# The Maximum Dissipation Principle in Rigid-Body Dynamics with Purely Inelastic Impacts

## TL;DR

This paper introduces a new impact model based on the maximum dissipation principle for rigid-body dynamics with inelastic impacts, resolving non-uniqueness issues present in Coulomb friction-based models.

## Contribution

It develops an alternative impact model that maximizes dissipation, providing unique solutions for single-contact impacts and integrating it into a time-stepping scheme.

## Key findings

- The maximum dissipation impact model resolves non-uniqueness in impact solutions.
- Analytic solutions are derived for single-contact impact problems.
- The model's macroscopic behavior is comparable to Coulomb friction in granular flow simulations.

## Abstract

Formulating a consistent theory for rigid-body dynamics with impacts is an intricate problem. Twenty years ago Stewart published the first consistent theory with purely inelastic impacts and an impulsive friction model analogous to Coulomb friction. In this paper we demonstrate that the consistent impact model can exhibit multiple solutions with a varying degree of dissipation even in the single-contact case. Replacing the impulsive friction model based on Coulomb friction by a model based on the maximum dissipation principle resolves the non-uniqueness in the single-contact impact problem. The paper constructs the alternative impact model and presents integral equations describing rigid-body dynamics with a non-impulsive and non-compliant contact model and an associated purely inelastic impact model maximizing dissipation. An analytic solution is derived for the single-contact impact problem. The models are then embedded into a time-stepping scheme. The macroscopic behaviour is compared to Coulomb friction in a large-scale granular flow problem.