Steady and transient sliding under rate-and-state friction

The physics of dry friction is often modelled by assuming that static and kinetic frictional forces can be represented by a pair of coefficients usually referred to as μ_s and μ_k, respectively. In this paper we re-examine this discontinuous dichotomy and relate it quantitatively to the more general, and smooth, framework of rate-and-state friction. This is important because it enables us to link the ideas behind the widely used static and dynamic coefficients to the more complex concepts that lie behind the rate-and-state framework. Further, we introduce a generic framework for rate-and-state friction that unifies different approaches found in the literature.We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μ_s should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law.Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μ_s over a persistence length before dropping to the value μ_k. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μ_s in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws.This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μ_s is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.     

Non-linear dynamics of a mechanical system with a frictional unilateral constraint

We analyze the dynamics of a two-dimensional system constituted by two masses subjected to elastic, gravitational and viscous forces and constrained by a moving frictional mono-lateral surface. The model exhibits a time-varying dynamics capable of reproducing the hopping phenomenon, an unwanted phenomenon observed in many applications such as the motion of a robotic arm on a surface or that of a wiper on a windscreen. The system dynamics, besides being affected by geometrical non-linearities, has a non-smooth nature due to the impact and friction laws involved in the model. The complexity of the resulting equations and of the transition conditions require the problem to be solved numerically. Various periodic motions are found and the effect of varying the system parameters, in particular the friction coefficient, is investigated. Finally, simulations are used to gain some insight the behavior of the windscreen wiper.     

Sliding modes of two interacting frictional interfaces

In the context of rate-and-state friction, we report an extensive analysis of stability of the quasi-static frictional sliding of two parallel interfaces dividing a linear elastic solid sheared at a constant rate. One possibility for the frictional sliding is that the interfaces slip at equal rates, a steady state described as symmetric. However a steady-state friction law that is non-monotonic allows the competing possibility of an asymmetric steady state in which the interfaces slide at different rates. A rate-and-state law that delivers such behaviour and agrees with the experimental results of Heslot et al. [1994. Creep, stick-slip, and dry-friction dynamics: experiments and a heuristic model. Phys. Rev. E 49, 4973-4988] is proposed. Analytical results combined with numerical investigations performed with the continuation package Auto and direct time integration are used to compile the complete picture of the many bifurcations that exist between the diverse steady and oscillatory sliding modes. In addition to the control parameters corresponding to the driving velocity and the stiffness of the medium, we find that the geometrical details of the steady-state friction law determine the occurrence and nature of bifurcations. Pitchfork bifurcations from the symmetric to asymmetric steady states coincide with the extrema of the friction law; Hopf bifurcations occur in the velocity weakening regime of the friction law. Torus and period-doubling bifurcations of periodic orbits also occur, and lead to complicated dynamics. We also present results of numerical computations that illustrate the complex and versatile dynamics of the two-interface system. We anticipate that the dynamics found in our model should be verifiable by experiments.     

Stabilizing effect of in-plane angular misalignment in 2DOF sliding system with in-plane anisotropic stiffness

Recently, it has been theoretically shown that in a 1DOF sliding system, the in-plane angular misalignment (referred to as the yaw angle misalignment (YAM)) has a stabilizing effect to suppress the self-excited vibration induced by the velocity-weakening friction. The YAM theory has been supported qualitatively and quantitatively by some experiments and numerical simulations. However, in some other experiments with another type of apparatuses, the suppression condition was qualitatively different from the theoretical prediction. Based on the above, in this study, the YAM theory has been extended to a 2DOF sliding system with in-plane anisotropic stiffness. Numerical simulation and eigenvalue analysis revealed that the YAM around 45° had a damping effect to suppress the self-excited vibration induced by the velocity-weakening friction, with no supplementary mechanical devices (such as dampers or actuators) to suppress the vibration, which was consistent with the previous experimental results.     

On oscillations of a frictional pendulum

Conditions are established under which a standard limit cycle occurs in the system under consideration, or the trajectory closes under the influence of a stagnation domain. It is pointed out that when the solution falls into the stagnation domain it makes no sense to use the asymptotic method because of a large error __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 104–112, February 2006.     

Propagation Regimes of Self-Supported Light-Detonation Waves

The light-detonation wave structure is investigated. It is shown that self-supported laser radiation absorption waves can propagate in the Jouguet detonation or undercompressed detonation regimes. The conditions of realization of these regimes are found numerically. It is shown that the undercompressed detonation regime is realized if the radiation flux is sufficiently powerful. In the case of a light-detonation wave this regime is theoretically detected and investigated for the first time.     

