Micromechanics-based hyperelastic constitutive modeling of magnetostrictive particle-filled elastomers

Micromechanics-based hyperelastic constitutive modeling ofmagnetostrictive particle-filled elastomers

H.M. Yin a, L.Z. Sun a,*, J.S. Chen b

a Department of Civil and Environmental Engineering and Center for Computer-Aided Design, The University of Iowa, Iowa City,

IA 52242-1527, USAb Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095-1593, USA

Received 4 December 2001; received in revised form 15 March 2002

Abstract

An effective hyperelastic constitutive model is developed for particle-filled elastomer composites based on the mi-

crostructural deformation and physical mechanism of the magnetostrictive particles embedded in the hyperelastic

elastomer matrix. Two types of loading conditions are considered––magnetic field and mechanical load. Magnetic

eigenstrains are prescribed on the particles due to effect of magnetostriction. The effective constitutive relation of the

composites during infinitesimal deformation can be established based on Eshelby’s micromechanics approach. Since

the elastomers normally exhibit finite hyperelastic deformation, the corresponding hyperelastic constitutive law of the

composites is constructed in terms of the strain energy densities of the constituents.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Elastomer composites; Magnetostriction; Micromechanics; Hyperelasticity; Constitutive relations; Strain energy

1. Introduction

Magnetostriction is a phenomenon that, whena substance is exposed to a magnetic field, itsdimensions change (Cullity, 1972; Jiles, 1991;Tremolet de Lacheisserie, 1993). The study ofmagnetostriction can be traced back to James P.Joule, who first discovered the magnetostrictionphenomenon of iron in 1842. Since then, a numberof experimental and theoretical works have beendone for magnetostrictive homogeneous materials.

In recent years, magnetostrictive particle-rein-forced heterogeneous composites have attractedmuch interest in automotive applications such asintelligent devices (Pinkerton et al., 1997; Bedn-arek, 1999; Carlson and Jolly, 2000) and field-dependent elastomeric components (Ginder et al.,1999, 2000, 2001). Corresponding mathematicalmodels have been developed. For example, Herbstet al. (1997) studied SmFe2/Al and SmFe2/Fecomposites and proposed a single-sphere model topredict the overall magnetostriction of the com-posites. Nan (1998) and Nan and Weng (1999)developed an analytical model based on Green’sfunction technique. Davis (1999) used finite ele-ment method to analyze the dependence of effectiveshear modulus of magnetorheological elastomers

Mechanics of Materials 34 (2002) 505–516

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*Corresponding author. Tel.: +1-319-384-0830; fax: +1-319-

335-5660.

E-mail address: [email protected] (L.Z. Sun).

0167-6636/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0167-6636 (02 )00178-3

mail to: [email protected]

on interparticle magnetic forces. Chen et al. (1999)investigated the effect of the elastic modulus ofthe matrix on magnetostriction of the composite.Armstrong (2000) presented an analysis of mag-netoelastic behavior of dilute magnetostrictiveparticle composites. It is noted that all these modelsare limited to infinitesimal strain problems. Whenthe matrix of a magnetostrictive composite is se-lected from elastic materials such as elastomers orsilicones, the deformation of the matrix can belarge under loading. Therefore, mechanisms of fi-nite deformation in the matrix should be includedin the constitutive relations of magnetostrictiveelastomer composites.

Elastomers filled with nonmagnetostrictive par-ticles (such as carbon black) have been studied interms of their finite hyperelastic behavior. Mullinsand Tobin (1965) provided an estimate of effectivedeformation for carbon-black filled vulcanizedrubbers by amplifying the elastic modulus fromthe composition of matrix and reinforcement. Fol-lowing Mullins–Tobin’s method, Bergstrom andBoyce (1999) used neo-Hookean relation in therubber to set the ratio of the averaged strain to themeasured overall strain, and then obtained a goodcomparison with experimental data. On the otherhand, Govindjee and Simo (1991) investigated themicrostructure of the filled elastomers and deriveda continuum model that can capture the strain-induced stress softening Mullins effect (cf., Mul-lins, 1969) during the large deformation. It is notedthat these models can only be applied to the par-ticle-filled composites composed of a nonlinearhyperelastic matrix and rigid particles.

