Fibrous composites of piezoelectric and piezomagnetic phases: Generalized plane strain with transverse electromagnetic fields

Fibrous composites of piezoelectric and piezomagnetic phases:

Generalized plane strain with transverse electromagnetic fields

Hsin-Yi Kuo

⇑

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 24 January 2014

Received in revised form 7 April 2014 Available online 20 April 2014 Keywords:

Magnetoelectricity Fibrous composite Generalized plane strain Transverse electromagnetic field

a b s t r a c t

This work presents a theoretical framework for solving the field distributions of a piezo-electric–piezomagnetic fibrous composite subjected to generalized plane strain with trans-verse electromagnetic fields. The matrix is infinite containing arbitrarily distributed circular cylinders, which may have different sizes and material properties. By introducing an eigenstrain corresponding to the electro-magneto-elastic effect, this coupling problem can be reduced to an equivalent plane elasticity problem. The classic work of Muskhelish-vili to obtain the elastic potential of a composite is generalized to the current multi-field multi-inclusion media. Several numerical examples are presented to demonstrate the effectiveness of the approach.

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1. Introduction

The magneto-electric (ME) coupling refers to the polar-ization induced by a magnetic field, or conversely the mag-netization induced by an electric field. It was first predicted byLandau and Lifshitz (1984)and observed by

Astrov (1960)andRado and Folen (1961)over fifty years

ago. This ME effect has recently drawn ever-increasing interest due to their potential applications as multifunc-tional devices including ME data storage and switching

(Spaldin and Fiebig, 2005), modulation of optical waves

(Fiebig, 2005), and electrically microwave phase shifters (Bichurin et al., 2002). However, the coupling is rather weak in a single-phase material even at low temperature, and this has motivated the study of composites of piezo-electric and piezomagnetic media. The ‘‘product property’’ causes the ME effect in this multiferroic composite: an applied electric field generates a deformation in the

piezoelectric phase, which in turn generates a deformation in the piezomagnetic phase, resulting a magnetization (Nan et al., 2008).

The promise of applications, and the indirect coupling through strain have also made ME composites the topic of a number of theoretical investigations. Among them,

Nan (1994), Srinivas and Li (2005) and Liu and Kuo (2012)estimated the effective properties of ME composites of non-dilute volume fractions by mean-field-type models.

Benveniste (1995)derived exact relations in a ME

compos-ite with cylindrical geometry. The analysis for local fields is available for simple microstructures such as a single ellip-soidal inclusion (Huang and Kuo, 1997; Li and Dunn, 1998), arbitrarily distributed or periodic arrays of fibrous ME composites (Kuo, 2011; Kuo and Pan, 2011; Kuo and Bhattacharya, 2013), and laminates (Kuo et al., 2010). In addition,Liu et al. (2004)andLee et al. (2005)used finite element method to address ME composites for general microstructures, while Aboudi (2001) and

Camacho-Montes et al. (2009)adopted the homogenization method.

However, much of this work uses approximate methods and models based on single inclusions to estimate the

http://dx.doi.org/10.1016/j.mechmat.2014.04.007

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Mechanics of Materials

effective properties of composites. Exact methods that pro-vides the detailed field distribution are limited to the med-ium subjected to the anti-plane shear with in-plane electromagnetic fields due to the complexity.

In a classic work,Muskhelishvili (1975)used the Kolo-sov–Muskhelishvili potentials with truncated Laurent ser-ies for elastic problems with circular boundaries. Analogous representations were employed byMcPhedran and Movchan (1994)for a pair and a square array of circu-lar elastic inclusions, byBuryachenko and Kushch (2006)

