Magnetoelectricity in coated fibrous composites of piezoelectric and piezomagnetic phases

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Magnetoelectricity in coated ﬁbrous composites of piezoelectric

and piezomagnetic phases

Hsin-Yi Kuo

⇑

, Cheng-You Peng

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 3 May 2012

Received in revised form 19 July 2012 Accepted 12 August 2012

Available online 5 October 2012

Keywords: Magnetoelectricity Coated ﬁbrous composites Piezoelectric

Piezomagnetic Micromechanics Finite element analysis

a b s t r a c t

This paper studies the effective magnetoelectric (ME) behavior of coated fibrous compos-ites made of piezoelectric and piezomagnetic phases. We employ a micromechancial model, the two-level recursive scheme together with the Mori–Tanaka method, to evaluate the ME effect of the composites. The magnitudes and trends of the solutions are in good agreement with the calculations by the finite element analysis. Based on this model, we find the optimal volume fractions of the inclusion, the ratio of the radii between the core and shell for maximum ME coupling. Further, we correlate the ME effect with the material parameters of the constituent phases and propose useful engineering guide to the develop-ment of new ME coated fibrous composites.

1. Introduction

Magnetoelectric (ME) materials, which coexist magnetic and electric orderings, have stimulated considerable scientiﬁc and technological interest in recent years for potential applications, such as ME data storage and switching, magnetic ﬁeld detectors, and electric control of magnetism (Eerenstein, Mathur, & Scott, 2006; Fiebig, 2005; Spaldin & Fiebig, 2005). How-ever, the ME effect in natural materials is rather weak and is often observed at low temperature (Astrov, 1960; Rado & Folen, 1961). Therefore, various researchers have turned to composites made of piezoelectric (PE) and piezomagnetic (PM) media to enhance the magnetoelectricity, as explained in recent reviews byNan, Bichurin, Dong, Viehland, and Srinivasan (2008)and

Srinivasan (2010). This much stronger ME effect could be realized using product property (Nan, 1994):

ME effect ¼ electric mechanical

mechanical magnetic :

It means that an applied magnetic ﬁeld causes an elastic strain in the piezomagnetic material, and this strain is translated into the electric polarization, hence electric ﬁelds, in the piezoelectric material, or vice versa.

The promise of applications have also made ME composites the topic of a number of theoretical studies (Nan et al., 2008;

Zheng et al., 2004). For example, the classical Eshelby’s equivalent inclusion approach and the Mori–Tanaka mean-ﬁeld

mod-el have been generalized to multiferroic composites byLi and Dunn (1998a, 1998b), Wu and Huang (2000), Huang and Zhou

(2004), Srinivas et al. (2006)andLiu and Kuo (2012). The analysis for local ﬁelds is available for simple microstructures such

http://dx.doi.org/10.1016/j.ijengsci.2012.08.002

⇑ Corresponding author.

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

Contents lists available atSciVerse ScienceDirect

International Journal of Engineering Science

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as a single inclusion (Huang & Kuo, 1997), laminates (Bichurin et al., 2003; Kuo et al., 2010; Liverts et al., 2010; Srinivas

et al., 2001), and periodic array of circular/elliptic ﬁbrous ME composites (Dinzart and Sabar, 2011; Kuo, 2011; Kuo and

Pan, 2011). Homogenization methods were also proposed for periodic ME ﬁbrous composites (Aboudi, 2001;

Camacho-Mon-tes et al., 2009), while numerical methods based on the ﬁnite element analysis have been developed to address ME

compos-ites with more general microstructures (Lee et al., 2005; Liu et al., 2004).

Recently, some three-phase multiferroic composites were made experimentally to enhance the ME coupling. Among

them,Nan et al. (2002)andNan et al., 2003 made a Terfenol-D/PZT/PVDF mixture and enhanced the ME coefﬁcient to

45 mV/cm.Dong et al. (2006)prepared a MnZnFe2O4/Terfenol-D/PZT laminate, and found the enhanced ME ﬁeld coefﬁcients

of up to 8–28 times of those of Terfenol-D/PZT counterpart. Gupta and Chatterjee (2009) prepared a three-phase

BaTiO3/CoFe2O4/PVDF particulate composite, and showed a maximum ME voltage around 26 mV/cmOe. Jadhav et al.

