Planar multiple-contact problems subject to unilateral and bilateral kinetic constraints with static Coulomb friction

This paper is concerned with contact problems. A planar multiple-contact problem subject to unilateral and bilateral kinetic constraints with static friction is studied using the complementarity method. First, this paper discusses the one-to-one correspondence of solutions of the contact problems of concern and of the corresponding complementarity models. An enhanced complementarity model is proposed by adding missed tangential acceleration constraints into previous complementarity models. Solutions of the proposed complementarity model and solutions of the contact problem are proven to exhibit one-to-one correspondence, which may not be guaranteed in the previous complementarity models. Then, this paper applies linear complementarity theory to investigate the properties of the solutions of the proposed complementarity model. For both unilaterally constrained contact problems and bilaterally constrained contact problems, the existence of solutions and boundedness of solutions are proven. Sufficient conditions for the uniqueness of solutions and finiteness of the number of solutions are also provided. Several numerical examples are given to show the non-uniqueness of solutions or the infiniteness of the number of solutions. Such phenomena demonstrate the non-smoothness of the contact problems discussed herein.     

Quasistatic Viscoelastic Contact with Normal Compliance and Friction

We prove the existence of a unique weak solution to the quasistatic problem of frictional contact between a deformable body and a rigid foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We establish the continuous dependence of the solution on the normal compliance function. Moreover, we prove the existence of a unique solution to the problem of sliding contact with wear. This revised version was published online in August 2006 with corrections to the Cover Date.     

直流磁场下销盘摩擦接触区的电磁感应现象

为了研究直流磁场对45钢销盘摩擦副的摩擦磨损特性的影响,根据磁场销盘摩擦试验机的结构和销盘摩擦副在摩擦过程中的实际接触情况,建立了二维微凸峰接触静态磁场和瞬态磁场有限元模型,分析了销盘摩擦接触区的电磁感应现象,得出以下结论:磁感应强度B在摩擦接触区分布不均,在微凸峰接触点区域的磁感应强度B值较大;摩擦试验中,在销盘磨痕和磨屑的微凸峰接触区将产生较高频率的动态磁化,同时在微凸峰上产生较大的感应电流,这些现象促进了销盘磨痕表面和磨屑的氧化.     

弹塑性微凸体侧向接触相互作用能耗

传统的结合面研究多基于光滑刚性平面与等效粗糙表面接触假设,忽略了结合面上微凸体侧向接触及相邻微凸体之间的相互作用,这导致理论模型与实际结合面存在较大出入.针对承受法向静、动态力的机械结合面,从微观上研究了微凸体侧向接触及相互作用的接触能耗.将法向静、动态力分解为法向分力和切向分力,获取弹性/弹塑性/塑性阶段考虑微凸体侧接触及相互作用的加、卸载法向分力-变形和切向分力-位移的关系.通过力的合成定理,从而获取加、卸载法向合力与总变形之间的关系,由于法向分力产生的塑性变形及切向分力产生的摩擦,导致加载、卸载法向合力-总变形曲线存在迟滞回线.通过对一个加、卸载周期内的法向合力-总变形曲线积分,获得一个周期的微凸体接触能耗,包括应变能耗及摩擦能耗.仿真分析表明:微凸体在3个阶段的能耗均随变形的增大而非线性增大.微凸体侧向接触角度越大,能耗越大,且在弹性阶段最为明显.在弹性阶段,仅存在侧向的摩擦能耗,故结合面在低载荷作用下必须采用双粗糙表面假设.在塑性阶段,由于微凸体接触能耗为应变能耗,且接触角对其能耗影响甚微,故结合面在大载荷作用下可采用单平面假设对其进行研究.相对于KE和Etsion模型,本文提出的模型与Bartier的实验结果更吻合.     

Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction

Summary This work considers a contact problem with friction involving one contact point and two degrees-of-freedom. The contacting structure is linear elastic. Two different models of contact interaction are considered, the classical Signorini unilateral contact law and a normal compliance law. Coulomb's law of friction is used. All possible so-called rate problems are solved, from which one concludes that the quasistatic problem may possess non-uniqueness and non-existence of solutions. In the case of the normal compliance law this can be explained by a softening structural response. For Signorini's law softening explains only some of the possible situations where non-uniqueness can occur.

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Beispiele der Nichteindeutigkeit und Nichtexistenz der Lösung quasistatischer Kontaktprobleme mit Reibung

Übersicht In dieser Arbeit wird ein Kontaktproblem mit Reibung behandelt, das einen Kontaktpunkt und zwei Freiheitsgrade einschließt. Die kontaktgebende Struktur ist linearelastisch. Zwei verschiedene Modelle der Kontaktwirkung sind berücksichtigt: Erstens das klassische einseitige Signorini-Kontaktgesetz und zweitens ein Gesetz für die Nachgiebigkeit in Normalenrichtung. Das Coulombsche Reibungsgesetz wird verwendet. Alle möglichen sogenannten Geschwindigkeitsprobleme werden gelöst, woraus geschlossen wird, daß das quasistatische Problem Nichteindeutigkeit und Nichtexistenz der Lösung besitzen kann. Im Fall des Nachgiebigkeitsgesetzes kann dieses als abfallende Struktursteifigkeit erklärt werden. Im Fall eines Signorini-Gesetzes erklärt dieses nur einige der möglichen Situationen, wo Nichteindeutigkeit auftreten kann.

