Non-local dynamic solution of two parallel cracks in a functionally graded piezoelectric material under harmonic anti-plane shear wave

Author’s Accepted Manuscript Non-local dynamic solution of two parallel cracks in a functionally graded piezoelectric material under harmonic anti-plane shear wave Hai-Tao Liu, Jian-Bing Sang, Zhen-Gong Zhou www.elsevier.com/locate/physe

PII: DOI: Reference:

S1386-9477(16)30681-6 http://dx.doi.org/10.1016/j.physe.2016.06.024 PHYSE12499

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 10 January 2016 Revised date: 23 June 2016 Accepted date: 27 June 2016 Cite this article as: Hai-Tao Liu, Jian-Bing Sang and Zhen-Gong Zhou, Non-local dynamic solution of two parallel cracks in a functionally graded piezoelectric material under harmonic anti-plane shear wave, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.06.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Non-local dynamic solution of two parallel cracks in a functionally graded piezoelectric material under harmonic anti-plane shear wave Hai-Tao Liua*, Jian-Bing Sanga, Zhen-Gong Zhoub a

School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, P. R. China

b

Center for Composite Materials and Structures, Harbin Institute of Technology, Harbin 150080, P. R. China

*

Corresponding author. [email protected]

Abstract This paper investigates a functionally graded piezoelectric material (FGPM) containing two parallel cracks under harmonic anti-plane shear stress wave based on the non-local theory. The electric permeable boundary condition is considered. To overcome the mathematical difficulty, a one-dimensional non-local kernel is used instead of a two-dimensional one for the dynamic fracture problem to obtain the stress and the electric displacement fields near the crack tips. The problem is formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces. Different from the classical solutions, that the present solution exhibits no stress and electric displacement singularities at the crack tips. Key words: two parallel crack; non-local theory; FGPM; anti-plane shear wave

1. Introduction With the application of the functionally graded piezoelectric material (FGPM) in engineering, the fracture analysis of FGPM has absorbed much attention in works [1-7]. For example, Li and Weng [2] first applied the concept of fracture mechanics on a finite crack in a functionally graded piezoelectric material strip. The electroelastic behavior of a Griffith crack in a functionally graded piezoelectric strip was analyzed by Ma et al. [4]. Mousavi and Paavola [9] studied the fracture behavior of a cracked functionally graded piezoelectric strip under antiplane mechanical and inplane electrical loading. However, based on the classical continuum theory, these solutions contain the stress and electric displacement singularities at the crack tips in above works. This is obviously not reasonable according 1

to the physical nature of the engineering. To overcome the stress singularity in the classical theory, the stress near the tips of a sharp line crack in an isotropic elastic plate subject to uniform tension, shear and anti-plane shear have been calculated by using the nonlocal theory in Eringen et al. [8], Eringen [9, 10]. The nonlocal elasticity theory contains the information about the forces between atoms, and introduces the internal length scale into the constitutive equations as a material parameter. The constitutive theory of nonlocal elasticity also developed [11] the elastic modulus is influenced by the microstructure of the material. Recently, some fracture problems in the piezoelectric material, the functionally graded material, the functionally graded piezoelectric material have been analyzed by Zhou and Wang [12, 13], Zhou and Wu [14], and Zhou et al. [15] using non-local theory and the Schmidt method [16, 17]. Liang et al. [18] considered the nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves. Allegri and Scarpa [19] dealt with the crack-tip stress fields in orthotropic bodies within the framework of Eringen’s nonlocal elasticity via the Green’s function method. Based on the nonlocal theory and Timoshenko beam theory, Ke and Wang [20] and Ke et al. [21] reported the linear and nonlinear vibrations of piezoelectric nanobeams. To our knowledge, the dynamic electroelastic fracture behavior of two parallel cracks in the FGPM has been not investigated based on the non-local theory and the Schmidt method in any open literature. In the present paper, the dynamic behavior of two parallel permeable cracks subjected to harmonic anti-plane shear stress waves in FGPM is investigated based on the non-local theory. The traditional concept of linear elastic dynamic fracture mechanics and the non-local theory are extended to include the functionally graded piezoelectric effects. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. Fourier transform is applied to the governing equations and a mixed boundary value problem is reduced to a pair of dual integral equations and the Schmidt method is used for numerical calculations. As expected, the present solutions of this paper do not contain the stress and electric displacement singularities at the crack tips.

2. Formulation of the problem Consider a FGPM with two parallel cracks of length 2l along x-axis and h is the distance between two parallel cracks, as shown in Fig.1. In the present paper, the harmonic anti-plane stress 2

wave is vertically incident. Let  be the circular frequency of the incident wave, w0( j ) ( x, y, t ) and ( j) 0( j ) ( x, y, t ) ( j  1, 2,3) are the mechanical displacement and electric potential;  zk 0 ( x, y, t ) and

Dk( 0j ) ( x, y, t ) (k  x, y , j  1, 2,3) are the anti-plane shear stress field and in-plane electric

displacement field, respectively. For convenience, it is noted that all quantities with superscript j ( j  1, 2,3) refer to the upper half plane 1, the layer 2 and the lower half plane 3, respectively.

