Acta Materialia 52 (2004) 5315–5330

www.actamat-journals.com

Constitutive modeling of piezoelectric polymer composites

G.M. Odegard *

Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA

Received 11 December 2003; received in revised form 21 July 2004; accepted 23 July 2004

Available online 3 September 2004

Abstract

A new modeling approach is proposed for predicting the bulk electromechanical properties of piezoelectric composites. The pro-

posed model offers the same level of convenience as the well-known Mori–Tanaka method. The electromechanical properties of four

piezoelectric polymer composite materials are predicted with the proposed, Mori–Tanaka, Self-consistent methods, and detailed

finite element analyses are conducted over full ranges of reinforcement volume fractions. The presented data offer a comprehensive

comparison of the four modeling approaches for a wide range of matrix and reinforcement electromechanical properties, reinforce-

ment geometry, and reinforcement volume fraction. By comparison with the finite element data, it is shown that the proposed model

predicts properties that are, in some cases, more accurate than the Mori–Tanaka and Self-consistent schemes.

� 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Micromechanics; Mori–Tanaka; Piezoelectricity; PZT; PVDF; Self-consistent

1. Introduction

Piezoelectric materials are excellent candidates for

use in sensors and actuators because of their ability to

couple electrical and mechanical energy. For some

applications, it is necessary to use composite materialsin which one or more of the constituents have piezoelec-

tric properties. To facilitate the design of these piezoe-

lectric composite systems, convenient and accurate

structure–property relationships must be developed.

Numerous attempts have been made to develop mod-

els to relate bulk electromechanical properties of com-

posite materials to the electromechanical properties of

individual constituents. Simple estimates, utilizing Voigtor Reuss-type approaches, have been used to predict

the behavior of a limited class of composite geometries

[1–4]. Upper and lower bounds for the electromechani-

cal moduli have been determined [5–8]. Finite element

analysis has also been used to predict electromechanical

1359-6454/$30.00 � 2004 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2004.07.037

* Tel.: +1 906 487 2329; fax: +1 906 487 2822.

E-mail address: gmodegar@mtu.edu (G.M. Odegard).

properties [9,10]. Even though finite element analysis has

the best potential for accurately predicting composite

properties for any composite geometry, the solutions

can be very expensive and time-consuming.

Several authors have extended Eshelby�s [11] classicalsolution of an infinite medium containing a single ellip-soidal inclusion to include piezoelectric constituents [12–

15]. Also referred to as the dilute solution, this approach

ignores the interactions of the inclusions that occur at fi-

nite inclusion volume fractions. Other studies [14,16–19]

have focused on the classical extensions of Eshelby�ssolution for finite inclusion volume fractions, i.e., the

Mori–Tanaka [20,21], Self-consistent [22,23], and Differ-

ential [24,25] approaches. Analytical solutions forspecific composite systems have also been determined

[26–32]. Even though the overall framework of these ap-

proaches provides estimates for a wide range of inclu-

sion sizes, geometries, and orientations, each of

these methods suffers from drawbacks associated with

accuracy and computational convenience.

In this paper, a model is proposed for predicting the

coupled electromechanical properties of piezoelectric

ll rights reserved.

5316 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

composites. This model is an extension of a technique

originally developed for predicting mechanical proper-

ties of composites by generalizing the Mori–Tanaka

and Self-consistent approaches [33]. First, the overall

constitutive modeling of piezoelectric materials is dis-

cussed, followed by a description of the proposed model.Finally, the electromechanical properties of four differ-

ent piezoelectric composite systems are predicted using

the proposed, Mori–Tanaka, Self-consistent, and finite

element models. The four piezoelectric composite sys-

tems used in this study were chosen to represent a wide

range of practical materials: a graphite/poly(vinylidene

fluoride) (PVDF) composite, a silicon carbide (SiC)/

PVDF particulate composite, a fibrous lead zirconatetitanate (PZT)/polyimide composite, and a PZT/

polyimide particulate composite.

2. Constituent materials

The matrix and inclusion constituents used in this

study were chosen such that the composite materialshad four combinations of piezoelectric constituents

and reinforcement geometries. The graphite/PVDF and

SiC/PVDF composites have a piezoelectric polymer ma-

trix with fiber and particle reinforcement, respectively.

The PZT/polyimide composites have a piezoelectric

inclusion with fiber and particle reinforcements.

PVDF is a orthotropic, semi-crystalline polymer

which exhibits a piezoelectric effect with an electric fieldapplied along the 3-axis. Typical electromechanical prop-

erties of PVDF are given in Table 1 (these properties were

supplied by NASA Langley Research Center). LaRC-SI

is a thermoplastic polyimide that was developed for aer-

ospace applications. The properties of LaRC-SI used in

this study correspond to the system with a 3% stoichio-

Table 1

Electromechanical properties of matrix and inclusion materials

Property PVDF LaRC-SI

C11 (GPa) 3.8 8.1

C12 (GPa) 1.9 5.4

C13 (GPa) 1.0 5.4

C22 (GPa) 3.2 8.1

C23 (GPa) 0.9 5.4

C33 (GPa) 1.2 8.1

C44 (GPa) 0.7 1.4

C55 (GPa) 0.9 1.4

C66 (GPa) 0.9 1.4

j1/j0 7.4 2.8

j2/j0 9.3 2.8

j3/j0 7.6 2.8

e15 (C/m2) 0.0 0.0

e31 (C/m2) 0.024 0.0

e32 (C/m2) 0.001 0.0

e33 (C/m2) �0.027 0.0

metric imbalance at room temperature [34] and are also

shown in Table 1. The PVDF polymer was reinforced

with both infinitely-long graphite fibers and spherical

SiC particles. The fibers were unidirectionally aligned

along the PVDF 1-axis. This alignment was chosen for

the modeling because of the desire to maintain a high le-vel of material compliance (therefore maximizing the pie-

zoelectric effect) in the transverse directions, while

providing reinforcement in the direction in which little

piezoelectric effect and maximum mechanical reinforce-

ment are required. The LaRC-SI polymer was reinforced

with both infinitely-long PZT-7A fibers and spherical

PZT-7A particles. PZT-7A is a ceramic that exhibits a

piezoelectric effect with electric fields applied along allthree principle axes. The PZT-7A fibers were unidirec-

tionally aligned with the fiber 3-axis as the fiber-length

axis. This alignment was chosen to maintain consistency

with previous analyses [8,14,19], which ultimately pro-

vides alignment of the fibers during the poling process

in the fabrication of these materials. All of the inclusion

electromechanical properties are given in Table 1.

