How does inertia affect the steady-shear rheology of disordered solids?

We study the finite-shear-rate rheology of disordered solids with molecular dynamics simulations in two dimensions. By systematically varying the damping strength $\zeta$, we identify two well defined flow regimes, separated by a thin crossover region. In the overdamped regime, the athermal rheology is governed by the competition between elastic forces and viscous forces, whose ratio gives the Weissenberg number $\mathrm{Wi}\propto\zeta\dot{\gamma}$; the macroscopic stress $\Sigma$ follows the frequently encountered Herschel-Bulkley law $\Sigma=\Sigma_{0}+k\sqrt{\mathrm{Wi}}$, with yield stress $\Sigma_{0}>0$. In the underdamped (inertial) regime, dramatic changes in the rheology are observed for low damping: the flow curve becomes nonmonotonic. This change is not caused by longer-lived correlations in the particle dynamics at lower damping; instead, for weak dissipation, the sample heats up considerably and proportional to the driving. By thermostatting more or less underdamped systems, we are able to link quantitatively the rheology to the kinetic temperature $T_k$, while the damping strength enters only indirectly by setting $T_k$.