PSI - Issue 2_B

K.B. Ustinov / Procedia Structural Integrity 2 (2016) 3439–3446 K.B. Ustinov / Structural Integrity Procedia 00 (2016) 000 – 000

3441

3

Near the crack tip the stress field possesses an integrable singularity .

yy xy        



 1/ 2 ,



O x

x

0

 



(4)

Using the result of works by Zlatin and Khrapkov (1885, 1986, 1990), Khrapkov (2001) where, in particular, the auxiliary problems for a half-plane and a strip were considered, the formulated problem has been reduced to a matrix Riemann problem of complex variable at the imaginary axis   p L  as follows

11 a a a a    21

  

1

  p + - F K F 

    p p

  p

12

K

p L 

,



,

(5)

1

 

22









a

/ p p p d 

/ h p h p h p d 

sin cos

sin cos







11

1

  a a p d h p d a p p p d           2 2 12 21 / / sin cos /

'/ 2





1

(6)









/ h p h p h p d 

sin cos

22

1





    h p h p 

2

2

2 p p d  ,

2

(2) E E /

(1)

(2)         ' 1 1

(1)

d

sin

sin

,

,









1

    ,0 ,0 x x

    ,0 ,0 (2) v x v x x u x u x       (1) (2) (1)

    ,0 ,0

 

   

    

    

(2)

0



  p

yy

 

px e dx 

F

px e dx 

  

F

p E 

0 

 2 1







  

 



xy

,

(7)

with conditions at key points (zero and infinity), which are followed from (4), (3)

  1 N Mp o p T o         

    

K  

1



 3/ 2

  p

  p

I



F

O p

p

, Re





 

F

, Re 0 p



 

2      II p K

,

(8)







0 x  and

0 x  , respectively, and, hence,

Here, according to (7) and (2) the integrands in (7) vanish for   p  F are analytical in the right (left) half-planes, respectively. For the limiting case h  formulae (6) should be replaced with     sin cos / sgn / a p p p d i p i    

functions

11

2 a a p d /

p L 

'/ 2

    



(9)

12

21









a

/ p p p d i  

p i

sin cos

sgn /





22

The main obstacle consists in factorization of the matrix coefficient   K p , i.e. its representation in the form       p p p  -1 + - K X X (10)

  p ± X are analytical and

  det 0 p  ± X in the left/right half-planes of p . Nowadays the

where functions

exact analytical solution for this problem is unknown; the following approaches may be suggested: