Shakedown limits for an oscillating,elastic rolling contact with Coulomb friction

We examine experimentally and theoretically the effect of frictional shakedown of a three-dimensional elastic rolling contact. Small oscillations of the local normal forces lead to incremental sliding processes within the area of contact. Consequently, this causes a macroscopic slip motion of the two contacting bodies. If the oscillation amplitude is sufficiently small, the frictional slip ceases after the first few loading periods and a safe shakedown occurs. Otherwise the slip motion is continued and the contact fails.     

Optimal shapes of contact interfaces due to sliding wear in the steady relative motion

The transient wear process at contact frictional interface of two elastic bodies in relative steady motion induces evolution of shape of the interface. A steady wear state may be reached with uniform wear rate and fixed contact surface shape. In this paper, the optimal contact shape is studied by formulating several classes of shape optimization problems, namely minimization of generalized wear volume rate, friction dissipation power and wear dissipation rate occurring in two bodies. The wear rule was assumed as a nonlinear dependence of wear rate on friction traction and relative sliding velocity, similar to the Archard rule. The wear parameters of two bodies may be different. It was demonstrated that different optimal contact shapes are generated depending on objective functional and wear parameters. When the uniform wear rate is generated at contact sliding surfaces, the steady state is reached. It was shown that in the steady state the wear parameters of two bodies cannot be independent of each other. The solution of nonlinear programming problem was provided by the iterative numerical procedure. It was assumed that the relative sliding velocity between contacting bodies results from translation and rotation of two bodies. In general, both regular and singular regimes of wear rate and pressure distribution may occur. The illustrative examples of drum brake, translating punch and rotating annular punch (disc brake) provide the distribution of contact pressure and wear rate for regular and singular cases associated with the optimality conditions. It is shown that minimization of the generalized wear dissipation rate provides solutions assuring existence of steady wear states.     

An Eulerian approach for simulating frictional heating in disc-pad systems

Thermal stresses as a result from frictional heating must be considered when designing disc brakes, clutches or other rotating machine components with sliding contact conditions. The rotational symmetry of the disc in these kind of applications makes it possible to model these systems using an Eulerian approach instead of a Lagrangian framework. In this paper such an approach is developed and implemented. The disc is formulated in an Eulerian frame where the convective terms are defined by the angular velocity. By utilizing the Eulerian framework, a node-to-node formulation of the contact interface is obtained, producing most accurate frictional heat power solutions. The energy balance of the interface is postulated by introducing an interfacial temperature. Both frictional power and contact conductances are included in this energy balance. The contact problem is solved by a non-smooth Newton method. By adopting the augmented Lagrangian approach, this is done by rewriting Signorini’s contact conditions to an equivalent semi-smooth equation. The heat transfer in the disc is discretized by a Petrov–Galerkin approach, i.e. the numerical difficulties due to the non-symmetric convective matrix appearing in a pure Galerkin discretization is treated by following the streamline-upwind approach. In such manner a stabilization is obtained by adding artificial conduction along the streamlines. For each time step the thermo-elastic contact problem is first solved for the temperature field from the previous time step. Then, the heat transfer problem is solved for the corresponding frictional power. In such manner a temperature history is obtained sequentially via the trapezoidal rule. In particular the parameter is set such that both the Crank–Nicolson and the Galerkin methods are utilized. The method seems very promising. This is demonstrated by solving a two-dimensional benchmark as well as a real disc brake system in three dimensions.     

