Granular Impact Model as an Energy-Depth Relation

TL;DR

This paper reformulates a nonlinear granular impact force law into a linear differential equation for kinetic energy versus depth, enabling closed-form solutions and improved data fitting, supported by new experimental results.

Contribution

It introduces a novel linear differential equation approach for granular impact models, facilitating analytical solutions and better experimental data analysis.

Findings

Derived closed-form solutions for velocity versus depth.

Demonstrated improved data fitting with the new model.

Presented new experimental results on drag force dependence.

Abstract

Velocity-squared drag forces are common in describing an object moving through a granular material. The resulting force law is a nonlinear differential equation, and closed-form solutions of the dynamics are typically obtained by making simplifying assumptions. Here, we consider a generalized version of such a force law which has been used in many studies of granular impact. We show that recasting the force law into an equation for the kinetic energy versus depth, K(z), yields a linear differential equation, and thus general closed-form solutions for the velocity versus depth. This approach also has several advantages in fitting such models to experimental data, which we demonstrate by applying it to data from 2D impact experiments. We also present new experimental results for this model, including shape and depth dependence of the velocity-squared drag force.