---
abstract: |
    In this paper the relaxed micromorphic model proposed in [@NeffRelaxed; @Ghiba] has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called *Cosserat couple modulus* $\mu_{c}$, which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology.

    **Keywords:** relaxed micromorphic continuum, Cosserat 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 such as phononic crystals and lattice structures can inhibit wave propagation in particular frequency ranges (see e.g. [@Vasseur; @Vasseur1]). Both phononic crystals and lattice structures are artificial materials which are characterized by periodic microstructures in which strong contrasts of the material properties can be observed. Also suitably ordered granular assemblies with defects (see e.g. [@Merkel; @Merkel1; @Kafesaki]) and composites ([@Economou]) have shown the possibility of exhibiting band-gaps in which wave propagation cannot occur under particular loading conditions. In all these cases, the basic features of the observed band-gaps are directly related to the presence of an underlying microstructure in which strong contrasts of the elastic properties may occur. Indeed, the rich dynamic behavior of such materials stems mainly from their hierarchical heterogeneity at the microscopic level which produces the mixing of the longitudinal and the transverse components of travelling waves. It is clear that the applications of such metamaterials would be very appealing for what concerns vibration control in engineering structures. These materials could be used as an alternative to currently used piezo-electric materials which are commonly employed for structural vibration control and for this reason are widely studied in the literature (see, among many others, [@Piezo1; @isola2; @isola6; @isola1; @isola3; @Piezo]).

Motivation and originality of the present work
----------------------------------------------

Different modelling efforts were recently made trying to account for the observed band-gaps in a reliable manner. The most common models are based on homogenization procedures on simple periodic microstructures with strong contrast giving rise to Cauchy-type equivalent continua. Often, it is remarked that the equivalent continua obtained by means of these homogenization techniques are able to predict band-gaps when the equivalent mass of the macroscopic system becomes negative (see e.g.[@BoutinNegM; @Huang0; @Huang]). Homogenized systems with negative equivalent mass can be successfully replaced by suitable generalized continua. Indeed, it is known that homogenization of strongly heterogeneous periodic systems may lead to generalized continua as equivalent macroscopic medium (see e.g.[@isolaHomog]). Some rare papers (see e.g. [@HuangMicrostr; @Huang; @Vasiliev]) can be found in the literature in which lattice models with enriched kinematics are used to model band-gaps without the need of introducing negative equivalent masses. Nevertheless, to the authors’ knowledge, a systematic treatment of band-gap modelling based on the spirit of micromorphic continuuum mechanics is still lacking and deserves attention. Indeed, the idea of using generalized continuum theories to describe microstructured materials needs to be fully developed in order to achieve a simplified modeling and more effective conception of engineering devices allowing to stop wave propagation in suitable frequency bands. Continuum models are, in fact, suitable to be used as input in finite element calculations, which are for sure one of the main basis of engineering design.

In this paper, we propose to use the relaxed micromorphic continuum model presented in [@NeffRelaxed] to study wave propagation in microstructured materials exhibiting exotic properties with respect to wave propagation. The used kinematics is enhanced with respect to the one used for classical Cauchy continua by means of supplementary kinematical fields with respect to the macroscopic displacement. The introduced supplementary kinematical fields allow to account for the presence of microstructure on the overall mechanical behavior of considered continua. The considered extended kinematics has a similar form as the one used to describe phenomena of energy trapping in internal degrees of freedom (see e.g. [@Carcaterra] and references there cited). However, the aforementioned papers deal with damped systems differently from what done in the present paper where only conservative systems are considered.

It is well known that the mechanical behavior of isotropic Mindlin-Eringen media is, in general, described by means of 18 elastic constants (see [@Mindlin; @EringenBook; @NeffRelaxed]). Nevertheless, the set of 18 parameters introduced by Mindlin in [@Mindlin] and Eringen in [@EringenBook] is not suitable to disclose the main features of micromorphic media mainly because of unavoidable computational difficulties. We propose here to consider the relaxed micromorphic model presented in [@NeffRelaxed] which only counts 6 parameters and which, nevertheless, is fully able to describe the main characteristic features of micromorphic continua. Indeed, in [@NeffRelaxed] we showed that the linear isotropic microvoid model, the linear Cosserat model and the linear microstretch model are special cases of our relaxed micromorphic model in dislocation format. A direct identification of the coefficients gave us that the coefficient of the Cowin-Nunziato theory [@CowinNunziato], the Mindlin-Eringen theory [@Mindlin; @EringenBook] and the microstretch theory [@EringenBook; @Iesan] can be expressed in terms of our constitutive coefficients (see also [@Neff; @constants; @NeffRelaxed]). Using these identifications one could compare all the qualitative results obtained using our relaxed micromorphic model with those already available in the literature (see for instance [@Mindlin; @EringenBook; @Ghiba; @Ghiba1; @Ghiba2; @Ghiba3; @Ghiba4; @Ghiba5; @Ghiba6] and references therein) concerning classical theories of elastic materials with microstructure.

