COMPUTATIONAL MECHANICS - uni- · PDF file7 Jun 2002 8:37 AR AR162-17.tex AR162-17.SGM LaTeX2e(2002/01/18) P1: GJB COMPUTATIONAL MECHANICS 439 Figure 2 Numerical methods referring

7 Jun 2002 8:37 AR AR162-17.tex AR162-17.SGM LaTeX2e(2002/01/18)P1: GJB10.1146/annurev.matsci.32.103101.153157

Annu. Rev. Mater. Res. 2002. 32:437–65doi: 10.1146/annurev.matsci.32.103101.153157

Copyright c© 2002 by Annual Reviews. All rights reserved

COMPUTATIONAL MECHANICS

Siegfried SchmauderStaatliche Materialprufungsanstalt (MPA) University of Stuttgart, Pfaffenwaldring 32,D-70569 Stuttgart, Germany; e-mail: [email protected]

Key Words multiscale modeling, finite element method, composites,micromechanical modeling, deformation analyses

■ Abstract Computational mechanics comprises all types of computer modelingof the mechanical behavior of materials. In this contribution we concentrate on newdevelopments in modeling based on the finite element method (FEM), especially de-formation analyses based on numerical homogenization techniques (self-consistentembedding procedure, matricity model), simulations of real microstructural cut-outs,damage analyses of artificial and real microstructures, and multiscale modeling as-pects. The limit flow stresses for transverse loading of metal matrix composites rein-forced with continuous fibers and for uniaxial loading of spherical particle reinforcedmetal matrix composites are investigated by recently developed embedded cell mod-els in conjunction with the finite element method. A fiber of circular cross sectionor a spherical particle is surrounded by a metal matrix, which is again embedded inthe composite material, with the mechanical behavior to be determined iteratively ina self-consistent manner. Stress-strain curves have been calculated for a number ofmetal matrix composites with the embedded cell method and verified with literaturedata of a particle reinforced Ag/58vol.%Ni composite and for a transversely loaded uni-axially fiber reinforced Al/46vol.%B composite. Good agreement has been obtainedbetween experiment and calculation, and the embedded cell model is thus found towell represent metal matrix composites with randomly arranged inclusions. System-atic studies of the mechanical behavior of fiber- and particle-reinforced compositeswith plane strain and axisymmetric embedded cell models are carried out to determinethe influence of fiber or particle volume fraction and matrix strain-hardening ability oncomposite strengthening levels. Results for random inclusion arrangements obtainedwith self-consistent embedded cell models are compared with strengthening levels forregular inclusion arrangements from conventional unit cell models. It is found that withincreasing inclusion volume fractions pronounced differences in composite strength-ening exist between all models. Finally, closed-form expressions are derived to predictcomposite strengthening for regular fiber arrangements and for realistic random fiber orparticle arrangements as a function of matrix hardening and particle volume fraction.The impact of the results on effectively designing technically relevant metal matrixcomposites reinforced by randomly arranged strong inclusions is emphasized. Atom-istic modeling such as Monte Carlo (MC) simulations and molecular dynamics (MD)methods, dislocation theoretical modeling, and continuum mechanical methods are ap-plied in order to provide insight into the mechanical behavior of materials. Simulationsare presented graphically in a systematic manner for different material systems and are

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compared with experimental results. Finally, it will be shown that the results can beused to predict the future behavior of materials presently in service and even to designnew materials.

INTRODUCTION

The microstructure (fixed by features such as the grain size, the presence of parti-cles, layers/coatings, impurities, and internal interfaces) frequently determines theproperties (such as strength, toughness, and fracture energy) of advanced materials.Because the microstructure is determined during production (by parameters suchas temperature, time, and pressure) and the properties are determined in applica-tion (e.g., implementation and life-time), the microstructure/property-relationship(Figure 1) has been the focus of many investigations in the recent past. This rela-tionship especially requires knowledge of how to link the different length scalesin modeling and characterizing these materials.

The linkage of modeling on the nanoscale (nm-length scale) with that on themacroscale (length scale of real specimens) is a current challenge in materialsscience. Despite cheaper and faster computer resources, it is still difficult to in-flate atomistic models of crystals to the size of specimens and components. Spe-cial problems of crack propagation have been successfully treated by dedicatedcoupled atomistic continuum methods (1, 2). Nevertheless, a general method toconnect theoretical calculations on the nanoscale with the macroscale is not avail-able. Instead, at present it seems more promising to restrict oneself to selectedmaterials problems and to connect different length scales by qualified physicallaws that describe corresponding materials behavior (3, 4). The terminology ofrelevant length scales, together with referring physical phenomena and methods,is illustrated in Figure 2 (5). These methods require knowledge on the atomisticlevel.

Figure 1 Microstructure/property-relationship.


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Figure 2 Numerical methods referring to different length scales.

Particulate-reinforced composites have become increasingly attractive in recentyears for their high-strength and creep-resistant properties. These materials canbe divided into three basic groups: (a) metal matrix composites (MMCs) suchas Al reinforced with SiC-particles or with B-fibers, (b) brittle matrix composites(BMCs) such as Al2O3 toughened with Al, and (c) interpenetrating microstructuralcomposites (IMCs) such as WC toughened with Co and others. Figure 3 depictsexamples of fiber and particulate-reinforced composites. It should be noted thatwhen continuum mechanical methods are applied, the relevant microstructural

Figure 3 Typical microstructures to be modelled in computational mechanics.(a) Al/B f fiber-reinforced MMC, (b) Ag/58.4vol.%Ni particle-reinforced MMC.


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features such as particle sizes or phase areas have to be large enough (in themicrometer regime or above) in order to be modeled appropriately.

Deformation Analysis of Metal Matrix Composites (MMCs)1

Problem

Metal matrix composites (MMCs) with strong inclusions are a relatively new classof materials, which due to a high strength/weight ratio, are potentially valuable inaerospace and transportation applications (6).

MMCs are defined as ductile matrix materials reinforced by brittle fibers orparticles. MMCs are frequently reinforced by continuous fibers that are aligned inorder to make use of the high axial fiber strength. However, the mechanical behaviorof these composites under transverse loading is less than their axial performance(7–12). On the other hand, it is well known, both from experiment and calculations(13) that details of transverse strengthening in uniaxially fiber-reinforced MMCsare a strong function of the fiber arrangement. Under external loading, the overallresponse of MMCs is elastic-plastic. To derive the mechanical behavior of MMCs,a micromechanical approach is usually employed using cell models, representingregular inclusion arrangements. As regular fiber spacings are difficult to achieve inpractice, most of the present fiber-reinforced MMCs contain aligned but randomlyarranged continuous fibers. Thus the accurate modeling of the mechanical behaviorof actual MMCs is very complicated in practice even if the fibers are aligned.

