Effect of the imperfect interface on the scattering of SH wave in a piezoelectric cylinder in a piezomagnetic matrix

Effect of the imperfect interface on the scattering of SH wave

in a piezoelectric cylinder in a piezomagnetic matrix

Hsin-Yi Kuo

⇑

, Shu-Han Yu

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 14 November 2013

Received in revised form 7 August 2014 Accepted 28 August 2014

Available online 7 October 2014 Keywords:

Multiferroic composite Scattering

Anti-plane shear wave Imperfect interface

a b s t r a c t

We propose an exact analysis for the scattering of an anti-plane shear wave by a piezoelec-tric circular cylinder in a piezomagnetic matrix with imperfect interfaces. Two typical imperfect interfaces are investigated: mechanically stiff and highly electromagnetic conducting interfaces, and mechanically compliant and weakly electromagnetic conduct-ing interfaces. We obtain the fields of scattered wave by means of series expansion, and show that whether the interface is a perfect contact or with imperfection, it is sufficient to invert a 4  4 matrix and an infinite number of 6  6 matrices to solve the involved unknowns. Numerical examples are presented to demonstrate the effect of the imperfec-tion on the directivity patterns, scattering cross-secimperfec-tions, and potential field distribuimperfec-tions. Results show that the mechanical or highly electric conducting imperfect interface has great effect on those values. Further, we observe a large low-frequency peak of the scattering cross-section for the composite with mechanical stiff imperfection.

Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Multiferroic composites consisting of piezoelectric and piezomagnetic phases exhibit a magneto-electric (ME) effect that is absent in each constituent. The ME effect, which is related to inducing an electric polarization by a magnetic field or conversely inducing a magnetization by an electric field, provides a variety of technological applications including magnetic field sensors, four-state memory cells, and energy harvesting devices, etc. This has motivated a number of experimental fabrications and theoretical predictions of ME composites. An extensive review of the literature and the state of the art can be found inFiebig (2005), Nan, Bichurin, Dong, Viehland, and Srinivasan (2008), Bichurin, Petrov, Averkin, and Liverts (2010), Srinivasan (2010), Ma, Hu, Li, and Nan (2011).

The coupling in the multiferroic composite is achieved through the product property: the applied electric field creates a deformation in the piezoelectric phase which in turn induces a deformation in the piezomagnetic phase thereby inducing a magnetic field. The product ME effect is a new property determined by the mechanical interaction between the two phases. Therefore, the interface is critical in achieving the giant magnetoelectricity, and has also made the composites with imperfect interfaces the topic of a number of theoretical investigations. For example,Bichurin, Petrov, and Srinivasan (2003) intro-duced an interface coupling parameter that defines the degree to which the deformation of the piezoelectric layer follows that of the magnetostrictive layer.Nan, Liu, and Lin (2003)adopted the Green’s function approach to study the interfacial bonding on the ME effect in the PZT-Terfenol-D laminated composite.Chang and Carman (2007) proposed a quasistatic

http://dx.doi.org/10.1016/j.ijengsci.2014.08.006

0020-7225/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author.

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

Contents lists available atScienceDirect

International Journal of Engineering Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j e n g s c i

model including shear lag and demagnetization effect for predicting the ME effect in ME laminates.Wang, Pan, and Roy (2007) and Pan, Wang, and Wang (2009)employed the complex variable approach and Mori–Tanaka method to derive the effective moduli of ME fibrous composites with soft and stiff imperfect interfaces, respectively.Kuo (2013)generalized the classic work ofRayleigh (1892)in a periodic conductive perfect composite to the coupled magneto-electro-elastic fibrous composites with imperfect interfaces.

Recently, the dynamic behavior of ME composites has received considerable attention. For instance,Du, Shen, Ye, and Yue (2004)examined the scattering of anti-plane shear waves by a cylindrical inhomogeneity in a magneto-electro-elastic matrix with partial debonding.Chen and Shen (2007)extended the work ofLevin, Michelitsch, and Gao (2002), who studied the elastic waves propagation in composites with piezoelectric fibers, to ME composites. They first solved the problem associated with a single cylindrical fiber, and then considered the problem associated with multiple fibers by employing the effective field approach.Soh and Liu (2006)studied the propagation of an interfacial shear horizontal (SH) wave in two bonded semi-infinite piezoelectric-piezomagnetic materials.Chen, Pan, and Chen (2007)presented an analytical treat-ment for the propagation of harmonic waves in magneto-electro-elastic multilayered plates.Liu, Fang, and Liu (2007) and Wang, Mai, and Niraula (2007) demonstrated the existence of a SH surface wave in a semi-infinite ME medium with hexagonal symmetry.Pang, Liu, Wang, and Zhao (2008) and Liu, Fang, Wei, and Zhao (2008)investigated the propagation of Rayleigh-type surface waves and Love waves in a piezoelectric-piezomagnetic layered half-space. Effects of the imperfect interface on the SH waves, interfacial SH waves, and reflection and transmission of planes waves in a multiferroic composites were also studied bySun, Ju, Pan, and Li (2011), Huang, Li, and Lee (2009), and Pang and Liu (2011), respectively.

Motivated by these advances, and in a departure from previous works, this research is devoted to the anti-plane shear wave scattering by a piezoelectric circular cylinder of infinite length, which is imperfectly bonded to a piezomagnetic matrix. Both the fiber and matrix are assumed to be transversely isotropic, which are the materials frequently used in applications. Note that due to the polarization (magnetization) of the piezoelectric (piezomagnetic) material, the phase cannot possess the center of symmetry which makes the components of the piezoelectric (piezomagnetic) tensors be zero. Two kinds of imper-fect interfaces are studied: (1) mechanically stiff and highly electromagnetic conducting interfaces, at which the potentials are continuous across the interface, while the normal component of flux undergoes a discontinuity which is proportional to the local surface Laplacian of the potential fields; and (2) mechanically compliant and weakly electromagnetic conducting interfaces, at which the normal flux are continuous, while the potentials are discontinuous at such contact. The jumps in potential components are further assumed to be proportional to their respective interface flux components.

The remainder of this paper is organized as follows. In Section2we formulate the governing equation for the anti-plane shear wave scattering by a piezoelectric circular cylinder in a piezomagnetic matrix. We obtain the solution in Section3. Two kinds of imperfect contacts are studied: mechanically stiff and highly electromagnetic conducting interfaces, and mechan-ically compliant and weakly electromagnetic conducting interfaces. Numerical examples are demonstrated in Section4

using composites of BaTiO3 and CoFe2O4. We study the effect of the imperfection on the directivity pattern, scattering

cross-section, and potential field distributions. 2. Formulation

Let us consider an unbounded piezomagnetic matrix containing an infinite long piezoelectric circular cylinder of radius a. Assume that each phase is transversely isotropic (i.e., has 6 mm symmetry) with the symmetry axes oriented with cylinders. We introduce a Cartesian coordinate system with x- and y-axes in the plane of the cross-section and z-axis along the axis of the cylinder. Assume that the cylinder is subjected to an incident anti-plane shear (SH) wave traveling in the positive x-direction.

The constitutive laws of the pth phase for the non-vanishing fields in a polar coordinate system can be recast in the compact form as (Wang, Pan et al., 2007)

RðpÞ_j ¼ LðpÞZðpÞ_j ; j ¼ r; h; ð2:1Þ

where for ease of the terminology, p = ‘‘e’’ (p = ‘‘m’’) refers to the piezoelectric (piezomagnetic) phase,

LðeÞ ¼ C44 e15 0 e15 

j

11 0 0 0 

l

11 0 B @ 1 C A ðeÞ ; LðmÞ ¼ C44 0 q15 0 

j

11 0 q15 0 

l

11 0 B @ 1 C A ðmÞ ; RðpÞ_j ¼

r

zj Dj Bj 0 B @ 1 C A ðpÞ ; ZðpÞr ¼ @UðpÞ @r ; Z ðpÞ h ¼ @UðpÞ r@h ; U ðpÞ_¼ w

u

w 0 B @ 1 C A ðpÞ : ð2:2Þ

Here

r

zj;Dj;Bjare the stress, electric displacement, and magnetic flux. C44;

j

11, and

l

11are, respectively, the elastic modulus,

dielectric permittivity, and magnetic permeability, while e15and q15are the piezoelectric and piezomagnetic coefficients.

