74Æ0.61 RSFRC/3cyl90.60.2830.87Æ0.150.88Æ0.18 1.98Æ0.58 1.71Æ0.68 46H.J.B€o hm et al./Computational Materials Science25(2002)42–53uniaxial macroscopic load.The stresses in individ-ualfibers closely aligned with the loading direction exceed the above phase averages to a considerable degree,compare Fig.5and the associated discus-sion.In contrast,variations of the average stresses evaluated for identical spherical particles are much smaller[37].By comparing the responses of unit cells to loads acting in different directions the anisotropy of the phase arrangements can be gauged,which, in turn,provides some information on how well the overall isotropy of shortfiber or particle ar-rangements are represented by the unit cells.Such differences in the Young’s moduli amounted to approximately1.1%for model PRC/sp,4.3%for model RSFRC/sph,4.1%for model RSFRC/cyl, and5.0%for the combination of three models with cylindrical reinforcements,RSFRC/3cyl.The good quality of the15-sphere models is in agreementwith numerical results from the literature [37,42,43,54,55]and with analytical estimates on the impact of the size of volume elements on the quality of predictions for the elastic moduli[56]. The somewhat less satisfactory behavior of the unit cells containing randomly oriented shortfi-bers is not surprising in view of the fact that the same number of inclusions is used to model ran-dom distributions of bothfiber position and ori-entation,with the latter having a major impact on the stresses acting on any givenfiber.3.2.Elastoplastic behaviorUnit cell results for the overall stress vs.strain behavior of SiC/Al MMCs subjected to uniaxial tensile loading up to an applied nominal stress of 450MPa are displayed in Fig.2,each of the curves being an average over the responses to loading in the three coordinate directions.Noticeably weaker strain hardening is predicted for the particle rein-forced model than for unit cells employing ran-domly orientedfibers and,among the latter, cylindricalfibers were found to give a considerably stiffer overall response than spheroidal ones.Be-cause the models RSFRC/sph and RSFRC/cyl are based on the same orientations and positions of the inclusions in the unit cell,the differences be-tween the two results are due to the shapes of the fibers.It may be noted that qualitatively similar behavior was reported for MMCs reinforced by aligned spheroidal and cylindricalfibers[57,58].The above trends are reflected in Table3,in which predictions for selected microscalefields in matrix and reinforcements are given.Although a tendency towards somewhat higher loading of the matrix in the particle reinforced model compared to the random shortfiber reinforced casesisTable3Unit cell predictions for the microscale elastoplastic responses of SiC/Al MMCs reinforced by particles or randomly oriented short fibers(n¼0:15nominal)under a uniaxial tensile load of450MParðmÞeqv (GPa)rðmÞ1(GPa)rðmÞm(GPa)eðmÞeqv;p(Â10À2)rðrÞeqv(GPa)rðrÞ1(GPa)PRC/sp0:44Æ0:040:46Æ0:210:17Æ0:1912:77Æ4:381:07Æ0:290:82Æ0:22 RSFRC/sph0:42Æ0:040:43Æ0:170:17Æ0:1610:49Æ4:481:49Æ0:580:96Æ0:54 RSFRC/cyl0:40Æ0:040:41Æ0:190:16Æ0:188:43Æ3:841:58Æ0:781:07Æ0:77 RSFRC/3cyl0:38Æ0:050:41Æ0:190:16Æ0:187:56Æ3:751:71Æ0:781:12Æ0:82H.J.B€o hm et al./Computational Materials Science25(2002)42–5347evident,the predicted phase averages and standard deviations of the equivalent stress r eqv,the maxi-mum principal stress r1and the mean stress r m in the matrix do not differ dramatically between these two groups of microgeometries.Because at this load level the matrix has yielded extensively, however,the small differences in stress translate into markedly different accumulated equivalentplastic strains in the matrix,eðmÞeqv;p .Fig.3compares the distributions of the accu-mulated equivalent plastic strains in the matrix in three parallel section planes as predicted for MMCs reinforced by identical randomly oriented spheroidal or cylindricalfibers(arrangements RSFRC/sph and RSFRC/cyl),respectively,sub-jected to an applied uniaxial load of450MPa acting in y-direction.The highly heterogeneous nature of the plastic strainfields is immediately apparent.In addition,the difference of some20%in the phase average of eðmÞeqv;p between the twofibershapes listed in Table3can be seen to be due to elevated plastic strains throughout the matrix in the case of the model reinforced by randomly oriented spheroids rather than to local effects in the vicinity of the reinforcements.Clear differences in the microstress distributions in the inclusions are evident in Table3when the material