# Surface coupling effects on contact mechanics: contact area and interfacial separation between an elastic solid and a hard substrate with randomly rough, self-affine fractal surfaces

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## Abstract

The objective of this study is to investigate both the contact area and the interfacial separation between two surfaces. Both surfaces are considered to be rough, one of them being elastic and the other one hard. The work is based on an extended version of Persson’s model of contact mechanics to study the behavior of the contact area, the interfacial separation and the pressure distribution. The results are compared with the case merely the hard substrate is rough. It is seen that introducing a roughness in the elastic surface decreases the real contact, if the surfaces are uncorrelated. A positive (negative) correlation increases (decreases) the real contact. A reverse pattern occurs for the width of the pressure distribution, as well as the interfacial separation (at equal pressures).

### Keywords

Self-affine fractal Cross-correlation Surface effects## Introduction

All of the surfaces occurring in nature and industry are rough, provided they are observed with sufficiently high magnifications (small length scales) [1, 2]. So, for two contacting solid surfaces, microscopically, there are many non-contact regions (the interfacial separation), and microscopic contact occurs only at a fraction of the macroscopic contact. This fraction of real contact, as well as the interfacial separation, are affected by the roughness of the surfaces, and play important roles in the mechanical properties of the system. The area of real contact characterizes the frictional properties of the contact, as well as the strength of adhesion and the amount of wear [3, 4, 5]. Some other phenomena are affected by the interfacial separation, among which are the heat transfer, the contact resistivity, lubrication, and sealing [5, 6, 7, 8]. The effect of the surface roughness on the area of real contact has been studied by two classes of analytical models. The first class involves multiasperity contact theories (originally formulated by Greenwood and Williamson (GW) [9, 10, 11, 12, 13]), where the contact between the surfaces is modeled as an ensemble of randomly distributed Hertzian contacts between the asperities. The second class is based on Persson’s model of contact mechanics [1, 2], where the probability distribution of the contact pressure is shown to be governed by a diffusive process in terms of the magnification at which the interface is observed. Numerical studies [14, 15, 16, 17] have shown that, in the case of non-adhesive contacts, when a flat elastic body is brought into contact with a rough surface, the real contact area increases proportional to the applied normal squeezing pressure (applied load). In [18], it has been shown that the GW-type theories predict linearity only for vanishingly small contact areas, corresponding to vanishingly small applied normal squeezing pressures. When the applied normal squeezing pressure is increased, the theoretical predictions rapidly deviate from the asymptotic linearity. This behavior is not seen in Persson’s model, which predicts linearity between contact area and applied normal squeezing pressure for real contact values of up to about 15–20 percent of nominal contact area. This is in agreement with some experimental and numerical results. Under full contact conditions, Persson’s model is exact, but in the case of partial contact, some numerical results [16, 19] regarding non-adhesive contacts between rough surfaces indicate an underestimation of the contact area in Persson’s model, while the results of the model still qualitatively agree with numerical calculations [20].

In all of these works, the area of real contact between a smooth elastic solid surface and a hard substrate with randomly rough surface has been studied. As stated earlier, however, there are essentially no surfaces which are smooth on atomic scales. Here, the elastic solid is assumed to have a rough surface as well, and the effect of roughness on the area of real contact and the interfacial separation is studied. The contacts are assumed to be frictionless and non-adhesive, and the roughness of both surfaces is assumed to be random. An extended version of Persson’s model of contact mechanics is used to investigate the area of real contact, as well as the interfacial separation between two surfaces.

The outline of the paper is as follows. In Sect. 2, Persson’s model of contact area and interfacial surface separation is reviewed. In Sect. 3, an extended version of Persson’s model of contact mechanics is used to calculate the contact area and the interfacial surface separation for randomly rough elastic solids and hard substrates with randomly rough surfaces in contact with each other. Numerical results corresponding to the randomly rough self-affine fractal surfaces are presented in Sect. 4. Sections 3 and 4 contain the main results. The novelty, which is described in these sections, is the introduction of a second rough surface and the investigation of the effects of both surfaces being rough and also the effect of their correlation. Section 5 contains the concluding remarks.

