Damien Vandembroucq

Extremal model for amorphous media plasticity.

An extremal model for the plasticity of amorphous materials is studied in a simple two-dimensional antiplane geometry. The steady state is analyzed through numerical simulations. Long-range spatial and temporal correlations in local slip events are shown to develop, leading to nontrivial and highly anisotropic scaling laws. In particular, the plastic strain is shown to concentrate statistically over a region which tends to align perpendicular to the displacement gradient. By construction, the model can be seen as giving rise to a depinning transition, the threshold of which (i.e., the macroscopic yield stress) also reveals scaling properties reflecting the localization of the activity.

Effective toughness of heterogeneous brittle materials

A heterogeneous brittle material characterized by a random field of local toughness Kc(x) can be represented by an equivalent homogeneous medium of toughness, Keff. Homogenization refers to a process of estimating Keff from the local field Kc(x). An approach based on a perturbative expansion of the stress intensity factor along a rough crack front shows the occurrence of different regimes depending on the correlation length of the local toughness field in the direction of crack propagation. A “weak pinning” regime takes place for long correlation lengths, where the effective toughness is the average of the local toughness. For shorter correlation lengths, a transition to “strong pinning” occurs leading to a much higher effective toughness, and characterized by a propagation regime consisting in jumps between pinning configurations.

Density hardening plasticity and mechanical ageing of silica glass under pressure: a Raman spectroscopic study

In addition to a flow, plastic deformation of structural glasses (in particular amorphous silica) is characterized by a permanent densification. Raman spectroscopic estimators are shown to give a full account of the plastic behavior of silica under pressure. While the permanent densification of silica has been widely discussed in terms of amorphous–amorphous transition, from a plasticity point of view, the evolution of the residual densification with the maximum pressure of a pressure cycle can be discussed as a density hardening phenomenon. In the framework of such a mechanical ageing effect, we propose that the glass structure could be labeled with the maximum pressure experienced by the glass and that the saturation of densification could be associated with the densest packing of tetrahedra only linked by their vertices.

Material-independent crack arrest statistics

The propagation of (planar) cracks in a heterogeneous brittle material characterized by a random field of toughness is considered, taking into account explicitly the effect of the crack front roughness on the local stress intensity factor. In the so-called strong-pinning regime, the onset of crack propagation appears to map onto a second-order phase transition characterized by universal critical exponents which are independent of the local characteristics of the medium. Propagation over large distances can be described by using a simple one-dimensional description, with a correlation length and an effective macroscopic toughness distribution that scale in a non-trivial fashion with the crack front length. As an application of the above concepts, the arrest of indentation cracks is addressed, and the analytical expression for the statistical distribution of the crack radius at arrest is derived. The analysis of indentation crack radii on alumina is shown to obey the predicted algebraic expression for the radius distribution and its dependence on the indentation load.

Electromagnetic wave scattering from conducting self-affine surfaces: an analytic and numerical study.

We derive an analytical expression for the scattering of an s-polarized plane wave from a perfectly conducting self-affine one-dimensional surface in the framework of the Kirchhoff approximation. We show that most of the results can be recovered by means of a scaling analysis. We identify the typical slope taken over one wavelength as the relevant parameter controlling the scattering process. We compare our predictions with direct numerical simulations performed on surfaces of varying roughness parameters and confirm the broad range of applicability of our description up to very large roughness. Finally we verify that a nonzero electrical resistivity, provided that it is small, does not invalidate our results.

Conformal Mapping on Rough Boundaries I: Applications to harmonic problems

The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this boundary. We introduce a conformal mapping technique that is tailored to this problem in two dimensions. An efficient algorithm is introduced to compute the conformal map for arbitrarily chosen boundaries. Harmonic fields can then simply be read from the conformal map. We discuss applications to ’equivalent’ smooth interfaces. We study the correlations between the topography and the field at the surface. Finally we apply the conformal map to the computation of inhomogeneous harmonic fields such as the derivation of Green function for localized flux on the surface of a rough boundary. Defining and computing effective properties of heterogeneous media is a subject which has been studied for a long time, and for which a number of powerful techniques have been developped. In most cases however, the heterogeneities are considered to lie in the bulk of the material. Another type of inhomogeneity is due to the random geometry of the surface on which boundary conditions are applied. This study focusses on this second type. We will thus consider homogeneous media which are limited by a rough surface or interface. Our purpose here is to introduce a very efficient way of solving harmonic problems in two-dimensional systems for any geometry of the boundary.

Frozen capillary waves on glass surfaces: an AFM study

Abstract.Using atomic force microscopy on silica and float glassnsurfaces, we give evidence that the roughness of melted glass surfacesncan be quantitatively accounted for by frozen capillary waves. In thisnframework the height spatial correlations are shown to obey anlogarithmic scaling law; the identification of this behaviour allows tonestimate the ratio kTF/πγ where k is the Boltzmannnconstant, γ the interface tension and TF the temperaturencorresponding to the “freezing” of the capillary waves. Variationsnof interface tension and (to a lesser extent) temperatures ofnannealing treatments are shown to be directly measurable from anstatistical analysis of the roughness spectrum of the glass surfaces.nn

Universal depinning force fluctuations of an elastic line: Application to finite temperature behavior

The depinning of an elastic line in a random medium is studied via an extremal model. The latter gives access to the instantaneous depinning force for each successive conformation of the line. Based on conditional statistics the universal and nonuniversal parts of the depinning force distribution can be obtained. In particular the singular behavior close to a (macroscopic) critical threshold is obtained as a function of the roughness exponent of the front. We show, moreover, that the advance of the front is controlled by a very tenuous set of subcritical sites. Extension of the extremal model to a finite temperature is proposed, the scaling properties of which can be discussed based on the statistics of depinning force at zero temperature. In particular a new temperature-dependent correlation length is shown to become relevant.

Material-independent crack arrest statistics: Application to indentation experiments

An extensive experimental study of indentation and crack arrest statistics is presented for four different brittle materials (alumina, silicon carbide, silicon nitride, glass). Evidence is given that the crack length statistics is described by a universal (i.e., material independent) distribution. The latter directly derives from results obtained when modeling crack propagation as a depinning phenomenon. Crack arrest (or effective toughness) statistics appears to be fully characterized by two parameters, namely, an asymptotic crack length (or macroscopic toughnes) value and a power law size-dependent width. The experimental knowledge of the crack arrest statistics at one given scale thus gives access to its knowledge at all scales.

Large-scale numerical simulations of ultrametric long-range depinning.

The depinning of an elastic line interacting with a quenched disorder is studied for long-range interactions, applicable to in-plane crack propagation or wetting. An ultrametric distance is introduced instead of the Euclidean distance, allowing for a drastic reduction of the numerical complexity of the problem. Based on large-scale simulations, two to three orders of magnitude larger than previously considered, we obtain a very precise determination of critical exponents which are shown to be indistinguishable from their Euclidean metric counterparts. Moreover, the scaling functions are shown to be unchanged. The choice of an ultrametric distance thus does not affect the universality class of the depinning transition and opens the way to an analytic real-space renormalization-group approach.