Simplified stiffness model for spherical rough contacts

Abstract

Obtaining a surface with negligible roughness is very expensive, time consuming and unnecessary. The influence of surface roughness on the contact stiffness is of great importance. The extra cost associated with unnecessary surface finish can be limited by eliminating the unnecessary machining operations beyond the required surface finish. In this article, a simplified solution is presented to calculate the stiffness of rough contact between the workpiece and spherical locator; also, the effect of surface roughness on the stiffness and deformation of rough spherical contact is studied for different applied loads to find an ‘economic roughness’ under machining forces.     

Validation of a simplified numerical contact model

Surface roughness tends to have a significant effect on how loads are transmitted at the contact interface between solid bodies. Most numerical contact models for analyzing rough surface contacts are computational demanding and more computationally efficient contact models are required. Depending on the purpose of the simulation, simplified and less accurate models can be preferable to more accurate, but also more complex, models. This paper discusses a simplified contact model called the elastic foundation model and its applicability to rough surfaces. The advantage of the model is that it is fast to evaluate, but its disadvantage is that it only gives an approximate solution to the contact problem. It is studied how surface roughness influences the errors in the elastic foundation solution in terms of predicted pressure distribution, real contact area, and normal and tangential contact stiffness. The results can be used to estimate the extent of error in the elastic foundation model, depending on the degree of surface roughness. The conclusion is that the elastic foundation model is not accurate enough to give a correct prediction of the actual contact stresses and contact areas, but it might be good enough for simulations where contact stiffness are of interest.     

A finite element-based model of normal contact between rough surfaces

Engineering surfaces can be characterized as more or less randomly rough. Contact between engineering surfaces is thus discontinuous and the real area of contact is a small fraction of the nominal contact area. The stiffness of a rough surface layer thus influences the contact state as well as the behavior of the surrounding system. A contact model that takes the properties of engineering surfaces into account has been developed and implemented using finite element software. The results obtained with the model have been verified by comparison with results from an independent numerical method. The results show that the height distribution of the topography has a significant influence on the contact stiffness but that the curvature of the roughness is of minor importance. The contact model that was developed for determining the apparent contact area and the distribution of the mean contact pressure could thus be based on a limited set of height parameters that describe the surface topography. By operating on the calculated apparent pressure distribution with a transformation function that is based on both height and curvature parameters, the real contact area can be estimated when the apparent contact state is known. The model presented is also valid for cases with local plastic flow in the bulk material.     

Plastic deformation in rough surface line contacts—a finite element study

The paper describes an elastic-plastic finite element (EPFE) analysis of line contact between a cylinder and rigid plane using commercial software. The range of loading demonstrates the transition from purely elastic to fully plastic contact behaviour, revealing the residual deformations and stress fields upon unloading. A multiple contact configuration was analysed in the form of sinusoidal roughness. Results obtained under elastic conditions were validated by comparison with theoretical solutions. This model was extended by replacing the sinusoidal surface with a real roughness profile. Modelling multiple contacts indicates the influence of adjacent surface “asperities” on contact pressure and residual stress distributions.     

A basic theoretical study of the temperature rise in sliding contact with multiple contacts

The objectives of this paper are to develop a theoretical solution for the temperature rise due to sliding contact between surfaces with multiple, interacting asperities and to use this solution to examine the effects of the important contact area and system parameters. A solution based on the Green's function method is developed for the basic problem of two half-space regions in sliding contact with any arbitrarily specified arrangement of rectangular asperities.Studies are conducted to demonstrate the effects of the contact area parameters, namely the number, size, spacing and orientation of the contacts, as well as sliding velocity. Results indicate that the contact temperatures are extremely sensitive to the number and relative spacing between contacts, where subdivision of a single contact into separated pieces significantly reduces the contact temperature rises. The orientation of the contacts relative to the sliding direction is shown to have only a small influence on temperature. The shape of the contacts also has only a small influence, except in the case of contact patches with large aspect ratios where significantly lower surface temperatures can occur. Sliding speed is shown to be extremely important in that increased speed causes both higher temperature levels and greater interaction between contacts due to the convective effect.The current paper is intended to describe the basic solution methodology for calculating temperature rises due to multiple, interacting contacts and to show some fundamental trends for a selected set of regularly arranged contact area distributions.     

