Platonic crystal

Platonic crystals are periodic structures which are designed to guide flexural wave energy through thin elastic plates.

The term platonic crystal is formed by analogy to photonic crystals, phononic crystals, and plasmonic crystals. The name emphasizes a thin plate formulation, which is governed by a fourth order partial differential equation, as opposed to a second order equation which governs other types of crystals. There are also strong connections between platonic crystals and metamaterials.

The study of platonic crystals is referred to as platonics, and does not refer to the teachings of Plato, or the Platonic solids. The term is now in common usage by multiple research groups in Australia, New Zealand, France, and the United Kingdom.

Applications

The types of platonic structures that have been examined include arrays of perforations,[1][2][3] arrays of pins,[4][5][6][7][8] arrays of point masses,[9] as well as periodic variations in the plate material itself.[10] Platonic crystals have been shown to exhibit a number of behaviours similar to photonic crystals, including negative refraction, beam splitting, and wave trapping.[2][7][8] They may also feature large stopbands where wave propagation is not possible through the crystal,[1] as well as cloaking near degenerate band surfaces.[11] Investigations into defective platonic crystals has also revealed strong energy localization effects within the defects, with high quality factors.[9][12][13]

Experimental work in platonics to date has shown promising results in cloaking[14] and flat lens focusing of flexural wave energy.[15]

References

  1. ^ a b A.B. Movchan; N.V. Movchan; R.C. McPhedran (2007), "Bloch-Floquet bending waves in perforated thin plates", Proceedings of the Royal Society A, 463 (2086): 2505–2518, Bibcode:2007RSPSA.463.2505M, doi:10.1098/rspa.2007.1886, S2CID 111291430
  2. ^ a b M. Farhat; S. Guenneau; S. Enoch; A.B. Movchan; G.G. Petursson (2010), "Focussing bending waves via negative refraction in perforated thin plates", Applied Physics Letters, 96 (8): 081909, Bibcode:2010ApPhL..96h1909F, doi:10.1063/1.3327813[dead link‍]
  3. ^ M. Farhat; S. Guenneau; S. Enoch (2010), "High directivity and confinement of flexural waves through ultra-refraction in thin perforated plates", EPL, 91 (5): 54003, Bibcode:2010EL.....9154003F, doi:10.1209/0295-5075/91/54003, S2CID 120154590
  4. ^ D.V. Evans; R. Porter (2007), "Penetration of flexural waves through a periodically constrained thin elastic plate in vacuo and floating on water", Journal of Engineering Mathematics, 58 (1): 317–337, Bibcode:2007JEnMa..58..317E, doi:10.1007/s10665-006-9128-0, S2CID 52238485
  5. ^ N.V. Movchan; R.C. McPhedran; A.B. Movchan; C.G. Poulton (2009), "Wave scattering by platonic grating stacks", Proceedings of the Royal Society A, 465 (2111): 3383–3400, Bibcode:2009RSPSA.465.3383M, doi:10.1098/rspa.2009.0301
  6. ^ M. H. Meylan; R.C. McPhedran (2011), "Fast and slow interaction of elastic waves with platonic clusters", Proceedings of the Royal Society A, 467 (2136): 3509–3529, Bibcode:2011RSPSA.467.3509M, doi:10.1098/rspa.2011.0234
  7. ^ a b S.G. Haslinger; N.V. Movchan; A.B. Movchan; R.C. McPhedran (2012), "Transmission, trapping and filtering of waves in periodically constrained elastic plates", Proceedings of the Royal Society A, 468 (2137): 76–93, Bibcode:2012RSPSA.468...76H, doi:10.1098/rspa.2011.0318
  8. ^ a b M.J.A. Smith; R.C. McPhedran; C.G. Poulton; M.H. Meylan (2012), "Negative refraction and dispersion phenomena in platonic clusters", Waves in Random and Complex Media, 22 (4): 435–458, Bibcode:2012WRCM...22..435S, doi:10.1080/17455030.2012.711495, S2CID 119831638
  9. ^ a b C.G. Poulton; A.B. Movchan; N.V. Movchan; R.C. McPhedran (2012), "Analytic theory of defects in periodically structured elastic plates", Proceedings of the Royal Society A, 468 (2140): 1196–1216, Bibcode:2012RSPSA.468.1196P, doi:10.1098/rspa.2011.0609
  10. ^ T. Antonakakis; R.V. Craster (2012), "High-frequency asymptotics for microstructured thin elastic plates and platonics", Proceedings of the Royal Society A, 468 (2141): 1408–1427, Bibcode:2012RSPSA.468.1408A, doi:10.1098/rspa.2011.0652
  11. ^ T. Antonakakis; R. Craster; S. Guenneau (2013), Moulding flexural waves in elastic plates lying atop a Faqir's bed of nails, arXiv:1301.7653, Bibcode:2013arXiv1301.7653A
  12. ^ R. C. McPhedran; A.B. Movchan; N.V. Movchan (2009), "Platonic crystals: Bloch bands, neutrality and defects", Mechanics of Materials, 41 (4): 356–363, doi:10.1016/j.mechmat.2009.01.005
  13. ^ M.J.A. Smith; R. Porter; T.D. Williams (2012), "The effect on bending waves by defects in pinned elastic plates", Journal of Sound and Vibration, 331 (23): 5087–5106, Bibcode:2012JSV...331.5087S, doi:10.1016/j.jsv.2012.06.013, hdl:1983/d65f8bc3-1d33-44b7-b3e3-f75aa60d1a01
  14. ^ N. Stenger; M. Wilhelm; M. Wegener (2012), "Experiments on elastic cloaking in thin plates", Physical Review Letters, 108 (1): 014301, Bibcode:2012PhRvL.108a4301S, doi:10.1103/physrevlett.108.014301, PMID 22304261, S2CID 35264750
  15. ^ M. Dubois; M. Farhat; E. Bossy; S. Enoch; S. Guenneau; P. Sebbah (2013), "Flat lens for time-domain focusing of elastic waves in thin plates", Applied Physics Letters, 103 (7): 071915, arXiv:1303.3022, Bibcode:2013ApPhL.103g1915D, doi:10.1063/1.4818716