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).     

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。     

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.     

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.     

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.     

Improved Elastic Contact Model Accounting for Asperity and Bulk Substrate Deformation

An improved elastic contact model for a single asperity system is proposed accounting for both the effects of bulk substrate and asperity deformations. The asperity contact stiffness is based on the Hertzian solution for spherical contact, and the bulk substrate stiffness on the solution of Hertzian pressure on a circular region of the elastic half-space. Depending on the magnitude of the applied load, as well as the geometrical and physical properties of the asperity and bulk materials, the bulk substrate could have considerable contribution to the overall contact stiffness. The proposed single asperity model is generalized using two parameters based on physical and geometrical properties, and is also verified using finite element analysis. A parametric study for a practical range of geometric and physical parameters is performed using finite element analysis to determine the range of validity of the proposed model and also to compare it with the Hertz contact model. The single asperity model is extended to rough surfaces in contact and the contact stiffness from the proposed model and the simpler Greenwood–Williamson asperity model are compared to experimental measurements.     

A New Approach to the Calculation of Flash Temperatures in Dry, Sliding Contacts

The work done in overcoming frictional resistance between sliding surfaces is transformed into heat at the separate, very small and very highly loaded asperity contacts that make up the real area of contact. The temperatures at these initial asperity contacts (the ‘flash temperatures’) are usually much higher than the bulk temperatures and, although of very short duration, can cause potentially major local changes at the surfaces, such as softening, chemical transformations, and local melting. A new approach to calculating these flash temperatures is presented, which relies on the solution, by finite-element analysis, of the three-dimensional equation for transient heat flow in an hemispherical asperity. A Design of Experiments (DoE) exercise showed that the major influential factors on flash temperature were the thermal conductivity, the product of friction coefficient and hardness, the velocity of sliding, and the radius and degree of wear/flattening of the hemisphere. The DoE analysis also produced predictions for the flash temperature, which agreed extremely well with the FE calculations. Response surfaces of flash temperatures are presented, which facilitate the determination of flash temperatures without resorting to FE analysis.     

Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts

Finite-element analyses are carried out to study the effects of friction on the contact and deformation behavior of sliding asperity contacts. In the analysis, on elastic-perfectly-plastic asperity is brought in contact with a rigid flat at a given normal approach. Two critical values of the normal approach are used to describe the asperity deformation. One is the approach corresponding to the point of initial plastic yielding, and the other at the point of full plastic flow. Additional variables used to characterize the deformation behavior include the shape and size of the plastic zone and the asperity contact size, pressure, and load capacity. Results from the finite-element analysis show that the two values of critical normal approach decrease significantly as the friction in the contact increases, particularly the approach that causes plastic flow of the asperity. The size of the plastically deformed zone is reduced by the friction when the contact becomes fully plastic. The reduction is very considerable with a high friction coefficient, and the plastic deformation is largely confined to a small thin surface layer. For a low friction coefficient, the contact size, pressure and load capacity of the asperity are not very sensitive to the friction coefficient. For a moderate friction coefficient, the contact pressure is reduced and the junction size increased; the load capacity of the asperity is not significantly affected due to the compensating effects of the pressure reduction and the junction growth. For a high friction coefficient, the pressure-junction compensation is not longer sufficient and the asperity load capacity is reduced. The degree of the friction effects on these contact variables depends on the applied force or the normal approach. Although the analyses are conducted using a line-contact model, the authors believe that the effects of friction in sliding asperity contacts of three-dimensional geometry are essentially the same and the same conclusions would have been reached. These results may provide some guidance to the modeling of rough surfaces in boundary lubrication, in which the asperity friction coefficient can be high and vary significantly both in time and from one micro-contact to another.     

