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.     

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.     

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.     

Rolling Element Bearing Contact Geometry Analysis

Rolling element bearings are subjected to a variety of loads during the operation of machinery. Raceway contact geometries should be designed and analyzed in a manner which accurately models internal contact stress distributions for these different load cycle conditions. To properly determine contact stresses, analyses should determine the orientation of rolling elements relative to the raceways through consideration of load, bearing alignment and bearing internal geometry. Since design loads are not always well defined and machinery upgrades may increase loads, contact geometry designs should have sufficient flexibility to handle conditions differing from the initial design loads. An analytical procedure with examples is discussed.     

箔带轧制的三维接触有限元分析

基于对三维多体接触问题有限元混合法基本原理及接触面的定解条件和判定条件的分析 ,本文将箔带轧机辊系及轧件组成的辊带系统视为三维多体接触问题 ,建立了接触有限元分析模型。对某厂四辊铝箔轧机初轧工况进行了数值计算 ,求得了辊缝内轧制压力的真实分布规律 ,充分揭示了箔带轧制过程中非轧制辊面存在压靠的事实。最后 ,针对来料板凸度对轧件板形的影响进行了有限元仿真分析 ,得出了与轧制规程图分析法一致的结论     

A Critique of Rolling Contact Fatigue Criteria for Bearing Rings with Hoop Stresses

Tensile residual and interference fit stresses not treated in classical bearing formulations are known to reduce bearing rolling contact fatigue (RCF) life. Recent modifications of such theory to account for these stresses have simply included them in the computation of a single yield stress type criterion—either maximum shear or equivalent stress. An alternative modification is proposed and demonstrated for fatigue crack initiation that recognizes the primary influence of the maximum range of shear stress but includes the effect of normal stress on the critical planes, as in other successful bulk fatigue criteria for multiaxial nonproportional stress cycle fatigue.     

An Improved Approach for 3D Rolling Contact Fatigue Simulations with Microstructure Topology

Several 2D and 3D numerical models have been developed to investigate rolling contact fatigue (RCF) by employing a continuum damage mechanics approach coupled with an explicit representation of microstructure topology. However, the previous 3D models require significant computational effort compared to 2D models. This work presents a new approach wherein efficient computational strategies are implemented to accelerate the 3D RCF simulation. In order to reduce computational time, only the volume that is critically stressed during a rolling pass is modeled with an explicit representation of microstructure topology. Furthermore, discontinuities in the subsurface stress calculation in the previously developed models for line and circular contact loading are removed. Additionally, by incorporating a new integration algorithm for damage growth, the fatigue damage simulations under line contact are accelerated by a factor of nearly 13. The variation in fatigue lives and progression of simulated fatigue spalling under line contact obtained using the new model were similar to the previous model predictions and consistent with empirical observations. The model was then extended to incorporate elastic–plastic material behavior and used to investigate the effect of material plasticity on subsurface stress distribution and shear stress–strain behavior during repeated rolling Hertzian line contact. It is demonstrated that the computational improvements for reduced solution time and enhanced accuracy are indispensable in order to conduct investigations on the effects of advanced material behavior on RCF, such as plasticity.     

二重分形特征表面弹塑性接触模型的研究

基于Bhushan的二重分形弹塑性接触模型，完全分析和推导出弹塑性接触点上接触线长度与总载荷的关系，对二重分形特征表面弹塑性接触模型作了进一步的完善。     

超越离合器滚动接触状态的三维有限元分析

基于超越离合器的CAD实体几何模型，建立了由摆杆、滚拄和星轮组成的摩擦传动组合体的三维有限元计算模型，进行了滚动接触状态分析，得到了各接触体的法向接触应力、切向接触应力和接触变形，并研究了接触体之间的摩擦系数对接触状态的影响规律，为超越离合器的磨损研究提供了依据。     

Study on Rolling Contact Fatigue in Hydrogen Environment at a Contact Pressure below Basic Static Load Capacity

A method to determine the shoulder height in an angular contact ball bearing using a 3D contact analysis is proposed. The load analysis was performed by calculating the distributions of internal loads and contact angles for each rolling element. From the results of the bearing load analysis and the contact geometry between the ball and raceways, 3D contact analyses using an influence function were conducted. The algorithm developed was applied to an angular contact ball bearing for an automotive wheel. The effects of axial load on contact pressure at the inner and outer raceways were evaluated and the critical axial loads in the present shoulder height were calculated. The critical shoulder heights were also determined when the bearing was subjected to a practical load for a steel ball bearing. The proposed methodology is generally applicable for the purpose of reducing the material cost and improving the efficiency of the bearing design process.     

A New Statistical Model of Contact Fatigue

A detailed derivation of a new statistical model of contact fatigue life followed by its qualitative and quantitative analysis are presented. The model is based on contact and fracture mechanics and statistical treatment of the initial distribution of material defect. The model assumptions and their validation as well as the model properties are discussed. A parametric study of the model is performed. A generalization of the model for the case of stochastic residual stress or other contact parameters is proposed. Some analytical formulas for calculation of contact fatigue are proposed and analyzed. The validation of the model and its applicability to calculation of bearing fatigue life and some particular data are considered. A reflection of the quality of bearing manufacturing process on the contact fatigue model is discussed.     

