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

127

3

It can be shown that the dimensionless mathematical model of the problem has the form

n

1   j

( )( *

* out

) ( ) ( , ) ( * * * * m z q z t i i

) I V F * in

* * * * q z t j

* *

ij

i 

 I V

( , )

(

),

1, , n

c t

z

i

 





(1)\

*

*

[1, ).      t z 1, 1

The dimensionless variables and functions in it are related to dimensional ones by means of the formulas

*

1

*

,    t a ) * 1 

1

*

,    * in 1 0  

1

* out

1









2(   z

)

,

2(  

,

,

,

z

a

t

 i

     







0

in 0

out 0

i

j

j

2 out

2[

( )   a r h z g z i out

( )]

1

( ( E t E t out in

)





( )

 

 

a h z

* *

( ) *

( ) *

 i

m z i

in

( ) z

,

,

,

c t







)

2 in

1





out

i

2 in

2 (1 a

) ( , ) q z t





* * * in

* * * i

i

i

( , )

,

( , ) 

, ( K t in    in

) ,

q z t

K t



   

in 0

( aE t in

)





in

( ( E E t out out

) ( ) ( in in E

) )





 

in  

* * * out

out

( , ) 

( K t out

,

) , 0

(2)

K t



 

   

out

out

E t



out  



in

*

*

t

t





* in

( ) *

* * * in

( , ) ( ) * f

,    f t V * out *

( ) *

* * * out

( , ) ( ) * f d

,    *

V

f t

K t

d

K t





1

1





1



( , ) * * 

-1

*

( ) *

( , ) ( ) * * * f d

, ,    i j n 1, * 

k z ij

ij

k z ij

F

( k z cyl

)

,

r

f z









1

in



1



* *

* *

, a z b a  

, 1      b

,

1,

,

[ , ),  

,

[1, ),

z

t

t





 



 

0

i

i

j

j

where q i ( z,t ) is an unknown radially distributed load in the interval [ a i , b i ] (contact stress under the i th bush, a i and b i are z -coordinates of the i th bush boundaries), i i i a b a   is length of the i th bush, ) 0.5( i i i a b    is the middle point of the contact area under the i th bush, i n i a a  1,2, min   ,  in and E in ( t ) are elastic moduli of the main tube (internal),  out and E out ( t ) are elastic moduli of the coating (outer thin layer), V in and V out are Volterra operators with creep kernels K in ( t ,  ) and K out ( t ,  ) for the main tube and coating, respectively, I is the identity operator, F j ( j =1,2,…, n ) are Fredholm operators with known kernel k cyl ( s ) of cylindrical problem (see, for example, Manzhirov and Chernysh (1991) and Arutyunyan and Manzhirov (1999)), calculated by the formula

2

)] , 2 A u Su k f u fk u kAuu uBufk u kCuuf u Duf ufk u r r u r k r f r u S u u u B u f u D u su du L u u L u k s r r r r r r r                     )] , 2 u k u B u () [K( )I() K()I( k u u () (1,) () (1,) ( ,), ( ) ( , ) ( ) (1, ) ( , ) ( ) , 2(1 ) in , ( , ) ( ) [ () (1, ) ()1] ) , ( ) cos( ( ) ( ) 2 4 2 1 in out 2 0 cyl

,

(3)

( ) [K( )I( ) K( )I( () [K( )I() K()I( 1 0 0 1 u k u k u u u k u k u u r r    

0

0

0

0

r

r

2

)] . 2

() [K()I( ) K( )I()], u k u C u  

k u u D u

1

0

0

1

1

1

1

1

r

r

r

r

Here we use modified Bessel functions of the first (I 0 ( u ) and I 1 ( u )) and second (K 0 ( u ) and K 1 ( u )) kind. Note that the creep kernel contains information about both the viscoelastic properties of the material and its aging. For example, it has the form             ( , ) / ( ) ( ) ( , ) 1 K t E E C t k k k k ( in,out  k ), where C k ( t ,  ) is the tensile creep function (Arutyunyan and Manzhirov (1999) or Arutyunyan (1952)). Having derived expressions for the functions ( , ) * * * q z t i from (1), we will be able to obtain the contact pressure distributions q i ( z , t ) under each bush using formulas (2). 3. Solution method and final formulas We pay attention to characteristic features of system (6). It consists of equations each of which contains both Volterra and Fredholm operators. Volterra operators * in V and * out V arise due to the aging and viscoelasticity of the