stiffness predictions for unidirectional short-fiber composites review and evaluation

Sti�ness predictions for unidirectional short-®ber composites:Review and evaluation

Charles L. Tucker III a,*, Erwin Liangb

aDepartment of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USAbGE Corporate Research and Development, Schenectady, NY 12301, USA

Received 9 July 1997; received in revised form 24 March 1998; accepted 22 June 1998

Abstract

Micromechanics models for the sti�ness of aligned short-®ber composites are reviewed and evaluated. These include the dilutemodel based on Eshelby's equivalent inclusion, the self-consistent model for ®nite-length ®bers, Mori±Tanaka type models,bounding models, the Halpin±Tsai equation and its extensions, and shear lag models. Several models are found to be equivalent tothe Mori±Tanaka approach, which is also equivalent to the generalization of the Hashin±Shtrikman±Walpole lower bound. The

models are evaluated by comparison with ®nite-element calculations which use periodic arrays of ®bers, and to Ingber and Papa-thanasiou's boundary element results for random arrays of aligned ®bers. The ®nite-element calculations provide E11, E22, �12, and�23 for a range of ®ber aspect ratios and packing geometries, with other properties typical of injection-molded thermoplastic matrix

composites. The Halpin±Tsai equations give reasonable estimates for sti�ness, but the best predictions come from the Mori±Tanakamodel and the bound interpolation model of Lielens et al. # 1999 Published by Elsevier Science Ltd. All rights reserved.

1. Introduction

This paper reviews and evaluates models that predictthe sti�ness of short-®ber composites. The overall goalof the research is to improve processing-property pre-dictions for injection-molded composites. The polymerprocessing community has made substantial progress inmodeling process-induced ®ber orientation, particularlyin injection molding, and these results are now routinelyused to predict mechanical properties. The purpose ofthis paper is to review the relevant micromechanics lit-erature, and to provide a critical evaluation of theavailable models. Real injection-molded compositesinvariably have misoriented ®bers of highly variablelength, but aligned-®ber properties are always calcu-lated as a prelude to modeling the more realistic situa-tion. Hence, we focus here on composites havingaligned ®bers with uniform length and mechanicalproperties. The modeling of composites with distribu-tions of ®ber orientation and ®ber length, and thetreatment of multiple types of reinforcement, will bediscussed in a subsequent paper [1].

In selecting models for consideration, we impose thegeneral requirements that each model must include thee�ects of ®ber and matrix properties and the ®bervolume fraction, include the e�ect of ®ber aspect ratio,and predict a complete set of elastic constants for thecomposite. Any model not meeting these criteria wasexcluded from consideration.

All of the models use the same basic assumptions:

. The ®bers and the matrix are linearly elastic, thematrix is isotropic, and the ®bers are either iso-tropic or transversely isotropic.

. The ®bers are axisymmetric, identical in shape andsize, and can be characterized by an aspect ratio `=d.

. The ®bers and matrix are well bonded at theirinterface, and remain that way during deforma-tion. Thus, we do not consider interfacial slip,®ber/matrix debonding or matrix micro-cracking.

Section 2 presents some important preliminary con-cepts, emphasizing strain-concentration tensors andtheir relationship to composite sti�ness. Section 3 thenreviews the various theories. Section 4 compares andevaluates the available models. We use ®nite elementcomputations of periodic arrays of short ®bers to pro-vide reference properties, since it has not proved possi-ble to create physical specimens with perfectly aligned

Composites Science and Technology 59 (1999) 655±671

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0266-3538(98)00120-1

* Corresponding author.

®bers. A subsequent paper [1] will compare model pre-dictions to experiments on well-characterized compo-sites with misaligned ®bers.

2. Preliminaries

2.1. Notation

Vectors will be denoted by lower-case Roman letters,second-order tensors by lower-case Greek letters, andfourth-order tensors by capital Roman letters. When-ever possible, vectors and tensors are written as boldfacecharacters; indicial notation is used where necessary.

A subscript or superscript f indicates a quantityassociated with the ®bers, and m denotes a matrixquantity. Thus, the ®bers have Young's modulus Ef andPoisson ratio �f , while the corresponding matrix prop-erties are Em and �m.

The symbol I represents the fourth-order unit tensor.C and S denote the sti�ness and compliance tensors,respectively, and � and " are the total stress and in®ni-tesimal strain tensors. Hence, the constitutive equationsfor the ®ber and matrix materials are:

�f � Cf"f ; �1�

�m � Cm"m: �2�

2.2. Average stress and strain

Let x denote the position vector. When a compositematerial is loaded, the pointwise stress ®eld ��x� andthe corresponding strain ®eld "�x� will be non-uniformon the microscale. The solution of these non-uniform®elds is a formidable problem. However, many usefulresults can be obtained in terms of the average stressand strain [2]. We now de®ne these averages.

Consider a representative averaging volume V.Choose V large enough to contain many ®bers, butsmall compared to any length scale over which theaverage loading or deformation of the composite varies.The volume-average stress � is de®ned as the average ofthe pointwise stress ��x� over the volume V

� � 1

V

�V

��x� dV: �3�

The average strain " is de®ned similarly.It is also convenient to de®ne volume-average stresses

and strains for the ®ber and matrix phases. To obtainthese, ®rst partition the averaging volume V into thevolume occupied by the ®bers Vf and the volume occu-pied by the matrix Vm. We consider only two-phasecomposites, so that

V � Vf � Vm: �4�

The ®ber and matrix volume fractions are simplyvf � Vf=V and vm � Vm=V and, since only ®bers andmatrix are present, vm � vf � 1.

The average ®ber and matrix stresses are the averagesover the corresponding volumes,

�f � 1

Vf

�Vf

��x� dV and �m � 1

Vm

�Vm

��x� dV: �5�

The average strains for the ®ber and matrix are de®nedsimilarly.

The relationships between the ®ber and matrixaverages and the overall averages can be derived fromthe preceding de®nitions; they are:

� � vf�f � vm�

m �6�" � vf"

f � vm"m: �7�

An important related result is the average strain theo-rem. Let the averaging volume V be subjected to surfacedisplacements u0�x� consistent with a uniform strain "0.Then the average strain within the region is

" � "0: �8�This theorem is proved [2] by substituting the de®nitionof the strain tensor " in terms of the displacement vectoru into the de®nition of average strain ", and applyingGauss's theorem. The result is

"ij � 1

V

�S

u0i nj � niu0j

� �dS �9�

where S denotes the surface of V and n is a unit vectornormal to dS. The average strain within a volume V iscompletely determined by the displacements on the sur-face of the volume, so displacements consistent with auniform strain must produce the identical value ofaverage strain. A corollary of this principle is that, if wede®ne a perturbation strain "C�x� as the di�erencebetween the local strain and the average,

"C�x� � "�x� ÿ " �10�then the volume-average of "C�x� must equal zero

"C � 1

V

�V

"C�x� dV � 0 �11�

The corresponding theorem for average stress alsoholds. Thus, if surface tractions consistent with a �0 areexerted on S then the average stress is

� � �0: �12�

2.3. Average properties and strain concentration

The goal of micromechanics models is to predict theaverage elastic properties of the composite, but eventhese need careful de®nition. Here we follow the directapproach [3]. Subject the representative volume V to

656 C.L. Tucker III, E. Liang /Composites Science and Technology 59 (1999) 655±671

surface displacements consistent with a uniform strain"0; the average sti�ness of the composite is the tensor Cthat maps this uniform strain to the average stress.Using Eq. (8) we have

� � C": �13�The average compliance S is de®ned in the same way,applying tractions consistent with a uniform stress �0

on the surface of the averaging volume. Then, using Eq.(12),

" � S�: �14�It should be clear that S � Cÿ1. Other authors de®nethe average sti�ness and compliance through the inte-gral of the strain energy over V; this is equivalent to thedirect approach [2,4].

