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The purpose of this study was to establish a method based on an iterative scheme to approximate the numerical solution obtained from finite elements analysis for an RVE in two and three dimensions based on the homogenization concept for the assessment of the effective properties. The bounds of Hashin–Shtrikman and Voigt–Reuss were considered in the iterative process based on an updating of the constitutive relations of these models respectively. In this study, by assumption, we took the particular case of the heterogeneous materials with several elastic isotopic phases. The output variables considered using the iterative process are the bulk, shear modulus and the thermal conductivity. We have found a fast convergence of the iterative solution to the numerical result with a suitable concordance between the two solutions at the final step.

International audience; The main goal of this paper is to predict the elastic modulus of partially intercalated and exfoliated polymer-clay nano-composites using numerical homogenization techniques based on the finite element method. The representative volume element was employed here to capture nano-composites microstructure, where both intercalated exfoliated and clay platelets coexisted together. The effective macroscopic properties of the studied microstructure are obtained with two boundary conditions: periodic boundary conditions and kinematic uniform boundary conditions. The effect of particle volume fractions, aspect ratio, number and distribution of particles and the type of boundary conditions are numerically studied for different configurations. This paper investigate also the performance of several classical analytical models as Mori and Tanaka model, Halpin and Tsai model, generalized self consistent model through their ability to estimate the mechanical properties of nano-composites. A comparison between simulation results of polypropylene clay nano-composites, analytical methods and experimental data has confirmed the validity of the set results.