Comptes Rendus
Second-order theory for nonlinear composites and application to isotropic constituents
Comptes Rendus. Mécanique, Volume 334 (2006) no. 10, pp. 575-581.

New prescriptions are proposed for the ‘reference’ fields in the context of the ‘second-order’ nonlinear homogenization method [P. Ponte Castañeda, Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory, J. Mech. Phys. Solids 50 (2002) 737–757], and are used to generate estimates for the effective behavior and first moments of the local fields in nonlinear composites. The new prescriptions yield simple, analytical expressions not only for the effective potentials, but also for the macroscopic stress-strain relation, as well as for the phase averages of the strain and stress fields. For illustrative purposes, ‘second-order’ estimates of the Hashin–Shtrikman type are provided for two-phase, transversely-isotropic composites with power-law phases, and are compared with exact results available for power-law, multiple-rank, sequential laminates. The agreement is found to be quite good for all ranges of nonlinearities and inclusion concentrations considered.

On utilise la méthode d'homogénéisation non linéaire proposée par Ponte Castañeda [P. Ponte Castañeda, Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory, J. Mech. Phys. Solids 50 (2002) 737–757], dite du « second ordre », pour générer des estimations pour le comportement effectif et les premiers moments des champs locaux dans des composites non-linéaires. Des expressions analytiques simples sont données non seulement pour les potentiels effectifs mais également pour la relation contrainte-déformation macroscopique, aussi bien que pour les moyennes par phase des champs de contrainte et de déformation. Des estimations du type de Hashin–Shtrikman sont données pour des composites biphasés, isotropes avec des phases suivant une loi puissance, et sont comparées aux résultats exacts disponibles pour les matériaux laminés. L'accord s'avère bon pour toutes les valeurs de la non-linéarité et de concentration d'inclusion considérées.

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DOI: 10.1016/j.crme.2006.06.006
Keywords: Computational solid mechanics, Homogenization, Nonlinear behavior, Ideal plasticity
Mot clés : Mécanique des solides numérique, Homogénéisation, Comportement non linéaire, Plasticité parfaite

Martín I. Idiart 1, 2; Kostas Danas 1, 2; Pedro Ponte Castañeda 1, 2

1 Laboratoire de mécanique des solides, CNRS UMR 7649, département de mécanique, École polytechnique, 91128 Palaiseau cedex, France
2 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA
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Martín I. Idiart; Kostas Danas; Pedro Ponte Castañeda. Second-order theory for nonlinear composites and application to isotropic constituents. Comptes Rendus. Mécanique, Volume 334 (2006) no. 10, pp. 575-581. doi : 10.1016/j.crme.2006.06.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.06.006/

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