An important related concept, ®rst introduced by Hill[2], is the idea of strain- and stress-concentration tensorsA and B. These are essentially the ratios between theaverage ®ber strain (or stress) and the correspondingaverage in the composite. More precisely:

"f � A"; �15��f � B�: �16�

A and B are fourth-order tensors and, in general, theymust be found from a solution of the microscopic stressor strain ®elds. Di�erent micromechanics models pro-vide di�erent ways to approximate A or B. Note that Aand B have both the minor symmetries of a sti�ness orcompliance tensor, but lack the major symmetry. Thatis,

Aijkl � Ajikl � Aijlk; �17�but in general,

Aijkl 6� Aklij: �18�For later use it will be convenient to have an alternatestrain concentration tensor A that relates the average®ber strain to the average matrix strain,

"f � A"m: �19�This is related to A by

A � A �1ÿ vf �I� vf Ah iÿ1

�20�

so the two forms are easily interchanged.Using equations now in hand, one can express the

average composite sti�ness in terms of the strain-con-centration tensorA and the ®ber andmatrix properties [2].Combining Eqs. (1), (2), (6), (7), (13) and (15), one obtains

C � Cm � vf Cf ÿ Cm� �A: �21�The dual equation for the compliance is

S � Sm � vf Sf ÿ Smÿ �

B: �22�

Eqs. (21) and (22) are not independent, since S � C� �ÿ1.Hence, the strain-concentration tensor A and the stress-concentration tensor B are not independent either. Thechoice of which one to use in any instance is a matter ofconvenience.

To illustrate the use of the strain-concentration andstress-concentration tensors, we note that the Voigtaverage corresponds to the assumption that the ®berand the matrix both experience the same, uniformstrain. Then "f � ", A � I, and from Eq. (21) the com-posite modulus is

CVoigt � Cm � vf Cf ÿ Cmÿ � � vfC

f � vmCm: �23�

Recall that the Voigt average is an upper bound on thecomposite modulus. The Reuss average assumes thatthe ®ber and matrix both experience the same, uniformstress. This means that the stress-concentration tensor Bequals the unit tensor I, and from (22) the compliance is

SReuss � Sm � vf Sf ÿ Smÿ � � vfS

f � vmSm: �24�

This represents a lower bound on the sti�ness of thecomposite.

3. Theories

3.1. Eshelby's equivalent inclusion

A fundamental result used in several di�erent modelsis Eshelby's equivalent inclusion [5,6]. Eshelby solved forthe elastic stress ®eld in and around an ellipsoidal par-ticle in an in®nite matrix. By letting the particle be aprolate ellipsoid of revolution, one can use Eshelby'sresult to model the stress and strain ®elds around acylindrical ®ber.

Eshelby ®rst posed and solved a di�erent problem,that of a homogeneous inclusion (Fig. 1). Consider anin®nite solid body with sti�ness Cm that is initiallystress-free. All subsequent strains will be measured fromthis state. A particular small region of the body will becalled the inclusion, and the rest of the body will becalled the matrix. Suppose that the inclusion undergoes

Fig. 1. Eshelby's inclusion problem. Starting from the stress-free state

(a), the inclusion undergoes a stress-free transformation strain "T (b).

Fitting the inclusion and matrix back together (c) produces the strain

state "C�x� in both the inclusion and the matrix.

C.L. Tucker III, E. Liang /Composites Science and Technology 59 (1999) 655±671 657

some type of transformation such that, if it were aseparate body, it would acquire a uniform strain "T withno surface traction or stress. "T is called the transfor-mation strain, or the eigenstrain. This strain might beacquired through a phase transformation, or by a com-bination of a temperature change and a di�erent ther-mal expansion coe�cient in the inclusion. In fact theinclusion is bonded to the matrix, so when the trans-formation occurs the whole body develops some com-plicated strain ®eld "C�x� relative to its shape before thetransformation. Within the matrix the stress �m is sim-ply the sti�ness times this strain,

�m�x� � Cm"C�x� �25�but within the inclusion the transformation strain doesnot contribute to the stress, so the inclusion stress is

�I � Cm "C ÿ "Tÿ �

: �26�The key result of Eshelby was to show that within anellipsoidal inclusion the strain "C is uniform, and isrelated to the transformation strain by

"C � E"T: �27�E is called Eshelby's tensor, and it depends only on theinclusion aspect ratio and the matrix elastic constants.A detailed derivation and applications are given byMura [7], and analytical expressions for Eshelby's ten-sor for an ellipsoid of revolution in an isotropic matrixappear in many papers [8±12]. The strain ®eld "C�x� inthe matrix is highly non-uniform [13], but this morecomplicated part of the solution can often be ignored.

The second step in Eshelby's approach is to demon-strate an equivalence between the homogeneous inclu-sion problem and an inhomogeneous inclusion of thesame shape. Consider two in®nite bodies of matrix, asshown in Fig. 2. One has a homogeneous inclusion withsome transformation strain "T; the other has an inclu-sion with a di�erent sti�ness Cf , but no transformationstrain. Subject both bodies to a uniform applied strain"A at in®nity. We wish to ®nd the transformation strain"T that gives the two problems the same stress andstrain distributions.

For the ®rst problem the inclusion stress is just Eq.(26) with the applied strain added,

�I � Cm "A � "C ÿ "Tÿ � �28�

while the second problem has no "T but a di�erentsti�ness, giving a stress of

�I � Cf "A � "Cÿ �

: �29�Equating these two expressions gives the transformationstrain that makes the two problems equivalent. UsingEq. (27) and some rearrangement, the result is

ÿ Cm � Cf ÿ Cmÿ �

E� �

"T � Cf ÿ Cmÿ �

"A: �30�Note that "T is proportional to "A, which makes thestress in the equivalent inhomogeneity proportional tothe applied strain.

3.2. Dilute Eshelby model

One can use Eshelby's result to ®nd the sti�ness of acomposite with ellipsoidal ®bers at dilute concentra-tions. Recall from Eq. (21) that to ®nd the sti�ness oneonly has to ®nd the strain-concentration tensor A. Todo this, ®rst note that for a dilute composite the averagestrain is identical to the applied strain,

" � "A; �31�since this is the strain at in®nity. Also, from Eshelby,the ®ber strain is uniform, and is given by

"f � "A � "C; �32�where the right-hand side is evaluated within the ®ber.Now write the equivalence between the stresses in thehomogeneous and the inhomogeneous inclusions, Eqs.(28) and (29),

Cf "A � "Cÿ � � Cm "A � "C ÿ "T

ÿ �; �33�

then use Eqs. (27), (31) and (32) to eliminate "T, "A and"C from this equation, giving

I� ESm Cf ÿ Cmÿ ��

"f � ": �34�Comparing this to Eq. (15) shows that the strain-con-centration tensor for Eshelby's equivalent inclusion is

AEshelby � I� ESm Cf ÿ Cmÿ �� ÿ1

: �35�This can be used in Eq. (21) to predict the moduli ofaligned-®ber composites, a result ®rst developed byRussel [14]. Calculations using this model to explore thee�ects of particle aspect ratio on sti�ness are presentedby Chow [15].

While Eshelby's solution treats only ellipsoidal ®bers,the ®bers in most short-®ber composites are much betterapproximated as right circular cylinders. The relation-ship between ellipsoidal and cylindrical particles wasconsidered by Steif and Hoysan [16], who developed avery accurate ®nite element technique for determining

Fig. 2. Eshelby's equivalent inclusion problem. The inclusion (a) with

transformation strain "T has the same stress �I and strain as the in-

homogeneity (b) when both bodies are subject to a far-®eld strain "A:


the sti�ening e�ect of a single ®ber of given shape. Forvery short particles, `=d � 4, they found reasonableagreement for E11 by letting the cylinder and the ellipsoidhave the same `=d. The ellipsoidal particle gave a slightlysti�er composite, with the di�erence between the tworesults increasing as the modulus ratio Ef=Em increased.Henceforth we will use the cylinder aspect ratio in placeof the ellipsoid aspect ratio in Eshelby-type models.

