The behavior of an elastic-perfectly plastic sinusoidal surface under contact loading

The goal of this paper is to provide the foundation for an analysis of contact between elastic-plastic solids, whose surface roughness is idealized with a Weierstrass profile. To this end, we conduct a parametric study of the plastic deformation and residual stress developed by the two-dimensional contact between a flat, rigid platen and an elastic-perfectly plastic solid with a sinusoidal surface. Our analysis shows that the general characteristics of the deformation can be characterized approximately by two parameters: α = a/λ, where a is the half-width of the contact and λ is the period of the surface waviness; ψ = E^*g/σ_Yλ, where E^* and σ_Y are the effective modulus and yield stress of the substrate, respectively, and g is the amplitude of the surface roughness. Depending on the values of these parameters, we identify eight general types of behavior for the asperity contacts: (a) elastic, elastic-plastic or fully plastic isolated Hertz type contacts; (b) elastic, or elastic-plastic non-Hertzian isolated contacts; and (c) elastic, elastic-plastic or fully plastic, interacting contacts. Relationships between contact pressure, contact size, effective indentation depth and residual stress are explored in detail in each regime of behavior. Implications on rough surface contacts are discussed.     

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.     

Coupled couple stress and surface roughness effects on elasto‐hydrodynamic contact

This paper deals with the combined effects of couple stress and surface roughness to inspect the elasto‐hydrodynamic performance of slider bearing systems. On the basis of the couple stress Stokes theory and homogenisation method, the homogenised generalised Reynolds equation including the slider bearing stationary surface deformation is derived. The total deformation include the deformation of smooth surface, taken into account by the elastic thin layer model, and the deformation of roughness corresponding to a sinusoidal normal displacement on an elastic half space of identical wavelength. The governing equations are discretised by the finite difference method, and the obtained algebraic equations are solved using the iterative overrelaxation Gauss–Seidel technique. The load‐carrying capacity and friction coefficient are presented for transverse, longitudinal and anisotropic roughness patterns for different values of the couple stress parameter in both rigid and deformable cases. The simulation results indicate that the interaction between couple stress, surface roughness and elastic deformation effects is significant. Copyright © 2013 John Wiley & Sons, Ltd.     

The contact of elastic regular wavy surfaces

A complete solution in closed form to the elastic contact of a one-dimensional sinusoidal surface with a flat surface was presented by Westergaard in 1939. This paper is concerned with the elastic contact of a two-dimensional sinusoidal surface with a flat. In this case the stress distribution within the elastic solids is three-dimensional. As the load is increased the contact areas change in shape from being circular to square and finally leave a circular region of no contact when the waves are almost squashed flat. The problem is solved in general using a numerical method due to Kalker, but asymptotic solutions in closed form have been found for light loads and also for heavy loads at which contact is almost complete. The variation of the mean separation with load, which determines the volume of the space trapped between the two surfaces, is also found.     

A new roughness parameter to evaluate the near-surface deformation in dry rolling/sliding contact

Within this work, a two-dimensional finite element model for rolling contact of a wheel on a rail is presented that accounts for the roughness of the contact surfaces. The rail material is modeled with elastic–plastic behavior. The maximum of the plastic shear strain is concentrated close to the surface of the rail and is mainly influenced by the surface roughness. A concept is proposed that demonstrates one crucial parameter of the roughness determines surface deformation (based on results of a sinusoidal roughness model). This roughness parameter depicts the ratio between asperity height and width. Numerical validation is achieved for predicting plastic shear strains in rough surfaces. The plastic shear strain is associated with surface damage, such as cracks and wear.     

Enhancement of a simplified model for maximum stress prediction

A simplified model for rapid estimation of maximum subsurface stress was previously reported by the authors. That model was found to predict the upper bound of the maximum von Mises stress in each layer below any surface with good accuracy. In this work, the model is enhanced by quantifying the difference between the predicted upper bound and the maximum stress for a specific rough surface. This is done by evaluating stress predictions for real rough surfaces, sinusoidal surfaces, ideal textured surfaces, and real rough surfaces with imposed computer generated texture. The difference between the predicted upper bound and the surface-specific maximum stress is found to be related to that surface’s roughness. The origin of this relationship is investigated in terms of apparent contact area. The enhanced simplified model provides an efficient means of estimating maximum stress in rough surface contact.     

