Elsevier

Wear 磨损

Volume 256, Issues 11–12, June 2004, Pages 1114-1127
第256卷，第11-12期，2004年6月，第1114-1127页
Wear

Finite element simulation and experimental validation of fretting wear
微动磨损的有限元模拟与实验验证

https://doi.org/10.1016/j.wear.2003.07.001Get rights and content 获取权限和内容

Abstract 摘要

A finite element-based method is presented for simulating both the fretting wear and the evolution of fretting variables with number of wear cycles in a cylinder-on-flat fretting configuration for application to aeroengine transmission components. The method is based on a modified version of Archard’s equation and is implemented within a commercial finite element code. Fretting tests are employed to determine the coefficient of friction (COF) and the wear coefficient applicable to the contact configuration and loading conditions. The wear simulation technique is incremental in nature and the total simulation time has been minimised via mesh and increment size optimisation. The predicted wear profiles have been compared with profilometer measurements of fretting test scars.
提出了一种基于有限元的方法来模拟微动磨损和微动变量的演变与磨损循环数在气缸上的平板微动配置应用于航空发动机传动部件。该方法是基于修改后的版本的Archard方程，并在商业有限元代码内实现。微动试验是用来确定摩擦系数（COF）和磨损系数适用于接触配置和负载条件。磨损模拟技术本质上是增量的，通过网格和增量大小优化，总模拟时间已最小化。预测的磨损轮廓进行了比较，轮廓仪测量微动试验疤痕。

Keywords 关键词

Wear
Fretting wear
Finite element
Contact
Friction
Simulation
Super CMV

磨损
微动磨损
有限元
接触
摩擦
仿真
Super CMV

Nomenclature 命名法

    a
    contact half-width 接触半宽
    A
    apparent contact area 表观接触面积
    b
    width of the flat specimen
    平板试样宽度
    cj
    minimum gap between the worn surfaces due to the wear of the jth increment
    由于第j个增量的磨损，磨损表面之间的最小间隙
    E
    Young’s modulus 杨氏模量
    h
    total wear depth 总磨损深度
    Δhi,j
    Δhi，j
    wear depth increment at node i for jth wear increment
    第j次磨损增量时节点i处的磨损深度增量
    Δhcrit
    Δh临界值
    critical wear depth increment
    临界磨损深度增量
    hm
    average wear scar depth 平均磨痕深度
    hmax
    maximum wear scar depth 最大磨痕深度
    H
    hardness of the material 材料硬度
    k
    dimensional Archard wear coefficient
    维Archard磨损系数
    kl
    local wear coefficient 局部磨损系数
    K
    non-dimensional Archard wear coefficient
    无因次Archard磨损系数
    Nt
    total number of wear cycles
    磨损循环总数
    ΔN
    increment in number of simulation wear cycles
    模拟磨损循环次数增量
    ΔNcrit
    ΔN临界值
    critical value of ΔN for stability
    ΔN稳定性临界值
    p0
    均p0
    maximum Hertzian contact pressure
    最大赫兹接触压力
    pi,j pi，j
    contact pressure at node i for jth wear increment
    第j次磨损增量时节点i处的接触压力
    p(x)
    p（x）
    contact pressure as a function of x-position
    作为x位置函数的接触压力
    P
    applied normal load 外加法向载荷
    R
    radius of contacting surface
    接触面半径
    si,j si，j
    contact slip during a quarter cycle at node i for jth wear increment
    对于第j个磨损增量，在四分之一周期期间在节点i处的接触滑动
    S
    total (accumulated) slip distance
    总滑移距离
    t
    time 时间
    T(t)
    T（t）
    frictional force at time t
    时间t时的摩擦力
    Tmax
    maximum frictional force during one fretting cycle
    一个微动循环期间的最大摩擦力
    V
    total wear volume 总磨损体积
    W
    wear scar width 磨痕宽度
    x,y
    x为oh
    rectangular co-ordinates 直角坐标
    yi,j yi，j
    y coordinate of node i at beginning of jth wear increment
    第j次磨损增量开始时节点i的y坐标

    Greek letters 希腊字母

    δ*
    applied stroke half-amplitude
    作用冲程半幅
    μ
    coefficient of friction 摩擦系数
    σx, σy
    σx，σy
    normal stresses on planes perpendicular to the x, y-axes
    垂直于x，y轴的平面上的正应力
    ν
    Poisson’s ratio 泊松比
    ω
    maximum penetration depth used in ABAQUS contact algorithm
    ABAQUS接触算法中使用的最大穿透深度

1. Introduction 1.介绍

Fretting wear is a surface degradation process in which removal of material is induced by small-amplitude oscillatory movement between contacting components, such as flexible couplings and splines, jointed structures and so on. Analysis of the mechanisms of this type of surface damage has been extensively developed [1], [2], [3], [4]. The main parameters affecting fretting wear are reported to be normal load, slip amplitude, frequency, contact geometry, surface roughness and material properties. The “fretting map” approach, established by Vingsbo and Soderberg [5] and Vincent et al. [6], has shown that fretting damage evolution depends strongly on the fretting regime. Debris is also a critical factor influencing fretting wear. It was reported that, once debris accumulates on the contact surfaces and forms a compact oxide layer, the wear rate is significantly reduced [4]. In recent years, Godet and co-workers [7], [8] developed the theories of third-body tribology and velocity accommodation mechanisms to explain the role of wear debris in specific fretting conditions.
微动磨损是一种表面退化过程，它是由柔性联轴器、花键、联接结构等接触部件之间的小幅振荡运动引起的材料去除，对这类表面损伤机理的分析已得到广泛的发展[1]、[2]、[3]、[4]。影响微动磨损的主要参数是法向载荷、滑动振幅、频率、接触几何形状、表面粗糙度和材料特性。由Vingsbo和Soderberg[5]和Vincent等人[6]建立的“微动图”方法表明微动损伤演化强烈依赖于微动机制。 磨屑也是影响微动磨损的重要因素。据报告，一旦碎屑在接触表面上积聚并形成致密的氧化层，磨损率就会显著降低[4]。近年来，Godet等人[7]、[8]发展了第三体摩擦学和速度调节机制理论，以解释磨屑在特定微动条件下的作用。
Compared with the development of qualitative understanding, quantitative assessment of fretting wear is less well advanced. One of the difficulties is the absence of a “universal” and well-formulated wear model [9]. In addition, it is not clear how to incorporate the effect of wear debris into such a quantitative model. For some situations, where wear debris is more easily eliminated from the contact area and more metal-to-metal contact is maintained, the influence of the debris can be reasonably neglected. In such cases, fretting wear can therefore be regarded as a purely contact-based wear problem. Analytical techniques for such problems, were developed by Korovchinsky [10], Galin [11], Galin and Korovchinsky [12], amongst others. Korovchinsky et al. [13] proposed an analytical method to simulate fretting wear under partial slip conditions for two-dimensional, initially, Hertzian configurations. In this model, Archard’s equation is applied locally to evaluate wear and the gap variation within the slip zones during one cycle. A stepwise procedure is then employed to calculate the evolution of the contact characteristics as a function of increasing numbers of wear cycles. Johansson [14] presented a finite element solution that incorporates a local implementation of Archard’s equation to evaluate the change of contact geometry and the associated changes in contact pressure. More recently, Oqvist [15] presented a study on the numerical simulation of mild wear between a cylindrical steel roller and a steel plate in a reciprocating contact configuration. Measurements of worn topographies were also obtained, and good correlation between the numerically predicted and experimentally measured worn profiles was established over about 1 million cycles. The numerical approach employed was based on earlier work by Podra and Andersson [16], however, Oqvist provided little detail on it.
与微动磨损定性认识的发展相比，微动磨损的定量评价还没有得到很好的发展。困难之一是缺乏一个“通用的”和良好制定的磨损模型[9]。此外，目前还不清楚如何将磨屑的影响纳入这样一个定量模型。在某些情况下，磨屑更容易从接触区域消除，并且保持更多的金属与金属接触，可以合理地忽略磨屑的影响。在这种情况下，微动磨损因此可以被视为一个纯粹的接触磨损问题。这些问题的分析技术，是由Korovchinsky[10]，Galin[11]，Galin和Korovchinsky[12]等人开发的。Korovchinsky等人 [13]提出了一种分析方法，用于模拟二维、初始赫兹配置的部分滑动条件下的微动磨损。在该模型中，Archard方程局部应用于评估一个循环期间滑动区内的磨损和差距变化。一个逐步的程序，然后采用计算的接触特性的演变作为一个功能的磨损循环次数的增加。Johansson[14]提出了一种有限元解，该解结合了Archard方程的局部实现，以评估接触几何形状的变化和相关的接触压力变化。最近，Oqvist[15]提出了一项关于圆柱形钢辊和往复接触配置中钢板之间轻度磨损的数值模拟研究。 磨损形貌的测量也得到了，并建立了良好的相关性之间的数值预测和实验测量的磨损轮廓超过约100万个周期。所采用的数值方法是基于Podra和Andersson的早期工作[16]，然而，Oqvist提供的细节很少。
In this paper, the development and initial validation of a detailed finite element model, which simulates the frictional contact behaviour of a cylinder-on-flat fretting test configuration, is first described. This finite element model is then employed, via geometrical updating, as the frictional contact solver for an incremental fretting wear simulation tool, which predicts the change of geometries (i.e. for both contact surfaces) and the associated evolution of salient fretting variables, that is, relative slip, contact pressure and sub-surface stresses, during the wear process. The geometrical updating is based on nodal wear depths computed using a modified version of Archard’s equation for sliding wear. Important aspects, such as mesh refinement and optimisation of numbers of wear cycles per increment, for minimisation of the total simulation time, are discussed. Experimental testing of the same fretting configuration using a high strength aerospace steel is also described. The tests employed a nitrided against non-nitrided contact pair, as commonly used in aeroengine splines, under a range of normal loads, giving rise to different fretting wear trends and thereby facilitating experimental validation and interpretation of the simulation results. This work forms the basis of a more general fretting wear simulation tool for complex three-dimensional geometries, with future incorporation of the effects of fretting debris.
在本文中，详细的有限元模型，它模拟的摩擦接触行为的圆柱上平微动测试配置的开发和初步验证，首先描述。该有限元模型，然后采用，通过几何更新，作为增量微动磨损模拟工具，预测的几何形状的变化（即两个接触表面）和相关的演变的显着微动变量，即相对滑移，接触压力和表面下的应力，在磨损过程中的摩擦接触求解器。几何更新是基于节点的磨损深度计算使用修改后的版本阿卡德的滑动磨损方程。重要的方面，如网格细化和优化的磨损循环次数每增量，最小化的总模拟时间，进行了讨论。还描述了使用高强度航空钢的相同微动构造的实验测试。 试验采用了氮化对非氮化接触对，如航空发动机花键中常用的，在一系列正常载荷下，引起不同的微动磨损趋势，从而促进实验验证和解释的模拟结果。这项工作的基础上形成了一个更一般的微动磨损模拟工具，为复杂的三维几何形状，未来纳入微动磨屑的影响。

