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.     

Regimes of frictional sliding of a spring-block system

In the context of rate-and-state friction, we revisit the crossover between the creep and inertial regimes in the dynamics of a spring-block system as observed and described in the dry friction experiment of Heslot et al. (1994) and Baumberger et al. (1994). We show that the transition between the quasi-static motion of a spring-block and its dynamic motion occurs at a lower sliding velocity than that which minimises the steady-state friction coefficient. We perform a weakly nonlinear stability analysis combined with numerical studies with the continuation package Auto. In particular, attention is focused on the change of nature the Hopf bifurcation from supercritical to subcritical, as observed by Heslot et al. Comparing the results obtained for different friction laws, we conclude that the weakly nonlinear analysis provides a possible criterion for distinguishing which friction laws may be physically relevant.     

Relaxation and Recovery in Hydrogel Friction on Smooth Surfaces

Background

Hydrogels are crosslinked polymer networks that can absorb and retain a large fraction of liquid. Near a critical sliding velocity, hydrogels pressed against smooth surfaces exhibit time-dependent frictional behavior occurring over multiple timescales. The origin of these dynamics is unresolved

Objective

Here, we characterize this time-dependent regime and show that it is consistent with two distinct molecular processes: sliding-induced relaxation and quiescent recovery.

Methods

Our experiments use a custom pin-on-disk tribometer to examine poly(acrylic acid) hydrogels on smooth poly(methyl methacrylate) surfaces over a variety of sliding conditions, from minutes to hours.

Results

We show that at a fixed sliding velocity, the friction coefficient decays exponentially and reaches a steady-state value. The time constant associated with this decay varies exponentially with the sliding velocity, and is sensitive to any precedent frictional shearing of the interface. This process is reversible; upon cessation of sliding, the friction coefficient recovers to its original state. We also show that the initial direction of shear can be imprinted as an observable “memory”, and is visible after 24 hrs of repeated frictional shearing.

Conclusions

We attribute this behavior to nanoscale extension and relaxation dynamics of the near-surface polymer network, leading to a model of frictional relaxation and recovery with two parallel timescales.

     

Subharmonic and grazing bifurcations for a simple bilinear oscillator

In this paper, subharmonic and grazing bifurcations for a simple bilinear oscillator, namely the limit discontinuous case of the smooth and discontinuous (SD) oscillator are studied. This system is an important model that can be used to investigate the transition from smooth to discontinuous dynamics. A combination of analytical and numerical methods is used to investigate the existence, stability and bifurcations of symmetric and asymmetric subharmonic orbits. Grazing bifurcations for a particular periodic orbit are also discussed and numerical results suggest that the bifurcations are discontinuous. We show via concrete numerical experiments that the dynamics of the system for the case of large dissipation is quite different from that for the case of small dissipation.     

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.     

The frequency response of dynamic friction: Enhanced rate-and-state models

The prediction and control of friction-induced vibration requires a sufficiently accurate constitutive law for dynamic friction at the sliding interface: for linearised stability analysis, this requirement takes the form of a frictional frequency response function. Systematic measurements of this frictional frequency response function are presented for small samples of nylon and polycarbonate sliding against a glass disc. Previous efforts to explain such measurements from a theoretical model have failed, but an enhanced rate-and-state model is presented which is shown to match the measurements remarkably well. The tested parameter space covers a range of normal forces (10–50 N), of sliding speeds (1–10 mm/s) and frequencies (100–2000 Hz). The key new ingredient in the model is the inclusion of contact stiffness to take into account elastic deformations near the interface. A systematic methodology is presented to discriminate among possible variants of the model, and then to identify the model parameter values.     

Numerical and experimental analysis of the bi-stable state for frictional continuous system

Unstable friction-induced vibrations are considered an annoying problem in several fields of engineering. Although several theoretical analyses have suggested that friction-excited dynamical systems may experience sub-critical bifurcations, and show multiple coexisting stable solutions, these phenomena need to be proved experimentally and on continuous systems. The present work aims to partially fill this gap. The dynamical response of a continuous system subjected to frictional excitation is investigated. The frictional system is constituted of a 3D printed oscillator, obtained by additive manufacturing that slides against a disc rotating at a prescribed velocity. Both a finite element model and an experimental setup has been developed. It is shown both numerically and experimentally that in a certain range of the imposed sliding velocity the oscillator has two stable states, i.e. steady sliding and stick–slip oscillations. Furthermore, it is possible to jump from one state to the other by introducing an external perturbation. A parametric analysis is also presented, with respect to the main parameters influencing the nonlinear dynamic response, to determine the interval of sliding velocity where the oscillator presents the two stable solutions, i.e. steady sliding and stick–slip limit cycle.

