Effect of asperity interactions on rough surface elastic contact behavior: Hard film on soft substrate

An improved elastic micro-contact model of rough surfaces accounting for asperity interactions is proposed. The contact behavior of a single asperity system is composed of a stiffer hemi-spherical asperity deformation and bellowing softer substrate deformation, which is then extended to rough surface contact including asperity interactions. Using the solution of substrate deformation, normal positions of individual asperities are adjusted during quasi-static contact, from which surface interactive forces are obtained. Analytical simulations are performed using the proposed rough surface contact model, whose results are compared to Greenwood-Williamson-based models and with experimental measurements.     

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.     

A numerical model of friction between rough surfaces

This paper describes a computational method to calculate the friction force between two rough surfaces. In the model used, friction results from forces developed during elastic deformation and shear resistance of adhesive junctions at the contact areas. Contacts occur between asperities and have arbitrary orientations with respect to the surfaces. The size and slope of each contact area depend on external loads, mechanical properties and topographies of surfaces. Contact force distribution is computed by iterating the relationship between contact parameters, external loads, and surface topographies until the sum of normal components of contact forces equals the normal load. The corresponding sum of tangential components of contact forces constitutes the friction force. To calculate elastic deformation in three dimensions, we use the method of influence coefficients and its adaptation to shear forces to account for sliding friction. Analysis presented in Appendix A gives approximate limits within which influence coefficients developed for flat elastic half-space can apply to rough surfaces. Use of the method of residual correction and a successive grid refinement helped rectify the periodicity error introduced by the FFT technique that was used to solve for asperity pressures. The proposed method, when applied to the classical problem of a sphere on a half-space as a benchmark, showed good agreement with previous results. Calculations show how friction changes with surface roughness and also demonstrate the method's efficiency.     

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.     

Elasto-Plastic Contact of Rough Surfaces

An isothermal elasto-plastic asperity contact model is developed and presented in this paper, which deals with micro plastic flows of materials and the influence of the elasto-plastic deformation of materials on the behavior of contacting surfaces. The model is solved with the incremental form of a simplex-type algorithm. The von Mises yield criterion is used to determine the onset of the plastic deformation. The effectiveness and validity of the model are studied through analyzing a Hertzian contact problem. Substrate stresses are calculated and differences are observed. Furthermore, the contact pressure, real area of contact, and average gap of real rough surfaces under the elastic, elastic-perfectly-plastic, and the elasto-plastic contact conditions are numerically investigated and the results are compared.     

基于真应力-应变关系的粗糙表面法向接触模型

提出一种粗糙表面的法向弹塑性接触分析的建模方法。基于微凸体的弹塑性有限元接触模型,分别研究了40Cr、45和Q235三种钢材料的微凸体与刚性平面的法向接触特性。有限元模型中采用三种材料的真应力-应变关系,考察了不同强化特性对微凸体接触性质的影响。建立了微凸体在弹性、弹塑性、塑性变形阶段统一的接触变量变化规律的表达式。在此基础上应用概率统计理论建立粗糙表面法向弹塑性接触模型。所建立的接触模型中微凸体接触变量的变化规律完全基于弹塑性有限元模型的计算结果,无需将微凸体的变形过程区分为不同的变形阶段,避免了接触变量在各阶段采用不同函数表达式带来的连续性和光滑性问题,以及在弹塑性阶段采用插值函数的随意性问题。通过与其他接触模型的计算结果相比较,证明了所提出接触模型的合理性。     

Microscopic asperity contact and deformation of ultrahigh molecular weight polyethylene bearing surfaces

The effect of the roughness and topography of ultrahigh molecular weight polyethylene (UHMWPE) bearing surfaces on the microscopic contact mechanics with a metallic counterface was investigated in the present study. Both simple sinusoidal roughness forms, with a wide range of amplitudes and wavelengths, and real surface topographies, measured before and after wear testing in a simple pin-on-plate machine, were considered in the theoretical analysis. The finite difference method was used to solve the microscopic contact between the rough UHMWPE bearing surface and a smooth hard counterface. The fast Fourier transform (FFT) was used to cope with the large number of mesh points required to represent the surface topography of the UHMWPE bearing surface. It was found that only isolated asperity contacts occurred under physiological loading, and the real contact area was only a small fraction of the nominal contact area. Consequently, the average contact pressure experienced at the articulating surfaces was significantly higher than the nominal contact pressure. Furthermore, it was shown that the majority of asperities on the worn UHMWPE pin were deformed in the elastic region, and consideration of the plastic deformation only resulted in a negligible increase in the predicted asperity contact area. Microscopic asperity contact and deformation mechanisms may play an important role in the understanding of the wear mechanisms of UHMWPE bearing surfaces.     

