Tribology Letters

, 67:107 | Cite as

An Extended Asymptotic Analysis for Elastic Contact of Three-Dimensional Wavy Surfaces

  • Ivan Y. TsukanovEmail author
Original Paper


An analytical approach for an extended asymptotic analysis of 3D wavy surfaces contact was developed on the basis of expansion of a double-sinusoidal surface in Fourier series, using cylindrical coordinates. The two different problems were considered: indentation of a double-sinusoidal non-periodic punch into an elastic half-space and a penny-shaped crack under action of non-axisymmetric pressure. The closed-form expressions for determining the load–area and the load–separation curves for the light and the high loads, considering virtual circular contact and non-contact areas, were obtained. The results were compared with existing analytical and numerical studies. They show that the mean contact characteristics at the light and the high loads mainly depend on the axisymmetric component of Fourier series, representing the wavy surface. These parameters can be calculated analytically with sufficient accuracy for a large range of applied pressures except transitional region. The relation between 2D and 3D solutions is also shown.


Elastic contact 3D wavy surface Double-sinusoidal surface Mean contact characteristics Asymptotic analysis 


x, y, z

Cartesian coordinates in the three-dimensional problem

r, θ, z

Cylindrical coordinates in the three-dimensional problem

x1, y1

Linear coordinates in the two-dimensional problem

t, φ, ξ

Auxiliary variables


Amplitude of a wavy surface


Period of a wavy surface


Radius of curvature of a wavy surface peak

E1, E2

Young’s moduli of materials of a wavy surface and a half-space

ν1, ν2

Poisson’s ratios of materials of a wavy surface and a half-space


Reduced modulus of elasticity


Initial gap function


Normal surface displacements


Contact pressure distribution in a spatial problem


Contact pressure distribution in a two-dimensional problem


Penetration depth of a rigid punch


Auxiliary function


Total load in a punch problem


Contact area radius in an axisymmetric problem, and contact half-length in a plane problem


Peak pressure in a punch problem


Amplitude pressure value for a complete contact state


Contact pressure distribution acting on a crack surface


Mean pressure

\(\bar{\delta }\)

Mean separation


Crack radius in a crack problem


Nominal contact area


Real contact area


Current separation



The study was partially supported by the Government program (Contract No. AAAA-A17-117021310379-5) and partially supported by RFBR (Grant No. 17-01-00352). The author is grateful to prof. Sergei A. Lychev for helpful discussion. The author is also grateful to the anonymous reviewers for their valuable comments.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of TribologyIshlinsky Institute for Problems in Mechanics RASMoscowRussia

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