# Dispersion of Waves in Micromorphic Media and Metamaterials

## Abstract

In this contribution we discuss the interest of using enriched continuum models of the micromorphic type for the description of dispersive phenomena in metamaterials. Dispersion is defined as that phenomenon according to which the speed of propagation of elastic waves is not a constant, but depends on the wavelength of the traveling wave. In practice, all materials exhibit dispersion if one considers waves with sufficiently small wavelengths, since all materials have a discrete structure when going down at a suitably small scale. Given the discrete substructure of matter, it is easy to understand that the material properties vary when varying the scale at which the material itself is observed. It is hence not astonishing that the speed of propagation of waves changes as well when considering waves with smaller wavelengths.

In an effort directed toward the modeling of dispersion in materials with architectured microstructures (metamaterials), different linear-elastic, isotropic, micromorphic models are introduced, and their peculiar dispersive behaviors are discussed by means of the analysis of the associated dispersion curves. The role of different micro-inertias related to both independent and constrained motions of the microstructure is also analyzed. A special focus is given to those metamaterials which have the unusual characteristic of being able to stop the propagation of mechanical waves and which are usually called band-gap metamaterials. We show that, in the considered linear-elastic, isotropic case, the relaxed micromorphic model, recently introduced by the authors, is the only enriched model simultaneously allowing for the description of non-localities and multiple band-gaps in mechanical metamaterials.

## Keywords

Dispersion Microstructure Metamaterials Enriched continuum models Relaxed micromorphic model Multi-scale modeling Gradient micro-inertia Free micro-inertia Complete band-gaps Nonlocal effects## Notes

### Acknowledgements

Angela Madeo thanks the Institut Universitaire de France (IUF) for financial support, INSA-Lyon for the funding of the BQR 2016 “Caractérisation mécanique inverse des métamatériaux: modélisation, identification expérimentale des paramètres et évolutions possibles,” as well as the CNRS-INSIS for the funding of the PEPS project.

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