On the speed of fast and slow rupture fronts along frictional interfaces

The transition from stick to slip at a dry frictional interface occurs through the breaking of the junctions between the two contacting surfaces. Typically, interactions between the junctions through the bulk lead to rupture fronts propagating from weak and/or highly stressed regions, whose junctions break first. Experiments find rupture fronts ranging from quasi-static fronts with speeds proportional to external loading rates, via fronts much slower than the Rayleigh wave speed, and fronts that propagate near the Rayleigh wave speed, to fronts that travel faster than the shear wave speed. The mechanisms behind and selection between these fronts are still imperfectly understood. Here we perform simulations in an elastic 2D spring--block model where the frictional interaction between each interfacial block and the substrate arises from a set of junctions modeled explicitly. We find that a proportionality between material slip speed and rupture front speed, previously reported for slow fronts, actually holds across the full range of front speeds we observe. We revisit a mechanism for slow slip in the model and demonstrate that fast slip and fast fronts have a different, inertial origin. We highlight the long transients in front speed even in homogeneous interfaces, and we study how both the local shear to normal stress ratio and the local strength are involved in the selection of front type and front speed. Lastly, we introduce an experimentally accessible integrated measure of block slip history, the Gini coefficient, and demonstrate that in the model it is a good predictor of the history-dependent local static friction coefficient of the interface. These results will contribute both to building a physically-based classification of the various types of fronts and to identifying the important mechanisms involved in the selection of their propagation speed.Comment: 29 pages, 21 figure

History-dependent friction and slow slip from time-dependent microscopic junction laws studied in a statistical framework

To study the microscopic origins of friction, we build a framework to describe the collective behaviour of a large number of individual micro-junctions forming a macroscopic frictional interface. Each micro-junction can switch in time between two states: A pinned state characterized by a displacement-dependent force, and a slipping state characterized by a time-dependent force. Instead of tracking each micro-junction individually, the state of the interface is described by two coupled distributions for (i) the stretching of pinned junctions and (ii) the time spent in the slipping state. We show how this framework represents an overarching structure for important models existing in the friction literature. We then use it to study systematically the effect of the time-scale that controls the duration of the slipping state. We first find the steady-state friction force as a function of the sliding velocity. As the framework allows for a whole family of micro-junction behaviour laws, we show how these laws can be chosen to obtain monotonic (strengthening or weakening) or non-monotonic velocity dependence at the macroscale. By then considering transient situations, we predict that the macroscopic static friction coefficient is strongly influenced by the way the interface was prepared, in particular by the slip dynamics of the previous sliding event. We also show that slow slip spontaneously occurs in the framework for a wide range of behaviour laws.Comment: 20 pages, 10 figure

Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces

The failure of the population of micro-junctions forming the frictional interface between two solids is central to fields ranging from biomechanics to seismology. This failure is mediated by the propagation along the interface of various types of rupture fronts, covering a wide range of velocities. Among them are so-called slow fronts, which are recently discovered fronts much slower than the materials' sound speeds. Despite intense modelling activity, the mechanisms underlying slow fronts remain elusive. Here, we introduce a multi-scale model capable of reproducing both the transition from fast to slow fronts in a single rupture event and the short-time slip dynamics observed in recent experiments. We identify slow slip immediately following the arrest of a fast front as a phenomenon sufficient for the front to propagate further at a much slower pace. Whether slow fronts are actually observed is controlled both by the interfacial stresses and by the width of the local distribution of forces among micro-junctions. Our results show that slow fronts are qualitatively different from faster fronts. Since the transition from fast to slow fronts is potentially as generic as slow slip, we anticipate that it might occur in the wide range of systems in which slow slip has been reported, including seismic faults.Comment: 35 pages, 5 primary figures, 6 supporting figures. Post-print version with improvements from review process include

The rainbow as a student project involving numerical calculations

It is a challenge to find interesting and realistic projects where numerical methods can be used to enhance student understanding of physical phenomena. We present such a project in which a group of students used numerical methods to analyze the physics of the rainbow. The project is suitable for students in an undergraduate physics course on the basic principles of geometrical optics. The central part of this paper is written by a group of students, and the introduction and discussion are written by their teacher. In this way both the students' and teacher's perspectives on using numerical methods are presented

Precursors to sliding and static friction threshold of heterogeneous frictional interfaces

Nous utilisons un modèle multi-échelles de la transition entre frottement statique et frottement dynamique, pour étudier la vitesse des fronts de rupture le long d'interfaces multi-contact étendues. Nous montrons que la vitesse des fronts est directement contrôlée par la vitesse de glissement associée, pour toute la gamme de vitesses explorée. Nous proposons ensuite un classement, basé sur les mécanismes en jeu, pour les différents types de fronts observés. Nous montrons finalement comment le coefficient de frottement statique local est contrôlé par l'histoire du glissement, au même endroit, mais lors de la rupture précédente de l'interface