Because Eshelby's solution only applies to a singleparticle surrounded by an in®nite matrix, AEshelby isindependent of ®ber volume fraction and the sti�nesspredicted by this model increases linearly with ®bervolume fraction. Modulus predictions based on Eqs.(35) and (21) should be accurate only at low volumefractions, say up to vf of 1%. The more di�cult pro-blem is to ®nd some way to include interactions between®bers in the model, and so produce accurate results athigher volume fractions. We next consider approachesfor doing that.

3.3. Mori±Tanaka model

A family of models for non-dilute composite materi-als has evolved from a proposal originally made byMori and Tanaka [17]. Benveniste [18] has provided aparticularly simple and clear explanation of the Mori±Tanaka approach, which we use here to introduce theapproach.

We have already introduced the strain-concentrationtensor in Eq. (15). Suppose that a composite is to bemade of a certain type of reinforcing particle, and that,for a single particle in an in®nite matrix, we know thedilute strain-concentration tensor AEshelby,

"f � AEshelby": �36�The Mori±Tanaka assumption is that, when manyidentical particles are introduced in the composite, theaverage ®ber strain is given by

"f � AEshelby"m: �37�That is, within a concentrated composite each particle`sees' a far-®eld strain equal to the average strain in thematrix. Using the alternate strain concentrator de®nedin Eq. (19), the Mori±Tanaka assumption can be re-stated as

AMT � AEshelby �38�Eq. (20) then gives the Mori±Tanaka strain con-centrator as

AMT � AEshelby �1ÿ vf �I� vfAEshelby

� �ÿ1: �39�

This is the basic equation for implementing a Mori±Tanaka model.

The Mori±Tanaka approach for modeling compositeswas ®rst introduced by Wakashima, Otsuka and Ume-kawa [19] for modeling thermal expansions of composites

with aligned ellipsoidal inclusions. (Mori and Tanaka'spaper [17] treats only the homogeneous inclusion pro-blem, and says nothing about composites). Mori±Tanaka predictions for the longitudinal modulus of ashort-®ber composite were ®rst developed by Taya andMura [8] and Taya and Chou [9], whose work alsoincluded the e�ects of cracks and of a second type ofreinforcement. Weng [20] generalized their method,and Tandon and Weng [11] used the Mori±Tanakaapproach to develop equations for the complete set ofelastic constants of a short-®ber composite. Tandon andWeng's equations for the plane-strain bulk modulus k23and the major Poisson ratio �12 must be solved iteratively.However, this iteration can be avoided by using an alter-ate formula for �12; details are given in Appendix A.

The usual development of the Mori±Tanaka model[8±11] di�ers somewhat from Benveniste's explanation.For an average applied stress �, the reference strain "0

is de®ned as the strain in a homogeneous body of matrixat this stress,

� � Cm"0: �40�Within the composite the average matrix strain di�ersfrom the reference strain by some perturbation ~"m,

"m � "0 � ~"m �41�A ®ber in the composite will have an additional strainperturbation ~"f , such that

"f � "0 � ~"m � ~"f �42�while the equivalent inclusion will have this strain plusthe transformation strain "T. The stress equivalencebetween the inclusion and the ®ber then becomes

Cf "0 � ~"m � ~"fÿ � � Cm "0 � ~"m � ~"f ÿ "T

ÿ �: �43�

Compare this to the dilute version, Eq. (33), noting that"A in the dilute problem is equivalent to �"0 � ~"m� here.The development is completed by assuming that theextra ®ber perturbation is related to the transformationstrain by Eshelby's tensor,

~"f � E"T �44�Combining this with Eqs. (41) and (42) reveals that Eq.(44) contains the essential Mori±Tanaka assumption:the ®ber in a concentrated composite sees the averagestrain of the matrix.

Some other micromechanics models are equivalent tothe Mori±Tanaka approach, though this equivalencehas not always been recognized. Chow [21] consideredEshelby's inclusion problem and conjectured that in aconcentrated composite the inclusion strain would bethe sum of two terms: the dilute result given by Eshelby(27) and the average strain in the matrix.

�"C�f � E"T � �"C�m �45�


This can be combined with the de®nition of the averagestrain from Eq. (7) to relate the inclusion strain �"C�f tothe transformation strain "T:

�"C�f � �1ÿ vf �E"T �46�Chow then extended this result to an inhomogeneityfollowing the usual arguments, Eqs. (28)±(35). Thisproduces a strain-concentration tensor

AChow � I� �1ÿ vf �ESm Cf ÿ Cmÿ �� ÿ1 �47�

which is equivalent to the Mori±Tanaka result (39).Chow was apparently unaware of the connection betweenhis approach and theMori±Tanaka scheme, but he seemsto have been the ®rst to apply theMori±Tanaka approachto predict the sti�ness of short-®ber composites.

A more recent development is the equivalent poly-in-clusion model of Ferrari [22]. Rather than use the strain-concentration tensor A, Ferrari used an e�ective Eshelbytensor E, de®ned as the tensor that relates inclusionstrain to transformation strain at ®nite volume fraction:

�"C�f � E"T: �48�Once E has been de®ned, it is straightforward to derive astrain-concentration tensor A and a composite modulus.

Ferrari considered admissible forms for E, given therequirements that E must (a) produce a symmetric sti�-ness tensor C, (b) approach Eshelby's tensor E asvolume fraction approaches zero, and (c) give a com-posite sti�ness that is independent of the matrix sti�nessas volume fraction approaches unity. He proposed asimple form that satis®es these criteria,

E � �1ÿ vf �E: �49�The combination of Eqs. (48) and (49) is identical toChow's assumption (46) and, for aligned ®bers of uni-form length, Ferrari's equivalent poly-inclusion model,Chow's model, and theMori±Tanaka model are identical.Important di�erences between the equivalent poly-inclusion model and the Mori±Tanaka model arise whenthe ®bers are misoriented or have di�erent lengths, atopic that will be addressed in a subsequent paper [1].

3.4. Self-consistent models

A second approach to account for ®nite ®ber volumefraction is the self-consistent method. This approach isgenerally credited to Hill [23] and Budiansky [24], whoseoriginal work focused on spherical particles and con-tinuous, aligned ®bers. The application to short-®bercomposites was developed by Laws and McLaughlin[25] and by Chou, Nomura and Taya [26].

In the self-consistent scheme one ®nds the propertiesof a composite in which a single particle is embedded inan in®nite matrix that has the average properties of thecomposite. For this reason, self-consistent models arealso called embedding models.

Again building on Eshelby's result for a ellipsoidalparticle, we can create a self-consistent version of Eq.(35) by replacing the matrix sti�ness and compliancetensors by the corresponding properties of the compo-site. This gives the self-consistent strain-concentrationtensor as

ASC � I� ES Cf ÿ Cÿ �� ÿ1

: �50�Of course the properties C and S of the embedding`matrix' are initially unknown. When the reinforcingparticle is a sphere or an in®nite cylinder, the equationscan be manipulated algebraically to ®nd explicitexpressions for the overall properties [23,24]. For short®bers this has not proved possible, but numerical solu-tions are easily obtained by an iterative scheme. Onestarts with an initial guess at the composite properties,evaluates E and then ASC from Eq. (50), and substitutesthe result into Eq. (21) to get an improved value for thecomposite sti�ness. The procedure is repeated using thisnew value, and the iterations continue until the resultsfor C converge.