3. Micromechanics modeling

3.1. Piezoelectric materials

There are three standard notation systems that are

commonly used to describe the constitutive modeling

of linear-piezoelectric materials. Using the conventionalindicial notation in which repeated subscripts are

summed over the range of i, j,m,n = 1,2,3, the constitu-

tive equations are

rij ¼ Cijmnemn þ enijEn;

Di ¼ eimnemn � jinEn;ð1Þ

Graphite fiber SiC particle PZT-7A

243.7 483.7 148.0

6.7 99.1 76.2

6.7 99.1 74.2

24.0 483.7 148.0

9.7 99.1 74.2

24.0 483.7 131.0

11.0 192.3 25.4

27.0 192.3 25.4

27.0 192.3 35.9

12.0 10.0 460.0

12.0 10.0 460.0

12.0 10.0 235.0

0.0 0.0 9.2

0.0 0.0 �2.1

0.0 0.0 �2.1

0.0 0.0 9.5

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5317

where rij, eij, Ei, and Di are the stress tensor, strain ten-

sor, electric field vector, and the electric displacement

vector, respectively. The quantities Cijmn, enij, and jinare the elastic stiffness tensor, the piezoelectric tensor,

and the permittivity tensor, respectively. The divergence

equations, which are the elastic equilibrium and Gauss�law, are, respectively,

rij;j ¼ 0;

Di;i ¼ 0;ð2Þ

where the subscripted comma denotes partial differenti-

ation. The gradient equations, which are the strain-displacement equations and electric field-potential, are,

respectively,

eij ¼1

2ðui;j þ uj;iÞ;

Ei ¼ �/;i;ð3Þ

where ui and / are the mechanical displacement and

electric potential, respectively.

In the modeling of piezoelectric materials, it is more

convenient to restate Eq. (1) so that the elastic and

electric variables are combined to yield a single consti-

tutive equation. This notation is identical to the con-

ventional indicial notation with the exception thatlower case subscripts retain the range of 1–3 and cap-

italized subscripts take on the range of 1–4, with

repeated capitalized subscripts summed over 1–4.

In this notation, Eq. (1) is

RiJ ¼ EiJMnZMn; ð4Þ

where Rij, EiJMn, and ZMn are, respectively,

RiJ ¼rij; J ¼ 1; 2; 3;

Di; J ¼ 4;

�ð5Þ

EiJMn ¼

Cijmn; J ;M ¼ 1; 2; 3;

enij; J ¼ 1; 2; 3; M ¼ 4;

eimn; J ¼ 4; M ¼ 1; 2; 3;

�jin; J ;M ¼ 4;

8>>><>>>:

ð6Þ

ZMn ¼emn; M ¼ 1; 2; 3;

En; M ¼ 4:

�ð7Þ

The piezoelectric constitutive equation can be further

simplified by expressing Eq. (4) in matrix notation

R ¼ EZ; ð8Þwhere the boldface indicates either a 9 · 9 matrix (E) ora 9 · 1 column vector (R,Z)

Rt ¼ r11 r22 r33 r23 r13 r12 jD1 D2 D3½ �;ð9Þ

t

Z ¼ e11 e22 e33 c23 c13 c12 jE1 E2 E3½ �; ð10Þ

. ð11Þ

In Eq. (11), C, e, and j denote the elastic stiffness ma-

trix, the piezoelectric constant matrix, and the permittiv-

ity matrix, respectively. The superscript t denotes a

matrix transposition. Note that cij = 2eij in order to keep

E a symmetric matrix. From Eqs. (8)–(11) the constitu-

tive equation for an orthotropic piezoelectric material is

;

ð12Þ

where the contracted Voigt notation is used. In Eq. (12),

the 3-axis is aligned with the principle direction of

polarization.

3.2. Electromechanical properties of composites

Using the direct approach [14,35,36] for the estimate

of overall properties of heterogeneous materials, the vol-ume-averaged piezoelectric fields of the composite with

a total of N phases are

�R ¼XNr¼1

cr�Rr; ð13Þ

�Z ¼XNr¼1

cr �Zr; ð14Þ

where cr is the volume fraction of phase r, the overbar

denotes a volume-averaged quantity, the subscript r de-

notes the phase, and r = 1 is the matrix phase. The con-stitutive equation for each phase is given by Eq. (8). For

a piezoelectric composite subjected to homogeneous

elastic strain and electric field boundary conditions,

Z0, it has been shown that �Z ¼ Z0 [16]. The constitutive

equation for the piezoelectric composite can be

expressed in terms of the volume-averaged fields

�R ¼ E�Z: ð15ÞThe volume-average strain and electric field in phase r is

�Zr ¼ Ar�Z; ð16Þ

where Ar is the concentration tensor of phase r, and

5318 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

XNr¼1

crAr ¼ I; ð17Þ

where I is the identity tensor. Combining Eqs. (13)–(17)

yields the electromechanical modulus of the composite

in terms of the constituent moduli

E ¼ E1 þXNr¼2

cr Er � E1ð ÞAr: ð18Þ

Various procedures exist for evaluating the concentra-

tion tensor. The most widely used approaches are the

Mori–Tanaka and Self-consistent schemes.

For the Mori–Tanaka approach, the concentration

tensor is

As ¼ Adilr c1Iþ

XNr¼2

crAdilr

" #�1

; ð19Þ

where Adilr is the dilute concentration tensor given by

Adilr ¼ Iþ SrE

�11 Er � E1ð Þ

� ��1: ð20Þ

In Eq. (20) Sr is the constraint tensor for phase r, which

is analogous to the Eshelby tensor used in determiningelastic properties of composite materials [11]. The con-

straint tensor is evaluated as a function of the lengths

of the principle axes of the reinforcing phase r, ari , andthe electromechanical properties of the surrounding

matrix

Sr ¼ f E1; ar1; ar2; a

r3

� �: ð21Þ

The complete expression for Eq. (21) is given elsewhere

[16]. While the Mori–Tanaka approach provides for a

quick and simple calculation of the bulk composite elec-

tromechanical properties, it has been shown that it

yields predicted mechanical properties that are relatively

low and high for composites with stiffer inclusions and

matrix, respectively [33]. This issue could possibly leadto less accurate estimations of the electromechanical

moduli, especially for relatively large inclusion volume

fractions [37–39].

In the Self-consistent scheme, the concentration ten-

sor is

Ar ¼ Iþ SrE�1 Er � Eð Þ

� ��1; ð22Þ

where E is the unknown electromechanical moduli of the

composite, and the constraint tensor, Sr, is evaluated as

a function of E and ari . Since the electromechanical mod-

uli of the composite appears in both Eqs. (22) and (18),

iterative schemes or numerical techniques are ultimatelyrequired for the prediction of the electromechanical

moduli of composites using the Self-consistent method.

This approach results in slow and complicated

calculations.

