Exact Solution for Depth of Impact Crater into Granular Bed

Abstract

When a sphere impacts a granular bed such as a sand bed, the penetration depth is often determined by empirical scaling laws or numerical simulations. This paper uses a unified force law for granular impact cratering to obtain a simple, explicit, and exact solution for the maximum penetration depth in terms of the Lambert function. The exact solution is confirmed with published laboratory data and numerical simulations; thus it can be used to determine the drag coefficients accurately.