壓電壓磁複合材料電磁耦合效應之提升

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壓電壓磁複合材料電磁耦合效應之提升

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計畫主持人：郭心怡

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計畫參與人員：郭祐旻、彭晟祐、凌毓翔、游舒含

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中 華 民 國 102 年 10 月 10 日

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500 字為限）

多鐵複合材料近十年來成為微觀力學的新興課題，然而過去的研究材料極化

方向假設與纖維方向同向，而對於任意極化方向與三相複合材料的課題，著

墨甚少。本研究探討極化方向與三相組合對於圓柱狀內含物複合材料的磁電

彈多重物理耦合場之影響，並預測最佳之磁電耦合效應所對應的極化方向或

組合方式，提供材料學家製造多鐵材料的設計依據。

E¤ective magnetoelectricity of coated …brous

composites of piezoelectric and piezomagnetic

phases

Hsin-Yi Kuo, Cheng-You Peng

Department of Civil Engineering,

National Chiao Tung University,

Hsinchu 30010, Taiwan

October 10, 2013

Abstract

This paper studies the e¤ective magnetoelectric behaviors of three-phase, core-shell-matrix …brous composites of piezoelectric and piezomagnetic phases. A micromechancial model, the two-level recursive scheme together with the Mori-Tanaka’s method, is proposed to investigate the magnetoelectricity of the coated …brous multiferroic composites. The magnitudes and trends of the solutions are in good agreement with the calculations by the …nite element analysis. Based on this micromechanical approach, we …nd that, for the case of PE/PM/PM (core/shell/matrix) multiferroic compoiste, with a coating

propriate for the inhomogeneity, the e¤ective magnetoelecitrc coupling can be enhanced many-fold as compared to the noncoated counterpart.

1

Introduction

This work is concerned with the magnetoelectric (ME) e¤ect of a coated …brous com-posite made of piezoelectric and piezomagnetic phases. ME materials are particular exciting since they posess the coupling between the electric and magnetic …elds. This make them appealing and promising for a wide range of applications, such as ME data storage and switching, magnetic …eld detectors, and electric control of magnetism, etc. (Fiebig, 2005; Spaldin and Fiebig, 2005; Eerenstein et al., 2006; ) However, the ME e¤ect in natural materials is rather weak and is often observed at low temperature (Astrov, 1960; Rado and Folen, 1961). Therefore, various researchers have turned to composites made of piezoelectric and piezomagnetic media to enhance the magneto-electricity, as explained in recent reviews by Nan et al. (2008) and Srinivasan (2010). This much stronger ME e¤ect could be realized using product properties: an applied magnetic …eld creates a strain in the piezomagnetic material which in turn creates a strain in the piezoelectric material, resulting in an electric polarization.

The promise of applications, and the indirect coupling through strain have also made ME composites the topic of a number of theoretical and experimental studies (Nan et al., 2008; Zheng et al., 2004). Among them, the classical Eshelby’s equivalent inclusion approach and the Mori-Tanaka mean-…eld model have been generalized to multiferroic composites by Li and Dunn (1998a, b), Huang (1998), Wu and Huang (2000), Huang and Zhou (2004) and Srinivas et al. (2006). The analysis for local …elds is available for simple microstructures such as a single inclusion (Huang and Kuo, 1997), laminates ( Srinivasan et al., 2001; Bichurin et al., 2003; Kuo et al., 2010), and

periodic array of circular/ellptic …brous ME composites (Kuo, 2010; Kuo and Pan, 2011; Dinzart and Sabar, 2011). Homogenization methods were also proposed for periodic ME …brous composites (Aboudi, 2001; Camacho-Montes et al., 2009), while numerical methods based on the …nite element analysis have also been developed to address ME composites with more general microstructures (Liu et al., 2004; Lee et al., 2005).

