Unified force law for granular impact cratering

TL;DR

This paper presents a unified model for granular impact cratering, demonstrating that the complex impact response can be understood as a combination of inertial drag and depth-dependent friction, supported by precise experiments.

Contribution

The authors develop a precise measurement method and show that granular impact dynamics can be described by a simple sum of inertial and frictional forces, unifying previous conflicting observations.

Findings

Impact force decomposes into inertial drag and frictional components.

Experimental data supports the unified force law across different impact conditions.

Granular impact behavior is fundamentally different from liquids and solids but can be simplified.

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 19th and 20th 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 us precision. For a 1-inch diameter steel sphere dropped from a wide range of heights into non-cohesive glass beads, we reproduce prior observations either as reasonable approximations or as limiting behaviours. Furthermore, we demonstrate that the interaction between projectile and 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, while fundamentally different from that of liquids and solids, can be simply understood.