Kinematic diffusion in quasi-static granular deformation

A statistical-mechanical model is proposed for the quasi-static deformation of granular assemblies, in which particle motion is decomposed into a mean -field contribution, given by the macroscopically imposed deformation, toge ther with fluctuations representing stochastic multiparticle mechanics. Thi s leads to the notion of kinematic diffusion and the postulate of a convect ion-diffusion (Fokker-Planck) equation for various configurational probabil ity distributions. Based on statistics obtained from numerical simulation o f a frictional-sphere assembly, self diffusivities and pair diffusivities a re derived for various homogeneous deformations, including 'cubical-triaxia l' strains as well as simple shear. Among the important findings are (i) di ffusive motions are found generally to be small relative to convection, sug gesting that the mean-field approximation should be quite accurate, and (ii ) pair correlations are weak, implying that two-particle and higher-order c luster diffusivities follow from single-particle diffusivities. Based on th e idea of negligible diffusion, a semi-theoretical model of granular plasti city with fabric evolution is proposed, as an extension of the exact mean-f ield model of Jenkins and Strack. It is concluded, however, that even weak diffusion effects might have important consequences for certain continuum p roperties, because of the influence on unstable equilibrium configurations. This is supported by comparison of various mean-field kinematic estimates of Reynolds dilatancy to a more accurate estimate obtained from the mechani cs simulation for a dense random packing of spheres.