Recently, some three-phase multiferroic composites were made experimentally to enhance the ME coupling. Nan et al. (2002, 2003) made a Terfenol-D/PZT/PVDF mixture, and the measured ME coe¢ cient was enhanced to 45mV/cm. Dong et al. (2006) prepared a MnZnFe2O4/Terfenol-D/PZT laminate, and found the enhanced

ME …eld coe¢ cients of up to 8-28 times of those of Terfenol-D/PZT counterpart. Gupta and Chatterjee (2009) prepared a three-phase BaTiO3/CoFe2O4/PVDF

par-ticulate composite, and showed a maximum ME voltage around 26mV/cmOe. Jadhav et al. (2009) prepared a Ni0:5Cu0:2Zn0:3Fe2O4/BaTiO3/PZT combination and

mea-sured a maximum ME coe¤cient of 975 V/cmOe. For theoretical investigations, Kuo (2010) and Kuo and Pan (2011) estimate the overall behavior of multiferroic com-posites with coated ciruclar/elliptic …brous under generalized anti-plane deformation. Dinzart and Sabar (2011) employed Green’s functions techniques, interfacial opera-tors, and Mori-Tanaka’s model for solving the magnet-electro-elastic coated inclusion problem.

In the work of Friebel et al. (2006), to estimate the overall property of viscoelastic composites with coated inclusions, they proposed a two-level recursive scheme and two-step method together with Mori-Tanaka or double-inclusion mean-…eld models as the homogenization method. Later, Kuo and Wu (2012) applied the two-level re-cursive scheme in conjunction with the Mori-Tanaka’s model to a core-shell-matrix particulate multiferroic composite. They showed that the solutions are in good

agree-ment with the prediction by the …nite eleagree-ment anaylsis. In this paper, we adopt the similar method to investigate the e¤ective property of the coated …brous composites made of piezoelectric and piezomagnetic phases.

This article is organized as follows: In Section 2, we formulate the basic equations for a piezoelectric-piezomagnetic composite and de…ne the e¤ective properties of the composite. In Section 3 we propose a micromechanical method to estimate the overall behavior of core-shell-matrix, three-phase multiferroic composites. We introduce the …nite element analysis in Section 4. Both methodologies are illustrated in Section 5. We study how the magnetoelectric voltage coe¢ cent depends on the radious ratio of the core and shell, volume fractions of the …ber phase, and material properties of constituent phases. Furthermore, we improve the ME coupling e¤ect by tuning the material parameters, and summarize a few useful design principles.

2

Problem statement

2.1

Basic equations

Let us consider a three-phase, coated …brous composite made of piezoelectric and piezomagnetic materials as shown in Figure 1. The cylinders are in…nitely long with …bers aligned in x3 direction. The composite is consisting of a continuous matrix

phase, m, in which there are embedded inhomogeneities of a circular shell phase, c, and a shell phase, s, which represents a layer of cpating that encapsulates each particle of the core phase. The radii of the core and coating are a and b, respectively, and the ratio between them is de…nsed as a=b: The general constitutive laws for

the rth phase are given by (see Alshits et al., 1992, for example) (r) ij = C (r) ijkl" (r) kl e (r) lijE (r) l q (r) lijH (r) l ; D(r)_i = e(r)_ikl"(r)_kl + _il(r)E_l(r)+ (r)_li H_l(r); B_i(r) = q_ikl(r)"(r)_kl + _il(r)E_l(r)+ (r)_il H_l(r); (2.1)

where ij; Di; Bi; "ij; Ei and Hi are the stress, electric displacement, magnetic ‡ux,

strain, electric …eld and the magnetic …eld, respectively. Cijkl is the elastic moduli;

eikl and qikl are the piezoelectric and piezomagnetic constants; il; il and il are the

dielectric permittivity, magnetic permeability and magnetoelectric coe¢ cient. The symmetry conditions satis…ed by the moduli are given by Nye (1985).

The strain "ij, electric …eld Ei, and magnetic …eld Hi are respectively de…ned by

the displacement ui; electric potential ', and magnetic potential via

"ij =

1

2(ui;j+ uj;i) ; Ei = ';i; Hi = ;i: (2.2)

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 intensity satisfy the following equilibrium equations

ij;j = 0; Di;i = 0; Bi;i= 0: (2.3)

These di¤erential equations can be solved, subject to suitable interface and boundary conditions. We assume that the interfaces are perfectly bonded, and therefore the

…eld quantities satisfy

[[ ijnj]] = 0; [[Dini]] = 0; [[Bini]] = 0;

[[ui]] = 0; [[']] = 0; [[ ]] = 0; (2.4)

where [[ ]] denotes the jump in some quantity across the interface, and ni is the unit

outward normal to the interface.

