A new 2D asperity model with interaction for studying the contact of multiscale rough random profiles

Although in principle simple and neat results are obtained with the classical Greenwood–Williamson (GW) model, linearity of real contact area and conductance with load, the GW definition of asperities as local maxima of the surface leads to paradoxical results for multiscale surfaces, as suspected already by Greenwood in a recent self-assessment of his theory, mainly because of interaction effects becoming increasingly important when the density of asperity grows.In the present paper, a new 2D asperity model, with interaction taken into account to the first order, is introduced for a periodic arrangement of asperities, using the Westergaard solution (rather than the isolated Hertzian solution) leading to a non-linear system of equations, which is easily solved iteratively. The use of Westergaard's solution also elegantly solves the problem of the unknown rigid body motion of 2D elasticity.Using the Weierstrass series profile, some example cases for fractals are discussed. The asperities are defined as parabolic functions, either near the GW “peaks”, or by the alternative Aramaki–Majumbdar–Bhushan (AMB) definition based on the geometrical intersection at a given separation. The former method (GW asperities used in the numerical code) gives quite accurate results, except for the contact area and small separations for the case of low fractal dimensions D, i.e. at very large bandwidth parameter. The latter method (AMB) gives generally no significant advantage, since results tend to be discontinuous and not necessarily the physics of the process is captured correctly. It is confirmed that interaction effects are the key missing ingredient of classical asperity models, but also that most quantities (like contact area, load and conductance) depend more on interaction for cases of a given separation, than for a given load. Also, despite the contact area does not depend much on interaction effects, for a given load, the conductance does.