Model for the resistance force acting on circular bodies in the imminence of rolling

TL;DR

This paper develops a mechanical model combining elasticity and Hertz contact mechanics to quantify the resistance force on circular bodies at the verge of rolling, filling a gap in understanding static friction for such shapes.

Contribution

It introduces a parameter-free model that predicts the critical force for initiating rolling based on physical properties, extending friction theory for circular bodies.

Findings

Provides a quantitative expression for the critical resistance force.

Confirms proportionality of resistance force with load.

Model depends only on measurable physical properties.

Abstract

The laws of friction are reasonably well understood for the case of blocks in contact with rough plane surfaces. However, as far as bodies with circular sections are concerned, the physics of friction becomes more involving and it is not possible to adopt a simple conceptual framework to explain all phenomena. In particular, there is no approach so far to the problem of the resistance force that opposes to circular bodies that remain at rest while acted upon by small forces. Here we fill this gap by introducing a mechanical model based on both the elasticity theory and Hertz contact mechanics. Our approach furnishes a quantitative expression for the critical force beyond which rest can no longer be maintained. Besides confirming the expected proportionality of the resistance force with the load, our result contains no free parameters and is expressed solely in terms of physical properties of the problem, such as the pressure of the body per unit of superficial area, a relation between the Young modulus of the surface and its Poisson ratio, and the symmetry of the contact.