A simple theory of asperity contact in elastohydro-dynamic lubrication

The Greenwood and Williamson theory of random rough surfaces in contact has been combined with established elastohydrodynamic theory to provide a theoretical approach to highly loaded lubricated contacts in which the load is shared between hydrodynamic pressure and asperity contact. It is shown that, provided a major part of the load is carried by elastohydrodynamic action, the separation between the two rough surfaces is given (to a first approximation) by the film thickness which would exist between two smooth surfaces under the same conditions of load, speed and lubricant. It then follows that the asperity pressure, both real and apparent, is determined primarily by the ratio of theoretical film thickness to the combined roughness of the two surfaces (h_o/σ). A corollary of this result is that an increase in total load, which has only a small influence on the film thickness, is carried by an increase in fluid pressure and only gives rise to a small increase in asperity contact pressure.     

A statistical model of elasto-plastic asperity contact between rough surfaces

This work models statistically elasto-plastic contact between two rough surfaces using the results of a previous finite element analysis of an elasto-plastic sphere in contact with a rigid flat. The individual asperity contact model used accounts for a varying geometrical hardness effect that has recently been documented in previous works (where geometrical hardness is defined as the uniform pressure found during fully plastic contact). The contact between real surfaces with known material and surface properties, such as the elastic modulus, yield strength, and roughness are modeled. The asperity is modeled as an elastic-perfectly plastic material. The model produces predictions for contact area, contact force, and surface separation. The results of this model are compared to other existing models of asperity contact. Agreement exists in some cases and in other cases it corrects flaws, especially at large deformations. The model developed by Chang, Etsion and Bogy is also shown to have serious flaws when compared to the others. This work also identifies significant limitations of the statistical models (including that of Greenwood and Williamson).     

Transient Heat Conduction Between Rough Sliding Surfaces

When two rough bodies slide against each other, asperities on the opposing surfaces interact with each other, defining a transient contact and heat conduction problem. We represent each body by a Greenwood and Williamson asperity model with a Gaussian height distribution of identical spherical asperities. The heat transfer during a typical asperity interaction is analyzed, and the results are combined with the height distributions to determine the mean heat flux and the mean normal contact pressure as functions of the separation between reference planes in the two surfaces. We find that the effective thermal conductance is an approximately linear function of nominal contact pressure, but it also increases with the square root of the sliding speed and decreases with the 3/4 power of the combined RMS roughness. The results can be used to define an effective thermal contact resistance and division of frictional heat in macroscale (e.g., finite element) models of engineering components, requiring as input only the measured roughness and material properties.     

Abrasive wear between rough surfaces in deep drawing

In tribology, many surface contact models are based on the assumption that surfaces are composed of a collection of small asperities of which the tips are equally sized and spherically shaped and have some kind of statistical height distribution. This approach was used in 1966 by Greenwood and Williamson and was successfully followed by many researchers during the following decades. The statistical representation of surface topography enables calculation of contact forces and asperity deformations with reasonable accuracy using well established equations. Although this approach has proven to be suitable for static contact situations, alternative representations of the surface topography are required when modelling abrasive wear. In the current work an elastoplastic contact model is developed in which a representation of the surface topography is obtained by best fit approximations of the micro-contacts, obtained from real, measured surface height data. In this deterministic surface representation, the tips of the contacting asperities are assumed to have an ellipsoidal shape. Given the material parameters and contact conditions, the load and deformation of a single asperity can be computed. Subsequently, the wear induced by each individual asperity is obtained by inserting its size and shape and the conditions into a “single asperity micro-abrasion model”. By summing the contributions of all individual asperities, the total abrasive wear volume is obtained. The results of the developed abrasive wear model are compared with results obtained using a statistical approach.     

