To Clarify Decision on Action of Concentrated Force to Elastic Quarter-Space for Arbitrary Poisson’s Ratio

Abstract

Previously, the author obtained an analytical solution for determining the vertical displacements of the homogeneous isotropic quarter-space face, which is affected by a vertical concentrated force. This expression gave an exact solution for determining the vertical displacements of a quarter-space face made of incompressible material and an approximate solution for a Poisson ratio different from 0.5. Later, in the published paper by S. V. Bosakov and P. D. Skachek “Action of Concentrated Force on 1/8 of Homogeneous Isotropic Space”, it was shown that by combining solutions for determining the vertical displacements of a quarter-space and a half-space from the action of concentrated forces, one can find vertical displacements for one-eighth face of a homogeneous isotropic space. The resulting expressions allow solving contact problems for non-classical domains in the form of a quarter of a space and one eighth of a space. Below, the author obtains the first approximation for displacements of a quarter-space face from the action of a vertical concentrated force. In this paper, the author gives the first approximation for determining the vertical displacements of a quarter-space face from the action of a vertical concentrated force, widely using the special approximation method developed in the works of V. M. Alexandrov and allowing to calculate successfully improper integrals. The constructed graphs show good results when determining displacements with a Poisson’s ratio different from 0.5 and equal to 0.5. It should be noted that the approach indicated in the paper by S. V. Bosakov and P. D. Skachek can be successfully used in determining all displacements of the faces of one-eighth face of a homogeneous isotropic sharpened forces tangent to the edge of a quarter-space and this will allow to solve contact problems taking into account friction forces in the contact zone of the beam or plate.