On a class of three-phase checkerboards with unusual effective properties

Richard V. Craster; Sébastien Guenneau; Julius Kaplunov; Evgeniya Nolde

doi:10.1016/j.crme.2011.03.016

On a class of three-phase checkerboards with unusual effective properties

Richard V. Craster ¹; Sébastien Guenneau ² ; Julius Kaplunov ³; Evgeniya Nolde ³

¹ Department of Mathematics, Imperial College London, London SW7-2AZ, UK
² Institut Fresnel, UMR CNRS 6133, University of Aix-Marseille, 13000 Marseille, France
³ Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 411-417.

Abstract
Résumé

We examine the band spectrum, and associated Floquet–Bloch eigensolutions, arising in a class of three-phase periodic checkerboards. On a periodic cell $[- 1, 1 [^{2}$ , the refractive index, n, is defined by $n^{2} = 1 + g_{1} (x_{1}) + g_{2} (x_{2})$ with $g_{i} (x_{i}) = r^{2}$ for $0 ⩽ x_{i} < 1$ , and $g_{i} (x_{i}) = 0$ for $- 1 ⩽ x_{i} < 0$ where $r^{2}$ is constant. We find that for $r^{2} > - 1$ the lowest frequency branch goes through origin with linear behaviour, which leads to effective properties encountered in most periodic structures. However, the case whereby $r^{2} = - 1$ is very unusual, as the frequency λ behaves like $\sqrt{k}$ near the origin, where k is the wavenumber. Finally, when $r^{2} < - 1$ , the lowest branch does not pass through the origin and a zero-frequency band gap opens up. In the last two cases, effective medium theory breaks down even in the quasi-static limit, while the high-frequency homogenization [R.V. Craster, J. Kaplunov, A.V. Pichugin, High-frequency homogenization for periodic media, Proc. R. Soc. Lond. Ser. A 466 (2010) 2341–2362] neatly captures the detailed features of band diagrams.

Nous étudions le spectre de bande associé aux modes de Floquet–Bloch dans une classe dʼéchiquiers périodiques. Sur une cellule de base $[- 1, 1 [^{2}$ , lʼindice de réfraction, n, est défini par $n^{2} = 1 + g_{1} (x_{1}) + g_{2} (x_{2})$ où $g_{i} (x_{i}) = r^{2}$ (une constante) pour $0 ⩽ x_{i} < 1$ , et $g_{i} (x_{i}) = 0$ pour $- 1 ⩽ x_{i} < 0$ . Pour $r^{2} > - 1$ , la première bande passe par lʼorigine avec un comportement linéaire, ce qui conduit à des propriétés effectives rencontrées dans la plupart des structures périodiques. En revanche, le cas $r^{2} = - 1$ est moins ordinaire, puisque la bande de fréquences acoustiques λ se comporte comme $\sqrt{k}$ au voisinage de lʼorigine, avec k le nombre dʼonde. Finallement, quand $r^{2} < - 1$ , la bande acoustique disparaît : la première bande ne passe plus par lʼorigine et une bande interdite à fréquence nulle apparaît. Dans ces deux derniers cas de figure, la théorie des milieux effectifs ne sʼapplique pas, alors que la théorie dʼhomogénéisation hautes fréquences [R.V. Craster, J. Kaplunov, A.V. Pichugin, High-frequency homogenization for periodic media, Proc. R. Soc. Lond. Ser. A 466 (2010) 2341–2362] reproduit avec précision les diagrammes de bandes.

Received: 2010-12-10
Accepted: 2011-03-11
Published online: 2011-05-20

DOI: 10.1016/j.crme.2011.03.016

Keywords: Waves, Homogenization, Negative refraction, Acoustic band
Mots-clés : Ondes, Homogénéisation, Réfraction négative, Bande acoustique

Author's affiliations:

Richard V. Craster ¹; Sébastien Guenneau ²; Julius Kaplunov ³; Evgeniya Nolde ³

¹ Department of Mathematics, Imperial College London, London SW7-2AZ, UK
² Institut Fresnel, UMR CNRS 6133, University of Aix-Marseille, 13000 Marseille, France
³ Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

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     author = {Richard V. Craster and S\'ebastien Guenneau and Julius Kaplunov and Evgeniya Nolde},
     title = {On a class of three-phase checkerboards with unusual effective properties},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {411--417},
     publisher = {Elsevier},
     volume = {339},
     number = {6},
     year = {2011},
     doi = {10.1016/j.crme.2011.03.016},
     language = {en},
}

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Richard V. Craster; Sébastien Guenneau; Julius Kaplunov; Evgeniya Nolde. On a class of three-phase checkerboards with unusual effective properties. Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 411-417. doi : 10.1016/j.crme.2011.03.016. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.03.016/

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