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We study the nominal (ensemble averaged) contact pressure p(x) acting on a cylinder squeezed in contact with an elastic half space with random surface roughness. The contact pressure is Hertzian-like for α<0.01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha < 0.01$$\end{document} and Gaussian-like for α>10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > 10$$\end{document}, where the dimensionless parameter α=hrms/δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = h_{\rm rms}/\delta $$\end{document} is the ratio between the root-mean-square roughness amplitude and the penetration for the smooth surfaces case (Hertz contact).

Cylinder-Flat Contact Mechanics with Surface Roughness