Abstract
The paper deals with localisation of flexural waves within gratings composed of either pinned points or rigid inclusions of finite radius in a structured plate. We study the filtering and resonant action of such systems. The effect of the finite size of inclusions on the dynamic localisation is analysed for the range of frequencies where only zeroth grating orders propagate. The structure of the resonant modes within gratings of inclusions is of special interest. In particular, we consider the circumstances under which such gratings can deliver for flexural waves a phenomenon similar to Electromagnetically Induced Transparency, where a resonant maximum of transmission is cut in two by a resonant minimum. We identify system designs which yield very high concentration of flexural fields within the interface that may lead to a further structural failure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramowitz M, Stegun IA (1965) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Reprint
Bertoldi K, Bigoni D, Drugan JW (2007a) Structural interfaces in linear elasticity. Part I: nonlocality and gradient approximations. J Mech Phys Solids 55:1–34
Bertoldi K, Bigoni D, Drugan JW (2007b) Structural interfaces in linear elasticity. Part II: effective properties and neutrality. J Mech Phys Solids 55:35–63
Bertoldi K, Bigoni D, Drugan JW (2007c) A discrete-fibers model for bridged cracks and reinforced elliptical voids. J Mech Phys Solids 55:1016–1035
Bigoni D, Gei M, Movchan AB (2008) Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from longwave asymptotics. J Mech Phys Solids 56:2494–2520
Bigoni D, Movchan AB (2002) Statics and dynamics of structural interfaces in elasticity. Int J Solids Struct 39:4843–4865
Born M, Wolf E (1959) Principles of optics. Pergamon Press, London
Botten LC, McPhedran RC, Lamarre JM (1985) Inductive grids in the resonant region: theory and experiment. Int J Infrared Millim Waves 6:511–575
Brun M, Guenneau S, Movchan AB, Bigoni D (2010) Dynamics of structural interfaces: filtering and focussing effects of elastic waves. J Mech Phys Solids 58(9):1212–1224
Evans DV, Porter R (2007) Penetration of flexural waves through a periodically constrained thin elastic plate in vacuo and floating on water. J Eng Maths 58:317–337
Farhat M, Guenneau S, Enoch S, Movchan AB (2009) Cloaking bending waves propagating in thin elastic plates. Phys Rev B 79:033102
Farhat M, Guenneau S, Enoch S (2010) High directivity and confinement of flexural waves through ultra-refraction in thin perforated plates. Euro Phys Lett 91:54003
Fleischhauer M, Imamoglu A, Marangos JP (2005) Electromagnetically induced transparency: optics in coherent media. Rev Mod Phys 77:633–673
Gei M, Movchan AB, Bigoni D (2009) Band-gap shift and defect-induced annihilation in prestressed elastic structures. J Appl Phys 105:063507
Haslinger SG, Movchan NV, Movchan AB, McPhedran RC (2012) Transmission, trapping and filtering of waves in periodically constrained elastic plates. Proc R Soc A 468:76–93
Movchan AB, Movchan NV, McPhedran RC (2007) Bloch-Floquet bending waves in perforated thin plates. Proc R Soc A 463:2505–2518
Movchan NV, McPhedran RC, Movchan AB, Poulton CG (2009) Wave scattering by platonic grating stacks. Proc R Soc A 465:3383–3400
Movchan NV, McPhedran RC, Movchan AB (2011) Flexural waves in structured elastic plates: mindlin versus bi-harmonic models. Proc R Soc A 467:869–880
McPhedran RC, Movchan AB, Movchan NV (2009) Platonic crystals: bloch bands, neutrality and defects. Mech Mater 41:356–363
Pelton EL, Munk BA (1979) Scattering from periodic arrays of crossed dipoles. IEEE Trans AP-27, 323–330
Poulton CG, McPhedran RC, Movchan NV, Movchan AB (2010) Convergence properties and flat bands in platonic crystal band structures using the multipole formulation. Waves Random Complex Media 20:702–716
Stenger N, Wilhelm M, Wegener M (2012) Experiments on elastic cloaking in thin plates. Phys Rev Lett 108:014301
Twersky V (1961) Elementary function representations of Schlomilch series. Arch Ration Mech Anal 8:323–332
Ulrich R, Tacke M (1973) Submillimeter wave-guiding on periodic metal structure. Appl Phys Lett 22:251–253
Acknowledgments
S.G.H. gratefully acknowledges the financial support of the Duncan Norman Charitable Trust through the Duncan Norman Research Scholarship. The participation of R.C.M. in this project was facilitated by a grant from the Research Centre in Mathematics and Modelling of the University of Liverpool; he also acknowledges support from the Australian Research Council through its Discovery Grants Scheme.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Haslinger, S.G., McPhedran, R.C., Movchan, N.V., Movchan, A.B. (2014). Localisation near defects and filtering of flexural waves in structured plates. In: Bigoni, D., Carini, A., Gei, M., Salvadori, A. (eds) Fracture Phenomena in Nature and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-04397-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-04397-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04396-8
Online ISBN: 978-3-319-04397-5
eBook Packages: EngineeringEngineering (R0)