It has been demonstrated [33] that a more general

form of the concentration tensor can be used for the pre-

diction of mechanical properties of composites. Extend-

ing this concept to the prediction of electromechanical

properties results in

Ar ¼ Iþ SrE�10 Er � E0ð Þ

� ��1; ð23Þ

where E0 is the electroelastic moduli of the reference

medium, and the constraint tensor is evaluated using

E0 and ari . Therefore, it is assumed that the reference

medium is the material that immediately surrounds the

inclusion for the evaluation of the constraint and con-

centration tensors. Naturally, the electroelastic moduliof the reference medium can have a wide range of val-

ues, however, it is most realistic to assume that they

are similar to the moduli of the overall composite, as

is the case in the Self-consistent method.

For convenience, a simple, yet accurate, estimation of

the overall electroelastic moduli can be chosen for the

reference medium so that the overall properties of the

piezoelectric composite can be calculated usingEqs. (18) and (23). Even though a simple and accurate

estimation of the reference medium means that the elec-

troelastic moduli can be calculated without Eqs. (18)

and (23), this framework allows for the computation

of the moduli for various inclusion sizes, geometries,

and orientations. The reference medium is approxi-

mated with a set of equations that are similar to the Hal-

pin–Tsai relation [40], which is extended here formultiple inclusions and piezoelectric composites

E0iJKl ¼ E1

iJKl

1þPN

r¼2griJKlcr

1�PN

r¼2griJKlcr

; ð24Þ

where

griJKl ¼EriJKl � E1

iJKl

EriJKl þ E1

iJKl

: ð25Þ

Eqs. (24) and (25) indicate that as c1!1 and cr!1,

E0iJKl ! E1

iJKl and E0iJKl ! Er

iJKl, respectively.

Eqs. (18) and (23)–(25) were used to calculate the

electromechanical properties for the four composite sys-tems for inclusion volume fractions ranging from 0% to

the maximum theoretical limits, which are about 90%

and 75% for fibrous and particulate composites, respec-

tively. The constraint tensor in Eq. (23) was evaluated

numerically using Gaussian quadrature [41]. The fibers

were modeled as infinitely long cylinders and the parti-

cles were modeled as spheres. Perfect bonding between

the inclusions and matrix was assumed.

4. Finite element analysis

Another approach to estimate the electromechanical

properties of piezoelectric composites is finite element

analysis of a representative volume element (RVE) of

the material. Whereas the methods of the previous

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5319

section provide relatively quick predictions by assuming

that the stress and strain fields inside the inclusions are

constant, finite element analysis predicts these fields in

the inclusion and matrix, and thus, provides a more real-

istic prediction to the overall electromechanical moduli

of the composite. This added accuracy comes at a price,however, since each independent property of the piezo-

electric composite (16 independent parameters are

shown in Eq. (12)) must be determined by a single finite

element analysis. In parametric studies where many

combinations of inclusion shape and volume fraction

must be considered, the finite element approach can be-

come very time-consuming and expensive. Therefore, in

this study, the finite element results are used to check theaccuracy of the modeling methods discussed in the pre-

vious section.

The finite element model was developed and exe-

cuted using ANSYS� 7.0. RVEs of fiber- and particu-

late-reinforced composites were meshed using 10-noded

electromechanical tetrahedral elements with 40 degrees

of freedom, three displacements and an electric poten-

tial at each node (SOLID98). The fibrous compositeRVE (Fig. 1) simulated a hexagonal packing arrange-

ment, with a maximum fiber volume fraction of about

90%. The particulate composite RVE (Fig. 2) had hex-

agonal packing in one plane with a maximum particle

volume fraction of about 60%. For each finite element

analysis, the desired volume fraction was obtained by

adjusting the dimensions of the RVE while keeping

the reinforcement size constant. The properties of thematerials are shown in Table 1. Additional reinforce-

Fig. 1. Finite element RVE

ment and matrix material were connected to each of

the eight faces of both the fibrous and particulate

RVEs to form the full finite element models (Figs. 1

and 2).

For homogeneous applied elastic strains and electric

fields, the displacements and voltages on the boundaryof the full finite element models were, respectively,

ui Bð Þ ¼ e0ijxj;

/ Bð Þ ¼ �E0i xi;

ð26Þ

where B indicates the boundary of the full finite element

model. A total of 16 boundary conditions were applied

to the finite element models for each combination of

material type and volume fraction. Each boundary con-

dition was used to predict one of the independent elec-

troelastic constants in Eq. (12). The electroelasticconstants and the corresponding applied strains, electric

fields, and the boundary conditions calculated using

Eq. (26) are listed in Tables 2–6. For each set of bound-

ary conditions, all unspecified strains and electric fields

in Tables 2–6 are zero. It is noted at this point that

the boundary conditions specified in Eq. (26) are often

referred to as kinematic boundary conditions. These

boundary conditions are not applied directly tothe boundary of the RVE. Instead, they are applied to

the boundary of the full finite element model. Therefore,

the resulting deformations of the RVE are not over-

constrained. Over-constrained RVE edges are a result

of applying the kinematic boundary conditions directly

to the boundary of the RVE [42].

of fiber composite.

Fig. 2. Finite element RVE of particle composite.

Table 2

Boundary conditions for axial stiffness components

Property Applied strain and

electric field

Displacements and

electric potential

Elastic energy

C11 e011 ¼ e0 u1(B) = e0x1 Ue ¼ V2C11 e0

� �2u2(B) = 0

u3(B) = 0

/(B) = 0

C22 e022 ¼ e0 u1(B) = 0 Ue ¼ V2C22 e0

� �2u2(B) = e0x2u3(B) = 0

/(B) = 0

C33 e033 ¼ e0 u1(B) = 0 Ue ¼ V2C33ðe0Þ2

u2(B) = 0

u3(B) = e0x3/(B) = 0

Table 3

Boundary conditions for plane-strain bulk moduli

Property Applied strain and

electric field

Displacements and

electric potential

Elastic energy

K23 e022 ¼ e033 ¼ e0 u1(B) = 0 Ue ¼ V2K23ðe0Þ2

u2(B) = e0x2u3(B) = e0x3/(B) = 0

K13 e011 ¼ e033 ¼ e0 u1(B) = e0x1 Ue ¼ V2K13ðe0Þ2

u2(B) = 0

u3(B) = e0x3/(B) = 0

K12 e011 ¼ e022 ¼ e0 u1(B) = e0x1 Ue ¼ V2K12ðe0Þ2

u2(B) = e0x2u3(B) = 0

/(B) = 0

5320 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

The elastic strain energy, dielectric energy, and elec-

tromechanical energy of a piezoelectric material are,

respectively,

Ue ¼Xn

m¼1

Ume ¼ V

2Cijkle

0ije

0kl;

Ud ¼Xn

m¼1

Umd ¼ V

2jijE0

i E0j ;

Uem ¼Xn

m¼1

Umem ¼ V

2eijke0jkE

0i ;

ð27Þ

where Um is the energy of element m, n is the total num-ber of finite elements in the RVE, and V is the volume of

the RVE. The energies where calculated for each ele-

ment in the RVE volumes for each set of boundary con-

ditions (Tables 2–6) applied to the full finite element

model boundary. The total energies of the RVEs weredetermined by summing the energies of each RVE ele-

ment, as indicated by the first equality in Eq. (27). The

corresponding elastic, dielectric, and piezoelectric con-

stants were subsequently calculated using the second

equality in Eq. (27).

