含橢圓纖維壓電壓磁複合材料之電磁耦合現象

行政院國家科學委員會專題研究計畫 成果報告

含橢圓纖維壓電壓磁複合材料之電磁耦合現象

研究成果報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 99-2221-E-009-053- 執 行 期 間 ： 99 年 08 月 01 日至 100 年 09 月 30 日 執 行 單 位 ： 國立交通大學土木工程學系（所） 計 畫 主 持 人 ： 郭心怡 計畫參與人員： 碩士班研究生-兼任助理人員：吳自勝 碩士班研究生-兼任助理人員：王勇量 碩士班研究生-兼任助理人員：彭晟祐 碩士班研究生-兼任助理人員：郭祐旻 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 100 年 10 月 16 日

Multicoated elliptic …brous composites of

piezoelectric and piezomagnetic phases

Abstract

A theoretical framework is developed to investigate the magnetoelectroelas-tic potential in a mulmagnetoelectroelas-ticoated ellipmagnetoelectroelas-tic …brous composite with piezoelectric and piezomagnetic phases. We generalize the classic work of Lord Rayleigh (1892) to obtain the electrostatic potential in ordered conductive composites and its ex-tension to a disordered system (Kuo and Chen, 2008; Kuo, 2010) to the current coupled magnetoelectroelastic multicoated elliptic composites. We combine the methods of complex potentials with a re-expansion formulae and the generalized Rayleigh’s formulation to obtain a complete solution of the multi-…eld many-inclusion problem. It is shown that the coe¢ cients of …eld expansions can be written in the form of an in…nite set of linear algebraic equations. Numerical results are presented for several con…gurations. We use this method to study BaTiO3-CoFe2O4 composites and …nd that, with appropriate coating, the

ef-fective magnetoelectric voltage coe¢ cient can be enhanced with one order of magnitude compared with their non-coating counterpart.

1

Introduction

Magnetoelectric materials, which induce the polarization by a magnetic …eld, or conversely induce the magnetization by an electric …eld, have been the focus of re-search due to their varieties of microstructural phenomena and macroscopic proper-ties. These make them promising for a wide range of applications, such as four-state memories, magnetic …eld sensors, and magnetically controlled opto-electric devices (Eerenstein et al., 2006; Nan et al., 2008). The study of magneto-electric coupling can be traced back to 1957 when Landau and Lifshitz (1984) showed the possibility of the coupling between the electric and magnetic …elds in a substance with a certain magnetic symmetry class. This was subsequently experimentally con…rmed in a single crystal Cr2O3 by Astrov (1960) and by Rado and Folen (1961) over …fty years ago.

However, this coupling is weak in single phase materials, and this has motivated the study of composites of piezoelectric and piezomagnetic media. The basic idea is to couple a piezoelectric and a piezomagnetic material using strain: an applied electric

…eld creates a strain in the piezoelectric material which in turn induces a deformation in the piezomagnetic material, resulting in a magnetic …eld.

A number of micromechanical models hence were proposed to predict the e¤ective moduli of multiferroic composites. For instance, Green’s function approach was used by Nan (1994) and Huang and Kuo (1997) to study a …brous composite consisting of Barium Titanate and Cobalt Ferrite. For such transversely isotropic …brous com-posites, Benveniste (1995) derived exact connections among e¤ective magnetoelec-troelastic moduli based on a formalism discovered by Milgrom and Shtrikman (1989). Particulate composites were investigated by Harshé et al. (1993) using a cubic model, while a homogenization micromechanical method was employed by Aboudi (2001). Eshelby’s equivalent inclusion approach and the mean …eld Mori-Tanaka model have been generalized to multiferroic composites by Li and Dunn (1998a, b), Huang (1998), Li (2000) , Wu and Huang (2000) and Srinivas et al. (2006). Frequency dependence of magnetoelectric coe¢ cients of multiferroic laminates was studied by Bichurin et al. (2001, 2010a, 2010b). For a good overview of the subject, the reader is referred to the review article by Nan et al. (2008).

In contrast to the variety of works on e¤ective moduli of the composites, the study on the …eld solutions appears to be a relatively rarely explored area. However, a method that provides the detailed …elds is useful to provide insights for developing better microstructures and more complex process such as the breakdown and failure (Li and Duxbury, 1989). Relevant works on the …eld distributions of magnetoelectric composites are not many. Lee et al. (2005) proposed a …nite element analysis based micromechanics approach to determine the overall behaviors and reveal concentration of stress, electric and magnetic …eld within the unit cells. Tang and Yu (2008, 2009) used the variational asymptotic method for predicting the e¤ective properties and local …eld distributions of smart materials. A two-scale asymptotic homogenization theory was adopted by Camacho-Montes et al. (2009) on the magnetoelectric coupling and cross-property connections in a two phase composite.

In a classic work, Lord Rayleigh computed the electrostatic potential for a con-ducting composite consisting of a periodic array of cylindrical or spherical inclusions. This was extended to arbitrary arrangements by Kuo and Chen (2008) and to el-liptic cylinders by Kuo (2010). These works concern single …elds. Later, Kuo and Bhattacharya (2010) generalized this methodology to electrostatic, magnetostatic and mechanical coupled …elds. In this paper, we extend this Rayleigh’s formulation to a multiferroic composite consisting of elliptic cylinders, speci…cally multicoated ellipses. Coating plays an important role in high-temperature systems and in various en-gineering applications. For instance, to reduce heat or stress concentration along the interface, interphase layers between the inclusions and the matrix are often intro-duced to act as thermal barrier. Graded materials can also be more damage-resistant than their conventional homogeneous counterpart (Suresh, 2001). Such interphase layer may have constant properties or spatially varying properties. Research into in-homogeneous multiferroics has primarily been con…ned to the study of bilayer and

multilayer structures. Among them, piezoelectric or piezomagnetic coe¢ cients are assumed linear variation in the thickness direction by Chen and Lee (2003), Petrov et al. (2008) and Petrov and Srinivasen (2008), while exponentially graded assumption is adopted by Pan and Han (2005) and Wang et al. (2009). Fracture behavior of functionally graded piezoelectric and piezomagnetic composites was studied by Feng and Su (2006), Li and Lee (2008), and Zhou and Chen (2008). To our knowledge, the subject of piezoelectric/piezomagnetic …brous composites with multicoated elliptic cylinders has not been examined in the literature before.

