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International Journal for Multiscale Computational Engineering
Impact-faktor: 1.016 5-jähriger Impact-Faktor: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Druckformat: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2012003105
pages 527-549


Duy Khanh Trinh
MINES ParisTech, Centre des materiaux, CNRS UMR 7633, BP 87, F−91003 Evry Cedex, France
Ralf Janicke
Ruhr-Universitat Bochum, Institut fur Mechanik-Kontinuumsmechanik, IA 3/28, Universitatsstr. 150, D−44780 Bochum, Germany
Nicolas Auffray
Laboratoire Modelisation et Simulation Multi-echelles (MSME), UMR 8208 CNRS, Universite Paris-Est Marne-la-Vallee, 5 Bd Descartes, D−77454 Marne-la-Vallee, France
Stefan Diebels
Universitat des Saarlandes, Lehrstuhl fuer Technische Mechanik, Postfach 1511 50, D−66041 Saarbrucken, Germany
Samuel Forest


Several extensions of standard homogenization methods for composite materials have been proposed in the literature that rely on the use of polynomial boundary conditions enhancing the classical affine conditions on the unit cell. Depending on the choice of the polynomial, overall Cosserat, second gradient, or micromorphic homogeneous substitution media are obtained. They can be used to compute the response of the composite when the characteristic length associated with the variation of the applied loading conditions becomes of the order of the size of the material inhomogeneities. A significant difference between the available methods is the nature of the fluctuation field added to the polynomial expansion of the displacement field in the unit cell, which results in different definitions of the overall stress and strain measures and higher order elastic moduli. The overall higher order elastic moduli obtained from some of these methods are compared in the present contribution in the case of a specific periodic two-phase composite material. The performance of the obtained overall substitution media is evaluated for a chosen boundary value problem at the macroscopic scale for which a reference finite element solution is available. Several unsatisfactory features of the available theories are pointed out, even though some model predictions turn out to be highly relevant. Improvement of the prediction can be obtained by a precise estimation of the fluctuation at the boundary of the unit cell.


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