In fact, the deformation mechanism of rein-forcement has a considerable effect on particle-filled elastomers. Recently, Bednarek (1999) foundthat magnetostrictive particle-filled elastomersshow some unique mechanical properties underthe magnetic field. Lanotte et al. (2001) furtherdiscussed the deformation mechanism of mag-netomechanical coupling for the magnetostric-tive composites. In this paper, a micromechanicalframework is proposed to investigate the effectivemechanical behavior of magnetostrictive particle-filled elastomer composites. Under infinitesimaldeformation, the magnetostriction in the particle istreated as a prescribed eigenstrains, and the overall

stress–strain relationship is derived by the Es-helby’s equivalent inclusion method (Eshelby,1957). For finite deformation, a strain energydensity function of the elastomer composites isderived based on Saint Venant–Kirchhoff assump-tion (Ciarlet, 1988; Belytschko et al., 2000) and thenonGaussian behavior (Treloar, 1975) of materi-als. The corresponding hyperelastic constitutivelaw of the composites is constructed in terms of theinvariant-based strain energy which combines thecontributions from both the particles and elas-tomer matrix.

The outline of this paper is as follows. In Sec-tion 2, we first established the microstructure ofthe particle-filled composites. Following Eshelby’sequivalent inclusion method, we investigated theeight-particle interaction and obtained the aver-aged eigenstrain. By using the homogeneous stressboundary condition, we derived the effective elas-tic constitutive relation of the composites. Con-sidering the magnetostriction as a prescribedeigenstrain, we further obtained the effectivemagnetostrictive deformation of the composites.Comparisons with the available experimental dataand existing models were made. In Section 3, weevaluated the effective free energy in the mag-netostrictive composites. We employed the con-cept of nonGaussian chain in the microstructureso that all particles are connected by the chains.We established the deformation relation betweenthe stretch in elastomer and the total stretch in theparticles. We further derived the energy densitiesin the particles and elastomer, respectively, inorder to develop the energy-based hyperelasticconstitutive relation of the composite. Finally, inSection 4, we presented numerical results andcomparisons based on the proposed model.

2. Linear elastic deformation model

Let us consider a two-phase composite con-sisting of an elastomer matrix (phase 0) and fer-romagnetic spherical particles (phase 1). Whenthe deformation is infinitesimal upon loading, thematrix and the particles can be treated as linearlyelastic materials with different elastic constants.For the problem of one particle embedded in an

506 H.M. Yin et al. / Mechanics of Materials 34 (2002) 505–516

infinite domain, Eshelby (1957) offered an analyt-ical solution through an equivalent inclusionmethod in which the total domain is assumed to bethe same material as the matrix but an eigenstrainis introduced in the particle domain to representthe inhomogeneity. This method has been widelyapplied in evaluating the effective elastic propertiesof composites (e.g., Mura, 1987; Nemat-Nasserand Hori, 1999). However, the direct interactionamong particles has been ignored since it is im-possible to obtain an exact solution.

To account for particle interaction, an approx-imate treatment is considered here. Let us assumethat the matrix and particle have the isotropicelasticity tensors as C0 and C1, respectively. More-over, for any given particle, only the interactionwith its eight neighboring particles with the sameradius is considered, as shown in Fig. 1.

2.1. Equivalent eigenstrains

Consider the existence of a uniform far-fieldstrain loading e1. According to Eshelby’s equiva-lent inclusion method, at point r inside the particleX0 (Fig. 1), the perturbed strain e0ðrÞ induced bythe inhomogeneity can be related to equivalent ei-genstrain e�ðrÞ by replacing the particles with thematrix material, following (Ju and Sun, 1999)

C1 : ½e1 þ e0ðrÞ � epeðrÞ ¼ C0 : ½e1 þ e0ðrÞ � e�ðrÞ;r 2 X0 ð1Þ

where epeðrÞ represents the prescribed magneto-strictive eigenstrain and X0 designates the do-main of particle being considered (see Fig. 1). Thesymbol ‘‘:’’ denotes the contraction between thefourth-rank tensor and the second-rank tensor. Inaddition, the perturbed strain e0ðrÞ can be derivedas

e0ðrÞ ¼ S : e�ðrÞ þXni¼1

ZXi

Gðr� r0Þ : e�ðr0Þdr0 ð2Þ

where S is the Eshelby’s tensor, and n is the totalnumber of particles surrounded to the consideredparticle X0. It is noted that the first term of righthand side of the above equation demonstrates theperturbation contribution from the particle itselfðX0Þ while the second term represents the effectsfrom material points r0 in the neighboring particlesðPn

