PSI - Issue 3

Nataly Vaysfeld et al. / Procedia Structural Integrity 3 (2017) 526–544 Author name / Structural Integrity Procedia 00 (2017) 000–000

532

7







I

 

1     k 4

  

1

k

are the regular kernels,

cos .   k i

f

R p

   k

i

k

2 I All integrals in system (17) should be integrated by parts with regard to a crack’s closeness condition     0,       j j j j then all equations of this system should be integrated with regard to variable .  One must

2    t    

,   

2

  , 

 t where   K x is full elliptical integral of 1-st order (obtaining this formula

0 00 W t

use that

K



  



t



is shown at App. A). After these transformations the system (17) takes the form           1 2 2 , , ,                                   j i N i ij ij j i i i j t t K dt S t R t t dt f C t t

(18)

, 1,     i    

N

i

 i

 j



     k k

  H p

I

d

  t

  t

  

0

1





where

,

4

cos ,   k i

G t

f

 i



 i



 

 

i

k

dt

  I

2

k

1

k



0   0  

x

x

x

  

  









  , 

     J x J t x sh

 ij

1

1

sech ,

,

S t

sh

 dx i j

    

 



 

0

0

i

j

i

j







x

x x

  

 





  , 

     J x J t x sh

 ii

1 2 sech  

1 ,  dx

S t

  th



0

0

i







     0   k I



K

1    k

  0    k k I



  ,

2

 ij

4 cos

cos     k i k j

,

R t

t

 

I

2

k

i C are the unknown constants. The transition to the new variables was done in the obtained system of integral equations 1 1 1 exp , exp , 2 ln ,                                   i i i i i i i t It reduces the interval of integration   ;   i i to the interval 

 1;1 .  After the transition the resulting system of

the integral equations will be following:   1 sech sech             K d

1 

 N



  , M L

    ,            d g  

  

 

i i C h



 

 



i

ij

ij

j

i

2

2

 i

 i

1

j



1

1





(19)

1 1, 1,      i N

  

   

  

  

     

  

1

1









     

where

exp

exp

,

  

i

i

i

i

2

 i

 i

  

   

   

   

   

  

  

1 1 2 2

1

1



   





 





  ,       i i j

exp

exp

, exp 

,

M

S

 i

ij

ij

j

 

 j

 

 i

 j

i

j