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.     

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.     

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.     

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.     

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 influence of surface roughness and the contact pressure distribution on friction in rolling/sliding contacts

A numerical contact model is used to study the influence of surface roughness and the pressure distribution on the frictional behaviour in rolling/sliding contacts. Double-crowned roller surfaces are measured and used as input for the contact analysis. The contact pressure distribution is calculated for dry static contacts and the results are compared with friction measurements in a lubricated rolling/sliding contact made with a rough friction test rig. The mean pressure is suggested as a parameter that can be used to predict the influence of surface roughness on the friction coefficient in such contacts. The results show two important properties of the friction coefficient for the friction regime studied in this paper: (1) there is a linear decrease in friction coefficient as a function of the slide-to-roll ratio, and (2) the friction coefficient increases linearly with increasing mean contact pressure up to a maximum limit above which the friction coefficient is constant. The absolute deviation of experimental results from the derived theory is for most cases within 0.005.     

Effect of real longitudinal surface roughness on lubrication film formation within line elastohydrodynamic contact

In many applications (e.g. roller, barrel or needle bearings) surface features exhibit longitudinal alignment to the direction of motion. These features are produced by surface finishing techniques in the circumferential direction and are associated with line or very wide elliptical contact geometries. In such a case, the contact length in the direction of motion is considerably shorter comparing its width and the effect of a longitudinal roughness could significantly influence the lubrication film formation. Recent experimental studies have indicated less severe effect of a longitudinal roughness on lubrication film formation in the comparisons with that observed with transversely orientated roughness caused by the inlet perturbation. Nevertheless, these experimental studies have been focused on the behaviour of artificially produced asperities within a circular contact. The quantitative experimental study of longitudinal real surface roughness within a line contact has not been realized yet. That is why, in this study, the line contact formed between a steel tapered roller and glass disc is observed within an optical test rig and the effects of real surface roughness on lubrication film formation are studied. Experiments carried out under pure rolling conditions have shown that the depth is the key parameter that influences the effect on the film thickness. If the roughness features are shallow, the lubrication film shape within the contact follows the shape of the surface closely. However, the groove having only about 800 nm in depth divided the line contact into two parts that behave as two separate line contacts. Such an effect can increase the risk of the wear of rubbing surfaces as the lubrication film thickness between the real machine components can be significantly lower than expected.     

Numerical and experimental study of multi-contact on an elastic half-space

This paper deals with the numerical and experimental study of multiple contacts at the surface of an elastic half-space. A two-scale iterative method is proposed for solving the problem. First a procedure that takes into account interaction gives the contact forces at the tips of the asperities from which the pressure distribution at the contact interface is then calculated using an iterative scheme. Numerical results in the case of two and seven spherical indentors show that the method is as accurate as classical methods and very time efficient even for close proximity contacts. Additionally contact forces and pressures between a rubber block and several spherical indenters were measured. The differences between experiments and theoretical predictions were below 10%. This means that the proposed method can be a reliable tool to model contact problems for which such an accuracy is enough.     

基于分形理论的滑动摩擦表面接触力学模型

依据分形理论,考虑微凸体变形特征及摩擦作用的影响建立滑动摩擦表面接触力学模型。采用一个三次多项式来表达弹塑性变形微凸体的接触压力与接触面积的关系,从而满足在变形状态转变临界点处的微凸体接触面积与接触压力转化皆是连续和光滑的条件。推导出滑动摩擦表面临界弹性变形微接触面积、临界塑性变形微接触面积、量纲一真实接触面积的数学表达式。理论计算结果表明,表面形貌一定时,真实接触面积随着载荷的增大而增大;载荷一定时,真实接触面积随着特征尺度系数的增大而减小,随着分形维数的增大先增大后减小;当表面较粗糙时,摩擦因数对真实接触面积的影响很小;随着表面光滑程度的增大,摩擦因数对真实接触面积的影响增大,真实接触面积随着摩擦因数的增大而增大,特别是当摩擦因数较大时,真实接触面积增大的幅度也较大。接触力学模型的建立,为研究滑动摩擦表面间的摩擦磨损性能提供了依据。     

