Alberto Rosso

Fractional Laplacian in bounded domains.

The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Significance Yield stress solids flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield. Here we propose a scaling description of the yielding transition that relates the flow curve, the statistics of the avalanches of plasticity observed at threshold, and the density of local zones that are about to yield. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. Numerical simulations on a simple elasto-plastic model find good agreement with our predictions. Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel–Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σc, the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, df, and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elasto-plastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions.

Dynamics below the Depinning Threshold in Disordered Elastic Systems

We study the steady-state low-temperature dynamics of an elastic line in a disordered medium below the depinning threshold. Analogously to the equilibrium dynamics, in the limit T-->0, the steady state is dominated by a single configuration which is occupied with probability 1. We develop an exact algorithm to target this dominant configuration and to analyze its geometrical properties as a function of the driving force. The roughness exponent of the line at large scales is identical to the one at depinning. No length scale diverges in the steady-state regime as the depinning threshold is approached from below. We do find a divergent length, but it is associated only with the transient relaxation between metastable states.

Width distribution of contact lines on a disordered substrate

We have studied the roughness of a contact line on a disordered substrate by measuring its width distribution, which characterizes the roughness completely. The measured distribution is in excellent agreement with the distribution calculated in previous works, extended here to the case of open boundary conditions. This type of analysis, which is performed here on experimental data, provides a strong confirmation that the Joanny-de Gennes model is not sufficient to describe the dynamics of the contact line at the depinning threshold.

Avalanche-size distribution at the depinning transition: A numerical test of the theory

We calculate numerically the sizes

Creep dynamics of elastic manifolds via exact transition pathways

S

Creep Motion of an Elastic String in a Random Potential

of jumps (avalanches) between successively pinned configurations of an elastic line

On the density of shear transformations in amorphous solids

(d=1)

Path-integral representation for quantum spin models : Application to the quantum cavity method and Monte Carlo simulations

or interface

Universal interface width distributions at the depinning threshold

(d=2)