for a matrix reinforced two linearly elastic isotropic aligned circular fibers, and byKushch et al. (2008)for the progressive damage in the fiber reinforced composite. This method was extended to investigate the multiple piezo-electric inclusions in a non-piezopiezo-electric matrix (Yang and Gao, 2010), and for a three-phase thermo-electro-mag-neto-elastic cylinder model (Tong et al., 2008). In addition, a Galerkin boundary integral method has also been devel-oped to address the elastic composites with multiple circu-lar cylinders (Mogilevskaya and Crouch, 2001), while Eshelby’s equivalent inclusion for a fibrous piezoelectric inhomogeneity was proposed byXiao and Bai (1999). In this paper, we generalize Muskhelishvili’s methodology to a fibrous composite made of piezoelectric and piezo-magnetic phases under generalized plane strain ð

e

0

13¼ 0;

e

023¼ 0;

e

033–0Þ with transverse electromagnetic

fields. Specifically we seek the stress and displacement dis-tributions of the composite.

The remainder of this paper is organized as follows. In Section2we formulate the equation for a piezoelectric– piezomagnetic composite under generalized plane strain with transverse electromagnetic fields. We show that the multi-field coupled problem can be reduced to an equiva-lent plane elastic problem with a corresponding uniform eigenstrain. In Section 3 we generalize the work of

Muskhelishvili (1975) to obtain a representation of the

solution. The basic idea is to express the stress and dis-placement via two complex potentials and expand each field in each medium in a series. We use this method to study selected systems with sufficient accuracy in Section4.

2. General framework

Let us consider an infinite medium containing N arbi-trarily distributed, parallel and separated circular cylinders (Fig. 1). The domain of the pth circular cylinder is denoted Vp; p ¼ 1; 2; . . . ; N, and the remaining matrix is denoted

Xm. We assume that the cylinders and the matrix are made

of distinct phases: transversely isotropic piezoelectric or piezomagnetic materials. A global Cartesian coordinate system is introduced with x1- and x2-axes in the plane of

the cross-section and x3-along the axes of the cylinders

(Fig. 1). The centers of the pth circular cylinder are desig-nated as Op, each of which may have different radii ap.

Assume that the composite is subjected to in-plane mechanical strain

e

0

11;

e

022 and

e

012 (or in-plane stress

r

0

11;

r

022and

r

012) at infinity and uniform strain

e

033, electric

field E0

3and magnetic field H 0

3in the x3-direction. It can be

shown that the general constitutive law for the non-vanishing

field quantities can be written in a compact form as (Benveniste, 1995)

r

11

r

22

r

33

r

12 D3 B3 0 B B B B B B B B B B @ 1 C C C C C C C C C C A ¼ C11 C12 C13 0 e31 q31 C12 C11 C13 0 e31 q31 C13 C13 C33 0 e33 q33 0 0 0 C66 0 0 e31 e31 e33 0 

j

33 k33 q31 q31 q33 0 k33 

l

33 0 B B B B B B B B B B @ 1 C C C C C C C C C C A

e

11

e

22

e

33 2

e

12 E3 H3 0 B B B B B B B B B B @ 1 C C C C C C C C C C A : ð2:1Þ

Here

r

ijand

e

ijare the stress and strain; Diand Eiare the

electric displacement and electric field; Biand Hi are the

magnetic flux and magnetic field, respectively. C11; C12; C13; C33, and C66 are the elastic moduli, e31 and

e33 are piezoelectric constants, q31 and q33 are

piezomag-netic constants, and

j

33;

l

33, and k33are the dielectric

per-mittivity, magnetic permeability and magnetoelectric coefficients, respectively.

The constitutive equation(2.1)are rather complicated. However, it is observed that

e

33; E3, and H3are constants

in the composite (Tong et al., 2008). Thus we can introduce an uniform eigenstrain field

e

_¼

_e

 11¼

e

 22

¼ ðC13

e

33þ e31E3þ q31H3Þ=ðC11þ C12Þ: ð2:2Þ

Substitution of Eq.(2.2)into Eq.(2.1)yields

r

11þ

r

22¼ 2K½ð

e

11þ

e

22Þ  2

e

;

r

22

r

11¼ 2

l e

ð22

e

11Þ;

