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

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## Abstract

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.

## Keywords

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

*x*,*y*,*z*Cartesian coordinates in the three-dimensional problem

*r*,*θ*,*z*Cylindrical coordinates in the three-dimensional problem

*x*_{1},*y*_{1}Linear coordinates in the two-dimensional problem

*t*,*φ*,*ξ*Auxiliary variables

- Δ
Amplitude of a wavy surface

- λ
Period of a wavy surface

*R*Radius of curvature of a wavy surface peak

*E*_{1},*E*_{2}Young’s moduli of materials of a wavy surface and a half-space

*ν*_{1},*ν*_{2}Poisson’s ratios of materials of a wavy surface and a half-space

*E*^{*}Reduced modulus of elasticity

*g*_{0}(*r*,*θ*)Initial gap function

*u*_{z}(*r*,*θ*)Normal surface displacements

*p*(*r*,*θ*)Contact pressure distribution in a spatial problem

*p*_{2D}(*x*_{1})Contact pressure distribution in a two-dimensional problem

- δ
Penetration depth of a rigid punch

*h*(*t*)Auxiliary function

*P*Total load in a punch problem

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

*p*_{max}Peak pressure in a punch problem

*p*^{*}Amplitude pressure value for a complete contact state

*p*_{c}(*r*,*θ*)Contact pressure distribution acting on a crack surface

- \(\bar{p}\)
Mean pressure

- \(\bar{\delta }\)
Mean separation

*b*Crack radius in a crack problem

*A*_{n}Nominal contact area

*A*_{r}Real contact area

*G*Current separation

## Notes

### Acknowledgements

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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