Elastic Contact Of Rough Surfaces Research Articles

Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales

This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others.

Analysis of Normal Elastic Contact Stiffness of Rough Surfaces Based on Ubiquitiform Theory

Abstract In this study, the normal stiffness of elastic contact between rough surfaces with asperities following Gaussian distribution is investigated using ubiquitiform theory, developed from fractal theory. In the generalized ubiquitiformal Sierpinski carpet model, the rough surface including contact asperity is controlled for, given the lower bound to scale invariance of rough surfaces. Considering the stiffness of a single asperity deduced from the Hertz contact model, we deduce the theoretical relation between the normal stiffness and the elastic contact of rough surfaces based on ubiquitiform theory. The results show that the normal contact stiffness of a rough surface increases as the normal load rises. If the ubiquitiformal complexity of a rough surface increases or the lower bound to scale invariance of a rough surface decreases, the normal contact stiffness of the rough surface should increase. The larger the ubiquitiformal complexity of a rough surface is, the more obvious the impact of the lower bound to scale invariance on the normal contact stiffness of the rough surface becomes. The results based on the ubiquitiformal model and the experimental results are in closer agreement. Therefore, the introduction of scale invariance is crucial to the surface contact problem.

Interaction of Acoustic Waves in The Media with Quadratically Bimodular Nonlinearity

We obtain the quadratically bimodular equation of state for a rod with cracks on the basis of a model of the crack as an elastic contact of rough surfaces of solids. The perturbation method is used to theoretically study the propagation and interaction of collinear longitudinal elastic (strong low-frequency and weak high-frequency) waves in such media. Expressions for the amplitudes of the secondary waves, namely, the second and the fourth harmonics of the strong wave, the second harmonic of the weak wave, and the combination-frequency waves are obtained.

Statistical contact model of rough surfaces: The role of surface tension

The sizes of most asperities on actual rough surfaces range from nanometers to microns. Recent investigations reveal that surface tension plays an important role in micro-/nano-sized contact. Therefore, in this paper, we address the influence of surface tension on the elastic contact of rough surfaces. Nayak's model is employed to describe rough surface with stochastic distribution of both heights and curvatures. As a fundament, we present the general relations of contact area and load as the function of indent depth for a single micro-/nano-sized asperity by accounting for surface tension. Based on the refined relations, we derive the general relations concerned in the contact of rough surfaces. Compared with the conventional multi-asperity contact models based on the Hertzian contact theory, the existence of surface tension decreases the real contact area and requires higher load to generate a given indent depth. It is found that the area of intimate contact is still approximately proportional to the external load, but the proportionality depends on the surface tension.

The role of the roughness spectral breadth in elastic contact of rough surfaces

We study frictionless and non-adhesive contact between elastic half-spaces with self-affine surfaces. Using a recently suggested corrective technique, we ensure an unprecedented accuracy in computation of the true contact area evolution under increasing pressure. This accuracy enables us to draw conclusions on the role of the surface’s spectrum breadth (Nayak parameter) in the contact area evolution. We show that for a given normalized pressure, the contact area decreases logarithmically with the Nayak parameter. By linking the Nayak parameter with the Hurst exponent (or fractal dimension), we show the effect of the latter on the true contact area. This effect, undetectable for surfaces with poor spectral content, is quite strong for surfaces with rich spectra. Numerical results are compared with analytical models and other available numerical results. A phenomenological equation for the contact area growth is suggested with coefficients depending on the Nayak parameter. Using this equation, the pressure-dependent friction coefficient is deduced based on the adhesive theory of friction. Some observations on Persson’s model of rough contact, whose prediction does not depend on Nayak parameter, are reported. Overall, the paper provides a unifying picture of rough elastic contact and clarifies discrepancies between preceding results.

Effect of fine-scale roughness on the tractions between contacting bodies

Persson's theory for the elastic contact of rough surfaces is modified to include the compliance associated with an interface force law such as the Lennard-Jones law. We determine the effect of adding a small packet of waves on the probability distribution function [PDF] of the local interfacial gap (including the effect of elastic deformation). This procedure is then used iteratively to develop an algorithm for determining the PDF for a rough surface with a prescribed power spectral density. The results show that for untruncated quasi-fractal surfaces, the PDF then converges at large wavenumber, in contrast to the result when only elastic deformation is taken into account.If the roughness is restricted to wavenumbers greater than a certain critical value, the algorithm predicts a converged relation between nominal traction and mean gap that can be regarded as a modified interfacial force law describing the influence of just the fine-scale roughness on the contact. In particular, the area under the resulting curve can be interpreted as a measure of interface energy as reduced by this roughness. The remaining macroscopic features of the surface can then be described using the JKR methodology in combination with this modified interface energy.

