PSI - Issue 3

Nataly Vaysfeld et al. / Procedia Structural Integrity 3 (2017) 526–544 Author name / Structural Integrity Procedia 00 (2017) 000–000

528

3

u

u





  

1 , 





,

G

u

G







 r



 

z







 z The boundary conditions for the solid cylinder (problem №1) are written in the form  r r

u

u



 

 

1 



 





,

, 0 0, 

, r H G 

|

0

 u r

  G

  u

 r

  z



 z H





 r r The boundary conditions for the hollow cylinder (problem №2) have the same presentations but are supplemented with the condition on the internal cylindrical surface: z  

u

 

  

1







, R z G 

0

u

 r







0





r r

 

 r R

0

Displacements is discontinuous on the cracks’ surfaces         , 0 , 0 , , 0 0,         j j j z j u r d u r d r r d 

, a r b j     

1, , N

j

j

here    j r are the unknown jumps of the displacements on the cracks’ branches,   0   j r outside the segment of cracks’ location. Let’s pass to the dimensionless coordinates 1 1 ,       rR zH and designate             1 1 1 0 , , , , , , , ,                        j j j j u u R H p q H R RH R R d H 1 1 , , 1, .        j j j j a R b R j N One must find the solution of the equation

2

1

    u

u



  1 , 0 0, 0        1    

2  

(1)

1

  u



 

 

2

 

  





with the boundary conditions

u







(2)

, 0 0,  

|

0

u



1

  



1 u u

 

  



  1 

 RG p

(3)





 

 

1





1

1

u

u

  

  

  

  

 

 

  1 

 RG p

(4)

,

0

u

u









 

 

1



  



Boundary conditions (2), (3) should be satisfied for the problem №1; boundary conditions (2) – (4) should be satisfied for the problem №2. The conditions on the cracks’ surfaces should also be satisfied