(2009)prepared a Ni0.5Cu0.2Zn0.3Fe2O4/BaTiO3/PZT combination and measured a maximum ME coefﬁcient of 975

l

V/cmOe.

For theoretical investigations on this part,Kuo (2011)andKuo and Pan (2011)estimate the overall behavior of multiferroic

composites with coated circular/elliptic ﬁbrous under generalized anti-plane deformation. Dinzart and Sabar (2011)

employed Green’s functions techniques, interfacial operators, and Mori–Tanaka’s model for solving the

magneto-electro-elastic coated inclusion problem. Later, Kuo and Wu (2012) proposed a micromechanical model, the two-level

recursive scheme in conjunction with the Mori–Tanaka method, to a core–shell-matrix particulate multiferroic composite. They showed that the solutions are in good agreement with the prediction by the ﬁnite element analysis. In the present study, we follow this similar idea to investigate the effective property of a coated ﬁbrous composites made of piezoelectric and piezomagnetic phases.

The plan of this article is organized as follows: in Section2, we formulate the basic equations for a piezoelectric-piezo-magnetic composite and define the effective properties of the heterogeneous media. In Section3we present a micromechan-ical approach to estimate the overall behavior of core–shell-matrix, three-phase multiferroic fibrous composites. We introduce the finite element analysis in Section4. Both methodologies are illustrated in Section5. We study how the ME voltage coefficient depends on the radius ratio of the core and shell, volume fractions of the fiber phase, and material prop-erties of constituent phases. Furthermore, we improve the ME coupling by tuning the material parameters, and summarize a few useful design principles.

2. Problem statement 2.1. Basic equations

Let us consider a coated fibrous composite made of piezoelectric and piezomagnetic materials as shown inFig. 1. The cylinders are infinitely long with fibers aligned in x3-direction. The composite is consisting of a continuous matrix phase,

m, in which there are embedded inhomogeneities of a circular core phase, c, and a shell phase, s, which represents a layer of coating that encloses the core phase. The radii of the core and coating are a and b, respectively, and the ratio between them is deﬁned as

c

a/b. The general constitutive laws for the rth phase are given by (seeAlshits et al., 1992, for example)

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r

ðrÞ ij ¼ C ðrÞ ijkl

e

ðrÞ kl e ðrÞ lijE ðrÞ l q ðrÞ lijH ðrÞ l ; DðrÞi ¼ e ðrÞ ikl

e

ðrÞ kl þ

j

ðrÞ il E ðrÞ l þ k ðrÞ li H ðrÞ l ; BðrÞi ¼ q ðrÞ ikl

e

ðrÞ kl þ k ðrÞ il E ðrÞ l þ

l

ðrÞ il H ðrÞ l ; ð2:1Þ

where

r

ij, Di, Bi,

e

ij, Eiand Hiare the stress, electric displacement, magnetic flux, strain, electric field and the magnetic field,

respectively. Cijklis the elastic moduli; eikland qiklare the piezoelectric and piezomagnetic constants;

j

il,

l

iland kilare the

dielectric permittivity, magnetic permeability and magnetoelectric coefﬁcient. 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 potential

u

,

and magnetic potential

w

via

e

ij¼

1

2ðui;jþ uj;iÞ; Ei¼ u;i; Hi¼ w;i: ð2:2Þ

On the other hand, the balance of linear momentum, Gauss’s law, and the condition of no magnetic poles give that the stress, electric displacement, and magnetic ﬂux satisfy the following equilibrium equations

r

ij;j¼ 0; Di;i¼ 0; Bi;i¼ 0: ð2:3Þ

These differential equations can be solved, subject to suitable interface and boundary conditions. We assume that the inter-faces are perfectly bonded, and therefore the ﬁeld quantities satisfy

srijnjt¼ 0; sDinit¼ 0; sBinit¼ 0;

suit¼ 0; s

u

t¼ 0; swt¼ 0;

ð2:4Þ where st denotes the jump in some quantity across the interface, and niis the unit outward normal to the interface.

