Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements

Computational Materials Science 25 (2002) 42–53 www.elsevier.com/locate/commatsci

Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements H.J. B€ ohm *, A. Eckschlager, W. Han

1

Christian Doppler Laboratory for Functionally Oriented Materials Design, Institute of Lightweight Structures and Aerospace Engineering, Vienna University of Technology, Gusshausstrasse 27-29, A-1040 Vienna, Austria

Abstract A multi-inclusion unit cell approach is employed to study the elastic and elastoplastic behavior of metal matrix composites reinforced by randomly oriented short fibers. Periodic arrangements of 15 identical fibers of spheroidal or cylindrical shape with an aspect ratio of 5 and a reinforcement volume fraction of 15% 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 strain fields are evaluated via the finite element method. In addition, a microgeometry containing 15 identical 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 the fibers or particles to brittle fracture. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 07.05.Tp; 62.20.Fe Keywords: Discontinuously reinforced metal matrix composites; Random fiber orientation; Unit cell models; Mechanical properties

1. Introduction Short fiber reinforced metal matrix composites (MMCs)––like other short fiber reinforced materials––may contain aligned, nonaligned or randomly oriented fibers. Most production and processing routes give rise to microgeometries in which the fibers are neither perfectly aligned nor * Corresponding author. Tel.: +43-1-58801-31712; fax: +431-58801-31799. E-mail address: [email protected] (H.J. B€ohm). URL: http://ilfb.tuwien.ac.at. 1 Present address: Alcan Mass Transportation Systems, CH8048 Zurich, Switzerland.

fully random, but show fiber orientation distributions 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 majority 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, modeling the thermoelastic and thermoelastoplastic

0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 4 8 - 3

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responses of aligned short fiber 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 oriented fibers. A number of authors proposed to modify Mori–Tanaka schemes to incorporate 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 extended into the elastoplastic range via secant plasticity schemes, compare e.g. [20,21]. Even though these methods produce reasonable results for many applications (among them elastic two-phase materials with randomly oriented reinforcements), they have the drawback that there are a number of situations in which they generate nonsymmetric elasticity matrices, see e.g. [22,23]. The reason for this behavior lies in the construction of Mori–Tanaka formalisms, which is based on aligned ellipsoidal arrangements of inclusions and is subject to an intrinsic incompatibility with nonaligned reinforcements. The Hashin–Shtrikman procedure of Ponte Casta~ neda and Willis [24] can resolve these difficulties for many microgeometries, but for strongly nonaligned fiber-like inclusions its unconditional 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 mean field description for randomly oriented ellipsoidal inclusions, the Kuster–Toks} oz model [26], is basically a dilute estimate. For recent discussions of the relationships between some of the above models see [27,28]. Berryman’s self consistent 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 thermomechanical 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 thermomechanical responses. Another group of models for describing nonaligned short fiber reinforced composites are based on the assumption that the contribution of a given

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fiber to the overall stiffness and strength depends solely on its orientation with respect to the applied load and on its length, interactions between neighboring fibers 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 one fiber orientation and, where appropriate, one fiber length. Neither of these approaches provides full overall elastic tensors. Planar multi-fiber unit cell models [35,36] for nonaligned composites have achieved an impressive level of development, but by definition introduce strong idealizations in terms of the spatial arrangements and orientations of the fibers. 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 restrictions. Studies using three-dimensional discrete micromodels of nonaligned short fiber reinforced composites have included work in which the finite 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 boundary element methods to study the elastic behavior of materials containing nonaligned reinforcements, see [40,41]. The present state of the art in theoretical descriptions of the thermomechanical behavior of composites reinforced by nonaligned short fibers suggests that the exploration of improved modeling approaches is of considerable interest. One promising way of doing this consists in extending finite-element-based multi-inclusion unit cell models recently developed for particle reinforced MMCs [37,42,43] to fiber-like inclusions. Within such a framework, the case of random fibers is more difficult to handle than other fiber orientation distributions in terms of generating and meshing appropriate microgeometries. Accordingly, the present study concentrates on unit cell models for describing the elastoplastic behavior of

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MMCs reinforced by randomly oriented short fibers.