Initially, the transverse mechanical behavior of a unidirectionally continuousfiber-reinforced composite (Al-B) with fibers of circular cross section was studiedby Adams (14) by adopting finite element cell models under plane strain con-ditions: A simple geometrical cell composed of matrix and inclusion material isrepeated by appropriate boundary conditions to represent a composite with a pe-riodic microstructure. The influence of different regular fiber arrangements on thestrength of transversely loaded B-fiber reinforced Al was analyzed in References(7, 10, 12). It was found that the same square arrangement of fibers represents twoextremes of strengthening: high-strength levels are achieved if the composite isloaded in a 0◦ direction of nearest neighbors, whereas the 45◦-loading directionis found to be weak for the same fiber arrangement. A regular hexagonal fiber ar-rangement lies somewhere between these limits (7, 10, 12, 15, 16). The transversemechanical behavior of a realistic fiber-reinforced composite containing about30 randomly arranged fibers was best described but significantly underestimatedby the hexagonal fiber model (10). The reason for the superiority of the hex-agonal over the square arrangement in describing random fiber arrangements liesin the fact that the stress invariants of the square arrangement agree only to firstorder with the invariants of the transversely isotropic material whereas the hexa-gonal arrangement does so up to the second order (15). Dietrich (11), using fibers

1Partly reprinted with permission from M. Dong, S. Schmauder. 1996Acta Mater. 44:2468–78.



of different diameters, numerically made a transversely isotropic square fiber-reinforced Ag/Ni composite material. A systematic study in which the fiber vol-ume fraction and the fiber arrangement effects were investigated resulted in thecreation of the simple model as seen in Reference (16).

The influence of fiber shape and clustering was numerically examined in somedetail by Llorca et al. (17), Dietrich (11), and Sautter (18). It was observed thatfaceted fiber cross sections lead to higher strengths compared with circular crosssections, except for such fibers that have facets predominantly oriented by 45◦ withrespect to the loading axis, in close agreement with findings in particle-reinforcedMMCs (19). Thus hindering of shear band formation within the matrix was foundto be responsible for strengthening with respect to fiber arrangement and fibershape (17). In References (13, 16, 20, 21), local distributions of stresses and strainswithin the microstructure have been identified as also strongly influenced by thearrangement of fibers. However, no agreement was found between the mechanicalbehavior of composites based on cell models with differently arranged fibers andexperiments with randomly arranged fibers loaded in transverse direction.

The overall mechanical behavior of a particle-reinforced composite, studiedwith axisymmetric finite element cell models by Bao et al. (22), represented uni-form particle distributions within an elastic-plastic matrix. Tvergaard (23) intro-duced a modified cylindrical unit cell containing one half of a single fiber to modelthe axial performance of a periodic square arrangement of staggered short fibers.Hom (24) and Weissenbek (25) have used three-dimensional finite elements tomodel different regular arrangements of short fibers and spherical as well as cylin-drical particles with relatively small volume fractions (f< 0.2). It was generallyfound that the arrangement of fibers strongly influences the different overall behav-ior of composites. When short fibers are arranged in a side-by-side manner, theyconstrain the plastic flow in the matrix, and the computed stress-strain responseof the composite in the fiber direction is stiffer than observed in experiments. Ifthe fibers in the model are overlapping, strong plastic shearing can develop inthe ligament between neighboring fibers, and the predicted load-carrying capa-city of the composite is closer to the experimental measurements. In Reference(26) the stress-strain curves based on FE-numerical solutions of axisymmetricunit cell models of MMCs are given in a closed form as a function of the mostimportant control parameters, namely, volume fraction, aspect ratio, and shape(cylindrical or spherical) of the reinforcement, as well as the matrix hardeningparameter.

One reason for the discrepancy between experiments and calculations basedon simple cell models in case of particle, whisker, and fiber-reinforced metalsis believed to be the unnatural constraint governing the matrix material betweeninclusion and simulation cell border (10, 20, 22–24, 27–29), which results in anunrealistic strength increase. The influence of thermal residual stresses in fiber-reinforced MMCs under transverse tension was studied in Reference (12) and foundto lead to significant strengthening elevations in contrast to findings in particulate-reinforced MMCs where strength reductions have been calculated (30).


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A limited study on the overall limit flow stress for composites with randomlyoriented disk-like or needle-like particles arranged in a packet-like morphologyis reported by Bao et al. (22). In References (31, 32), a modified Oldroyd modelhas been proposed to investigate analytically—and numerically the overall behav-ior of MMCs with randomly arranged brittle particles. Duva (33) has introducedan analytical model to represent a random distribution of non-interacting rigidspherical particles perfectly bonded in a power law matrix. The Duva model is self-consistent and should be valid particularly in the dilute regime of volume fractions,f < 0.2.

In this work, cell models are used to simulate (for a number of relevant param-eters) the transverse behavior of MMCs containing fibers in a regular square orhexagonal arrangement as well as the mechanical behavior of MMCs containingparticles in a regular arrangement. MMCs with randomly arranged inclusions aremodeled by a recently introduced self-consistent procedure with embedded cellmodels. This method of surrounding a simulation cell by an additional “equivalentcomposite material” was introduced in References (11, 13) for structures that areperiodical in loading direction; this method was later extended to non-periodictwo-dimensional (33–35) and three-dimensional composites (18, 32). The methodis known to remove the above-described unrealistic constraints of cell models.An initial comparison of two- and three-dimensional embedded cell models inthe case of perfectly plastic matrix material shows elevated strength levels for thethree-dimensional model (32), similar to composites with regularly arranged fibers(16).

The purpose of the analysis presented here is to investigate the mechanical be-havior of MMCs reinforced with regular or randomly arranged continuous fibersunder transverse loading, as well as the mechanical behavior of MMCs reinforcedwith regular or randomly arranged particles under uniaxial loading, and to system-atically study composite strengthening as a function of inclusion volume fractionand matrix hardening ability. The finite element method (FEM) is employed withinthe framework of continuum mechanics to carry out the calculations.

Model Formulation

A continuum mechanics approach is used to model the composite behavior. Theinclusion behaves elastically in all cases considered here, and its stiffness is muchhigher than that of the matrix, thus the inclusion can be regarded as rigid. In ad-dition, the continuous fibers of circular cross section and spherical particles areassumed to be well bonded to the matrix so that debonding or sliding at the inclu-sion/matrix interface is not permitted. The uniaxial matrix stress-strain behavioris described by a Ramberg-Osgood type of power law

σ = Eε ε ≤ ε

σ = σ0

[ε

ε0

]N

ε > ε0, 1.



whereσ andε are the uniaxial stress and strain of the matrix, respectively,σ 0 isthe tensile flow stress, the matrix yield strain is given asε0= σ 0/E, E is Young’smodulus, and N= 1/n is the strain-hardening exponent. Thus N= 0 correspondsto a non-hardening matrix.

