  We perform molecular dynamics simulations of model granular systems
undergoing boundary-driven planar shear flow in two spatial dimensions with the
goal of developing a more complete understanding of how dense particulate
systems respond to applied shear. In particular, we are interested in
determining when these systems will possess linear velocity profiles and when
they will develop highly localized velocity profiles in response to shear. In
previous work on similar systems we showed that nonlinear velocity profiles
form when the speed of the shearing boundary exceeds the speed of shear waves
in the material. However, we find that nonlinear velocity profiles in these
systems are unstable at very long times. The degree of nonlinearity slowly
decreases in time; the velocity profiles become linear when the granular
temperature and density profiles are uniform across the system at long times.
We measure the time $t_l$ required for the velocity profiles to become linear
and find that $t_l$ increases as a power-law with the speed of the shearing
boundary and increases rapidly as the packing fraction approaches random close
packing. We also performed simulations in which differences in the granular
temperature across the system were maintained by vertically vibrating one of
the boundaries during shear flow. We find that nonlinear velocity profiles form
and are stable at long times if the difference in the granular temperature
across the system exceeds a threshold value that is comparable to the glass
transition temperature in an equilibrium system at the same average density.
Finally, the sheared and vibrated systems form stable shear bands, or highly
localized velocity profiles, when the applied shear stress is lowered below the
yield stress of the static part of the system.
