Jean-Pierre Vilotte

Preavalanche instabilities in a granular pile.

We investigate numerically the transition between static equilibrium and dynamic surface flow of a 2D cohesionless granular system driven by a continuous gravity loading. This transition is characterized by intermittent local dynamic rearrangements and can be described by an order parameter defined as the density of critical contacts, i.e., contacts where the friction is fully mobilized. Analysis of the spatial correlations of critical contacts shows the occurrence of fluidized clusters which exhibit a power-law divergence in size at the approach of the stability limit. The results are compatible with recent models that describe the granular system during the static/dynamic transition as a multiphase system.

Percolation through self-affine surfaces

We study the percolation transition in a long-range correlated system: a self-affine surface. For all relevant physical cases (i.e. positive roughness exponents), it is found that the onset of percolation is governed by the largest wavelength of the height distribution, and thus self-averaging breaks down. Self-averaging is recovered for negative roughness exponents (i.e. power-law decay of the height pair correlation function) and, in this case, the critical exponents that characterize the transition are explicitly dependent on the roughness exponent above a threshold value. Below this threshold, the spatial correlations are no longer relevant. The problem is analytically investigated for a hierarchical network and by means of numerical simulations in two dimensions. Finally, we discuss the application of those properties to mercury porosimetry in cracks.

A dissipation‐based analysis of an earthquake fault model

We analyze the dynamics of a discrete dynamical model of earthquake faulting, derived from the Burridge-Knopoff model. The system is shown to exhibit a characteristic event size (L* = 2v2f l2) that separates two distinct regimes in the statistical distribution of event sizes and magnitudes. The dynamics of the system exhibits scaling laws that are in agreement with observed seismic laws. Influence of the frictional rate of dissipation, of the elastic stiffness coupling, and of the system size is investigated. The dynamics of the system is rather insensitive to the numerical treatment of the nonsmooth friction. In contrast, the exact form of the velocity weakening friction law is shown to have a major effect on the dynamics. For friction laws allowing local reversal backslipping, the distribution of the magnitudes does not exhibit a Gutenberg and Richter (GR) distribution, while by precluding backslipping a GR distribution is observed. Two populations of events can be characterized based on dissipation: weakly dissipative events that allow the mechanical energy of the system to increase; and strongly dissipative events that release a large fraction of the elastic potential energy of the system and introduce large stress heterogeneities. A coarse grain analysis in terms of the stored elastic energy and the magnitude of the disorder provides new interesting insights on the dynamics of the model. Weakly dissipative events, which reproduce the seismic laws, are shown to follow a deterministic evolution. A statistical criterion for the initiation of big dissipative events is proposed.

Some Physical Properties of Self-Affine Rough Surfaces

We study some examples where the self-affine nature of surfaces determines the scaling of transport properties, beyond the simple geometrical characterization. The first example deals with the permeability of two identical surfaces which have been translated with respect to each other along their mean orientation. The second example is the force displacement characteristic of two elastic solids limited by independent self-affine surfaces which are pressed against each other.

Velocity weakening friction: A renormalization approach

The frictional response when a two-dimensional elastic body, under a steady antiplane shearing, slides over a rigid frictional surface is solved using a real-space renormalization technique. The model is mapped onto iterations of the classical Burridge-Knopoff chain model. A velocity weakening friction law is assumed for the interactions between the rigid interface and the elastic slider. Through the renormalization approach, we specifically study the coupling between the bulk elastic response of the slider and the friction at the interface level. The frictional system is characterized by its mechanical response in terms of mean friction force opposed to a prescribed constant velocity. The response is shown to differ strongly from the interface friction law. Under renormalization, the response converges toward an apparent dynamical Coulombs friction law, with distinct static and dynamic coefficients. This result provides some new insight in the “ubiquity” of Coulombs friction law. On the base of this model, the experimental observability of velocity weakening friction laws and their scaling to seismically active system are discussed. The main result is that velocity weakening friction law is scale dependent. Within the assumptions of the model, the friction law at the scale of the system appears to be rather insensitive to the exact form of the friction at local scale on the interface.

Propagative macrodislocation modes in an earthquake Fault model

We investigate a 1D dynamical version of the Burridge-Knopoff model for earthquakes with a velocity-weakening friction law. Such system exhibits two types of solution: chaotic motion and solitary-wave propagation. The latter can be seen as propagative localized macrodislocations whose shape, size and velocity are found to exhibit very little fluctuations. This property leads to resonances in the mean friction force as a function of the control parameter Θ, defined as the product of the driving displacement rate by the size of the system. The number of solitary waves is proportional to the parameter Θ. Such solitary-wave solutions can be correlated to recent self-healing crack models for the dynamics of earthquake rupture.

Slip correlations on a creeping fault

Using a quasi-static three dimensional fault model which accounts for long range elastic interactions, we examine the influence of spatial heterogeneities of frictional strength on the slip distribution along a creeping fault. Slip fluctuates spatially because of pinning on local asperities. We show that three regimes of slip correlations exist. The first regime results in a uniform slip as in an homogeneous medium. On the contrary, slip in the second regime highly fluctuates and is controlled by heterogeneities of frictional strength. The third regime is intermediate and develops areas of high slip that are much bigger than the local asperity size (self-affine properties of the slip distribution). This particular regime illustrates the possible misinterpretation of low frequency slip data (e.g. interferometric and GPS data) in terms of structural or compositional properties along the fault.

A MODEL FOR GOUGE DEFORMATION : IMPLICATIONS FOR REMANENT MAGNETIZATION

Using a two-dimensional construction for space-filling bearings (SFB) recently proposed by Herrmann et al.[1990], as an idealized description of the texture of gouge-like material, we study the effect of grain rotations in a shear band. The SFB model is by construction self-similar and the disk-size distribution follows a power-law in good agreement with the observed particle-size distribution of natural gouges. The kinematics of the model is described by a single degree of freedom, i.e.. each disk can only rotate, with no slip at the contact points. We investigate the isothermal remanent magnetization (IRM) reduction during such a cataclastic shearing. The problem can be solved analytically for this model. The magnetization loss, due to a disorientation of the grains can be shown to be a power-law as a function of the shear. The exponent of the power law is simply related to the fractal dimension of the particle-size distribution in the gouge. This result may stimulate experimental investigations using IRM to gain new insight into the gouge texture and kinematics.

GEOMETRY OF CONTACT BETWEEN SELF-AFFINE SURFACES

We study the geometry of the contact between two rigid self-affine surfaces. We investigate the mean shape of the surface in the vicinity of the contact point as well as the probability distribution of apertures a as a function of the distance to the contact point. The latter reveals two distinct scaling regimes which are characterized. We show that as the two surfaces are shifted with respect to each other, the contact point on one surface undergoes a “Levy walk”. If u is the relative shift of the surfaces, the distance d between the two contact points (before and after the shift), scales as d ∝ uα where the exponent α is shown to be equal to the roughness exponent of the surfaces.

Some physical properties of the Burridge-Knopoff model

The dynamics of the discrete Burridge-Knopoff model, initially presented for earthquake rupture, is analyzed here as a function of the main physical parameters. This discrete array of massive spring-connected rigid blocks, subject to a velocity weakening friction, exhibits two distinct regimes of solution, depending on the ratio between the characteristic loading and propagative time scales. The first regime is often described as “chaotic”, while the second is characterized by propagative solitary waves. The transition between these two regimes is numerically studied, and shown to be the transition between two distinct macrospic frictional regimes : a chaotic stick-slip and a continuous creep. ...