On the Flamant problem for a half-plane loaded with a concentrated force

Abstract

The paper is concerned with the classical Flamant problem of the theory of elasticity. The classical solution of this problem obtained by A. Flamant in the nineteenth century specifies the stresses and the displacements induced in a half-plane by a concentrated force applied to the half-plane boundary in the normal direction. Being presented in numerous textbooks in the theory of elasticity, this solution provides results that can be qualified as paradoxical and which traditionally are left without proper comments in the literature. Particularly, the half-plane displacements are singular at the point of the force application and are infinitely increasing with a distance from this point. The displacement of the boundary in the boundary direction is not continuous and experiences a jump at the point of the force application. A boundary element under any force, irrespective of how low it is, rotates at the point of the force application by 90$^{\circ }$ which does not correspond to basic assumptions of the linear theory of elasticity; hence, the classical solution cannot be qualified as consistent. The consistent solution of the problem is constructed in the paper on the basis of the generalized theory of elasticity the equations of which are obtained for the solid element that has small but not infinitesimal dimensions. As a result, these equations provide regular solutions of the problems that are singular in the classical theory of elasticity. The obtained generalized solution of the Flamant problem demonstrates regular behavior of the displacements which are not singular at the point of the force application. The solution is supported by experimental results obtained for the plate of silicon rubber simulating the half-plane.