2. Experimental procedure

2.1. Material and specimens
2.1.材料和样本

The material used in this study was a high strength alloy steel usually referred to as Super CMV, which is employed in gas turbine aeroengine transmission components, such as spline couplings; the composition is shown in Table 1.
本研究中使用的材料是一种通常称为Super CMV的高强度合金钢，用于燃气涡轮机航空发动机传动部件，如花键联轴器;成分见表1。

Table 1. Composition of Super CMV (wt.%)
表1. Super CMV的组成（重量%）

CSiMnPSCrMo 莫NiVFe
0.35–0.43 0.35-0.430.1–0.35 0.1-0.350.4–0.7 0.4-0.7<0.007<0.0023.0–3.5 3.0-3.50.8–1.10 0.8-1.10<0.30.15–0.25 0.15-0.25Remainder 其余部分
The test materials were initially machined to specimen blanks slightly in excess of the required dimension of the specimens. The blanks were heated to 940 °C for 45 min followed by oil quenching. The surface hardness after quenching was around 700HV0.3. The blanks were then tempered at 570 °C for 2 h and 15 min, followed by air-cooling. The final hardness was 480–510HV0.3. The heat-treated blanks were finally machined and ground to obtain the desired shape and size for the crossed cylinder-on-flat fretting test arrangement. Afterwards, the flat specimens were nitrided to enhance the surface hardness to about 800HV0.3, following a practice commonly employed for improved wear performance of gas turbine aeroengine spline couplings. The cylindrical specimens were not nitrided.
最初将试验材料机加工成略微超过样本所需尺寸的样本坯件。将坯料加热至940 °C持续45分钟，然后油淬。淬火后的表面硬度约为700HV0.3。然后将坯件在570 °C下回火2小时15分钟，然后空气冷却。最终硬度为480-510HV0.3。最后对热处理后的坯料进行机加工和研磨，以获得所需的形状和尺寸，用于交叉圆柱-平板微动磨损试验装置。之后，平坦试样被氮化以将表面硬度提高到约800 HV 0.3，遵循通常用于改善燃气涡轮机花键联接件的磨损性能的实践。圆柱形试样未氮化。
Fig. 1a shows the topographical features of the flat specimen after nitriding. Fig. 1b is a cross-section view of the nitrided specimen. A ‘white layer’ with a thickness of about 6–7 μm can be seen. This layer has been identified as a heterogeneous mixture of γ′-(Fe4N) and ε-(Fe2–3N) phases and contains high internal stresses in the transitional regions between the various lattice structures [17]. The internal stresses make the white layer very brittle so that it can readily spall off during fretting wear, especially under high contact loads.
图1a所示为平面试样氮化后的形貌特征。图1b是氮化试样的横截面图。可以看到厚度约为6-7 μm的“白色层”。该层已被确定为γ′-（Fe4 N）和ε-（Fe2-3 N）相的异质混合物，并且在各种晶格结构之间的过渡区域中含有高内应力[17]。内应力使白色层非常脆，从而在微动磨损期间，特别是在高接触载荷下，其容易剥落。
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Fig. 1. SEM micrographs of nitrided flat specimen showing: (a) plan view; and (b) cross-section view, including white layer.
Fig. 1.氮化平坦试样的SEM显微照片，显示：（a）平面图;和（B）横截面图，包括白色层。

2.2. Fretting tests 2.2.微动磨损试验

The fretting tests were conducted using a crossed cylinder-on-flat arrangement as illustrated in Fig. 2. The diameter of the cylindrical specimens was 12 mm. The flat specimen was rigidly attached to the bed of the machine; the cylindrical specimen was driven by an electromagnetic vibrator. A linear variable displacement transducer mounted across the specimen holders monitored the applied stroke. The stroke was automatically maintained constant throughout the tests, except for the initial stages when manual adjustment was required. The normal load was applied by dead weight via a lever. During the tests, the tangential friction force was measured by strain gauges attached to the drive arm. More detail on this fretting rig can be found in [18]. The fretting conditions used in this study are summarised in Table 2.
如图2所示，使用平板上的交叉圆柱体布置进行微动磨损试验。圆柱形试样的直径为12 mm。将扁平试样刚性连接到机器的底座上;圆柱形试样由电磁振动器驱动。一个线性可变位移传感器安装在整个试样架监测所施加的冲程。除了需要手动调节的初始阶段外，在整个测试过程中冲程自动保持恒定。正常载荷通过杠杆由自重施加。在测试过程中，通过连接到驱动臂的应变计测量切向摩擦力。有关该微动试验台的更多详细信息请参阅[18]。本研究中使用的微动磨损条件总结见表2。
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Fig. 2. Schematic crossed round-against-flat specimen arrangement.
图2.示意性交叉圆对平试样排列。

Table 2. Fretting test conditions
表2.微动磨损试验条件

Normal load (N) 正常载荷（N）185, 500, 1670
Initial maximum Hertzian stress (MPa)336, 550, 1000
Stroke (μm) 行程（μm）50
Frequency (Hz) 频率（Hz）20
Total number of wear cycles
磨损循环总数		18000
Room temperature (°C) 室温（°C）14–20
Relative humidity (%) 相对湿度（%）40–50

3. Experimental results 3.实验结果

3.1. Coefficient of friction
3.1.摩擦系数

The coefficient of friction (COF), μ, is defined as the ratio of the measured maximum friction force amplitude during one cycle, Tmax, and the applied normal load P, as follows:(1)μ=TPFig. 3 shows the change of COF with the number of wear cycles under different (normal) contact loads. COF increases rapidly in the initial stages of testing, especially under 185 and 500 N normal loads. When the number of wear cycles exceeds 2000 cycles, COF tends towards a stable value. The initially low COF can be attributed to the presence of surface oxide films. Once the oxide films are eliminated, metal–metal and/or metal-wear particle interactions start, promoting a strong increase in COF. From Fig. 3, it is also found that the stable COF is reduced with an increase in normal load. Specifically, the COF for 185 N normal load is the highest at ≈0.8. It decreases to 0.75 and then 0.6 when the normal load is increased to 500 and 1670 N, respectively. This trend is consistent with previous observations [19], and has been attributed to the interfacial shear stress being a function of a constant value τ0 and a product of mean contact pressure p and the interfacial shear stress coefficient γ [20].
摩擦系数（COF）μ定义为一个循环期间测得的最大摩擦力幅度Tmax与施加的法向载荷P的比值，如下所示： (1)μ=TP 图3显示了在不同（法向）接触载荷下COF随磨损循环次数的变化。COF在测试的初始阶段迅速增加，特别是在185和500 N的正常载荷下。当磨损循环次数超过2000次时，COF趋于稳定值。最初的低COF可以归因于表面氧化膜的存在。一旦氧化膜被消除，金属-金属和/或金属-磨损颗粒相互作用开始，促进COF的强烈增加。从图3中还发现，稳定COF随着正常负载的增加而减小。具体而言，185 N正常载荷的COF最高，为0.8。 当正常载荷分别增加到500和1670 N时，它分别降低到0.75和0.6。该趋势与之前的观察结果一致[19]，并且归因于界面剪切应力是恒定值τ0的函数以及平均接触压力p和界面剪切应力系数γ的乘积[20]。
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Fig. 3. Coefficient of friction vs. number of fretting wear cycles. Stroke: 50 μm; frequency 20 Hz.
图三.摩擦系数与微动磨损循环次数。行程：50 μm;频率20 Hz。

