A Ball in a Groove

TL;DR

This paper analyzes the static equilibrium states of an elastic sphere in a groove under gravity and friction, identifying two regimes based on the groove's geometry and tilt, with implications for granular media mechanics.

Contribution

It introduces a contact mechanics model that distinguishes elastic and plastic regimes of a sphere in a groove, linking to granular media behavior.

Findings

Two regimes of equilibrium identified: elastic and plastic.

Elastic regime involves rolling at both contacts; plastic involves rolling and sliding.

Stress indeterminacy is resolved by the ball's response kinetics.

Abstract

We study the static equilibrium of an elastic sphere held in a rigid groove by gravity and frictional contacts, as determined by contact mechanics. As a function of the opening angle of the groove and the tilt of the groove with respect to the vertical, we identify two regimes of static equilibrium for the ball. In the first of these, at large opening angle or low tilt, the ball rolls at both contacts as it is loaded. This is an analog of the "elastic" regime in the mechanics of granular media. At smaller opening angles or larger tilts, the ball rolls at one contact and slides at the other as it is loaded, analogously with the "plastic" regime in the mechanics of granular media. In the elastic regime, the stress indeterminacy is resolved by the underlying kinetics of the ball response to loading.