Optimization of magnetoelectricity in multiferroic fibrous composites

Optimization of magnetoelectricity in multiferroic fibrous composites

Hsin-Yi Kuo

⇑

, Yong-Liang Wang

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 29 August 2011

Received in revised form 10 January 2012 Available online 24 March 2012 Keywords:

Magnetoelectricity Fibrous composites Crystallographic orientation Mori–Tanaka’s method Finite element analysis Optimization

a b s t r a c t

We propose a method to optimize the effective magnetoelectric voltage coefficient of fibrous composites made of piezoelectric and piezomagnetic phases. The optimization of magnetoelectricity is with respect to the crystallographic orientations and the volume frac-tion for the two materials. We show that the effective in-plane (a

E;11) and out-of-plane

(a

E;33) coupling constants can be enhanced many-fold at the optimal orientation compared

to those at normal orientation. For example, we show that the constants are 101 and 5 times larger for the optimal orientation of CoFe2O4fibers in a BaTiO3matrix of the

opti-mized volume fraction compared to the normal orientation, while they are 43 and 5 times larger for BaTiO3fibers in a CoFe2O4matrix. The predictions are in good agreement with

the finite element analysis.

Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Magnetoelectric (ME) materials, which show a polariza-tion induced by an applied magnetic field, or conversely, a magnetization induced by an applied electric field, have been the focus of recent research due to their coupling be-tween the electric and magnetic fields. This make them particularly appealing and promising for a wide range of applications, such as ME data storage and switching, mag-netic field detectors, and amplification and frequency con-version between the electric and magnetic fields (Fiebig, 2005). However, the ME effect in single phase materials is rather weak or cannot be observed at room temperature (Astrov, 1960; Rado and Folen, 1961). Composite materials, on the other hand, offer an alternative option for improve-ment of the ME coupling, as explained in recent reviews by Eerenstein et al. (2006) and Nan et al. (2008). This much stronger ME effect could be realized in a composite made

of piezoelectric and piezomagnetic/magnetostrictive

phases using product properties: an applied magnetic field creates a strain in the piezomagnetic/magnetostrictive material which in turn creates a strain in the piezoelectric material, resulting in an electric polarization.

A variety of models have been proposed to predict the effective magnetoelectroelastic moduli of the multiferroic composite. The estimates of the effective properties of ME composites are usually obtained by various

approxi-mate mean-field models (Nan, 1994; Benveniste, 1995;

Wu and Huang, 2000). The exact solutions for local fields are available for simple microstructures such as a single ellipsoidal inclusions (Huang and Kuo, 1997; Li and Dunn, 1998a), periodic arrays of circular/elliptic fibrous ME com-posites (Kuo, 2011; Kuo and Pan, 2011) and laminates (Srinivas et al., 2001; Bichurin et al., 2003), etc. A homoge-nization method was employed for calculating the

effec-tive properties of periodic ME fibrous composites

(Aboudi, 2001; Camacho-Montes et al., 2009), while

numerical methods based on the finite element analysis have also been developed to address ME composites with more general microstructures (Liu et al., 2004; Lee et al.,

2005). However, much of this theoretical development

limits itself to the situation where the poling direction (magnetic axis) of the piezoelectric (piezomagnetic) mate-rial is either normal to or along the layer (fiber) direction. Further, many of these works assume transverse isotropy or uniaxial symmetry.

In the work ofLi and Dunn (1998b), they used Eshelby’s pioneering approach to study the fields in and around inclu-sions and inhomogeneities in anisotropic solids exhibiting 0167-6636/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.mechmat.2012.03.005 ⇑ Corresponding author.

E-mail address:[email protected](H.-Y. Kuo).

Contents lists available atSciVerse ScienceDirect

Mechanics of Materials

full coupled-field behavior. Later,Li (2000a)developed a numerical algorithm to evaluate the magnetoelectroelastic Eshelby’s tensor for the general material symmetry and ellipsoidal inclusion shape. Recently, experiments byYang et al. (2006) and Wang et al. (2008)showed that single crys-tals are attractive and the effective ME coefficient of the laminate can depend sensitively on the crystallographic ori-entation of the material.Srinivas et al. (2006)developed a mean-field Mori–Tanaka model to calculate the ME cou-pling of matrix-based multiferroic composites, emphasing

the effects of shape and orientation distribution of second phase particles. In addition,Kuo et al. (2010)proposed a simple framework to optimize the effective magnetoelectric response of a piezoelectric-magnetostrictive bilayer. The es-sence of the concept is that the induced electric field in the piezoelectric phase could be increased if the orientation and volume fraction of the piezoelectric layer can be carefully chosen. They have used it to show that, for anisotropic materials as in single crystals, the optimal ME response is obtained for non-trivial orientations.

