couple modulus, wave band-gaps, phononic crystals, lattice structures.
author:
- 'Angela Madeo$^{1,5}$[^1], Patrizio Neff$^{2,5}$, Ionel-Dumitrel Ghiba$^{2,3}$, Luca Placidi$^{4,5}$, Giuseppe Rosi$^{5,6}$'
title: 'Wave propagation in relaxed micromorphic continua: modelling metamaterials with frequency band-gaps'
---

> “Sans la curiosité de l’ésprit, que serions-nous?” Marie Curie

1\. Université de Lyon-INSA, 20 Av. Albert Einstein, 69100 Villeurbanne Cedex, France.

2\. Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127 Essen, Germany.

3\. Department of Mathematics, University A.I. Cuza, Blvd. Carol I, no. 11, 700506 Iaşi, Romania; Octav Mayer Institute of Mathematics, Romanian Academy, 700505 - Iaşi; Institute of Solid Mechanics, Romanian Academy, 010141-Bucharest, Romania.

4\. Università Telematica Internazionale Uninettuno, Corso V. Emanuele II 39, 00186 Roma, Italy.

5\. International Center M&MOCS Mathematics and Mechanics of Complex Systems, Palazzo Caetani, Cisterna di Latina, Italy.

6\. Laboratoire de Modelisation Multi Echelle, MSME UMR 8208 CNRS, Université Paris-Est, 61 Avenue du Général De Gaulle, Créteil Cedex 94010, France

Introduction
============

Engineering metamaterials showing exotic behaviors with respect to wave propagation are recently attracting growing attention from the scientific community for what concerns both modelling and experiments. Indeed, there are multiple experimental evidences supporting the fact that engineering microstructured materials 