Investigation of the Stress State of a Composite in the Form of a Layer and a Half Space with a Longitudinal Cylindrical Cavity at Stresses Given on Boundary Surfaces

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DOI https://doi.org/10.15407/pmach2019.04.024
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 4, 2019 (December)
Pages 24-31
Cited by J. of Mech. Eng., 2019, vol. 22, no. 4, pp. 24-31

 

Author

Vitalii Yu. Miroshnikov, Kharkiv National University of Construction and Architecture (40, Sumska St., Kharkiv, 61002, Ukraine), e-mail: m0672628781@gmail.com, ORCID: 0000-0002-9491-0181

 

Abstract

An analytical-numerical approach to solving the spatial problem of the theory of elasticity for a half-space rigidly coupled to a layer is proposed. In the half-space, parallel to its boundaries, there is an infinite circular cylindrical cavity. Both the layer and half-space are homogeneous isotropic materials, different from each other. It is necessary to investigate the stress-strain state of the elastic bodies of both the layer and half-space. On both the surface of the cavity and upper boundary of the layer, stresses are given. On the flat surface of contact between the layer and half-space, conjugation conditions arise. The solution to the spatial problem of the elasticity theory is obtained by the generalized Fourier method with regard to both the system of Lamé equations in cylindrical coordinates associated with the cavity and Cartesian coordinates associated with both the layer and half-space. The infinite systems of linear algebraic equations obtained as a result of satisfying both the boundary and conjugation conditions are solved by a reduction method. As a result, displacements and stresses are obtained at different points of both the elastic layer and elastic half-space. The fulfillment of boundary conditions was reduced to 10-4 by using the selected reduction parameter for the given geometric characteristics. An analysis of the stress-strain state of both the layer and half-space with the given physical and geometric parameters has been carried out. Graphs of stresses at the boundary between the layer and half-space, on the surface of the cavity and upper boundary of the layer, as well as on the bridge between the cavity and boundary of the half-space are presented. The indicated stress graphs show that the maximum stresses are concentrated both on the surface of the cylindrical cavity and surface of the half-space. The proposed method can be used to calculate parts and components, underground structures and communications, whose design schemes coincide with the purpose of this paper. The stress analysis presented above can be used to select geometric parameters at the design stage, and the stress graphs at the boundary between the layer and half-space, to analyze the coupling strength.

 

Keywords: cylindrical cavity in a half-space, composite, Lamé equation, conjugation conditions, generalized Fourier method.

 

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References

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Received 28 August 2019

Published 30 December 2019