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Stiffness Predictions for

Unidirectional Short-Fiber Composites:Review and Evaluation

Charles L. Tucker III

Department of Mechanical and Industrial Engineering

University of Illinois at Urbana-Champaign

1206 W. Green St.

Urbana, IL 61801

Erwin Liang

GE Corporate Research and Development

Schenectady, NY 12301

September 29, 1998

To appear in Composites Science and Technology

Abstract

Micromechanics models for the stiffness of aligned short-fiber composites are reviewed

and evaluated. These include the dilute model based on Eshelby’s equivalent inclusion, the

self-consistent model for finite-length fibers, Mori-Tanaka type models, bounding models, the

Halpin-Tsai equation and its extensions, and shear lag models. Several models are found

to be equivalent to the Mori-Tanaka approach, which is also equivalent to the generalization

of the Hashin-Shtrikman-Walpole lower bound. The models are evaluated by comparison to

finite element calculations using periodic arrays of fibers, and to Ingber and Papathanasiou’s

boundary element results for random arrays of aligned fibers. The finite element calculations

provideE

1 1

,E

2 2

,

1 2

, and

2 3

for a range of fiber aspect ratios and packing geometries, with

other properties typical of injection-molded thermoplastic matrix composites. The Halpin-

Tsai equations give reasonable estimates for stiffness, but the best predictions come from the

Mori-Tanaka model and the bound interpolation model of Lielens et al.

To whom correspondence should be addressed


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1 Introduction

This paper reviews and evaluates models that predict the stiffness of short-fiber composites. The

overall goal of the research is to improve processing-property predictions for injection-molded

composites. The polymer processing communityhas made substantial progress in modeling process-

induced fiber orientation, particularly in injection molding, and these results are now routinely usedto predict mechanical properties. The purpose of this paper is to review the relevant micromechan-

ics literature, and to provide a critical evaluation of the available models. Real injection-molded

composites invariably have misoriented fibers of highly variable length, but aligned-fiber proper-

ties are always calculated as a prelude to modeling the more realistic situation. Hence, we focus

here on composites having aligned fibers with uniform length and mechanical properties. The

modeling of composites with distributions of fiber orientation and fiber length, and the treatment

of multiple types of reinforcement, will be discussed in a subsequent paper 1.

In selecting models for consideration, we impose the general requirements that each model

must include the effects of fiber and matrix properties and the fiber volume fraction, include the

effect of fiber aspect ratio, and predict a complete set of elastic constants for the composite. Any

model not meeting these criteria was excluded from consideration.All of the models use the same basic assumptions:

The fibers and the matrix are linearly elastic, the matrix is isotropic, and the fibers are either

isotropic or transversely isotropic.

The fibers are axisymmetric, identical in shape and size, and can be characterized by an

aspect ratio ` = d .

The fibers and matrix are well bonded at their interface, and remain that way during defor-

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

cracking.

Section 2 presents some important preliminary concepts, emphasizing strain-concentration ten-

sors and their relationship to composite stiffness. Section 3 then reviews the various theories.

Section 4 compares and evaluates the available models. We use finite element computations of

periodic arrays of short fibers to provide reference properties, since it has not proved possible to

create physical specimens with perfectly aligned fibers. A subsequent paper 1 will compare model

predictions to experiments on well-characterized composites with misaligned fibers.

2 Preliminaries

2.1 Notation

Vectors will be denoted by lower-case Roman letters, second-order tensors by lower-case Greek

letters, and fourth-order tensors by capital Roman letters. Whenever possible, vectors and tensors

are written as boldface characters; indicial notation is used where necessary.

A subscript or superscriptf

indicates a quantity associated with the fibers, andm

denotes

a matrix quantity. Thus, the fibers have Young’s modulus E f

and Poisson ratio f

, while the

corresponding matrix properties are E m

and m

.

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The symbol I represents the fourth-order unit tensor. C and S denote the stiffness and compli-

ance tensors, respectively, and and " are the total stress and infinitesimal strain tensors. Hence,

the constitutive equations for the fiber and matrix materials are

f

= C

f

"

f (1)

m

= C

m

"

m

(2)

2.2 Average Stress and Strain

Let x denote the position vector. When a composite material is loaded, the pointwise stress field

( x ) and the corresponding strain field " ( x ) will be non-uniform on the microscale. The solution

of these non-uniform fields is a formidable problem. However, many useful results can be obtained

in terms of the average stress and strain 2. We now define these averages.

Consider a representative averaging volume V . Choose V large enough to contain many fibers,

but small compared to any length scale over which the average loading or deformation of the

composite varies. The volume-average stress is defined as the average of the pointwise stress

( x )

over the volumeV

:

1

V

Z

V

( x ) d V

(3)

The average strain " is defined similarly.

It is also convenient to define volume-average stresses and strains for the fiber and matrix

phases. To obtain these, first partition the averaging volume V into the volume occupied by the

fibersV

f

and the volume occupied by the matrixV

m

. We consider only two-phase composites, so

that

V = V

f

+ V

m

(4)

The fiber and matrix volume fractions are simply v f

= V

f

= V and v m

= V

m

= V and, since only

fibers and matrix are present,v

m

+ v

f

= 1

.

The average fiber and matrix stresses are the averages over the corresponding volumes,

f

1

V

f

Z

V

f

( x ) d V a n d

m

1

V

m

Z

V

m

( x ) d V (5)

The average strains for the fiber and matrix are defined similarly.

The relationships between the fiber and matrix averages and the overall averages can be derived

from the preceding definitions; they are

= v

f

f

+ v

m

m (6)

" = v

f

"

f

+ v

m

"

m

(7)

An important related result is the average strain theorem. Let the averaging volume V be

subjected to surface displacements u 0 ( x ) consistent with a uniform strain " 0 . Then the average

strain within the region is

" = "

0 (8)

This theorem is proved2 by substituting the definition of the strain tensor " in terms of the dis-

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placement vector u into the definition of average strain " , and applying Gauss’s theorem. The

result is

"

i j

=

1

V

Z

S

u

0

i

n

j

+ n

i

u

0

j

d S (9)

whereS

denotes the surface of V

andn

is a unit vector normal tod S

. The average strain withina volume

V

is completely determined by the displacements on the surface of the volume, so dis-

placements consistent with a uniform strain must produce the identical value of average strain. A

corollary of this principle is that, if we define a perturbation strain " C ( x ) as the difference between

the local strain and the average,

"

C

( x ) " ( x ) ? " (10)

then the volume-average of " C ( x ) must equal zero:

"

C

=

1

V

Z

V

"

C

( x ) d V = 0 (11)

The corresponding theorem for average stress also holds. Thus, if surface tractions consistent with

a

0 are exerted onS

then the average stress is

=

0 (12)

2.3 Average Properties and Strain Concentration

The goal of micromechanics models is to predict the average elastic properties of the composite,

but even these need careful definition. Here we follow the direct approach3. Subject the represen-

tative volume V to surface displacements consistent with a uniform strain " 0 ; the average stiffness

of the composite is the tensor C that maps this uniform strain to the average stress. Using eqn (8)we have

= C " (13)

The average compliance S is defined in the same way, applying tractions consistent with a uniform

stress 0 on the surface of the averaging volume. Then, using eqn (12),

" = S (14)

It should be clear that S = C ? 1 . Other authors define the average stiffness and compliance through

the integral of the strain energy over V ; this is equivalent to the direct approach2,4.