Numerical Investigation on the Effects of Machining-Induced White Layer during Rolling Contact

It is well known that a thin phase-transformed white layer can be formed on component surfaces produced by hard machining. However, it is not clear as to how the white layer affects component performance, for example, in rolling contact fatigue. This study aims to determine the effects of white layer and associated residual stress on rolling contact stresses and strains. It is nearly impossible for an experimental study to identify the effects of white layer alone on rolling contact. Furthermore, small-scale contact stresses and strains (less than 30 μm) of the phase-transformed region are difficult to measure using the current experimental techniques. Therefore, a finite element analysis simulation model of rolling contact incorporating machining-induced surface integrity has been developed in this study. Three cases were investigated to decouple the effects of surface integrity factors: surface with white layer only, surface with residual stress only, and surface with white layer and residual stress. The simulation results show that distinct material properties of the white layer significantly influence the magnitudes and distributions of near-surface stresses and strains instead of those in the subsurface. Furthermore, it can be inferred that the white layer would affect near-surface fatigue damage instead of subsurface fatigue damage. The simulated near-surface fatigue damage mechanisms have been substantiated by the fatigue test data.     

直线滚动导轨结合部的静刚度解析

介绍了影响结合部静态特性的主要因素及接触理论,并通过非线性接触表面层理论对直线滚动导轨圆柱结合部的接触变形进行了解析计算,可为机械结构的静态特性提供结合部的边界条件。     

飞机柔性件铆装中的接触力建模与偏差分析

基于线弹性小变形假设,分析飞机柔性板件装配过程中的相互接触作用。提出了基于三角网格模型几何干涉分析的接触状态检测,采用部分析因设计方法（FractorialFactorialDesign）建立了接触力与接触节点间隙的线性关系,并在此基础上设计了接触平衡位置求解算法。通过与实验对比,验证了基于接触力模型的装配偏差分析模型预测柔性板件装配偏差的准确性。     

Influence of Mixed-Lubrication and Rough Elastic-Plastic Contact on the Performance of Small Fluid Film Bearings

A previously developed deterministic elastohydrodynamic (EHD) numerical model for small fluid film bearings functioning in the mixed lubrication regime is extended in this work by considering the rough contact. Several simplifying hypotheses are made: the shaft is considered rigid and smooth, turning at low speeds (isothermal regime), and the pad is assumed to have an elastic-perfectly-plastic behavior. The Reynolds equation is solved on a very fine mesh and the elasto-plastic pad deformation caused by the hydrodynamic pressure is taken into account. A deterministic active set-based method is used for determining the contact pressure, the contact extent, and the corresponding deformation. The results are presented for a partial journal bearing configuration, with a linear shaft velocity of 0.47 m/s under specific pressures varying up to 50 MPa. Two pad materials are considered, and the lubricant is either isoviscous or piezoviscous oil. The simulation results, presented as a function of the bearing specific pressure, include eccentricity, the film thickness, the friction torques, the contact extent, etc. Stribeck curves showing the evolution of the friction coefficient in the hydrodynamic and mixed lubrication regimes are also discussed.     

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 Wear Model of Plane Sliding Pairs Based on Fatigue Contact Analysis of Asperities

Wear modeling is essential to predict and improve wear resistance of machine parts. This article presents a fatigue wear model of plane sliding pairs under dry friction. The wear model is constructed through developing a dynamic contact model of surfaces and proposing a mean fatigue damage constant of asperities. It is simpler and more practical than existing fatigue wear models because it describes the quantitative relationship between the wear behaviors of the plane sliding pairs and the main factors including the load and sliding speed, material property, friction property, and surface topography of the pairs. Furthermore, the wear model can predict the wear of each component of the sliding pairs. Reasonability and applicability of the wear model are validated via pin-on-disc wear tests. The wear model is applicable to predict the wear of the plane sliding pairs, which is characterized by friction fatigue of contact surfaces. The wear model can also be used to guide the tribological design of sliding pairs in machinery.     

Surface and Subsurface Cracks in Rolling Contact Fatigue of Hardened Components

The competitive aspect of surface and subsurface fatigue crack propagation in hardened components subjected to rolling contact fatigue is highlighted, the former being greatly affected by the working conditions (in particular the presence of tangential stresses and lubricant), the latter depending mainly on the inclusions content and on the hardness profile. In order to determine which one of these kinds of damage is favoured, initial data consisting of contact load, rolling and sliding speed, theological properties of the lubricant, material hardness and inclusions content are necessary.

The concurrent role of asperities and Hertzian stress field in determining surface crack propagation is explained with the approach of the “quiescent zone.” calculating the stress intensity factors range in a contact cycle and considering the pumping effect of the fluid possibly present on the contact surface.

Inherent defects (especially oxides) are thought to be responsible for subsurface cracks origin and the Murakami formula for short cracks is extrapolated to describe their growth threshold, which also depends on the hardness and therefore on the depth in surface hardened components.

A crack propagation index is then defined as a ratio of applied to threshold stress intensity factor, both for surface and subsurface cracks. Evaluating this index for a general operating condition, it is possible to determine which damage mechanism is favoured, taking into account the decisive effect of the hardness profile.     

轨道结构参数对轮轨滚动接触应力影响

利用三维弹性体非Hertz滚动接触理论及数值程序CONTACT,并借助于弹性力学中的Bossinesq-Cerruti力1位移公式和Gauss数值积分方法,分析JM3型踏面轮对沿曲线轨道滚动接触时轨底坡、轨距和曲线半径等轨道参数对轮轨滚动接触斑最大切应力、等效应力、正压力和磨耗数的影响.数值结果表明,当内外轨底坡为1/20时,轮轨接触斑的应力、磨耗数及正压力分布达到最小值,其中最大切应力与等效应力可分别降低40.15%、39.37%;现行使用1/40轨底坡情况下轮轨接触斑正压力较大,建议对磨耗型车轮踏面进行优化设计.适当增加轨距能达到降低轮轨最大切应力、等效应力和正压力的效果.曲线工况下接触斑正压力值显著增加,曲线半径能改变轮轨接触斑粘滑区的分布且减小曲线半径值会增加接触斑的总滑动量,从而导致轮轨磨耗数的显著增加.