Elasticity Problem for a Layer with a Cylindrical Cavity Under Periodic Loading

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DOI https://doi.org/10.15407/pmach2025.03.042
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute of Power Machines and Systems
of National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 28, no. 3, 2025 (September)
Pages 42-50
Cited by J. of Mech. Eng., 2025, vol. 28, no. 3, pp. 42-50

 

Authors

Tetiana M. Aloshechkina, O. M. Beketov National University of Urban Economy in Kharkiv (17, Chornohlazivska str., Kharkiv, 61002, Ukraine), e-mail: atn4042@gmail.com, ORCID: 0000-0001-7234-1558

Natalia A. Ukrainets, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: n.ukrayinets@khai.edu, ORCID: 0000-0001-7406-5809

Vitalii Yu. Miroshnikov, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: v.miroshnikov@khai.edu, ORCID: 0000-0002-9491-0181

 

Abstract

In aerospace and mechanical engineering, elements that are loaded by periodic loads (periodic function) are used. In problems for a layer with cylindrical inhomogeneities, it is difficult to take such loads into account. Therefore, there is a need to develop a methodology for calculating the stress state for a layer with a cylindrical cavity and taking into account the boundary conditions in the form of a periodic function. In this paper, we propose a solution to the problem of elasticity theory for a layer with a cylindrical cavity within the framework of the generalized Fourier method. Stresses are given at the upper boundary of the layer and on the surface of the cylindrical cavity, and displacements are given at the lower boundary of the layer. The layer and cylindrical cavity are considered in different coordinate systems (Cartesian and cylindrical). The redistribution functions of the generalized Fourier method are applied to the Lamé equations. The problem is reduced to the sum of two solutions – an auxiliary problem and the main problem. Both problems are reduced to infinite systems of linear algebraic equations, which allow the application of the reduction method to them. After finding the unknowns in the auxiliary problem, the stresses at the geometric location of the cavity are found. The main problem is solved for the layer with the cavity, on which stresses obtained from the auxiliary problem are set with the reverse sign. The complete solution consists of the auxiliary and main problems. Having calculated all the unknowns, it is possible to obtain the stress-strain state at any point of the body with a given accuracy. Numerical analysis of the stress state showed high accuracy of the boundary conditions and dependence on periodic loading. Thus, the stresses sx and sz at the upper boundary of the layer have extremes in the places of maximum values sy and their negative values increase at the location of the cavity. At the same time, the stresses sx exceed the specified sy.

 

Keywords: periodic loading, layer with a cylindrical cavity, Lamé equation, generalized Fourier method.

 

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Received 12 March 2025

Accepted 20 May 2025

Published 30 September 2025