An important related concept, first introduced by Hill2, is the idea of strain- and stress-concentration

tensors A and B . These are essentially the ratios between the average fiber strain (or stress) and

the corresponding average in the composite. More precisely,

"

f

= A " (15)

f

= B (16)

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A and B are fourth-order tensors and, in general, they must be found from a solution of the

microscopic stress or strain fields. Different micromechanics models provide different ways to

approximate A or B. Note that A and B have both the minor symmetries of a stiffness or compliance

tensor, but lack the major symmetry. That is,

A

i j k l

= A

j i k l

= A

i j l k (17)

but in general,

A

i j k l

6= A

k l i j

(18)

For later use it will be convenient to have an alternate strain concentration tensor ^A that relates

the average fiber strain to the average matrix strain,

"

f

=

^

A "

m (19)

This is related to A by

A =

^

A

h

( 1 ? v

f

) I + v

f

^

A

i

? 1

(20)

so the two forms are easily interchanged.

Using equations now in hand, one can express the average composite stiffness in terms of the

strain-concentration tensor A and the fiber and matrix properties 2. Combining eqns (1), (2), (6),

(7), (13), and (15), one obtains

C = C

m

+ v

f

C

f

? C

m

A (21)

The dual equation for the compliance is

S = S

m

+ v

f

S

f

? S

m

B

(22)

Equations (21) and (22) are not independent, since S = C ? 1

. Hence, the strain-concentration

tensor A and the stress-concentration tensor B are not independent either. The choice of which one

to use in any instance is a matter of convenience.

To illustrate the use of the strain-concentration and stress-concentration tensors, we note that

the Voigt average corresponds to the assumption that the fiber and the matrix both experience the

same, uniform strain. Then " f = " , A = I , and from eqn (21) the composite modulus is

C

V o i g t

= C

m

+ v

f

C

f

? C

m

(23)

= v

f

C

f

+ v

m

C

m

Recall that the Voigt average is an upper bound on the composite modulus. The Reuss average

assumes that the fiber and matrix both experience the same, uniform stress. This means that the

stress-concentration tensor B equals the unit tensor I, and from (22) the compliance is

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εT

(a) (b) (c)

εC (x)

Figure 1: Eshelby’s inclusion problem. Starting from the stress-free state (a), the inclusion under-

goes 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.

S

R e u s s

= S

m

+ v

f

S

f

? S

m

(24)

= v

f

S

f

+ v

m

S

m

This represents a lower bound on the stiffness of the composite.

3 Theories

3.1 Eshelby’s Equivalent Inclusion

A fundamental result used in several different models is Eshelby’s equivalent inclusion5,6. Eshelby

solved for the elastic stress field in and around an ellipsoidalparticle in an infinitematrix. By letting

the particle be a prolate ellipsoid of revolution, one can use Eshelby’s result to model the stress

and strain fields around a cylindrical fiber.

Eshelby first posed and solved a different problem, that of a homogeneous inclusion (Fig. 1).

Consider an infinite solid body with stiffness C m that is initially stress-free. All subsequent strains

will be measured from this state. A particular small region of the body will be called the inclusion,

and the rest of the body will be called the matrix. Suppose that the inclusion undergoes some type

of transformation such that, if it were a separate body, it would acquire a uniform strain"

T

withno surface traction or stress. " T is called the transformation strain, or the eigenstrain. This strain

might be acquired through a phase transformation, or by a combination of a temperature change

and a different thermal expansion coefficient in the inclusion. In fact the inclusion is bonded to the

matrix, so when the transformation occurs the whole body develops some complicated strain field

"

C

( x ) relative to its shape before the transformation. Within the matrix the stress m is simply the

stiffness times this strain,

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Cm

C f

εT

ε A

(a) (b)

σ I

Figure 2: Eshelby’s equivalent inclusion problem. The inclusion (a) with transformation strain

"

T

has the same stress I

and strain as the inhomogeneity (b) when both bodies are subject to afar-field strain " A

m

( x ) = C

m

"

C

( x )

(25)

but within the inclusion the transformation strain does not contribute to the stress, so the inclusion

stress is

I

= C

m

"

C

? "

T

(26)

The key result of Eshelby was to show that within an ellipsoidal inclusion the strain"

C

is uniform,and is related to the transformation strain by

"

C

= E "

T (27)

E is called Eshelby’s tensor , and it depends only on the inclusion aspect ratio and the matrix elastic

constants. A detailed derivation and applications are given by Mura 7, and analytical expressions

for Eshelby’s tensor for an ellipsoid of revolution in an isotropic matrix appear in many papers8–12.

The strain field " C ( x ) in the matrix is highly non-uniform 13, but this more complicated part of the

solution can often be ignored.

The second step in Eshelby’s approach is to demonstrate an equivalence between the homoge-

neous inclusion problem and an inhomogeneous inclusion of the same shape. Consider two infinite

bodies of matrix, as shown in Fig. 2. One has a homogeneous inclusion with some transformation

strain " T ; the other has an inclusion with a different stiffness C f , but no transformation strain.

Subject both bodies to a uniform applied strain " A at infinity. We wish to find the transformation

strain " T that gives the two problems the same stress and strain distributions.

For the first problem the inclusion stress is just eqn (26) with the applied strain added,

I

= C

m

"

A

+ "

C

? "

T

(28)

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while the second problem has no " T but a different stiffness, giving a stress of

I

= C

f

"

A

+ "

C

(29)

Equating these two expressions gives the transformation strain that makes the two problems equiv-

alent. Using eqn (27) and some rearrangement, the result is

?

h

C

m

+

C

f

? C

m

E

i

"

T

=

C

f

? C

m

"

A (30)

Note that " T is proportional to " A , which makes the stress in the equivalent inhomogeneity pro-

portional to the applied strain.

3.2 Dilute Eshelby Model

One can use Eshelby’s result to find the stiffness of a composite with ellipsoidal fibers at dilute

concentrations. Recall from eqn (21) that to find the stiffness one only has to find the strain-

concentration tensor A. To do this, first note that for a dilute composite the average strain is iden-

tical to the applied strain,

" = "

A (31)

since this is the strain at infinity. Also, from Eshelby, the fiber strain is uniform, and is given by

"

f

= "

A

+ "

C (32)

where the right-hand side is evaluated within the fiber. Now write the equivalence between the

stresses in the homogeneous and the inhomogeneous inclusions, eqns (28) and (29),

C

f

"

A

+ "

C

= C

m

"

A

+ "

C

? "

T

(33)

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

h

I + E S

m

C

f

? C

m

i

"

f

= " (34)

Comparing this to eqn (15) shows that the strain-concentration tensor for Eshelby’s equivalent

inclusion is

A

E s h e l b y

=

h

I + E S

m

C

f

? C

m

i

? 1

(35)

This can be used in eqn (21) to predict the moduli of aligned-fiber composites, a result first devel-

oped by Russel14

. Calculations using this model to explore the effects of particle aspect ratio onstiffness are presented by Chow15.

While Eshelby’s solution treats only ellipsoidal fibers, the fibers in most short-fiber composites

are much better approximated as right circular cylinders. The relationship between ellipsoidal and

cylindrical particles was considered by Steif and Hoysan16, who developed a very accurate finite

element technique for determining the stiffening effect of a single fiber of given shape. For very

short particles, ` = d = 4 , they found reasonable agreement for E 1 1

by letting the cylinder and the

ellipsoid have the same ` = d . The ellipsoidal particle gave a slightly stiffer composite, with the

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difference between the two results increasing as the modulus ratio E f

= E

m

increased. Henceforth

we will use the cylinder aspect ratio in place of the ellipsoid aspect ratio in Eshelby-type models.

