PSI - Issue 3

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

542

17

1 

1  0

1  0

1 2

1 2

1

1 2

1 2

1 2

1 2

  

 

   





  2 











 ln 1











 

1

1

1

ln ln 2 1  t dt t

ln 2

1 . 

 d t

t

t

dt





 















   

 

1



Hence

1 

1

1 2

1 1 3 1 1, ; ; 1 2 2 1      z F z  

1 2

z z

  

 

  

      

  2 









 z d   

 

1

1

ln

2  

ln 2

1 . 

















 

1



When

1    z x

1 

1 1 3 1 1, ; ; 1 2 2 1      x F x  

1 2

dt

x x

  

  

  

      





  1

ln

2

ln 2

.

(32)

 t x





 





 

2

1

t



1



It leads from this constructed equality that for

0  n

1 2

  

  

   

 

  1

when

1 0    x , there is no singularity here too .

ln 2

0 L x



 

   

 

After the differentiation of constructed equality (32)

  

  

1 1 3 1 2 1 2 2 1 3         x F x 1, ; ;  

1 3 5 1 2, ; ; 1 2 2 1      x F x  

x x

x x

  

2

2





 









.

2

1

1

1 L x

x

  x



 



1 2

 2      x 1 

   L x 1

Hence, for

, when

1 0    x .

0  n

2

  0 L x are bounded when

Finally we show that for all n integrals

1 0   x , and when

1 0    x . The

asymptotic formula (26) is constructed for integrals   1 L x

Appendix C. Suppose for definiteness that two cracks r -d and l -d are located in one plane the integral equations (17) only formulas for the kernel’s are changed           2 1 1 0 , , 1 2 sec .                    rl lr x x x S t S t J x J t x sh sh x h dx The corresponding kernels of system (18) will be changed           0 0 0 , , 1 2 sec .                      rl lr x x x S t S t J x J t x sh sh h dx After extraction of the weakly convergent part one gets             0 0 00 00 , , , , , , , .               rr rr ll ll S t W t S t S t W t S t Summands   0 00

.      r l Тhen in the system of

,  W t haven’t singularities, because variables  and t are changed here on the nonintersecting  ;   l l . For their calculation one can use formula (1.12.31.1, Prudnikov et al (1983))  ;   r r and 

intervals 