Nonlocal Solutions of the Theory of Elasticity Problems for an Infinite Space Loaded with Concentrated Forces

Abstract

Two classical problems of the theory of elasticity are considered in the paper. The first is the Kelvin problem for an infinite space loaded with a concentrated force. The classical solution is singular and specifies an infinitely high displacement of the point of the force application which has no physical meaning. To obtain a physically consistent solution, the nonlocal theory of elasticity is used, which, in contrast to the classical theory, is based on the equations derived for an element of continuum that has small but finite dimensions, and allows one to obtain regular solutions for traditional singular problems. The equations of the nonlocal theory include an additional experimental constant, which has the dimension of length and cannot be determined for a space problem. Consequently, the second problem for an infinite plane loaded with two concentrated forces lying on the same straight line and acting in the opposite directions is considered. The classical solution of this problem is also singular and specifies an infinitely high elongation of the distance between the forces, irrespective of their magnitude. The solution of this problem is also obtained within the framework of the nonlocal theory of elasticity, which specifies a regular dependence of this distance on the forces magnitude. This solution also includes an additional constant which is determined experimentally for a plane problem.