PSI - Issue 81

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

110

2. Problem statement Let us consider an elastic bimaterial (Fig. 1), which consists of two half-spaces A and B . Isotropic materials of half-spaces are characterized by shear moduli , A B G G and Poisson’s ratios , A B   , respectively.

Fig. 1. Calculation scheme of the problem.

There is a non-ideal contact with slippage on the interface surface S 0 of the conjugation of half-spaces. The half-space B contains a single-periodic array of circular cracks located in a single plane perpendicular to the surface S 0 . Cracks occupy circular regions 1 , , , , 2, k n S S S k n   of radius а at a depth from the interface d 1 . The distances between the geometric centers of two neighboring defects d 2 are the same. Static tensile forces 3 3 0 N N N     act on the opposite surfaces of defects 1 , , , m S S m k n    . Fig. 1 shows the selection and arrangement of Cartesian coordinate systems. For the selected location of defects in the bimaterial and the type of loading, only the normal opening functions of the crack surfaces 31 3 3 3 ( ) ( ) ( ) ( ) 4 m m m u u u u            x x x x will be non-zero. The problem is reduced to solving differential equilibrium equations in terms of the displacement vectors 1 2 3 ( , , ) D D D D u u u u r :

r

u r

1

D

D

  u

0 ,

, D A B 

graddiv



(1)

3

1 2

 

D

with two groups of boundary conditions. The first group characterizes the contact conditions for sliding at the interface

( ) x

( ) , x

33 0   x ( ) B

( ) x

3 0 ( ) 0 , D i    x i

, , D A B S   x

B

A

A

,

1,2 ,

.

u

u



(2)

3 0

3 0

33 0

0

0

The second group of boundary conditions is formulated in the region of the location of cracks in the half-space B

3



3 1 ( )cos( , ) i x

x

31 1 x x N i  0 ,

1,3 ,

,

S







(3)

1

1

i

1

i



3



3 ( )cos( , ) m im x m

x

, x x N i 

1,3 ,

, S m k n     , 2, . m m





3

0

i

1

i



Also, the solutions of the boundary value problem (1) – (3) must vanish at infinity. 3. Boundary-integral formulation of the problem

The displacements and stresses at each point of the half-space D are caused by the displacements 0 D

j u of the interface surface