For simplicity, we write the above constitutive laws (2.1), strain-displacement (2.2) and equilibrium equations (2.3) can be rewritten in more compact form as (Alshits et al., 1992) iJ = LiJ M nZM n; ZM n = UM;n; iJ;i= 0; (2.5) where iJ = 8 > > > > < > > > > : ij; J = 1; 2; 3; Di; J = 4; Bi; J = 5; ZM n = 8 > > > > < > > > > : "mn; M = 1; 2; 3; En; M = 4; Hn; M = 5; UM = 8 > > > > < > > > > : um; M = 1; 2; 3; '; M = 4; ; M = 5: (2.6)

The magnetoelectroelastic moduli are expressed as LiJ M n= 8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : Cijmn; J; M = 1; 2; 3; eijn; M = 4; J = 1; 2; 3; qijn; M = 5; J = 1; 2; 3; eimn; J = 4; M = 1; 2; 3; in; J = 4; M = 4; in; J = 4; M = 5; qimn; J = 5; M = 1; 2; 3; in; J = 5; M = 4; in; J = 5; M = 5; (2.7)

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.

2.2

E¤ective moduli

In this study, we are interested in determining the overall properties of the multiferroic composites in terms of their microstructure. The macroscopic properties are de…ned in terms of average …elds,

h iJi = LiJ M nhZM ni ; (2.8)

where L denotes the macroscopic magnetoelectroelastic coe¢ cients of the heterge-neous material, and the angular brackets denote the average over the representative volume element (RVE; unit cell in the case of periodic composites),

h iJi = 1 V Z V iJdxi; hZM ni = 1 V Z V ZM ndxi:

Here, V is the area of RVE. Note that, although in each component, the magneto-electric coe¢ cient is zero, i.e., = 0;the coupling e¤ect may be non-zero.

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

Z_{M n}(r) = A(r)_{M nAbhZ}Abi ; (2.9)

where A(r)_{M nAb}is the generalized strain concentration tensor of the r-th phase, satisfying

N

X

r=1

A(r)_{M nAb} = IJ iAb; (2.10)

where IJ iAb is the fourth-order identity tensor. As a result, from the average

gen-eralized stress and strain theorems, the e¤ective moduli can be determined for a (N + 1)-phase composite as L_{iJ Ab} = L(m)_{iJ Ab}+ N X r=1 fr L(r)_{iJ M n} L(m)_{iJ M n} A(r)_{M nAb}: (2.11)

Here f is the volume fraction of the inclusion, and the superscripts m and r denote the matrix and the r-th phase, respectively. The concentration tensor can be determined by various micromechanical models.

3

Micromechanical approach

To estimate the e¤ectifve moduli of multiferroic composites, we …rst 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. The key assumption of the Mori-Tanaka method, which is essentially a mean-…eld method, is

that the average …eld in the rth inclusion of the heterogeneous material is equivalent to the …eld in a single partivle embedded in an in…nite mideium, with the unknown average …eld in the matrix applied at the boundary. This gives the e¤ective properties of the core-shell-matrix mutliferroic as Eq. (2.11). Here, the concentration tensor for the core (j = 1) or shell (j = 2) is

A(j)_{M nAb} = Adil (j)_{M nJ i} f(m)IJ iAb+ f(1)A dil (1) J iAb + f (2)_Adil (2) J iAb 1 ; j = 1; 2; (3.1) where Adil (j)_{M nJ i} =hIM nAb+ S (j) M nLk(L (m) LkiJ) 1 _L(j) iJ Ab L (m) iJ Ab i 1 : (3.2)

Here SM nAb 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 by (Li and Dunn, 1998b)

SM nAb = 1 8 LiJ Ab 8 > > > > < > > > > : R1 1 R2 0 [GmJ in(zi) + GnJ im(zi)] d d 3; M = 1; 2; 3; 2R1₁R₀2 G4J in(zi)d d 3; M = 4; 2R1₁R₀2 G5J in(zi)d d 3; M = 5: (3.3)