The plan of this article is organized as follows. First we consider a composite medium made of piezoelectric and piezomagnetic phases arranged in a microstruc-ture consisting of parallel elliptic cylinders in a matrix in Section 2. The phases are transversely isotropic and under anti-plane shear with in-plane electromagnetic …elds. In this situation, the …elds are decoupled in the interior of every phase, and the coupling between the …elds occurs only through the interface conditions (Kuo and Bhattacharya, 2010). We exploit this in Section 2.2 to obtain a representation of the solution. The basic idea is to follow Kuo (2010) and expand each …eld in each medium in a series. We consider periodic arrays in Section 2.3. In Section 3 we consider the case of multicoated elliptic cylinders. We show that a (6 6) array alone can math-ematically simulate the e¤ects of multiple coatings. We obtain e¤ective properties in Section 4, and signi…cantly show that the macroscopic properties depend solely on a single expansion coe¢ cient (amongst the in…nite). This methodology is illustrated in Section 5 using composites of BaTiO3 and CoFe2O4: We choose this material pair

for its practical potential and also because it enables comparison with previous work. We observe that the composite medium has a non-trivial magnetoelectric coupling even through the individual components do not. Further, we show that the ME co-e¢ cient can be enhanced with an order of magnitude if the BTO …ber is coated with Terfenol-D.

2

Multiple elliptic cylinders

2.1

Basic formulations

Let us consider an in…nite medium R3 _{containing N arbitrarily distributed, parallel}

and separated elliptic cylinders. The domain of the pth elliptic cylinder is denoted Vp; p = 1; 2; ; N; and the remaining matrix is denoted m: We assume that the

cylinders and the matrix are made of distinct phases. Further, we assume that each phase is either piezoelectric or piezomagnetic with transversely isotropic symmetry (i.e. has 6mm symmetry) about the …ber axes. We introduce a Cartesian coordinate system positioned at a selected point selected point O of the plane with the x and y axes in the plane of the cross-section and z along the axes of the cylinders (Fig. 1). The centroids of the pth elliptic cylinders are designated as Op; with Opxp and

has the major and minor semi-axis, l(p)x and l(p)y ; and the inter-foci distance is 2dp;

where d2 p = l

(p)2

x l(p)2y :The ellipses are well separated so that any two inclusions will

not get in touch with each other.

Let the composite be subjected to the anti-plane shear strain "zx; "zy;the in-plane

electric …elds Ex; Ey; and the magnetic …elds Hx; Hy at in…nity. Thus the

hetero-geneous material is in a state of anti-plane shear problem (Chen, 1993; Benveniste, 1995; Kuo and Bhattacharya, 2010) and can be described by

ux = uy = 0; uz = w (x; y) ;

' = ' (x; y) ;

= (x; y) ; (2.1)

where ux; uy; uzare the mechanical displacements along the x ; y ; and z axes,

and ' and are the electric and magnetic potentials, respectively.

The constitutive laws of the constituents for the non-vanishing …elds become 0 @ Dzjj Bj 1 A = 0 @ C44 e15 q15 e15 11 11 q15 11 11 1 A 0 @ "zj Ej Hj 1 A (2.2)

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

U j = LU VZjV; U; V = w; '; ; j = x; y; (2.3) where j = 0 @ Dzjj Bj 1 A ; L = 0 @ Ce1544 e1511 q1511 q15 11 11 1 A ; Zj = 0 @ "Ezjj Hj 1 A : (2.4) Here zj; Dj; Bj; "zj; Ej and Hj are the stress, electric displacement, magnetic ‡ux,

strain, electric …eld, and the magnetic …eld, respectively. C44; 11; 11and 11are the

elastic modulus, dielectric permittivity, magnetic permeability and magnetoelectric coe¢ cients. The shear strains "zxand "zy;in-plane electric …elds Ex; Ey;and in-plane

magnetic …elds Hx and Hy can be derived from the gradient of elastic displacement,

electric potential, and magnetic potential as follows: "zx = @w @x; "zy = @w @y; Ex = @' @x; Ey = @' @y; Hx = @ @x; Hy = @ @y: (2.5)

Further, the equilibrium equations, in the absence of body force, electric charge density and electric current density, are given by

@ zx @x + @ zy @y = 0; @Dx @x + @Dy @y = 0; @Bx @x + @By @y = 0: (2.6)

Substitution of Eq.(2.3) into Eq.(2.6) yields

C44r2w + e15r2' + q15r2 = 0;

e15r2w 11r2' 11r2 = 0;

q15r2w 11r2' 11r

2 _{= 0; (2.7)}

where r2 = @2_=@x2 _{+ @}2_=@y2 _{represents the two-dimensional Laplace operator for}

the variable x and y: Since L is a nonsingular matrix, generically we can completely decouple (2.7) into three independent Laplace equations,

r2w = 0; _r2' = 0; _r2 = 0 (2.8) in the interior of each phase.

