Shear banding, discontinuous shear thickening, and rheological phase transitions in athermally sheared frictionless disks

We report on numerical simulations of simple models of athermal, bidisperse, soft-core, massive disks in two dimensions, as a function of packing fraction $\phi$, inelasticity of collisions as measured by a parameter $Q$, and applied uniform shear strain rate $\dot\gamma$. Our particles have contact interactions consisting of normally directed elastic repulsion and viscous dissipation, as well as tangentially directed viscous dissipation, but no inter-particle Coulombic friction. Mapping the phase diagram in the $(\phi,Q)$ plane for small $\dot\gamma$, we find a sharp first-order rheological phase transition from a region with Bagnoldian rheology to a region with Newtonian rheology, and show that the system is always Newtonian at jamming. We consider the rotational motion of particles and demonstrate the crucial importance that the coupling between rotational and translational degrees of freedom has on the phase structure at small $Q$ (strongly inelastic collisions). At small $Q$ we show that, upon increasing $\dot\gamma$, the sharp Bagnoldian-to-Newtonian transition becomes a coexistence region of finite width in the $(\phi,\dot\gamma)$ plane, with coexisting Bagnoldian and Newtonian shear bands. Crossing this coexistence region by increasing $\dot\gamma$ at fixed $\phi$, we find that discontinuous shear thickening can result if $\dot\gamma$ is varied too rapidly for the system to relax to the shear-banded steady state corresponding to the instantaneous value of $\dot\gamma$.