An Analytical Solution for Elastic and Elastic-Plastic Contact Models

In tribology often a closed form solution for calculation of contact stress and real contact area is required for the purposes of, for example, developing wear maps and temperature profiles at asperities. In assuming a Gaussian distribution of asperity heights it is not possible to obtain an analytical solution for the contact load and real contact area for many analytical models such as those developed by Greenwood and Williamson (elastic model), Chang, et al. (elastic-plastic model) and Horng (elliptic elastic-plastic model). In this paper, two exponential functions have been derived from a fitting procedure applied to the numerical results of the Gaussian height distribution thus offering an analytical expression for the above three models. It has been demonstrated that the two exponential functions (φ₂* and φ₄*) can give a fair approximation to the contact load and the real contact area in the separation of 0 to 4σ. In addition, variations in plasticity index (ψ) and effective asperity radius (γ) do not significantly affect the approximated accuracy. The results obtained by the newly derived exponential functions have been compared with the exponential function φ₁*; suggested by Greenwood and Williamson, 1966 and it has been shown that use of φ₁* invariably gives a larger error than using two exponential functions over two ranges of separation distances.     

考虑蠕变特性的橡胶微凸体与金属表面接触特性分析

金属-橡胶接触广泛存在于密封结构中,密封接触表面上微凸体间的相互作用会直接影响整个密封界面的接触特性,进而影响其密封性能。基于粗糙密封界面的单个微凸体,考虑橡胶的蠕变特性,采用理论分析和仿真研究相结合的方式研究橡胶微凸体与金属表面的接触特性。通过橡胶蠕变特性的实验结果,构建橡胶蠕变计算模型;构建半球微凸体与金属平板间的有限元模型,进行考虑蠕变特性的仿真,分析其接触特性,并与Hertz接触理论的计算值进行对比。结果表明：在蠕变阶段,接触半径、法向变形量和最大等效蠕变应变均随蠕变时间的增加而增大,最大接触压力随蠕变时间增大而减小,这均可能导致密封性能的下降;随压力载荷的增大,接触半径、法向变形量、最大接触压力和最大等效蠕变应变均增大,但增大的趋势逐渐减小;橡胶微凸体与金属表面间的等效模量随蠕变时间的增加而减小,随压力载荷增大而增大。     

Analysis and Convenient Formulas for Elasto-Plastic Contacts of Nominally Flat Surfaces: Average Gap,Contact Area Ratio,and Plastically Deformed Volume

Interaction of nominally flat engineering surfaces that leads to a large contact area exists in many mechanical systems. Considering periodic similarity of surface geometry, a numerical three-dimensional elasto-plastic contact model can be used to simulate the contact behaviors of two nominally flat surfaces with the assistance of the continuous convolution and Fourier transform (CC-FT) algorithm. This model utilizes the analytical frequency response functions (FRF) of elastic/plastic responses of materials and provides contact performance results, including the average surface gap, the contact area ratio, and the volume of plastically deformed material, which may be defined as performance variables. Following the digital filtration technology, rough surfaces can be numerically generated with specified autocorrelation length and the first four orders of statistical moments. A group of contact simulations are conducted with various working conditions. The effects of topographic and material properties on the contact behaviors are discussed. With a multi-variables regression method, empirical formulas are developed for the performance variables as functions of surface statistical characteristics, material properties, a hardening parameter, and the applied load in terms of pressure.     

Modelling of static friction in rubber–metal contact

A static friction model for contact between rough rubber and metal surfaces is developed. This model is based on the contact of a viscoelastic–rigid asperity couple. Single asperity contact is modelled in such a way that the asperities stick together in a central region and slip over an annulus at the edge of the contact. The slip area increases with increasing tangential load. Consequently, the static friction force is the force when the slip area is equal to the contact area. Using the model, the traction distributions, contact area, tangential and normal displacement of two contacting asperities are calculated. The single asperity model is then extended to multi-asperity contact, suitable for rough surfaces. This model allows calculation of the above-mentioned parameters for two rough surfaces (a rubber and a metal one) subjected to normal and tangential loads. A parametric study will be presented. The results are qualitatively in good agreement with those found in literature.     

