D.H. Pahr

Vienna University of Technology, Vienna, Austria

S.M. Arnold

Glenn Research Center, Cleveland, Ohio

The Applicability of the Generalized Method

of Cells for Analyzing Discontinuously

Reinforced Composites

NASA/TM—2001-211165

November 2001
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National Aeronautics and

Space Administration

Glenn Research Center

D.H. Pahr

Vienna University of Technology, Vienna, Austria

S.M. Arnold

Glenn Research Center, Cleveland, Ohio

The Applicability of the Generalized Method

of Cells for Analyzing Discontinuously

Reinforced Composites

NASA/TM—2001-211165

November 2001
Available from

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Hanover, MD 21076

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Acknowledgments

Scholarship funding for the first author to complete this work was provided by the “Akademisch-soziale

Arbeitsgemeinschaft Österreichs (academic-social syndicate of Austria).”
The Applicability of the Generalized Method of Cells

for Analyzing Discontinuously Reinforced Composites

D.H. Pahr

Vienna University of Technology

A-1040 Vienna, Austria

and

S. M. Arnold

Glenn Research Center

Abstract

The paper begins with a short overview of the recent work done in the field of discontinuous reinforced

composites, focusing on the different parameters which influence the material behavior of discontinuous

reinforced composites, as well as the various analysis approaches undertaken. Based on this overview

it became evident, that in order to investigate the enumerated effects in an efficient and comprehensive

manner, an alternative approach to the computationally intensive finite-element based micro-mechanics

approach is required. Therefore an investigation is conducted to demonstrate the utility of utilizing the

generalized method of cells (GMC), a semi-analytical micromechanics-based approach, to simulate the

elastic and elastoplastic material behavior of aligned short fiber composites. The results are compared

with 1) simulations using other micromechanical based mean field models and finite element (FE) unit

cell models found in the literature given elastic material behavior, as well as 2) finite element unit cell and

a new semi-analytical elastoplastic shear lag model in the inelastic range. GMC is shown to definitely

have a window of applicability when simulating discontinuously reinforced composite material behavior.

1 Introduction

Metal matrix composites (MMCs) have received considerable attention over the past 30 years due to

their attractive specific strength, stiffness, fatigue and thermal properties[Clyne and Withers (1993)], [Kelly

and Zweben (Eds) (2000)], [Sinclair and Gregson (1997)], [Lloyd (1994)]. MMC materials can be generically classified by their type of reinforcement, which consists of either continuous fiber (identified herein

as CFMMC) or discontinuous particulate (termed herein as DRX) reinforcements. While both classes of

materials share common features at a fundamental level (e.g., matrix/reinforcement compatibility, interfacial properties sensitivity, architectural interaction (reinforcement size, shape and packing) and property

optimization, at a practical level dramatic differences can and do exist. For instance, CFMMC typically

have exceptionally high strength and stiffness capabilities that are inherently strongly anisotropic; whereas,

DRX material properties are typically more isotropic in nature with lower specific strength, yet have excellent stiffness capability. Clearly, failure to appreciate and consider explicitly the appropriate reinforcement

morphology can result in inaccurate performance assessments, as elaborated upon in the next section. The

combination of good transverse properties, relatively low cost, high workability and reasonable increases in

National Aero an utics and Space Administration

Cleveland, Ohio 44135

NASA/TM—2001-211165 1
performance over monolithic metallic alloys have made discontinuous reinforced materials the most commercially attractive system to date for many stiffness driven applications [Clyne and Withers (1993)], [Kelly and

Zweben (Eds) (2000)], [Miracle and Maruyama (2000)].

Although manufacturing issues and cost are probably the greatest impediments to the wide application of

these advanced materials, it is also clear that accurate design and life prediction tools are greatly needed to

facilitate the implementation of these developing materials. Although closure has not been reached regarding

the best models for use in design and life prediction, it has become evident that if a model is ever to serve a

purpose beyond that of basic research, it must fulfill several primary requirements. These include a significant

level of accuracy on both the macro and micro scales, computational efficiency, and compatibility with the

finite element method. Fulfillment of these requirements allows a model to serve the composite developer

by enabling quick and easy variation of composite parameters for material development and optimization

purposes. Likewise, those who attempt to utilize composites in structural design are well served if the model

is compatible with the finite element method. The generalized method of cells (GMC), originally developed

by Aboudi (1991, 1995), is a good choice for implementation into modeling tools for advanced composites,

given the requirements described above. Particularly, whenever microfield as well as macrofield quantities

will be needed - as is the case when attempting to account for inelastic material behavior in MMCs.

GMC is a fully analytical micromechanics model for multi-phased materials with arbitrary periodic

microstructures. It provides pseudo closed-form multiaxial constitutive equations for such materials, and

allows straightforward implementation of physically-based viscoplastic deformation models, as well as arbitrary failure and damage models for each phase. Further, recent independent advances have simplified the

implementation of GMC as an elemental constituent material model in finite element analysis [Arnold et al,

(1999)], and significantly increased the modelís computational efficiency [Pindera and Bednarcyk (1999)].

GMC has been implemented in the NASA Glenn Research Center comprehensive micromechanics analysis

code, MAC/GMC [Arnold et al, (1999)]. The code has many features that render it useful for design, deformation modeling, and life prediction for a wide range of materials. These features include the ability to

simulate general thermomechanical loading on composites whose geometries are represented by a library of

continuous and discontinuous repeating unit cells, a library of nonisothermal elastic/viscoplastic constitutive

models, fatigue damage analysis, yield surface analysis, laminate analysis, and interface modeling.

In the past, the various capabilities described above have been exercised extensively, however, the predominant material system examined has been that of the CFMMC [Aboudi(1996)], [Arnold, et. al. (1996)],

[Goldberg and Arnold (2000)], [Iyer et al. (2000)], [Bednarcyk and Arnold (2000a)], [Lissenden et al. (2000)],

[Bednarcyk and Arnold (2000b)]. Consequently, the primary objective of this current study is to assess the

applicability of GMC in predicting the elastic and inelastic (plastic) behavior of particulate reinforced composites (DRXs). The literature indicates that DRX behavior is significantly influenced by various characteristic parameters like fiber arrangements (staggered, non-staggered, cluster, random, packing type), fiber

volume fraction, fiber aspect ratios, fiber-matrix stiffness ratios and reinforcement type (cylinder, particle,

etc. ). A brief review of this is provided in the background section. Even though GMC is fully capable of modeling arbitrary multiphased composite architectures, to facilitate comparison with previous work

reported in the literature and limit the scope of this investigation, only one type of DRX is considered,

namely, the simulation of aligned short fiber composites with non-staggered cylindrical reinforcements and a

hexagonal packing arrangement. The need for such an assessment stems from the fact that although GMC

consists of analytical equations that are able to deal with both the homogenization and localization process

for composite materials in a very numerically efficient way, the methodís linear displacement field assumption

in combination with the imposition of displacement and traction boundary conditions in an average sense,

gives rise to an inherent lack of normal and shear component coupling. This lack of coupling is such that

application of a global normal (or shear) stress field will produce only a local normal (or shear) stress field.

Consequently, for cases (e.g., DRX materials) in which the load transfer mechanism between phases (e.g.,

matrix and fibers) is shear dominated, the lack of normal-shear coupling may become quite problematic.

In particular, in the case of an elastoplastic matrix regime placed between two short fibers, GMCís lack of

shear coupling requires both the fiber and matrix to carry the same normal stress component. This tends

to produce an overly compliant (soft) composite response. The results presented within demonstrate this

NASA/TM—2001-211165 2
fact, and more importantly characterize the window of applicability for the GMC method within the context

of discontinuously reinforced metallic composites. The introduction of an ìartificialî elastoplastic matrix

material between the particulate reinforcements, is proposed in an attempt to extend the applicability of

GMC. Although, providing more reasonable macro behavior, such a strategy still results in incorrect micromechanical stress fields within the unit cells, thereby limiting the usefulness of the current GMC model

when calculating failure or fatigue analyses.

With section 2, the paper begins by reviewing recent work done in the field of short fiber composites so

as to reveal the key parameters influencing DRX material behavior. Section 3 then articulates the various

analytical and numerical approaches employed and compared in this paper. Finally, section 4 presents a

comparison of GMC results with other analytical and numerical results within the elastic and elastoplastic

range, respectively.

