Closed-form solutions to the effective properties of fibrous magnetoelectric composites and their applications

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Closed-form solutions to the effective properties of ﬁbrous magnetoelectric

composites and their applications

L.P. Liu

a,b

_{, H.-Y. Kuo}

c,⇑

a

Department of Mechanical and Aerospace Engineering, Rutgers University, NJ 08854, USA b

Department of Mathematics, Rutgers University, NJ 08854, USA c

Department of Civil Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 6 October 2011

Received in revised form 8 May 2012 Available online 6 July 2012 Keywords:

Magnetoelectricity Periodic composites E-inclusion

Generalized anti-plane shear deformation Optimal design

a b s t r a c t

Magnetoelectric coupling is of interest for a variety of applications, but is weak in natural materials. Strain-coupled fibrous composites of piezoelectric and piezomagnetic materials are an attractive way of obtaining enhanced effective magnetoelectricity. This paper studies the effective magnetoelectric behaviors of two-phase multiferroic composites with periodic array of inhomogeneities. For a class of microstructures called periodic E-inclusions, we obtain a rigorous closed-form formula of the effective magnetoelectric coupling coefficient in terms of the shape matrix and volume fraction of the periodic E-inclusion. Based on the closed-form formula, we find the optimal volume fractions of the fiber phase for maximum magnetoelectric coupling and correlate the maximum magnetoelectric coupling with the material properties of the constituent phases. Based on these results, useful design principles are pro-posed for engineering magnetoelectric composites.

1. Introduction

Magnetoelectricity (ME) refers to the magnetization induced by an electric field, or conversely the polarization induced by a magnetic field. The ME effect has many important technological applications, ranging from large-area sensitive detection of mag-netic fields (Fiebig, 2005), magnetoelectric memory cells (Kumar et al., 2009), and to electrically controlled microwave phase shifters

(Bichurin et al., 2002). However, the ME coupling coefﬁcient is

barely noticeable for most single-phase materials in spite of recent discovery of gigantic magnetoelectric effects in TbMnO3 at

cryogenic temperature (Kimura et al., 2003). Therefore various researchers have turned to composites or nano-structured materi-als (Zheng et al., 2004; Fennie, 2008), as explained in recent reviews ofEerenstein et al. (2006) and Nan et al. (2008). The ‘‘product prop-erty’’ causes the ME effect in composites of piezoelectric (PE) and piezomagnetic (PM) materials: an applied electric ﬁeld generates a strain in the piezoelectric material which in turn induces a strain in the piezomagnetic material, resulting in a magnetization.

The promise of applications, and the indirect coupling through strain have also made ME composites the topic of a number of the-oretical and experimental investigations (Nan et al., 2008; Zheng et al., 2004). The estimates of the effective properties of ME com-posites of non-dilute volume fractions are usually obtained by

mean-field-type models (Nan, 1994; Srinivas and Li, 2005). Exact relations in a ME composite with cylindrical geometry were de-rived byBenveniste (1995). The analysis for local fields is available for simple microstructures such as a single ellipsoidal inclusion (Huang and Kuo, 1997; Li and Dunn, 1998a), periodic array of cir-cular/elliptic fibrous ME composites (Kuo, 2011; Kuo and Pan, 2011) and laminates (Kuo et al., 2010), etc. Numerical methods based on the finite element method have also been developed to address ME composites for general microstructures (Liu et al.,

2004; Lee et al., 2005), while homogenization methods were

pro-posed byAboudi (2001) and Camacho-Montes et al. (2009). In this paper we consider two-phase composites of piezoelectric (PE) materials and piezomagnetic (PM) materials and seek closed-form predictions of their effective properties by generalizing the uniformity property of ellipsoids to other geometries, namely, peri-odic E-inclusions. In the classic work ofEshelby (1957, 1961), he discovered that any uniform eigenstress on an ellipsoidal inclusion induces uniform strain on the inclusion in an inﬁnite homogeneous medium. This remarkable uniformity property of ellipsoids allows for rigorous closed-form solutions to inhomogeneous problems by the so-called equivalent inclusion method, which has been used to develop many important materials models concerning compos-ites, phase transformation, dislocations and cracks, etc. (Mura, 1987). However, since two or more ellipsoids do not enjoy the uni-formity property, analysis based on Eshelby’s solution and the equivalent inclusion method cannot account for the interactions between inhomogeneities, e.g., composites with non-dilute

http://dx.doi.org/10.1016/j.ijsolstr.2012.06.007

⇑ Corresponding author.