Because Eshelby’s solution only applies to a single particle surrounded by an infinite matrix,

A

E s h e l b y is independent of fiber volume fraction and the stiffness predicted by this model increases

linearly with fiber volume fraction. Modulus predictions based on eqns (35) and (21) should be

accurate only at low volume fractions, say up to v f of 1%. The more difficult problem is to findsome way to include interactions between fibers in the model, and so produce accurate results at

higher volume fractions. We next consider approaches for doing that.

3.3 Mori-Tanaka Model

A family of models fornon-dilute compositematerials has evolved from a proposal originally made

by Mori and Tanaka17. Benveniste18 has provided a particularly simple and clear explanation of

the Mori-Tanaka approach, which we use here to introduce the approach.

We have already introduced the strain-concentration tensor in eqn (15). Suppose that a com-

posite is to be made of a certain type of reinforcing particle, and that, for a single particle in an

infinite matrix, we know the dilute strain-concentration tensor A E s h e l b y ,

"

f

= A

E s h e l b y

" (36)

The Mori-Tanaka assumption is that, when many identical particles are introduced in the compos-

ite, the average fiber strain is given by

"

f

= A

E s h e l b y

"

m (37)

That is, within a concentrated composite each particle ‘sees’ a far-field strain equal to the average

strain in the matrix. Using the alternate strain concentrator defined in eqn (19), the Mori-Tanaka

assumption can be re-stated as^

A

M T

= A

E s h e l b y (38)

Equation (20) then gives the Mori-Tanaka strain concentrator as

A

M T

= A

E s h e l b y

h

( 1 ? v

f

) I + v

f

A

E s h e l b y

i

? 1

(39)

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

The Mori-Tanaka approach for modeling composites was first introduced by Wakashima, Ot-

suka and Umekawa19 for modeling thermal expansions of composites with aligned ellipsoidal

inclusions. (Mori and Tanaka’s paper 17 treats only the homogeneous inclusion problem, and says

nothing about composites). Mori-Tanaka predictions for the longitudinal modulus of a short-fiber

composite were first developed by Taya and Mura8 and Taya and Chou9, whose work also included

the effects of cracks and of a second type of reinforcement. Weng20 generalized their method, and

Tandon and Weng11 used the Mori-Tanaka approach to develop equations for the complete set of

elastic constants of a short-fiber composite. Tandon and Weng’s equations for the plane-strain bulk

modulus k 2 3

and the major Poisson ratio 1 2

must be solved iteratively. However, this iteration can

be avoided by using an alterate formula for 1 2

; details are given in Appendix A.

The usual development of the Mori-Tanaka model 8,9,11 differs somewhat from Benveniste’s

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explanation. For an average applied stress , the reference strain " 0 is defined as the strain in a

homogeneous body of matrix at this stress,

= C

m

"

0 (40)

Within the composite the average matrix strain differs from the reference strain by some perturba-tion ~ " m ,

"

m

= "

0

+

~

"

m (41)

A fiber in the composite will have an additional strain perturbation ~ " f , such that

"

f

= "

0

+

~

"

m

+

~

"

f (42)

while the equivalent inclusion will have this strain plus the transformation strain " T . The stress

equivalence between the inclusion and the fiber then becomes

C

f

"

0

+

~

"

m

+

~

"

f

= C

m

"

0

+

~

"

m

+

~

"

f

? "

T

(43)

Compare this to the dilute version, eqn (33), noting that " A in the dilute problem is equivalent to

( "

0

+

~

"

m

) here. The development is completed by assuming that the extra fiber perturbation is

related to the transformation strain by Eshelby’s tensor,

~

"

f

= E "

T (44)

Combining this with eqns (41) and (42) reveals that eqn (44) contains the essential Mori-Tanaka

assumption: the fiber in a concentrated composite sees the average strain of the matrix.

Some other micromechanics models are equivalent to the Mori-Tanaka approach, though this

equivalence has not always been recognized. Chow21 considered Eshelby’s inclusion problem and

conjectured that in a concentrated composite the inclusion strain would be the 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 definition of the average strain from eqn (7) to relate the inclusion

strain ( " C ) f to the transformation strain " T :

( "

C

)

f

= ( 1 ? v

f

) E "

T (46)

Chow then extended this result to an inhomogeneity following the usual arguments, eqns (28) to

(35). This produces a strain-concentration tensor

A

C h o w

=

h

I + ( 1 ? v

f

) E S

m

C

f

? C

m

i

? 1

(47)

which is equivalent to the Mori-Tanaka result (39). Chow was apparently unaware of the connec-

tion between his approach and the Mori-Tanaka scheme, but he seems to have been the first to

apply the Mori-Tanaka approach to predict the stiffness of short-fiber composites.

A more recent development is the equivalent poly-inclusion model of Ferrari 22. Rather than

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use the strain-concentration tensor A , Ferrari used an effective Eshelby tensor Ê , defined as the

tensor that relates inclusion strain to transformation strain at finite volume fraction:

( "

C

)

f

=

^

E "

T (48)

Once Ê has been defined, it is straightforward to derive a strain-concentration tensor A and a

composite modulus.

Ferrari considered admissible forms for Ê , given the requirements that Ê must (a) produce a

symmetric stiffness tensor C , (b) approach Eshelby’s tensor E as volume fraction approaches zero,

and (c) give a composite stiffness that is independent of the matrix stiffness as volume fraction

approaches unity. He proposed a simple form that satisfies these criteria,

^

E = ( 1 ? v

f

) E

(49)

The combination of eqns (48) and (49) is identical to Chow’s assumption (46) and, for aligned

fibers of uniform length, Ferrari’s equivalent poly-inclusion model, Chow’s model, and the Mori-

Tanaka model are identical. Important differences between the equivalent poly-inclusion model

and the Mori-Tanaka model arise when the fibers are misoriented or have different lengths, a topic

that will be addressed in a subsequent paper 1.

3.4 Self-Consistent Models

A second approach to account for finite fiber volume fraction is the self-consistent method. This

approach is generally credited to Hill 23 and Budiansky24, whose original work focused on spherical

particles and continuous, aligned fibers. The application to short-fiber composites was developed

by Laws and McLaughlin25 and by Chou, Nomura and Taya26.

In the self-consistent scheme one finds the properties of a composite in which a single particle

is embedded in an infinite matrix that has the average properties of the composite. For this reason,self-consistent models are also called embedding models.

Again building on Eshelby’s result for a ellipsoidal particle, we can create a self-consistent

version of eqn (35) by replacing the matrix stiffness and compliance tensors by the corresponding

properties of the composite. This gives the self-consistent strain-concentration tensor as

A

S C

=

h

I + E S

C

f

? C

i

? 1

(50)

Of course the properties C and S of the embedding ‘matrix’ are initially unknown. When the rein-

forcing particle is a sphere or an infinite cylinder, the equations can be manipulated algebraically

to find explicit expressions for the overall properties 23,24. For short fibers this has not proved pos-

sible, but numerical solutions are easily obtained by an iterative scheme. One starts with an initialguess at the composite properties, evaluates E and then A S C from eqn (50), and substitutes the

result into eqn (21) to get an improved value for the composite stiffness. The procedure is repeated

using this new value, and the iterations continue until the results for C converge.

An additional, but less obvious, change is that Eshelby’s tensor E depends on the ‘matrix’

properties, which are now transversely isotropic. Expressions for Eshelby’s tensor for an ellipsoid

of revolution in a transversely isotropic matrix are given by Chou, Nomura and Taya27 and by Lin

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and Mura28. With these expressions in hand one can use eqn (50) together with (21) to find the

stiffness of the composite. This is the self-consistent approach used for short-fiber composites 25,26.