An additional, but less obvious, change is that Eshel-by's tensor E depends on the `matrix' properties, whichare now transversely isotropic. Expressions for Eshelby'stensor for an ellipsoid of revolution in a transverselyisotropic matrix are given by Chou, Nomura and Taya[27] and by Lin and Mura [28]. With these expressionsin hand one can use Eq. (50) together with Eq. (21) to®nd the sti�ness of the composite. This is the self-con-sistent approach used for short-®ber composites [25,26].

A closely-related approach, called the `generalizedself-consistent model,' also uses an embeddingapproach. However, in these models the embeddedobject comprises both ®ber and matrix material. Whenthe composite has spherical reinforcing particles, theembedded object is a sphere of the reinforcementencased in a concentric spherical shell of matrix; this isin turn surrounded by an in®nite body with the averagecomposite properties. The generalized self-consistentmodel is sometimes referred to as a `double embedding'approach. For continuous ®bers the embedded object isa cylindrical ®ber surrounded by a cylindrical shell ofmatrix. The ®rst generalized self-consistent models weredeveloped for spherical particles by Kerner [29], and forcylindrical ®bers by Hermans [30]. Both of these paperscontain an error, which is discussed and corrected byChristensen and Lo [31]. While the generalized self-consistent model is widely regarded as superior to theoriginal self-consistent approach, no such model hasbeen developed for short ®bers.

3.5. Bounding models

A rather di�erent approach to modeling sti�ness isbased on ®nding upper and lower bounds for the com-posite moduli. All bounding methods are based on


assuming an approximate ®eld for either the stress orthe strain in the composite. The unknown ®eld is thenfound through a variational principle, by minimizing ormaximizing some functional of the stress and strain.The resulting composite sti�ness is not exact, but it canbe guaranteed to be either greater than or less than theactual sti�ness, depending on the variational principle.This rigorous bounding property is the attraction ofbounding methods.

Historically, the Voigt and Reuss averages were the®rst models to be recognized as providing rigorousupper and lower bounds [32]. To derive the Voigtmodel, Eq. (23), one assumes that the ®ber and matrixhave the same uniform strain, and then minimizes thepotential energy. Since the potential energy will have anabsolute minimum when the entire composite is inequilibrium, the potential energy under the uniformstrain assumption must be greater than or equal to theexact result, and the calculated sti�ness will be an upperbound on the actual sti�ness. The Reuss model, Eq.(24), is derived by assuming that the ®ber and matrixhave the same uniform stress, and then maximizing thecomplementary energy. Since the complementary energymust be maximum at equilibrium, the model provides alower bound on the composite sti�ness. Detailed deri-vations of these bounds are provided by Wu andMcCullough [33].

The Voigt and Reuss bounds provide isotropic results(provided the ®ber and matrix are themselves isotropic),when in fact we expect aligned-®ber composites to behighly anisotropic. More importantly, when the ®berand matrix have substantially di�erent sti�nesses thenthe Voigt and Reuss bounds are quite far apart, andprovide little useful information about the actual com-posite sti�ness. This latter point motivated Hashin andShtrikman to develop a way to construct tighterbounds.

Hashin and Shtrikman developed an alternate varia-tional principle for heterogeneous materials [34,35].Their method introduces a reference material, and basesthe subsequent development on the di�erences betweenthis reference material and the actual composite. Ratherthan requiring two variational principles, like the Voigtand Reuss bounds, their single variational principlegives both the upper and lower bounds by makingappropriate choices of the reference material. For anupper bound the reference material must be as sti� orsti�er than any phase in the composite (®ber or matrix),and for a lower bound the reference material must havea sti�ness less than or equal to any phase. In mostcomposites the ®ber is sti�er than the matrix, so choos-ing the ®ber as the reference material gives an upperbound and choosing the matrix as the reference materialgives a lower bound. If the matrix is sti�er than the®ber, the bounds are reversed. The resulting bounds aretighter than the Voigt and Reuss bounds, which can be

obtained from theHashin±Shtrikman theory by giving thereference material in®nite or zero sti�ness, respectively.

Hashin and Shtrikman's original bounds [35] apply toisotropic composites with isotropic constituents. Fre-quently the bounds are regarded as applying to compo-sites with spherical particles, though a ®ber compositewith 3-D random ®ber orientation must also obey thebounds.

Walpole re-derived the Hashin±Shtrikman boundsusing classical energy principles [36], and extended themto anisotropic materials [37]. Walpole also derivedresults for in®nitely long ®bers and in®nitely thin disksin both aligned and 3-D random orientations [38].

The Hashin±Shtrikman±Walpole bounds were exten-ded to short-®ber composites by Willis [39] and by Wuand McCullough [33]. These workers introduced a two-point correlation function into the bounding scheme,allowing aligned ellipsoidal particles to be treated.Based on these extensions, explicit formulae for alignedellipsoids were developed by Weng [40] and by Eduljeeet al. [41,42].

The general bounding formula, shown here in theformat developed by Weng, gives the composite sti�nessC as

C � vfCfQf � vmC

mQm� �

vfQf � vmQ

m� �ÿ1

; �51�where the tensors Qf and Qm are de®ned as

Qf � I� E0S0�Cf ÿ C0�� ÿ1and

Qm � I� E0S0�Cm ÿ C0�� ÿ1:

�52�

Here E0 is Eshelby's tensor associated with the proper-ties of the reference material, which has sti�ness C0 andcompliance S0.

When the matrix is chosen as the reference material,Eq. (51) gives a strain concentrator of

Alower � I� EmSm�Cf ÿ Cm�� ÿ1 �53�This result is labeled here as the lower bound, on thepresumption that the ®ber is sti�er than the matrix. Thecomposite sti�ness is found by substituting Alower intoEqs. (20) and (21). Eduljee and McCullough [41,42]argue that the lower bound provides the most accurateestimate of composite properties, and recommend it as amodel. Note that this lower bound prediction is iden-tical to the Mori±Tanaka model, Eq. (39) [20,40]. Thiscorrespondence lends theoretical support to the Mori±Tanaka approach, and guarantees that it will alwaysobey the bounds.

The other bound, found by using Eq. (51) with the®ber as the reference material, has a strain concentratorof

Aupper � I� EfSf �Cm ÿ Cf �� : �54�

Note that the Eshelby tensor Ef is now computed forinclusions of matrix material surrounded by the ®ber


material. Eq. (54) is labeled as the upper bound, pre-suming that the ®ber is sti�er than the matrix. An iden-tical result can be obtained from the Mori±Tanakatheory by assuming that ellipsoidal particles of thematrix material are embedded in a continuous phase ofthe ®ber material.

If the matrix is sti�er than the ®bers, then the right-hand sides of Eqs. (53) and (54) are unchanged but Eq.(53) becomes the upper bound and Eq. (54) becomes thelower bound. All of the preceding bounding formulaehave been given for two-component composites, but thetheory readily accommodates multiple reinforcements.

At ®ber volume fractions close to unity, the matrixsti�ness strongly in¯uences the composite sti�ness forthe lower bound/Mori±Tanaka models, despite the tinyamount of it that is present. Packing considerationssuggest that the only way to approach such high volumefractions is for the ®ber phase to become continuous,and Lielens et al. [43] suggest that at very high ®bervolume fractions the composite sti�ness should be muchcloser to the upper bound, or equivalently to the Mori±Tanaka prediction using the ®ber as the continuousphase. This insight prompted Lielens and co-workers topropose a model that interpolates between the upperand lower bounds, such that the lower bound dominatesat low volume fractions and the upper bound dominatesat high volume fractions (again presuming the ®ber isthe sti�er phase). They perform this interpolation on theinverse of the strain-concentration tensor A, producingthe predictive equation [43]

ALielens � �1ÿ f��Alower�ÿ1 � f�Aupper�ÿ1n oÿ1

: �55�

The interpolating factor f depends on ®ber volumefraction, and they propose

f � vf � v2f2

: �56�

This theory reproduces the lower bound and Mori±Tanaka results at low volume fractions, but is said togive improved results at reinforcement volume fractionsin the 40±60% range.