By studying dispersion relations in the proposed relaxed model, we show that wave propagation in such simplified micromorphic media is affected by the presence of microstructure in a quite controllable manner. More precisely, we show that, if only longitudinal waves are allowed to propagate in the considered medium (no displacements and micro-deformations allowed in the direction orthogonal to the direction of wave propagation), then the presence of band-gaps can be forecasted only with 5 parameters. The main features of these longitudinal band gaps are seen to be directly related to the absolute rotations of the microstructure. Nevertheless, the proposed 5-parameters relaxed model is not able to account for the presence of band-gaps when also transverse waves are allowed to propagate in the considered material. In order to treat the more general case, we hence consider a slightly generalized relaxed model by introducing only one extra elastic parameter, the so-called *Cosserat couple modulus* $\mu_{c}$, which is related to the relative deformation of the microstructure with respect to the macroscopic matrix. We hence end up with a relaxed micromorphic model with only 6 parameters (5+1) which is able to account for the prediction of band-gaps when generic waves can travel in the considered medium.

Moreover, we consider the so-called “classical micromorphic medium”, i.e. a micromorphic continuum in which the whole gradient of the micro-deformation tensor plays a role (instead of only its Curl). We show that also the classical micromorphic continuum is not suitable to account for the decription of band-gaps. Finally, a particular second gradient material is obtained as limit case of the proposed classical micromorphic continuum by letting $\mu_{e}\rightarrow\infty$ and $\mu_{c}\rightarrow\infty$. Also in this case, we will show that the existence of band-gaps cannot be accounted for.

We can summarize by saying that the proposed relaxed micromorphic model (6 elastic coefficients) accounts for the description of frequency band-gaps in microstructured media which are “switched on” by the only parameter $\mu_{c}$ related to relative rotations of the microstucture with respect to the solid macroscopic matrix. These band gaps cannot be predicted by means of classical micromorphic or Cosserat and second gradient media, as it will be extensively pointed out in the body of the paper. Moreover, band gaps are impossible to observe in the purely one-dimensional situation, since, among others, the Curl operator vanishes in this simplified case. This absence of band gaps is confirmed by the results found in [@berezowski] in which a one-dimensional micromorphic model is presented together with a detailed analysis of wave propagation.

We finally remark that there are some similarities between dispersion relations for micromorphic continua (including the proposed relaxed model) and dispersion relations of various type of plates models, see [@Victor1] where Kirchhoff, Mindlin, Reissner types of plates including consideration of rotatory inertia are investigated. In particular, for almost all models of plates there are optical branches of dispersion relations and linear asymptotes when the wave number tends to infinity. On the other hand, in the theory of plates there are no band gaps discovered within the considered models. Also porous media with strong contrast between the solid and fluid phases show a marked dispersive behavior (see e.g. [@Steeb; @Steeb1]), but band gaps are not observed in such media.

