Comptes Rendus
Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex
[Théorème de Bloch généralisé pour les métamatériaux visqueux : dispersion et propriétés effectives fondées sur les fréquences et nombres d'onde simultanément complexes]
Comptes Rendus. Physique, Volume 17 (2016) no. 5, pp. 565-577.

Les courbes de dispersion des matériaux périodiques amortis sont habituellement basées soit sur des fréquences réelles en fonction de nombres d'onde complexes, soit sur des nombres d'onde réels en fonction de fréquences complexes. Le premier cas correspond à la propagation d'ondes harmoniques, dont la fréquence d'excitation est imposée, et dont l'atténuation due à la dissipation survient uniquement dans l'espace, en même temps que l'atténuation spatiale due à la diffraction de Bragg. Le second cas concerne la propagation d'ondes libres dont l'atténuation est due à une perte d'énergie dans le temps, en plus de l'atténuation spatiale causée par la diffraction de Bragg. Dans cet article, nous développons un algorithme pour des systèmes unidimensionnels afin d'obtenir—pour le mouvement d'ondes libres amorties—les courbes de dispersion fondées sur des fréquences et des nombres d'onde qui sont autorisés à être simultanément complexes. Cette application généralisée du théorème de Bloch fournit une structure de bandes qui décrit pleinement tous les mécanismes d'atténuation, dans l'espace comme dans le temps. L'algorithme est appliqué à un metamatériau à résonance locale (masse incluse dans une masse) amorti de façon visqueuse. Une masse effective dépendant de la fréquence est également obtenue pour cette chaine infinie amortie.

It is common for dispersion curves of damped periodic materials to be based on real frequencies as a function of complex wavenumbers or, conversely, real wavenumbers as a function of complex frequencies. The former condition corresponds to harmonic wave motion where a driving frequency is prescribed and where attenuation due to dissipation takes place only in space alongside spatial attenuation due to Bragg scattering. The latter condition, on the other hand, relates to free wave motion admitting attenuation due to energy loss only in time while spatial attenuation due to Bragg scattering also takes place. Here, we develop an algorithm for 1D systems that provides dispersion curves for damped free wave motion based on frequencies and wavenumbers that are permitted to be simultaneously complex. This represents a generalized application of Bloch's theorem and produces a dispersion band structure that fully describes all attenuation mechanisms, in space and in time. The algorithm is applied to a viscously damped mass-in-mass metamaterial exhibiting local resonance. A frequency-dependent effective mass for this damped infinite chain is also obtained.

Publié le :
DOI : 10.1016/j.crhy.2016.02.009
Keywords: Damped waves, Complex dispersion, Complex band structure, Phononic crystals, Acoustic metamaterials, Periodic materials
Mot clés : Ondes amorties, Dispersion complexe, Structure de bandes complexe, Cristaux phononiques, Métamatériaux acoustiques, Matériaux périodiques

Michael J. Frazier 1 ; Mahmoud I. Hussein 1

1 Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80309-0429, USA
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Michael J. Frazier; Mahmoud I. Hussein. Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex. Comptes Rendus. Physique, Volume 17 (2016) no. 5, pp. 565-577. doi : 10.1016/j.crhy.2016.02.009. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2016.02.009/

[1] D.J. Mead Wave propagation in continuous periodic structures: Research contributions from Southampton, 1964–1995, J. Sound Vib., Volume 190 (1996), pp. 495-524

[2] M.I. Hussein; M.J. Leamy; M. Ruzzene Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook, Appl. Mech. Rev., Volume 66 (2014)

[3] Acoustic Metamaterials and Phononic Crystals (P.A. Deymier, ed.), Springer, Heidelberg, Germany, 2013

[4] Phononic Crystals: Fundamentals and Applications (A. Khelif; A. Adibi, eds.), Springer, New York, 2015

[5] V. Laude Phononic Crystals: Artificial Crystals for Sonic, Acoustic and Elastic Waves, De Gruyter, Berlin, 2015

