Effect of collisional elasticity on the Bagnold rheology of sheared frictionless two-dimensional disks

We carry out constant volume simulations of steady-state, shear-driven flow in a simple model of athermal, bidisperse, soft-core, frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. Focusing on the small strain rate limit, we map out the rheological behavior as a function of particle packing fraction $\varphi$ and a parameter $Q$ that measures the elasticity of binary particle collisions. We find a $Q^{*}\left(\varphi \right)$ that marks the clear crossover from a region characteristic of strongly inelastic collisions, $Q<Q^{*}$, to a region characteristic of weakly inelastic collisions, $Q>Q^{*}$, and give evidence that $Q^{*}\left(\varphi \right)$ diverges as $\varphi \to {\varphi }_J$, the shear-driven jamming transition. We thus conclude that the jamming transition at any value of $Q$ behaves the same as the strongly inelastic case, provided one is sufficiently close to ${\varphi }_J$. We further characterize the differing nature of collisions in the strongly inelastic vs weakly inelastic regions, and recast our results into the constitutive equation form commonly used in discussions of hard granular matter.

Figure 1

The elastic part of the Bagnold coefficients for (a) pressure $p$ and (b) shear stress $\sigma$ vs collision elasticity parameter $Q$ for different values of packing fraction $\varphi$, which goes from 0.835–0.60 as the curves go from top to bottom. Open symbols at all $\varphi$ are for a shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$; corresponding solid symbols at $\varphi =0.60$ and 0.835 are for $\dot{\gamma }{\tau }_e={10}^{-6}$. The absence of a dependence on $\dot{\gamma }{\tau }_e$ shows that results are in the hard-core limit.

Figure 2

The dissipative part of the Bagnold coefficients for (a) pressure $p$ and (b) shear stress $\sigma$ vs collision elasticity parameter $Q$ for different values of packing fraction $\varphi$. Open symbols connected by solid lines at all $\varphi$ are for a shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$; corresponding solid symbols connected by dashed lines, at $\varphi =0.60$ and 0.835, are for $\dot{\gamma }{\tau }_e={10}^{-6}$.

Figure 3

The kinetic part of the Bagnold coefficients for (a) pressure $p$ and (b) shear stress $\sigma$ vs collision elasticity parameter $Q$ for different values of packing fraction $\varphi$. In (a), $\varphi$ decreases as the curves go from top to bottom; in (b) $\varphi$ increases as the curves go from top to bottom. Open symbols at all $\varphi$ are for a shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$; corresponding solid symbols at $\varphi =0.60$ and 0.835 are for $\dot{\gamma }{\tau }_e={10}^{-6}$. The absence of a dependence on $\dot{\gamma }{\tau }_e$ shows that results are in the hard-core limit. Note the linear vertical scale in (b), which is necessary since $B_{\sigma }^{\mathrm{kin}}$ changes sign.

Figure 4

Relative contribution of the kinetic part to the total Bagnold coefficient: (a) $B_p^{\mathrm{kin}}/B_p$ and (b) $|B_{\sigma }^{\mathrm{kin}}|/B_{\sigma }$ vs $Q$ for different fixed packing fraction $\varphi$, which goes from 0.60–0.835 as the curves go from top to bottom. Open symbols at all $\varphi$ are for a shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$; solid symbols at $\varphi =0.60$ and 0.835 are for $\dot{\gamma }{\tau }_e={10}^{-6}$. Note, $B_p^{\mathrm{kin}}/B_p=n{T}_g/p$, with $T_g$ the granular temperature.

Figure 5

(a) Phase diagram in the $Q-\varphi$ plane, showing the crossover $Q^{*}\left(\varphi \right)$ that separates the region of strongly inelastic behavior from weakly inelastic behavior. We show values for $Q^{*}$ as determined independently from the Bagnold coefficients $B_p^{\mathrm{el}}, B_{\sigma }^{\mathrm{el}}$, and $B_p^{\mathrm{kin}}$ of Figs. 1 and 3; these are all found to agree. Solid line is a fit of $Q^{*}$, as obtained from $B_p^{\mathrm{el}}$, to the form $Q_0+c{({\varphi }_J-\varphi )}^{-x}$ with fixed ${\varphi }_J=0.84335$, and yields the value $x\approx 1.65$. (b) Exponents $q$ that determine the large $Q$ algebraic increase of the Bagnold coefficients $B_p^{\mathrm{el}}, B_{\sigma }^{\mathrm{el}}$, and $B_p^{\mathrm{kin}}$. In both panels the vertical dashed line locates the jamming transition at ${\varphi }_J=0.84335$. Results are from simulations with shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$.

Figure 6

Total Bagnold coefficients (a) $B_p$ and (b) $B_{\sigma }$, for pressure $p$ and shear stress $\sigma$ respectively, vs packing fraction $\varphi$, for different fixed values of the collision elasticity parameter $Q$; $Q$ increases from 0.1–500 as curves go from bottom to top. The vertical dashed line locates the jamming transition at ${\varphi }_J=0.84335$. Results are from simulations with shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$.

Figure 7

Total Bagnold coefficients (a) $B_p$ and (b) $B_{\sigma }$, for pressure $p$ and shear stress $\sigma$, respectively, vs ${\varphi }_J-\varphi$, for different fixed values of the collision elasticity parameter $Q$; $Q$ increases from 0.1–500 as curves go from bottom to top. We use ${\varphi }_J=0.84335$ from Ref. [5]. The bold straight lines, with slopes as indicated in the figure, denote the approximate power-law dependencies of our data closest to ${\varphi }_J$, for the smallest and largest values of $Q$ (these are not the true power-law divergences asymptotically close to ${\varphi }_J$; see text). Results are from simulations with shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$.