A deterministic model for line-contact partial elastohydrodynamic lubrication

A deterministic model for partial elastohydrodynamic lubrication (EHL) is presented in this paper. The modelling methodology adopts some of the concepts used in the stochastic modelling of partial EHL and some of the procedures for deterministic calculation of asperity pressures. The model is shown to be capable of simulating the basic process of asperity interaction and solid-to-solid contact within an EHL conjunction of rough surfaces. Deterministic results of transient partial EHL in line contacts are obtained when one pair or multiple pairs of asperities collide. The model may help to gain a fundamental understanding of the transient behaviour of asperity interactions in lubricated concentrated contacts of rough surfaces. Asperity pressures may be calculated more accurately than the conventional analyses under dry and static contact conditions. The work represents a first attempt in deterministic modelling of tribo-contacts operating in the mixed regime of micro-EHL and boundary lubrication. Future work will aim at developing more realistic models incorporating factors such as three-dimensional asperity contacts, asperity plastic deformation, thermal effects and the effect of tribo-chemistry.     

Elastic–plastic adhesive contact of rough surfaces based on plastic asperity concept

The paper describes an analysis of adhesive contact between rough surfaces with small-scale surface asperities using an elastic–plastic model of contact deformation based on fictitious plastic asperity concept developed by Abdo and Farhang [Int. J. Non-Linear Mech. 40 (2005) 495]. The model considers simultaneous occurrence of elastic and plastic behaviours for an asperity. The well-established elastic adhesion index and plasticity index are used to consider the different contact conditions that arise as a result of varying load and material parameters. The load-separation behaviour for different combinations of these parameters is obtained. Comparison with previous elastic–plastic model that was based on elastic-then-plastic assumption is made showing significant differences.     

Temperature Analysis in Lubricated Simple Sliding Rough Contacts

A temperature analysis of dry sliding fully plastic contact is extended to calculate the asperity temperatures between a sliding lubricated rigid smooth plane and a stationary elastic rough surface. First, surface roughness is generated numerically to have a Gaussian height distribution and a bilinear autocorrelation function. Lai and Cheng's elastic rough contact computer program is then used to determine the asperity contact loads and geometries of real contact areas. Assuming different frictional coefficients for shearing the lubricant film at the noncontact areas, shearing the surface film at the asperity contacts and shearing the oxide film as the asperity temperature exceeds a critical temperature, asperity temperature distributions can be calculated. Eight cases in Durkee and Cheng's scuffing tests of lubricated simple sliding rough contacts are simulated by using 20 computer-generated rough surfaces. The results show that scuffing is correlated to high-temperature asperities which are above the material-softening temperature.     

尺度相关的分形结合面法向接触刚度模型

基于分形理论,利用双变量Weierstrass-Mandelbrot函数模拟三维分形结合面,建立尺度相关的三维分形结合面法向接触刚度模型。推导出各等级微凸体发生弹性、弹塑性以及完全塑性变形的存在条件。确定结合面上各等级微凸体的面积分布密度函数,推导出法向接触刚度和法向接触载荷的解析表达式。计算结果表明：当结合面上的微凸体只能发生弹性变形,即自身等级小于弹性临界等级的微凸体,该部分微凸体引起的法向接触刚度和对应法向载荷关系呈非线性。当微凸体的等级大于弹性临界等级,在结合面接触过程中,微凸体弹性变形引起的法向接触刚度与对应的法向载荷关系为线性,非弹性变形引起的法向接触刚度与法向载荷关系为非线性。微凸体的等级范围对结合面的刚度影响较大,在相同的法向载荷作用下,高等级微凸体的结合面产生较高的法向接触刚度,即结合面越平整,结合面的法向刚度越高。     

