Critical scaling near jamming transition for frictional granular particles

The critical rheology of sheared frictional granular materials near jamming transition is numerically investigated. It is confirmed that there exists a true critical density which characterizes the onset of the yield stress and two fictitious critical densities which characterize the scaling laws of rheological properties. We find the existence of a hysteresis loop between two of the critical densities for each friction coefficient. It is noteworthy that the critical scaling law for frictionless jamming transition seems to be still valid even for frictional jamming despite using fictitious critical density values.

Figure 1

Snapshot of the scaled peculiar momentum field ${\mathit{p}}_i/\sqrt{\mathit{mT}}$ for the decreasing process (a) and the increasing process (b) for $\mu =2.0$, $\varphi =0.795$, $N=8000$, $a={10}^{0.1}$, and $\mathrm{\gamma ̇}=7.9\times {10}^{-6}\sqrt{k^{(n)}/m}$.

Figure 2

(a) Scaled shear stress $S^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ as a function of scaled shear rate ${\mathrm{\gamma ̇}}^{*}=\mathrm{\gamma ̇}\sqrt{k^{(n)}/m}$ for various values of packing fraction $\varphi$ for $\mu =2.0$, $N=8000$, and $a={10}^{0.1}$. (b) Scaled pressure $P^{*}=P/(k^{(n)}{\sigma }_0^{-1})$ as a function of ${\mathrm{\gamma ̇}}^{*}$ for various values of packing fraction $\varphi$ for $\mu =2.0$, $N=8000$, and $a={10}^{0.1}$.

Figure 3

Scaled shear stress $S^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ as a function of scaled shear rate ${\mathrm{\gamma ̇}}^{*}=\mathrm{\gamma ̇}\sqrt{k^{(n)}/m}$ for $\varphi =0.795$, $\mu =1.8$, and $a={10}^{0.1}$, with $N=8000$ and $16\hspace{0.5em}000$.

Figure 4

Scaled shear stress $S^{*}=S/(k^{(n)}{\sigma }_0)$ as a function of scaled shear rate ${\mathrm{\gamma ̇}}^{*}=\mathrm{\gamma ̇}\sqrt{k^{(n)}/m}$ for $\varphi =0.795$, $\mu =1.8$, and $N=8000$ with the shear rate step $a={10}^{0.1}$ characterized by the solid squares and $a={10}^{0.05}$ characterized by the open squares.

Figure 5

Scaled shear stress $S^{*}=S/(k^{(n)}{\sigma }_0)$ as a function of scaled shear rate ${\mathrm{\gamma ̇}}^{*}=\mathrm{\gamma ̇}\sqrt{k^{(n)}/m}$ for $\varphi =0.795$, $\mu =1.8$, $a={10}^{0.1}$, and $N=8000$ with the waiting time ${\tau }_{\mathrm{w}}=0.4{\mathrm{\gamma ̇}}^{-1}$ characterized by the open squares, $4.0{\mathrm{\gamma ̇}}^{-1}$ characterized by the solid squares, and $8.0{\mathrm{\gamma ̇}}^{-1}$ characterized by the open circles.

Figure 6

Jammed fraction $f$ as a function of $\varphi$ for $N=4000$ with $\mu =0.0$, 0.2, and $2.0$.

Figure 7

(a) The rescaled pressure $P^{*}$ with $P^{*}=P/(k^{(n)}{\sigma }_0^{-1})$ as a function of $\varphi$ for $N=8000$ by using the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-5}$, which we call “SL1”, the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-6}$, which we call “SL2”, and the QS and SC methods for $\mu =0.0$. (b) The rescaled pressure $P^{*}$ with $P^{*}=P/(k^{(n)}{\sigma }_0^{-1})$ as a function of $\varphi$ for $N=8000$ by using the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-5}$, which we call “SL1”, the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-6}$, which we call “SL2”, and the QS and SC methods for $\mu =2.0$. (c) The rescaled stress $S^{*}$ with $S^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ as a function of $\varphi$ for $N=8000$ by using the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-5}$, which we call “SL1”, the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-6}$, which we call “SL2”, and the SC methods for $\mu =0.0$. Note that we eliminate the data by SC. (d) The rescaled stress $S^{*}$ with $S^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ as a function of $\varphi$ for $N=8000$ by using the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-5}$, which we call “SL1”, the SL method with ${\mathrm{\gamma ̇}}^{*}=5\times {10}^{-6}$, which we call “SL2”, and the SC methods for $\mu =2.0$.