2 Background: Factors Influencing DRX Materials

The early work on the principles of strengthening in composites reinforced with discontinuous fibers,

whiskers or particulates carried out in the 1950ís and 1960ís is characterized by the use of relatively simple

models that do not explicitly take into account the actual inclusion distribution through interaction between

adjacent inclusions. Many models have been employed to study the effect of inclusion shape on the elastic

moduli, as well as subsequent inelastic response, and also the mechanism of stress transfer from the matrix

to a finite-length fiber. For instance, Eshelbyís analytical solution to the problem of an arbitrarily shaped

ellipsoidal inclusion embedded in an homogeneous material deformed by uniform tractions or displacements

at infinity has formed the basis for calculating the effective response of macroscopically homogeneous twophase composites using a number of approaches [Eshelby (1957)]. These approaches include the self-consistent

scheme which neglects the inclusion/matrix interaction in calculating stress fields in the inclusion phase [Hill

(1965)], and the Mori-Tanaka method (MTM) which takes this interaction into account in an approximate

fashion [Mori and Tanaka (1973)]. The problem of an array of ellipsoidal inclusions with different aspect

ratios embedded in an elastoplastic matrix can be treated using the Mori-Tanaka approach, as was done

by [Brown and Clarke (1975)] investigating the effect of inclusion shape on work hardening of metal matrix

composites. However, this method typically underestimates yielding and subsequent hardening effects due to

the use of mean stress and strain fields in the matrix phase, and treats all inclusion distributions on the same

footing so long as macroscopic homogeneity is preserved. Along similar lines, the so-called shear-lag analysis

has been employed to study the effectiveness of short fibers as reinforcement using the strength-of-materials

approach to analyze stress fields around and within a finite-length fiber embedded in a surrounding matrix

[Dow (1963)]. While this type of analysis helps to identify shearing of the matrix as the primary mechanism

of force transfer from the matrix to the fiber, and thus the critical fiber length over which the axial stress is

introduced into the fiber from both ends, it is based on a very simplified one-dimensional analysis of stress

fields which neglects the influence of morphology of surrounding fibers, among other things.

Despite the relatively long history of modeling the response of discontinuous fiber composites, only

recently have systematic investigations of the effects of reinforcement shape and arrangement been initiated

for these types of composites. Inclusion of the third dimension in the analysis of the elastic and inelastic

response of discontinuous fiber composites increases the number of variables several fold relative to the

analysis of continuous fiber composites. Thus, in addition to the arrangement and shape of the reinforcement

in the plane transverse to the loading direction, variables associated with the planes parallel to the loading

direction were included. These variables include the fiber aspect ratio, fiber spacing in the vertical and

horizontal directions, those associated with the effect of fiber clustering, and the extent of overlap between

adjacent columns of fibers. These added complexities typically require numerical solution procedures, such as

the finite-element analysis; particularly if complicated reinforcement shapes and arrangements are involved.

Finite-element investigations carried out in recent years have focused on separating the effects of inclusion

shape from the effects of inclusion packing array geometry on the overall elastic, elastoplastic and creep

behavior. Due to many different types of reinforcement employed in discontinuously-reinforced composites,

including whiskers, particulates, platelets as well as finite-length fibers, extensive analyses of inclusion shape

NASA/TM—2001-211165 3
have been conducted for this class of composites. Examples of inclusion shapes investigated include spherical

particles, circular cylinders with different aspect ratios, truncated circular (octagonal) cylinders, double-cone

particles, and ellipsoidal or disk-like particles(platelets). Similarly, different packing arrays studied include

hexagonal arrangements of inclusions in the plane of loading distributed in either regular (aligned) arrays or

off-set (staggered) arrays in the planes parallel and perpendicular to the loading direction so as to model and

assess the influence of distribution randomness that is typical in DRX composites. Inclusions such as disklike particles arranged in packet morphologies have also been considered, as have clustered arrangements.

These investigations have been driven in large part by the wide range of inclusion morphologies that result

from current material processing techniques, as well as by the wide range of shapes available for the inclusion

phase.

In order to reduce the complexity of a three-dimensional finite-element formulation in analyzing the

response of discontinuous fiber, whisker or particulate composites, various idealizations of the unit cell have

been employed by a number of researchers (e.g. clusters of whiskers or cylindrical particles in the plane

parallel to the applied load have been modeled using a plane strain idealization of rectangular platelets).

This effectively reduces the problem to that of a continuously reinforced composite subjected to loading

in the plane perpendicular to the long fiber direction. The results of such analysis for rectangular crosssections with different aspect ratios in the plane of loading should be easily deduced from the early analysis

of continuously reinforced composites with rectangularly shaped fibers [Ashton et al.(1969) ]. In the case of

a hexagonal array of inclusions in the plane transverse to the loading direction, the problem is often reduced

to an axisymmetric (i.e., two-dimensional) problem by approximating the unit cell using a circular cylinder

with different types of lateral boundary conditions to simulate the interaction with adjacent fibers (i.e., unit

cells). Using this model, different fiber arrangements in the plane parallel to the applied load have been

investigated, with varying amounts of overlap between vertical columns of adjacent fibers (e.g. [Christman

et al. (1989)], [Tvergaard (1990)], [Dragone and Nix (1960)], [Povirk et al. (1990)], [Yang et al. (1991)],

[Bao et al.(1991)], [Siegmund et al. (1992)]). In contrast to continuously reinforced composites, the inclusion

arrangement in discontinuously reinforced composites is inherently three-dimensional, so that the use of

plane models requires considerable caution. The results of fully three-dimensional finite-element analyses

of such periodic arrays are also available (e.g. [Levy and Papazian (1990)], [Weissenbek and Rammerstorfer

(1993)], [Abel et al.(1993)], [Weissenbek et al. (1993)]). They are, however rather restrictive in terms of

fiber arrangements and shapes, and tend to entail considerable computational costs. Both three-dimensional

and axisymmetric unit cell models have been used successfully for studying the nonlinear thermomechanical

behavior of aligned short reinforced composites.

The literature survey presented indicates significant influence of both inclusion shape and distribution on

the elastic, elastoplastic and creep response of discontinuous fiber composites. The extent of this influence for

a given fiber architecture depends on the inclusion content, inclusion/matrix material property mismatch, and

the direction of applied load with respect to the internal micro-structure. The influence of fiber architecture

on the response of DRX composites is significant and is based on a substantial number of investigations,

each of which was necessarily limited in scope due to the prevalent use of the finite-element approach in

modeling the response of the investigated architectures. Thus while a considerable body of knowledge

has been generated that sheds light on the effectiveness of different fiber architectures in strengthening

discontinuous fiber composites, considerably more systematic research is required to develop design guidelines

for optimization of material performance through fiber architecture manipulation.

However, it becomes clear, based on the above discussion, that in order to investigate the enumerated

effects in an efficient and comprehensive manner, including parametric studies involving fiber content and

material property mismatch variations, an alternative approach to the computationally intensive finiteelement based micromechanics approach is required. This is particularly true when inelastic (viscoplastic)

theories, which typically require computationally intensive integration algorithms, are employed to model

the response of metallic matrices. Therefore, the objective of this paper is to apply the computationally

efficient GMC method so as to determine its suitability for predicting the macroscopic response of aligned,

DRX materials.

NASA/TM—2001-211165 4
3 Modeling Approaches

All simulations conducted in this study consist of aligned short fiber composites, wherein the particulates

(wiskers) are idealized as non-staggered, cylindrical reinforcements, with a hexagonal packing arrangement.

Similarly other basic constraints imposed are as follows:

ï The inclusion properties are assumed isotropic and linear elastic, while the matrix is taken to be

isotropic elastoplastic with isotropic hardening.The material data for both the inclusion and matrix

are treated as temperature-independent and isotropic.

ï The inclusions are axisymmetric, identical in shape and size and can be characterized by an aspect

ratio, ar, that relate the length of an inclusion to its diameter.

ï Investigated model parameters are the fiber (inclusion) volume fraction (Vf = 0.2, 0.5), the fiber-matrix

stiffness ratio (sr = EF /EM = 3, 30) and the fiber aspect ratio (ar = l/d = 5, 15, 25).

ï The inclusions and matrix are well bonded at their interface and remain that way during deformation.

Thus, we do not consider interfacial slip, fiber-matrix debonding or matrix micro-cracking.

ï No time dependence (relaxation or creep) effects are included in the present analysis.

Note, obtaining experimental reference data for unidirectional short fiber composites appears to be problematic, as it has not proved possible to produce physical samples with perfectly aligned fibers. Consequently, in

this paper we avoid this additional complexity of orientation effect by using three-dimensional finite element

analysis of aligned short fiber composites as our reference standard, rather than experimental results.

3.1 Material Data

The material properties used for all calculations are given in Table 1. These properties are in the range

of typical fiber-reinforced engineering thermo-plastics (PMCís - Material 1) and metal matrix composites

(MMCís - Material 2), respectively. The following input data are needed for both components: Youngís

modulus E and Poissonís ratio ν. The yield stress σY and the hardening modulus EH are needed for the

matrix material only.

Table 1: Prototypical material properties used for all calculations.

Material 1 (sR = 30) Material 2 (sR = 3)

Property Fiber Matrix Fiber Matrix

E (GPa) 300. 10. 300. 100.

ν 0.17 0.33 0.17 0.33

σY (MPa) - 20. - 200.

EH (GPa) - 0.1 - 1.0

3.2 Analytical Models

In the past, numerous analytical models have been examined in an attempt to determine the ìbestî

(see e.g., [Tucker and Liang (1999)], [Aboudi (1991)]) available analytical model for predicting the elastic

stiffness behavior of aligned short fiber reinforced composites relative to reference FEA-model representations. Analytical models considered, were models such as: the Mori-Tanaka method (MTM)[Mori and

NASA/TM—2001-211165 5
(a) (b) (c)

Figure 1: Sketch of inclusions-matrix shapes for the applied aligned short fiber composite models. (a) MoriTanaka model, (b) Hashin-Shtrikman estimates of Ponte CastaÒeda and Willis, (c) GMC-model (ar = 5.0)

Tanaka (1973)], self-consistent models, bounding models (e.g., variable constraint model (VCM) and HashinShtrikman bounds and estimates), Halpin-Tsai equations and shear lag models (see Appendix B for details

of the elastoplastic extension used in this study). For example, the recent work of [Tucker and Liang (1999)]

recommended that the Mori-Tanaka model was the ìbestî choice for estimating the overall elastic stiffness

behavior of aligned short fiber composites.