Fig. 1. The fibrous composite configurations.

Motivated by these advances, in this paper we optimize the effective ME voltage coefficient of a multiferroic fibrous composite without any assumptions on the symmetry of the underlying materials and without any assumptions on the crystallographic orientations of the materials. We give the basic equations and Euler transformations regard-ing the magnetoelectroelasticity in Section2.1. In Section 2.2, we derive a micromechanical model for the multiferro-ic composites. We introduce the finite element analysis in Section2.3, which is used for comparison with the micro-mechanical approach. This methodology is illustrated in Section3using composites of cobalt ferrite (CoFe2O4) and barium titanate (BaTiO3). We show that the optimal orien-tations can be non-trivial and the enhancement to be many-fold over the normal orientations.

Table 1

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

Property BaTiO3 CoFe2O4

C11(GPa) 166 286 C12(GPa) 77 173 C13(GPa) 78 170 C33(GPa) 162 269.5 C44(GPa) 43 45.3 e15(C/m2) 11.6 0 e31(C/m2) -4.4 0 e33(C/m2) 18.6 0 q15(N/Am) 0 550 q31(N/Am) 0 580.3 q33(N/Am) 0 699.7 j11(nC2/Nm2) 11.2 0.08 j33(nC2/Nm2) 12.6 0.093 l11(lNs 2 /C2 ) 5 590 l33(lNs 2 /C2 ) 10 157 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.025 -0.02 -0.015 -0.01 -0.005 0

Volume Fraction of Inclusion

α * E, 1 1 (V/cmOe) MT SQU HEX CFO[001]/BTO[001] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Volume Fraction of Inclusion

α * E,3 3 (V/cmOe) MT SQU HEX CFO[001]/BTO[001]

a

b

Fig. 3. The ME voltage coefficients of the CFO fibers in a BTO matrix at the normal direction versus the fiber volume fraction. (a) In-plane ME voltage coefficienta

2. Model

2.1. Basic equations

Consider a perfectly bonded magnetoelectric circular fi-brous composite made of piezoelectric and piezomagnetic materials as shown inFig. 1. The response of the composite in a Cartesian frame with the x3direction normal to the plane can be described by the following general equations (Alshits et al., 1992)

rij

¼ Cijklekl elijEl qlijHl;

Di¼ eikleklþ

jil

Elþ kliHl;

Bi¼ qikl

ekl

þ kilElþ

l

ilHl;

ð1Þ

where

r

ijand

e

ijare the stress and strain; Diand Eiare the electric displacement and electric field vectors; Bi and Hi are the magnetic flux and magnetic field vectors; Cijkl is

the elastic stiffness (fourth-order tensor), eijkis the piezo-electric moduli (third-order), qijk is the piezomagnetic moduli (third-order),

j

ijis the permittivity (second-order),

l

ijis the permeability (second-order) and kijis the magne-toelectric coefficient (second-order). The summation con-vention is used. The symmetry conditions satisfied by the moduli are given byNye (1985).

The strain

e

ij, electric field Ei, and magnetic field Hiare respectively defined by the displacement ui, electric poten-tial

u

, and magnetic potential w via

eij

¼1

2 ui;jþ uj;i

 

; Ei¼ 

u

;i; Hi¼ w;i: ð2Þ

Here the comma in the subscript denotes partial derivative.