5. Results and discussion

The Young�s moduli, Y1, Y2, and Y3; shear moduli,

G23, G13, and G12; piezoelectric constants, e31, e32, e33;

Table 4

Boundary conditions for shear stiffness components

Property Applied strain and

electric field

Displacements and

electric potential

Elastic energy

C44 e023 ¼c0

2u1(B) = 0 Ue ¼ V

2C44ðc0Þ2

u2(B) = (c0/2)x3u3(B) = (c0/2)x2/(B) = 0

C55 e013 ¼c0

2u1(B) = (c0/2)x3 Ue ¼ V

2C55ðc0Þ2

u2(B) = 0

u3(B) = (c0/2)x1/(B) = 0

C66 e012 ¼c0

2u1(B) = (c0/2)x2 Ue ¼ V

2C66ðc0Þ2

u2(B) = (c0/2)x1u3(B) = 0

/(B) = 0

Table 5

Boundary conditions for dielectric constants

Property Applied strain and

electric field

Displacements and

electric potential

Dielectric

energy

j1/j0 E01 ¼ E0 u1(B) = 0 Ud ¼ V

2j1ðE0Þ2

u2(B) = 0

u3(B) = 0

/(B) = �E0x1

j2/j0 E02 ¼ E0 u1(B) = 0 Ud ¼ V

2 j2ðE0Þ2

u2(B) = 0

u3(B) = 0

/(B) = �E0x2

j3/j0 E03 ¼ E0 u1(B) = 0 Ud ¼ V

2j3ðE0Þ2

u2(B) = 0

u3(B) = 0

/(B) = �E0x3

Table 6

Boundary conditions for piezoelectric constants

Property Applied strain

and electric field

Displacements and

electric potential

Electromechanical

energy

e15 e013 ¼c0

2u1(B) = (c0/2)x3 Uem ¼ V

2e15c0E0

u2(B) = 0

E01 ¼ E0 u3(B) = (c0/2)x1

/(B) = �E0x1

e31 e011 ¼ e0 u1(B) = e0x1 Uem ¼ V2e31e0E0

u2(B) = 0

E03 ¼ E0 u3(B) = 0

/(B) = �E0x3

e32 e022 ¼ e0 u1(B) = 0 Uem ¼ V2e32e0E0

u2(B) = e0x2E03 ¼ E0 u3(B) = 0

/(B) = �E0x3

e33 e033 ¼ e0 u1(B) = 0 Uem ¼ V2e33e0E0

u2(B) = 0

E03 ¼ E0 u3(B) = e0x3

/(B) = �E0x3

0

5

10

15

20

25

30

0 10 20 30 40

Fiber volume

Yo

un

g's

mo

du

lus

(GP

a)

Finite element,Finite element,Finite element,Proposed, Y1

Proposed, Y2Proposed, Y3Mori-Tanaka, YMori-Tanaka, YMori-Tanaka, YSelf-ConsistenSelf-ConsistenSelf-Consisten

Finite element,Finite element,Finite element,Proposed, Y1

Proposed, Y2Proposed, Y3Mori-Tanaka, YMori-Tanaka, YMori-Tanaka, YSelf-ConsistenSelf-ConsistenSelf-Consisten

Fig. 3. Young�s moduli vs. fiber volume fra

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5321

and dielectric constants, j1/j0, j2/j0, and j3/j0; for thefour materials discussed in this paper are presented be-low. The subscripts of these quantities indicate the cor-

responding axes, as shown in Eq. (12), and the

permittivity of free space, j0, is 8.85 · 10�12 C/m2.

5.1. Graphite/PVDF fiber composite

The Young�s moduli of the graphite/PVDF compos-

ite are shown in Fig. 3 as a function of the graphite fibervolume fraction for the results obtained with the finite

element analysis, the proposed model discussed above,

50 60 70 80 9

fraction (%)

0

Y1 Y2 Y3

1

2

3

t, Y1

t, Y2

t, Y3

Y1 Y2 Y3

1

2

3

t, Y1

t, Y2

t, Y3

ction for graphite/PVDF composite.

5322 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

the Mori–Tanaka model, and the Self-consistent

method. For the Young�s modulus parallel to the

fiber-alignment direction, Y1, all four models predict

the same values for the entire range of fiber volume frac-

tions. For the two transverse moduli, Y2 and Y3, the

Mori–Tanaka and finite element models match very wellfor the entire range of fiber volume fractions, while the

proposed and Self-consistent models over-predict the

Young�s moduli for fiber volume fractions over 40%.

The shear moduli of this material for the entire range

of fiber volume fractions are shown in Fig. 4. For the

0

5

10

15

20

25

0 10 20 30 40

Fiber volum

Sh

ear

mo

du

lus

(GP

a)

Finite element, G23

Finite element, G13 G12Proposed, G 23

Proposed, G 13 G12Mori-Tanaka, G23

Mori-Tanaka, G13 G12Self-Consistent, G23

Self-Consistent, G13 G12

Finite element, G23

Finite element, G13 ≈ G12

≈Mori-Tanaka, G23

Mori-Tanaka, G13 ≈ G12Self-Consistent, G23

Self-Consistent, G13 ≈ G12

Fig. 4. Shear moduli vs. fiber volume frac

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0 10 20 30 40

Fiber volum

Pie

zoel

ectr

ic c

on

stan

t (C

/m2 )

Finite element, e31Finite element, e32Finite element, e33Proposed, e31Proposed, e32Proposed, e33

Finite element, e31Finite element, e32Finite element, e33Proposed, e31Proposed, e32Proposed, e33

Fig. 5. Piezoelectric constants vs. fiber volume

longitudinal shear moduli, G13 and G12, the proposed

model has a closer agreement with the finite element

model than the Mori–Tanaka and Self-consistent mod-

els have with the finite element model for fiber volume

fractions above 40%. For the transverse shear modulus,

G23, the proposed, Mori–Tanaka, and the Self-consistentall predict slight higher values than the finite element

model, with the Mori–Tanaka showing the closest

agreement.

The piezoelectric constants, e31, e32, and e33, are

shown in Fig. 5 as a function of the fiber volume frac-

50 60 70 80 9

e fraction (%)

0

tion for graphite/PVDF composite.

