On the origin of Amonton's friction law

Amonton's law states that the sliding friction force increases linearly with the load. We show that this result is expected for stiff enough solids, even when the adhesional interaction between the solids is included in the analysis. As a function of the magnitude of the elastic modulus E, one can distinguish between three regions: (a) for E>E₂, the area of real contact (and the friction force) depends linearly on the load, (b) for E₁<E<E₂, the area of real contact depends nonlinearly on the load but vanishes for zero load, and (c) for E<E₁ the area of real contact depends nonlinearly on the load and is non-vanishing at zero load. In this last case a finite pull-off force is necessary in order to separate the solids. Based on molecular dynamics calculations, we also discuss the pressure dependence of the frictional shear stress for polymers. We show that the frictional shear stress is independent of the normal pressure p₀ as long as p₀ is much smaller than the adhesional pressure p_ad, which depends on the atomic corrugation of the solid surfaces in the sliding interface. Finally, we discuss the origin of why the contact area between a soft elastic solid (e.g. rubber) and a flat substrate decreases from the JKR (adhesive contact) limit at zero or small sliding velocities, to the Hertz (non-adhesive) limit at high sliding velocities.