PSI - Issue 3

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

531

6

            I t The expression (14) was used during formulas (15), (16) construction. The formula (1.445(1), Gradshtein et al (1963)) was used to find the series       2 2 2 2 1 cos sin 1 sgn 1 1 sech . 4                                   k k j k j j j k k x x x ch ch x     j (the jump of stress through the branches of j -d crack) are the components of the displacement formulas (15) and (16). To find them one must use the conditions (6) and to demand the absence of the stress on cracks’ branches. Let’s start with Problem №2. The expression (15) should be substituted in condition (6) for a crack             1 1 1 2 1 1 0 2 1 | cos 1 2                                    j i j N k k k i j i j k k j I u H x p t t dt J x J t x sh G I   2 2 2 2 1 sech 2 cos cos                    k i j k k i k j K x x sh x dx     K t K I  2 2 1 2 1 .        k k k k k k I 4. Obtaining the integral equation system and its solution with the orthogonal polynomials method The unknown functions   1 K

     k

1     j j N j k 1   

 

I





2



  1      k k j I t 

 

0,   i

1, . N

I

t t dt





1

2



     x J x J tx 2

     J x J tx

Gradshtein et al (1963) is used for the extraction of singular

The identity



1

1

0

0

t

  

kernel. It leads to the system of the integral equations         0 , , ,          i N d W t t t dt S t R t

  j j





  t t dt G f 1  

  

  

(17)

,





00

i

ij

ij

j

i

 d t 

1

j



 i

, 1,       i i i N

where    W t J x J t x x dx is Weber-Schafheitlin discontinuous integral Beitman et al (1974), with singularity when ,   t   ,       0     

0   0  

x

x

x

  

 2   x







  , 

     J x J t x sh

1

1

sech ,

S t

sh

 dx i j



    

 

 

1

1

ij

i

j

i

j









x

x x

  

  





  , 

     J x J t x sh

2 x dx

1 2 sech  

1

S t

  th



1

1

ii

i







     1   k I

 

K

  1    k k I



  ,

2

3

2

4         cos cos

R t

t

ij

k

k i

k j

I

2

k

1

k