50 60 70 80 9

e fraction (%)

0

Mori-Tanaka, e31Mori-Tanaka, e32Mori-Tanaka, e33Self-Consistent, e31Self-Consistent, e32Self-Consistent, e33

Mori-Tanaka, e31Mori-Tanaka, e32Mori-Tanaka, e33Self-Consistent, e31Self-Consistent, e32Self-Consistent, e33

fraction for graphite/PVDF composite.

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5323

tion. The four models predict nearly equal values of e31and e32 over the entire range. For the piezoelectric con-

stant e33 the proposed and Self-consistent results over-

predict the finite element model, while the Mori–Tanaka

method shows good agreement with the finite element

model.The dielectric constants, j1/j0, j2/j0, and j3/j0, are

shown in Fig. 6. All four models predict identical values

for all three dielectric constants for the complete range

of fiber volume fractions.

6

7

8

9

10

11

12

0 10 20 30 40

Fiber volume

Die

lect

ric

con

stan

t

Finite elemeFinite elemeFinite elemeProposed, Proposed, Proposed,

Finite elemeFinite elemeFinite elemeProposed, κProposed, κProposed, κ

Fig. 6. Dielectric constants vs. fiber volume

0

5

10

15

20

25

30

35

0 10 20 30

Particle volu

Yo

un

g's

mo

du

lus

(GP

a)

Finite element, Y1Finite element, Y2Finite element, Y3Proposed, Y1Proposed, Y2Proposed, Y3Mori-Tanaka, Y1Mori-Tanaka, Y2Mori-Tanaka, Y3

Self-Consistent, Y1

Self-Consistent, Y2

Self-Consistent, Y3

Finite element, Y1Finite element, Y2Finite element, Y3Proposed, Y1Proposed, Y2Proposed, Y3Mori-Tanaka, Y1Mori-Tanaka, Y2Mori-Tanaka, Y3

Self-Consistent, Y1

Self-Consistent, Y2

Self-Consistent, Y3

Fig. 7. Young�s moduli vs. particle volum

5.2. SiC/PVDF particle composite

The Young�s moduli of the SiC/PVDF composite are

shown in Fig. 7 as a function of particle volume fraction.

At particle volume fractions of about 20% and lower, all

four models predict nearly identical moduli. At higherparticle volume fractions, the proposed model predicts

moduli that have closer agreement with the finite ele-

ment results than has the predicted values from the

Mori–Tanaka model. For particle volume fractions

50 60 70 80

fraction (%)

90

nt, 1/ 0nt, 2/ 0nt, 3/ 0

1/ 0

2/ 0

3/ 0

Mori-Tanaka, 1/ 0Mori-Tanaka, 2/ 0Mori-Tanaka, 3/ 0Self-Consistent, 1/ 0Self-Consistent, 2/ 0Self-Consistent, 3/ 0

nt, κ1/κ0nt, κ2/κ0nt, κ3/κ0

1/κ0

2/κ0

3/κ

Mori-Tanaka, κ1/κ0Mori-Tanaka, κ2/κ0Mori-Tanaka, κ3/κ0Self-Consistent, κ1/κ0Self-Consistent, κ2/κ0Self-Consistent, κ3/κ0

fraction for graphite/PVDF composite.

40 50 60 70

me fraction (%)

e fraction for SiC/PVDF composite.

5324 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

higher than 20%, the Self-consistent approach

significantly over-predicts the other three models.

The three shear moduli are shown in Fig. 8 for the en-

tire range of particle volume fractions. For all three

shear moduli, at volume fractions of 50% and less, the

Mori–Tanaka and finite element models have closeagreement, with the proposed model over-predicting

the shear moduli. For a volume fraction of 60%, the

shear moduli of the finite element model start increasing

dramatically, and the proposed model shows closer

agreement with the finite element model than does the

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30

Particle volum

Sh

ear

mo

du

lus

(GP

a)

Finite element, G23Finite element, G13Finite element, G12Proposed, G23Proposed, G13Proposed, G12Mori-Tanaka, G23Mori-Tanaka, G13Mori-Tanaka, G12Self-Consistent, G23Self-Consistent, G13Self-Consistent, G12

Finite element, G23Finite element, G13Finite element, G12Proposed, G23Proposed, G13Proposed, G12Mori-Tanaka, G23Mori-Tanaka, G13Mori-Tanaka, G12Self-Consistent, G23Self-Consistent, G13Self-Consistent, G12

Fig. 8. Shear moduli vs. particle volume

-0.035

-0.025

-0.015

-0.005

0.005

0.015

0.025

0 10 20 30

Particle vol

Pie

zoel

ectr

ic c

on

stan

t (C

/m2 )

Finite element, e31

Finite element, e32

Finite element, e33Proposed, e31Proposed, e32Proposed, e33

Finite element, e31

Finite element, e32

Finite element, e33Proposed, e31Proposed, e32Proposed, e33

Fig. 9. Piezoelectric constants vs. particle vol

Mori–Tanaka approach. For particle volume fractions

over 20%, the Self-consistent approach significantly

over-estimates all three shear moduli.

The piezoelectric constants are shown in Fig. 9 as a

function of particle volume fraction. For the constant

e31 the proposed model data matches the finite elementdata more closely than does the Mori–Tanaka and

Self-consistent approaches. For the constant e32 all four

models predict nearly identical values for the entire

range of particle volume fractions. For the piezoelectric

constant e33 the Mori–Tanaka approach exhibits the

40 50 60 70

e fraction (%)

fraction for SiC/PVDF composite.

40 50 60 70

ume fraction (%)

Mori-Tanaka, e31Mori-Tanaka, e32Mori-Tanaka, e33Self-Consistent, e31

Self-Consistent, e32

Self-Consistent, e33

Mori-Tanaka, e31Mori-Tanaka, e32Mori-Tanaka, e33Self-Consistent, e31

Self-Consistent, e32

Self-Consistent, e33

ume fraction for SiC/PVDF composite.

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5325

closest agreement with the finite element predictions,

especially for particle volume fractions above 30%.

The three dielectric constants for the composite are

shown in Fig. 10. Similar to the graphite/PVDF com-

posite, all four models predict identical values for all

three dielectric constants for the entire range of particlevolume fractions.