In addition to these di¤erential equations, we have to use interface conditions. We assume that the interfaces are perfectly bonded, and therefore the …elds satisfy

[[Zjtj]] = 0; [[ jnj]] = [[(LZj) nj]] = 0 (2.9)

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

tangent to the interface and nj is the unit outward normal to the interface, and the

repeated index j denotes summing over the components x; y. Since L is di¤erent in each phase, the …elds w; ' and are generally coupled by the interface equations.

2.2

Representation of the solution

We start by considering the case that the cylinders are homogeneous. We showed above that the …elds are decoupled in the interior of every phase, but are coupled at the interfaces. Therefore, we may follow Kuo (2010) and use a series expansion for each …eld in the interior of each phase and then obtain the coe¢ cients by enforcing the interface and boundary conditions.

Since w; ' and are harmonic, we can construct an analytic function (z) = U (z)+i U (z) ; of the complex variable z = x+i y; where U is the conjugate harmonic

function, related to U by the Cauchy-Riemann equation @U @x = @U @y ; @U @y = @U @x ; U = w; '; : (2.10) Further, the shape of the cross section of the cylinders de…nes elliptic coordinates: ( > 0; < )(Moon and Spencer, 1998)

z = x + i y = d cosh ! = d cosh ( + i ) (2.11) are the most appropriate system for the solution of Laplace’s equation.

We now consider a situation where the composite is subjected to a macroscopically uniaxial loading

wext = "zxx; 'ext = Exx; ext = Hxx; (2.12)

for constants "zx; Ex and Hx:We may rewrite this in short as

ext = z; (2.13)

where represents the appropriate …eld – the anti-plane deformation w, electric potential ', or magnetic potential –and = _R+ i I the corresponding applied

…eld –"zx; Ex or Hx:

We rewrite the governing equation, Eq. (2.8), in elliptic coordinates ( ; ) ;

r2 = 1 d2 _cosh2 _cos2 @2 @ 2 + @2 @ 2 = 0: (2.14) The potential …eld for the pth elliptic cylinder and its surrounding matrix can be expanded with respect to its centroid Op as (Yardley et al., 1999)

(p) i (zp) = 1 X n= 1 C_n(p)e n!p_{; C} (p) n = C (p) n (2.15)

for the inclusion, and

(p) m (zp) = 1 X n= 1 A_n(p)e n!p₊ 1 X n=1 B_n(p)e n!p_{; A} (p) n = A (p) n (2.16)

for the matrix. Here !p = p+i pis the local elliptic coordinate centered at the origin

of the pth ellipse, the subscripts i and m denote the inclusion and matrix, respectively. The coe¢ cients An(p) = A_nR(p)+ i A_nI(p); Bn(p) = B_nR(p)+ i B_nI(p) and Cn(p) = C_nR(p)+

i C_nI(p) are some complex unknowns to be determined. The superscripts p appearing in (2.15) and (2.16) indicate that the …elds are expanded with respect to the pth ellipse centroid.

We recall the interface conditions (2.9) which we rewrite as Re (p)_i @Vp = Re (p)_{m @V} p; (p) m np _@V p = (p)_i np @Vp (2.17) where w = ( zx; zy) ; ' = (Dx; Dy) ; = (Bx; By) ; (2.18)

@Vp : p = ap denotes the interface between the matrix and the pth elliptic cylinder,

and np is the unit outward normal of the interface @Vp:

Using the orthogonality properties of trigonometric functions, the interface con-ditions (2.17) provide a(p)_n = T(p)_an b(p)_n ; c(p)_n = T(p)_cn b(p)_n ; (2.19) and A0(p) = C (p) 0 ; where a(p)_n = 0 B @ Aw(p)_n A'(p)_n A (p)_n 1 C A ; b(p)n = 0 B @ B_nw(p) B_n'(p) B (p)_n 1 C A ; c(p)n = 0 B @ C_nw(p) C_n'(p) C_n (p) 1 C A ; (2.20)

T(p)_anR = L(m) L(p) 1 cosh napL(m)+ sinh napL(p)

e nap

sinh 2nap

;

T(p)_anI = L(m) L(p) 1 sinh napL(m)+ cosh napL(p)

e nap sinh 2nap ; T(p)_cnR = T(p)_anR+ e nap 2 cosh nap I; T(p)_cnI = T(p)_anI e nap 2 sinh nap I; (2.21) represents R; the real part, or I; the imaginary part of the coe¢ cients, and I is the 3 3identity tensor

We now need to relate the solutions to the applied boundary conditions. We do so by applying the Green’s second identity (Arfken and Weber, 2001) to the matrix domain m. This gives

Z m G (x; x0)_r02 m(x0) m(x0)r02G (x; x0) dA0 = Z @ m [G (x; x0)_r0 m(x0) m(x0)r0G (x; x0)] n0ds0; (2.22)

where the prime 0 denotes the operation in reference to the x0 coordinate, n0 is the outward unit normal to the matrix’s boundary @ m; dA0 represents the area element

for the x0 _{coordinate, ds}0 _{is the di¤erential arc length. Here G (x; x}0_{) is the free-space}

(x x0_{) is the Dirac-delta function. Following the procedure in Kuo (2010), it can}

be shown that Eq. (2.22) yields

m(z) = ext(z) + N X l=1 1 X n=1 B_m(l)e n!l_{: (2.23)}

This is the consistency equation which relates the external applied …elds to the local potential expansions. Note that the …eld identity (2.23) is written based on di¤erent coordinates.

