Frictional elastic contact with periodic loading

Quasi-static frictional contact problems for bodies of fairly general profile that can be represented as half planes can be solved using an extension of the methods of Ciavarella and Jäger. Here we consider the tangential traction distributions developed when such systems are subjected to loading that varies periodically in time. It is shown that the system reaches a steady state after the first loading cycle. In this state, part of the contact area (the permanent stick zone) experiences no further slip, whereas other points may experience periods of stick, slip and/or separation. We demonstrate that the extent of the permanent stick zone depends only on the periodic loading cycle and is independent of the initial conditions or of any initial transient loading phase. The exact traction distribution in this zone does depend on these factors, but the resultant of these tractions at any instant in the cycle does not. The tractions and slip velocities at all points outside the permanent stick zone are also independent of initial conditions, confirming an earlier conjecture that the frictional energy dissipation per cycle in such systems depends only on the periodic loading cycle. We also show that these parameters remain unchanged if the loading cycle is changed by a time-independent tangential force, provided this is not so large as to precipitate a period of gross slip (sliding).     

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.     

两自由度振动系统的斜碰撞分析

研究斜碰撞振动系统动力学的一个关键问题是对系统在碰撞前后的状态进行合理描述和正确计算．针对两弹性体斜碰撞问题，基于瞬间碰撞假设，提出了采用步进冲量来分析和求解斜碰撞前后的状态关系；并以弹簧摆和振子组成的两自由度斜碰撞振动系统为例，具体介绍了该算法如何实现．用解析方法讨论了该系统在斜碰撞过程中可能出现的各种力学现象，将冲量步进算法得到的数值解与解析结果进行对比，取得了完全一致的结果．该数值方法能适应多种斜碰撞问题的计算．     

Shakedown of coupled two-dimensional discrete frictional systems

Recent results have established that Melan's theorem can be applied to discrete elastic systems governed by the Coulomb friction law only when the normal contact reactions are uncoupled from the tangential (slip) displacements. For coupled systems, periodic loading scenarios can be devised which lead to either shakedown or cyclic slip depending on the initial condition. Here we explore this issue in the simplest coupled system involving two contact nodes. The evolution of the system ‘memory’ is conveniently represented graphically by tracking the instantaneous condition in slip-displacement space. The frictional inequalities define directional straight line constraints in this space that tend to ‘sweep’ the operating point towards the safe shakedown condition if one exists. However, if the safe shakedown region is defined by a triangle in which two adjacent sides correspond to the extremal positions of the two frictional constraints for the same node, initial conditions can be found leading to cyclic slip. The critical value of a loading parameter at which this occurs can be determined by requiring that three of the four constraint lines intersect in a point. Below this value, shakedown occurs for all initial conditions. Similar concepts can be extended to multi-node systems.     

Experimental investigation of laminar flow of a liquid with a free surface in triangular-shaped channels

Results are given of an experimental study of laminar flow of a liquid in triangular-shaped open channels with tangential frictional stress at the free surface. Experiments were carried out when the liquid flow in inclined triangular-shaped channels had Reynolds numbers R < 10 and the working range of Reynolds numbers of the approach air stream was R = (1.6–3.6)·10⁴. The data are presented in relative coordinates as a dependence of the hydraulic resistance coefficient of the liquid on the tangential frictional stress at the free surface. It is shown that with an increase of the tangential frictional stress the hydraulic resistance coefficient considerably increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 168–170, January–February, 1977.     

On the sliding instabilities at rough surfaces

In this paper, the onset of sliding between two elastic half-spaces in contact, subjected to a tangential force, is studied within the framework of critical phenomena. First, it is shown that the contact domain between two rough surfaces is a lacunar set and that the distribution of contact stresses is multifractal. By applying an increasing tangential force, under constant normal load, the so-called regime of partial-slip comes into play. However, the continuous and smooth transition to full sliding, predicted by the classical Cattaneo-Mindlin theory, is not confirmed by the experiments, which show marked frictional instabilities. A numerical multi-scale procedure is proposed, taking into account the redistribution of stress, consequent to partial-slip, among the contact areas at all scales. It is shown that the lacunarity of the contact domain delays the onset of instability, when compared to compact Euclidean domains. Independently of the assumptions made for the frictional behaviour at the scale of the asperities (Coulomb friction for meso-scale asperities, adhesion for micro-scales), renormalization permits the critical value of the tangential force which provides the instability to be found. Moreover, the multifractal analysis of the domains where the shear resistance is activated captures the size-scale effects on the friction coefficient, currently evidenced by the experiments.     