     

非局部摩擦在塑性加工中的应用

在几类金属塑性成形加工问题中,为了考虑金属材料表面微凸体在模具与工件之间的接触区上的非局部摩擦效应,采用Oden等提出的非局部摩擦模型,借助主应力法,建立了相应问题的单元体的积微分形式的力平衡方程,在简化的情况下,利用摄动法求得接触面上接触压力在非局部摩擦下的近似解,分析了影响接触压力非局部效应的各种因素。     

考虑粗糙结合面弹塑性接触的黏滑摩擦建模

提出一种同时考虑粗糙面上微凸体弹性变形和塑性接触的切向黏滑摩擦建模方法。采用Hertz弹性理论和Mindlin解描述弹性接触微凸体的切向载荷和相对变形的关系;采用AF(Abbott-Firstone)塑性理论和Fujimoto模型描述塑性接触微凸体切向载荷和相对变形的关系。再利用GW(Greenwood-Williamson)模型统计分析方法建立粗糙表面切向载荷和相对变形之间的关系。将模型与仅考虑微凸体弹性接触情况的模型进行对比,并研究了不同塑性指数对切向载荷和相对变形关系的影响。结果表明：与完全弹性接触模型相比,本文模型引入了塑性接触理论,能够更好地描述粗糙表面切向载荷和相对变形关系,并且考虑不同接触条件下弹性变形微凸体和塑性变形微凸体对切向接触载荷的贡献,在微滑移阶段,主要由弹性接触变形影响,而在进入宏观滑移阶段之后,切向行为主要由塑性变形影响。界面切向载荷由黏着和滑移接触作用共同决定,随着切向变形的增加,滑移接触力逐渐增加,而黏着接触力先增加后减少,反映了界面由微滑移逐渐向宏滑移演化的过程。随着塑性指数的增加,粗糙面上发生塑性接触的微凸体数目逐渐增加,切向黏滑行为主要受到塑性接触特征的控制。     

Transition from stick to slip in Hertzian contact with “Griffith” friction: The Cattaneo–Mindlin problem revisited

Classically, the transition from stick to slip is modelled with Amonton–Coulomb law, leading to the Cattaneo–Mindlin problem, which is amenable to quite general solutions using the idea of superposing normal contact pressure distributions – in particular superposing the full sliding component of shear with a corrective distribution in the stick region. However, faults model in geophysics and recent high-speed measurements of the real contact area and the strain fields in dry (nominally flat) rough interfaces at macroscopic but laboratory scale, all suggest that the transition from ‘static’ to ‘dynamic’ friction can be described, rather than by Coulomb law, by classical fracture mechanics singular solutions of shear cracks. Here, we introduce an ‘adhesive’ model for friction in a Hertzian spherical contact, maintaining the Hertzian solution for the normal pressures, but where the inception of slip is given by a Griffith condition. In the slip region, the standard Coulomb law continues to hold. This leads to a very simple solution for the Cattaneo–Mindlin problem, in which the “corrective” solution in the stick area is in fact similar to the mode II equivalent of a JKR singular solution for adhesive contact. The model departs from the standard Cattaneo–Mindlin solution, showing an increased size of the stick zone relative to the contact area, and a sudden transition to slip when the stick region reaches a critical size (the equivalent of the pull-off contact size of the JKR solution). The apparent static friction coefficient before sliding can be much higher than the sliding friction coefficient and, for a given friction fracture “energy”, the process results in size and normal load dependence of the apparent static friction coefficient. Some qualitative agreement with Fineberg's group experiments for friction exists, namely the stick–slip boundary quasi-static prediction may correspond to the arrest of their slip “precursors”, and the rapid collapse to global sliding when the precursors arrest front has reached about half the interface may correspond to the reach of the “critical” size for the stick zone.     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.     

Homogenization in non-linear dynamics due to frictional contact

This work is devoted to a study of the classical homogenization process and its influence on the behavior of a composite under non-linear dynamic loading due to contact and friction. First, the general problem of convergence of numerical models subjected to dynamic contact with friction loading is addressed. The use of a regularized friction law allows obtaining good convergence of such models. This study shows that for a dynamic contact with friction loading, the classical homogenization process, coupled with an homogenization of the frictional contact, enables replacing the entire heterogeneous model by a homogenized one. The dynamic part of the frictional contact must be homogenized by modifying the dynamic parameter of the friction law. Modification of the dynamic parameter of the friction law is function of the type and regime of instability. A calculation of a homogenized friction coefficient is presented in view to homogenizing the static part of the frictional contact when the friction coefficient is not constant over the contact surface. Finally matrix and heterogeneities stresses in the heterogeneous models are identified by using the relocalization process and a frictional contact dynamic analysis of a homogeneous model.     