r

12¼ 2

le

12 ð2:3Þ and

r

33¼ C13ð

e

11þ

e

22Þ þ C33

e

33 e33E3 q33H3; D3¼ e31ð

e

11þ

e

22Þ þ e33

e

33þ

j

33E3þ k33H3; B3¼ q31ð

e

11þ

e

22Þ þ q33

e

33þ k33E3þ

l

33H3; ð2:4Þ

where K ¼ ðC11þ C12Þ=2 is the in-plane bulk modulus, and

l

¼ ðC11 C12Þ=2 is the transverse shear modulus. It is

O

noted that the in-plane stress–strain fields(2.3)in the pie-zoelectric–piezomagnetic composite material is the same as that in the corresponding elastic material with an appropriate uniform eigenstrain field

e

 _{(2.2). Therefore,}

the multi-field coupled problem is reduced to an equiva-lent in-plane elastic problem with the eigenstrain

e

_{. Once}

the in-plane strain is determined, the out-of-plane stress, electric displacement, and magnetic flux are also deter-mined through(2.4).

3. Representation of the solution

We follow Kolosov–Muskhelishvili formulae to express the stress via two complex potentials

u

ðzÞ and wðzÞ as fol-lows (Muskhelishvili, 1975)

r

11þ

r

22¼ 2½

u

ðzÞ þ

u

0ðzÞ;

r

22

r

11þ 2i

r

12¼ 2 z½

u

00ðzÞ þ w0ðzÞ; ð3:1Þ

where the overbar represents the complex conjugate, and the prime denotes differentiation. In these equations, z ¼ x1þ ix2is complex coordinate of a point ðx1;x2Þ in the

global Cartesian coordinate system, and i ¼pffiffiffiffiffiffiffi1. The dis-placement u and resultant stress F are then obtained as

u ¼ u1þ iu2¼ 1 2

l

ju

ðzÞ  z

u

0ðzÞ  wðzÞ h i þ

e

_z; F ¼ F1þ iF2¼ i

u

ðzÞ þ z

u

0ðzÞ þ wðzÞ h iB A; ð3:2Þ

where

j

¼ 3  4

m

in plane strain;

j

¼ ð3 

m

Þ=ð1 þ

m

Þ in plane stress;

m

is Poisson’s ratio; F1and F2are the resultant

stress components from point A to point B along any arc, and ½B

A denotes the value difference between two points

A and B.

Further, we assume that the interfaces are perfectly bonded, and therefore the field quantities satisfy

umj@Vp¼ u ðpÞ i   _@V p ; Fmj@Vp¼ F ðpÞ i   _@V p ; ð3:3Þ

where @Vp¼ apeihp ap

r

p, and the subscripts m and i

denote the matrix and inclusion, respectively.

FollowingBuryachenko and Kushch (2006), the general solution for the matrix is the superposition of the external applied field and the disturbances induced by the inclu-sions. That is,

um¼ uextþ XN l¼1 uðlÞ m; Fm¼ Fextþ XN l¼1 FðlÞ m; ð3:4Þ where uext¼ 1 2

l

m

j

m

u

extðzÞ  z

u

0extðzÞ  wextðzÞ

h i þ

e

 mz; uðlÞ m ¼ 1 2

l

l

j

l

u

ðlÞmðzlÞ  zl

u

ðlÞ0mðzlÞ  wðlÞmðzlÞ h i ; Fext¼

u

extðzÞ þ z

u

0extðzÞ þ wextðzÞ;

FðlÞ

m ¼

u

ðlÞmðzlÞ þ zl

u

ðlÞ0mðzlÞ þ wðlÞmðzlÞ; ð3:5Þ

zl¼ z  Zl, and Zlis the center of the lth inclusion. On the

contrary, for the pth inclusion

uðpÞi ¼ 1 2

l

p

j

p

u

ðpÞi ðzpÞ  zp

u

ðpÞ0i ðzpÞ  wðpÞi ðzpÞ h i þ

e

 pzp; FðpÞi ¼

u

ðpÞ i ðzpÞ þ zp

u

ðpÞ0i ðzpÞ þ wðpÞi ðzpÞ: ð3:6Þ

We consider a situation where the composite is subjected to a homogeneous remote strain

e

0

ij. It follows from(3.5)1that

u

extðzÞ ¼

C

uz; wextðzÞ ¼

C

wz; ð3:7Þ where

C

u¼

l

m

e

011þ

e

022  

j

m 1 ;

