Critical behaviors of sheared frictionless granular materials near the jamming transition

Critical behaviors of sheared dense and frictionless granular materials in the vicinity of the jamming transition are numerically investigated. From the extensive molecular dynamics simulation, we verify the validity of the scaling theory near the jamming transition proposed by Otsuki and Hayakawa [Prog. Theor. Phys. 121, 647 (2009)]]. We also clarify the critical behaviors of the shear viscosity and the pair correlation function based on both a mean field theory and the simulation.

Figure 1

(Color online) Collapsed data of the shear rate dependence of the kinetic temperature $T$ in the polydisperse system $\left(N=2000\right)$ with $\Delta =3/2$ using the scaling law, Eq. (4), for $D=2$, 3, and 4. The dashed line, the dotted line and the solid line are proportional to $\dot{\gamma }$, ${\dot{\gamma }}^2$, and ${\dot{\gamma }}^{x_{\gamma }}$, respectively. The legends show the dimension $D$ and the volume fraction $\varphi$ as $\left(D,\varphi \right)$. The critical exponents estimated from Eq. (15) are $x_{\varphi }=9/2$ and $x_{\gamma }=14/11$.

Figure 2

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ for the polydisperse system $\left(N=2000\right)$ with $\Delta =3/2$ using the scaling law, Eq. (5), for $D=2$, 3, and 4. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the dimension $D$ and the volume fraction $\varphi$ as $\left(D,\varphi \right)$. The critical exponents estimated from Eq. (15) are $y_{\varphi }=3/2$ and $y_{\gamma }=6/11$.

Figure 3

(Color online) Collapsed data of the shear rate dependence of the pressure $P$ for the polydisperse system $\left(N=2000\right)$ with $\Delta =3/2$ using the scaling law, Eq. (6), for $D=2$, 3, and 4. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }^{\prime }}$, respectively. The legends show the dimension $D$ and the volume fraction $\varphi$ as $\left(D,\varphi \right)$. The critical exponents estimated from Eq. (15) are $y_{\varphi }^{\prime }=3/2$ and $y_{\gamma }^{\prime }=6/11$.

Figure 4

(Color online) Collapsed data of the shear rate dependence of the characteristic frequency $\omega$ for the polydisperse system $\left(N=2000\right)$ with $\Delta =3/2$ using the scaling law, Eq. (7), for $D=2$, 3, and 4. The dotted line and the solid line are proportional to $\dot{\gamma }$ and ${\dot{\gamma }}^{z_{\gamma }}$, respectively. The legends show the dimension $D$ and the volume fraction $\varphi$ as $\left(D,\varphi \right)$. The critical exponents estimated from Eq. (15) are $z_{\varphi }=3/4$ and $z_{\gamma }=3/11$.

Figure 5

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ for the bidisperse system $\left(N=4000\right)$ with $\Delta =1$ using the scaling law, Eq. (5), for $D=3$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends represent the volume fraction $\varphi$. The critical exponents estimated from Eq. (15) are $y_{\varphi }=1$ and $y_{\gamma }=2/5$.

Figure 6

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ for the monodisperse system $\left(N=4000\right)$ with $\Delta =1$ using the scaling law, Eq. (5), for $D=3$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the volume fraction $\varphi$. The critical exponents estimated from Eq. (15) are $y_{\varphi }=1$ and $y_{\gamma }=2/5$.

Figure 7

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ in the larger monodisperse system $\left(N=20000\right)$ with $\Delta =1$ using the scaling law, Eq. (5), for $D=3$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the volume fraction $\varphi$. The critical exponents estimated from Eq. (15) are $y_{\varphi }=1$ and $y_{\gamma }=2/5$.

Figure 8

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ in the monodisperse system $\left(N=2000\right)$ with the contact force given by Eqs. (16, 17) using the scaling law, Eq. (5) for $D=3$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the volume fraction $\varphi$. The critical exponents estimated from Eq. (15) with $\Delta =1$ are $y_{\varphi }=1$ and $y_{\gamma }=2/5$.

Figure 9

Plot of $y_{\gamma }$ versus $\Delta$ for the polydisperse system $\left(N=8000\right)$ with $D=2$.

Figure 10

Plot of $y_{\gamma }^{\prime }$ versus $\Delta$ for the polydisperse system $\left(N=8000\right)$ with $D=2$.

Figure 11

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ for the polydisperse system $\left(N=8000\right)$ with $\Delta =0.5$ using the scaling law, Eq. (5), for $D=2$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the volume fraction $\varphi$. The critical exponents are estimated from Eq. (15) as $y_{\varphi }=1/2$ and $y_{\gamma }=2/9$.

Figure 12

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ for the polydisperse system $\left(N=8000\right)$ with $\Delta =2.0$ using the scaling law, Eq. (5), for $D=2$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the volume fraction $\varphi$. The critical exponents are estimated from Eq. (15) as $y_{\varphi }=2$ and $y_{\gamma }=2/3$.

Figure 13

Plots of $\omega$ versus $\Phi$ of the polydisperse systems for $D=2$ and $\Delta =3/2$ with $\dot{\gamma }=5\times {10}^{-6}$, $5\times {10}^{-7}$, and $5\times {10}^{-8}$.

Figure 14

Plots of $\mu /\dot{\gamma }$ versus $\Phi$ of the two-dimensional polydisperse systems for and $\Delta =1$ and 3/2 with $\dot{\gamma }=5\times {10}^{-5}$, $5\times {10}^{-6}$, and $5\times {10}^{-7}$.

Figure 15

$\mu /\sqrt{T}$ as a function of $\varphi$ for the three-dimensional bidisperse system $\left(N=4000\right)$ with $\Delta =1$ and $\dot{\gamma }=5\times {10}^{-7}$, where the solid line and the dashed line are proportional to ${\left({\varphi }_{\mu }-\varphi \right)}^{-1}$ and ${\left({\varphi }_J-\varphi \right)}^{-3}$, respectively.

Figure 16

The snapshot of the two-dimensional monodisperse system $\left(N=2401\right)$ with $\Delta =1$, $k=1.0$, and $\eta =0.00225$. We use Eq. (2) for the viscous contact force. The shear rate is $\dot{\gamma }=5\times {10}^{-5}$, and the density is $\varphi =0.80$.

Figure 17

(Color online) Collapsed data of the shear rate dependence of the shear stress $S$ in the nearly elastic monodisperse system $\left(N=2000\right)$ with $\Delta =1$ using the scaling law, Eq. (5), for $D=3$. The dotted line and the solid line are proportional to ${\dot{\gamma }}^2$ and ${\dot{\gamma }}^{y_{\gamma }}$, respectively. The legends show the volume fraction $\varphi$. The critical exponents estimated from Eq. (15) are $y_{\varphi }=1$ and $y_{\gamma }=2/5$.

Figure 18

Structure factor $S\left(k\right)$ in the three-dimensional monodisperse system $\left(N=20000\right)$ with $\varphi =0.629,0.639,0.649$ and $\dot{\gamma }=5\times {10}^{-4}$ and $5\times {10}^{-5}$.

Figure 19

Pair correlation function $g\left(r\right)$ in the three-dimensional monodisperse system $\left(N=20000\right)$ with $\varphi =0.629,0.639,0.649$ and $\dot{\gamma }=5\times {10}^{-4}$ and $5\times {10}^{-5}$.

Figure 20

The first peak of pair correlation function $g\left(r\right)$ in the three-dimensional monodisperse system $\left(N=20000\right)$ with $\Delta =1$ and $\varphi =0.639$ for various shear rates $\dot{\gamma }$.

Figure 21

The height of the first peak $g_0$ of $g\left(r\right)$ as a function of $\dot{\gamma }$ for the three-dimensional monodisperse systems $\left(N=20000\right)$ with $\Delta =1$ for various densities $\varphi$. The critical density is ${\varphi }_J=0.639$. The solid line is proportional to ${\dot{\gamma }}^{-2/\left(\Delta +4\right)}$. The dashed line is proportional to ${\dot{\gamma }}^{-2/\Delta }$.

Figure 22

The plots of $g_0$ versus $\left|\Phi \right|$ in the jammed phase for the three-dimensional monodisperse system $\left(N=20000\right)$ with $\Delta =1$ for various shear rates $\dot{\gamma }$. The dashed guide line is proportional to ${\left|\Phi \right|}^{-1}$.

Figure 23

The plots of $g_0{\dot{\gamma }}^{2/\Delta }$ versus $\left|\Phi \right|$ in the unjammed phase for the monodisperse system $\left(N=20000\right)$ with $\Delta =1$ for various shear rates $\dot{\gamma }$. The dashed guide line is proportional to ${\left|\Phi \right|}^{4/\Delta }$.

Figure 24

(Color online) The scaling of the shear stress $S/{\left|\Phi \right|}^{y_{\varphi }}$ for the polydisperse system $\left(N=4000\right)$ with $\Delta =1$ using Eq. (33) as a function of the scaled shear rate $\dot{\gamma }/{\left|\Phi \right|}^{y_{\varphi }/y_{\gamma }}$ for ${10}^{-4}\le \dot{\gamma }\le {10}^{-1}$, $0.5\le \varphi \le 0.7$ with $D=3$.

Figure 25

(Color online) The scaling of the shear stress $S/{\left|\Phi \right|}^{y_{\varphi }}$ for the polydisperse system $\left(N=4000\right)$ with $\Delta =1$ using Eq. (33) as a function of the scaled shear rate $\dot{\gamma }/{\left|\Phi \right|}^{y_{\varphi }/y_{\gamma }}$ for $5\times {10}^{-7}\le \dot{\gamma }\le 5\times {10}^{-5}$, $0.624\le \varphi \le 0.652$ with $D=3$.

Figure 26

(Color online) The scaling of the shear stress $S/{\left|\Phi \right|}^{y_{\varphi }}$ for the polydisperse system $\left(N=4000\right)$ with $\Delta =1$ using Eq. (15) as a function of the scaled shear rate $\dot{\gamma }/{\left|\Phi \right|}^{y_{\varphi }/y_{\gamma }}$ for $5\times {10}^{-7}\le \dot{\gamma }\le 5\times {10}^{-5}$, $0.624\le \varphi \le 0.652$ with $D=3$.