A closely-related approach, called the ‘generalized self-consistent model,’ also uses an embed-

ding approach. However, in these models the embedded object comprises both fiber and matrix

material. When the composite has spherical reinforcing particles, the embedded object is a sphere

of the reinforcement encased in a concentric spherical shell of matrix; this is in turn surroundedby an infinite body with the average composite properties. The generalized self-consistent model

is sometimes referred to as a ‘double embedding’ approach. For continuous fibers the embedded

object is a cylindrical fiber surrounded by a cylindrical shell of matrix. The first generalized self-

consistent models were developed for spherical particles by Kerner29, and for cylindrical fibers

by Hermans30. Both of these papers contain an error, which is discussed and corrected by Chris-

tensen and Lo31. While the generalized self-consistent model is widely regarded as superior to the

original self-consistent approach, no such model has been developed for short fibers.

3.5 Bounding Models

A rather different approach to modeling stiffness is based on finding upper and lower bounds for the

composite moduli. All bounding methods are based on assuming an approximate field for either

the stress or the strain in the composite. The unknown field is then found through a variational

principle, by minimizing or maximizing some functional of the stress and strain. The resulting

composite stiffness is not exact, but it can be guaranteed to be either greater than or less than the

actual stiffness, depending on the variational principle. This rigorous bounding property is the

attraction of bounding methods.

Historically, the Voigt and Reuss averages were the first models to be recognized as providing

rigorous upper and lower bounds 32. To derive the Voigt model, eqn (23), one assumes that the

fiber and matrix have the same uniform strain, and then minimizes the potential energy. Since

the potential energy will have an absolute minimum when the entire composite is in equilibrium,the potential energy under the uniform strain assumption must be greater than or equal to the exact

result, and the calculated stiffness will be an upper bound on the actual stiffness. The Reuss model,

eqn (24), is derived by assuming that the fiber and matrix have the same uniform stress, and then

maximizing the complementary energy. Since the complementary energy must be maximum at

equilibrium, the model provides a lower bound on the composite stiffness. Detailed derivations of

these bounds are provided by Wu and McCullough 33.

The Voigt and Reuss bounds provide isotropic results (provided the fiber and matrix are them-

selves isotropic), when in fact we expect aligned-fiber composites to be highly anisotropic. More

importantly, when the fiber and matrix have substantially different stiffnesses then the Voigt and

Reuss bounds are quite far apart, and provide little useful information about the actual composite

stiffness. This latter point motivated Hashin and Shtrikman to develop a way to construct tighter

bounds.

Hashin and Shtrikman developed an alternate variational principle for heterogeneous materials34,35. Their method introduces a reference material, and bases the subsequent development on the

differences between this reference material and the actual composite. Rather than requiring two

variational principles, like the Voigt and Reuss bounds, their single variational principle gives both

the upper and lower bounds by making appropriate choices of the reference material. For an upper

11


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bound the reference material must be as stiff or stiffer than any phase in the composite (fiber or

matrix), and for a lower bound the reference material must have a stiffness less than or equal to

any phase. In most composites the fiber is stiffer than the matrix, so choosing the fiber as the

reference material gives an upper bound and choosing the matrix as the reference material gives a

lower bound. If the matrix is stiffer than the fiber, the bounds are reversed. The resulting bounds

are tighter than the Voigt and Reuss bounds, which can be obtained from the Hashin-Shtrikmantheory by giving the reference material infinite or zero stiffness, respectively.

Hashin and Shtrikman’s original bounds35 apply to isotropic composites with isotropic con-

stituents. Frequently the bounds are regarded as applying to composites with spherical particles,

though a fiber composite with 3-D random fiber orientation must also obey the bounds.

Walpole re-derived the Hashin-Shtrikman bounds using classical energy principles 36, and ex-

tended them to anisotropic materials37. Walpole also derived results for infinitely long fibers and

infinitely thin disks in both aligned and 3-D random orientations38.

The Hashin-Shtrikman-Walpole bounds were extended to short-fiber composites by Willis 39

and by Wu and 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 aligned ellipsoids were developed by Weng40 and by Eduljee et al. 41,42.

The general bounding formula, shown here in the format developed by Weng, gives the com-

posite stiffnessC

as

C =

h

v

f

C

f

Q

f

+ v

m

C

m

Q

m

i h

v

f

Q

f

+ v

m

Q

m

i

? 1

(51)

where the tensors Q f and Q m are defined as

Q

f

=

h

I + E

0

S

0

( C

f

? C

0

)

i

? 1

a n d Q

m

=

h

I + E

0

S

0

( C

m

? C

0

)

i

? 1

(52)

HereE

0

is Eshelby’s tensor associated with the properties of the reference material, which hasstiffness C 0 and compliance S 0 .

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

^

A

l o w e r

=

h

I + E

m

S

m

( C

f

? C

m

)

i

? 1

(53)

This result is labeled here as the lower bound, on the presumption that the fiber is stiffer than the

matrix. The composite stiffness is found by substituting ^A l o w e r into eqns (20) and (21). Eduljee

and McCullough41,42 argue that the lower bound provides the most accurate estimate of composite

properties, and recommend it as a model. Note that this lower bound prediction is identical to

the Mori-Tanaka model, eqn (39)20,40. This correspondence lends theoretical support to the Mori-

Tanaka approach, and guarantees that it will always obey the bounds.The other bound, found by using eqn (51) with the fiber as the reference material, has a strain

concentrator of ^

A

u p p e r

=

h

I + E

f

S

f

( C

m

? C

f

)

i

(54)

Note that the Eshelby tensor E f is now computed for inclusions of matrix material surrounded by

the fiber material. Equation (54) is labeled as the upper bound, presuming that the fiber is stiffer

than the matrix. An identical result can be obtained from the Mori-Tanaka theory by assuming

12


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that ellipsoidal particles of the matrix material are embedded in a continuous phase of the fiber

material.

If the matrix is stiffer than the fibers, then the right-hand sides of eqns (53) and (54) are un-

changed but eqn (53) becomes the upper bound and eqn (54) becomes the lower bound. All of

the preceding bounding formulae have been given for two-component composites, but the theory

readily accommodates multiple reinforcements.At fiber volume fractions close to unity, the matrix stiffness strongly influences the composite

stiffness for the lower bound/Mori-Tanaka models, despite the tiny amount of it that is present.

Packing considerations suggest that the only way to approach such high volume fractions is for

the fiber phase to become continuous, and Lielens et al. 43 suggest that at very high fiber volume

fractions the composite stiffness should be much closer to the upper bound, or equivalently to the

Mori-Tanaka prediction using the fiber as the continuous phase. This insight prompted Lielens and

co-workers to propose a model that interpolates between the upper and lower bounds, such that the

lower bound dominates at low volume fractions and the upper bound dominates at high volume

fractions (again presuming the fiber is the stiffer phase). They perform this interpolation on the

inverse of the strain-concentration tensor ^A , producing the predictive equation43

^

A

L i e l e n s

=

n

( 1 ? f )

^

A

l o w e r ? 1

+ f

^

A

u p p e r ? 1

o

? 1

(55)

The interpolating factor f depends on fiber volume fraction, and they propose

f =

v

f

+ v

2

f

2

(56)

This theory reproduces the lower bound and Mori-Tanaka results at low volume fractions, but is

said to give improved results at reinforcement volume fractions in the 40 to 60% range.

3.6 Halpin-Tsai Equations

The Halpin-Tsai equations44,45 have long been popular for predicting the properties of short-fiber

composites. A detailed review and derivation is provided by Halpin and Kardos46, from which we

summarize the main points.

