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.     

依据各向异性分形几何理论的固定结合部法向接触力学模型

基于各向异性分形几何理论,考虑微凸体变形特点、表面微凸体承受法向载荷的连续性和光滑性原理,以及区分微凸体分别处于弹性、塑性变形时的一个微凸体实际微接触面积,建立固定结合部法向接触力学模型。采用二变量Weierstrass-Mandelbrot函数模拟各向异性三维分形轮廓表面。推导出划分弹塑性区域的临界弹性变形微接触截面积、结合部量纲一法向载荷、结合部量纲一法向接触刚度的数学表达式。数值仿真结果表明：当表面形貌的分形维数、分形粗糙度一定时,真实接触面积随着结合部法向载荷的增大而增大;结合部法向接触刚度随着真实接触面积、结合部法向载荷、相关因子或材料特性参数的增大而变大;当分形维数由1变大时,结合部法向接触刚度随着分形维数的变大而增大;当分形维数增加到趋近于2时,结合部法向接触刚度有时却会随着分形维数的增加而降低。结合部法向接触力学模型的构建,有助于分析固定接触表面间的真实接触情况。     

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。     

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。     

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.     

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。     

Application of Fractal Contact Model in Dynamic Performance Analysis of Gas Face Seals

Fractal theory provides scale?independent asperity contact loads and assumes variable curvature radii in the contact analyses of rough surfaces, the current research for which mainly focuses on the mechanism study. The present study introduces the fractal theory into the dynamic research of gas face seals under face?contacting conditions. Structure?Function method is adopted to handle the surface profiles of typical carbon?graphite rings, proving the fractal con?tact model can be used in the field of gas face seals. Using a numerical model established for the dynamic analyses of a spiral groove gas face seal with a flexibly mounted stator, a comparison of dynamic performance between the Majumdar?Bhushan(MB) fractal model and the Chang?Etsion?Bogy(CEB) statistical model is performed. The result shows that the two approaches induce differences in terms of the occurrence and the level of face contact. Although the approach distinctions in film thickness and leakage rate can be tiny, the distinctions in contact mechanism and end face damage are obvious. An investigation of fractal parameters D and G shows that a proper D(nearly 1.5) and a small G are helpful in raising the proportion of elastic deformation to weaken the adhesive wear in the sealing dynamic performance. The proposed research provides a fractal approach to design gas face seals.     

Elastic and elastic-perfectly plastic analysis of an axisymmetric sinusoidal surface asperity contact

ABSTRACT

Closed-form finite-element empirical models are available for elastic and elastic–plastic spherical and sinusoidal contact. However, some of these models do not consider the effect of interaction with adjacent asperities or require extensive numerical resources because they employ a full 3-D model. Therefore this work has analysed and quantified the behaviour of an elastic and elastic- perfectly plastic axisymmetric sinusoidal surface in contact with a rigid flat for a wide range of material properties and different values of the amplitude to wavelength ratio from initial to complete contact (high load). The numerical results agreed well with the Hertz model and the Jackson–Green elastic–plastic spherical contact model at low loads. Empirical equations for elastic and also elastic-perfectly plastic cases are formulated for the contact pressure, contact area and surface separation. From the current analysis, it is found that it is not any single parameter, but different combinations of material properties and surface roughness that govern the whole contact behaviour. The critical value of the amplitude of the sinusoidal asperity below which it will deform completely elastically from initial to complete contact is established. At low values of amplitude normalized by the critical amplitude, it was found that the contact behaved similar to a spherical contact, with the average pressure (hardness) always remaining lower than three times the yield strength. However, at higher values the average pressure increased toward a value as high as six times the yield strength at complete contact. All of these equations should be useful in rough surface contact modelling, lubrication analysis, electrical contact modelling and in many other applications.     

基于域扩展因子和微凸体相互作用的结合面接触刚度模型研究

结合面接触刚度直接影响了机械设备的整机动态特性,为了建立更为准确的接触刚度模型,以分形几何理论为基础,利用单一微凸体承受局部载荷时的弹性变形特性,并基于域扩展因子引入微接触截面积分布函数,推导了考虑表面微凸体相互作用影响的结合面接触刚度分形模型。为了验证所提出模型的准确性,通过三维非接触式测量,获得了试验试样的表面轮廓数据,并根据结构函数法,计算了各个试样的表面分形参数,进而将理论接触刚度与试验结果对比分析,结果表明：法向接触刚度的增长速率与粗糙面表面临界接触面积有关,临界接触面积决定了结合面内的弹性变形占比。考虑微凸体相互作用后,所提出模型的预测曲线更加符合试验中法向载荷与接触刚度的关系。     

Fractal prediction model of thermal contact conductance of rough surfaces

The thermal contact conductance problem is an important issue in studying the heat transfer of engineering surfaces,which has been widely studied since last few decades,and for predicting which many theoretical models have been established.However,the models which have been existed are lack of objectivity due to that they are mostly studied based on the statistical methodology characterization for rough surfaces and simple partition for the deformation formats of contact asperity.In this paper,a fractal prediction model is developed for the thermal contact conductance between two rough surfaces based on the rough surface being described by three-dimensional Weierstrass and Mandelbrot fractal function and assuming that there are three kinds of asperity deformation modes:elastic,elastoplastic and fully plastic.Influences of contact load and contact area as well as fractal parameters and material properties on the thermal contact conductance are investigated by using the presented model.The investigation results show that the thermal contact conductance increases with the increasing of the contact load and contact area.The larger the fractal dimension,or the smaller the fractal roughness,the larger the thermal contact conductance is.The thermal contact conductance increases with decreasing the ratio of Young’s elastic modulus to the microhardness.The results obtained indicate that the proposed model can effectively predict the thermal contact conductance at the interface,which provide certain reference to the further study on the issue of heat transfer between contact surfaces.     