3.6. Halpin±Tsai equations

The Halpin±Tsai equations [44,45] have long beenpopular for predicting the properties of short-®bercomposites. A detailed review and derivation is pro-vided by Halpin and Kardos [46], from which we sum-marize the main points.

The Halpin±Tsai equations were originally developedwith continuous-®ber composites in mind, and werederived from the work of Hermans [30] and Hill [47].Hermans developed the ®rst generalized self-consistentmodel for a composite with continuous aligned ®bers(see Section 3.4). Halpin and Tsai found that three ofHermans' equations for sti�ness could be expressed in acommon form

P

Pm� 1� ��vf

1ÿ �vfwith � � �Pf=Pm� ÿ 1

�Pf=Pm� � 1: �57�

Here P represents any one of the composite moduli lis-ted in Table 1, and Pf and Pm are the correspondingmoduli of the ®bers and matrix, while � is a parameterthat depends on the matrix Poisson ratio and on theparticular elastic property being considered. Hermansderived expressions for the plane-strain bulk modulusk23, and for the longitudinal and transverse shear mod-uli G12 and G23. The � parameters for these propertiesare given in Table 1. Note that for an isotropic matrixkm � �Em=2�1� �m��1ÿ 2�m��.

Hill [47] showed that for a continuous, aligned-®bercomposite the remaining sti�ness parameters are givenby

E11� vfEf � vmEmÿ4 �f ÿ �m

1kfÿ 1

km

" #21

k23ÿ vf

kfÿ vm

km

� �; �58�

�12 � vf�f � vm�m� �f ÿ �m

1kfÿ 1

km

" #1

k23ÿ vf

kfÿ vm

km

� �: �59�

This completes Hermans' model for aligned-®ber com-posites; note that one must know k23 to ®nd E11 and �12.We now know that Hermans' result for G23 is incorrect,in that it does not satisfy all of the ®ber/matrix continuityconditions [3]. It is, however, identical to a lower bound

Table 1

Correspondence between Halpin±Tsai Equation (57) and generalized self-consistent predictions of Hermans [30] and Kerner [29]. After Halpin and

Kardos [46]

P Pf Pm � Comments

k23 kf km1ÿ�mÿ2�2m

1��mPlane strain bulk modulus, aligned ®bers

G23 Gf Gm1��m

3ÿ�mÿ4�2m Transverse shear modulus, aligned ®bers

G12 Gf Gm 1 Longitudinal shear modulus, aligned ®bers

K Kf Km2�1ÿ2�m�1��m

Bulk modulus, particulates

G Gf Gm7ÿ5�m

8ÿ10�mShear modulus, particulates


on G23 derived by Hashin [48]. Hermans' remainingresults are identical to Hashin and Rosen's compositecylinders assemblage model [49], so Hermans' k23, andthus his E11 and �12, are identical to the self-consistentresults of Hill [23].

The Halpin±Tsai form (57) can also be used toexpress equations for particulate composites derived byKerner [29], who also used a generalized self-consistentmodel. Table 1 gives the details. Kerner's result forshear modulus G is also known to be incorrect, butreproduces the Hashin±Shtrikman±Walpole lowerbound for isotropic composites, while Kerner's resultfor bulk modulus K is identical to Hashin's compositespheres assemblage model [50]. See Christensen and Lo[31] and Hashin [3] for further discussion of Kerner'sand Hermans' results.

To transform these results into convenient forms forcontinuous-®ber composites, Halpin and Tsai madethree additional ad hoc approximations:

. Eq. (57) can be used directly to calculate selectedengineering constants, with E11 or E22 replacing P.

. The � parameters in Table 1 are insensitive to �m,and can be approximated by constant values.

. The underlined terms in Eqs. (58) and (59) can beneglected.

In Eq. (58) the underlined term is typically negligible,and dropping it gives the familiar rule of mixtures forE11 of a continuous-®ber composite. However, drop-ping the underlined term in Eq. (59) and using a rule ofmixtures for �12 is not necessarily accurate if the ®berand matrix Poisson ratios di�er. Halpin and Tsai arguefor this latter approximation on the grounds that lami-nate sti�nesses are insensitive to �12.

In adapting their approach to short-®ber composites,Halpin and Tsai noted that � must lie between 0 and1.If � � 0 then Eq. (57) reduces to the inverse rule ofmixtures [46],

1

P� vf

Pf� vm

Pm�60�

while for � � 1 the Halpin±Tsai form becomes the ruleof mixtures,

P � vfPf � vmPm: �61�Halpin and Tsai suggested that � was correlated withthe geometry of the reinforcement and, when calculatingE11, it should vary from some small value to in®nity as afunction of the ®ber aspect ratio `=d. By comparingmodel predictions with available 2-D ®nite elementresults, they found that � � 2�`=d� gave good predic-tions for E11 of short-®ber systems. Also, they suggestedthat other engineering constants of short-®ber compo-sites were only weakly dependent on ®ber aspect ratio,and could be approximated using the continuous-®berformulae [45]. The resulting equations are summarizedin Table 2. The early references [44,45] do not mention

G23. When this property is needed the usual approach isto use the � value given in Table 1. While the Halpin±Tsai equations have been widely used for isotropic ®bermaterials, the underlying results of Hermans and Hillapply to transversely isotropic ®bers, so the Halpin±Tsai equations can also be used in this case.

The Halpin±Tsai equations are known to ®t somedata very well at low volume fractions, but to under-predict some sti�nesses at high volume fractions. Thishas prompted some modi®cations to their model.Hewitt and de Malherbe [51] proposed making � afunction of vf , and by curve ®tting found that

� � 1� 40v10f ; �62�gave good agreement with 2-D ®nite element results forG12 of continuous ®ber composites.

Nielsen and Lewis [52,53] focused on the analogybetween the sti�ness G of a composite and the viscosity� of a suspension of rigid particles in a Newtonian ¯uid,noting that one should ®nd �=�m � G=Gm when thereinforcement is rigid �Gf=Gm !1� and the matrix isincompressible. They developed an equation in whichthe sti�ness not only matches dilute theory at lowvolume fractions, but also displays G=Gm !1 as vf

approaches a packing limit vfmax. This leads to a mod-i®ed Halpin±Tsai form

P

Pm� 1� ��vf

1ÿ �vf ��vf�63�

with � retaining its de®nition from Eq. (57). Here thefunction �vf� contains the maximum volume fractionvf max as a parameter. is chosen to give the properbehavior at the upper and lower volume fraction limits,which leads to forms such as

�vf � � 1� 1ÿ vf max

v2f max

� �vf ; �64�

�vf � � 1

vf1ÿ exp

ÿvf

1ÿ �vf=vf max��

: �65�

The Nielsen and Lewis model improves on the Halpin±Tsai predictions, compared to experimental data for Gof particle-reinforced polymers [52] and to ®nite elementcalculations for G12 of continuous-®ber composites [53],using vf max values from 0.40 to 0.85.

Table 2

Traditional Halpin±Tsai parameters for short-®ber composites, used in

Eq. (57)

P Pf Pm � Comments

E11 Ef Em 2�`=d� Longitudinal modulus

E22 Ef Em 2 Transverse modulus

G12 Gf Gm 1 Longitudinal shear modulus

�12 Poisson ratio, = vf�f � vm�m

For G23 see Table 1.


