On the Linearity of Contact Area and Reduced Pressure

Abstract

Computer simulations, Persson theory, and dimensional analysis find that the relative contact area between nominally flat surfaces grows linearly with the reduced pressure $$p^*$$ at small loads, where $$p^*$$ is the ratio of the macroscopic pressure p to the contact modulus times the root-mean-square height gradient $$\bar{g}$$ . Here, we show that it also holds for Hertzian and other harmonic, axisymmetric indenters—as long as $$\bar{g}$$ is determined over the true contact area and p is defined as the load divided by an arbitrary but fixed reference area. For a Hertzian indenter, the value for the proportionality coefficient $$\kappa$$ turns out to be $$\kappa = 3\pi /\sqrt{32}$$ . The analysis explains why mathematically rigorous treatments of Greenwood–Williamson type models identify a sublinear dependence of contact area on load.