PSI - Issue 72

Taras Dalyak et al. / Procedia Structural Integrity 72 (2025) 13–19

15

Membrane forces and bending moments at infinity are absent:

 ( , ) x y .

0    y xy x N N N ,

0    xy x y M M M ,

(2)

The contact of the edges of the cuts is interpreted as the closure of their sharp edges in one of the front surfaces of the plate according to Kirchhoff's assumptions about a rigid normal. With this approach, the real two-dimensional contact region transforms into a line. In addition to the jumps of the normal rotation angles     i i x y w x w y i i       / / , [ ] [ ]   Displacement discontinuities occur [ ], [ ] i i x y u u in the middle surface of the plate. Let us write the boundary conditions for smooth contact along the line (Shatsky (1988), Shatskii (1989)), that correspond to this problem formulation:

i

[ ] |[ ]| 0   i i y y u h  ,

0 

( 1)   

] sgn[ 

i y N

M

m hN

,

,

y

y

y

i

i

i

 i i x y M

0  i y ,

0 

i i x L  ,

0

i i x y N

1,2  i .



,

,

(3)

2.2. Integral equations Taking into account the symmetry and antisymmetry of the displacement fields relative to the abscissa axis, it is possible to consider the boundary conditions only on one cut (for example, the lower one). For the conditions of antisymmetrical loading we have [ ] [ ] [ ] 2 1 y y y u u u   , [ ] [ ] [ ] 2 1 x x x u u u   , [ ] [ ] [ ] 2 1 y y y      , [ ] [ ] [ ] 2 1 x x x      . Then we can write the integral expressions of forces and moments on the crack line in terms of the derivatives of the unknown functions of jumps of displacement and rotation angles:

 1

1

 ) ( )  1

 2 ) ( )}

( ,0)

11 { ( K

  (  K 12

N t y

t f

t f

  d ,









1



 1  1

1

 ) ( )  1

 2 ) ( )}

( ,0)

21 { ( K

(

N t xy

t f

K

t f

  d













,

22



1



1

 3 ) ( ) 

 4 ) ( )}

( ,0)

33 { ( K

(

M t y

t f

K

t f

  d













,

34



1



 1

1

* xy M t

 3 ) ( ) 

 4 ) ( )}

( ,0)

43 { (

(

C K

t f

K

t f

  d

 











;

(4)

44



1



2

 2 2 2 2 2 4 4

 2 2 2 2 2 4 4

2

 2 2 2 2 2 4 4  2 2 2 2 2 4

1

         1 3 4

        1 1

1

         1 1 4

,

,

,

( ) 

( ) 

( ) 

( ) 

K

K

 K

K

 



 

11

12

21

22

2





     2 2 2 2 2 4        0 2 1 1 2 4

2

1

4

1

    0        1 1 2 4

4

( )

( )    K  0 12

( ) 

K

K



,

,

;

( ) 

( ) 

K

K

 

 

34

43

33

44





B

B

Da

Da

 

[ ] ( )   y u

( ) 2 

[ ] ( )   x u

( ) 3   f t

[ ] ( )   

( ) 4 

[ ] ( )   

( ) 1

f t

f t

f t

,

,

,

.





y

x

4

4

4

4