In the absence of body force, electric charge density and electric current density, the equilibrium equations are given by C44 e15 q15 e15 

j

11 0 q15 0 

l

11 0 B @ 1 C A ðpÞ

r

2w

r

2

u

r

2 w 0 B @ 1 C A ðpÞ ¼ 

qx

2_w 0 0 0 B @ 1 C A ðpÞ ; ð2:3Þ wherer2 ¼1 r @ @rþ @2 @r2þ_r12 @ 2

@h2represents the two-dimensional Laplace operator for the variable r and h, and

q

is the mass den-sity of the material. Note that throughout the paper, the time factor eixt_{, where}

_x

_{is the angular frequency, is suppressed. In}

addition, the quasi-static approximation for the electric and magnetic fields is used in the analysis. This is because the cor-responding characteristic velocity of the electromagnetic waves has 105_{times higher than that of the elastic waves.}

There-fore, we can neglect the electromagnetic field generated by the elastic field propagation (Levin et al., 2002). To proceed, we introduce two new functions

u

and w which are related to w;

u

, and

w

by

u

ðeÞ_¼

_u

ðeÞ_e ðeÞ 15

j

ðeÞ 11 wðeÞ_; wðmÞ¼ wðmÞ_q ðmÞ 15

l

ðmÞ 11 wðmÞ_{: ð2:4Þ}

Substitution of Eq.(2.4)into Eq.(2.1)yields the auxiliary constitutive laws:

RðpÞ_j ¼ LðpÞ_ZðpÞ j ; ð2:5Þ ZðpÞ r ¼ @UðpÞ @r ; Z ðpÞ h ¼ @UðpÞ r@h ; where LðeÞ_¼ eC44 e15 0 0 

j

11 0 0 0 

l

11 0 B @ 1 C A ðeÞ ; UðeÞ¼ w

u

w 0 B @ 1 C A ðeÞ ð2:6Þ

for the piezoelectric material, and

LðmÞ_¼ eC44 0 q15 0 

j

11 0 0 0 

l

11 0 B @ 1 C A ðmÞ ; UðmÞ¼ w

u

w 0 B @ 1 C A ðmÞ ð2:7Þ

for the piezomagnetic material. Further, the governing equation(2.3)decoupled into the Helmholtz and Laplace equations for the phase:

ð

r

2 þ k2eÞw ðeÞ_{¼ 0;}

_r

2

u

ðeÞ_{¼ 0;}

_r

2 wðeÞ¼ 0; ð

r

2þ k2mÞwðmÞ¼ 0;

r

2

u

ðmÞ_{¼ 0;}

_r

2 wðmÞ_{¼ 0; ð2:8Þ}

where k is the wave number defined by

k2e

q

ðeÞ

_x

2 eCðeÞ 44 ; eCðeÞ₄₄ CðeÞ44þ eðeÞ 15  2

j

ðeÞ 11 ; k2m

q

ðmÞ

_x

2 eCðmÞ 44 ; eCðmÞ₄₄  CðmÞ44 þ qðmÞ15  2

l

ðmÞ 11 ; ð2:9Þ with eCðpÞ

44 being the stiffened elastic constant.

3. Representation of the solution

Consider that the composite is subjected to an incident SH wave of unit amplitude propagating along the x-direction, and can be expanded as (Arfken & Weber, 2001)

eikmx_{¼ J}

0ðkmrÞ þ 2 X1 n¼1

where i is the imaginary number. The general solution to the Helmholtz and Laplace equations for the circular cylinder and its surrounding matrix can be expanded with respect to its center as

wðeÞ_{ðr; hÞ ¼ A}w 0J0ðkerÞ þ 2 X1 n¼1 inAw nJnðkerÞ cos nh;

u

ðeÞ_{ðr; hÞ ¼ A}u 0þ X1 n¼1 Aunðr=aÞ n cos nh; wðeÞðr; hÞ ¼ Aw0þ X1 n¼1 Aw nðr=aÞ n_{cos nh} ð3:2Þ

for the piezoelectric inclusion, and

wðmÞ_{ðr; hÞ ¼ e}ikmx_{þ B}w 0H0ðkmrÞ þ 2 X1 n¼1 inBw nHnðkmrÞ cos nh;

u

ðmÞ_{ðr; hÞ ¼}X 1 n¼1 Bunða=rÞ n cos nh; wðmÞ_{ðr; hÞ ¼}X 1 n¼1 Bw nða=rÞ n_{cos nh} ð3:3Þ

for the piezomagnetic matrix. Here r; hð Þ is the polar coordinate centered at the origin of the inclusion. JnðÞ and HnðÞ are,

respectively, the Bessel function of the first kind and the Hankel function of the first kind, both of order n. The coefficients Awn;A u n;A w n;B w n;Bun;B w

n, are unknown constants to be determined from the interface and boundary conditions. The dimension of

these coefficients are the same as the corresponding displacement w [L], electric potential [ML2A1_T3_{], and magnetic}

poten-tial [A].