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

0 45 90 135 180 0 45 90 135 180 -2.5 -2 -1.5 -1 -0.5 0 0.5 γi (deg) βi (deg) α * E, 1 1 (V /c m O e )

Normal = -0.0244V/cmOe

Max. = -2.4823V/cmOe

(

_α

,

_β

,

_γ

) = (

_α

,90

o

,90

o

)

f = 0.98

0 45 90 135 180 0 45 90 135 180 -2.5 -2 -1.5 -1 -0.5 0 0.5 γm (deg) β_m (deg) α * E,1 1 (V /c m O e )

Normal = -0.0244V/cmOe

Max. = -2.4823V/cmOe

(

_α

,

_β

,

_γ

) = (

_α

,90

o

,90

o

)

f = 0.98

a

b

Fig. 4. The in-plane ME voltage coefficient of the CFO fibers in a BTO matrix for various orientations of CFO and BTO. The subscripts i and m denote the inclusion and matrix, respectively. Note that this coefficient depends only on the Euler angles b andcand is independent ofa. The optimized constant occurs at both phases poled along the same direction.

rij;i

¼ 0; Di;i¼ 0; Bi;i¼ 0; ð3Þ along with the analogous interfacial conditions and appro-priate boundary conditions.

The constitutive laws, strain-displacement and equilib-rium equations can be rewritten in a more concise form as follows (Alshits et al., 1992)

R

iJ¼ LiJMnZMn; ZMn¼ UM;n;

R

iJ;i¼ 0; ð4Þ where

R

iJ¼

rij

; J ¼ 1; 2; 3; Di; J ¼ 4; Bi; J ¼ 5; 8 > < > : ZMn¼

emn

; M ¼ 1; 2; 3; En; M ¼ 4; Hn; M ¼ 5; 8 > < > : UM¼ um; M ¼ 1; 2; 3;

u

; M ¼ 4; w; M ¼ 5: 8 > < > : ð5Þ

The magnetoelectroelastic moduli are expressed as

LiJMn¼ Cijmn; J; M ¼ 1; 2; 3; eijn; M ¼ 4; J ¼ 1; 2; 3; qijn; M ¼ 5; J ¼ 1; 2; 3; eimn; J ¼ 4; M ¼ 1; 2; 3; 

jin

; J ¼ 4; M ¼ 4; kin; J ¼ 4; M ¼ 5; qimn; J ¼ 5; M ¼ 1; 2; 3; kin; J ¼ 5; M ¼ 4; 

l

_in; J ¼ 5; M ¼ 5; 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : ð6Þ

where the upper case subscript ranges from 1 to 5 and the lower case subscript ranges from 1 to 3. Repeated upper case subscripts are summed from 1 to 5.

The equations above refer the material properties to the fiber frame (Fig. 1). However, the material properties are commonly described in the crystallographic frame and we need to transform them to the fiber frame. To this end, let us denote the crystal frame with primes and intro-duce the rotation matrix aij. This is given in terms of the three Euler anglesð

a

;b;

c

Þ as follows (Arfken and Weber, 2001)

For the change of frame, the material parameters then fol-low the tensor transformation rules for second-, third- and fourth-order tensors

jij

¼ aimajnj0mn;

l

ij¼ aimajnl0mn; eijk¼ aimajnakoe0mno; qijk¼ aimajnakoq0mno; Cijkl¼ aimajnakoalpC0mnop; ð8Þ

where the primed quantities

j

0

ij;

l

0ij;e0ijk;q0ijk;C 0 ijkl

 

denote the material properties referred to the crystallographic frame.

2.2. Effective moduli and Mori–Tanaka’s approach

We are interested in the effective behavior for a situa-tion where we have a large number of inclusions. The effective material properties are defined in terms of aver-age fields, 0 45 90 135 180 0 45 90 135 180 -10 -5 0 5 10 γi (deg) γm (deg) α * E,3 3 (V /c m O e )

Normal = 1.2288V/cmOe

Max. = -6.2079V/cmOe

(

_α

,

_β

,

_γ

) = (

_α

,90

o

,

_γ

)

f = 0.94

Fig. 5. The out-of-plane ME voltage coefficient of the CFO fibers in a BTO matrix for various orientations of CFO and BTO. The subscripts i and m denote the inclusion and matrix, respectively. Note that this coefficient depends only on the Euler angles b andcand is independent ofa. The optimized constant occurs at both phases poled along the same direction.

