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.     

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.     

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

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

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。     

Plastic deformation in rough surface line contacts—a finite element study

The paper describes an elastic-plastic finite element (EPFE) analysis of line contact between a cylinder and rigid plane using commercial software. The range of loading demonstrates the transition from purely elastic to fully plastic contact behaviour, revealing the residual deformations and stress fields upon unloading. A multiple contact configuration was analysed in the form of sinusoidal roughness. Results obtained under elastic conditions were validated by comparison with theoretical solutions. This model was extended by replacing the sinusoidal surface with a real roughness profile. Modelling multiple contacts indicates the influence of adjacent surface “asperities” on contact pressure and residual stress distributions.     

The influence of surface roughness in elliptical contacts

The influence of surface roughness on contact behaviour is of great importance in many tribological situations. In the last decade several methods to calculate the pressure distribution and the real contact area in contacts between rough surfaces have been described. A problem arising for slender elliptical contacts, such as between gear teeth, roller and raceway, cam and follower, etc., is that the size of the contact is much greater than the size of the asperities. Accordingly the number of contact nodes necessary for an accurate solution to the problem becomes excessively large. This paper describes a method to calculate the influence of three-dimensional surface roughness in contacts that are very long in one direction. The method is based on restricting the calculations to a subpart of the real contact area, while the rest of the contact is taken into account by mirroring techniques. The results show that the real contact area is very sensitive to the amplitude of the roughness, while the waviness is less important. An equation is suggested from which the real area can be calculated if the smooth case contact parameters and two roughness parameters are known.     

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.     

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。     

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

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

An analysis of three-dimensional elasto-plastic sinusoidal contact

Researchers have developed many models to simulate the elasto-plastic contact of spheres. However, there does not appear to exist a closed-form analytical model for elasto-plastic three-dimensional sinusoidal contact. This work uses a finite element model (FEM) to characterize elasto-plastic sinusoidal contact. Although at initial contact the sphere and sinusoidal case are very similar and can both be described by the classic elastic Hertz contact case, once the contact is pressed past a certain range of deformation the two cases are very different. The model produces equations which can be used to approximately relate the area of contact to the contact pressure for elasto-plastic sinusoidal contact. The equations are fit to the FEM results and existing elastic solutions of sinusoidal contact. An empirical expression for the average pressure which causes complete contact between elasto-plastic sinusoidal contacts is also provided.     

Effect of transverse surface roughness and additives in TEHD contacts

Surface roughness effects in mixed rheological thermal EHL of rolling/sliding line contacts are investigated numerically. The surface roughness is assumed to be transverse and its profile is generated by a sinusoidal function defined in terms of its amplitude and wavelength. A homogeneous mixture of Newtonian base oil and power law fluid additive with varying volume fraction, viscosity ratio and power law index is used to represent polymer-modified oils. The velocity profile for the mixed rheological fluid model is obtained using perturbation method to derive Reynolds and mean temperature equations. It is found that the surface roughness effects on EHL characteristics are significantly modified due to the presence of polymeric fluid additives.     

The influence of surface roughness and the contact pressure distribution on friction in rolling/sliding contacts

A numerical contact model is used to study the influence of surface roughness and the pressure distribution on the frictional behaviour in rolling/sliding contacts. Double-crowned roller surfaces are measured and used as input for the contact analysis. The contact pressure distribution is calculated for dry static contacts and the results are compared with friction measurements in a lubricated rolling/sliding contact made with a rough friction test rig. The mean pressure is suggested as a parameter that can be used to predict the influence of surface roughness on the friction coefficient in such contacts. The results show two important properties of the friction coefficient for the friction regime studied in this paper: (1) there is a linear decrease in friction coefficient as a function of the slide-to-roll ratio, and (2) the friction coefficient increases linearly with increasing mean contact pressure up to a maximum limit above which the friction coefficient is constant. The absolute deviation of experimental results from the derived theory is for most cases within 0.005.     