In this study we will extend the scope of comparison to include the semi-analytical generalized method of

cells model (GMC) and compare GMC with the MTM, VCM, FEM results and Hashin-Shtrikman estimates

(HSE)1 of [Ponte CastaÒeda and Willis (1995)]. Benveniste (1987) has provided a particularly simple and

clear explanation of the Mori-Tanaka approach. Analytical bounds for the stiffness behavior can be obtained

from the variable constraint model (VCM), published by [Pedersen and Withers (1992)]. From a practical

point of view it is of interest that the lower (VCM) bound corresponds to the standard Mori-Tanaka results,

while the upper (VCM) bound can also be obtained from the Mori-Tanaka approach after a so called ìcolor

inversionî (i.e., exchanging the roles of inclusion and matrix). These bounds obtained from the Mori-Tanaka

approach also correspond to the Willis bounds in the case of aligned inclusions, compare for example, [Weng

(1990)].

It is important to remember that the basic assumption of the Mori-Tanaka method is that the aligned

ellipsoidal inclusions are surrounded by coaxial aligned ellipsoidal matrix domains in which both ellipsoids

have the same aspect ratio (Fig.1a) and that each inclusion (fiber) within a concentrated composite sees the

average strain of the matrix. Consequently, the MTM provides only macro (average) fields and properties

and therefore lacks the ability to address adequately problems dominated by microfield quantities, such as

damage and inelastic behavior analysis. Alternatively, GMC allows both homogenization and localization of

the associated stress and strain fields (see Appendix A), thereby providing not only macro stiffness properties

but also local eigenstrain field quantities. Note, that in the present investigation the GMC unit cell differs

from MTM as different fiber-unit cell aspect ratios (Figs.2 and 3) can be realized since the whole inclusion

(fiber) is surrounded by a matrix material layer of constant thickness (side-to-side and end-to-end distances

are equal) and labeled, e, subsequently. Because of this varying aspect ratio, HSE for the prediction of the

effective moduli of the elastic inhomogeneous material given in [Ponte CastaÒeda and Willis (1995)] are also

introduced (Fig.1b) to overcome the drawback of the original Mori-Tanaka method so that different fiber-unit

cell aspect ratios can be realized. Note, these Hashin-Shtrikman estimates correspond identically with the

Mori-Tanaka method for the case of aligned, coaxial inclusions with equal aspect ratios of the inclusion and

the surrounding matrix ellipsoids (see [Hu and Weng (2000)])

1Note, the Hashin-Shtrikman estimates differ from the Hashin-Shtrikman bounds, as [Ponte CastaÒeda and Willis (1995)]

and [Willis (1977)] removed the restriction of statistically isotropic overall material symmetry (used by both HS and Willis)

by separating the spatial distribution of inclusions from the inclusion shape. Also in the HSE, [Ponte CastaÒeda and Willis

(1995)] employed only a single reference material (e.g., the effective material like in self consistent models) instead of the two

used by [Willis (1977)].

NASA/TM—2001-211165 6
3.3 Generalized Method of Cells

Although, GMC is capable of analyzing any multiphased composite material with an arbitrary internal

microstructure and reinforcement shape (see Appendix A), the unit cell analyzed in this study is shown

in Fig. 2. Due to the periodic boundary conditions inherent to the GMC-model the whole RVE must be

modeled, instead of only a quarter of the unit cell (see the dashed rectangle in Fig. 2) as typically done

in the case of FEA representations where symmetric boundary conditions are applied. Apart from this

x3

x2

x1

df/2

fl

b

a

c

e

Figure 2: Aligned short fiber GMC unit cell model, comprised of 338 subcells.

fact, the unit cell dimensions were chosen similar to the dimensions of the FE-model (see Fig. 3). The fiber

was approximated with more than one subcell (sufficient for the longitudinal stiffness behavior) in order to

reduce the error in the transverse response (i.e., the difference between E22 and E33) and better approximate

the targeted transversely isotropic response behavior resulting from a hexagonal packing arrangement. The

cross sectional area of the approximated fiber is taken equal to the cross sectional area of the actual circular

fiber. These conditions, together with the known fiber diameter, fiber volume fraction, and the distance, e

(see Fig. 3) suffice to determine the dimensions of the unit cell for each fiber aspect ratio ar=lf /df . All

GMC analyses were realized with the computer code MAC/GMC [Arnold et al.(1999)] developed at the

NASA Glenn Research Center.

3.4 Finite Element Models

The finite element unit cell analyzed consisted of an identical periodic, three dimensional non-staggered

array of fibers (similar to that idealized with GMC) with the corresponding dimensions shown in Fig. 3. The

unit cell dimension were chosen such that b would be proportional to a (i.e., b = βa) through the constant

β. In the case of hexagonal packing, β = √3, and the distance between neighboring fiber ends (end-to-end

distance, equal to c − lf in Fig. 3) was set to the closest distance between two fibers (side-to-side distance,

equal to e in Fig. 3).

These conditions, together with the fiber diameter and volume fraction, suffice to determine the dimension

a, b and c for the repeating volume element (RVE) to be analyzed. Note that a new unit cell and its

corresponding 3D mesh are generated for each fiber aspect ratio (l/d) and fiber volume fraction (vf ).

NASA/TM—2001-211165 7
x3

x1

df

x2

/2

fl /2

a/2

c/2

b/2 e

GMC unit cell FE unit cell

e/2

a

b

c

Figure 3: Example representative volume element and finite element unit cell for aR = 5.0 and VF = 0.2

Stiffnesses within the elastic and plastic range of these unit cells were calculated using [ABAQUS (1998)].

Fully integrated, first order, solid elements2 were used, as this element type does not suffer from volumetric

locking, which is critical for accurate elastoplastic calculations. The mesh discretization containing 13,300

elements employed is shown in Fig. 3. This mesh density is approximately three times finer than that used

by [Weissenbek and Rammerstorfer (1993)] and was shown during the course of this work (by convergency

studies) to be sufficiently accurate for both elastic and inelastic analysis. Modeling of only one eighth of the

whole RVE (depicted on the left) was achieved due to the use of appropriate symmetric boundary conditions.

For example, when considering axial or transverse loading, symmetry requires all faces of the unit cell to

remain plane. Consequently, to determine E11 and ν12 the normal displacements of the front (x1 = 0), left

(x2 = a/2) and bottom (x3 = 0) faces of the unit cell were fixed. Then a master node was defined (the

intersection point of the back, right and top face) and linear constraint equations applied on the back, right

and top face such that the normal displacements of all nodes (except the master node) on these faces were

set equal to the normal displacements of the master node. The tangential displacements on all faces were left

unconstrained. The average (marco) stress was then computed from the reaction force of the master node

divided by the cross-sectional area normal to the reaction force of the investigated unit cell. Similarly, the

average (macro) strain was computed by dividing the displacement of the master node by the corresponding

length of the unit cell. For convenience, a one Newton load was applied to the unit cell.

Analogous conditions were used to load the unit cell in the x2-direction to determine E22 and ν23.

The fifth and final independent material parameter for transversal isotropic material, the longitudinal shear

modulus G12, could also be determined using the unit cell shown in Fig. 3, assuming the appropriate periodic

boundary conditions were applied to each face. This more complicated analysis was not undertaken at this

time.

4 Results

4.1 Elastic Material Behavior

All of the analytical micromechanics theories used in this study predict full transversely isotropic properties. Transverse isotropy about the x1 axis implies that the stiffness behavior is the same for any loading

2 In [ABAQUS (1998)] this is the element known as C3D8.

NASA/TM—2001-211165 8
direction in the 2-3 plane; consequently, this not only requires that E22 = E33, but also that

G23 = E22

2(1 + ν23)

. (1)

Therefore, only five independent stiffness constants (e.g., E11, E22, ν12, ν23 and G12) need be determined for

the case of transverse isotropy. Periodic unit cells with hexagonal packing should be transversely isotropic

as there are six planes of symmetry. However, since periodic boundary conditions were not applied to the

FE unit cell, we were unable to obtain G12 from the FE investigations. Consequently, we will limit our

comparison for the various approaches to the prediction of E11, E22, ν12 and ν23 only.

Results of the normalized axial modulus (x1- direction, scaled by the matrix modulus) obtained from the

VCM-model (MTM-model), Hashin-Shtrikman estimates, GMC-, and FE-model are shown in Fig. 4 (DRX

like materials with a lower stiffness ratio, e.g., sr = 3) and Fig. 5 (PMC like material, with a higher stiffness

ratio, e.g., sr = 30) for numerous fiber aspect ratios and two fiber volume fractions, a low (vf = 0.2) and

a high (vf = 0.5). The Hashin-Shtrikman-, GMC- and FE-models were evaluated only at discrete aspect

ratios; whereas, the VCM-result were computed for many different aspect ratios and thus displayed with a

solid line. Note, for high stiffness ratios only the lower VCM-bound (which corresponds to the MTM) was

plotted. From Figs. 4 and 5 it is apparent that the influence of fiber aspect ratio on E11 is significant (Max.

error of 35%) for a composite with a high stiffness ratio whereas relatively insignificant (Max. error of 3%)

for materials with a low stiffness ratio. In both cases the Hashin-Shtrikman estimate lay as expected within

the upper and lower VCM bounds. The GMC- and FE-results are partially out of the Mori-Tanka based

VCM-bounds due to the different inclusion topology (ellipsoidal versus cylindrical). Generally speaking,

the GMC-model underestimates the longitudinal stiffness, particularly for high stiffness ratios, although the

trend is captured quite nicely. Alternatively, for materials with low stiffness ratios, good agreement between

GMC and Hashin-Shtrikman estimates as compared with FE-results are observed given a wide range of

aspect ratios and fiber volume fractions.

