On the mathematical basis of solid friction

Jeffrey, Mike R.

doi:10.1007/s11071-015-2100-7

Cited by 9 publications

(8 citation statements)

References 48 publications

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“…This example contains a hidden term α(u 2 − 1) from the outset, as a simplification of a model of static friction in mechanical oscillators proposed in e.g. [22,25]. At x = 0, we let u take all values on the interval [−1, +1], giving the right-hand side of (4.1) as in (2.2).…”

Section: (A) Example Of a Single Discontinuitymentioning

confidence: 99%

Pausing for arbitrarily long times in dynamical systems

2019

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It is well known that continuity in dynamical systems is not sufficient to guarantee uniqueness of solutions, but less obvious is that non-uniqueness can carry internal structure useful to characterize a system's dynamics. The non-uniqueness that concerns us here arises when an isolated non-differentiability of a flow results in spatial or temporal ambiguity of solutions. Spatial ambiguity can render a flow set valued after a specific event, and non-trivial examples are increasingly being seen in models of switching occurring in electronic or biological systems. Temporal ambiguity can mean that the same spatial trajectory may be traversed in different times, making an arbitrarily long pause at the nondifferentiable point. We focus here on temporal indeterminacy and the extent to which it can be resolved. To investigate the typical forms, we take representative examples of the different conditions (non-differentiability, discontinuity or singularity) under which it occurs.

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Section: (A) Example Of a Single Discontinuitymentioning

confidence: 99%

Pausing for arbitrarily long times in dynamical systems

2019

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“…Let us briefly set out the methods used to analyze such a system from [26,27,30], which develop on methods in [19,52].…”

Section: ) With a Set Of Quantitiesmentioning

confidence: 99%

“…[15,42,21,55,56,45]. The possibility of including static friction or stiction as nonlinear (or 'hidden') terms of a piecewise-smooth model was introduced in [31,27].…”

Section: One Oscillatormentioning

confidence: 99%

Micro-Slip as a Loss of Determinacy in Dry-Friction Oscillators

Webber

Jeffrey

2019

Int. J. Bifurcation Chaos

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Dry-friction contacts in mechanical oscillators can be modeled using nonsmooth differential equations, and recent advances in dynamical theory are providing new insights into the stability and uniqueness of such oscillators. A classic model is that of spring-coupled masses undergoing stick-slip motion on a rough surface. Here, we present a phenomenon in which multiple masses transition from stick to slip almost simultaneously, but suffer a brief loss of determinacy in the process. The system evolution becomes many-valued, but quickly collapses back down to an infinitesimal set of outcomes, a sort of “micro-indeterminacy”. Though fleeting, the loss of determinacy means masses may each undergo different microscopic sequences of slipping events, before all masses ultimately slip. The microscopic loss of determinacy is visible in local changes in friction forces, and in creating a bistability of global stick-slip oscillations. If friction forces are coupled between the oscillators then the effect is more severe, as solutions are compressed instead onto two (or more) macroscopically different outcomes.

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“…We note, however, that the use of a first-order kinetics for the interfacial state evolution is mathematically justified whenever ϕ is interpretable as an n th statistical moment of the friction force [ 70 ]. On pure mathematical grounds, it has also been argued that piecewise-smooth descriptions of friction such as Coulomb’s need to be smoothed out by introducing an evolution equation for some hidden variable in order to account for any departure from steady sliding and hysteretic effects [ 18 ]. Originally, ad hoc choices of state evolution laws, such as ( 5.1 ), were motivated by fitting the frictional stress relaxation observed in response to sudden changes in driving velocity within ‘slide-hold-slide’ experiments [ 66 , 68 ].…”

Section: Rate-and-state Friction Modelmentioning

confidence: 99%

“…Using such an approach, quite sophisticated mathematical theories of non-smooth mechanics have developed [ 16 , 17 ], which can predict complex temporal behaviours at the macroscale (e.g. [ 18 , 19 ]).…”

Section: Introductionmentioning

confidence: 99%

A phase-plane analysis of localized frictional waves

2017

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Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

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On the mathematical basis of solid friction

Cited by 9 publications

References 48 publications

Pausing for arbitrarily long times in dynamical systems

Pausing for arbitrarily long times in dynamical systems

Micro-Slip as a Loss of Determinacy in Dry-Friction Oscillators

A phase-plane analysis of localized frictional waves

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