Notations
---------

Let $\mathbf{x},\mathbf{y}\in\mathbb{R}^{3}$ be two vectors and $\mathbf{X},\mathbf{Y}\in\mathbb{R}^{3\times3}$ be two second order tensors of components $x_{i},y_{i}$ and $X_{ij},Y_{ij}$, $i,j=\{1,2,3\}$, respectively. We let[^2] $\text{<}\mathbf{x},\,\mathbf{y}>_{\mathbb{R}^{3}}=x_{i}y_{i}$ and $\text{<}\mathbf{X},\,\mathbf{Y}>_{\mathbb{R}^{3\times3}}=X_{ij}Y_{ij}$ denote the scalar product on $\mathbb{R}^{3}$ and $\mathbb{R}^{3\times3}$ with associated vector norms $\left\Vert \mathbf{x}\right\Vert _{\mathbb{R}^{3}}^{2}=\text{<}\mathbf{x},\,\mathbf{x}>_{\mathbb{R}^{3}}$ and $\left\Vert \mathbf{X}\right\Vert _{\mathbb{R}^{3\times3}}^{2}=\text{<}\mathbf{X},\,\mathbf{X}>_{\mathbb{R}^{3\times3}}$, respectively. In what follows, we omit the subscripts $\mathbb{R}^{3}$ and $\mathbb{R}^{3\times3}$ if no confusion can arise. We define the standard divergence, gradient and curl operators for vectors and second order tensors respectively as[^3] $$\begin{gathered}
\mathrm{Div}\,\mathbf{x}=x_{i,i},\quad\left(\,\nabla\,\mathbf{x}\,\right)_{ij}=x_{i,j},\quad\left(\,\mathrm{Curl}\,\mathbf{x}\,\right)_{i}=x_{a,b}\,\epsilon_{iab},\\
\\
\left(\,\mathrm{Div}\,\mathbf{X}\,\right)_{i}=X_{ij,j},\quad\left(\,\nabla\,\mathbf{X}\,\right)_{ijk}=X_{ij,k},\quad\left(\,\mathrm{Curl}\,\mathbf{X}\,\right)_{ij}=X_{ia,b}\epsilon_{jab},\end{gathered}$$ where $\boldsymbol{\epsilon}$ is the standard Levi-Civita tensor.

For any second order tensor $\mathbf{X}\in\mathbb{R}^{3\times3}$ we introduce its symmetric, skew-symmetric, spheric and deviatoric part respectively as $$\mathrm{sym}\:\mathbf{X}=\frac{1}{2}\left(\mathbf{X}+\mathbf{X}^{T}\right),\quad\mathrm{skew}\:\mathbf{X}=\frac{1}{2}\left(\mathbf{X}-\mathbf{X}^{T}\right),\quad\mathrm{sph}\:\mathbf{X}=\frac{1}{3}\,\mathrm{tr}\,\mathbf{X}\:\mathds1,\quad\mathrm{dev}\,\mathbf{X}=\mathbf{X}-\mathrm{sph}\:\mathbf{X},$$ or equivalently, in index notation $$\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\left(\mathrm{sym}\:\mathbf{X}\right)_{ij}=\frac{1}{2}\left(X_{ij}+X_{ji}\right),\ \ \left(\mathrm{skew}\:\mathbf{X}\right)_{ij}=\frac{1}{2}\left(X_{ij}-X_{ji}\right),\ \ \left(\mathrm{sph}\:\mathbf{X}\right)_{ij}=\frac{1}{3}\, X_{kk}\:\delta_{ij},\ \ \left(\mathrm{dev}\,\mathbf{X}\right)_{ij}=X_{ij}-\left(\mathrm{sph}\:\mathbf{X}\right)_{ij},$$ where $\delta_{ij}$ is the Kronecher delta tensor which is equal to $1$ when $i=j$ and equal to $0$ if $i\neq j$. The *Cartan-Lie-algebra decomposition* for the tensor $\mathbf{X}$ is introduced as $$\mathbf{X}=\mathrm{dev\: sym}\,\mathbf{X}+\mathrm{skew}\,\mathbf{X}+\mathrm{sph}\,\mathbf{X}.\label{CartanLie}$$

Equations of motion in strong form
==================================

We describe the deformation of the considered continuum by introducing a Lagrangian configuration $B_{L}\subset\mathbb{R}^{3}$ and a suitably regular kinematical field $\boldsymbol{\chi}(\mathbf{X},t)$ which associates to any material point $\mathbf{X}\in B_{L}$ its current position $\mathbf{x}$ at time $t$. The image of the function $\boldsymbol{\chi}$ gives, at any instant $t$ the current shape of the body $B_{E}(t)$ which is often referred to as Eulerian configuration of the system. Since we will use it in the following, we also introduce the displacement field $\mathbf{u}(\mathbf{X},t)=\boldsymbol{\chi}(\mathbf{X},t)-\mathbf{X}$. The kinematics of the continuum is then enriched by adding a second order tensor field $\mathbf{P}(\mathbf{X},t)$ which accounts for deformations associated to the microstructure of the continuum itself. Hence, the current state of the considered continuum is identified by 12 independent kinematical fields: 3 components of the displacement field and 9 components of the micro-deformation field. Such a theory of a continuum with microstructure has been derived in [@Mindlin] for the linear-elastic case and re-proposed e.g. in [@EringenI; @Eringen; @II; @Forest0; @Forest] for the case of non-linear elasticity.

