A general method for homogenization of the electromagnetic properties of a heterogeneous periodic medium is developed, based on its microstructure. This method is inspired by micromechanics (Nemat-Nasser and Hori, 1999). Contrary to other conventional techniques, commonly used in electromagnetism to calculate the overall properties of composites, this microstructurally-based method does not require an explicit numerical solution of the Maxwell equations. We define the macroscopic field quantities as volume averages of the spatially variable fields, taken over a representative volume element (RVE), consisting of a unit cell of the periodic medium (Hill, 1963; Willis, 1981; Hashin, 1983; Nemat-Nasser, 1986). The boundary conditions are based on the Bloch representation of wave propagation in the heterogeneous media. Instead of explicitly solving the Maxwell equations, these equations are directly used in the averaging scheme. This distinguishes our method from others, where usually a known point-wise solution is used to obtain the average field quantities. The resulting constitutive relations therefore may be used to directly estimate the response of any heterogeneous periodic assembly of material constituents of given geometry and properties.
On développe une méthode générale d'homogénéisation des propriétés électromagnétiques des milieux périodiques hétérogènes fondée sur une base microstructurelle. Cette méthode est inspirée de la micromécanique (Nemat-Nasser and Hori, 1999). Contrairement à d'autres techniques conventionnelles, couramment utilisées en électromagnétisme pour calculer les propriétés globales des composites, cette méthode à base microstructurelle ne nécessite pas de solution numérique explicite des équations de Maxwell. Nous définissons les champs macroscopiques comme des moyennes volumiques des champs spatialement variables sur un volume représentatif élémentaire (VRE), qui consiste en une cellule de base du milieu périodique (Hill, 1963 ; Willis, 1981 ; Hashin, 1983 ; Nemat-Nasser, 1986). Les conditions aux limites reposent sur la représentation de Bloch de la propagation d'ondes dans le milieu hétérogène. Au lieu de résoudre explicitement les équations de Maxwell, ces équations sont directement utilisées dans l'opération de moyenne. Ceci distingue notre méthode d'autres qui utilisent généralement une solution connue point par point pour obtenir les champs moyens. Les équations constitutives résultantes peuvent par conséquent être utilisées pour estimer directement la réponse d'un assemblage périodique hétérogène de constituants matériels de géométrie et de propriétés données.
Mots-clés : Mécanique des solides numérique, Milieux périodiques, Propriétés électromagnétiques
Alireza V. Amirkhizi 1; Sia Nemat-Nasser 1
@article{CRMECA_2008__336_1-2_24_0,
author = {Alireza V. Amirkhizi and Sia Nemat-Nasser},
title = {Microstructurally-based homogenization of electromagnetic properties of periodic media},
journal = {Comptes Rendus. M\'ecanique},
pages = {24--33},
publisher = {Elsevier},
volume = {336},
number = {1-2},
year = {2008},
doi = {10.1016/j.crme.2007.10.012},
language = {en},
}
TY - JOUR AU - Alireza V. Amirkhizi AU - Sia Nemat-Nasser TI - Microstructurally-based homogenization of electromagnetic properties of periodic media JO - Comptes Rendus. Mécanique PY - 2008 SP - 24 EP - 33 VL - 336 IS - 1-2 PB - Elsevier DO - 10.1016/j.crme.2007.10.012 LA - en ID - CRMECA_2008__336_1-2_24_0 ER -
Alireza V. Amirkhizi; Sia Nemat-Nasser. Microstructurally-based homogenization of electromagnetic properties of periodic media. Comptes Rendus. Mécanique, Duality, inverse problems and nonlinear problems in solid mechanics, Volume 336 (2008) no. 1-2, pp. 24-33. doi : 10.1016/j.crme.2007.10.012. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.10.012/
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