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

  • Claude Boutin
  • Jean-Louis Auriault
  • Guy Bonnet
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


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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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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Claude Boutin
    • 1
  • Jean-Louis Auriault
    • 2
  • Guy Bonnet
    • 3
  1. 1.Université de Lyon, LGCB/LTDS CNRS, ENTPEVaulx-en-VelinFrance
  2. 2.3SR LabUniversité Grenoble Alpes-CNRSGrenobleFrance
  3. 3.Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208Université Paris-Est, CNRSMarne-la-ValléeFrance

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