Some symmetrical cavity problems for a hypoplastic granular material

The notion of hypo-elasticity originates from the work of Truesdell and involves a constitutive law expressing the stress rate as a properly invariant isotropic tensorial function of the stress- and strain-rate tensors. On the other hand hypoplasticity involves the same basic constitutive model as hypo-elasticity, except that the isotropic tensorial function is not necessarily a differentiable function of the strain-rate tensor. This non-differentiable dependence accommodates the known different behaviour of a granular material in compression and tension. Such constitutive models have, over the past decade, been successfully employed to account for the behaviour of various granular materials, including sand, soil and certain powders. The hypoplasticity model is inherently nonlinear, with the consequence that much of the progress to date has been predominantly of a numerical nature arising from its use as an incremental law. Here we examine certain symmetric dynamical cylindrical and spherical cavity problems, with a view to the determination of simple exact results. In the first part of the paper, assuming an infinite granular medium, we show that for any prescribed time-dependent pressure applied at the cavity wall, an exact stress profile may be determined, which satisfies the appropriate conditions at the cavity and at infinity. It is an exact solution in the sense that the underlying equations are properly satisfied, but the initial data for the stress and the void ratio, and the boundary data for the void ratio cannot be arbitrarily prescribed, and must take on those values generated by the stress profile. In the second part of the paper, we examine similarity solutions of cylindrical and spherical cavity problems and show that the governing equations admit a large number of such solutions, including some particularly simple power-law cases. Numerical results are given for the problem of an initially 'infinitesimally small' spherical cavity subjected to constant internal pressure.