Power Laws Variance Scaling of Boolean Random Varieties

International audienceLong fibers or stratifed media show very long range correlations. These media can be simulated by models of Boolean random varieties and their iteration. They show non standard scaling laws with respect to the volume of domains K for the variance of the local volume fraction: on a large scale, the variance of the local volume fraction decreases according to power laws of the volume of K. The exponent is equal to (n-k)/ n for Boolean varieties with dimension k in the space Rn : 2/3 for Boolean fibers in 3D, and 1/3 for Boolean strata in 3D. When working in 2D, the scaling exponent of Boolean fibers is equal to 1/2. From the results of numerical simulations, these scaling laws are expected to hold for the prediction of the effective properties of such random media

Variance scaling of Boolean random varieties

Long fibers or stratied media show very long range correlations. These media can be simulated by models of Boolean random varieties. We study for these models the non standard scaling laws of the variance of the local volume fraction with the volume of domains K: on a large scale, a the variance of the local volume fraction decreases with power laws of the volume of K. The exponent is equal to 2/3 for Boolean fibers in 3D, and 1/3 for Boolean strata in 3D. When working in 2D, the scaling exponent of Boolean fibers is equal to 1/2 . These laws are expected to hold for the prediction of the e¤ective properties of such random media from numerical simulations

Introduction to some basic random morphological models

The Boolean RF are a generalization of the Boolean RACS. Their construction based on the combination of a sequence of primary RF by the operation supremum or infimum, and their main properties (among which the supremum or infimum infinite divisibility) are given in the case of scalar RF built on a Poisson point process

Particle-by-Particle Reconstruction of Ultrafiltration Cakes in 3D from Binarized TEM Images

Transmission electron microscopy (TEM) imaging is one of the few techniques available for direct observation of the microstructure of ultrafiltration cakes. TEM images yield local microstructural information in the form of two-dimensional grayscale images of slices a few particle diameters in thickness. This work presents an innovative particle-by-particle reconstruction scheme for simulating ultrafiltration cake microstructure in three dimensions from TEM images. The scheme uses binarized TEM images, thereby permitting use of lesser-quality images. It is able to account for short- and long-range order within ultrafiltration cake structure by matching the morphology of simulated and measured microstructures at a number of resolutions and scales identifiable within the observed microstructure. In the end, simulated microstructures are intended for improving our understanding of the relationships between cake morphology, ultrafiltration performance, and operating conditions

Elastic behavior of composites containing Boolean random sets of inhomogeneities

International audienceThe overall mechanical response as well as strain and stress field statistics of an heterogeneous material made of two randomly distributed, linear elastic phases, are investigated numerically. The Boolean model of spheres is used to generate microstructures consisting of either porous or rigid inclusions, at any volume fraction of the phases. Stress and strain field integral ranges, or equivalently the representative volume element, are computed and linked to features of the field statistics, and to the microstructure geometry

Elastic and electrical behavior of some random multiscale highly-contrasted composites

International audienceThe role of a non-uniform distribution of heterogeneities on the elastic as well as electrical properties of composites is studied numerically and compared with available theoretical results. Specifically, a random model made of embedded Boolean sets of spherical inclusions (see e.g. Jean et al, 2007) serves as the basis for building simple two-scales microstructures of ``granular''-type. Materials with ``infinitely-contrasted'' properties are considered, i.e. inclusions elastically behave as rigid particles or pores, or as perfectly-insulating or highly-conducting heterogeneities. The inclusion spatial dispersion is controlled by the ratio between the two characteristic lengths of the microstructure. The macroscopic behavior as well as the local response of composites are computed using full-field computations, carried out with the "Fast Fourier Transform" method (Moulinec and Suquet, 1994). The entire range of inclusion concentration, and dispersion ratios up to the separation of length scales are investigated. As expected, the non-uniform dispersion of inhomogeneities in multi-scale microstructures leads to increased reinforcing or softening effects compared to the corresponding one-scale model (Willot and Jeulin, 2009); these effects are however still significantly far apart from Hashin-Shtrikman bounds. Similar conclusions are drawn regarding the electrical conductivity

