Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation
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International Applied Mechanics, Vol. 56, No. 3, May, 2020
STRESS STATE OF AN ELASTIC LAYER WITH A CYLINDRICAL CAVITY ON A RIGID FOUNDATION
V. Yu. Miroshnikov
The third basic problem of elasticity for a layer with a cylindrical cavity parallel to its surfaces is solved. Stresses on the cavity and the upper boundary of the layer are specified. On the lower boundary, the displacements are given. The solution to the spatial problem of elasticity is obtained by the generalized Fourier method for the Lame system of equations. The infinite systems of linear algebraic equations obtained by satisfying the boundary conditions are solved by the reduction method. As a result, the displacements and stresses are obtained at different points of the elastic body. We carried out a comparative analysis for the stress–strain state of the layer with a cylindrical cavity and without a cavity as well as for different distances to the lower boundary of the layer. Keywords: cylindrical cavity in a layer, Lame equation, generalized Fourier method, infinite systems of linear algebraic equations Introduction. When designing various structures and communications whise design model is a layer with a cylindrical cavity, it is necessary to assess the stress state of the body. An algorithm is needed that allows determining the stresses at any point of the elastic layer with the required accuracy and taking into account stress concentrators. The papers [3, 6] consider stationary problems of the diffraction of waves, and paper [16] addressed the determination of stresses in a layer with a cylindrical cavity based on the method of expansion in Fourier series. In these works, a plane problem is solved. A layer with a spherical or circular cylindrical cavity, stretched by radial forces at infinity, is considered in [13, 17–19]. Problems in which cavities are perpendicular to the layer surface are studied in [5, 7, 9, 15]. In the paper [2],, a numerical-analytical approach based on the image method was proposed to solve a two-dimensional boundary-value problem of the diffraction scattering of symmetric normal waves of longitudinal displacement for a layer with a cylindrical cavity. Problems for a hollow cylinder and a layer with a circular hole solved by the method of superposition of general solutions are considered in [4]. The analysis of the stress state of an elastic plate of finite dimensions containing a round hole was obtained, using the method of three-dimensional finite elements, in [22]. The generalized Fourier method [12] is used for problems with several boundary surfaces. Based on this method, the problems for a half-space in displacements [10], in stresses [11, 14, 20], and a mixed problem with conditions of contact type [21] are solved. In this paper, based on the generalized Fourier method, we consider a layer on a rigid foundation (displacements are specified on the lower boundary) with a cylindrical cavity parallel to the layer boundaries. 1. Problem Statement. An elastic homogeneous layer has a cylindrical cavity of radius R para
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