SOLUTION OF THE FIRST BASIC BOUNDARY VALUE PROBLEM OF THE PLANE ELASTICITY THEORY FOR A MULTILAYER BASE WITH ORTHOTROPIC LAYERS

Abstract

The paper deals with the problem of determining stresses and displacements at the points of a multilayer base consisting of orthotropic layers and coupled to a half-space. The external loads on the top layer are known, such that the deformation of the body becomes flat. At infinity, the stresses are zero. This paper presents a brief review of scientific studies that highlight methods and approaches to solving problems related to the theory of elasticity for studying the stress-strain state of multilayer bodies, plates, plates, and strips. The article formulates an algorithm for analytically solving the problem for a multilayer base, in which all the basic equations of the problem and boundary conditions are subjected to a direct Fourier transform. The stress function is found as a solution of the analog of a biharmonic differential equation in the space of transformants in the case of an orthotropic material. The relationships between the stress function transformant and the stress and displacement transformants are established. For each layer, four auxiliary functions are introduced that are associated with the stress and displacement transformants of points on the surface of the layers. From the conditions on the common boundaries between the layers, recurrent relations are constructed that express the auxiliary functions of the lower layer through the functions of the previous layer. By expressing the four auxiliary functions for the first layer, we can find similar functions for any layer using recurrent formulas. After substituting the found expressions into the stress and displacement transforms and applying the inverse Fourier integral transform, we obtain the true values of stresses and displacements at the points of the multilayer orthotropic base. The proposed algorithm takes into account the peculiarities of the properties of the orthotropic material and allows us to obtain analytical solutions of the stress-strain state in each layer of the base.