To proceed, we shift the origin of the expansions (2.23) to a …xed point, say Zp;

the centroid of the pth ellipse, by expanding the term e m!l _{as (Kushch et al., 2005)}

e n!l ₌ 1 X m= 1 lp nme m!p_{; (2.24)} with lp nm = ( 1) m n dl dlp n_X1 s=0 v_lp(n+m+2s) s X t=0 ( 1)s t (s t)! dp dlp m+2t Mnmt(dl; dp) (n + m + t + s 1)! (s t)! ; (2.25) where dlp dl+ dp; vlp Zlp=dlp+ q (Zlp=dlp)2 1 and Mnmt(dl; dp) = t X k=0 (dl=dp)2k k!(t k)!(k + n)!(m + t k)!: (2.26) Introducing (2.24) into (2.23), we have the expansion

(p) m; near(z) = Zp+ 1 X n= 1 B_n(p)+ b_n(p) e m!p_{; (2.27)} where b_n(p) = dp 2 n; 1+ N X l6=p 1 X m=1 B_m(l) lp_mn (2.28) valid for the domain within an ellipse centered in Zp with inter-foci distance 2dlp and

passing the pole of lth elliptic coordinate systems closest to Zp (Kushch et al., 2005):

Further, since z lies in the matrix domain, Equations (2.27) and (2.16) should be identical. This provides the condition

1 X n= 1 A_n(p)e n!p _{= Z} p+ 1 X n= 1 b_n(p)e n!p_{: (2.29)}

Taking the real part and the imaginary part of (2.29), we …nd the two conditions A_nR(p) = _RRe Zp I Im Zp n;0+ b (p) nR ; (2.30) and A_nI(p) = _I Re Zp+ RIm Zp n;0+ b (p) nI : (2.31)

Equations (2.30), (2.31) and (2.19)1 constitute an in…nite set of linear algebraic

equa-tions. Upon appropriate truncations of the expansions terms, we can determine the expansion coe¢ cients An(p); Bn(p); Cn(p). Here we make one further remark.

Remark The essential step of the framework is to establish the generalized Rayleigh’s identities, (2.30) and (2.31). We observe, however, that the derivation of the identities is independent of inclusions’expansions. In other words, these identi-ties can be applicable to inclusions with inhomogeneous constituents provided that the admissible …elds in the inclusions and the transition relations, similar to (2.19) can be constructed.

2.3

Periodic arrays

The analysis carried out in the previous section for the arbitrary system with a …nite number of cylinders may also be adapted for the case of a periodic array of cylinders. Here we concentrate on the rectangular lattice, and we sketch the outline of the derivation focussing on the di¤erences from the previous situation.

Let us …rst introduce a Cartesian coordinate system (x; y) positioned at the cen-troid O of one of the ellipses in a rectangular array, as shown in Figure 2. The sides of the rectangular cell parallel to the x and y coordinates are, respectively, denoted by and : The elliptic cylinders are of the same orientation, elliptic radius = a and inter-foci distance 2d: Uniform intensities Ex and Hx are applied along the positive x

axis, and an anti-plane shear deformation "zxis applied out of the xy plane. In terms

of elliptic coordinates, the general solution has the admissible form

i = 1 X n= 1 C_ne n!; C_n = C _n (2.32) for < a; and m = 1 X n= 1 A_ne n!+ 1 X n=1 B_ne n!; A_n = A _n; (2.33) for > a: The coe¢ cients A_n; B_n; and C_n are unknown complex constants to be determined from the interface and boundary conditions.

Analogous to (2.19), the continuity conditions at the interface will give constraints (2.19) between the coe¢ cients. Next, imposing the periodicity conditions analogous

to the boundary condition we did to derive (2.30) and (2.31), lead to generalized Rayleigh’s identities

A_n = b_n ; = R; I: (2.34) Here the quantities

b_n = d 2 n; 1+ 1 X m=1 B_m 1 X l6=o lo mn; (2.35) 1 X l6=o lo mn= ( 1) m n 1 X t=0 d 2 n+m+2t Mnmt(d; d) (n + m + 2t 1)!Sn+m+2t; (2.36)

where Mnmt( )is de…ned in (2.26), and

Sm =

X

l6=o

Z_l m; (2.37)

are the lattice sums characterizing the geometry of the periodic structure, and Zl is

the centroid of the lth cylinder when measured in the complex plane centered at the central point O: The index l runs over all cylinders’centers underlying the periodic array except the central one. Previous studies (Rayleigh, 1892) have reported that the sum S2 is conditionally convergent and its value depends upon the shape of the

exterior boundary of the array. A list of S2 for di¤erent values of = can be found

in Nicrovici and McPhedran (1996).

Equations (2.34) and (2.19)1constitute an in…nite set of linear algebraic equations.

Upon appropriate truncations of the expansion terms at a …nite order M , we can determine the expansion coe¢ cients An; Bn;and Cn:

3

Confocally multicoated elliptic cylinders

From the previous remark, we now consider that the inclusions are confocally mul-ticoated elliptic cylinders with the outer elliptic radius a(1)p ; p = 1; 2; ; N; where

N is the number of inclusions. We denote the matrix as phase 0; with transversely isotropic material parameters C₄₄(0); e(0)₁₅; q₁₅(0); (0)₁₁; (0)₁₁ and (0)₁₁:The multicoated cylin-der consists of a core, with radius p = a(M)p ; surrounded by (M 1)layers of

coat-ing. The jth layer of the coatings occupies the annulus Vp(j) : a(j+1)p _p a(j)p ;

j = 1; 2; ;_{M; in which V}p = V (1)

p [ Vp(2) [ [ Vp(M). Here the innermost core

is solid so that we have a(M+1)p = 0: We assume that the material properties of jth

constituent layer of the pth multicoated cylinder are C44(p;j); e (p;j) 15 ; q (p;j) 15 ; (p;j) 11 ; (p;j) 11 and (p;j)11 .