C

w¼

l

m

e

0 22

e

0 11þ 2i

e

0 12   : ð3:8Þ

Similarly if the composite is subjected to a homogeneous remote stress

r

0 ij,

C

u¼

r

0 11þ

r

0 22 4 ;

C

w¼

r

0 22

r

0 11þ 2i

r

0 12 2 : ð3:9Þ

In addition, the potential fields for the pth cylinder and its matrix can be expanded as

u

ðpÞ i ðzpÞ ¼ X1 n¼0 CðpÞn z n p; w ðpÞ i ðzpÞ ¼ X1 n¼0 DðpÞn z n p ð3:10Þ

for the inclusion, and

u

ðlÞ mðzlÞ ¼ X1 n¼1 AðlÞnz n l ; w ðlÞ mðzlÞ ¼ X1 n¼1 BðlÞnz n l ð3:11Þ

for the matrix. Here the coefficients AðlÞ n; B ðlÞ n; C ðlÞ n and D ðlÞ n

are some unknowns to be determined. The superscript p ðlÞ indicates that the fields are expanded with respect to the pth (lth) cylinder’s center ZpðZl).

To proceed, we shift the origin of the expansions to a fixed point, say Zp. For point z satisfying the inequality

jzpj < jZp Zlj, we can expand the term znl using the

bino-mial theorem as zn l ¼ X1 s¼0 ð1Þs n þ s  1 s   ðZp ZlÞnszsp X1 s¼0 Lplnsz s p: ð3:12Þ

Introducing (3.11) and (3.12) into (3.4), we have the expansions XN l¼1

u

ðlÞ mðzlÞ ¼ X1 n¼1 AðpÞ n znp þ X1 s¼0 AðpÞ sz s p; ð3:13Þ XN l¼1 wðlÞ mðzlÞ ¼ X1 n¼1 BðpÞn znp þ X1 s¼0 BðpÞsz s p ð3:14Þ and XN l¼1 zl

u

ðlÞ0mðzlÞ ¼ zp  X1 n¼1 nAðpÞ n zn1p þ X1 s¼0 sAðpÞ szs1p " # X 1 s¼0 AðpÞ szsp; ð3:15Þ

which are valid for the domain

Here AðpÞ s XN l – p X1 n¼1 AðlÞ nL pl ns; B ðpÞ s XN l – p X1 n¼1 BðlÞ nL pl ns; AðpÞs XN l – p X1 n¼1 AðlÞ nL pl nsðn þ sÞ jZp Zlj2 ðZp ZlÞ 2: ð3:17Þ

Substituting (3.10), (3.11) and (3.13)–(3.15) into the interface conditions(3.3), we obtain

