Steady shearing flows of deformable, inelastic spheres.

We extend models for granular flows based on the kinetic theory beyond the critical volume fraction at which a rate-independent contribution to the stresses develops. This involves the incorporation of a measure of the duration of the particle interaction before and after this volume fraction. At volume fractions less than the critical, the stress components contain contributions from momentum exchanged in collisions that are influenced by the particle elasticity. At volume fractions greater than the critical, the stress components contain both static contributions from particle elasticity and dynamic contributions from the momentum transfer associated with the release of elastic energy by the breaking of force chains. A simple expression for the duration of a collision before and after the critical volume fraction permits a smooth transition between the two regimes and predictions for the components of the stress in steady, homogeneous shearing that are in good agreement with the results of numerical simulations. Application of the theory to steady, inhomogeneous flows reproduces the features of such flows seen in numerical simulations and physical experiments.