Fibrous composites of piezoelectric and piezomagnetic phases

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Fibrous composites of piezoelectric and piezomagnetic phases

Hsin-Yi Kuo

a

, Kaushik Bhattacharya

b,⇑

a_{Department of Civil Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan} b

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

a r t i c l e

i n f o

Article history: Received 23 March 2010

Received in revised form 18 October 2012 Available online 4 January 2013 Keywords: Magnetoelectricity Fibrous composite Piezoelectric Piezomagnetic

a b s t r a c t

We propose a theoretical framework for evaluation of magnetoelectroelastic potentials in a fibrous composite with piezoelectric and piezomagnetic phases, motivated by the techno-logical desire for materials with large magnetoelectric coupling. We show that the problem with transversely isotropic phases can be decomposed into two independent problems, plane strain with transverse electromagnetic fields and anti-plane shear with in-plane electromagnetic fields. We then consider the second problem in detail, and generalize the classic work of LordRayleigh (1892)to obtain the electrostatic potential in an ordered conductive composite and its extension to a disordered system byKuo and Chen (2008)to the current coupled magnetoelectroelastic problem. We use this method to study BaTiO3– CoFe2O4composites and provide insights into obtaining large effective magnetoelectric coefficient.

1. Introduction

A variety of technological applications including mag-netic field sensors, electrically controlled microwave de-vices and magneto-electric memory cells have motivated the study of magneto-electric coupling in materials and composites (Eerenstein et al., 2006; Nan et al., 2008). The magneto-electric coupling was predicted by Landau and Lifshitz(1984)and observed byAstrov (1960)and by Rado and Folen(1961)over fifty years ago. The coupling is weak in monolithic materials, and this has motivated the study of composites of piezoelectric and piezomagnetic media. The idea is that the applied electric field causes a deforma-tion of the piezoelectric material which in turn induces a deformation in the piezomagnetic material thereby induc-ing a magnetic field.

The performance of a piezomagnetic/piezoeletric com-posite depends on the micro-geometry of the phases since one has to provide effective strain coupling and avoid elec-tromagnetic shielding. This has motivated a number of micromechanical models to predict the effective moduli

of multiferroic composites. For example, Nan(1994)and Huang and Kuo(1997)used the Green’s function method to study a ﬁbrous composite consisting of BaTiO3 and

CoFe2O4. For such transversely isotropic ﬁbrous

compos-ites,Benveniste (1995)derived exact connections among effective magnetoelectroelastic moduli based on a formal-ism discovered byMilgrom and Shtrikman (1989). Particu-late composites were investigated byHarshé et al. (1993)

using a cubic model and byLee et al. (2005)using finite element method. Eshelby’s approach and the mean field Mori–Tanaka model have been generalized to multiferroic composites by Li and Dunn (1998a,b), Huang (1998), Li (2000), Wu and Huang (2000), Huang and Zhou (2004) and Srinivas et al. (2006). Frequency dependence of mag-netoelectric coefficients of multiferroic laminates was studied byBichurin et al. (2003, 2005).Nan et al. (2008)

provide an extensive review of the literature and the state of the art.

However, much of this work uses approximate methods and models based on single inclusions, and focus on the effective properties of composites with somewhat uncon-trolled microstructure. There is a need for exact methods that can be used to evaluate these approximate methods. Further, a method that provides the detailed ﬁelds is useful

⇑Corresponding author.

E-mail address:[email protected](K. Bhattacharya).

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Mechanics of Materials

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to provide insights for developing better microstructures and more complex processes like dielectric breakdown and failure (Li and Duxbury, 1989). Similarly, detailed sta-tistical methods require the fields associated with multiple particles (Fassi-Fehri et al., 1989). Furthermore, recent ad-vances in synthesis allows the fabrication of composites with highly controlled microstructure. For example,Ren et al. (2008)have recently used a diblock copolymer pre-cursors to produce a self-assembled hexagonal array of CFO nanofibers in a PZT matrix. Therefore, there is a need for obtaining the fields and properties of composites with controlled microstructure. All of these motivate the cur-rent work.

