Rapid indentation of a pre-stressed hyper-elastic half-space: comparison of axially symmetric and plane strain cases

Abstract

Transient indentation of a pre-stressed hyper-elastic half-space by either a rigid smooth cone or by a rigid smooth wedge of infinite length is considered. The former represents the basis for an axially symmetric problem, the latter, for one of plane strain. The half-space is modeled as an isotropic compressible neo-Hookean material, and both cases are treated as the superposition of infinitesimal deformations upon (possibly) finite deformations due to pre-stress. The equations for the superposed deformations exhibit the usual anisotropy induced by pre-stress, but exact expressions for the full infinitesimal fields, as well as for the dilatational, rotational and Rayleigh wave speeds in the deformed configuration, are obtained. These demonstrate some effects of pre-stress: in particular, critical tensile pre-stress levels exist beyond which negative Poisson effects arise. Critical compressive pre-stresses also exist beyond which Rayleigh waves disappear and contact zone expansion at constant sub-critical rates cannot occur. However, the critical compressive pre-stress level is lower for the plane strain case and, indeed, this case is found to be in many respects more sensitive than the axially symmetric case to pre-stress.