Uniform shock waves in disordered granular matter

The confining pressure $P$ is perhaps the most important parameter controlling the properties of granular matter. Strongly compressed granular media are, in many respects, simple solids in which elastic perturbations travel as ordinary phonons. However, the speed of sound in granular aggregates continuously decreases as the confining pressure decreases, completely vanishing at the jamming-unjamming transition. This anomalous behavior suggests that the transport of energy at low pressures should not be dominated by phonons. In this work we use simulations and theory to show how the response of granular systems becomes increasingly nonlinear as pressure decreases. In the low-pressure regime the elastic energy is found to be mainly transported through nonlinear waves and shocks. We numerically characterize the propagation speed, shape, and stability of these shocks and model the dependence of the shock speed on pressure and impact intensity by a simple analytical approach.

Figure 1

Piston compression experiments. A continuous compression of the system by a piston moving at velocity $u_P$ leads to the formation of a pressure front (shock wave) traveling at a velocity $v_S$. The shock compression process can be seen as a fast transition between two states (initial and shocked) of the granular system.

Figure 2

Phenomenology of shock waves. Propagation of fronts as seen through the profiles of the (a) averaged pressure $P$ in units of the individual grain's Young modulus $E^{*}$ and (b) averaged particle velocity $u$ normalized with the piston velocity $u_P$. The front moves in the direction of the arrows. Different curves corresponds to profiles at different times and the (red) lines are phenomenological fits. (c) Position of the front as a function of time. Note that at long times the front moves linearly with time (red line), with a propagation speed $v_S$.

Figure 3

Shock waves in a granular crystal. (a) Cartoons of the pressure oscillations as a consequence of the in-phase motion of crystalline planes. Dark grays (shades of red) indicate regions of higher pressure. (b) Propagation of the shock wave as seen through the particle velocity profiles. The front travels from left to right. (c) Shock profiles in the presence of viscosity, for viscosity coefficients above (thick black line) and below (thin gray line) a critical value.

Figure 4

Propagation speed. (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 ${\delta }_0$. The inset shows the variation of the sound speed $c_0$ with the initial overlap ${\delta }_0$ (data) and the scaling $c_0\sim {\delta }_0^{1/4}$ (line). (b) Normalized plot obtained by rescaling $v_S$ with $c_0$ and $u_P$ with $u_P^{☆}$. The dashed line indicates the power law $v_S\sim u_P^{1/5}$ characteristic of a sonic vacuum. The inset shows the variation of $u_P^{☆}$ with the initial overlap ${\delta }_0$ (data) and the scaling $u_P^{☆}\sim {\delta }_0^{5/4}$ (line).

Figure 5

Typical shock profiles for (a) weak shock and (b) strong shock. The continuous lines represent phenomenological fits.

Figure 6

Shape characterization of the shock profiles. (a) Scheme of the definition of shock's width and asymmetry. Variation of (b) shock width and (c) asymmetry with the Mach number for the different studied shocks. The symbols are the same as in Fig. 4.

Figure 7

(a) Schematic of a general shock Hugoniot curve (shock adiabat). (b) Numerically obtained shock Hugoniot for systems with different initial pressures as obtained from the jump conditions. Dashed lines are guides to the eyes. The symbols are the same as in Fig. 4.

Figure 8

Dyakov's stability analysis. Note that all the shock waves belong to the stable region of the phase diagram. The symbols are the same as in Fig. 4.

Figure 9

(a) Cartoon of the roughness in the pressure front. (b) Evolution of the front's roughness as a function of time for different Mach numbers. Note that when the nonlinearity of the wave is large, the roughness reaches a plateau at a small value.

Figure 10

Effective potential surface. For $v_S>c$ the initial state becomes a local maximum allowing the description of traveling perturbations.

Figure 11

Steady-state solutions of the nonlinear wave equation. (a) Potential well for a given propagation speed $v_S$. (b) Solitary wave solutions obtained in the absence of viscosity. (c) Shock wave solutions obtained for viscous dissipation.