For convenience, we rewrite the constitutive laws(2.1)in the matrix notation as (Alshits et al., 1992)

R¼ LZ; ð2:5Þ with R¼

r

D B 2 6 4 3 7 5; Z ¼

e

E H 2 6 4 3 7 5; L ¼ C et _qt e j kt q k l 2 6 4 3 7 5: ð2:6Þ

Here the superscript t denotes the transpose of the matrix. 2.2. Effective moduli

In this study, we are interested in determining the overall properties of multiferroic composites. The macroscopic prop-erties are deﬁned in terms of average ﬁelds,

hRi ¼ LhZi; ð2:7Þ

where L⁄ _{denotes the macroscopic magnetoelectroelastic coefﬁcients of the heterogeneous material, and the angular}

brackets denote the average over the representative volume element (RVE). Note that, although in each component, the magnetoelectric coefﬁcient is zero, i.e., k = 0, the coupling effect k⁄_{may be non-zero.}

Due to the linearity, the generalized strain in the rth phase for a matrix-based multiphase multiferroic composite is given by (Srinivas et al., 2006)

Zr¼ ArhZi; ð2:8Þ

where Aris the generalized strain concentration tensor of the rth phase, satisfyingPNr¼1Ar¼ I; where I is the fourth-order

identity tensor. As a result, from the average generalized stress and strain theorems, the effective moduli can be determined for a (N + 1)-phase composite as

L

¼ Lmþ

XN r¼1

frðLr LmÞAr: ð2:9Þ

Here fris the volume fraction of the rth phase. The concentration tensor can be determined by various micromechanical

models, which will be discussed in the following section. 3. Micromechanical approach

To estimate the effective moduli, L⁄

, of multiferroic composites, we ﬁrst turn to the direct Mori–Tanaka method, which approximates the coated particle problem using a composite with distinct particles representing the core and shell phases

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(Fisher and Brinson, 2001). This gives the effective properties of the core–shell-matrix mutliferroic as Eq.(2.9), with the concentration tensor for the core (j = 1) and shell (j = 2)

Aj¼ A dil j fmI þ fcA dil 1 þ fsA dil 2 1 ; j ¼ 1; 2: ð3:1Þ Here Adilj ¼ I þ SjL1m Lj Lm h i1 ; ð3:2Þ

Sjis the magnetoelectroelastic Eshelby tensor, which is a function of the magnetoelectroelastic moduli of matrix, the shape

and orientation of the jth inclusion, and is described by (Li and Dunn, 1998b)

SMnAb¼ 1 8pLiJAb R1 1 R2p 0 ½GmJinðzjÞ þ GnJimðzjÞdhdn3; M ¼ 1; 2; 3; 2R₁1 R₀2pG4JinðzjÞdhdn3; M ¼ 4; 2R₁1 R₀2pG5JinðzjÞdhdn3; M ¼ 5: 8 > > < > > : ð3:3Þ

In the above equation, zi= ni/ai(no summation on i), aiis the semi-axis of size, and n1and n2can be expressed in terms of n3

and h by n1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n23 q cos h and n2¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n23 q

sin h. In addition GMJin¼ ziznK1MJðzjÞ, where K1MJ is the inverse of KJR= ziznLiJRn.Li

and Dunn (1998a)have obtained the closed-form expressions of magnetoelectroelastic Eshelby’s tensors for the aligned

elliptic cylinder inclusion in a transversely isotropic medium. For the coated ﬁbrous composites with arbitrary crystal sym-metry as we discussed in this work, we resort to Gauss quadrature numerical method to calculate SMnAb. The integral(3.3)

then is approximated by the weighted sum of function values at certain integration points (Li, 2000).

However, we will show later that this prediction deviates largely from that determined by the ﬁnite element analysis. Therefore, the direct Mori–Tanaka method is not good in estimating the coupling constants. We now turn to another ap-proach, the two-level recursive scheme with the Mori–Tanaka technique. The basic concept of the two-level recursive scheme is that the matrix sees coated particles that are themselves composites. This procedure was ﬁrst used to predict the behavior of viscoelastic composites containing multiphases of coated inclusions (Friebel et al., 2006). At the deepest level, each coated particulate inclusion is seen as a two-phase composite, which, once homogenized, plays the role of a homogeneous inclusion for the matrix material (highest level).

Further, at each level, we employ the Mori–Tanaka approach in prediction the effective moduli of the corresponding two-phase composite. Based on this model, at the deepest level, the coated inclusions are seen as a two-two-phase composite with effective moduli L sc¼ Lsþ fc fi ðLc LsÞAc: ð3:4Þ

Here, the subscripts c, s, and i represent core, shell and inclusion (core plus shell), respectively. The concentration tensor Ac

can be determined as Ac¼ fiAdilc fsI þ fcAdilc

1

; ð3:5Þ

with the dilute concentration tensor Adilc given by

Adilc ¼ I þ ScL1s ðLc LsÞ

h i1

: ð3:6Þ

Here Scis the generalized Eshelby tensor for the core phase, which is a function of the property of the shell, and the shape

and orientation of the core phase.