Properties of dynamic rupture and energy partition in a solid with a frictional interface

We study properties of dynamic ruptures and the partition of energy between radiation and dissipative mechanisms using two-dimensional in-plane calculations with the finite element method. The model consists of two identical isotropic elastic media separated by an interface governed by rate- and state-dependent friction. Rupture is initiated by gradually overstressing a localized nucleation zone. Different values of parameters controlling the velocity dependence of friction, the strength excess parameter and the length of the nucleation zone, lead to the following four rupture modes: supershear crack-like rupture, subshear crack-like rupture, subshear single pulse and supershear train of pulses. High initial shear stress and weak velocity dependence of friction favor crack-like ruptures, while the opposite conditions favor the pulse mode. The rupture mode can switch from a subshear single pulse to a supershear train of pulses when the width of the nucleation zone increases. The elastic strain energy released over the same propagation distance by the different rupture modes has the following order: supershear crack, subshear crack, supershear train of pulses and subshear single pulse. The same order applies also to the ratio of kinetic energy (radiation) to total change of elastic energy for the different rupture modes. Decreasing the dynamic coefficient of friction increases the fraction of stored energy that is converted to kinetic energy. General considerations and observations suggest that the subshear pulse and supershear crack are, respectively, the most and least common modes of earthquake ruptures.     

The equilibrium state of a structure subject to frictional contact

A structure in frictional contact subject to static loads has not, in general, a unique static equilibrium state. This is because the state, displacements and contact forces, depend on the load history of the structure.In cases where the exact load history is not known it would be of interest to find a state that is in some sense likely and define this as the equilibrium state. In this paper, it is assumed that the state with the smallest potential energy is the most likely one. The implication of this definition of likely state is analysed and shows that the resulting problem basically can be seen as a generalization of the frictionless contact problem to structures where no frictionless state is possible, i.e. structures where non-zero friction forces are necessary to satisfy force equilibrium.The results of several numerical experiments are given. The structures in the experiments are trusses and structures modelled by the finite element method. Both a sequential quadratic programming method and an enumeration method are used to solve the likely-state problem.     

Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation

The problem of thermoelastic contact mechanics for the coating/substrate system with functionally graded properties is investigated, where the rigid flat punch is assumed to slide over the surface of the coating involving frictional heat generation. With the coefficient of friction being constant, the inertia effects are neglected and the solution is obtained within the framework of steady-state plane thermoelasticity. The graded material exists as a nonhomogeneous interlayer between dissimilar, homogeneous phases of the coating/substrate system or as a nonhomogeneous coating deposited on the substrate. The material nonhomogeneity is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The Fourier integral transform method is employed and the formulation of the current thermoelastic contact problem is reduced to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure. Numerical results include the distributions of the contact pressure and the in-plane component of the surface stress under the prescribed thermoelastic environment for various combinations of geometric, loading, and material parameters of the coated medium. Moreover, in order to quantify and characterize the singular behavior of contact pressure distributions at the edges of the flat punch, the stress intensity factors are defined and evaluated in terms of the solution to the governing integral equation.     

On partial sliding along a planar crack: the case of a circular sliding zone

Summary A problem of partial sliding along a planar crack with a local drop in frictional resistance is investigated. A sliding zone initiates in the area of reduced friction, and then propagates as the applied shear load is monotonously increased. The problem is formulated in general terms, and then solved for the case when sliding spreads as a penny-shaped zone. Conditions under which the front of the zone stays circular during sliding are analyzed. It is observed that the axisymmetry of the profile of frictional resistance does not necessarily guarantee uniform propagation of sliding in the radial direction. The circular shape becomes the most favorable growth condition only if the shear modes are related in a certain way. The problem is studied based on the criterion of propagation that stress intensity factors(SIFs) for II and III modes vanish on the boundary of the sliding zone. The singular integrals in expressions for the SIFs are reduced to non-singular ones. Analytical solutions are derived for a number of special cases where the radius of the sliding zone is related to the applied shear load.This work is supported by the National Science Foundation through grant DGE-0209543 to the University of New Mexico.     

Thermo-electro-mechanical contact behavior of a finite piezoelectric layer under a sliding punch with frictional heat generation

The thermal contact problem of a piezoelectric strip with heat supply generated by the frictional tangential traction under the action of a rigid sliding punch is investigated. The inertial effects are considered. It is convenient to introduce the Galilean transform. Whole cases of the root distribution of the corresponding characteristic equation are detailed. Appropriate fundamental solutions that can lead to real solutions of the thermo-electro-mechanical quantities are derived for the piezoelectric governing equation. The stated problem is reduced to Cauchy singular integral equation of the second kind finally. Numerical results are also presented. The solutions have a reduced dependence on the material properties. The singular behaviors at the edges of the punch are revealed. The stress distribution and temperature distribution above the punch with the variations of the relative sliding speed, the frictional coefficient and the thickness are plotted. The effects of the material constants on the stress distribution and temperature distribution above the punch are presented.     