i¼1 XiÞ. The fourth-rank Green function G reads

Gijklðr� r0Þ ¼ 1

8pð1� m0Þr3½ð1� 2m0Þ

� ðdikdjl þ dildjk � dijdklÞþ 3m0ðdiknjnl þ dilnjnk þ djkninl

þ djlninkÞ þ 3dijnknl

þ 3ð1� 2m0Þdklninj � 15ninjnknlð3Þ

with r ¼ kr� r0k, n ¼ ðr� r0Þ=kr� r0k, and m0 isthe Poisson ratio of the matrix.

As an approximation, only the closest eightneighboring particles ðXi; i ¼ 1; 2; . . . ; 8Þ are con-sidered to interact with the particle X0. Substitu-tion of Eq. (2) into Eq. (1) yields

� A : e�ðrÞ þ B : epeðrÞ

¼ e1 þ S : e�ðrÞ þX8

i¼1

ZXi

Gðr� r0Þ : e�ðr0Þdr0

ð4Þwhere the fourth-rank elastic-phase ‘‘mismatchtensors’’ A ¼ ðC1 � C0Þ�1 � C0 and B ¼ ðC1 �C0Þ�1 � C1.

Performing the volume average on Eq. (4) andconsidering the same eigenstrains in each particle,we obtain the following equation

Fig. 1. Eight-particle interaction model of composites for in-

finitesimal deformation.

H.M. Yin et al. / Mechanics of Materials 34 (2002) 505–516 507

�A : e�h iX0þ B : epeh iX0

¼ e1 þ S : e�h iX0

þ g : e�h iX0ð5Þ

where the symbol h�iX0denotes the volume average

in X0 and the interaction term

g ¼ 1

X0

X8

i¼1

ZX0

ZXi

Gðr� r0Þdr0 dr ð6Þ

The above equation can be further simplified as

gijkl ¼8q3ð14q2 � 5Þ45ð1� m0Þ

� ½ð1� 5dIKÞdijdkl þ dikdjl þ dildjk ð7Þ

with q ¼ r=b where r signifies the radius of parti-cles and b denotes the center-to-center spacingbetween particle X0 and its neighbor particle Xi

(i ¼ 1; 2; . . . 8). It is noted that Mura’s (1987) ten-sorial indicial notation is followed in the aboveequation; i.e., upper-case indices has the samerepresentation as the corresponding lower-caseones but are not summed (Ju and Sun, 2001; Sunand Ju, 2001).

Through lengthy but straightforward deriva-tion, the averaged eigenstrain can be obtained as

e�h iX0¼ �C : e1

h� B : epeh iX0

ið8Þ

where

Cijkl ¼�m

ð2w�MÞð2wþ 3m�MÞ

�þ M

2wð2w�MÞ dIK

�dijdkl

þ 1

4wdikdjl

�þ dildjk

ð9Þ

with

m ¼ 1

30ð1� m0Þa þM

5

w ¼ 1

30ð1� m0Þb þM

5; M ¼ 40ð14q2 � 5Þq3

45ð1� m0Þ

a ¼ 10ð1� m0ÞK0

K1 � K0

�� l0

l1 � l0

�þ 2ð5m0 � 1Þ

b ¼ 15ð1� m0Þl0

l1 � l0

þ 2ð4� 5m0Þ

It is noted that K0, K1 are the bulk moduli of thematrix and particle, respectively, and l0, l1 aretheir shear moduli.

From the geometrical setting of the composite,the volume fraction of particles / can be calcu-lated as

/ ¼83pr3

2bffiffi3

p� �3

¼ 5:4414q3 ð10Þ

Furthermore, since the eigenstrains are thesame for all particles, X0 is replaced by X in fol-lowing sections for simplicity.

