Determination of the size of the representative volume element for random composites: statistical and numerical approach |
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| Affiliation: | 1. Centre des Matériaux/UMR 7633, Ecole des Mines de Paris/CNRS, B.P. 87, 91003 Evry, France;2. Centre de Morphologie Mathématique, Ecole des Mines de Paris, 35, Rue St-Honoré, 77305 Fontainebleau, France;1. Laboratoire de Mécanique des Structures et Matériaux, LMSM, Université H. Lakhdar, Batna, Algeria;2. Laboratoire de Mécanique de Lille, LML, CNRS/UMR 8107, Université Lille 1, Villeneuve d''Ascq, France;1. Department of Aeronautics, Imperial College London, South Kensington Campus, SW7 2AZ, UK;2. Mechanical Methods, Rolls-Royce Plc, PO Box 31, DE24 8BJ Derby, UK;1. Department of Mechanical Engineering, University of Massachussetts Lowell, 197 Riverside St., Lowell, MA 01854, USA;2. Institute of Textile Technology, RWTH Aachen University, Otto-Blumenthal Str. 1, 52074 Aachen, Germany;3. Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe Str. 1, 52074 Aachen, Germany;1. Institute of Structural Analysis & Antiseismic Research, National Technical University of Athens, 9 Iroon Polytechneiou, Zografou Campus, 15780 Athens, Greece;2. Institute of Structural Analysis & Dynamics of Structures, Department of Civil Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece |
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| Abstract: | 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. 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 applied to a specific random microstructure, namely a two-phase three-dimensional Voronoı̈ mosaic. Finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity. The volumes are subjected to homogeneous strain, stress 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 are 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 wanted 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. |
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