Winkler partial slip solution for harmonic oscillations in steady rolling contact problems

A Winkler model (Kalker’s simplified theory) is adopted for solving analytically partial slip rolling contact problem in the first order perturbation form of small periodic oscillations of generally both normal and tangential load about a steady state. At present, only numerical investigations exist for this problem, with various approximations to deal with the transient effects (often, simply neglected), and particularly the effect of varying normal load and hence contact area, has not been investigated in detail, despite the problem of corrugation is essentially driven by the change of normal load.The linear perturbation analysis is used to obtain closed form expressions for the receptances of the tangential load. Also, similar expressions are obtained for the energy dissipation, which is correlated with the local wear.     

基于边界元法求解三维弹性摩擦接触问题

边界元法求解三维摩擦接触问题,其中一个关键点在于如何确定滑移方向。即当出现相对滑移时,滑移方向如何确定。当前常采用的方法是,粘结点利用切向面力得到滑移方向,滑移点利用切向相对位移得到滑移方向。不过该方法难以保证收敛性。针对这一问题,本文采用滑移方向预测技术得到滑移方向。即以后出现相对滑移时,滑移方向采用预测技术中得到的滑移方向。由于摩擦接触问题和历史加载相关,本文采用增量法求解。不同摩擦系数下的数值结果都证明了本文算法的有效性和收敛性及滑移方向预测技术的有效性。     

A hysteretic model of localized frictional contacts with instrumental stiffness

The steady-state dynamic response of a single-degree-of-freedom system comprising both a hysteretic element and a spring is considered. The Hertz–Cattaneo–Mindlin theoretical framework for modeling of local tangential contact with friction is applied in conjunction with the Masing model of hysteresis to describe the hysteretic behavior of the multiple localized frictional contact interface. The steady-state tangential displacement amplitude of a rigid body under harmonic tangential force excitation is approximately determined by means of the equivalent linearization technique, based on the harmonic balance principle. A special attention is paid to the evaluation of the frictional damping and the determination of the backbone curve of the Masing model from the dissipation-amplitude relation.

     

Relations between twin and slip in parent lattice due to kinematic compatibility at interfaces

The relationships between a slip system in the parent lattice and its transform by twinning shear are considered in regards to tangential continuity conditions on the plastic distortion rate at twin/parent interface. These conditions are required at coherent interfaces like twin boundaries, which can be assigned zero surface-dislocation content. For two adjacent crystals undergoing single slip, relations between plastic slip rates, slip directions and glide planes are accordingly deduced. The fulfillment of these conditions is investigated in hexagonal lattices at the onset of twinning in a single slip deforming parent crystal. It is found that combinations of slip system and twin variant verifying the tangential continuity of the plastic distortion rate always exist. In all cases, the Burgers vector belongs to the interface. Certain twin modes are only admissible when slip occurs along an 〈a〉 direction of the hexagonal lattice, and some others only with a 〈c + a〉 slip. These predictions are in agreement with EBSD orientation measurements in commercially pure Ti sheets after plane strain compression.     