Recently Ingber and Papathanasiou [54] tested theHalpin±Tsai equation and its modi®cations againstboundary element calculations of E11 for aligned short®bers. They found the Nielsen modi®cation to be betterthan the original Halpin±Tsai form. Hewitt and deMalherbe's form could be adjusted to ®t data for anysingle `=d, but was not useful for predictions over arange of aspect ratios. These results are discussed fur-ther in Section 4.

3.7. Shear lag models

Historically, shear lag models were the ®rst micro-mechanics models for short-®ber composites [55], aswell as the ®rst to examine behavior near the ends ofbroken ®bers in a continuous-®ber composite [56,57].Despite some serious theoretical ¯aws, shear lag modelshave enjoyed enduring popularity, perhaps due to theiralgebraic simplicity and their physical appeal.

Classical shear lag models only predict the long-itudinal modulus E11, so they do not meet our criterionof predicting a complete set of elastic constants. How-ever, we include them here because of their historicalimportance and their widespread use. One could obtaina complete sti�ness model by using the shear lag pre-diction for E11 and some continuous-®ber model (suchas Hermans') for the remaining elastic constants. If the®ber is anisotropic then its axial modulus should beused in the shear lag equations.

Following Cox [55], the shear lag analysis focuses ona single ®ber of length ` and radius rf , which is encasedin a concentric cylindrical shell of matrix having radiusR. The ®ber is aligned parallel to the z axis, as shown inFig. 3. Only the axial stress �11 and axial strain "11 areof interest, and Poisson e�ects are neglected so that�f11 � Ef"

f11. The outer cylindrical surface of the matrix

is subjected to displacement boundary conditions con-sistent with an average axial strain "11, and one solvesfor the ®ber stress �f

11�z�. (More rigorously, �f11�z� is the

average stress over the ®ber cross-section at z.) Axialequilibrium of the ®ber requires that

d�f11

dz� ÿ 2�rz

rf; �66�

where �rz is the axial shear stress at the ®ber surface.The key assumption of shear lag theory is that �rz isproportional to the di�erence in displacement wbetween the ®ber surface and the outer matrix surface:

�rz�z� � H

2�rfw�R; z� ÿ w�rf ; z�� ; �67�

where H is a constant that depends on matrix propertiesand ®ber volume fraction. Solving Eq. (66) for �f

11�z�and applying boundary conditions of zero stress at the®ber ends gives an average ®ber stress of

�f11 � Ef"11 1ÿ tanh ��`=2�

��`=2��

�68�

with

�2 � H

�r2fEf

: �69�

It is convenient to rewrite this as an expression for theaverage ®ber strain,

"f11 � �`"11; �70�where �` is a length-dependent `e�ciency factor',

�` � 1ÿ tanh ��`=2��`=2�

� �: �71�

Note that �` is a scalar analog of the strain-concentrationtensor A de®ned in Eq. (15), and �1=�� is a characteristiclength for stress transfer between the ®ber and the matrix.

Cox [55] found the coe�cient H by solving a secondidealized problem. The concentric cylinder geometry ismaintained, but the outer cylindrical surface of thematrix is held stationary and the inner cylinder, which isnow rigid, is subjected to a uniform axial displacement.An elasticity solution for the matrix layer then gives

H � 2�Gm

ln �R=rf � : �72�

Rosen [56,57] simpli®ed this part of the problem byassuming that the matrix shell was thin compared to the®ber radius, �Rÿ rf� � rf , obtaining

H � 2�Gm

�R=rf � ÿ 1: �73�

Rosen's approximation gives an error in H of 10% atvf � 0:60, with much larger errors at lower volumefractions, and we will not consider it further.

It remains to choose the radius R of the matrix cylin-der, and the exact choice is important. Several choiceshave been used, all of which can be written in the form

R

rf�

��KR

vf

r; �74�

where KR is a constant that depends on the assumptionused to ®nd R. Table 3 summarizes the choices for KR.Fig. 3. Idealized ®ber and matrix geometry used in shear lag models.


Cox [55] assumed a hexagonal packing, and chose R asthe distance between centers of nearest-neighbor ®bers(Fig. 4(a)). It seems more realistic to let R equal half ofthe distance between nearest neighbors (Fig. 4(b)), achoice labeled `hexagonal' in Table 3. Rosen [56,57],and later Carman and Reifsnider [58], chose r2f =R

2 � vf

so that the concentric cylinder model in Fig. 3 wouldhave the same ®ber volume fraction as the composite.This is the same R as the composite cylinders model ofHashin and Rosen [49]. More recently, Robinson andRobinson [59,60] assumed a square array of ®bers, andchose R as half the distance between centers of nearestneighbors (Fig. 4(c)) [61]. Each of these choices gives asomewhat di�erent dependence of �` on ®ber volumefraction, with larger values of KR producing lowervalues of E11.

Shear lag models are usually completed by combiningthe average ®ber stress in Eq. (68) with an averagematrix stress to produce a modi®ed rule of mixtures forthe axial modulus

E11 � �`vfEf � �1ÿ vf �Em: �75�However, the matrix stress in this formula is not con-sistent with the basic concepts of average stress andaverage strain. Note that Eq. (7) must hold for "11, asfor any other component of strain. Combining this withEq. (70) to ®nd the average matrix strain, and followingthrough to ®nd the composite sti�ness (with Poissone�ects neglected), gives a result that is consistent withboth the assumptions of shear lag theory and the basicconcepts of average stress and strain

E11 � �`vfEf � �1ÿ �`vf �Em �76�� Em � vf �Ef ÿ Em��`:

This equation is an exact scalar analog of the generaltensorial sti�ness formula, eqn (21). For the cases in thispaper, the di�erence between Eqs. (75) and (76) is small,and we will use the classical shear lag result (75) whentesting the models.

A model by Fukuda and Kawata [62] for the axialsti�ness of aligned short-®ber composites is closelyrelated to shear lag theory. They begin with a 2-D elas-ticity solution for the shear stress around a single slen-der ®ber in an in®nite matrix. The usual shear lagrelation, Eq. (66), is used to transform this into anequation for the ®ber stress distribution, which is thenapproximated by a Fourier series. The coe�cients of atruncated series are evaluated analytically using Galer-kin's method. This is a dilute theory, in which modulusvaries linearly with ®ber volume fraction.

Like any shear lag theory, Fukuda and Kawata'stheory predicts that E11 approaches the rule of mixturesresult as the ®ber aspect ratio approaches in®nity. Butfor short ®bers Fukuda and Kawata's theory givesmuch lower E11 values than shear lag theory. In Fukudaand Kawata's theory, the ratio of ®ber strain to matrixstrain is governed by the parameter �`=d��Em=Ef�. Incontrast, for shear lag theory, Eq. (71), the governingparameter is �`=2, which is proportional to�`=d� ��Em=Ef

p. Thus, for high modulus ratio and low

aspect ratio, Fukuda and Kawata's theory tends tounder-predict E11. For this reason we do not pursuetheir theory further.

4. Tests and comparisons

Obtaining reference data for unidirectional short-®bercomposites presents a problem. Accurate experimentaldata is not available, since it has not proved possible toproduce physical samples with perfectly aligned ®bers.The best that can be done experimentally is to makesamples with partially aligned ®bers, though even inthose samples the ®bers may be clustered or bundledtogether in some unspeci®ed way [42]. Any comparisonbetween the properties of such samples and predictionsnecessarily includes both the model for aligned-®bercomposites and the model for ®ber orientation e�ects.

In this paper we avoid this complication by usingthree-dimensional ®nite element models of alignedshort-®ber composites, rather than experimental results,as the reference data. This necessitates the assumptionof a periodic arrangement of the ®bers, but all of themicromechanics models are su�ciently vague about thegeometric arrangement of the ®bers that they admitperiodic geometries. We also compare the theories tosome boundary element results for random arrays ofaligned ®bers [54].