Elastoplastic contact mechanics model of rough surface based on fractal theory

Because the result of the MB fractal model contradicts with the classical contact mechanics, a revised elastoplastic contact model of a single asperity is developed based on fractal theory. The critical areas of a single asperity are scale dependent, with an increase in the contact load and contact area, a transition from elastic, elastoplastic to full plastic deformation takes place in this order. In considering the size distribution function, analytic expression between the total contact load and the real contact area on the contact surface is obtained. The elastic, elastoplastic and full plastic contact load are obtained by the critical elastic contact area of the biggest asperity and maximun contact area of a single asperity. The results show that a rough surface is firstly in elastic deformation. As the load increases, elastoplastic or full plastic deformation takes place. For constant characteristic length scale G, the slope of load-area relation is proportional to fractal dimension D. For constant fractal dimension D, the slope of load-area relation is inversely proportional to G. For constant D and G, the slope of load-area relation is inversely proportional to property of the material ϕ, namely with the same load, the material of rough surface is softer, and the total contact area is larger. The contact mechanics model provides a foundation for study of the friction, wear and seal performance of rough surfaces.     

Mathematical slip-line field solutions for ploughing a hard particle over a softer material

Wear mechanisms and friction in metals can be investigated by the analysis of the unit event represented by the interaction of a hard particle or asperity with a softer surface. Effective friction is the result of the interaction of many such asperities which constitute the roughness of the harder of the solid surfaces. Three types of plastic deformation at the metal surface can be identified: ploughing, edge formation and chip formation. Each mode of plastic deformation can be analysed using the slip-line field plasticity theory which requires as inputs the geometry of the hard particle and some information on the interface between the harder and the softer surfaces. The classical and the recent chord solution by Oxley assumes a sharp edge sliding against a metal surface but does not consider a curved roughness profile. However, the profiles of real asperities are more like waves with rounded summits. In the present work a new model for the asperities interaction is shown, using the slip-line field theory to calculate the friction forces, depth of sheared layer, average contact pressure and friction coefficient for a cylindrical hard particle sliding over a softer surface. The theoretical results are presented as friction graphs and maps in which the regions of elastic deformations are shown using the Hertz theory while the region of plastic strains is obtained from the present analysis. Present model results are in good agreement with experimental data obtained by Busquet et al. and are quite different from the Oxley chord model for sliding a circular particle.     

A Theoretical Analysis of Mixed Lubrication by Macro Micro Approach: Part I—Results in a Gear Surface Contact

Mixed lubrication is a key to bring the performance analysis to the failure analysis in most tribological components. A macro-micro approach to mixed lubrication has been developed in the present model. The relation between the average contact pressure and the average gap for a typical rough contact patch is first determined numerically in micro scale. Using this relation, the average gap, average oil-film pressure, and average contact pressure in a mixed-lubricated elastohydrodynamic contact can be solved simultaneously in macro scale by treating the contact to be smooth. The total pressure is assumed a superposition of average asperity contact pressure and lubricant pressure. The new approach is simple, efficient and robust, and covers entire range of the load ratio, from unity (dry contact) to zero (full-film EHL). In addition, it can be used for a wide range of operating conditions and on a much larger contact area with a much less computing time than deterministic simulation of mixed lubrication. Implementation of the Fast Fourier Transform (FFT) allows for a rapid calculation of the elastic deformation and asperity con/act pressure. As a demonstration to this new approach, a parametric study of dimensionless speed, load and contact shape on the load ratio and gap ratio was conducted for a gear rough surface of the load ratio, from unity (dry contact) to zero (full-film EHL). In addition, it can be used for a wide range of operating conditions and on a much larger contact area with a much less computing time than deterministic simulation of mixed lubrication. Impletmentation of the Fast Fourier Transform (FFT) allows for a rapid calculation of the elastic deformation and asperity contact pressure. As a demonstration to this new approach, a parametric study of dimensionless speed, load and contact shape on the load ratio and gap ratio was conducted for a gear rough surface     

The Subsurface Stress Field Created by Three-Dimensionally Rough Bodies in Contact with Traction

A numerical simulation technique for calculating the complete subsurface stress field for three-dimensionally rough bodies in sliding contact is described. The stresses are calculated using real digitized three-dimensional surface profiles. The effects of the surface roughness and the sliding friction are presented. Using an existing contact simulation code, the digitized surfaces are mathematically pressed together and the real areas of contact and the asperity pressures are calculated. The surfaces are assumed to remain elastic throughout the contact simulation process. The shear forces at the asperity contact interfaces are assumed to be proportional to their calculated normal pressures. The subsurface stresses are then determined with these known normal and tangential forces at the surface.     

