Multicoated elliptic fibrous composites of piezoelectric and piezomagnetic phases

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Multicoated elliptic ﬁbrous composites of piezoelectric

and piezomagnetic phases

Hsin-Yi Kuo

Department of Civil Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan

a r t i c l e

i n f o

Article history:

Received 8 December 2010

Received in revised form 28 January 2011 Accepted 17 February 2011

Available online 12 March 2011 Keywords:

Magnetoelectricity Piezoelectric Piezomagnetic Elliptic ﬁbrous composite Multicoated

a b s t r a c t

A theoretical framework is developed to investigate the magnetoelectroelastic potential in a multicoated elliptic fibrous composite with piezoelectric and piezomagnetic phases. We generalize the classic work of Rayleigh (1892) to obtain the electrostatic potential in ordered conductive composites and its extension to a disordered system (Kuo, 2010; Kuo & Chen, 2008) to the current coupled magnetoelectroelastic multicoated elliptic compos-ites. We combine the methods of complex potentials with a re-expansion formulae and the generalized Rayleigh’s formulation to obtain a complete solution of the multi-field many-inclusion problem. It is shown that the coefficients of field expansions can be written in the form of an infinite set of linear algebraic equations. Numerical results are presented for several configurations. We use this method to study BaTiO3–CoFe2O4composites and find that, with appropriate coating, the effective magnetoelectric voltage coefficient can be enhanced with one order of magnitude compared to their non-coating counterpart.

1. Introduction

Magnetoelectric materials, which induce the polarization by a magnetic field, or conversely induce the magnetization by an electric field, have been the focus of research due to their varieties of microstructural phenomena and macroscopic prop-erties. These make them promising for a wide range of applications, such as four-state memories, magnetic field sensors, and magnetically controlled opto-electric devices (Eerenstein, Mathur, & Scott, 2006; Nan, Bichurin, Dong, Viehland, &

Srinivasan, 2008). The study of magneto–electric coupling can be traced back to 1957 whenLandau and Lifshitz (1984)

showed the possibility of the coupling between the electric and magnetic ﬁelds in a substance with a certain magnetic sym-metry class. This was subsequently experimentally conﬁrmed in a single crystal Cr2O3byAstrov (1960)and byRado and

Folen (1961)over 50 years ago. However, this coupling is weak in single phase materials, and thus has motivated the study

of composites of piezoelectric and piezomagnetic media. The basic idea is to couple a piezoelectric and a piezomagnetic material using strain: an applied electric ﬁeld creates a strain in the piezoelectric material which in turn induces a deforma-tion in the piezomagnetic material, resulting in a magnetic ﬁeld.

A number of micromechanical models hence were proposed to predict the effective moduli of multiferroic composites. For instance, Green’s function approach was used byNan (1994) and Huang and Kuo (1997)to study a ﬁbrous composite consisting of Barium Titanate and Cobalt Ferrite. For such transversely isotropic ﬁbrous composites, Benveniste (1995)

derived exact connections among effective magnetoelectroelastic moduli based on a formalism discovered by Milgrom

and Shtrikman (1989). Particulate composites were investigated byHarshé, Dougherty, and Newnham (1993)using a cubic

model, while a homogenization micromechanical method was employed byAboudi (2001). Eshelby’s equivalent inclusion approach and the mean ﬁeld Mori–Tanaka model have been generalized to multiferroic composites by Li and Dunn

E-mail address:[email protected]

Contents lists available atScienceDirect

International Journal of Engineering Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j e n g s c i

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(1998a, 1998b), Huang (1998), Li (2000), Wu and Huang (2000) and Srinivas et al. (2006). A two-scale asymptotic homog-enization theory was adopted byCamacho-Montes, Sabina, Bravo-Castillero, Guinovart-Díaz, and Rodríguez-Ramos (2009)

on the magnetoelectric coupling and cross-property connections in a two-phase multiferroic composite. For a good overview of the subject, the reader is referred to the review article byNan et al. (2008).

In a classic work, Lord Rayleigh computed the electrostatic potential for a conducting composite consisting of a periodic array of cylindrical or spherical inclusions. This was extended to arbitrary arrangements byKuo and Chen (2008)and to ellip-tic cylinders byKuo (2010). These works concern single fields. Later,Kuo and Bhattacharya (submitted for publication) gen-eralized this methodology to electrostatic, magnetostatic and mechanical coupled fields. In this paper, we extend this Rayleigh’s formulation further to a multiferroic composite consisting of elliptic cylinders, specifically multicoated ellipses. Coating plays an important role in high-temperature systems and in various engineering applications. For instance, to re-duce heat or stress concentration along the interface, interphase layers between the inclusions and the matrix are often introduced to act as thermal barrier. Graded materials can also be more damage-resistant than their conventional homoge-neous counterpart (Suresh, 2001). Such interphase layer may have constant properties or spatially varying properties. Research into graded multiferroics has primarily been confined to bilayer and multilayer structures. Among them, piezoelec-tric or piezomagnetic coefficients are assumed linear variation in the thickness direction byChen and Lee (2003), Petrov,

Srinivasan, and Galkina (2008) and Petrov and Srinivasan (2008), while exponentially graded assumption is adopted by

Pan and Han (2005) and Wang et al. (2009). To our knowledge, the subject of piezoelectric/piezomagnetic ﬁbrous composites

with multicoated elliptic cylinders has not been examined in the literature before.

