A Winkler solution for the axisymmetric Hertzian contact problem with wear and finite element method comparison

Contact problems with wear are often modelled according to the Reye-Archard law that applies locally to the wearing parts. In the transient regime, for geometries where the contact area cannot be assumed to be constant, a simple solution is possible when using the Winkler simplifying assumption. Here, we obtain such a solution in the axisymmetric contact problem, for an initially Hertzian geometry. Also, we explore the possibility to improve the solution by assuming that the Winkler constant adapts to the changing size of the contact. The correction is relevant in intermediate regimes before the solution tends to a 'rigid' asymptotic regime, independent of the elastic modulus. Comparison with a full finite element method simulation shows that the error in either contact area or peak pressure tends to be reduced from the initial error intrinsic in the Winkler assumption; however, the improvement remains small.