In the above equation, zi = i=ai (no summation on i), ai is the semi-axis of size

and 1 and 2 can be expressed in terms of 3 and by 1 =

p

1 2₃cos and

2 =

p

1 2₃sin : In addition GM J in = ziznKM J1(z), where K 1

M J is the inverse of

KJ R= ziznLiJ Rn: 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 …brous composites with arbitrary crys-tal symmetry, we resort to Gauss quadrature numerical method to calculate SM nAb.

certain integration points (Li, 2000a).

However, we will show later that this prediction deviates largely from those de-termined by the …nite element analysis. Therefore, the direct Mori-Tanaka method is not good in estimating the coupling constants. We now turn to another appraoch, the two-level recursive scheme in conjunction with the Mori-Tanaka technique. The two-level recursive scheme is based on the idea that the matrix sees coated particles that are tmeselves composite. This procedure was …rst used to predict the behavior of viscoelastic composites containing multiple phases of coated inclusions. As illus-trated in Fig. 2, each coated particle inclusion is seen (deepest level) as a two-phase composite, which, once, homogenized, plays the role of a homogeneous inclusion for the matrix material (highest level).

Further, at each levle, we employ the Mori-Tanaka appraoch in predictin the e¤ective moduli of te the corresponding two-phase composites. Using this model, at the deepest level, the coated inlcusions are seen as a two-phase composite with e¤ective moduli L_{iJ Ab}(sc) = L(s)_{iJ Ab}+f (c) f(i) L (c) iJ M n L (s) iJ M n A (c) M nAb: (3.4)

Here, the superscripts c; s; and i represent core, shell and inclusion (core plus shell), respectively. The concentration tensor A(c)_{M nAb} can be determined as

A(c)_{M nAb} = f(c)Adil (c)_{M nJ i} f(s)IJ iAb + f(c)A dil (c) J iAb

1

; (3.5)

with the dilute concentration tensor Adil (c)_{M nJ i} given by

Adil (c)_{M nJ i} =hIM nAb+ S (c) M nLk(L (s) LkiJ) 1 _L(c) iJ Ab L (s) iJ Ab i 1 : (3.6)

At the highest level, the e¤ective coated …bers plya the role of reinfocements and, similarly, we have the e¤ective behavior

L_{iJ Ab} = L(m)_{iJ Ab}+ f(i) L_{iJ M n}(sc) L(m)_{iJ M n} A(sc)_{M nAb}: (3.7)

Again the concentration tensor can be dtermined as

A(sc)_{M nAb} = Adil (sc)_{M nJ i} f(m)IJ iAb+ f(c)Adil (sc)_{J iAb} 1

; (3.8)

with the dilute concentration tensor

Adil (sc)_{M nJ i} = h

IM nAb+ S_{M nLk}(sc) (L(m)_LkiJ) 1 L_{iJ Ab}(sc) L(m)_{iJ Ab}

i 1

: (3.9)

Here, S_{M nLk}(sc) , is the generalized Eshelby tensor for e¤ecitve coated particles, which is a function of the moduli of the matrix and the shape and orientation of the coated …bers (coreplus shell).

4

Finite element method

In this section, we introduce the …nite element method which is used for comparison with the above micromechanical solutions. We …rst choose an appropriate represen-tative volume element (RVE), a periodic unit cell, which captures the major features of the underlying microstructure. There are …ve possible ways of packing cylinders in regular arrays in two dimensions (See Kittel, 2005, for instance). Here we concentrate on the two lattices, square and hexagonal arrangements (Fig. 2). A square packing is more frequently employed than a hexagonal packing in the literature, and in the case of conduction, square symmetry and transverse isotropy become identical (Perrins

et al., 1979.). However, in our case of magnetoelectroelasticity, it lacks the trans-verse isotropy in that most unidirectional composites possess owing to the random distribution of …bers in the matrix over the cross-section perpendicular to …bers (Li, 2000).???(material symmetry 3mm)

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

the electric potential ' and the magnetic potential 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