For clarity we limit our comparisons to the modelslisted in Table 4. For the shear lag model we show

Table 3

Values for KR used in Eq. (74) for shear lag models

Fiber packing KR

Cox 2�=��3p

=3.628

Composite cylinders 1=1.000

Hexagonal �=2��3p

=0.907

Square �=4=0.785

Fig. 4. Fiber packing arrangements used to ®nd R in shear lag models.

(a) Cox. (b) Hexagonal. (c) Square.


results only for the square array, noting that this choicefor R gives the highest sti�ness. The models which arenot shown are: the dilute Eshelby model, which is lim-ited to small volume fractions; the Hashin±Shtrikman±Walpole lower bound, which is identical to the Mori±Tanaka model; and the upper bound, which is notclaimed to be useful by itself.

4.1. Finite element modeling

Using the ®nite element method we analyzed twotypes of periodic, three-dimensional arrays of ®bers,which we call regular and staggered arrays. The repre-sentative volume elements (RVE's) are shown in Fig. 5.The unit cell dimensions were chosen with b � �a,where � is a constant. We used both � � 1 to obtainsquare packing, and � � ��

3p

which gives hexagonalpacking. For the regular ®bers the distance betweenneighboring ®ber ends (equal to 2cÿ ` in Fig. 5(a)) wasset to 0:538` for square packing and 0:136` for hex-agonal packing. For the staggered arrays the distancealong each ®ber that is overlapped by neighboring ®berswas set at a ®xed percentage of the ®ber length: 65% forsquare packing and 76% for hexagonal packing. Theseconditions, together with the ®ber diameter and volumefraction, su�ce to determine the dimensions a, b and c foreach RVE. Note that a new RVE and its corresponding3-D mesh are generated for each ®ber aspect ratio.

Sti�nesses of these RVEs' were calculated usingABAQUS [63]. Twenty-node isoparametric elementswere used, and a sample mesh is shown in Fig. 6. The

analysis was geometrically nonlinear but the appliedstrain was 0.5%, so the results are in the region of linearbehavior. For axial or transverse loading, symmetryrequires all faces of the RVE to remain plane. Todetermine E11 and �12 we ®xed the normal displace-ments of the back, left, and bottom faces of the RVE;required the right and top faces to remain plane andparallel to the coordinate axes (using multi-point con-straints); and displaced the front face uniformly in thex1 direction. The tangential displacements on all faceswere unconstrained. The average stress was computedfrom the reaction force in the loading direction, dividedby the cross-sectional area of the RVE. Average strainswere computed from the initial and deformed dimen-sions of the RVE. Analogous conditions were used toload the RVE in the x2 direction to determine E22 and�23. The longitudinal shear modulus G12 could in prin-ciple be determined using these same RVEs', but thatcalculation requires a complicated application of peri-odic boundary conditions and we did not undertake it.

All of the micromechanics theories reviewed herepredict transversely isotropic properties. Transverseisotropy about the x1 axis implies that the tensile modulusis the same for any loading direction in the 2±3 plane.This not only requires that E22 � E33, but also that

1

G23� 2

E22�1� �23�: �77�

RVEs' with hexagonal packing should also be transver-sely isotropic and obey these same relationships. How-ever, for square packing the properties are only

Table 4

Models selected for comparison

Model Comments

Halpin±Tsai Eq. (57) and Table 2

Nielsen Eqs. (63), (64) and (57b) and Table 2

Mori±Tanaka Eqs. (39), (35) and (21)

Lielens Eqs. (55), (56), (53), (54), (20) and (21)

Self-Consistent Eqs. (50) and (21)

Shear Lag Eqs. (75), (71), (69), (72) and (74) and Table 3

Fig. 5. Representative volume elements used in the ®nite element calculations. (a) Regular array; the bold lines show the RVE. (b) Staggered array.

Fig. 6. Example ®nite element mesh for a staggered, hexagonal array.


guaranteed to be orthotropic. That is, calculations forsquare packing will always give E22 � E33, but theresults will not necessarily obey Eq. (77) nor will thetransverse modulus necessarily be the same for otherloading directions in the 2±3 plane. Here we simplyreport �23 and E22 for loading in the x2 direction, anddo not explore the other orthotropic constants forsquare packing.

The material properties used in the ®nite element cal-culations (Table 5) are typical of ®ber-reinforced engi-neering thermoplastics. All of the moduli are scaled bythe matrix modulus.

4.2. Results and discussion

Fig. 7 compares the theoretical and ®nite elementresults for longitudinal modulus E11. The strong in¯u-ence of ®ber aspect ratio on E11 is apparent, and all ofthe theories exhibit a similar S-shaped curve, asymptot-ing to the same rule-of-mixtures value at high aspectratio. However, the various theories give quite di�erentvalues for very short ®bers, and rise at di�erent rates.

The di�erent packing arrangements create some scat-ter in the ®nite element results, but the scatter is smallfor `=d58. For `=d44 the scatter is signi®cant. This isnot surprising, since the properties of particulate-rein-forced composites are known to be very sensitive to thepacking arrangement. The high E11 values for the hex-agonal staggered array probably occur because ourrules for forming this particular type of RVE tend to

create long `chains' of nearly-touching particles parallelto the x1 axis, with a high degree of axial overlap. Whileall of the ®nite element results are equally `true,' webelieve the lower ®nite element values are more repre-sentative of the actual packing and the actual sti�ness ofcomposites with very short ®bers.

Comparing models to ®nite element data for E11, theHalpin±Tsai equation is accurate for very short ®bers,but falls below the data for longer ®bers. The Nielsenmodel improves on the Halpin±Tsai predictions for thevery short ®bers, but is still below the data for longer®bers. A better ®t in the higher aspect ratio range isprovided by the Mori±Tanaka and Lielens models,which are only slightly di�erent from one another atthis volume fraction. These models are good over mostof the data range. The self-consistent results are usuallyhigh, while the shear lag model is good for the longer®bers but too low for very short ®bers. This latterbehavior is not surprising, since shear lag theory treatsthe ®ber as a slender body. Using any of the othervalues for R in the shear lag model shifts the curve tothe right, moving the predictions away from the data.

Results for transverse modulus E22 are shown inFig. 8. The ®nite element data again have moderatescatter. Fiber aspect ratio has little e�ect on the trans-verse modulus, though some of the packing geometriesshow a slight dip at low aspect ratio. Interestingly, theshape and location of this dip are matched by the mod-els that use the Eshelby tensor. Note that the Halpin±Tsai and Nielsen models contain no dependence onaspect ratio for E22. Shear lag models do not predictE22.

Most of the models do a good job of predicting E22,with the Mori±Tanaka and Lielens models being themost accurate. The Halpin±Tsai result is slightly higherthan most of the data, while the Nielsen model notice-ably over-predicts this property. For comparison theupper bound result falls well above the data, with anasymptote of E22=Em � 3:59 at high aspect ratio.

Fig. 7. Theoretical predictions and ®nite element results for E11.

Table 5

Material properties used in ®nite element calculations

Property Fiber Matrix

E 30 1

� 0.20 0.38

vf 0.20

`=d 1, 2, 4, 8, 16, 24, 48

Fig. 8. Theoretical predictions and ®nite element results for E22.


