Analytical determination of friction resistance as a function of normal load and geometry of surface irregularities

A mathematical modeling and simulation of friction during steady state sliding of metals, based on the upper-bound approach, is demonstrated. The existence of wedge-shaped protrusions on the tool surface is assumed. Pressing these protrusions onto the workpiece and sliding the tool along the workpiece produces asperities on the surface of the workpiece. These asperities move in a wave-like motion along the surface layer and cause plastic deformation through a specified depth under the surface. This plastic deformation combines with local friction between the tool and the workpiece along the asperity interface to produce resistance to sliding. The relation between the normal pressure and the sliding resistance is established for the entire range of pressure levels from zero to infinity. The apparent Coulomb coefficient of friction for lower levels of normal pressure and the constant friction factor for excessive load levels are determined. The transition region from Coulomb coefficient of friction to constant friction factor also becomes clear. A mathematical determination is obtained by means of a force equilibrium considering the concept of a contact surface friction ratio. The force of resistance to sliding is related both to the geometry of the asperity of the surface of the tool and to the constant friction factor, which is used for measuring a local frictional force along the interface of each asperity.     

Numerical and Experimental Studies on Tribological Behaviors of Cu-Based Friction Pairs from Hydrodynamic to Boundary Lubrication

When studying the tribological behaviors of a Cu-based friction pair in different lubrication regimes, calculation of the real contact area of asperity contacts is crucial but difficult. In this work, a mixed lubrication model in plane contacts is developed, and pin-on-disc tests are carried out. The real contact area ratio, load sharing ratio, and friction coefficient are investigated. Effects of sliding velocity, temperature, and pressure are considered. The results show that when the maximum contact area ratio is about 14.6%, the load sharing ratio of asperity contacts is about 95%. The friction coefficient obviously increases from less than 0.04 to about 0.15 as the regime changes from hydrodynamic to boundary lubrication. Asperities have a significant influence on the local lubrication of a Cu-based friction pair, and the action of hydrodynamic pressure cannot be ignored.     

基于分形理论的滑动摩擦表面接触力学模型

依据分形理论,考虑微凸体变形特征及摩擦作用的影响建立滑动摩擦表面接触力学模型。采用一个三次多项式来表达弹塑性变形微凸体的接触压力与接触面积的关系,从而满足在变形状态转变临界点处的微凸体接触面积与接触压力转化皆是连续和光滑的条件。推导出滑动摩擦表面临界弹性变形微接触面积、临界塑性变形微接触面积、量纲一真实接触面积的数学表达式。理论计算结果表明,表面形貌一定时,真实接触面积随着载荷的增大而增大;载荷一定时,真实接触面积随着特征尺度系数的增大而减小,随着分形维数的增大先增大后减小;当表面较粗糙时,摩擦因数对真实接触面积的影响很小;随着表面光滑程度的增大,摩擦因数对真实接触面积的影响增大,真实接触面积随着摩擦因数的增大而增大,特别是当摩擦因数较大时,真实接触面积增大的幅度也较大。接触力学模型的建立,为研究滑动摩擦表面间的摩擦磨损性能提供了依据。     

Effects of Interfacial Strength and Roughness on the Static Friction Coefficient

A finite element model is used to simulate sliding inception of a rigid flat on a deformable sphere under combined normal and tangential loading. Sliding inception is treated as the loss of tangential contact stiffness under combined effects of plasticity, crack propagation and interfacial slip. Energy dissipation distribution is used to quantify the relative contribution of these mechanisms on the increased compliance during tangential loading. Materials with different strength and toughness properties, and varying local interface conditions ranging from fully adhered to finite friction, are studied to relate variations in plastic deformations, crack and slip to the sliding inception. For fully adhered contact condition, crack and fracture toughness have no effect on sliding inception, with plasticity, the dominant failure mechanism. A measure of recoverable strain (yield strength to Young’s modulus ratio) is found to be the most influential parameter in sliding inception. Interfacial slip is expectedly the dominant mechanism for sliding inception for lower coefficient of friction, modeling lubricated contacts. Interplay of plasticity and interfacial slip is found to govern the onset of sliding for higher local friction coefficients. Furthermore, the single asperity results are incorporated in a statistical model for nominally flat contacting rough surfaces under combined normal and tangential loading to investigate the stochastic effects due to surface roughness and material property uncertainties. The results show that the static coefficient of friction strongly depends on the normal load, material properties, local interfacial strength and roughness parameters.     

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.     

