Sliding Viscoelastic Contacts: The Role of Adhesion, Boundary Conditions, and Finite Geometry

In this study, we investigate the tangential sliding of a rigid Hertzian indenter on a viscoelastic substrate, a problem of practical interest due to the crucial role that sliding contacts play in various applications involving soft materials. A finite element model is developed, where the substrate is modelled using a standard linear viscoelastic model with one relaxation time, and adhesion is incorporated using a Lennard-Jones potential law. We propose an innovative approach to model tangential sliding without imposing any lateral displacement, thereby enhancing the numerical efficiency. Our goal is to investigate the roles of adhesive regimes, boundary conditions (displacement and force-controlled conditions), and finite thickness of the substrate. Results indicate significant differences in the system's behaviour depending on the boundary conditions and adhesion regime. In the short-range adhesion regime, the contact length L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} initially increases with sliding speed before decreasing, showing a maximum at intermediate speeds. This behaviour is consistent with experimental observations in rubber-like materials and is a result of the transition from small-scale to large-scale viscous dissipation regimes. For long-range adhesion, this behaviour disappears and L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} decreases monotonically with sliding speed. The viscoelastic friction coefficient mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} exhibits a bell-shaped curve with its maximum value influenced by the applied load, both in long-range and short-range adhesion. However, under displacement control, mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} can be unbounded near a specific sliding speed, correlating with the normal force crossing zero. Finally, a transition towards a long-range adhesive behaviour is observed when reducing the thickness t of the viscoelastic layer, which is assumed to be bonded to a rigid foundation. Moreover, the friction coefficient reduces when t tends to zero. These findings provide insights into the viscoelastic and adhesive interactions during sliding, highlighting the critical influence of boundary conditions on contact mechanics.