2. Three-dimensional multi-inclusion unit cell models A 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 iteratively adjusting appropriately chosen ‘‘starting’’ phase arrangements until the required phase distribution statistics (which do not have to be uniform) are achieved. In contrast, RSA schemes in their basic form sequentially add inclusions to a volume by randomly generating candidate inclusion positions, which are accepted if a reinforcement placed there does not overlap any previously accepted inclusion and are rejected otherwise. Their main drawback is a tendency to show geometrical frustration when trying to attain high reinforcement volume fractions, i.e. the process cannot be continued once ‘‘free’’ positions capable of accommodating further inclusions become scarce or are lacking. All of these approaches can be adapted to handle statistically distributed fiber 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 distributed fiber orientations, and for the periodicity of the volume elements. Whereas the geometrical 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 considerable computational resources. In order to allow a direct assessment of fiber shape effects, special phase arrangements were generated which can be used with either spheroidal or cylindrical fibers 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 contains 15 fibers of equal size and aspect ratio a ¼ 5 at a total reinforcement volume fraction of n ¼ 0:15. In addition, a cell with 15 identical spherical particles of the same reinforcement volume fraction is displayed in Fig. 1(c). In all three cases the minimum distance between neighboring inclusions was set at 0.0075 times the side length of the unit cell (which corresponds to about

Fig. 1. Periodic unit cells with randomly positioned reinforcements in the form of (a) 15 randomly oriented identical spheroidal short fibers (a ¼ 5, arrangement RSFRC/sph), (b) 15 randomly oriented identical cylindrical short fibers (a ¼ 5, arrangement RSFRC/cyl) and (c) 15 identical spherical particles (a ¼ 1, arrangement PRC/sp). The reinforcement volume fraction is n ¼ 0:15 in all cases.

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5.6% 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 SiC fibers embedded in an elastoplastic matrix of Al2618-T4, which is described by a J2 plasticity model. A modified Ludwik strain hardening law was used for the matrix, which describes the actual flow stress ry in terms of the initial yield stress ry;0 and the accumulated equivalent plastic strain eeqv;p as n

ry ¼ ry;0 þ hðeeqv;p Þ ;

ð1Þ

where h and n are the hardening coefficient and the hardening exponent, respectively. Initially stressfree 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 Table 1, where E denotes the Young’s modulus and m the Poisson’s ratio. The unit cells were meshed with 10-node tetrahedra using the preprocessor code PATRAN V.8.0 (MacNeal–Schwendler Corp., Los Angeles, CA, 1998), element counts approaching 100,000. The elastoplastic responses of the unit cells under uniaxial tensile loading were evaluated with the finite element program ABAQUS/Standard V.5.8 (Hibbitt, Karlsson and Sorensen Inc., Pawtucket, RI, 1998), load controlled geometrically nonlinear analyses being used. Multi-point displacement constraints were employed to implement the periodic boundary conditions [48] and modified tetrahedral elements (3D10M) were used to avoid volume locking in fully yielded matrix regions. The number of nodes in the unit cells with 15 randomly positioned fibers exceeded 130,000. Although microgeometries with a higher number of reinforcements are clearly desirable for describing the complex phase arrangements of actual materials,

45

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 integrals of functions to be approximated by summing up the appropriate functions values at the integration points weighted by the volumes associated with the integration points. This algorithm was also used to evaluate Weibull-type fracture probabilities [49] for the reinforcements following the definition
 m  Z 1 r1 ðxÞ ðjÞ Pfr ¼ 1  exp  dV : V0 V ðjÞ :r1 >0 rf ð2Þ Here r1 ðxÞ stands for the distribution of maximum principal stress within the jth inclusion, V ðjÞ : r1 > 0 denotes the part of the inclusion in which r1 is tensile, and V0 is a reference volume that was chosen equal to the volume of a single reinforcement. The material behavior of the fibers or particles enters via the Weibull modulus m and the characteristic strength rf . The values used for these parameters are listed in Table 1. For an indepth discussion of issues related to using Weibull fracture probabilities for modeling the failure behavior of brittle inclusions embedded in a ductile matrix, see [50].

