A Summary of Asymptotic Finite Deformation Results for a Point Load on a Hyperelastic Half-Space

Abstract

The nonlinearly elastic Boussinesq problem uses the exact equations of finite elasticity to mathematically model the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary. For this core problem of elasticity and engineering, the 1885 linear elasticity solution of Boussinesq is still used in a variety of applications despite the fact that the linear solution predicts physically unrealistic behavior in the primary region of interest beneath the load. In this paper, we aim to summarize two recent SIAM Journal of Applied Mathematics papers (D. A. Polignone Warne and P. G. Warne SIAM J. Appl. Math. 62 107-128 (2001) and SIAM J. Appl. Math. at press (2002)). These two papers develop asymptotic analyses for the nonlinearly elastic tensile point load problem for a half-space composed of either a general incompressible or compressible material, respectively. We also wish to present several new closed-form asymptotic solutions in the case of a tensile point load acting on an incompressible half-space. Finally, we comment on our approach to treating the complementary problem involving a compressive point load. Here we summarize the governing equations, conservation laws, hypotheses, and asymptotic tests needed to determine whether an isotropic hyperelastic material can support a finite deflection under a tensile point load. A variety of particular constitutive models in nonlinear elasticity are tested, and it is found that a material must be sufficiently stiff in order to support the load. Thus, it follows that many of the well-known strain-energy models for compressible hyperelastic materials proposed in the literature are unable to do so. For models which may sustain a tensile point load, we determine either the full asymptotic solution or the remaining equations and conditions for this solution. For classes of material models motivated by studies of Beatty and Jiang, we solve the resulting boundary value problem numerically in a compressible hyperelasticity setting, and we derive some new results including several closed-form explicit asymptotic solutions for a family of incompressible material models.