Transient response of a Mode III interface crack between piezoelectric layer and functionally graded orthotropic layer

Accepted Manuscript

Transient response of a Mode III interface crack between piezoelectric layer and functionally graded orthotropic layer Jeong Woo Shin , Tae-Uk Kim PII: DOI: Reference:

S0020-7683(16)30018-X 10.1016/j.ijsolstr.2016.03.033 SAS 9121

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

7 December 2015 19 March 2016 31 March 2016

Please cite this article as: Jeong Woo Shin , Tae-Uk Kim , Transient response of a Mode III interface crack between piezoelectric layer and functionally graded orthotropic layer, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.03.033

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ACCEPTED MANUSCRIPT

Transient response of a Mode III interface crack between piezoelectric layer and functionally graded orthotropic layer

Jeong Woo Shina,* and Tae-Uk Kimb Korea Aerospace Research Institute, 169-84 Gwahak-ro, Yuseong-gu, Daejeon,

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a,*

34133, Republic of Korea. TEL : +82-42-860-2026 / E-mail : [email protected] b

Korea Aerospace Research Institute, 169-84 Gwahak-ro, Yuseong-gu, Daejeon, 34133,

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Republic of Korea.

Abstract

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In this study, transient response analysis of a Mode III interface crack between a piezoelectric layer and a functionally graded orthotropic material (FGOM) layer is

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conducted using integral transform techniques. Material properties of the FGOM layer change continuously along the layer thickness and two layers are connected weak-

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discontinuously. Laplace and Fourier transforms are applied to solve the problem, and

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the problem is then expressed by a Fredholm integral equation of the second kind. It is found that the followings are beneficial to impede transient fracture of the interface

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crack between piezoelectric layer and FGOM layer: a) an increase of shear moduli of the FGOM from the interface to the lower free surface; b) an increase of the gradient of shear moduli, a decrease of ratio of shear moduli, the electric boundary condition EBC I of the piezoelectric layer in the case of increase of shear moduli of the FGOM from the interface to the lower free surface; c) an increase of thickness of the layers.

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Keywords : Functionally graded orthotropic material (FGOM); Piezoelectric material; Interface; Crack; Transient response; Dynamic stress intensity factor (DSIF)

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1 Introduction

Ever since piezoelectric materials having an electro-mechanical coupling effect were discovered, they have been widely used in various engineering fields. In particular, piezoelectric actuators and sensors for smart structures and structural health monitoring

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are a popular research area in the aerospace engineering field. Bonding of piezoelectric material to different type of materials such as metal and composite materials, is typical. However, high stress occurs at the bonding surface and the bonding structures often fail.

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Recently, attention has been paid to functionally graded materials (FGMs) which can reduce the high stress at the interface as well as be applied in high temperature

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environments. FGMs have inhomogeneous characteristics because their material properties change gradually. Because of the nature of the techniques used in processing,

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graded materials are seldom isotropic. Thus, in studying the mechanics of many of the

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graded materials, an appropriate model would be an inhomogeneous orthotropic elastic continuum (Ozturk and Erdogan (1997)).

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The fracture behavior at the interface between piezoelectric materials and the

functionally graded orthotropic materials (FGOM) is important for practical usage. Cracks in their interface may experience complex behavior because of their piezoelectric effect, inhomogeneity, and orthotropy. Dag et al. (2004) studied interface crack problem between an FGOM coating and a homogeneous orthotropic material (HOM) substrate using analytical and finite element

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techniques. Anti-plane transient fracture analysis of the FGMs with weak/infinitesimal interface was conducted by Li et al. (2006). Zhou et al. (2010) analyzed a partially insulated interface crack between an FGOM coating and an HOM substrate under heat flux supply. Ding et al. (2014) investigated interface crack behavior for an HOM strip sandwiched

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between two different FGOMs subjected to thermal and mechanical loading.