In a classic work, LordRayleigh (1892) computed the electric potential for a conducting composite consisting of a periodic array of inclusions (cylinders and spheres). This was extended to arbitrary arrangements byKuo and Chen (2007, 2008). These works concern single fields. In this paper, we generalize this methodology to multiple coupled fields, specifically electrostatic, magnetostatic and mechanical.

We consider a composite medium made of piezoelectric and piezomagnetic phases arranged in a microstructure consisting of parallel cylinders in a matrix in Section2. We follow the work of Chen (1993) (also Benveniste, 1995; Zheng and Chen, 1999 Camacho-Montes et al., 2009) and observe in Section 2.1 that if the phases are transversely isotropic, then the general problem can be decomposed into two independent problems, plane strain with transverse electromagnetic fields and anti-plane shear with in-plane electromagnetic fields. We then focus on the latter problem for much of paper. We notice in Sec-tion2.2that the coupling between the fields occurs only through the interface conditions. We exploit this in Sec-tion2.3to obtain a representation of the solution. The ba-sic idea is to followKuo and Chen (2008)and expand each field in each medium in a series.

While we have developed a direct strategy to solve the coupled problem, we note that there is an alternate strat-egy. Milgrom and Shtrikman (1989) (also, Benveniste, 1997; Chen, 1997; Milgrom, 1997; Milton., 2001) show that it is possible to transform the coupled problem to the uncoupled problem for a two-phase composite. The transformed problem could then be solved by the approach ofKuo and Chen (2007, 2008). However, this transform approach can not be generalized to N-phase composites for N > 2. Furthermore the solution is in the transformed do-main, and therefore it is difficult to develop insights into the field. The direct approach we develop here overcomes these difficulties.

We specialize to periodic arrays in Section3. We obtain effective properties in Section 4, and significantly show that the macroscopic properties depend solely on a single expansion coefficient (amongst the infinite).

This methodology is illustrated in Section5using com-posites of BaTiO3(BTO) and CoFe2O4(CFO). 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 nontrivial magnetoelectric coupling even though the individual components do not. Further, we observe signiﬁcant difference between

com-posites with BTO ﬁbers in a CFO matrix and its complement.

We briefly comment on the first problem – the plane elasticity with transverse electromagnetic fields – in Sec-tion6and show the opportunity for extremely large mag-netoelectric coupling.

2. Arbitrary arrangement of circular cylinders 2.1. General setting

Let us consider an inﬁnite medium R3_{containing N}

arbi-trarily distributed, parallel and separated circular cylinders (Fig. 1). The domain of the pth circular cylinder is denoted Vp;p ¼ 1; 2; . . . ; N, and the remaining matrix is denotedXm.

We assume that the cylinders and the matrix are made of distinct phases.1 _{Further, we assume that each phase is}

ðrÞ_HðrÞ_;

where

r

, D, B,

e

, E and H are the stress, electric displace-ment, magnetic flux, strain, electric field, and the magnetic field respectively. C is the fourth-order tensor of elastic moduli, and

j

,

l

and k are the second order tensors of dielectric permittivity, magnetic permeability and magne-toelectric coefﬁcients. e and q are piezoelectric and piezo-magnetic constants.

Now assume that each phase is transversely isotropic (i.e., has 6 mm symmetry) with the symmetry axes ori-ented with the cylinders. We introduce a Cartesian coordi-nate system with the x- and y-axes in the plane of the cross-section and z-along the axes of the cylinders. In the Voigt notation the properties C, e, q,

j

,

l

, and k are given byNye (1985):

Fig. 1. The cross-section of the ﬁber composite.

1

Later we shall specialize to a two-phase situation where all the cylinders belong to one phase and the matrix to another.