3. Discussion of results In the following, results pertaining to spherical reinforcements are marked by the prefix PRC and results for composites reinforced by randomly oriented short fibers of aspect ratio a ¼ 5 are marked by the prefix RSFRC. Predictions obtained from the matching pair of unit cells shown

Table 1 Material parameters used for the elastoplastic Al2618-T4 matrix (modified Ludwik hardening law) and the elastic SiC reinforcements

Al2618-T4 matrix SiC reinforcement

E (GPa)

m

ry;0 (MPa)

h (MPa)

n

m

rf (GPa)

70 450

0.30 0.17

184 –

723 –

0.49 –

– 3

– 1.0

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in Fig. 1(a) and (b), which contain 15 identical randomly oriented spheroidal or cylindrical fibers each, are denoted as RSFRC/sph and RSFRC/cyl, respectively. Data generated with a unit cell containing 15 identical spherical particles, compare Fig. 1(c), are designated as PRC/sp and predictions obtained from a set of three different unit cells containing 15 randomly 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 behavior In Table 2 predictions for the macroscale and, where applicable, microscale elastic responses of statistically isotropic SiC/Al MMCs subjected to uniaxial unit loads are compared. The homogenized Young’s moduli are denoted by E and the overall Poisson’s ratios by m . To allow an assessment of the microfields, the von Mises equivalent stresses reqv and the maximum principal stresses r1 are given in terms of phase averages  standard deviations for both matrix and reinforcements, which are indicated by superscripts (m) and (r), respectively. In addition to the unit cell predictions a number of analytical results are listed. The Hashin–Shtrikman bounds [30] for isotropic composites (HSB), which are combined with Zimmerman’s [51] procedure for bounding m , hold for materials reinforced by particles and by randomly oriented short fibers. The three-point bounds for

materials reinforced by identical noninterpenetrating spheres [52] (PRC/3PB) and the generalized self-consistent scheme [53] (PRC/GSCS) apply to the particle reinforced unit cell. As specific approaches for materials reinforced by randomly oriented short fibers, the modified Mori–Tanaka procedure of Benveniste [16] (RSFRC/MTM), Berryman’s self-consistent scheme [29] (RSFRC/ SCS) and the Kuster–Toks} oz 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 short fiber 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 oriented short fibers, good agreement was found between the unit cell predictions and the analytical results. Whereas no major differences are evident for the microstress fields in the matrix, both the phase averages and widths (described by the standard deviations in Table 2) of the distributions of the equivalent and maximum principal stresses in the reinforcements are significantly higher for the nonaligned short fibers than for the particles. The interpretation of the phase averaged stresses for the fibers, however, is not straightforward because the loads acting on a given fiber depend markedly on its orientation with respect to the

Table 2 Analytical and numerical predictions for overall and microscale elastic responses of SiC/Al MMCs reinforced by particles or randomly oriented short fibers (n ¼ 0:15 nominal) under a uniaxial tensile load of 1 MPa (see text for identification of models)

HSB PRC/3PB PRC/GSCS PRC/sp RSFRC/MTM RSFRC/SCS RSFRC/KTM RSFRC/sph RSFRC/cyl RSFRC/3cyl

ðmÞ

ðrÞ

E (GPa)

m

rðmÞ eqv (MPa)

r1

(MPa)

rðrÞ eqv (MPa)

r1 (MPa)

87.6–106.1 87.9–89.2 87.8 87.9 89.8 91.2 90.3 89.4 90.0 90.6

0.246–0.305 0.283–0.287 0.286 0.286 0.285 0.284 0.285 0.285 0.284 0.283

– – 0.89 0.90  0.16 0.86 0.85 0.86 0.89  0.16 0.88  0.15 0.87  0.15

– – 0.91 0.91  0.20 0.88 0.87 0.88 0.89  0.18 0.89  0.18 0.88  0.18

– – 1.64 1.69  0.20 1.78 1.88 1.81 1.86  0.46 1.92  0.52 1.98  0.58

– – 1.53 1.56  0.20 1.67 1.76 1.70 1.70  0.55 1.74  0.61 1.71  0.68

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uniaxial macroscopic load. The stresses in individual fibers closely aligned with the loading direction exceed the above phase averages to a considerable degree, compare Fig. 5 and the associated discussion. 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 short fiber or particle arrangements are represented by the unit cells. Such differences in the Young’s moduli amounted to approximately 1.1% for model PRC/sp, 4.3% for model RSFRC/sph, 4.1% for model RSFRC/cyl, and 5.0% for the combination of three models with cylindrical reinforcements, RSFRC/3cyl. The good quality of the 15-sphere models is in agreement with 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 short fibers is not surprising in view of the fact that the same number of inclusions is used to model random distributions of both fiber position and orientation, with the latter having a major impact on the stresses acting on any given fiber. 3.2. Elastoplastic behavior Unit 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 450 MPa are displayed in Fig. 2, each of the curves being an average over the responses to loading in