In order to treat the imperfect interface effect, we first resort to a more general three-phase composite of a similar dis-tribution in which the inclusions possess a concentric elastic coating of thickness t and material parameter Lc¼ diagðC44;

j

11;

l

11Þ (Hashin, 2001; Miloh & Benveniste, 1999; Torquato & Rintol, 1995.) By passing to the limit that

t ! 0 and that either L1

c ! 0 (mechanically stiff and highly electromagnetic conducting interfaces) or Lc! 0 (mechanically

soft and weakly electromagnetic conducting interfaces), we recover the distribution of interest in which the interfacial prop-erty is characterized by the parameters

a

and b given by:

a

¼ lim t!0 L1 c !0 tLc ð Þ ¼

a

w _{0 0} 0

a

u ₀ 0 0

a

w 0 B @ 1 C A ð3:4Þ

for the mechanically stiff and highly electromagnetic conducting case, and

b¼ lim t!0 Lc !0 tL 1 c   ¼ bw 0 0 0 bu 0 0 0 bw 0 @ 1 A ð3:5Þ

for the mechanically soft and weakly electromagnetic conducting case.

We first consider that the interface is mechanically stiff and highly electromagnetic conducting, that is,

RðmÞ r   @V R ðeÞ r   @V¼

a

D

sU ðeÞ_ @V; U ðmÞ_ @V¼ U ðeÞ_ @V; ð3:6Þ whereDs¼r12 @ 2

@h2is the surface Laplace operator, and @V : r ¼ a denotes the interface between the matrix and the circular

cylinder. The case where

a

¼ 0 corresponds to a perfect interface, whereas

a

1_{¼ 0 describes an isoexpansion and}

equipoten-tial interface.

Using the orthogonality properties of trigonometric and Bessel (Hankel) functions, the interface conditions(3.6)provide

K0u0¼

v

0; ð3:7Þ where K0¼ H0ðkmaÞ J0ðkeaÞ 0 0 0 e ðeÞ 15 jðeÞ 11 J0ðkeaÞ 1 0 q ðmÞ 15 lðmÞ 11 H0ðkmaÞ 0 0 1 eCðmÞ44kmH00ðkmaÞ eCðeÞ44keJ00ðkeaÞ 0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ;

0.02 0.04 0.06 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 0.5 0.05 0.1 0.15 0.2 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 1 0.1 0.2 0.3 30 210 60 240 90 270 120 300 150 330 180 0 k_ma= 3 0.2 0.4 0.6 0.8 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 5 1 2 3 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 10 5 10 15 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 20

0.1 0.2 0.3 0.4 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 0.5 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 1 1 2 3 4 30 210 60 240 90 270 120 300 150 330 180 0 k_ma= 3 1 2 3 4 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 5 2 4 6 8 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 10 5 10 15 20 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 20

0.01 0.02 0.03 0.04 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 0.5 0.02 0.04 0.06 0.08 0.1 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 1 0.1 0.2 0.3 30 210 60 240 90 270 120 300 150 330 180 0 k_ma= 3 0.2 0.4 0.6 0.8 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 5 0.5 1 1.5 2 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 10 2 4 6 8 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 20

0.05 0.1 0.15 0.2 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 0.5 0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 1 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 k ma = 3 1 2 3 4 5 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 5 2 4 6 8 10 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 10 5 10 15 20 30 210 60 240 90 270 120 300 150 330 180 0 k_ma = 20

u0¼ Bw0 Aw0 Au0 Aw 0 0 B B B B @ 1 C C C C A;

v

0¼ J0ðkmaÞ 0 qðmÞ₁₅ lðmÞ 11 J0ðkmaÞ e CðmÞ 44kmJ00ðkmaÞ 0 B B B B B @ 1 C C C C C A ð3:8Þ for n ¼ 0, while Knun¼

