Granular Impact at High Mach Number

How do dynamic stresses propagate in granular material after a high-speed impact? This occurs often in natural and industrial processes. Stress propagation in a granular material is controlled by the inter-particle force law, $f$, in terms of particle deformation, $\delta$, often given by $f\propto\delta^{\alpha}$, with $\alpha>1$. This means that a linear wave description is invalid when dynamic stresses are large compared to the original confining pressure. With high-speed video and photoelastic grains with varying stiffness, we experimentally study how forces propagate following an impact and explain the results in terms of the nonlinear force law (we measure $\alpha\approx 1.4$). The spatial structure of the forces and the propagation speed, $v_f$, depend on a dimensionless parameter, $M'=t_cv_0/d$, where $v_0$ is the intruder speed at impact, $d$ is the grain diameter, and $t_c$ is a binary collision time between grains with relative speed $v_0$. For $M'\ll 1$, propagating forces are chain-like, and the measured $v_f \propto d/t_c\propto v_b(v_0/v_b)^\frac{\alpha-1}{\alpha+1}$, where $v_b$ is the bulk sound speed. For larger $M'$, the force response has a 2D character, and forces propagate faster than predicted by $d/t_c$ due to collective stiffening of a packing.