47

Fig. 2. Predicted overall uniaxial tensile stress–strain responses of SiC/Al MMCs ðn ¼ 0:15Þ reinforced by spherical particles (PRC/sp, dash-dotted line), randomly oriented spheroidal short fibers of aspect ratio a ¼ 5 (RSFRC/sph, solid line) and randomly oriented cylindrical short fibers of aspect ratio a ¼ 5 (RSFRC/cyl, dotted line).

the three coordinate directions. Noticeably weaker strain hardening is predicted for the particle reinforced model than for unit cells employing randomly oriented fibers and, among the latter, cylindrical fibers were found to give a considerably stiffer overall response than spheroidal ones. Because the models RSFRC/sph and RSFRC/cyl are based on the same orientations and positions of the inclusions in the unit cell, the differences between 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 cylindrical fibers [57,58]. The above trends are reflected in Table 3, in which predictions for selected microscale fields 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 short fiber reinforced cases is

Table 3 Unit cell predictions for the microscale elastoplastic responses of SiC/Al MMCs reinforced by particles or randomly oriented short fibers (n ¼ 0:15 nominal) under a uniaxial tensile load of 450 MPa

PRC/sp RSFRC/sph RSFRC/cyl RSFRC/3cyl

ðmÞ

ðrÞ

rðmÞ eqv (GPa)

r1

(GPa)

rðmÞ m (GPa)

2 eðmÞ eqv;p (10 )

rðrÞ eqv (GPa)

r1 (GPa)

0:44  0:04 0:42  0:04 0:40  0:04 0:38  0:05

0:46  0:21 0:43  0:17 0:41  0:19 0:41  0:19

0:17  0:19 0:17  0:16 0:16  0:18 0:16  0:18

12:77  4:38 10:49  4:48 8:43  3:84 7:56  3:75

1:07  0:29 1:49  0:58 1:58  0:78 1:71  0:78

0:82  0:22 0:96  0:54 1:07  0:77 1:12  0:82

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evident, the predicted phase averages and standard deviations of the equivalent stress reqv , the maximum principal stress r1 and the mean stress rm 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 equivalent plastic strains in the matrix, eðmÞ eqv;p . Fig. 3 compares the distributions of the accumulated equivalent plastic strains in the matrix in three parallel section planes as predicted for MMCs reinforced by identical randomly oriented spheroidal or cylindrical fibers (arrangements RSFRC/sph and RSFRC/cyl), respectively, subjected to an applied uniaxial load of 450 MPa acting in y-direction. The highly heterogeneous nature of the plastic strain fields is immediately apparent. In addition, the difference of some 20% in the phase average of eðmÞ eqv;p between the two fiber shapes listed in Table 3 can 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 Table 3 when the material reinforced by particulates is compared to the ones containing randomly oriented short fibers. As in the elastic range, the fibers can be seen to be subjected to significantly higher average stresses than particles of the same volume and volume fraction, and the fluctuations in the loads carried by individual inclusions are significantly higher for the fibers than for the particles. In addition, the predictions show a clear tendency for cylindrical fibers 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, Weibull fracture probabilities under uniaxial loading evaluated according to Eq. (2) are lower for reinforcement by particles compared to randomly oriented fibers, see Table 4, where the average, minimum and maximum fracture probabilities enðrÞ ðrÞ countered in the unit cell are given as Pfr;avg , Pfr;min ðrÞ and Pfr;max , respectively. Again, marked fluctuations between the results for individual fibers are

evident, which can be correlated to the orientation of the fibers with respect to the loading direction. Fig. 4 displays the Weibull fracture probabilities under a uniaxial tensile load of 450 MPa acting in y-direction predicted for directly comparable arrangements of spheroidal or cylindrical fibers. While fracture probabilities in excess of 0.9 and 0.7, respectively, can be seen to occur in the same fibers in both cases, which indicates the dominant influence of fiber orientation, Pfr 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 oriented short fibers is obtained by subjecting some of these fibers to very high loads, giving rise to a tendency towards an increased vulnerability to reinforcement failure compared to particle reinforced materials. It is well known that the stress fields within a given fiber in a composite reinforced by nonaligned short fibers depend strongly on its orientation 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 averages. Fig. 5 displays such inclusion averages and the associated standard deviations of the maximum principal stresses as functions of the angle between the fibers and the direction of the applied load evaluated for arrangement RSFRC/sph for loads of 1 MPa (left) and 450 MPa (right), i.e. in the elastic and elastoplastic ranges, respectively. Horizontal lines represent the ensemble averages ðrÞ of r1 listed in Tables 2 and 3 as well as the corresponding standard deviations. As expected the highest loads are carried by fibers that are nearly aligned with the applied uniaxial stress, but fibers 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 the fibers. While the general trends shown in Fig. 5 are in broad agreement with the ideas underlying the Fukuda–Kawata theory and laminate approximation approaches, considerable variations in the inclusion averages between fibers subtending sim-

H.J. B€ohm et al. / Computational Materials Science 25 (2002) 42–53

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Fig. 3. Predicted distributions of the accumulated equivalent plastic strain eðmÞ eqv;p in the matrix of SiC/Al MMCs ðn ¼ 0:15Þ reinforced by randomly oriented spheroidal (left, arrangement RSFRC/sph) and cylindrical (right, arrangement RSFRC/cyl) short fibers of aspect ratio a ¼ 5 subjected to a uniaxial load of 450 MPa acting in y-direction.

Table 4 Unit cell predictions for Weibull fracture probabilities of the reinforcements in SiC/Al MMCs reinforced by particles or randomly oriented short fibers (n ¼ 0:15 nominal) subjected to a uniaxial tensile load of 450 MPa ðrÞ

PRC/sp RSFRC/sph RSFRC/cyl RSFRC/3cyl

ðrÞ

ðrÞ

Pfr;avg

Pfr;min

Pfr;max

0.46 0.51 0.67 0.65

0.27 0.20 0.25 0.21

0.94 1.0 1.0 1.0

ilar 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 perturbations 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 ¼ 0:15. These effects were found to be less pronounced for spheroidal fibers of the same orientations, aspect ratio and volume fraction, and

Fig. 4. Predicted Weibull fracture probabilities of the fibers of SiC/Al MMCs ðn ¼ 0:15Þ reinforced by randomly oriented spheroidal (left, arrangement RSFRC/sph) and cylindrical (right, arrangement RSFRC/cyl) short fibers of aspect ratio a ¼ 5 subjected to a uniaxial tensile load of 450 MPa acting in y-direction.

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Fig. 5. Predicted averages of the maximum principal stress in the individual fibers of arrangement RSFRC/cyl (15 randomly oriented cylindrical short fibers, n ¼ 0:15) at uniaxial tensile loads of 1 MPa (left) and 450 MPa (right) plotted as functions of the angles subtended between fibers and loading direction. The horizontal lines represent the ensemble averages and standard deviations listed in Tables 2 and 3, respectively.

much smaller variations were again predicted for arrangements of spherical particles. Interestingly, in the elastoplastic range the mean stress in fibers subtending angles of more than approximately 65° with the applied uniaxial loads was found to be compressive. Such behavior is typically associated with generalized plane strain analyses used for studying transversely loaded composites containing continuous aligned fibers, see e.g. [37]. Generally, it may be concluded from Fig. 5 that phase averages of microstresses, which are quite useful in assessing the predictions of unit cell models for materials reinforced by particles or aligned fibers, are a somewhat blunt instrument for describing the microfields in composites reinforced by nonaligned fibers. In a similar vein, extreme caution must be exercised in using phase averaged stresses and strains obtained by mean field methods for studying reinforcement failure in composites with randomly oriented fibers, because the averages tend to be dominated by fibers subtending a high angle to the loading direction, whereas a small number of highly loaded fibers are subjected to much more demanding conditions. The tendency for the predicted overall behavior obtained from unit cells containing 15 randomly oriented fibers to depend on the loading direction, which was discussed for the elastic behavior, was evident to a considerably stronger degree in the elastoplastic range. Together with the deviations

between results obtained from one unit cell, RSFRC/cyl and the ensemble average from three fiber arrangements, RSFRC/3cyl, this indicates that studies involving microgeometries containing higher numbers of randomly oriented fibers are highly desirable to improve the statistical significance of the present results.

4. Conclusions A unit cell based approach for studying the elastic and elastoplastic response of discontinuously reinforced statistically isotropic MMCs at moderate reinforcement volume fractions and fiber aspect ratios was presented. It uses RSA algorithms to generate periodic microgeometries that correspond to composites reinforced by randomly oriented identical short fibers of spheroidal or cylindrical shape or by identical spherical particles. The finite element method was employed for evaluating the microscale stresses and strains in these phase arrangements. Clear differences between the overall and microscale responses of randomly short fiber reinforced materials on the one hand and particle reinforced composites on the other hand were demonstrated. Direct assessments of fiber shape effects were carried out and it was shown that an assumption used by some micromechanical models, viz. that the contributions to stiffness and strength of composites depend only

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on the angle between fiber and loading direction, may give rise to relatively coarse approximations, especially for elastoplastic matrix behavior. The approach proposed above can be directly applied to other thermomechanical load cases and is well suited for studying fiber arrangements with general orientation distribution and aspect ratio distribution functions. Extensions to higher reinforcement volume fractions, higher fiber aspect ratios, and more irregular fiber shapes appear to be perfectly feasible with refined methods. The main drawback of the present modeling strategy is its restriction to a low number of reinforce-ments per unit cell on account of the substantial computational requirements, which limits the statistical representativeness of the phase arrangements. Acknowledgements The authors wish to acknowledge the financial support of the Christian Doppler Research Society and numerous discussions with Prof. F.G. Rammerstorfer. References [1] L. Nguyen Thanh, M. Suery, Microstructure and compression behavior in the semisolid state of short-fibrereinforced A356 aluminium alloys, Mater. Sci. Engng. A196 (1995) 33–44. [2] S.Q. Wu, Z.S. Wei, S.C. Tjong, The mechanical and thermal expansion behavior of an Al–Si alloy composite reinforced with potassium titanate whiskers, Compos. Sci. Technol. 60 (2000) 2873–2880. [3] D.H. Allen, J.W. Lee, The effective thermoelastic properties of whisker-reinforced composites as functions of material forming parameters, in: G.J. Weng, M. Taya, H. Abe (Eds.), Micromechanics and Inhomogeneity, Springer, New York, 1990, pp. 17–40. [4] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc. London A 241 (1957) 376–396. [5] P. Ponte Casta~ neda, P. Suquet, Nonlinear composites, in: E. van der Giessen, T.Y. Wu (Eds.), Advances in Applied Mechanics 34, Academic Press, New York, 1998, pp. 171– 302. [6] G.P. Tandon, G.J. Weng, The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites, Polym. Compos. 5 (1984) 327–333.

