PSI - Issue 81
Ivan Shatskyi et al. / Procedia Structural Integrity 81 (2026) 129–134
131
2.2. Integral Equations of the Problem
To construct the solution of problem (1) – (3), we apply the method of singular integral equations. Using fundamental solutions of biharmonic equations and accounting for the periodicity of the problem, we write integral representations for forces and moments on the crack line through derivatives of unknown jump functions (Savruk, 1981):
(
)
(
)
B
x
Da
x
l
l
(6)
( ,0)
cot
[ ] ( ) , u d
( ,0) M x m
cot
[ ] ( ) , d
y N x
y
y
y
4
4
d
d
d
d
l
l
where
1
h
h
2
( ) , B E z dz D
( ) E z z dz a ,
(3 )(1 ).
(7)
2
h
h
1
We assume that the jump [ ] y is sign-constant over the entire crack length. From contact conditions (2), a relationship between the derivatives of jumps follows:
[ ]( ) [ ]( )sgn([ ]( )). y y y u x h x x
(8)
Then, after substituting the integral representations into boundary conditions (2), we obtain an integral equation with a singular Hilbert kernel:
(1 )
(
)
Da
x
l
(9)
cot
[ ] ( )
, d m x l l ( , ),
y
4
d
d
l
where
h
2 ( ) 1 3
h E z dz
2 2
Bh
h
.
(10)
h
Da
( ) E z z dz
h
For a symmetrical three-layer structure, expression (10) takes the form:
3 3 (1 ) (1 ) 3(1 )
E E
E
outer
inner
.
(11)
3
E
outer
inner
2.3. Analytical Solution
The solution of integral equation (9) with the additional condition [ ]( ) 0 y l has the form:
x
tan
4
m
2
l
[ ] ( ) x
,
(12)
y
(1 )
Da
x
2
2
2
2
cos
tan
tan
2
l
where 2 / [0,1) l d is a dimensionless parameter characterizing the mutual arrangement of cracks. Integrating expression (12), we find the jump of the normal rotation angle:
4
1
m
x
2
2
2
2
[ ]( ) x
Atanh cos
tan
tan
,
(13)
y
(1 )
2
Da
l
From the kinematic contact condition, we find the jump of normal displacement:
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