J2 flow theory of plasticity with isotropic hardening is employed with a vonMises yield criterion to characterize the rate-independent matrix material. Thevon Mises equivalent stress and strain are given as

σv =√

3J2 =√

3

2si j si j

εv = 1

1+ ν√

3J ′2 =1

1+ ν

√3

2ei j ei j , 2.

wheresi j = σi j −σkk/3, ei j = εi j −εkk/3 andν is Poisson’s ratio. In the analyticalapproach, the metal matrix is considered incompressible, thus Poisson’s ratio ofthe matrix will become 0.5 after reaching the yield stress. However, in realityPoisson’s ratio of the composite remains below 0.5 as the matrix starts yielding. Itchanges from the elastic valueν to the limit value 0.5 with an increasing yieldingzone in the matrix.

The Ramberg-Osgood type of power law hardening is assumed to be valid forthe matrix described in terms of von Mises equivalent stress and strain;

σv = σv0

[εv

εv0

]N

, 3.

with the following relations between stress and strain under uniaxial loading andvon Mises equivalent stress and strain:

(a) in the case of two-dimensional plane strain condition for continuous fiberreinforced composites (

√3/2≈ 0.866, 2/

√3≈ 1.1547)

σ =√

3

2σv with σ0 =

√3

2σv0

ε = 2√3εv with ε0 = 2√

3εv0; 4.

(b) in the case of three-dimensional axisymmetrical condition for particle rein-forced composites

σ = σv with σ0 = σv0

ε = εv with ε0 = εv0. 5.

The global mechanical response of the composite under external loading ischaracterized by the overall stress ¯σ as a function of the overall strain ¯ε. Moreover,to describe the results in a consistent way, the reference axial yield stressσ 0 andyield strainε0 of the matrix, as defined in Equation 4 for the two-dimensional case


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and in Equation 5 for the three-dimensional case, will be taken to normalize theoverall stress and strain of the composite, respectively.

Following Bao et al. (22), the composite-containing hard inclusions will nec-essarily harden with the same strain hardening exponent, N, as the matrix for thecase of hard inclusions, when strains are in the regime of fully developed plas-tic flow. At sufficiently large strains, the composite behavior is then describedby

σ = σN

[ε

ε0

]N

, 6.

whereσN is called the asymptotic reference stress of the composite, which can bedetermined by normalizing the composite stress by the stress in the matrix at thesame overall strain ¯ε (Equation 1), as indicated in Figure 4:

σN = σ0

[σ (ε)

σ (ε)

]for ε À ε0. 7.

Figure 4 Composite strengthening.



Unit Cell Models

Before introducing the self-consistent embedded cell model, two regularly alignedcontinuous fiber arrangements, i.e., square and hexagonal arrangements, and tworegular particle arrangements, i.e., primitive cubic and hexagonal arrangements,are considered, as shown in Figures 5a–d.

It is well known that the repeating unit cells 1 (or 3) and 2 in Figure 5a canbe extracted from the regular array of uniform continuous fibers to model exactlythe composite with square fiber arrangement under 0◦ and 45◦ transverse loading,respectively, whereas the repeating unit cells 6 and 7 (or 8) can be taken fromFigure 5b to model exactly a hexagonal fiber arrangement under 0◦ and 30◦ trans-verse loading, respectively, if the appropriate boundary conditions are introduced

Figure 5 Modeling of unidirectional continuous fiber-reinforced composites with(a) square and (b) hexagonal fiber arrangements under transverse loading conditions,and particle-reinforced composites with: (c) primitive cubic and (d) hexagonal particlearrangements.


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Figure 5 (Continued)

and symmetry conditions are used. Moreover, the unit cells 4, 9, 10 can also rep-resent geometrically regular arrangements.

For further simplification, the modified unit cells 5, 11, 12 (circular and el-liptically shaped cell models) may be derived from the cells 3, 4, 8, 9, and 10.An ellipsoidal unit cell has been used previously (37) and shown to possess verycomplicated boundary conditions. However, as can be seen below, all these unitcells can be employed in the embedding method to model the mechanical behaviorof composites with random fiber arrangement.

The results of the unit cell models (for three different regular fiber arrange-ments with non-hardening matrix) show that the composite-strengthening levelsare quite different, especially at high volume fractions of fibers, with the compos-ite strengthening for square arrangements under 0◦ loading being highest and forsquare arrangements under 45◦ loading being lowest. Further results about thesethree regular fiber arrangements using unit cell models are found in Reference (16).A comparison of the stress-strain curves for the composite Al/46vol.%B showsthat the stress-strain curve from random fiber packing (10) lies between the curvesfrom square unit cell modeling under 0◦ loading and hexagonal unit cell modeling.The curves from square unit cell modeling under 45◦ loading are even lower. Asmentioned above, the strength of composites with randomly arranged inclusionscan not be described by modeling regular inclusion arrangements. Rather, a newmodel that describes approximately the geometry and the mechanical behavior ofreal composites with randomly arranged inclusions must be employed if modelswith many inclusions have to be avoided.



For particle-reinforced composites, the primitive cubic particle arrangementscan be modeled exactly by a conventional cell model with appropriate symmetryand boundary conditions (Figure 5c); however, the hexagonal arrangement has tobe simulated by approximate cell models (Figure 5d) (22). The primitive cubicand axisymmetric unit cells have been used herein to model two representatives ofregular particle arrangements. Further simplifications for three-dimensional mod-eling are spherical unit cells shown in Figure 5c,d, which are used for embeddedcell modeling.

Embedded Cell Models

In the present work, two-dimensional and three-dimensional self-consistent em-bedded cell models are applied to model the mechanical behavior of compos-ites with random continuous fiber and particle arrangements. Figure 6 describesschematically a typical plane strain (two-dimensional) embedded cell model witha volume fraction of f= (d/D)2 or an axisymmetric (three-dimensional) embeddedcell model with a volume fraction of f= (d/D)3, where instead of using fixed orsymmetry boundary conditions around the fiber/matrix or particle/matrix cell, theinclusion/matrix cell is embedded in an equivalent composite material with the

Figure 6 Finite element mesh for an embedded cell model.


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mechanical behavior to be determined iteratively in a self-consistent manner. Ifthe dimension of the embedding composite is sufficiently large compared with thatof the embedded cell, e.g., L/D= 5 as used here, the external geometry boundaryconditions introduced around the embedding composite are almost without influ-ence on the composite behavior of the inner embedded cell. Indeed, there exists nodifference in the calculated results for all external geometry boundary conditionsconsidered here for uniaxial loading.

