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We perform molecular dynamics simulations of model granular systemsundergoing boundary-driven planar shear flow in two spatial dimensions with thegoal of developing a more complete understanding of how dense particulatesystems respond to applied shear. In particular, we are interested indetermining when these systems will possess linear velocity profiles and whenthey will develop highly localized velocity profiles in response to shear. Inprevious work on similar systems we showed that nonlinear velocity profilesform when the speed of the shearing boundary exceeds the speed of shear wavesin the material. However, we find that nonlinear velocity profiles in thesesystems are unstable at very long times. The degree of nonlinearity slowlydecreases in time; the velocity profiles become linear when the granulartemperature and density profiles are uniform across the system at long times.We measure the time $t_l$ required for the velocity profiles to become linearand find that $t_l$ increases as a power-law with the speed of the shearingboundary and increases rapidly as the packing fraction approaches random closepacking. We also performed simulations in which differences in the granulartemperature across the system were maintained by vertically vibrating one ofthe boundaries during shear flow. We find that nonlinear velocity profiles formand are stable at long times if the difference in the granular temperatureacross the system exceeds a threshold value that is comparable to the glasstransition temperature in an equilibrium system at the same average density.Finally, the sheared and vibrated systems form stable shear bands, or highlylocalized velocity profiles, when the applied shear stress is lowered below theyield stress of the static part of the system.Comment: 11 pages, 14 figure

Stabilization of nonlinear velocity profiles in athermal systems undergoing planar shear flow