A multiscale analysis of elastic contacts and percolation threshold for numerically generated and real rough surfaces
Introduction
Contact mechanics between surfaces has wide implications in engineering systems: lots of technologically important aspects, like contact stiffness and electrical [1], [2] or thermal contact resistance [3], [4], [5], [6], [7], [8], contact dissipation [9], [10], are strongly influenced by the micromechanical characteristics of the contact process. Also, the wide diffusion of seals as technical devices to isolate chambers and to minimize flow between them makes their design important. Calculating leakage accurately is not a simple matter because the surface roughness at the seal interface spans a wide range of length scales [11]. In all these applications it is crucial to evaluate with accuracy the real contact area and the effective gap distribution between the contacting surfaces.
The earliest pioneering attempt to investigate the multiscale nature of elastic contact between rough surfaces is that of Archard in 1956 [12]. Afterwards, Greenwood and Williamson introduced asperity-based models [13], [14], [15], [16], where the distribution of contacting asperities is replaced by a distribution of Hertzian asperities with equivalent height and curvature. Later models have used random process theory to make the asperity curvature depending on their heights or have resorted to an apparently different approach that uses fractal theories to recognize more directly the multiscale nature of most real surfaces [17]. Although the results obtained with asperity contact theories are of practical interest, the multi-asperity models neglect interactions and coalescing of contact spots. This problem becomes more and more important when approaching full contact conditions. For this reason, in [18] a new approach has been proposed, in which the summits interaction and the coalescing of asperities are taken into account and a very good agreement with the results obtained by fully numerical approaches is found.
Persson [19] has proposed a different approach, which gives the exact solution in full contact and an approximate but accurate solution in partial contact situations. Starting from the full contact solution, where the surfaces are assumed smooth, roughness is progressively added, thus generating contact pressure random fluctuations and gap formations. Persson argues that when adding an increment of roughness corresponding to an increment of magnification, the probability density function of the contact pressure must satisfy a diffusion equation. This allows us to take into account the interaction between the contact spots and to provide exact predictions in full-contact conditions. In the case of partial contact, the initial version of the theory provides approximate predictions [19]. However, further developments of the theory [20] give almost exact solution, provided that a universal correcting factor is introduced properly in Persson's calculation of elastic energy stored at the interface. It is worth noticing that Persson's theory predicts linearity between contact area and load on a larger range of applied loads in agreement with experiments, although the slope of the relation differs from the predictions of multi-asperity contact theories and from numerically calculated values (see e.g. [21], [22]).
In this paper, starting from a numerical method recently developed by the authors [23], we investigate the effect, on the contact between a linear elastic half-space and a rough self-affine fractal rigid surface, of the wavelength numbers used to describe the surface, in order to analyze the contact behavior at different scales. Moreover, we show that self-affine fractal surfaces give a very good representation of real surfaces from the point of view of contact mechanics, and some preliminary calculations about percolation related phenomena, involving multiscale aspects, are also presented.
In this respect, if two surfaces in contact had exactly the same shape, e.g. if there was no roughness, no fluid would leak through their interface; actually, because of the presence of roughness, the contact between sealing surfaces is imperfect, and the fluid can find a percolating path leaking between two chambers at different pressures. Many theories have been developed to predict the rate of percolating flow [24], [25], [26], [27], [28], [29]. However, despite recent progresses, traditional approaches and industrial procedures for seal systems are still based on approximate treatments which usually neglect the effect of elastic deformation on the contacting surfaces [30], [31], [32]. Such approaches usually predict a percolation threshold lying at a relative contact area , overestimating the real value. In fact, there are many simulations [33], [34], [35], [36], [37] suggesting that elastic contacts may percolate below , in agreement with the results presented in this paper.
Section snippets
Mathematical formulation
In this section we present only the fundamental aspects of the mathematical formulation for the problem sketched in Fig. 1, i.e. the adhesionless contact between a periodic numerically generated isotropic randomly rough rigid surface and a linear elastic half-space. The interested reader is referred to Ref. [23] for more details.
The elastic displacement in any point of the half-space can be considered as a sum of two terms: the first is equal to the average displacement and the second
Effect of scales number on contact area
Fig. 2 shows the effect of the wavelengths number N on the dependence of the real contact area A, normalized with respect to the nominal one A0, on the dimensionless contact pressure , where is the composite Young's modulus of the material. Results refer to self-affine fractal surfaces with spectral components q ranging between and . For each surface seven different realizations have been considered and the ensemble average of the calculated results has
Conclusions
We have investigated the contact between a linear elastic solid and isotropic randomly rough surfaces. We have found that the statistical properties of self-affine rough surfaces affect the contact area vs load relation only through the moment m2. We find also that this holds true at high contact pressures, where the contact area/load relation is non-linear. Our calculations also show that, due to the very large number of roughness length scales, which may cover even 6 orders of magnitude of
Acknowledgement
The authors thank the financial support of the Italian Ministry of Education, Universities and Research, within the project PON01 02238.
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