To investigate the influence of the geometrical shape of the matrix phase onthe overall behavior of the composite, different shapes of cross section of theembedded cell were chosen with circularly shaped continuous fiber surrounded bya circular, square, elliptical, or rectangular metal matrix. Typical FE meshes andcorresponding symmetry and boundary conditions are given in Figure 6, where acircular fiber or a spherical particle is surrounded by a circular (two-dimensional)or spherical (three-dimensional) shaped metal matrix, which again is embeddedin the composite material with the mechanical behavior to be determined.

Iterative Modeling Procedure

Under axial displacement loading at the external boundary of the embedding com-posite (Figure 6), the overall response of the inner embedded cell can then beobtained by averaging the stresses and strains in the embedded cell or alternativelythe reaction forces and displacements at the boundary between the embedded celland the surrounding volume.

The embedding method is a self-consistent procedure that requires several it-erations, as shown in Figure 7. An initially assumed stress-strain curve (iteration 0in Figure 7) is first assigned to the embedding composite, in order to perform thefirst iteration step. An improved stress-strain curve of the composite (iteration 1)is obtained by analyzing the average mechanical response of the embedded cell.This procedure is repeated until the calculated stress-strain curve from the em-bedded cell is almost identical to that from the previous iteration. The conver-gence of the iteration occurs typically at the fifth iteration step, as illustrated inFigure 7.

It has been found from systematic studies that convergence of the iteration to thefinal stress-strain curve of the composite is independent of the initial mechanicalbehavior of the embedding composite (iteration 0). From an arbitrary initial stress-strain curve of the embedding composite, the required composite response can bereached after 4 to 5 iterations for all cases.

The LARSTRAN finite element program (38) was employed using 8-nodedplane strain elements (two-dimensional), as well as axisymmetric bi-quadrilateralelements (three-dimensional) generated with the help of the pre- and postprocess-ing program PATRAN (39). A DEC-Alpha work station 3000/300L was used tocarry out the calculations, which typically took∼30–50 min to obtain a stress-straincurve with∼100–50 loading steps in one iteration loop.



Figure 7 Iterative modeling procedure: stress-strain curves for differentsteps.

Comparison with Experiments

An example of the composite Al/46vol.%B with random fiber packing (10) hasbeen selected to verify the embedded cell model. This MMC is a 6061-O aluminumalloy reinforced with unidirectional cylindrical boron fiber of 46% volume frac-tion. The room temperature elastic properties of the fibers are Young’s modulus,E(B)= 410 GPa, and Poisson’s ratio,ν(B)= 0.2. The experimentally determinedmechanical properties of the 6061-O aluminum matrix are Young’s modulus,E(Al) = 69 GPa, Poisson’s ratio,ν(Al) = 0.33, 0.2% offset tensile yield strength,σ 0= 43 MPa, and strain-hardening exponent, N= 1/n= 1/3.

Figure 8a shows a comparison of the stress-strain curves of the compositeAl/46vol.%B under transverse loading from simulations of a real microstructuretogether with results from different cell models. The stress-strain curve from theembedded cell model employed in this review shows good agreement with thatfrom the calculated random fiber packing in the elastic and plastic regime, whichlies between the curves from square unit cell modeling under 0◦ loading andhexagonal unit cell modeling.

Furthermore, the stress-strain curve from another experiment (32) on the com-posite Ag/58vol.%Ni with random particle arrangement (Young’s modulus, E(Ni)=199.5 GPa, E(Ag)= 82.7 GPa, Poisson’s ratio,ν(Ni)= 0.312,ν(Ag)= 0.367, and yieldstrength,σ (Ni)

0 = 193 MPa,σ (Ag)0 = 64 MPa) has been compared in Figure 8b with


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Figure 8 Comparison of the mechanical behavior (a) of an Al-46vol.%B fiber-reinforced composite (N= 1/3, f= 0.46) under transverse loading from different mod-els; (b) of a Ag-58vol.%Ni particulate composite from an embedded cell model andexperiment.

that from the self-consistent embedded cell model. Good agreement in the regimeof plastic response is obtained, although the Ni-particles in the experiment werenot perfectly spherical.

These results indicate that the embedded cell model can be used to successfullysimulate composites with random inclusion arrangements and to predict the elastic-plastic composite behavior.

Geometrical Shape of Embedded Cell

As mentioned above, different shapes of the cross section of the embedded cellmodel with a circular shaped fiber, as shown in Figure 9a, are also taken into



Figure 9 Embedded cell models: influence of (a) different matrixshapes on (b) stress-strain curves for an Al-46vol.%B (N= 1/3,f= 0.46) composite.

account to investigate the influence of the geometrical shape of the embeddedcells on the overall behavior of the composite. The stress-strain curves of allembedded cell models with different geometrical shapes, are plotted in Figure 9b.With the exception of the square - 45◦ embedded cell model, the stress-straincurves are very close for all embedded cell shapes, namely, square - 0◦, circular,rectangular - 0◦, rectangular - 90◦, elliptic - 0◦, and elliptic - 90◦.

From the calculated results of the embedded cell models localized flows havebeen found around the hard fiber with preferred yielding at 45◦. Because of the


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special geometry of the square - 45◦ embedded cell model, with the cell boundaryparallel to the preferred yielding at 45◦, the overall stresses of the composite withsuch a geometrical cell shape are therefore reduced so that a relatively lowerstress-strain curve can be obtained from the modeling.

The almost identical responses of the other embedded cell models indicatethat, excluding the special shape of matrix with 45◦ cell boundaries, the pre-dicted mechanical behavior of fiber-reinforced composites under transverse load-ing are independent of the modeling shape of the embedded composite cell.That allows us to employ any embedded cell shape to model the mechanicalbehavior of the composite with random fiber arrangement. In the following sys-tematic studies, the circularly shaped embedded cell model with circular crosssection of rigid fiber are used to predict the general transverse mechanical be-havior of the composite with random fiber arrangement. In the same way, aspherically shaped embedded cell model containing a spherical particle will beused to predict the axial mechanical behavior of composites with random particlearrangement.

Parametric Studies with Embedded Cell Models

Composite strengthening is dependent on fiber and particle arrangement, inclusionvolume fraction, and the matrix strain-hardening exponent. The effect of fiber andparticle arrangement is best demonstrated by considering different cell models. Theeffect of the inclusion volume fraction has been taken into account by applyingdifferent ratios of the circular (two-dimensional) and spherical (three-dimensional)matrix and the inclusion in the embedded cell model. The effect of the matrix strain-hardening exponent has been investigated by changing the parameter, N, of thematerial law for the matrix in Equation 1. Some results from the systematic studieswith embedded cell models are given in Figures 10–13.

The influence of matrix strain-hardening is shown in Figure 10aand Figure 11a.The predicted transverse (two-dimensional) and axial (three-dimensional) overallstress-strain curves (normalized by yield stress and yield strain of the matrix,respectively) for the case of strain-hardening exponents between N= 0.0 and 0.5are depicted in Figure 10a (two-dimensional) for a fiber volume fraction of f= 0.5and in Figure 11a (three-dimensional) for a particle volume fraction of f= 0.4.At sufficiently large strains (e.g., at ¯ε = 10ε0) the normalized overall stressesapproach constant values, i.e., composite strengthening levels, as illustrated on theright-hand side of Figure 10a and Figure 11a for ε = 10ε0. The strength of thecomposite is seen to increase with N, and similar trends are found for all particlevolume fractions, f (Figure 11b and Figure 13).

The dependence of composite strengthening on inclusion volume fraction ob-tained by the self-consistent embedded cell modeling is shown in Figure 10b fora non-hardening matrix with continuous fiber reinforcement and in Figure 11b fora strain-hardening matrix (N= 0.2) with particle reinforcement. For comparison,the corresponding values of composite strengthening for regular fiber and parti-cle arrangements taken from unit cell modeling and for two approximate models,



Figure 10 Normalized stress-strain curves from embedded cell models (two-dimensional) for different N-values (f= 0.5) and (b) composite strengthening fromdifferent cell models (two-dimensional) (N= 0).

namely Duva’s model (33) and the modified Oldroyd model (31, 32) are also drawnas a function of fiber and particle volume fraction in these figures, respectively. Adetailed discussion of this comparison is given in Reference (32).

Duva’s model is a self-consistent model and valid particularly in the diluteregime, f< 0.2. Composite strengthening by the Duva model is given as


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Figure 11 Normalized overall composite stress-strain curves from axisymmetric em-bedded cell models (three-dimensional) for different N-values (f= 0.4) and (b) com-posite strengthening for matrix strain-hardening N= 0.2 from different cell models(three-dimensional).



Figure 12 Comparison of composite strengthening by continu-ously aligned fibers from different two-dimensional cell models:embedded cell model, and square and hexagonal unit cell model.

σN

σ0= (1− f )−(2.11N+0.39) . 8.

Composite strengthening levels of the modified Oldroyd model are taken fromReference (32).

A summary of dependence of composite strengthening on inclusion volumefraction and on matrix strain-hardening for randomly arranged continuous fibersand particles is depicted in Figure 12 and Figure 13a for matrix strain-hardeningexponents N= 0.0–0.5 and compared with predictions from two-dimensional andthree-dimensional unit cell models. In addition, a comparison with predictions fromDuva’s model and the modified Oldroyd model for the case of particle reinforcedcomposites is shown in Figure 13b.

For continuous fiber-reinforced composites with a non-hardening matrix, thehexagonal arrangement provides slightly higher transverse composite strengthen-ing over the square arrangement at volume fractions f< 0.5, whereas the randomarrangement from the embedded cell model supplies an intermediate value be-tween the two regular arrangements except at volume fractions of 0.38< f < 0.5(Figure 12). At volume fractions 0.38< f < 0.5, the random arrangement from theembedded cell model possesses the lowest composite strengthening lever. Con-versely, the composite strengthening of the square arrangement is higher than thatof the hexagonal arrangement for volume fractions f> 0.4. For the composite witha small exponent of strain-hardening matrix, N< 0.2, similar relations among thethree arrangements can also be seen, but the cross points of the three curves be-come closer and closer to f= 0 with increasing N. For the composite with a higher


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Figure 13 Comparison of composite strengthening by spherical particles from differ-ent three-dimensional cell models: (a) embedded cell model, and primitive cubic andaxisymmetric unit cell model and (b) embedded cell model, Duva model, and Oldroydmodel.



exponent of strain-hardening matrix N> 0.2, the values of composite strengthen-ing are almost the same for the three arrangements considered at volume fractionsf < 0.2. At volume fractions f> 0.2, the square arrangement again provides thehighest strengthening, the random arrangement the middle, and the hexagonalarrangement the lowest strengthening. The comparison of the composite strength-ening in Figure 12 for three continuous uniaxial fiber arrangements is summarizedas this: In the case of low strain-hardening of the matrix, the composites withhexagonal fiber arrangement are stronger than those with random fiber arrange-ment, which are, however, at low volume fractions stronger than those with randomarrangement. With the increasing of strain-hardening of the matrix the compositesof three arrangements act similarly at low volume fraction. At higher inclusion vol-ume fractions, the difference of the composite strengthening becomes larger, withsquare arrangement the highest level, the random arrangement the intermediate,and the hexagonal arrangement the lowest level.

The stress and strain distributions in a typical embedded cell model (two-dimensional) show localized flow with preferred yielding at 45◦ direction aroundthe hard fiber that extends into the embedding composite in the vicinity of theembedded cell. It characterizes the local response of the composite under trans-verse loading. However, the stresses and strains are distributed homogeneously inthe embedding composite far from the embedded cell. This represents the overallbehavior of the corresponding composite. In the case of random fiber arrangement,the situation around every single fiber is, of course, more complicated, but it can beconsidered, in general, as a unit cell surrounded with a corresponding compositeequivalent, which represents the average overall behavior of the real composite.

By comparing the particle-reinforced composites in Figure 13a,b, it can be seenthat the composite strengthening levels for the primitive cubic, the axisymmetricunit cell models, and the self-consistent axisymmetric embedded cell models arevery close at low particle volume fractions, f, and low matrix strain-hardening ex-ponents, N. With increasing particle volume fraction f and matrix strain-hardeningexponent N, the strengthening level of the composites increases for all the models.However, the primitive cubic unit cell provides the highest composite strength,whereas the self-consistent axisymmetric embedded cell predicts the softest com-posite response among these three models. This effect can be explained by consid-ering the constraint of neighboring particles through the necessary strict boundaryconditions of the models with regularly arranged particles. As the particle volumefraction increases, the distance between the particles decreases until they touchat a volume fraction of f= 0.5236 for the primitive cubic cell model (primitivecubic particle arrangement), and a volume fraction of f= 2/3 for the axisymmetriccell model (hexagonal particle arrangement). In the self-consistent axisymmetricembedded cell model, the particles remain surrounded by the matrix up to anextreme volume fraction of f= 1, as seen in Figure 6. The equivalent compositesurrounding the embedded inclusion/matrix cell provides less constraint on the in-clusion/matrix cell compared with conventional unit cell models. In most practicalrelevant composites, the inclusions are randomly arranged, and on average, no suchrestricting constraints exist as in the primitive cubic and hexagonal arrangements.