5.3. PZT-7A/polyimide fiber composite

The Young�s moduli of the PZT-7A composite are

shown in Fig. 11 for the entire range of fiber volume

6

7

8

9

10

0 10 20 30

Particle volum

Die

lect

ric

con

stan

t

Finite elemeFinite elemeFinite elemeProposed, Proposed, Proposed,

Finite elemeFinite elemeFinite elemeProposed, κProposed, κProposed, κ

Fig. 10. Dielectric constants vs. particle volu

0

10

20

30

40

50

60

70

80

0 10 20 30 40

Fiber volume

Yo

un

g's

mo

du

lus

(GP

a)

Finite element, Y1 = Y2Finite element, Y3Proposed, Y1 = Y2Proposed, Y3Mori-Tanaka, Y1 = Y2Mori-Tanaka, Y3Self-Consistent, Y1 = Y2Self-Consistent, Y3

Finite element, Y1 = Y2Finite element, Y3Proposed, Y1 = Y2Proposed, Y3Mori-Tanaka, Y1 = Y2Mori-Tanaka, Y3Self-Consistent, Y1 = Y2Self-Consistent, Y3

Fig. 11. Young�s moduli vs. fiber volume fra

fractions. For the longitudinal Young�s modulus, Y3,

all three models predict the same values over the com-

plete range of volume fractions. For the transverse

Young�s moduli, Y1 and Y2, the Mori–Tanaka and finite

element models have close agreement up to a fiber vol-

ume fraction of 80%. At a fiber volume fraction of90%, the proposed model exhibits the closest agreement

to the finite element model. The Self-consistent method

significantly over-predicts the finite element model at

fiber volume fractions above 50%.

The shear moduli of the composite are plotted as a

function of fiber volume fraction in Fig. 12. For the

40 50 60 70

e fraction (%)

nt, 1/ 0nt, 2/ 0nt, 3/ 0

1/ 0

2/ 0

3/ 0

Mori-Tanaka, 1/ 0Mori-Tanaka, 2/ 0Mori-Tanaka, 3/ 0Self-Consistent, 1/ 0Self-Consistent, 2/ 0Self-Consistent, 3/ 0

nt, κ1/κ0nt, κ2/κ0nt, κ3/κ0

1/κ0

2/κ0

3/κ0

Mori-Tanaka, κ1/κ0Mori-Tanaka, κ2/κ0Mori-Tanaka, κ3/κ0Self-Consistent, κ1/κ0Self-Consistent, κ2/κ0Self-Consistent, κ3/κ0

me fraction for SiC/PVDF composite.

50 60 70 80 9

fraction (%)

0

ction for PZT-7A/LaRC-SI composite.

Finite element, G23 = G13Finite element, G12Proposed, G23 = G13Proposed, G12Mori-Tanaka, G23 = G13Mori-Tanaka, G12Self-Consistent, G23 = G13Self-Consistent, G12

Finite element, G23 = G13Finite element, G12Proposed, G23 = G13Proposed, G12Mori-Tanaka, G23 = G13Mori-Tanaka, G12Self-Consistent, G23 = G13Self-Consistent, G12

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80

Fiber volume fraction (%)

Sh

ear

mo

du

lus

(GP

a)

90

Fig. 12. Shear moduli vs. fiber volume fraction for PZT-7A/LaRC-SI composite.

5326 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

longitudinal shear moduli, G23 and G13, and the trans-

verse shear modulus, G12, the proposed, Mori–Tanaka,

and Self-consistent models over-predict the finite element

model for the entire range of volume fractions. While the

Mori–Tanaka model exhibits the closest agreement with

the finite element model, the Self-consistent significantlyever-estimates the finite element model data.

The piezoelectric constants of this material are shown

in Fig. 13. For the constants e31 = e32 and e33 all four

models predict very similar values for the entire range

of fiber volume fractions. For e15 both the proposed

and Mori–Tanaka models closely agree with the finite

-2

0

2

4

6

8

10

12

0 10 20 30 40

Fiber volum

Pie

zoel

ectr

ic c

on

stan

t (C

/m2 )

Finite element, e15

Finite element, e31 = e32

Finite element, e33Proposed, e15Proposed, e31 = e32

Proposed, e33

Mori-Tanaka, e15

Mori-Tanaka, e31 = e32

Mori-Tanaka, e33Self-Consistent, e15Self-Consistent, e31 = e32

Self-Consistent, e33

Finite element, e15

Finite element, e31 = e32

Finite element, e33Proposed, e15Proposed, e31 = e32

Proposed, e33

Mori-Tanaka, e15

Mori-Tanaka, e31 = e32

Mori-Tanaka, e33Self-Consistent, e15Self-Consistent, e31 = e32

Self-Consistent, e33

Fig. 13. Piezoelectric constants vs. fiber volume

element model for the entire range of volume fractions.

The Self-consistent method significantly overestimates

e15 for the fiber volume fractions above 30%.

The dielectric constants are shown in Fig. 14 as a

function of fiber volume fraction. For the transverse die-

lectric constants, j1/j0 and j2/j0, the Mori–Tanakamodel predicts the finite element model data better than

does the proposed and Self-consistent models for fiber

volume fractions above 50%. For the longitudinal die-

lectric constant, j3/j0, the four models predict nearly

identical values over the entire range of volume

fractions.

50 60 70 80 9

e fraction (%)

0

fraction for PZT-7A/LaRC-SI composite.

Finite element, 1/ 0 = 2/ 0Finite element, 3/ 0Proposed, 1/ 0 = 2/ 0Proposed, 3/ 0Mori-Tanaka, 1/ 0 = 2/ 0Mori-Tanaka, 3/ 0Self-Consistent, 1/ 0 = 2/ 0Self-Consistent, 3/ 0

Finite element, κ1/κ0 = κ2/κ0Finite element, κ3/κ0Proposed, κ1/κ0 = κ2/κ0Proposed, κ3/κ0Mori-Tanaka, κ1/κ0 = κ2/κ0Mori-Tanaka, κ3/κ0Self-Consistent, κ1/κ0 = κ2/κ0Self-Consistent, κ3/κ0

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80

Fiber volume fraction (%)

Die

lect

ric

con

stan

t

90

Fig. 14. Dielectric constants vs. fiber volume fraction for PZT-7A/LaRC-SI composite.

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5327

5.4. PZT-7A/polyimide particle composite

The Young�s moduli of the PZT-7A/LaRC-SI partic-

ulate composite are shown in Fig. 15 as a function of

particle volume fraction. For both the transverseYoung�s moduli, Y1 and Y2, and the longitudinal

Young�s modulus, Y3, the proposed model agrees closely

with the finite element model for the entire range of par-

ticle volume fractions considered. The Mori–Tanaka

and Self-consistent models significantly under-predict

Finite element, Y1 = Y2Finite element, Y3Proposed, Y1 = Y2Proposed, Y3Mori-Tanaka, Y1 = Y2Mori-Tanaka, Y3Self-Consistent, Y1 = Y2Self-Consistent, Y3

Finite element, Y1 = Y2Finite element, Y3Proposed, Y1 = Y2Proposed, Y3Mori-Tanaka, Y1 = Y2Mori-Tanaka, Y3Self-Consistent, Y1 = Y2Self-Consistent, Y3

0

10

20

30

40

50

60

0 10 20 30

Particle volum

Yo

un

g's

mo

du

lus

(GP

a)

Fig. 15. Young�s moduli vs. particle volume fr

and over-predict, respectively, the finite element data

for volume fractions above 50%.

The shear moduli of this material for the range of vol-

ume fractions are shown in Fig. 16. For the longitudinal

shear moduli, G23 and G13, and the transverse shearmodulus, G12, the proposed, Mori–Tanaka, and self-

consistent models over-predict the finite element model

data for the entire range of considered particle volume

fractions, with the Mori–Tanaka exhibiting the closest

agreement. For particle volume fractions above 20%,

40 50 60 70

e fraction (%)

action for PZT-7A/LaRC-SI composite.

0

5

10

15

20

25

0 10 20 30 40 50 60 70

Particle volume fraction (%)

Sh

ear

mo

du

lus

(GP

a)

Finite element, G23 = G13Finite element, G12Proposed, G23 = G13Proposed, G12Mori-Tanaka, G23 = G13

Mori-Tanaka, G12Self-Consistent, G23 = G13Self-Consistent, G12

Finite element, G23 = G13Finite element, G12Proposed, G23 = G13Proposed, G12Mori-Tanaka, G23 = G13

Mori-Tanaka, G12Self-Consistent, G23 = G13Self-Consistent, G12

Fig. 16. Shear moduli vs. particle volume fraction for PZT-7A/LaRC-SI composite.

5328 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

the Self-consistent model significantly over-predicts

both the longitudinal and transverse shear moduli.The piezoelectric constants are shown in Fig. 17 as a

function of particle volume fraction. For all four con-

stants, e15, e31 = e32, and e33, the finite element, pro-

posed, and Mori–Tanaka models show close

agreement up to a particle volume fraction of 40%.

For the constants e15 and e32 the Mori–Tanaka model

has the closest agreement with the finite element model

in the particle volume fraction range between 40% and50%. At a particle volume fraction of 60%, the proposed

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 10 20 30

Particle volu

Pie

zoel

ectr

ic c

on

stan

t (C

/m2 )

Finite element, e15

Finite element, e31 = e32

Finite element, e33

Proposed, e15

Proposed, e31 = e32

Proposed, e33

Mori-Tanaka, e15

Mori-Tanaka, e31 = e32

Mori-Tanaka, e33

Self-Consistent, e15

Self-Consistent, e31 = e32

Self-Consistent, e33

Finite element, e15

Finite element, e31 = e32

Finite element, e33

Proposed, e15

Proposed, e31 = e32

Proposed, e33

Mori-Tanaka, e15

Mori-Tanaka, e31 = e32

Mori-Tanaka, e33

Self-Consistent, e15

Self-Consistent, e31 = e32

Self-Consistent, e33

Fig. 17. Piezoelectric constants vs. particle volum

model exhibits the closest agreement with the finite ele-

ment model. For e31 = e32 the proposed model showsthe closest match with the finite element model for par-

ticle volume fractions above 40%. For particle volume

fractions above 20%, the Self-consistent results are dra-

matic and do not appear to closely predict any of the

piezoelectric constants.

The dielectric constants of the material are shown

in Fig. 18. For the transverse dielectric constants, j1/j0and j2/j0, and the longitudinal dielectric constant,j3/j0, the predicted values from the Mori–Tanaka

40 50 60 70

me fraction (%)

e fraction for PZT-7A/LaRC-SI composite.

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70

Particle volume fraction (%)

Die

lect

ric

con

stan

t

Finite element, 1/ 0 = 2/ 0

Finite element, 3/ 0

Proposed, 1/ 0 = 2/ 0

Proposed, 3/ 0

Mori-Tanaka, 1/ 0 = 2/ 0

Mori-Tanaka, 3/ 0

Self-Consistent, 1/ 0 = 2/ 0

Self-Consistent, 3/ 0

Finite element, κ1/κ0 = κ2/κ0

Finite element, κ3/κ0

Proposed, κ1/κ0 = κ2/κ0

Proposed, κ3/κ0

Mori-Tanaka, κ1/κ0 = κ2/κ0

Mori-Tanaka, κ3/κ0

Self-Consistent, κ1/κ0 = κ2/κ0

Self-Consistent, κ3/κ0

Fig. 18. Dielectric constants vs. particle volume fraction for PZT-7A/LaRC-SI composite.

G.M. Odegard / Acta Materialia 52 (2004) 5315–5330 5329

model agree with the finite element model up to a par-

ticle volume fraction of about 50%. Above that value,

the Mori–Tanaka model under-predicts the finite ele-

ment data for j1/j0 and j2/j0. At that point, the pro-posed model exhibits better agreement with the finite

element model. For particle volume fractions above

20%, the Self-consistent approach significantly overes-

timates both transverse and longitudinal dielectric

constants.

6. Conclusions

A new modeling approach has been proposed for pre-

dicting the bulk electromechanical properties of piezoe-

lectric composites. The proposed model offers the samelevel of convenience as the Mori–Tanaka method, that

is, it does not require iterative or numerical schemes

for obtaining the predicted properties, as is required

with the Self-consistent and Differential schemes. The

electromechanical properties of four piezoelectric poly-

mer composite materials were predicted with the pro-

posed, Mori–Tanaka, Self-consistent methods, and

detailed finite element analyses for a wide range of ma-trix and reinforcement electromechanical properties,

geometries, and volume fractions. The four piezoelectric

composite materials considered were: a graphite/PVDF

composite, a SiC/PVDF particulate composite, a fibrous

PZT-7A/LaRC-SI composite, and a PZT-7A/LaRC-SI

particulate composite.

It was shown that the proposed model yields

predicted properties that were, in some cases, moreaccurate than the Mori–Tanaka and Self-consistent

schemes. In particular, the proposed model exhibits

equal or closer agreement with the finite element

model than does the Mori–Tanaka and Self-consistent

schemes for the prediction of several electromechanicalproperties. For the PVDF matrix composites these

properties include the longitudinal shear and

longitudinal Young�s moduli and all dielectric con-

stants for the graphite/PVDF composite; and all

Young�s moduli, all shear moduli (for volume frac-

tions above 50%), the piezoelectric constants e31 and

e32 and all dielectric constants for the SiC/PVDF com-

posite. For the PZT-reinforced composites these in-clude the longitudinal Young�s modulus, the

piezoelectric constants e33 and e15 and the longitudinal

dielectric constant of the fibrous PZT-7A/LaRC-SI

composite; and all of the Young�s moduli, all of the

piezoelectric constants, and the longitudinal dielectric

constant (for volume fractions above 60%) of the par-

ticulate PZT-7A/LaRC-SI composite. Based on these

results, the choice of the most accurate model (be-tween the proposed, Mori–Tanaka, and Self-consistent

methods) for a specific piezoelectric composite mate-

rial should be based on the constituent properties

and the geometry and volume fraction of the

inclusions.

