Notion of representative volume element for heterogeneous materials: statistical and numerical approach
Résumé
The Representative Volume Element (RVE) plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties. A quantitative definition of its size is proposed in this work using a numerical and statistical approach. A RVE size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume $V$ of microstructure that one is able to consider. It is shown to depend on the investigated morphological or physical
property, the contrast in the properties of the constituents, and their volume fractions. The methodology is developed on a specific random microstructure, namely a two-phase three-dimensional Voronoi mosaic and applied to a real two-phase heterogeneous material from food industry. Large scale finite element simulations of volumes of
different sizes are performed in the case of linear elasticity (thermal conductivity respectively), using parallel computing. The volumes are subjected to homogeneous strain (gradient of temperature respectively),
stress (heat flux respectively) at the boundary or periodic
boundary conditions. The effective properties can be determined for large volumes and a small number of realizations. Conversely, smaller volumes can be used providing that a sufficient number of realizations is
considered. A bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions. The variance of computed apparent properties for each volume size is used to define the
precision of the estimation. The key-notion of integral range is introduced to relate this error estimation and the definition of the RVE size. For given precision and number of realizations, one is able to provide a minimal volume size for the computation of effective properties. The results can also be used to predict the minimal number of
realizations that must be considered for a given volume
size in order to estimate the effective property for a given precision. The RVE sizes found for elastic and thermal properties, but also for a geometrical property like volume fraction, are compared. A general comparison of the elastic and thermal properties of three different microstructures is given for Voronoi mosaics, two real material from food industry and another virtual model, a boolean model of hexagonal prismatic rods and plates. Computational homogenization technique is used to predict the effective properties from 3D confocal images of real
samples. An analysis of the percolation strain fields in deformed samples is proposed to select stiffer or higher conductive products. The present work can be regarded as a first step towards a computational approach of the design of microstructures for wanted overall properties. The aim is to explore new morphologies that can lead to unexpected properties like outstanding stiffness or conductivity, or controlled compliance.
property, the contrast in the properties of the constituents, and their volume fractions. The methodology is developed on a specific random microstructure, namely a two-phase three-dimensional Voronoi mosaic and applied to a real two-phase heterogeneous material from food industry. Large scale finite element simulations of volumes of
different sizes are performed in the case of linear elasticity (thermal conductivity respectively), using parallel computing. The volumes are subjected to homogeneous strain (gradient of temperature respectively),
stress (heat flux respectively) at the boundary or periodic
boundary conditions. The effective properties can be determined for large volumes and a small number of realizations. Conversely, smaller volumes can be used providing that a sufficient number of realizations is
considered. A bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions. The variance of computed apparent properties for each volume size is used to define the
precision of the estimation. The key-notion of integral range is introduced to relate this error estimation and the definition of the RVE size. For given precision and number of realizations, one is able to provide a minimal volume size for the computation of effective properties. The results can also be used to predict the minimal number of
realizations that must be considered for a given volume
size in order to estimate the effective property for a given precision. The RVE sizes found for elastic and thermal properties, but also for a geometrical property like volume fraction, are compared. A general comparison of the elastic and thermal properties of three different microstructures is given for Voronoi mosaics, two real material from food industry and another virtual model, a boolean model of hexagonal prismatic rods and plates. Computational homogenization technique is used to predict the effective properties from 3D confocal images of real
samples. An analysis of the percolation strain fields in deformed samples is proposed to select stiffer or higher conductive products. The present work can be regarded as a first step towards a computational approach of the design of microstructures for wanted overall properties. The aim is to explore new morphologies that can lead to unexpected properties like outstanding stiffness or conductivity, or controlled compliance.
Notion de volume élémentaire représentatif pour les matériaux hétérogènes : approche statistique et numérique.