Narita and Shindo (1999) studied interface crack in bonded layers of piezoelectric and HOM strips under antiplane shear. Their study showed that the effect of electroelastic interactions on the stress intensity factor and the energy release rate can be highly

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significant. An anti-plane moving interface crack between a piezoelectric and two HOM layers was analyzed by Lee et al. (2002). This group reported that dynamic stress intensity factor (DSIF) depends on the ratio of stiffness, thickness, the crack length, and the

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magnitude and direction of electrical loads as well as crack speed. Kwon and Meguid (2002) analyzed a central crack normal to the interface between a rectangular piezoelectric

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ceramic and two rectangular HOMs under electrical and mechanical loading. The results revealed the field intensity factors and energy release rate are considerably affected by the

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electrical crack boundary conditions. The Mode-III problem of an interface crack between

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two dissimilar homogeneous piezoelectric layers under mechanical and electrical impacts was analyzed by means of integral transform method in Gu et al. (2002). Meguid and Zhao

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(2002) studied the interface crack problem of bonded piezoelectric and elastic half-space under transient electromechanical loads. Kwon and Lee (2003) examined the steady state dynamic behavior of an eccentric crack moving at constant velocity in a piezoelectric ceramic layer bonded between two HOM layers. DSIF, dynamic energy release rate (DERR) and the crack sliding displacement are presented to show the effects of the crack propagation speed, the crack length, and the electro-mechanical coupling coefficient. Feng

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et al. (2011) investigated the problem of multiple cracks on the interface between a piezoelectric layer and an HOM substrate. They suggested the optimal stiffness ratio of the HOM substrate in order to prevent interfacial fracture, which is significant for the design and assessment of smart structures. Problem of an anti-plane crack in the interface of

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functionally graded piezoelectric and homogeneous piezoelectric materials was analyzed by Dai and Chong (2014). Bayat et al. (2015) presented an analytical model for the analysis of an HOM strip with piezoelectric coating weakened by multiple defects. The analysis showed the stress intensity factor and hoop stress are dependent on the imperfect bonding

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coefficient, the defect geometry, and the material properties of the HOM substrate. The fracture problem for a medium composed of a cracked piezoelectric strip with FGOM coating was studied by Bagheri et al. (2015).

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Despite all of these existing studies, a solution for the transient response of an interface crack between piezoelectric layer and FGOM layer has not been presented to date.

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In this study, a transient response analysis of a Mode III interface crack between piezoelectric layer and FGOM layer is conducted. The material properties of the FGOM

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layer change continuously along the thickness. Weak-discontinuous interface condition

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is adopted (Li et al. (2006)). Applying Laplace and Fourier transforms makes the problem a dual integral equation. Then the dual integral equation is transformed into a

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Fredholm integral equation of the second kind. Numerical analyses on the DSIF are carried out to show the effect of piezoelectricity of the piezoelectric layer, orthotropy and inhomogeneity of FGOM layer, and thickness of layers.

2 Problem statement and methods of solution

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We consider an interface crack between the homogeneous piezoelectric layer and FGOM layer subjected to anti-plane shear Heaviside step impact loading, as shown in Fig. 1. It is assumed that mechanical loading is applied only. Cartesian coordinates

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( x, y, z ) are fixed to the center of the crack. The piezoelectric layer poled with z -axis and occupies the region,    x  , 0  y  h1 . And the FGOM layer occupies the region,    x  ,  h2  y  0 . Their thicknesses in the z -direction are infinite in order to make a state of anti-plane shear. The crack is located along the interface line

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(  a  x  a, y  0 ). Only the right-hand half layers are considered due to the symmetry in geometry and loading.

We assume that shear moduli and density of the FGOM layer change continuously

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along the thickness and are expressed as follows (Ozturk and Erdogan (1997)):

(2)

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c55  c550 exp( y)

(1)

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c44  c440 exp( y)

   0 exp( y)

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(3)

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where c44 , c55 , and  are shear moduli and material density of the FGOM, respectively. c440 , c550 , and  0 are shear moduli and material density at the interface, respectively,  is the non-homogeneous material constant. It is assumed that the shear modulus ( c440 ) and material density (  0 ) of the piezoelectric layer are the same as those of the interface line.