2.2. Effective elasticity of composites

The overall (effective) stress tensor of the com-posites is defined as

hriV ¼ 1

V

ZV

rðrÞdr ð11Þ

Considering the boundary condition in the farfield e1, the averaged stress can be further ex-pressed as (Mura, 1987)

hriV ¼ C0 : e1 ð12ÞOn the other hand, the local strain field of

composites can be expressed in terms of localstress by using Eshelby’s method (Ju and Sun,1999)

eðrÞ ¼ C�10 : rðrÞ þ e�ðrÞ; in Vp ðparticle domainÞ

C�10 : rðrÞ; in Vm ðmatrix domainÞ

ð13Þ

Therefore, the averaged strain tensors over theparticle domain X and the whole composite do-main V can be written, respectively, as

heiX ¼ e1 þ ðSþ gÞ : e�h iX ð14Þ

and

heiV ¼ e1 þ / e�h iX ð15ÞCombining Eqs. (15) and (8), we obtain

e1 ¼ ðI� /CÞ�1: heiV�

� /C : B : epeh iX

ð16Þ

In the absence of prescribed magnetostrictiveeigenstrain (epe ¼ 0), the effective strain and stressof the composites can be simplified as


heiV ¼ ðI� /CÞ � C�1 � A�1 : heiXhriV ¼ C0 : ðI� /CÞ�1

: heiVð17Þ

The corresponding effective stiffness tensor of thecomposite reads

C ¼ C0 : ðI� /CÞ�1 ð18ÞIt is noted that the above stiffness tensor of thecomposites is obtained by considering the directparticle interactions. This result recovers Nemat-Nasser and Hori’s (1999) formula (Eq. 8.1.8) if theinteraction term is ignored.

When the composites are subjected only tomagnetostrictive eigenstrain (e1 ¼ 0), the aver-aged strains heiX and heiV are then shown as

heiX ¼ ðSþ gÞ : C : B : epeh iX ð19Þand

heiV ¼ /C : B : epeh iX ð20Þ

2.3. Effective magnetostriction of composites

Magnetostriction is measured by the fractionalchange in length k ¼ Dl=l (Clark, 1980). The valueof k measured at magnetic saturation is called thesaturation magnetostriction ks. For the isotropic,homogenous magnetostrictive material, the defor-mation due to saturated magnetostriction has theform of

ksiðhÞ ¼ 32ksðcos2 h � 1

3Þ ð21Þ

where h denotes the angle between the magneti-zation direction and the measurement direction.

For magnetostrictive composites, for simplicity,let us assume that the magnetization directionfollows one of the three principal directions of thecube, say x1-direction. The magnetostrictive ei-genstrains are

epe11 ¼ ks; epe22 ¼ epe33 ¼ �12ks; epeij ¼ 0 for i 6¼ j

ð22Þ

From Eq. (20), the effective magnetostriction ofthe composites can be calculated as

heijiV ¼ /2w�M

l1

l1 � l0

epeII dij ð23Þ

or the normalized effective magnetostriction of thecomposite reads

k� ¼ /

2w�Ml1

l1 � l0

ð24Þ

Fig. 2 shows the comparison between the aboveprediction and the reported results of magneto-striction for SmFe2/Al and SmFe2/Fe compositesby Herbst et al. (1997). The material constantsused are obtained from Herbst et al.’s paper; i.e.,elastic Young’s moduli for SmFe2, Al and Fe are55, 62, and 208/130 GPa, respectively. Elasticshear moduli for SmFe2, Al and Fe are 20, 23.3,and 78.2/47.5 GPa, respectively. It is shown thatgood agreement between the present predictionand experimental data is obtained. It is further il-lustrated that the present model accounts for boththe volume fraction of particles and elastic con-stants of matrix and reinforcement, whereasHerbst et al.’s analytical model is not sensitive tothose material constants.

3. Finite-deformation model

Magnetostrictive particle-reinforced elastomersmay display finite, nonlinear deformation under

Fig. 2. Comparisons of the effective magnetostriction with the

experimental data and Herbst et al.’s model (Herbst et al.,

1997).


loading. Many hyperelastic constitutive models forhomogeneous elastomers are developed in terms ofinvariant-based strain energy function (see Trel-oar, 1975), including phenomenological modelssuch as neo-Hookean, Mooney-Rivlin and Ogden.Statistical mechanics offers another way to explaindeformation mechanisms of elastomers from theconcept of molecular chains. It is shown that theGaussian chain theory yields the same strain en-ergy density as neo-Hookean model (Treloar,1975). To study the finite stretch problem, inves-tigators have built networks based on the non-Gaussian chain theory.

