Greenwood And Williamson Research Articles

Contact interface damping effect on internal friction and rotordynamic instability

Rotating machinery shafting is typically comprised of multiple components that are assembled to ensure reliable operation, proper alignment minimal vibration. However, conventional rotordynamics approximates the shafting as a single, continuous member, neglecting contact interfaces between the components. The presence of an interface can induce microslip, which generates internal friction that may cause instability and machinery failure. A novel approach of modeling the interface viscous damping effect on rotordynamics is proposed by combining a GW (Greenwood and Williamson) contact model with the Yoshimura damping model. All structural components are modeled using 3D solid finite elements. Modal damping ratio is utilized to identify the instability onset speed (IOS). The results show that internal friction has a destabilizing effect on whirl motion above the first critical speed, but has a stabilizing effect on the motion below the first critical speed. The destabilizing effect can be reduced by increasing the bearing damping, however excessive bearing damping can drive the effective damping towards negative values. Increasing the number of interfaces reduces stability while increasing an interface preload prevents microslip, and increases stability. Lastly, smoother surfaces at the interfaces increase the IOS.

Adhesive contact of randomly rough surfaces: experimental and numerical investigations

The contact mechanics of soft matters is strongly affected by short-range adhesive interactions, which can lead to large deformations and contact instabilities.In this work, we present both experimental and numerical investigations of the adhesive contact between soft elastic bodies with a Greenwood and Williamson (GW)-like roughness. To investigate the coalescence of neighbour contact spots, surfaces have been designed with overlapping spherical asperities. Normal contact experiments are carried out by using a home-built device. Numerical simulations are performed with the Interacting and Coalescing Hertzian Asperities (ICHA) model, conveniently modified to take account of adhesion according to the Johnson, Kendall & Roberts (JKR) theory.

The shear stiffness criterion for rock joints considering rock wear behaviour

Rock is a material that is affected by wear, and the curvature of the asperities on a rock joint surface increases with the degree of wear after shearing. Based on the Greenwood and Williamson (GW) model, a new model considering the change of asperity curvature is proposed to explain the wear behaviour of rock joints. First, the shear stiffness formula for a joint surface is derived when the asperity curvature is constant, which shows that the shear stiffness increases with increase of asperity curvature. According to the Mohr–Coulomb criterion, the yield position of a single asperity under normal force and tangential friction force is discussed. Then, the critical normal force for a single asperity at a specific friction coefficient is obtained, which shows that the normal force corresponds to the curvature radius of the asperity. A rough surface model with multi-level curvature radius is proposed. With increase of normal force, the higher-order asperities gradually fail and the curvature radius become larger. A specific pressure value excites a specific radius of curvature, and the larger the pressure, the larger the radius of curvature. The relation between the normal force and the curvature radius is proposed and a shear stiffness formula considering the change of curvature radius of the asperity is derived. The proposed model is verified on the basis of the published experimental results. The calculation results of the proposed model can reflect the test results well: for a given joint surface, with increase in normal force the joint surface gradually becomes smooth; for different joint surfaces, with increase in roughness, the joint surface is more easily smoothed.

Exact Spectral Moments and Differentiability of the Weierstrass-Mandelbrot Fractal Function

Abstract Fractal mathematics using the Weierstrass-Mandelbrot (WM) function has spread to many fields of science and engineering. One of these is the fractal characterization of rough surfaces, which has gained ample acceptance in the area of contact mechanics. That is, a single mathematical expression (the WM function) contains characteristics that mimic the appearance of roughness. Moreover, the “roughness” is “similar” across large dimension scales ranging from macro to nano. The field of contact mechanics is largely divided into two schools of thought: (1) the roughness of real surfaces is essentially random, for which stochastic treatment is appropriate, and (2) surface roughness can be reduced to fractal mathematics using fractal parameters. Under certain mathematical constraints, the WM function is either stochastic or deterministic. The latter has the appeal that it contains no randomness, so fractal mathematics may offer closed-form solutions. Spectral moments of rough surfaces still apply to both approaches, as these represent physical metrology properties of the surface standard deviation, slope, and curvature. In essence, spectral moments provide a means of data reduction so that other physical processes can subsequently be applied. It is well known, for example, that the contact model of rough surfaces, by Greenwood and Williamson (GW), depends on parameters that are direct outcomes of these moments. Despite the vast amount of publications on the WM function dedicated to surfaces, two papers stand out as originators, where the others mostly rework their results. These two papers, however, contain some omissions and approximations that may lead to gross errors in the estimation of the spectral moments. The current work revisits these papers and adds information, but departs in the mathematical treatment to derive exact expressions for the said moments. Moreover, it is said that the WM function is nondifferential. That is also revisited herein, as another approach to derive the spectral moments depends on such derivatives. First, the complete mathematical treatment of the WM function is made, then the spectral moments are derived to yield exact forms, and finally, examples are given where the physical meanings of the approximate and exact moments are discussed and their values are compared. Numerical procedures will be introduced for both, and the effectiveness of the computational effort is discussed. One numerical procedure is particularly effective for any digitized signal, whether that originates from analytical functions (e.g., WM) or real surface measurements.

An experimental method for quantitative analysis of real contact area based on the total reflection optical principle**Project supported by the National Natural Science Foundation of China (Grant No. 11872033) and the Beijing Natural Science Foundation, China (Grant No. 3172017).

The simulation of real contact area between materials is foundationally important for the contact mechanics of mechanical structures. The Greenwood and Williamson (GW) model and the Majumdar (MB) model are the basic models in this field, which are widely accepted and proven to be valid in many experiments and engineering. Although the contact models have evolved considerably in recent years, the verifications of the models are most based on the indirect methods such as electrical conductivity and contact stiffness, because of the lack of effective methods to directly measure the variation of contact surface. In this paper, the total reflection (TR) method is introduced into the verification of contact models. An experiment system based on TR method is constructed to measure the real contact area of two PMMA specimens. The comparison analysis between the results of experiment and models suggests that the experiment result has the same trend with simulation, the MB model has better agreement with the experimental result because this method can take into account the variation of radius and the merging of asperities, while the GW model has a huge deviation because of the dependence on resolution and the lack of considering the variation of radius and asperity’s merging process. Taking the interaction of asperities into account could give a better result that is closer to the experiment. Our results and analysis prove that the experimental methods in this paper could be used as a more direct and valid method to quantitatively measure the real contact area and to verify the contact models.

Investigation into the Normal Contact Stiffness of Rough Surface in Line Contact Mixed Elastohydrodynamic Lubrication

ABSTRACT In this work, the statistical asperity microcontact models in combination with the acoustic spring model and the load sharing concept are utilized to study the interfacial normal contact stiffness for a rough surface in line contact elastohydrodynamic lubrication (EHL). Two different statistical microcontact models of Greenwood and Williamson (GW) and Kogut and Etsion (KE) are employed to derive the normal contact stiffness expressions for a dry rough line contact considering the purely elastic contact and the multiple regimes elastic–elastoplastic–fully plastic contact, respectively. The liquid film stiffness is calculated based on the relationship between film thickness and bulk modulus of the lubricant. The lubricant film thickness equations are employed in conjunction with the load sharing concept and the empirical formulas for the maximum contact pressure in a dry rough contact are fitted for the GW model and the KE model, to evaluate the relationship between film thickness and motion velocity for the purely elastic GW microcontact model and the multiregime KE microcontact model, respectively. The comparison with experimental results shows that the KE model predicts closer total contact stiffness results than the GW model. The stiffness contributions from the solid asperity contact and lubricant film are obtained and effects of surface roughness, applied load, motion velocity, and type of lubricant on the normal contact stiffness are analyzed.

Statistical model of rough surface contact accounting for size-dependent plasticity and asperity interaction

The work by Greenwood and Williamson (GW) has initiated a simple but effective method of contact mechanics: statistical modeling based on the mechanical response of a single asperity. Two main assumptions of the original GW model are that the asperity response is purely elastic and that there is no interaction between asperities. However, as asperities lie on a continuous substrate, the deformation of one asperity will change the height of all other asperities through deformation of the substrate and will thus influence subsequent contact evolution. Moreover, a high asperity contact pressure will result in plasticity, which below tens of microns is size dependent, with smaller being harder. In this paper, the asperity interaction effect is taken into account through substrate deformation, while a size-dependent plasticity model is adopted for individual asperities. The intrinsic length in the strain gradient plasticity (SGP) theory is obtained by fitting to two-dimensional discrete dislocation plasticity simulations of the flattening of a single asperity. By utilizing the single asperity response in three dimensions and taking asperity interaction into account, a statistical calculation of rough surface contact is performed. The effectiveness of the statistical model is addressed by comparison with full-detail finite element simulations of rough surface contact using SGP. Throughout the paper, our focus is on the difference of contact predictions based on size-dependent plasticity as compared to conventional size-independent plasticity.