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CðrÞ_¼ C11 C12 C13 0 0 0 C12 C11 C13 0 0 0 C13 C13 C33 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44 0 0 0 0 0 0 C66 0 B B B B B B B B @ 1 C C C C C C C C A ðrÞ ; eðrÞ_¼ 0 0 0 0 e15 0 0 0 0 e15 0 0 e31 e31 e33 0 0 0 0 B @ 1 C A ðrÞ ; qðrÞ_¼ 0 0 0 0 q15 0 0 0 0 q15 0 0 q31 q31 q33 0 0 0 0 B @ 1 C A ðrÞ ; jðrÞ_¼ j11 0 0 0 j11 0 0 0 j33 0 B @ 1 C A ðrÞ ; lðrÞ_¼ l11 0 0 0 l11 0 0 0 l33 0 B @ 1 C A ðrÞ ; kðrÞ_¼ k11 0 0 0 k11 0 0 0 k33 0 B @ 1 C A ðrÞ : ð2Þ Consistent with known material properties, the magneto-electric coupling coefﬁcients kðrÞ _{is negligible though we}

do not explicitly use this fact here.

To obtain the effective properties of this medium, we need to solve for equilibrium equations

r

¼ 0;

r

D ¼ 0;

r

B ¼ 0; ð3Þ

along with the analogous interfacial conditions and appro-priate boundary conditions.

We followChen (1993)(alsoBenveniste, 1995; Zheng and Chen, 1999; Camacho-Montes et al., 2009) and show in the appendix that for this cylindrical geometry and transversely isotropic material symmetry, the problem splits naturally into two independent problems:

Plane strain and transverse electromagnetic ﬁelds

u ¼ uxðx; yÞ uyðx; yÞ 0 0 B @ 1 C A; E ¼ 0 0 Ezðx; yÞ 0 B @ 1 C A; H ¼ 0 0 Hzðx; yÞ 0 B @ 1 C A; ð4Þ

Anti-plane shear and in-plane electromagnetic ﬁelds

u ¼ 0 0 uzðx; yÞ 0 B @ 1 C A; E ¼ Exðx; yÞ Eyðx; yÞ 0 0 B @ 1 C A; H ¼ Hxðx; yÞ Hyðx; yÞ 0 0 B @ 1 C A: ð5Þ

Therefore, it is sufficient to treat each of these problems. In this work, we largely focus on the second, i.e., anti-plane shear with in-plane electromagnetic fields with brief com-ments on the first in Section6.

11 k11 q15 k11

l

11 0 B @ 1 C A

e

zj Ej Hj 0 B @ 1 C A ð7Þ

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

R

j U¼ LUWZjW;

U

;

W

¼ w;

u

;w;j ¼ x; y; ð8Þ 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: ð9Þ

u

l

115 2 w¼ 0; ð12Þ where r2 ¼ @2=@x2_{þ @}2

=@y2 _{represents the}

two-dimen-sional Laplace operator for the variable x and y. Since L is a nonsingular matrix, generically we can decouple (12)

into three independent Laplace equations,

r

2

w ¼ 0;

r

2

u

¼ 0; and

r

2

w¼ 0 ð13Þ

in the interior of each phase. In other words, the three ﬁelds – displacement, electrostatic potential and magneto-static potential – are completely decoupled in the interior of each phase.

In addition to these differential equations, we have to use interface and boundary conditions. We assume that the inter-faces are perfectly bonded, and therefore the ﬁelds satisfy

½½Rjnj

¼ ½½ðLZjÞnj ¼ 0; ½½Zjtj

¼ 0 ð14Þ

where ½½ denotes the jump in some quantity across the interface, n is the unit normal to the interface and t is

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the unit tangent to the interface, and the repeated index j denotes summing over the components x; y. Since L is dif-ferent in the two phases, the ﬁelds w;

u

and w are generally coupled by the interface equations.

2.3. Representation of the solution

In the anti-plane shear problem, we showed above that the fields are decoupled in the interior of every phase, but are coupled at the interfaces. Therefore, we may followKuo and Chen (2008)and use a series expansion for each field in the interior of each phases and then obtain the coeffi-cients by enforcing the interface and boundary conditions. We consider a situation where the composite is sub-jected to a macroscopically uniaxial loading

wext¼

e

zxx;

u

ext¼ Exx; wext¼ Hxx; ð15Þ

for constants

e

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

U

ext¼ ZxUx; ð16Þ

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

u

, or magnetic potential w– and Zx

U the corresponding applied ﬁeld –

e

zx;Ex, or

Hx.