For magnetostrictive elastomers, although theelastomer matrix undergoes large deformation,particle reinforcement may still be deformed in therange of infinitesimal strain since the reinforce-ment is usually much stiffer than the elastomermatrix. In the present model, the nonGaussianchain theory is used to evaluate the energy densityin the matrix while infinitesimal deformation the-ory is adopted to derive the energy density in theparticles.

3.1. Geometric analysis

Elastomeric materials are chemically made upof long-chain molecules with a network of freelyrotating links that form a network. When particlesare embedded in, the chain network will be brokenand most of chains will re-connect to particles toform the main network of the composites as mi-crostructurally illustrated in Fig. 3. For simplicity,elastomer chains are assumed to link the centralparticle to eight neighboring particles. Motivatedby Arruda and Boyce’s (1993) assertion, the com-posites considered are always stretched in theprincipal frame, as described by the three principalstretches k1, k2 and k3.

As shown in Fig. 4, the deformed vector b

pointing from the center of central particle to itsneighboring particle is expressed as

b ¼ bffiffiffi3

p ðk1iþ k2jþ k3kÞ ð25Þ

Deformation vector r inside the particle is similarlyshown in terms of the three principal stretches k̂k1,k̂k2, k̂k3 of the particle

r ¼ rffiffiffi3

p ðk̂k1iþ k̂k2jþ k̂k3kÞ ð26Þ

Moreover, the deformed vector c along thesame direction with the vector b has the relation-ship with the principal stretches ~kk1, ~kk2, ~kk3 of thematrix as

c ¼ b� 2rffiffiffi3

p ð~kk1iþ ~kk2jþ ~kk3kÞ ð27Þ

From the geometric setting b ¼ cþ 2r, the fol-lowing relationship among the principal stretchescan be derived

~kki ¼ki � 2qk̂ki

1� 2qð28Þ

3.2. Effective free energy and hyperelastic constit-utive law

For magnetostrictive particle-filled elastomers,the effective deformation is induced by mag-netostriction in particle and external mechanicalloading. To yield the maximum effect of the mag-netostriction in the composites, the prescribedmagnetic eigenstrains should be consistent with theprincipal stretch directions of the composites. The

Fig. 3. Chain network between magnetostrictive particles em-

bedded in the elastomer matrix.


particle magnetostriction eigenstrain tensor is de-noted as

Epe ¼ks1 � 1 0 00 ks

2 � 1 00 0 ks

3 � 1

24 35 ð29Þ

The averaged strain tensor of particles due tomagnetostriction is further obtained from Eq. (19)asbEE0 ¼ ðSþ gÞ � C � B : Epe ð30Þand, from Eq. (20), the corresponding effectivestrain tensor of the composites reads

E0 ¼ /C � B : Epe ð31ÞUpon external mechanical loading, the total

stretches of the composites are considered as finitedeformation. The deformation gradient of thecomposites F in the principal frame can be writtenas

Fij ¼ kIdij ð32ÞBased on Saint Venant–Kirchhoff assumption

on hyperelastic materials (see, e.g., Ciarlet, 1988;

Belytschko et al., 2000), the relationship betweenthe deformation rate bDD of particles and the finite-deformation rate D of the composites can be es-tablished from infinitesimal theory (Eq. (17)) as

bDD ¼ A � C � ðI� /CÞ�1: D ð33Þ

where

Dij ¼ _FFikF �1kj ¼

_kkI

kI

dij ð34Þ

Through simplification, Eq. (33) can be furtherexpressed as

bDDij ¼ ðv1DII þ v2DmmÞdij ð35Þ

where

v1 ¼1

2w�M � /l0

l1 � l0

v2 ¼1

3ð2wþ 3m�M � /ÞK0

K1 � K0

� 1

3ð2w�M � /Þl0

l1 � l0

ð36Þ

The Lagrangian strain bEE of particles can bederived by integration upon the above equation as

bEEij ¼ bEE0ij þ v1 ln

kI

k0I

þ v2 ln

k1k2k3

k01k

02k

03

!dij ð37Þ

where the stretches with superscripts 0 denote theinitial deformation caused by the magnetostrictionof particle.

