Multiscale Surfaces and Amontons' Law of Friction

The theories of Archard, Greenwood, and more recently Persson all predict that the area of actual contact between two rough surfaces will be approximately proportional to the applied normal force, though these authors make different assumptions and approach the problem from different points of view. Here we discuss the nature of these approximations and show that the common conclusion follows from the multiscale nature of the surface profile. In particular, it is shown that whatever assumption is made about the nature of friction on the microscale, the macroscale frictional behaviour will always approximate Amontons' law if the surface has sufficient multiscale content.