PSI - Issue 33

O. Pozhylenkov et al. / Procedia Structural Integrity 33 (2021) 385–390 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 3 The elastic rectangular domain 0 ≤ ≤ , 0 ≤ ≤ , 0 ≤ < ∞ meets a load at the upper face of the domain, the lower side is under condition of ideal contact, the left side and the right side are under conditions of ideal contact too ( , , ) = − ( , ), ( , , ) = 0 ( , 0, ) = 0, ( , 0, ) = 0 (0, , ) = 0, (0, , ) = 0 ( , , ) = 0, ( , , ) = 0 (1) It is required to estimate the stress state of the rectangular domain 0 ≤ ≤ , 0 ≤ ≤ , 0 ≤ < ∞ satisfying the boundary conditions (1) and the equilibrium equations ′′ ( , , ) + ∗∗ ( , , ) + 0 ( ′′ ( , , ) + ′∗ ( , , )) = 1 12 2 ( , , ) 2 ′′ ( , , ) + ∗∗ ( , , ) + 0 ( ∗∗ ( , , ) + ′∗ ( , , )) = 1 22 2 ( , , ) 2 (2) Here we use next notation ( , ) = ( , ) , ( , ) = ( , ) , ′ ( , ) = ( , ) , ∗ ( , ) = ( , ) , 0 = 1− 1 2 . We consider the case of the harmonic vibrations, so all functions are presented in the following form ( , , ) = ( , ), ( , , ) = ( , ), ( , ) = ( ) 2.2 Problem solving The Fourier`s transforms are applied to the equations (2) with following scheme: ( ( ) ( ) ) = ∫ ( ( , ) ∗ sin( ) ( , ) ∗ cos( ) ) 0 , = ( − 1 2 ) (3) It leads to the homogeneous system of the ordinary differential equations in the transform`s domain: 387

′′ ( ) − 2 ( ) + 1 12 2 ( ) − 0 2 ( ) − 0 ′ ( ) = 0 ′′ ( ) − 2 ( ) + 0 ′′ ( ) + 1 22 2 ( ) + 0 ′ ( ) = 0

(4)

Boundary conditions (1) are reformulated in the terms of the displacements: