Computations of dilatancy and yield surfaces for assemblies of rigid frictional spheres

This paper is concerned with the Reynolds dilatancy and shear strength of idealized granular media. The first part of the paper offers a calculation of dilatancy in rigid-sphere assemblies based on the theoretical estimate previously proposed by J. D. Goddard (Proc. 11th Int. Congress of Rheology (ed. P. Moldenaers and R. Kuenings; Elsevier, Amsterdam 1992) 141-142) as a generalization of the classical work of O. Reynolds (Phil. Mag. 20 (1885) 469-481). This new estimate yields an analytical expression for the dilatancy of two-dimensional isotropic assemblies of disks with arbitrary size distribution. Only a few special cases of three-dimensional assemblies could be treated analytically, but a stochastic (Monte Carlo) calculation is readily implemented. The estimates of the present method are roughly three times those of Reynolds, and it is argued that the two types of estimate represent opposite bounds for the dilatancy and strength of frictionless sphere assemblies. The second part of the paper summarizes results from a detailed particle-mechanics (discrete-element) computer simulation of rigid-sphere assemblies, based on the recently developed quasi-static method of Zhuang, Didwania and Goddard (J. Comput. Phys. 121 (1995) 331-346). Dilatancies and yield cones are computed for various mono-and polydisperse assemblies of rigid frictional spheres subject to simple monotonic deformations, of the type encompassed by the 'cubical triaxial' test of soil mechanics. At large plastic strains, the computed yield surfaces bear a striking resemblance to the empirical Lade-Duncan yield surface of soil mechanics, which is adopted as a convenient referential basis for presenting the simulations. Many of the computed yield cones exhibit a lack of convexity at small strain, which may suggest a plastic instability that is suppressed in the current simulations, a matter proposed for further investigation. The results presented here suggest that frictional-sphere assemblies may mimic the shape and evolution of yield surfaces at small confining pressures for certain real non-cohesive granular media, if not the absolute magnitudes of yield strength.