Issue 45

O. Reut et alii, Frattura ed Integrità Strutturale, 45 (2018) 183-190; DOI: 10.3221/IGF-ESIS.45.16



1

( , ) r     

r L r 

( , ) 

( , ) 

n u r

n

c

n

2

2









1

1

( , ) ) ) r  

( , ) 

( ( , ) ( r r      r



( , ) r 





n v r

in

sin

n

n

n

2

1

1 ( sin ) (  



( , ) ) ) r  



( , ) ( r r    

 

( , ) 

( , ) r 

n w r

in r

(14)

n

n

n

2

1

1  

2



( , ) 2 ( r  

* 

( , ) r   

( , ) 



( , ) )), 

G q

r

n v r

n u r

(



n

n



1 r u r  

1





( , ) r 

( , ) ) 

( , ) ), 





( ( G r r v r

rn

n

n





( , ) r 



( , ) 



( , ) 



cos in ec v r 

( , ) ) 

r

( G w r

ctgw w r



n

n

n

n

( , ) n v r  



( , ) n r 

In the formulae (14) the jump

should be excluded. The volume expansion’s transformation

is

2

q r 

( , ) 

( , ) r   

expressed through the displacements transformations to realize it. It is proved that the equality

is true

n

n

( , ) n v r  

is derived

also. As a result, the jump

n v r 

1 2 r r u r  (



2

1



( , ) 

( , ) r     rq

( , ) ) 

ctg v r 

( , ) 



( , ) 





in

n w r

sin

n

n

n

Hence, from the formula for the normal stress jump (14), the jump of the scalar wave potential is expressed only through the given jumps of the stress and displacements

1 G r  







1

1

1



0 

( , ) (2 ) r 

( , ) 

r ctg v r 

( , ) 

( , ) ) 

( sin ) 

( , ) 

 







( r r u r

in r

n w r



n

n

n

n



( , ) n r 

The formula is constructed for the jump

where it is expressed through the jumps of the displacements and

1

stress only

1 G r 



1

( , )   



( , ) 2 (sin ) r in   

( , ) 2  

ctg w r 

( , ) 

L r

n v r

. (15)



c

n

n

n

1

2

As it seen this formula is the differential equation with regard of unknown jump 1 ( , ) n r  

. To solve this equation a

x r q

 was done for Eqn. (15) and integral transformation (6) was applied to both part of the equation.

change of variable

The unknown jump was written in the transformation’ domain



  

 

( , ) 

0  

0  

( , ) 

n w r



n

1

s

  1 Gs 

( )    





 



 

( ) K d

ctg

( ) K d

2







n

i

i

1

1 4





  

2



(16)

  

0  

( , ) 

n v r



1



2 (sin ) in 

 

( ) K d



i



The inverse integral Kantorovich-Lebedev transformation is applied to the obtained expression (16), and final solution is derived in the form

188