Because of the incident wave is a harmonic anti-plane shear stress wave, all field quantities of ( j) ( j) w0( j ) ( x, y, t ) , 0( j ) ( x, y, t ) ,  zk 0 ( x, y, t ) and Dk 0 ( x, y, t ) can be assumed as follows： ( j) ( j) [ w0( j ) ( x, y, t ) , 0( j ) ( x, y, t ) ,  zk 0 ( x, y, t ) , Dk 0 ( x, y, t )] 

( j) [w( j ) ( x, y) ,  ( j ) ( x, y) ,  zk ( x, y) , Dk( j ) ( x, y )]eit

(1)

In what follows, the time dependence of eit will be suppressed but understood. 2.1 Boundary conditions As discussed in [22], the crack surface can be assumed to be in perfect contact. So the permeable condition will be enforced in this work. Here, the standard superposition technique is used in the present paper. Therefore, the boundary conditions can be expressed as follows:

 yz(1) ( x, h  )   yz(2) ( x, h  )   0 , x  l  (1)  (2)   yz ( x, h )   yz ( x, h ), x  l  (1)  (2)   Dy ( x, h )  Dy ( x, h ), x  0  (1)  (2)   ( x, h )   ( x, h ), x  0  w(1) ( x, h  )  w(2) ( x, h  ), x  l 

(2)

 yz(2) ( x, 0 )   yz(3) ( x, 0 )   0 , x  l  (2)  (3)   yz ( x, 0 )   yz ( x, 0 ), x  l  (2)  (3)   Dy ( x, 0 )  Dy ( x, 0 ), x  0  (2)  (3)   ( x, 0 )   ( x, 0 ), x  0  w(2) ( x, 0 )  w(3) ( x, 0 ), x  l 

(3)

w( j ) ( x, y)   ( j ) ( x, y)  0 , ( x 2  y 2 )1 2   ( j =1, 2, 3)

where  0 is a magnitude of the incident wave.

Fig.1. 3

(4)

2.2 Basic equations In the absence of body forces and free charges, the basic equations of dynamic non-local anti-plane FGPM are

  xz( j ) ( x, y )  yz( j ) ( x, y )     ( y ) 2 w( j ) ( x, y )   x  y   ( j) Dy( j )  Dx  x  y  0 

(5)

 kz( j ) ( X )  [c '44 ( X '  X ) w,(kj ) ( X ' )  e '15 ( X '  X ),(k j ) ( X ' )]dV ( X ') V  , ( k  x, y )  ( j) ( j) ( j) D ( X )  [ e ' ( X '  X ) w ( X ' )   ' ( X '  X )  ( X ' )] dV ( X ')  k ,k 11 ,k V 15

(6)

where

  ( y) 2 w( j ) ( x, y)eit   ( y)

 2 w0( j ) ( x, y, t )  2 ( w( j ) ( x, y)eit )   ( y ) t 2 t 2

and

 ( y) is

the

material density. The only difference from the classical electric-elastic theory and the nonlocal electric-elastic theory is in the stress and the electric displacement constitutive equation (6) in which the stress  zk( j ) ( X ) and the electric displacement Dk( j ) ( X ) at a point X depends on w,(kj ) ( X ) and

,(k j ) ( X ) , at all points X of the body. For the FGPM, there exist only three material parameters, c' 44 ( X '  X ) , e'15 ( X '  X ) and '11 ( X '  X ) . The integrals in Eq. (6) are over the volume V of the body enclosed within a surface V . As discussed in [23], the forms of c' 44 ( X '  X ) ,

e'15 ( X '  X ) and '11 ( X '  X ) are assumed as the follows: [c '44 ( X ' X ), e '15 ( X ' X ),  '11 ( X ' X )]  [c44 ( y), e15 ( y), 11 ( y)] ( X ' X )

(7)

where  ( X ' X ) is known as the influence function. Crack problems in a FGPM do not appear to be analytically tractable for arbitrary variations of material properties. Usually, ones try to generate the forms of a FGPM for which the problems become tractable. The variation forms of a FGPM can be assumed as the power function [24], the exponential function [25, 26], the piecewise linear function [27] and the piecewise exponential function [28], etc. Similar to the treatment of the crack problem for the isotropic functionally graded material [25, 26], we assume the material properties are described by [c44 ( y), e15 ( y), 11 ( y),  ( y)]  [c440 , e150 , 110 , 0 ]e y

(8) 4

where c440 , e150 , 110 and  0 are the shear modulus, the piezoelectric coefficient, the dielectric parameter and the mass density along y  0 , respectively.  is the functionally graded parameter and is a positive or negative constant. Substituting Eqs. (7) and (8) into Eq. (6) yield

 kz( j ) ( X )   ( X '  X ) kz( j ) ( X ' )dV ( X ' )   V , ( k  x, y )  ( j) c( j )  Dk ( X )    ( X '  X ) Dk ( X ' )dV ( X ' ) V 

(9)

 kz( j )  e y (c440 w,(kj )  e150,(k j ) )  where  c ( j ) y ( j) ( j)   Dk  e (e150 w,k  110,k )

(10)

The expression (10) is the classical constitutive equations of FGPM.