The plan of this article is organized as follows. First we consider a composite medium made of piezoelectric and piezo-magnetic phases arranged in a microstructure consisting of parallel elliptic cylinders in a matrix in Section2. The phases are transversely isotropic and under anti-plane shear with in-plane electromagnetic fields. In this situation, the fields are decou-pled in the interior of every phase, and the coupling between the fields occurs only through the interface conditions (Kuo &

Bhattacharya, submitted for publication). We exploit this in Section2.2to obtain a representation of the solution. The basic

idea is to followKuo (2010)and expand each field in each medium in a series. We consider periodic arrays in Section2.3. In Section3we consider the case of multicoated elliptic cylinders. We show that a (6 6) array alone can mathematically sim-ulate the effects of multiple coatings. We obtain effective properties in Section4, and significantly show that the macroscopic properties depend solely on a single expansion coefficient (amongst the infinite). This methodology is illustrated in Section5

using composites of BaTiO3and CoFe2O4. We choose this material pair for its practical potential and also because it enables comparison with previous work. We observe that the composite medium has a non-trivial magnetoelectric coupling even through the individual components do not. Further, we show that the ME coefﬁcient can be enhanced with an order of mag-nitude if the BaTiO3ﬁber is coated with Terfenol-D.

2. Multiple elliptic cylinders 2.1. Basic formulations

Let us consider an inﬁnite medium R3_{containing N arbitrarily distributed, parallel and separated elliptic cylinders. The} domain of the pth elliptic cylinder is denoted Vp, p = 1, 2, . . . , N, and the remaining matrix is denotedXm. We assume that the cylinders and the matrix are made of distinct phases. Further, we assume that each phase is either piezoelectric or piezomagnetic with transversely isotropic symmetry (i.e. has 6 mm symmetry) about the ﬁber axes. We introduce a Cartesian coordinate system positioned at a selected point O of the plane with the x- and y-axes in the plane of the cross-section and z- along the axes of the cylinders (Fig. 1). The centroids of the pth elliptic cylinders are designated as Op, with Opxpand Opypaxes are directed along the major and minor axes of the ellipse. Each of ellipse has the major and minor semi-axis, lðpÞx and l

ðpÞ

y , and the inter-foci distance is 2dp, where d2p¼ l ðpÞ2

x l

ðpÞ2

y . The ellipses are well separated so that any two inclusions will not get in touch with each other.

Let the composite be subjected to the anti-plane shear strain

e

zx;

e

zy;the in-plane electric fields Ex; Ey;and the magnetic fields Hx; Hyat infinity. Thus the heterogeneous material is in a state of anti-plane shear problem (Benveniste, 1995; Chen,

1993; Kuo & Bhattacharya, submitted for publication) and can be described by

ux¼ uy¼ 0; uz¼ wðx; yÞ;

u

¼

u

ðx; yÞ;

w¼ wðx; yÞ;

ð2:1Þ

where ux, uy, uzare the mechanical displacements along the x-, y-, and z-axes, and

u

and

w

are the electric and magnetic potentials, respectively.

The constitutive laws of the constituents for the non-vanishing ﬁelds become

r

zj Dj Bj 0 B @ 1 C A ¼ C44 e15 q15 e15

j

11 k11 q15 k11

l

11 0 B @ 1 C A

e

zj Ej Hj 0 B @ 1 C A; ð2:2Þ

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where j denotes the component x, y. We can write this compactly as

R

Uj ¼ LUVZVj; U; V ¼ w;

u

;w; j ¼ x; y; ð2:3Þ where Rj¼

r

zj Dj Bj 0 B @ 1 C A; L ¼ C44 e15 q15 e15

j

11 k11 q15 k11

l

11 0 B @ 1 C A; Zj¼

e

zj Ej Hj 0 B @ 1 C A: ð2:4Þ

Here

r

zj, Dj, Bj,

e

zj, Ejand Hjare the stress, electric displacement, magnetic flux, strain, electric field, and the magnetic field, respectively. C44,

j

11,

l

11and k11are the elastic modulus, dielectric permittivity, magnetic permeability and magnetoelectric coefﬁcients. The shear strains

e

zxand

e

zy, in-plane electric ﬁelds Exand Ey, and in-plane magnetic ﬁelds Hxand Hycan be de-rived from the gradient of elastic displacement, electric potential, and magnetic potential as follows:

e

zx¼ @w @x;