Accordingly, the strain energy of particles canbe obtained as

Wp ¼kp

2ðv1

�þ 3v2Þ

2 þ lpð2v1v2 þ 3v22Þ�

� ln2 k1k2k3

k01k

02k

03

þ lpv21 ln2 k1

k01

þ ln2 k2

k02

þ ln2 k3

k03

!ð38Þ

For the homogeneous elastomer matrix, Arrudaand Boyce’s (1993) chain model is adopted here toobtain the free energy Wm for the matrix, namely

Wm ¼ l0

6

Z ~II1

~II01

‘�1~II1=21ffiffiffiffiffiffiffi3N

p" # ffiffiffiffiffiffiffi

3Np

~II1=21

d~II1 ð39Þ

Fig. 4. Eight-particle interaction model of composites for finite

deformation.


where l0 is the shear modulus and N is the numberof chain segment links of elastomers. The term

‘�1½~II1=21 =ffiffiffiffiffiffiffi3N

p is the inverse Langevin function

defined as ‘ð�Þ ¼ cothð�Þ � 1=ð�Þ. In addition,

eI1I1 ¼ ekikiekiki ð40Þ

with

~kki ¼ki � 2qðbEEII þ 1Þ

1� 2qð41Þ

Following Govindjee and Simo (1991), themacroscopic free energy W of the composites isconstructed by considering the strain energies fromboth the particles and the matrix. For the unit cellof composites with total volume V0 in Fig. 4, theeffective strain energy density W can be derivedas

W ¼ 8ðb� 2rÞwchain

V0

þ VparticleWp

V0

ð42Þ

where wchain is the chain strain energy density perunit length while Wp is the strain energy density ofthe particles. For the elastomer matrix, since theenergy density Wm per unit volume is related to thechain energy wchain as Wm ¼ 8bwchain=V0, the totalenergy of the composites can be further derivedas

W ¼ ð1� 2qÞWm þ /Wp ð43ÞOnce the total energy of the elastomer com-

posites is derived, the hyperelastic stress–strainrelations can be developed based on the conven-tional finite-deformation theory. For example, theeffective Cauchy stress r of the composite has thefollowing dependence on the effective Lagrangianstrain E:

rij ¼1

jFj FirFjsoWoErs

ð44Þ

Therefore, a hyperelastic finite-deform constitutivemodel is developed for magnetostrictive particle-reinforced elastomer composites. This model isformulated in Eqs. (38), (39), (43) and (44) basedon a micromechanics approach. The proposedmodel allows one to estimate the nonlinear hy-perelastic stress–strain curves of the magneto-strictive composites.

4. Numerical results and discussion

Stress–stretch curves under uniaxial loading areoften referred to as important indicators of themechanical performance of materials. To illustratethe capability of proposed micromechanics-basedmodel, let us first consider the case of uniaxialstress loading of the composites with a saturatedmagnetostriction.

Predictions of the proposed method are illus-trated in Fig. 5, showing the true stress–stretchcurves of magnetostrictive elastomer composites.It is noted that the mechanical uniaxial loadingdirection is the same as that of the saturatedmagnetic filed. The selected material constants ofthe matrix are the shear modulus l0 ¼ 1 MPa, thePoisson’s ratio m0 ¼ 0:5, and the number of chainsegment links N ¼ 10. It is shown from Fig. 5(a)that the volume fraction of magnetostrictive par-ticles has a significant effect on the hyperelasticresponse of the composites. The stiffness of thecomposites becomes higher with the increase ofthe particle concentration. Magnetostriction isinitiated from the small strain region of the stress–stretch curves. Fig. 5(b) shows that the hyper-elasticity of the composites becomes stiffer as theelastic modulus of reinforcement increases. Theeffect of prescribed magnetostrictive eigenstrainsis exemplified in Fig. 5(c). It is shown that themagnetostriction strongly affects the overall tensilehyperelastic deformation. At the same stress level,the composite with magnetostriction exhibits largedeformation.

Few experimental data have been published onthe magnetostriction and its influence to the fi-nite hyperelastic response of elastomer compos-ites. Mullins and Tobin (1965) and Bergstrom andBoyce (1999) conducted experimental investiga-tions on rigid particle-filled elastomers withoutmagnetostrictions. As a special case, the presentmodel is also able to predict the nonlinear finite-deformation behavior of particle-filled compositesby dropping the prescribed eigenstrains. Compar-isons are performed between the present predictionand the experimental results from Bergstrom andBoyce (1999) and Mullins and Tobin (1965), asshown in Figs. 6 and 7, respectively. In addition,analytical results from both the Govindjee and


Simo’s (1991) model and the Bergstrom and Boy-ce’s (1999) model are also presented for compari-sons. It is noted that the input material constantsused in the presented model are consistent with theexperimental results and Bergstrom and Boyce’s(1999) predictions, as indicated in the figures. Bothof Figs. 6 and 7 show that the present model agreeswell with those experimental results, especially inthe large stretch range. Bergstrom and Boyce’sanalytical results are similar to the present model,

while Govindjee and Simo’s predictions underes-timate the mechanical response of filled elasto-mers.

