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.     

Point contact surface fatigue in zirconia ceramics

The surface deformation and fragmentation behaviour of three zirconia ceramics have been studied by using unlubricated metallic repeated point contact loading at room temperature to investigate the possibilities of cyclic fatigue effects. All tests were conducted on a purpose built computer-controlled apparatus. The zirconias studied were ceria stabilized tetragonal polycrystalline, magnesia partially stabilized, and single crystal calcia stabilized. 120° steel cones were cyclically loaded against the flat, polished zirconia counterfaces, and the damage was observed and analysed as a function of the number of cycles, up to a total of 5 × 10⁵ cycles, for loads of 19.6 ± 9.8 N. The ground tips of the cones plastically deformed during the initial loading cycle to produce a flattened end which conformed with the zirconia counterface. The contact pressures were in the range 4 to 10 GPa. In all cases plastic deformation was observed in the zirconias within, and adjacent to, the contact areas. The degree of plastic deformation increased with increasing number of cycles. After approximately 1 × 10⁴ cycles, localized cracking was induced at the peripheries of the contact zones, which gradually increased in extent until after 5 × 10⁵ cycles there was extensive fragmentation. No material transfer, i.e. metal onto ceramic, or vice versa, was generally observed until after the surface had become rough as a result of the fracturing.     

An analysis of three-dimensional elasto-plastic sinusoidal contact

Researchers have developed many models to simulate the elasto-plastic contact of spheres. However, there does not appear to exist a closed-form analytical model for elasto-plastic three-dimensional sinusoidal contact. This work uses a finite element model (FEM) to characterize elasto-plastic sinusoidal contact. Although at initial contact the sphere and sinusoidal case are very similar and can both be described by the classic elastic Hertz contact case, once the contact is pressed past a certain range of deformation the two cases are very different. The model produces equations which can be used to approximately relate the area of contact to the contact pressure for elasto-plastic sinusoidal contact. The equations are fit to the FEM results and existing elastic solutions of sinusoidal contact. An empirical expression for the average pressure which causes complete contact between elasto-plastic sinusoidal contacts is also provided.     

Prediction of subsurface stress in elastic perfectly plastic rough components

Understanding and anticipating the effects of surface roughness on subsurface stress in the design phase can help ensure that performance and life requirements are satisfied. One approach used to address this problem is to simulate contact between digitized real, machined surfaces, and then analyze the predicted subsurface stress field. Often, elastic-perfectly plastic contact models are used in these simulations because of their relative computational efficiency. Reported here is an analysis of the magnitude and location of maximum stress predicted using an elastic-perfectly plastic model. Trends are identified which then enable estimation of the upper bound of the simulation results based on surface discretization, operating conditions, and material properties. These estimations can be used as an effective and efficient tool for rapid prediction of maximum subsurface stress in real surface contact.     

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.     

Role of nanometer roughness on the adhesion and friction of a rough polymer surface and a molecularly smooth mica surface

Friction and adhesion measurements between surfaces of cross-linked, stiff polymers of varying roughness against smooth, bare mica surfaces were carried out in dry air as well as in the presence of lubricating oil. The nominal (macroscopic) contact area varies with the applied load according to the Johnson, Kendall and Roberts (JKR) theory, yet shows significant hysteresis due to the irreversibility arising from the loading/unloading curves of multiple asperities. Upon introducing the oil between the surfaces, the critical shear stress is reduced to zero due to the elimination of the adhesion force. However, the effect is less noticeable on the friction coefficient. Lastly, the effect of increasing the (RMS) roughness was greatest over the first few nanometers due to the diminution of the adhesion-controlled contribution to the friction, after which a further increase in roughness had less dramatic effects. A model is presented to account for the observed adhesion hysteresis during repeated loading/unloading cycles of purely elastically deforming rough surfaces. Bruno Zappone and Kenneth J. Rosenberg made equal contributions.     