Instability of thermoelastic contact for two half-planes sliding out-of-plane with contact resistance and frictional heating

Thermoelastic contact is known to show instabilities when the heat transmitted across the interface depends on the pressure, either because of a pressure-dependent thermal contact resistance R(p) or because of frictional heating due to the product of friction coefficient, speed, and pressure, fVp. Recently, the combined effect of pressure-dependent thermal contact resistance and frictional heating has been studied in the context of simple rod models or for a more realistic elastic conducting half-plane sliding against a rigid perfect conductor “wall”. Because R(p) introduces a non-linearity even in full contact, the “critical speed” for the uniform pressure solution to be unstable depends not just on material properties, and geometry, but also on the heat flux and on pressure.Here, the case of two different elastic and conducting half-planes is studied, and frictional heating is shown to produce significant effects on the stability boundaries with respect to the Zhang and Barber (J. Appl. Mech. 57 (1990) 365) corresponding case with no sliding. In particular, frictional heating makes instability possible for a larger range of prescribed temperature drop at the interface including, at sufficiently high speeds, the region of opposite sign of that giving instability in the corresponding static case. The effect of frictional heating is particularly relevant for one material combinations of the Zhang and Barber (J. Appl. Mech. 57 (1990) 365) classification (denominated class b here), as above a certain critical speed, the system is unstable regardless of temperature drop at the interface.Finally, if the system has a prescribed heat flow into one of the materials, the results are similar, except that frictional heating may also become a stabilizing effect, if the resistance function and the material properties satisfy a certain condition.     

L2 Well-Posedness of Planar Div-Curl Systems

Criteria for the existence and uniqueness of solutions of div-curl boundary value problems on bounded planar regions with nice boundaries are developed. The boundary conditions to be treated include prescribed normal component of the field, tangential component of the field and disjoint combinations of these conditions. Under natural assumptions on the data, when either tangential or normal components are given on the whole boundary, weak (finite-energy) solutions exist if and only if a compatibility condition holds. If the region is simply connected this solution is unique. When the region is multiply connected, there is a finite-dimensional family of solutions. The dimension of the solution space is the Betti number of the region. The problem is well-posed with a unique solution when certain line integrals are further prescribed. L ² estimates of the solutions are given. If mixed tangential, and normal, components of the field are specified on different parts of the boundary, no compatibility condition is required for solvability. In general, though, there is considerable non-uniqueness of solutions. Well-posedness is recovered by specifying certain line integrals. L ² estimates of the solutions are given. The dimensionality of the solution space depends on the topology of the boundary data. These results depend on certain weighted orthogonal decompositions of L ² vector fields on the region which are related to classical Hodge-Weyl decomposition results.     

含摩擦与碰撞平面多刚体系统动力学线性互补算法

基于接触力学理论和线性互补问题的算法, 给出了一种含接触、碰撞以及库伦干摩擦, 同时具有理想定常约束(铰链约束) 和非定常约束(驱动约束) 的平面多刚体系统动力学的建模与数值计算方法. 将系统中的每个物体视为刚体, 但考虑物体接触点的局部变形, 将物体间的法向接触力表示成嵌入量与嵌入速度的非线性函数,其切向摩擦力采用库伦干摩擦模型. 利用摩擦余量和接触点的切向加速度等概念, 给出了摩擦定律的互补关系式; 并利用事件驱动法, 将接触点的黏滞-滑移状态切换的判断及黏滞状态下摩擦力的计算问题转化成线性互补问题的求解. 利用第一类拉格朗日方程和鲍姆加藤约束稳定化方法建立了系统的动力学方程, 由此可降低约束的漂移, 并可求解该系统的运动、法向接触力和切向摩擦力, 还可以求解理想铰链约束力和驱动约束力. 最后以一个类似夯机的平面多刚体系统为例, 分析了其动力学特性, 并说明了相关算法的有效性.      

