Dissipation and Rheology of Sheared Soft-Core Frictionless Disks Below Jamming

We use numerical simulations to investigate the effect that different models of energy dissipation have on the rheology of soft-core frictionless disks, below jamming in two dimensions. We find that it is not necessarily the mass of the particles that determines whether a system has Bagnoldian or Newtonian rheology, but rather the presence or absence of large connected clusters of particles. We demonstrate the key role that tangential dissipation plays in the formation of such clusters and in several models find a transition from Bagnoldian to Newtonian rheology as the packing fraction $\varphi$ is varied. For each model, we show that appropriately scaled rheology curves approach a well defined limit as the mass of the particles decreases and collisions become strongly inelastic.

Figure 1

Dimensionless pressure $P^{\text{el}}$ vs dimensionless strain rate $\dot{\gamma }{\tau }_0$ or $\dot{\gamma }{\tau }_e$ for different values of $Q$ for the four dissipative models of Eqs. (2)–(5). Left hand column is for packing fraction $\varphi =0.60$; right hand column is for $\varphi =0.82$, close below jamming. For each value of $Q$, several different choices of $m_s$ and $k_d$ were used. Straight lines indicate algebraic behaviors, with power law as indicated by the neighboring number.

Figure 2

Average contact number $z$ vs $\varphi$ at different strain rates $\dot{\gamma }$ for models (a) CD, (b) ${\mathrm{CD}}_n$, and (c) ${\mathrm{CD}}_t$ at $Q=0.1$ and (d) CD at $Q=10$. Insets show the fraction of states $f_p$ with percolating connected clusters.