For particle-reinforced elastomer composites,reinforcement particles in general are much stifferthan the matrix. Therefore, many available modelsignore the contribution of the strain energy fromthe particles for simplicity. However, the presentmodel shows that, when the volume fraction of theparticles is large or the elastic modulus of particles

Fig. 5. Stress–stretch curves of magnetostrictive composites: (a) effect of volume fraction of particles, (b) effect of shear modulus of

particles, (c) effect of magnetostrictive eigenstrain.


is not significantly large compared with that ofthe matrix, the strain energy in the particles shouldbe considered. Fig. 8 illustrates the effect of shearmodulus of reinforcement. It is shown that themodulus of particles has strong effect on the stress–stretch responses of the composites. Bergstrom andBoyce’s model with rigid particles would overes-timate the mechanical responses of filled elastom-ers if the material contrast ratio of particles tomatrix were less than 20.

Mechanical responses under other loadingconditions are also calculated. Fig. 9 shows thepredictions of nonlinear mechanical behavior ofelastomer composites under uniaxial compression,biaxial compression and pure shear from the pre-sent model and Bergstrom–Boyce’s model. It isfound that, for small stretch, the predictions of the

Fig. 6. Comparisons of composite stress–stretch curves with

experimental data and other models without magnetostriction:

(a) / ¼ 0:25, (b) / ¼ 0:15.

Fig. 7. Comparisons of stress–stretch relation with experi-

mental data and the other model without magnetostriction.

Fig. 8. Comparisons of stress–stretch for different l1=l0 with-

out magnetostriction.


two models are close. For finite deformation,however, the present prediction is different fromthat of Bergstrom and Boyce’s model, since latteris based on neo-Hookean model, which generallyyields a smaller stress prediction for the largestretch. The biaxial compression yields the largestdifference, while good agreement exists in the uni-axial compression.

5. Conclusions

In this paper, a nonlinear hyperelastic constit-utive model is developed for magnetostrictiveparticle-reinforced elastomer composites, based onthe microstructural deformation and physicalmechanism of the magnetostrictive particles em-bedded in the hyperelastic elastomer matrix. Twotypes of loading conditions are considered––satu-rated magnetic field and mechanical load. Mag-netic eigenstrains are prescribed on the particlesdue to effect of magnetostriction. Direct particleinteraction is taken into account, to accommodatethe highly concentrated volume fractions of rein-forcement. The proposed hyperelastic model is ableto characterize the overall nonlinear elastic stress–stretch relations of the composites under generalthree-dimensional loading. Model formulationsare implemented to investigate the mechanical re-

sponses of the composites. Comparisons are con-ducted among the present results, other existingmodels, and the experimental data to illustratethe performance of the proposed micromechanicsframework.

It is noted that the proposed model is applicablefor magnetostrictive particle-filled elastomersunder general three-dimensional mechanical load-ing conditions. However, the applied magneticfield is assumed to be uniaxial. Local deformationmechanisms of interacting particles have been ta-ken into consideration, while the magnetically di-polar interactions between particles are ignored.Future research work on effects of dipolar forces iswarranted to improve the magnetomechanicalmodeling for elastomer composites.

Acknowledgements

This work is sponsored by the National ScienceFoundation under grant number CMS-0084629.The support is gratefully acknowledged. The au-thors wish to thank Dr. M.C. Lee of Delphi Au-tomotive Systems, Inc. for his encouragement inthis work. The authors also thank Dr. J.M. Ginderand Dr. L.C. Davis from Ford Motor Companyand Dr. J.F. Herbst and Dr. F.E. Pinkerton fromGeneral Motors Corporation for their commentsand suggestions.

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Micromechanics-based hyperelastic constitutive modeling of magnetostrictive particle-filled elastomers