Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces

Abstract

The rupture of dry frictional interfaces occurs through the propagation of fronts breaking the contacts at the interface. Recent experiments have shown that the velocities of these rupture fronts range from quasi-static velocities proportional to the external loading rate to velocities larger than the shear wave speed. The way system parameters influence front speed is still poorly understood. Here we study steady-state rupture propagation in a one-dimensional (1D) spring-block model of an extended frictional interface, for various friction laws. With the classical Amontons-Coulomb friction law, we derive a closed-form expression for the steady-state rupture velocity as a function of the interfacial shear stress just prior to rupture. We then consider an additional shear stiffness of the interface and show that the softer the interface, the slower the rupture fronts. We provide an approximate closed form expression for this effect. We finally show that adding a bulk viscosity on the relative motion of blocks accelerates steady-state rupture fronts and we give an approximate expression for this effect. We demonstrate that the 1D results are qualitatively valid in 2D. Our results provide insights into the qualitative role of various key parameters of a frictional interface on its rupture dynamics. They will be useful to better understand the many systems in which springblock models have proved adequate, from friction to granular matter and earthquake dynamics.

Minimal model for slow, sub-Rayleigh, supershear, and unsteady rupture propagation along homogeneously loaded frictional interfaces

“…At lowᾱ andτ , the model predicts the existence of slip pulse solutions, in agreement with the experimental observation that slip pulses occur when the prestress is low compared to the static friction threshold [10]. Forᾱ = 0, the model only predicts super-shear rupture [36]. For non-zeroᾱ, sub-Rayleigh and slow rupture can occur.…”

“…Slip pulses have been reproduced using discrete [29] or continuum models assuming either a Coulomb [30], regularized Coulomb [31][32][33] or state-andrate [34,35] friction laws. Models of cracks are ubiquitous, featuring super-shear [24,36,37], sub-Rayleigh [24-26, 28, 38], slow [26,28,39,40] or quasi-static [26,41,42] fronts. Note that front speed has been shown to depend on many features of the frictional system, including slip history [25,28], interaction between different fault planes [43], the shape of the high speed branch of the friction law [44], and spatial heterogeneities in stress or constitutive parameters [45][46][47][48][49].…”

Continuum contact models for coupled adhesion and friction

“…1) The friction force is proportional to the applied normal load. (Amontons' 1st law) 3 2) The friction force is independent of the apparent contact area. (Amontons' 2nd law)…”

“…In many engineering applications it is sufficient to consider the coefficient of sliding friction, µ, as a constant parameter for the material pair at the interface; see e.g. [23,112,3,125]. Depending on the application, however, µ may be affected by the sliding velocity [47,122], contact pressure [104], temperature [47], or microscopic time scales [113].…”

“…The second proposed traction-separation law is inspired by extended Amontons' law (3). Importantly, that law was originally formulated in terms of force resultants and average tractions.…”