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The boundary value problem is simplified if the out-of-plane displacement and the inplane electric fields are considered only such that:

u zP  w P ( x, y, t )

E xP  E xP ( x, y, t ) ,

u xF  u yF  0 ,

(4)

E yP  E yP ( x, y, t ) ,

E zP  0

u zF  w F ( x, y, t )

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u xP  u yP  0 ,

(5) (6)

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where u kP and u kF ( k  x, y, z ) are the displacements and EkP is the electric fields. Superscript P and F denote the piezoelectric layer and FGOM layer, respectively.

 zyF ( x, y, t )  c44w,Fy

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 zxF ( x, y, t )  c55w,Fx ,

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In this case, the constitutive equation becomes:

(7) (8)

D Pj ( x, y, t )  e150w,Pj  d110,Pj

(9)

E Pj ( x, y, t )   ,Pj

(10)

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CE

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 zjP ( x, y, t )  c440w,Pj  e150,Pj

where  zj and D j ( j  x, y ) are the stress components and electric displacements,

respectively.  , e150 and d110 are the electric potential, the piezoelectric constant, and the dielectric permittivity measured at a constant strain, respectively. The dynamic anti-plane governing equations for FGOM and piezoelectric material become expressed as:

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c550

 2 wF  2 wF w F  2 wF  c   c   440 440 0 x 2 y 2 y t 2

c440 2 w P  e150 2 P   0

(11)

 2 wP t 2

(12)

e150 2 wP  d110 2 P  0

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(13)

where  2   2 x 2   2 y 2 .

 P P 

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A new function  P is introduced as follows (Li and Mataga (1996)):

e150 P w d110

(14)

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By substituting Eq. (14) for Eqs. (12) and (13), the dynamic governing equations are

 2 wF  2 wF w F 1  2 wF     2 2 y x 2 y 2 c2F t 1  2 wP 2 2 c2P t

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 2 wP 

(15)

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2

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transformed into the following equations:

(16)

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 2 P  0

(17)

where

 

c550 c440

(18)

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c440

(19)

0

0 c  , 0

2

e  0  c440  150 d110

P 2

(20)

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c2F 

and c550 c440 is the ratio of shear moduli. c2F and c2P are the shear wave velocity of the FGOM and the piezoelectric material, respectively.  0 is the piezoelectric stiffened elastic constant at the interface.

x

2

 2 w P* 



 2 wF* y

p2 c

P2 2

2

y

w P*



p2 c

F2 2

wF*

CE 

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(24)

1 ci * w ( x, y, p) exp( pt )dp 2i ci 

 P* ( x, y, p)  0  P ( x, y, t ) exp(  pt )dt  P ( x, y, t ) 

(22)

(23)

w* ( x, y, p)  0 w ( x, y, t ) exp(  pt )dt

w ( x, y, t ) 

(21)

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 2 P*  0

where



w F *

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 2 wF*

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

2

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The Laplace transforms of Eqs. (15) to (17) are the following forms:

(25) (26)

1 ci P*  ( x, y, p) exp( pt )dp 2i ci

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(27)

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e150 P* w d110

 P*   P* 

(28)

The superscript * denotes the Laplace transform.

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The Fourier transforms are applied to Eqs. (21) to (23), and the out-of-plane displacement and the electric potential are given as follows:

w F * ( x, y, p) 

 A (s, p) exp(q1 y)  A2 (s, p) exp(q2 y)cos(sx)ds  0 1

(29)

w P* ( x, y, p) 

 A (s, p) exp(y)  A4 (s, p) exp(y)cos(sx)ds  0 3

(30)

 P* ( x, y, p) 

2 e150   A (s, p) exp( y)  A4 (s, p) exp(y)cos(sx)ds  d110 0 3

2

2



0  B1 (s, p) exp( sy)  B2 (s, p) exp( sy)cos(sx)ds 

(31)

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



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2



 2

 g 



(32)

2

2

(33)

4

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2

, q2   

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q1   

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where

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g   2s2 

 s  2

p2

c2F

(34)

2

p2 c2P

(35)

2

A1 , A2 , A3 , A4 , B1 and B2 are the unknowns to be solved.

