Unifying linear proportionality between real contact area and load in rough surface contact

A long-standing debate and challenge in contact mechanics is to confirm the linearity between the real contact area and load on rough surfaces as well as its proportionality. Here, we first theoretically prove the linearity between the real contact area and load on rough surfaces by considering an infinite number of surface asperities. The mechanism for such linearity is that the applied force on each "small region" on the rough surface is directly proportional to the area of the region, resulting in a statistical proportionality between the total load and area. This explanation is confirmed via Green's function molecular dynamics (GFMD) simulations. On this basis, we develop a novel framework of surface slope-based multi-asperity contact model. The proportionality between the contact load and area is governed by the elastic property, mean absolute slope, and shape coefficient of the contact surface over the pressed depth. The elastic contacts of single-scale and multiscale rough surfaces are investigated using the developed contact model and GFMD. The shape coefficient of rough surfaces predicted by numerical simulations closely resembles that of surfaces with symmetric parabolic asperities. This work not only sheds light on the physical mechanism underlying the linearity between the contact area and load on rough surfaces but also provides a theoretical foundation for designing and evaluating surface contact and friction performance in micro- and nano-engineering systems.