At the highest level, the effective coated ﬁbers play the role of reinforcements and, similarly, we have the effective behavior

L

¼ Lmþ fiðLsc LmÞAsc: ð3:7Þ

Again the concentration tensor can be determined as Asc¼ Adilsc fmI þ fcAdilsc

1

; ð3:8Þ

with the dilute concentration tensor Adil sc ¼ I þ SscL1m L sc Lm h i1 : ð3:9Þ

Here Sscis the generalized Eshelby tensor for effective coated ﬁbers, which is a function of the moduli of the matrix, and the

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4. Finite element method

In this section, we introduce the finite element method which is used for comparison with the above micromechanical solutions. 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 in regular arrays in two dimensions (seeKittel, 2005, for instance). Here we concentrate on the two lattices, square and hexagonal arrangements (Fig. 2).

Further, due to the periodicity in the composite structure, the displacement, ui, electric potential,

u

, and the magnetic

potential,

w

, in any point of the unit cell can be expressed in terms of those at an equivalent point in another RVE such that the periodic boundary conditions

U

ðd; x2;x3Þ ¼

U

ðd; x2;x3Þ þ h

U

;1i2d;

U

ðx1;d; x3Þ ¼

U

ðx1;d; x3Þ þ h

U

;2i2d;

U

ðx1;x2;dÞ ¼

U

ðx1;x2;dÞ þ h

U

;3i2d

ð4:1Þ

are satisﬁed for a square lattice. HereUis the component of ui,

u

, or

w

, and 2d is the length of the unit cell. The comma in the

subscript denotes the partial derivative. Similarly, the periodic boundary conditions for a hexagonal lattice are

U

ðd; x2;x3Þ ¼

U

ðd; x2;x3Þ þ h

U

;1i2d;

U

x1; ffiffiffi 3 p d; x3 ¼

U

x1; ffiffiffi 3 p d; x3 þ h

U

;2i2 ffiffiffi 3 p d;

U

ðx1;x2;dÞ ¼

U

ðx1;x2;dÞ þ h

U

;3i2d: ð4:2Þ

In order to evaluate the effective coefﬁcients of the above periodic multiferroic composite, the strain,

e

ij, electric ﬁeld, Ei,

and magnetic ﬁeld states, Hiare applied individually to the unit cell. The periodic 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 hU,ii in Eqs.( 4.1)or

(4.2), all other components are made equal to zero. Then each effective constant can be determined by(2.7). We perform the

three-dimensional ﬁnite element analysis using the software COMSOL Multiphysics. 5. Results and discussion

As a numerical example, we take a composite made of PE cores coated PM shell in a PM matrix. For the piezoelectric material, we ﬁrst choose the widely used BaTiO3(BTO) ceramic as the core phase. For the piezomagnetic material we choose

CoFe2O4(CFO) as the shell phase while Terfenol-D (TD) as the matrix phase. They are all transversely isotropic, i.e. with 6 mm

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symmetry. For convenience, we denote the composite as BTO/CFO/TD. The independent material constants of these constit-uents are given inTable 1in Voigt notation, where the x1x2plane is isotropic and the poling direction/magnetic axis is along

the x3-direction.

In this study, a material property of particular interest is the ME voltage coefﬁcient

a

E;ij¼ kij=

j

ij(no summation), where

k_ij

j

ij

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

a

E;ij;which relates the overall electric field generated in the composite with the applied magnetic field, is the figure of merit

for magnetic ﬁeld sensors.