We rewrite the governing equation, Eq.(13) in polar coordinates r; hð Þ, 52

_U

_¼@ 2

U

n ;B

UðpÞ n ; . . . ;F

UðpÞ

n are some unknowns to be determined.

The superscripts p in(18) and (19)indicate that the ﬁelds that are expanded with respect to the pth cylinder center.

We recall the interface conditions (14) which we re-write as

U

ðpÞ m _@V p ¼

U

ðpÞ_i @Vp ; ðRUÞ ðpÞ m np _@V p ¼ Rð UÞ ðpÞ i np _@V p ð20Þ where Rw¼

r

zx;

r

zy;Ru¼ D x;Dy;Rw¼ B x;By; ð21Þ

@Vp:rp¼ apdenotes the interface between the matrix and

the pth circular cylinder, and npis the unit outward normal

of the interface @Vp.

Using the orthogonality properties of trigonometric functions, the conditions(20)provide

aðpÞ n ¼ a2np T ðpÞ_bðpÞ n ;cðpÞn ¼ a2np T ðpÞ þ I bðpÞn ; ð22Þ dðpÞn ¼ a 2n p T ðpÞ_eðpÞ n ;f ðpÞ n ¼ a 2n p T ðpÞ_{þ I} eðpÞ n ; ð23Þ and AUðpÞ 0 ¼ C UðpÞ 0 , where aðpÞ n ¼ AwðpÞn AuðpÞ n AwðpÞ n 0 B B @ 1 C C A; bðpÞn ¼ BwðpÞ n BuðpÞ n BwðpÞn 0 B B @ 1 C C A; cðpÞn ¼ CwðpÞn CuðpÞ n CwðpÞn 0 B B @ 1 C C A; ð24Þ dðpÞn ¼ DwðpÞn DuðpÞ n DwðpÞ n 0 B B @ 1 C C A; eðpÞn ¼ EwðpÞn EuðpÞ n EwðpÞ n 0 B B @ 1 C C A; fðpÞn ¼ FwðpÞn FuðpÞ n FwðpÞ n 0 B B @ 1 C C A; ð25Þ TðpÞ ¼ L ð Þm LðpÞ1 Lð Þm þ LðpÞ ; ð26Þ

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 identity (Arfken and Weber, 2001) to the matrix domainXm. This gives

0G x; xð 0Þ n0_ds0 ; ð27Þ

where the prime 0 denotes the operation in reference to the x0_{coordinate, n}0_{is the outward unit normal to the matrix’s}

boundary @Xm;dA0 represents the area element for the x0

coordinate, ds0 is the differential arc length. Here G x; x_ð 0_Þ

point z satisfying the inequality z zp

< zp zl

, we can then expand the term z zð lÞm using the binomial

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z zl ð Þm¼X 1 s¼0 1 ð Þs m þ s 1 s _{z z} p s zp zl mþs: ð30Þ

Introducing(30)into(29), we have the expansion

U

ðpÞ m;nearð Þ ¼ Zx x URezpþ ZxURe z z p þX 1 m¼1 BUðpÞ m Re z zp m EUðpÞm Im z zp m h i þX l–p X1 s¼0 X1 m¼1 1 ð Þs m þ s 1 s BU lð Þ m Re z zp s zp zl mþs E U lð Þ m Im z zp s zp zl mþs " # ð31Þ

valid for the domain

z zp < min zp zl ; for l ¼ 1; 2; . . . ; N; p – l: ð32Þ

Since the expansion(31)are valid for points satisfying the condition(32), which generally applies to points near the pth inclusion, this expansion will be referred to as a near-ﬁeld expansion, denoted asUðpÞm;nearð Þ. Further, sincex

x lies in the matrix domain, Eqs.(31) and (19)should be identical. This provides the condition