[6] M.M. Sigalas; E.N. Economou Elastic and acoustic wave band structure, J. Sound Vib., Volume 158 (1992), pp. 377-382

[7] M.S. Kushwaha; P. Halevi; L. Dobrzynski; B. Djafari-Rouhani Acoustic band structure of periodic elastic composites, Phys. Rev. Lett., Volume 71 (1993), pp. 2022-2025

[8] Z. Liu; X. Zhang; Y. Mao; Y.Y. Zhu; Z. Yang; C.T. Chan; P. Sheng Locally resonant sonic materials, Science, Volume 289 (2000), pp. 1734-1736

[9] Z.Y. Liu; C.T. Chan; P. Sheng Three-component elastic wave band-gap material, Phys. Rev. B, Volume 65 (2002)

[10] G. Wang; X.S. Wen; J.H. Wen; L.H. Shao; Y.Z. Liu Two-dimensional locally resonant phononic crystals with binary structures, Phys. Rev. Lett., Volume 93 (2004)

[11] Y. Pennec; B. Djafari-Rouhani; H. Larabi; J.O. Vasseur; A.-C. Ladky-Hennion Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate, Phys. Rev. B, Volume 78 (2008)

[12] T.T. Wu; T.C. Huang; Z.G. Tsai; T.C. Wu Evidence of complete band gap and resonances in a plate with periodic stubbed surface, Appl. Phys. Lett., Volume 93 (2008)

[13] J. Li; C.T. Chan Double-negative acoustic metamaterial, Phys. Rev. E, Volume 70 (2004)

[14] Y. Ding; Z. Liu; C. Qiu; J. Shi Metamaterial with simultaneously negative bulk modulus and mass density, Phys. Rev. Lett., Volume 99 (2007)

[15] X. Ao; C.T. Chan Negative group velocity from resonances in two-dimensional phononic crystals, Waves Random Complex Media, Volume 20 (2010), pp. 276-288

[16] X.N. Liu; G.K. Hu; G.L. Huang; C.T. Sun An elastic metamaterial with simultaneously negative mass density and bulk modulus, Appl. Phys. Lett., Volume 98 (2011)

[17] M.I. Hussein; M.J. Frazier Metadamping: an emergent phenomenon in dissipative metamaterials, J. Sound Vib., Volume 332 (2013), pp. 4767-4774

[18] I. Antoniadis; D. Chronopoulos; V. Spitas; D. Koulocheris Hyper-damping properties of a stiff and stable linear oscillator with a negative stiffness element, J. Sound Vib., Volume 346 (2015), pp. 37-52

[19] Y.Y. Chen; M.V. Barnhart; J.K. Chen; G.K. Hu; C. Sun; G.L. Huang Dissipative elastic metamaterials for broadband wave mitigation at subwavelength scale, Compos. Struct., Volume 136 (2016), pp. 358-371

[20] B.L. Davis; M.I. Hussein Nanophononic metamaterial: Thermal conductivity reduction by local resonance, Phys. Rev. Lett., Volume 112 (2014)

[21] J.W.S. Rayleigh The Theory of Sound, vol. 1, Macmillan and Co., London, 1877

[22] T.K. Caughey; M.E.J. O'Kelly Classical normal modes in damped linear dynamic systems, J. Appl. Mech. – Trans. ASME, Volume 32 (1965), pp. 583-588

[23] S. Adhikari Damping modelling using generalized proportional damping, J. Sound Vib., Volume 293 (2005), pp. 156-170

[24] S. Adhikari; A.S. Phani Experimental identification of generalized proportional viscous damping matrix, J. Vib. Acoust., Volume 131 (2009)

[25] J. Woodhouse Linear damping models for structural vibration, J. Sound Vib., Volume 215 (1998), pp. 547-569

[26] S. Adhikari; J. Woodhouse Identification of damping: part 1, viscous damping, J. Sound Vib., Volume 243 (2001), pp. 43-61