Contact of surface asperities in wear

Stresses have been examined under the elliptical Hertzian contact area, which appear during a real contact between two asperities in the process of sliding friction. The stresses are found according to the Mises-Hencky criterion and characterize the ability of the material to change its shape. Relationships have been found of such a maximum stress under the surface of the contact area depending on the geometric parameters of the contact area, on the friction coefficient and the maximum pressure in the centre of contact area. The processing of the calculation data was conducted according to a special program on a computer and yielded an approximate relationship to calculate the maximum reduced stress. The distribution of dangerous stresses under the contact surface has been shown. The most critical points may be both within the surface layer of the material and on the contact area itself.     

Experimental investigation of contact behaviour in grinding

The nature of the wheel and workpiece contact in grinding has a strong effect on the temperature, force and surface integrity as well as wheel wear in grinding. An applied power source method was used to measure the real contact length. The interpretation of contact length measurements is discussed. Experimental measurements of contact length in grinding show that the contact length is much larger than the geometrical contact length. This difference is most significant in fine grinding and in the sparkout stage of plunge grinding. The difference also increases when the table speed is increased. The contact length in wet grinding is longer than the contact length in dry grinding. The contact length when grinding cast iron is shorter than the contact length when grinding mild steel. It is found that grinding geometry, grinding force and the roughness of the grinding wheel have independent effects on the contact length. The newly developed contact length model by the authors describes these effects quantitatively. These results also show the importance of the roughness factor, R_r, for analysis of the contact behaviour in grinding processes.     

Film-forming capability in rough surface EHL investigated using contact resistance

Highly loaded machine elements such as gears and cams have a non-smooth surface topography that is created during manufacturing. It is well known that the film-building properties of such surfaces may be different from those that are perfectly smooth. The capability to form a separating film may also be altered in time due to run-in phenomena. In this study, a smooth steel ball was loaded against rough steel discs and run under pure rolling as well as sliding conditions. Several different steel surfaces were tested under nominal EHL conditions, where the contact was monitored by means of its electrical resistance and capacitance. Each surface was first run in for 15 min, followed by a sweep-in speed determining the lift-off curve. Electrical contact measurements were continuously conducted during run in as well as lift-off. Fully formulated gear oil and its base fluid were used as test lubricants. Results show that run in of a surface seems to be a competition between conformation of surface topography and tribofilm formation. At the tested conditions, the formation of a tribofilm is dependent on the initial surface topography and is created rapidly causing less metal–metal contact. This film also seems to effectively prohibit changes in surface topography causing less structural run in than expected.     

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。     

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.     

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.     

Influence of surface roughness on normal-sliding lubrication

Contact between machine components can involve normal, sliding and rolling motion, either singly or in combination. Combined normal and sliding motion, which occur for example in the meshing of gear teeth and in heavily-loaded rolling elements, can present problems for lubrication. The purpose of the present experimental study was to investigate how surface roughness affects the lubricant film characteristics under conditions of combined normal and sliding motion. The experimental arrar gement consisted of a rotating roller which impacted a stationary ball in the presence of a lubricant. Under the same conditions of normal surface approach, increasing the surface roughness significantly decreased the level of roller sliding that could occur without breakdown of the lubricating film. This behaviour was similar to a step function. Of the several surface roughness parameters investigated, only those which involved the maximum peak-to-valley height correlated well with experimental results. In general, surface roughness had a greater effect on oil film breakdown than did either viscosity or load.     

A simple datum for measurement of the Abbott curve of a profile and its first derivative

The methods used in some recent work to predict boundary friction from surface slopes require that the variation of rms slope with bearing area be calculated: an Abbott curve of the first derivative of the profile. It is shown that this curve can be distorted if a conventional filter is used to set the datum, in the same way as the Abbott curve itself. Better results can be obtained for both the standard Abbott curve and the Abbott curve of slopes if an envelope through the peaks of the profile is used as the datum, rather than a digital filter.     

