PSI - Issue 28

K. Mysov et al. / Procedia Structural Integrity 28 (2020) 352–357 Author name / Structural Integrity Procedia 00 (2019) 000–000

354

3

, , r R               are free from stresses   1 0 0 0 r r R r R G w r w           

The spherical crack’s branches

(1.5)

, , r R               is non-zero

The displacement’s jump over crack’s branches





               1 0, 0, w R

(1.6)

, w R w R 

here   1   is unknown jump. It is required to determine the displacement satisfying boundary conditions (1.1), (1.3), (1.4), conditions on the crack (1.5), (1.6) and the torsion equation 2 2 2 2 2 2 (sin ) ( ) sin sin w w w r w r q t              , (1.7) where   , w r w       , q c   is the wave number and / c G   , is the shear wave speed. The problem is axisymmetric, thus variable  is omitted in all next formulas.

2. Problem solution The problem is solved as the superposition of continuous and discontinuous solutions       1 2 , , , w r w r w r     

(2.1)

here the continuous solution 

 1 , , w r   was found in Mysov and Vaysfeld (2019) and for arbitrary function   F 

has the form

1 2

        k   F J qr Y qb J qb Y qr J qa Y qb J qb Y qa               k   2 k k k

1

(cos ) 

P

1

a

 

  , 

k P 

1 w r

,

(2.2)

l 

r       

sin



(cos ) 

0 1

k



 

 

 

 



k

k

k

k

k



  F d   

  k J x   and

  k Y x   are Bessel’s functions,

 

1 0 sin (cos ) k P  

1 ( ) k P x  is

where  is found from (1.2),

,

k F

, k  are the roots of the transcendental equation

associated Legendre’s function of the first kind,

1 / 2

k k     

1  k

(cos ) 

P



(2.3)

1 ctg (cos ) 0 P   









k

  

To find discontinuous solution one must apply Legendre integral transform:       1 sin , cos w r w r P d      

0 

k

k

(2.4)

 1 !         1 2 k



k

 

  , 

  w r P  ,1 k k k



   k

1

cos ,

w r







 1 !

,1

k



0

k



and a variation of the Hankel integral transform by generalized scheme Popov (1992)       w rI r w r dr   

0 

1 2

k

k



k



(2.5)

0  

1 2

     1 w r  

   k    I r w d 

k



1 2

k

k



to correspondence (1.7) and receive the solution in transform domain



    



   



,

,

I

R w R

I

R w R







1 2

1 2

k

k

R

 

k

k





 w R  

,

(2.6)

k

2

2



q