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From the constitutive Eqs. (7) to (10), the shear stress components and the electric displacement are as follows:

 q1 A1 (s, p) exp(q2 y)  q2 A2 (s, p) exp(q1 y)cos(sx)ds  0 2

 xzF * ( x, y, p)   c550

 e150

sA1 (s, p) exp( q2 y)  A2 (s, p) exp( qi y)sin( sx)ds  0 2



  A3 (s, p) exp(y)   A2i (s, p) exp(y)cos(sx)ds  0 2

2



DyP* ( x, y, p)   d110



0  sB1 (s, p) exp( sy)  sB2i (s, p) exp( sy)cos(sx)ds 

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 yzP* ( x, y, p)   0



 sB1 (s, p) exp(sy)  sB2 (s, p) exp( sy) cos(sx)ds  0 2

(36)

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 yzF * ( x, y, p)  c440



(37)

(38)

(39)

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The boundary conditions in the Laplace transform domain can be expressed as:

 yzP* ( x,0 , p)   yzF* ( x,0 , p)  0 p

(40)

(a  x  )

(41)

 yzP* ( x,0  , p)  yzF* ( x,0  , p)

(a  x  )

(42)

 yzP* ( x, h1 , p)  yzF* ( x,h2 , p)  0

(0  x  )

(43)

EBC I : DyP* ( x,0  , p)  DyP* ( x, h1 , p)  0

(0  x  )

(44)

EBC II : E yP* ( x,0 , p)  E yP* ( x, h1 , p)  0

(0  x  )

(45)

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(0  x  a )

AC

CE

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wP* ( x,0 , p)  wF* ( x,0 , p)

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where  0 is the uniform shear traction. The crack surfaces are loaded by the antiplane shear impact only. Two kind of electric boundary conditions, EBC I and EBC II, are considered. No electric displacement is applied at surfaces ( y  0 , y  h1 ) of the

piezoelectric layer for the EBC II.

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piezoelectric layer for the EBC I and no electric field is applied at surfaces of the

From the continuity and edge loading conditions of Eqs. (42) to (45), the following

A1 ( s, p) 

q2 exp( 2h2 ) A2 ( s, p) q1

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relationships are evaluated among the unknowns:

(46)

A3 ( s, p)  

c440q2 (1  exp( 2h2 )) A2 ( s, p) c44 (1  exp( 2h1 ))

(47)

A4 ( s, p)  

c440q2 (1  exp( 2h2 )) exp( 2h1 ) A2 ( s, p) c44 (1  exp( 2h1 ))

(48)

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0  B1 ( s, p)   e150 q2 (1  exp( 2h2 )) A2 ( s, p) d s ( 1  exp(  2 sh )) 110 1 

( EBC II )

(49)

( EBC I ) ( EBC II )

(50)

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0  B2 ( s, p)   e150 q2 (1  exp( 2h2 )) exp( 2sh1 ) A2 ( s, p) d  110 s(1  exp( 2sh1 ))

( EBC I )

AC

where

 0 c44   c440

( EBC I )

(51)

( EBC II )

A dual integral equation is evaluated from the mixed boundary conditions of Eqs. (40) and (41) as the following form:

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



0

s F ( s, p) M ( s, p) cos ( sx)ds 

  0 c44  c440 2 p c44c440

M (s, p) cos(sx) ds  0

(a  x  )

where

1 c44  c440 s 

(52)

   q1q2 (1  exp( 2 h2 ))(1  exp( 2 h1 ))    c44 (q1  q2 exp( 2 h2 ))(1  exp( 2h1 ))  c440q1q2 (1  exp( 2 h2 ))(1  exp( 2h1 )) 

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F ( s, p ) 

(0  x  a )

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0

(53)

 c  (q  q exp( 2 h2 ))(1  exp( 2h1 ))  c440q1q2 (1  exp( 2 h2 ))(1  exp( 2h1 ))  M ( s, p)   44 1 2  A2 ( s, p) (54) c44 q1 (1  exp( 2 h1 ))  

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Then, the dual integral equation of Eq. (52) can be transformed to a Fredholm integral

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equation of the second kind using the new function * ( ) , as follows:

a

M ( s, p)  0  * ( , p) J 0 ( s ) d

PT

(55)

CE

where J 0 is the zero-order Bessel function of the first kind.