RESEARCH ON MODELING AND NUMERICAL METHOD OF FREE STANDING BODY ON PLANAR RIGID MULTIBODY DYNAMICS

采用非光滑多体系统动力学的方法研究浮放物体与基础平台组成的多体系统,建立其非光滑接触的动力学方程与数值算法.浮放物体由主体部分和支撑腿组成,其间通过含黏弹性阻力偶的转动铰连接.支撑腿与基础平台间的接触力简化为接触点的法向接触力和摩擦力,采用扩展的赫兹接触力模型描述接触点的法向接触力,采用库伦干摩擦模型描述其摩擦力.采用笛卡尔坐标系下的位形坐标作为系统的广义坐标.首先,将基础平台运动看作非定常约束,用第一类拉格朗日方程建立系统的动力学方程,并采用鲍姆加藤约束稳定化的方法解决违约问题.随后给出基于事件驱动法和线性互补方法的数值算法.当相对切向速度为零时,构造静滑动摩擦力的正负余量和正、负向加速度的互补关系,从而将接触点黏滞——滑移切换的判断以及静滑动摩擦力的计算转化为线性互补问题进行求解,并采用Lemke算法求解线性互补问题.最后,通过数值仿真选择合适的步长;通过仿真结果说明浮放物体运动中存在的黏滞-滑移切换现象以及基础平台运动、质心位置对浮放物体运动的影响.     

Viscous stress distribution over a wavy gas–liquid interface

Viscous stress contributes to momentum transfer between two phases, which plays an important role in both industrial applications and environmental processes. Near a wavy interface, the flow is modulated and produces a spatially non-uniform normal and tangential viscous stress. This study presents measurements of these stresses at a liquid–gas interface populated with two-dimensional millimeter scale waves performed with multiphase particle image velocimetry. Large datasets enable conditional phase-averaging of the data based on wave steepness, which increases the precision of the results and allows statistical analysis. For the first time at this scale, the spatial distribution of normal and tangential viscous stress is obtained for a large range of wave steepness (ak = 0–1, with a the amplitude and k the wavenumber). As the steepness increases, the mean shear stress over a wavelength decreases in magnitude, while the normal viscous stress increases. These trends are linear for ak < 0.6, and correlations are proposed. At ak > 0.7, flow separation is observed in the gas phase near the troughs and drastically alters the viscous stress distribution.     

基于共轭梯度法和快速傅立叶变换的粗糙表面微动接触数值研究

实际工程表面多为粗糙表面,研究粗糙表面的表面形貌对微动接触中压力和应力的影响具有重要意义。本文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,算法核心是利用共轭梯度法CGM(Conjugate gradient method)计算微动接触中的表面压力及切向应力并利用快速傅里叶变换FFT(Fast Fourier transform)加快计算速度。然后,对单峰表面、正弦表面和随机粗糙表面的接触进行数值研究。结果表明,表面幅值对切向载荷-位移曲线以及接触过程中的能量耗散有影响,表面幅值越大,相同切向载荷作用下产生的切向位移越大,能量耗散也越大。     

Modélisation des problèmes d'impact avec dissipation d'énergie par frottement

The bi-potential method has been successfully applied for the modeling of frictional contact problems in static cases. This paper presents the application of this method for dynamic analysis of impact problems with multiple deformable bodies. A first order algorithm is applied for numerical integration of the time-discretized equation of motion. The numerical results show clearly the physical energy dissipation introduced by frictional effects between the solids in contact. To cite this article: B. Magnain et al., C. R. Mecanique 333 (2005).     

A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.     

基于Iwan模型的接合面切向响应建模

针对工程中常见预紧力作用下的搭接接头,研究其在小幅切向位移激励时的切向位移响应问题,为此提出一种新的基于实际表面形貌和材料性能参数的滑移力密度分布函数.应用该分布函数得到搭接接头切向响应本构模型,并获得单位加载周期内的迟滞曲线和能量耗散值, 通过与已出版的实验结果相对比,发现得到的模拟值与实验结果吻合, 证明该模型的合理性.在此基础上利用该分布函数研究了接合面切向位移与切向力、切向接触刚度及能量耗散之间的关系,结果表明: 建立的模型能很好地描述接合面间切向力与切向位移之间的关系,临界滑移力函数开始迅速上升, 到达最大值后迅速收敛到零;切线力与切向位移之间表现出非线性特性, 随着切向位移的增大,切向接触刚度表现出"软化"现象;初始切向刚度与法向载荷、粗糙度参数及塑性指数有关, 对于确定的接触表面,法向力越大, 初始切向刚度越大; 初始切向刚度同样也随着塑性指数的增大而增大.