  There have been many attempts to derive continuum models for dense granular
flow, but a general theory is still lacking. Here, we start with Mohr-Coulomb
plasticity for quasi-2D granular materials to calculate (average) stresses and
slip planes, but we propose a "stochastic flow rule" (SFR) to replace the
principle of coaxiality in classical plasticity. The SFR takes into account two
crucial features of granular materials - discreteness and randomness - via
diffusing "spots" of local fluidization, which act as carriers of plasticity.
We postulate that spots perform random walks biased along slip-lines with a
drift direction determined by the stress imbalance upon a local switch from
static to dynamic friction. In the continuum limit (based on a Fokker-Planck
equation for the spot concentration), this simple model is able to predict a
variety of granular flow profiles in flat-bottom silos, annular Couette cells,
flowing heaps, and plate-dragging experiments -- with essentially no fitting
parameters -- although it is only expected to function where material is at
incipient failure and slip-lines are inadmissible. For special cases of
admissible slip-lines, such as plate dragging under a heavy load or flow down
an inclined plane, we postulate a transition to rate-dependent Bagnold
rheology, where flow occurs by sliding shear planes. With different yield
criteria, the SFR provides a general framework for multiscale modeling of
plasticity in amorphous materials, cycling between continuum limit-state stress
calculations, meso-scale spot random walks, and microscopic particle
relaxation.