Dynamic analysis of a disc brake under frictional and thermomechanical internal loading

The study of the vibratory response of a mechanical system as complex as a disc brake needs to consider the complexity of the problem induced by the coupling of tribological, thermomechanical and dynamical effects. Experimental consideration are discussed here for two set ups at the full scale of the disc brake and at a local scale focussed on the third body interface. A numerical model with thermomechanical and dynamical couplings is then presented, followed by a substantial discussion.     

Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system

Many dynamical systems are subject to some form of non-smooth or discontinuous nonlinearity. One eminent example of such a nonlinearity is friction. This is caused by the fact that friction always opposes the direction of movement, thus changing sign when the sliding velocity changes sign. In this paper, a structure with friction-based seismic base isolation is regarded. Seismic base isolation can be employed to decouple a superstructure from the potentially hazardous surrounding ground motion. As a result, the seismic resistance of the superstructure can be improved. In this case study, the base isolation system is composed of linear laminated rubber bearings and viscous dampers and nonlinear friction elements. The nonlinear dynamic modelling of the base-isolated structure with the aid of constraint equations, is elaborated. Furthermore, the influence of the dynamic characteristics of the superstructure and the nonlinear modelling of the isolation system, on the total system’s dynamic response, is examined. Hereto, the effects of various modelling approaches are considered. Furthermore, the dynamic performance of the system is studied in both nonlinear transient and steady-state analyses. It is shown that, next to (and in correlation with) transient analyses, steady-state analyses can provide valuable insight in the discontinuous dynamic behaviour of the system. This case study illustrates the importance and development of nonlinear modelling and nonlinear analysis tools for non-smooth dynamical systems.     

复合表面径向滑动轴承的概念和性能预报

近年来的研究发现固液界面的滑移可以减小表面摩擦,但也会造成流体动力效应下降甚至消失.本文提出了复合表面滑动轴承的概念,轴套表面由具有不同吸附和滑移特性的复合表面组成,发现复合表面轴承比普通轴承有许多优点.通过改变轴套表面的滑移特性可以改变和优化轴承的各种性能,例如摩擦系数、承载力、润滑剂流量、承载角等.数值解表明,在轴承高压区改变轴套表面滑移特性,轴承的整体性能会有大幅度提高.例如,本文给出的初步优化设计方案使得摩擦系数降低50%以上,同时承载力可提高20%,并且承载角可以降低33%.本文提出的设计理念不但可用于设计出具有更优异特性的径向滑动轴承,而且可以设计出具有复合表面的轴向滑动轴承或滑块轴承.复合表面滑动轴承在降低轴承摩擦、提高承载能力方面有很大的空间可以探索.     

Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat stamp profile.     

Non-axisymmetric oscillations of a hemispherical drop

Natural oscillations of a hemispherical drop on a solid substrate are considered. The Hocking condition is used to take into account the contact angle dynamics. The natural oscillations attenuate due to dissipation on the contact line and the degeneracy in the azimuthal number is eliminated. Forced oscillations of the drop are studied for tangential vibrations of the substrate. For the case of a fixed contact line, the amplitude of the forced oscillations grows without bound near the fundamental requency. In other cases, the amplitude is finite.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 8–20. Original Russian Text Copyright © 2004 by Lyubimov, Lyubimova, Shklyaev.     

Nonlinear dynamics of an elastic rod with frictional impact

A model is presented for the impact with friction of a flexible body in translation and rotation. This model consists of a system of nonlinear differential equations which considers the multiple collisions as well as frictional effects at the contacting end, and allows one to predict the rigid and elastic body motion after the impact. The kinetic energy is derived by utilizing a generalized velocity field theory for elastic solids. The model uses a dry coefficient of friction and a nonlinear contact force. We introduce a finite number of vibrational modes to take into account the vibrational behavior of the body during impact. The vibrations, the multiple collisions, and the angle of incidence angle, are found to be important factors for the kinematics of frictional impact. Analytical and experimental results were compared to establish the accuracy of the model.     

Slip-shakedown analysis of a system of circular beams in frictional contact

In automotive components, the cumulative microslip phenomenon is often observed for engine assemblies. This phenomenon results in an accumulation of the relative slips in a preferred tangential direction on the contact interface of two solids under cyclic loadings. A significant relative displacement may occur and leads to the assembly failure. In particular, a global rotation of the bearing shell may result from this mechanism of cumulated slips in conrod big end systems. To discuss this rotation problem, a model of two circular beams in frictional contact and submitted to a periodical rotating load is considered here. The aim is to give some simplified estimates of the critical rotation load based on a slip-shakedown analysis. The discussion holds for Tresca friction and can be extended to Coulomb friction under the assumption of small coupling. The static and kinematic slip-shakedown approaches are discussed. The obtained analytical results are shown to be in agreement with the finite element computations.