[27] A.S. Phani; J. Woodhouse Viscous damping identification in linear vibration, J. Sound Vib., Volume 303 (2007), pp. 475-500

[28] E. Tassilly Propagation of bending waves in a periodic beam, Int. J. Eng. Sci., Volume 25 (1987), pp. 85-94

[29] R.S. Langley On the forced response of one-dimensional periodic structures: vibration localization by damping, J. Sound Vib., Volume 178 (1994), pp. 411-428

[30] V. Laude; Y. Achaoui; S. Benchabane; A. Khelif Evanescent Bloch waves and the complex band structure of phononic crystals, Phys. Rev. B, Volume 80 (2009)

[31] V. Romero-García; J.V. Sánchez-Pérez; L.M. Garcia-Raffi Propagating and evanescent properties of double-point defects in sonic crystals, New J. Phys., Volume 12 (2010)

[32] R.P. Moiseyenko; V. Laude Material loss influence on the complex band structure and group velocity in phononic crystals, Phys. Rev. B, Volume 83 (2011)

[33] E. Andreassen; J.S. Jensen Analysis of phononic bandgap structures with dissipation, J. Vib. Acoust., Volume 135 (2013)

[34] D.J. Mead A general theory of harmonic wave propagation in linear periodic systems with multiple coupling, J. Sound Vib., Volume 27 (1973), pp. 235-260

[35] F. Farzbod; M.J. Leamy Analysis of Bloch's method in structures with energy dissipation, J. Vib. Acoust., Volume 133 (2011)

[36] M. Collet; M. Ouisse; M. Ruzzene; M.N. Ichchou Floquet–Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems, Int. J. Solids Struct., Volume 48 (2011), pp. 2837-2848

[37] S. Mukherjee; E.H. Lee Dispersion relations and mode shapes for waves in laminated viscoelastic composites by finite difference methods, Comput. Struct., Volume 5 (1975), pp. 279-285

[38] R. Sprik; G.H. Wegdam Acoustic band gaps in composites of solids and viscous liquids, Solid State Commun., Volume 106 (1998), pp. 77-81

[39] M.I. Hussein Theory of damped Bloch waves in elastic media, Phys. Rev. B, Volume 80 (2009)

[40] M.I. Hussein; M.J. Frazier Band structure of phononic crystals with general damping, J. Appl. Phys., Volume 108 (2010)

[41] A.S. Phani; M.I. Hussein Analysis of damped Bloch waves by the Rayleigh perturbation method, J. Vib. Acoust., Volume 135 (2013)

[42] J.D. Achenbach Wave Propagation in Elastic Solids, North-Holland, London, 1999

[43] B.R. Mace; E. Manconi Modelling wave propagation in two-dimensional structures using finite element analysis, J. Sound Vib., Volume 318 (2008), pp. 884-902

[44] E. Manconi; B.R. Mace Estimation of the loss factor of viscoelastic laminated panels from finite element analysis, J. Sound Vib., Volume 329 (2010), pp. 3928-3939

[45] M.I. Hussein; M.J. Frazier; M.H. Abedinnassab Microdynamics of phononic materials (S. Li; X.-L. Gao, eds.), Handbook of Micromechanics and Nanomechanics, Pan Stanford Publishing, 2013 (Chapter 1)

[46] H.H. Huang; C.T. Sun; G.L. Huang On the negative effective mass density in acoustic metamaterials, Int. J. Eng. Sci., Volume 47 (2009), pp. 610-617

[47] F. Farzbod; M.J. Leamy Analysis of Bloch's method and the propagation technique in periodic structures, J. Vib. Acoust., Volume 133 (2011)

[48] G.W. Milton; J.R. Willis On modifications of Newton's second law and linear continuum elastodynamics, Proc. R. Soc. A, Volume 463 (2007), pp. 855-880

[49] S. Nemat-Nasser; A. Srivastava Negative effective dynamics mass-density and stiffness: Micro-architecture and phononic transport in periodic composites, AIP Adv., Volume 1 (2011)

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