Characterization of fretting fatigue crack initiation processes in CR Ti–6Al–4V

A study was conducted to quantify fretting fatigue damage and to evaluate the residual fatigue strength of specimens subjected to a range of fretting fatigue test conditions. Flat Ti–6Al–4V specimens were tested against flat Ti–6Al–4V fretting pads with blending radii at the edges of contact. Fretting fatigue damage for two combinations of static average clamping stress and applied axial stress was investigated for two percentages of total life. Accumulated damage was characterized using full field surface roughness evaluation and scanning electron microscopy (SEM). The effect of fretting fatigue on uniaxial fatigue strength was quantified by interrupting fretting fatigue tests, and conducting uniaxial residual fatigue strength tests at R=0.5 at 300 Hz. Results from the residual fatigue strength tests were correlated with characterization results.While surface roughness measurements, evaluated in terms of asperity height and asperity spacing, reflected changes in the specimen surfaces as a result of fretting fatigue cycling, those changes did not correspond to decreases in residual fatigue strength. Neither means of evaluating surface roughness was able to identify cracks observed during SEM characterization. Residual fatigue strength decreased only in the presence of fretting fatigue cracks with surface lengths of 150 μm or greater, regardless of contact condition or number of applied fretting fatigue cycles. No cracks were observed on specimens tested at the lower stress condition. Threshold stress intensity factors were calculated for cracks identified during SEM characterization. The resulting values were consistent with the threshold identified for naturally initiated cracks that were stress relieved to remove load history effects.     

粗糙表面弹性微动接触数值研究

由于实际工程表面多为粗糙表面,这里研究了粗糙表面对微动接触中压力和切向应力的影响。研究接触过程中法向载荷保持不变,切向载荷为周期性的交变载荷。首先,建立接触算法和模型,其算法核心是利用共轭梯度法(CGM)计算微动接触中的表面压力及切向应力并使用快速傅里叶变换(FFT)加快计算速度。然后,在验证算法正确的基础上,分析正弦和非高斯粗糙表面接触的压力和切向应力的分布,通过对光滑与粗糙表面的研究对比,表明:(1)在正弦表面接触切向应力分布呈现尺寸效应;(2)在非高斯表面接触中,切向应力分布跟光滑表面形状类似;同时由于粗糙峰存在,粗糙表面下的切向应力比光滑表面下的要大,研究粗糙表面微动接触对实际工程具有重要意义。     

复合铣削轨迹对表面质量的影响

复合铣削一般由多个旋转运动和多个移动复合而成。通过齐次变换矩阵法得到了复合铣削轨迹方程和复合铣削轨迹，探究了复合铣削轨迹对表面质量的影响，结果表明：复合铣削的包络面的表面粗糙度明显低于未包络面，复合铣削的未包络面与普通端铣的表面粗糙度相差不大；进给量对复合铣削的未包络面的表面粗糙度影响不大，而包络面的表面粗糙度随进给量增大而增大；未包络面和包络面的表面粗糙度Ra均随着切削速率比k的增大而减小；复合铣削的残余应力比普通端铣的残余应力小，而复合铣削中的包络面的残余应力比未包络面的小。     

A non-local fatigue approach to quantify Ti-10V-2Fe-3Al fretting cracking process: Application to grinding and shot peening

This paper presents an experimental study on the tribological behaviour and cracking response of a Ti-10V-2Fe-3Al titanium alloy under fretting loading with a cylinder on plane configuration. Three types of surfaces were investigated: a polished one considered as the reference, a ground one and a shot peened surface. Surfaces were compared with respect to residual stress, hardness and roughness. The first step of this study was to determine sliding conditions and coefficient of friction of the three contact types. Next, fretting tests under stabilized partial slip regime were carried out to investigate crack nucleation and propagation. Results show that whatever surface roughness or residual stress in the material, tribological behaviour is the same. These latter confirm that sliding condition and coefficient of friction in partial slip regime is due to material effect and not to roughness or surface hardness. Then, residual stress induced by grinding or shot peening have no influence on the crack nucleation threshold under fretting solicitation because crack nucleation is only induced by a sufficient tangential loading. The crack nucleation threshold is formalized by applying the Crossland criterion taking into account the stress gradient and the ensuing “size effect”. As expected, cracks propagation is influenced by residual stress under the surface. Compared to the reference case, for a same loading parameters set, residual stress induced by grinding is not sufficient to decrease the crack length reached whereas effects of shot peening decrease highly these latter. So, there is a threshold of residual stress from which residual stresses are useful against cracking.     