v

n; ð3:9Þ Kn¼

2inHnðkmaÞ 0 0 2inJnðkeaÞ 0 0

0 1 0 2in e ðeÞ 15 jðeÞ 11 JnðkeaÞ 1 0 2in q ðmÞ 15 lðmÞ 11 HnðkmaÞ 0 1 0 0 1 2ineCðmÞ44kmH0nðkmaÞ 0 nqðmÞ 15 a 2ineCðeÞ44keJ0nðkeaÞ 2in

a

w n2 a2JnðkeaÞ neðeÞ 15 a 0 0 

j

ðmÞ 11 0 2i n

a

ueðeÞ15 jðeÞ 11 n aJnðkeaÞ

a

un a 

j

ðeÞ 11 0 0 0

l

ðmÞ 11 0 0

l

ðeÞ 11 

a

wn a 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; un¼ Bw n Bun Bw n Awn Aun Aw n 0 B B B B B B B B B @ 1 C C C C C C C C C A ;

v

n¼ 2in JnðkmaÞ 0 qðmÞ₁₅ lðmÞ 11 JnðkmaÞ eCðmÞ 44kmJ0nðkmaÞ 0 0 0 B B B B B B B B B B @ 1 C C C C C C C C C C A ð3:10Þ

for n ¼ 1; 2; 3; . . . ; 1. Here the prime0_{denotes the derivative with respect to the variable in the parenthesis.}

Next, we consider that the interface is mechanically soft and weakly electromagnetic conducting, i.e.,

RðmÞr   @V¼ R ðeÞ r   @V; U ðmÞ_ @V U ðeÞ_ @V¼ bR ðeÞ r   @V: ð3:11Þ

The case where b ¼ 0 corresponds to a perfect interface, whereas b1_{¼ 0 describes a completely debonded and electric}

charge-free (insulating) interface. Analogous to the previous case, the interface conditions(3.11) give constraints (3.7) and (3.9), but with K0and Knreplaced by

0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 k ma Scattering cross-section Perfect α=diag(1,0,0) α=diag(0,1,0) β=diag(1,0,0)

Fig. 5. Scattering cross-sections of the scattered shear wave for a perfect interface (solid line ‘‘-’’), for a stiff interface (dashed line ‘‘- -’’), for a highly electric conducting interface (dashed-dotted line ‘‘-’’), and for a compliant interface (dotted line ‘‘. . .’’).

0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 |w| (m) k_ma=0.5 k_ma=1.0 k_ma=3.0 0.005 0.01 0.015 30 210 60 240 90 270 120 300 150 330 180 0 |_ε 0φ| (C/m) k_ma=0.5 k ma=1.0 k_ma=3.0 0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 |μ₀ψ| (N/A) k_ma=0.5 k_ma=1.0 k_ma=3.0

a

b

c

Fig. 6. Angular distribution of the absolute value of vertical displacement, electric potential and magnetic potential for a cylinder with a perfect contact versus h for various kma.

2 4 6 30 210 60 240 90 270 120 300 150 330 180 0 |w| (m) k ma=0.5 k ma=1.0 k_ma=3.0 0.01 0.02 0.03 0.04 0.05 30 210 60 240 90 270 120 300 150 330 180 0 |ε₀φ| (C/m) k_ma=0.5 k ma=1.0 k_ma=3.0 2 4 6 30 210 60 240 90 270 120 300 150 330 180 0 |μ₀ψ| (N/A) k_ma=0.5 k ma=1.0 k_ma=3.0

a

b

c

Fig. 7. Angular distribution of the absolute value of vertical displacement, electric potential and magnetic potential for a cylinder with a stiff interface a¼ diagð1; 0; 0Þ versus h for various kma.

0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 |w| (m) k_ma=0.5 k_ma=1.0 k_ma=3.0 0.001 0.002 0.003 0.004 0.005 30 210 60 240 90 270 120 300 150 330 180 0 |_ε 0φ| (C/m) k_ma=0.5 k_ma=1.0 k_ma=3.0 0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 |_μ0ψ| (N/A) k_ma=0.5 k ma=1.0 k_ma=3.0

c

b

a

Fig. 8. Angular distribution of the absolute value of vertical displacement, electric potential and magnetic potential for a cylinder with a highly electric conducting interfacea¼ diagð0; 1; 0Þ versus h for various kma.