E-mail address:hykuo@mail.nctu.edu.tw(H.-Y. Kuo).

Contents lists available atSciVerse ScienceDirect

International Journal of Solids and Structures

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inhomogeneities. To overcome this limitation, mean-ﬁeld-type Mori–Tanaka models have been developed to address multiferroic composites byLi and Dunn (1998a,b), Huang (1998), Li (2000), Wu and Huang (2000) and Srinivas et al. (2006). In addition, a phase-ﬁeld method based on a generalized Eshelby’s equivalency princi-ple is proposed for arbitrary microstructures (Ni et al., 2010).

Following the work ofEshelby (1957), Liu et al. (2007, 2008)

have recently found a periodic generalization of ellipsoids called periodic E-inclusions (also called Vigdergauz microstructures in two dimensions). Periodic E-inclusions share partially the unifor-mity property of ellipsoids: a uniform dilatational eigenstress on the periodic E-inclusions induces uniform strain on the periodic E-inclusions for isotropic materials. Since it is not the ellipsoid per se but its uniformity property that is being used in the classic analysis based on Eshelby’s solution, we extend the argument of equivalent inclusion method for ellipsoidal inclusions to periodic E-inclusions and achieve explicit closed-form solutions to the effective properties of the composites and local fields. This strategy has been used to predict the effective properties of conductive composites (Liu, 2009) and elastic composites (Liu et al., 2008). Here we present the detailed calculations for composites of PE and PM materials. Aiming to improve the magnetoelectric coupling of the composite, we further study how the effective ME voltage coefficient, the figure of merit of ME materials, depends on the vol-ume fraction, the topology of microstructures and the material properties of constituent phases. In particular, we find the optimal volume fraction of the fiber phase for maximum effective voltage coefficient and draw a few useful design principles, which are sum-marized in Section5.

The paper is organized as follows. In Section2we formulate the governing equation for a periodic piezoelectric–piezomagnetic composite and define the effective properties of the composite. In Section 3 we introduce the periodic E-inclusion and derive the closed-form formula of the effective properties of a composite with a periodic E-inclusion microstructure. In Section4we study how the magnetoelectric voltage coefficient depends on volume fractions of the fiber phase and material properties of constituent phases. Finally we summarize a few useful design principles in Section5.

2. Problem statement

We consider a composite consisting of a periodic array of paral-lel and separated prismatic cylinders as sketched inFig. 1. The cyl-inders and the matrix are made of distinct phases: transversely isotropic piezoelectric or piezomagnetic materials. A Cartesian coordinate system is introduced with the xy-axes in the plane of the cross-section and z-axis along the axes of the cylinders. Let Y be a unit cell in the xy-plane andX Y denote the cross-section of the cylinder in this unit cell.

Assume that the composite be subjected to anti-plane shear strains

e

zx;

e

zy, in-plane electric ﬁelds Ex; Ey and magnetic ﬁelds

Hx; Hyat inﬁnity. It can be shown that the composite is in a state

of generalized anti-plane shear deformation and can be described by (Benveniste, 1995)

r

zj; Dj; Bj;

e

zj; Ej, and Hj(j ¼ x; y) are the stress,

electric displacement, magnetic flux, strain, electric field, and the magnetic field, respectively. The materials constants C44;

j

11;

l

11

and k11 are the elastic modulus, dielectric permittivity, magnetic

permeability and ME coefﬁcient, while e15and q15are the

jYj Z

Y

F L xð Þð

r

u þ FÞ dx: ð7Þ

3. The closed-form solutions

A closed-form analytical solution to Eq.(5)is not anticipated for general microstructure. Nevertheless, for periodic E-inclusions we solve Eq. (5) by the well-known Eshelby equivalent inclusion

Fig. 1. Conﬁguration of the ﬁbrous composite: (a) the overall composite and (b) a unit cell in the xy-plane withXbeing one phase and Y nXbeing the other phase.