1 2

l

m

j

m Cuðap

r

pþZpÞþ X1 s¼1 AðpÞ s asp

r

sp þ X1 s¼0 AðpÞ sa s p

r

s p " # ( Cu ap

r

pþZp   ap

r

p  X1 s¼1 sAðpÞs as1p

r

sþ1p þ X1 s¼0 sAðpÞ_sas1 p

r

sþ1p ! þX 1 s¼0 AðpÞ_sas p

r

sp Cw a1p

r

pþZp  X 1 s¼1 BðpÞ s asp

r

p X 1 s¼0 BðpÞ sa s p

r

sp ) þ

e

 m ap

r

pþZp   ¼ 1 2

l

p

j

p X1 s¼0 CðpÞs asp

r

spap

r

p X1 s¼0 sCðpÞs as1p

r

sþ1p  X1 s¼0 DðpÞs asp

r

sp " # þ

e

 pap

r

p ð3:18Þ and

C

uðap

r

pþ ZpÞ þ X1 s¼1 AðpÞ s asp

r

sp þ X1 s¼0 AðpÞ sa s p

r

sp þ

C

u ap

r

pþ Zp   þ ap

r

p  X1 s¼1 sAðpÞ s as1p

r

sþ1p þ X1 s¼0 sAðpÞ sas1p

r

sþ1p ! X 1 s¼0 AðpÞ sa s p

r

sp þ

C

wðap

r

1p þ ZpÞ þ X1 s¼1 BðpÞ s asp

r

sp þX 1 s¼0 BðpÞ sa s p

r

sp ¼X 1 s¼0 CðpÞ s a s p

r

s pþ ap

r

p X1 s¼0 sCðpÞ s as1p

r

sþ1p þX 1 s¼0 DðpÞs a s p

r

s p : ð3:19Þ

Equating the coefficients of

r

s

p and

r

sp of(3.18) and (3.19)gives 1 2

l

_m ð

j

m 1Þ

C

u AðpÞ1 h i apds1 n þ ð

j

m 1Þ

C

uZp

C

wZp 2AðpÞ2a 2 pþ A ðpÞ 0 B ðpÞ 0 h i ds0 þ

j

mAðpÞsa s pþ ðs  2ÞA ðpÞ s2a sþ2 p Hðs  3Þ  B ðpÞ s a s p Hðs  1Þ o þ

e

 mðapds1þ Zpds0Þ ¼ 1 2

l

p

j

pCðpÞs a s p C ðpÞ 1 apds1 ð2CðpÞ2 a 2 pþ D ðpÞ 0 Þds0 h i þ

e

 papds1; s P 0; ð3:20Þ 

C

wapds1þ

j

mAðpÞs asp  ðs þ 2ÞA ðpÞ ðsþ2Þa sþ2 p þ A ðpÞ sa s p B ðpÞ sa s p ¼ 

l

m

l

p ðs þ 2ÞCðpÞ_sþ2asþ2 p þ D ðpÞ s asp h i ; s P 1 ð3:21Þ and ð2

C

uþ AðpÞ1Þapds1þ ð2

C

uZpþ

C

wZpþ 2AðpÞ2a 2 p A ðpÞ 0þ B ðpÞ 0Þds0 þ AðpÞsa ðsÞ p  ðs  2ÞA ðpÞ s2a sþ2 p Hðs  3Þ þ B ðpÞ s a s p Hðs  1Þ ¼ CðpÞs a s pþ C ðpÞ 1 apds1þ ð2CðpÞ2 a 2 pþ D ðpÞ 0 Þds0; s P 0; ð3:22Þ

C

wapds1þ AðpÞs asp þ ðs þ 2ÞA ðpÞ ðsþ2Þasþ2p  AðpÞsa s pþ B ðpÞ sa s p ¼ ðs þ 2ÞCðpÞ_sþ2asþ2 p þ D ðpÞ s asp; s P 1; ð3:23Þ

where dijis the Kronecker delta and HðÞ is the unit step

function.

After some algebra, Eqs.(3.20)–(3.23)can be arranged as

K

ðpÞ1 A ðpÞ s a2sp þ ðs þ 2ÞA ðpÞ ðsþ2Þa 2 p A ðpÞ sþ B ðpÞ s ¼ 

C

wds1; s P 1; ð3:24Þ

K

ðpÞ₃ BðpÞ 1 a 2 p þ ReA ðpÞ 1¼

K

ðpÞ 4 

C

u; s ¼ 1; ð3:25Þ AðpÞsþ

K

ðpÞ 2 ðs  2ÞA ðpÞ s2a 2sþ2 p  B ðpÞ s a 2s p h i ¼ 0; s P 2 ð3:26Þ and CðpÞ 0 ¼

c

p

j

pþ 1

c

1 p þ

j

m 

C

uZpþ AðpÞ0  þ 2

l

mZp

e

m h i þ1 

c

p

j

pþ 1

C

uZpþ

C

wZpþ 2AðpÞ2a 2 p A ðpÞ 0þ B ðpÞ 0  ; ð3:27Þ CðpÞ 1 ¼

c

p 2

c

pþ

j

p 1 Re

P

þ i

c

p

j

pþ 1 Im

P

; ð3:28Þ CðpÞ s ¼ A ðpÞ s ðs  2ÞA ðpÞ s2a2sþ2p þ B ðpÞ s a2sp ; s P 2; ð3:29Þ DðpÞ 0 ¼ 2

