Granular impact and the critical packing state

Impact dynamics during collisions of spheres with granular media reveal a pronounced and non-trivial dependence on volume fraction \phi. Post impact crater morphology identifies the critical packing state \phi_{cps}, where sheared grains neither dilate nor consolidate, and indicates an associated change in spatial response. Current phenomenological models fail to capture the observed impact force for most \phi; only near \phi_{cps} is force separable into additive terms linear in depth and quadratic in velocity. At fixed depth the quadratic drag coefficient decreases (increases) with depth for \phi < \phi_{cps} (\phi > \phi_{cps}). At fixed low velocity, depth dependence of force shows a Janssen-type exponential response with a length scale that decreases with increasing \phi; and is nearly constant for \phi > \phi_{cps}.