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method. Below we ﬁrst present a brief description of periodic E-inclusions and then solution to Eq.(5).

3.1. Periodic E-inclusions: existence and property

Motivated by the broad applications of Eshelby’s solutions in a variety of materials models,Liu et al. (2007, 2008)generalized the geometric shape of ellipsoids according to their uniformity property in the context of Newtonian potential problem, i.e., the Newtonian potential / : Rn

! R (n P 2 is the dimension of space) induced by a homogeneous ellipsoidXsatisﬁes the overdetermined problem:

r

2/¼

v

X on R n_;

rr

/¼ Q on

X

; j

r

/j ! 0 as jxj ! þ1; 8 > < > : ð8Þ

where

v

Xis the characteristic function ofX, equal to one onXand

vanishing otherwise, and Q is a nonnegative symmetric n n ma-trix with TrðQ Þ ¼ 1. In analogy with Eq.(8), a periodic E-inclusion in a unit cell Y Rn_{is deﬁned as a domain}_X_{such that the solution}

to the potential problem (Liu et al., 2008)

r

2

/¼ f

v

X on Y;

periodic boundary conditions on @Y; (

ð9Þ

satisﬁes the overdetermined condition

rr

/¼ ð1 f ÞQ on

X

; ð10Þ

where f ¼ jXj=jYj is the volume fraction of the inclusion. The termi-nology ‘‘E-inclusion’’ arises from the associations with ‘‘Eshelby’’, ‘‘Ellipsoid’’ and ‘‘Extremal’’ properties of such geometries.

The overdetermined condition (Eq.(10)) places strong restric-tions on the domainX. The existence of periodic E-inclusions can be established by considering a simple variational inequality ( Fried-man, 1982): I½/ ¼ min u2U I½

u

:¼ Z Y 1 2j

ru

j 2 þ f

u

dx ; ð11Þ

v

_YnX_C

r

2

n

v

XC on Y;

rr

/¼

rr

n on

X

C;

periodic boundary conditions on @Y: 8

> < >

: ð12Þ

If, in particular, one chooses a quadratic obstacle n¼ 1f₂ ðx c0Þ Q ðx c0Þ with c0being the center of the unit cell

Y, comparing Eq.(12)with Eq.(9) and (10)one concludes that the coincident set XC is precisely a periodic E-inclusion, i.e.XC¼X.

The interested reader is referred toLiu et al. (2008)for details of the above existence proof.

Geometrically, the shape of a periodic E-inclusion in Rn_is

pre-scribed by the scalar volume fraction f, the symmetric shape matrix Q 2 Rnn_{and the unit cell Y associated to the periodicity. In the}

di-lute limit the shape matrix Q coincides with the demagnetization matrix of an ellipsoid in the study of ferromagnetics and is deter-mined by the aspect ratios and orientations of the ellipsoid. In two dimensions, explicit parameterizations of periodic E-inclu-sions are available for a rectangular unit cell (Vigdergauz, 1988;

Grabovsky and Kohn, 1995; Liu et al., 2007) and examples of

peri-odic E-inclusions in the unit cell ½0; 1:5 ½0; 1 are shown inFig. 2

for isotropic shape matrix Q ¼ I=2 and volume fractions from 0:1 to 0:7. FromFig. 2we see that a two-dimensional periodic E-inclu-sion of isotropic shape matrix is roughly a circle at a low volume fraction, say, 0:1, and a rounded rectangle of roughly the same aspect ratio as the unit cell at a high volume fraction, say, 0:7. For more general unit cells and in three dimensions, periodic E-inclusions can be constructed by solving the above variational inequality(11)and numerical calculations show similar qualitative dependence of the shape on the volume fraction (Liu et al., 2007, 2008).

3.2. Applications to magnetoelectric composites

We now solve Eq.(5)by the equivalent inclusion method for periodic E-inclusions. To this end, we ﬁrst consider the associated homogeneous inclusion problem

r

LðmÞ

r

u þ R

_v

X

h i

¼ 0 on Y; periodic boundary conditions on @Y; (

ð13Þ

where R_{2 R}32_{is the ‘‘eigenstress’’. We remark that the physical}

interpretations of Eqs. (5) and (13)are different from the classic Eshelby inclusion problem in elasticity, though their forms appear to be the same. Further, the applied periodic boundary conditions in Eqs. (5) and (13) take into account the interactions between the inclusions which are neglected or phenomenologically ac-counted for by the analysis based on Eshelby’s solution.

