Abstract
The objective of this chapter is to introduce the topic of damping in the context of both its modeling and its effects in phononic crystals and acoustic metamaterials. First, we provide a brief discussion on the modeling of damping in structural dynamic systems in general with a focus on viscous and viscoelastic types of damping (Sect. 6.2) and follow with a non-exhaustive literature review of prior work that examined periodic phononic materials with damping (Sect. 6.3). In Sect. 6.4, we consider damped 1D diatomic phononic crystals and acoustic metamaterials as example problems (keeping our attention on 1D systems for ease of exposition as in previous chapters). We introduce the generalized form of Bloch’s theorem, which is needed to account for both temporal and spatial attenuation of the elastic waves resulting from the presence of damping. We also describe the transformation of the governing equations of motion to a state-space representation to facilitate the treatment of the damping term that arises in the emerging eigenvalue problem. Finally, the effects of dissipation (based on the two types of damping models considered) on the frequency and damping ratio band structures are demonstrated by solving the equations developed for a particular choice of material parameters.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
A.S. Phani, J. Woodhouse, N.A. Fleck, Wave propagation in two-dimensional periodic lattices. J. Acoust. Soc. Am. 119, 1995–2005 (2006)
A.S. Nowick, B.S. Berry, Anelastic Relaxation in Crystalline Solids (Academic, New York, 1972)
C.W. Bert, Material damping: an introductory review of mathematic measures and experimental techniques. J. Sound Vib. 29, 129–153 (1973)
J. Woodhouse, Linear damping models for structural vibration. J. Sound Vib. 215, 547–569 (1998)
J.W.S. Rayleigh, Theory of Sound (Macmillan and Co., London, 1878)
Y.C. Fung, P. Tong, Classical and Computational Solid Mechanics (World Scientific Publishing Co., Singapore, 2001)
Y. Yong, Y.K. Lin, Propagation of decaying waves in periodic and piecewise periodic structures of finite length. J. Sound Vib. 129, 99–118 (1989)
I.E. Psarobas, Viscoelastic response of sonic band-gap materials. Phys. Rev. B 64, 012303 (2001)
P.W. Mauriz, M.S. Vasconcelos, E.L. Albuquerque, Acoustic phonon power spectra in a periodic superlattice. Phys. Status Solidi B 243, 1205–1211 (2006)
M.P. Castanier, C. Pierre, Individual and interactive mechanisms for localization and dissipation in a mono-coupled nearly-periodic structure. J. Sound Vib. 168, 479–505 (1993)
Y. Liu, D. Yu, H. Zhao, J. Wen, X. Wen, Theoretical study of 2D PC with viscoelasticity based on fractional derivative models. J. Phys. D Appl. Phys. 41, 065503 (2008)
F. Bloch, Über die quantenmechanik der electron in kristallgittern. Z. Phys. 52, 555–600 (1928)
D.J. Mead, A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. J. Sound Vib. 27, 235–260 (1973)
S. Mukherjee, E.H. Lee, Dispersion relations and mode shapes for waves in laminated viscoelastic composites by finite difference methods. Comput. Struct. 5, 279–285 (1975)
X. Zhang, Z. Liu, J. Mei, Y. Liu, Acoustic band gaps for a 2D periodic array of solid cylinders in viscous liquid. J. Phys. Condens. Mater. 15, 8207–8212 (2003)
B. Merheb, P.A. Deymier, M. Jain, M. Aloshyna-Lesuffleur, S. Mohanty, A. Baker, R.W. Greger, Elastic and viscoelastic effects in rubber-air acoustic band gap structures: a theoretical and experimental study. J. Appl. Phys. 104, 064913 (2008)
R.S. Langley, On the forced response of one-dimensional periodic structures: vibration localization by damping. J. Sound Vib. 178, 411–428 (1994)
E. Tassilly, Propagation of bending waves in a periodic beam. Int. J. Eng. Sci. 25, 85–94 (1987)
R.P. Moiseyenko, V. Laude, Material loss influence on the complex band structure and group velocity in phononic crystals. Phys. Rev. B 83, 064301 (2011)
F. Farzbod, M.J. Leamy, Analysis of Bloch’s method in structures with energy dissipation. J. Vib. Acoust. 133, 051010 (2011)
R. Sprik, G.H. Wegdam, Acoustic band gaps in composites of solids and viscous liquids. Solid State Commun. 106, 77–81 (1998)
M.I. Hussein, Theory of damped Bloch waves in elastic media. Phys. Rev. B 80, 212301 (2009)
M.I. Hussein, M.J. Frazier, Band structures of phononic crystals with general damping. J. Appl. Phys. 108, 093506 (2010)
M. J. Frazier, M. I. Hussein, Dissipative Effects in Acoustic Metamaterials, in Proceedings of Phononics 2011, Paper Phononics-2011-0172, Santa Fe, New Mexico, USA, May 29–June 2, 2011, pp. 84–85
N. Wagner, S. Adhikari, Symmetric state-space method for a class of nonviscously damped systems. AIAA J. 41, 951–956 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hussein, M.I., Frazier, M.J. (2013). Damped Phononic Crystals and Acoustic Metamaterials. In: Deymier, P. (eds) Acoustic Metamaterials and Phononic Crystals. Springer Series in Solid-State Sciences, vol 173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31232-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-31232-8_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31231-1
Online ISBN: 978-3-642-31232-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)