Elastic contact of rough surfaces: A simple criterion to make 2D isotropic roughness equivalent to 1D one

We analyse the periodic contact between an elastic half-space and two types of rough substrates: (i) a perfect isotropically rough rigid substrate (2D isotropic roughness), and (ii) a perfect anisotropically rough rigid substrate, i.e. a substrate with roughness in only one direction (1D roughness). The analysis is carried out with the aid of proprietary codes, that we have developed (both in real and Fourier space) to deal with this type of contacts. Of course, 1D contacts differ from 2D isotropic contacts. However our results and theoretical arguments suggest a possible criterion to make 2D contacts equivalent to 1D ones from the point of view of contact area and separation calculations. The rule consists in replacing the 2D Power Spectral Density (PSD) of the isotropic surface into an equivalent 1D PSD. Interestingly the trasformation rule does not depend on the statistical properties of the surface roughness, hence seems to have a universal character for isotropic surfaces.