An asymptotic finite deformation analysis for an isotropic compressible hyperelastic half-space subjected to a tensile point load

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary, using the exact equations of elasticity. For this core problem of elasticity and engineering, the 1885 linear elasticity solution of Boussinesq is still used in a variety of applications. In [SIAM J. Appl. Math., 62 (2001), pp. 107-128], we addressed the case of a tensile point load under the constraint of incompressible finite elasticity. Here we consider an analogous asymptotic analysis of this problem within the context of compressible finite elasticity. Asymptotic tests are developed to determine whether an isotropic hyperelastic material can support a finite deflection under a tensile point load. The results are then applied to a variety of particular constitutive models for compressible nonlinearly elastic materials. It is found that, for many of the well-known strain energy models for compressible hyperelastic materials proposed in the literature, a tensile point load cannot be supported. For models which may sustain a tensile point load, we determine the remaining equations and conditions for the asymptotic solution, and numerically compute this solution for a particular case.