Shocks near Jamming

Nonlinear sound is an extreme phenomenon typically observed in solids after violent explosions. But granular media are different. Right when they jam, these fragile and disordered solids exhibit a vanishing rigidity and sound speed, so that even tiny mechanical perturbations form supersonic shocks. Here, we perform simulations in which two-dimensional jammed granular packings are dynamically compressed and demonstrate that the elementary excitations are strongly nonlinear shocks, rather than ordinary phonons. We capture the full dependence of the shock speed on pressure and impact intensity by a surprisingly simple analytical model.

Figure 1

Snapshots of the piston-compression simulation. A massive piston moves to the right at a constant velocity $u_P$, resulting in the formation of a compression front traveling at a speed $v_S$. Color indicates the local pressure at each grain. The average particle overlap is ${\delta }_0$ to the right of the front and ${\delta }_S>{\delta }_0$ to the left.

Figure 2

(a) Profiles of a shock wave at different times obtained by averaging the particle velocity in the $\widehat{y}$ direction (symbols). The lines shows the fits of the fronts to an empirical fit formula. (b) Oscillatory velocity profile of a shocklike wave propagating through a hexagonal array. The inset shows the in-phase motion of the crystalline planes that gives rise to the pressure oscillations. (c) Representative snapshots of the focusing and flattening of an initially curved front generated by a sinusoidal piston (time progresses from left to right).

Figure 3

(a) Speed of the front $v_S$ versus particle velocity $u_P$ measured in units of $v_g$, the sound speed within the grain, for decreasing particle overlap ${\delta }_0$. (b) Same plot as in (a) but with $v_S$ normalized by $v_S(0)$ and $u_P$ normalized by the crossover particle speed $u_P^{*}$: $v_S(0)$ and $u_P^{*}$ are indicated in (a). The dashed line indicates the power law $v_S\sim u_P^{1/5}$ characteristic of a sonic vacuum. The solid line indicates the theory developed here to describe the universal transition from weakly to strongly nonlinear waves in systems close to jamming. (c) Variation of $v_S(0)$ and $u_P^{*}$ with distance to the jamming transition parametrized by the initial average overlap ${\delta }_0$. The dashed line indicates the power law $v_S(0)\sim {\delta }_0^{1/4}$ and the continuous line the power law $u_P^{*}\sim {\delta }_0^{5/4}$. (d) Variation of the kinetic energy with potential energy in dimensionless units—same color code as in (a),(b). The dashed line indicates the linear relationship observed for strong shocks.

Figure 4

(a) Simulations of an ordered chain with small viscosity display large coherent oscillations in the front profile (black line) [26]. If the viscosity is large enough, one obtains a homogeneous shock profile, shown as a light gray (green) line, similar to the profile in Fig. 2a. (b) The presence of an effective viscosity will induce the oscillation of the particle (represented by thin black lines) towards the bottom of the potential $W(\delta )$. If the viscosity is large enough, the particle can move directly to the minimum without performing any oscillations; see the light gray (green) trajectory corresponding to the homogeneous shock profile of (a).