3. The dual integral equation Substituting Eq. (9) into Eq. (5) and using the Green-Gauss theorem leads to 







 

 ( x ' x , y ' y ){c440 [ 2 w( j ) ( x ', y ')  

w( j ) ( x ', y ') ]  e150 [ 2 ( j ) ( x ', y ') y '

l  ( j ) ( x ', y ') ]}dx ' dy '   ( x ' x , h)[ yz(1) ( x , h  )   yz(2) ( x , h  )]dx ' l y '



(11)

l

   ( x ' x , 0)[ yz(2) ( x , 0 )   yz(3) ( x , 0 )]dx '   0 2 w( j ) ( x, y ) l









 

 ( x ' x , y ' y ){e150 [ 2 w( j ) ( x ', y ')  

w( j ) ( x ', y ') ]  110 [ 2 ( j ) ( x ', y ') y '

l  ( j ) ( x ', y ')  ]}dx ' dy '    ( x ' x , h)[ Dyc (1) ( x , h  )  Dyc (2) ( x , h  )]dx ' l y '

(12)

l

   ( x ' x , 0)[ Dyc (2) ( x , 0 )  Dyc (3) ( x , 0 )]dx '  0 l

where 2   2 / x2   2 / y 2 is the two-dimensional Laplace operator. Here the surface integral may be dropped since the mechanical displacement and the electric displacement fields vanish at infinity. As

discussed

in

[8],

it

can

be

obtained

that

[ yz(1) ( x, h )   yz(2) ( x, h )]  0

,

[ yz(2) ( x, 0 )   yz(3) (x , 0 )]  0 , [ Dyc (1) ( x , h )  Dyc (2) ( x , h )]  0 and [ Dyc (2) ( x ,0 )  Dyc (3) ( x ,0 )]  0 . Hence the line integrals in Eqs. (11) and (12) vanish. The general solution of Eqs. (11) and (12) are identical to

5







w( j ) ( x ', y ') 2 ( j)   ( x ' x , y ' y ){c440[ w ( x ', y ')   y ' ]  e150[  ( x ', y ') 

2

( j)

 ( j ) ( x ', y ')  ]}dx ' dy '   0 2 w( j ) ( x, y ) y ' 







 

 ( x ' x , y ' y ){e150 [2 w( j ) ( x ', y ')  

(13)

w( j ) ( x ', y ')  ( j ) ( x ', y ') ]  110 [2 ( j ) ( x ', y ')   ]}dx ' dy '  0 (14) y ' y '

almost everywhere. What now remains is to solve the integrodifferential Eqs. (13) and (14) for the anti-plane mechanical displacement w( j ) ( x, y) and the electric potential  ( j ) ( x, y) . It is assumed that the non-local interaction in the y -direction is ignored [29]. In view of our assumptions, it can be given as

 ( x ' x , y ' y )  0 ( x ' x ) ( y ' y) where  0 ( x ' x ) 

1



(15)

(  / a) exp[(  / a) 2 ( x ' x) 2 ] in which 

is a constant and can be

determined by experiment and a is the characteristic length. In the present paper, a is taken as the lattice parameter. Substituting Eq. (15) into Eqs. (13) and (14), and using the Fourier transform with x can be given as follow:

 0 ( s){c440 [

d 2 w( j ) ( s, y ) 2 ( j ) w( j ) (s, y ')  s w ( s , y )   ] dy 2 y '  2 ( j ) ( s, y) 2 ( j )  ( j ) ( s, y ') e150 [  s  ( s, y )   ]}   0 2 w( j ) ( s, y) 2 y y '

e150 [

(16)

 2 w( j ) ( s, y) 2 ( j ) w( j ) ( s, y ')  2 ( j ) ( s, y) 2 ( j )  ( j ) ( s, y ')  s w ( s , y )   ]   [  s  ( s , y )   ]0 110 y 2 y ' y 2 y ' (17)

A superposed bar indicates the Fourier transform. Because of the symmetry, it suffices to consider the problem for x  0, y   . The general solutions of Eqs. (16) and (17) satisfying Eq. (4) are

2   (1) w ( x , y )  A1 ( s )e11 y cos( sx)ds   0   ,y0    (1) ( x, y )  e150 w(1) ( x, y )  2 B1 ( s )e  21 y cos(sx)ds 110  0   6

(18)

2   (2) w ( x, y )   [ A2 ( s)e 11 y  B2 ( s)e 12 y ]cos( sx)ds   0  ,0  y  h   e 2   21 y   22 y (2) (2) 150  ( x, y )  w ( x, y )   [C2 ( s)e  D2 ( s)e ]cos( sx)ds 110  0  

(19)

2   (3) w ( x , y )  A3 ( s )e12 y cos( sx)ds   0   ,y0    (3) ( x, y )  e150 w(3) ( x, y )  2 B3 ( s )e  22 y cos( sx)ds 110  0  

(20)

where A1 ( s) , B1 ( s) , A2 ( s) , B2 ( s) , C2 ( s) , D2 ( s) , A3 ( s) and B3 ( s) are unknown functions and

 12 

to

be determined

by the boundary conditions.

   2  4[ s 2   2 / c12 0 ( s)]

0  c440 

2 2 e150

110

,  21 

   2  4s 2 2

 11 

,  22 

   2  4[ s 2   2 / c12 0 ( s)] 2

   2  4s 2 2

,

, c1  0 / 0 ,

.

Substituting Eqs. (18)-(20) into Eq. (10), it can be obtained

 (1) 2e y   ( x , y )   [   A ( s)e11 y  e150 21 ( s) B1 ( s)e  21 y ]cos( sx)ds  yz  0 0 11 1  y   D c (1) ( x, y )  2110e  21B1 ( s)e  21 y cos( sx)ds y  0  

(21)

 (2) 2e y    11 y   12 B2 ( s)e 12 y ]  e150 [ 21C2 ( s)e  21 y   22 D2 ( s)e  22 y ]}cos( sx) ds  yz ( x, y )    0 {0 [ 11 A2 ( s)e (22)  y  2  e   y   y c (2)  D ( x, y )  110 0 [ 21C2 (s)e 21   22 D2 (s)e 22 ]cos(sx)ds  y 

 (3) 2e y   ( x , y )   [   A ( s)e 12 y  e150 22 B3 ( s)e  22 y ]cos( sx)ds  yz  0 0 12 3  y   D c (3) ( x, y )  2110 e  22 B3 ( s)e 22 y cos( sx)ds y  0  