Analyzing Elastic-Plastic Real Rough Surface Contact in Running-in

A numerical method is presented for evaluating the elastic-elastic contact of real rough surface contacts during running-in. For the surface contact, an elastic-plastic model based on the variational method is applied to analyze the pressure distribution and contact area of worn surfaces during running-in. In conjunction with the classical statistic model of Greenwood and Williamson, the numerical result showed that the plasticity index Ψ was decreased to one in the elastic range as running-in proceeded. In comparison with the Hertzian solution, the influence of the asperities is very significant on the pressure distribution, thereafter causing a higher peak value of contact pressure. For the subsurface, the interior stress from the von Mises criterion was calculated to evaluate the subsurface stress field subject to both normal and tangential forces. In the calculated of the interior stress, the total stress is decomposed into a fluctuating component and a smooth component. The fluctuating part is solved by using FFT from the concept of the convolution theorem while the smooth part is obtained directly by analytical solution. Calculations of contact area and subsurface stress on experimentally produced surfaces whose topography has been determined using an atomic force microscope and friction coefficient front sliding have been carried out. The results showed that asperities and friction coefficient gave rise to stress increase in the near-surface stress field and produced a high stress zone towards the surface. As a result, transverse asperity cracking was produced. The calculations and supporting experimental evidence clearly confirmed that the reduction of peak pressure during running-in decreased the plastic deformation of contact.     

Elastoplastic contact mechanics model of rough surface based on fractal theory

Because the result of the MB fractal model contradicts with the classical contact mechanics, a revised elastoplastic contact model of a single asperity is developed based on fractal theory. The critical areas of a single asperity are scale dependent, with an increase in the contact load and contact area, a transition from elastic, elastoplastic to full plastic deformation takes place in this order. In considering the size distribution function, analytic expression between the total contact load and the real contact area on the contact surface is obtained. The elastic, elastoplastic and full plastic contact load are obtained by the critical elastic contact area of the biggest asperity and maximun contact area of a single asperity. The results show that a rough surface is firstly in elastic deformation. As the load increases, elastoplastic or full plastic deformation takes place. For constant characteristic length scale G, the slope of load-area relation is proportional to fractal dimension D. For constant fractal dimension D, the slope of load-area relation is inversely proportional to G. For constant D and G, the slope of load-area relation is inversely proportional to property of the material ϕ, namely with the same load, the material of rough surface is softer, and the total contact area is larger. The contact mechanics model provides a foundation for study of the friction, wear and seal performance of rough surfaces.     

尺度相关的分形结合面法向接触刚度模型

基于分形理论,利用双变量Weierstrass-Mandelbrot函数模拟三维分形结合面,建立尺度相关的三维分形结合面法向接触刚度模型。推导出各等级微凸体发生弹性、弹塑性以及完全塑性变形的存在条件。确定结合面上各等级微凸体的面积分布密度函数,推导出法向接触刚度和法向接触载荷的解析表达式。计算结果表明：当结合面上的微凸体只能发生弹性变形,即自身等级小于弹性临界等级的微凸体,该部分微凸体引起的法向接触刚度和对应法向载荷关系呈非线性。当微凸体的等级大于弹性临界等级,在结合面接触过程中,微凸体弹性变形引起的法向接触刚度与对应的法向载荷关系为线性,非弹性变形引起的法向接触刚度与法向载荷关系为非线性。微凸体的等级范围对结合面的刚度影响较大,在相同的法向载荷作用下,高等级微凸体的结合面产生较高的法向接触刚度,即结合面越平整,结合面的法向刚度越高。     

Development of a wear equation for polymer-metal sliding in terms of the fatigue and topography of the sliding surfaces

A wear equation has been derived using the concept of fatigue failure due to asperity interactions in the contact region between sliding bodies. One of the three principal stresses that arise in the contact zone under the effect of a normal as well as a tangential load is of tensile nature. It is this principal stress that has been considered to be responsible for the initiation and propagation of fatigue cracks. It is assumed that the deformation in the contact zone is of elastic nature and that both the contacting surfaces are covered with asperities that have spherical tips. The wear equation involves the asperity height distribution φ(z). The particular distribution for a sliding situation is determined from experimental studies of the topography of sliding surfaces. The wear equation indicates that the wear rate depends upon the fatigue properties of the weaker material, normal load, sliding speed, coefficient of friction, moduli of elasticity of the contacting materials, asperity density, asperity radius of curvature and the distribution and standard deviation of asperity heights. The variation of wear with these parameters as indicated by the wear equation is in agreement with the experimental studies already reported in the literature.     

