The contact problem for a periodic cluster of microcontacts

The three-dimensional contact problem of the linear theory of elasticity is investigated for a system consisting of a large number of small punches, periodically arranged within a limited area on the boundary of an elastic semi-infinite body. The problem is investigated by a method which combines the averaging method and the method of matched asymptotic expansions. It is assumed that the ratios of the diameters of the actual contact areas to the distance between them are small, and each such ratio for neighbouring contact areas is in proportion to the ratio of the dimension of the periodic cell to the diameter of the nominal contact area. The asymptotic form of the doubly periodic contact problem for an elastic half-space is constructed. An approximate solution, in explicit form, is constructed for circular and elliptic contact areas. In the first case, the results agree with the well-known solution in the literature, obtained by another method.