## The contact area, and the interfacial surface separation

*L*, the length scale is \(\lambda \), and the wave numbers

*q*and \(q_L\) correspond to the length scale and the size of the system, respectively:

*p*is the nominal squeezing pressure and

*E*and \(\upsilon \) being the elastic modulus and the Poisson’s ratio of the elastic block, respectively.

*C*(

*q*) is the auto-spectral density function [23] of the hard randomly rough substrate. Denoting the actual (microscopic) and the nominal (macroscopic) contact areas by

*A*and \(A_0\), respectively, the relative contact area is

*p*required to produce this separation, an elastic energy is stored in the block. Denoting this by \(U_\mathrm {el}\), one arrives at

*C*(

*q*) is the auto-spectral density function of the hard randomly rough surface. In cases where the applied normal squeezing pressure

*p*is small, the surface asperities do not fully penetrate the elastic block and only a partial contact is realized. So, the full contribution of the auto-spectral density function is not received by the elastic energy. In (9), this has been addressed through the factor \(\gamma \,P(q)\), where

*P*(

*q*) is the relative contact area for elastic nonadhesive contact and is given by [1, 21]

*p*is [22, 24].

## An extension to the case of two randomly rough surfaces in contact with each other

As stated before, usually both of surfaces which are in contact with each other are rough. Here, Persson’s model of contact mechanics is extended to such cases.

### The contact area

*C*with the autocorrelation corresponding to \((h_2-h_1)\), that is [27, 28]

*G*is to be put in (6) and (7), to obtain the contact area. Here, \(\eta \) is considered to be a constant (independent of \(\varvec{q}\)), and the results for this simple case are presented. Special cases are \(\eta =0\) (uncorrelated surfaces), \(\eta =+1\) (completely positive correlated surfaces), or \(\eta =-1\) (completely negative correlated surfaces).

### The interfacial surface separation

*C*substituted according to (22) [27, 28]:

## Numerical results

*H*is the Hurst exponent, and it is assumed that \(q_a\) is much larger than \(q_L\), which usually is. For the hard substrate and the elastic block, these values have been used.

### Contact area

In these curves, the circle curve is for the case where only the substrate is rough. The asterisk, solid, and dotted curves correspond to the cases where both surfaces are rough and uncorrelated (\(\eta =0\)), completely positively correlated (\(\eta =1\)), and completely negatively correlated (\(\eta =-1\)), respectively. Figure 2 shows that both surfaces being rough, but uncorrelated, results in a decrease in the contact area, compared to the case of only one rough surface. If both surfaces are rough, and they are correlated, depending on the sign of the correlation, an increase or decrease in the values of the contact area is resulted, compared to the case of two uncorrelated surfaces and the case of only one rough surface. A positive correlation (\(\eta =1\)) increases the contact area between the two surfaces, so that the pressure distribution vanishes in a smaller normalized pressure, as seen from Fig. 2. For a negative correlation \(\eta =-1\), however, the contact area is decreased compared to the case of uncorrelated surfaces and the case of only rough surface, so that the pressure distribution vanishes in larger pressure. It is seen that when both surfaces are rough but uncorrelated, the width of the pressure distribution is larger compared to the case of only one rough surface. A positive (negative) correlation results in a decrease (an increase) of the width of the pressure distribution.

### Interfacial surface separation

## Concluding remarks

An extension of Persson’s model of contact mechanics was used to study the contact area and the interfacial separation, when the elastic solid and the hard substrate are both rough. It was seen that when the two surfaces are rough but uncorrelated, the real contact decreases compared to the case where only the substrate is rough, effectively the roughness has been increased. It was also shown that when the surfaces are correlated, a positive correlation increases the real contact area compared to the case of no correlation, while a negative correlation decreases the real contact area compared to the case of no correlation. A reverse pattern is seen for the width of the pressure distribution, as well as the interfacial separation (at equal pressures): making both surfaces rough but uncorrelated increases these, and a positive (negative) correlation results in a decrease (an increase) in these.

## Notes

### Acknowledgements

This work was supported by the research council of the Alzahra University.

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