Stress state of a rectangular domain with the mixed boundary conditions

Abstract

The problem stated at the paper is not new, and some analytical methods and a lot of numerical methods are used to solve it. The approach proposed in the paper Popov G., Vaysfeld N. (2011) is based on the integral transform method and reduce the stated problem to a singular integral equation. This paper is continuation of paper Pozhylenkov O. V. (2019), where a rectangular domain with conditions of ideal contact at the lateral sides was considered. The novelty of the presented paper consists of a new statement of the problem when the conditions of the second main elasticity problem are given at the lateral sides. With the help of the Fourier transform, the one-dimensional vector boundary problem in the transform’s domain is derived. The solution of the homogeneous problem is found using matrix differential calculations, the fundamental solution matrix is constructed in the form of the contour integral, which is found using the residue theorem. As a result, we get the non-homogeneous one-dimensional problem, so the final solution is found with help of Green’s matrix Popov G., Abdimanapov S., Efimov V. (1999), which is constructed as a combination of the fundamental basis solution matrix. The solution is constructed as a superposition of the homogeneous and non-homogeneous solutions and contains the unknown derivatives of the displacement. The aim is to find the unknown function, which must satisfy the boundary condition on the upper edge of the domain. It leads to the singular integral equation which is solved with the help of the orthogonal method. The stress state of a domain was investigated depending on load properties and domain size.