Comment on Leonardo da Vinci’s Friction Experiments: An Old Story Acknowledged and Repeated

It is very impressive to read the most recently published reproduction of Leonardo da Vinci’s friction experiments [1]. In Table 1 of this paper, the authors reported measured average values of static friction coefficient for various roughness levels of three types of wood surfaces. These values are between 0.25 and 0.29 for rough (Ra [ 3000 nm) surfaces, and they increase up to 0.72 for smooth (Ra = 200 nm) surfaces. Moreover, the lowest friction coefficient value of 0.25 is in agreement with da Vinci’s friction experiments from 500 years ago. In these experiments, da Vinci also found that the friction coefficient is independent of the apparent contact area and the applied load. An interesting question is whether such effects of surface roughness on friction coefficient can be predicted by a theoretical model. A first hint in this direction was probably provided by Greenwood and Williamson [2] in 1966. In this seminal paper, a model (known as the GW model) for the elastic contact of rough surfaces is provided, showing that the real contact area is independent of the apparent contact area and is linearly proportional to the applied load. A plasticity index having the general form w = (E/H)(r/r), which was first introduced in the GW model, provides a measure of the plasticity level of the rough surface contact. In this index, E and H indicate modulus of elasticity and hardness, respectively, while r and r represent, respectively, standard deviation of asperity heights and average tip curvature of the asperities of the rough surface contact. The softer and rougher the rough surface is, the larger the plasticity index is. The first attempt to theoretically predict static friction of contacting rough surfaces was probably made by Chang et al. in 1988 [3]. In this model, it was shown that at low to moderate values of the plasticity index the static friction coefficient l decreases with increasing applied load, but it becomes practically independent of the applied load as the plasticity index becomes large enough. Due to the oversimplifying assumptions made in this first crude model, the values of the static friction coefficient were severely underestimated already at w = 2.5. The model of Ref. [3] has been gradually improved over the years by refining its original assumptions [4] and gaining further insight on how to handle a single spherical asperity under combined normal and tangential loading [5]. The static friction coefficient model of Ref. [5] was further validated experimentally in Ref. [6]. Finally, based on the above improvements, Cohen et al. [7] presented a static friction model to cover plasticity index values up to w = 8, and more recently, Li et al. [8] extended this model up to w = 32. The results of Ref. [8] are shown in Fig. 1 in solid lines (in comparison with those from Ref. [7] in dashed line). An empirical relation of the static friction coefficient l as a function of the dimensionless applied load and plasticity index that was offered in [8] is given by Eq. (1) in the form

A self-consistent model for the elastic contact of rough surfaces

The interaction of asperities plays an important role in the contact of rough surfaces. This paper develops a self-consistent contact model, in which the effect of asperity interaction is accounted for by applying the mean pressure around a representative asperity. Based on the Boussinesq’s solution of a point force acting on an elastic half-plane, the problem is transformed into a singular integral equation. Using the Gauss–Legendre quadrature formula, we solve the integral equation numerically. The results demonstrate that when the ratio between the real contact area and the nominal one is small, the effect of asperity interaction is negligible and the present mode coincides with the Greenwood–Williamson model. However, when the area ratio gets larger, the interaction of asperities becomes prominent. For a given ratio between the real contact area to the nominal one, the self-consistent contact model predicts a higher load than the Greenwood–Williamson model, in agreement with relevant experimental results.

Elastic contact of rough surfaces: A simple criterion to make 2D isotropic roughness equivalent to 1D one

We analyze the periodic contact between an elastic half-space and two types of rough substrates: (i) a perfect isotropically rough rigid substrate (2D isotropic roughness), and (ii) a perfect anisotropically rough rigid substrate, i.e. a substrate with roughness in only one direction (1D roughness). The analysis is carried out with the aid of proprietary codes, that we have developed (both in real and Fourier space) to deal with this type of contacts. Of course, 1D contacts differ from 2D isotropic contacts. However, our results and theoretical arguments suggest a possible criterion to make 2D contacts equivalent to 1D ones from the point of view of contact area and separation calculations. The rule consists in replacing the 2D power spectral density (PSD) of the isotropic surface into an equivalent 1D PSD. Interestingly the transformation rule does not depend on the statistical properties of the surface roughness, hence seems to have a universal character for isotropic surfaces.

