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

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 x and z at the upper boundary of the layer have extremes in the places of maximum values y and their negative values increase at the location of the cavity. At the same time, the stresses x exceed the specified y.