A New Definition of the Representative Volument Element in Numerical Homogenization Problems and its Application to the Performance Evaluation of Analytical Homogenization Models.

RESUME Le Volume Elementaire Representatif (VER) joue un role important dans la mecanique des materiaux composites dans le but de determiner leurs proprietes effectives. L'homogeneisation numerique permet de calculer avec precision les proprietes effectives des composites etant donne que le VER soit utilise. Dans le cadre d'homogeneisation numerique, le VER refere a un ensemble de volumes elementaires du materiau qui produisent, en moyenne, les proprietes effective du composite. Plusieurs definitions quantitatives du VER, detaillant les methodes de sa determination, sont trouvees dans la litterature. Dans cette etude, les differentes definitions du VER sont evaluees vis a vis leurs estimations des proprietes effectives du composite. L'etude est realisee sur une microstructure elastique bi-phasee, les composites renforces par des fibres aleatoirement dispersees. Des simulations par la methode des elements finis a grand nombre de degres de liberte sur des volume de taille differentes sont parallelisees sur des serveurs de calculs. Les volumes elementaires du materiau sont generees virtuellement et assujetties a des conditions aux rives periodiques. Il est demontre que les definitions populaires du VER, basees sur la convergence des proprietes moyennes, produisent des proprietes erronees. Une nouvelle definition du VER est introduite basee sur les variations statistiques des proprietes issues des volumes elementaires. Cette definition est demontree apte a estimer avec grande precision les proprietes effectives des composites a fibres aleatoires. De plus, la nouvelle definition permet de calculer les proprietes effectives a un VER plus petit que celui necessaire selon les autres definitions, mais aussi de reduire considerablement le nombre de simulations necessaires pour determiner le VER. Les proprietes effectives calcules par homogeneisation numeriques sont comparees a celles des modeles d'homogeneisation analytiques. Les comparaisons sont realisees sur une large gamme de rapports de forme des fibres (allant jusqu'a 120), de contraste des proprietes (allant jusqu'a 300) pour des fractions volumiques a moins de 20%. Le modele de Mori-Tanaka, ainsi que les methodes d'homogeneisation par deux etapes de Mori-Tanaka/Voigt et Lielens/Voigt produisent des estimations precises des proprietes effectives des composites a fibres aleatoirement dispersees et ce jusqu'au rapport de forme de 100. Ceci permet d'utiliser ces modeles analytiques dans un calcul rapide et de grande precision des proprietes effectives de ce type de composites. Pour des fibres de rapports de forme depassant 100, une saturation de la rigidite effective est observee dans les resultats numeriques et aucun des modeles analytiques ne produit des estimations precises des proprietes effectives pour cette gamme de microstructures. ----------ABSTRACT The Representative Volume Element (RVE) plays a central role in the mechanics of composite materials with respect to predicting their effective properties. Numerical homogenization delivers accurate estimations of composite effective properties when associated with a RVE. In computational homogenization, the RVE refers to an ensemble of random material volumes that yield, by an averaging procedure, the effective properties of the bulk material, within a tolerance. A large diversity of RVE quantitative definitions, providing computational methods to estimate the RVE size, are found in literature. In this study, the ability of the different RVE definitions to yield accurate effective properties is investigated. The assessment is conducted on a specific random microstructure, namely an elastic two-phase three dimensional composite reinforced by randomly oriented fibers. Large scale finite element simulations of material volumes of different sizes are performed on high performance computational servers using parallel computing. The materials volumes are virtually generated and subjected to periodic boundary conditions. It is shown that most popular RVE definitions, based on convergence of the properties when increasing the material volume, yields inaccurate effective properties. A new RVE definition is introduced based on the statistical variations of the properties computed from material volumes. It is shown to produce more accurate estimations of the effective properties. In addition, the new definition produced RVE that are smaller in size than that of other RVE definitions ; also the number of necessary finite element simulations to determine the RVE is substantially reduced. The computed effective properties are compared to that of analytical models. The comparisons are performed for a wide range of fibers aspect ratios (up to 120), properties contrast (up to 300) and volume fractions only up to 20% due to computational limits. The Mori-Tanaka model and the two-step analytical methods of Mori-Tanaka/Voigt and Lielens/Voigt are best suited for estimating the effective elastic properties of composites reinforced by randomly oriented fibers of aspect ratio up to 100. This finding allows the future utilization of those analytical models for very quick and accurate effective properties calculations for this class of composites. For aspect ratios larger than 100, a stiffening saturation occurs and no analytical model delivers accurate estimations of the effective properties. The present work can be regarded as an awareness flag regarding the validity of currently utilized RVE determination techniques. Moreover, this study provides a first step towards optimization of the RVE determination methods for computing accurate effective properties. The aim is to explore new RVE definitions that yield smaller volumes and have lower determination costs, while yielding accurate effective properties.