Non-local and numerical formulations for dry sliding friction and wear at high velocities

Severe contact stress problems generate high temperature and create thermomechanical gouging and wear due to high velocity sliding between two materials staying in contact. In order to improve the facilitation of the design of particular components and improve performance of these engineering applications, it is necessary to better understand the physical behavior of high speed environment. As presented here this environment is made up of two components in contact. Therefore, basing on the experimental approach ( [Lodygowski, 2010] and [Lodygowski et al., submitted for publication]) the major consideration of this paper is aimed to develop an experimental/theoretical model for the material constitutive behavior in order to better characterize and predict the internal failure surrounding the gouging and wear events.This research is to be carried out in two stages. First, by investigating the phenomenon of wear and later it will be extended to incorporate gauging problems. The principle of virtual power is used by introducing the contributions from damage and its corresponding gradients as a measure of micro motion of damage within the bulk. In addition two internal state variables are introduced on the frictional contact interface, one measuring the tangential slip and another measuring the wear. By using these internal state variables together with displacement and temperature, the constitutive model is formulated with state laws based on the free energies and the complimentary laws based on the dissipation potentials. The proposed theoretical model is implemented as user defined subroutine VUMAT in the explicit finite element code ABAQUS to analyze the structural response of the ultra high speed sliding experiment between Steel and VascoMax steel at Ecole de’Nationale Institut der Mechanic, at Metz France.This model provides a potential feature for enabling one to relate the non-local continuum plasticity and damage of the bulk material to friction and wear at the contact interfaces. The findings of this research effort is invaluable in providing a multiscale material model and numerical procedure that will be used within a hydrocode to better facilitate the design components of the severe contact stress applications.     

The frictional sliding response of elasto-plastic materials in contact with a conical indenter

Over the past decade, many computational studies have explored the mechanics of normal indentation. Quantitative relationships have been well established between the load–displacement hysteresis response and material properties. By contrast, very few studies have investigated broad quantitative aspects of the effects of material properties, especially plastic deformation characteristics, on the frictional sliding response of metals and alloys. The response to instrumented, depth-sensing frictional sliding, hereafter referred to as a scratch test, could potentially be used for material characterization. In addition, it could reproduce a basic tribological event, such as asperity contact and deformation, at different length scales for the multi-scale modeling of wear processes. For these reasons, a comprehensive study was undertaken to investigate the effect of elasto-plastic properties, such as flow strength and strain hardening, on the response to steady-state frictional sliding. Dimensional analysis was used to define scaling variables and universal functions. The dependence of these functions on material properties was assessed through a detailed parametric study using the finite element method. The strain hardening exponent was found to have a greater influence on the scratch hardness and the pile-up height during frictional sliding than observed in frictionless normal indentation. When normalized by the penetration depth, the pile-up height can be up to three times larger in frictional sliding than in normal indentation. Furthermore, in contrast to normal indentation, sink-in is not observed during frictional sliding over the wide range of material properties examined. Finally, friction between indenter and indented material was introduced in the finite element model, and quantitative relationships were also established for the limited effects of plastic strain hardening and yield strength on the overall friction coefficient. Aspects of the predictions of computational simulations were compared with experiments on carefully selected metallic systems in which the plastic properties were systematically controlled. The level of accuracy of the predicted frictional response is also assessed by recourse to the finite element method and by comparison with experiment.     

A slip-line plasticity analysis of sliding friction of rough surfaces exhibiting self-affine (fractal) behavior

A plasticity analysis of sliding friction of rough (fractal) surfaces sliding against smooth surfaces was developed based on a slip-line model of a rigid spherical asperity (wear particle) plowing and cutting through a soft semi-infinite medium. Solutions of the fraction of fully plastic asperity microcontacts responsible for the evolution of friction and energy dissipation were obtained in terms of the total normal load (global interference), interfacial adhesion characteristics, topography (fractal) parameters of the hard surface, and elastic–plastic material properties of the soft surface. This was accomplished by incorporating the slip-line model of a single microcontact into a friction analysis of sliding surfaces demonstrating multi-scale roughness. Numerical results provide insight into the effects of global interference (normal load), fractal parameters (surface roughness) of the hard surface, interfacial shear strength (adhesion), and material properties of the soft surface on plastic deformation at the microcontact level, global coefficient of friction, and frictional energy dissipated during sliding.     

