PSI - Issue 81
Ivan Zvizlo et al. / Procedia Structural Integrity 81 (2026) 109–115
111
1 В j u from crack opening. So, we describe the displacement in the half-space D by Newtonian
S 0 points and the displacements potentials in the following form:
0 ( ) x
( ) x
( ) , x
, ,
D
D
B
u
u
u
D A B
(4)
0 0
1 0
j
j
DB j
2
x
( ) x
0 ( ) x
0 ( ) x
( ) x
D
D
D
D
u
H
H
H
30
0 0
3
3 0
j
j
j
2 x x D j 0
2(1 )
x
x
30
(3 )0 j
30
1 2
0 ( ) x
( ) , x
1,2 ,
D
D
H
H
j
D
(5)
3 0
j
2(1 )
x
x
30
30
D
1 2
x
2
( ) x
0 ( ) x
D
D
u
H
30
D
30 0
i
2(1 )
2(1 )
x
D i x x
1
i
0
0 30
D i
dS
x
ξ
3 0 H H x ( ) , D
0 ( ) x
( ) ξ
,
D
D
u
30
20
0
j
j
0 x ξ
2(1 )
x
30
D
S
0
dS
dS
ξ
ξ
( ) x
( )
( )
,
1,3 .
B
B
B
B
B
ξ D
ξ D
u
u
u
j
1 0
31
310
3
3 0
j
j
m
j m
x
ξ
0 m x ξ
2
m
10
S
S
1
m
0 D j u characterize the displacement of the interface surface S 0 points of the component D in the
Here, the unknown densities direction of the coordinate axes
0 0 j O x . The differential operators
D
3 0 ,
1, m
B
l
have the following structure
j l
1 2
3 4
x
D
.
B
3 0 l
B
B
3 0
1
2
3
2
j l
j
j
j
j
2(1 )
2(1 )
2(1 )
x
x
x
x
x
3 0 l
2 0 l
3 0 l
1 0 l
3 0 l
B
B
B
After substituting (4) and (5) into Hooke ’ s law relation, we obtain integral representations for the stress components in the form
G
3 0 x ( ) D j
30 0 x ( ) D j DB j D
( ) , x
D
31 0
1
D
3
2
30 0 x ( ) D j
0 ( ) x
D
x
H
30
20
D
j
2 x x
2 x
0 30
(3 )0 j
j
3
2
3 0 ( ) H x x D j
( ) , x
1,2 ,
D
x
H
j
(6)
30
30 20
3 0
D
x x x
x x
x
10 20 30
10 0
0
j
x
3
330 0 x ( ) D x
0 ( ) x
( ) , x
20 1
D
D
H
x
H
30 20
30
3 0
i
x
1 0
i
30
i
dS
dS
ξ
ξ
31 0 x ( ) B j u
( )
( )
,
, .
B
B
B
B
ξ Χ
ξ Χ
u
m n k
31
310
3
3 0
j
m
j m
x
ξ
0 m x ξ
2
m
10
S
S
1
m
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