Effects of Inertia on the Steady-Shear Rheology of Disordered Solids

We study the finite-shear-rate rheology of disordered solids by means of molecular dynamics simulations in two dimensions. By systematically varying the damping strength $\zeta$ in the low-temperature limit, we identify two well-defined flow regimes, separated by a thin (temperature-dependent) 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 $\mathrm{\Sigma }$ follows the frequently encountered Herschel-Bulkley law $\mathrm{\Sigma }={\mathrm{\Sigma }}_0+k\sqrt{\mathrm{Wi}}$, with yield stress ${\mathrm{\Sigma }}_0>0$. 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 due to, and in proportion to, the driving. By thermostating more or less underdamped systems, we are able to link quantitatively the rheology to the kinetic temperature and the shear rate, rescaled with Einstein’s vibration frequency.

Figure 1

Athermal flow curves $\mathrm{\Sigma }(\mathrm{Wi},T_0=0)$ in the overdamped regime $Q\lesssim 1$, for various combinations $[\zeta ,m]$: $[\zeta =1,m=1]$ (⋄), [10,1] (□), [10,0.1] (△), and [1,0.1] (▿). The solid line represents Eq. (2). A flow curve at $T_0=0.2$, $Q=1$ [1,1] ($⧫$) is also shown. The thin dashed line is a best fit to Eq. (3). Inset: $\mathrm{\Sigma }$ vs $\dot{\gamma }$.

Figure 2

Flow curves $\mathrm{\Sigma }(Q,\mathrm{Ei},0)$ of athermal underdamped systems and $\mathrm{\Sigma }[Q^{\prime },\mathrm{Ei},T_K(Q,\mathrm{Ei})]$ of their thermostated counterparts (see text). Symbols are listed in Table 1. Thin dashed lines are the best fit to Eq. (3), $\mathrm{\Sigma }=0.69+2\sqrt{\mathrm{Ei}}-0.17T_K^{2/3}\ln (0.4T_K^{5/6}/\mathrm{Ei}{)}^{2/3}$, where $T_K=0.15Q\mathrm{Ei}$.

Figure 3

Shear contribution $T_{\dot{\gamma }}=T_K-T_0$ to the kinetic temperature measured in underdamped samples vs $Q\mathrm{Ei}$. (▿) Data at $T_0=0.2$. The line represents $T_{\dot{\gamma }}=0.15Q\mathrm{Ei}$.