Persson's Theory of Purely Normal Elastic Rough Surface Contact: A Tutorial Based on Stochastic Process Theory

TL;DR

This paper provides a detailed tutorial on Persson's theory of rough surface contact, offering a new derivation based on stochastic process theory to clarify its fundamental assumptions and improve understanding.

Contribution

It introduces a novel derivation of Persson's theory using stochastic process theory, clarifying its assumptions and providing a self-contained explanation.

Findings

New derivation based on Markov process assumption

Clarification of diffusion equation and boundary conditions

Enhanced understanding for non-experts

Abstract

Persson's theory of purely normal rough surface contact was developed two decades ago during the study of tire-road interaction, and gradually became one of the dominant approaches to study the solid-solid interaction between rough surfaces. Contrary to its popular applications in various cross-disciplinary fields, the fundamental study of Persson's theory of contact attracted little attention from the tribology and contact mechanics communities. As far as the authors know, many researchers struggle to understand the derivation of the theory. Few attempts have been made to clarify the oversimplified derivation provided by Persson (Persson, 2001). The present work provides a detailed tutorial on Persson's theory, which does not simply follow the commonly adopted derivation initiated by Persson. A new derivation is given based on stochastic process theory, assuming that the variation of the random contact pressure with respect to scale is a Markov process. We revisit the essential assumptions utilized to derive the diffusion equation, boundary conditions, drift and diffusion coefficients, and closed-form results. This tutorial can serve as a self-consistent introduction for solid mechanicians, tribologists, and postgraduate students who are not familiar with Persson's theory, or who struggle to understand it.