Once the used kinematics has been made clear, we can introduce the action functional of the considered micromorphic system as $$\mathcal{A}=\int_{0}^{T}\int_{B_{L}}\left(T-W\right)d\mathbf{X}\, dt$$ where $T$ and $W$ are the kinetic and potential energies of the considered system respectively. Denoting by $\rho$ and $\text{\ensuremath{\eta}}$ the macroscopic and microscopic mass densities respectively, we choose the kinetic energy of the system to be $$\begin{gathered}
T=\frac{1}{2}\,\rho\left\Vert \,\mathbf{u}_{t}\,\right\Vert ^{2}+\frac{1}{2}\,\eta\left\Vert \,\mathbf{P}_{t}\,\right\Vert ^{2},\label{Kinetic}\end{gathered}$$ where we denote by a subscript $t$ partial derivative with respect to time. We remark that, since the micro-deformation tensor $\mathbf{P}$ is dimensionless, the micro-density $\eta$ has the dimensions of a bulk density ($Kg/m^{3}$) times the square of a length. This means that, if $\rho'$ is the true density (per unit of macro volume) of the material constituting the underlying microstructure of the medium one can think to write the homogenized density $\eta$ as (see also [@Mindlin]) $$\eta=d^{2}\:\rho',$$ where we denoted by $d$ the characteristic length of the microscopic inclusions[^4].

On the other hand, we will specify the choice of the strain energy density $W$ in the next subsections by discussing the cases of the relaxed micromorphic continuum introduced in [@NeffRelaxed] and of the classical micromorphic continuum.

The relaxed micromorphic continuum
----------------------------------

The strain energy density for the relaxed micromorphic continuum can be written as

$$\begin{aligned}
W & =\mu_{e}\left\Vert \,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\lambda_{e}}{2}\left(\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right)^{2}+\mu_{h}\left\Vert \,\mathrm{sym}\,\mathbf{P}\,\right\Vert ^{2}+\frac{\lambda_{h}}{2}\left(\mathrm{tr}\,\mathbf{P}\right)^{2}\label{KinPot}\\
 & +\mu_{c}\left\Vert \,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\alpha_{c}}{2}\left\Vert \,\mathrm{\mathrm{Cur}l}\,\mathbf{P}\,\right\Vert ^{2},\nonumber\end{aligned}$$

where all the introduced elastic coefficients are assumed to be constant. This decomposition of the strain energy density, valid in the isotropic, linear-elastic case, has been proposed in [@NeffRelaxed; @Ghiba] where well-posedness theorems have also been proved. It is clear that this decomposition introduces a limited number of elastic parameters and we will show how this may help in the physical interpretation of these latter. Positive definiteness of the potential energy implies the following simple relations on the introduced parameters $$\mu_{e}>0,\qquad\mu_{c}>0,\qquad3\lambda_{e}+2\mu_{e}>0,\qquad\mu_{h}>0,\qquad3\lambda_{h}+2\mu_{h}>0,\qquad\alpha_{c}>0.\label{DefPos}$$ One of the most interesting features of the proposed strain energy density is the reduced number of elastic parameters which are needed to fully describe the mechanical behavior of a micromorphic continuum. Indeed, each parameter can be easily related to specific micro and macro deformation modes. In the following, we use a strengthened set of requirements which implies (\[DefPos\]), namely $$\mu_{e}>0,\qquad\mu_{c}>0,\qquad2\lambda_{e}+\mu_{e}>0,\qquad\mu_{h}>0,\qquad2\lambda_{h}+\mu_{h}>0,\qquad\alpha_{c}>0.\label{DefPos-1}$$

