Double shearing and double rotation: A generalisation of the plastic potential model in the mechanics of granular materials

Two important models for rigid-plastic flows of granular materials are the non-associated plastic potential flow rule and the double shearing model. Both models generalise the equations governing metal plasticity and both, unlike the well-posed metal plasticity equations, are ill-posed in the sense that solutions to initial value problems do not depend continuously on the initial data. For the non-associated plastic potential model ill-posedness is a consequence of the non-coincidence of the velocity and stress characteristics, whereas for the double shearing model the presence of the material time rate of change of the inclination of the major principal axis of stress gives rise to a spatially elliptic system. A recently proposed model represents a generalisation of both the plastic potential and double shearing model and which incorporates a non-coaxial term in which the stress rotation rate of the double shearing model is replaced by a Cosserat intrinsic spin. The new model preserves the characteristic structure of metal plasticity, i.e. the characteristics exist and the non-coaxial term can be chosen so as to give rise to coincident stress and velocity characteristics.