Comptes Rendus
On a class of three-phase checkerboards with unusual effective properties
Comptes Rendus. Mécanique, Volume 339 (2011) no. 6, pp. 411-417.

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 n2=1+g1(x1)+g2(x2) with gi(xi)=r2 for 0xi<1, and gi(xi)=0 for 1xi<0 where r2 is constant. We find that for r2>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 r2=1 is very unusual, as the frequency λ behaves like k near the origin, where k is the wavenumber. Finally, when r2<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 n2=1+g1(x1)+g2(x2)gi(xi)=r2 (une constante) pour 0xi<1, et gi(xi)=0 pour 1xi<0. Pour r2>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 r2=1 est moins ordinaire, puisque la bande de fréquences acoustiques λ se comporte comme k au voisinage de lʼorigine, avec k le nombre dʼonde. Finallement, quand r2<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.

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Accepted:
Published online:
DOI: 10.1016/j.crme.2011.03.016
Keywords: Waves, Homogenization, Negative refraction, Acoustic band
Mot clés : Ondes, Homogénéisation, Réfraction négative, Bande acoustique

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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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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