can be expressed as (p;j) ₌ 1 X n= 1 A_n(p;j)e nwp₊ 1 X n=1 B_n(p;j)e nwp_{; A} (p;j) n = A (p;j) n ; (3.1)

where An(p;j)= A_nR(p;j)+i A_nI(p;j)and Bn(p;j)= B_nR(p;j)+i B_nI(p;j)are unknown complex

constants to be determined. Note that the potential at _{! 0 should be …nite and} thus we can set

B_n(p;M) = 0: (3.2) We consider that the interfaces are perfectly bonded, the potential and the normal component of ‡ux are continuous across the interfaces,

Re (p;j 1) p=a (j) p = Re (p;j) p=a (j) p ; (p;j 1) n(j)p p=a (j) p = (p;j) n(j)p p=a (j) p : (3.3)

These continuity conditions lead to a(p;j 1)_n b(p;j 1)_n ! = k(p;j)_n a (p;j) n b(p;j)_n ! ; = R; I; j = 1; 2; ;_M; (3.4) where a(p;j)_n = 0 B @ Aw(p;j)_n A'(p;j)_n A (p;j)_n 1 C A ; b(p;j)n = 0 B @ B_nw(p;j) B_n'(p;j) B_n (p;j) 1 C A ; (3.5) k(p;j)_nR 2 cosh na (j) p I e na (j) p _I 2 sinh na(j)p L(j 1) e na (j) p _L(j 1) ! 1 2 cosh na(j)p I e na (j) p _I 2 sinh na(j)p L(j) e na (j) p _L(j) ! ; k(p;j)_nI 2 sinh na (j) p I e na (j) p _I 2 cosh na(j)p L(j 1) e na (j) p _L(j 1) ! 1 2 sinh na(j)p I e na (j) p _I 2 cosh na(j)p L(j) e na (j) p _L(j) ! ; (3.6) and I is the 3 3 identity matrix. Now, repeated use of (3.4) gives

a(p;0)_n b(p;0)_n ! = K(j;n)_p a (p;j) n b(p;j)_n ! ; j = 1; 2; ;_M; (3.7) where K(j;n)_p k(p;1)_n k(p;2)_n k(p;j)_n : (3.8)

For j = M, we have a(p;0)_n b(p;0)_n ! = K(M;n)_p a (p;M) n b(p;M)_n ! : (3.9)

Further, according to (3.2), (3.9) implies that a(p;0)_n =hK(M;n)_p i 11 h K(M;n)_p i 1 21 b(p;0)_n : (3.10) Here h K(M;n)_p i

11represents the upper-left (3 3)submatrix of K (M;n) p and h K(M;n)_p i 21

is the lower-left (3 3) submatrix of K(M;n)_p : The formulation implies that he e¤ect of the multiple coatings can be incorporated through a recurrence procedure and is solely represented by a (6 6) array alone. We mention that once we construct the admissible …eld (3.1) and the transition relation (3.4) in the inhomogneneous inclusions, we can follow the remaining generalized Rayleigh’s framework proposed in previous section to determine the potential distribution.

4

E¤ective moduli

We are interested in the e¤ective behavior for a situation where we have a large number of cylinders. The e¤ective material properties are de…ned in terms averaged …elds,

h ji LjhZji ; no summation, (4.1)

where the angular brackets denote the area averages over the representative volume element (unit cell in the case of periodic composites)

h ji = 1 V Z V jdx; hZji = 1 V Z V Zjdx; (4.2)

and Lj denotes the e¤ective magnetoelectroelastic parameters of the composite. Note

that since the microstructure is nonsymmetric, the responses of the composite along x and y axes are di¤erent:

L_x= 0 @ C₅₅ e₁₅ q₁₅ e_{15 11 11} q_{15 11 11} 1 A ; L_y = 0 @ C₄₄ e₂₄ q₂₄ e_{24 22 22} q_{24 22 22} 1 A : (4.3) Let the composite is subjected to uniform intensities "zx; Ex; and Hx along

is a gradient and applying the divergence theorem. We obtain:

Z_x = _R: (4.4)

Next, to …nd _x ; we again use the divergence theorem and the equilibrium condition (including the interface conditions) to obtain

x = 1 V Z V xdx = 1 V Z V r x dx =1 V Z @V x m nds; (4.5)

where is de…ned in (2.18). We then use the expansions (2.16) for the …elds to obtain 1 V Z @V x Z m nds = R dB_1R (4.6) where Zw = ("zx; "zy) ; Z' = (Ex; Ey) ; Z = (Hx; Hy) : (4.7)

Putting (4.5) and (4.6) together, and recalling the constitutive relation (2.2) for the matrix, we obtain 0 @ h_hDzxxii hBxi 1 A = 0 @ Ce1544 e1511 q1511 q15 11 11 1 A (m)0 B B @ "zx dBw 1R Ex dB_1R' Hx dB_1R 1 C C A : (4.8) Putting together (4.1) and (4.8) and noting that the coe¢ cients B₁ depend linearly on the applied …eld R; we obtain set of equations for the e¤ective property Lx:

We can determine this by applying di¤erent loading combinations between "zx; Ex

and Hx: Similarly we can determine Ly by applying di¤erent loading combinations

between "zy; Ey and Hy:

5

Results and discussion

In order to have a better understanding for the theoretical results above, we …rst perform a numerical computation for a two-phase transversely isotropic piezoelectric-piezomagnetic composite with 6mm material symmetry about the …ber axis. Specif-ically we consider a composite of BaTiO3 and CoFe2O4 which has been studied by

other researchers. We consider the square arrays, and two cases: BTO …bers in a CFO matrix and BTO coated Terfenol-D …bers in a CFO matrix. The independent material constants of these constituents are given in Table 1, where the xy plane is isotropic and the …ber axis is along the z direction. Note that in all materials magnetoelectric coe¢ cients are zero, i.e. 11= 0:

the contours of displacement, electric potential and magnetic potential for a square array with an applied magnetic …eld. The ratio of the semi-axes lx and ly is 1:2:

The magnetic …eld induces a mechanical stress in the CFO which in turn results in an electric displacement in the BTO …ber. The magnetic …eld is attracted by the BTO since it has a smaller magnetic permeability. Further, Figure 3 (d-f) show the potential contours for BTO …bers coated Terfenol-D in a CFO matrix for a square array with an applied magnetic …eld. The ratio of semi-axis lx between BTO and

Terfenol-D is 0:7: Since the magnetic permeability of Terfenol-D is almost the same as that of BTO, the magnetic potential is similar to its homogeneous counterpart. However, in this case the contours of vertical displacement and the electric potential have dramatically di¤erence from those with the homogeneous …ber. The …elds in the …ber are now not uniform, and there are …eld concentrations at the BTO and Terfenol-D interfaces.

We now turn to e¤ective moduli of the composite. Figure 4 shows the e¤ective elastic, dielectric, magnetic, piezoelectric, piezomagnetic and magnetoelectric moduli for this composite. We assume that the inclusions are circular cylinders, i.e., lx ! ly,

and the inclusions are in a square array. Therefore there is no distinction between Lx

and L_y: The e¤ective moduli vary nonlinearly with volume fraction, and the curve stops at f = ₄ when the inclusions touch. The magnetoelectric coe¢ cient is non-zero for every non-non-zero volume fraction, then reaches a maximum before dropping just as the …bers are close to touching. Further, Figure 4 also compares the e¤ective moduli with those predicted by Benveniste (1995) who used the composite cylinder assemblage (CCA) model. In CCA, there is no upper limit on the volume fraction since one can have …bers with various sizes. Still, the overall magnitudes and trends agree well between our periodic and his CCA. In addition, our numerical results ful…l the compatibility conditions given in Eq. (21) of the work by Benveniste (1995). Figure 5 shows the e¤ective moduli for the BTO …bers coated Terfenol-D in a CFO matrix. The ratio of the radius of BTO and Terfenol-D is 0:8: We …nd that the magnetoelectric coe¢ cient 11 has dramatically enhanced, and the enhancement is

increased as the particles touch.

We …nally turn to the magnetoelectric voltage coe¢ cient, which is the important …gure of merit for magnetic …eld sensors. It relates the overall electric …eld that is generated in the composite when it is subjected to a magnetic …eld. It combines the coupling and dielectric coe¢ cients, and is de…ned by

11= 11 11

: (5.1)

Figure 6 shows how this coe¢ cient depends on the …ber volume fraction for the above two cases. Note that there is a qualitative di¤erence between the case of BTO …bers in CFO and BTO …bers coated Terfenol-D in the CFO matrix. In the former, the maximum coe¢ cient is for intermediate volume fraction of f = 0:35 where

CFO matrix, the maximum is attained as the …bers near close touching at f = 0:725 where 11 = 0:5732V/cmOe, which is one order of magnitude enhancement of the

coupling coe¢ cient.

6

Concluding remarks

In summary, we have extended Rayleigh’s formalism on periodic conductive com-posites to a magnetoelectroelastic composite consisting of arbitrarily distributed or periodic arrays of elliptic cylinders under anti-plane shear deformation, in-plane elec-tric …eld and in-plane magnetic intensities. The cylinders can be homogeneous or confocally multicoated. Expressions for the elastic, electric and magnetic potentials for the cylinders and the matrix are derived, and used to compute the e¤ective mod-uli. This extension is a hybrid technique: the admissible potentials for the matrix and inclusions are expanded in complex planes, while the interface conditions are di-rectly satis…ed by using elliptic coordinates. It is shown that the e¤ective properties solely depend on one particular constant B1R among the in…nite number of

expan-sion coe¢ cients. Finally, as a practical example, explicit numerical calculations for …eld distributions and the magnetoelectric e¤ects in BaTiO3-CoFe2O4 and BaTiO3

-Terfenol-D-CoFe2O4composites are presented and discussed. This example shows the

important di¤erence between the case of BTO …bers in a CFO matrix from the case of BTO …bers coated Terfenol-D in a CFO matrix. We expect that these results will be bene…cial as design tools for functionally graded tunable composites.

References

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[7] Bichurin, M. I., Petrov, V. M., Averkin, S. V., Liverts, E., 2010b. Present status of theoretical modeling the magnetoelectric e¤ect in magnetostrictive-piezoelectric nanostructures. Part II: Magnetic and magnetoacoustic resonance ranges. J. Appl. Phys. 107, 053905.

[8] Camacho-Montes, H., Sabina, F. J., Bravo-Castillero, J., Guinovart-Díaz, R., Rodríguez-Ramos R., 2009. Magnetoelectric coupling and cross-property con-nections in a square array of a binary composite. Int. J. Eng. Sci. 47, 294-312. [9] Chen, T. Y., 1993. Piezoelectric properties of multiphase …brous composites:

Some theoretical results. J. Mech. Phys. Solids 41, 1781-1794.

[10] Chen, W. Q., Lee, K. Y., 2003. Alternative state space formulations for magneto-electric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates. Int. J. Solids Struct. 40, 5689-5705.

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[16] Kuo, H. -Y., Chen, T., 2008. Electrostatic …elds of an in…nite medium containing arbitrarily positioned coated cylinders. Int. J. Eng. Sci. 46, 1157-1172.

[17] Kuo, H. -Y., 2010. Electrostatic interactions of arbitrarily dispersed multicoated elliptic cylinders. Int. J. Eng. Sci. 48, 370-382.

[18] Kuo, H. -Y., Bhattacharya, K., 2010. Fibrous composites of piezoelectric and piezomagnetic phases. Submitted.

