Mechanical origin of Amonton’s law in nanoscale friction

The widely used Amonton’s law, which describes the common observation that dry friction between two surfaces is proportional to the normal load, has yet to be explicitly derived since its discovery from Leonardo da Vinci era. Here, Amonton’s law is explicitly extrapolated as the friction coefficient $\mu =\mathrm{\Delta }{F}_{\mathrm{N}}/\left(k\mathrm{\Delta }x\right)$ via concise perturbation theory analysis of the frictional energy landscape parametrized by the load. This reveals that Amonton’s law may arise from the competition between the normal stiffness (k) and the interfacial normal force fluctuation during sliding (ΔF_N/Δx). The predicted friction coefficient is quantitatively validated through both atomic force microscopy (AFM) experiments and atomistic simulations of frictional systems ranging from bulk to layered materials. Unlike the traditional framework of recondite rough contact theory, this study concisely extrapolates Amonton’s law and friction coefficient and traces their microscopic origins to the inherent mechanical properties, thereby enabling their rational design.