Mechanics of discrete contact

An approach based on the method of averaging is proposed to solve the discrete contact problems for bodies with given macro- and microgeometry. It divides the problem analysis into two scales of sizes. In macroscale the nominal (continuous) contact region is considered, and the integral equation to determine the nominal contact pressure is reduced. This equation includes the additional displacement function which depends on the nominal contact pressure and the microgeometry parameters at the fixed point of the nominal contact region. This function can be determined from the consideration of the discrete contact problem in the microscale for the particular models of surface microgeometry. The periodic contact problem for an elastic half-space has been solved to find the additional displacement function taken into account the interaction between contact spots. The analysis of the system of equations obtained for the periodic problem makes it possible to state the principle of localization which is used to solve the problems in micro- and in macroscale. The influence of the microgeometry parameters on the real and nominal contact characteristics is analyzed.     

Coalescence and breakup of contact areas: Effects on surface temperatures

In a study of the mechanisms by which thin polymeric films can prevent or delay the onset of fretting corrosion, experimental observations were made of the apparent real area of contact and temperatures generated by friction in a dry-sliding system consisting of stationary polymer-coated steel balls loaded against a vibrating sapphire disk. Five different polymers were used in the original study at vibrating frequencies ranging from 100 to 200 Hz and amplitudes from 20 to 100 μm; but this paper focuses on only one of these—polystyrene coated steel balls in contact with sapphire. Surface temperatures generated by friction were measured using an infrared microscope system. A photomacro/video technique was developed to view the fretting contact interface and to measure the size and distribution of the real areas of contact. The experiments revealed several complex patterns and unusual phenomena. In one example of behavior, a number of small contact patches would suddenly coalesce into one larger contact patch and then break up again into a similar collection of separate patches. This coalescence and breakup occurred at a regular frequency which was much lower than the oscillating frequency. In addition, significant surface temperature spikes corresponding to the occurrence of coalesced areas were observed.A general thermal model previously developed was used to theoretically predict the temperatures corresponding to the experimental conditions [Furey MJ, Vick B, Foo SJ, Weick BL. A theoretical and experimental study of surface temperatures generated during fretting. In: Proceedings of Japan international tribology conference, Nagoya, Japan, 29 October 1990–1 November 1990, vol. II, pp. 809–14; Vick B, Furey MJ. A basic theoretical study of the temperature rise in sliding contact with multiple contacts. Tribology International 2001;34:823–29]. The thermal model consists of a sliding pair of any material combination with three-dimensional and transient conditions. The key feature is the contact area, which is modeled as a collection of rectangular patches in which each patch can have any specified size, shape, position, and time duration. In this way, each contact has a unique start and finish time and the entire collection of contacts can evolve with time in any specified manner. This provides the flexibility to model everything from hard, brittle surfaces such as ceramics to softer, deformable surfaces such as polymers.Using the changes in the apparent real area of contact as observed in the experiments, the theoretical model predicted surface temperatures in close agreement with experimental values. The results of this study show not only that the area of contact is complex and dynamically changing, but that the surface temperatures produced are extremely sensitive to the real area of contact. Although the fundamental mechanisms for the observed phenomena of breakup, coalescence and motion of contact areas are unknown, the study is important since it illustrates the connection between areas of contact and surface temperature—a key unknown which influences physical and chemical behavior in tribological processes.     

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.     

快速点磨削周边磨削层模型及参数

为深入研究快速点磨削机理及工艺,根据快速点磨削的技术与几何学特征,建立点磨削周边接触层及参数的数学模型,对砂轮和工件的等效速度和直径、磨削参数进行理论分析。在已建立快速点磨削接触层及参数的理论模型基础上,推证计及点磨削变量角度和磨削深度的砂轮周边理论接触宽度的计算公式,并对超薄快速点磨 削砂轮周边理论接触宽度和表面粗糙度进行数值仿真。结果表明：与普通外圆磨削不同,砂轮周边与工件实际接触宽度并不恒等于砂轮宽度,点磨削变量角度和磨削深度显著影响砂轮周边的实际接触宽度与工件表面粗糙度 数值。