The behavior of an elastic-perfectly plastic sinusoidal surface under contact loading

The goal of this paper is to provide the foundation for an analysis of contact between elastic-plastic solids, whose surface roughness is idealized with a Weierstrass profile. To this end, we conduct a parametric study of the plastic deformation and residual stress developed by the two-dimensional contact between a flat, rigid platen and an elastic-perfectly plastic solid with a sinusoidal surface. Our analysis shows that the general characteristics of the deformation can be characterized approximately by two parameters: α = a/λ, where a is the half-width of the contact and λ is the period of the surface waviness; ψ = E^*g/σ_Yλ, where E^* and σ_Y are the effective modulus and yield stress of the substrate, respectively, and g is the amplitude of the surface roughness. Depending on the values of these parameters, we identify eight general types of behavior for the asperity contacts: (a) elastic, elastic-plastic or fully plastic isolated Hertz type contacts; (b) elastic, or elastic-plastic non-Hertzian isolated contacts; and (c) elastic, elastic-plastic or fully plastic, interacting contacts. Relationships between contact pressure, contact size, effective indentation depth and residual stress are explored in detail in each regime of behavior. Implications on rough surface contacts are discussed.     

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.     

Influence of surface roughness on friction and scuffing behaviour of cast iron under sparse lubrication

Friction and scuffing behaviour of grey cast iron as influenced by the surface roughness under sparse lubrication conditions is studied. The studies are carried out on a three-shoe-on-disc machine under the conditions of parallel sliding. The experiments were conducted at three oil supply rates of 0.10, 0.22 and 0.36 μg/cm² per contact obtained through a mist oiling system and with specimens of cast iron shoes of different roughness values, Ra, between 0.04 and 2.0 μm against a 0.55% C steel disc. In a step load procedure, the friction torque at the end of each load step and the scuffing load are the major parameters measured. Results of friction and scuffing behaviour as a function of roughness and oil supply rates have been discussed.     

A simplified elliptic model of rough surface contact

Nayak's analysis of isotropic surface roughness as a random field has been extended to show that most summits are only mildly ell, the most common ratio of principal summit curvatures being near 2:1. Also from Nayak's theory, the distribution of the geometric-mean summit curvature with height has been obtained. By using an approximate solution for elliptical Hertzian contacts based on the geometric-mean summit curvature, the full elliptical solution of Bush, Gibson and Thomas can be reproduced more conveniently. Their values for the area of contact are accurately reproduced, and it is argued that the present values for load and contact pressure are more plausible: unlike the original numerical values, the present values converge smoothly to the BGT asymptote . Once again, it is found that elastic contact models can explain the proportionality between contact area and load, although at realistic loads the proportionality is merely very good, not exact. The model shows that a plasticity index ψm≡(E^*/H)σm (closely related to Mikic's index) can be used to predict the behaviour of surfaces in contact.     

The effect of surface roughness on contact and stress states of spur gears

The contact characteristics of spur gears are analysed, taking into consideration real surface micro-geometry, and using numerical algorithms. The contact pressure distribution, contact area, and rigid body rotation of the gears have been calculated for different types of gear surface according to machining: shaping, grinding, or milling. Finally, the sub-surface stress state is evaluated for different pressure distributions, by finite element method.     

Elastoplastic finite element analyses of an upsetting-sliding test for the determination of friction at medium and high contact pressure

The upsetting-sliding test is a friction test employed for the determination of the tangential stress-contact pressure relationship at a specimen-tool or target part-contactor contact surface. It is used in addition to the analysis of forming sequences or mechanical design. The specimen, or target part, used in the test is the same that the one involved in the studied forming sequence so that the physical and chemical characteristics of the contact involved by the test and sequence are the same. In the test, the contact pressure is exerted by an inclined front surface indenter which takes the place of the tool or the contactor. Then, the indenter is moved in contact with the generator line surface of the specimen or the target part. The computation of the tangential stress, contact pressure and effective plastic strain firstly with respect to the penetration, sizing length and front face angle and, secondly, with respect to the frictional variations are performed with a three-dimensional finite element model. An initial guide for the determination of the indenter length and front angle and indenter penetration in the specimen is proposed in order to define the most realistic tangential stress-contact pressure curves.