e

zy¼ @w @y; Ex¼ @

u

@x; Ey¼ @

u

@y; Hx¼ @w @x; Hy¼ @w @y: ð2:5Þ

Further, the equilibrium equations, in the absence of body force, electric charge density and electric current density, are given by @

r

zx @x þ @

r

zy @y ¼ 0; @Dx @x þ @Dy @y ¼ 0; @Bx @x þ @By @y¼ 0: ð2:6Þ

Substitution of Eq.(2.3)into Eq.(2.6)yields

C44

r

2w þ e15

r

2

u

þ q15

r

2 w¼ 0; e15

r

2w

j

11

r

2

u

k11

r

2w¼ 0; q15

r

2 w k11

r

2

u

l

11

r

2 w¼ 0; ð2:7Þ

x

y

x

p

y

p

O

p

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wherer2_{= @}2_/@x2_{+ @}2_/@y2_{represents the two-dimensional Laplace operator for the variable x and y. Since L is a nonsingular} matrix, generically we can completely decouple(2.7)into three independent Laplace equations,

r

2_{w ¼ 0;}

r

2

_u

¼ 0;

r

2

w¼ 0 ð2:8Þ

in the interior of each phase.

In addition to these differential equations, we have to use interface conditions. We assume that the interfaces are per-fectly bonded, and therefore the ﬁelds satisfy

sZjtjt¼ 0; s½Rjnjt¼ sðLZjÞnjt¼ 0 ð2:9Þ where s t denotes the jump in some quantity across the interface, tjis the unit tangent to the interface and njis the unit outward normal to the interface, and the repeated index j denotes summing over the components x, y. Since L is different in each phase, the ﬁelds w,

u

and

w

are generally coupled by the interface equations.

2.2. Representation of the solution

We start by considering the case that the cylinders are homogeneous. We showed above that the fields are decoupled in the interior of every phase, but are coupled at the interfaces. Therefore, we may followKuo (2010)and use a series expansion for each field in the interior of each phase and then obtain the coefficients by enforcing the interface and boundary conditions.

Since w,

u

and

w

are harmonic, we can construct an analytic functionU(z) = U(z) + iU⁄

(z), of the complex variable z = x + iy, where U⁄_{is the conjugate harmonic function, related to U by the Cauchy-Riemann equation}

@U @x¼ @U @y ; @U @y¼ @U @x ; U ¼ w;

u

;w: ð2:10Þ

Further, the shape of the cross section of the cylinders deﬁnes elliptic coordinates: (

l

> 0,

p

< h 6

p

)

z ¼ x þ iy ¼ d cosh

x

¼ d coshð

l

þ ihÞ ð2:11Þ

are the most appropriate system for the solution of Laplace’s equation.

We now consider a situation where the composite is subjected to a macroscopically uniaxial loading

wext¼

e

zxx;

u

ext¼ Exx; wext¼ Hxx; ð2:12Þ for constant

e

zx;Exand Hx. We may rewrite this in short as

U

ext¼

C

Uz; ð2:13Þ

whereUrepresents the appropriate ﬁeld – the anti-plane deformation w, electric potential

u

, or magnetic potential

w

– and

CU_¼_CU

Rþ iC

U

I the corresponding applied ﬁeld –

e

zx; Exor Hx.

We rewrite the governing equation, Eq.(2.8), in elliptic coordinates (

l

, h),

r

2

U

¼ 1 d2ðcosh2

l

cos2_hÞ @2

U

@

l

2þ @2

U

@h2 ! ¼ 0: ð2:14Þ

The potential ﬁeld for the pth elliptic cylinder and its surrounding matrix can be expanded with respect to its centroid Opas

U

ðpÞ_i ðzpÞ ¼ X1 n¼1 CUðpÞn e nxp_; _CUðpÞ n ¼ C UðpÞ n ð2:15Þ

for the inclusion, and

U

ðpÞmðzpÞ ¼ X1 n¼1 AUðpÞ n enxpþ X1 n¼1 BUðpÞ n enxp; A UðpÞ n ¼ A UðpÞ n ð2:16Þ

for the matrix. Here

x

p=

l

p+ ihpis the local elliptic coordinate centered at the origin of the pth ellipse, the subscripts i and m denote the inclusion and matrix, respectively. The coefﬁcients AUðpÞ

n ¼ A UðpÞ nR þ iA UðpÞ nI ; B UðpÞ n ¼ B UðpÞ nR þ iB UðpÞ nI and CUðpÞ n ¼ C UðpÞ nR þ iC UðpÞ

nI are some complex unknowns to be determined. The superscripts p appearing in(2.15) and (2.16) indi-cate that the ﬁelds are expanded with respect to the pth ellipse centroid.