Fig. 3shows how the ME voltage coefﬁcient depends on the inclusion volume fraction, fi, and the ratio of radii,

c

, for the

BTO/CFO/TD three-phase multiferroic composite. 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 ME coupling is mediated by the elastic interaction. In the micromechanical approach, there is no upper limit on the volume fractions, since Mori–Tana-ka’s model is a mean-field theory. On the other hand, the finite element analysis is estimated for discrete volume fractions and stops around fi=

p

/4 and fi¼

p

=2

ffiffiffi 3 p

for the square and hexagonal arrays, respectively, when the inclusions begin to touch each other. The prediction of the two-level recursive scheme together with the Mori–Tanaka’s approach is in good agreement with the results of the ﬁnite element analysis. The maximum ME voltage coefﬁcient

a

E;11 is 7.8359 V/cmOe

at volume fraction fi= 0.86 with

c

= 0.7 (Fig. 3(a)). On the other hand, the maximum

a

E;33is 1.3425 V/cmOe at volume

fraction fi= 1 with

c

= 0.3, which corresponds to the two-phase composite BTO/CFO (Fig. 3(b)). 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 model allows the inclusion to fulﬁll the matrix.

Fig. 3(a) and (b) also compare with the effective ME voltage coefﬁcients predicted byKuo and Pan (2011)who used

mul-tipole expansion technique. Still, the overall magnitudes and trends agree well among predictions based on the microme-chanical model, ﬁnite element analysis, and Kuo and Pan’s model. Further,Fig. 3compares the overall moduli with those calculated by the direct Mori–Tanaka method for the case

c

= 0.8. It is observed that the prediction deviates largely from those determined by the ﬁnite element analysis. Therefore, the direct Mori–Tanaka method is not good in estimating the coupling constants, although calculations show that they evaluate elastic stiffness well.

Finally, for comparison,Fig. 3also shows the effective moduli of the composite made by the corresponding two-phase medium (BTO/TD). It shows that the ME voltage coefﬁcients in the coated ﬁbrous composite can be indeed increased com-pared to this two-phase counterpart.

Next, we study how the effective ME voltage coefﬁcient depends on the elastic moduli, CPEand CPM, dielectric permittivity,

j

PEand

j

PM, and magnetic permeability,

l

PEand

l

PM, of the PE and PM materials, piezoelectric constant, ePE, of the PE

mate-rial, and piezomagnetic coefﬁcient, qPM, of the PM material. For ease of comparison, we choose the material properties of BTO

and CFO as the reference and deﬁne the normalized materials properties of the PE and PM phases as Cr;CoreI ¼ CPEðCBTOÞ1; Cr;ShellI ¼ CPMðCCFOÞ1; Cr;MatrixI ¼ CPMðCCFOÞ1

and, likewise, are er,Core, qr,Shell, qr,Matrix,

j

r,Core,

j

r,Shell,

j

r,Matrix,

l

r,Core,

l

r,Shell,

l

r,Matrix. Note that all the components of the

mate-rial constant are magniﬁed simultaneously for simplicity. Below we numerically compute the ME voltage coefﬁcients

a

E;11

Table 1

Material parameters of BaTiO3 (eFunda), CoFe2O4(Li and Dunn, 1998b), Terfenol-D (Engdahl, 2000; Nan et al., 2001), LiNbO3(eFunda), and PZT-5J (eFunda:Nan et al., 2001).

Property BaTiO3 CoFe2O4 Terfenol-D LiNbO3 PZT-5 J

C11(GPa) 150.37 286 8.541 203 82.3 C12(GPa) 65.63 173 0.654 52.9 34.1 C13(GPa) 65.94 170.3 3.91 74.9 30.2 C33(GPa) 145.52 269.5 28.3 243 59.8 C44(GPa) 43.86 45.3 5.55 59.9 21.3 C66(GPa) 43.37 56.5 18.52 74.9 24.1 C14(GPa) 0 0 0 8.99 0 C56(GPa) 0 0 0 8.985 0 j11(nC2/Nm2) 9.87 0.08 0.05 0.39 14.53 j33(nC2/Nm2) 11.08 0.093 0.05 0.26 10.12 l11(lNs2/C2) 5 590 8.644 5 1.26 l33(lNs2/C2) 10 157 2.268 10 1.26 e15(C/m2 ) 11.4 0 0 3.7 14.26 e16(C/m2 ) 0 0 0 2.534 0 e21(C/m2 ) 0 0 0 2.538 0 e31(C/m2 ) 4.32 0 0 0.19 10.45 e33(C/m2₎ _17.36 ₀ ₀ _1.31 _16.58 q15(N/Am) 0 550 155.56 0 0 q31(N/Am) 0 580.3 5.7471 0 0 q33(N/Am) 0 699.7 270.1 0 0

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and

a

E;33and their dependence on the normalized material properties of core (PE), shell (PM), and matrix (PM) phases. These

results give important guidelines for practical designs of ME coated ﬁbrous composites.

