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

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.