a11 a12 a13 a21 a22 a23 a31 a32 a33 0 B @ 1 C A ¼

cos

c

cos b cos

a

 sin

c

sin

a

cos

c

cos b sin

a

þ sin

c

cos

a

 cos

c

sin b  sin

c

cos b cos

a

 cos

c

sin

a

 sin

c

cos b sin

a

þ cos

c

cos

a

sin

c

sin b

sin b cos

a

sin b sin

a

cos b

0 B @ 1 C A: ð7Þ

R

iJ

  ¼ L

iJMnhZMni; ð9Þ

where the angular brackets denote the average over the representative volume element (unit cell in the case of periodic composites), and L

iJMn denotes the effective magnetoelectroelastic parameters of the composite. Due to the linearity, the generalized strain in the inclusion for a two-phase composite is (Srinivas et al., 2006)

ZMn¼ AMnAbhZAbi; ð10Þ

where AMnAbis the generalized strain concentration tensor of the inclusion. As a result, the effective moduli can be determined for a two-phase composite as

L iJAb¼ L ðmÞ iJAbþ f L ðiÞ iJMn L ðmÞ iJMn   AMnAb: ð11Þ

Here f is the volume fraction of the inclusion, and the superscripts m and i denote the matrix and inclusion, respectively.

The concentration tensor AMnAb can be determined by the Mori–Tanaka’s approach as

AMnAb¼ AdilMnJi ð1  fÞIJiAbþ fA dil JiAb

h i1

; ð12Þ

with the dilute concentration tensor Adil_MnAbgiven by

AdilMnAb¼ IMnAbþ SMnLkðLðmÞLkiJÞ 1 LðiÞiJAb L ðmÞ iJAb   h i1 ; ð13Þ

where SMnAb is the magnetoelectroelastic Eshelby tensor, which is a function of the magnetoelectroelastic moduli of matrix, the shape and orientation of the inclusion, and is described byLi and Dunn (1998b).

SMnAb¼ 1 8

p

LiJAb R1 1 R2p 0 GmJinðzÞ þ GnJimðzÞ dhdn3; M ¼ 1; 2; 3; 2R_11 R₀2pG4JinðzÞdhdn3; M ¼ 4; 2R_11 R02pG5JinðzÞdhdn3; M ¼ 5: 8 > > < > > : ð14Þ In the above equation, zi¼ ni=ai(no summation on i), aiis the semi-axis of size and n1 and n2 can be expressed in

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2.5 -2 -1.5 -1 -0.5 0

Volume Fraction of Inclusion

α * E,1 1 ( V /c mOe ) MT SQU HEX CFO[010]/BTO[010] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -7 -6 -5 -4 -3 -2 -1 0

Volume Fraction of Inclusion

α * E,3 3 (V /c m O e ) MT SQU HEX CFO[010]/BTO[010]

a

b

Fig. 6. The optimal ME voltage coefficients of the CFO fibers in a BTO matrix for various fiber volume fraction. (a) In-plane ME voltage coefficienta E;11. (b) Out-of-plane ME voltage coefficienta

terms of n3 and h by n1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  n2 3 q cos h and n2¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  n2 3 q

sin h. In addition GMJin¼ ziznK1MJðzÞ, where K1

MJ is the inverse of KJR¼ ziznLiJRn. Li and Dunn (1998a) have obtained the closed-form expressions of magneto-electroelastic Eshelby’s tensors for the aligned elliptic cyl-inder inclusion in a transversely isotropic medium. However, for the piezoelectric and piezomagnetic materi-als with arbitrary poling direction and magnetic axes as discussed in this work, we resort to Gauss quadrature numerical method to calculate SMnAb. The integral(14)then is approximated by the weighted sum of function values at certain integration points (Li, 2000a).

2.3. Finite element analysis

In this section we introduce the finite element analysis which is used for comparison with the Mori–Tanaka’s approach. We first choose an appropriate representative

volume element (RVE), a periodic unit cell, which captures the major features of the underlying microstructure. There are five possible ways of packing cylinders a regular array in two dimensions (SeeKittel, 2005for instance). Here we concentrate on the two lattices, rectangular and hexagonal arrays (Fig. 2).

Because of the periodicity in the composite structure, the displacement ui, the electric potential

u

and the mag-netic potential w in any point of the unit cell can be ex-pressed in terms of those at an equivalent point in another RVE such that the periodic boundary conditions

UMðd; x2;x3Þ ¼ UMðd; x2;x3Þ þ U M;12d;

UMðx1;d; x3Þ ¼ UMðx1;d; x3Þ þ U M;22d;

UMðx1;x2;dÞ ¼ UMðx1;x2;dÞ þ U M;32d;

ð15Þ

are satisfied for a square array. Here UM is defined in(5) and 2d is the length of the unit cell. Similarly, the periodic boundary conditions for a hexagonal array are

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

Volume Fraction of Inclusion

α * E, 1 1 (V/cmOe) MT SQU HEX BTO[001]/CFO[001] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 α * E, 3 3 (V/cmOe)

Volume Fraction of Inclusion

MT SQU HEX BTO[001]/CFO[001]

a

b

Fig. 7. The ME voltage coefficients of the BTO fibers in a CFO matrix at the normal direction versus the fiber volume fraction. (a) In-plane ME voltage coefficienta

0 45 90 135 180 0 45 90 135 180 -1.5 -1 -0.5 0 0.5 γ_i (deg) β_i (deg) α * E,1 1 (V /c mO e)

Normal = -0.0306V/cmOe

Max. = -1.3384V/cmOe

(

_α

,

_β

,

_γ

) = (

_α

,69

o

,90

o

)

f = 0.34

0 45 90 135 180 0 45 90 135 180 -1.5 -1 -0.5 0 0.5 γ_m (deg) β_m (deg) α * E,1 1 (V /c mO e)

Normal = -0.0306V/cmOe

Max. = -1.3384V/cmOe

(

_α

,

_β

,

_γ

) = (

_α

,69

o

,90

o

)

f = 0.34

a

b

Fig. 8. The in-plane ME voltage coefficient of the BTO fibers in a CFO matrix for various orientations of BTO and CFO. The optimized constant occurs at both phases poled along the same direction.

0 45 90 135 180 0 45 90 135 180 -6 -4 -2 0 2 4 6 γi (deg) γm (deg) α * E, 3 3 (V /c m O e )

Normal = 1.1494V/cmOe

Max. = -5.7986V/cmOe

(

_α

,

_β

,

_γ

) = (

_α

,90

o

,

_γ

)

f = 0.06

Fig. 9. The out-of-plane ME voltage coefficient of the BTO fibers in a CFO matrix for various orientations of BTO and CFO. The optimized constant occurs at both phases poled along the same direction.

UMðd; x2;x3Þ ¼ UMðd; x2;x3Þ þ UM;1   2d; UM x1; ffiffiffi 3 p d; x3   ¼ UM x1; ffiffiffi 3 p d; x3   þ UM;2   2pffiffiffi3d; UMðx1;x2;dÞ ¼ UMðx1;x2;dÞ þ UM;3   2d: ð16Þ

In order to determine the effective properties of the multif-erroic composite, the strain, electric field, and magnetic field states are applied individually to the unit cell. The boundary conditions have to be applied to the unit cell in such a way that, apart from one component of the strain, electric field, or magnetic field UM;i

 

in Eq.(15)for square arrays or(16)for hexagonal arrays, all other components are made equal to zero. Then each effective coefficient can be determined by(9). We perform the finite element analysis using the software COMSOL Multiphysics.

3. Numerical results and optimization

We consider two systems of interest. For the piezoelec-tric material, we choose the widely used BaTiO3ceramic,

while we choose CoFe2O4 as the piezomagnetic phase

which has been studied by other researchers. Both of them are with 6 mm symmetry. We consider square and hexag-onal arrays in finite element analysis, and both cases, i.e., both CFO fibers in a BTO matrix and BTO fibers in a CFO matrix. The independent material constants of these con-stituents are given in Table 1 in Voigt notation, where the x1x2 plane is isotropic and the poling direction/mag-netic axis is along the x3-direction.

