Stéphane Roux

Disorder and fracture

Sixteen papers (lectures) from the June 1989 Institute are grouped in sections covering tools, diffusion-limited aggregation model, statistical models of fracture, rheology and fracture, and materials and applications. The papers range from theoretical concepts to practical applications. Each sectio

Annealed Model for Breakdown Processes

We consider a stochastic model of breakdown processes, similar to the dielectric breakdown model, but where the connectivity of the crack is not required. Starting with a homogeneous medium, at each time step a bond is broken at random with a probability proportional to the current it carries raised to a power η. For η 2, one large cluster of broken bonds generated in the breakdown process is much bigger than the other clusters and it has the scaling properties of the clusters of the dielectric breakdown model of Niemeyer et al.

Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media

We present a spatial renormalization group algorithm to handle immiscibletwo-phase flow in heterogeneous porous media. We call this algorithmFRACTAM-R, where FRACTAM is an acronym for Fast Renormalization Algorithmfor Correlated Transport in Anisotropic Media, and the R stands for relativepermeability. Originally, FRACTAM was an approximate iterative process thatreplaces the L × L lattice of grid blocks, representing the reservoir,by a (L/2) × (L/2) one. In fact, FRACTAM replaces the original L× L lattice by a hierarchical (fractal) lattice, in such a way thatfinding the solution of the two-phase flow equations becomes trivial. Thistriviality translates in practice into computer efficiency. For N=L ×L grid blocks we find that the computer time necessary to calculatefractional flow F(t) and pressure P(t) as a function of time scales as τ∼ N1.7 for FRACTAM-R. This should be contrasted with thecomputational time of a conventional grid simulator τ ∼N2.3. The solution we find in this way is an accurateapproximation to the direct solution of the original problem.

Minimal Path on the Hierarchical Diamond Lattice

We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Lévys distributions with a power-law decay at-∞, and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtain the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.

Inverse Identification of the Loading Applied by a Tire on a Landing Gear Wheel

This study aims at identifying the loading applied by a tire on a landing gear wheel for an inflation case. A full scale test instrumented via stereo-DIC (Digital Image Correlation) and strain gages is performed. A 3D finite element model of the wheel is developed and a parameterization of the tire-rim loading is proposed based on model reduction techniques. This parameterization is further used for an inverse identification of the loading parameters. This approach leads to a simpler and more robust problem that can easily be extended to more complex service loadings.

Non-local and non-linear problems in the physics of disordered media

Many concepts issued from the field of statistical physics have enriched our understanding of the physical properties of random materials. This text outlines some of these recent developments. We will recall some basic characteristics of percolation and will extend them to problems displaying non-local ordering and to problems where the existence of threshold behaviors at the local scale lead to strong heterogeneous behaviors in systems with an initial small disorder.