# On the Contact Area and Mean Gap of Rough, Elastic Contacts: Dimensional Analysis, Numerical Corrections, and Reference Data

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## Abstract

The description of elastic, nonadhesive contacts between solids with self-affine surface roughness seems to necessitate knowledge of a large number of parameters. However, few parameters suffice to determine many important interfacial properties as we show by combining dimensional analysis with numerical simulations. This insight is used to deduce the pressure dependence of the relative contact area and the mean interfacial separation \(\Updelta \bar{u}\) and to present the results in a compact form. Given a proper unit choice for pressure *p*, i.e., effective modulus *E* ^{*} times the root mean square gradient \(\bar{g}\), the relative contact area mainly depends on *p* but barely on the Hurst exponent *H* even at large *p*. When using the root mean square height \(\bar{h}\) as unit of length, \(\Updelta \bar{u}\) additionally depends on the ratio of the height spectrum cutoffs at short and long wavelengths. In the fractal limit, where that ratio is zero, solely the roughness at short wavelengths is relevant for \(\Updelta \bar{u}\). This limit, however, should not be relevant for practical applications. Our work contains a brief summary of the employed numerical method Green’s function molecular dynamics including an illustration of how to systematically overcome numerical shortcomings through appropriate finite-size, fractal, and discretization corrections. Additionally, we outline the derivation of Persson theory in dimensionless units. Persson theory compares well to the numerical reference data.

## Keywords

Contact mechanics Surface roughness Relative contact area Mean gap## Notes

### Acknowledgements

We thank the Jülich Supercomputing Centre for computing time on JUGENE, JUQUEEN, and JUROPA. MHM also thanks DFG for support through Grant No. Mu 1694/5-1.

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