The solution to Eq.(13) is closely related with the following simple potential problem (Eq.(9)). To see this, by Fourier transfor-mations of Eq.(9)we ﬁnd that

rr

/ðxÞ ¼ X k2Knf0g k k jkj2 ^

v

XðkÞ expðik xÞ

calculations reveal that

Fig. 2. Periodic E-inclusions (Vigdergauz structures) with unit cell ½0; 1:5 ½0; 1 and isotropic shape matrix Q ¼ I=2. From inward to outward, the volume fraction of the inclusion increases from 0:1 to 0:7.

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LðmÞpiqjkikj¼ AðmÞpqjkj 2 ; NðkÞ ¼ 1 jkj2ðA ðmÞ Þ1: ð16Þ

Comparing Eq.(14)with Eq.(15), we conclude that

r

u ¼ ðAðmÞÞ1R

rr

/ on Y: ð17Þ

We emphasize that the above relation between the solution to the system of equations Eq.(13)and the scalar potential problem (Eq.

(9)) holds for any inclusionX.

Further, we assume the inclusionXis a periodic E-inclusion with shape matrix Q and volume fraction f. From the deﬁnition of periodic E-inclusions discussed above, the solution to Eq.(9)

for a periodic E-inclusion satisﬁes the overdetermined condition (Eq.(10)). By Eqs.(17) and (10), we conclude that the periodic E-inclusion has the Eshelby uniformity property for the homogeneous periodic problem(13)in the sense that the ﬁeldru is uniform in-side the inclusionX, and is given by

r

u ¼ ð1 f ÞRR _on

_X

_; _ð18Þ

where the components of the tensor R : R32

! R32are given by

Rpiqj¼ AðmÞ

1

pqQij ðp; q ¼ 1; 2; 3; i; j ¼ 1; 2Þ: ð19Þ

Here the reader is cautioned thatru being uniform on periodic E-inclusions for any applied ‘‘eigenstress’’ R_{depends on a property of}

tensor LðmÞ_{, i.e., the matrix NðkÞ is independent of k upon being}

mul-tiplied by a scalar factor jkj2. From this viewpoint, periodic E-inclu-sions does not enjoy the full uniformity property as ellipsoids, as shown inLiu (2010)by the complex variable method.

We now consider the inhomogeneous problem(5). Following the equivalent inclusion method we claim that the solution to Eq.(5)is identical to that of Eq.(13)if the average applied ﬁeld F for Eq.(5)and the ‘‘eigenstress’’ R_{for Eq.}₍₁₃₎_{are related by}

D

LF ¼ ð1 f Þ

D

LRR_R_{¼ ½ð1 f Þ}

D

LR IIR_; _ð20Þ

whereDL ¼ LðmÞ

Lð Þi, and II : R32_{! R}32_{is the identity mapping.}

To see this, we first notice that a solution to Eq.(13)with uniform field insideX(cf. Eq.(18)) satisfies Eq.(5)inside the matrix Y nX

since they are the same equations, and inside the inclusionXsince

ru is uniform onX. Further, on the interface @Xwe ﬁnd that Eq.(5)

requires the interfacial conditions

½LðiÞð

r

uðxÞ þ FÞ LðmÞ

ð

r

uðxþÞ þ FÞn ¼ 0 on @

X

; ð21Þ

where n is the outward normal on @X, and x (xþ) denotes the boundary value approached from inside (outside)X. Similarly, Eq.

(13)implies the interfacial conditions

½LðmÞ

r

uðxÞ þ R_LðmÞ

r

uðxþÞn ¼ 0 on @

X

: ð22Þ

A brief and straightforward algebraic calculation shows that if Eq.

