**Abstract**

This chapter reviews advanced models for solving the normal contact problem of two elastic bodies with rough boundaries. Starting from the fundamental formulation of Greenwood and Williamson, an extension is proposed with details on the possible algorithmic implementation to consider the interactions between asperities. A second multi-scale-based approach, considering the self-affine nature of the rough surface, also known as Persson's theory, is briefly discussed. As a third method, special attention is given to review the standard Boundary Element Method (BEM). Finally, all the mentioned methods are applied to a rough gold surface measured by Atomic Force Microscope (AFM) and the evolution of the real contact area with loading is analyzed. The aim of this contribution is to present the basic guidelines to tackle the problem of contacting rough surfaces, accounting for the real surface topography.

**Keywords:** contact mechanics, roughness, elastic contact, true contact are, atomic force microscopy

## **1. Introduction**

Among the most complicated problems in mechanics, there is the contact modeling of rough surfaces where one seeks to predict the stresses, strains, and true contact area. These predictions play a very important role in studying many phenomena such as friction [1], wear [2], thermal [3] and electrical conductivity [4], sealing [5], squeal [6, 7], etc. Understanding the micromechanical characteristics of these phenomena leads to a robust design of mechanical systems by increasing their performances. For example, in micro-systems with brittle behavior such as (i) radio frequency micro-electro-mechanical-systems (RF MEMS) with silicon-to-Silicon (Si-to-Si) contacts [8], (ii) micro-turbines using Si or Polydimethylsiloxane (PDMS)-based micro-valve and [9] (iii) Si-to-Si wafer bonding [10–12], the performances and the reliability of the device rely on the quality of the contact [13–17]. An accurate prediction of the pressure and the true contact area leads to a good

dimensioning of the adhesion forces and thus, to an increase in the reliability of components.

As part of the design process of these systems, the contact problem is addressed by assuming that the contact interfaces are perfectly flat. This assumption neglects the notion of roughness, which often leads to an overestimation of the contact surface and an underestimation of the contact pressure. In reality, every surface, even mirror polished, still exhibits roughness at the micro- and nano-scales. A rough surface can be seen as the superposition of several wavelengths, forming hills (asperities or local maxima) and valleys (local minima), and it is fully described by two statistical functions: (i) the height probability density function (HPDF) and (ii) the power spectral density function (PSDF). The former describes the randomness of the rough surface, while the latter quantifies the contribution of the different wavelengths to the surface topography. Both HPDF and PSDF are widely used to characterize engineering surfaces [18–20] as well as to generate numerically artificial surfaces [21, 22].

When two rough surfaces come in contact, the mechanical load is borne by the top of the highest asperities. Thus, the real contact area is only a small fraction of the nominal area. The first and most popular theory that elucidated this point was made in the 1960s by the pioneers' Greenwood and Williamson (GW) [23]. In such theory, the rough surface is represented by a set of spherically shaped asperities, with the same radius of curvature and whose height varies randomly following a given probability density function. Then, the Hertz theory of one elastic sphere contacting a rigid flat plane is extended to the entire surface. In the same period, Bush et al. [24] proposed a refined model approximately equivalent to its counterpart (i.e. GW theory). The special point with respect to GW theory is that Bush et al. assumed that asperities have a paraboloidal shape. Then, they used the random process theory [25] to describe the statistics of the rough surface. Other models quite similar to those described above can be found in Refs. [26–28]. Using the above theories, it has been found that, at a low squeezing load, the predicted true contact area *A* increases linearly with the applied load.

The aforementioned contact models assume that all the contacting asperities behave independently. Thus, the interaction between the different contact region are neglected (i.e. each asperity is locally deformed without inducing a displacement of the neighboring asperity). This last assumption is not met for medium to high loading cases (i.e. nearly full contact case) since the contacting asperities become tighter and hence, the contribution of the surface deflections produced by each of them, at the level of the neighboring micro-contacts, becomes significant [29, 30]. On the other hand, O'Callaghan et al. [31] and Hendriks et al. [32] demonstrated experimentally that asperities can interact strongly and even coalesce at low loads questioning the reliability of the classic asperity-based models. To consider the interaction between asperities, authors in the literature have proposed several models based on GW theory. For instance, Paggi and Ciavarella [33, 34] included interaction between asperities by means of semi-analytical modeling to simulate the effect of the bulk. In the same context, Afferrante et al. [35] have introduced a multi-asperity model that takes into account the coalescence between two contacting asperities. Despite all these advanced models, the semi-analytical approach remains very approximate compared to numerical methods such as the finite or the boundary element method [36].

