Unified force law for granular impact cratering

Abstract

Experiments on the low-speed impact of solid objects into granular media have been used both to mimic geophysical events and to probe the unusual nature of the granular state of matter. Observations have been interpreted in terms of conflicting stopping forces: product of powers of projectile depth and speed; linear in speed; constant, proportional to the initial impact speed; and proportional to depth. This is reminiscent of high-speed ballistics impact in the nineteenth and twentieth centuries, when a plethora of empirical rules were proposed. To make progress, we developed a means to measure projectile dynamics with 100 nm and 20 s precision. For a 1-inch-diameter steel sphere dropped from a wide range of heights into non-cohesive glass beads, we reproduce previous observations either as reasonable approximations or as limiting behaviours. Furthermore, we demonstrate that the interaction between the projectile and the medium can be decomposed into the sum of velocity-dependent inertial drag plus depth-dependent friction. Thus, we achieve a unified description of low-speed impact phenomena and show that the complex response of granular materials to impact, although fundamentally different from that of liquids and solids, can be simply understood. ©2007 Nature Publishing Group.