(22)is satisfied andruðxÞ is given by Eq.(18), then Eq.(21)is sat-isfied as well for any average applied field F satisfying Eq.(20). We henceforth conclude that the solution to the homogeneous problem

(13)is indeed a solution to the inhomogeneous problem(5)if the uniformity property (Eq. (18)) holds and the algebraic relation (Eq.(20)) is satisﬁed.

To calculate the effective tensor of the composite, by Eqs.(7) and (18)we ﬁnd that the effective tensor Le_satisﬁes

F LeF ¼ 1 Y j j Z Y F ðLðmÞ

D

L

v

XÞð

r

u þ FÞdx ¼ F LðmÞ_{F f F}

D

L½ð1 f ÞRR_{þ F:}

By Eq.(20)we rewrite the above equation as

F Le_{F ¼ F L}ðmÞ_{F þ f F R}_:

Further, it can be shown that the tensor ð1 f ÞDLR II is invertible for generic cases and the above equation implies

Le¼ LðmÞþ f ½ð1 f Þ

D

LR II1

D

L; ð23Þ

which is our closed-form formula of the effective properties for two-phase composites of PM and PE materials.

A few remarks are in order regarding Eq.(23). First, it is a rigor-ous closed-form prediction to the effective properties of periodic composites of PE and PM materials with microstructures being periodic E-inclusions and there is no phenomenological parame-ters in Eq.(23). Also, we do not need to compute the generalized Eshelby tensor which is usually quite time consuming in the classic analysis based on the Eshelby’s works. Second, the assumption of unit cell Y being rectangular is not essential since there exist corre-sponding periodic E-inclusions for any unit cell with any given po-sitive semi-deﬁnite shape matrix Q with TrðQ Þ ¼ 1 and volume fraction f 2 ð0; 1Þ. If we send the shape matrix Q to a degenerate matrix with eigenvalues f0; 1g, the inclusion degenerates to a lam-inate regardless of the unit cell Y and Eq.(23)recovers the formula for simple laminated composites. Third, the anisotropy of the effec-tive tensor Le _{is determined by the anisotropy of microstructure}

(i.e., the shape matrix Q ) and the anisotropy of the materials. As illustrated inFig. 2, the aspect ratios of the inclusions alone cannot determine the anisotropy of the microstructure (i.e., the shape ma-trix Q ). Another geometric feature, particularly important for peri-odic composites of any microstructure at high volume fractions, is the unit cell Y or equivalently the inter-distance and inter-orienta-tion between one inclusion and its neighbors. Eq.(23)offers a prac-tical and simple way to characterize the anisotropy of the microstructure from the measured anisotropy of one kind of effec-tive properties, e.g., the effeceffec-tive electric conductivity, which in turn can be used to predict other effective properties including the effective ME tensors. Finally, we may use Eq.(23) to design the anisotropy of the microstructure according to the desired anisotropy of the effective ME composites in applications.

4. Applications

Below we apply the closed-form solution (Eq.(23)) to the de-sign of ME composites. For simplicity we will assume the micro-structure is isotropic in the sense that the shape matrix Q ¼ I=2. A material property of particular interest is the ME voltage coefﬁ-cient

a

E;11¼ ke11=

j

e11, where k

e

11 (

j

e11) is the effective ME coupling

coefﬁcient (dielectric permittivity) of the composite. The effective ME voltage coefﬁcient

a

E;11relates the overall electric ﬁeld

gener-ated in the composite with the applied magnetic field and is the figure of merit for magnetic field sensors.

As a ﬁrst example, we choose the widely used BaTiO3(BTO) as

the piezoelectric phase and CoFe2O4(CFO) as the piezomagnetic

phase. Both BTO and CFO are transversely isotropic, i.e. with 6 mm symmetry. The independent material constants are listed

inTable 1in Voigt notation, where the xy plane is isotropic and

the fiber axis is along the z-direction. Note that in all materials the ME coefficient k11¼ 0. We consider both cases: BTO fibers in

a CFO matrix and CFO ﬁbers in a BTO matrix.

Fig. 3 shows how the ME voltage coefﬁcient depends on the

volume fraction of the inclusion. The ME voltage coefficient is non-zero for every non-zero volume fraction of the inclusion even though this coefficient is zero for each constituent phase. This reflects the magnetoelectric coupling is mediated by the elastic interaction and implies that there is an optimal volume fraction for the desired maximum ME voltage coefficient. Fig. 3a shows the maximum (absolute value) ME voltage coefficient occurs at the volume of fopt¼ 0:35 with

a

E;11¼ 0:0306 V=cmOe in the case

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(absolute value) ME voltage coefﬁcient occurs at the volume of fopt¼ 0:98 with

a

E;11¼ 0:0245 V=cmOe in the case of CFO ﬁbers

in a BTO matrix. Figs. 3a and b also compare with the effective ME voltage coefﬁcients predicted byKuo (2011)who used multi-pole expansion technique and byBenveniste (1995)who employed the composite cylinder assemblage (CCA) model. InKuo (2011), the curve stops at f ¼

p

=4 when the inclusions begin to touch each other. Still, the overall magnitudes and trends agree well among predictions based on the closed-form solutions for periodic E-inclusions, Kuo’s model, and Beveniste’s CCA, and in particular Benveniste’s CCA gave the same predictions as the present closed-form solutions. Further, we verify our results with the

Fig. 4. The contour plots of the maximum effective ME voltage coefﬁcientsaI

E;11versus different material parameters for composite of PE fibers in a PM matrix. The unit for ME voltage coefficient is 0:0306 V=cmOe and the horizontal and vertical axes represent: (a) normalized elastic constants bC44;PEand bC44;PM; (b) normalized piezoelectric coefficient of PE phase ê15;PEand normalized piezomagnetic coefficient of PM phase ^q15;PM; (c) normalized dielectric permittivities ^j11;PEand ^j11;PM; (d) normalized magnetic permeabilities ^l11;PEand ^l11;PM.

Fig. 3. The predicted ME voltage coefficients versus volume fractions: (a) BTO fibers in a CFO matrix and (b) CFO fibers in a BTO matrix. In both (a) and (b), the solid line ‘‘—’’ is based on the presented closed-form solution for periodic E-inclusions with shape matrix Q ¼ I=2, i.e., Eq.(23); the dotted line ‘‘. . .’’ is fromBenveniste (1995); the dashed line ‘‘– –’’ is fromKuo (2011).

Table 1

Material parameters of BaTiO3a, CoFe2O4a, P(VDF-TrFE)band Terfenol-D/epoxyc. Property BaTiO3 CoFe2O4 P(VDF-TrFE) Terfenol-D/epoxy

C44(N/m2) 43.0G 45.3G 0.256G 13.6G e15(C/m2) 11.6 0 0.015 0 q15(N/A m) 0 550 0 108.3 j11(C2/N m2) 11:2 109 _{0:08 10}9 _{0:07 10}9 _{0:05 10}9 l11(Ns2/C2) 5 106 _{590 10}6 _{1:26 10}6 _{5:4 10}6 k11(Ns/VC) 0 0 0 0 a Li and Dunn (1998b). b Nan et al. (2001b). c Liu et al. (2003, 2004).

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compatibility relations given in Eq.(21)proposed in the work of

Benveniste (1995). These exact universal connections are derived

based on a formalism discovered by Milgrom and Shtrikman

(1989), and are independent of the details of the microgeometry

and of the particular choice of the averaging model. Again, they are in good agreement.

Next, we study how the effective ME voltage coefﬁcient de-pends on the elastic moduli C44;PEand C44;PM, dielectric

permittivi-ties

j

11;PE and

j

11;PM, magnetic permeabilities

l

11;PE and

l

11;PM of

the PE and PM materials, piezoelectric coefﬁcient e15;PE of the PE

material, and piezomangetic coefﬁcient q15;PMof the PM material.

a

I

E;11by Eq.(23)and its dependence of the normalized material

properties of PE and PM phases. These results give important guide-lines for practical designs of ME composites of PE and PM materials.