Analytical and Numerical Models for Tangential Stiffness of Rough Elastic Contacts

The paper considers elastic contact of rough surfaces and develops a simple analytical expression for the stiffness of the contact under tangential loading, which predicts that the contact stiffness is proportional to normal load and independent of Young’s Modulus. The predictions of this model are compared to a full numerical analysis of a rough elastic contact of finite size. The two approaches are found to be in good agreement at low loads, when the asperity spacing is large, but the numerical approach predicts much lower stiffnesses at medium and high loads. It is shown that the overall stiffness cannot exceed that of the equivalent smooth contact, and a simple means of modifying the analytical approach is proposed and validated.

Adhesive Elastic Contact of Rough Surfaces with Power-Law Axisymmetric Asperities

The contact model for rough surfaces with power-law axisymmetric asperities in the presence of adhesion is developed. The extended JKR adhesive contact model for power-law axisymmetric asperities, denoted as JKR-n, developed by Zheng and Yu (2007,“Using the Dugdale Approximation to Match a Specific Interaction in the Adhesive Contact of Elastic Objects,” J. Colloid Interface Sci., 310, pp. 27–34) is utilized to investigate the adhesive contact of rough surfaces. The JKR-n adhesive contact model generalizes the most adopted JKR model for spherical objects of n = 2. This work compares the effect of surface roughness on the adhesion force for rough surfaces with various power-law axisymmetric asperities. It is found that shapes of the asperities influence the pull-off forces greatly during the separation of rough surfaces. A general adhesion parameter that includes the shape index of asperities is proposed, and it can be used to characterize the adhesion performance of rough surfaces.

Closed‐Form Equations for Contact Force and Moment in Elastic Contact of Rough Surfaces

It is reasonable to expect that, when two nominally flat rough surfaces are brought into contact by an applied resultant force, they must support, in addition to the compressive load, an induced moment. The existence of a net applied moment would imply noneven distribution of contact force so that there are more asperities in contact over one region of the nominal area. In this paper, we consider the contact between two rectangular rough surfaces that provide normal and tangential contact force as well as contact moment to counteract the net moment imposed by the applied forces. The surfaces are permitted to develop slight angular misalignment, and thereby contact moment is derived. Through this scheme, it is possible to also define elastic contribution to friction since the half‐plane tangential contact force on one side of an asperity is no longer balanced by the half‐plane tangential force component on the opposite side. The elastic friction force, however, is shown to be of a much smaller order than the contact normal force. Approximate closed‐form equations are found for contact force and moment for the contact of rough surfaces.

Elastic Contact of Rough Surfaces Subject to Combined Force and Moment

In this paper we consider the contact between two rectangular rough surfaces that provide normal and tangential contact forces, as well as contact moment, to counteract the net moment imposed by the applied forces. The surfaces are permitted to develop a slight angular misalignment, and thereby contact moment is derived. Through this scheme it is possible to also define elastic contribution to friction, since the half-plane tangential contact force on one side of an asperity is no longer balanced by the half-plane tangential force component on the opposite side. The elastic friction force, however, is shown to be of a much smaller order than the contact normal force.

On the influence of surface roughness on real area of contact in normal, dry, friction free, rough contact by using a neural network

A model previously developed at LTU was used in order to perform numerical simulations of normal, dry, friction free, linear elastic contact of rough surfaces. A variational approach was followed and the FFT-technique was used to speed up the numerical solution process. Five different steel surfaces were measured using a Wyko optical profilometer and several 2D profiles were taken. The real area of contact and the pressure distribution over the contact length were calculated for all the 2D profiles. A new slope parameter was defined. An artificial neural network was applied to determine the relationship between the roughness parameters and the real area of contact. The trained model was able to capture the dependence of the real area of contact on the roughness parameters. The ability of the neural network to generalize on unseen data was tested. The neural network was able to prove the correlation between the roughness parameters and the real area of contact.