The contact behaviour of surfaces of small nominal area

In treating the problem of surface contact it is usually assumed that the asperity radius remains constant during the deformation process. Such an assumption may be valid for the contact of surfaces of large nominal area subjected to moderate normal loads; however, with surfaces of small area where the population of asperities is small the amount of asperity deformation is expected to be relatively large and the displaced material may play an effective role in determining the contact behaviour. In this paper the asperity radius is assumed to increase with deformation and analyses are presented for determining the variation of normal approach with load for rough flat surfaces. The results of such analyses suggest that the deformation process produces increasingly suffer surfaces and are shown to have better agreement with experiment than those results obtained from analyses based on a constant asperity radius.     

基于JKR模型的粗糙圆柱表面线黏着接触分析*

利用粗糙平面接触模型,假定表面单个微凸体的接触采用JKR黏着接触模型,同时考虑圆柱体表面的整体变形,建立了粗糙圆柱表面线黏着接触模型,推导出表面等效压力分布方程。把压力方程量纲一化,采用修正Newton-Raphson法对方程进行迭代求解,计算出粗糙圆柱表面存在表面力作用下的等效压力分布曲线。结果表明外载荷不小于零时,接触中心压力为正,微凸体被压缩;而接触边缘处压力为负,微凸体被拉伸,表明黏着区域主要分布在接触边缘。同时计算出接触半宽随外载荷的变化曲线,当外载荷为拉伸力并大于某一临界值时,表面分开。并且与经典的接触模型进行了对比,发现低载时模型之间的差别较大,而载荷比较大时趋于一致。     

渗氮钢粗糙表面的弹塑性接触研究

利用弹塑性有限元和单纯形法求解弹塑性接触模型，分别模拟了屈服强度呈梯度变化的渗氮钢、未经处理的匀质材料和硬涂层材料粗糙表面的弹塑性接触行为。与未经处理的匀质材料相比，渗氮钢可承受更大接触载荷。在相同载荷作用下，渗氮钢表面粗糙峰接触面积较小，平均间距较大，接触体内材料不易发生屈服，从而显著提高接触性能。和硬涂层材料相比，渗氮钢接触体内等效von Mises应力分布平缓，没有应力突变。最后讨论了渗氮层和硬涂层的厚度对粗糙表面接触特性的影响。     

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.     

Contact mechanics of rough surfaces in tribology: multiple asperity contact

Contact modeling of two rough surfaces under normal approach and with relative motion is carried out to predict real area of contact and surface and subsurface stresses affecting friction and wear of an interface. When two macroscopically flat bodies with microroughness come in contact, the contact occurs at multiple asperities of arbitrary shapes, and varying sizes and heights. Deformation at the asperity contacts can be either elastic and/or elastic-plastic. If a thin liquid film is present at the interface, attractive meniscus forces may affect friction and wear. Historically, statistical models have been used to predict contact parameters, and these generally require many assumptions about asperity geometry and height distributions. With the advent of computer technology, numerical contact models of 3-D rough surfaces have been developed, particularly in the past decade, which can simulate digitized rough surfaces with no assumptions concerning the roughness distribution. In this article, a comprehensive review of modeling of multiple-asperity contacts in dry and wet conditions is presented. Contact models for homogeneous and layered, elastic and elastic-plastic solids with and without tangential loading are presented. The models reviewed in this paper fall into two groups: (a) analytical solutions for surfaces with well-defined height distributions and asperity geometry and (b) numerical solutions for real surfaces with asperities of arbitrary shape and varying size and height distributions. Implications of these models in friction and wear studies are discussed. This revised version was published online in June 2006 with corrections to the Cover Date.