The frequency response of dynamic friction: A new view of sliding interfaces

Any attempt to predict friction-induced vibration, such as brake squeal in vehicle systems, requires as input some information about the dynamic friction force at the sliding interface. If the concern is only with the threshold of instability based on linearised theory, the quantity required is a type of frictional frequency response function over the range of possible unstable frequencies. This quantity has not previously been measured, and this paper reports the development and testing of a novel tribometer to measure it. Based on the concept of a pin-on-disc machine, controlled fluctuations in sliding speed are applied using an actuator and the resulting fluctuations in friction force are monitored in amplitude and phase. Sample results are shown, revealing parameter dependence, which is in some cases strikingly different from that of the mean friction force. The device has potential for providing necessary data on friction materials and also for discriminating between rival proposed theoretical models for dynamic friction.     

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.     

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.     

A thermodynamics-based cohesive model for interface debonding and friction

A constitutive model for interface debonding is proposed which is able to account for mixed-mode coupled debonding and plasticity, as well as further coupling between debonding and friction including post-delamination friction. The work is an extension of a previous model which focuses on the coupling between mixed-mode delamination and plasticity. By distinguishing the interface into two parts, a cracked one where friction can occur and an integral one where further damage takes place, the coupling between frictional dissipation and energy loss through damage is seamlessly achieved. A simple framework for coupled dissipative processes is utilised to derive a single yield function which accurately captures the evolution of interface strength with increasing damage, for both tensile and compressive regimes. The new material model is implemented as a user-defined interface element in the commercial package ABAQUS and is used to predict delamination under compressive loads in several test cases.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

On the frequency behaviour,stability and isolation properties of dry friction oscillators

This paper proposes a new approach to the frequency responses of one-degree-of-freedom oscillators subject to periodic excitations in presence of mixed dry-viscous friction. The idea is to get free from the analysis of one fixed system by letting the physical parameters cover their own whole ranges and investigating the various behavioural aspects of wide classes of oscillators. The existence, uniqueness and stability of the steady-state solutions are analysed in detail, assuming different coefficients of static and sliding friction. The possible arising of motions characterized by anti-periodic asymmetry or multi-stick oscillations is enlightened and maps of the system behaviour are presented. A study on transmissibility shows the favourable features of dry friction isolators in the high-frequency range.     

The effect of friction on the vibro-impact locomotion system: modeling and dynamic response

Meccanica - This paper shows a new study on the effect of various dry and isotropic friction levels on the progression and dynamic response of a vibro-impact locomotion system. An experimental...     

Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation

A thermal-mechanical multiresolution continuum theory is applied within a finite element framework to model the initiation and propagation of dynamic shear bands in a steel alloy. The shear instability and subsequent stress collapse, which are responsible for dynamic adiabatic shear band propagation, are captured by including the effects of shear driven microvoid damage in a single constitutive model. The shear band width during propagation is controlled via a combination of thermal conductance and an embedded evolving length scale parameter present in the multiresolution continuum formulation. In particular, as the material reaches a shear instability and begins to soften, the dominant length scale parameter (and hence shear band width) transitions from the alloy grain size to the spacing between micro-voids. Emphasis is placed on modeling stress collapse due to micro-void damage while simultaneously capturing the appropriate scale of inhomogeneous deformation. The goal is to assist in the microscale optimization of alloys which are susceptible to shear band failure.     

The role of friction on sharp indentation

Frictional effects on sharp indentation of strain hardening solids are examined in this paper. The results of finite element simulations in a wide range of solids allow us to derive two simplified equations, accounting for the influence of the friction coefficient on hardness. Comparisons between the simulations and instrumented micro-indentation experiments are undertaken to ensure the validity of the former to metallic materials. Quantitative estimates of the role of friction on the development of pileup and sinking-in around the contact boundary are also given in the paper. These results provide a physical insight into the plastic flow features of distinctly different solids brought into contact with sharp indenters. Overall, the investigation shows that the amount of pileup can be used to set the range of validity of the two hardness equations indicated above. Friction has the largest influence on the contact response of solids exhibiting considerable piling-up effects (whose parameter , see text for details), whereas materials developing moderate pileup or sinking-in are less sensitive to friction. Finally, a methodology is devised to assess the influence of the friction coefficient on mechanical properties extracted through indentation experiments.     

