First Basic Problem of Elasticity Theory for a Composite Layer with Two Thick-Walled Tubes

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DOI
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. 27, no. 4, 2024 (December)
Pages 40-50
Cited by J. of Mech. Eng., 2024, vol. 27, no. 4, pp. 40-50

 

Authors

Oleksandr Yu. Denshchykov, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: Alex_day@ukr.net, ORCID: 0009-0008-2385-5841

Valentyn P. Pelykh, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: venator.verba@gmail.com, ORCID: 0009-0007-5301-6697

Yaroslav V. Hrebeniuk, National Aerospace University “Kharkiv Aviation Institute” (17, Vadyma Manka str., Kharkiv, 61070, Ukraine), e-mail: i.grebeniuk@khai.edu, ORCID: 0009-0004-6032-7125

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

The spatial problem of elasticity theory for a fiber composite in the form of a layer with two thick-walled cylindrical tubes is solved. Stresses are given on the flat surfaces of the layer and on the inner surface of the tubes. The solution to the problem is presented in the form of Lame equations in different coordinate systems, where the layer is considered in a Cartesian system and the tubes in local cylindrical ones. To combine the basic solutions in different coordinate systems, the generalized Fourier method is used. Satisfying the boundary conditions and conjugation conditions between the layer and the tubes, an infinite system of integro-algebraic equations is formed, which is reduced to linear algebraic equations of the second kind, and the reduction method is applied. After finding the unknowns, it is possible to obtain the stress-strained state at any point of the elastic combined bodies using the generalized Fourier method to the basic solutions of the problem. According to the results of numerical studies, it can be stated that the problem can be solved with a given accuracy, which depends on the order of the system of equations and has a rapid convergence of solutions to the exact one. Numerical analysis of the stressed state was considered with a variation of the distance between the tubes. The obtained graphs of the distribution of internal stresses in the tubes and the layer. The results show an inverse relationship between the magnitude of stresses and the distance between the tubes. In addition to the absolute value of stresses, changes in the character of the diagrams and the sign are possible. The proposed method of solution can be applied in the design of a layer with tubes. The obtained stressed-deformed state makes it possible to preliminarily evaluate the geometric parameters of the structure. Further development of the research topic is necessary for a model where tubes are combined with other types of inhomogeneities.

 

Keywords: fibrous composite, generalized Fourier method, Lamé equation.

 

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Received 31 May 2024

Published 30 December 2024