On Hypo-elastic Analogues of the Dilatant Double-sliding Model

Constitutive relations for two-dimensional deformations of cohesionless granular materials are considered. The granular materials are assumed to be isotropic and rate-independent. One of the current trends in modelling the behaviour of granular materials is to study constitutive relations of the hypo-elastic The general hypo-elastic constitutive relation, originally proposed by Tr~esdell ,~ is a generalization of the theory of nonlinear elasticity and states that the stress rate is determined by the current stress and the rate of deformation. The stress rate depends linearly on the rate of deformation. Tokuoka*-lo showed that the theory of hypoelasticity and the classical theory of plasticity are similar. He demonstrated that hypo-elastic constitutive equations can possess a yield surface, and also derived the corresponding flow rule. A critical comparison of the theory of hypo-elasticity and the classical theory of plasticity was given by Mroz." Mehrabadi and Cowin" proved the existence of hypo-elastic analogues of Spencer's double-sliding model13* l4 and its generalization for dilatant materials.' Spencer's model is a well-known model which is especially relevant for large, continued deformations of cohesionless granular materials. Here a constructive derivation is given of the conditions that must be fulfilled by hypo-elastic analogues of the dilatant double-sliding model. Tensile stresses are reckoned positive, as is the usual convention in continuum mechanics. Let u denote the velocity vector and T the stress tensor. The rate of deformation tensor D is defined by