PSI - Issue 80

Miroslav Hrstka et al. / Procedia Structural Integrity 80 (2026) 471–492 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000 – 000

477

7

where

0 0 0 1 0 0 0 0 0

   

    

Q

= 

The complex potentials f ( z ) are defined as

( ) ( ) ( ) z z             

( ) z

(

)

, z z r =

cos

sin , 

f

i

=

+



(19)

in which

( )

( )

( )







,

,

.

z z v =

z z v =

z z v =







(20)

1

2

3

The displacements and stress have the form



( ) ( ) z z A Z v A Z w T LZ v L Z w * * * * * * * * * * ,    = + = + u

,

(21)

where the matrices A * and L * are expressed by

1

1

        

        

0

i 

i

−

4

4

Gi

Gi

0  −  1 1 0 0 0 2 − i i       

1

1

1 2

*

*

0 ,

A

L



=

= 

(22)

4

4

Gi

Gi

0

0

a

33

( 12 12 44 2 E E E C C C +

ˆ

' 22    = , 3

44 E G C = and

3 4   = − for plane strain and

.

with

 =

a

)

33

*  Z are

The complex functions

    

    

i  

e

0

0

     

    

r



0 0 0 

) z z z z 

(

) 1   −

(

i

*

1

i  





2 e ir  

−

0 . 

(23)

sin

e

Z

z

r

= −

= −



i  



0

0

0

0

e

z

r

 

Note that the upper left 2×2 matrix describes pure isotropic elasticity, while the third diagonal element describes the electric field. Complex conjugation of the function (23) leads to