粗糙度各向异性对润滑接触面摩擦的影响

齿轮、轴承、凸轮等重载接触副的性能受表面粗糙度的显著影响。高负载情况下的摩擦因数与润滑接触面粗糙度的各向异性相关。测量的表面粗糙度可以分解为一系列具有不同波长、幅值的正弦表面粗糙度,因此,考虑各向异性正弦表面粗糙度,构建粗糙表面点接触瞬态弹性流体动力润滑(TEHL)模型,提出基于多重网格算法的粗网格构造新方法,提高粗糙表面润滑问题求解的稳健性。研究表面粗糙度各向异性对高负载情况下摩擦因数的影响规律。结果表明,粗糙度的各向异性影响接触面压力、油膜厚度分布、粗糙度形变量,从而影响摩擦因数。提出一个组合函数来量化粗糙度各向异性对摩擦因数的影响,表明全膜润滑到混合润滑的过渡不仅与载荷、速度等工况参数相关,还与粗糙度各向异性相关。     

A numerical investigation of the sinusoidal model for elastic layers in line contact

Distributions of normal stresses and surface deformations, induced when an elastic layer of finite thickness is indented by a frictionless rough rigid flat or cylindrical indenter, are calculated numerically. It is assumed that the punch has a sinusoidal roughness superimposed on its nominal profile. Two cases will be examined, namely when the elastic layer is either bonded to a rigid backing or resting on a frictionless rigid backing (unbonded). Chebyshev polynomials of the first kind T_n(x) are utilized to model both the unknown pressure and the given deformation over the contact area. The governing elasticity equation is thereby reduced to a finite set of linear equations and hence a complete solution is found. The present numerical method is simple, accurate and valid in the full range of Poisson's ratio 0 v 0.5. Moreover, a set of semi-analytical solutions for the contact pressure is obtained for both thin unbonded and bonded elastic layers. The numerical results compared favourably with the asymptotic solutions. The effects of the layer thickness, layer compressibility and roughness amplitude parameters on the contact stresses and deformations are considered.     

Failure analysis of rolling bearings by X-ray measurement of residual stress

Residual stress measurements were carried out for the failure analyses of rolling bearings. According to the survey of more than one hundred examples, some change in residual stress was observed in most cases.Compressive residual stresses peak below the surface and if present, represent the position of maximum shearing stress due to normal contact pressure. Unexpected overloading is confirmed by this method.However, bearings often fail without such overloading. Early failures were observed to be associated with the residual stress changes near the surface. The influences of sliding, vibration and surface roughness on residual stress changes are discussed.     

Three-dimensional modeling of elasto-plastic sinusoidal contact under time dependent deformation due to stress relaxation

In the current work, the effect of stress relaxation in contact between sinusoidal surfaces is studied using FE simulations. There are a few works on the elastic and elasto-plastic contact between sinusoidal surfaces, but the transient effects such as creep and stress relaxation are not considered in these works. Stress relaxation causes significant change in the contact area and pressure between the contacting surfaces. The Garofalo formula is used to model the transient behavior of stress relaxation. The results for the contact area and contact pressure are presented and discussed. Empirical equations are developed to predict contact area and pressure by fitting to the FEM results. The equations are dependent on the initial surface separation, aspect ratio, and the Garofalo constants.     

Microscopic asperity contact and deformation of ultrahigh molecular weight polyethylene bearing surfaces

The effect of the roughness and topography of ultrahigh molecular weight polyethylene (UHMWPE) bearing surfaces on the microscopic contact mechanics with a metallic counterface was investigated in the present study. Both simple sinusoidal roughness forms, with a wide range of amplitudes and wavelengths, and real surface topographies, measured before and after wear testing in a simple pin-on-plate machine, were considered in the theoretical analysis. The finite difference method was used to solve the microscopic contact between the rough UHMWPE bearing surface and a smooth hard counterface. The fast Fourier transform (FFT) was used to cope with the large number of mesh points required to represent the surface topography of the UHMWPE bearing surface. It was found that only isolated asperity contacts occurred under physiological loading, and the real contact area was only a small fraction of the nominal contact area. Consequently, the average contact pressure experienced at the articulating surfaces was significantly higher than the nominal contact pressure. Furthermore, it was shown that the majority of asperities on the worn UHMWPE pin were deformed in the elastic region, and consideration of the plastic deformation only resulted in a negligible increase in the predicted asperity contact area. Microscopic asperity contact and deformation mechanisms may play an important role in the understanding of the wear mechanisms of UHMWPE bearing surfaces.     