     

氧化锆陶瓷/碳纤增强聚醚醚酮在水润滑下的摩擦磨损特性研究

在材料端面摩擦试验机上对氧化锆陶瓷与碳纤增强聚醚醚酮(CFRPEEK)配副在水润滑条件下的摩擦磨损特性进行了试验研究,探讨了滑动速度和接触压力对材料摩擦磨损的影响规律.发现氧化锆陶瓷与CFRPEEK配副在水润滑条件下的摩擦系数随速度增加而减小,在速度较低时,存在明显的磨合过程;速度较高时,摩擦系数较小且随滑动过程变化很小,CFRPEEK的磨损率随速度变化不大.压力为0.4和0.5 MPa时,CFRPEEK的摩擦系数和磨损率均较小,但当压力达到0.8 MPa时,摩擦系数显著增加且剧烈振荡,并发生严重磨损.CFRPEEK的磨损机理主要是黏着磨损,氧化锆陶瓷磨损的主要机理是应力引起的点蚀.     

Instability of frictional contact states in infinite layers

The research reported in this paper is focused on the instability of equilibrium and steady sliding states of elastic orthotropic layers in the presence of unilateral obstacles with Coulomb friction with emphasis on a divergence type instability called directional instability of frictional contact states that cannot occur in isotropic layers. Analytic expressions and numerical solutions are provided for the instability mode and for the coefficient of friction at the onset of instability. A parametric study is done to investigate how this coefficient of friction and the instability mode vary with changes of the system parameters. For certain combinations of material data, significantly low coefficients of friction were required for the onset of instability. A finite element model that approximates the continuum and a lumped model that captures some of the features of the continuum are presented.     

微液滴在PDMS软基体表面的动态摩擦行为研究

通过固液界面摩擦力测试装置研究了微液滴在PDMS软基体表面运动时的动态摩擦学行为,并对微液滴体积、滑动速度及软基体力学性能对固液界面动态摩擦行为的影响进行了分析. 结果表明:微液滴在软基体表面运动时表现出最大静摩擦力和动态摩擦力. 最大静摩擦力与微液滴黏度和速度梯度呈正比,动态摩擦力与微液滴体积、滑动速度和基体力学性能有关. 随着微液滴体积的增加,三相接触线长度增加,动态摩擦力增加;随着相对滑动速度增加,三相接触线长度及接触角滞后增加,动态摩擦力增加;随着软基体弹性模量降低,固液界面黏附力增加,固液界面运动能量耗散增加,动态摩擦力增加. 研究结果可为PDMS软基体表面微液滴的精确驱动和运动参数优化提供理论指导,也可进一步丰富固液界面摩擦理论.      

The stability of a top rotating on a rough horizontal plane

In the present paper the stability of permanent rotation of a symmetric top rotating on a slightly rough horizontal plane about the erect polar or transverse axis is discussed via the perturbation method. As a perturbation factor, the frictional force is an arbitrary nonlinear function of the sliding velocity. The same stability criterion as that in the Contensou-Magnus linear theory is obtained, but the hypothesis of the linearity for friction can be omitted. It shows that the direction of the sliding velocity at the contact point is an important factor, which influences the stability of the top. And then an explicit physical explanation is given.     

