PSI - Issue 33

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

516 8

( 0, ) r r       G G I ( 0, ) r r

matrix ( ) r С was found



1



  

  

1





S

S

 Y Y  ( ) r R p

R

 

 

 С Y Y   ( ) r ( ) r



( ) r

( ) r





p

p

p

The element form of Green matrix function (16) is derived

      

   

1

1

g g

g g

11

12

  

r

, 1

1

1

21

22

   r

G

( , )

  

   

2

2

g g

g g



11

12

   

r

,

 

2

2



21

22

The solution (14) can be written in a form

1    1 

  

2

1

0

( ) ( , ) ( ) g r r dr g r r dr U          ( , ) ( ) ( )

U

11

11

p

p





(18)

  

2

1

0

( ) W g r r dr g r r dr W          ( , ) ( ) ( , ) ( ) ( )

21

21

p

p





where functions 2

1

2

1

( , ), ( , ), ( , ) g r g r g r g r     are given in the Appendix B. ( , ),

11

11

21

21

5. Inverse integral transform and the case of steady oscillations. After applying to the solution (18) the inverse Laplace and finite cos- , sin- transforms

 

 

  n    ( )cos

  ( )sin ,       n

( , ) U U

( ) 2

,

( , ) 2   

U

W

W

    

0

p

p

p

p

p

n

n





1

1

n

n





displacements of the initial problem (5 - 8) take a form

j i

 

  

1  0

1

2

a

a



 





  p h d   

  n  

  



, ,   

co

s

U

2 p F







1

p



2

4

i 

G

G



1

n



j i

 

(19)

    

  

    



 



  n     2 cos 

  ,

  , dr g r  1 11

p dr e dp   

2 g r 11

( ) r  

( ) r

 



   

 

1

n



1

