On an Incompressible Nonlinearly Elastic Half-Space under a Compressive Point Load

Abstract

The problem of determining the deformation produced in an elastic half space due to a point force normal to the undeformed boundary has been a subject of considerable interest in the linear theory of elas-ticity and in a variety of engineering and geotechnical applications for well over a century. Recently, this inherently nonlinear problem has been posed within the framework of the nonlinear theory of elasticity for tensile point loads and results stemming from the exact governing equations have been obtained aymptotically In this paper, we consider a compressive point load and address this core problem within the context of incompressible finite elasticity. An asymptotic analysis of the exact governing equations is carried out, and asymptotic tests providing a simple way to determine if an isotropic homogeneous hyperelastic material is capable of sustaining a compressive point load are developed. The results are then applied to several particular constitutive models for incompressible nonlinearly elastic materials, and contrasts with the tensile load problem are made.