Friction Force During Lubricated Steady Sliding of a Rigid Cylinder on a Viscoelastic Substrate

We study the friction force during lubricated sliding of a rigid cylindrical indenter against a viscoelastic substrate in the iso-viscous visco-elasto-hydrodynamic lubrication (VEHL) regime. The substrate is represented by a foundation model. The solution is controlled by three dimensionless parameters. The first of these, λ, measures the time for the indenter to move one contact zone relative to the viscoelastic relaxation time; the second is the ratio of the long time to short time compliance of the substrate, c∞/c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{\infty } /c_{0}$$\end{document}; the third parameter, β, is the ratio of average fluid flow rate to the sliding velocity. Although our solution works well for the full range of parameters, we focus on the “Hertz” regime (β >>1) where practically all the fluid in the contact region is squeezed out. This regime is quite common in soft contact lubrication problems and presents significant numerical difficulties. Our analysis gives insight into why these numerical difficulties arise. The friction force can be decomposed into two parts, one due to viscoelastic dissipation and the other from hydrodynamics. Although these two are generally coupled, in the Hertz limit, an important result is that the viscoelastic portion of the friction force can be well approximated by the solution of the corresponding “dry” sliding problem, in which there is no lubricating fluid layer. This provides a simple way to decouple the hydrodynamic portion of the friction force from the viscoelasticity of the substrate. We study how hydrodynamic pressure and film thickness vary with the controlling dimensionless parameters. Scaling laws for these relationships are given in closed form.