Dimensionless Numbers and Master Curves for Sliding Friction from the Kelvin-Voigt Viscoelasticity of Solids

Abstract

Recent advances in computational methods have enabled realistic numerical simulations for the sliding friction of viscoelastic solids (e.g., rubber friction by the finite element method in the macroscale and soft matter friction by the molecular dynamics method in the nanoscale). However, the factor analysis of accumulated numerical results is generally not easy as their models include many system parameters. Here we theoretically examine the sliding friction with the minimum number of system parameters: the sliding friction between a rigid probe of various shapes and a viscoelastic foundation that consists of the Kelvin-Voigt elements. The analytical solutions for the master curves of the friction coefficient are described by using dimensionless numbers under two boundary conditions (i.e., a constant indentation depth and a constant contact load). Based on the analytical solutions, we discuss how the bell-shaped velocity dependence of the friction coefficient appears under a constant contact load without bell-shaped rheology: The keys are the vertical lift of the probe and the change in the inlet slope due to the vertical lift. The analytical solutions (exact and approximate) obtained here would help the factor analysis to some extent for several numerical simulations in computer-aided science and engineering.

Keywords

• sliding friction
• viscoelasticity
• dimensionless analysis
• master curves
• analytical solutions
• velocity dependence of friction