3.2. Wear results 3.2.磨损结果

It is well known that the limited wear damage under fretting is difficult to measure. In the current study, the wear extent was evaluated from two-dimensional surface profiles traced using a SURFCOM 200 scanning stylus profilometer. In general, a horizontal magnification of 85× and a vertical magnification of either 1000× or 2000× were employed. The unworn surfaces were used as the reference surface. Compact oxide debris was removed from the wear scar surfaces before profiling using 3% HCl solution. There were two objectives to the profilometry measurements. First, to obtain appropriate estimates of the wear coefficient for input to the wear simulation tool, and second, to obtain worn profiles for validation of the numerically predicted profiles, as described later in Sections 5 and 6. This section describes how the wear coefficient was determined and presents a summary of the worn profiles for subsequent comparison against the numerical predictions. The simulation tool of Section 5 requires a wear coefficient, kl, expressed as the wear per unit local slip per unit local contact pressure. From experimental results, it is only possible to directly measure the wear coefficient per unit displacement per unit normal load. The latter is the Archard wear coefficient, which is defined by:(2)k=VSPwhere, V is the total wear volume, P the applied normal load and, for fretting wear, S, the total accumulated displacement, is equal to 4Nδ∗, where δ* is the displacement amplitude (i.e. half-stroke) and Nt is total number of wear cycles.
众所周知，微动磨损下的有限磨损损伤难以测量。在本研究中，使用SURFCOM 200扫描触针轮廓仪跟踪二维表面轮廓，评价磨损程度。一般情况下，水平放大倍数为85×，垂直放大倍数为1000×或2000×。未磨损的表面用作参考表面。在使用3%HCl溶液进行仿形之前，从磨痕表面去除致密的氧化物碎片。轮廓测量有两个目的。首先，获得磨损系数的适当估计值以输入到磨损模拟工具，其次，获得磨损轮廓以验证数值预测轮廓，如稍后在第5节和第6节中所述。 本节描述了如何确定磨损系数，并总结了磨损曲线，以便随后与数值预测进行比较。第5节的模拟工具需要一个磨损系数k，表示为单位局部滑动/单位局部接触压力的磨损。从实验结果来看，只能直接测量每单位位移每单位法向载荷的磨损系数。后者是Archard磨损系数，其定义为：其中，V是总磨损体积，P是施加的法向载荷，对于微动磨损，S是总累积位移，等于其中δ*是位移幅度（即半行程），Nt是磨损循环的总数。
Direct calculation of the desired wear coefficient requires knowledge of the local contact slips and local contact pressures. As these are not readily measurable a modified form of Eq. (2) was used to determine an averaged wear coefficient from the measured wear profiles using:(3)k=Wbh4δ×NPwhere W is the wear scar width, b the width of the flat specimen and hm is the average wear scar depth, with respect to x-coordinate. It should be appreciated that this wear coefficient is averaged across a range of contact pressures and slips, as well as across an appreciable number of wear cycles, and is thus not ideal. However, its use is necessary until the wear simulation tool is sufficiently developed to enable the local contact pressures and slips to be calculated from the measured wear profiles.
直接计算所需的磨损系数需要了解局部接触滑移和局部接触压力。由于这些是不容易测量的修改形式的方程。(2)其中W是相对于x坐标的磨痕宽度，B是平坦试样的宽度，hm是相对于x坐标的平均磨痕深度。应当理解，该磨损系数是在接触压力和滑动的范围内以及在相当数量的磨损循环内平均的，因此是不理想的。然而，在磨损模拟工具充分发展到能够从测量的磨损轮廓计算局部接触压力和滑动之前，它的使用是必要的。
The two-dimensional wear profiles in the fretting (x) direction were roughly similar across the width of the flat specimen and along the complementary length of the cylindrical specimen, so that, for present purposes, a representative measured profile is employed to obtain the necessary data for wear coefficient calculation. Fig. 2 shows that the width of the flat specimen, b, is 10 mm. The two-dimensional surface profiles for the flat and cylindrical specimens are shown in Fig. 4, corresponding to the case of 185 N normal load. The mean wear depth of the measured scar, hm, is evaluated by averaging a series of ten discrete values of wear depth, hi, at different positions along the scar, from the profilometer trace. A maximum wear depth for the flat specimen, hmax, is also obtained from the measured wear scar profile.
在微动磨损（x）方向上的二维磨损曲线在扁平试样的整个宽度方向和沿着圆柱形试样的互补长度方向上大致相似，因此，出于目前的目的，采用代表性的测量曲线来获得磨损系数计算所需的数据。图2显示了扁平试样的宽度B为10 mm。扁平和圆柱形试样的二维表面轮廓如图4所示，对应于185 N法向载荷的情况。测量的疤痕的平均磨损深度hm，通过平均一系列的10个离散值的磨损深度h，在不同的位置沿着疤痕，从轮廓仪轨迹进行评估。 还从测量的磨痕轮廓获得了平坦试样的最大磨损深度hmax。
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Fig. 4. Measured two-dimensional surface profiles of unworn and worn flat and cylindrical specimen for 185 N normal load case.
见图4。在185 N法向载荷情况下，测量未磨损和磨损的扁平和圆柱形试样的二维表面轮廓。

The wear coefficients for the cases of Table 2 are presented in Fig. 5. The results are seen to vary with normal load. With respect to the flat specimen, the wear coefficient increases as normal load is increased from 185 to 500 N, and decreases when the normal load is further increased to 1670 N. However, the coefficients for the cylindrical specimens remain almost constant with respect to normal load variation. The effect of normal load on the wear coefficient is attributed to: (a) the presence of the white layer on the surface of the flat specimens, which is brittle and fragments easily under high normal loads thus contributing to the measured wear rates; and (b) the different tribological behaviour of debris under different contact loads. More detailed interpretations on this matter will be presented in future work.
表2所示情况下的磨损系数见图5。可以看出，结果随正常载荷而变化。对于平板试样，当法向载荷从185 N增加到500 N时，磨损系数增加，当法向载荷进一步增加到1670 N时，磨损系数减小。然而，圆柱形试样的系数相对于法向载荷变化几乎保持不变。法向载荷对磨损系数的影响归因于：（a）平坦试样表面上白色层的存在，该白色层在高法向载荷下是脆性的并且容易破碎，从而有助于测量的磨损率;以及（B）在不同接触载荷下碎片的不同摩擦学行为。关于这一问题的更详细解释将在今后的工作中提出。
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Fig. 5. Wear coefficients for the different normal loads.
图五.不同法向载荷的磨损系数。

4. FE model for fretting contact
4.微动接触有限元模型

4.1. Development of the FE model
4.1.有限元模型的开发

The basis for the fretting wear prediction tool described in the next section is a detailed, two-dimensional, finite element model of the fretting test geometry. The general purpose, non-linear code, ABAQUS [21], was employed for the FE modelling, to facilitate generalisation of the present approach to more complex applications. The finite element model is shown in Fig. 6, where the radius of the cylinder is the same as the cylindrical specimen in the fretting tests. Two-dimensional, four-node, plane strain (linear) elements are employed throughout. The mesh (element size) in the contact area is very fine (about 10 μm) to capture the complicated variation of surface and sub-surface stresses and relative slip. The sharp transition from coarse mesh, remote from the contact region, to fine mesh, in the contact region, is achieved via multi-point constraints (MPCs). This is necessary to achieve the correct balance of detailed contact region mesh refinement, for modelling microscopic wear depth increments and for accurate prediction of the salient fretting variables, and minimal CPU time for the total wear simulation, e.g. 18,000 cycles. As discussed below, a typical incremental number of wear cycles is about 30, each increment requiring one combined normal and tangential loading analysis, so that a total of 18,000 wear cycles requires about six hundred individual FE analyses. Optimisation of the mesh has led to reductions in total wear simulation time from about 6 days to less than 1 day. The contact surface interaction is defined via the contact pair approach in ABAQUS, which uses the master–slave algorithm to enforce the contact constraints. The nodes on the slave surface are permitted to penetrate the master surface by a user-controlled maximum penetration depth, ω. The cylindrical surface is chosen as the slave contact surface and a ω value of approximately 5 μm is found to be satisfactory. The basic Coulomb friction model with isotropic friction is employed. The frictional contact conditions are introduced via the Lagrange multiplier approach, rather than the more approximate penalty method, in order to enforce exact sticking (zero slip) constraints between the bodies when the equivalent shear stress is less than the critical shear stress. During the loading scheme, linear constraint equations are employed to ensure uniform displacement of the nodes on the top surface of the cylinder. The bottom of the flat substrate is restrained from movement in the x and y directions. The elastic modulus and Poisson’s ratio of both the cylinder and flat are taken as 200 GPa and 0.3, respectively.
下一节中描述的微动磨损预测工具的基础是微动试验几何形状的详细二维有限元模型。通用非线性代码ABAQUS[21]用于FE建模，以便于将本方法推广到更复杂的应用。有限元模型如图6所示，其中圆柱体的半径与微动磨损试验中的圆柱体试样相同。整个计算采用二维、四节点、平面应变（线性）单元。接触区域的网格（单元尺寸）非常细（约10 μm），以捕捉表面和亚表面应力以及相对滑移的复杂变化。通过多点约束实现了从远离接触区域的粗网格到接触区域的细网格的急剧过渡。 这是必要的，以实现正确的平衡，详细的接触区域网格细化，建模微观磨损深度增量和准确预测的显着微动磨损变量，和最小的CPU时间的总磨损模拟，如18,000个周期。如下所述，磨损循环的典型增量数量约为30，每个增量需要一个组合法向和切向载荷分析，因此总共18，000个磨损循环需要约600个单独的FE分析。网格的优化导致总磨损模拟时间从约6天减少到不到1天。接触面的相互作用是通过接触对的方法在ABAQUS中定义，它使用主从算法来执行接触约束。允许从表面上的节点以用户控制的最大穿透深度ω穿透主表面。 选择圆柱形表面作为从接触表面，发现ω值约为5 μm是令人满意的。采用库仑摩擦模型，考虑各向同性摩擦。通过拉格朗日乘子方法引入摩擦接触条件，而不是更近似的罚函数法，以便在等效剪应力小于临界剪应力时，强制执行机构之间的精确粘附（零滑移）约束。在加载过程中，采用线性约束方程，以确保圆柱体顶面节点的位移均匀。平坦基板的底部被限制在x和y方向上移动。圆柱体和平板的弹性模量和泊松比分别取为200 GPa和0.3。
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Fig. 6. Finite element model for cylinder-on-flat configuration.
见图6。圆柱-平板结构的有限元模型。