C

uZpþ

C

wZpþ AðpÞ0þ B ðpÞ 0 C ðpÞ 0  A ðpÞ 0 þ 2 AðpÞ2 C ðpÞ 2  a2 p; ð3:30Þ DðpÞs ¼

C

wds1þ AðpÞs a2sp þ ðs þ 2ÞA ðpÞ ðsþ2Þa 2 p AðpÞs þ BðpÞs ðs þ 2ÞC ðpÞ sþ2a2p ð3:31Þ for s P 1. Here

c

p¼

l

p=

l

m;

P

¼ ð

j

mþ 1Þ

C

uþ AðpÞ1  þ 2

l

m

e

m

e

p  ;

K

ðpÞ₁ ¼1 þ

j

m

c

p 1 

c

p ;

K

ðpÞ₂ ¼

c

pþ

j

p

j

m

c

p

j

p ;

K

ðpÞ₃ ¼ 2

c

pþ

j

p 1 2

j

p 1 

c

pð

j

m 1Þ h i;

K

ðpÞ₄ ¼ 2

l

m

c

pð

e

 m

e

pÞ

j

p 1 

c

pð

j

m 1Þ : ð3:32Þ

Eqs. (3.24)–(3.26) and their conjugates constitute an infinite set of linear algebraic equations. Upon appropri-ate truncations of the expansion terms, we can determine the expansion coefficients AðpÞn and BðpÞn . Substituting them

back to (3.27)–(3.31), we can determine the remaining coefficients CðpÞ

n and D ðpÞ

n . After they are determined, all

the complex potentials of the matrix and inclusions are known, and then all the field variables can be easily obtained from Eqs. (3.1) and (3.2). For instance, the stress fields in the polar coordinate are given by the transformation

r

rr¼

r

11þ

r

22 2 þ

r

11

r

22 2 cos 2h þ

r

12sin 2h;

r

hh¼

r

11þ

r

22 2 

r

11

r

22 2 cos 2h 

r

12sin 2h;

r

rh¼ 

r

11

r

22 2 sin 2h þ

r

12cos 2h: ð3:33Þ

4. Numerical results and discussion

Below we perform a numerical computation for stress and displacement fields of composites of various

0 20 40 60 80 100 120 140 160 180 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 θ (deg) σθθ (P a ) 0 20 40 60 80 100 120 140 160 180 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 θ (deg) σθθ (P a ) 0 20 40 60 80 100 120 140 160 180 -6 -4 -2 0 2 4 6 8 θ (deg) σθθ (P a ) 0 20 40 60 80 100 120 140 160 180 -2 0 2 4 6 8 10 12 14 16x 10 8 θ (deg) σθθ (P a ) 0 20 40 60 80 100 120 140 160 180 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 θ (deg) σθθ (P a ) 0 20 40 60 80 100 120 140 160 180 -200 -150 -100 -50 50 100 θ (deg) σθθ (P a )

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2. The hoop stressrhhin the matrix along the interface for discrete values ofc¼li=lmunder six different remote loading cases (two PE inclusions in a