Homogenization in finite thermoelasticity

A homogenization framework is developed for the finite thermoelasticity analysis of heterogeneous media. The approach is based on the appropriate identifications of the macroscopic density, internal energy, entropy and thermal dissipation. Thermodynamical consistency that ensures standard thermoelasticity relationships among various macroscopic quantities is enforced through the explicit enforcement of the macroscopic temperature for all evaluations of temperature dependent microscale functionals. This enforcement induces a theoretical split of the accompanying micromechanical boundary value problem into two phases where a mechanical phase imposes the macroscopic deformation and temperature on a test sample while a subsequent purely thermal phase on the resulting deformed configuration imposes the macroscopic temperature gradient. In addition to consistently recovering standard scale transition criteria within this framework, a supplementary dissipation criterion is proposed based on alternative identifications for the macroscopic temperature gradient and heat flux. In order to complete the macroscale implementation of the overall homogenization methodology, methods of determining the constitutive tangents associated with the primary macroscopic variables are discussed. Aspects of the developed framework are demonstrated by numerical investigations on model microstructures.     

Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: Role of the homogenization scheme

This paper deals with micromechanical analysis of anisotropic damage and its coupling with friction in quasi brittle materials. The anisotropic model is formulated in the framework of Eshelby-based homogenization methods. The emphasis is put on the study of effects of spatial distribution of microcracks and their interactions. Microcracks closure effects as well as coupling between damage evolution and frictional sliding on closed cracks lips are taken into account. The interaction of sliding and damage evolution is addressed by performing a global thermodynamic analysis on two macroscopic criteria established in the paper. The role of the homogenization scheme is discussed in detail through various applications.     

磁头/磁盘滑动接触下磁盘温度及热退磁临界条件的研究

采用二维轴对称有限元模型计算磁头/磁盘滑动接触下,铝质磁盘的稳态温度和热应力场以及热退磁临界条件.结果表明:磁盘温度在极短时间内升至摩擦稳态值,然后缓慢线性升高到最终稳态值;经过充分热传导和热交换后磁盘的温度梯度较小,此时磁层内的热应力集中分布于磁盘固定端边缘附近;磁盘的稳态温度和热应力均随速度增大而增大,且载荷越大其值增大越快;热应力小于1.2 GPa时所对应的速度和载荷为安全工况;温升大于373 K时所对应的工况将导致磁盘退磁.     

Frictional energy dissipation in contact of nominally flat rough surfaces under harmonically varying loads

If the nominal contact tractions at an interface are everywhere below the Coulomb friction limit throughout a cycle of oscillatory loading, the introduction of surface roughness will generally cause local microslip between the contacting asperities and hence some frictional dissipation. This dissipation is important both as a source of structural damping and as an indicator of potential fretting damage. Here we use a strategy based on the Ciavarella-Jäger superposition and a recent solution of the general problem of the contact of two half spaces under oscillatory loading to derive expressions for the dissipation per cycle which depend only on the normal incremental stiffness of the contact, the external forces and the local coefficient of friction. The results show that the dissipation depends significantly on the relative phase between the oscillations in normal and tangential load—a factor which has been largely ignored in previous investigations. In particular, for given load amplitudes, the dissipation is significantly larger when the loads are out of phase. We also establish that for small amplitudes the dissipation varies with the cube of the load amplitude and is linearly proportional to the second derivative of the elastic compliance function for all contact geometries, including those involving surface roughness. It follows that experimental observations of less than cubic dependence on load amplitude cannot be explained by reference to roughness alone, or by any other geometric effect in the contact of half spaces.     

不同表面粗糙度的摩擦副润滑状态的 Stribeck曲线研究

利用Stribeck润滑曲线理论分析和薄膜润滑试验考察了不同表面粗糙度的钢球与圆盘点接触及钢滑块与圆盘面接触摩擦副的润滑状态，通过改变圆盘转速获得了包含薄膜润滑状态的Stribeck曲线．结果表明：在Stribeck曲线上可以划分出薄膜润滑状态，其位于摩擦系数谷底附近；薄膜润滑的产生及其区间大小同圆盘表面粗糙度密切相关；圆盘表面粗糙度较小时更易形成薄膜润滑状态，而圆盘表面粗糙度较大时薄膜润滑状态不明显；光滑表面对应的Stribeck曲线谷底较宽，相应的薄膜润滑区间亦较宽．     

Évolution des propriétés élastiques en poroplasticité finie

The formulation of poroplasticity at large strains requires to clarify the influence of plasticity on the poroelastic properties. This effect is modelled within the framework of a micro–macro approach which takes into account the geometry changes of the microstructure. It is shown that the coupling between elasticity and plasticity introduces an additional term in the rate-type form of the macroscopic state equation which yields a modified form of the Jaumann rate.     

