PSI - Issue 65
Anvar Chanyshev et al. / Procedia Structural Integrity 65 (2024) 56–65 Anvar Chanyshev / Structural Integrity Procedia 00 (2024) 000–000
61
6
(6). Substituting (16) into (6) leads to the following dependencies of functions i G on i F ( 1,2...4) i :
/
13 a a
a a a
2
G a
a
i F
23
12
33
23
, ( 1,2...4) i .
(17)
i
11
13
2
2
i
i
i
i
Expressions for all stresses are constructed by applying the solution (16). For instance, the expressions for the stresses x according to (3), are as follows:
G
F
G G
G
F F
F
3 . 3 1 2 4 1 2 4 11 1 a F F F F a 2 3 4 12 13 1 2 3 4 1 2 3 4 1 2 3 4 a G G G G
x
3. Problem solution Let us formulate the boundary problem to determine yet unknown functions 1 F , 2 F , 3 F , 4 F using the boundary conditions (1), (2). To solve it, we substitute (16) into (3), (3) into (1), (2), and obtain the following system of equations for functions 1 F , 2 F , 3 F , 4 F :
a
a
a
a
a
1 F a
2 F a
3 F a
F
23 1
23
23
23
12
12
12
12
4
2
3
4
a
a
a
a
23 a
1 G a
2 G a
3 G a
G
0,
22 1 a
22
22 3
22 4
23
23
23
4
2
a
a
a
a
1 F a
2 F a
3 F a
F
33 1
33
33 3
33 4
(18)
13
13
13
13
4
2
a
a
a
a
33 a
1 G a
2 G a
3 G a
G
0,
23 1
23
23 3
23
33
33
33
4
2
4
1 F F F F f x G G G G x 2 3 4 ( ), ( ),
1
2
3
4
and from (17) it follows that
/
13 a a
a a a
2
G a
a
F
23
12
33
23
, ( 1,2...4) i .
(19)
i
i
11
13
2
2
i
i
i
i
Solving (18), (19) with respect to the functions 1 F , 2 F , 3 F , 4 F Next, we replace this argument by the corresponding arguments
, we obtain these functions as functions of x .
/ i x y ,
1, ..., 4 i . It should be taken into
, x y u u from (16) must be real. Therefore, we should focus on the real parts of the
account, that the functions
functions presented in the right part of (16) (Lekhnitsky 1977). It should be noted that if there be multiple roots of equation (11), the solutions
x u , y u will be given in the form:
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