UM(d; x2; x3) = UM( d; x2; x3) +hUM;1i 2d;

UM(x1; d; x3) = UM(x1; d; x3) +hUM;2i 2d; (4.1)

UM(x1; x2; d) = UM(x1; x2; d) +hUM;3i 2d;

are satis…ed for the square lattice. Here UM is de…ned in (2.6) 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

UM(d; x2; x3) = UM( d; x2; x3) +hUM;1i 2d; UM x1; p 3d; x3 = UM x1; p 3d; x3 +hUM;2i 2 p 3d; (4.2) UM(x1; x2; d) = UM(x1; x2; d) +hUM;3i 2d:

In order to evaluate the e¤ective coe¢ cients of the above periodic multiferroic composite, the strain "ij, electric …eld Ei, and magnetic …eld states Hi are 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 …eld, or magnetic …eld hUM;ii in Eq. (4.1), all other components are made equal to zero.

Then each e¤ective constant be determined by (2.8). We perform the …nite 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 …rst 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 transversily isotropic, i.e., with 6mm symmetry. For convenience, we denote the com-posite as BTO/CFO/TD. The independent material constants of these constituents are given in Table 1 in Voigt notation, where the x1x2plane is isotropic and the poling

direction/magnetic axis is along the x3-direction. Note that in all materials, the ME

coe¢ cients are zero, i.e. ij = 0:

In our study, we are particularly interested in the e¤ective magnetoelectric (ME) response. The induced voltage is proportional to the applied magnetic …eld and the constant of proportionality is the e¤ective ME voltage coe¢ cient. It combines the coupling and dielectric coe¢ cients, and is de…ned by

E;ij = ij= ij; no summation. (5.1)

Figure 3 shows how the ME voltage coe¢ cients depend on both the inclusion volume fraciotn, fi; and the ratio of radii, ; for the BTO/CFO/TD three-phase

multiferroic composite. for this composite. In the micromechanical approach, there is no upper limit on the volume fractions, since Mori-Tanaka’s model is a mean-…eld theory. On the other hand, the …nite element analysis is estimated for discrete volume

fractions and stops around f = =4 and f = =2p3 for the square and hexagonal arrays, respectively, when the inclusions touch. The ratio of the radius between the circular …br and the coating shell is de…ned as a=b: It is obvious that if = 0; then a = 0: In other words, there is no …ber phase. On the other hand, if = 1; it means that there is no coating shell. The prediction of the Mori-Tanaka’s approach is in good agreement with the result of the …nite element analysis. The maximum ME voltage coe¢ cient _E;11 is xxxxV/cmOe at volume fraction f = 0:xx, while the maximum _E;33 = xxxxV/cmOe at volume fraction f = 0:xx: 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 ful…ll the matrix. In addition a square array lacks the transversely isotropy that this composite possesses (Li, 2000b).

Further, Fig. 3 compares the overall moduli with those predicted by the direct Mori-Tanaka method for the case = 0:8: It is observed that the prediction devi-ates largely from those determined by the …nite element analysis. Therefore, the direct Mori-Tanak method is not good in estimating the coupling constants, although calculations show that they evaluate elastic sti¤ness well.

Finally, Fig. 3(a) also compares the e¤ective moduli with the prediction by Kuo and Pan (2011). Kuo and Pan considered multiferroic composites with coated circular …bers under anti-plane shera with in-plane electric-magnetic …elds.

We now turn to study how the e¤ective ME voltage coe¢ cient depends on the elastic moduli, CP E and CP M; dielectric permittivities, P E and P M;and magnetic

permeabilites, P E and P M; of the PE and PM materials, piezoelectric constant,

eP E; of the PE material, and piezomagnetic coe¢ cent, qP M; of the PM material.

reference and de…ne hte normalized materials properties of the PE and PM phases as

Cr;CoreI= CP E(CBT O) 1; Cr;ShellI= CP M(CCF O) 1; Cr;M atrixI= CP M(CCF O) 1;

and, likewise, are er;Core;qr;Shell;qr;M atrix; r;Core; r;Shell; r;M atrix; r;Core; r;Shell; r;M atrix:

Note that all the compnents of the material constant are magnifuied simultanesouly for simplicity. Belowm we numerically compute the ME voltage coe¢ cients _E;11 and

E;33 and their dependence on the normalized material properties of core (PE), shell

(PM), and materix (PM) phases.

