PSI - Issue 16

Mykola Stashchuk et al. / Procedia Structural Integrity 16 (2019) 252–259 Mykola Stashchuk et al. / Structural Integrity Procedia 00 (2019) 000 – 000

256

5

Equation (12) is realized if

Thus, the value of critical pressure is:

2 4 2 

Bp   

0.

2 .

B  

cr p

(13)

The same value of critical pressure was established by Cottrell (1958).

3. Stress intensity factor of dislocation crack

Stress intensity factor is one of important parameter in fracture mechanics. Expressions for stress intensity factors are received using complex potentials     Ф , z z  .

2 p z l 

B    

2 p

1

 

  z       z





.

 

 

 4 1



4

z z l 

z z l 

If in the right dislocation crack tip (Fig. 2) the transfer is done to the polar coordinate system according to the scheme 1 z z l   , 1 i z re   , then when 1 1 z l  potentials 1 1 1 ( ) ( ) z z l     , 1 1 1 ( ) ( ) z z l     will be:

β 2

l

l

K

K

β 2

β 2

  

  

i



0 ( )  

0 ( ).

1 1 Ф ( ) z

1 1    ( ) z

e O r

i

O r

cos

sin





(14)

I

I

r

r

2

2

where 0 ( ) O r – limited value, and the stress intensity factor in the right tip of the dislocation crack (Fig. 2) equals:

p l  

B

μ

l I

K

.

(15)

2 2 π(1 ) 

l 

( i z z re  

Analogous for the left crack tip (Fig. 2)

β

)

1

β 2

β 2   i

0

0

K

K

β 2

β 2

  

i



0 ( )  

0 ( ).

1 1 Ф ( ) z

1 1    ( ) z

e O r

e

i

O r

cos

sin





(16)

I

I

 

r

r

2

2

where

p l  

B

μ

0

K

.

(17)

I

2 2 π(1 ) 

l 

Comparison of (15) and (17) shows that stress intensity factors for the dislocation crack without internal pressure are opposite in sign. In this particular case it is agreed with the results obtained by Stashchuk and Dorosh (2016). From a physical point of view value σ( ) x in the point 0 x  should be limited, just as it was analyzed by Gupta and Erdogan (1974) and Savruk (1981). It also follows from (17). Function σ( ) x in the point 0 x  has the peculiarities of order lower then 1 r . So it is accepted that 0 I 0 K  . Under this condition

B

μ

p

.



(18)

π(1 ) l  