A mean-field model for the rheology and the dynamical phase transitions in the flow of granular matter

Based on a large set of experiments and numerical simulations, it has been recently shown ( GDR MiDi ( collective work), Eur. Phys. J. E, 14 ( 2004) 341) that dense granular flows are well described by a local rheology: the ratio of the shear stress tau to the normal stress P is an increasing function of the properly rescaled shear rate I. We propose a mean field model for this quasi-local constitutive relation and the phase diagram of granular matter, based on the motion of single spherical grain on an array of identical grains. The model recovers a '' solid- liquid '' transition that is controlled by potential trapping, as well as a subcritical '' liquidgas '' transition governed by the restitution coefficient. The system presents a '' triple point '' above which the grain directly leaves the static equilibrium to enter the gaseous regime. In the liquid regime, the relation between force and velocity is found to be almost independent of the microscopic parameters, friction and restitution coefficient. The dynamics is dominated by potential trapping and leads to a constitutive relation of the form tau/P = mu(d) + delta mu exp (- I-d/ I), in close agreement with experimental and numerical results. This rheology is only quasi-local as the inertial number is redefined, introducing the effective number of grains N involved in a collision: I =gamma root Nm/Pd.