A numerical algorithm for the solution of two-dimensional rough contact problems

In this paper, a numerical algorithm is developed to solve the elastic contact problem accurately for two-dimensional rough surfaces. A first version of the method gives a full numerical solution for the discrete problem with all the details of the profile included, and the second version simulates approximately the roughness on a smaller scale with the presence of a non-linear elastic layer (as in the classical Winkler foundation model). In the literature, usually the solution of line contact is given by assuming displacements relative to a datum point, to overcome the difficulty that in two dimensions the displacements are undefined to an arbitrary constant. In the present work, the compliance matrix of the elastic half-plane is calculated starting from a self-equilibrated load distribution with periodic boundary conditions. Some examples are shown to validate the methods. Finally, the method is applied to discuss previous results by the present authors on rough contact problems defined by Weierstrass series profiles, and a discussion follows. In particular, it is found that the Winkler non-linear layer model is surprisingly useful for evaluating the electrical conductance, since (at least in the limited case of two superposed sinusoids) it does not require the wavelength and amplitude of the microscopic component of roughness to be much smaller than the macroscopic component. Some aspects of the mutual role of various components of roughness in the compliance and conductance are elucidated by means of example cases.