Effect of Mach number on granular impacts

When an object strikes a granular material, its momentum and energy are transferred to the grains and dissipated. An important dimensionless parameter in such impacts is $M$, the ratio of the intruder speed, $v_0$, to a typical granular sound speed, $c$. In many previous studies, $M$ has been very small, $M\sim 10^{-2}$. In this regime, the granular force on the intruder is dominated by a $v^2$ drag term, leading to a smooth, monotonic deceleration of the intruder. To probe the regime closer to $M\sim 1$, we perform experiments (and matching simulations) with granular materials comprised of photoelastic disks of varying stiffness, where softer particles allow us to reduce the granular sound speed. As we increase $M$, we reach a regime for which the intruder dynamics are no longer described by $v^2$ drag, but rather show a shock-like front which behaves elastically in response to the impact. Surprisingly, for the higher $M$ impacts ($M\sim 10^{-1}$), penetration depth is greatly reduced compared to the smaller $M$ impacts ($M\sim 10^{-2}$), and the intruder typically rebounds temporarily, before coming to rest. We understand the transition from $v^2$ drag to damped elastic behavior in terms of grain-grain collision time compared to the time for the intruder to move one grain size.