0.5 1 1.5 2 30 210 60 240 90 270 120 300 150 330 180 0 |w| (m) k_ma=0.5 k ma=1.0 k_ma=3.0 0.002 0.004 0.006 0.008 0.01 30 210 60 240 90 270 120 300 150 330 180 0 |ε₀φ| (C/m) k_ma=0.5 k_ma=1.0 k_ma=3.0 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 |μ₀ψ| (N/A) k_ma=0.5 k_ma=1.0 k ma=3.0

a

b

c

Fig. 9. Angular distribution of the absolute value of vertical displacement, electric potential and magnetic potential for a cylinder with a complaint interface b¼ diagð1; 0; 0Þ versus h for various kma.

K0¼

H0ðkmaÞ bweCðeÞ44keJ00ðkeaÞ þ J0ðkeaÞ 0 0

0 e ðeÞ 15 jðeÞ 11 J0ðkeaÞ 1 0 q ðmÞ 15 lðmÞ 11 H₀ðkmaÞ 0 0 1 eCðmÞ44kmH00ðkmaÞ eC ðeÞ 44keJ00ðkeaÞ 0 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ; ð3:12Þ Kn¼ 2inHnðkmaÞ 0 0 2i n bwCeðeÞ 44keJ0nðkeaÞ þ2inJnðkeaÞ bw ne ðeÞ 15 a 0 0 1 0 2in e ðeÞ 15 jðeÞ 11 JnðkeaÞ b unjðeÞ 11 a þ1 0 2in q ðmÞ 15 lðmÞ 11 HnðkmaÞ 0 1 0 0 b w nlðeÞ11 a þ1 2ineCðmÞ 44kmH0nðkmaÞ 0 nqðmÞ 15 a 2i n_eCðeÞ 44keJ0nðkeaÞ neðeÞ 15 a 0 0 

j

ðmÞ 11 0 0 

j

ðeÞ 11 0 0 0

l

ðmÞ 11 0 0

l

ðeÞ 11 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : ð3:13Þ

4. Numerical results and discussion

As a numerical example, we apply our solution to the BaTiO3(BTO, piezoelectric)-CoFe2O4(CFO, piezomagnetic)

multif-erroic composite. Both of them are transversely isotropic. The independent material constants of BTO are C44¼ 43  109N=m2;e15¼ 11:6 C=m2;

j

11¼ 11:2  109C2=Nm2;

l

11¼ 5  10

6

Ns2=C2;

q

¼ 6:02  103kg=m3_{, while those}

of CFO are C44¼ 45:3  109N=m2;q15¼ 550 N=Am;

j

11¼ 0:08  109C2=Nm2;

l

11¼ 590  10

6_Ns2_=C2_;

_q

¼ 5:20 103kg=m3_{(Cannas, Falqui, Musinu, Peddis, & Piccaluga, 2006; Wang, Pan et al., 2007). Here the xy plane is an isotropic plane}

and the poling direction/magnetic axis is along the z-direction. Further, the following dimensionless imperfect parameters have been chosen for computation in the analysis:

a

w_¼

_a

w_=ðCðeÞ

44aÞ;

a

u¼ 

a

u=ð

j

ðeÞ 11aÞ;

a

w¼ 

a

w=ð

l

ðeÞ 11aÞ; b w ¼ bwCðeÞ 44=a; bu¼ bu

j

ðeÞ 11=a, and b w ¼ bw

l

ðeÞ 11=a.

In our calculation, the series in Eqs.(3.2) and (3.3)are truncated at n ¼ N with a relative error less than 1%. It is observed that N is a monotonically increasing function of the normalized frequency kma. That is, the smaller truncated number of N

can be adopted when the wave frequency is smaller. 4.1. Directivity pattern

The directivity patterns of the scattered waves are of interest in practical applications. The scattered component in the mechanical displacement wðmÞ_{in the far field has the following asymptotic behavior (Abramowitz & Stegun, 1972; Wang,}

Pan et al., 2007) wðmÞ s ¼ ffiffiffiffiffiffiffiffiffiffiffi 2

p

kmr s ei kðmrp4Þ Bw 0þ 2 X1 n¼1 inBwne inp 2cosðnhÞ " # ; r ! 1; ð4:1Þ

where the subscript s denotes the scattering field in the matrix.

