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

Random Image Models for Microstructure Analysis and Simulation

A wide spectrum of Random Function models, based on the theory of Random sets, are introduced to simulate single or multivariate signals as encountered in the common practice of electron probe microscopy. These models are built in three steps, combining the choice of a family of primary random functions and of Poisson varieties in the n-dimensional space for their implantation. For electron microscopy images, they can describe the following situations: - topography (as obtained from stereo pair images in fractography) simulated by Boolean and by alternate sequential random functions; - thick slices (as in the case of Transmission Electron Microscope (TEM) and Scanning Transmission Electron Microscope (STEM) specimens) for the dilution random functions; - perspective views (e.g. secondary electron images in the SEM from non planar samples, such as powder samples) for the Dead Leaves model; - multispectral mappings on polished sections. Their main strength is to enable the estimation of parameters (namely the statistical properties of the structural unit made of primary random functions, and the density of its implantation in space) from simple operations and measurements on grey level images based on Mathematical Morphology, without any segmentation of images. This purpose is illustrated by the main properties of the models with reference to electron microscopy and microprobe situations, and by 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

Towards gigantic RVE sizes for 3D stochastic fibrous networks

The size of representative volume element (RVE) for 3D stochastic fibrous media is investigated. A statistical RVE size determination method is applied to a specific model of random microstructure: Poisson fibers. The definition of RVE size is related to the concept of integral range. What happens in microstructures exhibiting an infinite integral range? Computational homogenization for thermal and elastic properties is performed through finite elements, over hundreds of realizations of the stochastic microstructural model, using uniform and mixed boundary conditions. The generated data undergoes statistical treatment, from which gigantic RVE sizes emerge. The method used for determining RVE sizes was found to be operational, even for pathological media, i.e., with infinite integral range, interconnected percolating porous phase and infinite contrast of propertie

Random Walk Based Stochastic Modeling of 3D Fiber Systems

published in Physical Review E, publisher's version available on http://link.aps.org/doi/10.1103/PhysRevE.83.041804For the simulation of fiber systems, there exist several stochastic models: systems of straight non-overlapping fibers, systems of overlapping bending fibers or fiber systems created by sedimentation. However, there is a lack of models providing dense, non-overlapping fiber systems with a given random orientation distribution and a controllable level of bending. We introduce a new stochastic model in this paper, that generalizes the force-biased packing approach to fibers represented as chains of balls. The starting configuration is modeled using random walks, where two parameters in the multivariate von Mises-Fisher orientation distribution control the bending. The points of the random walk are associated with a radius and the current orientation. The resulting chains of balls are interpreted as fibers. The final fiber configuration is obtained as an equilibrium between repulsion forces avoiding crossing fibers and recover forces ensuring the fiber structure. This approach provides high volume fractions up to 72.0075%

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

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