The Halpin-Tsai equations were originally developed with continuous-fiber composites in mind,

and were derived from the work of Hermans 30 and Hill47. Hermans developed the first generalized

self-consistent model for a composite with continuous aligned fibers (see Section 3.4). Halpin and

Tsai found that three of Hermans’ equations for stiffness could be expressed in a common form:

P

P

m

=

1 + v

f

1 ? v

f

w i t h =

( P

f

= P

m

) ? 1

( P

f

= P

m

) + 1

(57)

Here P represents any one of the composite moduli listed in Table 1, and P f

and P m

are the

corresponding moduli of the fibers and matrix, while is a parameter that depends on the matrix

Poisson ratio and on the particular elastic property being considered. Hermans derived expressions

for the plane-strain bulk modulus k 2 3

, and for the longitudinal and transverse shear moduli G 1 2

and

G

2 3

. The

parameters for these properties are given in Table 1. Note that for an isotropic matrix

13


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Table 1: Correspondence between Halpin-Tsai equation (57) and generalized self-consistent pre-

dictions of Hermans30 and Kerner29. After Halpin and Kardos46.

P P

f

P

m

Comments

k

2 3

k

f

k

m

1 ?

m

? 2

2

m

1 +

m

plane strain bulk modulus, aligned fibers

G

2 3

G

f

G

m

1 +

m

3 ?

m

? 4

2

m

transverse shear modulus, aligned fibers

G

1 2

G

f

G

m

1 longitudinal shear modulus, aligned fibers

K K

f

K

m

2 ( 1 ? 2

m

)

1 +

m

bulk modulus, particulates

G G

f

G

m

7 ? 5

m

8 ? 1 0

m

shear modulus, particulates

k

m

=

E

m

2 ( 1 +

m

) ( 1 ? 2

m

)

.

Hill47 showed that for a continuous, aligned-fibercomposite the remaining stiffness parameters

are given by

E

1 1

= v

f

E

f

+ v

m

E

m

? 4

2

4

f

?

m

1

k

f

?

1

k

m

3

5

2

1

k

2 3

?

v

f

k

f

?

v

m

k

m

!

(58)

1 2

= v

f

f

+ v

m

m

+

2

4

f

?

m

1

k

f

?

1

k

m

3

5

1

k

2 3

?

v

f

k

f

?

v

m

k

m

!

(59)

This completes Hermans’ model for aligned-fiber composites; note that one must know k 2 3

to find

E

1 1

and 1 2

. We now know that Hermans’ result for G 2 3

is incorrect, in that it does not satisfy all

of the fiber/matrix continuity conditions3. It is, however, identical to a lower bound on G 2 3

derived

by Hashin48. Hermans’ remaining results are identical to Hashin and Rosen’s composite cylinders

assemblage model49, so Hermans’k

2 3

, and thus hisE

1 1

and

1 2

, are identical to the self-consistent

results of Hill23.

The Halpin-Tsai form (57) can also be used to express equations for particulate composites

derived by Kerner29, who also used a generalized self-consistent model. Table 1 gives the details.

Kerner’s result for shear modulus G is also known to be incorrect, but reproduces the Hashin-

Shtrikman-Walpole lower bound for isotropic composites, while Kerner’s result for bulk modulus

K is identical to Hashin’s composite spheres assemblage model50. See Christensen and Lo31 and

Hashin3 for further discussion of Kerner’s and Hermans’ results.

To transform these results into convenient forms for continuous-fiber composites, Halpin and

Tsai made three additional ad hoc approximations:

Equation (57) can be used directly to calculate selected engineering constants, with E 1 1

or

E

2 2

replacing P .

The parameters in Table 1 are insensitive to m

, and can be approximated by constant

values.

The underlined terms in eqns (58) and (59) can be neglected.

14


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Table 2: Traditional Halpin-Tsai parameters for short-fiber composites, used in eqn (57). For G 2 3

see Table 1.

P P

f

P

m

Comments

E

1 1

E

f

E

m

2 ( ` = d ) longitudinal modulus

E

2 2

E

f

E

m

2 transverse modulus

G

1 2

G

f

G

m

1 longitudinal shear modulus

1 2

Poisson ratio, =v

f

f

+ v

m

m

In eqn (58) the underlined term is typically negligible, and dropping it gives the familiar rule

of mixtures for E 1 1

of a continuous-fiber composite. However, dropping the underlined term in

eqn (59) and using a rule of mixtures for 1 2

is not necessarily accurate if the fiber and matrix

Poisson ratios differ. Halpin and Tsai argue for this latter approximation on the grounds that

laminate stiffnesses are insensitive to 1 2 .In adapting their approach to short-fiber composites, Halpin and Tsai noted that must lie

between 0 and 1 . If = 0 then eqn (57) reduces to the inverse rule of mixtures 46,

1

P

=

v

f

P

f

+

v

m

P

m

(60)

while for = 1 the Halpin-Tsai form becomes the rule of mixtures,

P = v

f

P

f

+ v

m

P

m

(61)

Halpin and Tsai suggested that was correlated with the geometry of the reinforcement and, when

calculatingE

1 1

, it should vary from some small value to infinity as a function of the fiber aspect

ratio` = d

. By comparing model predictions with available 2-D finite element results, they found

that = 2 ( ` = d )

gave good predictions forE

1 1

of short-fiber systems. Also, they suggested that

other engineering constants of short-fiber composites were only weakly dependent on fiber aspect

ratio, and could be approximated using the continuous-fiber formulae 45. The resulting equations

are summarized in Table 2. The early references 44,45 do not mention G 2 3

. When this property is

needed the usual approach is to use the value given in Table 1. While the Halpin-Tsai equations

have been widely used for isotropic fiber materials, the underlying results of Hermans and Hill

apply to transversely isotropic fibers, so the Halpin-Tsai equations can also be used in this case.

The Halpin-Tsai equations are known to fit some data very well at low volume fractions, but to

under-predict some stiffnesses at high volume fractions. This has prompted some modifications totheir model. Hewitt and de Malherbe 51 proposed making a function of v

f

, and by curve fitting

found that

= 1 + 4 0 v

1 0

f

(62)

gave good agreement with 2-D finite element results for G 1 2

of continuous fiber composites.

Nielsen and Lewis52,53 focused on the analogy between the stiffness G of a composite and the

viscosity of a suspension of rigid particles in a Newtonian fluid, noting that one should find

15


17/34

=

m

= G = G

m

when the reinforcement is rigid ( G f

= G

m

! 1 ) and the matrix is incompressible.

They developed an equation in which the stiffness not only matches dilute theory at low volume

fractions, but also displays G = G m

! 1 as v f

approaches a packing limit v f m a x

. This leads to a

modified Halpin-Tsai formP

P

m

=

1 + v

f

1 ? ( v

f

) v

f

(63)

with retaining its definition from eqn (57). Here the function ( v f

) contains the maximum

volume fraction v f m a x

as a parameter. is chosen to give the proper behavior at the upper and

lower volume fraction limits, which leads to forms such as

( v

f

) = 1 +

1 ? v

f m a x

v

2

f m a x

!

v

f

(64)

( v

f

) =

1

v

f

"

1 ? e x p

? v

f

1 ? ( v

f

= v

f m a x

)

! #

(65)

The Nielsen and Lewis model improves on the Halpin-Tsai predictions, compared to experimen-tal data forG

of particle-reinforced polymers 52 and to finite element calculations forG

1 2

of

continuous-fiber composites53, using v f m a x

values from 0.40 to 0.85.

