Frustrated bearings

TL;DR

This paper investigates frustrated bearing systems in two and three dimensions, revealing a transition from sharp interfaces to fragmented structures with minimum dissipation, driven by the system's degrees of freedom.

Contribution

It introduces the first analysis of frustrated bearing states in three dimensions, showing the emergence of complex, non-sharp minimal dissipation configurations.

Findings

In 2D, a sharp minimum cut separates bearing states.

In 3D, intermediate, fragmented states are energetically favored.

Three-dimensional bearing states have four degrees of freedom.

Abstract

In a bearing state, touching spheres (disks in two dimensions) roll on each other without slip. Here we frustrate a system of touching spheres by imposing two different bearing states on opposite sides and search for the configurations of lowest energy dissipation. If the dissipation between contacts of spheres is viscous (with random damping constants), the angular momentum continuously changes from one bearing state to the other. For Coulomb friction (with random friction coefficients) in two dimensions, a sharp line separates the two bearing states and we show that this line corresponds to the minimum cut. Astonishingly however, in three dimensions, intermediate bearing domains, that are not synchronized with either side, are energetically more favorable than the minimum-cut surface. Instead of a sharp cut, the steady state displays a fragmented structure. This novel type of state of minimum dissipation is characterized by a spanning network of slipless contacts that reaches every sphere. Such a situation becomes possible because in three dimensions bearing states have four degrees of freedom.