PSI - Issue 81

Ivan Zvizlo et al. / Procedia Structural Integrity 81 (2026) 109–115

112

2 2 x x      is the two-dimensional Laplace operator, the differential operators 3 0 B 2 2 20 10 20

j l Χ have the following

Here

structure

3

2

3







X

X

(1 2 )

,

,

B

B

x

x



  



13 0 l

3 0 l

23 0 l

3 0 l

B

2

1 0 2 0 3 0 l l l x x x   

x x

x x

 

 

1 0 2 0 l l

1 0 3 0 l l

3

2

2





x 

X

2   

.

B

x



33 0 l

3 0 2 l

B

2

2

1 0 3 0 l l    x x x



1 0 l

2 0 l

The boundary value problem is solved in two stages. In the first stage, we satisfy the conditions (2) of non-ideal contact using the integral representations (4) ‒ (6) for displacements and stresses. In this case, the boundary conditions (2) are transformed into a system of two-dimensional integral equations of the convolution type, defined over an infinite surface S 0 . Applying the direct and inverse two-dimensional Fourier transforms alternately to the variables 10 20 , х х , we obtain a representation of the unknown densities 0 D j u  in terms of the unknown normal opening function of the crack surfaces 31 B u  . For the case of the half-space B

1     2         3 4 E E E E

   

    

( ) ( ) ( ) ξ ξ ξ

B

u u u



10

   

1



( ) F

(

)

.

B  

ξ η

J

d dS 

 

(7)

20

3 4 0 

η

2

4





0



B

 

30

Here 0 ( ) J y is the Bessel function of the first kind of zeroth order of a real argument у ; the matrix F components and the E matrix-column have the form

2

2





2 F g g   

0 ,

,

,

F F F F

F F g  

   

13

14

23

24

22

1

2

12

21

2

2





1

1 2

2







2 F g g   

3 , F g F b g F g     , 2 3 3   31 5 34 4 33

,

,

,

32 F g

 

22

1

2

5

2 2







1

2

(1 )(   

2 3 b b g g g  2 1 ) , , B

(1 )(1 ) ,

, g g b g g bb  

,

g

  

1

3

4

3 2

5

3 1 2

A

A

B

1 2

G

G

 

3 2 3 b b     ( A

,

,

,

)(1 )(1 ) ,

b

b

b







B

A

B

1

2

3

B

2(1 ) 

1

1





B

A

B

dS

dS

 





ξ

ξ

( )

( )

,

1,3 ,

B

B

B

B

ξ Χ

ξ Χ

E

u

u

j

 

 



(8)

31

310

3

3 0

j

j

m

j m

x

ξ

0 m x ξ 



2

m



10

S

S

1

m

dS

dS

 





ξ

ξ

( )

( )

.

B

B

B

B

ξ D

ξ D

E u  

u

 

4

31

3310

3

33 0 m

m

x

ξ

0 m x ξ 



2

m



10

S

S

1

m

B S , which, considering

At the second stage of solving the problem, we satisfy the boundary conditions (3) on the crack surface 1 u u   x x are transformed to a two-dimensional boundary integral equation (BIE) of the form Eq. (6) and ( )

( ) m

31

3

dS

dS

 

2   



ξ

ξ

( ) ξ

( ) ξ

B

B

u

u

  



(9)

31

2

31

1 x ξ 

1 m x ξ 

2

m



S

S

1

1