PSI - Issue 65
Anvar Chanyshev et al. / Procedia Structural Integrity 65 (2024) 56–65 Anvar Chanyshev / Structural Integrity Procedia 00 (2024) 000–000
60
5
y
g f x
2
.
(15)
1
x
y
1
By solving (15), we find the function g as the sum of the solution to the homogeneous equation and the partial solution of the inhomogeneous equation in the form:
2 1 F x 2 1 y y g f x
1 1 / f x y
/
1
.
, where
F
1
2 1
Substituting the function
1 F into (15) we obtain that:
1
2
y
y
1 F x
2 F F 1
1 F f x 1
2
.
1
1
x
y
1
1
1
Then, using the definition of the function g , we obtain the following differential equation:
y x
y
y
2 1 F x u f x
3
4
.
x
y
2
1
4 x y
Similarly, we introduce the function
, which satisfies the equation
u
y
y
2 f x 1 F x
3 y
.
x
2
1
The solution to this equation is obtained as follows:
3 f x 2 1 F x 3 2 1 y y y F x
3 2
y
.
, where
2 F f x 2
/ 1
2
Thus the general solution of the equilibrium equations system (6) in terms of displacements can be given as:
1 2 F x 3 F x 4 F x 1 2 3 4 1 2 3 4 1 2 3 4 x y y y y y u F x y y y y u G x G x G x G x
,
(16)
,
where 1 2 3 4 , , , are roots of the characteristic equation (11). In this paper they are assumed to be different. The distinctive feature of the stated problem is that, despite the fact that summands of functions , x y u u have different arguments, they are all reduced to the argument x on the half-plane boundary 0 y , making it easier to solve the problem. Functions i F and i G ( 1,2...4) i presented in (16) are not independent. They must satisfy each of the equations
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