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

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• 作者：Richard V. Craster^a ; ^{r.craster@imperial.ac.uk"" rel=""nofollow} ; Sé ; bastien Guenneau^b ; ^{sebastien.guenneau@fresnel.fr"" rel=""nofollow} ; Julius Kaplunov^c ; ^{julius.kaplunov@brunel.ac.uk"" rel=""nofollow} ; Evgeniya Nolde^c ; ^{evgeniya.nolde@brunel.ac.uk"" rel=""nofollow}
• 关键词：Waves ; Homogenization ; Negative refraction ; Acoustic band
• 刊名：Comptes Rendus Mecanique
• 出版年：2011
• 出版时间：June 2011
• 年：2011
• 卷：339
• 期：6
• 页码：411-417
• 全文大小：325 K

文摘

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 g_i(x_i)=r² for 0x_i<1, and g_i(x_i)=0 for −1x_i<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 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.