PSI - Issue 81
Borys Shelestovskyi et al. / Procedia Structural Integrity 81 (2026) 162–169
166
0 G = = G
( ) , 2 rz r h G = ( ) , 2 zz r h G = 0
( )
( ) ( ) 1 1 J r d
( ) ( ) 2 1 , J r d
,0 2
rz r
,
( )
( ) ( ) 3 0 , J r d
( ) ( ) 4 0 , J r d
,0 2
zz r
0
0
0
0
1
1
( ) 2
( ) ( ) , rz r r h J r dr
( ) 1
( ) ( ) ,0 rz r r J r dr
=
=
,
,
1
1
2
2
G
G
1
1
( ) ( ) ( ) ,0 zz r r J r dr (18) As a result, we obtain a system of relatively unknown ( ) A , ( ) B , ( ) C , ( ) D where coefficients are the Hankel images of the stresses rz and zz , which are obtained from a partial solution of the problem. 3 0 0 2 G = , ( ) ( ) ( ) , zz r r h J r dr 4 0 0 2 G = .
0
0
*
1
0
(
) ( exp
) ( ) 0 p r J r dr 2 2
( ) 3
( ) ( ) ,0 zz r r J r dr
( ) 1 0, =
( ) 2 0, =
2 3
2
.
r
p r
=
=−
− + −
−
0
2
1
G
−
Or
*
exp
2
2
1
0
( ) = ( )
2
.
=−
− +
−
3
4
2
2
2
1 2 −
4
4
p
p
p
(19)
After solving the system, we get:
(
)
( ) ( ) ( ) R f ch h =− − ( ) ( ) 2 1 ,
( ) A R f
( ) ( ) h sh h = + ( ) 2
,
B
( ) ( ) ( ) ( ) ( ) 1 , C R f ch h = −
( ) ( ) ( ) ( ) , D R f sh h =−
( ) 1
exp
2
2
1
( )
( )
,
R
* 1 0 = m
2 − −
.
=
f
−
(20)
2
2
2
sh h h +
2
4
4
p
p
p
The final ratio (8) can be written as follows:
*
G
0
(
(
)
( ) ( ) ( 2 R f
) ( ) ( ) sh z h sh z J r − −
) − 0
=
+
rr
(
1
0
2
1
p
1
(
)
(
)
(
)
(
)
( ) (
)
( ( ) sh h z sh z − + )
( ( ) z ch z h ch z − − )
( )
( ) J r d 1
1 2
,
hch z
J r
+
− −
+
−
0
r
*
G
0
(
(
)
( ) ( ) ( 2 R f
) ( ) ( ) sh z h sh z J r − −
) − 0
=
+
(
1
0
2
1
p
1
,
(
)
(
)
(
)
(
)
( ) (
)
( ( ) sh h z sh z − + )
( ( ) z ch z h ch z − − )
( ) 1 J r d
1 2
hch z
+
− −
+
r
*
G
0
) ) ( ) 0
(
(
)
(
( ) ( ) ( R f
) ( ) sh z h sh z − −
( ) − + ( ) hch z
(
)
) − 0
, zchzch zh J rd − −
=
zz
(
1
2
1
p
*
G
0
(
) ( ) ) ( ) 0 , zsh zh shzJ rd − − ( )
( ) ( ) + ( ) ( R f hsh z
) − 0
=
rz
(
1
2
1
p
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