[19] Kushch, V. I., Shmegera, S. V., Buryachenko, V. A., 2005. Interacting elliptic inclusions by the method of complex potentials. Int. J. Solids Struct. 42, 5491-5512.

[20] Landau, L. D., Lifshitz, E. M., 1984. Electrodynamics of Continuous Media. Pergamon Press, New York, pp.119.

[21] Lee, J., Boyd IV, J. G., Lagoudas, D. C., 2005. E¤ective properties of three-phase electro-magneto-elastic composites. Int. J. Eng. Sci. 43, 790-825.

[22] Li, J. Y., 2000. Magnetoelectroelastic multi-inclusion and inhomogeneity prob-lems and their applications in composite materials. Int. J. Eng. Sci. 38, 1993-2001. [23] Li, J. Y., Dunn, M. L., 1998a. Micromechanics of magnetoelectroelastic compos-ite materials: average …elds and e¤ective behaviour. J. Intel. Mat. Syst. Struct. 9, 404-416.

[24] Li, J. Y., Dunn, M. L., 1998b. Anisotropic coupled-…eld inclusion and inhomo-geneity problems. Philos. Mag. A 77, 1341-1350.

[25] Li, Y. S., Duxbury, P. M., 1989. From moduli scaling to breakdown scaling: A moment spectrum analysis. Phys. Rev. B 40, 4889-4897.

[26] Li, Y. D., Lee, K. Y., 2008. Anti-plane crack intersecting the interface in a bonded smart structure with graded magnetoelectroelastic properties. Theor. Appl. Fract. Mec. 50, 235-242.

[27] Liu, Y. X., Wan, J. G., Liu, J. -M., Wen, C. W., 2003. Numerical modeling of magnetoelectric e¤ect in composite structure. J. Appl. Phys. 94, 5111-5117. [28] Liu, G., Nan, C.-W., Cai, N., Lin, Y., 2004. Dependence of giant magnetoelectric

e¤ect on interfacial bonding for multiferroic laminate composites of rare-earth-iron alloys and lead-zirconate-titanate. J. Appl. Phys. 95, 2660-2664.

[29] Milgrom, M., Shtrikman, S., 1989. Linear response of two-phase composites with cross moduli: Exact universal relations. Phys. Rev. A 40, 1568-1575.

[30] Moon, P., Spencer, D. E., 1998. Field Theory Handbook. Springer-Verlag, Berlin, pp.17.

[31] Nan, C. -W., 1994. Magnetoelectric e¤ect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B 50, 6082-6088

[32] Nan, C.-W., Bichurin, M. I., Dong, S, Viehland, D., Srinivasan, G., 2008. Mul-tiferroic magnetoelectric composites: Historical perspective, status, and future directions. J. Appl. Phys. 103, 031101.

[33] Nicrovici, N. A., McPhedran, R. C., 1996. Transport properties of arrays of elliptical cylinders. Phys. Rev. E 54, 1945-1957.

[34] Pan, E., Han, F., 2005. Exact solution for functionally graded and layered magneto-electro-elastic plates. Int. J. Eng. Sci. 43, 321-339.

[35] Petrov, V. M. Srinivasan, G., 2008. Enhancement of magnetoelectric coupling in functionally graded ferroelectric and ferromagnetic bilayers. Phys. Rev. B 78, 184421.

[36] Petrov, V. M., Srinivasan, G., Galkina, T. A., 2008. Microwave magnetoelectric e¤ects in bilayers of single crystal ferrite and functionally graded piezoelectric. J. of Appl. Phys. 104, 113910.

[37] Rado, G. T., Folen, V. J., 1961. Observation of the magnetically induced mag-netoelectric e¤ect and evidence for antiferromagnetic domains. Phys. Rev. Lett. 7, 310-311.

[38] Rayleigh, L., 1892. On the in‡uence of obstacles arranged in rectangular order upon the properties of a medium. Philos. Mag. 34, 481-502.

[39] Srinivas, S., Li, J. Y., Zhou, Y. C., Soh, A. K., 2006. The e¤ective magneto-electroelastic moduli of matrix-based multiferroic composites. J. Appl. Phys. 99, 043905.

[40] Suresh, S., 2001. Graded materials for resistance to contact deformation and damage. Science 292, 2447-2451.

[41] Tang, T., Yu, W., 2008. Variational asymtotic homogenization of heterogeneous electromagnetoelastic materials. Int. J. Eng. Sci. 46, 741-757.

[42] Tang, T., Yu, W., 2009. Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method. Smart Mater. Struct. 18, 125026.

[43] Wang, X., Pan, E., Albrecht, J. D., Feng, W. J., 2009. E¤ective properties of multilayered functionally graded multiferroic composites. Compos. Struct. 87, 206-214.

[44] Wu, T. -L., Huang, J. -H., 2000. Closed-form solutions for the magnetoelectric coupling coe¢ cients in …brous composites with piezoelectric and piezomagnetic phases. Int. J. Solids Struct. 37, 2981-3009.

[45] Yardley, J. G., McPhedran, R. C., Nicorovici, N. A., Botten, L. C., 1999. Addi-tion formulas and the Rayleigh identity for arrays of elliptical cylinders. Phys. Rev. E 60, 6068-6080.

[46] Zhou, Z. -G., Chen, Z. -T., 2008. Basic solution of a Mode-I limited-permeable crack in functionally graded piezoelectric/piezomagnetic materials. Int. J. Solids Struct. 45, 2265-2296.

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

Terfenol-D (Liu et al., 2003, Liu et al., 2004)

Figure 1: The cross-section of the …ber composite. Figure 2: A schematic representation of a unit cell.

