PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471

460

6

T. Profant et al/ Structural Integrity Procedia 00 (2023) 000 – 000

Fig. 1. The traction free and isolated/conducted faces of the semi-infinite crack of the LEFM problem as the outer solution , out I u or , out II u .

r x  

   =

out

out

0, for

and

t

n σ

= 

=

(16)

,

0

where ( ) 0, 1 =  n . The same argument can be put to the polarization P , which reduces the boundary condition (5) 6 to the insulating or conducting form

out

0, 0.  =   =  

E E







0 (17) The outer solution obeying the boundary conditions (16) and (17) is equivalent to the solution describing the crack tip field in the classical elasticity under the plain strain conditions with exponent =1⁄2 , i.e. for the case of the symmetric mode I crack tip opening is ( ) ( ) ( ) 1 1 , 2 2 1 3 2 5 8 cos cos , out I I K u r     −   = −  −   for and .   =  r x out r

    

 



r

4

2

2



(18)

1 2

1 2

K

1 2

3 2

  

−

( ) r 

( ) 2 

(

)

, out I

I





7 8 sin 

sin

u





=

− −

+

 

4



and for the case of the antisymmetric mode II

1 2

1

K

1 2

3 2

     

  

( ) ( 2 2  −

( ) r 

)

, out II

II

 

3sin ,  

5 8 sin 

u

=

− +

+



r

4



(19)

1 2

1

K

1 2

3 2

   

( ) ( 2 2  −

( ) r 

)

, out II

II

 

7 8 cos 

3cos

.

u



=

− +

+

 

4



Moreover, the outer electric potential is zero valued for both mode of loading:

, out I out II   = = ,

0.

(20)

The derivation of (20) from boundary conditions (17) is discussed in our paper (Profant et al, 2023).