PSI - Issue 59

Hryhorii Habrusiev et al. / Procedia Structural Integrity 59 (2024) 494–501 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

497

4

B

3 



  ,0 r





  J r d 0

0  

(11)

;

33 c s s



 

 

2

  

0

zz

      1 0 m s s

  ,0

  B J r d    ; 2 2 0

(12)

z u r

0

 





    2 s s sh h sssh hch h h ss                0 s s 

2 e e      1 2 h

4

h





    

0

.



4      1 h e  

4

he



2

h





0

0

Satisfying condition (2) using expression (11), we will have

B

3 







  J r d 0

0  

0,  

.

33 c s s 

a r

 

2

  

0

Extend the last equation over the entire interval 0 r   :

B

3 







  J r d 0

    x r a r   , 0 r   ,

0  

(13)

33 c s s 

 



2

  

0

where   r  is the unit Heaviside function. The unknown function   x r determines the distribution of contact stresses under the indenter. Taking into account its continuity and equality to zero on the boundary of contact domain (for r a  ), we can represent it as a partialsum of series:

n       

N



    ,0 x r

, 0 r a   ,

(14)

r 

0 a J r a n

 

z

z

1

n



where n  ,   0 0 n J   and n a are unknown coefficients. Applying the inversion formula of the Hankel integral transformation to relation (13), with regard for expression (14), we arrive at an expression       2 2 1 33 0 N n n n B a c s s          ,     0 0 0 n a n J r a r J r dr             . 1, n N  are positive roots of the Bessel function

Substituting last relation into equation (12), obtain



 

 c s s   1 33 m s s



N

n    

  ,0

      J r d    

u r

k a



,

k



0

1

0

z

n

1

1 0

n



0

Satisfying condition (7) and using last expression, we will have



N





1 0   n

      J r J a d r              

1 n k a 

.

0

0

n





0 q rJ r a  and integrating the result with respect to r from 0 to a , obtain

Multiplying last expression by