The Contact of Elastic Regular Wavy Surfaces Revisited

Abstract

We revisit the classic problem of an elastic solid with a two-dimensional wavy surface squeezed against an elastic flat half-space from infinitesimal to full contact. Through extensive numerical calculations and analytic derivations, we discover previously overlooked transition regimes. These are seen in particular in the evolution with applied load of the contact area and perimeter, the mean pressure and the probability density of contact pressure. These transitions are correlated with the contact area shape, which is affected by long range elastic interactions. Our analysis has implications for general random rough surfaces, as similar local transitions occur continuously at detached areas or coalescing contact zones. We show that the probability density of null contact pressures is nonzero at full contact. This might suggest revisiting the conditions necessary for applying Persson’s model at partial contacts and guide the comparisons with numerical simulations. We also address the evaluation of the contact perimeter for discrete geometries and the applicability of Westergaard’s solution for three-dimensional geometries.

Appendices

Appendix 1: Probability Density Function of a Wavy Surface

To obtain the PDF of the contact pressure for the case of full contact

$$\begin{aligned} \tilde{p} = 1+\cos \left( 2\pi x/\lambda \right) \cos \left( 2\pi y/\lambda \right) \end{aligned}$$

we use the following observation. If for a given pressure, we express the iso-pressure curve as $y=y(x,\tilde{p})$, then the area between two iso-curves $y(x,\tilde{p})$ and $y(x,\tilde{p}+\hbox {d}p)$ would be proportional to the increment of the probability density of the contact pressure $P(\tilde{p},\tilde{p}+\hbox {d}p)\hbox {d}p$ (see Fig. 12

Fig. 12

Schematic figure explaining the computation method for the probability density function of a wavy surface. Iso-pressure curves $y(x,\tilde{p})$ and $y(x,\tilde{p}+\hbox {d}p)$ at different pressures $\tilde{p}$, the area between curves is proportional to the probability density increment $P(\tilde{p},\tilde{p}+\hbox {d}p)\hbox {d}p$

). At the full contact $\tilde{p} \in [0,2]$. Then, the iso-pressure curve is given by:

$$\begin{aligned} y=\frac{\lambda }{2\pi }\arccos \left( \frac{\tilde{p}-1}{\cos (2\pi x/\lambda )}\right) . \end{aligned}$$

(18)

The area under this curve in the range $x,y\in [0;\lambda /4]$, i.e., for $\tilde{p}\in [1;2]$ may be expressed as:

$$\begin{aligned} A(\tilde{p}) = \int \limits _{0}^{\frac{\lambda }{2\pi }\arccos (\tilde{p}-1)} \frac{\lambda }{2\pi }\arccos \left( \frac{\tilde{p}-1}{\cos (2\pi x/\lambda )}\right) \,\hbox {d}x \end{aligned}$$

(19)

The differential of the probability density is then

$$\begin{aligned} P(\tilde{p},\tilde{p}+\hbox {d}p) \hbox {d}p = \frac{A(\tilde{p})-A(\tilde{p}+\hbox {d}p)}{A_0}, \end{aligned}$$

where $A_0=\lambda ^2/16$ is the considered area. In the limit $\hbox {d}p\rightarrow 0$ we obtain

$$ P(\tilde{p}) = -\frac{1}{A_0}\frac{{\hbox{d}}A(\tilde{p})}{{\hbox{d}} {\tilde{p}}} $$

Substituting the integral form (19) in this expression enables us to evaluate this derivative using the differentiation under the integral sign¹²

$$\begin{aligned} \frac{{\hbox{d}}A({\tilde{p}})}{{\hbox{d}}{\tilde{p}}}&= \frac{\lambda }{2\pi }\int \limits _{0}^{\frac{\lambda }{2\pi }\arccos (\tilde{p}-1)} \frac{\hbox {d}x}{\cos (2\pi x/\lambda )\sqrt{1-\frac{(\tilde{p}-1)^2}{\cos ^2(2\pi x/\lambda )}}} \\&= -\frac{\lambda }{2\pi \sqrt{2\tilde{p} -\tilde{p}^2}}\int \limits _{0}^{\frac{\lambda }{2\pi }\arccos (\tilde{p}-1)} \frac{\hbox {d}x}{\sqrt{1-\frac{\sin ^2(2\pi x/\lambda )}{2\tilde{p} - \tilde{p}^2}}} \\&= -\frac{\lambda ^2}{4\pi ^2\sqrt{2\tilde{p} -\tilde{p}^2}}\int \limits _{0}^{\arccos (\tilde{p}-1)} \frac{d\varphi }{\sqrt{1-\frac{\sin ^2(\varphi )}{2\tilde{p} - \tilde{p}^2}}} \\&= -\frac{\lambda ^2}{4\pi ^2\sqrt{2\tilde{p} -\tilde{p}^2}}\;{{\mathrm {F}}}\left( \arccos (\tilde{p}-1),\frac{1}{\sqrt{2\tilde{p} - \tilde{p}^2}}\right) , \end{aligned}$$

(20)

where ${\mathrm{F}}(l,k)=\int \limits _0^l 1/\sqrt{1-k^2\sin ^2(\varphi )}\,\hbox {d}\varphi$ is the incomplete elliptic integral of the first kind. So the probability density of the contact pressure for $\tilde{p} \in [1;2]$ or equivalently for $\{x,y\}\in [0;\pi /4]$ is

$$\begin{aligned} P(\tilde{p}) = \frac{4}{\pi ^2}\frac{{\mathrm {F}}\left( \arccos (\tilde{p}-1),1/\sqrt{2\tilde{p} - \tilde{p}^2}\right) }{\sqrt{2\tilde{p} - \tilde{p}^2}} \end{aligned}$$

To extend it to a periodic domain $x,y\in \mathbb R$ and for $\tilde{p}\in [0;2]$, one needs simply to take the absolute value of the argument in $\arccos$ and divide the PDF by a factor of two

$$\begin{aligned} P(\tilde{p}) = \frac{2}{\pi ^2}\frac{{\mathrm {F}}\left( \arccos (|1-\tilde{p}|),1/\sqrt{2\tilde{p} - \tilde{p}^2}\right) }{\sqrt{2\tilde{p} - \tilde{p}^2}} \end{aligned}$$

(21)

This expression is depicted in Fig. 13 and compared with numerically evaluated probability density of a wavy surface computed on the grid of $2\times 10^8$ equally spaced points in the region $\{x,y\}\in [0;\pi /4]$ using 500 bins.

Fig. 13

Comparison of the analytically evaluated (21) and numerically evaluated ($2\times 10^8$ grid points per $\lambda /4\times \lambda /4$ and 500 bins) probability density function of a wavy surface

Appendix 2: Data

In Table 1 some numerical results are presented (pressure, area and perimeter) as well as coefficients of Gielis formula (5) that fit the corresponding area shape.

Table 1 Evolution of the contact area fraction $A^{\prime }$ and corrected perimeter $S^{\prime }$ (see Eq. (7)) with increasing pressure $p^{\prime }=p_0/p^*$; $q,k,\rho _0$ are the coefficients of Gielis formula (5) that approximate the shape of the contact perimeter, coefficients are found by the least mean error fit