Figure 8

(a) The rescaled pressure $P^{*}$ with $P^{*}=P/(k^{(n)}{\sigma }_0^{-1})$ as a function of $\varphi -{\varphi }_S(\mu )$ for $N=8000$ with $\mathrm{\gamma ̇}=5.0\times {10}^{-6}$. (Inset) The rescaled pressure $P^{*}$ near ${\varphi }_S(\mu )-\varphi =0$ for $\mu =0.2$, 0.4, and 0.8. (b) $S^{*}/A$ with $S^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ for $\mathrm{\gamma ̇}=5.0\times {10}^{-6}\sqrt{k^{(n)}/m}$ and $N=8000$ as a function of $\varphi -{\varphi }_S(\mu )$ in the solid branch, where $A$ is a constant that depends only on $\mu$, that is, $A=1.0$, 0.4, 0.3, 0.3, and $0.3$ for $\mu =0.0$, 0.2, 0.4, 0.8, and $2.0$, respectively. (Inset) $S^{*}/A$ near ${\varphi }_S(\mu )-\varphi =0$ for $\mu =0.2$, 0.4, and 0.8.

Figure 9

The critical fractions ${\varphi }_C(\mu )$, ${\varphi }_S(\mu )$, and ${\varphi }_L(\mu )$ as a function of $\mu$. The amount $\alpha$ of the hysteresis-dependence in $\mu -\varphi$ plane is plotted with a gray scale. The region of the hysteresis loop, which is defined by the region where $\alpha >0.1$, is enclosed by thin solid lines.

Figure 10

Jammed fraction $f$ as a function of $\varphi$ for $\mu =2.0$ with $N=1000$, 2000, and $4000$ obtained from the simulation using the QS method with $\Delta \gamma ={10}^{-6}$ and $E_{\mathrm{th}}={10}^{-7}{k}^{(n)}{\sigma }_0^2$.

Figure 11

$Z-Z_c(\mu )$ for $\mathrm{\gamma ̇}=5.0\times {10}^{-6}\sqrt{k^{(n)}/m}$ as a function of $\varphi -{\varphi }_S(\mu )$ in the solid branch. Inset shows $Z_c(\mu )$ as a function of the friction coefficient $\mu$.

Figure 12

(a) $S^{*}/(B{\mathrm{\gamma ̇}}^{*2})$ with $S^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ and ${\mathrm{\gamma ̇}}^{*}=\mathrm{\gamma ̇}\sqrt{k^{(n)}/m}$ for ${\mathrm{\gamma ̇}}^{*}=2.0\times {10}^{-7}$ and $N=30\hspace{0.5em}000$ as a function of ${\varphi }_L(\mu )-\varphi$ in the liquid branch, where $B$ is a constant that depends only on $\mu$, that is, $B=1.0$, 0.5, 0.25, and $0.25$, for $\mu =0.0$, 0.4, 1.2, and $2.0$, respectively. (b) $P^{*}/(B{\mathrm{\gamma ̇}}^{*2})$ with $P^{*}=S/(k^{(n)}{\sigma }_0^{-1})$ and ${\mathrm{\gamma ̇}}^{*}=\mathrm{\gamma ̇}\sqrt{k^{(n)}/m}$ for ${\mathrm{\gamma ̇}}^{*}=2.0\times {10}^{-7}$ and $N=30\hspace{0.5em}000$ as a function of ${\varphi }_L(\mu )-\varphi$ in the liquid branch, where $B$ is a constant that depends only on $\mu$, that is, $B=1.0$, 0.9, 0.25, and $0.25$, for $\mu =0.0$, 0.4, 1.2, and $2.0$, respectively.

Figure 13

$S/P$ in the solid branch as a function of $\varphi -{\varphi }_S$ for $\mathrm{\gamma ̇}=5.0\times {10}^{-6}\sqrt{k^{(n)}/m}$ and $N=8000$.

Figure 14

$S/P$ in the liquid branch as a function of ${\varphi }_L(\mu )-\varphi$ for $\mathrm{\gamma ̇}=2.0\times {10}^{-7}\sqrt{k^{(n)}/m}$ and $N=30\hspace{0.5em}000$.

Figure 15

$S/P$ in the solid branch as a function of $\mu$ for $\varphi =0.850$, $\mathrm{\gamma ̇}=5.0\times {10}^{-6}\sqrt{k^{(n)}/m}$, and $N=8000$.