AC

By inserting Eq. (55) into Eq. (52), a Fredholm integral equation of the second kind is as follows:

a

* ( , p)  0 K ( , , p) * ( , p) d 

  0 c44  c440 2 p c44c440

where

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(56)

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K ( , , p)  0 sF ( s, p)  1 J 0 ( s ) J 0 ( s ) ds 

(57)

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The following dimensionless variables and function are introduced for numerical analysis:

S  sa ;    a

(58)

  a ;   a *   0 c44  c440  (, p)  ( , p)  2 p c44c440  *

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(59)

(60)

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By substituting Eqs. (58) to (60) for Eq. (56), the normalized Fredholm integral

1

ED

equation of the second kind can be obtained in the following form:

(61)

CE

PT

 * (, p)  0 L(, , p)  * (, p) d  

where

AC

  S   L(, , p)   0 S  F  , p   1 J 0 ( S ) J 0 ( S ) dS  a  

S 1 c 44 F ( , p)  a S

  h   h    Q1Q2 1  exp   2 2  1  exp   2 1    a   a   c 440         h2   h1   h     c 44  Q1  Q2 exp   2  1  exp   2    c 440Q1Q2 1  exp   2 2 a   a  a      

(62)    h1        1  exp   2    a      

(63)

Q1   

  ; Q2    2 2

(64)

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2  G  4 2

c

1



c440 c2P pa 1 P 2

pa

(66)



2

(67)



2

* K III ( p)  lim

x a 

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The DSIF is defined in the following form:

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0

G   2S 2 

  S2 

(65)

2 ( x  a)  yzF * ( x,0, p)

(68)

where K III* ( p) is DSIF in the Laplace transform domain.

K III* ( p)  lim

c44c440 2  s F ( s, p) M ( s, p) cos( sx)ds c44  c440  0

2 ( x  a)

(69)

PT

x a 

ED

M

From Eq. (36), (46), and (68), the DSIF is evaluated in the following form:

CE

By inserting Eq. (55) into Eq. (69), the DSIF is as follow:

AC

K III* ( p)  lim 2 ( x  a) x a

 d a  * ( , p) d  0 x2   2  dx a   0  * ( , p) 0 sF ( s, p)  1J 0 ( s ) cos(sx) ds d 2 c44c440  c44  c440



(70)

By ignoring non-singular term, applying integration by parts, and substituting Eqs. (58) to (60) for Eq. (70), the DSIF can be obtained in the form:

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* K III ( p) 

1  0  a  * (1, p) p

(71)

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From the inverse Laplace transform of Eq. (71), the DSIF in the physical domain can be obtained in the form:

K III (t )   0  a M (t )

where

* 1 ci  (1, p) exp( pt )dp 2 i ci p

(73)

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M (t ) 

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(72)

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in which the function  * (1, p) can be calculated from Eq. (61).

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3 Discussion

In order to check the accuracy and correctness of our results, two comparisons are

considerd. Eq. (60) can be reduced to the solution of Kwon and Lee (2002) by ignoring inhomogeneity (   0 ). Our result, then, can also be reduced to the solution of homogeneous and isotropic infinite elastic body containing a crack by ignoring

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piezoelectriciy, inhomogeneity, orthotropy ( e150  0 ,   0 , c440  c550 ), and having infinite thickness ( h1 , h2   ). Fig. 2 shows the normalized DSIF for homogeneous and isotropic infinite elastic body with the present work and the isotropic result of Zhang (2003). This comparison shows good agreement. These two comparisons imply the

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accuracy and correctness of our results.