The original GW theory has also been criticized since it deals with special cases where the height distribution follows a Gaussian or exponential distribution. Especially, it is well admitted by the scientific community that the roughness can appear at many scales exactly like the fractal patterns, but down to a wavelength

#### *A Review on the Contact Mechanics Modeling of Rough Surfaces in the Elastic Regime… DOI: http://dx.doi.org/10.5772/intechopen.102358*

that corresponds to, perhaps, some lattice constant. The use of GW theory with the "fractal" characteristic has been questioned by many authors, in particular Persson [37]. The latter comes out with an ingenious idea, completely different from the asperity-based model, to tackle the scaling effect on realistic rough surfaces [37–39]. In contrast to asperity-based models, Persson's theory considers the stress probability distribution as a function of the surface resolution. The idea is to determine the evolution of the contact stress density function when the PSDF is extended to cover a wide frequency range. Initially, the problem is solved by assuming a full-contact assumption (i.e. smooth surfaces assumption) under initial uniform pressure *p*0. Then, step by step, the roughness is added according to the self-affine character carried by the PSDF which leads to the variations in the contact pressure [40]. According to Persson, the contact stress distribution satisfies a diffusion-like equation. Persson's approach was also criticized by the leaders of the asperity-based models. They argued that Persson's proof is not rigorous while deriving the diffusion equation [40]. Hence, the predicted contact area, using Persson's theory, is smaller compared to the available results in the literature.

In parallel with analytical models, several research groups have proposed numerical simulations reproducing the behavior of rough surfaces in contact. Among these methods, there is the finite element method (FEM) introduced for the first time by Hyun et al. [41]. This category of method is free of any assumption. The only disadvantage it presents is the convergence which is related to (i) the numerical treatement (the penalty method for instance) and (ii) the discretization error. Indeed, to have significant results it is necessary that the size of the discretization is much smaller than the shortest wavelength of the rough surface [5]. However, the finest mesh leads to a prohibitive computational cost. The latest point was clearly demonstrated by Yastrebov et al. [36]. It was shown that the numerical error of the discretization depends on two ratios; (i) Δ*x=λ<sup>s</sup>* and (ii) *λl=L* where Δ*x* is the distance between the mesh points, *λ<sup>s</sup>* and *λ<sup>l</sup>* refer to the shortest and the longest wavelength of the surface PSDF, respectively. *L* is the length of the representative elementary volume (i.e. the simulation box). Authors in [36] argued that previous ratios should tend towards zero for an accurate prediction and a good representation of the mechanical behavior of each asperity. However, if these criteria are no longer met, in particular the second ratio criterion, it is necessary to conduct several stochastic simulations so that the average prediction is relevant. To overcome these drawbacks, authors developed more attractive numerical techniques allowing to discretize only the rough surface but not the bulk. These methods are based on the Green Fuctions (GF) (fundamental solution) and were able to refine the mesh in order to perform more accurate simulation and thus, predict a reliable contact area—pressure law [42–44]. The first development of efficient Boundary Element Method (BEM) solvers goes back to the 1990s when Brandt et al. [45] suggested the multi-level multi-summation (MLMS) scheme allowing to simulate realistic rough surface within a reasonable computational cost. In the same context, Venner et al. [46] combined the MLMS solver with the full multi-grid (FMG) iteration method to solve elasto-hydrodynamically lubricated contact problem. It should be noted that the coupled scheme MLMS-FMG solver failed to converge when solving the dry contact problem with roughness [47] since the corresponding convergence theorems do not handle problems with multiple inequality constraints. In order to enhance the efficiency of the numerical scheme and to optimize the CPU cost, Polonsky et al. [42] enhanced MLMS solver by using a iteration scheme based on the conjugate gradient (CG) method showing good convergence properties for a 3D rough surface with large numbers of nodes (approximately 10<sup>6</sup> nodes). Nogi et al. [48] and Polonsky et al. [49] have considered the Fast Fourier Transform (FFT) for solving rough contact problems for both homogeneous and layered solids. They demonstrated that the FFT-based method can increase dramatically the computational speed. Recently, Bemporad et al. [44] introduced the Non-Negative Least Squares (NNLS) algorithm with a suitable warm start based on accelerated gradient projections (GPs). The results in [44] clearly prove that the NNLS-GPs solver is faster than the previously mentioned solvers. It should be noted that the standard BEM is based on the linear elasticity assumption for a homogeneous medium. Hence, it can not handle the contact problem of rough boundaries including geometrical, interfacial, and material nonlinearities such as large deformation, adhesion, or plasticity. Sometimes, BEM can be adjusted to tackle the abovementioned nonlinearities [50–53] but for convenience in this type of problem, authors preferred FEM instead of BEM [54, 55].

This review paper provides a comprehensive comparison of the abovementioned models with particular attention to the standard Boundary Element Method (BEM), which is based on the Green functions. The concern of the present study is limited to solve the frictionless rough contact problem of micro-systems with purely brittle behavior (i.e. in the elastic regime). Thus, the chapter is organized as follows. First, we start to recall and describe 4 theories, namely (i) the original GW model in its continuous and discrete form, (ii) the modified GW to take into account interaction between the contacting asperities, (iii) Persson theory, and (iv) the BEM. Then, the methods will be applied to a 1 μm-thick gold rough surfaces. The first one is measured by the atomic force microscopy (AFM) technique while the others are numerically generated using a suitable algorithm taking as an input the statistical functions of the measured Au rough surface (i.e. HDF and PSDF). Finally, the chapter will be closed with an analysis and discussion of the predicted contact area and pressure.
