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Inner Resonance in Media Governed by Hyperbolic and Parabolic Dynamic Equations. Principle and Examples

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Generalized Models and Non-classical Approaches in Complex Materials 1

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 89))

Abstract

This chapter deals with the modeling and design of inner resonance media, i.e. media that present a local resonance which has an impact on the overall dynamic behaviour. The aim of this chapter is to provide a synthetic picture of the inner resonance phenomena by means of the asymptotic homogenization method (Sanchez-Palencia, 1980). The analysis is based on the comparative study of a few canonical realistic composite media. This approach discloses the common principle and the specific features of different inner resonance situations and points out their consequences on the effective behavior. Some general design rules enabling to reach such a specific dynamic regime with a desired effect are also highlighted. The paper successively addresses different materials

  • having different behaviours and inner structures as elastic composites, reticulated media, permeable rigid and elastic media,

  • undergoing phenomena governed either by momentum transfer or/and mass transfer,

  • in which the inner resonance mechanisms can be highly or weakly dissipative,

  • in situation of inner resonance or inner anti-resonance.

The results related to different physical behaviours show that inner resonance requires a highly contrasted microstructure. It constrains the resonant constituent to respond in a forced regime imposed by the non resonant constituent. Then, the effective constitutive law is determined by this latter while the resonating constituent acts as an atypical source term in the macroscopic balance equation. It is established that inner resonance governed by momentum (resp. mass) balance yields unconventional mass (resp. bulk modulus). Furthermore, inner-resonance in media characterized by hyperbolic or parabolic dynamic equations can be handled in a similar manner, leading however to strongly distinct effective features.

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Acknowledgements

This work was performed within the framework of the Labex CeLyA of Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-11-IDEX-0007).

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Authors and Affiliations

  1. Université de Lyon, LGCB/LTDS CNRS, ENTPE, 5513, Vaulx-en-Velin, France

    Claude Boutin

  2. 3SR Lab, Université Grenoble Alpes-CNRS, F-38000, Grenoble, France

    Jean-Louis Auriault

  3. Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208, Université Paris-Est, CNRS, Marne-la-Vallée, France

    Guy Bonnet

Authors
  1. Claude Boutin
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  2. Jean-Louis Auriault
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  3. Guy Bonnet
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Correspondence to Claude Boutin .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität, Magdeburg, Germany

    Holm Altenbach

  2. Université Pierre et Marie Curie, Institut Jean Le Rond d’Alembert, Paris, France

    Joël Pouget

  3. Université Pierre et Marie Curie, Institut Jean Le Rond d’Alembert, Paris, France

    Martine Rousseau

  4. Université Pierre et Marie Curie, Institut Jean Le Rond d'Alembert, Paris Cedex 05, France

    Bernard Collet

  5. Institut Jean le Rond d'Alembert, Université Pierre et Marie Curie, Paris, France

    Thomas Michelitsch

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Boutin, C., Auriault, JL., Bonnet, G. (2018). Inner Resonance in Media Governed by Hyperbolic and Parabolic Dynamic Equations. Principle and Examples. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 1. Advanced Structured Materials, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-72440-9_6

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  • DOI: https://doi.org/10.1007/978-3-319-72440-9_6

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  • Online ISBN: 978-3-319-72440-9

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