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Formulations of periodic contact problems for an elastic half-plane and an elastic half-space interacting with a rigid body, having regular microgeometry, and a method for their approximate solution based on the localization principle are proposed. General relations, connecting contact characteristics of the interface (contact pressure distribution and dependence of the real contact area on the nominal pressure) with a single asperity shape and the distance between them, are obtained. The examples, illustrating the use of the obtained approximate relations for the contact characteristics analysis in the case of wavy and wedged profiles, are presented. The comparison of the obtained results with the available exact solutions is carried out. It was established that the approximate dependences coincide with the exact solution up to high values of the nominal pressures. New approximate solutions of 2D contact problems for a periodic system of parabolic asperities with single and double contact segments within a period are derived. It is also shown that the ratio of the contact zone size to the distance between asperities, at which the interaction effect becomes significant, only slightly depends on asperities shape.

The interaction effects, arising at partial contact of rigid multisinusoidal wavy surface with an elastic half-plane, are considered in the assumption of continuous contact configuration. The analytical exact and asymptotic solutions for periodic and nonperiodic contact problems for wavy indenters are derived. Continuous contact configuration, appearing at small ratios of amplitude to wavelength for cosine harmonics, leads to continuous oscillatory contact pressure distribution and oscillatory relations between mean pressure and a contact length. Comparison of periodic and nonperiodic solutions shows that long-range elastic interaction between asperities does not depend on a number of cosine wavelengths.

The physical effects associated with the shape and the scale of regular wavy surface asperities are investigated analytically. A special periodic analytical function, which is a generalization of a sine wave and allows to describe waviness of arbitrary smooth shape is suggested. The formulation and solution of the plane problem of elastic contact of a wavy surface with a half-plane is considered. Asperities of two-scale levels are taken into account—regular waviness with arbitrary shape (small-scale asperities) and regular sine-shaped roughness (large-scale asperities). The obtained pressure distribution for an arbitrary shaped one-scale wave is a generalization of the Westergaard’s solution for a sine wave. The results show that the shape of asperities has significant influence on pressure distribution over the entire range of contact lengths. It is also shown that the elastic coupling of adjacent asperities and asperities of different scales increases the nonlinearity of the contact interaction. But for the small loads the problem can be approximately reduced to linear, and the contact area fraction can be obtained directly from the geometry of contacting surfaces.

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