Figs. 4 and 5show the contours of the normalized effective ME voltage coefﬁcients

a

E;11=

a

0E;11and

a

E;33=

a

0E;33of a PE/PM/

PM composite at the inclusion volume fraction fi= 0.5 and the ratio of the radii

c

= 0.8, where the ME voltage coefﬁcients

a

0

E;11¼ 0:0336 V=cmOe and

a

0E;33¼ 0:925 V=cmOe of BTO/CFO/CFO composite are chosen as the unit for ease of comparison.

InFigs. 4(a) and5(a), the vertical and horizontal axes represent the normalized elastic constants of core and shell phases,

respectively, while the variation of matrix’s elastic constant is shown by different subplots (a-1,a-2,a-3). The other material

Fig. 3. The predicted ME voltage coefficient vs the volume fraction of inclusion, fi, and radius ratio,c, of a BTO/CFO/TD coated fibrous composite: (a) ME voltage coefficienta

E;11and (b) ME voltage coefﬁcientaE;33. In both (a) and (b), solid lines are based on the two-level recursive scheme with the Mori– Tanaka model, while discrete point symbols are based on the ﬁnite element analysis.

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constants are fixed as those of BTO for PE phase or CFO for PM phase. It is observed that the ME voltage coefficient increases when any one of the core, shell or matrix’s elastic constant decreases. Therefore, softer PE and PM materials are preferred for improving the ME voltage coefficients of PE/PM/PM three-phase fibrous composites.Figs. 4(b) and5(b) show the contours of the relative effective ME voltage coefficient versus the piezoelectric and piezomagnetic constants in linear scale. For a fixed

Fig. 4. The predicted in-plane ME voltage coefﬁcient vs different material parameters. The composite is made of PE core phase, PM shell and matrix phases. The volume fraction of inclusion fi= 0.5, and the radius ratioc= 0.8. The normalized ME voltage coefﬁcienta

E;11=a 0

E;11vs (a) normalized elastic constants Cr,Core, Cr,Shelland Cr,Matrix; (b) normalized piezoelectric coefﬁcient of PE core er,Coreand piezomagnetic coefﬁcient of PM shell qr,Shelland matrix qr,Matrix; (c) normalized dielectric permittivityjr,Core,jr,Shellandjr,Matrix; (d) normalized magnetic permeabilitylr,Core,lr,Shellandlr,Matrix.

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normalized piezoelectric coefﬁcient, er,Core, the absolute ME voltage coefﬁcient

a

E;11decreases ﬁrst and increases after certain

minimum as the shell or matrix’s piezomagnetic coefficient increases. However, for fixed normalized piezomagnetic coeffi-cients of the shell and matrix phases and as the piezoelectric coefficient increases, the coupling increases first and decreases after certain optimal. Therefore, nontrivial optimal piezoelectric coefficient and lower or higher piezomagnetic constants are preferred for improving the ME effect

a

E;11. On the other hand, for the ME voltage coefﬁcient

a

E;33; the behavior is quite

different (Fig. 5(b)). It is observed that the ME voltage coefficient increases when any of the core’s piezoelectric constant, shell or matrix’s piezomagnetic constant increases. Therefore, higher piezoelectric coefficient or piezomagnetic coefficient

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are preferred for improving the ME voltage coefficient in the axial direction.Figs. 4(c) and5(c) show the contours of the relative ME coupling versus the normalized electric permittivity of PE and PM phases. We observe that the smaller the core’s permittivity, the larger the ME voltage coefficients. For the PM shell or matrix’s permittivity, however, they only slightly influence the ME effect.Figs. 4(d) and5(d) show the contours of the relative ME voltage constants versus the normalized

Fig. 5. The predicted out-of-plane ME voltage coefﬁcient vs different material parameters. The composite is made of PE core phase, PM shell phase and PM matrix phase. The volume fraction of inclusion fi= 0.5, and the radius ratioc= 0.8. The normalized ME voltage coefﬁcienta_E;33=_a0

E;33vs (a) normalized elastic constants Cr,Core, Cr,Shelland Cr,Matrix; (b) normalized piezoelectric coefﬁcient of PE core er,Coreand piezomagnetic coefﬁcient of PM shell qr,Shelland matrix qr,Matrix; (c) normalized dielectric permittivityjr,Core,jr,Shellandjr,Matrix; (d) normalized magnetic permeabilitylr,Core,lr,Shellandlr,Matrix.