PM matrix): (a) uniform tensionr0

11¼ 1, (b) uniform tensionr022¼ 1, (c) pure shearr012¼ 1, (d) uniform vertical straine033¼ 1, (e) uniform transverse

electric field E0

3¼ 1 and (f) uniform transverse magnetic field H 0

remote loadings. The numerical calculations are first verified with the analytical solution for a composite of a piezoelectric inclusion in an elastic matrix. Our results of stress distributions agree very well with those of the solution proposed by Xiao and Bai (1999). Next we consider the particular case of a piezoelectric– piezomagnetic composite including a single inclusion with the center O, i.e. Z1¼ 0. The only non-zero

coeffi-cients are A1¼ 

K

11

C

wa2; B1¼ 2a2

j

mþ 1 ð Þ

C

uþ 2

l

m

e

m

e

i   2 þlm lið

j

i 1Þ 

C

u " # ; B3¼ 

K

11

C

wa4; C1¼ ð

j

mþ 1Þ

C

uþ 2

l

m

e

m

e

i   2 þlm lið

j

i 1Þ ; D1¼ ð1 

K

11 Þ

C

w: ð4:1Þ 0 20 40 60 80 100 120 140 160 180 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 θ (deg) σrr (P a )

(a) (b)

(c) (d)

(e) (f)

0 20 40 60 80 100 120 140 160 180 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ (deg) σrr (P a ) 0 20 40 60 80 100 120 140 160 180 -1.5 -1 -0.5 0 0.5 1 1.5 2 θ (deg) σrr (P a ) 0 20 40 60 80 100 120 140 160 180 -13 -11 -9 -7 -5 -3 -1 1 3 5 7 θ (deg) σrr (P a ) 0 20 40 60 80 100 120 140 160 180 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 θ (deg) σrr (P a ) 0 20 40 60 80 100 120 140 160 180 -20 0 20 40 60 80 100 120 140 160 180 200 220 θ (deg) σrr (P a )

Fig. 3. The normal contact stressrrrin the matrix along the interface for discrete values ofc¼li=lmunder six different remote loading cases (two PE

inclusions in a PM matrix): (a) uniform tensionr0

11¼ 1, (b) uniform tensionr022¼ 1, (c) pure shearr012¼ 1, (d) uniform vertical straine033¼ 1, (e) uniform

transverse electric field E0

3¼ 1 and (f) uniform transverse magnetic field H 0

Therefore, the complex potentials

u

mðzÞ ¼ A1z1;

wmðzÞ ¼ B1z1þ B3z3 ð4:2Þ

for the matrix, and

u

iðzÞ ¼ C1z;

wiðzÞ ¼ D1z ð4:3Þ

for the inclusion.

Substitutions of(4.2) and (4.3)into Eq.(3.1)1, we obtain

the stress field in the fiber

ð

r

11þ

r

22Þi¼ 4

ð

j

mþ 1Þ

C

uþ 2

l

mð

e

m

e

iÞ

2 þlm

lið

j

i 1Þ

; ð4:4Þ

which covers the known result byTong et al. (2008). We now turn to the case of two inclusions of unit radius and with identical material properties placed on the x1-axis and jZ1 Z2j ¼ 4. The inclusion is piezoelectric

(PE), while the matrix is piezomagnetic (PM). A state of

plane stress is assumed. The hoop stress is presented in

Fig. 2for six different remote loading cases. Poisson’s ratio for both the inclusions and matrix is the same with

m

i¼

m

m¼ 1=3, but the shear modulus of the inclusions is

different from that of the matrix. Circumferential stresses

r

hh are plotted for

c

¼

l

i=

l

m¼ 0; 1=3; 1; 3; 1. The

remaining material constants are: e31¼ 4:32 C=m2;

e33¼ 18:6 C=m2;

j

33¼ 11:8  109C2=Nm2;

l

33¼ 10  10 6

Ns2_=C2

for the PE phase, and q31¼ 580:3 N=Am; q33¼

699:7 N=Am;

j

33¼ 0:093  109C2=Nm2;

l

33¼ 157  10 6

Ns2_=C2_{for the PM phase. The results inFig. 2(a)–(c) agree}

within the plotting accuracy with results from Yu and

Sendeckyj (1974) and Mogilevskaya and Crouch (2001)

for the pure elastic problem. This is because

e

_{¼ 0 for}

these cases. The loading case for vertical strain and trans-verse electromagnetic fields (Fig. 2(d)–(f)) is new. It is interesting to observe that, in contrast to the stress con-centration around the holes subjected to in-plane loadings, the hoop stress is zero around the holes subjected to the external applied vertical strain and transverse electromag-netic fields. On the other hand, there is no concentration for the normal contact stress

r

rr around the holes under

all kinds of remote loadings (Fig. 3). Further, the stress concentration in a multiferroic composite may aggravate or alleviate by adjusting the magnitude and sign of various loading combinations.