We recall the interface conditions(2.9)which we rewrite as

Re

U

ðpÞ_i @Vp ¼ Re

U

ðpÞm @Vp; ðR U ÞðpÞm np _@V p ¼ ðRUÞðpÞi np _@V p ð2:17Þ where Rw¼ ð

r

zx;

r

zyÞ; Ru¼ ðDx;DyÞ; Rw¼ ðBx;ByÞ; ð2:18Þ

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@Vp:

l

p= apdenotes the interface between the matrix and the pth elliptic cylinder, and npis the unit outward normal of the interface @Vp.

Using the orthogonality properties of trigonometric functions, the interface conditions(2.17)provide

aðpÞnK¼ T ðpÞ anKb ðpÞ nK; c ðpÞ nK¼ T ðpÞ cnKb ðpÞ nK; ð2:19Þ and AUðpÞ 0 ¼ C UðpÞ 0 ;where aðpÞnK¼ AwðpÞnK AuðpÞ nK AwðpÞnK 0 B @ 1 C A; bðpÞnK¼ BwðpÞnK BuðpÞ nK BwðpÞnK 0 B @ 1 C A; cðpÞnK¼ CwðpÞnK CuðpÞ nK CwðpÞnK 0 B @ 1 C A; ð2:20Þ TðpÞanR¼ L ðmÞ LðpÞ 1 cosh napLðmÞþ sinh napLðpÞ _enap sinh 2nap ; TðpÞanI¼ L ðmÞ LðpÞ 1 sinh napLðmÞþ cosh napLðpÞ _enap sinh 2nap ; TðpÞ cnR¼ T ðpÞ anRþ enap 2 cosh nap I; TðpÞ cnI¼ T ðpÞ anI enap 2 sinh nap I; ð2:21Þ

Krepresents R, the real part, or I, the imaginary part of the coefﬁcients, and I is the 3 3 identity tensor.

We now need to relate the solutions to the applied boundary conditions. We do so by applying the Green’s second iden-tity (Arfken & Weber, 2001) to the matrix domainXm. This gives

Z Xm Gðx; x0_Þ

_r

02

U

mðx0Þ

U

mðx0Þ

r

02Gðx; x0Þ h i dA0¼ Z @Xm Gðx; x0_Þ

_r

0

U

mðx0Þ

U

mðx0Þ

r

0Gðx; x0Þ n0_ds0 ; ð2:22Þ

where the prime0 _{denotes the operation in reference to the x}0_{coordinate, n}0_{is the outward unit normal to the matrix’s} boundary @Xm, dA0represents the area element for the x0 coordinate, ds0is the differential arc length. Here G(x; x0) is the free-space Green’s function for Laplace operator satisfyingr2_{G(x; x}0_{) = d(x x}0_{), where d(x x}0_{) is the Dirac-delta function.} Following the procedure inKuo (2010), it can be shown that Eq.(2.22)yields

U

mðzÞ ¼

U

extðzÞ þ XN l¼1 X1 n¼1 BUðlÞm e nxl_: _ð2:23Þ

This is the consistency equation which relates the external applied ﬁelds to the local potential expansions. Note that the ﬁeld identity(2.23)is written based on different coordinates.

To proceed, we shift the origin of the expansions(2.23)to a ﬁxed point, say Zp, the centroid of the pth ellipse, by expand-ing the term emxl _as_{Kushch, Shmegera, and Buryachenko (2005)}

enxl_¼ X 1 m¼1

g

lp nmemxp; ð2:24Þ with

g

lp nm¼ ð1Þ m n dl dlp n_X1 s¼0

v

ðnþmþ2sÞ lp Xs t¼0 ð1Þst ðs tÞ! dp dlp mþ2t Mnmtðdl;dpÞðn þ m þ t þ s 1Þ! ðs tÞ! ; ð2:25Þ where dlp dlþ dp;

v

lp Zlp=dlpþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðZlp=dlpÞ2 1 q and Mnmtðdl;dpÞ ¼ Xt k¼0 ðdl=dpÞ 2k k!ðt kÞ!ðk þ nÞ!ðm þ t kÞ!: ð2:26Þ

Introducing(2.24)into(2.23), we have the expansion

U

ðpÞm;nearðzÞ ¼

C

U_Z pþ X1 n¼1 BUðpÞ n þ b UðpÞ n emxp_; _ð2:27Þ where bUðpÞn ¼

C

Udp 2dn;1þ XN l–p X1 m¼1 BUðlÞ m

g

lp mn ð2:28Þ

valid for the domain within an ellipse centered in Zpwith inter-foci distance 2dlpand passing the pole of lth elliptic coordi-nate systems closest to Zp(Kushch et al., 2005). Further, since z lies in the matrix domain, Eqs.(2.27) and (2.16)should be identical. This provides the condition

X1 n¼1 AUðpÞn e nxp_¼

_C

U_Z pþ X1 n¼1 bUðpÞn e nxp_: _ð2:29Þ

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Taking the real part and the imaginary part of(2.29), we ﬁnd the two conditions AUðpÞnR ¼

C

U RReZp

C

UIImZp dn;0þ bUðpÞnR ; ð2:30Þ and AUðpÞ nI ¼

C

U IReZpþ

C

URImZp dn;0þ b UðpÞ nI : ð2:31Þ

Eqs.(2.30), (2.31) and (2.19)1constitute an inﬁnite set of linear algebraic equations. Upon appropriate truncations of the

expansions terms, we can determine the expansion coefﬁcients AUnðpÞ; B

UðpÞ

n ; C

UðpÞ

n . Here we make one further remark. Remark. The essential step of the framework is to establish the generalized Rayleigh’s identities,(2.30) and (2.31). We observe, however, that the derivation of the identities is independent of inclusions’ expansions. In other words, these identities can be applicable to inclusions with inhomogeneous constituents provided that the admissible ﬁelds in the inclusions and the transition relations, similar to(2.19)can be constructed.

2.3. Periodic arrays

The analysis carried out in the previous section for the arbitrary system with a ﬁnite number of cylinders may also be adapted for the case of a periodic array of cylinders. Here we concentrate on the rectangular lattice, and we sketch the out-line of the derivation focussing on the differences from the previous situation.

Let us ﬁrst introduce a Cartesian coordinate system (x, y) positioned at the centroid O of one of the ellipses in a rectangular array, as shown inFig. 2. The sides of the rectangular cell parallel to the x and y coordinates are, respectively, denoted by

a

and b. The elliptic cylinders are of the same orientation, elliptic radius

l

= a and inter-foci distance 2d. Uniform intensities Ex and Hxare applied along the positive x axis, and an anti-plane shear deformation

e

zxis applied out of the xy plane. In terms of elliptic coordinates, the general solution has the admissible form

U

i¼ X1 n¼1 CU nenx; C U n ¼ C U n ð2:32Þ for

l

< a, and

U

m¼ X1 n¼1 AUne nx_þX 1 n¼1 BUne nx_; _AU n¼ A U n; ð2:33Þ

for

l

> a. The coefﬁcients AU

n; B

U

n;and C

U

nare unknown complex constants to be determined from the interface and boundary conditions.

Analogous to(2.19), the continuity conditions at the interface will give constraints(2.19)between the coefﬁcients. Next, imposing the periodicity conditions analogous to the boundary condition we did to derive(2.30) and (2.31), lead to general-ized Rayleigh’s identities

AUnK¼ b U

nK;

K

¼ R; I: ð2:34Þ

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Here the quantities bUn¼

C

Ud 2dn;1þ X1 m¼1 BU m X1 l–o

g

lo mn; ð2:35Þ X1 l–o

g

lo mn¼ ð1Þ m nX 1 t¼0 d 2 nþmþ2t Mnmtðd; dÞðn þ m þ 2t 1Þ!Snþmþ2t; ð2:36Þ where Mnmt() is deﬁned in(2.26), and

Sm¼

X

l–o

Zml ; ð2:37Þ

are the lattice sums characterizing the geometry of the periodic structure, and Zlis the centroid of the lth cylinder when mea-sured in the complex plane centered at the central point O. The index l runs over all cylinders’ centers underlying the periodic array except the central one. Previous studies (Rayleigh, 1892) have reported that the sum S2is conditionally convergent and its value depends upon the shape of the exterior boundary of the array. A list of S2for different values of

a

/b can be found in

Nicrovici and McPhedran (1996).

Eqs.(2.34) and (2.19)1constitute an inﬁnite set of linear algebraic equations. Upon appropriate truncations of the

expan-sion terms at a ﬁnite order M, we can determine the expanexpan-sion coefﬁcients AUn; B

U

n, and C

U

n.

3. Confocally multicoated elliptic cylinders

From the previous remark, we now consider that the inclusions are confocally multicoated elliptic cylinders with the out-er elliptic radius að1Þp ; p ¼ 1; 2; . . . ; N; where N is the number of inclusions. We denote the matrix as phase 0, with trans-versely isotropic material parameters Cð0Þ

44; e ð0Þ 15; q ð0Þ 15;

j

ð0Þ 11;

l

ð0Þ 11 and k ð0Þ

11. The multicoated cylinder consists of a core, with radius

l

p¼ a

ðMÞ

p , surrounded by ðM 1Þ layers of coating. The jth layer of the coatings occupies the annulus VðjÞ

p :a ðjþ1Þ

p 6

l

p6a

ðjÞ

p; j ¼ 1; 2; . . . ; M; in which Vp¼ Vð1Þp [ Vð2Þp [ [ VðMÞp . Here the innermost core is solid so that we have aðMþ1Þ

p ¼ 0. We assume that the material properties of jth constituent layer of the pth multicoated cylinder are Cðp;jÞ44 ; e ðp;jÞ 15 ; qðp;jÞ 15 ;

j

ðp;jÞ 11 ;

l

ðp;jÞ 11 and k ðp;jÞ 11 .

The admissible potentials in each constituent layer of the multicoated inclusion can be expressed as

U

ðp;jÞ_¼ X 1 n¼1 AUðp;jÞn e nxp_þX 1 n¼1 BUðp;jÞn e nxp_; _AUðp;jÞ n ¼ A Uðp;jÞ n ; ð3:1Þ where AUðp;jÞ n ¼ A Uðp;jÞ nR þ iA Uðp;jÞ nI and B Uðp;jÞ n ¼ B Uðp;jÞ nR þ iB Uðp;jÞ

nI are unknown complex constants to be determined. Note that the po-tential at

l

?0 should be ﬁnite and thus we can set

BUðp;MÞ

n ¼ 0: ð3:2Þ

We consider that the interfaces are perfectly bonded, the potential and the normal component of ﬂux are continuous across the interfaces,

Re

U

ðp;j1Þ lp¼aðjÞp ¼ Re

U

ðp;jÞ lp¼aðjÞp ; RU ðp;j1Þ nðjÞ p _l p¼a ðjÞ p ¼ ðRU_Þðp;jÞ nðjÞ p _l p¼a ðjÞ p : ð3:3Þ

These continuity conditions lead to

aðp;j1Þ nK bðp;j1ÞnK ! ¼ kðp;jÞnK aðp;jÞ nK bðp;jÞnK ! ;

K

¼ R; I; j ¼ 1; 2; . . . ; M; ð3:4Þ where aðp;jÞnK ¼ Awðp;jÞnK AunKðp;jÞ Awðp;jÞnK 0 B B @ 1 C C A; bðp;jÞnK ¼ Bwðp;jÞnK BunKðp;jÞ Bwðp;jÞnK 0 B B @ 1 C C A; ð3:5Þ kðp;jÞnR 2 cosh naðjÞ pI ena ðjÞ pI 2 sinh naðjÞpLðj1Þ ena ðjÞ p _Lðj1Þ 0 @ 1 A 1 2 cosh naðjÞ pI ena ðjÞ p I 2 sinh naðjÞpLðjÞ ena ðjÞ p _LðjÞ 0 @ 1 A; kðp;jÞ nI 2 sinh naðjÞpI ena ðjÞ p_I 2 cosh naðjÞ pLðj1Þ ena ðjÞ pLðj1Þ 0 @ 1 A 1 2 sinh naðjÞp I ena ðjÞ p_I 2 cosh naðjÞ p LðjÞ ena ðjÞ pLðjÞ 0 @ 1 A; ð3:6Þ

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and I is the 3 3 identity matrix. Now, repeated use of(3.4)gives aðp;0Þ nK bðp;0ÞnK ! ¼ Kðj;nÞpK aðp;jÞ nK bðp;jÞnK ! ; j ¼ 1; 2; . . . ; M; ð3:7Þ where Kðj;nÞ pK k ðp;1Þ nK k ðp;2Þ nK k ðp;jÞ nK: ð3:8Þ For j ¼ M, we have aðp;0ÞnK bðp;0ÞnK ! ¼ KðM;nÞpK aðp;MÞnK bðp;MÞnK ! : ð3:9Þ

Further, according to(3.2), (3.9), implies that

aðp;0ÞnK ¼ K ðM;nÞ pK h i 11 K ðM;nÞ pK h i1 21b ðp;0Þ nK : ð3:10Þ

Here ½KðM;nÞ_p_K ₁₁represents the upper-left (3 3) submatrix of KðM;nÞ_p_K and ½KðM;nÞ_p_K ₂₁is the lower-left (3 3) submatrix of KðM;nÞ_p_K . The formulation implies that the effect of the multiple coatings can be incorporated through a recurrence procedure and is solely represented by a (6 6) array alone. We mention that once we construct the admissible ﬁeld(3.1)and the transition relation(3.4)in the inhomogneneous inclusions, we can follow the remaining generalized Rayleigh’s framework proposed in previous section to determine the potential distribution.

4. Effective moduli

We are interested in the effective behavior for a situation where we have a large number of cylinders. The effective mate-rial properties are deﬁned in terms averaged ﬁelds,

hRji LjhZji; no summation; ð4:1Þ

where the angular brackets denote the area averages over the representative volume element (unit cell in the case of peri-odic composites) hRji ¼ 1 V Z V Rjdx; hZji ¼ 1 V Z V Zjdx; ð4:2Þ and L

j denotes the effective magnetoelectroelastic parameters of the composite. Note that since the microstructure is non-symmetric, the responses of the composite along x and y-axes are different:

L x¼ C55 e15 q15 e 15

j

11 k 11 q 15 k11

l

11 0 B B @ 1 C C A; Ly¼ C44 e24 q24 e 24

j

22 k 22 q 24 k22

l

22 0 B B @ 1 C C A: ð4:3Þ

Let the composite be subjected to uniform intensities

e

zx; Ex;and Hxalong the positive x- axis. We can compute the average Zxby noting that each component is a gradient and applying the divergence theorem. We obtain:

hZU xi ¼

C

U

R: ð4:4Þ

Next, to ﬁnd hRU

xi, we again use the divergence theorem and the equilibrium condition (including the interface condi-tions) to obtain h

R

U xi ¼ 1 V Z V

R

U xdx ¼ 1 V Z V

r

ðxRU_{Þdx ¼}1 V Z @V xðRU_Þ m n ds; ð4:5Þ

whereRUis deﬁned in(2.18). We then use the expansions(2.16)for the ﬁelds to obtain

1 V Z @V xðZU Þm n ds ¼

C

U R

p

dBU1R

a

b ; ð4:6Þ where Zw¼ ð

e

zx;

e

zyÞ; Zu¼ ðEx;EyÞ; Zw¼ ðHx;HyÞ: ð4:7Þ Putting(4.5) and (4.6)together, and recalling the constitutive relation(2.2)for the matrix, we obtain

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Table 1

Material parameters of BaTiO3, CoFe2O4(Li & Dunn, 1998a), and Terfenol-D (Liu et al., 2003; Liu et al.,

2004).

Property BaTiO3 CoFe2O4 Terfenol-D

C44(N/m2) 43 109 45.3 109 13.6 109 e15(C/m2) 11.6 0 0 q15(N/Am) 0 550 108.3 j11(C2/Nm2) 11.2 109 0.08 109 0.05 109 l11(Ns2/C2) 5 106 590 106 5.4 106 k11(Ns/VC) 0 0 0

a

b

c

d

e

f

Fig. 3. Potential contours for a square array composite ðf ¼ 0:2; ezx¼ 0; Ex¼ 0; Hx¼ 1C=msÞ (a)–(c) BTO ﬁbers embedded in a CFO matrix, (d)–(f) BTO

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h

r

zxi hDxi hBxi 0 B @ 1 C A ¼ C44 e15 q15 e15

j

11 k11 q15 k11

l

11 0 B @ 1 C A ðmÞ

e

zx pdBw 1R ab Ex pdBu 1R ab Hx pdBw 1R ab 0 B B B B @ 1 C C C C A: ð4:8Þ

Putting together(4.1) and (4.8)and noting that the coefﬁcients BU

1Rdepend linearly on the applied ﬁeldC

U

R;we obtain set of equations for the effective property L

x. We can determine this by applying different loading combinations between

e

zx; Ex and Hx. Similarly we can determine Lyby applying different loading combinations between

e

zy; Eyand Hy.

a

b

c

d

e

f

Fig. 4. Effective moduli of a composite of BTO fibers in a CFO matrix versus fiber volume fractions (a) effective elastic modulus, (b) effective dielectric permittivity, (c) effective magnetic permeability, (d) effective piezoelectric modulus, (e) effective piezomagnetic modulus, (f) effective magnetoelectric coefficient.

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a

b

c

d

e

f

Fig. 5. Effective moduli of a composite of BTO fibers in a CFO matrix versus the aspect ratio and fiber volume fractions (a) effective elastic modulus, (b) effective dielectric permittivity, (c) effective magnetic permeability, (d) effective piezoelectric modulus, (e) effective piezomagnetic modulus, (f) effective magnetoelectric coefficient.

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5. Results and discussion

To have a better understanding for the theoretical results above, we perform a numerical computation for two- and three-phase transversely isotropic piezoelectric–piezomagnetic composites with 6 mm material symmetry about the ﬁber axis. For the piezoelectric material, we consider the widely used BaTiO3(BTO). For the peizomagnetic material, we consider CoFe2O4 (CFO) as well as the Terfenol-D alloy (TD). The cylinders are arranged in a square array. The independent material constants

a

b

c

d

e

f

Fig. 6. Effective moduli of a composite of BTO fibers coated Terfenol-D in a CFO matrix versus fiber volume fractions (a) effective elastic modulus, (b) effective dielectric permittivity, (c) effective magnetic permeability, (d) effective piezoelectric modulus, (e) effective piezomagnetic modulus, (f) effective magnetoelectric coefficient.

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of these constituents are given inTable 1, where the xy plane is isotropic and the ﬁber axis is along the z-direction. Note that in all materials magnetoelectric coefﬁcients are zero, i.e. k11= 0.

We begin with a composite of BTO fibers in a CFO matrix.Fig. 3(a–c) show the contours of displacement, electric potential and magnetic potential with an applied magnetic field. The ratio of the semi-axes lxand lyis 1.2. The magnetic field induces a mechanical stress in the CFO which in turn results in an electric displacement in the BTO fiber. The magnetic field is attracted by the BTO since it has a smaller magnetic permeability. Further,Fig. 3(d–f) show the potential contours for BTO fibers coated TD in a CFO matrix with an applied magnetic field. The ratio of semi-axis lxbetween BTO and TD is 0.7. Since the mag-netic permeability of TD is almost the same as that of BTO, the magmag-netic potential is similar to its homogeneous counterpart. However, in this case the contours of vertical displacement and the electric potential have dramatically difference from those with the homogeneous fiber. The fields in the fiber are now not uniform, and there are field concentrations at the BTO and TD interfaces.

