PSI - Issue 40

Kirill E. Kazakov et al. / Procedia Structural Integrity 40 (2022) 201–206 Kirill E. Kazakov / Structural Integrity Procedia 00 (2022) 000 – 000

203 3

   

   

   

t

 a

2 out

in     r

  

2(1

)

) ( ( , ) ( , )     

in E K t q z

( , )

1







0  

2 in

(1

( E t h z q z t ) ( ) in 

( , )  q t d

d

k z













c

)

( E t out

)









in

in

in

in

out

a



(1)

   

t

 a

in     r

  

( , )

out K t

)    out



0  

[ , ], t q d d g z r h z z a a ( )], ( ) [    ( , )      

.

k z







c

in

in

0

(

E



out

a



where q ( z , t ) is unknown function describing contact pressures, E out ( t ) and E in ( t ) are elastic moduli of main layer and inner coating, that change over time due to aging and viscoelasticity of materials;  out and  in are Poisson ’ s ratios of the main layer and coating that are considered constant; K out ( t ,  ) and K in ( t ,  ) are known tensile creep kernels of the corresponding layers (Arutyunyan and Manzhirov (1999) or Arutyunyan (1952)),  out and  in are time moments of their production; h in ( z ) is coating thickness, r in is inner radius of main (outer) layer, which equal to outer radius of coating (inner layer), g ( z ) is variable outer radius of insert, which width is equal to 2 a ;  0 is time moment of starting the interaction of insert and pipe; k c ( s ) is known kernel of cylindrical problem (it takes into account, among other things, the thickness of the main layer of pipe h out ; see, for example, works by Manzhirov and Chernysh (1988) or Arutyunyan and Manzhirov (1999)) : ) , ( ) cos( ( ) 0 c    su du u k s L u

where

2 [ ( , ) ( ) u f k u D u k k u C u    1 2 2 ( )]

( )

,

L u



1

1

r

r

r

( )

S u

(1, )

f u

4 2 f k u k u A u u f k u B u k u f u C u f u f k u D u k r        2 2 ( , ) ( ) 2 2 (1, ) ( ) (1, ) ( , ) ( ), 2 1 1 1 1 ( , ) ( ) r r r r r r

( )

S u



in 2(1 ) ( , )

2 , ( ) I ( )K ( ) I ( )K ( ), r f r u u r k r r A u      ( ) I ( )K ( ) I ( )K ( ), C u k u u u k u D u u k u k u u      out in 1 0 0 0 0 1 0 ( ) I ( )K ( ) I ( )K ( ). r r u k u k u u , ( ) I  r r u k u k u u B u 1 1 0 ( )K ( ) I ( )K ( ),

1

0

1

1

0

1

1

1

1

1

r

r

r

r

Here I 0 ( u ), I 1 ( u ), K 0 ( u ), K 1 ( u ) are modified Bessel functions of first and second kind. It is assumed in this model that coating thickness much smaller than insert width and inner radius of pipe main layer. Equation (1) contains both integrals with constant limits and integrals with variable limits. Moreover, this equation includes functions h in ( z ) and g ( z ), which describe coating thickness and the shape (outer radius) of the insert and, therefore, can be rapidly changing. 3. Conversion of model to dimensionless form Produce replacement in equation (1) of variables by formulas