Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or "optimal system" of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically.