A Linear Elasticity Theory to Analyze the Stress State of an Infinite Layer with a Cylindrical Cavity under Periodic Load

Abstract

The design of parts of machines, mechanisms, structures and foundations, particularly in the aerospace industry, is closely related to the definition of the stress state of the body. The accuracy of determining the stress state is the key to optimizing the use of materials. Therefore, it is important to develop methods to achieve such goals. In this work, the second main spatial problem of the elasticity theory is solved for a layer with a longitudinal cylindrical cavity with periodic displacements given on the surface of the layer. The solution of the problem is based on the generalized Fourier method for a layer with a cylindrical cavity. To take into account periodic displacements, an additional problem is applied with the expansion of the solution for a layer (without a cavity) in the Fourier series. The general solution is the sum of these two solutions. The problem is reduced to an infinite system of linear algebraic equations, which is solved by the reduction method. As a result, the stress-strain state of the layer on the surface of the cavity and isthmuses from the cavity to the boundaries of the layer was obtained. The conducted numerical analysis has a high accuracy for fulfilling the boundary conditions and makes it possible to assert the physical regularity of the stress distribution, which indicates the reliability of the obtained results. The method can be applied to determine the stress-strain state of structures, whose calculation scheme is a layer with a cylindrical cavity and a given periodic displacement. Numerical results make it possible to predict the geometric parameters of the future structure.