In order to validate the accuracy of the unworn model, comparisons are made with the well-known analytical solutions for the Hertzian stress distributions [22]. The contact pressure distribution is given:(4)p(x)=p01−x2a2The half-width of the contact area, a, and the maximum contact pressure, p0, are, respectively, given by:(5)a=4PRπE∗1/2(6)p0=PE∗πR1/2where P is the applied normal load and E* is the composite modulus of the two contacting bodies. For plane strain conditions, the latter is given by:(7)E∗=1−(v)2E+1−(v)2E−1where Ef, Ec are the elastic moduli and vc, vf are the Poisson’s ratios of the flat and cylindrical bodies, respectively. R is the relative curvature given by:(8)R=1R+1R−1where Rf and Rc are the radii of the contacting surfaces. Fig. 7a shows a comparison of p(x) from Eq. (4) and the corresponding FE-predicted distribution using the mesh of Fig. 6. Elasticity theory [22] allows the sub-surface x and y direction normal stresses along the y-axis (principal stresses) to be derived:(9)σx=−p0a{(a2+2y2)(a2+y2)−1/2−2y}(10)σy=−p0a(a2+y2)−1/2Fig. 7b shows a comparison between the latter stress distributions and the corresponding FE predictions. The finite element model is seen to give excellent agreement with the analytical solution for all three variables, thus indicating satisfactory mesh refinement.
为了验证未磨损模型的准确性，将其与赫兹应力分布的已知解析解进行比较[22]。接触压力分布由下式给出：接触面积的半宽度a和最大接触压力p0分别由下式给出：其中P是施加的法向载荷，E*是两个接触体的复合模量。对于平面应变条件，后者由下式给出：其中Ef、Ec分别是扁平体和圆柱体的弹性模量，vc、vf分别 R是由下式给出的相对曲率：其中Rf和Rc是接触表面的半径。图7a示出了来自等式7b的p（x）的比较。(4)以及使用图6的网格的对应FE预测分布。弹性理论[22]允许导出沿y轴沿着的次表面x和y方向的正应力（主应力）： (9)σx=−p0a{(a2+2y2)(a2+y2)−1/2−2y} (10)σy=−p0a(a2+y2)−1/2 图7 b显示了后者应力分布与相应FE预测之间的比较。 有限元模型被认为是给所有三个变量的分析解决方案非常一致，从而表明令人满意的网格细化。
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Fig. 7. Comparison of Hertzian problem from FE prediction and analytical solution for: (a) distribution of contact pressure; and (b) sub-surface stress along Y-axis under a normal load of 1200 N.
见图7。来自FE预测和解析解的赫兹问题的比较：（a）接触压力分布;和（B）在1200 N法向载荷下沿着Y轴的亚表面应力。

4.2. Fretting contact under combined normal and tangential loading
4.2.法向和切向载荷联合作用下的微动接触

In the fretting wear tests, the crossed cylinder-against-flat specimens of Fig. 2 are subjected to fixed normal contact loading with superimposed cyclic tangential displacement, for the test data shown in Table 2. This loading cycle must therefore be implemented within the FE model in order to permit wear calculations. Referring to Fig. 6, the loading history for the FE model is as follows: a normal load is applied to Point A in the y direction, in one analysis step, and then a periodic x-displacement of amplitude δ* is imposed at Point B, in two subsequent analysis steps, introducing an oscillatory tangential friction force, T(t), on the contacting surfaces.
在微动磨损试验中，图2中的交叉圆柱体对平面试样承受固定法向接触载荷，并叠加循环切向位移，试验数据见表2。因此，必须在FE模型中执行此加载循环，以便进行磨损计算。参考图6，FE模型的加载历史如下：在一个分析步骤中，沿y方向向点A施加法向载荷，然后在两个后续分析步骤中，在点B施加振幅为δ* 的周期性x位移，从而在接触表面上引入振荡切向摩擦力T（t）。
The contact pressure distribution under fretting conditions is consistent with the Hertzian solution, since the distribution is independent of the tangential load or displacement when the contacting bodies have the same Young’s modulus [22]. Note that this is not the case in the presence of wear as described in Section 6. It is well known that under fretting conditions, the actual relative slip between contacting components differs more significantly from the applied tangential stroke or displacement than under, for example, sliding conditions. It is the relative slip that is employed for fretting wear prediction, rather than the applied stroke, even though applied stroke is conventionally employed for wear coefficient evaluation from test data. This is due to the direct availability of the applied stroke data and the difficulty of estimating relative slip for a given test configuration. It is not generally possible to obtain the slip distribution analytically for displacement-controlled conditions, although analytical solutions are available for some simplified configurations under load control [22]. In any case, the objective of the present work is to develop a general tool for application to complex geometries such as spline teeth. Fig. 8 shows the FE-predicted relationship between contact slip, for an applied (at point A) bulk displacement of 2.5 μm, under a range of normal loads. The results demonstrate the evolution from the gross slip regime under low normal loads, e.g. 600–1000 N, to the partial slip regime under higher normal loads, e.g. 1200–1600 N; in both cases, it is clear that relative slip at the contact interface is significantly less than the applied displacement. The relative slip varies with horizontal position along the contact interface, being lower in magnitude at the centre than at the edges of contact. This spatial variation of slip gives rise to a corresponding variation of local wear depth with horizontal position, as described below.
微动条件下的接触压力分布与赫兹解一致，因为当接触体具有相同的杨氏模量时，该分布与切向载荷或位移无关[22]。请注意，如第6节所述，存在磨损的情况并非如此。众所周知，在微振磨损条件下，接触部件之间的实际相对滑动与所施加的切向行程或位移的差异比在例如滑动条件下更显著。微动磨损预测采用的是相对滑动，而不是施加的行程，尽管施加的行程通常用于根据试验数据评估磨损系数。这是由于直接可用的应用行程数据和估计相对滑移的困难，为一个给定的测试配置。 通常不可能在位移控制条件下通过解析法获得滑移分布，尽管在载荷控制条件下，某些简化配置的解析解可用[22]。在任何情况下，本工作的目的是开发一种通用的工具，应用到复杂的几何形状，如花键齿。图8显示了在一系列法向载荷下，对于2.5 μm的施加（A点）体位移，接触滑移之间的有限元预测关系。结果表明，从低法向载荷（如600-1000 N）下的总滑移状态到较高法向载荷（如1200-1600 N）下的部分滑移状态的演变;在这两种情况下，很明显，接触界面处的相对滑移明显小于施加的位移。 相对滑动随着沿接触界面的水平位置沿着而变化，在接触的中心处的幅度小于在接触的边缘处的幅度。滑动的这种空间变化引起局部磨损深度随水平位置的相应变化，如下所述。
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Fig. 8. Comparison of slip distributions for a range of normal loads, with the same applied displacement of 2.5 μm.
见图8。在相同的2.5 μm施加位移条件下，一系列法向载荷下的滑移分布比较。