In our study, we are particularly interested in the effec-tive magnetoelectric (ME) response. The induced voltage is proportional to the applied magnetic field and the constant of proportionality is the effective ME voltage coefficient. It

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2.5 -2 -1.5 -1 -0.5 0 0.5

Volume Fraction of Inclusion

α * E,1 1 ( V /c m O e ) MT SQU HEX BTO(0o,69o,90o)/CFO(0o,69o,90o) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 -5 -4 -3 -2 -1 0

Volume Fraction of Inclusion

α * E,3 3 ( V /c m O e ) MT SQU HEX BTO[100]/CFO[100]

a

b

Fig. 10. The optimal ME voltage coefficients of the BTO fibers in a CFO matrix for various fiber volume fraction. (a) In-plane ME voltage coefficienta E;11. (b) Out-of-plane ME voltage coefficienta

combines the coupling and dielectric coefficients, and is defined by

a

 E;ij¼ k  ij=

j

ij; ð17Þ

where there is no summation for the repeated indices. We seek to optimize this ME voltage coefficient with respect to the crystallographic orientation of the materials. Specifi-cally we consider the in-plane (

a



E;11) and out-of-plane (

a



E;33) coupling constants. However, this is a highly nonlin-ear problem, therefore we resort to a brute-force approach where we create a fine grid of Euler angles and exhaus-tively compare the values on this grid.

3.1. Piezomagnetic fibers in a piezoelectric matrix

To check the correctness of our model, we first perform a numerical computation for CFO fibers in a BTO matrix with 6 mm material symmetry about the fiber axis.Fig. 3

shows the ME voltage coefficients for this composite. The finite element analysis is estimated for discrete volume fractions and stops around f ¼

p

=4 and f ¼

p

=2pffiffiffi3 for the square and hexagonal arrays, respectively, when the inclusions touch. The prediction of the Mori-Tanaka’s ap-proach is in good agreement with the result of the finite element analysis. The maximum ME voltage coefficient

a



E;11is 0:0244 V/cmOe at volume fraction f ¼ 0:98, while

the maximum

a



E;33¼ 1:2288 V/cmOe at volume fraction f ¼ 0:94. Note that the results of the hexagonal array are closer to the Mori–Tanaka’s estimation than those of the square array. This is because a hexagonal array is a closed packing structure, and the Mori–Tanaka’s model allows the inclusion to fulfill the matrix. In addition a square array lacks the transversely isotropy that this composite pos-sesses (Li, 2000b).

We now turn to the optimization of this composite. For each orientation, we follow the procedure developed in

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 0 100 200 300 400 500

Volume Fraction of Inclusion

α * E, 2 3 (V/c m O e ) MT SQU HEX BTO(0o,69o,90o)/CFO(0o,69o,90o) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -14 -12 -10 -8 -6 -4 -2 0 2

Volume Fraction of Inclusion

α * E,3 2 ( V /c m O e ) MT SQU HEX BTO(0o,69o,90o)/CFO(0o,69o,90o)

a

b

Fig. 11. The off-diagonal ME voltage coefficients of the BTO fibers in a CFO matrix for various fiber volume fraction. (a) ME voltage coefficienta E;23. (b) ME voltage coefficienta

Section2to obtain the magnetoelectric voltage coefficient. The reference volume fraction is f ¼ 0:98 for calculating optimal

a



E;11, while it is chosen as 0:94 when calculating optimal

a



E;33since these happen to be optimal at the nor-mal cut. The orientation of both materials are arbitrary.

Fig. 4 shows the ME voltage coefficient

a



E;11 with re-spect to the crystallographic orientation of CFO and BTO. It happens to be optimal when the poling direction of pie-zoelectric phase coincides with the magnetic axis of the piezomagnetic phase. We observe that the maximum of 2:4823 V/cmOe occurs at Euler angles ð

a

;b;

c

Þ ¼ ð

a

;90_;90_{Þ, where}

_a

_{is arbitrary. This degeneracy of} opti-mal orientation reflects the 6 mm symmetry. Further, if

a

¼ 0, it is equivalent to the poling direction/magnetic axis

along ½010. Significantly, the optimized value of

2:4823 V/cmOe is almost one hundred and one times higher than 0:0244 V/cmOe, which is the value of the normal cut where the c axis of the CFO and BTO is along the fiber axis.

We show how the ME voltage coefficient

a



E;33depends

on its orientation in Fig. 5. The maximum value is

6:2079 V/cmOe at the optimal orientation ð

a

;b;

c

Þ ¼

a

;90_;

_c

ð Þ of both phases, and this is as much as five times

higher than the value of 1:2288 V/cmOe at the normal cut. Fig. 6shows the effect of volume fraction f on the ME voltage coefficients. The piezoelectric phase is poled along one of the optimized directions, sayð

a

;b;

c

Þ ¼ ð0; 90;90_Þ or equivalently 010½ , and the piezomagnetic phase is along the same optimized magnetic axis. The maximum value is obtained at piezoelectric material almost vanish at volume fraction f ¼ 0:98 and 0:92 for ME voltage coefficient

a



E;11 and

a



E;33, respectively. The maximum value of

a

E;11 is

2:4823 V/cmOe while that of

a



E;33 is 6:2357 V/cmOe

both of these evaluated at their optimal orientations. 3.2. Piezoelectric fibers in a piezomagnetic matrix

We now turn to the composite made of BTO fibers in a CFO matrix. Similarly, we begin with the case of the mate-rial symmetry about the fiber axis, i.e. along 001½ . The

maximum

a



E;11is 0:0306 V/cmOe at f ¼ 0:34 and

a

E;33is 1:1494 V/cmOe at f ¼ 0:06 at their normal orientation (Fig. 7).

Figs.8 and 9 show the magnetoelectric voltage coeffi-cient

a



E;11and

a

E;33as a function of orientation for the case where the volume fraction is corresponding to their opti-mal value at the noropti-mal cut. We find that the maximum coupling coefficient is 1.3384 V/cmOe with ð

a

;b;

c

Þ ¼

a

;69_;90

ð Þ or

a

;111_;90

ð Þ for

a



E;11. The plot forð

a

;111; 90_{Þ is similar toFig. 8but with 180}_{reverse with respect} to b. For

a



E;33, the maximum value is 5:7986 V/cmOe with

a

;90;

c

ð Þ. If we choose

a

¼ 0 and

c

¼ 0, the optimized

direction is equivalent to 100½ .

Fig. 10shows the effect of fibrous volume fraction on the ME voltage coefficients. For the optimized volume fraction, the numbers are 1:3441 V/cmOe and 5:8250 V/cmOe ðf ¼ 0:08Þ, respectively. All of these are evaluated at their respective optimal orientation. Note that although the dif-ference between the results of finite element analysis and Mori–Tanaka’s method is larger inFig. 10(a), the trend is similar for both methods. One reason of the deviation is

that because the ME voltage coefficient is an indirect calcu-lated value through Eq.(17). The effective permittivity

j



11 approaches to zero hence is sensitive when calculating

a



E;11. Further, the magnetoelectric coefficient k11 of this case has larger difference between the two approaches.

Finally, we observe that there are off-diagonal elements of

a



E;23and

a

E;32when the poling direction/magnetic axis is at the orientationð

a

;b;

c

Þ ¼_ð

a

;69_;90_{Þ.Fig. 11shows how} these coefficients depend on the volume fraction.

Remark-ably, the maximum

a



E;23 is 469:6768 V/cmOe ðf ¼ 0:25Þ, while that of

a



E;32is 5:7340 V/cmOe ðf ¼ 0:50Þ.

4. Concluding remarks

In this work, we have proposed a theoretical framework to compute the effective magnetoelectric response of a piezoelectric–piezomagnetic fibrous composite. We have used it to show that, for anisotropic materials as in single crystals, the optimal ME response is obtained for non-triv-ial orientations. For the CFO fibers in a BTO matrix, the highest in-plane magnetoelectric voltage coefficient

a



E;11 at its optimized crystallographic orientation is 2.4823 V/ cmOe, which is 101 times larger than that of a fibrous com-posite made with the normal cut type CFO and BTO single crystals. The out-of-plane ME voltage coefficient,

a



E;33, on the other hand, can be increased around five times to 6:2079 V/cmOe. For the BTO fibers in a CFO matrix, the in-plane and out-of plane ME voltage coefficients can be increased around 43 times and 5 times respectively com-pared to the normal orientation. The dependence of the magnetoelectric voltage coefficient with respect to the vol-ume fraction f was also determined when both phases poled along the optimized direction. The coefficients var-ied with the volume fraction and were optimized when the piezoelectric phase approaches zero for the case of CFO fibers in a BTO matrix. Finally, the results are com-pared to finite element analysis and show good agreement. Acknowledgement

We gratefully acknowledge Professor J. Y. Li for the Fortran code of Eshelby tensor. We are grateful to the financial support of National Science Council, Taiwan, under Contract No. NSC 100-2628-E-009-022-MY2. References

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