A modified torsional kolsky bar for investigating dynamic friction

This paper introduces an experiment to investigate dry sliding resistance of frictional interfaces at normal pressures up to 100 MPa, slip speeds up to 10 m/s and slip distances of approximately 10 mm. This new apparatus involves a novel modification of the conventional torsional Kolsky bar apparatus, employed extensively in the past for investigating high strain rate behavior of engineering materials. The new experimental configuration represents a significant improvement over conventional tribology experiments because it uses elastic torsional waves with a superimposed static compressive force to control the interfacial traction. Moreover, the apparatus allows critical frictional parameters such as the interfacial sliding resistance, slip speeds and slip without the use of transducers at the frictional interface. The usefulness of the device is demonstrated by presenting results of high-speed friction on 6061-T6 Al/1018 steel and Carpenter Hampden tool steel/7075-T6 Al tribo pairs.     

高速运动条件下玄武岩摩擦系数的实验研究

在MM－1000型摩擦磨损试验机上采用不同试验条件对玄武岩试件进行了高速摩擦试验研究,结果表明：摩擦表面载荷、运动速度以及摩擦速度以及摩擦表面特性等因素均对摩擦系数产生影响;摩擦系数随载荷的增大呈现减小趋势;对于饱和吸水试件,摩擦系数随运动速度增大持续降低;在其它试验条件下,饱和吸水试件的动摩擦系数比干燥试件明显要低。     

Wave-induced dynamics of a particle on a thin circular plate

In this paper, the dynamics of a particle placed on a thin circular plate carrying circumferential harmonic travelling wave is studied. Coulomb friction is used to model the particle–surface interaction. Distinct regions on the plate surface are identified where either of the three phases of particle motion, namely jumping, sliding and sticking, occurs. Also, the effect of wave frequency and the plate geometry on these regions is studied. Interestingly, there exists an optimum plate thickness for which the region of sliding is maximum. At certain wave frequencies, from the numerical simulations within sticking and sliding regions, it is observed that the average particle motion spirals inwards towards the plate centre. Such an average motion is observed whenever the particle is placed initially with a zero velocity relative to the plate surface. The Gedanken experiments discussed herein provide cogent explanations to all the observed average (slow) dynamics and are also found to be useful in predicting the slow dynamics of the particle a priori, that is, before the actual numerical simulations. The particle’s velocity couples the radial and tangential sliding friction components and is found to be the key physical feature that explains the observed behaviour. Also, it is observed that the plate surface excited by circumferential travelling waves can provide acoustic lubrication to a particle by reducing the limiting force required to move it relative to the surface. The methods discussed in this paper can be extended to study the dynamics of a group of particles (granular materials) and extended rigid bodies, interacting with such surface waves.

     

Analytical period-3 motions to chaos in a hardening Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.     

摩擦力和摩擦振动的分形行为研究

在不同的摩擦磨损试验机上提取了摩擦磨损过程中摩擦力和摩擦振动的时间序列信号,采用关联维数方法研究了摩擦力和摩擦振动的分形行为．结果表明：摩擦力和摩擦振动信号具有分形特征;随着摩擦磨损过程的进行,信号分形维数的变化出现规律性的递增或递减;对于“收敛”或磨合磨损过程,不同阶段摩擦信号的分形维数趋于增大;对于“发散”的摩擦磨损过程,不同阶段的摩擦信号的关联维数趋于减小．摩擦力和摩擦振动的分形维数的变化规律同摩擦磨损过程中表面形貌分形维数的变化规律相似。     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Investigation of Painlevé’s paradox and dynamic jamming during mechanism sliding motion

The paradox of Painlevé occurs when the dynamics of a sliding rigid body has solution inconsistency. When approaching a configuration of inconsistency, the contact forces and accelerations may grow unbounded in finite time, a scenario which is called dynamic jamming. Painlevé’s paradox was originally formulated for a sliding rod with uniform mass distribution, for which inconsistency occurs only under unrealistically high contact friction conditions. This paper shows that when the sliding rod is replaced by a sliding mechanism, the dynamic jamming predicted by Painlevé’s paradox can occur under relatively low friction. The paper proposes a particular mechanism, called IPOS, which consists of an inverted pendulum hinged to a plate that slides on an inclined plane (IPOS = Inverted Pendulum On a Slider). The IPOS mechanism can reach solution inconsistency under practical friction conditions, and can thus be used to experimentally validate the theoretical predictions. The conditions under which dynamic jamming occurs in the IPOS mechanism dynamics are explicitly derived, and numerical simulations illustrate the feasibility of dynamic jamming experiments.