5. Wear simulation method
5.磨损模拟方法

5.1. Wear model 5.1.磨损模型

Following the hypothesis of Stowers and Rabinowicz [23] and as implemented by Johansson [14], it is assumed here that fretting wear can be evaluated by applying Archard’s equation to local contact conditions along a differential width of the contact interface. Archard’s equation for sliding wear is normally expressed as [24]:(11)VS=KPHwhere K is the dimensionless wear coefficient and H is the hardness (MPa) of the material. P, S and V have been defined in Section 3. In order to simulate the evolution of the contact surface profiles with wear cycles, it is necessary to determine the wear depth locally as a function of horizontal contact position, x, at each contact node of the finite element model. Therefore, for an infinitesimally small apparent contact area, dA, the increment of wear depth, dh, associated with an increment of sliding distance, dS, is determined. This can be obtained by applying Eq. (11) locally to the area dA and for the increment of sliding distance, dS:(12)dVdS=KdPHThen, dividing both sides by dA, the following equation is obtained:(13)dVdAdS=KdPHdAThe dP/dA term is the local contact pressure, p(x), while dV/dA is the required increment of local wear depth, dh, noting that h is a function of both horizontal position x and the total local slip distance, S. The following equation is thus obtained for the prediction of the increment of local wear depth:(14)dhdS=kp(x)where the quantity K/H is replaced here by kl, the local wear coefficient. Unfortunately, the authors are not aware of any existing method for estimating kl. Consequently, as indicated earlier in Section 3, the best available alternative is to measure an average value across the complete contact width. Thus, in the present study it is assumed that kl≈k. The implication of Eq. (14) is that the incremental wear depth at a given point on the contact is proportional to the local wear coefficient, the local contact pressure and the local increment of slip distance. In the numerical prediction of fretting wear described below, which may involve either gross slip or partial slip situations, S is assumed to be the total local slip distance between the contacting surfaces. Note again that k, and thus kl, is obtained using the applied test stroke, which will naturally give an under-estimate of the wear volume, since the local contact interface slip is always less than the applied bulk displacement. The degree of under-estimation can be expected to decrease with decreasing normal load for a given applied stroke, due to the diminishing difference between applied stroke and contact slip.
根据Stowers和Rabinowicz[23]的假设以及Johansson[14]的实施，此处假设微动磨损可通过将Archard方程应用于沿接触界面不同宽度的局部接触条件来评估。Archard的滑动磨损方程通常表示为[24]： (11)VS=KPH 其中K是无因次磨损系数，H是材料的硬度（MPa）。P、S和V已在第3节中定义。为了模拟接触表面轮廓随磨损循环的演变，有必要在有限元模型的每个接触节点处局部确定作为水平接触位置x的函数的磨损深度。 因此，对于无穷小的表观接触面积，DA，磨损深度的增量，DH，与滑动距离的增量，DS，确定。这可以通过应用Eq. (11)对于滑动距离的增量dS：#1然后，将两侧除以dA，得到以下等式： (13)dVdAdS=KdPHdA dP/dA项是局部接触压力p（x），而dV/dA是局部磨损深度的所需增量dh，注意h是水平位置x和总局部滑动距离S的函数。 因此，获得了用于预测局部磨损深度增量的以下等式：其中，量K/H在此由局部磨损系数k代替。不幸的是，作者不知道任何现有的方法来估计k。因此，如前面第3节所述，最好的替代方法是测量整个接触宽度的平均值。因此，在本研究中，假设k=k。方程式的含义。(14)在接触面上给定点处的磨损深度增量与局部磨损系数、局部接触压力和局部滑移距离增量成正比。 在下文描述的微动磨损的数值预测中，可能涉及总滑动或部分滑动情况，S被假定为接触表面之间的总局部滑动距离。再次注意，使用所施加的测试冲程获得k，从而获得k，这自然会低估磨损体积，因为局部接触界面滑移总是小于所施加的体积位移。对于给定的施加行程，由于施加行程和接触滑动之间的差异减小，可以预期低估的程度随着法向载荷的减小而减小。

5.2. Wear modelling procedure
5.2.磨损建模程序

An automated, incremental, wear simulation tool has been developed based on the wear prediction equation of the above section. This section describes this development for the cylinder-on-flat test arrangement. An important aspect of this work is the use of the ABAQUS commercial finite element code; its general capabilities facilitate extension to other contact geometries, such as flat-on-flat or spline tooth flanks. Fig. 9 shows a flowchart of the numerical procedure that forms the basis of the simulation tool. Once the finite element model of the initial, unworn geometry has been generated, the program can be run for any specified number of wear cycles to predict the corresponding worn surface profiles and the evolution of surface and sub-surface contact variables. The simulation tool consists of an interaction between a special-purpose Fortran program and ABAQUS, whereby the FE model is incrementally updated, as described below, according to the calculated wear depths, based on the local contact pressure and local slip results of the FE analyses.
基于上一节的磨损预测方程，开发了一个自动化、增量式磨损模拟工具。本节介绍了气缸平放试验装置的发展。这项工作的一个重要方面是使用ABAQUS商业有限元代码;其通用功能有助于扩展到其他接触几何形状，例如平面对平面或花键齿面。图9示出了形成模拟工具的基础的数值过程的流程图。一旦生成了初始未磨损几何形状的有限元模型，该程序就可以运行任意指定数量的磨损循环，以预测相应的磨损表面轮廓以及表面和次表面接触变量的演变。 仿真工具由专用Fortran程序和ABAQUS之间的交互组成，根据有限元分析的局部接触压力和局部滑移结果，根据计算出的磨损深度，如下所述，逐步更新有限元模型。
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Fig. 9. Flow chart of the numerical method for modeling fretting wear.
见图9。微动磨损数值模拟方法流程图。

The initial parameters required for the wear simulation include contact geometry, material properties, normal load, applied stroke and coefficient of friction, all of which are defined within the FE model, along with the governing parameters for wear modelling, namely wear coefficient kl, the total number of wear cycles Nt and the increment in number of wear cycles per step, ΔN.
磨损模拟所需的初始参数包括接触几何形状、材料特性、法向载荷、施加的冲程和摩擦系数，所有这些都在FE模型中定义，沿着磨损建模的控制参数，即磨损系数k、磨损循环总数Nt和每步磨损循环数增量ΔN。
The total number of wear cycles, Nt, is discretised into n wear increments and the increment of wear depth, Δh, at each contact node on each surface is then calculated incrementally, for the specified value of ΔN. The choice of a suitable value for ΔN is important for both the stability of the simulation and the resulting computational time, as further discussed in Section 7.
将磨损循环总数Nt离散为n个磨损增量，然后针对指定的ΔN值，递增计算每个表面上每个接触节点处的磨损深度增量Δh。选择合适的ΔN值对于模拟的稳定性和计算时间都很重要，如第7节中进一步讨论的。
Specifically, for the jth wear increment, at a given node i on either contact surface, the contact pressure, pi,j, and slip per cycle, 4si,j, are determined using the FE model of Section 4. It is assumed that pi,j and si,j are constant within a given increment. The increments of wear depth for the flat and cylindrical contact surfaces at node i in the jth increment are then given by(15)Δhi,j=k×4si,j×ΔN×pi,j(16)Δhi,j=k×4si,j×ΔN×pi,jwhere the superscripts f and c represent the flat and cylindrical surfaces, respectively. Hence, the updated vertical coordinate, yi,j+1f, of node i at the start of the j+1th wear increment for the flat surface is given by:(17)yi,j+1=yi,j−Δhi,jand the corresponding updated vertical coordinate for the cylindrical surface is given by:(18)yi,j+1=yi,j+Δhi,j−cjThe cj term is the amount by which the FE model of the cylindrical specimen needs to be moved down vertically, i.e. a rigid body movement, to ensure that the contact surfaces are initially in contact at the beginning of the new wear increment. This term is calculated as follows:(19)cj=min((yi,j+Δhi,j)−(yi,j−Δhi,j))It represents the minimum gap between the worn surfaces due to the wear of the jth increment. Repetition of these calculations for each increment of the total number of wear cycles then achieves the required wear simulation.
具体而言，对于第j个磨损增量，在任一接触表面上的给定节点i处，使用第4节的FE模型确定接触压力pi，j和每循环滑移4si，j。假设pi，j和si，j在给定增量内是常数。在第j个增量中，节点i处的平坦和圆柱形接触表面的磨损深度增量由 (15)Δhi,j=k×4si,j×ΔN×pi,j (16)Δhi,j=k×4si,j×ΔN×pi,j 给出，其中上标f和c分别表示平坦和圆柱形表面。 因此，在第j+1次磨损增量开始时，平面的节点i的更新垂直坐标y，f由下式给出： (17)yi,j+1=yi,j−Δhi,j ，圆柱形表面的相应更新垂直坐标由下式给出： (18)yi,j+1=yi,j+Δhi,j−cj cj项是圆柱形试样的FE模型需要垂直向下移动的量，即刚体移动，以确保接触表面在新的磨损增量开始时最初接触。该项计算如下： (19)cj=min((yi,j+Δhi,j)−(yi,j−Δhi,j)) 它表示由于第j个增量的磨损而导致的磨损表面之间的最小间隙。对磨损循环总数的每个增量重复这些计算，然后实现所需的磨损模拟。
One of the most important challenges for the general application of the above method to complex three-dimensional components, such as aeroengine spline couplings, e.g. see [25], where fretting wear assessment is critical to component design optimisation, is minimisation of the wear simulation times. This is achieved here via a number of different aspects, including mesh optimisation (see above) and wear increment optimisation (see below). Assessment of the slip distributions during the different stages of the tangential force-displacement cycle has established that the total slip distance for the complete cycle can be satisfactorily estimated from the slip distance, si,j, corresponding to the application of the positive tangential displacement only. Although initial development work was based on slip estimates obtained from incremental simulation of both the positive and negative tangential displacements, the use of si,j gives a significant saving of 67% on simulation time. This saving is simply due to the observation that it is not necessary to simulate the negative tangential displacement increment to get accurate contact slip estimates for the complete cycle, which in turn results from the near geometrical symmetry of the connection, even in the worn condition.
上述方法一般应用于复杂三维部件（如航空发动机花键联轴器，例如参见[25]，其中微动磨损评估对部件设计优化至关重要）的最重要挑战之一是磨损模拟时间的最小化。这是通过许多不同的方面来实现的，包括网格优化（见上文）和磨损增量优化（见下文）。在切向力-位移循环的不同阶段期间的滑移分布的评估已经确定，对于完整循环的总滑移距离可以从滑移距离si，j满意地估计，对应于仅正切向位移的应用。 虽然最初的开发工作是基于从正负切向位移的增量模拟中获得的滑移估计，但使用si，j可以显著节省67%的模拟时间。这种节省仅仅是由于观察到不需要模拟负切向位移增量来获得完整循环的精确接触滑动估计，这反过来又是由于连接的接近几何对称性，即使在磨损条件下。

