2006 Magneto-elastic.. - UCLA Engineering

ARTICLE IN PRESS 

Journal of the Mechanics and Physics of Solids 

54 (2006) 975–1003 

www.elsevier.com/locate/jmps 

Magneto-elastic modeling of composites containing 

chain-structured magnetostrictive particles 

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

a Department of Civil and Environmental Engineering and Center for Computer-Aided Design, 

The University of Iowa, Iowa City, IA 52242, USA 

b Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 

Urbana, IL 61801, USA 

c Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USA 

d Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, USA 

Received 2 May 2005; received in revised form 21 September 2005; accepted 22 November 2005 

Abstract 

Magneto-elastic behavior is investigated for two-phase composites containing chain-structured 

magnetostrictive particles under both magnetic and mechanical loading. To derive the local magnetic 

and elastic fields, three modified Green’s functions are derived and explicitly integrated for the 

infinite domain containing a spherical inclusion with a prescribed magnetization, body force, and 

eigenstrain. A representative volume element containing a chain of infinite particles is introduced to 

solve averaged magnetic and elastic fields in the particles and the matrix. Effective magnetostriction 

of composites is derived by considering the particle’s magnetostriction and the magnetic interaction 

force. It is shown that there exists an optimal choice of the Young’s modulus of the matrix and the 

volume fraction of the particles to achieve the maximum effective magnetostriction. A transversely 

isotropic effective elasticity is derived at the infinitesimal deformation. Disregarding the interaction 

term, this model provides the same effective elasticity as Mori–Tanaka’s model. Comparisons of 

model results with the experimental data and other models show the efficacy of the model and 

Corresponding author. Department of Civil and Environmental Engineering, University of California, Irvine, 

CA 92697, USA. Tel.: +1 949 824 8670; fax: +1 949 824 2117. 

E-mail address: lsun@uci.edu (L.Z. Sun). 

0022-5096/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. 

doi:10.1016/j.jmps.2005.11.007

976 


H.M. Yin et al. / J. Mech. Phys. Solids 54 (2006) 975–1003 

suggest that the particle interactions have a considerable effect on the effective magneto-elastic 

properties of composites even for a low particle volume fraction. 

r 2005 Elsevier Ltd. All rights reserved. 

Keywords: Microstructure; Elastic material; Particulate reinforced composites; Magnetostriction; Magneto-elastic 

modeling 

1. Introduction 

In the past few years, applications of magnetic particle-filled composites as a class of 

smart materials have attracted a good deal of attention from engineers and researchers (Jin 

et al., 1992; Sandlund et al., 1994; Duenas and Carman, 2000; Borcea and Bruno, 2001). 

Composites manufactured by brittle magnetic particles and a strong matrix allow the 

magnetic properties of the particles to be maintained while keeping overall mechanical 

performance adjustable (Sandlund et al., 1994). Composites made of ferromagnetic 

particles and a soft matrix belong to another specific class of smart materials where the 

mechanical properties can be changed under different magnetic environments. These 

composites have been applied to automotive components for isolation of car noise, 

vibration, and harshness (Ginder et al., 2000). 

In recent years it has been found that fabricating the composite under an applied electric 

or magnetic field transforms the randomly dispersed particles into chains parallel to the 

direction of the applied field due to the interaction force (Halsey, 1992; Sandlund et al., 

1994; Ginder et al., 1999). Upon curing the matrix material, the chain structure is locked 

into place so that the chain-like microstructures in the composites are obtained. When a 

magnetic field is further applied, an effective magnetostriction has been observed due to 

the particle’s magnetostriction and the magnetic interaction force between particles. 

In general, magnetostriction is relatively small for most typical ferromagnetic materials 

such as iron, nickel, and cobalt, and saturation magnetostriction is of the order 10 5 –10 6 

(Bozorth, 1951; Bednarek, 1999). In the 1960s, researchers were surprised by the 

measurement of giant magnetostriction in the range of 0.2–0.6% for some rare earth 

crystals like Terfenol-D measured at a variety of temperatures (McKnight, 2002). 

Sandlund et al. (1994) measured the effective magnetostriction, elastic moduli, and 

magneto-mechanical coupling factor for Terfenol-D composites. Herbst et al. (1997) 

fabricated magnetostrictive composites consisting of SmFe 2 embedded in an Fe or Al 

matrix and proposed a single-sphere model to predict overall magnetostriction of the 

composites. Chen et al. (1999) examined Terfenol-D particles filled in various kinds of 

matrices and investigated the effect of the matrix’s elastic modulus on effective 

magnetostriction. Guo et al. (2001) also studied the effect of the matrix’s elastic modulus 

on the magneto-mechanical coupling and effective magnetostriction of Terfenol-D 

composites. Duenas and Carman (2000) evaluated the magnetostrictive response of 

Terfenol-D composites and evaluated the effective elasticity by Voight and Reuss 

approximations (Mura, 1987). Anjanappa and Wu (1997) and Friedmann et al. (2001) 

showed some unique applications of magnetostrictive particulate composites as actuators 

or sensors. Nan (1998) and Nan and Weng (1999) developed an analytical model based on 

the Green’s function technique. Since this model is based on the solution for one particle 

with magnetostriction embedded in the matrix, they are unable to take into account either


H.M. Yin et al. / J. Mech. Phys. Solids 54 (2006) 975–1003 977 

particle distribution or interaction between particles. Armstrong (2000) studied composites 

containing a large number of well-distributed ellipsoidal magnetostrictive particles and 

presented an analysis of magneto-elastic behavior based on Eshelby’s theory. Feng et al. 

(2003) extended the double-inclusion model to predict magneto-elastic behavior of these 

composites. 

In the above-mentioned models, magnetic force between particles is not considered, and 

only the magnetostriction of magnetostrictive particles is accounted for in the effective 

magnetostriction. Thus, these models are only applicable to composites whose matrix 

moduli are sufficiently high or whose magnetic permeability ratio of the particle phase to 

the matrix phase is very small. However, for some magnetostrictive composites like 

Terfenol-D filled composites where the matrix elastic modulus may not so be high enough 

and the particles’ relative magnetic permeability can also reach as high as 15 times as the 

matrix (McKnight, 2002), the magnetic forces between particles cannot be simply 

disregarded since they may play an important role on the effective magneto-elastic 

behavior of the magnetostrictive composites. 

To study the effect of magnetic force on magneto-mechanical behavior of these 

composites, Davis (1992) applied the classical point dipole force (Coulson, 1961) to 

consider pair-wise interaction between particles. Gast and Zukoski (1989) and Klingenberg 

and Zukoski (1990) used the multipole expansion method to derive the interaction force 

for the general case in which the center–center line is at an arbitrary angle with respect to 

the external field, and this approach provided an approximate solution with three 

dimensionless functions generated by curve fitting. Chen et al. (1991) numerically solved 

local interaction for a chain of spherical particles in electrorheological fluid from 

Rayleigh’s theory (Rayleigh, 1892). The numerical results showed that this model has good 

agreement with experimental data but is obviously different from Klingenberg and 

Zukoski’s (1990) model. In addition, Davis (1992) used the finite element method to solve 

the interaction force and also discovered this difference. 

The objective of this paper is to investigate effective magneto-elastic behavior of 

composites containing chain-structured magnetostrictive particles. Here, the particles are 

assumed to be ideal soft magnetic materials, i.e. the residual magnetic field is always zero 

when the external magnetic field is removed. For simplicity, we assume that applied 

magnetic and elastic loading is in the linear range, both the particle and matrix phases have 

isotropic magnetic and elastic properties, and the magnetic permeability for each phase is 

assumed to be independent to the mechanical loading such that the magnetic permeability 

and elasticity of the particles and matrix are treated constant. For the magnetostatic theory 

to be applicable, magnetic loading is assumed to be at a low frequency and the size of the 

particles is much smaller than the wavelength of the magnetic field. Given magnetic and 

mechanical loading, we can solve the averaged stress and strain in the particle and matrix 

phases. From the relation between mechanical responses and magnetic or mechanical 

loading, we can derive the effective magnetostriction and elasticity of composites. 

The rest of this paper is organized as follows. In Section 2, Green’s functions for 

magnetic and elastic problems are reviewed, and the modified Green’s functions for the 

homogeneous infinite medium including a spherical inclusion with prescribed magnetization, 

body force, and strain are derived. Using the explicit integral of these Green’s 

function, the local magnetic and elastic fields can be obtained. In Section 3, Green’s 

functions and Eshelby’s equivalent inclusion method are employed in micromechanical 

analysis of chain-structured composites. A representative volume element (RVE)

978 



containing a chain of infinite particles is used to solve the averaged magnetic fields and 

elastic fields for composites under magnetic and mechanical loading with consideration of 

particle interactions. The effective magnetostriction due to the particle’s magnetostriction 

and the magnetic interaction force are shown in Section 4. Under the infinitesimal 

deformation, effective elasticity is proved to be transversely isotropic. Some comparisons 

of numerical solution with experimental data and other models and parametric analysis are 

also presented in this section. Concluding remarks are given in Section 5. 

