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

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.

Appendices

Appendix 1: Solutions for a Two-Dimensional Sinusoidal Surface

For a two-dimensional sinusoidal surface y₁(x₁) = Δ(1 − cos(2πx₁/λ)), the contact pressure distribution, corresponding to a periodic contact plane-strain problem, was first obtained by Westergaard [10]:

$$p(x_{1} ) = \frac{{\sqrt 2 \pi E^{*} \Delta }}{\lambda }\cos \left( {\pi x_{1} /\lambda } \right)\sqrt {\cos \left( {2\pi x_{1} /\lambda } \right) - \cos \left( {2\pi a/\lambda } \right)} .$$

This expression is a solution of the integral equation with a Hilbert kernel:

$$\frac{{E^{*} }}{2}\frac{{\partial g_{0} (x_{1} )}}{{\partial x_{1} }} = \frac{1}{2\pi }\int\limits_{ - a}^{a} {p(\xi )} \,\cot \frac{{x_{1} - \xi }}{2}d\xi .$$

The relation between mean pressure and contact length is [11]

$$\bar{p} = p^{*} \sin^{2} \left( {\pi a/\lambda } \right) = \frac{{p^{*} }}{2}\left( {1 - \cos \left( {2\pi a/\lambda } \right)} \right),$$

which coincides with the expression describing wavy surface shape. The inverse relation in-turn corresponds to bearing length of a sinusoidal profile.

The relation between current separation and mean pressure was obtained first by Kuznetsov [34].

$$G = 1 - \left( {\frac{{\bar{p}}}{{p^{*} }}} \right)\left( {1 - \ln \left( {\frac{{\bar{p}}}{{p^{*} }}} \right)} \right).$$

For the non-periodic two-dimensional problem, i.e., sinusoidal punch problem, the integral equation with Cauchy kernel is used:

$$\frac{{E^{*} }}{2}\frac{{\partial g_{0} (x_{1} )}}{{\partial x_{1} }} = \frac{1}{2\pi }\int\limits_{ - a}^{a} {\frac{p(\xi )}{{x_{1} - \xi }}} \,d\xi .$$

The analytical solution can be obtained in terms of Fourier series [33], or more directly with the use of series of Chebyshev polynomials:

$$p(x_{1} ) = \frac{{E^{*} \Delta }}{\lambda }\sqrt {1 - \left( {\frac{{x_{1} }}{a}} \right)^{2} } \sum\limits_{ \, k = 0}^{\infty } {( - 1)^{k} J_{2k + 1} \left( {\frac{2\pi a}{\lambda }} \right)U_{2k} \left( {\frac{{x_{1} }}{a}} \right)} ,$$

where U_i(x) is a Chebyshev polynomial of the second kind with a degree i; J_i(x) is a Bessel function of the first kind and of the integer order i.

Appendix 2: Asymptotic Equations of Johnson, Greenwood, and Higginson for Three-Dimensional Sinusoidal Problem

For the sufficiently low loads, the shape of three-dimensional sinusoidal punch can be considered as quadratic. With the use of the Hertzian contact theory, the following equations for real contact area and mean separation can be obtained [12]:

$$\frac{{A_{r} }}{{A_{n} }} = \pi \left( {\frac{3}{8\pi }\frac{{\bar{p}}}{{p^{*} }}} \right)^{{\frac{2}{3}}} .$$

$$\bar{\delta } = \frac{1}{2}\Delta \left( {3\pi^{2} \left( {\bar{p}/p^{*} } \right)} \right)^{{\frac{2}{3}}} - \Delta \left( {\bar{p}/p^{*} } \right)\left( {4\ln \left( {\sqrt 2 + 1} \right)} \right).$$

For the sufficiently high loads, the shape of contact pressure distribution acting on a crack surface can be considered as quadratic. With the use of Sneddon’s theory for pressurized penny-shaped cracks, the following can be obtained [12]:

$$1 - \frac{{A_{r} }}{{A_{n} }} = \frac{3}{2\pi }\left( {1 - \frac{{\bar{p}}}{{p^{*} }}} \right).$$

$$\bar{\delta } = \Delta - \frac{16\Delta }{{15\pi^{2} }}\left( {\frac{3}{2}} \right)^{{\frac{3}{2}}} \left( {1 - \frac{{\bar{p}}}{{p^{*} }}} \right)^{{\frac{5}{2}}} .$$