Jamming Transition as Probed by Quasistatic Shear Flow

We study the rheology of amorphous packings of soft, frictionless particles close to jamming. Implementing a quasistatic simulation method we generate a well-defined ensemble of states that directly samples the system at its yield stress. A continuous jamming transition from a freely flowing state to a yield-stress situation takes place at a well-defined packing fraction, where the scaling laws characteristic of isostatic solids are observed. We propose that long-range correlations observed below the transition are dominated by this isostatic point, while those that are observed above the transition are characteristic of dense, disordered elastic media.

Figure 1

Trajectory taken by quasistatic simulations in state space of shear stress $\sigma$ and volume fraction $\varphi$. (Dotted lines) contours of constant strain rate $\dot{\gamma }$. The simulation corresponds to $\dot{\gamma }\to 0$ and thus follows the yield-stress line, ${\sigma }_y(\varphi )$. Previous approaches either probe linear elasticity of the solid (decompression at $\sigma =0$) or the steady-state flow of the fluid.

Figure 2

Stress-strain relation for a sample of 2500 particles at two different volume fractions, $\varphi =0.8470$, 0.8433.

Figure 3

Number of jamming events (defined as states where stress first exceeds a threshold value, ${\sigma }_{\mathrm{thresh}}=2\times {10}^{-7}$ [20]) as function of volume fraction $\varphi$ and system size $N$. Inset: Finite-size scaling indicates a vanishing width and a diverging height as $N\to \infty$.

Figure 4

Contact number $z$ as function of pressure (left inset) and volume-fraction $\varphi$ (main panel). Symbols are as in Fig. 3. Right inset: scaling of gap width, $\delta z_{\mathrm{gap}}\equiv 3.8-z$, with system size for $\varphi =0.844$, 0.843.

Figure 5

Nonaffine displacements $\Delta$ ($N=900$) taken from: average over all configurations (red plus), restricted average over flowing states (circles), jammed states (square), and jammed states without plastic events (triangles). (Small closed symbols) flowing or jammed states at $N=1600$.

Figure 6

Correlation function $G(y)$ ($N=2500$) for various volume fractions above and below ${\varphi }_c$. For $\varphi ⪅{\varphi }_c$ correlations do not decay to zero, probably due to the periodicity of the simulation box. Inset: Length scale $\xi$ as defined by the position of the minimum (for two different system sizes). Dashed lines are $\sim \delta {\varphi }^{-0.8}$ and $\sim \delta {\varphi }^{-1}$ (${\varphi }_c=0.8433$).