AUðpÞ 0 þ X1 n¼1 AUðpÞ n Re z zp n þ DUðpÞn Im z zp n h i ¼ Zx URezpþ ZxURe z zp þX l–p X1 s¼0 X1 m¼1 1 ð Þs m þ s 1 s BU lð Þ m Re z zp s zp zl mþs E U lð Þ m Im z zp s zp zl mþs " # : ð33Þ

Taking the real part and the imaginary part of(33), we ﬁnd the two conditions

AUðpÞn X l–p X1 m¼1 1 ð Þn m þ n 1 n BU lmð ÞRe zp zl mn EU lmð ÞIm zp zl mn h i ¼ ZxURezpdn;0þ ZxUdn;1; ð34Þ and DUðpÞ n þ X l–p X1 m¼1 1 ð Þn m þ n 1 n BU lð Þ m Im zp zl mn þ EU lð Þ m Re zp zl mn h i ¼ 0: ð35Þ

Eqs.(34), (35), (22)1, and(23)1constitute an inﬁnite set of

linear algebraic equations. Upon appropriate truncations of the expansion terms, we can determine the expansion coefﬁcients AUnðpÞ;BUnðpÞ; . . . ;FUnðpÞ.

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. Rel-evant works in magnetoelectric composites with periodic conﬁgurations include, for example,Aboudi (2001), who developed a homogenization method for the prediction of the effective properties of magneto-electro-thermo-elastic

composites and the results are in good agreement with those of the Mori–Tanaka model.Lee et al. (2005)proposed a ﬁnite element analysis based micromechanics approach to determine the effective material properties of three-phase electromagnetoelastic composites. A variational asymptotic method has been used byTang and Yu (2008, 2009)to construct a fully coupled micromechanics model for prediction of effective behaviours and local ﬁelds of smart composites.Camacho-Montes et al. (2009)adopted a two-scale asypmtotic homogenization method to deter-mine the overall behaviour of magnetoelectric coupling and cross-property connections in a square array of a bin-ary composite.

There are five possible ways of packing cylinders in reg-ular arrays in two dimensions (see Kittel, 1986, for in-stance). Here we concentrate on the two lattices, rectangular and hexagonal. It is known that in the case of elasticity, a hexagonal arrangement of circular fibers re-sults in effective transverse isotropy; on the other hand a square arrangement results in general in square symmetry (Li., 2000). It turns out however that in the case of conduc-tion square symmetry and transverse isotropy become identical, and both kinds of arrangements give therefore rise to the same overall symmetry (Perrins et al., 1979). This statement is also correct for piezomagnetoelectricity under anti-plane shear with in-plane electromagnetic fields which is the case of our study. We sketch the outline of the derivation focussing on the differences from the pre-vious situation. Finally, we limit ourselves to the case of anti-plane shear with in-plane electromagnetic fields.

Let us ﬁrst introduce a Cartesian coordinate system x; y

ð Þ positioned at the center O of one of the cylinders in a square or a hexagonal array, as shown inFig. 2. The ra-dius of the cylinders is a and we may assume unit distance between the centers of neighboring cylinders without loss of generality. Uniform intensities Ex and Hx are applied

along the positive x axis, and an anti-plane shear deforma-tion

e

zx is applied out of the xy plane. In terms of polar

coordinates, the general solution has the admissible form

U

i¼ CU0þ X1 n¼1 CUnr n_{cos nh} _ð36Þ for r < a, and

U

m¼ AU0þ X1 n¼1 AU nrnþ B U nrn cos nh ð37Þ

for r > a. The coefﬁcients AU_n;BU

n, and C

U

nare unknown

con-stants to be determined from the interface and boundary conditions. Note that the sine terms that would be present in a general expansion are missing since we impose a uni-axial loading along the x direction. Further,Uðr; hÞ has to be antisymmetric with respect to the y axis, and thus only terms with an odd number are included. Finally, for a hexagonal lattice, all terms in which n is a multiple of three are disallowed (Perrins et al., 1979).