The Contact of Elastic Regular Wavy Surfaces Revisited

We revisit the classic problem of an elastic solid with a two-dimensional wavy surface squeezed against an elastic flat half-space from infinitesimal to full contact. Through extensive numerical calculations and analytic derivations, we discover previously overlooked transition regimes. These are seen in particular in the evolution with applied load of the contact area and perimeter, the mean pressure and the probability density of contact pressure. These transitions are correlated with the contact area shape, which is affected by long range elastic interactions. Our analysis has implications for general random rough surfaces, as similar local transitions occur continuously at detached areas or coalescing contact zones. We show that the probability density of null contact pressures is nonzero at full contact. This might suggest revisiting the conditions necessary for applying Persson’s model at partial contacts and guide the comparisons with numerical simulations. We also address the evaluation of the contact perimeter for discrete geometries and the applicability of Westergaard’s solution for three-dimensional geometries.     

Elastic Sinusoidal Wavy Surface Contact Under Full Stick Conditions

A purely normal contact problem of an elastic half-space with a three-dimensional periodic sinusoidal wavy surface and a rigid flat under the full stick condition is studied. The contacting points from mating surfaces have zero relative tangential displacement under the full stick condition. The scope of this study is restricted to a special case where the entire contact interface is in contact (referred to as complete contact) under the full stick condition. Complete contact is defined as when there are no gaps remaining between the surfaces. The corresponding state of stress of the half-space is derived analytically. According to the state of stress, we find (1) an analytical solution for the average pressure required to cause complete contact, (2) the location of the global maxima of the von Mises stress and (3) the critical magnitude of the waviness amplitude below which the plastic yielding of the half-space will never occur before the initiation of complete contact. The results are also compared with the solution under the perfect slip condition. We find that the location of the maximum von Mises stress may occur either on the contact interface or beneath it depending on the value of Poisson’s ratio.     

Cold compaction of an array of cylindrical fibres

The biaxial compaction of a square array of circular cylinders is studied using slip-line field, upper bound and finite element methods. Densification is assumed to occur by plastic deformation at the contacts. It is found that contact–contact interaction has a softening effect on the local indentation pressure at each contact. The yield surfaces resulting from hydrostatic and closed-die compaction are constructed at various stages of densification: the size and shape of the yield surface depend upon the loading history and upon the relative density of the compact. The finite element predictions suggest the formation of a vertex at the loading point for the entire densification process in the case of isostatic compaction. On the other hand, a vertex at the loading point is formed only for a relative density D0.85 in the case of closed-die compaction.     

An Integrated Force Probe and Quartz Crystal Microbalance for High-Speed Microtribology

We have developed a technique for measuring frictional forces and contact areas, over a wide range of applied loads, at microscopic contacts reaching high sliding speeds near 1 m/s. Our approach is based on integrating two stand-alone methods: nanoindentation and quartz crystal microbalance (QCM). Energy dissipation and lateral contact stiffness are monitored by a transverse shear quartz resonator, while a spherical indenter probe is loaded onto its surface. Variations in these two quantities as functions of shear amplitude, with the normal load held fixed, reveal a transition from partial to full slip at a critical amplitude. Average values of both the threshold force for full slip and the kinetic friction during sliding are determined from these trends, and the contact area is inferred from the lateral stiffness at low shear amplitudes. Measurements are performed at loads ranging from 5 µN to 8 mN using an electrostatically actuated indenter probe. For the materials chosen in this study, we find that the full slip threshold force is about a factor of two larger than kinetic friction. The forces increase sublinearly with load in close correspondence with the contact area, and the shear strengths are found to be relatively insensitive to pressure. The threshold shear amplitude scales in proportion to the contact radius. These results demonstrate that the probe–QCM technique is a versatile and full-featured platform for microtribology in the speed range relevant to practical applications.     

机械结合面法向动态接触刚度理论模型与试验研究

机械结合面经常可能在动态条件下工作,使得结合面的接触状态偏离静态工作状态,导致接触刚度发生改变。为了揭示动态接触刚度的变化规律,考虑两个粗糙结合表面上单个微凸体由弹性变形向弹塑性变形以至最终向完全塑性变形转化的接触过程,建立一个振动周期内各变形阶段微凸体的平均接触刚度模型;在此基础上,基于高斯分布假设,建立整个粗糙结合表面的动态接触刚度解析模型。该模型揭示了接触面压、振动频率和相对位移振幅对动态接触刚度的影响规律,并与试验结果和静态接触刚度计算结果进行了比较。研究表明:法向动态接触刚度和静态接触刚度与接触面压之间的关系基本一致,但有一定偏离。这种偏离程度随动态位移幅值和振动频率的增加而分别呈线性和非线性增大。