On a class of three-phase checkerboards with unusual effective properties - Archive ouverte HAL
Article Dans Une Revue Comptes Rendus Mécanique Année : 2011

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

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 A behaves like 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.

Dates et versions

hal-00647943 , version 1 (03-12-2011)

Identifiants

Citer

Richard Craster, Sébastien Guenneau, Julius Kaplunov, Evgeniya Nolde. On a class of three-phase checkerboards with unusual effective properties. Comptes Rendus Mécanique, 2011, 339 (6), pp.411-417. ⟨10.1016/j.crme.2011.03.016⟩. ⟨hal-00647943⟩
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