Solid friction from stick–slip down to pinning and aging

Abstract

We review the present state of understanding of solid friction at low velocities and for systems with negligibly small wear effects. We first analyze in detail the behavior of friction at interfaces between macroscopic hard rough solids, whose main dynamical features are well described by the Rice–Ruina rate and state-dependent constitutive law. We show that it results from two combined effects: (i) the threshold rheology of nanometer-thick junctions jammed under confinement into a soft glassy structure and (ii) the geometric aging, i.e. slow growth of the real area of contact via asperity creep interrupted by sliding. Closer analysis leads to identifying a second aging-rejuvenation process, at work within the junctions themselves. We compare the effects of structural aging at such multicontact, very highly confined, interfaces with those met under different confinement levels, namely boundary lubricated contacts and extended adhesive interfaces involving soft materials (hydrogels, elastomers). This leads us to propose a classification of frictional junctions in terms of the relative importance of jamming and adsorption-induced metastability.

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Acknowledgments

We are particularly grateful to J.R. Rice, K.L. Johnson and Y. Bréchet for encouraging our first steps in the field of friction, and letting us benefit liberally from their vast knowledge of mechanics and material science. We are deeply indebted to P. Nozières, B. Perrin, B. Velicky and F. Heslot for their precious contributions at various stages of our work on this subject, as well as to L. Bureau and O. Ronsin for a long-lasting collaboration, and fruitful exchanges on these and related subjects.

Notes

¹This means, in particular, that we limit ourselves to systems and sliding regimes such that wear and frictional self-heating play a negligible role.

²A detailed historical survey will be found in Dowson's book ‘History of Tribology’ 1.

³On the basis of geometric arguments, one estimates that H ≃ 3Y 14.

⁴The statistical properties of a self-affine surface z(x,y) are invariant under the scaling transformation , where D _f is the fractal dimension.

⁵Power law fits with small exponents are also possible though less easily amenable to physical interpretation.

⁶More precisely, for an asymmetric interface, is on the order of the yield stress of the softer material.

⁷In uniaxial loading, σ _Y  = Y, while for a sphere–sphere contact σ _Y  = H ≃ 3Y 13.

⁸Strictly speaking the value of σ_s which appears in this expression should be that at the driving velocity (see section 2.3.3. and 30).

⁹This approximation, which neglects wear, is validated by the observed stability of the frictional characteristics of an MCIs over slid lengths as large as a few 10 cm 33 – a distance over which we can estimate the number of contact configuration renewals to 10⁴⁻⁵.

¹⁰The opposite situation, which we do not consider here, is met at contacts between atomically flat cleaved surfaces in ultrahigh vacuum, under moderate pressures. In this case, as shown by Hirano et al. 35, lattice commensuration plays an essential part, full discommensuration leading to superlubricity, i.e. vanishing friction.

¹¹A rough estimate for the corresponding ratio is , with the average microcontact radius, h the junction thickness.

¹²Note, however, that with MCIs this creep is certainly amplified by the geometric rejuvenation (weakening) associated with sliding.

¹³Note, however, that the T-range in these experiments is small.

¹⁴This model was first developed by Caroli and Noziéres 41. Although their formal results directly apply here, let us point out that the original physical interpretation – that the elementary mechanically unstable units were the interasperity microcontacts as a whole – was not correct 42. Let us insist again that the relevant instabilities do not occur on the micrometric scale but, within the inter-asperity junction, on the nanometric scale.

¹⁵The corrections resulting from imperfect adiabatic adaptation are easily shown to be of relative order (V/c)^2/3 41.

¹⁶This expression assumes, implicitly, that the drive is slow enough for the instantaneous jumping rate to be that for the non-advected system.

¹⁷In the absence of shear localization.

¹⁸To which extent and how this density depends on is an important though still an unsettled question.

¹⁹The poor accuracy on Ω_c was due in particular to the difficulty of extrapolating measurements necessarily performed at finite amplitude to the continuous bifurcation while, in this case, non linearities develop very fast 52.

²⁰The expansion parameter is found to be ε∼ A/μ₂ with , so that for the RR model (μ₂ = 0 ) the expansion explodes. For PMMA, μ₂ is measured to be ∼ 10⁻³, hence ε∼ 10 : non-linearities develop fast in the SS regime.

²¹Choosing blocks much longer in the transverse than in the longitudinal direction (i.e. in Burridge–Knopoff-like models, compressive stiffnesses much larger than shear ones) may lead to dynamical artefacts: for the so-called small events which involve only a few blocks, the associated elastic fields only affect depths, on the order of their lateral extent, much smaller than the block height. It is then illegitimate to neglect internal block degrees of freedom.

²²Since glass is much harder than PMMA, no ploughing of the track by the slider asperities takes place.

²³As the absolute value of Σ_r is not measurable accurately, the absolute value of σ_s cannot be accessed in this configuration.

²⁴As can be seen on figure 24, the log-slope of μ_s associated with structural aging is on the order of a few 10⁻³, typically from 10 to 4 times smaller than the aging slopes for rough/rough MCIs. This leads one to conclude that, in the latter case, the growth of μ_s, though dominated by the geometric effect, does contain a small contribution due to structural aging.

²⁵Viasnoff and Lequeux 75 have found that applying to a colloidal glass, aging under stress-free conditions, an oscillating shear stress of finite duration does result in a strong perturbation of the aging process. However, it is important to note that, in their experiment, the stress amplitude is larger than the yield stress of the material, which certainly results in some non-stationary flow. This contrasts with the situation of 61, where the applied waiting stress does not provoke any sliding.

²⁶For some fluids made of short-chain molecules, e.g. hexadecane 77, the solid-like response only builds up after a finite amount of shear-induced sliding – suggesting that, in this case, the initial sliding helps ordering.

²⁷Whether or not such a discontinuous transition exhibits hysteresis upon increasing/decreasing V is not documented, to our knowledge.

²⁸Note that, while the shear modulus is controlled by the elasticity of the loose polymer network, the undrained bulk modulus is that of the solvent, so hydrogels can be considered incompressible.

²⁹It is worth mentioning that Rubinstein et al. 89 have recently observed that sliding between purely elastic solids also sets in via propagation of interfacial cracks, but in the elastic case these propagate at velocities in the sonic range. The low values of V _tip for gelatin are therefore, in contrast, a signature of poroelasticity.

³⁰Interfacial viscous dissipation is neglected in the VC analysis. As already mentioned, it should come into play, especially in the V -weakening regime V > V _m, where it is certainly needed to interpret the observed stick–slip behavior.

³¹Except possibly for some elastomers under low normal stress and in contact with highly passivated substrates.

³²When established sliding is not steady, but occurs via stick–slip, static aging results in the growth of the first peak with stopping time.

³³We assume that, as is commonly the case, K is much smaller than the interfacial stiffness. Finite interfacial compliance effects are evaluated in 53.