PSI - Issue 33

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

518 10

1  

0  

  

2( ) n

1( ) n r dr g r

( , ; ) ( )    

( , ; ) ( ) r    

( , ; ) ( ) ; r dr   

F r

r dr g

11

11

n

a a G G

 



(1)

2 p F 

( ; )      

( ; )  

f

n

8

det

1

n



The solution to the integral equation is represented in the series expansion



 

( ) L r 

r

 

( )  

( )

r r e

(22)

m m

0

m



here m  - unknown constants, - Laguerre polynomials. After completing the orthogonalization, i.e. multiplying the equation by m L r  ( ) ( )

( ) ( ) k e L      and integrating it in the

[0, )   , the infinite system of algebraic equations of the first kind is derived

interval



, A f k      m mk k

0,1, 2,...

(23)

0

m

where

( ) e L d            ( ; ) ( ) k f

1 0 0    n    

( )     ( ) ( ) ( ) F r r e L r L drd       ( , ; )( ) r m k n

A

f



,

mk

k

0

To find the value of  , the mechanical sense of unknown function ( ) r  should be analysed. Considering the formula of tangential stress rz G U W z r              and taking into account boundary condition in (6) ( ,0, ) 0, W    the relation 0 0 rz z z G U z       is obtained. Therefore ( , 0) p U r    ( ) rz r   . Analysing the type of singularity that appear to be in the mixed problems for a wedge (Uflyand, 1968), when an angle equals / 2  , it was found that  depends on Poisson’s ratio  : for 1/ 4   0.2552   ; for 1/ 3   0.3100   . After solving the integral equation (21) the unknown function (22) should be substituted into expressions for displacements (20). It leads to the final solution to the given problem (5 - 8) in the case of steady-state oscillations

2

a

a





p  





  n  

  

 ;  

, ;      

cos

U

F



2 1

n

4

det

G G

1

n



    

    



 

N

N 



  n   2 cos

   n

   r r e L r dr      ( ) , ; r

   n

   r r e L r dr      ( ) , ; r

1 

2

1

g

g





  



11

11

m

m

m

m

1

0

0

n

m

m









2

a





p  





  n  

  

 ,  

, ;   

sin

W

F





* 2

(24)

n

det

G

1

n



    

    



 

N

N 



  n   2 sin

   n

   r r e L r dr      ( ) , ; r

   n

   r r e L r dr      ( ) , ; r

1 

2

1

g

g





  



21

21

m

m

m

m

1

0

0

n

m

m