1 10 100

Aspect Ratio, aR

1

1.5

2

2.5

Axial Youngs Modulus Ratio, E 11/E

M

VF=0.5

VF

=0.2

VCM L.B.

VCM U.B.

HS-Estimates

FE-Results

GMC-Results

Figure 4: Comparison of axial modulus ratios for sr = EF

EM = 3

The physical reason for the difference between the FEA-results and those of GMC can be inferred from

Fig. 6, where the longitudinal stress distributions for a load in the x1-direction are shown. In both FEA

analyses shown in Fig. 6, the geometry, boundary conditions, and applied forces are the same. Only the

stiffness ratio between matrix and fiber constituent material is different. Evidently, these stress contours

NASA/TM—2001-211165 9
1 10 100

Aspect Ratio, aR

0

5

10

15

20

Axial Youngs Modulus Ratio, E 11/E

M

VF=0.5

VF

=0.2

MTM

HS-Estimates

FE-Results

GMC-Results

Figure 5: Comparison of axial modulus ratios for sr = EF

EM = 30

indicate that an increase in the stiffness ratio leads to a decrease in the matrix loading and conversely to

an increase in the fiber loading (see maximum and minimum stress values). Similarly, the difference of the

ìmeanî matrix stress in the volume of material between the two fiber ends as compared with the ìmeanî fiber

stress becomes greater as the stiffness ratio is increased. Considering, the GMC representation, the stresses

are constant and equal along a longitudinal fiber-matrix row. Therefore the error for lower stiffness ratios

is decreased, yet when the stiffness ratio increases so too does the error. Figure 6 also gives an explanation

as too why the GMC stiffness results are slightly under the FE-results in Fig. 4, as the GMC assumptions

lead to an unloading of the ìstiffî fiber and to an additional loading of the ìsoftî matrix.

Figures 7 and 8 illustrate the normalized transverse modulus results obtained from the various analytical

and numerical methods considered. Here as one would expect, the fiber aspect ratio has little, if any,

effect on the transverse modulus Also, in general the GMC results compare favorably (maximum error of

approximately 5%) with the FE-results throughout. For both stiffness ratios, the Hashin-Shtrikman estimates

are equal to the lower VCM bound (which in turn are equal to the original MTM-results). Noteworthy is the

fact that the GMC-results are much closer to the FE-results for the case of high stiffness ratios, low aspect

ratios, and high fiber volume fractions than the other analytical estimates. Similarly, as observed for the

longitudinal stiffness the qualitative trend of the FE results is better captured by GMC.

To help explain this observation, we examine the calculated transverse stress distribution obtained from

FEA, as shown in Fig. 9. Note, only the case for the low stiffness ratio (sr = 3) is shown in Fig. 9 as

the FE-results for the higher stiffness ratios appear similar. Relative to the FE-results for the longitudinal

loading case, the difference in the ìmeanî matrix stress of the volume of material between the two fibers

in the loading direction and that of the ìmeanî fiber stresses for the transverse loading case are small.

Consequently, the inaccuracy caused by having similar stress states in a given row or column of subcells

within GMC is small; thus explaining the lack of influence the stiffness ratio has on the GMC predictions

relative to those determined using an FEA unit cell.

Figures 10 and 11 show a comparison of the axial Poissonís ratios based on the various evaluation methods.

Again the comparison between FEA and GMC results are quite favorable, irrespective of aspect ratio and

stiffness ratio; the difference being approximately a maximum of 5.2%. Again, the GMC results follow the

qualitative trend of the FE-results much better than either the Hashin Shtrikman estimates or MTM-results,

especially for the case of high stiffness ratios and low fiber aspect ratios.

NASA/TM—2001-211165 10

1 10 100

Aspect Ratio, aR

1

1.2

1.4

1.6

1.8

2

2.2

Transverse Youngs Modulus Ratio, E 22/E

M

VF=0.5

VF

=0.2

VCM L.B.

VCM U.B.

HS-Estimates

FE-Results

GMC-Results

Figure 7: Comparison of transverse modulus ratios for sr = 3

1 10 100

Aspect Ratio, aR

1

1.5

2

2.5

3

3.5

Transverse Youngs Modulus Ratio, E 22/E

M

VF=0.5

VF

=0.2

MTM

HS-Estimates

FE-Results

GMC-Results

Figure 8: Comparison of transverse modulus ratios for sr = 30

NASA/TM—2001-211165 12

1 10 100

Aspect Ratio, a R

0.2

0.25

0.3

0.35

0.4

Axial Poisson Ratio, 12

MTM: VF=0.5

FE-Results: V F=0.5

HS-Estimates: V F=0.5

GMC-Results: V F=0.5

1 10 100

Aspect Ratio, a R

0.2

0.25

0.3

0.35

0.4

Axial Poisson Ratio, 12

MTM: V=0.2

FE-Results: V F=0.2

HS-Estimates: V F=0.2

GMC-Results: V F=0.2

Figure 11: Comparison of axial Poisson ratios for sr = 30

Similarly, results for the in-plane Poissonís ratio ν23 appear in Figs. 12 and 13. Once again, in general

the GMC results are slightly above those produced with FEA. The Hashin-Shtrikman estimates being closer

to the FE-results than the GMC results. The discrepancy between methods being more evident with higher

fiber volume fraction than higher stiffness ratios. Even so, the maximum error between the GMC predictions

and those of FEA is at most 4.8%.

4.2 Elastoplastic Material Behavior

In this section the elastoplastic stiffness behavior of aligned short-fiber composites will be analyzed using

primarily the GMC and the finite element method. A comparison of the resulting predictions should confirm

whether or not the GMC is capable of accurately simulating the elastoplastic stiffness behavior of DRX

materials. Additionally, results obtained from a semi-analytical elastoplastic shear lag model (see Appendix

B) are included as well to help illuminate any differences between the methods. The investigated model

parameters are similar to those of the elastic investigation, with the exception that only two fiber aspect

ratios (ar = 5 and 25) are examined. Furthermore we restrict ourselves to the case of longitudinal (x1−

direction) loading so as to allow the inclusion of shear lag results.

Figure 14 shows the effective longitudinal stress-strain curve produced by the GMC (dotted and dashed

line), FE unit cell (solid line), and shear lag (symbols) models for the special case of Vf = 0.2, ar = 5, sr = 3.

Clearly, a large difference in the longitudinal stiffness response exists, depending upon the analysis method

employed, with the GMC model predictions being significantly softer. The question before us is whether or

not the lack of shear coupling inherent in the GMC method is the primary cause for the softer elastoplastic

behavior. To address this question, we introduced a semi-analytical shear-lag-model (and shear-free model) as

described in Appendix B. A comparison of the shear-lag (open circles) and shear-free (open square symbols)

models with those of FE and GMC are also shown in Fig. 14.

Evidently, the 8-subcell (1 fiber subcell surrounded by 7 matrix subcells) GMC results and the shear-free

model results agree extremely well; whereas, those determined using the elastoplastic shear-lag model agree

with those coming from the FE unit cell model. Consequently, it may be concluded that:

1. The difference between the GMC- and FE-simulations is directly related to the lack of normal-shear

coupling inherent in the GMC formulation.

NASA/TM—2001-211165 14
1 10 100

Aspect Ratio, a R

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Transverse Poisson Ratio, 23

VF=0.2

VF

=0.5

MTM

HS-Estimates

FE-Results

GMC-Results

Figure 12: Comparison of transverse Poisson ratios for sr = 3

1 10 100

Aspect Ratio, a R

0.35

0.4

0.45

0.5

Transverse Poisson

Ratio, 

MTM: VF=0.2

HS-Estimates: V F=0.2

FE-Results: V F=0.2

GMC-Results: V F=0.2

1 10 100

Aspect Ratio, a R

0.35

0.4

0.45

0.5

Transverse Poisson

Ratio, 

MTM: VF=0.5

HS-Estimates: V F=0.5

FE-Results: V F=0.5

GMC-Results: V F=0.5

Figure 13: Comparison of transverse Poisson ratios for sr = 30

NASA/TM—2001-211165 15
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

200

400

600

Effective Longitudinal Stress, eff

FE-Model

GMC: 8 Subcells

GMC: 338 Subcells

Shear Lag Model

Shear Free Model

Figure 14: Comparison of FE-, GMC-, Shear-lag-, Shear-free-model results (sr = 3, Vf = 0.2, ar = 5)

2. The standard (without any modifications) GMC-model is not well suited for predicting the elastoplastic

response behavior of DRX materials, as the GMC-model predicts an overly soft plastic response.

3. The semi-analytical elastoplastic shear-free and shear-lag models introduced in Appendix B appear

fully capable of predicting both the GMC- and FEA-results. The primary disadvantage of this shearlag model being its îuniaxialî nature, that is, it can only provide the effective modulus (response) in

a single direction.

