Hypoplasticity theory for granular materials—I: Two-dimensional plane strain exact solutions

Abstract

Numerous theories have been developed in order to gain a better understanding of the behaviour of granular materials. One such theory, originally developed by the Institute for Soil and Rock Mechanics at the University of Karlsruhe over the last decade or so, is the rational continuum mechanical theory termed hypoplasticity. This theory involves a constitutive law for which the stress-rate is a properly invariant isotropic tensorial function of the stress- and strain-rate tensors, but possesses a non-differentiable dependence on the strain-rate tensor. From a practical perspective, it would be highly desirable to determine simple solutions of hypoplasticity applying to a range of fundamental problems, such as gravity flow in a two-dimensional wedge-shaped hopper. Although this is the original motivation of this study, the complexity of the theory appears to preclude the determination of simple analytical solutions, such as the classical solution of Jenike applying to the Coulomb–Mohr granular solid. In this paper, we undertake a mathematical investigation to determine solutions for two-dimensional steady quasi-static plane strain compressible gravity flow for hypoplastic granular materials. 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 solution is not a trivial matter. Three-dimensional axially symmetric solutions analogous to those given in this present study are presented in a companion paper, part II.