We now turn to effective moduli of the composite.Fig. 4shows the effective elastic, dielectric, magnetic, piezoelectric, piezomagnetic and magnetoelectric moduli for this composite. We assume that the inclusions are circular cylinders, i.e., lx?ly, and the inclusions are in a square array. Therefore there is no distinction between Lxand L

y. The effective moduli vary nonlinearly with volume fraction, and the curve stops at f ¼p

4when the inclusions touch. The magnetoelectric coefficient is non-zero for every non-zero volume fraction, then reaches a maximum before dropping just as the fibers are close to touch-ing. Further,Fig. 4also compares the effective moduli with those predicted byBenveniste (1995)who used the composite cylinder assemblage (CCA) model and by Camacho-Montes et al. (2009)who employed the asymptotic homogenization method. In CCA, there is no upper limit on the volume fraction since one can have fibers with various sizes. Still, the overall magnitudes and trends agree well among our periodic, Benveniste’s CCA, and Camacho-Montes et al.’s homogenization method. In addition, our numerical results fulfil the compatibility conditions given in Eq. (21) of the work byBenveniste (1995).

Fig. 5shows how the effective properties L

xof the composite depend on the ﬁbers’ aspect ratio lx/lyand volume fraction f. The value of the aspect ratio begins at4f

pand terminates at4fpwhen the inclusions touch in y- and x-direction, respectively. It is observed that, for a fixed volume fraction, the effective elastic modulus and magnetoelectric coefficient decrease when the aspect ratio increases. While the effective dielectric permittivity, magnetic permeability, piezoelectric and piezomagnetic moduli increase monotonously. Furthermore, for a fixed aspect ratio, the effective elastic modulus, dielectric permittivity, piezoelectric, and magnetoelectric coefficients increase with increasing volume fraction, while the effective magnetic perme-ability and piezomagetic coefficient decrease when f increases. Interestingly, the trends of the dielectric permittivity and pie-zoelectric constant are similar, and those of the magneitc permeability and piezomagnetic coefficients are similar as well. In

Fig. 6, we show the effective moduli for the BTO ﬁbers coated TD in a CFO matrix. The ratio of the radius of BTO and TD is 0.8.

We ﬁnd that the magnetoelectric coefﬁcient k

11has dramatically enhanced, and the enhancement is increased as the parti-cles touch.

We finally turn to the magnetoelectric voltage coefficient, which is the important figure of merit for magnetic field sen-sors. It relates the overall electric field that is generated in the composite when it is subjected to a magnetic field. It combines the coupling and dielectric coefficients, and is defined by

a

11¼ k 11

j

11 : ð5:1Þ

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Fig. 7shows how this coefficient depends on the fiber volume fraction for the above two- and three-phase circular cylinder cases. Note that there is a qualitative difference between the case of BTO fibers in CFO and BTO fibers coated TD in the CFO matrix. In the former, the maximum coefficient is for intermediate volume fraction of f = 0.35 where

a

11¼ 0:0306 V=cmOe. In contrast, in the case of BTO ﬁbers coated TD in the CFO matrix, the maximum is attained as the ﬁbers near close touching at f = 0.725 where

a

11¼ 0:5732 V=cmOe, which is one order of magnitude enhancement of the coupling coefﬁcient.

6. Concluding remarks

In summary, we have extended Rayleigh’s formalism on periodic conductive composites to a magnetoelectroelastic composite consisting of arbitrarily distributed or periodic arrays of elliptic cylinders under anti-plane shear deformation, in-plane electric ﬁelds and in-plane magnetic intensities. The cylinders can be homogeneous or confocally multicoated. Expressions for the elastic, electric and magnetic potentials for the cylinders and the matrix are derived, and used to compute the effective moduli. This extension is a hybrid technique: the admissible potentials for the matrix and inclusions are ex-panded in complex planes, while the interface conditions are directly satisﬁed by using elliptic coordinates. It is shown that the effective properties solely depend on one particular constant BU

1Ramong the infinite number of expansion coefficients. Finally, as a practical example, explicit numerical calculations for field distributions and the magnetoelectric effects in BTO-CFO and BTO-TD-CFO composites are presented and discussed. This example shows the important difference between the case of BTO fibers in a CFO matrix from the case of BTO fibers coated TD in a CFO matrix. We expect that these results will be beneficial as design tools for functionally graded tunable composites.

Acknowledgments

This work was supported by the National Science Council, Taiwan, under Contract No. NSC 98-2221-E-009-095. References

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