In addition to the pseudo-square3 fiber (8-subcell) GMC representation (dotted line) in Fig. 14, the

typical (338 subcell) circular fiber unit cell (dashed line) is shown in Fig. 14. Clearly, a discretization effect

is observed and is a result of the piecewise plastification of the matrix subcells between the two fiber ends

in the case of the finer GMC-model. This effect results in a softer response within the initial plastic range

(i.e., from 0.2 - 0.5% strain range). After plastification of all matrix subcells the effective hardening slope of

both GMC unit cells are essentially the same.

To verify the above conclusions, simulations with other aspect ratios, volume fractions and stiffness ratios

were conducted with the results being displayed in Figs. 15 - 18. The semi-analytical shear-free model is not

shown in these figures as its results always agreed with the GMC (pseudo square) simulations. The fitting

parameter ψ of the semi-analytical shear-lag-model is found to be only a function of the fiber volume fraction

i.e. ψ = ψ(Vf ) and takes a value of approximately 2 for Vf = 0.2 and approximately 2.5 for Vf = 0.5. The

dashed lines in all figures represent the effective stress-strain response of the standard circular GMC-model.

All figures confirm the above conclusions.

Some characteristic features of the effective finite element stress-strain curves in Figs. 15 - 18 are evident;

for example, increasing either the aspect ratio or volume fraction significantly hardens the composite response,

particularly for high stiffness ratio materials. The elastoplastic stiffness response may be partitioned into

three ranges. Figure 19 shows these ranges as a function of effective longitudinal strain.

Range 1, delineates the domain where linear elastic behavior is manifested. This is followed by range 2

in which a large variation in the hardening slope takes place and range 3 in which approximately a constant

3 In GMC the geometry of the subcell representing the fiber is square; however in actuality since all fields are taken at the

centroid of the subcell only the ratio of subcell to total length is important, no corners are felt. Thus the square fiber is really

a pseudo-square.

NASA/TM—2001-211165 16
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

200

400

600

800

Effective Longitu

dinal Stress, eff

aR=25

aR=5

FE-Models

Shear Lag Models

GMC-Models

a =25 R

aR =5

aR =5

a =25 R

Figure 15: Comparison of FE-, GMC- and Shear-lag-model results for sr = 3 and Vf = 0.2

0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

100

200

300

400

500

600

Effective Lon

gitu

dinal Stress, eff

aR=25

aR=5

FE-Models

Shear Lag Models

GMC-Models

aR =5

a =25 R

a =25 R

aR =5

Figure 16: Comparison of FE-, GMC- and Shear-lag-model results sr = 3 and Vf = 0.5

NASA/TM—2001-211165 17
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

50

100

150

Effective Longitu

dinal Stress, eff

aR=25

aR=5

FE-Models

Shear Lag Models

GMC-Models

aR =5

a =25 R

aR =5

a =25 R

Figure 17: Comparison of FE-, GMC- and Shear-lag-model results sr = 30 and Vf = 0.2

0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

100

200

300

400

500

600

Effective Longitu

dinal Stress, eff

aR=25

aR=5

FE-Models

Shear Lag Models

GMC-Models

Figure 18: Comparison of FE-, GMC- and Shear-lag-model results sr = 30 and Vf = 0.5

NASA/TM—2001-211165 18
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

200

400

600

Effective Longitudinal Stress, eff

Shear Lag Model

4

2

3

1

Range 2Range 1 Range 3

2

5

4

1

3

Number of yielded beams

Figure 19: Semi-analytical shear-lag model, characteristic ranges and number of yielded truss elements

(sr = 3, Vf = 0.2, ar = 5)

effective hardening slope is observed. More detailed investigations into the flow behavior show that at the

beginning of range 2 the matrix material between the two fiber ends begins to yield first (see Fig. 19, point

1). After which the whole matrix yields (point 2). During the longitudinal loading step the matrix from the

side-to-side region tries to move to the end-to-end region. This material flow leads to a considerable shear

loading of the side-to-side matrix interphase (between fiber and matrix). In the plastic range large shear

deformations of the interphase start at the fiber ends (point 3), then grow over the fiber and prevent any

additional loading of the fiber. In the case of the semi-analytical shear-lag-model this means that one by

one each non-linear truss element will start to yield. All trusses are plastic at point 4 (see Fig. 19), which is

the beginning of range 3. Note, the stair-stepped dashed line in Fig. 19 indicates the piecewise yielding of,

altogether, five truss elements.

Based on these results, the question remaining is whether the GMC can be appropriately modified in

order to predict accurately the elastoplastic behavior of DRX materials. Considering why the GMC fails,

it becomes clear that if the matrix between the two fiber ends yields, the fiber is no longer capable of

carrying any additional load beyond the matrix since both the matrix and fiber have the same stress state.

This continuity of specific stress components along a given row or column of subcells is a direct result of the

inherent lack of normal and shear coupling with GMC. Given this fact, a simple modification to GMC would

be the introduction of an artificial matrix material for all subcells within the region between the two fiber

ends. This new artificial matrix material must be chosen so that the resulting effective stress-strain curve

produced corresponds with the reference data (be it experimental or analytical). Of course this strategy

possesses the drawback that the obtained micro-mechanical GMC stress state is no longer based on any

physical meaning; i.e., fracture, damage or fatigue analyses could no longer be performed confidently when

using the modified GMC-results, as the microfield quantities would be even more questionable than before.

As a first attempt, the artificial matrix material was assumed to behave as purely elastic, with the

associated Youngís modulus of the artificial matrix material being that of the matrix itself. Clearly, such an

assumption results in the overestimation (at least well within the plastic range) of the elastoplastic response

curve as illustrated in Fig. 20. This response prediction is non-conservative. In contrast, it is shown that the

NASA/TM—2001-211165 19
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

200

400

600

Effective Longitudinal Stress, eff

FE-Model

GMC: Standard

GMC: modified, bilinear

GMC: modified, pure elastic

Figure 20: Comparison of FE-, GMC-, and GMC-modified simulations (sr = 3, Vf = 0.2, ar = 5).

standard GMC-model provides a lower bound (conservative response prediction) for the effective elastoplastic

behavior, thus suggesting that an artificial bilinear elastoplastic material might provide a more accurate

simulation. For such a material definition, four material parameters would be required as input; they are

1) Poissonís ratio, 2) Youngís modulus, 3) yield stress, and 4) the plastic hardening slope. With the yield

stress and hardening slope being the best candidates for modification from that of the matrix material, since

the elastic range is already accurately predicted, the parameters (E,ν) can be taken to be that of the matrix

material. Here we restrict ourselves to the modification of the hardening slope alone (keeping the yield

the same as the of the matrix) and thereby attempt to obtain the unknown artificial hardening slope E∗

H

from the consideration of a simple one dimensional spring model (with each spring corresponding to a given

subcell) which is shown in Fig. 21.

The following discussion is limited to range 3 in Fig. 19 (i.e., the range where the effective hardening

slope is approximately constant). Assume an incremental load, ∆F, is applied to the spring assemblage in

Fig. 21. This results in an incremental displacement, ∆u, that depends on both the geometry and material

properties of the spring assemblage, that is,

∆F = 1

lf

Ef Af

+ l−lf

E∗

HAf

+

1

l

EH(A−Af )

∆u, (2)

where A is the total cross sectional area of the considered unit cell and Af the fiber cross sectional area.

EH and E∗

H are the hardening slope and artificial hardening slope, respectively. Alternatively, the global

longitudinal stiffness behavior can be described by

∆F = Eeff

H A

l ∆u (3)

where Eeff

H is the effective longitudinal hardening slope. The combination of Eqs. (2) and (3) , and the

introduction of the following abbreviations,

s

f

r,H = Ef

EH

, seff

r,H = Eeff

H

EH

, lr = lf

l , Vf = Af lfAl (4)

NASA/TM—2001-211165 20
l

lf

EH

 F

 u

Af Ef

A

EH

EH

cial Matrix, * Matrix,

Matrix, Fiber, 

Figure 21: Spring assemblage model

lead to the final equation for the artificial hardening slope:

E∗

H

EH

=

s

f

r,H(1 − lr) s

eff

r,H + Vf

lr − 1

Vf (

sf

r,T

lr − 1) + lr(1 − s

eff

r,H )

. (5)

Note, the only unknown in this equation is the effective longitudinal hardening slope Eeff

H which for example

may be determined from the semi-analytical shear-lag model (see Appendix B) or obtained from some other

reference data.

A comparison of the FE-, GMC-, GMC-bilinear-modified-model response histories are shown in Figs. 22-

25. Evidently, as one might expect, the GMC-model, modified by using an artificial bilinear matrix material

definition in the intermediate subcell, results in a more accurate longitudinal simulation than the previous

standard GMC-model (see Fig. 14 and that modified using an artificial elastic matrix material definition).

With this approach, reasonably accurate results can be produced when range 2 is small, typically for high fiber

aspect ratios and low stiffness ratios. Alternatively, for situations involving high stiffness ratio materials,

low volume fractions and high aspect ratios (e.g., ar = 25, see Fig. 24) simulations using the modified

GMC approach give the largest error. This is due to the fact that these conditions induce the smallest area

(highly loaded) subcell between fibers and thus maximizes the flow between inclusions; thereby producing

the most compliant overall response. Clearly, only a small difference exists between FE and GMC results in

the extensive plastic range. This is due to the fitting procedure utilized for the simple spring model. Note

that, again all response histories are conservative when compared with FEA simulations.

