Hypoplasticity theory for granular materials-II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.