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magnetic permeability in logarithmic scale. For the in-plane ME voltage coefﬁcient

a

E;11;we observe that increasing the PE

core’s or PM shell’s magnetic permeabilities largely enhances the coupling constant, and on the contrary, increasing the PM matrix’s magnetic permeability lowers the ME voltage coefﬁcient. Therefore, a large magnetic permeability of the PE core and PM shell and a small magnetic permeability of the PM matrix phase are preferred for improving the in-plane ME voltage coefﬁcient

a

E;11: However, the out-of-plane coupling constant

a

E;33 is almost independent of magnetic permeability

(Fig. 5(d)). For clearness, the contours are shown as the change rate of the ME coefﬁcient,

a

E;33

a

0E;33

=

a

0 E;33. Fig. 5. (continued)

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Motivated by the above study, we study a ME composite of LiNbO3(LNO, 3 m symmetry), CoFe2O4, and Terfenol-D as the

core, shell, and matrix phases (Fig. 6(a)), since LNO has lower dielectric permittivity and the matrix TD has lower elastic stiffness and magnetic permeability. The material constants of LNO are listed in Table 1. For this ME composite, the

maximum is attained at the volume fraction fi= 0.86 and ratio of the radius

c

= 0.8 where ME voltage coefﬁcient

a

E;11¼ 44:9393 V=cmOe. For the out-of-plane ME voltage coefﬁcient

a

E;33;we choose a composite made of PZT/TD/CFO.

This is because PZT (6 mm symmetry) has lower elastic constant and higher piezoelectric coefﬁcient, while the shell TD is softer. The maximum occurs at the volume fraction fi= 0.06 and ratio of the radius

c

= 1.0 with coupling constant

a

E;33¼ 4:6134 V=cmOe.

Fig. 6. The predicted ME voltage coefficient vs the volume fraction of the inclusion, fi, and radius ratio,c, of a (a) LNO/CFO/TD coated fibrous composite; (b) PZT/TD/CFO coated fibrous composite. In both (a) and (b), lines are based on the two-level recursive scheme with the Mori–Tanaka model, while discrete point symbols are based on the finite element analysis.

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6. Concluding remarks

In this work, a micromechanical approach, the two-level recursive scheme in conjunction with the Mori–Tanaka model, is adopted to estimate the effective moduli of a coated fibrous composites made of PE and PM phases. We have used it to study the dependence of a particular material property of interest, the ME voltage coefficient, on the volume fraction of the inclu-sion, the ratio of the radii between the core and shell, and the material parameters of the PE and PM phases. The results are compared with the finite element analysis and the semi-analytical method proposed byKuo and Pan (2011). The magnitudes and trends among them are in good agreement. In addition, for a PE/PM/PM coated fibrous composite with fixed volume fraction and radius ratio, we show that softer materials are desirable for improving the ME coupling. Further, for the in-plane ME voltage coefficient

a

E;11, it is desirable to have larger (smaller) magnetic permeability in the PM shell (PM matrix), and

smaller dielectric permittivity, but larger magnetic permeability in the PE core. Besides, there exists optimal values of the piezoelectric constant. On the other hand, for the out-of-plane ME voltage coefﬁcient

a

E;33, it is desirable to have smaller

dielectric permittivity of PE phase, and higher piezoelectric (piezomagnetic) coefﬁcient of the PE (PM) phase. The magnetic permeability has no effect on the ME coupling

a

E;33.

In the past decades, some three-phase composites consisting of piezoelectric and magnetostrictive/piezomagnetic phases have been investigated in experiments. However, those composites are either in the form of laminates (Dong et al., 2006) or particulate composites (Gupta and Chatterjee, 2009; Jadhav et al., 2009; Nan et al., 2002, 2003). To the best of authors’ knowledge, there is no known experimental result for core–shell-matrix ME ﬁbrous composites. Therefore, we believe that this framework and principles will stimulate new experimental works and shed new light on magnetoelectric coated ﬁbrous composites.

Acknowledgement

We are glad to acknowledge the ﬁnancial support from the National Science Council, Taiwan, under Contract No. NSC 100–2628-E-009–022-MY2.

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