### Comparison with Mindlin and Eringen models

It can be checked that the proposed strain energy density (\[KinPot\]) represents a particular case of the strain energy density proposed by Mindlin (cf. Eq. (5.5) of [@Mindlin]). Indeed, considering that our micro-strain tensor is the transposed of the one introduced by Mindlin ($P_{ij}=\psi_{ji}$) and using the substitutions (cf. also [@Neff_Forest_jel05]): $$\begin{gathered}
\mu=\mu_{h},\quad\lambda=\lambda_{h},\quad b_{1}=\lambda_{e}+\lambda_{h},\quad b_{2}=\mu_{e}+\mu_{h}+\mu_{c},\quad b_{3}=\mu_{e}+\mu_{h}-\mu_{c},\quad g_{1}=-\lambda_{h},\quad g_{2}=-\mu_{h},\label{Identification1}\\
\nonumber \\
a_{10}=\alpha_{c},\quad a_{14}=-\alpha_{c},\qquad a_{1}=a_{2}=a_{3}=a_{4}=a_{5}=a_{8}=a_{11}=a_{13}=a_{15}=0,\label{Identification2}\end{gathered}$$ we are able to recover that our relaxed model can be obtained as a particular case of Mindlin’s one. The relaxed strain energy density (\[KinPot\]) is not positive definite in the sense of Mindlin and Eringen, but it gives rise to a well posed model (see [@NeffRelaxed]). It is evident that the representation of the strain energy density (\[KinPot\]) is more suitable for applications than Mindlin’s one due to the reduced number of parameters. Indeed, if we consider all the terms with the space derivatives of $\mathbf{P}$ to be vanishing ($\alpha_{c}=0$ in our model and $a_{i}=0$ in Mindlin) we have complete equivalence of the two models when using the parameters identification (\[Identification1\]). We can hence start noticing that we use in this model only 5 parameters instead of Mindlin’s 7 parameters. Things become even more interesting when looking at the terms in the energy which involve derivatives of the microstrain tensor $\mathbf{P}$. Indeed, our relaxed model only provides 1 extra parameter, $\alpha_{c}$, instead of Mindlin’s 11 independent parameters ($a_{1},\: a_{2},\: a_{3},\: a_{4},\: a_{5},\: a_{8},\: a_{10},\: a_{11},\: a_{13},\: a_{14},\: a_{15}$). We claim that the proposed relaxed model is able to catch the basic features of observable material behaviours of materials with microstructure.

When considering Eringen model for micromorphic continua (see [@EringenBook] p. 273 for the strain energy density), we can recover that our relaxed model can be obtained from Eringen’s one by setting $$\begin{gathered}
\mu=\mu_{e}-\mu_{c},\quad\lambda=\lambda_{e},\quad\tau=\lambda_{e}+\lambda_{h},\quad\nu=-\lambda_{e},\quad\eta=\mu_{e}+\mu_{h}-\mu_{c},\quad\sigma=\mu_{c}-\mu_{e},\quad k=2\mu_{c},\label{Identification1-1}\\
\nonumber \\
\tau_{7}=\alpha_{c},\quad\tau_{9}=-\alpha_{c},\qquad\tau_{1}=\tau_{2}=\tau_{3}=\tau_{4}=\tau_{5}=\tau_{6}=\tau_{8}=\tau_{10}=\tau_{11}=0.\label{Identification2-1}\end{gathered}$$ Also in this case, we replace Eringen’s 7 parameters with only 5 parameters and, when considering terms with derivatives of the microstrain tensor $\mathbf{P}$, we have only one additional parameter $\alpha_{c}$ instead of Eringen’s 11 parameters.

