RESEARCH OF THE STRESS-STRAIN STATE OF THE ORTHOTROPIC HALF-PLANE UNDER THE PLANAR DEFORMATION CONDITIONS

Abstract

The first basic boundary problem of the elasticity theory connected with the determination of the stress-strain state of the orthotropic half-plane under the planar deformation conditions has been considered. The stresses at the boundary y = 0 are known. The stresses tend to zero at infinity. It is necessary to determine the stress and displacement at any point of the half-plane. A brief overview of the scientific works which highlight the methods and approaches to solving the problems of the theory of elasticity, the strength as for the determination of stresses and deformations in orthotropic bodies, in particular plates, slabs, and beams is provided. The solution of the given boundary problem for the orthotropic half-plane is sought in the transformant space of the one-dimensional integral Fourier transformation. At the same time, all the main equations of the problem and the boundary conditions are subjected to direct one-dimensional integral Fourier transformation. The solution of the formulated planar problem is based on the construction of the Fourier transform of the stress function, which satisfies the corresponding analogue of the biharmonic differential equation in the transformant space for the case of an orthotropic material. The form of the transformant of the stress function depends on the values of the orthotropic material elastic constants, namely on the values of the roots of the characteristic equation obtained in the transformant space. One of three possible cases have been considered. The relations between the transformant of the stress function and the transformants of stresses and displacements are established. The transformants of the stress function are expressed in terms of four auxiliary functions, which are related to the loads on the surface of the half-plane. We find two auxiliary functions from the boundary condition ( y = 0 ). The infinity conditions make it possible to establish the connection between two found auxiliary functions and the other two functions. After substituting the found expressions into transformants of stresses and displacements and applying the inverse integral Fourier transformation, we obtain the true values of stresses and displacements at the points of the orthotropic half-plane. The solutions for the specific cases have been obtained and the numerical results have been analyzed. The obtained calculations demonstrate the adequacy of the results and the logic of using the chosen method for the solution of the given problem.