More detailed studies were not performed at this time using a more complex artificial matrix material

definition, since the actual usefulness of such an ad-hoc approach is debatable due to the lack of any real

physical meaning. Furthermore, a new higher order GMC theory (appropriately termed - high fidelity

GMC), which provides the necessary normal and shear coupling, has recently been developed by [Aboudi et.

al.(2001)]. This new formulation, although more computationally demanding than the standard or modified

GMC (but significantly faster than the FEA approach) should provide the required balance between speed

and accuracy when attempting to simulate DRX material behavior. Consequently, further analysis using

the GMC will be reserved until later when the theoretical extension of the present elastic continuously

reinforced high fidelity GMC version (see [Aboudi et. al.(2001)]) to that of discontinuous and inelastic

material behavior, is accomplished.

NASA/TM—2001-211165 21
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

200

400

600

800

Effective Longitudinal Stress, eff

modified GMC-Models

FE-Models

aR=25

aR=5

Figure 22: Comparison of FE- and GMC-modified model (sr = 3, Vf = 0.2)

0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

500

1000

1500

2000

2500

Effective Longitudinal Stress, eff

aR=25 aR=5

modified GMC-Models

FE-Models

Figure 23: Comparison of FE- and GMC-modified model simulations (sr = 3, Vf = 0.5)

NASA/TM—2001-211165 22
0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

50

100

150

200

Effective Longitudinal Stress, eff

modified GMC-Models

FE-Models

aR=25

aR=5

Figure 24: Comparison of FE- and GMC-modified model simulations (sr = 30, Vf = 0.2)

0 0.005 0.01 0.015 0.02

Effective Longitudinal Strain, eff

0

100

200

300

400

500

600

Effective Longitudinal Stress, eff

aR=25

aR=5

modified GMC-models

FE-Models

Figure 25: Comparison of FE- and GMC-modified model simulations (sr = 30, Vf = 0.5)

NASA/TM—2001-211165 23
5 Conclusions

This paper has focused on evaluating the predictive ability of the generalized method of cells, in the

context of discontinuously reinforced composite (DRX) material behavior; specifically aligned short fiber

composites. GMC simulations were compared with: i) simulations using other micromechanical based mean

field models and finite element (FE) unit cell models found in the literature for the elastic range, as well

as ii) finite element unit cell model and a new semi-analytical elastoplastic shear lag model in the inelastic

range. Results indicated that the GMC is fully capable of accurately analyzing the elastic behavior of DRX

materials. However, in the case of inelastic behavior, the standard GMC approach, due to the inherent lack

of normal-shear coupling, was shown to be unsuitable for predicting the longitudinal elastoplastic behavior

of aligned short fiber reinforced composites. However, a simple modification of the GMC-model, through

the introduction of an îartificialî bilinear matrix material definition within the subcell(s) residing between

inclusion (fiber) ends was shown to improve the overly conservative GMC predictions within the elastoplastic

regime significantly. This ad-hoc modification to the GMC unit cell definition would allow reasonably

accurate multiaxial simulations to be made. The case when both the stiffness and fiber aspect ratios are

high being the least accurate. Finally, it was shown herein that an elastoplastic shear lag model can

provide identical uniaxial results to that of the FE unit cell calculations. For truly consistent and accurate

results it is recommended that a high fidelity micromechanics approach be used when examining inelastic

discontinuously reinforced composite behavior.

References

ABAQUS (1998). Standard Userís Manual, Version 5.8. Hibbit, Karlsson & Sorensen, Inc., Pawtucket, RI,

USA.

Abel, C. A., Weissenbek, E. and Rammerstorfer, F. G. (1993). Influence of damage on the mechanical

behavior of short fiber reinforced MMCs with strong and weak interfaces. ZAMM 73, pp. T423óT426.

Aboudi, J. (1991) Mechanics of Composite Materials, A Unified Micromechanical Approach, Studies in

Applied Mechanics, 29, Elsevier.

Aboudi, J. (1995). Micromechanical analysis of thermo-inelastic multiphase short-fiber composites. Comp.

Eng. 5, pp. 839ó850.

Aboudi, J. (1996). Micromechanical analysis of composites by the method of cells - Update. Appl. Mech.

Rev. 49, No. 10, pp. S83óS91.

Aboudi, J., Pindera, M.J. and Arnold, S.M. (2001) Thermoelastic Higher-Order Theory for Periodic

Multiphase Materials. Jnl. Of Applied Mech., to appear 2001

Arnold, S. M.; Pindera, M. J.; and Wilt, T.E. (1996) Influence of Fiber Architecture On The Inelastic

Response of Metal Matrix Composites. Int. Jnl. of Plasticity, 12, No. 4, pp. 507-545.

Arnold, S. M., Bednarcyk, B. A., Wilt, T. E. and Trowbridge, D. (1999). Micromechanical analysis code

with generalized method of cells (MAC/GMC), user guide: version 3.0. NASA/TM-1999-209070, Cleveland,

USA.

Ashton, J. E., Halpin, J. C. and Petit, P. H. (1969). Primer on composite materials: Analysis. Technomic

Publishing Co, Stamford, Conn. pp. 77ó81.

NASA/TM—2001-211165 24
Bao, G., Hutchinson, J. W. and McMeeking, R. M. (1991). Particle reinforcement of ductile matrices

against plastic flow and creep. Acta Metall. Mater. 39, pp. 1871ó1882.

Bednarcyk, B. A., and Arnold, S. M. (2000a) A New Local Failure Model With Application to the Longitudinal Tensile Behavior of Continuously Reinforced Titanium CompositesNASA/TM 2000-210027.

Bednarcyk, B. A., and Arnold, S. M. (2000b) A New Local Debonding Model With Application to the

Transverse Tensile and Creep Behavior of Continuously Reinforced Titanium Composites NASA/TM 2000-

210029

Benveniste, Y. (1987). New approach to the application of Mori-Tanakaís theory in composite materials.

Mech. Mat. 6, pp. 147ó157.

Brown, L. M. and Clarke, D. R. (1975). Work hardening due to internal stresses in composite materials.

Acta Metall. 23, pp. 821ó830.

Christman, T., Needleman, A. and Suresh, S. (1989). An experimental and numerical study of deformation

in metal-ceramic composites. Acta Metall. Mater. 37, pp. 3029ó3050.

Cox, H. L. (1952). The elasticity and strength of paper and other fibrous materials. Brit. J. Appl. Phys 3,

pp. 72ó79.

Clyne, T.W. and Withers, P.J. (1993) An Introduction To Metal Matrix Composites, Cambridge

University Press.

Dow, N. F. (1963). No. R63-SD61 . Technical report, General Electric Report.

Dragone, T. L. and Nix, W. D. (1960). Geometric factors affecting the internal stress distribution and high

temperature creep rate of discontinuous fiber reinforced metals. Acta Metall. Mater. 38, pp. 1941ó1953.

Eshelby, D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems.

Proc. Royal Soc. (London) A 241, pp. 376ó396.

Goldberg, R.K and Arnold, S. M. (2000) A Study of Influencing Factors on the Tensile Response of a

Titanium Matrix Composite with Weak Interfacial BondingNASA/TM 2000-209798.

Hill, R. (1965). A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, pp. 213ó222.

Hu, G. K. and Weng, G. J. (2000). The connections between the double-inclusion model and the Ponte

Castaneda-Willis, Mori-Tanaka, and Kuster-Toksoz models. Mech. Mat. 32, pp. 495ó503.

Iyer, S.K., Lissenden, C.J., and Arnold, S.M. (2000) Local and overall flow in composites predicted by

micromechanics Composites: Part B, 31, pp. 327-343.

Kelly, A. and Zweben, C. (Eds) (2000) Comprehensive Composite Materials, Volume 3: Metal Matrix

Composites ISBN: 0-080437214, Pergamon.

Levy, A. and Papazian, J. M. (1990). Tensile properties of short fiber-reinforced SiC/Al composites - Part

II: Finite-element analysis. J. Appl. Mech. 21A, pp. 411ó420.

Lissenden, C.J., Arnold, S.M., and Iyer S.K. (2000) Flow/damage Surfaces for Fiber-Reinforced Metals

having Different Periodic Microstructures Int. J. Plasticity, 16, No. 9, pp. 1049-1074.

Lloyd, D.J. (1994) Particle Reinforced Aluminium and Magnesium Matrix Composites, Int. Material

Reviews, 39, No. 1, pp. 1-23

Miracle, D.B. and Maruyama, B. (2000) Metal Matrix Composites for Space Systems: Current Uses and

Future Opportunities, Proc. National Space and missile Materials Symposium, Ed., M. Stropki, Dayton,

OH: Anteon Corp., pgs 25.

NASA/TM—2001-211165 25
Mori, T. and Tanaka, K. (1973). Average stress in the matrix and average elastic energy of materials with

misfitting inclusions. Acta Metall. 21, pp. 571ó574.

Paley, M. and Aboudi, J. (1992). Micromechanical analysis of composites by the generalized cells model.

Mech. Mat. 14, pp. 127ó139.