To investigate the effects of piezoelectricity and thickness of the piezoelectric layer, and inhomogeneity, orthotropy, and thickness of the FGOM layer on DSIF, numerical analyses are conducted. Numerical analysis for DSIF is performed by Gaussian

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quadrature formulas. The inverse Laplace transform of DSIF is carried out by the numerical method proposed by Miller and Guy (1966). Material properties at the

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interface are assumed as follows (Narita and Shindo (1999)):

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c440  2.3  1010 N / m2 , e150  17.0 C / m 2 , d110  150.4 1010 C / Vm

A general characteristic of the curves for the transient response behavior is observed

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from Figs. 3 to 8. DSIF rapidly increases with time, reaches its peak value, then

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decreases and approaches its static value.

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. 3 displays the variation of the normalized DSIF K III (t )  0  a against the normalized time c2P t / a with the various normalized non-homogeneous material constant  for EBC I. Shear moduli of the FGOM layer decrease from the interface to the lower free surface ( y  h2 ) when the normalized non-homogeneous material constant is a positive value because the value of the y-axis is negative. On the contrary, shear moduli of the FGOM layer increase from the interface to the lower free surface

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when the normalized non-homogeneous material constant is a negative value. Peak values of DSIF in the case of the normalized non-homogeneous material constant being negative are smaller than those of DSIF in the case where the normalized nonhomogeneous material constant is positive.

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The peak value of DSIF increases as the normalized non-homogeneous material constant increases in the case of the normalized non-homogeneous material constant being a positive value. In contrast, the peak value of DSIF decreases as the absolute value of the normalized non-homogeneous material constant increases in the case where

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the normalized non-homogeneous material constant is a negative value.

Thus, by way of a summary, the following are beneficial to impede the transient fracture of interface crack between piezoelectric layer and FGOM layer:

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 shear moduli of the FGOM layer increase from the interface to the lower free surface;

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 the gradient of shear moduli increases when shear moduli of the FGOM layer increase from the interface to the lower free surface;

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 the gradient of shear moduli decreases when shear moduli of the FGOM layer

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decrease from the interface to the lower free surface.

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Fig. 4 presents the variation of the normalized DSIF K III (t )  0  a against the normalized time c2P t / a with the various ratios of shear moduli c550 c440 for normalized non-homogeneous material constant   1.0 and EBC I. The peak value of DSIF decreases as the ratio of shear moduli of the FGOM layer increases in the case of the normalized non-homogeneous material constant being a positive value. Increase of ratio of shear moduli c550 c440 in the case of a decrease of the shear moduli from the

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interface to the lower free surface is beneficial to impede the transient fracture of the interface crack between piezoelectric layer and FGOM layer. Fig. 5 shows the variation of the normalized DSIF K III (t )  0  a against the

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normalized time c2P t / a with the various ratio of shear moduli c550 c440 for normalized non-homogeneous material constant   1.0 and EBC I. A different trend is observed from that of Fig. 4. The peak value of normalized DSIF increases as the ratio of shear moduli of the FGOM layer increases in the case of the increase of the

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shear modulus from the interface to the lower surface. Decrease of the ratio of shear moduli c550 c440 in the case of the increase of the shear moduli from the interface to the lower free surface is beneficial to impede transient fracture of an interface crack between piezoelectric layer and FGOM layer.

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Similar trends, as shown in Fig. 3 to 5, were observed in EBC II.

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Different trends are observed between EBC I and II in Fig. 6. Fig. 6 presents the variation of the normalized DSIF K III (t )  0  a against the normalized time c2P t / a

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with the various ratios of shear moduli c550 c440 and normalized non-homogeneous

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material constant  for EBC I and II. The peak values of DSIF for EBC I are larger than those of DSIF for EBC II when the non-homogeneous material constant is positive.