Analytical and numerical analysis of frictional damage in quasi brittle materials

Frictional sliding and crack growth are two main dissipation processes in quasi brittle materials. The frictional sliding along closed cracks is the origin of macroscopic plastic deformation while the crack growth induces a material damage. The main difficulty of modeling is to consider the inherent coupling between these two processes. Various models and associated numerical algorithms have been proposed. But there are so far no analytical solutions even for simple loading paths for the validation of such algorithms. In this paper, we first present a micro-mechanical model taking into account the damage-friction coupling for a large class of quasi brittle materials. The model is formulated by combining a linear homogenization procedure with the Mori–Tanaka scheme and the irreversible thermodynamics framework. As an original contribution, a series of analytical solutions of stress–strain relations are developed for various loading paths. Based on the micro-mechanical model, two numerical integration algorithms are exploited. The first one involves a coupled friction/damage correction scheme, which is consistent with the coupling nature of the constitutive model. The second one contains a friction/damage decoupling scheme with two consecutive steps: the friction correction followed by the damage correction. With the analytical solutions as reference results, the two algorithms are assessed through a series of numerical tests. It is found that the decoupling correction scheme is efficient to guarantee a systematic numerical convergence.     

Elastoplastic formulation for friction with orthotropic anisotropy and rotational hardening

In various fields of engineering, it is important to clarify the frictional sliding behavior over a wide scale. In this study, we formulate an anisotropic friction model with the orthotropy and rotation of a sliding surface based on the elastoplastic theory. This model can also describe preliminary microscopic sliding and rate-dependent frictional response. Further, basic experimental results of anisotropic frictional sliding can be pertinently represented by the present model. We also employ this model with the finite element method and analyze typical frictional contact problems. We then demonstrate the effect of anisotropy parameters on the numerical results.     

A Frictional Heat Model of Planetary Roller Screw Mechanism Considering Load Distribution

Planetary roller screw (PRS), with higher thrust, higher load capacity, and higher speed, is the best choice of the transmission component of the servo system. However, spinning sliding of rollers and support bearings can cause frictional moments and frictional heat, which is an undesirable phenomenon. Besides, frictional heat will further result in high temperature that causes deterioration of lubrication and eventually lead to destruction of the mechanism. Therefore, it is important to predict frictional moments which result in frictional heat. In order to predict the magnitude of frictional heat of PRS mechanism and study the influence of structural parameters of thread and operating conditions on frictional heat, first, a frictional moment model of bearings is built, and frictional moments models of PRS considering the elastic hysteresis of material, the spinning sliding of the rollers, the viscosity of lubricating oil and the differential sliding of thread raceways are established in this paper, respectively. Second, heat generation models of bearing and PRS are presented, respectively. Finally, relationships between frictional heat in terms of operating conditions of PRS, contact angle, and helix angle of roller thread are investigated. The achievements of this project will provide theoretical basis for the design of PRS with lower frictional moments and higher transmission efficiency.     