2. Basic theories of micromechanics 

Micromechanics encompasses mechanics related to microstructures of composites and it 

studies the effective behavior of composites. The method employed is a continuum theory 

based on local fields at the microscale. Due to material inhomogeneities, the local fields are 

significantly disturbed even under a macroscopically uniform loading. Green’s functions 

are widely used to solve this class of problems. Green’s function is an integral kernel that 

has been used to solve an inhomogeneous differential equation with boundary conditions. 

The Green’s function for the infinite domain is a scalar function for magnetic problem and 

a second-order tensorial function for elastic problem (Kro¨ner, 1990). 

Modified Green’s functions are defined as the derivatives of the conventional Green’s 

functions. Although Green’s functions have been one of the most powerful tools in the past 

century to solve temperature, electric, magnetic and elastic fields (Morse and Feshbach, 

1953; Mura, 1987; Jackson, 1999) in infinite medium with point sources, the method was 

not widely used in material science until Eshelby (1957, 1959) obtained an explicit solution 

for an ellipsoidal inhomogeneity within an isotropic matrix. Since then, Mura (1963, 1987), 

Kro¨ner (1972, 1990), and Mazilu (1972) have done extensive work in this field. Willis 

(1965), Indenbom and Orlov (1968), Mura and Kinoshita (1971), andPan and Chou 

(1976) have extended Green’s functions to anisotropic materials. Recently, Wang (1992), 

Chen (1993), Nan (1994), Dunn (1994), Michelitsch (1997), Huang et al. (1998), Gao and 

Fan (1998), and Li (2002) have obtained Green’s functions for piezomagnetic and 

piezoelectric solids. 

In this work, we consider magnetostrictive inhomogeneities distributed in a nonmagnetic 

matrix. Under magnetic and mechanical loading, the inhomogeneities are magnetized, 

magnetic forces are induced between the particles, and the elastic fields are also 

significantly disturbed due to the stiffness mismatch between the particles and matrix. In 

this section, modified Green’s functions are derived for three conditions: (1) a magnetic 

field caused by a magnetization, (2) a displacement field caused by a body force (magnetic 

force), and (3) a displacement field induced by a prescribed inelastic strain (magnetostriction), 

which is also called eigenstrain. The integrals of modified Green’s functions over a 

spherical domain can be explicitly derived and these integrals can be employed to obtain 

the local magnetic and elastic fields. 

2.1. Magnetic field caused by magnetization 

Green’s theorem reads (Jackson, 1999): 

Z 

Z 

ður 2 v vr 2 uÞ dr ¼ u qv v qu 

dr, (1) 

qn qn 

V 

qV

where u and v are functions with continuous second order derivatives in a region V with 

surface qV. The above equation transfers an integral over a domain to one over the 

boundary. We know for function v ¼ 1=jr r 0 j that 

r 2 v ¼ 4pdðr r 0 Þ. (2) 

Thus, using Eqs. (1) and (2), we can write 

uðr 0 Þ¼ 1 Z 

r 2 u 

4p V jr r 0 j dr þ 1 Z 

1 

u 

4p qV jr r 0 j 2 þ 1 

qu 

dr. (3) 

jr r 0 j qn 

For linear isotropic magnetic materials, the magnetic problem is reduced to finding a 

solution to the following Poisson’s equation with boundary conditions (Reitz et al., 1979): 

r 2 UðrÞ ¼ qM kðrÞ 

, (4) 

qr k 

where U is the scalar magnetic potential, and M k is the magnetization. For numerable 

magnetic particles filled in the infinite nonmagnetic domain, the magnetic field falls off 

faster than 1=jrj for r !1 (Jackson, 1999). Considering the identity of Eq. (3) in the 

magnetic potential governed by Eq. (4) with the boundary integral vanished, the magnetic 

potential can be expressed as 

Z 

UðrÞ ¼ Gðr; r 0 Þ qM kðr 0 Þ 

V qr 0 dr 0 , (5) 

k 

where Green’s function Gðr; r 0 Þ describes the magnetic potential at point r due to the 

magnetization at point r 0 in the infinite domain. Here, we have 

Gðr; r 0 1 

Þ¼ 

4pjr r 0 j . (6) 

Because M i ðrÞ ¼0 for r !1, Eq. (5) can be rewritten as 

Z 

qGðr; r 0 Þ 

UðrÞ ¼ 

M k ðr 0 Þ dr 0 , (7) 

V qr k 

where qGðr; r 0 Þ=qr 0 k ¼ qGðr; r0 Þ=qr k is used. 

The magnetic field is related to the magnetic potential as 

H i ¼ U ;i . (8) 

Then, the substitution of Eq. (7) into Eq. (8) yields 

Z 

H i ðrÞ ¼ G m ik ðr; r0 ÞM k ðr 0 Þ dr 0 , (9) 

V 

where the modified Green’s function is 



G m ik ðr; r0 Þ¼ q2 Gðr; r 0 Þ 

. (10) 

qr i qr k 

Consider one spherical magnetic particle with radius ‘‘a’’ occupying domain O with 

uniform magnetization M 0 i embedded in the infinite nonmagnetic domain V. Outside O, 

the magnetization is zero. Then, from Eq. (9), the local magnetic field reads 

H i ðrÞ ¼D m ik ðr r0 ÞM 0 k . (11)

980 



Here D m ik ðr r0 Þ¼ R O Gm ik ðr; r0 Þ dr 0 can be written in explicit form as 

( 

1 

D m ij ðr r0 3 

Þ¼ 

r3 ðd ij 3n i n j Þ for r4a; 

1 

3 d (12) 

ij for rXa; 

where r ¼jr r 0 j, r ¼ a=r, and n i ¼ðr i r 0 i Þ=r, r0 is the center of the particle. For a linear 

isotropic magnetic particle with magnetic permeability m m 1 filled in the infinite nonmagnetic 

domain under a far field H 0 i , the magnetic constitutive law provides 

M i ðrÞ ¼ mm 1 m m 0 

m m H i ðrÞ, (13) 

0 

where the magnetic permeability in vacuum reads 

m m 0 ¼ 4p 10 7 Ns 2 =C 2 . (14) 

Here N, s and C are units for force, time and electric charge, respectively. It is noted that 

the magnetic permeability for nonmagnetic materials is the same as that in vacuum. Using 

Eshelby’s equivalent inclusion method (Eshelby, 1957, 1959), we can simulate the material 

mismatch by introducing a distributed magnetization on the particle domain. The total 

domain is therefore treated as a homogeneous material with thermal conductivity m m 0 

subjected to a uniform field and a prescribed magnetization on the particle domain. The 

magnetization is solved based on the equivalent condition that the magnetic induction in 

the real particle with permeability m m 1 is same as that in the equivalent particle with 

permeability m m 0 . This method has been used to solve the thermal conduction problem 

(Hatta and Taya, 1986; Yin et al., 2005). From Eqs. (11) and (13), we can solve 

magnetization of the particle as 

M i ¼ 3 mm 1 m m 0 

m m 1 þ H 0 

2mm i (15) 

0 

and the magnetic field can be expressed in two parts: the far field and the perturbed field 

from the magnetization of the particle, i.e., 

H i ðrÞ ¼H 0 i þ D m ij ðr r0 ÞM j , (16) 

which is same as the solution obtained by the variable separation method (von Hippel, 

1954). 

