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

TL;DR

This paper presents a mathematically friendly tutorial that recasts Persson's theory of purely normal elastic rough-surface contact as a stochastic-process problem. By treating the scale evolution of the contact pressure within the contact area as a Markov process, it derives the Chapman–Kolmogorov equation and, under mean-field homogenization, the diffusion (Fokker–Planck) equation with drift $B_1(\zeta)=0$ and diffusion $B_2(\zeta)=dV/d\zeta$, where $V(\zeta)$ is the scale-dependent pressure variance. The authors provide closed-form expressions for the PDF $P_0(p,\zeta)$, the relative contact area $A^*(p̄,\zeta)$, and the elastic energy $U_{el}(p̄,\zeta)$, and discuss practical aspects such as fudge factors, compounded CK, and the zero-contact-area paradox. The tutorial aims to clarify fundamental assumptions, bridge gaps between tribology and stochastic-process theory, and offer a self-contained framework for applying these methods to rough-surface contact problems.

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.

Figures (6)

• Figure 1: Graphical illustration of cross-sectional view of deformed interface when subjected to uniform compressive normal traction $\bar{p}$ and contact pressure distribution $p({\bf x})$ along line $y = 0$.
• Figure 2: Evolution of: (a) nominally flat, isotropic, self-affine rough surface topography with respect to $\zeta$; (b) PDF of contact pressure $P(p, \xi)$ with respect to $\xi$ as predicted by Persson's theory.
• Figure 3: Piecewise representation of PDF $P(p, \zeta)$, including Dirac delta function $\delta(p)$ (purple arrow), and probability density function $P_0(p > 0,\zeta)$ (red solid line).
• Figure 4: Graphical illustration of transition probability density $P(p, \zeta + \Delta \zeta | p', \zeta)$.
• Figure 5: (a) Graphical illustration of Gaussian distribution in Eq. \ref{['eq:Gauss_Distribution']}, its mirror image about $p = 0$, and $P_0(p, \zeta)$ in Eq. \ref{['eq:P0_solution']}; (b) Variation of $A^*(\bar{p}, \zeta)$ with $\bar{p}/\sqrt{V}$ predicted by Eq. \ref{['eq:Area_Persson']}.