### Governing equations

Imposing the first variation of the action functional to be vanishing (i.e. $\delta\mathcal{A}=0$), integrating by parts a suitable number of times and considering arbitrary variations $\delta\boldsymbol{\chi}$ and $\delta\mathbf{P}$ of the basic kinematical fields, we obtain the strong form of the bulk equations of motion of considered system which read $$\begin{aligned}
\rho\,\mathbf{u}_{tt} & =\mathrm{Div}\left[2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1+2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right],\nonumber \\
\label{eq:bulk-mod-3}\\
\eta\,\mathbf{P}_{tt} & =2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1-2\,\mu_{h}\,\mathrm{sym}\,\mathbf{P}-\lambda_{h}\mathrm{tr}\,\mathbf{P}\:\mathds1\nonumber \\
 & +2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)-\alpha_{c}\,\mathrm{\mathrm{Cur}l}\left(\mathrm{\mathrm{Cur}l}\,\mathbf{P}\right)\nonumber\end{aligned}$$ or equivalently, in index notation[^5] $$\begin{aligned}
\rho\,\ddot{u}_{i} & =\mu_{e}\left(u_{i,jj}-P_{ij,j}+u_{j,ij}-P_{ji,j}\right)+\lambda_{e}\left(u_{j,ji}-P_{jj,i}\right)\delta_{ij}+\mu_{c}\left(u_{i,jj}-P_{ij,j}-u_{j,ij}+P_{ji,j}\right),\nonumber \\
\label{eq:bulk-mod-3-1}\\
\eta\,\ddot{P}_{ij} & =\mu_{e}\left(u_{i,j}-P_{ij}+u_{j,i}-P_{ji}\right)+\lambda_{e}\left(u_{k,k}-P_{kk}\right)\delta_{ij}-\mu_{h}\left(P_{ij}+P_{ji}\right)-\lambda_{h}P_{kk}\delta_{ij}\nonumber \\
 & +\mu_{c}\left(u_{i,j}-P_{ij}-u_{j,i}+P_{ji}\right)+\alpha_{c}\left(P_{ij,kk}-P_{ik,jk}\right).\nonumber\end{aligned}$$

The classical micromorphic continuum
------------------------------------

From here on, we call classical micromorphic continuum a medium the energy of which is given by $$\begin{aligned}
W & =\mu_{e}\left\Vert \,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\lambda_{e}}{2}\left(\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right)^{2}+\mu_{h}\left\Vert \,\mathrm{sym}\,\mathbf{P}\,\right\Vert ^{2}+\frac{\lambda_{h}}{2}\left(\mathrm{tr}\,\mathbf{P}\right)^{2}\label{KinPot-1-1}\\
 & +\mu_{c}\left\Vert \,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\alpha_{g}}{2}\left\Vert \nabla\,\mathbf{P}\right\Vert ^{2}.\nonumber\end{aligned}$$

### Comparison with the Mindlin and Eringen models

Analogously to what done for the relaxed case, we can recover that the classical micromorphic continuum can be seen as a particular case of Mindlin’s model by means of the parameter identification (\[Identification1\]) to which one must add: $$a_{10}=\alpha_{g},\qquad a_{1}=a_{2}=a_{3}=a_{4}=a_{5}=a_{6}=a_{7}=a_{8}=a_{9}=a_{11}=a_{12}=a_{13}=a_{14}=a_{15}=0.$$ Analogously, the classical micromorphic continuum can be obtained from Eringen’s model by means of the parameter identification (\[Identification1-1\]) to which one must add the conditions $$\tau_{7}=\alpha_{g},\qquad\tau_{1}=\tau_{2}=\tau_{3}=\tau_{4}=\tau_{5}=\tau_{6}=\tau_{8}=\tau_{9}=\tau_{10}=\tau_{11}=0.$$

### Governing equations

The equations of motion are the same as Eqs. (\[eq:bulk-mod-3\]), except for the gradient term in the second equation: $$\begin{aligned}
\rho\,\mathbf{u}_{tt} & =\mathrm{Div}\left[2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1+2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right],\nonumber \\
\label{eq:bulk-mod-3-2-1}\\
\eta\,\mathbf{P}_{tt} & =2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1-2\,\mu_{h}\,\mathrm{sym}\,\mathbf{P}-\lambda_{h}\mathrm{tr}\,\mathbf{P}\:\mathds1\nonumber \\
 & +2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\alpha_{g}\:\mathrm{Div}\left(\nabla\mathbf{P}\right).\nonumber\end{aligned}$$ The second equation can hence be rewritten in index notation as $$\begin{aligned}
\eta\,\ddot{P}_{ij} & =\mu_{e}\left(u_{i,j}-P_{ij}+u_{j,i}-P_{ji}\right)+\lambda_{e}\left(u_{k,k}-P_{kk}\right)\delta_{ij}-\mu_{h}\left(P_{ij}+P_{ji}\right)-\lambda_{h}P_{kk}\delta_{ij}\label{EqBulkClass}\\
 & +\mu_{c}\left(u_{i,j}-P_{ij}-u_{j,i}+P_{ji}\right)+\alpha_{g}P_{ij,kk}.\nonumber\end{aligned}$$

Plane wave propagation in micromorphic media
============================================

