Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07

a 



1       2 c c

3       c c



  y x Y x   

 









, G x f 

d  

2 Y x

(10)

( )



1

4 0

    1 2 , Y x Y x are the system of fundamental matrix solutions, , i c i 

are known constants, 

 , G x  is the

where

1, 4

Green’s matrix function [29]. The expression (10) can be rewritten in scalar form







3

  u x Y x c Y x c     11 12 1 1 ( )  1

  Y x c 11

  Y x c 12

 G x 11

   '   

0 











d

,

2 2

3 2

4





1

c

c



12

1

1

 

 

 

 



G

1 1

1 1



   '   

 11 B G x 

     



     

12

0 



























x

d

d

B

x

d

(11)

,

sin

,

sin

,

1

1





c

c

0

0

c

c

11

1

1

3 1      G

 

 

 

1 1



     

 12 B G x 

     















B

x

d

d

cos

,

cos

,

2

2



c

c

0

0







3

  v x Y x c Y x c     21 22 1 1 ( )  1

  Y x c 21

  Y x c 22

 G x 21

   '   

0 











d

,

2 2

3 2

4





1

c

c



22

1

1

 

 

 

 



G

1 1

1 1



   '   

 21 B G x 

     



     

22

0 



























x

d

d

B

x

d

(12)

,

sin

,

sin

,

1

1





c

c

0

0

c

c

21

1

1

3 1      G

 

 

 

1 1



     

 22 B G x 

     















B

x

d

d

cos

,

cos

,

2

2



c

c

0

0

here a   in the second case. The inverse transformations were applied to the formulae (11)-(12), and the substitution of the displacement functions in the boundary conditions         , 0 , , 0 0, , 0 0 y xy y x p x x B x B         reduce to the system of the singular integral equations.        , , G x d  ij ij x   , and upper limit of the integrals 1 a   in the first case and

S OLVING OF THE SIE SYSTEM FOR THE TWO CASES

T





*

* 2 



c  

c

2





0 1

in the integrals with the limits 0 and  , and









he changing of the variable

in the



c c 

1 0





I  

c and 1

c were done to pass the integration interval

integrals with the limits 0

. Similar changes were done

1;1

1

in the other equations. We first consider in details the second case. SSIE is written in the form

1 

          

 

  

1

    

     Z x d K x r x x I      , , 





0

1

x 





1

1 

1  1 

     1

  d K x x I x     1 0,

(13)

1





1

1 

     2

  d K x x I x     2 0,

1





1

86