Coupled effects of surface roughness and flow rheology on elastohydrodynamic lubrication

The coupled effects of surface roughness and flow rheology on elastohydrodynamic lubrication (EHL) circular contact problems are analyzed and discussed. The averaged type Reynolds equation utilizing the average flow model on the interactions between couple stress fluids and surface roughness, the elastic deformation equation, the viscosity–pressure and density–pressure relations equations, and the force balance equation are solved numerically by the multilevel multi-integration (MLMI) algorithm to calculate the pressure distributions and film thickness shapes. The results show that the transverse type roughness and standard deviation of composite roughness enhance the pressure and film thickness in the central contact region. Moreover, the longer the characteristic length of the couple stress fluids is, the smaller the pressure distribution is in the central contact region and the greater the film thickness is in all regions.     

Method and surface roughness aspects for the design of DLC coatings

The design of coatings for highly loaded component contacts, such as bearings, gears and valve train components involves several important factors, including load, friction, lubrication, surface characteristics and material parameters. This paper presents an investigation of the influence of the material, coating thickness and surface roughness on tensional stress levels for coatings that are more compliant than the substrate material. Specifically the effect of multiple asperity contact is studied in three dimensions. The simulation is based on a finite element model where the load is applied as several interacting Hertzian pressure distributions.The results show that the surface structure, in combination with the elastic properties of the coating, has a large influence on the tensional stress level in the coating. The highest tensional stress level in the coating occurs when contact spots almost overlap neighbouring cells and at the same time the size of the contact spots is in the same order of magnitude as the coating thickness.     

Fretting Wear Behavior of Laser Peened Ti-6Al-4V

This work deals with the influence of laser peening on the fretting wear behavior of Ti-6Al-4V. Laser peening was carried out on Ti-6Al-4V. The laser-peened surface was characterized by transmission electron microscopy. Surface roughness, nanoindentation hardness, residual stress, and tensile properties of the material in both laser-peened and unpeened conditions were determined. Fretting wear tests were conducted at different normal loads using a ball-on-flat contact geometry. Laser peening resulted in the formation of nanocrystallites on the surface and near-surface regions, increased hardness, and compressive residual stress. Laser peening did not affect the tensile properties and surface roughness significantly. There was no considerable difference between the values of the tangential force coefficient of laser-peened and unpeened samples. The fretting scar size, wear volume, and wear rate of laser-peened specimens were lower than those of unpeened samples. This may be attributed to an increase in surface hardness due to strain hardening and grain refinement at the surface and near-surface regions, higher compressive residual stress, and higher resistance to plastic deformation of laser-peened samples.     

Stress history in rolling–sliding contact of rough surfaces

An investigation into the non-Hertzian, elastic stress history, due to the contact of two rough surfaces is presented. A complex evolution of stress is produced whose magnitude and rate depend strongly upon the roughnesses and speeds of the contacting bodies. The key features of the stress fields are illustrated by plots of stress versus time and horizontal distance, for a range of depths and for various contact conditions. The stresses near the surface are many times higher than in an equivalent smooth contact and the roughness on thecounterface generates a moving stress field which, when sliding is present, greatly increases the number of cycles of stress during each passage of the contact. This may account, in part, for the observation that the rolling fatigue life of hard steels declines more rapidly with sliding speed for rough, than for smooth surfaces and suggests that counterface roughness is especially important in determining the fatigue life.     

考虑表面粗糙度的面齿轮齿面接触应力分析

根据齿轮啮合原理,得出了面齿轮的齿面方程,求得面齿轮齿面曲率。结合赫兹接触理论,推导了点接触面齿轮传动接触点的接触应力及应力沿齿面的分布规律,从齿根到齿顶,齿面接触应力先增大后减小,在靠近齿面中点处达到最大值。由粗糙表面接触理论,分析了面齿轮齿面微观弹塑性变形时的接触面积,并得到粗糙齿面接触时面齿轮齿面接触应力及其分布。对比分析了几种不同粗糙度条件下面齿轮齿面接触应力的变化规律,结果表明:齿面接触应力随表面粗糙度的增大而增大,与齿根处相比,齿顶接触应力受表面粗糙度影响更大。文中的分析可为面齿轮磨损及润滑机理的研究提供依据。