2.2. Strain field caused by body force 

For linearly isotropic materials, the mechanical problem is reduced to finding a solution 

to the following equation with boundary conditions: 

C ijkl u k;lj ðrÞþf i ðrÞ ¼0, (17) 

where C ijkl ¼ ld ij d kl þ m e ðd ik d jl þ d il d jk Þ is the isotropic stiffness tensor with the Lame 

constants l and m e , u k ðrÞ is the displacement field and f i ðrÞ is the body force. The elastic 

Green’s function tensor reads (Mura, 1987) 

G ij ðr; r 0 Þ¼ 1 

4pm e jr 

d ij 

r 0 j 

1 

16pm e ð1 

q 2 jr r 0 j 

, (18) 

nÞ qr i qr j

which satisfies 

C ijkl G kp;lj ðr; r 0 Þ¼ d ip dðr r 0 Þ. (19) 

Here, n is Poisson’s ratio. Using this Green’s function, we can find that the displacement 

for an infinite homogeneous isotropic domain with local body force is written as 

Z 

u i ¼ G ij ðr; r 0 Þf j ðr 0 Þ dr 0 . (20) 

V 

Here, we use a boundary condition that the far field strains are zero. Based on the 

kinematic relation between strains and displacements, strains can be written as 

Z 

e ij ¼ G f ijk ðr; r0 Þf k ðr 0 Þ dr 0 , (21) 

V 


G f ijk ðr; r0 Þ¼ 1 2 ½G ik;jðr; r 0 ÞþG jk;i ðr; r 0 ÞŠ. (22) 

Consider an infinite domain V with shear modulus m e and Poisson’s ratio n containing a 

spherical domain O with a body force f 0 i , the local strain field is 

e ij ðrÞ ¼D f ijk ðr r0 Þf 0 k , (23) 

where D f ijk ðr 

D f ijk ðr 

r0 Þ¼ R O Gf ijk ðr; r0 Þ dr 0 . This integral can also be written in explicit form as 

8 

" # 

r 2 a ð5 10n þ 3r 2 Þðd ik n j þ d jk n i Þ 

>< 

60m e ð1 nÞ þð 5 þ 3r 2 Þd ij n k þ 15ð1 r 2 for r4a; 

Þn i n j n k 

r 

>: 

30m e ð1 nÞ ½ð4 5nÞðd ikn j þ d jk n i Þ d ij n k Š for rXa: 

r0 Þ¼ 

Note that the strain in O is linearly distributed. 

2.3. Strain field caused by eigenstrain 

The concept of eigenstrain was first introduced by Eshelby (1957) and was initially 

named by Mura (1987) to describe local inelastic strain, such as thermal expansion, phase 

transformation, initial strains, plastic strains and misfit strains. Magnetostriction is also a 

kind of eigenstrain. In the following, Eshelby’s solution for a spherical inclusion with 

eigenstrain embedded in the infinite domain is reviewed. 

For linearly isotropic elastic material with local magnetostriction, the mechanical 

problem without body force is reduced to finding a solution to the following equation with 

boundary conditions: 

C ijkl ½u k;lj ðrÞ e kl;jðrÞŠ ¼ 0. (25) 

Using the same elastic Green’s function tensor Eq. (18), we can obtain the displacement 

field as 

Z 

u i ¼ C jlmn e mn ðr0 ÞG ij;l ðr; r 0 Þ dr 0 , (26) 

V 



(24)

982 



where the boundary condition with the far field strain being zero is used. Using the minor 

symmetry of C ijmn ¼ C jimn , Eq. (26) can be rewritten as 

u i ¼ 1 Z 

C jlmn e mn 

2 

ðr0 Þ½G ij;l ðr; r 0 ÞþG il;j ðr; r 0 ÞŠ dr 0 . (27) 

V 

From the kinematic relation, strains can be written as 

Z 

e ij ¼ G e ijmn ðr; r0 ÞC mnkl e kl ðr0 Þ dr 0 , (28) 

V 


G e ijmn ðr; r0 Þ¼ 1 4 ½G im;njðr; r 0 ÞþG in;mj ðr; r 0 ÞþG jm;ni ðr; r 0 ÞþG jn;mi ðr; r 0 ÞŠ. (29) 

In contrast to Kro¨ner’s (1990) original definition, which only has minor symmetry, we 

can find that this modified Green’s function has both minor and major symmetries, i.e. 

G e ijmn ðr; r0 Þ¼G e jimn ðr; r0 Þ¼G e mnij ðr; r0 Þ. As can be seen in the subsequent chapters, this change 

provides convenience in numerical operations as well as carrying a clear physical meaning. 

If in the infinite domain V with shear modulus m e and Poisson’s ratio n there is only a 

spherical domain O with an eigenstrain e 0 

ij , the local strain field can be written as 

e ij ðrÞ ¼ D e ijmn ðr r0 ÞC mnkl e 0 

kl , (30) 

where D e ijkl ðr r0 Þ¼ R O Ge ijkl ðr; r0 Þ dr 0 . This integral can also be written in explicit form 

(Mura, 1987; Ju and Sun, 1999) as 

D e ijkl ðr 

8 

r0 Þ 

r 3 

>< 60m e ð1 

¼ 

2 

ð5 3r 2 Þd ij d kl ð5 10n þ 3r 2 3 

Þðd ik d jl þ d il d jk Þ 

15ð1 r 2 Þðd ij n k n l þ d kl n i n j Þ 

nÞ 15ðn r 2 6 Þðd ik n j n l þ d il n j n k þ d jk n i n l þ d jl n i n k Þ 7 

4 

5 

þ15ð5 7r 2 Þn i n j n k n l 

for r4a; 

1 

>: 

30m e ð1 nÞ ½d ijd kl ð4 5nÞðd ik d jl þ d il d jk ÞŠ for rpa: 

We can find that the strain field in the particle domain is still uniform. 

3. Micromechanical analysis 

Consider a two-phase chain-structured particulate composite containing magnetostrictive 

particles and the nonmagnetic matrix with isotropic permeability m m 1 and m m 0 , and 

isotropic elasticity C 1 and C 0 , respectively, as seen in Fig. 1(a). Here m m 0 is defined in Eq. 

(14); C 1 may be written as C 1 ijkl ¼ l 1d ij d kl þ m e 1 ðd ikd jl þ d il d jk Þ with the Lame constants l 1 

and m e 1 ; C0 is also with the Lame constants l 0 and m e 0 

. The chains are randomly dispersed in 

the composite and have the same direction. The radius of particles is a, and the distance 

between two neighboring particles in one chain is assumed to be equal, denoted by b. Due 

to a bunching effect when fabricating the composite under a high magnetic field, the 

distance between two neighboring chains is generally much larger than b and r 0 ¼ a=b is 

close to 1 2 . A uniform magnetic field H0 i and a uniform stress field s 0 ij applied on the 

ð31Þ



H 0 X 0 

D 

(a) 

Ω 8 

Ω 6 

D 

Ω 4 

Ω 2 

x3 

a 

Ω 0 

Ω 1 

b 

x 2 

x 1 

Ω 3 

Ω 5 

(b) 

Ω 7 

Fig. 1. Chain structured composite under a uniform external magnetic field. (a) Overall microstructure; (b) RVE 

of the neighborhood of the material point X 0 . 

boundary of the composite are considered in this investigation of the magneto-elastic 

response. Because the particle spacing in the same chain is much smaller than the distance 

between chains, the particle interactions in the same chain is dominant. Thus, we only

984 



consider the interactions of particles from the same chain and disregard those from 

particles in other chains. It should be noted that, when the volume fraction of particles is 

much higher, the effect of the chain spacing may become an important factor and should 

be considered in the future. In this section, we will study the averaged magnetic field, stress 

field and strain field considering the particle interactions. 

3.1. Averaged magnetic field 

From Maxwell’s equations and boundary conditions, the magnetostatic field in the 

composite satisfies 

B i;i ðrÞ ¼0 for r 2 D, (32) 

and 

H i ðrÞ ¼H 0 i for r 2 qD (33) 

½B þ i ðrÞ B i ðrÞŠl i ¼ 0 for r 2 qO p , (34) 

where superscripts + and denote the outside and inside surface on the interface, l i is the 

outward unit normal vector, the subscript D is the overall domain of the composite, and O p 

is the domain of the pth particle ðp ¼ 1; 2; ...; NÞ. Here the total number of particles N can 

be infinite. In the following, we use subscripts O and M to denote the domains of the 

particle and matrix phases, respectively. The overall domain D ¼ O [ M. The local 

magnetic field and induction satisfy 

B i ðrÞ ¼mðrÞH i ðrÞ, (35) 

where mðrÞ ¼m m 1 for r on particles and mðrÞ ¼mm 0 for r on the matrix. Thus, Eq. (33) can be 

rewritten as 

B i ðrÞ ¼m m 0 H0 i for r 2 qD. (36) 

Using the following identity 

B i ðrÞ ¼ðr i B j ðrÞÞ ;j B j;j ðrÞr i , (37) 

along with Eq. (32), we can write the averaged internal induction as 

2 

3 

hB i i D ¼ 1 

Z 

4 

V D 

Zq P r 

N i B j ðrÞl j dr þ r i B j ðrÞl j dr5. (38) 