Towards stochastic discrete element modelling of spherical particles with surface roughness: A normal interaction law

The current work is the first attempt towards establishing a stochastic discrete element modelling framework by developing a normal contact interaction law based on the classic Greenwood and Williamson (GW) model for spheres with rough surfaces. Two non-dimensional forms of the model that have a substantial impact on the computational efficiency are discussed and the theoretical relationship between the GW model and the Hertzian model for smooth spheres is formally established. Due to the inter-dependence between the contact pressure and deformation distributions in the model, a Newton–Raphson based iterative solution procedure is proposed to effectively and accurately obtain the contact force in terms of the overlap and two surface roughness parameters. The related key components of the procedure are addressed in detail. The numerical results obtained are first validated and then curve-fitted to derive an empirical formula as a new normal interaction law for spheres with surface roughness. The explicit nature of the new interaction law makes it readily be incorporated into the current discrete element modelling framework. A simple example is presented to illustrate the effect of surface roughness on the packing behaviour of a particle assembly.

Effect of roughness on surface force distributions measured by newly developed surface force apparatus with ultra-high accuracy

A pull-off force between a sphere and a flat plate is precisely investigated using a newly developed surface force apparatus (SFA). In this system, (1) a pull-off force between a spherical glass probe and a sample plate is measured in vacuum, (2) the probe is directly pulled off from the sample by an electromagnetic force, and (3) the pull-off force and displacement of the probe are measured with ultra-high resolution of 0.4 nN and 0.3 nm, respectively, which are electrical noise levels. The pull-off process, which appears as a part of force curve, is clearly measured by this system, and pull-off force is measured with high reproducibility and accuracy. Pull-off force distributions on flat surfaces of Si, SrTiO3, glassy carbon, and diamond-like carbon are measured. It is shown that despite the differences between these materials, for all four for them, the distribution strongly depends on the surface roughness, such that the relative standard deviation of pull-off force is proportional to the surface roughness ( Ra and Rz). The Greenwood and Williamson (GW) model is then used for the analysis of the pull-off force. Comparing with SFA experiments, it is shown that the pull-off force and separation displacement can be predicted by the GW model and that the pull-off force strongly depends on the standard deviation of surface height, causing the broad distribution of pull-off force at rough surfaces.

Friction Reduction in Lubricated Rough Contacts: Numerical and Experimental Studies

Combining the contact model of elastic-layered solid with the concept of asperity contact in elastohydrodynamic lubrication (EHL), a mixed-lubrication model is presented to predict friction coefficient over rough surfaces with/without an elastic-layered medium under entire lubrication regimes. Solution of contact problems for elastic-layered solids is presented based upon the classical model of Greenwood and Williamson (GW) in conjunction with Chen and Engel's analysis. The effects of the Young's modulus ratio of the layer to substrate and the thickness of the layer on the elastic real area of contact and contact load for a fixed dimensionless separation are studied using the proposed method, which is used for the asperities having contact with an elastic coating. Coefficient of friction with elastic-layered solids in boundary lubrication is calculated in terms of Rabinowicz's findings and elastic-layered solutions of Gupta and Walowit. The effect of rough contacts with an elastic layer on friction coefficient in lubrication regimes has been analyzed. Variations in plasticity index ψ significantly affect friction coefficients in boundary and mixed lubrications. For a large value of ψ, the degree of plastic contact exhibits a stronger dependence of the mean separation or film thickness than the roughness, and for a small value of ψ, the opposite result is true. The effect of governing parameters, such as inlet oil viscosity at ambient pressure, pressure–viscosity coefficient, combined surface roughness, and El/E2 on friction coefficient, has been investigated. Simulations are shown to be in good agreement with the experimental friction data.

Modeling of the Friction Behavior in Metal Forming Process Considering Material Hardening and Junction Growth

In various plastic forming processes of metals, friction has been revealed to play an important role in the determination of the material flow, fracture, and surface quality. The precise description of friction behavior is thus a critical issue for the accurate prediction and analysis of these formability indicators. Generally, the friction behavior is inevitably affected by material hardening and junction growth. However, few of the previous models have taken both of them into consideration, especially for the nonlinear hardening materials. In this study, the classical contact model was modified to include the power-law hardening material, and the general friction law combined with Tabor's equation was employed to estimate the friction stress with the junction growth of asperities. An asperity-based friction model for rough surfaces in metal forming process was then obtained by summarizing the normal and tangential stresses of all the asperities on the surface using Greenwood and Williamson (GW) method. The model was validated by comparing to the finite element (FE) results and the experimental results. And its comparison with Kogut and Etsion (KE) model and Cohen's model revealed a wider range of application for the present model. It was also found to be able to predict the friction coefficient and the real contact area of nonlinear hardening materials under various contact conditions. This work is helpful to understand the friction behavior and further guide the simulation and optimization of forming processes.

Partial-slip frictional response of rough surfaces.

If two elastic bodies with rough surfaces are first pressed against each other and then loaded tangentially, sliding will occur at the boundary of the contact area while the inner parts may still stick. With increasing tangential force, the sliding parts will expand while the sticking parts shrink and finally vanish. In this paper, we study the fractions of the contact area, tangential force and tangential stiffness, associated with the sticking portion of the contact area, as a function of the total applied tangential force up to the onset of full sliding. For the numerical analysis randomly rough, fractal surfaces are used, with the Hurst exponent H ranging from 0.1 to 0.9. Numerical simulations by boundary element method are compared with an analytical analysis in the framework of the Greenwood and Williamson (GW) model. In both cases, a universal linear dependency between the real contact area fraction in stick condition and the applied tangential force is found, regardless of the Hurst exponent of the rough surfaces. Regarding the dependence of the differential tangential stiffness on the tangential force, a linear relation is found in the GW case. For randomly rough surfaces, a nonlinear relation depending on H is derived.

Effects of Bending Moments and Pretightening Forces on the Flexural Stiffness of Contact Interfaces in Rod-Fastened Rotors

The rod-fastened rotor (RFR) is comprised of a series of discs clamped together by a central tie rod or several tie rods on the pitch circle diameter. The equivalent flexural stiffness of contact interfaces Kc in the RFR is the key concern for accurate rotor dynamic performance analysis. Each contact interface was modeled as a bending spring with a stiffness of Kc and a hinge in this study. The contact states of the contact interfaces, which depend on the pretightening forces and bending moments (static), have effects on Kc. The approach to calculating Kc in two contact states is presented. The first contact state is that the whole zone of the contact interface is in contact; Kc is determined by the contact layer, which consists of asperities of the contact surfaces. Hertz contact theory and the Greenwood and Williamson (GW) statistical model are used to calculate the equivalent flexural stiffness of the contact layer Kcc. The second contact state is that some zones of the contact interface are separated (when the bending moment is relatively large); the equivalent flexural stiffness of the rotor segment Ksf (not including Kcc) decreases, as the material in the separated zone has no contribution to the bending load-carrying capacity of the rotor. The strain energy, which is calculated by the finite element method (FEM), is used to determine Ksf. The stiffness Ksf is equivalent to the series stiffness of the discs of the rotor segment with flexural stiffness of Kd and a spring with bending stiffness of Kcf in the location of the contact interface, so Kc is equal to the series stiffness of Kcc and Kcf in the second contact state. The results of a simplified RFR indicate that, for a fixed pretightening force, Kcc decreases with bending moments in the first contact state, whereas increases with bending moments in the second contact state. In addition, Kcf and Kc decrease abruptly with the increase of bending moments in the second contact state when the rotor is subjected to a relatively large pretightening force. Finally, the multipoint exciting method was used to measure the modal parameters of the experimental RFR. It is found that the experimental modal frequencies decrease as the pretightening force decreases.

On the Modeling of Elastic Contact between Rough Surfaces

The contact force and the real contact area between rough surfaces are important in the prediction of friction, wear, adhesion, and electrical and thermal contact resistance. Over the last four decades various mathematical models have been developed. Built on very different assumptions and underlying mathematical frameworks, model agreement or effectiveness has never been thoroughly investigated. This work uses several measured profiles of real surfaces having vastly different roughness characteristics to predict contact areas and forces from various elastic contact models and contrast them to a deterministic fast Fourier transform (FFT)-based contact model. The latter is considered “exact” because surfaces are analyzed as they are measured, accounting for all peaks and valleys without compromise. Though measurement uncertainties and resolution issues prevail, the same surfaces are kept constant (i.e., are identical) for all models considered. Nonetheless, the effect of the data resolution of measured surface profiles will be investigated as well. An exact closed-form solution is offered for the widely used Greenwood and Williamson (GW) model (Greenwood and Williamson, Proceedings of the Royal Society of London A, vol. 295, pp. 300–319), along with an alternative definition of the plasticity index that is based on a multiscale approach. The results reveal that several of the theoretical models show good quantitative and qualitative agreement among themselves, but though most models produce a nominally linear relationship between the real contact area and load, the deterministic model suggests otherwise in some cases. Regardless, all of the said models reduce the complicated surface profiles to only a few key parameters and it is therefore unrealistic to expect them to make precise predictions for all cases.