Contact analysis of three-dimensional rough surfaces under frictionless and frictional contact

A methodology for surface and sub-surface stress calculation of nominally flat on flat rough surface contact has been developed. This methodology is applicable for both large area contact (Hertzian contact) and small area of asperity contact (point load contact) with and without surface friction. A total of nine rough surfaces are generated by the computer with specified standard deviation of surface heights, σ, of 0.3, 1.0 and 3.0 nm, and correlation length, β^*, of 0.1, 0.5 and 0.9 μm. Under the typical applied load at the magnetic head slider-disk interface, small numbers of contact points are obtained and the deformation is purely elastic. Since these contact points are scattered and isolated, asperity contact behaves like point load contact. As β^* becomes larger, more adjacent points will be in contact at a certain contact spot and this is especially true at small σ. All the cases of flat on flat rough surface contact yield maximum von Mises stress on and near the surface at both frictionless and frictional contacts; no local maximum occurs in the sub-surface. In general, the friction effect in the vicinity of contact point is to increase the stress magnitude, while outside this region it also alters the stress distribution. For a surface of small β^* and large σ at high load of 1000 times of the nominal pressure at the head-disk interface, the contact pressure reaches the hardness at a few contact points and plastic deformation takes place in the near surface.     

考虑硬度变化的结合面法向接触刚度分形模型*

基于分形几何理论,考虑微凸体因应变硬化而造成弹塑性变形阶段硬度随变形量变化而变化,建立结合面第一、第二弹塑性变形阶段单次加载刚度分形模型。推导出在计入硬度变化的情况下,单个微凸体在弹塑性变形阶段法向接触刚度与接触面积之间的关系式,进而得出结合面在弹塑性变形阶段法向接触刚度与接触面积、接触载荷之间量纲为一的关系式,并通过仿真分析得出相关参数对结合面法向接触刚度的影响。仿真结果显示:考虑硬度变化时,结合面量纲一法向接触刚度的值与法向实际接触载荷、实际接触面积之间存在关系;结合面法向接触刚度随着分形维数D的增大而增大;分形维数一定时,结合面法向接触刚度随表面长度尺度参数G值增大而增大。     

Fatigue Life Reduction in Mixed Lubricated Elliptical Contacts

Highly loaded ball and rolling element bearings are often required to operate in the mixed elastohydrodynamic lubrication regime in which surface asperity contact occurs simultaneously during the lubrication process. Predicting performance of components operating in this regime is important as the high asperity contact pressures can significantly reduce the fatigue life of the interacting components. Rolling contact fatigue is one of the most dominant causes of failure of components operating in mixed lubrication regime. Contact fatigue begins with the initiation of microscopic fatigue cracks in the rolling contact surfaces or within the sub-surface regions due to cyclic shear stresses. Investigation of mixed lubrication effects on performance of machine components is of significant importance in order to understand and enhance their load carrying capacity. This article investigates the effects of mixed lubrication and surface roughness on machine components performance. Results from a mixed lubrication model are utilized to investigate the effects of different operating conditions on fatigue life of the components. Simple rough surfaces consisting of single hemispherical bump as well as complex rough surfaces consisting of a numerically generated 3D rough surface operating under mixed lubrication conditions are studied and results presented. The stress-based Ioannides and Harris model incorporating the fatigue limit is used to evaluate the fatigue life variation. Fast Fourier Transform (FFT) technique is used to significantly reduce the time required for the computation of internal stresses.     

Real area of contact and boundary friction in metal forming

Upper-bound models for asperity flattening on a workpiece surface undergoing bulk plastic deformation are developed. It is found that the effective hardness of the surface can be greatly reduced by the presence of underlying plastic flow. Theoretical predictions of the variation of real area of contact with strain show excellent agreement with experiments using model asperities in rolling. Friction models which allow for the reduction in effective hardness are developed for cases in which roughness is concentrated on either the workpiece or tooling.     

硬度影响压痕弹塑性行为的有限元分析

以具有不同硬度40Cr为研究对象,根据集中载荷下的接触模型和赫兹理论,计算了压痕接触半径和压痕附近弹性区域的表面局部接触应力,并采用有限元法,分析硬度物理量对压痕弹塑性行为、局部接触应力、卸载后保留在内部的残余应力的影响,探讨压痕参数、压痕接触应力、残余应力与硬度之间的关系以及载荷增加时它们的发展.结果表明,相同载荷下塑性隆起量、压痕接触半径、压痕量和塑性区范围随着硬度值的提高而减小,弹性回弹量、最大接触应力和残余应力随硬度提高而增加;压痕周围处接触应力和残余应力、其分布范围和塑性区域随载荷的增加而增加.