Figure 3: Potential contours for a square array composite (f = 0:2; "zx = 0;

Ex = 0; Hx = 1C=ms) (a)-(c) BTO …bers embedded in a CFO matrix, (d)-(f) BTO

…bers coated Terfenol-D embedded in a CFO matrix, (a), (d) Vertical displacement (m), (b), (e) Electric potential (V ), (c), (f) Magnetic potential (C=s).

Figure 4: E¤ective moduli of a composite of BTO …bers in a CFO matrix versus …ber volume fractions (a) E¤ective elastic modulus, (b) E¤ective dielectric permit-tivity, (c) E¤ective magnetic permeability, (d) E¤ective piezoelectric modulus, (e) E¤ective piezomagnetic modulus, (f) E¤ective magnetoelectric coe¢ cient.

Figure 5: E¤ective moduli of a composite of BTO …bers coated Terfenol-D in a CFO matrix versus …ber volume fractions (a) E¤ective elastic modulus, (b) E¤ective dielectric permittivity, (c) E¤ective magnetic permeability, (d) E¤ective piezoelectric modulus, (e) E¤ective piezomagnetic modulus, (f) E¤ective magnetoelectric coe¢ -cient.

Figure 6: E¤ective magnetoelectric voltage coe¢ cient of the composite versus the …ber volume fraction.

2

nd

International Conference on Material Modelling

incorporating the 12th European Mechanics of

Materials Conference

第二屆國際材料模擬會議

暨

第十二屆歐洲材料力學會議

服務機關：國立交通大學土木工程學系

姓名職稱：郭心怡 助理教授

前往國家：法國

Ecole des Mines de Paris

出國期間：100/08/31~09/02

國科會補助專題研究計畫項下出席國際學術會議心得

報告

日期： 99 年 9 月 13 日

一、參加會議經過

國際材料模擬會議（International Conference on Material Modelling）

為每兩年舉辦之會議，今年為第二屆並與歐洲材料力學會議合併舉行。

今年投稿及與會者眾，約七百人次。會議自8/31 起至9/2 間於法國巴黎

礦業大學Ecole des Mines de Paris舉行，本人的報告於8/31 第一場次第一

位舉行，並受邀擔任該場次之Session Chair；報告進行順利，也能理解

並回答提問。此次會議為歐洲大陸會議，且場次眾多並無任何參觀觀光

行程，純為學術交流與思想激盪。

計畫編

號

NSC 99－2221－E－009－053－

計畫名

稱

含橢圓纖維壓電壓磁複合材料之電磁耦合現象

出國人

員姓名

郭心怡

服務機

構及職

稱

國立交通大學土木工程學

系助理教授

會議時

間

100 年 8 月 31

日至

100 年 9 月 2

日

會議地

點

法國

Ecole des Mines de Paris

會議名

稱

(中文)第二屆國際材料模擬會議暨第十二屆歐洲材料力學

會議

(英文) 2

nd

International Conference on Material Modelling

incorporating the 12th European Mechanics of Materials

Conference

發表論

文題目

(中文)磁電鍍膜纖維複合材料之等效行為

(英文)Effective behavior of coated fibrous magnetoelectric

composites

二、與會心得

國際材料模擬會議雖因主辦者主要為歐洲國家（第一屆於德國Dortmund

舉行）以致與會者多為歐洲大陸之學者或研究生，然由於以材料模擬為

主題的會議甚少，國際間材料模擬領域內的許多學者因此皆會與會以彼

此交流。藉由聆聽報告、問答互動，拓展大家的研究視野，促使自己保

有研究熱忱與反思能力。

此次會議於Ecole des Mines de Paris舉行，地處巴黎拉丁區，位於盧

森堡公園旁，附近有多所大學，學術風氣鼎盛，且離巴黎聖母院僅約

一公里。另會議所提供的餐點及設備雖然簡單樸實，但已足以使會議

流暢進行。

三、建議

由於中午用餐時間需外出自行用餐，雖可便於瀏覽巴黎景緻，然卻

減少與會人員間之交流機會，甚為可惜，建議午餐仍應由大會統籌辦

理。

四、攜回資料名稱及內容

1. 會議行程

國科會補助計畫衍生研發成果推廣資料表

日期:2011/09/13

國科會補助計畫

計畫名稱: 含橢圓纖維壓電壓磁複合材料之電磁耦合現象 計畫主持人: 郭心怡 計畫編號: 99-2221-E-009-053- 學門領域: 結構應力

無研發成果推廣資料

99 年度專題研究計畫研究成果彙整表

計畫主持人：郭心怡 計畫編號：99-2221-E-009-053-計畫名稱：含橢圓纖維壓電壓磁複合材料之電磁耦合現象 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 1 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 4 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次

其他成果

(

無法以量化表達之成 果如辦理學術活動、獲 得獎項、重要國際合 作、研究成果國際影響 力及其他協助產業技 術發展之具體效益事 項等，請以文字敘述填 列。) 無 成果項目 量化 名稱或內容性質簡述 測驗工具(含質性與量性) 0 課程/模組 0 電腦及網路系統或工具 0 教材 0 舉辦之活動/競賽 0 研討會/工作坊 0 電子報、網站 0 科 教 處 計 畫 加 填 項 目 計畫成果推廣之參與（閱聽）人數 0

國科會補助專題研究計畫成果報告自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）

、是否適

合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

■達成目標

□未達成目標（請說明，以 100 字為限）

□實驗失敗

□因故實驗中斷

□其他原因

說明：

2. 研究成果在學術期刊發表或申請專利等情形：

論文：■已發表 □未發表之文稿 □撰寫中 □無

專利：□已獲得 □申請中 ■無

技轉：□已技轉 □洽談中 ■無

其他：（以 100 字為限）

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以

500 字為限）