Analogous to(22), the continuity conditions at the inter-face will give constraints(22)between the coefﬁcients.

Next, imposing the periodicity conditions analogous to imposing the boundary condition we did to derive(34), leads to a generalized Rayleigh’s identity

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AUnþ X1 m¼1 m þ n 1 n SmþnBUm¼ Z x Udn;1: ð38Þ

Here the quantities

Sm¼

X

l–o

Rezm

l ð39Þ

are the lattice sums characterizing the geometry of the periodic structure, and zl is the center of the lth cylinder

when measured at the central point O. The index l runs over all cylinders’ centers underlying the periodic array ex-cept the central one. A list of non-zero normalized lattice sums for square and hexagonal arrays can be found in Ber-man and Greengard (1994).

Eqs. (38) and (22)1 constitute an inﬁnite set of linear

algebraic equations. Upon appropriate truncations of the expansion terms, we can determine the expansion coefﬁ-cients AU n, B U n, and C U n. 4. Effective moduli

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

Rj

LD EZj ; ð40Þ

where the angular brackets denote the average over the representative volume element (unit cell in the case of periodic composites) Rj ¼1 V Z V Rjdx; D EZj ¼1 V Z V Zjdx; ð41Þ

and L_{denotes the effective magnetoelectroelastic}

param-eters of the composite.

We can compute the average Zj _{by noting that each}

component is a gradient and applying the divergence the-orem. We obtain ZxU ¼ Zx U: ð42Þ Next, to ﬁnd RxU

, we again use the divergence theorem and the equilibrium condition (including the interface con-ditions) to obtain:

R

x U ¼1 V Z V

R

x Udx ¼ 1 V Z V

r

xRð UÞdx ¼1 V Z @V x Rð UÞm nds; ð43Þ

where RUis deﬁned in(21). We then use the expansions

(19)for the ﬁelds to obtain

1 V Z @V x Zð UÞm nds ¼ Z x U 2 XN l¼1 a2 l flBU l1ð Þ; ð44Þ where Zw¼

e

zx;

e

zy;Zu¼ E x;Ey;Zw¼ H x;Hy; ð45Þ

and flis the volume fraction of phase l deﬁned as fl¼

p

a2l=V

for square arrays and is2p_ﬃﬃ 3 p a2

l=V for hexagonal arrays.

Putt-ing(43) and (44)together, and recalling the constitutive relation(7)for the matrix, we obtain

r

zx h i Dx h i Bx h i 0 B @ 1 C A ¼ C44 e15 q15 e15

j

11 k11 q15 k11

l

11 0 B @ 1 C A m ð Þ

e

zx 2 XN l¼1 a2 l flBw l1ð Þ Ex 2 XN l¼1 a2 l flBu1ð Þl Hx 2 XN l¼1 a2 l flBw1ð Þl 0 B B B B B B B B B B @ 1 C C C C C C C C C C A : ð46Þ Putting together(40) and (46)and noting that the coefﬁ-cients BU

1depend linearly on the applied ﬁeld ZxU, we obtain

set of equations for the effective property L. We can deter-mine this by applying different loading combinations be-tween

e

zx;Exand Hx.

5. Numerical results and discussion

In order to have a better understanding for the theoret-ical results above, we perform a numertheoret-ical computation for a two-phase transversely isotropic piezoelectric-piezo-magnetic composite with 6mm material symmetry about the ﬁber axis. Speciﬁcally we consider a composite of BaTiO3 and CoFe2O4 which has been studied by other

researchers. We consider square and hexagonal arrays,

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and both cases, i.e., both BTO fibers in a CFO matrix and CFO fibers in a BTO matrix. The independent material con-stants of these constituents are given inTable 1, where the xy plane is isotropic and the unique axis is along the z direction. Note that in both materials magnetoelectric coefficients are zero, i.e. k11¼ 0.