Numerical Modelling of the Dynamics of the Onset of Sliding

Nous utilisons un modèle multi-échelles de la transition entre frottement statique et frottement dynamique, pour étudier la vitesse des fronts de rupture le long d'interfaces multi-contact étendues. Nous montrons que la vitesse des fronts est directement contrôlée par la vitesse de glissement associée, pour toute la gamme de vitesses explorée. Nous proposons ensuite un classement, basé sur les mécanismes en jeu, pour les différents types de fronts observés. Nous montrons finalement comment le coefficient de frottement statique local est contrôlé par l'histoire du glissement, au même endroit, mais lors de la rupture précédente de l'interface

1D model of precursors to frictional stick-slip motion allowing for robust comparison with experiments

We study the dynamic behaviour of 1D spring-block models of friction when the external loading is applied from a side, and not on all blocks like in the classical Burridge-Knopoff-like models. Such a change in the loading yields specific difficulties, both from numerical and physical viewpoints. To address some of these difficulties and clarify the precise role of a series of model parameters, we start with the minimalistic model by Maegawa et al. (Tribol. Lett. 38, 313, 2010) which was proposed to reproduce their experiments about precursors to frictional sliding in the stick-slip regime. By successively adding (i) an internal viscosity, (ii) an interfacial stiffness and (iii) an initial tangential force distribution at the interface, we manage to (i) avoid the model's unphysical stress fluctuations, (ii) avoid its unphysical dependence on the spatial resolution and (iii) improve its agreement with the experimental results, respectively. Based on the behaviour of this improved 1D model, we develop an analytical prediction for the length of precursors as a function of the applied tangential load. We also discuss the relationship between the microscopic and macroscopic friction coefficients in the model.Comment: 13 pages, 14 figures, accepted in Tribology Letter

Modelling the onset of frictional sliding: Rupture fronts, slow slip, and time-dependent junction laws

Friction is scientifically interesting and technologically important. We can characterize friction well, but even the friction force between macroscopic surfaces of known chemistry and topography under known loading conditions cannot yet be predicted from the bottom up. A major obstacle to predicting frictional properties is to link the macroscopic observations to the behavior of the myriad microscopic connections that make up the interaction. The onset of frictional sliding occurs through the breaking of the contacts that were keeping the interface stuck. Recent experiments performed with high spatial and temporal resolution show that rupture nucleates at weak or highly stressed points and propagates outwards from there. Understanding how the rupture travels is an important step towards understanding friction. This thesis presents simulations and theory aimed at improving our understanding of this onset of sliding in dry friction systems. The principal model combines 2D elasticity with an asperity level description of the interface and reproduces and explains many of the experimental results. Analytical calculations provide additional insights

Modelling the onset of dynamic friction : Importance of the vertical dimension

The last decade has seen major advances in the experimental study of the onset of dynamic friction. Optical methods give access to the sliding interface before and during sliding onset, enabling characterisation of the local response to external shear. Treating the sliding interface as an extended system, experimentalists have probed the evolution of slip with high spatial resolution. Cameras operating at the order of 100 kHz have enabled direct study of the fast crack/rupture fronts associated with the transition from local pinning to shear displacement. The spatiotemporal resolution goes beyond the phenomenological descriptions of the global frictional response, i.e. the net resistance to shear motion. Models of the friction of extended systems have a longstanding history in the earthquake community and as models of global friction motivated by the microscopic formation and breaking of contacts. Some of these models have been adapted to the study of sliding onset and investigated numerically. However, their quantitative predictive power has been poor. In this thesis I study deterministic spring--block models of an elastic slider under dry friction. I apply Amontons--Coulomb friction at the block level. First, I study a one-dimensional model and investigate the length of precursors as a function of the driving force. Analytical expressions for point and uniform driving are found and shown to be in excellent agreement with simulation results. Qualitative agreement with experiments is demonstrated. The effect of a friction-induced torque is studied for uniform driving, and the output form the model is compared to a recently proposed theory. I then study a two-dimensional model that includes the direction of sliding and the direction out of the sliding plane, the vertical. By comparison to the one-dimensional model, I show that successful prediction of the experimental results depends crucially on accurate representation of the forces associated with elastic deformations of the slider. This can be obtained in the two-dimensional model if realistic boundary conditions are applied. The statics of sliding onset are the measures that correspond to the states the sliding system comes to rest in, for example the arrest point of a local slip zone, the interfacial shear and normal stress profiles and the length and number of precursors. The dynamics of sliding onset are the rapid time dynamics, for example the speed of the front of a growing slip zone. The statics are reproduced remarkably well in my two-dimensional model, while the dynamics still lack important features of the experimental results. This indicates that the statics depend only weakly on the dynamics, and that they could be studied independently of the fast time evolution. Conversely, the dynamics, although not reproducing the range of experimental observations, are shown to depend strongly on the local stresses and the details of the friction law, i.e. cannot be predicted independently of the statics