6. Results 6.结果

6.1. Predicted wear profiles
6.1.预测磨损轮廓

The above approach has been applied for the loading conditions of Table 2. The coefficient of friction and the wear coefficient for each load are given in Section 3 and these values are employed in the respective wear simulations. Fig. 10 shows the FE-predicted evolution of the contact surface profiles with increasing fretting wear cycles, for the case of 185 N normal load. It is found that as fretting wear proceeds; a wear scar develops in the flat surface while the shape of the round surface is also modified. The contact tends towards conforming, with similar radii on both surfaces.
上述方法已应用于表2的加载条件。第3节中给出了每种载荷的摩擦系数和磨损系数，这些值用于相应的磨损模拟。图10显示了在185 N法向载荷情况下，接触表面轮廓随微动磨损循环次数增加的FE预测演变。结果发现，随着微动磨损的进行，在平坦表面上形成磨痕，而圆形表面的形状也被修改。接触倾向于一致，在两个表面上具有相似的半径。
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Fig. 10. Predicted wear profiles for: (a) cylindrical; and (b) flat specimens under 185 N normal load case, for different numbers of wear cycles N.
见图10。在185 N正常载荷情况下，不同磨损循环次数N下，（a）圆柱形和（B）扁平样本的预测磨损曲线。

The numerically predicted worn surface profiles of the flat specimens after 18,000 cycles have been compared with experimental results for all three cases and are presented in Fig. 11. The predicted values of scar width and maximum wear depth, together with corresponding experimental results, are presented in Table 3. For the low normal load case of 185 N, the predicted and measured results correlate closely. For the 500 and 1670 N cases, the wear scar widths are seen to be over-predicted, by 34 and 16%, respectively, while the maximum wear depths are under-predicted, by 44 and 25%, respectively.
在18，000次循环后，对所有三种情况下的平板试样的磨损表面轮廓进行了数值预测，并与实验结果进行了比较，如图11所示。表3列出了疤痕宽度和最大磨损深度的预测值以及相应的实验结果。对于185 N的低法向载荷情况，预测结果和测量结果密切相关。对于500和1670 N的情况下，磨痕宽度被认为是过度预测，分别为34%和16%，而最大磨损深度被低估，分别为44%和25%。
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Fig. 11. Comparison of FE prediction and experimental results for worn surface profiles of the flat specimen, after 18,000 wear cycle at: (a) 185 N normal load; (b) 500 N normal load; and (c) 1670 N normal load. Stroke is 50 μm.
见图11。在18，000次磨损循环后，在（a）185 N法向载荷;（B）500 N法向载荷;和（c）1670 N法向载荷下，平板试样磨损表面轮廓的FE预测和实验结果的比较。行程为50 μm。

Table 3. Comparison of FE prediction and experimental results for wear scars on the flat specimen after 18,000 wear cycle. Stroke is 50 μm
表3. 18，000次磨损循环后平板试样上磨痕的FE预测和实验结果的比较。行程为50 μm

Normal load (N) 正常载荷（N）Scar width (mm) 疤痕宽度（mm）Maximum wear depth (μm) 最大磨损深度（μm）
Measured 测量Predicted 预测Measured 测量Predicted 预测
1850.540.522.93.0
5000.590.7915.08.4
16700.750.8715.611.7
Fig. 12 shows a comparison between the wear volumes predicted using: (i) the current wear simulation tool, which employs local contact slip; and (ii) Archard’s equation, directly, which uses the applied stroke. The wear volume is plotted against the number of fretting cycles for the three values of normal load. For the 185 N load, the Archard equation approach gives very similar results to the wear simulation approach, whereas under the higher loads the disparity is seen to increase with increasing load. For the 1670 N case, the difference is about 15%. Note that the Archard wear coefficient is calculated in terms of the worn surface profile at 18,000 wear cycles. Thus, it is apparent that the local contact slip-based approach gives rather good wear volume prediction for low normal loads but under-estimates the wear volumes for high normal loads. This is further discussed below.
图12示出了使用以下各项预测的磨损体积之间的比较：（i）采用局部接触滑动的当前磨损模拟工具;以及（ii）直接使用所施加的冲程的Archard方程。对于三个正常载荷值，磨损体积相对于微动循环次数作图。对于185 N载荷，Archard方程方法给出了与磨损模拟方法非常相似的结果，而在较高载荷下，可以看到差异随着载荷的增加而增加。对于1670 N的情况，差异约为15%。注意，Archard磨损系数是根据18，000次磨损循环的磨损表面轮廓计算的。因此，很明显，基于局部接触滑移的方法对于低法向载荷给出了相当好的磨损体积预测，但是对于高法向载荷低估了磨损体积。下文对此作进一步讨论。
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Fig. 12. Comparison of predicted wear volumes and Archard’s solutions under different normal loads.
见图12。不同法向载荷下预测磨损体积和Archard解决方案的比较。

6.2. Evolution of contact variables
6.2.接触变量的演变

Fig. 13 shows the FE-predicted evolution of the half contact width with increasing wear cycles, for the three load cases of Table 2. The predicted width for the unworn geometry is consistent with the Hertzian solution. However, as fretting wear proceeds, the width is obviously modified. There is a rapid increase in contact width over the first 2000 cycles, followed by a gradual reduction in the rate of increase. For subsequent cycles, the contact width continues to increase but at a slower rate.
图13显示了对于表2的三种载荷情况，随着磨损循环次数的增加，FE预测的半接触宽度的演变。未磨损几何形状的预测宽度与赫兹解一致。然而，随着微动磨损的进行，宽度明显修改。在前2000个周期内，接触宽度迅速增加，随后增加率逐渐降低。对于随后的循环，接触宽度继续增加，但速度较慢。
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Fig. 13. Evolutions of half contact width with increasing number of wear cycles, N, for the different normal loads.
图十三.不同法向载荷下半接触宽度随磨损循环次数N增加的演变。

The contact pressure distribution evolves concomitantly with the changes of contact width. Fig. 14 shows the FE-predicted evolution of the contact pressure distribution with increasing wear cycles, for the 185 N normal load; these distributions correspond to the zero tangential displacement position in the fretting cycle. The peak pressure continuously decreases as the distribution along the contact width finally tends towards uniform. The variation of peak pressure for all three cases is demonstrated in Fig. 15. There is a dramatic reduction in peak pressure during the first thousand cycles, followed by a much more gradual reduction over the subsequent 17,000 cycles. After 18,000 cycles, the peak pressures for all three cases are predicted to have reduced to less than 30% of their initial values.
接触压力分布随接触宽度的变化而变化。图14显示了对于185 N法向载荷，接触压力分布随磨损循环增加的FE预测演变;这些分布对应于微动循环中的零切向位移位置。随着接触宽度分布的沿着逐渐趋于均匀，峰值压力不断减小。所有三种情况下的峰值压力变化如图15所示。在第一个1000个循环期间，峰值压力急剧降低，随后在随后的17，000个循环中逐渐降低。在18，000次循环后，预计所有三种情况下的峰值压力均已降至其初始值的30%以下。
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Fig. 14. Predicted evolution of contact pressure with increasing number of wear cycles, N, under 185 N normal load case.
见图14。在185 N正常载荷情况下，预测接触压力随磨损循环次数N增加的演变。

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Fig. 15. Evolutions of peak pressure with increasing number of wear cycles, N, for the different normal loads.
图15.峰值压力随不同法向载荷下磨损循环次数N增加的演变。