The coefficient of proportionality κ between real contact area and load, with new asperity models

Abstract Most recent numerical works on fractal surfaces have simply compared the low load limit of the coefficient of proportionality κ of the relationship between real contact area and load. In particular, that provided by Persson's theory, and that obtained from the Bush, Gibson and Thomas (BGT-A) asperity contact theory, which is a generalized form of the Greenwood and Williamson (GW) one. The two theories differ only by a numerical constant κ = 8 / π ≈ 1.6 vs κ = 2 π ≈ 2.5 , but neither of the two provide an accurate prediction, Persson's value being generally too low, and BGT-A's limit being only valid for extremely large separations. A detailed numerical comparison using a range of generated fractal surfaces permits to compare the existing models, finding for example that bandwidth is more important than Gaussianity of the surfaces. Then, we propose two new theoretical equations generalizing GW and BGT to take into account interaction effects in an approximate way (GW-I and BGT-I, respectively), which significantly improve the accuracy of original asperity models. Further, as a practical alternative to the tribologist, we suggest a new very simple discrete form of the GW model (called GW-RI) whose accuracy is similar to BGT-I, but with much lower computational cost, comparable to analytical solutions since the latter require the evaluation of the variance of the profile slopes, σ m 2 , with a surface defined at a given set of points. The GW-RI model additionally avoids an ambiguity over how to define numerically the variance of the profile slopes, σ m 2 .

A slightly corrected Greenwood and Williamson model predicts asymptotic linearity between contact area and load

The author presents a very simple and slightly corrected version of the well-known Greenwood and Williamson (GW) model which closely follows the predictions of more complicated and computationally expensive theory such as the Bush, Gibson and Thomas (BGT) theory [Bush, A.W., Gibson, R.D., Thomas, T.R., 1975. Wear 35, 87]. This new model (which I call GW modified in the following) still treats the asperities of the rough surface as spherical cups, but, this time, the curvature of spheres is not constant and instead depends on the asperity height. The GW modified theory is, in particular, able to predict, in the limiting case of large separations, the same asymptotic linearity between contact area and load as in the BGT theory. This, in turn, proves that the BGT asymptotic linear area–load relation is not a consequence of having taken into account that the contact between the asperities is actually elliptic and, therefore, of having included the spread of asperity curvature at a given height (which incidentally makes the treatment very complicated), but a consequence of having included only the influence of asperity heights on the curvature distribution of the summits. I also give a simple explanation for why the GW modified model and the BGT theory follow exactly the same asymptotic behavior. Indeed, I show that the surface summits can be treated as perfectly spherical cups (all those at the same height having the same radius of curvature) as their height is increased to very large values. In fact, in the asymptotic limit of large separation, the mean curvature of summits is shown to increase proportionally to the summit height, whereas the difference of the two principal curvatures approaches a finite constant value. The consequence of this is that the Hertzian contact between the approaching elastic (initially flat) half-space and the summits exactly resemble that between an elastic half-space and a sphere.

Asperity contact theories: Do they predict linearity between contact area and load?