Estimation of tortuosity and reconstruction of geodesic paths in 3D

This work is licensed under a Creative Commons Attribution-NonCommercial License. Full text also available at http://www.ias-iss.org/ojs/IAS/article/view/941/870International audienceThe morphological tortuosity of a geodesic path in a medium can be defined as the ratio between its geodesic length and the Euclidean distance between its two extremities. Thus, the minimum tortuosity of all the geodesic paths into a medium in 2D or in 3D can be estimated by image processing methods using mathematical morphology. Considering a medium, the morphological tortuosities of its internal paths are estimated according to one direction, which is perpendicular to both starting and ending opposite extremities of the geodesic paths. The used algorithm estimates the morphological tortuosities from geodesic distance maps, which are obtained from geodesic propagations. The shape of the propagated structuring element used to estimate the geodesic distance maps on a discrete grid has a direct influence on the morphological tortuosity and has to be chosen very carefully. The results of our algorithm is an image with pixels p having a value equal to the length of the shortest path containing p and connected to two considered opposite boundaries A and B of the image. The analysis of the histogram of the morphological tortuosities gives access to their statistical distribution. Moreover, for each tortuosity the paths can be extracted from the original image, which highlights the location of them into the sample. However, these geodesic paths have to be reconstructed for further processing. The extraction, because applying a threshold on the tortuosities, results in disconnected components, especially for highly tortuous paths. This reconstruction consists in reconnecting these components to the geodesic path linking the two opposite faces, by means of a backtracking algorithm

Percolation d'agrégats multi-échelles de sphères et de fibres – Application aux nanocomposites

International audienceRESUME: Les nanocomposites comportant des sphères de noir de carbone ou des nanotubes de carbone de découverte récente [1] permettent d'élaborer des composites présentant des propriétés mécaniques, électriques et chimiques remarquables, essentiellement grâce à leur seuil de percolation très faible. L'agencement spatial des charges présente généralement plusieurs échelles: agrégats de nanoparticules, et zones de répulsion entre agrégats. Nous présentons une méthode de construction rapide et efficace, permettant de simuler la microstructure de matériaux composites de ce type, et d'estimer leur seuil de percolation. Cette méthode permet de simuler une distribution aléatoire 3D multi-échelle de sphères, ou de sphéro-cylindres de facteur de forme variable et d'orientation non uniforme, correspondant à des situations rencontrées dans les composites. A fraction volumique de charges donnée, il est possible d'abaisser significativement leur seuil de percolation, et d'optimiser les propriétés de nanocomposites. La percolation joue un rôle crucial concernant les propriétés macroscopiques effectives des matériaux composites hétérogènes. Ce rôle est dotant plus fort lorsque les constituants présentent un fort contraste de propriétés. Ces matériaux peuvent avoir une structure complexe de par leur processus de fabrication faisant intervenir un mélange non homogène des constituants. Leur morphologie présente alors plusieurs échelles de répartition des charges, comme par exemple des regroupements en agrégats, ou des zones complètement vide de charges. Nous présenterons une méthode rapide et efficace permettant de modéliser ces structures complexes et d'estimer leur seuil de percolation. Nous l'appliquerons à des modèles aléatoire multi-échelles de sphères et de sphéro-cylindres, ainsi qu'à des matériaux de structure complexe dont les charges sont modélisables par des sphères. Nous donnerons des estimations des seuils de percolation pour des distributions homogènes et non homogènes, que nous comparerons à d'autres méthodes analytiques et numériques

Regionalized Random Germs by a Classification for Probabilistic Watershed Application: Angiogenesis Imaging Segmentation

International audienceNew methods are presented to generate random germs regionalized by a previous classification in order to use probabilistic watershed on hyperspectral images. These germs are much more efficient than the standard uniform random germs