reinforced by particulates is compared to the ones containing randomly oriented shortfi-bers.As in the elastic range,thefibers can be seen to be subjected to significantly higher average stresses than particles of the same volume and volume fraction,and thefluctuations in the loads carried by individual inclusions are significantly higher for thefibers than for the particles.In ad-dition,the predictions show a clear tendency for cylindricalfibers to be more highly loaded than spheroidal ones occupying the same positions and having the same orientation as well as comparable geometrical parameters.As a consequence,Wei-bull fracture probabilities under uniaxial loading evaluated according to Eq.(2)are lower for rein-forcement by particles compared to randomly orientedfibers,see Table4,where the average, minimum and maximum fracture probabilities en-countered in the unit cell are given as PðrÞfr;avg ,PðrÞfr;minand PðrÞfr;max ,respectively.Again,markedfluctua-tions between the results for individualfibers are evident,which can be correlated to the orientation of thefibers with respect to the loading direction.Fig.4displays the Weibull fracture probabili-ties under a uniaxial tensile load of450MPa acting in y-direction predicted for directly comparable arrangements of spheroidal or cylindricalfibers. While fracture probabilities in excess of0.9and 0.7,respectively,can be seen to occur in the same fibers in both cases,which indicates the dominant influence offiber orientation,P fr tends to be higher for cylindrical reinforcements in most other cases. Taken together,the results show that the enhanced stiffness of MMCs reinforced by randomly ori-ented shortfibers is obtained by subjecting some of thesefibers to very high loads,giving rise to a tendency towards an increased vulnerability to reinforcement failure compared to particle rein-forced materials.It is well known that the stressfields within a givenfiber in a composite reinforced by non-aligned shortfibers depend strongly on its orien-tation with respect to the applied load[31].For studying such effects,‘‘inclusion averages’’were computed for the stresses acting in each individual fiber by evaluating the appropriate volume aver-ages.Fig.5displays such inclusion averages and the associated standard deviations of the maxi-mum principal stresses as functions of the angle between thefibers and the direction of the applied load evaluated for arrangement RSFRC/sph for loads of1MPa(left)and450MPa(right),i.e.in the elastic and elastoplastic ranges,respectively. Horizontal lines represent the ensemble averages of rðrÞ1listed in Tables2and3as well as the cor-responding standard deviations.As expected the highest loads are carried byfibers that are nearly aligned with the applied uniaxial stress,butfibers subtending high angles with the applied load also show considerable maximum principal stresses and carry appreciable loads;this effect is most marked in the elastic case.The directions of the maximum principal stresses,of course,are in general not aligned with the directions of thefibers.While the general trends shown in Fig.5are in broad agreement with the ideas underlying the Fukuda–Kawata theory and laminate approxi-mation approaches,considerable variations in the inclusion averages betweenfibers subtending sim-48H.J.B€o hm et al./Computational Materials Science25(2002)42–53ilar angles to the loading direction are evident and,especially in the elastoplastic range,the variations of the microstresses within each fiber tend to be substantial.These results indicate that perturba-tions of the stress fields due to the presence of neighboring fibers play a considerable role even at the relatively low fiber volume fraction of n ¼0:15.These effects were found to be less pro-nounced for spheroidal fibers of the same orien-tations,aspect ratio and volume fraction,andFig.4.Predicted Weibull fracture probabilities of the fibers of SiC/Al MMCs ðn ¼0:15Þreinforced by randomly oriented spheroidal (left,arrangement RSFRC/sph)and cylindrical (right,arrangement RSFRC/cyl)short fibers of aspect ratio a ¼5subjected to a uniaxial tensile load of 450MPa acting in y-direction.Table 4Unit cell predictions for Weibull fracture probabilities of the reinforcements in SiC/Al MMCs reinforced by particles or randomly oriented short fibers (n ¼0:15nominal)subjected to a uniaxial tensile load of 450MPaP ðr Þfr ;avgP ðr Þfr ;min P ðr Þfr ;max PRC/sp 0.460.270.94RSFRC/sph 0.510.20 1.0RSFRC/cyl 0.670.25 1.0RSFRC/3cyl0.650.211.0H.J.B €o hm et al./Computational Materials Science 25(2002)42–5349。