(23)

According to boundary conditions Eqs. (2) and (3), it can be obtained that  yz(1) ( x, h )   yz(2) ( x, h ) ,

 yz(2) ( x,0 )   yz(3) ( x,0 ) , Dyc (1) ( x , h )  Dyc (2) ( x , h ) and Dyc (2) ( x ,0 )  Dyc (3) ( x ,0 ) . We have 0{ 11[ A1 (s)  A2 (s)]e11h   12 B2 (s)e12h }  e150{ 21[ B1 (s)  C2 (s)]e 21h   22 D2 (s)e 22h }  0

(24)

0 [ 11 A2 (s)   12 B2 (s)   12 A3 (s)]  e150[ 21C2 (s)   22 D2 ( s)   22 B3 ( s)]  0

(25)

7

 21[ B1 (s)  C2 (s)]e h   22 D2 (s)e 21

22 h

0

(26)

 21C2 (s)   22 D2 (s)   22 B3 (s)  0 (27) To solve the problem, the jumps of displacements across the crack surfaces are defined as follows: (1)  (2)    f1 ( x)  w ( x, h )  w ( x, h )  (2)  (3)    f 2 ( x)  w ( x, 0 )  w ( x, 0 )

(28)

Taking Eqs. (18) and (20) into Eq. (28), applying the boundary conditions Eqs. (2) and (3) and the Fourier transform, we obtain f1 (s)  A1 (s)e11h  A2 (s)e11h  B2 (s)e12h

(29)

f 2 (s)  A2 (s)  B2 (s)  A3 (s)

(30)

e150

f1 ( s)  B1 ( s)e 21h  C2 ( s)e 21h  D2 ( s)e 22 h  0

(31)

e150

f 2 ( s)  C2 ( s)  D2 ( s)  B3 ( s)  0

(32)

110 110

By solving Eqs. (24)-(27) and (29)-(32) with unknown functions A1 ( s) , B1 ( s) , A2 ( s) , B2 ( s) , C2 ( s) , D2 ( s) , A3 ( s) and B3 ( s) , substituting these solutions into Eq. (26), applying  from Eq.

(19), the boundary conditions Eqs. (2) and (3) and the Fourier transform, we have: a s  2e h   4  2  yz(2) ( x, h)  e [ g1 ( s) f1 ( s)  g 2 ( s) f 2 ( s)]cos( sx) ds   0   0  a2 s2  (2) 2   4 2 [ g3 ( s) f1 ( s)  g 4 ( s) f 2 ( s)]cos( sx) ds   0  yz ( x, 0)  0 e   2  2   f1 (s) cos(sx)ds  0 ,  f2 (s) cos(sx)ds  0 , x  l 2 2





0

(33)

(34)

0

a s  2e150e h   4  2  21 22 (2)  D y ( x, h )  e [ f1 ( s)  e  21h f 2 ( s)]cos( sx)ds  0      21 22  a2 s2  (2) 2e150   4  2  21 22 D ( x , 0)  e [e 22 h f1 ( s)  f 2 ( s)]cos( sx)ds  y  0   21   22  2 2

where

g1 ( s) 

2 0 11 12  21 22e150   11   12 ( 21   22 )110

,

8

g 2 ( s) 

(35)

2 0 11 12e r h  21 22 e r h e150   11   12 ( 21   22 )110 11

21

,

g3 ( s ) 

2 2 0 11 12er h  21 22 er h e150    21 22e150 , g 4 ( s)  0 11 12  .   11   12 ( 21   22 )110  11   12 ( 21   22 )110 12

22

The dual-integral equations (33)-(34) must be solved to determine the unknown functions f1 ( s) and f 2 ( s) .

4. Solution of the dual integral equations For overcoming the difficult, the Schmidt method [16, 17] is used to solve the dual-integral equations (33)-(34). The jumps of displacements across the crack surfaces are represented by the following series: 1 1 ( , ) x  x 2 12 2 2  an P2 n  2 ( )(1  2 ) , 0  x  l f1 ( x)   l l n 1 0, x  l 

(36)

1 1 ( , ) x  x 2 12 2 2 b P ( )(1  ) , 0 xl  n 2n2 f 2 ( x )   l l2 n 1 0, x  l 

(37)

( 12 , 12 )

where an and bn are unknown coefficients to be determined and Pn

( x) is a Jacobi

polynomial [30]. The Fourier transforms of Eqs. (36)-(37) are [31] 

 1 1 f1 ( s)   anGn J 2 n1 ( sl ) , f 2 ( s)   bnGn J 2 n1 ( sl ) s s n 1 n 1

(38)

1 (2n  ) 2 , ( x) and J ( x) are the Gamma and Bessel functions of order where Gn   (1) n 1 n (2n  2)!

n , respectively. Taking Eq. (38) into Eqs. (33)-(34), and Eq. (34) can be automatically satisfied. Then the remaining Eq. (33) is reduced to 

G  n 1



n 0

G  n 1

a2 s2

1  4 2 e [g1 ( s)an  g 2 ( s)bn ]J 2 n 1 ( sl ) cos( sx)ds   0e   h / 2 s





n 0

(39)

a2 s2

1  4 2 e [g3 ( s)an  g 4 (s)bn ]J 2 n 1 (sl ) cos(sx)ds   0 / 2 s

(40)

For a large s, the integrands of Eqs. (39) and (40) are almost decreases exponentially. For briefly, Eqs. 9

(39) and (40) can be rewritten as 



n 1

n 1

 an En* ( x)   bn Fn* ( x)  U 0 ( x) , 0  x  l 



 a G ( x)   b H n 1

n

* n

n 1

n

* n

(41)

( x)  V0 ( x) , 0  x  l

(42)

where E n* ( x) , Fn* ( x) , Gn* ( x) , H n* ( x) , U 0 ( x) and V0 ( x) are known functions. These unknown coefficients a n and bn from Eqs. (41) and (42) can be solved by the Schmidt method [16,17].

5. Numerical calculations and discussion From works [12-15, 18], it can be seen that the Schmidt method is performed satisfactorily if the first ten terms of infinite series to Eqs. (41) and (42) are retained. These coefficients a n and bn are known, the entire stress and the electric displacement fields are obtained. However, in fracture mechanics, it is important to determine the stresses  yz(2) ( x, h) and  yz(2) ( x, 0) and the electric displacements Dy(2) ( x, h) and Dy(2) ( x, 0) in the vicinity of the crack tips. In the present work,

 yz(2) ( x, h) and  yz(2) ( x, 0) and Dy(2) ( x, h) and Dy(2) ( x, 0) along the crack line can be expressed as a s  1  2 2e h  (2)  yz1   yz ( x, h)  Gn  e 4  [g1 ( s)an  g 2 ( s)bn ]J 2 n 1 ( sl ) cos( sx)ds  0 s  n 1   a2 s2  1   2 (2) 4 2 [g3 ( s)an  g 4 ( s)bn ]J 2 n 1 ( sl ) cos( sx)ds  yz 2   yz ( x, 0)   Gn 0 e  n 1 s  2 2

(43)

a s    2 2e150 e h   21 22 (2)  D y1  D y ( x , h )  Gn  e 4  [an  e  21hbn ]J 2 n 1 ( sl ) cos( sx)ds  0  s( 21   22 )  n 1  2 2 a s    2 2e150   21 22 (2) Gn  e 4  [e 22h an  bn ]J 2 n 1 ( sl ) cos( sx)ds  Dy 2  Dy ( x, 0)   0  n 1 s( 21   22 )  2 2

(44)

The fracture behavior of crack propagation is investigated with important practical significance, especially by using non-local theory. Crack propagation is predicted using the strain energy criterion. According to the previous work [32], the volume energy density function can be expressed as

10

dW near the crack tips dV

dW (r , ) dW ( x, y) 1 (2) w(2) 1 (2) w(2) 1 (2) (2) 1 (2) (2)    xy   xy  Dx Ex  Dy E y dV dV 2 x 2 y 2 2

(45)

where x  l  r cos , y  r sin  ( r is the polar radius and  is the polar angle as shown in Fig. 1.). Ex(2) and E y(2) are the electric field intensity , i.e., Ex(2)  

 (2)  (2) , E y(2)   . At the case, x y

the angular functions of dynamic stress and dynamic electric displacement fields around crack tip are given in the present paper. For a /   0 , the semi-infinite integration and the series in Eqs. (43) and (44) are convergent for any variable x. Equations (43) and (44) give finite stresses and electric displacements all along y  0 , so there is no stress and electric displacement singularities at the crack tips. However, for a /   0 , the classical stress and electric displacement singularities are shown at the crack tips. At  l  x  l ,

 yz(1) /  0 is very close to negative unity. For x  l ,  yz(2) /  0 possesses finite value diminishing from a finite value at x  l to zero at x   . Assume that the piezoelectric material is made of PZT-4 with the material properties listed in Table 1 [33]. The results of the dynamic stress and the dynamic electric displacement fields are discussed in Figs. 2-19.

Table 1

(1) When a /   0 , the semi-infinite integration and series in Eqs. (43) and (44) are convergent for any variable x. The dynamic stress and the dynamic electric displacement fields give finite values along the crack line. Contrary to the classical piezoelectric theory solution, it is found that no stress and electric displacement singularities are present at the crack tips, as shown in Figs. 2-6. The maximum stress does not occur at the crack tips, but slightly away from it in Fig. 4. This phenomenon has been thoroughly clarified in [34]. The distance between the maximum stress point and the crack tips is very small, and it depends on the lattice parameter, the crack length, the distance between two parallel cracks, the functionally graded parameter and the circular frequency of the incident waves. (2) Figs. 7 and 8 plot the results of the dynamic stress and the dynamic electric displacement fields at the crack tips tend to decrease with the increase of the lattice parameter a /  l . For the dynamic 11

electric displacement field, it has the same changing tendency as the dynamic stress field and the amplitude value of the dynamic electric displacement filed is very small as shown in Fig. 8. (3) Figs. 9 and 10 examine the dynamic stress and the dynamic electric displacement fields at the crack tips tend to increase with increase of the crack length

l reach a peak, and then decrease in

magnitude. (4) Figs. 11 and 12 depict the dynamic stress and the dynamic electric displacement fields at the crack tips increase with increase of the distance between two parallel cracks h / l . This phenomenon is called crack shielding effect [35]. (5) For 0  l  1.6 , the dynamic stress field and the dynamic electric displacement field at the crack tips increase with increase of the functionally graded parameter l reach a peak, then decrease in magnitude, as shown in Figs. 13 and 14. It can be obtained that the functionally graded parameter and the crack length have the same trend. Hence, the stress field and the electric displacement field can reach a maximum value by changing the functionally graded parameter and the length of the crack. (6) From results in Figs. 15 and 16, the dynamic stress field and the dynamic electric displacement field at the upper crack tips increase with increase of the circular frequency of the incident waves

l / c1 until reaching the first peak value at l / c1  0.48 , then they decrease with increase in magnitude until reaching the minimum value at l / c1  1.32 . Therefore, the dynamic stress field and the dynamic electric displacement field at the lower crack tips increase with increase of the circular frequency of the incident waves l / c1 until reaching the first peak value at l / c1  0.27 , then they decrease with increase in magnitude until reaching the minimum value at l / c1  0.89 , and they increase again until reaching the second peak value at l / c1  1.32 . The first peak value is larger than the second peak value. This phenomenon may be cased by the electro-elastic and the functionally graded coupling effects and the high circular frequency of the incident waves. (7) In Figs. 17 and 18, the values of angular functions of dynamic stress and dynamic electric displacement fields at the crack tips are increasing when the polar angle  is rising, and then decrease in magnitude. It is observed that the values of angular functions of dynamic stress and dynamic electric displacement locate at the position in the direction of   90o . Meanwhile, the 12

volume energy density function dW / dV is calculated against the polar distance and the polar angle as shown in Fig. 19. The volume energy density function dW / dV at the crack tips is increasing when the polar angle  is rising, and then decrease in magnitude. It also can be seen obviously that the value of the normalized energy density function locates at the position in the direction of   90o , which means the crack is likely to expand along this direction.

Figs. 2-19.

6. Conclusion This paper investigates the dynamic fracture problem of two parallel permeable cracks subjected to harmonic anti-plane shear stress waves in functionally graded piezoelectric material based on the non-local theory. As expected, these solutions in this work do not contain the stress and the electric displacement singularities at the crack tips. The influences of the lattice parameter, the crack length, the distance between two parallel cracks, the functionally graded parameter and the circular frequency of the incident waves on the stress and the electric displacement fields are discussed in details. It can be obtained that the solution yields a finite hoop stress at the crack tips, thus allows us to use the maximum stress as a fracture criterion.

Acknowledgements This work was supported by the National Natural Science Foundation of China (11272105 and 11572101).

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strip, Compos. Struct. 69 (2005) 294–300. [5] Y.D. Li, K.Y. Lee, Anti-plane fracture analysis for the weak-discontinuous interface in a non-homogeneous piezoelectric bi-material structure, Eur. J. Mech. A-Solids 28 (2009) 241– 247. [6] Y.S. Ing, J.H. Chen, Dynamic fracture analysis of an interfacial crack in a two-layered functionally graded piezoelectric strip, Theor. Appl. Fract. Mech. 63-64 (2013) 40–49. [7] S.M. Mousavi, J. Paavola, Analysis of cracked functionally graded piezoelectric strip, Int. J. Solids Struct. 50 (2013) 2449–2456. [8] A.C. Eringen, C.G. Speziale, B.S. Kim, Crack tip problem in non-local elasticity, J. Mech. Phys. Solids 25 (1977) 339-346. [9] A.C. Eringen, Linear crack subject to shear, Int. J. Fract. 14 (1978) 367-379. [10] A.C. Eringen, Linear crack subject to anti-plane shear, Eng. Fract. Mech. 12 (1979) 211-219. [11] D.G.B. Edelen, Non-local field theory. In: Eringen, A.C. (Ed.), Continuum Physics, Vol. 4 (Academic Press, New York, 1976), pp. 75-104. [12] Z.G. Zhou, B. Wang, S.Y. Du, Investigation of anti-plane shear behavior of two collinear permeable cracks in a piezoelectric material by using the non-local theory, ASME J. Appl. Mech. 69 (2002) 388-390. [13] Z.G. Zhou, B. Wang, Non-local theory solution of two collinear cracks in the functionally graded materials, Int. J. Solids Struct. 43 (2006) 887–898. [14] Z.G. Zhou, L.Z. Wu, Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials, Int. J. Eng. Sci. 44 (2006) 1366–1379. [15] Z.G. Zhou, P.W. Zhang, L.Z. Wu, Investigation of the behavior of a mode-I crack in functionally graded materials by non-local theory, Int. J. Eng. Sci. 45 (2007) 242–257. [16] P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1958), pp. 828-930. [17] W.F. Yau, Axisymmetric slipless indentation of an infinite elastic cylinder, SIAM J. Appl. Math. 15 (1967) 219–227. [18] J. Liang, S.P. Wu, S.Y. Du, The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves, Acta Mech. Sin. 23 (2007) 427–435. [19] G. Allegri, F. L. Scarpa, On the asymptotic crack-tip stress fields in nonlocal orthotropic, Int. J. 14

Solids Struct. 51 (2014) 504-515. [20] L.L. Ke, Y.S. Wang, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory, Smart Mater. Struct. 21 (2012) 025018. [21] L.L. Ke, Y.S. Wang, Z.D. Wang, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Compos. Struct. 94 (2012) 2038–2047. [22] A.K. Soh, D.N. Fang, K.L. Lee, Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading, Eur. J. Mech. A-Solids 19 (2000) 961–977. [23] A.C. Eringen, B.S. Kim, Relation between non-local elasticity and lattice dynamics, Cryst. Lat. Defects 7 (1977) 51-57. [24] V.P. Plevako, Equilibrium of a nonhomogeneous half-plane under the action of forces applied to the boundary, Appl. Math. Mech. 37 (1973) 858–866. [25] M. Ozturk, F. Erdogan, Axisymmetric crack problem in bonded materials with a graded interfacial region, Int. J. Solids Struct. 33 (1996) 193-219. [26] Y.F. Chen, F. Erdogan, The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate, J. Mec. Phys. Solids 44 (1996) 771–787. [27] L.L. Ke, Y.S. Wang, Two-dimensional sliding frictional contact of functionally graded materials. Eur. J. Mech, A/Solids, 26 (2007) 171–188. [28] L.C. Guo, N. Noda, Modeling method for a crack problem of functionally graded materials with arbitrary properties—piecewise-exponential model, Int. J. Solids Struct. 44 (2007) 6768–6790. [29] J. L. Nowinski, On non-local theory of wave propagation in elastic plates, ASME J. Appl. Mech. 51 (1984) 608–613. [30] I. S. Gradshteyn, and I. M. Ryzhik, Tables of Integral, Series and Products (Academic Press, New York, 1980), 1159-1161. [31] A. Erdelyi, Tables of Integral Transforms, Vol. 1 (McGraw-Hill, New York, 1954), pp. 34-89. [32] J.Z. Zuo, G.C. Sih, Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics, Theor. Appl. Fract. Mech. 34 (2000) 17–33. [33] H.M. Huang, H.J. Shi, Y.J. Yin. Muti-cracks problem for piezoelectric materials strip subjected to dynamic loading. Mech. Res. Commun. 29 (2002) 413-424. [34] A.C. Eringen, Interaction of a dislocation with a crack, J. Appl. Phys. 54 (1983) 6811-6817. [35] N.I. Shbeeb, W.K. Binienda, Analysis of an interface crack for a functionally graded strip 15

sandwiched between two homogeneous layers of finite thickness, Eng. Fract. Mech. 64 (1999) 693–720. Fig. 1. Geometry and coordinate system for two parallel cracks

Fig. 2. The stress along the upper crack line versus x for l =1.0, h / l  1.0 , l  0.4 , a /  l =0.001 and l / c1  0.3 . Fig. 3. The stress along the lower crack line versus x for l =1.0, h / l  1.0 , l  0.4 , a /  l =0.001 and

l / c1  0.3 . Fig. 4. The locally enlarged graph of figure 2 near the upper crack tip Fig. 5. The electric displacement along the upper crack line versus x for h / l  1.0 , l =1.0, l  0.4 , a /  l =0.001 and l / c1  0.3 . Fig. 6. The electric displacement along the lower crack line versus x for l =1.0, h / l  1.0 , l  0.4 , a /  l =0.001 and

l / c1  0.3 .

Fig. 7. The stresses at the crack tips versus a /  l for l =1.0, h / l  1.0 , l  0.4 and

l / c1  0.3 . Fig. 8. The electric displacements at the crack tips versus a /  l for l =1.0, h / l  1.0 , l  0.4 and l / c1  0.3 . Fig. 9. The stresses at the crack tips versus l for l =1.0, h / l =1.0 , l  0.4 , a /  l  0.001 and

l / c1  0.3 . Fig. 10. The electric displacement at the crack tips versus l for a /  l  0.001 , h / l  1.0 , l  1.0 ,

l  0.4 and l / c1  0.3 . Fig. 11. The stresses at the crack tips versus h / l for l =1.0, l  0.4 , a /  l  0.001 and

l / c1  0.3 . Fig. 12. The electric displacement at the crack tips versus h / l for a /  l  0.001 , l =1.0,

l  0.4 and l / c1  0.3 . Fig. 13. The stresses at the crack tips versus l for l =1.0, a /  l  0.001 , h / l  1.0 and

l / c1  0.3 . 16

Fig. 14. The electric displacements at the crack tips versus l for l =1.0, a /  l  0.001 ,

h / l  1.0 and l / c1  0.3 . Fig. 15. The stresses at the crack tips versus l / c1 for l =1.0, a /  l  0.001 , h / l  1.0 and

l  0.4 . Fig. 16. The electric displacements at the crack tips versus l / c1 for l =1.0, a /  l  0.001 ,

h / l  1.0 and l  0.4 . Fig. 17. The stresses at the crack tips versus  for l =1.0, r / l  0.0001 , a /  l  0.001 ,

l / c1  0.3 , h / l  1.0 and l  0.4 . Fig. 18. The electric displacements at the crack tips versus  for l =1.0, r / l  0.0001 , a /  l  0.001, l / c1  0.3 , h / l  1.0 and l  0.4 .

Fig. 19. The volume energy density function dW / dV at the crack tips versus  for l =1.0,

r / l  0.0001 , a /  l  0.001 , l / c1  0.3 , h / l  1.0 and l  0.4 .

Table 1 Material properties of PZT-4. c440 ( 1010 N/m2 )

e150 ( C/m2 )

110 ( 1010 C/Vm )

 0 ( kg/m3 )

2.56

12.7

64.6

7500

Highlights 

Dynamic interaction of two parallel cracks in a FGPM is investigated under a harmonic anti-plane shear stress wave.



The present solution exhibits no stress and electric displacement singularities at the crack tips.



We can use the maximum stress and electric displacement as a fracture criterion.

17

y

r

1

θ r

h

θ

l

2

x

l

3

Fig. 1. Geometry and coordinate system for two parallel cracks

30 25

yz1/0

20 15 10 5 0 -5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x/l Fig. 2. The stress along the upper crack line versus x for l =1.0,

h / l  1.0 , l  0.4 , a /  l =0.001 and l / c1  0.3 .

25

yz2/0

20 15 10 5 0 -5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x/l Fig. 3. The stress along the lower crack line versus x for l =1.0,

h / l  1.0 , l  0.4 , a /  l =0.001 and l / c1  0.3 .

18

30

yz2/0

25 20 15 10 5 0 -5 0.99

1.00

1.01

1.02

x/l Fig. 4. The locally enlarged graph of figure 2 near the upper crack tip -7

1.0x10

-8

8.0x10

-8

Dy1/0

6.0x10

-8

4.0x10

-8

2.0x10

0.0 -8

-2.0x10

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x/l Fig. 5. The electric displacement along the upper crack line versus x for

h / l  1.0 , l =1.0, l  0.4 , a /  l =0.001 and l / c1  0.3 . -7

1.0x10

-8

Dy2/0

8.0x10

-8

6.0x10

-8

4.0x10

-8

2.0x10

0.0 -8

-2.0x10

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x/l Fig. 6. The electric displacement along the lower crack line versus x for

l =1.0, h / l  1.0 , l  0.4 , a /  l =0.001 and l / c1  0.3 .

19

24

 yz1 /  0  yz 2 /  0

yz/0

20 16 12 8 4 0.000

0.004

0.008

0.012

a / l Fig. 7. The stresses at the crack tips versus a /  l for l =1.0, h / l  1.0 , l  0.4 and

l / c1  0.3 .

-8

9.00x10

Dy1 /  0 Dy 2 /  0

-8

7.50x10

-8

4.50x10

-8

3.00x10

-8

1.50x10 0.000

0.004

0.008

0.012

a / l Fig. 8. The electric displacements at the crack tips versus

a /  l for l =1.0, h / l  1.0 , l  0.4 and l / c1  0.3 . 34

 yz1 /  0  yz 2 /  0

32

yz/0

Dy/0

-8

6.00x10

30 28 26 24 22 1.0

1.2

1.4

1.6

1.8

2.0

l Fig. 9. The stresses at the crack tips versus l for l =1.0,

20l and l / c  0.3 . h / l =1.0 , l  0.4 , a /  a l / 0.001 1

-7

1.2x10

-7

Dy/0

1.1x10

-7

1.0x10

-8

9.0x10

Dy1 /  0 Dy 2 /  0

-8

8.0x10

-8

7.0x10

1.0

1.2

1.4

1.6

1.8

2.0

l Fig. 10. The electric displacement at the crack tips versus l for

a /  l  0.001 , h / l  1.0 , l  1.0 ,a / ll 0.4 and l / c1  0.3 .

25

yz/0

24 23

 yz1 /  0  yz 2 /  0

22 21 20

0.5

1.0

1.5

2.0

2.5

3.0

h/l Fig. 11. The stresses at the crack tips versus h / l for l =1.0,

a / l

l  0.4 , a /  l  0.001 and l / c1  0.3 . -8

9.0x10

-8

8.7x10

-8

Dy/0

8.4x10

-8

8.1x10

-8

7.8x10

Dy1 /  0 Dy 2 /  0

-8

7.5x10

-8

7.2x10

0.5

1.0

1.5

2.0

h/l

2.5

3.0

21

Fig. 12. The electric displacement at the crack tips versus h / l

a / l

for a /  l  0.001 , l =1.0, l  0.4 and

l / c1  0.3 .

28

 yz1 /  0  yz 2 /  0

yz/0

26 24 22 20

-1.6

-0.8

0.0

0.8

1.6

l Fig. 13. The stresses at the crack tips versus  l for l =1.0,

a /  l  0.001 , h / l  1.0 and l / c1  0.3 .

Dy1 /  0 Dy 2 /  0

-8

9.6x10

-8

Dy/0

9.0x10

-8

8.4x10

-8

7.8x10

-8

7.2x10

-1.6

-0.8

0.0

0.8

l

1.6

Fig. 14. The electric displacements at the crack tips versus  l for l =1.0, a /  l  0.001 , h / l  1.0 and

l / c1  0.3 .

22

25

 yz1 /  0  yz 2 /  0

yz/0

20 15 10 5 0

0.0

0.4

0.8

1.2

1.6

l / c1 Fig. 15. The stresses at the crack tips versus

l / c1 for l =1.0,

a /  l  0.001 , h / l  1.0 and l  0.4 .

-8

9.00x10

Dy1 /  0 Dy 2 /  0

-8

7.50x10

Dy/0

-8

6.00x10

-8

4.50x10

-8

3.00x10

-8

1.50x10

0.0

0.4

0.8

1.2

1.6

l / c1 Fig. 16. The electric displacements at the crack tips versus for l =1.0, a /  l  0.001 , h / l  1.0 and l  0.4 .

23

l / c1

25

 yz1 /  0  yz 2 /  0

yz/0

20 15 10 5 0 0

20

40



60

80

100

Fig. 17. The stresses at the crack tips versus  for l =1.0, r / l  0.0001 ,

a /  l  0.001 , l / c1  0.3 , h / l  1.0 and l  0.4 .

-8

Dy/0

8.0x10

Dy1 /  0 Dy 2 /  0

-8

6.0x10

-8

4.0x10

-8

2.0x10

0.0 0

20

40



60

80

100

Fig. 18. The electric displacements at the crack tips versus  for l =1.0, r / l  0.0001 ,

a /  l  0.001 , l / c1  0.3 , h / l  1.0 and l  0.4 .

24

dW/dV/[(dW)0]

8

dW1 /dV/[(dW)0 ] dW2 /dV/[(dW)0 ]

6 4 2 0 0

20

40



60

80

100

Fig. 19. The volume energy density function dW / dV at the crack tips versus  for

l =1.0, r / l  0.0001 , a /  l  0.001 , l / c1  0.3 , h / l  1.0 and l  0.4 .

25