Math. Proc. Camb. Phil. Soc. (1990), 108, 153 153 Printed in Great Britain Three-dimensional constitutive equations for rigid/perfectly plastic granular materials BY J. OSTROWSKA-MACIEJEWSKA Institute of Fundamental Technological Research, Warsaw, Poland AND D. HARRIS Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 IQD (Received 22 November 1989) Abstract A three-dimensional constitutive equation governing the flow of an isotropic rigid/perfectly plastic granular material is presented. The equation relates the strain-rate tensor to the Cauchy stress tensor and to the co-rotational rate of the Cauchy stress. It contains scalar functions of the scalar invariants involving the stress, stress-rate and strain-rate tensors together with parameters which charac- terize the material. The model generalizes the double-shearing model and its relationship to existing theories is demonstrated. 1. Introduction The rigid/perfectly-plastic model for the quasi-static flow of a granular material comprises the stress equilibrium equations, a yield criterion and a set of constitutive equations relating a measure of the deformation (characterized by the velocity gradient tensor, see Section 3) to the state of stress. The equations may be considered to define an ideal granular material and it is generally accepted that the equations governing the stress field give a reasonably good approximation to the behaviour of a real granular material. However, there is no such consensus concerning the constitutive equations governing the flow. Indeed, it is an open question whether or not the flow of real granular materials can be described in terms of a rigid/perfectly- plastic model. Ultimately this question must be resolved by comparison of theory with experiment. Existing theories have all been found wanting in one regard or another but the difficulty of the measurement of quantities characterizing the flow of granular materials has prevented any decisive results concerning current theories. It is against this background that the present contribution is made and it concerns the mathematical formulation of constitutive equations governing general three- dimensional flows. A by-product of the formulation is to bring a unity to the existing theories, thereby enabling comparisons and contrasts to be made both between experiment and theory and between competing theories. The equations are a generalization of a number of existing models, opening up the possibility of encompassing a wider range of flows and materials than has previously been possible.