Fig. 4 shows the hoop stress in the matrix along the

interface under an uniform remote tension

r

0

11¼ 1 for a

PM matrix containing three PE inclusions. The material properties are the same as those for the two-inclusion case. Again, the results agree within the plotting accuracy with the result fromMogilevskaya and Crouch (2001) by the Galerkin boundary integral method for the pure elastic problem.

Finally, in Fig. 5 we demonstrate the displacement contours for a PM matrix containing five PE inclusions under an uniform remote tension

r

0

11¼ 1. Each of the

inclusions has different material properties and radii (

l

1=

l

m¼ 4;

l

2=

l

m¼ 6;

l

3=

l

m¼ 8;

l

4=

l

m¼ 10;

l

5=

l

m¼ 12. a1¼ 1:5; a2¼ 1:25; a3¼ 1; a4¼ 0:75; a5¼ 0:5. O1¼ u_x -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 x 10-11 1 2 3 4 5 u_y -5 0 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 0 1 x 10-11 1 2 3 4 5

Fig. 5. The displacement contours for a PM matrix containing five PE inclusions under an uniform remote tensionr0

11¼ 1. Phase properties are

l1=lm¼ 4;l2=lm¼ 6;l3=lm¼ 8;l4=lm¼ 10;l5=lm¼ 12. The radii are a1¼ 1:5; a2¼ 1:25; a3¼ 1; a4¼ 0:75; a5¼ 0:5. The centers of the inclusions

locate at O1¼ ð2:5; 1:5Þ; O2¼ ð2:5; 0Þ; O3¼ ð1; 2Þ; O4¼ ð1; 3Þ, and O5¼ ð3:5; 3:5Þ, respectively.

0 60 120 180 240 300 360 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 θ (deg) σθθ (P a ) γ → ∞ γ =3 γ =1/3 γ =1 γ =0 1 ₃ 1 3 θ

Fig. 4. The hoop stressrhhin the matrix along the interface for discrete

values ofc¼li=lmunder uniform tensionr011¼ 1 (three PE inclusions in

ð2:5; 1:5Þ; O2¼ ð2:5; 0Þ; O3¼ ð1; 2Þ; O4¼ ð1; 3Þ; O5¼

ð3:5; 3:5Þ. The remaining material properties are the same as those for the two-inclusion case). We observe that, in contrast to the generalized anti-plane case (Kuo

and Bhattacharya, 2013) the displacement fields inside

the inclusions are nonlinear with respect to z. Thus the higher-order terms play an important role in the in-plan deformation.

5. Concluding remarks

In summary, we have extended Muskhelishvili’s formu-lation on an elastic composite with circular boundaries to a magneto-electro-elastic composite consisting of multiple cylinders under generalized plane strain with transverse electromagnetic intensities. We reduce the multi-field cou-pling problem to an equivalent in-plane elasticity problem by introducing an uniform eigenstrain corresponding to the magneto-electro-elastic effect. The admissible poten-tials for the inclusions and matrix are calculated within sufficient accuracy for several configuration under differ-ent loading cases. Numerical results are compared with the previous known solutions and are shown in good agreement. It is observed that, in contrast to the stress con-centration around the holes subjected to in-plane loadings, the stress is zero around the holes under the external applied vertical strain or transverse electromagnetic fields. In addition, the stress concentration in a piezoelectric– piezomagnetic composite may aggravate or alleviate by adjusting the magnitude and sign of various loading combinations.

Acknowledgment

This work was supported by the National Science Coun-cil Taiwan, under Contract No. NSC 102-3332-E-009-087. References

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