On the elastic contact of rough surfaces: Numerical experiments and comparisons with recent theories

Some numerical experiments are conducted for studying the decrease of the elastic contact area in the elastic contact of fractal random surfaces when adding components of roughness of progressively smaller wavelengths. In particular, Fourier and Weierstrass random series are used, and a recent accurate and efficient method developed by the authors is used, involving superpositions of overlapping triangles. Some comparisons are made using two recent theories, that of Ciavarella et al. published in 2000 on the deterministic Weierstrass fractal profile, and that of Persson published in 2001 on random generic contact. We show that both theories tend to underpredict the contact area by a significant (and similar) factor in these representative cases in the region of light loads (partial contact), where the non-linearities of the contact mechanics are not included in neither of the formulations. In Persson's theory case, the discrepancy is particularly large at high fractal dimensions of the profile, where in theory the numerical experiments should be more closely reproducing a true Gaussian process. The Ciavarella et al. “Archard-like” theory, is only accurate when the parameter γ (the ratio of successive wavelengths) is unrealistically large. However, we only tested the Ciavarella et al. theory in the simplified “Hertzian approximation” form assuming partial contact at the peaks of contact, although we don’t expect the full version to improve dramatically the results.

Solving Elastic Contact Between Rough Surfaces as an Unconstrained Strain Energy Minimization by Using CGM and FFT Techniques

An accurate, robust, and efficient method for elastic contact of rough surfaces is presented. The method converts the contact problem into an unconstrained energy minimization problem. The equality load equilibrium constraint and the inequality complementary constraints are placed in the objective function and enforced simultaneously in search of a feasible solution. Conjugate gradient method is used in conjunction with FFT technique to solve the unconstrained energy minimization problem. To overcome periodical extension error associated with discrete Fourier transform when solving a nonperiodical contact problem, the solution is decomposed into two parts: a smooth part and a fluctuation part. The smooth part is solved analytically and the fluctuation part is solved by the proposed method. The net load or surface traction of the fluctuation part is set to zero to minimize the periodical extension error. Numerical examples demonstrate that the proposed method is accurate, robust, and efficient.

Numerical Analysis for the Elastic Contact of Real Rough Surfaces

The elastic contact of rough surfaces and the subsurface stresses caused by the contact have been analyzed by means of a numerical model based on fast Fourier transforms (FFT) and minimization of complementary energy. The elastic contact has been modeled mathematically as a linear complementarity problem and solved by a robust algorithm, Conjugate Gradient Method, while the force-displacement relation is determined through a FFT approach. After solving for the pressure distribution, the subsurface stress field is obtained by calculating the stresses due to the application of a point force, and then integrating over the contact region. In comparison with the matrix based method published in recent years, the numerical approach presented in this study is more efficient, more stable and requires less memory. It has great potential in application to problems with general contact geometry and three-dimensional surface roughness. The results show that high frequency roughness could lead to very sharp impulses and significant oscillations in pressure distribution. The amplitude and location of the maximum shear stress in the subsurface region are constantly changing when the contacting rough surfaces are in relative motion. Presented at the 53rd Annual Meeting in Detroit, Michigan May 17–21, 1998

Mid-life scuffing failure in automotive cam-follower contacts

There are several practically important engineering systems that fail by scuffing long after the initial running-in has been completed. A common feature of these systems is repeated contact between the same points on the mating surfaces. The processes leading up to such mid-life scuffing failures have been examined by monitoring the development of cam follower surface roughness and wear at regular intervals during a series of valve train wear tests in a fired engine. By the application of both statistical and numerical models for the elastic contact of rough surfaces, the following stages in the scuffing process have been identified: (a) surface roughening in the mild-wear regime, which progressively increased maximum asperity contact pressures until (b) the elastic shakedown limit was exceeded, causing plastic deformation wear, accelerated roughening and enlargement of valleys, leading eventually to (c) a transition to high rates of wear, probably resulting from hydrodynamic pressure loss and oil-film collapse.

On the contact and adhesion of rough surfaces

The elastic contact of rough surfaces has been studied by Greenwood and Williamson [Proc. R. Soc. London, Ser. A 295, 300 (1966)] for Hertzian contacts and by Fuller and Tabor [Proc. R. Soc. London, Ser. A 345, 327 (1975)] for Johnson-Kendall-Roberts contacts. The theory for Deryagin-Muller-Toporov contacts is proposed here and compared with previous results. An extra load due to adhesion forces acting around the contacts appears, which can account for the friction under negative load and the increase in the apparent friction coefficient on clean surfaces.

The Elastic Contact of Rough Surfaces and its Importance in the Reduction of Wear

The paper considers the conditions for the purely elastic contact of rough surfaces and shows that by considering the effect of the truncation of the asperity distribution, such elastic contact is greater at low nominal pressures than has been hitherto recognized. Experimental studies to define the limits of purely elastic contact confirm the theoretical predictions. It is further shown that purely elastic contact may be obtained up to loads of some 10 000 N/mm2 by using ceramic coatings of titanium nitride deposited by the plasma assisted vapour deposition process.