Mapping of Two-Dimensional Contact Problems on a Problem with a One-Dimensional Parametrization

We discuss a possible generalization of the ideas of the method of dimensionality reduction (MDR) for the mapping of two-dimensional contact problems (line contacts). The conventional formulation of the MDR is based on the existence and uniqueness of a relation between indentation depth and contact radius. In two-dimensional contact problems, the indentation depth is not defined unambiguously, thus another parametrization is needed. We show here that the Mossakovskii-Jäger procedure of representing a contact as a series of incremental indentations by flat-ended indenters can be carried out in two-dimensions as well. The only available parameter of this process is, however, the normal load (instead of indentation depth as in the case of threedimensional contacts). Using this idea, a complete solution is obtained for arbitrary symmetric two-dimensional contacts with a compact contact area. The solution includes both the relations of force and half-width of the contact and the stress distribution in the contact area. The procedure is generalized for adhesive contacts and is illustrated by solutions of a series of contact problems.