PSI - Issue 33

Anna Fesenko et al. / Procedia Structural Integrity 33 (2021) 509–527 Author name / Structural Integrity Procedia 00 (2019) 000 – 00

514 6

2

2

2

   

   

p

1 1       *

2  

1

*

1



1



M

( )

 





2

2

2

M

det

p

2      

1 1

 

*

*

1



  

     

     

     

  

2

2

2

2

2

2

2

2

M

det

i

p

i

p

i

p

i    

p

1 1

1 1

            

 

 

*

*

*

*

Further, with the help of the equality (13), one can be convinced that the solution of the matrix equation (12) is

1



1



Y

( , ) H M   

( )

( )

d  

 

2

C i



where C is the closed loop covering the origin and two poles of the first multiplicity 2 2 2 1 2 * * 1 , i p i p           lying in the upper half-plane. Applying the methods of contour integration, the matrix is derived using the residue theorem

 (1) H i  (1) H i



 (1) H i  (1) H i 0



2

    

    

    

    



1 1 i    

 



 (1) H i 1



 (1) H i



i

1 *

 



2 1

2

* 1

2

 

1

1

1

* 1

1

1



Y

( )

 



2

 

1





2

2

2

2

p

p

*

i

1 1    





 (1) H i



 (1) i H i 1 0



* 0

2

2

1 1



  



 

* 0

1

1

2

where

          2 2 2 1 * 2 * , p

p

1 2

1

Taking real and imaginary part of the matrix ( )  Y , increasing and decreasing, when  , solutions of the homogeneous matrix equation (12) are constructed









        

    

2



    

    

I

I

 

1 1

   1

  1



  * 1 I

I

 

1 *

2 1

2

* 1

2

1

1

   

1

1



R

 



Y

( )

2 *

 

1











p

2

2

p

p

   1



I

I

 

* 0

2

0

2

  1

  1

1

2 *

2

      p I I

 

1 1

2

* 0

0









    

 2 1 *

    

    

2 1 K

* 1 K

 

1 1

   1

  1



  * 1 K

K

 

2

2

1

   

1

1



S



   ( ) 1

Y

2 *

 

1











p

2

2

     1 * 0 K



p

p

K

2

0

2

       2 * 0 1 * K

   1

1

2 p K

 

1 1

2

0

0,1 m  .

( ) m I z are Infeld functions,

( ) m K z are the Macdonald functions,

where

Matrices are singular and regular in zero. Basis matrix can be constructed in the form (Popov et al., 1999)  p p Y Y    ( ), ( ) R S

       0 1 ( ) ( ) ( ) R S p p Ψ Y С Y С

0 1 , С С are found after satisfying the condition 

   ( ) 0 U Ψ , so

where matrix coefficients