Elastic Contact Mechanics of Randomly Rough Surfaces: An Assessment of Advanced Asperity Models and Persson’s Theory

In this work, we discuss important improvements of asperity models. Specifically, we assess the predictive capabilities of a recently developed multiasperity model, which differs from the original Greenwood and Williamson model by (i) including the coupling between the elastic fields generated by each contact spot, and (ii) taking into account the coalescence among the contact areas, occurring during the loading process. Interaction of the elastic field is captured by summing the contributions, which are analytically known, of the elastic displacements in a given point of the surface due to each Hertzian-like contact spot. The coalescence is instead considered by defining an equivalent contact spot in such a way to guarantee conservation of contact area during coalescence. To evaluate the accuracy of the model, a comparison with fully numerical ‘exact’ calculations and Persson’s contact mechanics theory of elastic rough surfaces is proposed. Results in terms of contact area versus load and separation versus load show that the three approaches give almost the same predictions, while traditional asperity models neglecting coalescence and elastic coupling between contact regions are unable to correctly capture the contact behavior. Finally, very good results are also obtained when dealing with the probability distribution of interfacial stresses and gaps.     

The contact behavior of elastic/plastic non-Gaussian rough surfaces

A three-dimensional contact analysis was conducted to investigate the contact behavior of elastic--perfectly plastic solids with non-Gaussian rough surfaces. The effect of skewness, kurtosis and hardness on contact statistics and the effect of skewness and kurtosis on subsurface stress are studied. Non-Gaussian rough surfaces are generated by the computer with skewness, Sk, of −0.3, 0.0 and 0.3, and kurtosis, K, of 2.0, 3.0 and 4.0. Contact pressures and subsurface stresses are obtained by contact analysis of a semi-infinite solid based on the use of influence functions and patch solutions. Variation of fractional elastic/plastic contact area, maximum contact pressure and interplanar separation as a function of applied load were studied at different values of skewness and kurtosis. Contact pressure profiles, von Mises stresses, tensile and shear stress contours as a function of friction coefficient were also calculated for surfaces with different skewness and kurtosis. In this study, it is observed that surfaces with Sk = 0.3 and K = 4 in the six surfaces considered have a minimum contact area and maximum interplanar separation, which may provide low friction and stiction. The critical material hardness is defined as the hardness at which severe level of plastic asperity deformation corresponding to the Greenwood and Williamson’s cut-off A _plastic/A _real = 0.02 occurs for a given surface and load condition. The critical material hardness of surfaces with Sk = 0.3 and K = 4 is higher than that of other surfaces considered.     

Evaluation of the real contact area in three-body dry friction by micro-thermal analysis

Many tribological properties and wear mechanisms occurring on the micro-and nanoscale are strongly controlled by the so-called real contact area (A_r) which is a small fraction of the nominal or apparent contact area (A_a). The determination of A_r is often based on either (i) a geometrical approach describing the real geometry of contacting surfaces or (ii) a mechanical approach involving contact mechanics and physical-mechanical properties. In addition some experimental methods have also been attempted but they generally do not take into account the presence of third body at the interface—i.e. the wear debris trapped within the contact. In this paper we propose an experimental approach to estimate the dynamic real contact area from the operating parameters (F_n, v, T) and the tribological responses (μ, F_t) in presence of third body. A scanning thermal microscope (SThM) is used for determining both the thermal conductivity of the third body and the relationship between the contact temperature and the thermal power really dissipated at the micro-asperity level. These results are combined with a thermal model of the macro-tribocontact for computing the real contact area and the real contact pressure. Validation of these results is carried out using a classical Greenwood Williamson model and finite element models built from the real AFM maps.     

The Effect of Scale-Dependent Hardness on Elasto-Plastic Asperity Contact between Rough Surfaces

Statistical methods are used to model elasto-plastic contact between two rough surfaces using a recent finite element model of elasto-plastic hemispherical contact and also recent advances in strain gradient modeling. The elasto-plastic hemispherical contact model used to model individual asperities accounts for a varying hardness effect due to deformation of the contact geometry that has been documented by other works. The strain gradient model accounts for changes in hardness due to scaling effects. The contact between surfaces with hypothetical material and surface properties, such as the elastic modulus, yield strength, and roughness are modeled. A model is also constructed to consider a variable asperity contact radius to evaluate if the strain gradient model will affect it differently. The models produce predictions for contact area, contact force, and surface separation. The strain gradient effects decrease the real area of contact and increase the average contact load in comparison to the model without these effects. The strain gradient model seems to have a larger influence on the predictions of contact load and area than does considering a variable asperity contact radius for the cases considered in this work.     

