Pressure distribution and critical exponent in statically jammed and shear-driven frictionless disks

We numerically study the distributions of global pressure that are found in ensembles of statically jammed and quasistatically sheared systems of bidisperse, frictionless disks at fixed packing fraction $\varphi$ in two dimensions. We use these distributions to address the question of how pressure increases as $\varphi$ increases above the jamming point ${\varphi }_J$, $p\sim |\varphi -{\varphi }_J{|}^y$. For statically jammed ensembles, our results are consistent with the exponent $y$ being simply related to the power law of the interparticle soft-core interaction. For sheared systems, however, the value of $y$ is consistent with a nontrivial value, as found previously in rheological simulations.

Figure 1

Histograms of total system pressure $p$ for several different packing fractions $\varphi$ for the RAND ensemble with harmonic soft-core interactions. The number of particles is (a) $N=512$ and (b) $N=1024$. Solid lines represent fits to Eq. (16) holding $y=1$ fixed.

Figure 2

Histograms of total system pressure $p$ for several different packing fractions $\varphi$ for the RAND ensemble with Hertzian soft-core interactions. The number of particles is (a) $N=512$ and (b) $N=1024$. Solid lines represent fits to Eq. (16) holding $y=3/2$ fixed.

Figure 3

Histograms of total system pressure $p$ for several different packing fractions $\varphi$ for the QS ensemble (quasistatic shearing) with harmonic soft-core interactions. The number of particles is (a) $N=512$ and (b) $N=1024$. Solid lines represent fits to Eq. (16) holding $y=1.1$ fixed.

Figure 4

Values of (a) $(\varphi -\mu )K^{1/y}$ and (b) $w{K}^{1/y}$ vs $\varphi$, for systems with $N=512$ and 1024 particles, as obtained from fits of the RAND-harmonic data of Fig. 1 to Eq. (16), keeping $y=1$ fixed. The solid line in (a) is the best linear fit, while the solid line in (b) is the best fit to a constant.

Figure 5

Values of (a) $(\varphi -\mu )K^{1/y}$ and (b) $w{K}^{1/y}$ vs $\varphi$, for systems with $N=512$ and 1024 particles, as obtained from fits of the RAND-Hertzian data of Fig. 2 to Eq. (16), keeping $y=3/2$ fixed. The solid line in (a) is the best linear fit, while the solid line in (b) is the best fit to a constant.

Figure 6

Values of (a) $(\varphi -\mu )K^{1/y}$ and (b) $w{K}^{1/y}$ vs $\varphi$, for systems with $N=512$ and 1024 particles, as obtained from fits of the QS-harmonic data of Fig. 3 to Eq. (16), keeping $y=1.1$ fixed. The solid line in (a) is the best linear fit, while the solid line in (b) is the best fit to a constant.