During the last few years, the scientific community has been debating about which theory of contact between rough surfaces can be considered as the most accurate. The authors have been attracted by such a discussion and in this paper try to give their personal thought and contribution to this debate. We present a critical analysis of the principal contact theories of rough surfaces. We focus on the multiasperity contact models (which are all based on the original idea of Greenwood and Williamson (GW) [1966. Proc. R. Soc. London A 295, 300]), and also briefly discuss a relatively recent contact theory developed by Persson [2001. J. Chem. Phys. 115, 3840]. For small loads both asperity contact models and Persson's theory predict a linear relation between the area of true contact and the applied external load, but the two theories differ for the constant of proportionality. However, this is not the only difference between the two approaches. Indeed, we show that the fully calculated predictions of asperity contact models very rapidly deviates from the predicted linear relation already for very small and in many cases unrealistic vanishing applied loads and contact areas. Moreover, this deviation becomes more and more important as the PSD breadth parameter α (as defined by Nayak) increases. Therefore, the asymptotic linear relation of multiasperity contact theories turns out to be only an academic result. On the contrary, Persson's theory is not affected by α and shows a linear behavior between contact area and load up to 10–15% of the nominal contact area, i.e. for physical reasonable loads. The authors also prove that, at high separation, all multiasperity contact models, which take into account the influence of summit curvature variation as a function of summit height, necessarily converge to a (slightly) improved version of the GW model, which, therefore, remains one of the most important milestones in the field of contact mechanics of rough surfaces.

Inclusion of “interaction” in the Greenwood and Williamson contact theory

Recent direct implementation of asperity theories is reinterpreted here to formulate an improved version of the Greenwood and Williamson (GW) theory with the inclusion of interaction between asperities. This is achieved by treating the contact pressures as uniformly distributed over the apparent contact area and the resulting deformation as uniform. The correction is equivalent to an increase of the effective separation of the mean planes by a quantity proportional to the nominal pressure, resulting in a reduction of the “real” area of contact and of total load for a given separation. However, the area–load relationship is unchanged. The correction effectively depends on the ratio between the nominal pressure and the elastic modulus multiplied by the ratio between the size of the nominal contact area and standard deviation of the asperity heights. For contacts much larger than the size of roughness, uniform interaction effects would be dominant at relatively modest pressures (particularly for soft materials). This also means that the effect of interaction is unlimited. However, the only significant change is in the prediction of gas-tightness, it is harder to seal a large area than a small one. The modification of the theory has a significant effect on stiffness and conductance. Indeed, a parallel is drawn between this correction and the “clustering” terms of resistance in the Holm–Greenwood formulae for a cluster of circular spots. Finally, numerical contact simulations using Weierstrass–Mandelbrot (WM) surfaces show a general agreement with the improved theory but also significant scatter for low load levels. Taking into account the effect of asperity interaction, the improved GW theory is now able to predict the numerically obtained contact response for intermediate load levels.

A multi-scale model for contact between rough surfaces

This work describes a non-statistical multi-scale model of the normal contact between rough surfaces. The model produces predictions for contact area as a function of contact load, and is compared to the traditional Greenwood and Williamson (GW) and Majumdar and Bhushan (MB) rough surface contact models, which represent single-scale statistical and fractal analyses, respectively. The current model incorporates the effect of asperity deformations at multiple scales into a simple framework for modeling the contact between nominally flat rough surfaces. Similar to the “protuberance upon protuberance” theory proposed by Archard, the model considers the effect of having smaller asperities located on top of larger asperities in repeated fashion with increasing detail down to the limits of current measurement techniques. The parameters describing the surface topography (areal asperity density and asperity radius) are calculated from an FFT performed of the surface profile. Thus, the model considers multi-scale effects, which fractal methods have addressed, while attempting to more accurately incorporate the deformation mechanics into the solution. After the FFT of a real surface is calculated, the computational resources needed for the method are very small. Perhaps surprisingly, the trends produced by this non-statistical multi-scale model are quite similar to those arising from the GW and MB models, but seem largely unaffected by the sampling resolution at the employed surface data.

Contact of Rough Surfaces With Asymmetric Distribution of Asperity Heights

The Greenwood and Williamson (GW) statistical approach of characterizing rough surfaces is extended to include asymmetric distribution of asperity heights using the Weibull distribution. A key parameter that is used to characterize asymmetry is the skewness, and the corresponding Weibull parameters are investigated for a range of practical skewness values. The Weibull distribution is then adopted to model the asperity heights, and once normalized, is used to calculate the contact load, real area of contact and number of contacting asperities using the CEB elastic-plastic model of an equivalent rough surface in contact with a smooth plane. The effect of skewness on different levels of surface roughness, ranging from very smooth surfaces encountered in microtribological applications to rougher surfaces encountered in macrotribological applications is investigated, and also compared to the symmetric Gaussian case. Also, to allow for closed-form solution of the contact equations, simpler exponential distributions are curved-fitted to the contact side of the Weibull distribution, and the analytical results are favorably compared with the numerical results using the Weibull distribution.