A modified fractal microcontact model developed for asperity heights with variable morphology parameters

The cross-sections formed by the contact asperities of two rough surfaces at an interference are actually island-shaped, rather than having the commonly assumed circular or elliptic contour. These island-shaped contact area contours show fractal behavior with fractal dimension D_s of the two-dimensional profile. The three-dimensional surface fractal dimension for the topography of asperity heights is defined as D and the topothesy is defined as G. In Mandelbrot's study, the relationship between D and D_s was given as D = D_s + 1 if these two fractal dimensions are obtained before contact deformation. In the present study, D, G, and D_s are considered to be varying with the mean separation between two contact surfaces. The D–D_s relationships for the contacts at the elastic, elastoplastic, and fully plastic deformation regimes are derived and the inceptions of the elastoplastic and fully plastic deformation regimes are redefined using the equality of two expressions established in two different ways for the number of contact spots (N). A revised elastic–plastic contact model of a single fractal asperity is also proposed. The revised model shows that a fractal asperity behaves according to classical contact mechanics, but not those predicted by the MB model. The contact parameters, including the total force and the real contact area, were evaluated when the size distribution functions (n) for the three deformation regimes were available. The results indicate that both the D and D_s parameters in these deformation regimes increased with increasing mean separation.     

Effect of surface roughness on contact between solid surfaces

The analysis of the mechanism of contact between two solids was carried out considering the distribution of the radii of curvature of asperity peaks. The analytical results show that the mean radius of curvature of asperity peaks has a considerable effect on the nature of the deformation of contact asperities, i.e. whether the contact is plastic or elastic, and more effect on the real area of contact than the variation of the distribution of the radii of curvature.The radii of curvature at the asperity peaks and the real area of contact between two smooth surfaces were measured for comparison with the theoretical results. The results for isotropic surfaces produced by buffing and sandpaper agree with the theory; the real area of contact increases with decreasing surface roughness.     

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.     

Analyzing Elastic-Plastic Real Rough Surface Contact in Running-in

A numerical method is presented for evaluating the elastic-elastic contact of real rough surface contacts during running-in. For the surface contact, an elastic-plastic model based on the variational method is applied to analyze the pressure distribution and contact area of worn surfaces during running-in. In conjunction with the classical statistic model of Greenwood and Williamson, the numerical result showed that the plasticity index Ψ was decreased to one in the elastic range as running-in proceeded. In comparison with the Hertzian solution, the influence of the asperities is very significant on the pressure distribution, thereafter causing a higher peak value of contact pressure. For the subsurface, the interior stress from the von Mises criterion was calculated to evaluate the subsurface stress field subject to both normal and tangential forces. In the calculated of the interior stress, the total stress is decomposed into a fluctuating component and a smooth component. The fluctuating part is solved by using FFT from the concept of the convolution theorem while the smooth part is obtained directly by analytical solution. Calculations of contact area and subsurface stress on experimentally produced surfaces whose topography has been determined using an atomic force microscope and friction coefficient front sliding have been carried out. The results showed that asperities and friction coefficient gave rise to stress increase in the near-surface stress field and produced a high stress zone towards the surface. As a result, transverse asperity cracking was produced. The calculations and supporting experimental evidence clearly confirmed that the reduction of peak pressure during running-in decreased the plastic deformation of contact.     

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

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.     

On the Relation of Load to Average Gap in the Contact Between Surfaces with Longitudinal Roughness

A computer simulation model for the contact between longitudinally-oriented rough surfaces has been formulated. This model closely duplicates the actual surf ace contact deformation behavior by taking into account the elastic interactions between the asperities. There were no assumptions made about the shapes, or any deformation behavior of the asperities, except for their obeying the laws of elasticity. The plastic deformations on the high asperity peaks were taken into account by setting a ceiling on their contact pressures at the material hardness value. The simulations used real surface profiles which were digitized from unworn circumferentially ground steel surfaces. Each pair of these profiles was mathematically combined to form an equivalent rough profile pressing against an infinitely rigid flat and having the appropriately adjusted elastic modulus. A total of 28 different pairs of profiles were used in the simulations. Each contacting pair was subjected to 30 different load levels and the local contact pressures and deformations were calculated. The contact simulations yielded some important mathematical relationships between parameters, such as the real area of contact, average gap, and average asperity load through statistical curve fitting. Two analytical functions were generated to relate the average load to average gap and the real area of contact to load.     