Fig. 16b shows the variation of predicted contact pressure distribution for the set of discrete displacements, δA, δB, δC, δD and δE, as shown in Fig. 16a, during the applied tangential displacement cycle of the 18,000th wear cycle. It is clear that there is a significant variation of contact pressure at any given position along the contact width during this cycle. Initial development work on the present wear simulation approach was based on the use of such instantaneous local pressures and the corresponding local slip values for the wear depth calculations. Comparison against wear simulation results based on the use of a ‘representative’ local contact pressure and corresponding local slip value, where the ‘representative’ value was taken as the zero tangential displacement value, established that the results agreed to within 5% for the 18,000 wear cycle case. The use of this ‘representative’ pressure improves the computational efficiency significantly, reducing the simulation time by approximately 60%. Consequently, the approach of Section 5 is based on the use of the ‘representative’ contact pressure.
图16 B示出了在第18，000个磨损循环的所施加的切向位移循环期间，如图16 a所示的离散位移组δA、δB、δC、δD和δE的预测接触压力分布的变化。很明显，在该循环期间，在沿沿着接触宽度的任何给定位置处存在接触压力的显著变化。本磨损模拟方法的初始开发工作是基于使用这种瞬时局部压力和相应的局部滑动值进行磨损深度计算。 与基于使用“代表性”局部接触压力和相应局部滑移值（其中“代表性”值被视为零切向位移值）的磨损模拟结果进行比较，确定了18，000次磨损循环情况下的结果在5%以内。这种“代表性”压力的使用显着提高了计算效率，将模拟时间减少了约60%。因此，第5节的方法是基于使用“代表性”接触压力。
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Fig. 16. (a) variation of the applied tangential displacement with time and (b) variation of contact pressure with the applied tangential displacement, during 18,000th wear cycle under 185 N normal load case.
见图16。(a)施加的切向位移随时间的变化，以及（B）接触压力随施加的切向位移的变化。

Fig. 17 shows the evolution of contact slip, corresponding to the positive tangential displacement point of the fretting cycle (point δB of Fig. 16a), from the initial unworn case to the worn case, after 18,000 fretting cycles. Note the significant increase in contact width from less than 0.1 to about 0.5 mm, as also shown in Fig. 13. The slip amplitude, which is predicted to increase by less than 1 μm, is, however, only negligibly affected by the wear damage. This small increase is attributed to the decrease in the contact pressure.
图17显示了接触滑移的演变，对应于微动循环的正切向位移点（图16 a的点δB），从最初的未磨损情况到磨损情况，经过18，000次微动循环。注意，接触宽度从小于0.1 mm显著增加到约0.5 mm，也如图13所示。然而，滑动幅度，预计将增加不到1 μm，仅受磨损损伤的影响可以忽略不计。这种小的增加归因于接触压力的降低。
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Fig. 17. Predicted change of relative slip distribution due to wear damage under 185 N normal load case.
图17.在185 N正常载荷情况下，由于磨损损伤，相对滑移分布的预测变化。

7. Discussion 7.讨论

The advantage of the present approach to fretting wear simulation is that it is based on the use of a commercial FE code and the implementation of the wear simulation tool only requires the development of an additional program to interact with the FE code, via modification of the model input and processing of the analysis output, to apply the wear depth equations of Section 5; although such a program also needs some built-in intelligence, as described in this paper. The use of a commercial code as the frictional contact solver part of the tool facilitates generalisation to more complex components, as mentioned above, such as three-dimensional contact geometries and couplings, both for industrial application, to predict the wear-limited service life of such components, and also for academic research purposes, to assess, for example, the interaction between fretting wear and crack nucleation in laboratory test configurations. The method is obviously flexible and can be easily transferred into in-house design procedures or other FE codes to assess wear damage and associated changes in stress field. The use of a commercial code also facilitates extension to simulate related failure phenomena, such as fracture mechanics and crack growth.
本方法微动磨损模拟的优点是，它是基于商业FE代码的使用和磨损模拟工具的实施只需要开发一个额外的程序与FE代码进行交互，通过修改模型输入和处理的分析输出，应用第5节的磨损深度方程，虽然这样的程序也需要一些内置的智能，如本文所述。使用商业代码作为工具的摩擦接触求解器部分有助于推广到更复杂的部件，如上所述，例如三维接触几何形状和联接器，用于工业应用，以预测这些部件的磨损限制使用寿命，并且还用于学术研究目的，以评估，例如，在实验室测试配置中微动磨损和裂纹成核之间的相互作用。 该方法具有很大的灵活性，可以很容易地移植到内部设计程序或其他有限元程序中，以评估磨损损伤和应力场的相关变化。商业代码的使用也有利于扩展到模拟相关的故障现象，如断裂力学和裂纹扩展。
One of the most important aspects dealt with throughout the paper is the need for optimisation with respect to computational cost without sacrificing accuracy. The three main techniques related to this and described so far are: (i) mesh optimisation, via mesh refinement MPCs and other judicious mesh refinement; (ii) the use of symmetry of the tangential force-displacement loop with respect to contact slip, to circumvent simulation of the complete tangential part of each fretting cycle; (iii) the use of ‘representative’ contact pressure and slip values in place of the instantaneous values throughout the tangential cycle.
在整个文件中处理的最重要的方面之一是需要优化计算成本，而不牺牲精度。到目前为止，与此相关的三种主要技术是：（i）通过网格细化MPC和其他明智的网格细化进行网格优化;（ii）使用切向力-位移环相对于接触滑移的对称性，以避免模拟每个微动循环的完整切向部分;（iii）在整个切向循环中使用“代表性”接触压力和滑动值代替瞬时值。
Another obvious method of reducing the simulation time is to employ a larger value for ΔN. However, it is found that if ΔN exceeds a critical value, ΔNcrit, the results become unstable, as shown in Fig. 18. In this case, the instability occurs after only 200 cycles, but clearly the consequences are greater if significant computational time has been invested in a greater number of simulated cycles. Similar stability problems were also found in the numerical approaches proposed by Johansson [14] and Oqvist [15]. ΔNcrit is found to depend on a number of different input parameters, including normal load, stroke or slip, and wear coefficient. For example, for the case of a normal load of 1200 N, with a COF of 0.6, a stroke of 20 μm and a wear coefficient of 1×10−8 MPa−1, ΔNcrit is approximately equal to 30. However, it is not satisfactory to have to determine ΔNcrit iteratively for every load case. Fortunately, the stability problem is more directly interpreted in terms of a maximum allowed wear depth per increment, Δhcrit, which is independent of contact load, stroke/slip and wear coefficient and which, if exceeded, is the cause of the instability, via the contact algorithm of the FE code. In a given wear simulation, due to the decreasing contact pressure and the negligible change of contact slip with increasing N, the value of Δh correspondingly decreases continuously. The approach recommended here is to determine Δhcrit for one load case and then to determine the stable ΔN for other load case simulations, using trial and error, by comparing the corresponding initial Δh to Δhcrit. Fig. 19 shows how Δhcrit can be determined for one load case. For the chosen load conditions, wear simulations are carried out with a series of decreasing ΔN values, until instability occurs; it has been found that if instability is to occur, it will do so within the first ten wear increments. An ΔN value is thus obtained for which instability does not occur and the Δh value corresponding to this ΔN is then taken as Δhcrit. To ensure against instability for other loading conditions, it is then only necessary to check the initial value of Δh, for a chosen ΔN, against Δhcrit. The latter approach can prevent instability but at the cost of increased computation. Oqvist [15] suggested a simple method to balance the simulation time and the stability, by introducing a varying ΔN. This method has also been successfully implemented here.
另一个明显的减少模拟时间的方法是采用更大的ΔN值。然而，发现如果ΔN超过临界值ΔNcrit，则结果变得不稳定，如图18所示。在这种情况下，不稳定性仅在200个循环后发生，但如果在更多数量的模拟循环中投入大量计算时间，则显然后果更大。Johansson[14]和Oqvist[15]提出的数值方法也存在类似的稳定性问题。发现ΔNcrit取决于许多不同的输入参数，包括法向载荷、行程或滑移以及磨损系数。例如，对于1200 N的正常载荷的情况，COF为0。6，行程为20 μm，磨损系数为1×10−8 MPa−1时，ΔNcrit约等于30。然而，对于每个载荷情况，必须迭代地确定ΔNcrit是不令人满意的。幸运的是，稳定性问题可以更直接地解释为单位增量的最大允许磨损深度Δhcrit，它与接触载荷、行程/滑动和磨损系数无关，如果超过这个值，则会通过有限元程序的接触算法导致不稳定性。在给定的磨损模拟中，由于接触压力随N的增加而减小，接触滑移量随N的变化可以忽略不计，Δh值相应地连续减小。 这里推荐的方法是先确定一种负荷工况的Δhcrit，然后通过比较相应的初始Δh和Δhcrit，采用试错法确定其他负荷工况模拟的稳定ΔN。图19显示了如何确定一种载荷情况下的Δhcrit。对于选定的载荷条件，使用一系列递减的ΔN值进行磨损模拟，直到发生不稳定性;已经发现，如果发生不稳定性，则会在前十个磨损增量内发生。因此，获得不发生不稳定性的ΔN值，然后将对应于该ΔN的Δh值作为Δhcrit。 为了确保在其他载荷条件下不发生不稳定性，只需针对选定的ΔN，对照Δh临界值检查Δh的初始值。后一种方法可以防止不稳定性，但以增加计算为代价。Oqvist[15]提出了一种简单的方法，通过引入变化的ΔN来平衡模拟时间和稳定性。这种方法在这里也得到了成功的应用。
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Fig. 18. Effect of incremental number of fretting cycles per solution step, ΔN on (a) the worn profile of the flat specimen and (b) contact pressure. Total simulation number of cycles 200, coefficient of friction 0.6, normal load 120 N and stroke 20 μm.
见图18。每个溶液步骤的微振磨损循环次数增量ΔN对（a）平板试样的磨损轮廓和（B）接触压力的影响。总模拟循环次数200，摩擦系数0.6，正常载荷120 N，行程20 μm。