Figure 8

Macroscopic friction $\mu \equiv \sigma /p$ vs $\varphi$ for different fixed values of $Q$; $Q$ increases from 0.1–500 as curves go from top to bottom. The vertical dashed line locates the jamming transition at ${\varphi }_J=0.84335$. Results are from simulations with shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$.

Figure 9

Schematic of the collision of two particles $i$ and $j$. (a) Initiation of contact, where ${90}^{\circ }<\theta <{270}^{\circ }$, and (b) breaking of contact, where $-{90}^{\circ }<\theta <{90}^{\circ }$. Here ${\mathbf{r}}_{ij}\equiv {\mathbf{r}}_i-{\mathbf{r}}_j$ and ${\mathbf{v}}_{ij}\equiv {\mathbf{v}}_i-{\mathbf{v}}_j$.

Figure 10

Histograms $\mathcal{P}\left(\theta \right)$ vs collision angle $\theta$, at initiation and breaking of particle contacts (see Fig. 9 for definition of $\theta$), at $\varphi =0.78$. Results are shown for different values of the collision elasticity parameter $Q$. (a) $Q=500$, (b) $Q=50$, (c) $Q=20$, (d) $Q=10$, (e) $Q=5$, and (f) $Q=0.1$. The crossover $Q^{*}\approx 7.35$ at this value of $\varphi$. Dotted blue line in (a) is a fit to $\mathcal{P}\left(\theta \right)=C|cos(\theta \left)\right|$. Results are for the shear strain rate $\dot{\gamma }{\tau }_e={10}^{-5}$.

Figure 11

(a) Average duration of collision ${\tau }_{\mathrm{dur}}/{\tau }_e$, from time of initiation of contact to time of breaking of contact, vs collision elasticity parameter $Q$. (b) Ratio of average collision rate to strain rate, ${\nu }_{\mathrm{coll}}/\dot{\gamma }$, vs $Q$. In both panels results are shown for packing fraction $\varphi =0.78$ and shear strain rates $\dot{\gamma }{\tau }_e={10}^{-4}, {10}^{-5}$, and ${10}^{-6}$. The crossover $Q^{*}\approx 7.35$, separating strongly from weakly inelastic regions, is denoted by the vertical arrow.

Figure 12

Comparison of average particle contact number $\langle Z\rangle$ (open symbols) with $2{\tau }_{\mathrm{dur}}{\nu }_{\mathrm{coll}}$ (solid symbols) vs $Q$ at packing fraction $\varphi =0.78$ for strain rates $\dot{\gamma }{\tau }_e={10}^{-4}, {10}^{-5}$, and ${10}^{-6}$.

Figure 13

(a) Average contact number $\langle Z\rangle$ vs strain rate $\dot{\gamma }{\tau }_e$ at $\varphi =0.78$, for different values of $Q$. The dashed line has slope of unity, indicating a linear relation at large $Q$. (b) $\langle Z\rangle /\dot{\gamma }{\tau }_e$ vs $Q$ for $\dot{\gamma }{\tau }_e={10}^{-5}$ (open symbols) and ${10}^{-6}$ (solid symbols) at several different values of $\varphi$. Vertical arrows indicate the location of the crossover $Q^{*}$ at each different $\varphi$.

Figure 14

(a) Packing fraction $\varphi$ vs inertial number $I=\dot{\gamma }/\sqrt{p/m_0}$ for various values of $Q$. Straight lines are linear fits to the data. (b) Same as panel (a) but plotted as $\varphi -{\varphi }_J$ vs $I$ on a log-log scale, where ${\varphi }_J=0.84335$ is taken from Ref. [5]. Solid lines correspond to power-law relations ${\varphi }_J-\varphi \propto I^{a_{\mathrm{eff}}}$ with $a_{\mathrm{eff}}=1.5$ and 0.6 as shown. Symbols in panel (b) correspond to the legend in (a). Results are for a strain rate of $\dot{\gamma }{\tau }_e={10}^{-5}$.

Figure 15

Macroscopic friction $\mu =\sigma /p$ vs the inertial number $I=\dot{\gamma }/\sqrt{p/m_0}$ for various $Q$. Results are for a strain rate of $\dot{\gamma }{\tau }_e={10}^{-5}$. The dashed line is a fit to the form $\mu ={\mu }_J+c_{\mu }{I}^{b_{\mathrm{eff}}}$, with fixed ${\mu }_J=0.093$ from Ref. [5], and gives the value $b_{\mathrm{eff}}\approx 0.46$.

Figure 16

The relative anisotropy in pressure, $\delta p/p$, vs $Q$ for different values of packing fraction $\varphi$. The shear strain rate is $\dot{\gamma }{\tau }_e={10}^{-5}$ and the system has $N=1024$ particles. The value of $\varphi$ increases as the curves go from top to bottom.

Figure 17

The relative difference between deviatoric shear stress and off-diagonal stress, $({\sigma }_{\mathrm{dev}}-\sigma )/\sigma$, vs $Q$ for different values of packing fraction $\varphi$. The shear strain rate is $\dot{\gamma }{\tau }_e={10}^{-5}$ and the system has $N=1024$ particles. The value of $\varphi$ increases as the curves go from top to bottom.