Figure 4 shows the ME voltage coe¢ cient _E;11with respect to the crystallographic orientation of CFO and BTO. It happens be optimal when the poling direction of piezoelectric phase coincides with the magnetic axis of the piezomagnetic phase. We observe that the maximum of 2:4823V/cmOe occurs at Euler angles ( ; ; ) = ( ; 90 ; 90 );where is arbitrary. This degeneracy of optimal orientation re‡ects the 6mm symmetry. Further, If = 0, it is equivalent to the poling direction/magnetic axis along [010]. Signi…cantly, the optimized value of 2:4823V/cmOe is almost one hundred and one times higher than 0:0244V/cmOe, which is the value of the normal cut where the c axis of the CFO and BTO is along the …ber axis.

Figure 5

Motivated by the above study, we do a similar calculation for LiNbO3 (LNO),

CoFe2O4, and Terfenol-D as the core, shell, and matrix phases, since LNO has lower

dielectric permittivity and the matrix TD has lower elastic sti¤ness and magnetic per-meability. The material constants of LNO are listed in Table I. Figure 6 shows the ME voltage coe¤cients, volume fraction, and ratio of radii dependence of LNO/CFO/TD. Signi…cantly, the maximum values are enhanced to xxx V/cmOe and xx V/cmOe for

6

Concluding remarks

We have proposed a micromechanical model, the two-level recursive scheme in con-junction with the Mori-Tanaka’model, to compute the e¤ective magnetoelectric re-sponse of a core-shell-matrix, three-phase, …brous composites made of piezoelectric and pizomagnetic phases. The results are compared wiht …nite element analysis and the semi-analytical method proposed by Kuo (2010) and Kuo and Pan (2011). The magnitudes and trends among them are in good agreement. We have used it to show that, for the

Acknowledgments

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

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Table 1: Material parameters of BaTiO3 and CoFe2O4 (Li and Dunn, 1998a)

Figure 1: The …brous composite con…gurations.

Figure 2: A schematic representation of a unit cell. (a) A square array. (b) A hexagonal array.

Figure 3: The ME voltage coe¢ cients of the CFO …bers in a BTO matrix at the normal direction versus the …ber volume fraction. (a) In-plane ME voltage coe¢ cient

E;11. (b) Out-of-plane ME voltage coe¢ cient E;33:

Figure 4: The in-plane ME voltage coe¢ cient of the CFO …bers 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 coe¢ cient depends only on the Euler angles and and is independent of : The optimized constant occurs at both phases poled along the same direction.

Figure 5: The out-of-plane ME voltage coe¢ cient of the CFO …bers 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 coe¢ cient depends only on the Euler angles and and is independent of : The optimized constant occurs at both phases poled along the same direction.

Figure 6: The optimal ME voltage coe¢ cients of the CFO …bers in a BTO matrix for various …ber volume fraction. (a) In-plane ME voltage coe¢ cient _E;11. (b) Out-of-plane ME voltage coe¢ cient _E;33:

Figure 7: The ME voltage coe¢ cients of the BTO …bers in a CFO matrix at the normal direction versus the …ber volume fraction. (a) In-plane ME voltage coe¢ cient

E;11. (b) Out-of-plane ME voltage coe¢ cient E;33:

Figure 8: The in-plane ME voltage coe¢ cient of the BTO …bers in a CFO matrix for various orientations of BTO and CFO. The optimized constant occurs at both phases poled along the same direction.

Figure 9: The out-of-plane ME voltage coe¢ cient of the BTO …bers in a CFO matrix for various orientations of BTO and CFO. The optimized constant occurs at both phases poled along the same direction.

Figure 10: The optimal ME voltage coe¢ cients of the BTO …bers in a CFO matrix for various …ber volume fraction. (a) In-plane ME voltage coe¢ cient _E;11. (b) Out-of-plane ME voltage coe¢ cient _E;33: