PSI - Issue 51

S. Zhao et al. / Procedia Structural Integrity 51 (2023) 69–75

72 4

S. Zhao et al./ Structural Integrity Procedia 00 (2022) 000–000

where ( ) j z  are the functions analytic in the whole complex plane except the crack region  1 ( , ) x b b   ,  3 0 x  ;

1,3,5 j  ;

1 3 5 [ , , ] j j j S S S  S are, T j

5 j j m S  , 5

1 j j m iS  , 1

1 j j n Y  , 1

3 j j n iY  , 3

5 j j n iY  , 5

j j  Y S ρ ;

j  and

)   ρ ρ and ρ is constructed of the matrix G of T T

respectively, the eigenvalues and eigenvectors of the matrix (

dimension 5 5  by crossing out the second and fourth rows and columns; ( ) m A , ( ) m B are matrixes similar to eponymous matrixes defined in Suo et al. (1992) ( (1) 1   G B D ,

(1) (2) 1 (1) ( )    D A A B B , 1 m  corresponds to the upper (2)

material and 2 m  – to the lower one). When obtaining relations (8) and (9), the continuity of stresses, electric displacements and electric potential along the whole material interface were taken into account. Due to this fact fourth row and column were excluded from the matrix G and also the case 4 j  from Eqs. (8) and (9). Due to (8) the conditions at infinity for the function ( ) j z  can be written in the form

(10)

1 (1 ) (  

( ) | z    

)

   



im

m H

 

1

5

j

z

j

j

j

Satisfying the interface conditions (6) by using Eq. (8), one arrives at the following problem of the linear relationship:     1 1 0 j j j x x        for   1 , x a a   , (11)

with the conditions at infinity (10). According to Muskhelishvili (1975) the solution of this problem has the following form

,

(12)

( )  

( )(

)( 2 ) i z ib       

z X z

j

j

j

j

j

ln

j 

1





1/2  

1/2  

i

i





,

,

,

(1 )    ,

where

( ) ( j X z z a  

)

(

)

j r     



j 



1 j j m r   /

z a 

5 m H j





r

j 



j

j

j

j

2



j

  1,3,5 j  . Substituting the solution (12) into the conjugate of Eq. (8) and integrated (9) we get

, (13)

(1)

(1) m H x 15 33 1

(1) im x  11 13 1

( ,0) x

( ,0)

( ,0)

( )

J x









33 1

1 1

, for 1 ( , ) x a a   (14)

(1)

(1) m H x 55 33 1

( ,0) x

( ,0)

( )

J x







33 1

5 1

, (15)

11 1 1 nux inux inWx   13 3 1 15 3 1 ( ) ( ) ( )

( )

x 



1 1

for 1 ( , ) x a a   . (16)

53 3 1 n u x n W x  55 3 1 ( ) ( )

( )

x 



5 1

The expressions of 1 1 ( ) J x , 5 1 ( ) J x , 1 1 ( ) x  and 5 1 ( ) x  are found with use of (12), but here only their asymptotic expressions are important and they are as follows:   1 1 0.5 1 1 0 11 12 1 ( ) | ( ) i x a J x q iq x a         , 1 5 1 0 3 1 ( ) | / x a J x q x a     , (17)

 







 

0.5

i



1 

| x a x

,

| x a x

0 4     , 1 q a x

(18)

q iq a x 



0     21



5 1

1 1

22

1

1

1