We begin with a composite of BTO ﬁbers in a CFO ma-trix.Fig. 3shows the effective elastic, dielectric, magnetic, piezoelectric, piezomagnetic and magnetoelectric moduli for this composite. They vary nonlinearly with volume fraction, and the curves stop at f ¼

p

=4 and f ¼

p

=2pffiffiffi3 for the square and hexagonal arrays respectively, when the inclusions touch. The magnetoelectric coefficient is non-zero for every (non-zero) volume fraction even though this coefficient is zero for each component. This reflects the coupling of the various fields across the boundary. Further, it initially increases with increasing volume fraction, then reaches a maximum before dropping just as the fibers are close to touching. To gain insight into this behavior, we plot the the contours of displacement, electric potential and magnetic potential for a square array inFig. 5(a)–(c) with an applied magnetic field. The magnetic field induces a mechanical stress in the CFO which in turn results in an electric displacement in the BTO fiber. The effective electric displacement in the horizontal direction depends on the average along the vertical direction. Thus, the effective electric displacement depends on the span of the fiber in the vertical direction. This is why the ME coefficient starts at zero and increases with volume fraction. The magnetic field is attracted by the BTO (since it has a smaller mag-netic permeability) and thus the scaling deviates from being proportional to the span and is close to linear ini-tially. Further, as the particles come close to touching, there is very little CFO to induce stress and thus the ME coefficient drops dramatically.

Finally,Fig. 3also compares the effective moduli with those predicted byBenveniste (1995)who used the com-posite cylinder assemblage (CCA) model. 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 between our periodic and his CCA. Fur-thermore, the results of our analysis fulﬁl the compatibility conditions given in Eq.(21)ofBenveniste (1995).

We now turn to the composite of CFO fibers in a BTO matrix.Fig. 4shows the effective moduli for this compos-ite. Again, magnetoelectric coefficient is non-zero for every (non-zero) volume fraction even though this coefficient is zero for each component. However, in this case it is mono-tone increasing with a sharp rise as the particles are close

to touching. Further, we verify our results with the com-patibility relation (21) proposed in Benveniste (1995). Again, they are in good agreement. Fig. 5(d)–(f) show the potential contours for an applied magnetic field. Now, the magnetic field is expelled by the fibers giving rise to a displacement which deforms the matrix to in-duce an electric displacement. The amount of deformation it can cause in the matrix increases with volume fraction, and this is reflected in the magnetoelectric coef-fecient. Further, it increases dramatically as the particles touch.

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

a

E11¼ k

11=

j

11: ð47Þ

Fig. 6shows how this coefficient depends on the vol-ume fraction for the various cases. Note that there is a qualitative difference between the case of BTO fibers in CFO and its complement. In the former, the maximum coefficient is for an intermediate volume fraction of f ¼ 0:35 where

a

E11¼ 0:0306 V/cm Oe independent of the

square or hexagonal geometry. In contrast, in the case of CFO fibers in the BTO matrix, the maximum is attained as the fibers begin to touch. These trends are similar to those of the magnetoelectric coefficient described before, and follow from the same reasons.

6. Plane strain with transverse electromagnetic fields We briefly discuss the other problem described in(4), and the potential for using it for large effective magneto-electric coefficient. Consider a situation where the average normal stress as well as the average electric displacement along the fibers are zero

r

zz

h i ¼ 0; hDzi ¼ 0: ð48Þ

The constitutive Eqs.(2)specialized to the current setting (see(A.1) and (A.2)) then implies that

e

zz between the two equations above, we

yy he33i qh 33i C33 h i þ kh 33i Hz: ð52Þ Table 1

Material parameters of BaTiO3and CoFe2O4(Li and Dunn, 1998b).

Property BaTiO3 CoFe2O4

C44(N/m2) 43 109 _{45:3 10}9 e15(C/m2) 11.6 0 q15(N/A m) 0 550 j11(C2/N m2) 11:2 109 _{0:08 10}9 l11(N s 2 /C2 ) 5 106 590 106 k11(N s/VC) 0 0

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Fig. 3. Effective moduli of a composite of BTO fibers in a CFO matrix versus BTO fiber volume fractions: (a) effective elastic modulus, (b) effective dielectric permittivity, (c) effective magnetic permeability, (d) effective piezoelectric modulus, (e) effective piezomagnetic modulus, and (f) effective magnetoelectric coefficient.

