Issue 48
A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70
The matrices
Φ are searched with the help of the matrices (32)
2 0 N Q P
(34)
Φ I
After substitution of the expressions (33) into the equations (34), uniting these results, the fundamental matrix is constructed in the form
1 N z z (
N z
)
e
*
( , ) z
(35)
0
Φ
2 N z ( )
1
N z
(
)
N
4
*
According to the formula (29) the Green’s matrix has the form
1 N z z ( ) (
(11) (12) (21) (22)
N z
)
e
*
( , ) z
(36)
0
0
G
2 N z ( )
1
N z
(
)
N
ND
4
2
*
N
where components are shown at Appendix 4. Finally, the solution of the vector boundary problem (24) in Fourier’s transform domain is constructed with the formulas ( ) , , 1, 2 ij i j
h
0
N z
1 ( ) ( T
2 z N T d ) ( )
βα w ( ) z
e
N z
0
βα
*
βα
N
4
βα T h
( )
h
i b
ND 0
GD
a e
cos
0
(11) 2 (12) ( ) ( ) T N T d
F N z
F N z
( , )
( , )
βα
βα
1
1
D
2
2
2
N
N
N
(37)
h
0
N z
) ( )
1
( ) T d
βα Z z
e
( N z T
N z
( )
0
βα
*
βα
4
βα T h
( )
h
i b
ND 0
GD
a e
cos
0
(21) 2 (22) ( ) ( ) T N T d
F N z
F N z
( , )
( , )
βα
βα
2
2
D
2
2
2
N
N
N
where
1
1
F N z z h shN h z h z shN h z N F N z N z h chN h z N h z chN h z ) ( ) ( ) ( ) 1 1 ( , ) ( ) ( ) ( ( , ) ) ( ) (
) chN h z shN h z ( (
)
1 2
1
) shN h z shN h z ( (
)
1 2
2
0
2 2 N D sh Nh Nh 4
D ERIVING THE DISPLACEMENTS AND STRESS FORMULAS IN THE TRANSFORM DOMAIN AND THEIR INVERSION
where S
o, the transformation of the solution is constructed in the form of superposition
3
3
w ( ), k
Z z k
w ( ) z
z Z z
( )
( ),
k
k
1
1
778
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