Abstract
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.
Highlights
Inhomogeneous frictional sliding between solid bodies is ubiquitous and characterized by multiple spatiotemporal scales, compelling and practical examples being earthquake mechanics [1] or brake squeal [2]
A possible interpretation of these observations could be in terms of a new frictional slip mode that we describe in the present paper as a stick pulse
We first provide a few comments on the necessity of our spinodal law, contrasting it with the use of monotonic or unregularized friction laws
Summary
Inhomogeneous frictional sliding between solid bodies is ubiquitous and characterized by multiple spatiotemporal scales, compelling and practical examples being earthquake mechanics [1] or brake squeal [2]. The validity domain of such simple models is at best questionable [9], and theoretical studies tend to be limited to either point contact or to where surfaces in contact are assumed to behave homogeneously, see [10,11,12,13,14,15] for notable exceptions Such approximations have proved useful in situations where friction is regarded merely as a loss mechanism on an otherwise macroscopic motion. We note that a variety of solitary waves in the form of slip pulses or fronts were numerically found within the one-dimensional continuum limit of the Burridge–Knoppoff model [51], either with velocity-dependent non-smooth friction laws [52,53,54,55,56] or the Ruina rate-and-state law [31].
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