D 

p¼1 O p 

q P N 

p¼1 O p 

Considering the boundary conditions in Eqs. (33) and (34), the averaged internal 

induction is expressed as 

hB i i D ¼ m m 0 H0 i . (39) 

Alternatively, from the definition of volume average, we can write 

hB i i D ¼ fm m 1 hH ii O þð1 fÞm m 0 hH ii M (40) 

and 

hH i i D ¼ fhH i i O þð1 fÞhH i i M . (41)

To obtain a particle’s magnetic field, an RVE containing a chain of particles seen 

in Fig. 1(b) is introduced in the neighborhood of a material point X 0 marked in gray in 

Fig. 1(a). Without any loss of generality, a local coordinate is employed with the origin at 

X 0 . The x 3 -axis is along the direction of the chains. The particle centered at the origin is 

numbered the 0th particle, and the other particles are ordered from 1st to Nth. The 

number of particles N in the chain can be very large or infinite. Each particle is centered at 

r i . Because particles have a much smaller size than the composite, we can assume that the 

chain of the particles in the RVE is embedded in the infinite domain. Magnetic loading is 

transported from the boundary to the RVE through the matrix in the similar way to a far 

field, which is assumed to be the averaged magnetic field of the matrix. Thus, the magnetic 

field on the particle can be simulated by the solution for a chain of particle in the infinite 

domain under a far field strain hHi M . Similarly to Eq. (16), the magnetic field over the 

RVE is written in two parts: 

H i ðrÞ ¼hH i i M þ XN 

p¼0 

Z 

O p 

G m ij ðr; r0 ÞM j ðr 0 Þ dr 0 . (42) 

We use the averaged field on the 0th particle to represent a particle’s averaged field. By 

performing volume average of Eq. (42) over the domain of the 0th particle and by 

considering the same averaged magnetization in each particle, the following equation is 

reached: 

hH i i O ¼hH i i M þ mm 1 m m 0 

m m 0 

1 X N 

4=3pa 3 

p¼0 

ZO 0 

Z 

Using Eq. (12), we can simplify the above equation as 

in which 

O P 

G m ij ðr; r0 Þ dr 0 dr hH j i O . (43) 

hH i i M ¼ T ij hH j i O , (44) 

T ij ¼ 1 

m m 1 m m 0 

4pa 3 m m 0 



X N 

p¼0 

ZO 0 

Z 

O P 

G m ij ðr; r0 Þ dr 0 dr ¼ 1 a ð1 þ 2:4042b d I37:2126bÞd ij , 

where a ¼ 3m m 0 =ðmm 1 þ 2mm 0 Þ and b ¼½ðmm 1 m m 0 Þr3 0 Š=ðmm 1 þ 2mm 0 Þ. During the derivation, the 

identity P 1 

i¼1 1=i3 ¼ 1:2021 is used. It is noted that Mura’s (1987) tensorial indicial 

notation is followed in the above equation; i.e., upper-case indices have the same 

representation as the corresponding lower-case ones but are not summed. 

Combining Eqs. (39)–(44) yields the averaged magnetic fields in the particles, matrix and 

overall composite as 

1 

hH i i O ¼ f mm 1 

m m d ij þð1 fÞT ij H 0 j , (46) 

0 

(45) 

1 

hH i i M ¼ f mm 1 

m m T ij 1 þð1 fÞd ij H 0 j (47) 

0

986 

and 

" # 1 

fÞT ij 

hH i i D ¼ d ij f mm 1 m m 0 

m m 0 



f mm 1 

m m d ij þð1 

0 

H 0 j . (48) 

Due to the higher magnetic permeability of particles, from the continuity condition in Eq. 

(34), the particle magnetic field should be lower than the applied magnetic field on the 

boundary, so the overall averaged magnetic field in Eq. (48) is smaller than the applied 

magnetic field. 

3.2. Averaged stress field 

In the composite, the internal stress should satisfy the equilibrium and boundary 

conditions, as follows: 

and 

s ij;j ðrÞþt i ðrÞ ¼0 for r 2 D (49) 

s ij ðrÞ ¼s 0 ij for r 2 qD, (50) 

½s þ ij ðrÞ s ij ðrÞŠl j ¼ 0 for r 2 qO p , (51) 

where t i is the body force. Taking into account the following formula 

s ij ðrÞ ¼ðs ik ðrÞr j Þ ;k s ik;k ðrÞr j (52) 

along with Eq. (49), we can rewrite the averaged stress as 

2 

3 

hs ij i D ¼ 1 

Z 

Z 

4 

V D 

Zq P s 

N ik ðrÞl k r j dr þ P s 

D O N ik ðrÞl k r j dr þ t i ðrÞr j dr5. 

p¼1 

p 

q O p¼1 

p 

D 

(53) 

Because the gravity only produces an initial deformation, the contribution to the 

constitutive behavior can be disregarded. Only magnetic force is taken into account. With 

the consideration of boundary conditions in Eqs. (50) and (51), the averaged internal stress 

can be obtained as 

hs ij i D ¼ s 0 ij þ sm ij , (54) 

where s m ij ¼ 1=V D 

RD t iðrÞr j dr is the stress caused by magnetic force, and t i ðrÞ is the local 

magnetic body force at r. The magnetic force vanishes in the matrix, whereas in a particle 

there is an interaction magnetic force induced by any other particles. For any pair of 

particles, the magnetic interaction force can be simulated by the averaged field of two 

particles embedded in the matrix. The local magnetic field for two particle embedded in the 

matrix is solved by Eshelby’s equivalent inclusion method (Yin and Sun, 2005a). For the 

inhomogeneous magnetic field, the magnetic body force can be solved by (Pao, 1978) 

f i ðrÞ ¼m m 0 M kðrÞH e i;kðrÞ, (55)

where H e i denotes the external magnetic field. With a lengthy but straightforward 

derivation, we can obtain 

f i ðr I 

r J Þ¼ 3mm 0 ð11mm 1 þ 4mm 0 Þb2 ~r 4 

ð2m m 1 þ 3mm 0 Þ a ðd ij ~n k þ d ik ~n j þ d jk ~n i 

5 ~n i ~n j ~n k ÞhH j i M hH k i M 

þ 3mm 0 ð11mm 1 þ 4mm 0 Þb3 ~r 7 

ð2m m 1 þ 3mm 0 Þ a ðd ij ~n k þ d ik ~n j 2d jk ~n i 8 ~n i ~n j ~n k ÞhH j i M hH k i M 

þ Oð~r 10 Þ. 

Here f i ðr I ; r J Þ is the interaction force of the Ith particle due to the Jth particle, hH i i M is the 

matrix’s averaged magnetic field in Eq. (47), ~r ¼ a=jr I r J j,and ~n i ¼ðr I i r J i Þ=jrI r J j. 

When I ¼ J, f i ðr I ; r J Þ¼0. Here, we can find that the magnetic interaction force is rapidly 

attenuated with the order Oðjr I r J j 4 Þ. Considering that the distance between chains is 

much larger than b, we disregard the interaction with the particles of other chains. By 

ergodicity of particles in Fig. 1(b), we can write 

s m ij 

¼ V O 

V D 

X N 

X N 

I¼0 J¼0 

ð56Þ 

f i ðr I r J Þr I j , (57) 

where N is the largest number of the particle of the RVE in Fig. 1(b), and it can be 

infinitely large. Considering f i ðr I r J Þ¼ f i ðr J r I Þ, we can write 

s m ij 

¼ V O 

2V D 

X N 

X N 

I¼0 J¼0 

f i ðr I r J Þðr I j r J j Þ. (58) 

Based on geometric symmetry of a particle located in the chain containing infinite 

particles, we can rewrite the above equation as 

s m ij ¼ f 2 

X N 

J¼0 



f i ðr 0 r J Þðr 0 j r J j Þ, (59) 

where f ¼ NV O =V D is used. Finally, substituting Eq. (56) into Eq. (59) and truncating the 

higher order terms yields 

s m ij ¼ 3fmm 0 ð11mm 1 þ 4mm 0 Þðmm 1 m m 0 Þ2 

ð2m m 1 þ 3mm 0 Þðmm 1 þ hH k i M hH l i M 

2mm 0 

2 

Þ2 

1:2021r 3 0 ðd 3 

il ~n j ~n k þ d ik ~n j ~n l þ d kl ~n i ~n j 5 ~n i ~n j ~n k ~n l Þ 

6 

 

þ mm 1 m m 7 

4 

0 

m m 1 þ 1:0173r 6 

2mm 0 ðd 5, ð60Þ 

il ~n j ~n k þ d ik ~n j ~n l 2d kl ~n i ~n j 8 ~n i ~n j ~n k ~n l Þ 

0 

where the identities P 1 

i¼1 1=i3 ¼ 1:2021 and P 1 

i¼1 1=i6 ¼ 1:0173 are used. It is noted that the 

stress s m ij may be an asymmetric tensor when the magnetic field is not parallel to the chain’s 

direction due to a distributed angular momentum in the composite caused by the magnetic 

force (Brown, 1966; Pao, 1978). Consequently, the stress s m ij can be decomposed into two 

parts: 

s m ij ¼ s m ðijÞ þ sm ½ijŠ , (61)

988 



where the first is the symmetric part that induces a strain in the composite, and the second 

is the antisymmetric part which is balanced by the external mechanical loading s 0 ij . Thus, 

the overall averaged stress is still symmetric. 