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Fig. 19. Influence of incremental wear depth, Δh, on computational stability.
图19.磨损深度增量Δh对计算稳定性的影响。

The comparisons of Fig. 11, Fig. 12 and Table 3 show that, as the normal load increases, the FE-based approach under-estimates the wear volume, by under-predicting the maximum wear depth and over-predicting the wear scar width. An explanation is that there is less difference between the applied displacement and the calculated contact slip for low normal loads. The fact that contact slip decreases with increasing normal load under the same applied stroke has been schematically demonstrated in Fig. 8, and, since it was necessary to employ applied stroke for estimation of the wear coefficient (Section 3), the wear damage based on contact slip naturally under-estimates wear volume. For the low normal load, this under-estimation is small so that close correlation is obtained. The depth under-predictions for high normal loads are considered to be due to debris effects, caused by the experimentally-observed debris retention on the cylindrical surface, which in turn causes increased contact pressure and decreased contact slip, and thus increased wear depth at the centre of the contact scar.
图11、图12和表3的比较表明，随着法向载荷的增加，基于FE的方法通过低估最大磨损深度和高估磨痕宽度而低估了磨损体积。一种解释是，对于低法向载荷，施加的位移和计算的接触滑移之间的差异较小。在图8中示意性地展示了在相同的施加行程下，接触滑动随着法向载荷的增加而减小的事实，并且由于必须采用施加行程来估计磨损系数（第3节），因此基于接触滑动的磨损损坏自然会低估磨损量。 对于低的正常负载，这种低估是小的，从而获得密切的相关性。高法向载荷的深度低于预测值被认为是由于实验观察到的圆柱形表面上的碎屑滞留引起的碎屑效应，这反过来又导致接触压力增加和接触滑动减少，从而增加了接触疤痕中心的磨损深度。
Fig. 20 shows a comparison of the measured and predicted force-displacement loops for the 18,000th cycle for the 185 N case. Two main differences are observed between the measured and predicted loops. The first is larger inclination of the measured hysteresis loop, reflecting the tangential compliance of the system. The second difference is that the width of the measured loop is smaller than that predicted. Preliminary calculations have established that applied normal and tangential loads do not give rise to von Mises stresses in excess of the yield stress, so that these differences cannot be attributed to plastic yield. According to Vincent [26], the loop width can be regarded as being approximately equal to an averaged measure of the half-cycle relative slip. This suggests, referring to Fig. 20, that the measured slip value is less than the FE-predicted value. This difference can be explained in terms of the displacement accommodation of the fretting system. The velocity accommodation model proposed by Godet [8] shows that in a real (experimental) fretting system, the imposed displacement 2δ* can be partially accommodated by deformation of the contacting bodies, the third body (debris) and the associated interfaces. This suggests that the FE model employed needs to be enhanced in order to simulate more realistically the fretting conditions. Additional work is required to understand and simulate the mechanical behaviour of the debris and its effect on displacement (velocity) accommodation. This will permit better matching of the predicted and measured T–δ loops and thus also quantitative identification of the effect of debris on fretting wear damage.
图20显示了185 N情况下第18，000次循环的实测和预测力-位移环的比较。两个主要的差异之间观察到的测量和预测的循环。第一个是测量的磁滞回线的较大倾斜，反映了系统的切向柔度。第二个不同之处是，测量的环路宽度小于预测的环路宽度。初步计算表明，施加的法向和切向载荷不会引起超过屈服应力的von Mises应力，因此这些差异不能归因于塑性屈服。根据Vincent[26]，环宽度可被视为近似等于半周相对滑移的平均测量值。这表明，参考图。 20，测量的滑移值小于FE预测值。这种差异可以用微动系统的位移调节来解释。Godet[8]提出的速度调节模型表明，在真实的（实验）微动系统中，施加的位移2δ*可以通过接触体、第三体（碎屑）和相关界面的变形部分地调节。这表明，所采用的有限元模型需要加强，以更真实地模拟微动条件。需要开展更多的工作，以了解和模拟碎片的机械行为及其对位移（速度）调节的影响。 这将允许预测的和测量的T-δ环更好地匹配，从而也定量识别碎片对微动磨损损伤的影响。
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Fig. 20. Comparison of the FE-predicted and experimental measured tangential force vs. displacement loops for 18,000th wear cycle with a normal load of 185 N and an applied stroke of 50 μm.
图20.在185 N法向载荷和50 μm施加行程下，第18，000次磨损循环的FE预测和实验测量切向力与位移环的比较。

Ultimately, following this future work on incorporation of debris effects, it is anticipated that comparisons between simulation results and measured results will permit estimation of the local wear coefficient, kl. The availability of such a local wear coefficient, independent of contact geometry and operating conditions, will facilitate general application to new contact geometries and conditions.
最后，在今后就纳入碎片影响开展这项工作之后，预计模拟结果与测量结果之间的比较将有助于估计局部磨损系数k。这种局部磨损系数的可用性，独立于接触几何形状和操作条件，将促进新的接触几何形状和条件的一般应用。
The present paper has considered gross slip situations only. The issues of partial slip, where the contact region is divided into the stick and slip zones is dealt with in [27].
本文只考虑了总滑移的情况。部分滑移的问题，其中接触区域被划分为粘着和滑移区，在[27]中进行了处理。

8. Conclusions 8.结论

An incremental method for fretting wear simulation, based on a modified Archard equation, has been applied to a series of gross slip cylinder-on-flat tests on a high strength alloy steel for aeroengine applications. The measured and predicted worn surface profiles were found to correlate well for the low normal load case. Under high normal loads, the predicted maximum wear depth was under-estimated and the width of the scar was over-estimated. The differences are attributed to the use of stroke for wear coefficient calculation and the effect of debris, which was not modelled.
基于修正的Archard方程，采用增量法对航空发动机用高强度合金钢的微动磨损进行了数值模拟。测量和预测的磨损表面轮廓被认为是相关的低正常负载的情况下。在高正常载荷下，预测的最大磨损深度被低估，疤痕的宽度被高估。这些差异归因于使用冲程进行磨损系数计算和未建模的碎片的影响。
During the first 1000 wear cycles, the half contact width increases significantly, by about 100%, while the peak contact pressure decrease dramatically to about 40% of the initial peak Hertzian value; subsequent changes in these variables are at a slower rate. The contact pressure distribution was shown to evolve to a uniform distribution across the increased contact width. The slip between the contacting bodies was shown to increase slightly with wear, concomitant with the decreasing contact pressure.
在前1000个磨损循环期间，半接触宽度显著增加，增加约100%，而峰值接触压力急剧降低至初始峰值赫兹值的约40%;这些变量的后续变化速率较慢。结果表明，随着接触宽度的增加，接触压力分布逐渐趋于均匀。接触机构之间的滑动被证明是略有增加的磨损，伴随着接触压力的降低。
A number of techniques for minimisation of the total simulation time were identified including: (i) mesh optimisation; (ii) the informed use of ‘representative’ contact pressure and slip values, to circumvent simulation of the full fretting cycle; and (iii) optimization of ΔN, the increment in number of simulation wear cycles. A critical incremental wear depth technique for avoidance of stability problems associated with incorrect choice of ΔN has been presented.
确定了一些用于最小化总模拟时间的技术，包括：（i）网格优化;（ii）知情使用“代表性”接触压力和滑动值，以避免模拟整个微动磨损循环;以及（iii）优化ΔN，模拟磨损循环次数的增量。本文提出了一种临界磨损深度增量技术，以避免因ΔN选择不当而引起的稳定性问题。
Differences between the measured and predicted tangential force-displacement loops, and the associated slip values, have been interpreted in terms of the displacement accommodation effects of debris.
测量的和预测的切向力-位移环之间的差异以及相关的滑移值，已被解释为碎片的位移调节效应。

Acknowledgements 确认

The authors would like to thank Rolls-Royce plc for financial assistance, and Thomas R. Hyde and Nina Banerjee of Rolls-Royce plc for helpful discussions.
作者要感谢劳斯莱斯公司的财政援助，和托马斯R。Hyde和Rolls-Royce plc的Nina Banerjee进行了有益的讨论。

References

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