On the Control of the Friction Force

This paper tackles the problem of controlling the Coulomb friction force in order to achieve damping characteristics which can be advantageous in engineering applications, particularly in the area of vibration control, for example, vehicle suspensions, rotating machinery foundations or earthquake protection systems. The control schemes employed belong to the family of variable structure controllers (VSC), a class of robust control algorithms, namely sliding mode control (SMC) and switched state feedback (SSF). The schemes perform force tracking control, aiming at making the friction force track a reference force in order to reduce the acceleration experienced by a suspended mass. The plant investigated is a 2-degree of freedom system and in this study represents a quarter car. The controller performances are investigated both numerically and experimentally.     

The effect of friction reduction in the presence of in-plane vibrations

A reduction of friction by vibrations has been observed in various experiments. This effect can be applied to actively control frictional forces by modulating vibrations. Moreover, common methods of controlling friction rely on lubricants and suitable material combinations. The superimposition of vibrations can further reduce the friction force. This study presents a theoretical approach based on the Dahl friction model that describes the friction reduction observed in the presence of the tangential vibrations at an arbitrary angle. Analysis results indicated that the tangential compliance should be considered in modeling the effect of vibrations in reducing friction. At any vibration angle, the tangential compliance of the contacts reduces the friction reduction effect. The vibrations parallel to the macroscopic velocity are most effective for friction reduction.     

A planar rod model with flexible cross-section for the folding and the dynamic deployment of tape springs: Improvements and comparisons with experiments

A planar rod model with flexible cross-section has been recently proposed in literature (Guinot et al., 2012). This model is especially suitable for the modeling of tape springs, which develop localized folds due to the flattening of the cross-section. Starting from a complete non-linear elastic shell model, original kinematics assumptions (inspired from the elastica model) have been made to describe the important in-plane changes of the cross-section shape. In the present work, the choice of the position of the rod reference line is discussed. This choice plays an important role in the overall behavior because of the large changes of the cross-section shape. We show that the model published in Guinot et al. (2012) can be improved by considering the centerline as the rod reference line. This enhanced model is then validated through quantitative comparisons with experimental results of dynamic deployments taken from literature.     

The dynamic behavior of MEMS arch resonators actuated electrically

In this paper, we investigate the dynamic behavior of clamped-clamped micromachined arches when actuated by a small DC electrostatic load superimposed to an AC harmonic load. A Galerkin-based reduced-order model is derived and utilized to simulate the static behavior and the eigenvalue problem under the DC load actuation. The natural frequencies and mode shapes of the arch are calculated for various values of DC voltages and initial rises. In addition, the dynamic behavior of the arch under the actuation of a DC load superimposed to an AC harmonic load is investigated. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response of the arch due to DC and small AC loads. Results of the perturbation method are compared with those obtained by numerically integrating the reduced-order model equations. The non-linear resonance frequency and the effective non-linearity of the arch are calculated as a function of the initial rise and the DC and AC loads. The results show locally softening-type behavior for the resonance frequency for all DC and AC loads as well as the initial rise of the arch.     

On the long-term response of elastic-perfectly plastic solids to dynamic cyclic loads

It is shown that the long-term response of an elastic-perfectly plastic solid subjected to dynamic actions cyclically varying in time is characterized by stresses, plastic strain rates and velocities that are all periodic with the same period of the external actions, and are in perfect analogy with the quasi-static case; on the other hand, plastic strains and displacements are in general nonperiodic (except in case of alternating plasticity) and may increase indefinitely (except when elastic or plastic shakedown occurs). Besides, the work performed by the external actions in the steady cycle equals the work performed by the elastic stresses (i.e. pertaining to the elastic response of the body to the same actions) through the plastic strain rates.