Asperity micro-contact models as applied to the deformation of rough line contact

Different statistical micro-contact models including Greenwood–Williamson (GW), Chang–Etsion–Bogy (CEB), Zhou–Maietta–Chang (ZMC), Kogut–Etsion (KE) and Jackson–Green (JG) are employed together with the bulk deformation of the bounding solids to predict dry rough line-contact characteristics such as the apparent pressure profile, contact width and real area of contact. The approach involves solving the micro-contact models and separation formulas simultaneously. Comparison of different contact models reveals that the use of elastic–plastic micro-contact models predicts a lower maximum normal pressure and a greater contact width and real contact area compared to the GW model. Further, based on the results of numerical simulations, useful relationships are provided for the prediction of the maximum contact pressure, contact width, real area of contact and pressure distribution.     

The real area of contact and compliance of rough cylinders in compression

Bearing area analysis has been used to study the real area of contact and compliance of rough turned steel cylinders in compression. Calculations show that the elastic real area of contact is very small compared to the plastic real area of contact, and that local compliance due to flattening of asperity tips is a small proportion of the total compliance obtained from experiments. The fact that increased load brings more and more new asperities under load rather than enlarging the contact spots leads to a rather simple load-compliance relation for a rough cylinder, viz., W' = N_h · K₁δⁿ, where W₀ = K₁δⁿ defines the load-compliance relation of the individual asperities, and N_h represents the number of asperities bearing the load.     

新的粗糙表面弹塑性接触模型

提出一种新型的粗糙表面弹塑性微观接触模型.该模型的建立基于接触力学理论和接触微凸体由弹性变形向弹塑性变形及最终向完全塑性变形的转化皆是连续和光滑的假设.研究单个微凸体在载荷逐渐增加时的变形规律,并重点推出弹塑性变形区间的接触方程.在此基础上应用概率统计理论导出了粗糙表面的接触载荷、平均分离和实际接触面积之间的数学关系式.在不同的塑性指数和载荷条件下,该模型与GW弹性模型和CEB弹塑性模型就实际接触面积和法向距离的预测结果进行了对比.结果表明,在同样塑性指数和载荷条件下比GW模型预测的实际接触面积大但法向距离小,且两者的差距随塑性指数和载荷的增加而增大.因此该模型的预测结果更加符合人们的试验观察和直觉,能够更加科学和合理地描述两个粗糙表面的微观和宏观接触状态.     

两弹塑性接触粗糙表面的严格解析解

GW干接触模型存在5个缺陷。当相糙峰高度的概率密度为Gauss分布时,给出了数学期望接触点数、总接触面积、总载荷、总电导的严格解析解,采用软件Maple计算了抛物柱面函数。实例计算表明接触点数与载荷近似成凸弧形直线正比例关系;量纲一的间距是联系接触点数、总接触面积、总载荷、总电导的纽带,它依赖名义压应力,但对名义压应力的变化不敏感;接触面积与载荷很接近直线正比例关系,在弹性接触条件下存在一个准弹性接触硬度,接触压应力等于准弹性接触硬度,名义面积对接触面积几乎无影响。     

Development of a wear equation for polymer-metal sliding in terms of the fatigue and topography of the sliding surfaces

A wear equation has been derived using the concept of fatigue failure due to asperity interactions in the contact region between sliding bodies. One of the three principal stresses that arise in the contact zone under the effect of a normal as well as a tangential load is of tensile nature. It is this principal stress that has been considered to be responsible for the initiation and propagation of fatigue cracks. It is assumed that the deformation in the contact zone is of elastic nature and that both the contacting surfaces are covered with asperities that have spherical tips. The wear equation involves the asperity height distribution φ(z). The particular distribution for a sliding situation is determined from experimental studies of the topography of sliding surfaces. The wear equation indicates that the wear rate depends upon the fatigue properties of the weaker material, normal load, sliding speed, coefficient of friction, moduli of elasticity of the contacting materials, asperity density, asperity radius of curvature and the distribution and standard deviation of asperity heights. The variation of wear with these parameters as indicated by the wear equation is in agreement with the experimental studies already reported in the literature.