Substituting(4.1)into(3.3)2,3and(2.4)2, the stress

r

ðmÞzr;s, electric potential

u

ðmÞs and magnetic potential wðmÞs in the far field

induced by the scattered shear wave are given by

r

ðmÞ zr;s¼ ikmeCðmÞ44w ðmÞ s ;

u

ðmÞ s ¼ 0; w ðmÞ s ¼ qðmÞ 15

l

ðmÞ 11 wðmÞ s ; r ! 1: ð4:2Þ

Therefore, the far field behaviors of the stress, mechanical displacement, and magnetic potential in the piezomagnetic matrix at the far field are related. The far field directivity pattern of the scattered shear wave is defined byLiu, Wu, and Ying (2000)

D hð Þ ¼ Bw0þ 2 X1 n¼1 inBw nei np 2cos nh_{ð Þ}          ; ð4:3Þ

which exhibits the angular distribution of the absolute value of the amplitude of the stress component

r

ðmÞ

zr at a large

0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 |w| (m) k ma=0.5 k ma=1.0 k_ma=3.0 0.005 0.01 0.015 30 210 60 240 90 270 120 300 150 330 180 0 |_ε 0φ| (C/m) k_ma=0.5 k_ma=1.0 k ma=3.0 50 100 150 30 210 60 240 90 270 120 300 150 330 180 0 |μ₀ψ| (N/A) k ma=0.5 k ma=1.0 k_ma=3.0

c

b

a

Fig. 10. Angular distribution of the absolute value of vertical displacement, electric potential and magnetic potential for a cylinder with a weakly magnetic conducting interface b ¼ diagð0; 0; 1Þ versus h for various kma.

Fig. 1shows the directivity patterns of the scattered waves for a perfectly bonded cylinder (

a

¼ 0; b ¼ 0) under six differ-ent frequencies kma ¼ 0:5; 1; 3; 5; 10; 20.Figs. 2 and 3represent, respectively, the corresponding results for a stiff interface

with

a

¼ diagð1; 0; 0Þ and those for a highly electric conducting interface with

a

¼ diagð0; 1; 0Þ.Fig. 4plots the results for a compliant interface with b ¼ diagð1; 0; 0Þ. We do not show the plots for a highly magnetic conducting interface ð

a

¼ diagð0; 0; 1ÞÞ, and the weakly electromagnetic conducting interface ðb ¼ diagð0; 1; 0Þ; b ¼ diagð0; 0; 1ÞÞ since we observe that there are only minor effect of

a

w_;bu_{, and b}w

on the directivity pattern.

From these figures, we observe that for the mechanical imperfection interface (Figs. 2 and 4), the directivity pattern becomes complicated and concentrated at h ¼ 0 as the frequency increases, which means that the directivity pattern of the shadow side ðh ¼ 0Þ of the piezoelectric cylinder is more sensitive than that of the incident side ðh ¼

p

Þ. However, for the perfect interface as shown inFig. 1, the maximum of the directivity pattern occurs at the incident side at low frequency, while the pattern becomes complicated and concentrated along the opposite of the incident wave direction at high wavenumbers. For the highly electric conducting interface (Fig. 3), the maximum of the directivity pattern first occurs at the incident side at low frequency. As the frequency increases, the maximum of the pattern occurs symmetrically with respect to the x-axis and eventually merged along h ¼ 0. All the above imperfections significantly alter both the shape and size of the directivity pattern, and these multiple peaks results from the interference caused by the incident and reflected waves.

4.2. Scattering cross-section

The scattering cross-section of the shear wave for the cylinder is the ratio of the total energy flow carried outwards by the scattered wave to the energy flow of the incident wave through a normal area that is equal to the cross-section area of the scatterer, which is defined byLiu et al., 2000

Q ¼2 B w 0   2 þ 2P1n¼1i n Bwn   2 kma : ð4:4Þ

Fig. 5is the scattering cross-section of the scattered shear wave for a perfect interface (solid line ‘‘-’’), for a stiff interface (dashed line ‘‘- -’’), for a highly electric conducting interface (dashed-dotted line ‘‘-’’), and for a compliant interface (dotted line ‘‘. . .’’). For the other imperfect parameters, they have the same results for that of the perfect contact case. The curves are calculated in steps of kma ¼ 0:1 and N ¼ 60. The frequency range is kma ¼ 0 to 40. The most striking feature of the scattering

cross-section curves is the existence of a sequence of maximum and minimum for the scattered shear wave when the inter-face is a mechanical imperfect contact. Further, a large low-frequency peak occurring at kma ¼ 1, which corresponds to the

resonance scattering, can appear for a mechanically stiff imperfect interface with

a

¼ diagð1; 0; 0Þ. We will see how this effect influence the field distribution in the following subsection. We further remark that except there exists small fluctuation at small frequency, the scattering cross-section for a perfect contact and remaining kinds of interface imperfection is in general a monotonically increasing function of the normalized frequency kma.

