The small-scale topography of surfaces critically affects the contact area of solids and thus the forces acting between them. Although this has long been known, only recent advances made it possible to reliably model interfacial forces and related quantities for surfaces with multiscale roughness. This article sketches both recent and traditional approaches to their mechanics, while addressing the relevance of nonlinearity and nonlocality arising in soft- and hard-matter contacts. Graphical abstract: [Figure not available: see fulltext.].

Surfactant molecules, known as organic friction modifiers (OFMs), are routinely added to lubricants to reduce friction and wear between sliding surfaces. In macroscale experiments, friction generally decreases as the coverage of OFM molecules on the sliding surfaces increases; however, recent nanoscale experiments with sharp atomic force microscopy (AFM) tips have shown increasing friction. To elucidate the origin of these opposite trends, we use nonequilibrium molecular dynamics (NEMD) simulations and study kinetic friction between OFM monolayers and an indenting nanoscale asperity. For this purpose, we investigate various coverages of stearamide OFMs on iron oxide surfaces and silica AFM tips with different radii of curvature. We show that the differences between the friction-coverage relations from macroscale and nanoscale experiments are due to molecular plowing in the latter. For our small tip radii, the friction coefficient and indentation depth both have a nonmonotonic dependence on OFM surface coverage, with maxima occurring at intermediate coverage. We rationalize the nonmonotonic relations through a competition of two effects (confinement and packing density) that varying the surface coverage has on the effective stiffness of the OFM monolayers. We also show that kinetic friction is not very sensitive to the sliding velocity in the range studied, indicating that it originates from instabilities. Indeed, we find that friction predominately originates from plowing of the monolayers by the leading edge of the tip, where gauche defects are created, while thermal dissipation is mostly localized in molecules toward the trailing edge of the tip, where the chains return to a more extended conformation.

This paper presents a nanoscale-inspired continuum model to capture the coupling of adhesion and friction in contact-mechanics problems. The method relies on Green's function molecular dynamics to calculate the elastic body fields and on a phenomenological mixed-mode coupled cohesive-zone model to describe the interplay between normal and tangential tractions, i.e. adhesion and friction. While the presented formulation is applicable to linearly elastic solids with generic surface roughness, the focus of our analysis is on the indentation of an array of circular rigid punches into a flat, deformable solid. Our results show that the coupling between adhesion and friction leads to an increase in the contact size and a decrease in the pull-off load.

The Green's function molecular dynamics (GFMD) method for the simulation of incompressible solids under normal loading is extended in several ways: shear is added to the GFMD continuum formulation and Poisson numbers as well as the heights of the deformed body can now be chosen at will. In addition, we give the full stress tensor inside the deformed body. We validate our generalizations by comparing our analytical and GFMD results to calculations based on the finite-element method (FEM) and full molecular dynamics simulations. For the investigated systems we observe a significant speed-up of GFMD compared to FEM. While calculation and proof of concept were conducted in two-dimensions only, the methodology can be extended to the three-dimensional case in a straightforward fashion.