Pindera, M-J, and Bednarcyk, B.A., (1999) An Efficient Implementation of the Generalized Method of

Cells for Unidirectional, Multi-Phased Composites With Complex Microstructures Composites Part B, 30,

No. 1, pp. 87-105.

Pedersen, O. B. and Withers, P. J. (1992). Iterative estimates of internal stresses in short-fiber metal matrix

composite. Phil. Mag. A65, pp. 1217ó1233.

Ponte CastaÒeda, P. and Willis, J. R. (1995). The effect of spatial distribution on the effective behavior of

composite materials and cracked media. J. Mech. Phys. Solids 43, pp. 1919ó1951.

Povirk, G. L., Needleman, A. and Nutt, S. R. (1990). An analysis of residual stress formation in whiskerreinforced Al-SiC composites. J. Mat. Sci. and Eng. A125, pp. 129ó140.

Siegmund, T., Weissenbek, E., Fischer, F. D. and Rammerstorfer, F. G. (1992). Micromechanical consideration of topological aspects regarding thermocyclic behavior. in The Processing, Properties and Applications

of Metallic and Ceramic Materials II, pp. 971ó976.

Sinclair, I. and Gregson, P.J. (1997) Structural Performance of Discontinuous Metal Matrix Composites,

Mat. Sci. and Tech., 13, pp. 709-726.

Tucker, C. L. and Liang, E. (1999). Stiffness predictions for unidirectional short-fiber composites: Review

and evaluation. Comp. Sci. and Tech. 59, pp. 655ó671.

Tvergaard, V. (1990). Analysis of tensile properties for a whisker-reinforced metal-matrix composite. Acta

Metall. Mater. 38, pp. 185ó194.

Weissenbek, E., Bˆhm, H. J. and Rammerstorfer, F. G. (1993). Microgeometrical effects on the elastoplastic

behavior of particle reinforced MMCs. In: Proc. of 9th Int. Conf. on Composite Materials (ICCM/9 í93),

July 12ó16, 1993, Madrid, Spain.

Weissenbek, E. and Rammerstorfer, F. G. (1993). Influence of the fiber arrangement on the mechanical

and thermo-mechanical behavior of short fiber reinforced mmcs. Acta Metall. Mater. 41, pp. 2811ó2843.

Weng, G. J. (1990). The theoretical connection between Mori-Tanaka theory and Hashin-Shtrikman-Walpole

bounds. Int. J. Eng. Sci. 28, pp. 1111ó1120.

Willis J. R., (1977). Bounds and Self-Consistent Estimates for the Overall Properties of Anisotropic Composites; J. Mech. Phys. Sol. 25, 185-202.

Yang, J., Pickard, S. M., Cady, C., Evans, A. G. and Mehrabian, R. (1991). The stress-strain behavior of

aluminum matrix composites with discontinuous reinforcements. Acta Metall. Mater. 39, pp. 1863ó1869.

NASA/TM—2001-211165 26
6 Appendix A: Generalized Method of Cells Formulation

GMC can predict the elastic and inelastic thermomechanical response of both continuous and discontinuous multiphased composite materials with an arbitrary internal microstructure and reinforcement shape.

It is a continuum-based micromechanics model that provides closed-form expressions for the macroscopic

composite response in terms of the properties, size, shape, distribution, and response of the individual constituents or phases that make up the material4 . These constituent materials can be represented using any

elastic and/or inelastic deformation and life (e.g., continuum damage mechanics fatigue model) model. The

periodic nature of composites typically allows identification of a repeating unit cell that can be used as a

building block to construct the entire composite. The properties of this unit cell are thus representative of

the properties of the entire assemblage once the subcell dimensions and appropriate subcell materials are

selected. Thus, unidirectional long-fiber composites, short-fiber composites, porous materials and laminated

materials can all be modeled as special cases.

6.1 Model Description

Consider a composite material with a periodic structure whose repeating volume element consists of

Nα × Nβ × Nγ rectangular parallelepiped subcells. The volume of each one of the subcells is dαhβ%γ, where

α, β and γ are running indices: α = 1,...,Nα ; β = 1,...Nβ ; γ = 1,...,Nγ, in the x1 , x2 and x3 -

directions, respectively. The total volume of the repeating volume element is dh% where d = Nα

α=1 dα ,

h = Nβ

β=1 hβ , % = Nγ$=1 %γ. In Fig. 26, an example is shown for a repeating volume element with Nα = 2,

Nβ = 3 and Nγ = 3.

l

x3

x1 x2

l2

h1

h3

h2

l3

l1 d 2

1







d

d h

Subcell 1





 



Figure 26: Triply periodic repeating unit cell, illustrating subcell dimension nomenclature.

By an approximate micromechanical analysis of the detailed interactions of the subcells of the repeating

volume element, overall constitutive relations which govern the effective behavior of the multiphase elastoplastic composite, can be established. This analysis relies on the requirements that static equilibrium of the

materials in the various subcells are ensured, and that continuity of the displacements and tractions between

neighboring subcells within the repeating volume element, as well as between neighboring repeating volume

elements is satisfied on an average basis.

4Note, each of the subcells can be comprised, in general, by an elastic-viscoplastic temperature -dependent material.

NASA/TM—2001-211165 27
As the average behavior of the composite is sought, it is sufficient to consider a first order theory in which

the displacements u(αβγ)

i in the subcells are expanded linearly in terms of the distances from the center of

each subcell, i.e., in terms of the local coordinates xØ

(α)

1 , xØ

(β)

2 , and xØ

(γ)

3 . Thus, the first order expansion in

the subcell (αβγ) is represented as,

u(αβγ)

i = w(αβγ)i (x)+Øx(α)

1 φ(αβγ)

i + Øx(β)

2 χ(αβγ)

i + Øx(γ)

3 ψ

(αβγ)

i i = 1, 2, 3 (A-1)

where w(αβγ)

i (x) are the displacement components at the center of the subcell, and φ(αβγ)i , χ(αβγ)i and

ψ(αβγ)

i are micro-variables that characterize the linear dependence of the displacement u(αβγ)i on the local

coordinates xØ

(α)

1 , xØ

(β)

2 , xØ

(γ)

3 . In Eq. (A-1) and the sequel, repeated Greek letters do not imply summation.

Note that due to the linearity of Eq. (A-1) , static equilibrium of the material within the subcell (αβγ) is

ensured.

The components of the small strain tensor are given by

*

(αβγ)

ij = 1

2

(∂iu(αβγ)

j + ∂ju(αβγ)i ) i, j = 1, 2, 3 (A-2)

where ∂1 = ∂/∂xØ

(α)

1 , ∂2 = ∂/∂xØ

(β)

2 and ∂3 = ∂/∂xØ

(γ)

3 .

The volume averaged total strains and stresses in the composite are expressed, respectively, as

Ø*ij = 1

dh%

Nα

α=1

Nβ

β=1

Nγ

γ=1

dαhβ%γ Ø*

(αβγ)

ij (A-3)

and

σØij = 1

dh%

Nα

α=1

Nβ

β=1

Nγ

γ=1

dαhβ%γ σØ(αβγ)

ij (A-4)

with the relationship between the averages of the stress σØ(αβγ)

ij , total strain Ø*

(αβγ)

ij , inelastic strain Ø*

I(αβγ)

ij ,

and thermal strain Ø*

T(αβγ)

ij = α(αβγ)ij ∆T ( where α(αβγ)ij is the coefficient of thermal expansion tensor, and ∆T

is the temperature deviation T − TR from a reference temperature TR) being given by the classic Hookean

constitutive equation

σØ(αβγ)

ij = C(αβγ)

ijk$ (Ø*

(αβγ)

k$ − Ø*

I(αβγ)

k$ − Ø*

T(αβγ)

k$ ) (A-5)

where C(αβγ)

ijk$ is the elastic stiffness tensor of the material.

It has been shown ([Paley and Aboudi (1992)] and [Aboudi(1995)]) that by employing the displacement

and traction continuity conditions at the interfaces between the subcells of the repeating volume element,

and at the interfaces between neighboring repeating volume elements, it is possible to eliminate the microvariables and obtain, via a smoothing operation (homogenization), a set of continuum equations that model

the overall behavior of the multiphase short-fiber composite. This was achieved by establishing relationships

which connect the microstrains at the subcells to the total overall macrostrains in the composite via the

appropriate concentration tensors, A and D.

6.2 Overall Thermomechanical Constitutive Law

Given these concentration matrices, A(αβγ) and D(αβγ), expressions for the average total strain and

stress, respectively, in each subcell can be constructed; that is,

Ø!(αβγ) = A(αβγ)

Ø! + D(αβγ)

(!I

s + !Ts ) (A-6)

and

σØ(αβγ) = C(αβγ) A(αβγ)

Ø! + D(αβγ)

(!I

s + !Ts ) − (Ø!I(αβγ) + Ø*

T(αβγ)

) (A-7)

NASA/TM—2001-211165 28
in terms of the uniform overall strain Ø! (the applied macrostrain) and the subcell inelastic and thermal

strains, where

A = AM

AG

−1 0

J , D = AM

AG

−1 AM

0

and AM involves the elastic properties C(αβγ) of the subcell material, AG the geometric dimensions of the

repeating cell only, and J is a matrix containing the imposed average (composite) strains. Note, A(αβγ)

and D(αβγ) are square matrices of 6NαNβNγ order. Similarly, notice that the same matrices D(αβγ) are

operating on both the inelastic and thermal strains of the subcells.

