Dynamics of inelastically colliding spheres with Coulomb friction: Relaxation of translational and rotational energy

TL;DR

This paper studies how inelastic rough spheres with Coulomb friction cool down over time, revealing different long-term energy states depending on collision parameters and introducing effects of surface roughness and cohesion.

Contribution

It provides a detailed analysis of the asymptotic energy states of inelastic rough spheres with Coulomb friction, highlighting the effects of collision parameters and additional forces.

Findings

For high normal restitution and large friction, rotational and translational energies decay proportionally.

For low restitution and friction, rotational energy remains constant while translational energy decays.

Surface roughness and cohesion introduce intermediate regimes with constant rotational energy.

Abstract

We investigate the free cooling of inelastic rough spheres in the presence of Coulomb friction. Depending on the coefficients of normal restitution $\epsilon$ and Coulomb friction $\mu$, we find qualitatively different asymptotic states. For nearly complete normal restitution ($\epsilon$ close to 1) and large $\mu$, friction does not change the cooling properties qualitatively compared to a constant coefficient of tangential restitution. In particular, the asymptotic state is characterized by a constant ratio of rotational and translational energies, both decaying according to Haff's law. However, for small $\epsilon$ and small $\mu$, the dissipation of rotational energy is suppressed, so that the asymptotic state is characterized by constant rotational energy while the translational energy continues to decay as predicted by Haff's law. Introducing either surface roughness for grazing collisions or cohesion forces for collisions with vanishing normal load, causes the rotational energy to decay according to Haff's law again in the asymptotic long-time limit with, however, an intermediate regime of approximately constant rotational energy.