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However, peak values of DSIF for EBC I are smaller than those of DSIF for EBC II when the non-homogeneous material constant is negative. According to the electrical boundary conditions of the piezoelectric layer, the peak value of DSIF increases or decreases. The followings are beneficial to impede transient fracture of an interface crack between piezoelectric layer and FGOM layer:  the electrical boundary condition EBC I in the case of increase of shear moduli

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of the FGOM layer from the interface to the lower free surface  the electrical boundary condition EBC II in the case of decrease of shear moduli of the FGOM layer from the interface to the lower free surface

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Fig. 7 and Fig. 8 present the variation of the normalized DSIF K III (t )  0  a with the various normalized thicknesses of piezoelectric and FGOM layers. The peak value of normalized DSIF always decreases with the increase of normalized thickness of the

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piezoelectric layer and the FGOM layer regardless of the non-homogeneous material constant. Increase of the thickness of piezoelectric and FGOM layers is also beneficial to impede dynamic fracture of an interface crack of FGOM. Similar trends, as shown in

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4 Conclusions

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Fig. 7 and Fig. 8, were also observed for EBC II.

The transient response of an interface crack between piezoelectric layer and FGOM

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layer under anti-plane shear impact loading was analyzed. Shear moduli and mass density of the FGOM layer change continuously along the thickness. Laplace and

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Fourier transforms are applied in order to solve the problem. Numerical analysis on the Fredholm integral equation was performed. It can be concluded that the following areas are helpful to impede the transient fracture of an interface crack between piezoelectric layer and FGOM layer:

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

increase of shear moduli of the FGOM layer from the interface to the lower free surface;



increase of the gradient of shear moduli in the case of increase of shear moduli of the FGOM layer from the interface to the lower free surface; decrease of the ratio of shear moduli c550 c440 in the case of increase of shear

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

moduli of the FGOM layer from the interface to the lower free surface; 

the electrical boundary condition EBC I of the piezoelectric layer in the case of increase of shear moduli of the FGOM layer from the interface to the lower free

increase of thickness of the layers.

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

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surface;

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Acknowledgments

This work was supported by the Civil-Military Technology Cooperation Program

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funded by the Ministry of Trade, Industry and Energy, Republic of Korea

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93-101. Bayat, J., Ayatollahi, M., Bagheri, R., 2015. Fracture analysis of an orthotropic strip with imperfect piezoelectric coating containing multiple defects. Theoret. Appl. Fract. Mech. 77, 41-49.

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492. Li, S., Mataga, P.A., 1996. Dynamic crack propagation in piezoelectric materials-part I. electrode solution. J. Mech. Phys. Solids 44, 1799-1830. Li, Y.D., Jia, B., Zhang, N., Tang, L.Q., Dai. Y., 2006. Dynamic stress intensity factor of

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Jacobi polynomials. SIAM J. Numer. Anal. 3, 624-635. Narita, F., Shindo, T., 1999. The interface crack problem for bonded piezoelectric and

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orthotropic layers under antiplane shear loading. Int. J. Fract. 98, 87-101.

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Figure Lists

Fig. 1 A Mode III interface crack between piezoelectric layer and functionally

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graded orthotropic material layer : geometry and loading

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Fig. 2 Comparison of the normalized DSIF K III (t )  0  a with normalized time

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c2P t / a for homogeneous and isotropic infinite elastic body

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Fig. 3 Variation of the normalized DSIF K III (t )  0  a with normalized time c2P t / a

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for various  (EBC I, c550 c440  0.5 , h1 / a  h2 / a  10.0 )

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Fig. 4 Variation of the normalized DSIF K III (t )  0  a with normalized time c2P t / a

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for various c550 c440 at   1.0 (EBC I, h1 / a  h2 / a  10.0 )

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Fig. 5 Variation of the normalized DSIF K III (t )  0  a with normalized time c2P t / a

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for various c550 c440 at   1.0 (EBC I, h1 / a  h2 / a  10.0 )

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Fig. 6 Variation of the normalized DSIF K III (t )  0  a with normalized time c2P t / a

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for various c550 c440 and  (EBC I and EBC II, h1 / a  h2 / a  10.0 )

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Fig. 7Variation of the normalized DSIF K III (t )  0  a with normalized time c2P t / a

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for various h1 / a and h2 / a at   1.0 (EBC I, c550 c440  0.5 )

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Fig. 8 Variation of the normalized DSIF K III (t )  0  a with normalized time c2P t / a

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for various h1 / a and h2 / a at   1.0 (EBC I, c550 c440  0.5 )

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