PSI - Issue 3

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

529

4



  u



   

(5)

,  

0  

,  

0  

,

1,

u

j

N



j

j

j

u



(6)

0 | 0,

1,

  j

N

   j



2. The construction of one-dimensional problem and its solution To get the one-dimensional boundary value problem one must use finite integral Fourier transformation with regard to variable 

1  0

 



  









 





(7)

, sin        d u , ,

2

sin ,    

2 1 .  k

u

u

u







k

k

k

k

k

2

1

k



The application of transformation (7) to the equation(1)and boundary conditions (3), (4) (the boundary conditions(2) will be satisfied during it) accordingly to the generalized scheme Popov (1982) with regard to the conditions on the cracks (5) leads to the one dimensional boundary value problem             1 2 2 2 1 cos , 0 1 1                              N k k k k k j j j u u (8)       1 1 0 1 1 , sin           k k k k k u u RG p p p d (9)         1 1 1 , 0           k k k k k u u RG p u u (10) The general solution of the equation (1) for the problem №1 has the form



1    j j N k j k cos    





  

    ,   t

2

(11)

,

u

k A I

t t dt



    

1

k

k

k

k

j

For the Problem №2 the general solution is represented by the formula



1    j j N k j k cos    









  

    ,   t

2

(12)

,

u

k A I

k B K

t t dt



 



    

1

1

k

k

k

k

k

j

 

 

1 1 , I x K x are modified Bessel functions of first order,

, k k A B are the integration constants. The

where

fundamental function Kamke (1976) of equation (8) has the form



  1       k k I K t   1 k k t

 

, 0 , 0

1 1

K

       t



  ,       t

1

k



1 t it is easy to check that this function satisfies the homogeneous equation (9) , is bounded when   I

0,   on each of

the segments   0; t ,   ;1 , t and is continuous.

  ,   t



has the jump 1 , t

k

during the passing across line

.   t It is possible to represent

Its derivative



fundamental function (12) in another form with the help of the formula (6.541, Gradshtein et al (1963))