PSI - Issue 50

Kirill E. Kazakov et al. / Procedia Structural Integrity 50 (2023) 125–130 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

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4

pipe and coating materials. If the materials were elastic, these operators would vanish. The appearance of Fredholm operators * ij F is due to the fact that the thickness of the pipe is comparable to the widths of the bushes and the radii of the pipe. In this case, the stress-strain state changes not only under the bushes. Moreover, the mutual influence of the bushes is manifested. System (1) also contains two sets of functions depending on z *: the functions m i ( z *) whose dependence on z * is associated with the variable coating thickness; and the functions of   i ( z *) which change due to the fact that the shapes of the bushes and the coating are different in the non-deformed state (the surfaces are not conformal). In the general case, the coating thickness and the bushes inner shapes could be described by rather complicated functions. It means that the functions m i ( z *) and   i ( z *) can turn out to be rapidly changing. In this case, the standard known approaches (Fourier series, orthogonal polynomials, etc.) lead to essential errors (Kazakov and Kurdina (2020)). Moreover, even if the functions m i ( z *) and   i (z*) are constants, in the standard approaches, it is required to solve an infinite system of linear integral equations with variable limits of integration. These features show that to solve such a problem in the cases where either the contacting surface shapes described by “bad” functions or the materials of the pipe or the coating are not elastic, it is necessary to use special methods. It should be noted that system (1) of integral equations of the problem under study is similar to that obtained by Kazakov and Parshin (2019) for the plane case (see formula (4)). Moreover, the kernel of the Fredholm operator k pl ( s ) in the plane problem has the same properties as the kernel of the Fredholm operator k cyl ( s ) in the current problem. In both problems, the function L ( u ) has the following asymptotics: ( ) 1 lim   L u u , ( ) 0 lim 0   L u u , ( )] const [ lim 1 0    u L u u . Therefore, the approach to solving the current problem will be the same. Let us consider

the main points of the approach described in the paper of Kazakov and Parshin (2019). First, it is necessary to introduce new unknown functions according to the formula

( ) ( ) ( ) * * * * * m z z i i 

c t 

~ ( , ) * * q z t i

* * * i

( ) *

m z i

( , )

q z t

,





* ( ) (   

* 1 * * 1 out ) [ ( )]   c t

I V

, i n 1,  . Then the system of integral equations (1) will take a new form

c t

where

( 1 1 *    z ,

[1, ) *   t ):

~ δ

~

~ ( , ) ( * * z t

( )( *

* out I V q 

) ~ ( , ) ( * * z t

) I V Fq * in

) ( ) c t I V D  * in

* 1/2 * 

( ). * z

( ) z

c t

(4)

 

 



We write these equations in operator form. Here the following notation is introduced:

~

~ ( , ) 1 * * q z t

     

     

    

,     

    

,     

1 *

1 *

1/

( )

0

( ) z

m z



  

~ δ

~ ( , ) * * z t q

( ) * z

1/2 *

D



,

( ) z















~

~ ( , ) * * q z t n

( ) * z

( ) *

n

m z n

0

1/



(5)

* * * j

( ) ( ) ( , ) * * * *   j i j i i

 m z z j

m z m k z i ij

~

~ Ff

( ) ( )

n

n

1



1   

( ) * z

*

( ) * z

( ) *

.   *

i

ij

F

f

,

d









1

, 1 

j

i j



~ δ

Here i i is the identity n -dimensional vector. Note that the term

) ( ) c t I V D  * in

* 1/2 * 

( ) * z

(  

( ) z



is “good” due to the

smoothness of the kernels F ij * . We obtain a new mixed operator equation with n “bad” functions m i ( z *). We will construct the solution of equation (4) in the Hilbert space L 2 ([ – 1,1], V ) for which we will construct a special basis   1,2, , ; 0,1,2, * { ( )}   i nm i m z p by orthonormalizing the following system of linearly independent vector functions on [-1,1]:

{ / 1 i

/ ( ), 1 * * 1 i

( ),( ) / * 2 1 1 * i

( ), , / 1 *  n i

( ), m z z n * *

( ),( ) / * 2 * n i

( ), } * 

n

m z z n

m z n

i

/

m z z

m z z

m z

.