References

[1] Newnham RE, Skinner DP, Cross LE. Connectivity and piezo-

electric–pyroelectric composites. Mater Res Bull 1978;13:525–36.

[2] Banno H. Recent developments of piezoelectric ceramic products

and composites of synthetic rubber and piezoelectric ceramic

particles. Ferroelectrics 1983;50:329–38.

5330 G.M. Odegard / Acta Materialia 52 (2004) 5315–5330

[3] Chan HLW, Unsworth J. Simple model for piezoelectric ceramic/

polymer 1–3 composites used in ultrasonic transducer applica-

tions. IEEE Trans Ultrason Ferr 1989;36:434–41.

[4] Smith WA, Auld BA. Modeling 1–3 composite piezoelectrics:

thickness-mode oscillations. IEEE Trans Ultrason Ferr

1991;38:40–7.

[5] Olson T, Avellaneda M. Effective dielectric and elastic constants

of piezoelectric polycrystals. J Appl Phys 1992;71:4455–64.

[6] Bisegna P, Luciano R. Variational bounds for the overall

properties of piezoelectric composites. J Mech Phys Solids

1996;44:583–602.

[7] Hori M, Nemat-Nasser S. Universal bounds for effective piezo-

electric moduli. Mech Mater 1998;30:1–19.

[8] Li JY, Dunn ML. Variational bounds for the effective moduli of

heterogeneous piezoelectric solids. Philos Mag A 2001;81:903–26.

[9] Gaudenzi P. On the electromechanical response of active

composite materials with piezoelectric inclusions. Comput Struct

1997;65:157–68.

[10] Poizat C, Sester M. Effective properties of composites with

embedded piezoelectric fibres. Comput Mater Sci 1999;16:89–97.

[11] Eshelby JD. The determination of the elastic field of an ellipsoidal

inclusion, and related problems. Proc Roy Soc London, Ser A

1957;241:376–96.

[12] Wang B. Three-dimensional analysis of an ellipsoidal inclusion in

a piezoelectric material. Int J Solids Struct 1992;29:293–308.

[13] Benveniste Y. The determination of the elastic and electric fields

in a piezoelectric inhomogeneity. J Appl Phys 1992;72:1086–95.

[14] Dunn ML, Taya M. Micromechanics predictions of the effective

electroelastic moduli of piezoelectric composites. Int J Solids

Struct 1993;30(2):161–75.

[15] Chen T. An invariant treatment of interfacial discontinuities in

piezoelectric media. Int J Eng Sci 1993;31:1061–72.

[16] Dunn ML, Taya M. An analysis of piezoelectric composite

materials containing ellipsoidal inhomogeneities. Proc Roy Soc

London, Ser A 1993;443:265–87.

[17] Chen T. Micromechanical estimates of the overall thermoelectro-

elastic moduli of multiphase fibrous composites. Int J Solids

Struct 1994;31:3099–111.

[18] Huang JH, Kuo WS. Micromechanics determination of the

effective properties of piezoelectric composites containing spa-

tially oriented short fibers. Acta Mater 1996;44:4889–98.

[19] Fakri N, Azrar L, El Bakkali L. Electroelastic behavior modeling

of piezoelectric composite materials containing spatially oriented

reinforcements. Int J Solids Struct 2003;40:361–84.

[20] Mori T, Tanaka K. Average stress in matrix and average elastic

energy of materials with misfitting inclusions. Acta Metall

1973;21:571–4.

[21] Benveniste Y. A new approach to the application of mori-tanaka�stheory in composite materials. Mech Mater 1987;6:147–57.

[22] Hill R. A self-consistent mechanics of composite materials. J

Mech Phys Solids 1965;13:213–22.

[23] Budiansky B. On the elastic moduli of some heterogeneous

materials. J Mech Phys Solids 1965;13:223–7.

[24] McLaughlin R. A study of the differential scheme for composite

materials. Int J Eng Sci 1977;15:237–44.

[25] Norris AN. A differential scheme for the effective moduli of

composites. Mech Mater 1985;4:1–16.

[26] Dunn ML. Electroelastic green�s functions for transversely

isotropic piezoelectric media and their application to the solution

of inclusion and inhomogeneity problems. Int J Eng Sci

1994;32(1):119–31.

[27] Wang B. Effective behavior of piezoelectric composites. Appl

Mech Rev 1994;47:S112–21.

[28] Dunn ML, Wienecke HA. Green�s functions for transversely

isotropic piezoelectric solids. Int J Solids Struct 1996;33:4571–81.

[29] Chen T. Effective properties of platelet reinforced piezocompos-

ites. Composites, Part B 1996;27:467–74.

[30] Dunn ML, Wienecke HA. Inclusions and inhomogeneities in

transversely isotropic piezoelectric solids. Int J Solids Struct

1997;34(27):3571–82.

[31] Mikata Y. Determination of piezoelectric eshelby tensor in

transversely isotropic piezoelectric solids. Int J Eng Sci

2000;38:605–41.

[32] Mikata Y. Explicit determination of piezoelectric Eshelby tensors

for a spheroidal inclusion. Int J Solids Struct 2001;38:7045–63.

[33] Dvorak GJ, Srinivas MV. New estimates of overall properties of

heterogeneous solids. J Mech Phys Solids 1999;47:899–920.

[34] Nicholson LM, Whitley KS, Gates TS, Hinkley JA. Influence of

molecular weight on the mechanical performance of a thermo-

plastic glassy polyimide. J Mater Sci 2000;35(24):6111–22.

[35] Hill R. Elastic properties of reinforced solids: some theoretical

principles. J Mech Phys Solids 1963;11:357–72.

[36] Dvorak GJ, Benveniste Y. On transformation strains and uniform

fields in multiphase elastic media. Proc Roy Soc London, Ser A

1992;437:291–310.

[37] Christensen RM. A critical evaluation for a class of microme-

chanics models. J Mech Phys Solids 1990;38:379–404.

[38] Christensen RM, Schantz H, Shapiro J. On the range of validity

of the Mori–Tanaka method. J Mech Phys Solids 1992;40:69–73.

[39] Benedikt B, Rupnowski P, Kumosa M. Visco-elastic stress

distributions and elastic properties in unidirectional composites

with large volume fractions of fibers. Acta Mater

2003;51(12):3483–93.

[40] Halpin JC, Tsai SW. Environmental factors in composites design.

AFML-TR-67-423 1967.

[41] Davis PJ, Polonsky I. Numerical interpolation, differentiation and

integration. In: Abramowitz M, Stegun IA, editors. Handbook of

mathematical functions with formulas, graphs, and mathematical

tables. New York: Wiley; 1972. p. 875–924.

[42] Sun CT, Vaidya RS. Prediction of composite properties from a

representative volume element. Compos Sci Technol

1996;56:171–9.