Comptes Rendus
Microstructurally-based homogenization of electromagnetic properties of periodic media
Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 24-33.

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.

Published online:
DOI: 10.1016/j.crme.2007.10.012
Keywords: Computational solid mechanics, Periodic media, Electromagnetic properties
Mot clés : Mécanique des solides numérique, Milieux périodiques, Propriétés électromagnétiques

Alireza V. Amirkhizi 1; Sia Nemat-Nasser 1

1 Center of Excellence for Advanced Materials, University of California, San Diego, CA 92093-0416, USA
@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  - 
%0 Journal Article
%A Alireza V. Amirkhizi
%A Sia Nemat-Nasser
%T Microstructurally-based homogenization of electromagnetic properties of periodic media
%J Comptes Rendus. Mécanique
%D 2008
%P 24-33
%V 336
%N 1-2
%I Elsevier
%R 10.1016/j.crme.2007.10.012
%G en
%F CRMECA_2008__336_1-2_24_0
Alireza V. Amirkhizi; Sia Nemat-Nasser. Microstructurally-based homogenization of electromagnetic properties of periodic media. Comptes Rendus. Mécanique, 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/

[1] T. Mura Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, The Netherlands, 1987

[2] S. Nemat-Nasser; M. Hori Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier Science B.V., Amsterdam, 1999

[3] M.L. Dunn; M. Taya Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites, Int. J. Solids Struct., Volume 30 (1993), pp. 161-175

[4] Y. Benveniste On the micromechanics of fibrous piezoelectric composites, Mech. Mater., Volume 18 (1994), pp. 183-193

[5] G.J. Dvorak; Y. Benveniste On micromechanics of inelastic and piezoelectric composites (T. Tatsumi; E. Watanabe; T. Kambe, eds.), Theoretical and Applied Mechanics, Elsevier Science B.V., Amsterdam, 1997

[6] T. Chen Further correspondence between plane piezoelectricity and generalized strain elasticity, Proc. Roy. Soc. London A, Volume 554 (1998), pp. 873-884

[7] J.R. Willis Variational principles and operator equations for electromagnetic waves in inhomogeneous media, Wave Motion, Volume 6 (1984), pp. 127-139

[8] R. Hill Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, Volume 11 (1963), pp. 357-372

[9] J.R. Willis Variational and related methods for the overall properties of composites (C.S. Yih, ed.), Advances in Applied Mechanics, vol. 21, Academic Press, New York, 1981

[10] Z. Hashin Analysis of composite materials—A survey, J. Appl. Mech., Volume 50 (1983), pp. 481-505

[11] S. Nemat-Nasser Overall stresses and strains in solids with microstructure (J. Gittus; J. Zarka, eds.), Modeling Small Deformations of Polycrystals, Elsevier Applied Science, Amsterdam, 1986

[12] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978

[13] A. Sihvola Electromagnetic Mixing Formulas and Applications, Institute of Electrical Engineers, London, 1999

[14] G.W. Milton The Theory of Composites, Cambridge University Press, Cambridge, UK, 2002

[15] F.N.H. Robinson Macroscopic Electromagnetism, Pergamon Press, Oxford, UK, 1973

[16] J.B. Pendry Photonic band structures, J. Mod. Opt., Volume 41 (1994), pp. 209-229

[17] J.B. Pendry Calculating photonic band structure, J. Phys.: Condens. Matter, Volume 8 (1996), pp. 1085-1108

[18] A.V. Amirkhizi, S. Nemat-Nasser, Numerical calculation of electromagnetic properties including chirality parameters for uniaxial bianisotropic media, Smart Mater. Struct. 17 (2008)

[19] D.R. Smith; J.B. Pendry Homogenization of metamaterials by field averaging, J. Opt. Soc. Am. B, Volume 23 (2006), pp. 391-403

[20] S. Wolfram The Mathematica Book, Wolfram Media/Cambridge University Press, Champaign, IL/Cambridge, UK, 1999

[21] Ansoft, 2001. Ansfot HFSS 8.0 User Documentation. Ansoft Corporation, Pittsburgh, PA

[22] S.C. Nemat-Nasser; A.V. Amirkhizi; T.A. Plaisted; J.B. Isaacs; S. Nemat-Nasser Structural composites with integrated electromagnetic functionality, Proc. SPIE, Volume 4698 (2002), pp. 237-245

[23] J.C. Maxwell Garnett Colours in metal glasses and metal films, Philos. Trans. R. Soc. London A, Volume 203 (1904), pp. 385-420

[24] J.D. Jackson Classical Electrodynamics, John Wiley and Sons Inc., New York, 1999

[25] I.V. Lindell; A.H. Sihvola; S.A. Tretyakov; A.J. Viitanen Electromagnetic Waves in Chiral and Bi-Isotropic Media, Artech House, Boston, 1994

[26] S. Nemat-Nasser, A.V. Amirkhizi, Effective electromagnetic properties of periodic media, in: Progress in Electromagnetics Research Symposium PIERS 2002, July 1–5, 2002, Cambridge, MA

Cited by Sources:

Comments - Policy