Analysis of a Rigid Cylinder Rolling over a Linear Elastic Half-space in the Full-slip Regime

TL;DR

This paper develops an analytical and numerical solution for the deformation caused by a rigid cylinder rolling with full-slip over an elastic half-space, using Wiener-Hopf techniques and spectral methods.

Contribution

It introduces a novel analytical solution for the full-slip regime of a rolling cylinder on an elastic half-space, employing a matrix Wiener-Hopf approach and advanced numerical methods.

Findings

Solution successfully models the full-slip contact regime.

Numerical implementation demonstrates accurate contact region determination.

Method can be extended to similar contact mechanics problems.

Abstract

This paper provides an analytical solution for the deformation of an elastic half-space caused by a cylindrical roller. The roller is considered rigid, and is forced into the half space and rolls across its surface, with contact modelled by Coulomb friction. In general, portions of the surface of the roller in contact with the half space may slip across the surface of the half space, or may stick to it. In this paper, we consider only the regime where all of the rollers contact surface is slipping. This results in a mixed boundary value problem, which is formulated as a $2\times 2$ matrix Wiener-Hopf problem. The exponential factors in the Wiener-Hopf matrix allows a solution by following the iterative method of Priddin, Kisil, and Ayton (Phil. Trans. Roy. Soc. A 378, p. 20190241, 2020) which is implemented numerically by computing Cauchy transforms using a spectral method following Slevinsky and Olver (J. Comput. Phys. 332, pp. 290-315, 2017). The limits of the contact region are located a posteriori by applying an optimisation method. The solution is illustrated with several examples, and numerical code to compute the solutions in general is included in the supplementary material.