458 SCHMAUDER

For this reason, the self-consistent axisymmetric embedded cell model is believedto be the most realistic approximation to the geometry of real composites contain-ing randomly arranged spherical particles.

At low particle volume fractions (f< 0.2) and low matrix strain-hardening abil-ity (N < 0.2), the composite strengthening levels from the present self-consistentaxisymmetric embedded cell model and the Duva model are similar. With increas-ing particle volume fraction, the difference of composite strengthenings betweenthese two models is significant for low matrix strain-hardening, but for N∼ 0.5,the two strengthenings are comparable. The modified Oldroyd model provides anarrow distribution of composite strengthening with respect to N of matrix strain-hardening and an almost linear strength dependence on particle volume fraction.Good agreement between the self-consistent axisymmetric embedded cell modeland the modified Oldroyd model exists only at low volume fractions and highstrain-hardening rates (N∼ 0.5) of the matrix.

From above comparisons between the models it is clear that at low particlevolume fractions (f< 20%) the strengthening of metal matrix composites re-inforced with randomly or regularly arranged particles can be predicted with theaxisymmetric unit cell model, the embedded cell model, or the simple Duva model.However, for higher particle volume fractions, the self-consistent embedded cellmodel should be applied to obtain the overall mechanical response of technicallyrelevant composites reinforced by randomly arranged spherical particles.

Strengthening Model

The strength of MMCs reinforced by hard inclusions under external mechanicalloading has been shown to increase with inclusion volume fraction and strain-hardening ability of the matrix for all inclusion arrangements investigated. Fromthe numerical simulations, a strengthening model for aligned, continuous fiber-reinforced MMCs with random, square (0◦), and hexagonal arrangements, aswell as for spherical particle-reinforced MMCs with random, primitive cubic,and hexagonal arrangements, can be derived as a function of the inclusion volumefraction, f, and the strain-hardening exponent, N, of the matrix:

σN = σ0

[(1− f

c1(2+ N)

)−(c2N+c3)

− c4

(f + N

5

)], 9.

whereσ 0 is the matrix yield stress, and c1, c2, c3, and c4 are constants (summarizedin Table 1).

Equation 9 represents best fits to the calculated composite strengthening valuesσN for matrix strain-hardening exponents N in the limit of 0.0<N< 0.5 for square0◦, hexagonal, and random fiber arrangements, respectively, and fiber volumefractions f in the range of 0.0< f < 0.7. A comparison of this strengtheningmodel (Equation 9) for random fiber arrangements with the values calculatedby using self-consistent embedded cell models shows close agreement with anaverage error of 1.25% and a maximum error of 6.95% (36). Equation 9 is alsoavailable for matrix strain-hardening exponents N in the limits of 0.0<N< 0.5 for



TABLE 1 Constants for strengthening models

c1 c2 c3 c4

Two-dimensionalSelf-consistent embedded cell model 0.361 1.59 0.29 0.1(random fiber arrangement)

Square unit cell model (0◦) 0.405 2.35 0.65 0.22Hexagonal unit cell model 0.305 1.3 0.05 0.0

Three-dimensionalSelf-consistent axisymmetric embedded 0.45 2.19 0.84 0.53cell model (random particle arrangement)

Axisymmetric unit cell model 0.38 2.5 0.7 0.66Primitive cubic unit cell model 0.34 2.3 0.65 0.5

self-consistent axisymmetric embedded cell models (particle volume fractions f inthe range of 0.05< f < 0.65, with an average error of 1.59% and a maximum errorof 6.68% for the extreme case f= 0.05, N= 0.5), axisymmetric unit cell models(particle volume fraction f in the range of 0.05< f < 0.55, with an average errorof 1.22% and a maximum error of 6.18% for the extreme case f= 0.55, N= 0.5),and for primitive cubic unit cell models (particle volume fraction f in the range of0.05< f < 0.45, with an average error of 1.43% and a maximum error of 6.38%for the extreme case f= 0.05, N= 0.5).

Conclusions

The transverse elastic-plastic response of metal matrix composites reinforced withunidirectional continuous fibers and the overall elastic-plastic response of metalmatrix composites reinforced with spherical particles have been shown to dependon the arrangement of reinforcing inclusions, as well as on inclusion volume frac-tion, f, and matrix strain-hardening exponent, N. Self-consistent axisymmetricembedded cell models have been employed to predict the overall mechanical be-havior of metal matrix composites reinforced with randomly arranged continuousfibers and spherical particles perfectly bonded in a power law matrix. The embed-ded cell method is a self-consistent scheme that typically requires 4 to 5 iterationsto obtain consistency of the mechanical behavior in the embedded cell and the sur-rounding equivalent composite. Experimental findings on both an aluminum matrixreinforced with aligned but randomly arranged boron fibers (Al/46vol.%B) and asilver matrix reinforced with randomly arranged nickel inclusions (Ag/58vol.%Ni)compare favorably with those predicted by embedded cell models and are foundto be in close agreement. The strength of composites with aligned but randomlyarranged fibers can not be properly described by conventional fiber/matrix unit cellmodels, which simulate the strength of composites with regular fiber arrangements.

Systematic studies were carried out for predicting composite limit flow stressesfor a wide range of the parameters f and N. The results for random three-dimensional


460 SCHMAUDER

particle arrangements were then compared with regular three-dimensional parti-cle arrangements by using axisymmetric and primitive cube unit cell models.The numerical results were also compared with those from the Duva model andfrom the modified Oldroyd model. The strength of composites at low particle vol-ume fractions have been shown to be in close agreement, except for the modifiedOldroyd model. With increasing particle volume fractions, f, and strain hardeningof the matrix, N, the strength of composites with randomly arranged particles cannot be properly described by conventional particle/matrix unit cell models, becausethese parameters are able to predict the strength of only composites with regularparticle arrangements.

Finally, a strengthening model for randomly or regularly arranged continuousfiber and particle reinforced composites under axial loading is derived, providinga simple guidance for designing the mechanical properties of technically relevantmetal matrix composites: For any required strength level, Equation 9 will providethe possible combinations of f and N. Thus for the near future, strong impactof the present work on the development of new particle-reinforced metal matrixcomposites is expected.

MATRICITY MODEL

Although inclusion-type microstructures are typical for dilute systems with phasevolume fractions between 0 and 30%, interpenetrating microstructures are typicallyin the composition range of 30 to 70%. Modeling of the mechanical behavior ofthese composites where both phases are connected throughout the material wasachieved only recently (40–44).

