PSI - Issue 13

Ivan Shatskyi et al. / Procedia Structural Integrity 13 (2018) 1476–1481 Author name / Structural Integrity Procedi 00 (2018) 000 – 000

1479

4

1 k l     N l k 1 k l     N l 1 k l     k N l 1 k l     k N l 1 k l     k N l k k k

Da





43 K x nk

44 K x nk

0

( , )[ ] ( ) n y    

( , )[ ] ( ) n x     

n n d p C  



,

1 1, n n x L L n N    ; 1 ,

4



k

k

k

B





11 nk

12 nk

( , )[ ] ( ) n y   

( , )[ ] ( ) n x  

K x u

K x u

d  

0





,

4



k

k

B





21 K x u 

22 K x u 

( , )[ ] ( ) n y  

( , )[ ] ( ) n x 

d  

0





,

nk

nk

4



k

k

Da





33 nk K x

34 nk K x

0

( , )[ ] ( ) n y    

( , )[ ] ( ) n x     

d m 



,

n

4



k

k

k

Da





43 K x nk

44 K x nk

0

( , )[ ] ( ) n y    

( , )[ ] ( ) n x     

n n d p C  



,

2 1, n n x L L n N N     . 1 ,

(6)

4



k

k

k

Additional conditions are imposed on the unknown functions: [ ]( ) 0, [ ]( ) 0, [ ]( ) 0, u l u l l         (7) Thus, the boundary problem (1) – (5) for the system of N 1 cracks and N – N 1 slits is settled to the system of N singular integral equations (6) with additional conditions (7). [ ]( ) 0, l   [ ] ( ) 0, n n w l   1, . n N  n n n n y n x n y n x n

3. Results and discussions

Solution to the problem (5), (6) is obtained through the method of small parameter and mechanical quadratures (Savruk (1981)). Here we given an example of simulation of the interference of two parallels crack and slit. Let us assume that the infinite plate is weakened by a dyad of two parallel rectilinear defects – the crack and the slit ( 2 N  , 1 1 N  ), each 2 l long, with distance d between centres. We consider that the plate is under the influence of a uniform bending on infinity with the moment m . For this occasion we obtained approximate analytical and numeric solutions to the problem, depending on the parameter 2 / l d   , which characterizes mutual location of the defects. Figure 1 illustrates graphic dependencies of the normalized forces ( 1 2 , K K ) and moment’s ( 3 4 , K K ) intensity factors 0.3   . The curves are shown in pairs – numeric and analytical solutions are reflected respectively in solid and dashed lines. We should note that taking into account the interaction of edges on one of the two parallel defects leads to increase in the concentration of stresses in the neighbourhood of the tips of slits, while not taking into account the contact on the slit reinforces the concentration of the stress fields in the neighbourhood of the peaks of the crack. There are significant changes to the contact forces on the line of the crack depending on the type of the second defect. In case of a slit, contact forces weaken abruptly with the approach of the defects in the centre of the crack (Fig. 2, а ), which is similar to problems of interaction of cracks with the free edge of the plate (Dalyak et al. (2003)) or with a hole free from stress (Slobodyan (2005)). At the same time for two parallel contact cracks they, on the contrary, increase (Fig. 2, b ) ( Shats’kyi and Dalyak (2000)). 1 1 (| | ) K hK m l    , 2 2 (| | ) K hK m l     , 3 3 ( ) K K m l    , 4 4 ( ) K K m l     around the tips l x   on the parameter  for various combinations of cracks and slits obtained at