Fig. 4shows the contours of the maximum ME voltage

coefﬁ-cients

a

I

E;11of PE ﬁbers (over volume fraction f) in a PM matrix at

the optimal ﬁbrous volume fraction fopt, where the maximum ME

voltage coefﬁcients

a

E;11;BTO in CFO¼ 0:0306 V=cmOe of BTO ﬁbers

in a CFO matrix is chosen as the unit for the ME voltage coefﬁcient

a

I

E;11for ease of comparison. The optimal volume fractions of PE

phase foptvary from 0.28 to 0.64, whose exact values can be easily

computed by numerically maximizing the effective ME voltage coefficient over f 2 ½0; 1 (cf. Eq.(24)). InFig. 4a the horizontal and vertical axes represent the normalized elastic constants of PE and PM phases in logarithmic scale, respectively. It is observed that the ME voltage coefficient increases when either the fiber or ma-trix’s elastic constant decreases. Therefore, softer PM and PE mate-rials are preferred for improving the ME voltage coefficients of composites of PE fibers in a PM matrix.Fig. 4b shows the contours of the maximum ME voltage coefficients

a

I

E;11versus the

piezoelec-tric and piezomagnetic constants in linear scale. For a fixed piezo-electric coefficient e15, the ME voltage coefficient increases

monotonically as the piezomagnetic coefficient q₁₅increases. How-ever, for a fixed normalized piezomagnetic coefficient q15 and as

the piezoelectric coefﬁcient e15 increases, the ME voltage

coefﬁ-cient increases ﬁrst and decreases after certain optimal e15.

Fig. 5. The contour plots of the maximum effective ME voltage coefﬁcientsaI

E;11versus different material parameters for composite of PM fibers in a PE matrix. The unit for ME voltage coefficient is 0:0245 V=cmOe and the horizontal and vertical axes represent: (a) normalized elastic constants bC44;PEand bC44;PM; (b) normalized piezoelectric coefficient of PE phase ê15;PEand normalized piezomagnetic coefficient of PM phase ^q15;PM; (c) normalized dielectric permittivities ^j11;PEand ^j11;PM; (d) normalized magnetic permeabilities ^l11;PEand ^l11;PM.

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Therefore, a large piezomagnetic coefﬁcient q15 but a nontrivial

optimal piezoelectric coefﬁcient e15 are preferred for improving

the ME voltage coefficients of composites of PE fibers in a PM ma-trix.Fig. 4c shows the contours of the maximum ME voltage coef-ficient

a

I

E;11versus the normalized electric permittivities of PE and

PM phases in logarithmic scale. We observe that smaller PE per-mittivity

j

11;PEgives rise to larger ME voltage coefﬁcient. However,

the PM permittivity

j

11;PM does not inﬂuence ME effect much.

Fig. 4d shows the contours of the maximum ME voltage coefﬁcient

a

I

E;11versus the normalized magnetic permeabilities of the PE and

PM phases in logarithmic scale. We observe that increasing the PE’s magnetic permeability largely enhances the ME voltage coef-ficient, and on the contrary, increasing the PM’s magnetic perme-ability lowers the ME voltage coefficient. Therefore, a large magnetic permeability of the PE phase and a small magnetic per-meability of the PM phase are preferred for improving the ME volt-age coefficient for composites of PE fibers in a PM matrix.

We now turn to the case of PM ﬁbers (with isotropic shape ma-trix Q ¼ I=2) in a PE mama-trix.Fig. 5shows the contours of the max-imum ME voltage coefﬁcients

a

I

E;11at the optimal ﬁbrous volume

fraction fopt, where the maximum ME voltage coefﬁcients

a

E;11;CFO in BTO¼ 0:0245 V=cmOe of CFO ﬁbers in a BTO matrix is

cho-sen as the unit for the ME voltage coefﬁcient for ease of compari-son. The optimal volume fractions fopt of PM phase are also

computed by numerically maximizing the effective ME voltage coefficient over f 2 ½0; 1 (cf. Eq.(24)). FromFig. 5a we observe that the elastic constant of the PE phase has a much stronger influence on the ME voltage coefficient than that of the PM phase. Again, soft PM and PE phases are preferred for improving the ME voltage coef-ficient. We also notice that the optimal volume fraction of the PM phase foptis roughly a constant of 0.98 though the elastic constants

of the PM and PE phases change orders of magnitude. FromFig. 5b we observe that the ME voltage coefﬁcient increases monotonically as the piezomagnetic coefﬁcient q15of the PM phase increases and

there is an optimal piezoelectric coefﬁcient e15 of PE phase for

maximum ME voltage coefﬁcient of composites of PM ﬁbers in a PE matrix. We also notice that the optimal volume fraction of the inclusion foptis roughly a constant of 0.98.Fig. 5c and d shows that

to improve the ME voltage coefficient of composites of PM fibers in a PE matrix, we shall engineer the PM fibers and PE matrix such that the electric permittivity

j

11;PMof the PM phase is enhanced

and the magnetic permeability

l

11;PMis reduced, and on the

con-trary, the electric permittivity

j

11;PE of the PE phase is reduced

and the magnetic permeability

l

11;PEis enhanced. The optimal

vol-ume fraction foptvaries from 0.92 to 0.98 for cases shown inFig. 5c

and d.

Motivated by the above study, we study ME composites of P(VDF-TrFE) and Terfenol-D/epoxy TD/epoxy since they have much

lower elastic constants, electric permittivity, and magnetic permeability. Further, a particulate ME composite made of P(VDF-TrFE) and TD was also studied byNan et al. (2001a,b)which shows that the ﬂexible composite exhibits markedly larger cou-pling effect. For P(VDF-TrFE) in a TD/epoxy matrix, the maximum is attained at volume fraction f ¼ 0:34 where ME voltage coefﬁ-cient

a

E;11¼ 0:1051 V=cmOe (Fig. 6a). For TD/epoxy in a

P(VDF-TrFE) matrix, the maximum occurs at the volume fraction f ¼ 0:87 where the coupling effect

a

E;11¼ 0:9221 V=cmOe

(Fig. 6b). Both of them are around 3.5 times enhancement of the

coupling coefﬁcients compared to their BTO/CFO counterparts.

5. Summary and discussion

The coexistence of magnetic and electric ordering and their interaction in magnetoelectric materials have stimulated consider-able scientiﬁc and technological interest in recent years for poten-tial applications in actuators, sensors and storage devices. By considering a simple model of periodic two-phase composites of piezoelectric and piezomagnetic materials, we derive a closed-form solution to the effective properties of the composite in terms of material properties of the constituent phases and simple geo-metric parameters: the volume fraction f of the ﬁber phase and the shape matrix Q which characterizes the anisotropy of the microstructure. The predicted effective properties are realizable by microstructures of periodic E-inclusions.

Based on this closed-form solution, we study the dependence of a particular material property of interest, the ME voltage coeffi-cient, on the volume fraction of the fiber phase and the material properties of the PE and PM phases. In particular, we obtain the fol-lowing design principles for ME fibrous composites of PE and PM phases:

(1) There exists an optimal volume fraction for maximum ME voltage coefﬁcient which can be obtained by maximizing Eq. (24) over volume fraction f 2 ð0; 1Þ. This is probably the most important conclusion of our study since the vol-ume fraction is the easiest controllable design parameters. (2) Softer materials are desirable for improving the ME voltage

coefﬁcient.

(3) For composites of PE fibers in a PM matrix and PM fibers in a PE matrix (cf.Figs. 4 and 5), it is desirable to have larger piezomagnetic coefficient but smaller magnetic permeabil-ity in the PM phase, smaller electric permittivpermeabil-ity but larger magnetic permeability in the PE phase. Further, there exists an optimal value of the piezoelectric coefficient of the PE fibers for maximum ME voltage coefficient.

Fig. 6. The predicted ME voltage coefficients. P(VDF-TrFE) is the fiber phase and TD/epoxy is the matrix phase for (a). TD/epoxy is the fiber phase and P(VDF-TrFE) is the matrix phase for (b).

(8)

(4) The dielectric permittivity of PM phase has a much stronger effect on the ME voltage coefficient for composites of PM fibers in a PE matrix than for composites of PE fibers in a PM matrix (cf.Figs. 4c and 5c) and is preferably large.

Acknowledgments

We are grateful to acknowledge the supports under Grant Nos.: NSC 99–2221-E-009–053 (H.-Y. Kuo) and NSF CMMI-1101030 (L.P. Liu). This work was completed while LL held a position at the Department of Mechanical Engineering, University of Houston; the support and hospitality of the department is gratefully acknowledged.

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