  We study the steady plane shear flow of a dense assembly of frictional,
inelastic disks using discrete simulation and prescribing the pressure and the
shear rate. We show that, in the limit of rigid grains, the shear state is
determined by a single dimensionless number, called inertial number I, which
describes the ratio of inertial to pressure forces. Small values of I
correspond to the quasi-static regime of soil mechanics, while large values of
I correspond to the collisional regime of the kinetic theory. Those shear
states are homogeneous, and become intermittent in the quasi-static regime.
When I increases in the intermediate regime, we measure an approximately linear
decrease of the solid fraction from the maximum packing value, and an
approximately linear increase of the effective friction coefficient from the
static internal friction value. From those dilatancy and friction laws, we
deduce the constitutive law for dense granular flows, with a plastic Coulomb
term and a viscous Bagnold term. We also show that the relative velocity
fluctuations follow a scaling law as a function of I. The mechanical
characteristics of the grains (restitution, friction and elasticity) have a
very small influence in this intermediate regime. Then, we explain how the
friction law is related to the angular distribution of contact forces, and why
the local frictional forces have a small contribution to the macroscopic
friction. At the end, as an example of heterogeneous stress distribution, we
describe the shear localization when gravity is added.