Recently Ingber and Papathanasiou 54 tested the Halpin-Tsai equation and its modifications

against boundary element calculations of E 1 1

for aligned short fibers. They found the Nielsen

modification to be better than the original Halpin-Tsai form. Hewitt and de Malherbe’s form could

be adjusted to fit data for any single ` = d , but was not useful for predictions over a range of aspect

ratios. These results are discussed further in Section 4.

3.7 Shear Lag Models

Historically, shear lag models were the first micromechanics models for short-fiber composites55, as well as the first to examine behavior near the ends of broken fibers in a continuous-fiber

composite56,57. Despite some serious theoretical flaws, shear lag models have enjoyed enduring

popularity, perhaps due to their algebraic simplicity and their physical appeal.

Classical shear lag models only predict the longitudinal modulusE

1 1

, so they do not meet

our criterion of predicting a complete set of elastic constants. However, we include them here

because of their historical importance and their widespread use. One could obtain a complete

stiffness model by using the shear lag prediction forE

1 1

and some continuous-fiber model (such

as Hermans’) for the remaining elastic constants. If the fiber is anisotropic then its axial modulus

should be used in the shear lag equations.

Following Cox55, the shear lag analysis focuses on a single fiber of length ̀ and radius r f

,

which is encased in a concentric cylindrical shell of matrix having radius R . The fiber is aligned

parallel to the z axis, as shown in Fig. 3. Only the axial stress 1 1

and axial strain " 1 1

are of interest,

and Poisson effects are neglected so that f

1 1

= E

f

"

f

1 1

. The outer cylindrical surface of the matrix

is subjected to displacement boundary conditions consistent with an average axial strain " 1 1

, and

one solves for the fiber stress f

1 1

( z ) . (More rigorously, f

1 1

( z ) is the average stress over the fiber

16


18/34

rf R

l

z

Figure 3: Idealized fiber and matrix geometry used in shear lag models.

cross-section at z .) Axial equilibrium of the fiber requires that

d

f

1 1

d z

= ?

2

r z

r

f

(66)

where r z

is the axial shear stress at the fiber surface. The key assumption of shear lag theory is

that r z

is proportional to the difference in displacement w between the fiber surface and the outer

matrix surface:

r z

( z ) =

H

2 r

f

w ( R ; z ) ? w ( r

f

; z ) (67)

whereH

is a constant that depends on matrixpropertiesand fibervolume fraction. Solving eqn (66)

for

f

1 1

( z )

and applying boundary conditions of zero stress at the fiber ends gives an average fiber

stress of

f

1 1

= E

f

"

1 1

"

1 ?

t a n h ( ` = 2 )

( ` = 2 )

#

(68)

with

2

=

H

r

2

f

E

f

(69)

It is convenient to rewrite this as an expression for the average fiber strain,

"

f

1 1

=

`

"

1 1

(70)

where ̀ is a length-dependent ‘efficiency factor’,

`

=

"

1 ?

t a n h ( ` = 2 )

( ` = 2 )

#

(71)

Note that ̀ is a scalar analog of the strain-concentration tensor A defined in eqn (15), and ( 1 = )

is a characteristic length for stress transfer between the fiber and the matrix.

17


19/34

Table 3: Values for K R

used in eqn (74) for shear lag models.

Fiber packing K R

Cox 2 = p

3 = 3.628

Composite Cylinders1

= 1.000Hexagonal = 2

p

3 = 0.907

Square = 4 = 0.785

Cox55 found the coefficientH

by solving a second idealized problem. The concentric cylinder

geometry is maintained, but the outer cylindrical surface of the matrix is held stationary and the

inner cylinder, which is now rigid, is subjected to a uniform axial displacement. An elasticity

solution for the matrix layer then gives

H =

2 G

m

l n ( R = r

f

)

(72)

Rosen56,57 simplified this part of the problem by assuming that the matrix shell was thin compared

to the fiber radius, ( R ? r f

) r

f

, obtaining

H =

2 G

m

( R = r

f

) ? 1

(73)

Rosen’s approximation gives an error in H of 10% at v f

= 0 6 0 , with much larger errors at lower

volume fractions, and we will not consider it further.

It remains to choose the radiusR

of the matrix cylinder, and the exact choice is important.

Several choices have been used, all of which can be written in the form

R

r

f

=

s

K

R

v

f

(74)

where K R

is a constant that depends on the assumption used to find R . Table 3 summarizes the

choices for K R

. Cox55 assumed a hexagonal packing, and chose R as the distance between centers

of nearest-neighbor fibers (Fig. 4a). It seems more realistic to let R equal half of the distance

between nearest neighbors (Fig. 4b), a choice labeled ‘hexagonal’ in Table 3. Rosen56,57, and later

Carman and Reifsnider58, chose r 2f

= R

2

= v

f

so that the concentric cylinder model in Fig. 3 would

have the same fiber volume fraction as the composite. This is the same R as the composite cylinders

model of Hashin and Rosen49. More recently, Robinson and Robinson59,60 assumed a square array

of fibers, and chose R as half the distance between centers of nearest neighbors (Fig. 4c)61. Each

of these choices gives a somewhat different dependence of

̀ on fiber volume fraction, with larger

values of K

R

producing lower values of E

1 1

.

Shear lag models are usually completed by combining the average fiber stress in eqn (68) with

an average matrix stress to produce a modified rule of mixtures for the axial modulus:

E

1 1

=

`

v

f

E

f

+ ( 1 ? v

f

) E

m

(75)

18


20/34

(a) (b) (c)

RR

R

Figure 4: Fiber packing arrangements used to find R in shear lag models. (a) Cox. (b) Hexagonal.

(c) Square.

However, the matrix stress in this formula is not consistent with the basic concepts of average

stress and average strain. Note that eqn (7) must hold for " 1 1 , as for any other component of strain.Combining this with eqn (70) to find the average matrix strain, and following through to find the

composite stiffness (with Poisson effects neglected), gives a result that is consistent with both the

assumptions of shear lag theory and the basic concepts of average stress and strain:

E

1 1

=

`

v

f

E

f

+ ( 1 ?

`

v

f

) E

m

(76)

= E

m

+ v

f

( E

f

? E

m

)

`

This equation is an exact scalar analog of the general tensorial stiffness formula, eqn (21). For the

cases in this paper, the difference between eqns (75) and (76) is small, and we will use the classical

shear lag result (75) when testing the models.

A model by Fukuda and Kawata62 for the axial stiffness of aligned short-fiber composites is

closely related to shear lag theory. They begin with a 2-D elasticity solution for the shear stress

around a single slender fiber in an infinite matrix. The usual shear lag relation, eqn (66), is used

to transform this into an equation for the fiber stress distribution, which is then approximated by

a Fourier series. The coefficients of a truncated series are evaluated analytically using Galerkin’s

method. This is a dilute theory, in which modulus varies linearly with fiber volume fraction.

Like any shear lag theory, Fukuda and Kawata’s theory predicts thatE

1 1

approaches the rule

of mixtures result as the fiber aspect ratio approaches infinity. But for short fibers Fukuda and

Kawata’s theory gives much lower E 1 1

values than shear lag theory. In Fukuda and Kawata’s

theory, the ratio of fiber strain to matrix strain is governed by the parameter ( ` = d ) ( E m

= E

f

) . In

contrast, for shear lag theory, eqn (71), the governing parameter is ` = 2

, which is proportional to( ` = d )

q

E

m

= E

f

. Thus, for high modulus ratio and low aspect ratio, Fukuda and Kawata’s theory

tends to underpredict E 1 1

. For this reason we do not pursue their theory further.

19


21/34

Table 4: Models selected for comparison.

Model Comments

Halpin-Tsai eqn (57) and Table 2

Nielsen eqns (63), (64), (57b) and Table 2Mori-Tanaka eqns (39), (35), and (21)

Lielens eqns (55), (56), (53), (54), (20), and (21)

Self-Consistent eqns (50) and (21)

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

4 Tests and Comparisons

Obtaining reference data for unidirectional short-fiber composites presents a problem. Accurate

experimental data is not available, since it has not proved possible to produce physical samples with

perfectly aligned fibers. The best that can be done experimentally is to make samples with partially

aligned fibers, though even in those samples the fibers may be clustered or bundled together in

some unspecified way42. Any comparison between the properties of such samples and predictions

necessarily includes both the model for aligned-fiber composites and the model for fiberorientation

effects.

In this paper we avoid this complication by using three-dimensional finite element models

of aligned short-fiber composites, rather than experimental results, as the reference data. This

necessitates the assumption of a periodic arrangement of the fibers, but all of the micromechanics

models are sufficiently vague about the geometricarrangement of thefibers that they admit periodic

geometries. We also compare the theories to some boundary element results for random arrays of

aligned fibers54

.For clarity we limit our comparisons to the models listed in Table 4. For the shear lag model we

show results only for the square array, noting that this choice for R gives the highest stiffness. The

models which are not shown are: the dilute Eshelby model, which is limited to small volume frac-

tions; the Hashin-Shtrikman-Walpole lower bound, which is identical to the Mori-Tanaka model;

and the upper bound, which is not claimed to be useful by itself.

4.1 Finite Element Modeling

Using the finite element method we analyzed two types of periodic, three-dimensional arrays of

fibers, which we call regular and staggered arrays. The representative volume elements (RVE’s)

are shown in Fig. 5. The unit cell dimensions were chosen with b = a , where is a constant. Weused both = 1 to obtain square packing, and =

p

3 which gives hexagonal packing. For the

regular fibers the distance between neighboring fiber ends (equal to 2 c ? ̀ in Fig. 5a) was set to

0 5 3 8 ̀ for square packing and 0 1 3 6 ̀ for hexagonal packing. For the staggered arrays the distance

along each fiber that is overlapped by neighboring fibers was set at a fixed percentage of the fiber

length: 65% for square packing and 76% for hexagonal packing. These conditions, together with

the fiber diameter and volume fraction, suffice to determine the dimensions a , b and c for each

20


22/34

Table 5: Material properties used in finite element calculations.

Property Fiber Matrix

E 30 1

0.20 0.38v

f

0.20

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

RVE. Note that a new RVE and its corresponding 3-D mesh are generated for each fiber aspect

ratio.

Stiffnesses of these RVE’s were calculated using ABAQUS 63. Twenty-node isoparametric ele-

ments were used, and a sample mesh is shown in Fig. 6. The analysis was geometrically nonlinear

but the applied strain was 0.5%, so the results are in the region of linear behavior. For axial or

transverse loading, symmetry requires all faces of the RVE to remain plane. To determine E 1 1

and

1 2 we fixed the normal displacements of the back, left, and bottom faces of the RVE; requiredthe right and top faces to remain plane and parallel to the coordinate axes (using multi-point con-

straints); and displaced the front face uniformly in the x 1

direction. The tangential displacements

on all faces were unconstrained. The average stress was computed from the reaction force in the

loading direction, divided by the cross-sectional area of the RVE. Average strains were computed

from the initial and deformed dimensions of the RVE. Analogous conditions were use to load the

RVE in thex

2

direction to determineE

2 2

and

2 3

. The longitudinal shear modulusG

1 2

could

in principle be determined using these same RVE’s, but that calculation requires a complicated

application of periodic boundary conditions and we did not undertake it.

All of the micromechanics theories reviewed here predict transversely isotropic properties.

Transverse isotropy about the x 1

axis implies that the tensile modulus is the same for any loading

direction in the 2–3 plane. This not only requires that E 2 2

= E

3 3

, but also that

1

G

2 3

=

2

E

2 2

( 1 +

2 3

)

(77)

RVE’s with hexagonal packing should also be transversely isotropic and obey these same relation-

ships. However, for square packing the properties are only guaranteed to be orthotropic. That is,

calculations for square packing will always give E 2 2

= E

3 3

, but the results will not necessarily

obey eqn (77) nor will the transverse modulus necessarily be the same for other loading directions

in the 2–3 plane. Here we simply report 2 3

and E 2 2

for loading in the x 2

direction, and do not

explore the other orthotropic constants for square packing.

The material properties used in the finite element calculations (Table 5) are typical of fiber-reinforced engineering thermoplastics. All of the moduli are scaled by the matrix modulus.

4.2 Results and Discussion

Figure 7 compares the theoretical and finite element results for longitudinal modulus E 1 1

. The

strong influence of fiber aspect ratio onE

1 1

is apparent, and all of the theories exhibit a similar

21


23/34

a

b

cl/2

a

b

cl/2

(a) (b)

1

3

2

Figure 5: Representative volume elements used in the finite element calculations. (a) Regular

array; the bold lines show the RVE. (b) Staggered array.

Figure 6: Example finite element mesh for a staggered, hexagonal array.

22


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0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1 10 100 1000

Halpin-Tsai

Nielsen

Mori-Tanaka

Lielens

Self-consistent

Shear lag, square

FE, Sqr. Reg.

FE, Sqr. Stag.

FE, Hex. Reg.

FE, Hex. Stag.

M o d u l u s R a t i o ,

E 1 1

/ E m

Aspect Ratio, l/d

Figure 7: Theoretical predictions and finite element results for E 1 1

.

S-shaped curve, asymptoting to the same rule-of-mixtures value at high aspect ratio. However, the

various theories give quite different values for very short fibers, and rise at different rates.

The different packing arrangements create some scatter in the finite element results, but the

scatter is small for ` = d 8 . For ` = d 4 the scatter is significant. This is not surprising, since

the properties of particulate-reinforced composites are known to be very sensitive to the packingarrangement. The high E

1 1

values for the hexagonal staggered array probably occur because our

rules for forming this particular type of RVE tend to create long ‘chains’ of nearly-touching par-

ticles parallel to the x 1

axis, with a high degree of axial overlap. While all of the finite element

results are equally ‘true,’ we believe the lower finite element values are more representative of the

actual packing and the actual stiffness of composites with very short fibers.

Comparing models to finite element data for E 1 1

, the Halpin-Tsai equation is accurate for very

short fibers, but falls below the data for longer fibers. The Nielsen model improves on the Halpin-

Tsai predictions for the very short fibers, but is still below the data for longer fibers. A better fit in

the higher aspect ratio range is provided by the Mori-Tanaka and Lielens models, which are only

slightly different from one another at this volume fraction. These models are good over most of

the data range. The self-consistent results are usually high, while the shear lag model is good forthe longer fibers but too low for very short fibers. This latter behavior is not surprising, since shear

lag theory treats the fiber as a slender body. Using any of the other values for R in the shear lag

model shifts the curve to the right, moving the predictions away from the data.

Results for transverse modulus E 2 2

are shown in Fig. 8. The finite element data again have

moderate scatter. Fiber aspect ratio has little effect on the transverse modulus, though some of

the packing geometries show a slight dip at low aspect ratio. Interestingly, the shape and location

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0.0

0.5

1.0

1.5

2.0

1 10 100 1000

Halpin-TsaiNielsenMori-TanakaLielensSelf-consistent

FE, Sqr. Reg.FE, Sqr. Stag.FE, Hex. Reg.FE, Hex. Stag.