Data for the Poisson ratios �12 and �23 appear inFigs. 9 and 10. The Nielsen and Halpin±Tsai results for�12 are identical, so only the Halpin±Tsai curve isshown. Both Poisson ratios show a moderate depen-dence on aspect ratio and some sensitivity to packinggeometry. The shape of this dependence is similar for allbut the regular hexagon array and is matched qualita-tively by several models, but the quantitative match isnot as good. For �12 the constant value provided by theHalpin±Tsai equations is at least as good a match to thedata as the models that show some variation. However,the Halpin±Tsai and Nielsen models substantially over-predict �23, while the other models do very well on thisproperty, especially at the higher aspect ratios. Theerror in the Halpin±Tsai value results from a combina-tion of a slightly high prediction for E22 (Fig. 8) and aslightly low prediction for G23 (not shown here), thee�ects combining through Eq. (77).

One weakness of the ®nite element calculations is thatthey require the assumption of a regular, periodicpacking arrangement of the ®bers. Calculations that do

not have this limitation have been recently reported byIngber and Papathanasiou [54]. These workers used theboundary element method to calculate E11 for randomarrays of aligned ®bers. Each model typically contained100 ®bers, and results from ten such models were aver-aged to produce each data point. We tested their resultsagainst the various theories, and also performed a lim-ited number of ®nite element calculations for compar-ison purposes. The boundary element results are forrigid ®bers (Ef=Em � 1) and an incompressible matrix(�m � 0:5), but our ®nite element calculations and the-oretical results use Ef=Em � 106 and �m � 0:49 to avoidnumerical di�culties in some of the models.

Fig. 11 shows the results for E11 versus volume frac-tion for `=d � 10. The boundary element data are mostaccurately matched by the Lielens and Nielsen models,though the Halpin±Tsai and Mori±Tanaka models arenot bad. The self-consistent model predicts much highersti�nesses than the other models and than the boundaryelement data. So far these results are consistent with ourprevious comparisons.

What is surprising about Fig. 11 is that the ®nite ele-ment results fall so far above the boundary elementresults, and above the theories that work so well inother cases. Since the ®nite element data fall closer tothe self-consistent model, it is tempting to think thatthey support the accuracy of this model. But we believeit more likely that these results are revealing the sensitivityof sti�ness to the packing arrangement of the ®bers.

Other researchers have noted that gathering short®bers into bundles or clusters tends to reduce E11 com-pared to evenly dispersed ®bers [42]. In the boundaryelement calculations of Ingber and Papathanasiou theinter-®ber spacing is random, and hence uneven, sothere is a modest clustering e�ect. In contrast, our ®niteelement models impose a uniform inter-®ber spacing,

Fig. 9. Theoretical predictions and ®nite element results for �12.

Fig. 10. Theoretical predictions and ®nite element results for �23.

Fig. 11. Models compared to boundary element predictions of E11 for

random arrays of rigid cylinders by Ingber and Papathanasiou [54],

and to ®nite element calculations with Ef=Em � 106, all for `=d � 10.


and so represent an unusually even dispersion of ®bers.Our ®nite element models also maximize the axial over-lap between neighboring ®bers. It seems that these geo-metric e�ects have the greatest in¯uence on compositesti�ness when the ®bers are rigid. We believe that theboundary element calculations are more representativeof real composite behavior than the ®nite element cal-culations in Fig. 11.

Fortunately the in¯uence of ®ber packing is muchsmaller for the Ef=Em ratios typical of polymer-matrixcomposites. Note that the two di�erent ®nite elementresults for vf � 0:20 and rigid ®bers in Fig. 11 are farapart from one another, but in Fig. 7 where Ef=Em � 30the same packing geometries give nearly identical resultsat `=d � 8. This lends support to the idea that ®berpacking is important mainly when the ®bers are extre-mely sti� compared to the matrix, and supports the®nite element results in Figs. 7±10 as a meaningful testof the micromechanics theories.

5. Conclusions

Our goal is to identify the best model for predictingthe sti�ness of aligned short-®ber composites. Amongthe models that we tested, the self-consistent approachtends to over-predict E11 at high volume fractions,though it gives good predictions for other elastic con-stants. The Halpin±Tsai model, for long a standard forthis problem, gives reasonable results for all the elasticconstants except �23, and its E11 values are low formoderate-to-high aspect ratios. Nielsen and Lewis'smodi®cation of Halpin±Tsai improves the ®t to Ingberand Papathanasiou's boundary element data for E11,but it does not substantially improve the ®t to our E11

data and it substantially worsens the prediction of E22.The Mori±Tanaka and Lielens models give much betterpredictions than Halpin±Tsai for �23, and slightly betterpredictions for all the other properties. Our ®nite ele-ment data does not allow us to choose between theMori±Tanaka and Lielens models, since the di�erencesbetween their predictions are small for the volume frac-tions we examined. Shear lag models can give goodpredictions for E11 for aspect ratios greater than 10,provided one makes the proper choice of R, but thepredictions for shorter ®bers are too low.

Our results con®rm that the Halpin±Tsai equationsprovide reasonable estimates for the sti�ness of short-®ber composites, but they indicate that the Mori±Tanaka model is more accurate. The bound interpola-tion model of Lielens et al. may improve on the Mori±Tanaka model for higher ®ber volume fractions ormodulus ratios, but for injection-molded composites thedi�erence is small. We recommend the Mori±Tanakamodel as the best choice for estimating the sti�ness ofaligned short-®ber composites.

Acknowledgements

Funding to the University of Illinois was provided byThe General Electric Company and General MotorsCorporation. This work was conducted in support ofthe Thermoplastic Engineering Design (TED) Venture,a Department of Commerce Advanced TechnologyProgram administered by the National Institute ofStandards and Technology. The authors are grateful toMr. C. Matthew Dunbar of Hibbitt, Karlsson & Sor-ensen, Inc. for his assistance with mesh generation andthe ®nite element analysis, and to Dr. T. D. Papatha-nasiou of the University of South Carolina for makingavailable the detailed data from his recent paper.

Appendix A. Mori±Tanaka predictions without iteration

Tandon and Weng [11] derive explicit expressions forthe elastic constants of a short-®ber composite using theMori±Tanaka approach. Their formulae for the plane-strain bulk modulus k23 and the major Poisson ratio �12are coupled, and must be solved iteratively. Thisappendix presents a way to obtain the same resultswithout iteration. For brevity we use the notation ofTandon and Weng's paper and refer to equations fromthat paper by numbers like (T1).

To determine E11 an average stress �11 is applied,with all other �ij � 0. The reference strains, Eq. (40), are

"011 � �11=E0 "022 � "033 � ÿ�012�11=E0 �A:1�and the average composite properties E11 and �12 arede®ned from

"11 � �11=E11; �A:2��12 � ÿ"22="11: �A:3�

Combining Eqs. (A.2) and (A.3) gives the major Poissonratio,

�12 � E11ÿ"22�11

� �: �A:4�

Tandon and Weng give E11 (T25) as

E11 � E0

1� c A1�2�0A2

A

; �A:5�

where A, A1, etc. are auxiliary constants given in theirpaper and c is the ®ber volume fraction. We now needto ®nd "22. First, (T19) relates the transformation strain"�22 to the reference strains,

"�22 �2A3"

011 � �A4 � A5A�"022 � �A4 ÿ A5A�"033

2A: �A:6�

Substitute Eq. (A.1) into this and simplify, ®nding


"�22 ��11AE0�A3 ÿ �0A4� �A:7�

The average strain "22 is related to the reference strainsand the transformations strain by (T11)

"22 � "022 � c"�22: �A:8�Substitute Eqs. (A.1) and (A.6) into this to ®nd

"22 � ÿ �11E0

�0 ÿ c�A3 ÿ �0A4�

A

� �: �A:9�

Combine this with Eqs. (A.4) and (A.5) to get thedesired result

�12 � �0Aÿ c�A3 ÿ �0A4�A� c�A1 � 2�0A2� : �A:10�

Now �12 can be found using this equation instead of(T37). The result is then substituted into (T36) to ®ndk23, and no iteration is required.

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