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.     

Elastic contact between a cylindrical surface and a flat surface - A non-Hertzian model of multi-asperity contact

Contact between curved rough bodies is an important engineering problem. The paper addresses the problem in its simplest form where a smooth rigid cylinder presses down an elastic half space bounded by a plane of uniformly spaced cylindrical asperities. Keeping the separation between the bodies unchanged the problem is inverted and solved using the method of complex variables. As the asperities deform as well as move as rigid bodies, contact lengths and positions develop non-symmetrically with respect to the initial axes of symmetry of the asperities. The resulting local contact pressures are non-Hertzian and the normal load for a given contact area is greater than that estimated using a priori Hertzian pressure profiles.     

Constitutive equation for friction with transition from static to kinetic friction and recovery of static friction

A high friction coefficient is first observed as a sliding between bodies commences, which is called the static friction. Then, the friction coefficient decreases approaching the lowest stationary value, which is called the kinetic friction. Thereafter, if the sliding stops for a while and then it starts again, the friction coefficient recovers and a similar behavior as that in the first sliding is reproduced. In this article the subloading-friction model with a smooth elastic–plastic sliding transition [Hashiguchi, K., Ozaki, S., Okayasu, T., 2005. Unconventional friction theory based on the subloading surface concept. Int. J. Solids Struct. 42, 1705–1727] is extended so as to describe the reduction from the static to kinetic friction and the recovery of the static friction. The reduction is formulated as the plastic softening due to the separations of the adhesions of surface asperities induced by the sliding and the recovery is formulated as the viscoplastic (creep) hardening due to the reconstructions of the adhesions of surface asperities during the elapse of time under a quite high actual contact pressure between edges of asperities.     

Solvability of Static Contact Problems with Coulomb Friction for Orthotropic Material

We calculate an exact upper bound for the magnitude of the coefficient of friction that ensures the existence of a solution to a static contact problem with Coulomb friction. The approach is based on a general existence result that is valid under the assumption that the coefficient of friction is bounded by a certain constant depending on the constants in two special trace type estimates for a half space domain. We calculate these constants for orthotropic material and two space dimensions with the help of a representation for a partial Fourier transform of the solution to the corresponding system of elasticity equations. The result is compared to the formula for general anisotropic material. The new bound for orthotropic material is significantly larger than the old one for general material, if the material is close to an isotropic material with Poisson ration greater than zero. For some cases the new bound can be even larger than one.      

APPLICATION OF NONLOCAL FRICTION IN SEVERAL KINDS OF PLASTIC FORMING PROBLEMS

Introduction Frictionaffectsformingforce,energy,toollifeandtheproductquality(surfacefinish,internalstructure,productlife,etc)inplasticformingprocesses.Thefrictionphenomenon iscomplexformanyparametersinteractwitheachanotherandaffecttheformingprocess.Fricti…     

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.     

固定接触界面法向静弹性刚度

基于Hertz接触理论推导了两个微凸体之间互相作用的法向接触静弹性刚度.根据修正后的一个微接触点的平截面积尺寸分布,给出了界面的总法向接触静弹性条件刚度、总条件载荷的解析解.将法向静弹性刚度的解析解嵌入到有限元软件中,获得整机的理论模态.通过实验对解析解进行了定量验证.以机床结合部为研究对象,在理论振型与实验振型一致的...     

Contact with a Corner for Nonlinearly Elastic Rods

In elastic contact problems it is usually required that the contact force has to be directed normally to the contact surface in the absence of friction. For an obstacle with nonsmooth surface this gives infinitely many normal directions at an edge or at a corner. For the case where a nonlinearly elastic rod under terminal loads is hanging over a needle, it is shown that the balance equations supplemented with such a normality condition have a continuum of solutions. Moreover, an additional contact condition is derived from a corresponding variational problem by means of special inner variations that preserve the shape of the rod. This way one is finally lead to a unique solution at least locally.     

粘着摩擦系数的分形几何研究

计及作用于接触斑点上的切向力，通过比较作用于接触斑点上的法向弹性载荷与法向塑性载荷，确定了区分弹性接触与塑性接触区域的临界接触斑点面积．总的粘着摩擦系数被表示为弹性接触区与塑性接触区的粘着摩擦系数的组合．假设屈服压力及局部粘着摩擦系数不依赖于接触斑点且等于塑性接触区中的平均值，则总的粘着摩擦系数可用简单的表达式描述．分形几何参数及归一接触面积对于粘着摩擦系数的效应已通过算例表明，研究中，分别考虑了忽略与计及接触斑点的微粒间的相互作用，两种情况的结果完全不同．