In order to take the mutual interpenetration of the phases into account, a newparameter, matricity M (0<M < 1), was introduced. This parameter can be easilyobtained by quantitative metallography in a conventional scanning electron micro-scope (SEM). The self-consistent embedded cell model was extended, as shownin Figure 14, in order to take both cases, phaseα circumventing phaseβ and viceversa, into account. It is easily observed that in the case shown [ZrO2/NiCr8020,zirconia ceramic and alloy Ni(80vol.%) Cr(20vol.%)], a wide variety of possiblemechanical behaviors exists. Thus by computational mechanics, interpenetratingmicrostructures of the ceramic/metal type have a strong potential for further de-velopment and applications. Moreover, the elastic and thermal properties of thesecomposites can be derived from the matricity model as well.

FAILURE INITIATION ON THEMICROMECHANICAL LEVEL

Intensive shear banding can be taken as a first failure indicator in real microstruc-tures. Models of real and artificial microstructures on the mesoscale containingapproximately 100 ceramic SiC-particles in an Al metal matrix have been set up and



Figure 15 A bitmap of the undeformed structures with experimentally ob-tained edge displacements used as input for FE-calculations.

carefully analyzed (45). The distribution of the particles was found to have strongaffect on the formation and intensities of shear bands between the particles, as wellas on the concentration of stress maxima in the vicinity of the particles. A more de-tailed microstructural analysis with experimental boundary conditions (Figure 15)demonstrates that the calculated shear band concentrations are in goodagreement with experimental results from surface deformation measurements(Figure 16).

Micromechanical aspects of crack extension in multiphase materials are es-sential in understanding fracture characteristics on a macroscopic scale. Typically,hard second phase particles fracture or decohere in the course of externally applied


462 SCHMAUDER

loading before the matrix between these particles fails and macroscopic crackadvance is observed. Some of these phenomena are discussed (46) for the duc-tile/brittle systems WC/Co, Al/SiC, and Al/Si from a micromechanical viewpoint.In order to simulate local fracture, a criterion by Rice & Tracey of discrete voidformation was applied (47). Simulated and experimentally observed crack pathshave been shown to be in good agreement for an Al/20vol.%SiC MMC containingSiC-particles (48).

More recently, failure initiation in high-speed steels was analyzed for a numberof carbide arrangements (49, 50). The energy consumed during crack propagationwithin the microstructure was found to be strongly dependent on the arrangement ofthe carbides, giving preference to fine microstructures over coarse microstructures(Figure 17).

MULTISCALE MODELING

Scientists have long been interested in understanding macroscopic materials prop-erties from atomistic observations. This possibilty was first realized when it wasobserved that formation of nanometer-sized Cu-precipitates plays an essential rolein the embrittlement of steels. As a first step, the formation of these precipitates wasdescribed by an energetically based Monte Carlo method (51). A single vacancyis simulated to perform diffusion, and vacancy concentration is then adjusted toreality by adjusting the time of simulation. In this type of modeling, even res-olution and re-precipitation of Cu atoms during annealing and cooling down canbe simulated (52, 53).

The average size of these tiny precipitates is used in order to predict the materialsstrengthening based on MD simulations of the interaction of a dislocation with aprecipitate and on continuum mechanical dislocation theory. Both results are ingood agreement with experimental strengthening levels of Cu-containing steelsthat have been annealed for a long time. Finally, these results are used as inputfor damage mechanical analyses (54). Again the predicted J-integral versus crackadvancement curves coincide with experimental results (Figure 18), thus providinga sound basis for the prediction of the future mechanical behavior of Cu-containingsteels during ageing.

SUMMARY

Herein we have focused on new developments in modeling based on the finiteelement method (FEM), especially on deformation analyses based on numeri-cal homogenization techniques (self-consistent embedding procedure, matricitymodel) and on simulations of real microstructural cut-outs, on damage analyses ofartificial and real microstructures, and on multiscale modeling aspects. Atomisticmodeling such as Monte Carlo simulations and the molecular dynamics method,



dislocation theoretical modeling, and continuum mechanical methods have beendiscussed in light of the mechanical behavior of a number of different materialssuch as MMCs, BMCs, and IPMs, as well as microalloyed steels. Simulation re-sults have been presented graphically in a systematic manner for different materialsystems and compared with experimental results. Finally, it was shown that theresults can be used to predict the future behavior of materials that are already inservice and even to design new materials.

ACKNOWLEDGMENTS

This work was supported by several programs of the Deutsche Forschungsgemein-schaft (DFG): (a) the Research Group Investigation of the deformation behaviorof heterogeneous materials by direct combination of experiment and computation,subprojects Schm 746/16-1, RI 339/15-1, and FI 686/1-1; (b) by the Schwerpunk-tprogramm Gradientenwerkstoffe projects Schm 746/12-1 and Schm 746/12-2;and (c) by the Schwerpunktprogramm, Erweiterung der Formgebungsgrenzen beider Umformung, project Schm 746/25-1; and by the German Bundesministeriumfur Bildung, Wissenschaft, Forschung und Technologie (BMBF) under grant No1501029. This support is gratefully acknowledged. I especially thank my co-workers at MPA P. Binkele, and Drs. P. Kizler, E. Soppa, L. Mishnaevsky, andK. Zhu who provided the figures for this contribution.

The Annual Review of Materials Researchis online athttp://matsci.annualreviews.org

LITERATURE CITED

1. Kohlhoff S, Schmauder S. 1989. InAtom-istic Simulation of Materials: Beyond PairPotentials, ed. V Vitek, DJ Srolovitz, pp.411–18. New York: Plenum

2. Kohlhoff S, Gumbsch P, Fischmeister HF.1991.Philos. Mag. A64:851–78

3. Zohdi T, Oden J, Rodin G. 1996.Comp.Methods Appl. Mech. Eng.138:273

4. Thirteenth US National Congress of Ap-plied Mechanics (US-NCAM), June 21–26. 1998. University of Florida, AbstractBook, ISBN 0–9652609

5. Kizler P, Uhlmann D, Schmauder S. 2000.Nucl. Eng. Design196:175–83

6. Harrington WC Jr. 1993. InMechani-cal Properties of Metaloorganic Composi-tions, ed. S Ochiai, pp. 759–73. New York:Dekker

7. Brockenbrough JR, Suresh S. 1990.ScriptaMetall. Mater.24:325–30

8. Rammerstorfer FG, Fischer FD, BöhmHJ. 1990. InIUTAM/IACM Symp. Discre-tization Methods in Structural Mechan-ics, Vienna, Austria, 1989, ed. G Kuhn,H Mang, pp. 393–404. Berlin/Heidelberg:Springer