E 2 2

/ E m

Aspect Ratio, l/d

Figure 8: Theoretical predictions and finite element results for E 2 2

.

of this dip are matched by the models that use the Eshelby tensor. Note that the Halpin-Tsai and

Nielsen models contain no dependence on aspect ratio for E 2 2

. Shear lag models do not predict

E

2 2

.

Most of the models do a good job of predicting E 2 2

, with the Mori-Tanaka and Lielens models

being the most accurate. The Halpin-Tsai result is slightly higher than most of the data, while theNielsen model noticeably over-predicts this property. For comparison the upper bound result falls

well above the data, with an asymptote of E 2 2

= E

m

= 3 5 9 at high aspect ratio.

Data for the Poisson ratios 1 2

and 2 3

appear in Figs. 9 and 10. The Nielsen and Halpin-Tsai

results for 1 2

are identical, so only the Halpin-Tsai curve is shown. Both Poisson ratios show

a moderate dependence on aspect ratio and some sensitivity to packing geometry. The shape of

this dependence is similar for all but the regular hexagon array and is matched qualitatively by

several models, but the quantitative match is not as good. For 1 2

the constant value provided by

the Halpin-Tsai equations is at least as good a match to the data as the models that show some

variation. However, the Halpin-Tsai and Nielsen models substantially over-predict 2 3

, while the

other models do very well on this property, especially at the higher aspect ratios. The error in the

Halpin-Tsai value results from a combination of a slightly high prediction for E 2 2 (Fig. 8) and aslightly low prediction for G

2 3

(not shown here), the effects combining through eqn (77).

One weakness of the finite element calculations is that they require the assumption of a regular,

periodic packing arrangement of the fibers. Calculations that do not have this limitation have

been recently reported by Ingber and Papathanasiou 54. These workers used the boundary element

method to calculate E 1 1

for random arrays of aligned fibers. Each model typically contained 100

fibers, and results from ten such models were averaged to produce each data point. We tested

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0.20

0.25

0.30

0.35

0.40

1 10 100 1000

Halpin-TsaiMori-TanakaLielensSelf-consistentFE, Sqr. Reg.FE, Sqr. Stag.FE, Hex. Reg.FE, Hex. Stag.

P o i s s o n R a t i o , ν

1 2

Aspect Ratio, l/d

Figure 9: Theoretical predictions and finite element results for 1 2

.

0.30

0.40

0.50

0.60

0.70

1 10 100 1000

Halpin-TsaiNielsenMori-TanakaLielensSelf-consistentFE, Sqr. Reg.FE, Sqr. Stag.

FE, Hex. Reg.FE, Hex. Stag.

P o i s s o n R a t i o , ν 2 3

Aspect Ratio, l/d

Figure 10: Theoretical predictions and finite element results for 2 3

.

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0

5

10

15

20

0.00 0.10 0.20 0.30

Halpin-TsaiNielsenMori-TanakaLielensSelf ConsistentBE, randomFE, Sqr. Reg.FE, Sqr. Stag.


E 1 1 /

E m

Fiber Volume Fraction, vf

Figure 11: Models compared to boundary element predictions of E 1 1

for random arrays of rigid

cylinders by Ingber and Papathanasiou 54, and to finite element calculations with E f

= E

m

= 1 0

6 ,

all for` = d = 1 0

.

their results against the various theories, and also performed a limited number of finite element

calculations for comparison purposes. The boundary element results are for rigid fibers (E

f

= E

m

=

1 ) and an incompressible matrix ( m

= 0 5 ), but our finite element calculations and theoretical

results use E f

= E

m

= 1 0

6 and m

= 0 4 9 to avoid numerical difficulties in some of the models.

Figure 11 shows the results for E 1 1

versus volume fraction for ` = d = 1 0 . The boundary element

data are most accurately matched by the Lielens and Nielsen models, though the Halpin-Tsai and

Mori-Tanaka models are not bad. The self-consistent model predicts much higher stiffnesses than

the other models and than the boundary element data. So far these results are consistent with our

previous comparisons.

What is surprising about Fig. 11 is that the finite element results fall so far above the boundary

element results, and above the theories that work so well in other cases. Since the finite element

data fall closer to the self-consistent model, it is tempting to think that they support the accuracy of

this model. But we believe it more likely that these results are revealing the sensitivity of stiffness

to the packing arrangement of the fibers.

Other researchers have noted that gathering short fibers into bundles or clusters tends to reduce

E

1 1

compared to evenly dispersed fibers 42. In the boundary element calculations of Ingber and

Papathanasiou the inter-fiber spacing is random, and hence uneven, so there is a modest clustering

effect. In contrast, our finite element models impose a uniform inter-fiber spacing, and so represent

an unusually even dispersion of fibers. Our finite element models also maximize the axial overlap

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between neighboring fibers. It seems that these geometric effects have the greatest influence on

composite stiffness when the fibers are rigid. We believe that the boundary element calculations

are more representative of real composite behavior than the finite element calculations in Fig. 11.

Fortunately the influence of fiber packing is much smaller for the E f

= E

m

ratios typical of

polymer-matrix composites. Note that the two different finite element results for v f

= 0 2 0 and

rigid fibers in Fig. 11 are far apart from one another, but in Fig. 7 where E f = E m = 3 0 the samepacking geometries give nearly identical results at

` = d = 8

. This lends support to the idea that

fiber packing is important mainly when the fibers are extremely stiff compared to the matrix, and

supports the finite element results in Figs. 7–10 as a meaningful test of the micromechanics theo-

ries.

5 Conclusions

Our goal is to identify the best model for predicting the stiffness of aligned short-fiber composites.

Among the models that we tested, the self-consistent approach tends to over-predict E 1 1

at high

volume fractions, though it gives good predictions for other elastic constants. The Halpin-Tsaimodel, long a standard for this problem, gives reasonable results for all the elastic constants except

2 3

, and its E 1 1

values are low for moderate-to-high aspect ratios. Nielsen and Lewis’s modification

of Halpin-Tsai improves the fit to Ingber and Papathanasiou’s boundary element data for E 1 1

, but

it does not substantially improve the fit to our E 1 1

data and it substantially worsens the prediction

of E 2 2

. The Mori-Tanaka and Lielens models give much better predictions than Halpin-Tsai for

2 3

, and slightly better predictions for all the other properties. Our finite element data does not

allow us to choose between the Mori-Tanaka and Lielens models, since the differences between

their predictions are small for the volume fractions we examined. Shear lag models can give good

predictions forE

1 1

for aspect ratios greater than 10, provided one makes the proper choice of R

,

but the predictions for shorter fibers are too low.

Our results confirm that the Halpin-Tsai equations providereasonable estimates for the stiffness

of short-fiber composites, but they indicate that the Mori-Tanaka model is more accurate. The

bound interpolation model of Lielens et al. may improve on the Mori-Tanaka model for higher

fiber volume fractions or modulus ratios, but for injection-molded composites the difference is

small. We recommend the Mori-Tanaka model as the best choice for estimating the stiffness of

aligned short-fiber composites.

Acknowledgments

Funding to the University of Illinois was provided by The General Electric Company and GeneralMotors Corporation. This work was conducted in support of the Thermoplastic Engineering De-

sign (TED) Venture, a Department of Commerce Advanced Technology Program administered by

the National Institute of Standards and Technology. The authors are grateful to Mr. C. Matthew

Dunbar of Hibbitt, Karlsson & Sorensen, Inc. for his assistance with mesh generation and the finite

element analysis, and to Dr. T. D. Papathanasiou of the University of South Carolina for making

available the detailed data from his recent paper.

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