It was shown by Dvorak and Benveniste (1992) and Dvorak (1992) that for any representative volume

element under a uniform overall strain Ø! and temperature change ∆T, which contains a piecewise uniform

distribution of thermal and inelastic fields (eigenstrains) associated with the applied loading Ø! , the averages

of the local strains can be expressed in terms of the mechanical and eigenstrain concentration tensors. These

tensors depend on the local elastic moduli, and on the shape and volume fraction of the phases and are

therefore constant. Equation (A-6) is consistent with this representation of the average subcell strain, where

the constant mechanical and eigenstrain concentration tensors are given by A(αβγ) and D(αβγ)

, respectively.

Consequently, the following effective elastoplastic thermomechanical law of the composite can be established

σØ = B∗(Ø! − Ø!I − Ø!T ) (A-8)

where the effective elastic stiffness tensor, B∗, of the composite is given by

B∗ = 1

dh%

Nα

α=1

Nβ

β=1

Nγ

γ=1

dαhβ%γC(αβγ)A(αβγ) (A-9)

and the composite inelastic strain tensor is defined as

Ø!I = −B∗−1

dh%

Nα

α=1

Nβ

β=1

Nγ

γ=1

dαhβ%γC(αβγ)(D(αβγ)*

I

s − Ø!I(αβγ)

) (A-10)

and the average thermal strain tensor as

Ø!T = −B∗−1

dh%

Nα

α=1

Nβ

β=1

Nγ

γ=1

dαhβ%γC(αβγ)(D(αβγ)!T

s − Ø!T(αβγ)

)) (A-11)

with subcell inelastic, *I

s, and thermal, *Ts , strains being defined as:

!I

s = (Ø!I(111),..., Ø!I(NαNβNγ )

) (A-12)

!T

s = (Ø!T(111),...,Ø*

T(NαNβNγ )

) (A-13)

The effective coefficient of thermal expansion vector, α∗ of the composite, is given by

α∗ = Ø!T /∆T (A-14)

The above micromechanically established overall thermo-inelastic constitutive law (A-8) is valid for any

type of thermomechanical loading (i.e., any combination of normal, shear and thermal loadings). A significant

advantage of this constitutive law stems from the fact that it does not rely on any symmetry conditions that

may exist under certain types of applied loadings. Thus in the implementation of this law, the question as

to whether such symmetry conditions exist or not is irrelevant.

NASA/TM—2001-211165 29

7 Appendix B: Semi-analytical Axisymmetric Elastoplastic Shear

Lag Model

Given the elastoplastic investigations under taken in this study, a simple shear-lag model for aligned

short fiber composites is introduced. As mentioned in the background section, shear lag models were the

first micromechanics models considered for aligned short-fiber composites. Despite some serious theoretical

flaws (e.g., the ability to predict only the longitudinal modulus E11 within the elastic range), shear lag models

have enjoyed enduring popularity; probably due to their algebraic simplicity and their physical appeal. In

this paper, we have purposefully limited our investigation to the longitudinal elastoplastic stiffness behavior

so that the proposed modified shear lag model can be appropriately utilize.

Following Cox (1952), the shear lag analyses are focussed on a single fiber of length lf and radius rf ,

which is encased in a concentric cylindrical shell of radius R. Such a configuration then leads to the key

assumption of a shear lag model, which is that the shear stress between fiber and matrix (τ rz, r, φ,z...

cylindrical coordinate system) is proportional to the difference in displacement between the fiber surface and

the outer matrix surface, i.e.,

τ rz(z) = H

2πrf

w(R, z) − w(rf , z) (B-1)

where H is a constant and can be written as

H = 2πGm

ln(R/rf ) − 1

. (B-2)

It remains to choose the radius R of the outer matrix cylinder. Several choices have been used (see e.g. [Tucker

and Liang (1999)]). For our purposes it appears most realistic to let R be equal to a radius corresponding

to the width of the FE or GMC unit cell. The above equations are based on a one-dimensional analysis of

a fiber surrounded by a matrix layer (i.e., the Poisson effects are neglected (taken to be zero)) and assumes

linear elastic material behavior. To extend this analysis into the elastoplastic regime of aligned short fiber

composites, the existing shear-lag model must be modified so as to:

1. account for elastoplastic material behavior and

2. the appropriate Poisson effect.

The elastoplastic material behavior is simply introduced by replacing Gm by the plastic tangent shear

modulus GH (ìshear hardening slopeî) and an incremental formulation of Eq. (B-1), i.e.,

dτ rz(z) = H

2πrf

dw(R, z) − dw(rf , z) (B-3)

with

H = 2πGH

ln(R/rf ) − 1

. (B-4)

This new ìshear hardening slopeî can then be obtained from the Prandtl-Reuss-equation. For a pure

shear load in the r-z-plane this equation can be written as

d*rz = 1

2Gm

dτ rz +

3

2

dτ rz

τ rzEH

τ rz (B-5)

and leads finally to the shear hardening slope

GH = dτ rz

d*rz

= ψ GmEH

EH + 3Gm

= const (B-6)

NASA/TM—2001-211165 31
Matrix Axial-symmetry

Fiber

III

I II

IV

Symmetry

IIIa

IIIb

IVa

IVb

IVcIIIc

1 1'

2 2'

3 3'

Figure 27: Sketch of the semi-analytical shear lag model

where GH is constant because EH is a constant. Note, within the plastic range a fitting parameter ψ is

also introduced. This fitting parameter will account for effects which can not be captured by such a simple

model.

The Poisson effect is taken into account by considering an axisymmetric model (shown in Fig. 27 using

an exploded view) with appropriate boundary conditions on the concentric cylindrical matrix shell.

The model posses the following characteristics:

1. The cross section of the axisymmetric model is divided into four regions (I, II, III, IV ). Further,

regions III and IV are subdivided into three subregions each, e.g., IIIa, IIIb, IIIc and IV a, IV b,

IV c (see Fig. 27).

2. Within all regions only normal stress is allowed (i.e., shear stresses are zero) such that rectangular

cross sections of individual regions remain rectangular after loading.

3. The longitudinal stress are constant within regions I and II and are piecewise constant within regions

III and IV (i.e., constant within each subregion).

4. The points 1-1í, 2-2í, etc. are connected with non-linear truss elements which characterize the shear

forces between the fiber and the matrix.

5. The stiffness behavior of each non-linear truss element, i, is based on Eq. (B-1), and follows from

Fi

thruss = τ rz 2πrf l

i

τ = H w(R, z) − w(rf , z) l

i

τ = Citruss w(R, z) − w(rf , z) (B-7)

where, e.g., for i = 1, l

1

τ is the distance between points 1 and 2 in Fig. 27 and represents a kind of

shear-influence length for truss 1. C1

truss represents the stiffness of truss 1 between points 1-1í. The

material behavior of the truss element is bilinear and is characterized by a ìyield forceî (which is based

on Trescaís yield condition for pure shear within the interface) and can be written as

Fi

truss,Y = σY

2

2πrf l

i

τ (B-8)

NASA/TM—2001-211165 32
as well as by a stiffness hardening slope which can be written as

Ci

truss,H = ψH liτ . (B-9)

The number of introduced truss elements depends on the desired aspect ratio. For ar = 5 three truss

elements were used, whereas for ar = 25 six trusses were applied.

6. Finally, in addition to the symmetry conditions shown in Fig. 27, symmetry is also required on the top

and right outer surface of the axisymmetric model (see Fig. 27).

The introduced semi-analytical shear lag model is solved numerically using the finite element method.

The so called semi-analytical shear-free model is obtained by assuming that the truss elements are

characterized by zero stiffness. Such a model represents a standard GMC-model.

NASA/TM—2001-211165 33
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E–13026

WU–708–72–42–00

38

The Applicability of the Generalized Method of Cells for

Analyzing Discontinuously Reinforced Composites

D.H. Pahr and S.M. Arnold

Micromechanics; Elastic; Plastic; Metal matrix composites; Analysis

Unclassified -Unlimited

Subject Categories: 24 and 39 Distribution: Nonstandard

D.H. Pahr, Institute of Lightweight Structures and Aerospace Engineering, Vienna University of Technology,

A–1040 Vienna, Austria, and S.M. Arnold, NASA Glenn Research Center. Responsible person, S.M. Arnold,

organization code 5920, 216–433–3334.

The paper begins with a short overview of the recent work done in the field of discontinuous reinforced composites,

focusing on the different parameters which influence the material behavior of discontinuous reinforced composites, as

well as the various analysis approaches undertaken. Based on this overview it became evident, that in order to investigate the enumerated effects in an efficient and comprehensive manner, an alternative approach to the computationally

intensive finite-element based micromechanics approach is required. Therefore, an investigation is conducted to

demonstrate the utility of utilizing the generalized method of cells (GMC), a semi-analytical micromechanics-based

approach, to simulate the elastic and elastoplastic material behavior of aligned short fiber composites. The results are

compared with (1) simulations using other micromechanical based mean field models and finite element (FE) unit

cell models found in the literature given elastic material behavior, as well as (2) finite element unit cell and a new semianalytical elastoplastic shear lag model in the inelastic range. GMC is shown to definitely have a window of applicability when simulating discontinuously reinforced composite material behavior.

November 2001