9. Evans AG. 1991.Mater. Sci. Eng. A143:63–76

10. Brockenbrough JR, Suresh S, WieneckeHA. 1991.Acta Metall. Mater.39:735–51

11. Dietrich C. 1993. Mechanisches Verhal-ten von Zweiphasenwerkstoffen: Nume-rische und experimentelle Untersuchungenzum Einfluss der Gefügegeometrie,VDI-Fortschrittsberichte, Reihe 18, Nr. 128,Dusseldorf: VDI. pp. 138


464 SCHMAUDER

12. Nakamura T, Suresh S. 1993.Acta Metall.Mater.41:1665–81

13. Dietrich C, Poech MH, Schmauder S,Fischmeister HF. 1993. InVerbundwerk-stoffe und Werkstoffverbunde, ed. G Leon-hardt, G Ondracek, pp. 611–18. Oberursl,Germany: DGM-Informationsgesellschaft

14. Adams D. 1970.J. Composite Mater.4:310–28

15. Jansson S. 1992.Int. J. Solids Struct.29:2181–200

16. Zahl DB, Schmauder S. 1994.Acta Metall.Mater.42:2983–97

17. Llorca J, Needleman A, Suresh S. 1991.Acta Metall. Mater.39:2317–35

18. Sautter M. 1995. Modellierung des Verfo-mungsverhaltens mehrphasiger Werkstoffemit der Methode der finiten Elemente,VDI-Fortschrittsberichte, Reihe 5, Nr. 398,Dusseldorf: VDI. pp. 197

19. Povirk GL, Stout MG, Bourke M, Gold-stone JA, Lawson AC, et al. 1992.ActaMetall. Mater.40:2391

20. Bohm H, Rammerstorfer FG, WeissenbekE. 1993.Comput. Mater. Sci.1:177–94

21. Bohm HJ, Rammerstorfer FG, Fischer FD,Siegmund T. 1994.ASME J. Eng. Mater.Technol.116:268–73

22. Bao G, Hutchinson JW, McMeekingRM. 1991.Acta Metall. Mater.39:1871–82

23. Tvergaard V. 1990.Acta Metall. Mater.38:185–194

24. Hom CL. 1992.J. Mech. Phys. Solids40:991–1008

25. Weissenbek E. 1993.Finite element mod-elling of discontinuously reinforced metalmatrix composites.PhD. thesis. Techn.Univ. Vienna. 120 pp.

26. Li Z, Schmauder S, Wanner A, DongM. 1995.Scripta Metall. Mater.33:1289–94

27. Christman T, Needleman A, Suresh S.1989.Acta Metall. Mater.37:3029–50

28. Suquet PM. 1993. InMECAMAT 93, Int.Seminar on Micromechan. Mater.p. 361.Paris: Editions Eyrolles

29. Thebaud F. 1993. Vers l’introduction de

l’endommagement dans la prévision glob-ale du comportement de composites à ma-trice metallique. PhD. thesis.Ecole Poly-technique, Palaiseau, France. 269 pp.

30. Zahl DB, McMeeking RM. 1991.Acta Met-all. Mater.39:1171–22

31. Poech MH. 1992.Scripta Metall. Mater.27:1027–31

32. Farrissey L, Schmauder S, Dong M, SoppaE, Poech MH, McHugh P. 1995.Comp.Mater. Sci.15:1–10

33. Duva JM. 1984.ASME J. Eng. Mater. Tech-nol. 106:317

34. Sautter M, Dietrich C, Poech MH, Sch-mauder S, Fischmeister HF. 1993.Comput.Mater. Sci.1:225–33

35. Zahl DB, Schmauder S. 1994.Comput.Mater. Sci.3:293–99

36. Dong M, Schmauder S. 1996.Comput.Mater. Sci.5:53–66

37. Jarvstrat N. 1993.Comput. Mater. Sci.1:203–12

38. LASSO Engineering Association, Marko-mannenstr. 11,70771 Leinfelden-Echter-dingen, Germany

39. PDA Engineering, 2975 Redhill Avenue,Costa Mesa, CA. 92626

40. Lessle P, Dong M, Soppa E, SchmauderS. 1997. InVortragstexte der Tagung Ver-bundwerkstoffe und Werkstoffverbunde, ed.K Friedrich, pp. 765–70. Oberurs, Ger-many: DGM-Informationsgesellschaft

41. Lessle P, Dong M, Soppa E, Schmauder S.1998.Scripta Mater.38:1327–32

42. Lessle P, Dong M, Schmauder S. 1999.Comp. Mater. Sci.15:455–65

43. Schmauder S, Weber U. 2001.Arch. Appl.Mech.71:182–92

44. Schmauder S, Weber U, Hofinger I,Neubrand A. 1999.Tech. Mech.19:313–20

45. Soppa E, Schmauder S, Fischer G, The-sing J, Ritter R. 1999.Comp. Mater. Sci.16:323–32

46. Schmauder S. 2001. Micromechanics ofcrack extension in multiphase materials. InElsevier’s Encyclopedia of Materials, ed.Elsevier Science.



47. Rice JR, Tracey DM. 1969.J. Mech. Phys.Solids17:201–17

48. Schmauder S, Wulf J, Steinkoppff Th,Fischmeister HF. 1996. InIUTAM Symp.Micromechanics of Plasticity and Dam-age of Multiphase Materials, ed. A Pin-eau, A Zaoui, pp. 255–62. The Netherlands:Kluwer

49. Mishnaevsky L Jr, Lippmann N, Schmau-der S. 2001. InProceedings of IMECE2001, Int. Mech. Eng. Congr. Exp., Novem-ber 11–16, New York

50. Mishnaevsky L Jr, Lippmann N, Schmau-der S. 2002.J. Plast.In press

51. Soisson F, Barbu A, Martin G. 1996. ActaMater. 44:3789–800

52. Binkele P, Schmauder S. 2000. InWorkshop ‘Kompetenzerhaltung in derKerntechnik’, May 25, 2000. Bonn,Germany

53. Schmauder S, Binkele P. 2002.Comp.Mater. Sci.In press

54. Kizler P, Uhlmann D, Schmauder S. 2000.Nucl. Eng. Design196:175–83

17 Jun 2002 8:17 AR AR162-17-COLOR.tex AR162-17-COLOR.SGM LaTeX2e(2002/01/18)P1: GDL

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COMPUTATIONAL MECHANICS - uni- · PDF file7 Jun 2002 8:37 AR AR162-17.tex AR162-17.SGM LaTeX2e(2002/01/18) P1: GJB COMPUTATIONAL MECHANICS 439 Figure 2 Numerical methods referring