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.     

Transition from elastic to plastic deformation as asperity contact size is increased

Contacts between a clean sodium chloride pyramidal shaped asperity and a plane NaCl surface have been investigated by molecular dynamics simulations. For small contacts, a few atoms across, the asperity jumped to contact and behaved elastically as normal load was applied. Then, when the force was reversed to detach the asperity, brittle failure occurred without any damage to the crystalline materials. However, as the contact size of the asperity was increased to 6 × 6 atoms in area, the mechanism of detachment was seen to alter. The jump to contact was elastic and damage free, but the separation could not be achieved elastically, but required plastic deformation, giving extensive energy dissipation and severe damage as edge defects propagated through the asperity. Above this contact size, plastic flow was dominant. However, there is clearly a further transition back to elastic fracture once the asperity becomes large enough for Griffith-type cracking to propagate above 1 μm in size, since large sodium chloride contacts are known to be brittle above the micrometre scale, depending on the presence of crack initiating defects.     

基于分形几何的表面微观形貌模拟及粘着弹性接触计算研究

基于Ausloos和Berman提出的推广的W-M函数对具有分形特征的粗糙表面进行仿真模拟,分析了函数中与尺度无关的特征参数对表面微观形貌的物理意义。同时,基于Yan和Maugis的理论研究,用模拟的分形表面建立了考虑表面效应的弹性接触模型,通过数值方法对整个过程进行迭代求解,得到了两接触面在不同的接触条件下各个接触斑点上的载荷分布和真实接触面积以及接触斑点的数量和尺寸。由于真实接触面积的尺寸敏感地反应表面微观几何形貌的变化,因此该方法为研究粘着机制和减小微尺度粘着效应提供了思路。     

粗糙表面微凸体对液黏传动的影响

为研究液黏传动过程中粗糙表面的承载特性,将分形理论引入到两粗糙表面摩擦过程之中,分析传动过程中混合摩擦和边界摩擦两阶段的微凸体承载过程,考虑微凸体弹塑性变形,对M-B模型进行修正,建立修正的微凸体承载模型。建立基于修正M-B模型的微凸体承载模型。通过数值仿真得到有效面积系数、分形参数对液黏调速离合器传动过程的影响规律;对修正的微凸体承载模型的计算结果与M-B模型的计算结果进行对比分析。结果表明:微凸体接触载荷和传递转矩随着面积比的增大而增大,当有效面积系数与尺度系数增大时,接触载荷与传递转矩均有所增大;分形维数为1.5时,微凸体接触载荷与传递转矩最小且随面积比的变化最为缓慢;在整个接触区域内,弹性变形区域的面积、接触载荷以及传递转矩最大,其次是弹塑性变形区域,塑性变形区域最小;考虑弹塑性变形时,微凸体接触载荷与传递转矩均有所下降;修正M-B模型和M-B模型间的修正系数范围在25%以内,修正系数随着有效面积系数、尺度系数的增大而增大,随着分形维数的增大而减小。     

ELASTIC-PLASTIC ADHESION MODEL FOR SINGLE ASPERICAL ASPERITY MICROCONTACT

The adhesion of single asperity contacting with a rigid flat is investigated. The microcontact model of the deformable asperity is established utilizing fractal geometry, which makes the resuRed adhesion model to relate with the surface characteristics that the asperity belongs to. The Dugdale approximation is utilized to consider the adhesive interaction within and outside the contact area. Then the model for solving the elastic-plastic adhesion of single asperity is presented by combing the Maugis-Dugdale（MD） model. To illustrate the necessity of considering the plastic deformation in microcontact, simulations of the relationship between the adhesive contact load and the interference of the asperity are performed. The result shows that the presented model is more suitable for the solution of the elastic-plastic microcontact of spherical asperity due to intermolecular adhesive interactions.     

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.     

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.     

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.     

基于分形理论的圆弧齿轮滑动摩擦接触力学模型

考虑到圆弧齿线圆柱齿轮传动接触之间的滑动摩擦与微凸体的连续性变形,结合分形理论和Hertz接触理论建立圆弧齿线圆柱齿轮的滑动摩擦接触力学模型,通过模型数值分析与ANSYS WORKBENCH分析的最大接触应力结果对比,证明该模型所反映圆弧齿线圆柱齿轮接触应力状态的正确性。该模型中,载荷与真实接触面积之间关系不仅与分形维数和特征尺度系数有关,还与齿轮节点曲率和齿轮齿线半径有关。同时,理论计算表明,分形维数一定时,真实接触面积随着载荷的增大而增大;载荷一定时,真实接触面积随着分形维数的增大先增大后减小,随着特征尺度系数的增大而减小;摩擦因数对真实接触面积的影响不大。该模型的建立为圆弧齿线圆柱齿轮工作状态的研究及强度分析提供了理论依据。     

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.