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[7] P.J. Withers, W.M. Stobbs, O.B. Pedersen, The application of the Eshelby method of internal stress determination to short fibre metal matrix composites, Acta Metall. 37 (1989) 3061–3084. [8] G. Li, P. Ponte Casta~ neda, The effect of particle shape and stiffness on the constitutive behavior of metal matrix composites, Int. J. Solids Struct. 30 (1993) 3189– 3209. [9] D.C. Lagoudas, A.C. Gavazzi, H. Nigam, Elastoplastic behavior of metal matrix composites based on incremental plasticity and the Mori–Tanaka averaging scheme, Comput. Mech. 8 (1991) 193–203. [10] J.L. Lorca, A. Needleman, S. Suresh, An analysis of the effects of matrix void growth on deformation and ductility in metal ceramic composites, Acta Metall. Mater. 39 (1991) 2317–2335. [11] H.J. B€ ohm, F.G. Rammerstorfer, E. Weissenbek, Some simple models for micromechanical investigations of fiber arrangement effects in MMCs, Comput. Mater. Sci. 1 (1993) 177–194. [12] A. Levy, J.M. Papazian, Elastoplastic finite element analysis of short-fiber-reinforced SiC/Al composites. Effects of thermal treatment, Acta Metall. Mater. 39 (1991) 2255–2266. [13] M.S. Ingber, T.D. Papathanasiou, A parallel-supercomputing investigation of the stiffness of aligned short-fiberreinforced composites using the boundary element method, Int. J. Num. Meth. Engng. 40 (1997) 3477–3491. [14] L.J. Walpole, On bounds for the overall elastic moduli of inhomogeneous systems––I, J. Mech. Phys. Sol. 14 (1966) 151–162. [15] J.R. Willis, Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. Mech. Phys. Sol. 25 (1977) 185–202. [16] Y. Benveniste, A new approach to the application of Mori– Tanaka’s theory in composite materials, Mech. Mater. 6 (1987) 147–157. [17] Y. Takao, T.W. Chou, M. Taya, Effective longitudinal Young’s modulus of misoriented short fiber composites, J. Appl. Mech. 49 (1982) 536–540. [18] H.E. Pettermann, H.J. B€ ohm, F.G. Rammerstorfer, Some direction dependent properties of matrix–inclusion type composites with given reinforcement orientation distributions, Composites 28B (1997) 253–265. [19] B. Mlekusch, Thermoelastic properties of short-fibre-reinforced thermoplastics, Compos. Sci. Technol. 59 (1999) 911–923. [20] A. Bhattacharyya, G.J. Weng, The elastoplastic behavior of a class of two-phase composites containing rigid inclusions, Appl. Mech. Rev. 47 (1994) S45–S65. [21] M.L. Dunn, H. Ledbetter, Elastic–plastic behavior of textured short-fiber composites, Acta Mater. 45 (1997) 3327–3340. [22] Y. Benveniste, Some remarks on three micromechanical models in composite media, J. Appl. Mech. 57 (1990) 474– 476.

52

H.J. B€ohm et al. / Computational Materials Science 25 (2002) 42–53

[23] M. Ferrari, Asymmetry and the high concentration limit of the Mori–Tanaka effective medium theory, Mech. Mater. 11 (1991) 251–256. [24] P. Ponte Casta~ neda, J.R. Willis, The effect of spatial distribution on the effective behavior of composite materials and cracked media, J. Mech. Phys. Sol. 43 (1995) 1919–1951. [25] M. Hori, S. Nemat-Nasser, Double-inclusion model and overall moduli of multi-phase composites, Mech. Mater. 14 (1993) 189–206. [26] G.T. Kuster, M.N. Toks} oz, Velocity and attenuation of seismic waves in two-phase media: I. Theoretical formulation, Geophysics 39 (1974) 587–606. [27] J.G. Berryman, P.A. Berge, Critique of two explicit schemes for estimating elastic properties of multiphase composites, Mech. Mater. 22 (1996) 149–164. [28] G.K. Hu, G.J. Weng, The connections between the doubleinclusion model and the Ponte Casta~ neda–Willi, Mori– Tanaka, Mori–Tanaka, and Kuster–Toksoz models, Mech. Mater. 32 (2000) 495–503. [29] J.G. Berryman, Long-wavelength propagation in composite elastic media, II. Ellipsoidal inclusions, J. Acoust. Soc. Am. 68 (1980) 1820–1831. [30] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Sol. 11 (1963) 127–140. [31] H. Fukuda, K. Kawata, On Young’s modulus of short fiber composites, Fibre Sci. Technol. 7 (1974) 207–222. [32] K. Jayaraman, M.T. Kortschot, Correction to the Fukuda–Kawata Young’s modulus theory and the Fukuda– Chou strength theory for short fibre-reinforced composite materials, J. Mater. Sci. 31 (1996) 2059–2064. [33] J.C. Halpin, N.J. Pagano, The laminate approximation for randomly oriented fibrous composites, J. Compos. Mater. 3 (1969) 720–724. [34] S.Y. Fu, B. Lauke, The elastic modulus of misaligned short-fiber-reinforced polymers, Compos. Sci. Technol. 58 (1998) 389–400. [35] W.M.G. Courage, P.J.G. Schreurs, Effective material parameters for composites with randomly oriented short fibers, Comput. Struct. 44 (1992) 1179–1185. [36] S. Ghosh, Z. Nowak, K.H. Lee, Tesselation-based computational methods for the characterization and analysis of heterogeneous microstructures, Compos. Sci. Technol. 57 (1997) 1187–1210. [37] H.J. B€ ohm, W. Han, Comparisons between three-dimensional and two-dimensional multi-particle unit cell models for particle reinforced MMCs, Modell. Simul. Mater. Sci. Engng. 9 (2001) 47–65. [38] M. Dong, S. Schmauder, T. Bidlingmaier, A. Wanner, Prediction of mechanical behavior of short fiber reinforced MMCs by combined cell models, Comput. Mater. Sci. 9 (1997) 121–133. [39] N.J. Sørensen, A. Needleman, V. Tvergaard, Threedimensional analysis of creep in a metal–matrix composite, Mater. Sci. Engng. A158 (1992) 129–137.