Alternatively, from the definition of volume average, we can write 

hs ij i D ¼ fC 1 ijkl ðhe kli O e ms 

kl Þþð1 fÞC0 ijkl he kli M (62) 

and 

he ij i D ¼ fhe ij i O þð1 fÞhe ij i M , (63) 

where e ms 

ij is the particle’s magnetostriction, which depends on the particle’s magnetization. 

Especially, when the applied magnetic field is large enough, the magnetostriction is 

saturated on the particle and can be regarded as a material constant (Bozorth, 1951). In 

this paper, we assume magnetostriction to be uniform in each particle. When 

magnetostriction is less than the saturation level, for simplicity, it is assumed to vary 

linearly with the particle’s averaged magnetization (Cullity, 1972). The detailed discussions 

will be given later. 

3.3. Averaged strain field 

To solve the particle’s averaged strain, the RVE in Fig. 1(b) is again used. External 

loading is transferred to the RVE through matrix material in the similar way as a far field 

strain, which is assumed to be the averaged strain of the matrix. Thus, the strain field in the 

particle can be obtained by the solution for a chain of particles in the infinite medium 

under a far field strain he ij i M . If the particles have the same elasticity as the matrix, the 

strain field should come from three parts: (1) the initial strain in the matrix, which can be 

assumed to be the averaged strain in the matrix, denoted by hei M , (2) the strain induced by 

the magnetic force from other particles, denoted by e mp 

ij 

particles, denoted by e ms 

ij , namely, 

ē ij ðr 0 Þ¼he ij i M þ e mp 

ij ðr 0 Þ 

X N 

K¼0 

D e ijmn ðrK 

, and (3) the magnetostriction of 

r 0 ÞC 0 mnkl ems kl , (64) 

where the strain e mp 

ij ðr 0 Þ, which is caused by all the magnetic forces between particles, can be 

written as 

e mp 

ij ðr 0 Þ¼ XN X N 

D f ijk ðrI r 0 Þf k ðr I r J Þ. (65) 

I¼0 J¼0 

Based on the same procedure as that in Eqs. (57) and (58), e mp 

ij ðr 0 Þ can be written as 

e mp 

ij ðr 0 Þ¼ 1 X N X N 

½D f ijk 

2 

I¼0 J¼0 

r 0 Þ D f ijk ðrJ r 0 ÞŠf k ðr I r J Þ. (66) 

Because P N 

I¼0 Df ijk ðrI r 0 Þ¼ P N 

J¼0 Df ijk ðrJ r 0 Þ are two absolutely convergent series for 

N !1, the summation in Eq. (66) is zero. Physically, for a chain containing infinite 

particles, the local strain field caused magnetic forces is only affected by the neighboring 

particles due to the absolute convergence of the series. Based on the symmetry of particle 

distribution, the resultant magnetic force on each particle vanishes, which further makes

the local strain field zero. Eq. (64) can then be rewritten as 

ē ij ðr 0 Þ¼he ij i M 

X N 

K¼0 

D e ijmn ðrK 

r 0 ÞC 0 mnkl ems kl . (67) 

However, the particles have different elasticity property from the matrix. Based on 

Eshelby’s equivalent inclusion method (Mura, 1987), the strain at point r in the 0th particle 

due to the material mismatch can be obtained by introducing an eigenstrain in the particle 

domain, which satisfies 

C 1 ijkl ½ē klðrÞ e ms 

kl þ e 0 kl ðrÞŠ ¼ C0 ijkl ½ē klðrÞ e ms 

kl þ e 0 kl ðrÞ e klðrÞŠ, (68) 

where the perturbed strain e 0 ij ðrÞ is caused by the distributed eigenstrain e ijðrÞ, i.e. 

e 0 ij ðrÞ ¼ 

XN 

K¼0 

D e ijmn ðrÞC0 mnkl e kl . (69) 

Taking the volume average of Eq. (68) and combining Eqs. (67)–(69), we can solve the 

averaged eigenstrain as 

he ij i O ¼ 

e ms 

ij ðC 0 ijmn Þ 1 ðDC 1 

mnkl 

þðC 0 ijmn Þ 1 ðDC 1 

mnst 

D 0 mnst 

D 0 mnkl 

¯D mnkl Þ 1 he kl i M 

¯D mnst Þ 1 ðDC 1 

stpq þ C0 stpq Þ 1 C 0 pqkl ems kl , 

where D 0 ijkl describes the perturbed strain caused by 0th particle itself as 

D 0 ijkl ¼ De ijkl ð0Þ ¼ 1 

30m e 0 ð1 n 0Þ ½d ijd kl ð4 5nÞðd ik d jl þ d il d jk ÞŠ (71) 

and ¯D ijkl renders the interaction from all other particles in the same chain as 

¯D ijkl ¼ 1 X N Z 

4=3pa 3 G 

p¼1 

ZO e ijkl ðr; r0 Þ dr 0 dr. (72) 

0 O P 

Considering the geometry of the microstructure, we can explicitly write the above equation 

as 

¯D ijkl ¼ R 1 IK d ijd kl þ R 2 IJ ðd ikd jl þ d il d jk Þ, (73) 

where 



R 1 IK ¼ 1:202r3 0 

6m e 0 ð1 n 0Þ ½1 1:035r2 0 3ð1 1:725r 2 0 Þðd I3 þ d K3 Þþð15 36:23r 2 0 Þd I3d K3 Š 

ð70Þ 

ð74Þ 

R 2 IJ ¼ 1:202r 3 0 

6m e 0 ð1 n 0Þ ½1 2n 0 þ 1:035r 2 0 þ 3ðn 0 1:725r 2 0 Þðd I3 þ d J3 ÞŠ. (75) 

Here, the identities P 1 

i¼1 1=i3 ¼ 1:2021 and P 1 

i¼1 1=i5 ¼ 1:0369 are used. 

Substituting Eq. (70) into Eq. (69), we can solve a particle’s averaged strain as 

he ij i O ¼ DCijmn 1 ðDC mnkl 

1 

¯D mnkl Þ 1 he kl i M 

D 0 mnkl 

ðD 0 ijmn þ ¯D ijmn ÞðDC 1 

mnst 

D 0 mnst 

¯D mnst Þ 1 DC 1 

stpq C1 pqkl ems kl . 

ð76Þ

990 



Combining Eqs. (62), (63), and (76) provides the averaged strain of the composites as 

he ij i D ¼ T 1 ijkl ðC0 klmn Þ 1 hs mn i D þ T 2 ijkl ems kl , (77) 

where 

and 

T 1 ijkl ¼fI ijkl þ fðC 0 ijmn Þ 1 ½DC 1 

mnkl ð1 fÞðD 0 mnkl þ ¯D mnkl ÞŠ 

T 2 ijkl ¼ f½fDC ijpq 1 C1 1 

pqmn þð1 fÞðDCijpq 

Considering Eq. (54), we can obtain 

s 0 ij ¼ C0 ijkl ðT 1 klmn Þ 1 ðhe mn i D 

D 0 ijpq 

1 g 1 (78) 

¯D ijpq ÞC 0 pqmn Š 1 DC 1 

mnst C1 stkl . (79) 

T 2 mnpq ems pq Þ sm ij . (80) 

4. Results and discussion 

In the last section, we solved the averaged magnetic fields and elastic fields for the 

particles, matrix, and overall composites. In this section, we investigate the effective 

magneto-elastic behavior of chain-structured composites for a given magnetic and 

mechanical loading. 