4.3. Mechanical displacement, electric potential, and magnetic potential

Fig. 6is the angular distribution of vertical displacement, electric potential and magnetic potential for a cylinder with a perfect contact versus h for various kma.Figs. 7–10show the corresponding plots for a stiff interface with

a

¼ diagð1; 0; 0Þ

(Fig. 7), for a highly electric conducting interface with

a

¼ diagð0; 1; 0Þ (Fig. 8), for a compliant interface with b¼ diagð1; 0; 0Þ (Fig. 9), and for a weakly magnetic conducting interface with b ¼ diagð0; 0; 1Þ (Fig. 10). Similarly we do not show the figures for a highly magnetic conducting interface ð

a

¼ diagð0; 0; 1ÞÞ, and for a weakly electric conducting inter-face ðb ¼ diagð0; 1; 0ÞÞ since we observe that there are only minor effect of

a

w_{and b}u_{on the potential distribution.}

FromFigs. 6(a)–10(a)and6(c)–9(c), we observe that at kma ¼ 0:5, the distributions of displacement wj j and magnetic

potential

l

0w



  around the cylinder are almost uniform except those with the mechanical stiff imperfect interface (Fig. 7). In addition, multiple peaks and larger value appear along the circumference of the piezoelectric cylinder at kma ¼ 1 in

Fig. 7, which correspond to the resonance scattering shown inFig. 5. Further, we observe that the distributions of the electric potentialj

e

0

u

j are different from those of the displacement and magnetic potential. Each distribution has two lobes and is

symmetric with respect to both x- and y- axes at kma ¼ 0:5 and 1.0 (Figs. 6(b), 8(b)–10(b)). Finally, it is observed the

magnetic potential is extremely large for a cylinder with a weakly magnetic conducting interface (Fig. 10(c)), although the other two potentials are the same as those of the perfect case.

5. Concluding remarks

In summary, we have presented an exact analysis to the anti-plane shear wave scattering by a piezoelectric fiber in a piezomagnetic matrix with imperfect interfaces. Both mechanically stiff and highly electromagnetic conducting interfaces, and mechanically soft and weakly electromagnetic conducting interfaces are considered. Our analyses show that whether the interface is a perfect contact or with imperfection, it is sufficient to invert a 4  4 matrix and an infinite number of

6  6 matrices to solve the involved unknowns. Comparing with the results of the perfect contact case, the mechanical imperfection or highly electric conducting imperfect interface has great influence on the value and distribution of the direc-tivity pattern, scattering cross-section, mechanical displacement, and electromagnetic potential, while the weakly magnetic conducting interface has only influence on the value and distribution of the magnetic potential. The other imperfect param-eters, i.e., the highly (weakly) magnetic (electric) conducting imperfection has really minor effect on these figures of merits. We also observe a large low-frequency peak of the scattering cross-sections with mechanical stiff imperfection, and a sequence of small high-frequency peak for that with mechanical compliant imperfection. We note that for clarity we show the case of a piezoelectric fiber in a piezomagnetic matrix in the paper. However, based on this framework, it can be easily extended to its complimentary counterpart of a piezomagnetic fiber in a piezoelectric matrix, and can also be easily extended to take into account different kinds of imperfect interface conditions. The present theoretical framework provides a general guideline for the bonding interface of the piezoelectric and piezomagnetic phases under dynamic loading. Further, while in the present scattering problem, only a single piezoelectric cylinder is considered, the corresponding wave scattering by a cluster of piezoelectric cylinders is also of interest and forms the subject of future study.

Acknowledgment

We are pleased to thank Prof. Chih-Yu Kuo at the Academia Sinica, Taiwan, for helpful discussion. We also gratefully acknowledges the support of the National Science Council, Taiwan, through Grant NSC 102-2221-E-009-087.

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