Simulated configurational temperature of particles and a model of constitutive relations of rapid-intermediate-dense granular flow based on generalized granular temperature

In gas–solid flat-base spout bed with a jet, the flow of particles must go through an intermediate regime where both kinetic/collisional and frictional contributions play a role. In this paper, the statistical framework is proposed to define the generalized granular temperature which sums up the configurational temperature and translational granular temperature. The configurational temperature, translational and rotational granular temperatures of particles are simulated by means of CFD-DEM (discrete element method) in a 3D flat-base spout bed with a jet. The configurational temperatures of particles are calculated from instantaneous overlaps of particles. The translational and rotational granular temperatures of particles are calculated from instantaneous translational and angular velocities of particles. Roughly, the simulated translational and rotational granular temperatures increase, reach maximum, and then decrease with the increase of solids volume fractions. However, the configurational temperature increases with the increase of solids volume fractions. At high solid volume fraction, the predicted configurational temperatures are larger than the translational and rotational granular temperatures, indicating that the rate of energy dissipation do contributes by contact deformation of elastic particles. The generalized granular temperature is proposed to show the relation between the variance of the fluctuation velocity of deformation and the variance of the translational fluctuation velocity of particles. The constitutive relations of particle pressure, viscosity, granular conductivity of fluctuating energy and energy dissipation in rapid-intermediate-dense granular flows are correlated to the generalized granular temperature. The variations of particle pressure, shear viscosity, energy dissipation and granular conductivity are analyzed on the basis of generalized granular temperature in a flat-base spout bed with a jet. The axial velocities of particles predicted by a gas–solid two-fluid model of rapid-intermediate-dense granular flows agree with experimental results in a spout bed.     

FRICTIONAL TEMPERATURE ANALYSIS OF TWO-DIMENSIONAL ELASTO-PLASTIC WHEEL-RAIL SLIDING CONTACT WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES 1)

轮轨滑动接触温升的准确预测对于轮轨的磨耗和疲劳研究均具有重要意义. 目前的轮轨温升解析或半解析模型通常考虑Hertz弹性接触压力分布和单一材料属性的温度相关性, 与实际的轮轨传热状态尚有一定偏差, 因此在轮轨滑动温升计算模型中考虑接触压力的塑性修正和多种材料属性的温度相关性, 有望提高温升预测结果的准确性. 基于弹塑性接触理论, 同时考虑热导率、比热容和摩擦系数的温度相关性, 通过基尔霍夫变换方法以热导率温度相关性函数的积分作为待求量, 将复杂的非线性Fourier导热方程转化成含单个变系数的简单偏微分方程形式, 从而构建了一种不限制材料温度相关性函数形式的统一隐式差分求解格式, 分别讨论了对流换热系数、法向载荷、蠕滑率以及行车速度对钢轨表面滑动温升的影响. 结果表明, 当列车高速行驶时, 对流换热系数对轮轨滑动温升的影响甚微; 蠕滑率和行车速度的增大, 均会引起摩擦功率的增大, 进而导致钢轨表面温度的升高; 钢轨表面滑动温升的峰值随法向载荷的增大而近似线性上升. 此外, 在轮轨滑动温升计算模型中考虑材料属性的温度相关性可有效避免对滑动温升的过分高估, 且摩擦系数的温度相关性对温升的影响要显著强于热导率和比热容.     

Stochastic semi-Lagrangian micro–macro calculations of liquid crystalline solutions in complex flows

A general method for the simulation of complex flows of liquid crystalline polymers (LCPs) using a stochastic semi-Lagrangian micro–macro method is introduced. The macroscopic part uses a spatial-temporal second order accurate semi-Lagrangian algorithm, where ideas from the finite element and natural element methods are mixed in order to compute average quantities. The microscopic part employs a stochastic interpretation of the Doi–Hess LCP model, which is discretized with a second order Richardson extrapolated Euler–Maruyama scheme.The new method is validated and tested using the benchmark problem of flow between rotating eccentric cylinders. In a decoupled analysis, a discussion on the sensibility of the scalar order parameter to the macroscopic flow is offered. For the coupled situation, the proposed method predicts disclinations at certain regions of the geometry, as well as an accentuated abatement of the flow as the strength of the micro–macro interaction increases. Further examples are provided at different Peclet and concentration numbers to gain insight on the behavior of complex flows of LCPs in the eccentric cylinder geometry.The generality and robustness of the method, as well as its accurate prediction of LCP behavior under complex flows are main features of the implementation.