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0 0.2 0.4 0.6 0.8 1 4.3e+010 4.4e+010 4.5e+010 4.6e+010 4.7e+010 Square array Hexagonal array Derived from CCA

CoFe2O4 volume fraction

C44 * (N/m 2) 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 Square array Hexagonal array Derived from CCA

CoFe2O4 volume fraction e15 * (C/m 2) 0 0.2 0.4 0.6 0.8 1 0 2e-009 4e-009 6e-009 8e-009 1e-008 1.2e-008 Square array Hexagonal array Derived from CCA

κ11 * (C 2/Nm 2) 0 0.2 0.4 0.6 0.8 1 -1200 -800 -400 0 400 800 Square array Hexagonal array Derived from CCA

q15 * (N/Am) 0 0.2 0.4 0.6 0.8 1 -0.0008 -0.0004 0 0.0004 0.0008 0.0012 Square array Hexagonal array Derived from CCA

μ11 * (Ns 2 /C 2 ) 0 0.2 0.4 0.6 0.8 1 -5e-012 0 5e-012 1e-011 1.5e-011 2e-011 2.5e-011 Square array Hexagonal array Derived from CCA

-λ11

* (Ns/VC)

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 4. Effective moduli of a composite of CFO fibers in a BTO matrix versus CFO 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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We now assume further that the average planar strain is zero (alternately, we can proceed exactly the same if the effective planar stress is zero). Then, strain depends line-arly on Ezand Hz, and thus, we can write

33i : ð55Þ

In particular, we concentrate on the denominator. Notice that only A depends on the microgeometry and the planar moduli where as the rest of the terms do not. Thus, it is possible to tune the microgeometry to reduce the denom-inator to get extremely large coupling.

We may use the methodology described in this paper to compute A and B. However, in contrast to anti-plane shear, plane strain elasticity is a vectorial problem and thus the method is signiﬁcantly more difﬁcult to implement. This is the topic of current work and will be presented elsewhere.

7. Concluding remarks

In summary, we have extended Rayleigh’s formulation on periodic conductive composites to a magnetoelectro-elastic composite consisting of arbitrarily distributed or periodic arrays of cylinders under anti-plane shear defor-mation, in-plane electric and in-plane magnetic intensities. Expressions for the elastic, electric and magnetic potentials for the cylinders and the matrix are derived, and used to compute the effective moduli. It is shown that the effective properties solely depend on one particular constant BU1

among the infinite number of expansion coefficients. Final-ly, as a practical example, explicit numerical calculations for field distributions and the magnetoelectric effects in BaTiO3–CoFe2O4composites are presented and discussed.

This example shows the important difference between the case of BTO ﬁbers in a CFO matrix from its

comple-ment. The present theoretical framework provides a general guideline for the selection of the best combination with an efﬁcient coupling of piezoelectric and piezomag-netic properties. It can also provide a rigorous basis against which several approximate micromechanical models can be compared.

Acknowledgment

11Ex

j

11Ey

j

33Ez 2 6 4 3 7 5 þ k11Hx k11Hy k33Hz 2 6 4 3 7 5; ðA:2Þ 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 Square array Hexagonal array

BaTiO3 volume fraction -αE1 1 * (V/cmOe) 0 0.2 0.4 0.6 0.8 1 0 0.005 0.01 0.015 0.02 0.025 Square array Hexagonal array

CoFe2O4 volume fraction -αE1

1

* (V/cmOe)

(a)

(b)

Fig. 6. Effective magnetoelectric voltage coefficient of the composite versus the fiber volume fraction: (a) BTO fibers in a CFO matrix, (b) CFO fibers in a BTO matrix.

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33Hz 2 6 4 3 7 5: ðA:3Þ

xy;xþ

r

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