4.1. Effective magnetostriction 

If a magnetic field H 0 i is applied with the absence of mechanical load, there exists an 

effective stress s m ij in the composite as seen in Eq. (60), and an effective strain or 

magnetostriction can be derived from Eq. (77) as 

e m ij ¼ T 1 ijkl ðC0 klpq Þ 1 s m pq þ T 2 ijkl ems kl , (81) 

where Eq. (54) has been used. From the above equation, we can see that the effective 

magnetostriction e m ij is induced by the magnetic interaction force between particles as well 

as the particle magnetostriction. As we know, the effective magnetic stress s m ij in Eq. (60) is 

in general asymmetric except in some special cases where the magnetic field is parallel or 

perpendicular to the chain’s direction. Note that an asymmetric stress in a free body is not 

physical, and in fact, this situation will never happen. For instance, when a free chainstructured 

composite presents in a magnetic field with a small angle between the directions 

of chains and the magnetic field, the magnetic force will cause the composite to rotate until 

the chains have the same direction as the magnetic field to reach equilibrium. That is why 

this type of materials is also called ‘‘active’’ materials. Thus, a chain-structured composite 

will not have an asymmetric magnetic stress unless a balancing external mechanical 

loading exists. By virtue of this fact, we will only consider the effective magnetostriction for 

the composite with the chain’s direction parallel to the magnetic field. 

For pure magnetostrictive materials, magnetostriction is experimentally measured by the 

fractional change in length l ¼ Dl=l (Clark, 1980). The value of l measured at magnetic 

saturation is called the saturation magnetostriction l s . Typically, the saturated 

magnetostriction is a material constant. However, it may change with ambient 

temperatures and stress states in a nonlinear manner. It is noted that, in this work, we 

do not consider the effect of temperatures and stress on particle’s magnetostriction and



magnetization. We assume that particle’s magnetostriction changes linearly with the 

particle’s averaged magnetization (Cullity, 1972) before the magnetostriction is saturated 

at the fixed saturated magnetostriction. In the following, we set the external magnetic field 

in the same direction as the chains (x 3 direction). For isotropic magnetostrictive particles, 

the magnetostriction due to magnetic field in the x 3 direction has the form (Yin et al., 2002) 

lðyÞ ¼ 3 2 lðhH 3i O Þðcos 2 1 

y 

3Þ, (82) 

where y denotes the angle between the magnetization direction and the measurement 

direction and lðhH 3 i O Þ is measured in the direction of magnetic field. From the linear 

assumption, this can be written as 

lðhH 3 i O Þ¼ ls 

H s jhH 3i O j, (83) 

where H s is particle’s magnetic field when its magnetostriction is saturated. Then, the 

magnetostrictive eigenstrains are 

l s 

e ms 

33 ¼ ls 

H s jhH 3i O j; e ms 

11 ¼ ems 22 ¼ 1 2 H s jhH 3i O j; e ms 

ij ¼ 0 for iaj. (84) 

From Eq. (84), we can see that the volumetric strain of particles caused by 

magnetostriction is zero, which yields an incompressible deformation. Substituting the 

above equation into Eq. (81), we get the effective magnetostriction of the composite. 

Fig. 2 shows the effective magnetostriction changing with the magnetic loading and the 

comparison with Duenas and Carman’s (2000) experimental data. In the experiment, 

700.00 

λ s =0.00075, H s =9.5*10 4 , µ m 0 =4π*10 -7 , E 0 =3*10 9 , v 0 =0.45 

µ m 1 =3µ m 0 , E 1 =35*10 9 , v 1 =0.25, φ=0.4, ρ 0 =0.499 

600.00 

Effective magnetostriction (ppm) 

500.00 

400.00 

300.00 

200.00 

100.00 

0.00 

Experimental data 

Proposed model 

-0.4 -0.2 0.0 0.2 0.4 

Flux density (T) 

Fig. 2. Effective magnetostriction in the direction of magnetic field vs. flux density in comparison to the 

experimental data (Duenas and Carman, 2000).

992 



a magnetostrictive composite containing chain-structured Terfenol-D in an epoxy matrix is 

used and the Young’s moduli and Poisson’s ratios for the particles and matrix are 

E 1 ¼ 35 GPa, n 1 ¼ 0:25, E 0 ¼ 3 GPa, and n 0 ¼ 0:45, respectively. The volume fraction of 

particles is 40% and particle’s center–center distance is very close and thus we use 

r 0 ¼ 0:499. The relative magnetic permeability of Terfenol-D is in the range of 3–15 

(McKnight, 2002). Here we use m m 1 ¼ 3mm 0 . Based on Wun-Fogle and colleagues’ (1999) 

measurements, the saturation magnetostriction of Terfenol-D is 750 ppm when the applied 

magnetic field reaches 2000 Oe. Then, the particle’s magnetic field is calculated as 1200 Oe 

by Eq. (16), and l s ¼ 0:00075 and H s ¼ 9:5 10 4 A=m are also obtained. To solve the flux 

density, the effective permeability of the composite in the chains’ direction is calculated as 

1:689m m 0 (Yin, 2004). In Fig. 2, we find that the effective magnetostriction is still 

comparable to particle’s saturation magnetostriction in this microstructure. Since it is 

assumed that the magnetostriction changes linearly with the particle’s magnetic field up to 

the fixed saturated magnetostriction and that Terfenol-D has a constant magnetic 

permeability during the magnetic loading, it is shown that the proposed model provides a 

nearly linear prediction of the effective magnetostriction before it is saturated. Although 

this is a reasonable approximation under small magnetic fields, a rigorous consideration of 

the nonlinear magnetic behavior of Terfenol-D is ultimately needed for more general cases. 

In addition, the particles’ magnetostriction is also sensitive to stresses. In the 

experiments of Wun-Fogle et al. (1999), it is found that Terfenol-D has a considerably 

larger saturation magnetostriction under a compressive stress. For instance, the saturated 

magnetostriction can reach 1200 ppm under the prestress at 5.3 MPa. When a composite is 

subjected to a magnetic field, the particle stress comes from the constraint of matrix to the 

particle magnetostriction and the magnetic force between particles. The particle stress 

should be compressive, so particle’s saturated magnetostriction is underestimated. 

Consequently, it is shown in Fig. 2 that the proposed model underestimates when 

compared with the experimental results for the case of high magnetic flux density (larger 

than 0.2 T). If the magnetostriction and magnetic permeability of particles changing with 

stress is exactly characterized, the proposed linear model is still applicable in an 

incremental way, in which the loading is divided into many small steps and in each step the 

linear behavior is considered with updated magnetostriction and magnetic permeability. 

In Fig. 2, due to the very high modulus of the matrix, and the small magnetic 

permeability in Terfenol-D, the magnetic force does not provide a considerable 

contribution to the effective magnetostriction. Fig. 3 illustrates the effect of the Young’s 

modulus of the matrix and the magnetic permeability of the particles on the effective 

magnetostriction. In Fig. 3(a), the Young’s modulus of the matrix varies in the range of 

3 MPa to 30 GPa sampled by five values. Since magnetic force causes a considerable 

contractive deformation when the Young’s modulus of the matrix is small, we see that the 

effective magnetostriction decreases when the Young’s modulus is reduced from 

300–3 MPa. On the other hand, when the Young’s modulus is large, the matrix provides 

a strong constraint for the particles’ magnetostriction, and thus the effective magnetostriction 

also decreases as the Young’s modulus is increased from 300 MPa to 30 GPa. Thus, 

there exists an optimal choice of the matrix Young’s modulus that maximizes effective 

magnetostriction of the composite. However, this is contrary to the prediction of Herbst et 

al. (1997) in that the matrix’s stiffness has no effect on the overall magnetostriction. In 

addition, when the Young’s modulus of the matrix is at 3 MPa, the effective 

magnetostriction no longer changes linearly with the applied magnetic field. This is




600 

500 

400 

300 

200 

100 

λ s =0.00075, H s =9.5*10, µ 0 m =4π*10 -7 , v 0 =0.45 

4 

µ m 1 =3µ m 0 , E 1 =35*10 9 , v 1 =0.25, φ =0.4, ρ 0 =0.499 

E 0 =3MPa 

E 0 

=30MPa 

E 0 

=300MPa 

E 0 

=3GPa 

E 0 =30GPa 

(a) 

0 

0 40 80 120 160 

Applied magnetic field (KA/m) 

λ s =0.00075, H s =9.5*10, µ 0 m =4π*10 -7 , v 0 =0.45 

4 

E 0 =3*10 8 , E 1 =35*10 9 , v 1 =0.25, φ =0.4, ρ 0 =0.499 


600 

500 

400 

300 

200 

100 

µ m m 

1 

=3µ 0 

µ m m 

1 

=6µ 0 

µ m m 

1 

=9µ 0 

µ m m 

1 

=12µ 0 

µ m m 

1 

=15µ 0 

(b) 

0 

0 40 80 120 160 


Fig. 3. Effective magnetostriction in the direction of magnetic field vs. applied magnetic field for (a) the different 

matrix’s Young’s moduli and (b) the different particles’ magnetic permeabilities. 

because the magnetic force has a second order effect on the effective magnetostriction as 

seen in Eq. (60). In Fig. 3(b), the particles’ relative magnetic permeability varies in the 

range of 3–15 sampled by five values while the Young’s modulus of the matrix is fixed at 

300 MPa. It is apparent since the effective magnetostriction reduces as the particles’

994 



magnetic permeability increases due to the increasing magnetic force between particles. 

