PSI - Issue 33

519 11

Anna Fesenko et al. / Procedia Structural Integrity 33 (2021) 509–527 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

The normal stress of the layer is derived with the help of displacements (24)     0 , ; , ;            

(25)

    

    



 G r r e L r dr           ( ) , ; r

N

N 



cos   n 





 

1 



( ) G r r e L r dr    r

, ;     





3 2  

G

2

1

m

m

m

m

1

a





1

0

0

n

m

m









where





;  1



1( ) n

1 1( ) n

1( ) n

2( ) n

1 2( ) n

2( ) n

, ;  

, ;  





G r

g

g

g

G r

g

g

g









n  









n  

1 3

1 3













;

2

11

11

21

11

11

21

















2





11 ( ) ( )   

12 ( ) ( ) 

1( ) 11 n

( , ; ) r  

g

11 c r

21 c r





  ;









( )

C r

2



 22 ( ) ( ) r s r r s    11





2( ) n

( , ; ) r  

11 ( ) ( ) 

12 ( ) ( ) 

21 ( ) ( ) r r 

 

g

11 c r

12 r c r 







11

12





( )   C r



 r s





r r s r r  

21 ( ) ( ) 

22 ( ) ( ) 

11 ( ) ( ) 

( ) ( ) ;  

11 c r

21 r c r 





12

11

12

11 ( ),         are represented in Appendix C. Here dashes denote the derivative. , ;      corresponds to the situation when conditions of ideal contact are set on the upper ( ), ( ), ( ) s s     0 12 11 12

Expressions for

The normal stress

and bottom layer’s faces

 ;  

  n 



4  







0     , ;

cos

p

p

n  



3





(26)

n







2 0

det

1

n



n

  ; n    and det n , as well as the method of obtaining them are described in detail in the

The form of functions

Appendix D.

7. Discussion and numerical results.

0   ,     1 was investigated, depending on

The normal stress (25), (26) on the lower face of the layer

different characteristics: Poisson's ratio   1/3 or   1/4 ; ratio of cavity ’s radius to layer thickness 1/ 2   , 1/ 5   ; different cases of natural oscillation frequencies  1; 2; 5; 7 ; three types of acting across cavity ’s surface load       2 2 1 1 1 1 2 4 2 4 1) ( ) ; 2) ( ) 2 ; 3) ( ) sin 10 ; p p p p p p              The possibility of an appearance of tensile stress on the lower face of the layer was considered. The graphs of normal stress (26) on lower layer’s face are presented on Figures 2-4 when both layer’s faces are under condition of smooth contact with foundation. Here functions   ; n    and det n were chosen depending on the interval of membership n . On Figure 2 comparison of normal stress is carried out depending on Poisson ratio with the same 1/ 2   and load corresponds to parabola with branches going down ( p 1). Greater stress appear in the material with greater Poisson’s ratio. Stress graphs are represented as a wave oscillating near the  axes. Visualising zones of stretching stress signalize about lifting the lay er’s face that lying on the foundation without friction . With the help of / a h   ,