Analysis of rough line contacts operating under mixed elastohydrodynamic lubrication conditions

Heavily loaded machine elements, such as gears, usually operate in the mixed lubrication regime. Surface roughness has a significant effect on the pressure distribution, the subsurface stress field, and the friction coefficient. Based on the superposition of a dry rough and a fully flooded smooth contact, a mixed lubrication model has been developed. The roughness profile is assumed to be known. Surface deformation is calculated by taking into account the pressure distribution that is built up by asperity contacts, asperity interactions, and lubricant flow. Thermal and sliding effects are incorporated into the analysis. Non‐Newtonian lubricant behaviour is considered by using a power‐law rheological model. The pressure distribution, subsurface stress field, and friction coefficient were calculated from the model at several points along the contact path for an FZG type C gear pair. It was shown that a significant part of the load is carried by the contacting asperities. The position of the maximum shear stress is very close to the surface.     

An upper-bound analysis of friction in metal forming

In the analysis of metal forming processes, a knowledge of friction is important, especially when the microstructure evolution and criteria for limiting phenomena are predicted by numerical simulation. The friction wave model has been studied by several researchers. Their analyses are mainly based on the assumption that there is no plastic deformation of the bulk material. However, it is necessary to clarify the influence of bulk material deformation on the surface asperity deformation. This paper deals with the development of a friction wave model by considering the influence of bulk material on the surface asperity deformation. The situation of rough tool—smooth workpiece (RT—SW) contact during forming process has been investigated. Based on this condition, an admissible velocity field is constructed for the upper bound analysis. The relationship between the normal pressure and the sliding resistance is established over a large range of pressure. The role of surface roughness, bulk displacement and bulk strain on metal forming friction is analysed.     

Contact analysis of three-dimensional rough surfaces under frictionless and frictional contact

A methodology for surface and sub-surface stress calculation of nominally flat on flat rough surface contact has been developed. This methodology is applicable for both large area contact (Hertzian contact) and small area of asperity contact (point load contact) with and without surface friction. A total of nine rough surfaces are generated by the computer with specified standard deviation of surface heights, σ, of 0.3, 1.0 and 3.0 nm, and correlation length, β^*, of 0.1, 0.5 and 0.9 μm. Under the typical applied load at the magnetic head slider-disk interface, small numbers of contact points are obtained and the deformation is purely elastic. Since these contact points are scattered and isolated, asperity contact behaves like point load contact. As β^* becomes larger, more adjacent points will be in contact at a certain contact spot and this is especially true at small σ. All the cases of flat on flat rough surface contact yield maximum von Mises stress on and near the surface at both frictionless and frictional contacts; no local maximum occurs in the sub-surface. In general, the friction effect in the vicinity of contact point is to increase the stress magnitude, while outside this region it also alters the stress distribution. For a surface of small β^* and large σ at high load of 1000 times of the nominal pressure at the head-disk interface, the contact pressure reaches the hardness at a few contact points and plastic deformation takes place in the near surface.     

Elastic–plastic spherical contact under combined normal and tangential loading in full stick

The behavior of an elastic–plastic contact between a deformable sphere and a rigid flat under combined normal and tangential loading with full stick contact condition is investigated theoretically. Sliding inception is treated as a plastic yield failure mechanism, which allows static friction modeling under highly adhesive conditions. Several contact parameters such as: junction tangential stiffness, static friction force and static friction coefficient are extensively investigated. The phenomenon of junction growth and the evolution of the plastic zone in the contact region are briefly described. It is found that at low normal dimensionless loads the static friction coefficient decreases sharply with increasing normal load, in breach with the classical laws of friction. As the normal load further increases the static friction coefficient approaches a constant value that is about 0.3 for many material properties combinations.     

Deformation and wear behaviour of the junction in quasi-scratch friction

The initial scratching of soft metals by relatively hard metallic asperities involves considerable plastic deformation and wear of the harder metal. Thus the penetration effect on friction is reduced successively as sliding proceeds, leading to the shearing type of friction. Such a transition state of sliding can be defined as a quasi-scratch friction process because ploughing precedes the steady sliding condition.The deformation and wear behaviour at a friction junction was investigated using model experiments between a mild steel conical rider and a flat copper surface. Changes in geometry of the rider and pile-up of the flat metal were examined metallographically and with a microscope. It was found that a stable value of the friction force is determined from the geometric shape of the junction attained after the completion of transient sliding and the effect of initial asperity shape on the friction force becomes insignificant.     