However, it is expected that the effect of particles’ magnetic permeability on the effective 

magnetostriction will not be so significant when the matrix is stiffer. 

Fig. 4 shows the effect of the volume fraction and the center–center distance of particles 

on the effective magnetostriction of composites. In Fig. 4(a), we observe that when the 

particles’ volume fraction is larger than 20%, the effective magnetostrictions for four cases 

are very similar before the effective magnetostriction is saturated. Especially, 30% and 

40% volume fraction composites produce the largest magnetostriction in the linear range, 

which is contrary to the general understanding that the greater magnetostrictive particles’ 

volume fraction will lead to a greater effective magnetostriction of the composite. Duenas 

and Carman (2000) also observed this interesting phenomenon in their experiment, which 

was interpreted to be caused by thermal residual stresses during the curing process (Duenas 

and Carman, 2000). In the current work, it is found that the phenomenon may still exist in 

magnetostrictive composites even without thermal residual stresses. In Fig. 4(b), the 

volume fraction is set at 0.10. With the increase of the ratio of a particle’s radius to its 

center–center distance, the chain structure in the composite becomes more apparent. We 

find that the effective magnetostriction is significantly larger for r 0 ¼ 0:5 than that for 

r 0 ¼ 0:3. It is noted that because the topological structure of composites can greatly 

change the magnetic field on the particle, the effective magnetostrictions are saturated at 

the different applied magnetic fields for both figures. 

4.2. Effective elasticity 

When a mechanical loading is applied with the absence of a magnetic loading, from the 

relation between the loading and the response as hs ij i D ¼ ¯C ijkl he kl i D , we can derive the 

effective elasticity of the composite. From Eq. (77), we have 

¯C ijmn ¼ C 0 ijkl ðT 1 klmn Þ 1 . (85) 

Substituting Eq. (78) into Eq. (85) yields 

¯C ijkl ¼ C 0 ijkl þ f½DC ijkl 1 ð1 fÞðD 0 ijkl þ ¯D ijkl ÞŠ 

1 . (86) 

From Eqs. (86) and (73), it is apparent that effective stiffness depends on the material 

properties of the particles and matrix, as well as the volume fraction of the particles and 

the ratio r 0 . The term ¯D includes the interactions from all other particles in the same chain. 

If we drop this term, Eq. (86) is reduced to 

¯C ijkl ¼ C 0 ijkl þ f½DC ijkl 1 ð1 fÞD 0 ijkl Š 1 , (87) 

which is the same as the result of Mori–Tanaka’s model (Nemat-Nasser and Hori, 1999). 

We can see that Mori–Tanaka’s model does not consider the effect of microstructure and 

direct interaction between particles. 

Although chains have the same direction, they can be randomly distributed in the 

composite. The microstructure has transversely isotropic symmetry along the chain’s 

direction. We can show that the effective elasticity tensor ¯C in Eq. (86) is also transversely 

isotropic. The proof, along with the explicit form of the components of ¯C, is given in the 

Appendix.



4 

λ s =0.00075, H s =9.5*10, µ m 0 =4π*10 -7 , v 0 =0.45 

E 0 =3*10 8 , µ m 1 =3µ m 0 , E 1 =35*10 9 , v 1 =0.25, ρ 0 =0.499 


700 

600 

500 

400 

300 

200 

100 

φ=0.1 

φ=0.2 

φ=0.3 

φ=0.4 

φ=0.5 

(a) 

0 

0 40 

80 

120 160 


λ s =0.00075, H s =9.5*10, µ m 0 =4π*10 -7 , v 0 =0.45 

E 0 =3*10 8 , µ m 1 =3µ m 0 , E 1 =35*10 9 , v 1 =0.25, ρ 0 =0.1 

4 


ρ 0 

=0.3 

300 ρ 0 =0.4 

ρ 0 

=0.5 

200 

100 

(b) 

0 

0 

40 

80 

120 160 


Fig. 4. Effective magnetostriction in the direction of magnetic field vs. applied magnetic field for (a) the different 

particle volume fractions and (b) the ratios of a particle’s radius to its center–center distance. 

To assess the validity of the proposed model, Fig. 5 illustrates the comparisons to 

Duenas and Carman’s (2000) experimental data as well as Mori–Tanaka’s model, Voigt 

and Reuss’ (Mura, 1987) upper and lower bounds. Mechanical properties of materials are 

the same as those in Fig. 2. InFig. 5, we see that the upper bound provides a higher

996 



E 0 =3GPa, v 0 =0.45, E 1 =35GPa, v 1 =0.25, ρ 0 =0.499 

Effective elastic moduli (GPa) 

20 

18 

16 

14 

12 

10 

8 

6 

The upper bound 

The proposed model 

Experimental data 

Mori-Tanaka's model 

The lower bound 

4 

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 

Volume fraction φ 

Fig. 5. Effective elastic modulus in the direction of chains vs. volume fraction in comparison to the experimental 

data (Duenas and Carman, 2000), Mori–Tanaka’s model, Voigt and Reuss’ bounds. 

E 0 =3GPa, v 0 =0.45, E 1 =35GPa, v 1 =0.25, ρ 0 =0.499 

Effective elastic moduli (GPa) 

20 

16 

12 

8 

4 

C 1111 

C 1122 

C 1133 

C 3333 

C 2323 

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 


Fig. 6. Five independent elastic constants vs. volume fraction. 

estimate, the lower bound has an underestimation, and the proposed model yields the best 

prediction compared to the experimental data. Because Mori–Tanaka’s model does not 

consider particle interaction, it only provides an isotropic elasticity, which significantly 

underestimates effective elastic properties even when the particle’s volume fraction is quite 

small because particle orientations are so close in a chain.



The overall material properties are shown in Fig. 6 where five independent material 

constants for the transversely isotropic effective elasticity of the chain-structured 

composite are displayed. A comparison between C 3333 and C 1111 confirms that the elastic 

modulus in the chain’s direction is larger than the one perpendicular to the direction. Thus, 

this microstructure can greatly improve the effectiveness of materials in applications where 

the primary loading direction to the composites is known. Thus, in the following we will 

focus on the elastic properties of composites in the direction of the chains. It is noted that 

when the particle’s volume fraction approaches 50%, because the distance between chains 

is also small, the chain structure will be fuzzy and the advantage of this microstructure 

cannot be fully achieved. 

Although a chain-structured composite has a transversely isotropic symmetry with five 

independent elastic constants, one can still experimentally define the Young’s modulus 

and the shear modulus in the corresponding directions by the properties measured from a 

x 3 - directional uniaxial test and a simple shear test in the x 1 2x 3 or x 2 2x 3 plane, 

respectively, as 

Ē ¼ C 3333 

2C 2 1133 

C 1111 þ C 1122 

; ¯m ¼ C 2323 . (88) 

Fig. 7 shows how the effective Young’s modulus and shear modulus vary with volume 

fraction. With an increase of Young’s modulus in the particles, the effective elastic moduli 

also increase. Special cases are illustrated for rigid or void particles. In fact, when 

E 1 =E 0 o0:1, the effective elastic moduli do not change much so that particle’s can be 

approximated as void. Similarly when E 1 =E 0 4100, the effective elastic moduli are so 

similar that particle’s can be assumed rigid. 

To study the effect of the ratio of particle’s radius to the particle center–center distance 

r 0 , we fix the particle’s volume fraction as f ¼ 0:05, and vary r 0 from 0.2 to 0.5. Fig. 8 

shows that the effective Young’s moduli rapidly increase along with r 0 but the effective 

shear moduli slowly decrease. Thus, although composites become harder in the direction of 

the chains, as particles move closer, the capability to resist shear is reduced. In Fig. 8(a),we 

can see that when E 1 =E 0 gets larger, the increment in the effective Young’s moduli with the 

increase of r 0 gets larger. For E 1 =E 0 ¼ 100, the effective Young’s moduli increase as high 

as 1 3 for a r 0 that increases from 0.2 to 0.5. In contrast, for E 1 =E 0 p2, the effective Young’s 

moduli only have a very marginal increase associated with r 0 .InFig. 8(b), the change of 

the effective shear modulus is not as significant as that of the effective Young’s modulus. 