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Sommario Per un solido elastico perfettamente plastico soggetto ad azioni cicliche dinamiche si mostra che la risposta a lungo termine è caratterizzata da tensioni, deformazioni plastiche incrementali e velocità tutte periodiche con lo stesso periodo delle azioni esterne, in analogia di quanto avviene nel caso quasi-statico; per contro le deformazioni plastiche e gli spostamenti sono in generale non periodici (tranne nel caso di plasticità alternata) e possono crescere indefinitamente (tranne nel caso di adattamento elastico o plastico). Inoltre il lavoro compiuto dalle azioni esterne in un ciclo stazionario risulta eguale al lavoro delle tensioni elastiche (cioè ottenute come risposta puramente elastica del solido alle stesse azioni) attraverso le deformazioni plastiche incrementali.

     

Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation

The universal (i.e. independent of the constitutive equations) thermodynamic driving force for coherent interface reorientation during first-order phase transformations in solids is derived for small and finite strains. The derivation is performed for a representative volume with plane interfaces, homogeneous stresses and strains in phases and macroscopically homogeneous boundary conditions. Dissipation function for coupled interface (or multiple parallel interfaces) reorientation and propagation is derived for combined athermal and drag interface friction. The relation between the rates of single and multiple interface reorientation and propagation and the corresponding driving forces are derived using extremum principles of irreversible thermodynamics. They are used to derive complete system of equations for evolution of martensitic microstructure (consisting of austenite and a fine mixture of two martensitic variants) in a representative volume under complex thermomechanical loading. Viscous dissipation at the interface level introduces size dependence in the kinetic equation for the rate of volume fraction. General relationships for a representative volume with moving interfaces under piece-wise homogeneous boundary conditions are derived. It was found that the driving force for interface reorientation appears when macroscopically homogeneous stress or strain are prescribed, which corresponds to experiments. Boundary conditions are satisfied in an averaged way. In Part 2 of the paper [Levitas, V.I., Ozsoy, I.B., 2008. Micromechanical modeling of stress-induced phase transformations. Part 2. Computational algorithms and examples. Int. J. Plasticity (2008)], the developed theory is applied to the numerical modeling of the evolution of martensitic microstructure under three-dimensional thermomechanical loading during cubic-tetragonal and tetragonal-orthorhombic phase transformations.     

The formation of tabular compaction-band arrays: Theoretical and numerical analysis

The bifurcation analysis of compaction banding is extended to the formation of a tabular discrete compaction-band array. This analysis, taken together with the results of finite-difference simulations, shows that the bifurcation results in the formation of intermittent loading (elastic-plastic) and unloading (elastic) bands. The obtained analytical solution relates the spacing parameter χ (the ratio between the band thickness to the band-to-band distance) to all constitutive and stress-state parameters. Both this solution and numerical models reveal strong dependence of χ on the hardening modulus h: χ increases with h reduction. The band thickness in the numerical models is mesh dependent, but in terms of mesh-zone-size varies only from ∼2 to 4 depending on the constitutive parameters and independently on the mesh resolution. The thickness of the “elementary” compaction bands in real granular materials is equal to a few grain sizes. It follows that one grid zone in the numerical models corresponds approximately to one grain in the real material. The numerical models reproduce both discrete and continuous propagating compaction banding observed in the rock samples. These phenomena were shown to be dependent on the evolution of h and the dilatancy factor with deformation.     

Entropic stabilization of austenite in shape memory alloys

Martensitic transformations (MTs) are the key phenomena responsible for the remarkable properties of Shape Memory Alloys (SMAs). Recent Density Functional Theory (DFT) electronic structure calculations have revealed that the austenite structure of many SMAs is a saddle-point of the material's potential energy landscape. Correspondingly, the austenite is unstable and thus unobservable at zero temperature. Thus, the observable high temperature austenite structure in many SMAs is entropically stabilized by nonlinear dynamic effects.