A numerical method for the calculation of lubricant pressures in bearings with mixed lubrication

A model for mixed lubrication, assuming that the total normal load applied to the plane of the lubricated surfaces is carried partly by the hydrodynamic action of the lubrication film and partly by asperity contacts and that the total friction force between the lubricated surfaces is partly due to viscous friction and partly to asperity contacts, was used to develop a numerical solution for pressure distribution in a bearing experiencing mixed lubrication. The geometry treated and the pressure distribution obtained were for a simple slider bearing, but the method could easily be extended to other shapes. The model is based on measured roughness of a real surface. Real load carrying capacity and drag can therefore be determined since they are related directly to bearing pressure distribution     

热力耦合下微接触点塑性应变沿深度变化分析

在考虑粗糙实体弹塑性变形、热力耦合、微凸体间相互作用和摩擦热流耦合等影响下,运用有限元法数值模拟具有三维分形特性的粗糙面与刚性平面间滑动摩擦过程,分析了粗糙实体接触凸点塑性变形随深度变化情况。发现：在速度的突变和闪点温度形成时,摩擦接触表层等效塑性应变增大明显;在这一摩擦表层,过不同接触点的纵向剖面塑性应变沿深度分布不同：有的是接触表面塑性变形最大,有的是在接触微凸体表面下某一深度塑性变形最严重,而接触凸点表面的塑性应变稍小些。这与相关文献用SEM研究干摩擦后金属摩擦表层变形照片后发现的结果一致。滑动摩擦过程中,金属粗糙摩擦接触表层塑性变形的不断累积,将会导致材料表层中的夹杂或微观缺陷周围萌生微孔和裂纹源。     

Contact width and compliance between cylinders and rough plates

The contact mechanism between a cylinder and a rough plate is theoretically analysed for mixed, elastic and plastic contacts of asperities. The analysis leads to the result that the contact pressure, the contact width and the compliance between the cylinder and plate differ considerably from those calculated from the Hertz equation and the Lundberg equation when the surface roughness in contact is greater and the normal load is lower. It is also found that the difference between the calculated contact width and the compliance based on mixed asperity contacts and those based on elastic or plastic asperity contacts is small. To confirm the analysed results, the contact width between the cylinder and the rough steel or rough copper plate was measured by means of evaporated carbon and lamp black film coatings on the rough surfaces. The compliance between the surfaces was also measured using differential transformers. Little difference was found between the analysed results and the experimental results.     

The residual stress distribution in a plastically deformed model asperity

When rough metallic surfaces come into contact, plastic deformation may occur locally, even at the lightest loads. This plastic deformation is thought to be an important element in a wide range of contact failure mechanisms, including fatigue and nearly all forms of wear.In this paper a simple model of asperity plastic deformation is presented. The model is based on slip line field theory and is used to calculate residual and full-load stress distributions at fully plastic asperity contacts for normal and moderate tangential loads.Measurements of surface residual stress were carried out using two different techniques on a range of plastic contacts of various materials and geometries. The results show agreement with the main predictions of the theory.     

Dependence of polymer sliding friction on normal load and contact pressure

The sliding friction of bulk polymers was studied varying the normal load, contact pressure and sliding velocity. The variation of the area of apparent contact A with normal load W was also measured both under the sliding and unloaded conditions. For the sliding condition A ∝ W, while for the unloaded condition A ∝ Wⁿ where n is less than unity. The friction measurements were performed on a tribometer in the low load range and on a lathe using a strain gage dynamometer in the high load range. It was found that the coefficient of friction depends upon the velocity and pressure and the variation can be explained by the adhesion theory of friction in the light of the conditions at the interface. The measurement of sliding friction in an extrusion process shows that the coefficient of friction decreases with contact pressure and the interface friction shear stress is almost equal to the bulk shear strength of the material. All of these findings support the adhesion theory of friction for polymeric materials.     

传热界面真实接触面积计算与分析

在热流通过两相互接触材料尤其是金属材料的界面时，真实接触面积是界面传热的一个主要影响因素。当承受大应力的两接触体之间具有相对滑动或相对滑动趋势时，粗糙表面在压力和粘着力及剪切力的作用下接触粗糙峰发生弹性、弹一塑性或完全塑性变形，真实接触面积与压力之间的关系随变形机制而发生变化，在力的作用下材料的变形机制由表面微观几何形貌和力学性质决定。计算表明，单个粗糙峰接触面积与载荷的关系受变形机制的影响，粘着力对接触面积的影响可以忽略，表面相对滑动将增加真实接触面积。