It is noted that when the matrix is a very compliant material like rubber and when the 

particle’s relative magnetic permeability is high, the magnetic interaction forces between 

particles change sensitively with respect to the mechanical loading, and it provides a 

contribution to the effective elasticity. Here we do not consider this extreme case for 

magnetostrictive composites, because generally magnetostrictive materials have a small 

relative magnetic permeability. Furthermore, to effectively use the particles’ magnetostriction, 

as seen in Fig. 3(a), we should not use matrix material with high compliant. In this 

case, the magnetic interaction forces will not change severely to a mechanical loading that 

lead to infinitesimal deformation, and the effective elasticity of magnetostrictive 

composites will stay constant. On the contrary, for ferromagnetic composites, 

ferromagnetic particles typically have relative magnetic permeability as high as 10 4 and 

the matrix is made of rubber-like materials. The magneto-mechanical behavior is fully

998 



Effective Young's modulus E/E 0 

5.0 

4.5 

4.0 

3.5 

3.0 

2.5 

2.0 

1.5 

1.0 

rigid particles 

E 1 

/E 0 

=100 

E 1 

/E 0 

=10 

E 1 

/E 0 

=1.1 

E 1 

/E 0 

=0.1 

voids 

v 0 =0.45, v 1 =0.3, ρ 0 =0.45 

(a) 

0.5 

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 


Effective shear modulus µ/E 0 

0.8 

0.6 

0.4 

0.2 

rigid particles 

E 1 

/E 0 

=100 

E 1 /E 0 =10 

E 1 /E 0 =1.1 

E 1 

/E 0 

=0.1 

voids 

v 0 =0.45, v 1 =0.3, ρ 0 =0.45 

(b) 

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 


Fig. 7. Effective elastic moduli vs. volume fraction. (a) Young’s modulus; (b) the shear modulus. 

coupled and magnetic-dependent elasticity behavior is commonly observed (Yin and Sun, 

2005b). 

5. Conclusions 

A three-dimensional micromechanics-based elastic model is proposed to investigate magnetoelastic 

behavior of two-phase composites containing chain-structured magnetostrictive particles



Effective Young's modulus E/E 0 

1.4 

1.3 

1.2 

1.1 

1.0 

v 0 =0.45, v 1 =0.3, φ=0.05 

E 1 /E 0 =100 

E 1 

/E 0 

=10 

E 1 

/E 0 

=5 

E 1 

/E 0 

=2 

E 1 

/E 0 

=0.5 

E 1 /E 0 =0.1 

(a) 

0.9 

0.20 0.25 0.30 0.35 0.40 0.45 0.50 

Ratio of radius to center-center distance ρ 0 

Effective shear modulus µ/E 0 

0.42 

0.40 

0.38 

0.36 

0.34 

0.32 

v 0 =0.45, v 1 =0.3, φ=0.05 

E 1 /E 0 =100 E 1 

/E 0 

=10 

E 1 /E 0 =5 

E 1 /E 0 =2 

E 1 /E 0 =0.5 E 1 /E 0 =0.1 

(b) 

0.20 0.25 0.30 0.35 0.40 0.45 0.50 

Ratio of radius to center-center distance ρ 0 

Fig. 8. Effective elastic moduli vs. ratio of a particle’s radius to its center–center distance. (a) Young’s modulus; 

(b) the shear modulus. 

under both magnetic and mechanical loading. To derive the local magnetic and elastic 

fields, the Green’s function technique is used. Three modified Green’s functions are derived 

and explicitly integrated for the infinite domain containing a spherical inclusion with a 

prescribed magnetization, body force, and eigenstrain. Introducing the representative 

volume element containing a chain of infinite particles, we derive the averaged magnetic 

field, stress field and strain field in both phases of a chain-structured magnetostrictive 

particle filled composite. From the relation between the overall averaged strain and 

the magnetic/mechanical loading, we derive effective magnetostriction and elasticity

1000 



properties. It is found that there exists an optimal choice of the matrix Young’s modulus 

and the particle volume fraction that maximizes effective magnetostriction. In this 

composite model, a transversely isotropic effective elasticity is obtained at the infinitesimal 

deformation. Disregarding the interaction term, this model provides the same effective 

elasticity as Mori–Tanaka’s model. The numerical simulations and the comparisons to 

experiments and other models show the efficacy of the proposed model for prediction of 

effective magneto-elastic properties of chain-structured composites. Our results also 

suggest that the composite with a smaller particle center–center distance offers a 

considerable advantage in amplifying the effective magnetostriction and strengthening the 

effective Young’s modulus in the chain’s direction. 

Acknowledgements 

This work is sponsored by the National Science Foundation under Grant number CMS- 

0084629 and DMR-0113172. The support is gratefully acknowledged. 

Appendix. Details of effective elasticity 

Substituting Eq. (73) along with Eqs. (74) and (75) into Eq. (86), we can explicitly write 

effective elasticity ¯C as 

where 

¯C ijkl ¼ðl 0 þ L IK Þd ij d kl þðm e 0 þ M IJÞðd ik d jl þ d il d jk Þ, 

(A.1) 

and 

K ¼ f D 

0 

d 2 2 d 1 ðd 3 þ 2d 4 þ 4d 5 Þ d 2 1 

2 d 1 ðd 3 þ 2d 4 þ 4d 5 Þ 2d 4 ðd 1 þ d 2 Þ 

d 2 2 d 1 ðd 3 þ 2d 4 þ 4d 5 Þ d 2 2 d 1 ðd 3 þ 2d 4 þ 4d 5 Þ 2d 4 ðd 1 þ d 2 Þ 

B 

C 

@ 

A 

d 

2d 4 ðd 1 þ d 2 Þ 2d 4 ðd 1 þ d 2 Þ 2d 2 2 d 1 ðd 3 þd 4 Þ d 4 ðd 2 þd 3 Þ 

4 d 4þ2d 5 

ðA:2Þ 

0 

1 

1 1 1 

d 4 d 4 d 4 þ d 5 

M ¼ f 1 1 1 

4 d 4 d 4 d 4 þ d , (A.3) 

5 

B 

C 

@ 1 1 1 A 

d 4 þ d 5 d 4 þ d 5 d 4 þ 2d 5 

in which 

2 

d 1 þ 2d 4 d 1 d 1 þ d 2 

3 

6 

D ¼ det4 

d 1 d 1 þ 2d 4 d 1 þ d 2 

7 

5, 

d 1 þ d 2 d 1 þ d 2 d 1 þ 2d 2 þ 2d 4 þ 4d 5



d 1 ¼ 

l 1 l 0 

2ðm e 1 

m e 0 Þ½2ðme 1 

m e 0 Þþ3ðl 1 l 0 ÞŠ 

1 f 

30m e 0 ð1 n 0Þ 

ð1 1:035r 2 0Þð1 fÞw, 

d 2 ¼ 3ð1 1:725r 2 0Þð1 fÞw, 

d 3 ¼ ð15 36:225r 2 0Þð1 fÞw, 

1 

d 4 ¼ 

4ðm e 1 

m e 0 Þ þ ð1 fÞð4 5n 0Þ 

30m e 0 ð1 n 0Þ 

d 5 ¼ 3ðn 0 1:725r 2 0Þð1 fÞw, 

þð1 2n 0 þ 1:035r 2 0Þð1 fÞw, 

w ¼ 1:202r3 0 

6m e 0 ð1 n 0Þ . 

A tensor with the form of X ijkl ¼ c 1 IK d ijd kl þ c 2 IJ ðd ikd jl þ d il d jk Þ has the transversely 

isotropic symmetry along x 3 axis if and only if it should satisfy two conditions (Cowin and 

Mehrabadi, 1995): major and minor symmetries, 

g 1 IJ ¼ g1 JI ; g2 IJ ¼ g2 JI (A.4) 

and rotational symmetry, 

g 1 11 þ 2g2 11 ¼ g1 22 þ 2g2 22 ; g1 13 ¼ g1 23 ; g2 13 ¼ g2 23 ; g2 12 ¼ 1 2 ðg1 11 þ 2g2 11 g 1 12Þ. (A.5) 

Substituting Eqs. (A.1)–(A.3) into Eqs. (A.4) and (A.5), we can see that the effective 

elasticity satisfies the requirements and is a transversely isotropic tensor. Here, the term w 

signifies the interaction between particles. With w ¼ 0, Eq. (A.1) is reduced to 

Mori–Tanaka’s model. 

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