[40] P.K. Banerjee, D.P. Henry, Elastic analysis of threedimensional solids with fiber inclusions by BEM, Int. J. Sol. Struct. 29 (1992) 2423–2440. [41] M.B. Bush, An investigation of two- and three-dimensional models for predicting the elastic properties of particulateand whisker-reinforced composite materials, Mater. Sci. Engng. A154 (1992) 139–148. [42] H.J. B€ ohm, A. Eckschlager, W. Han, Modeling of phase arrangement effects in high speed tool steels, in: F. Jeglitsch, R. Ebner, H. Leitner (Eds.), Tool Steels in the Next Century, Montanuniversit€at Leoben, 1999, pp. 147– 156. [43] W. Han, A. Eckschlager, H.J. B€ ohm, Three-dimensional multi-particle arrangement effects on the mechanical behavior and damage initiation of particle reinforced composites, Compos. Sci. Technol. 61 (2001) 1581–1590. [44] A.A. Gusev, P.J. Hine, I.M. Ward, Fiber packing and elastic properties of a transversely random unidirectional glass/epoxy composite, Compos. Sci. Technol. 60 (2000) 535–541. [45] M. Rintoul, S. Torquato, Reconstruction of the structure of dispersions, J. Colloid Interf. Sci. 186 (1997) 467– 476. [46] T. Antretter, Micromechanical Modeling of High Speed Steel, VDI–Verlag, Reihe 18, Nr. 232, D€ usseldorf, 1998. [47] J. LLorca, C. Gonzalez, Microstructural factors controlling the strength and ductility of particle reinforced metal matrix composites, J. Mech. Phys. Sol. 46 (1998) 1–28. [48] H.J. B€ ohm, A Short Introduction to Basic Aspects of Continuum Micromechanics, CDL–FMD Report 3–1998, TU Wien, Vienna, 1998. [49] W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech. 18 (1951) 293–297. [50] K. Wallin, T. Saario, K. T€ orr€ onen, Fracture of brittle particles in a ductile matrix, Int. J. Fract. 32 (1987) 201– 209. [51] R.W. Zimmerman, Hashin–Shtrikman bounds on the Poisson ratio of a composite material, Mech. Res. Comm. 19 (1992) 563–569. [52] S. Torquato, Random heterogeneous media: Microstructure and improved bounds on effective properties, Appl. Mech. Rev. 44 (1991) 37–75. [53] R.M. Christensen, K.H. Lo, Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Sol. 27 (1979) 315–330. [54] A.A. Gusev, Representative volume element size for elastic composites: A numerical study, J. Mech. Phys. Sol. 45 (1997) 1449–1459. [55] T.I. Zohdi, A model for simulating the deterioration of structural-scale material responses of microheterogeneous solids, in: D. Gross, F.D. Fischer, E. van der Giessen (Eds.), Proceedings of the Euromech Colloqium 402––Micromechanics of Fracture Processes, TU Darmstadt, 1999, pp. 91–92. [56] J. Drugan, J.R. Willis, A micromechanics-based nonlocal constitutive equation and estimates of representative vol-

H.J. B€ohm et al. / Computational Materials Science 25 (2002) 42–53 ume element size for elastic composites, J. Mech. Phys. Sol. 44 (1996) 497–524. [57] G. Bao, J.W. Hutchinson, R.M. McMeeking, Particle reinforcement of ductile matrices against plastic flow and creep, Acta Metall. Mater. 39 (1991) 1871–1882.

53

[58] G.L. Povirk, M.G. Stout, M. Bourke, J.A. Goldstone, A.C. Lawson, M. Lovato, S.R. MacEwen, S.R. Nutt, A. Needleman, Thermally and mechanically induced residual stresses in Al–SiC composites, Acta. Metall. Mater. 40 (1992) 2391–2412.