In our study of wave propagation in considered micromorphic media, we will limit ourselves to the case of plane waves travelling in an infinite domain. With this end in mind, we can suppose that the space dependence of all the introduced kinematical fields is limited only to the component $X$ of $\mathbf{X}$ which we also suppose to be the direction of propagation of the considered wave. It is immediate that, according to the Cartan-Lie decomposition for the tensor $\mathbf{P}$ (see Eq. (\[CartanLie\])), the component $P_{11}$ of the tensor $\mathbf{P}$ itself can be rewritten as $P_{11}=P^{D}+P^{S}$ where we set $$P^{S}:=\frac{1}{3}\left(P_{11}+P_{22}+P_{33}\right),\qquad P^{D}:=\left(\mathrm{dev\: sym}\,\mathbf{P}\right)_{11}.\label{SpherDev}$$ We also denote the components 12 and 13 of the symmetric and skew-symmetric part of the tensor $\mathbf{P}$ respectively as $$\left(\mathrm{sym}\:\mathbf{P}\right)_{1\xi}=P_{(1\xi)},\qquad\left(\mathrm{skew}\:\mathbf{P}\right)_{1\xi}=P_{\left[1\xi\right]},\quad\xi=1,2.\label{SymSkew}$$ We finally introduce the last new variable $$P^{V}=P_{22}-P_{33}.\label{VolumeVar}$$

The relaxed micromorphic continuum
----------------------------------

We want to rewrite the equations of motion (\[eq:bulk-mod-3-1\]) in terms of the new variables (\[SpherDev\]), (\[SymSkew\]) and, of course, of the displacement variables $u_{i}$. Before doing so, we introduce the quantities[^6] $$\begin{gathered}
c_{m}=\sqrt{\frac{\alpha_{c}}{\eta}},\qquad c_{s}=\sqrt{\frac{\mu_{e}+\mu_{c}}{\rho}},\qquad c_{p}=\sqrt{\frac{\lambda_{e}+2\mu_{e}}{\rho}},\nonumber \\
\nonumber \\
\omega_{s}=\sqrt{\frac{2\left(\mu_{e}+\mu_{h}\right)}{\eta}},\qquad\omega_{p}=\sqrt{\frac{\left(3\lambda_{e}+2\mu_{e}\right)+\left(3\lambda_{h}+2\mu_{h}\right)}{\eta}},\qquad\omega_{r}=\sqrt{\frac{2\mu_{c}}{\eta}},\label{Definitions}\\
\nonumber \\
\omega_{l}=\sqrt{\frac{\lambda_{h}+2\mu_{h}}{\eta}},\qquad\omega_{t}=\sqrt{\frac{\mu_{h}}{\eta}}.\nonumber\end{gathered}$$ With the proposed new choice of variables and considering definitions (\[Definitions\]) we are able to rewrite the governing equations as different uncoupled sets of equations, namely:

-   A set of three equations only involving longitudinal quantities:

$$\begin{aligned}
\ddot{u}_{1} & =c_{p}^{2}u_{1,11}-\frac{2\mu_{e}}{\rho}\, P_{,1}^{D}-\frac{3\lambda_{e}+2\mu_{e}}{\rho}\, P_{,1}^{S},\nonumber \\
\nonumber \\
\ddot{P}^{D} & =\frac{4}{3}\,\frac{\mu_{e}}{\eta}u_{1,1}+\frac{1}{3}c_{m}^{2}P_{,11}^{D}-\frac{2}{3}c_{m}^{2}P_{,11}^{S}-\omega_{s}^{2}P^{D},\label{Longitudinal}\\
\nonumber \\
\ddot{P}^{S} & =\frac{3\lambda_{e}+2\mu_{e}}{3\eta}u_{1,1}-\frac{1}{3}c_{m}^{2}P_{,11}^{D}+\frac{2}{3}c_{m}^{2}P_{,11}^{S}-\omega_{p}^{2}P^{S},\nonumber\end{aligned}$$

-   Two sets of three equations only involving transverse quantities in the $k$-th direction, with $\xi=2,3$:

$$\begin{aligned}
\ddot{u}_{\xi} & =c_{s}^{2}u_{\xi,11}-\frac{2\mu_{e}}{\rho}\, P_{\left(1\xi\right),1}+\frac{\eta}{\rho}\omega_{r}^{2}P_{\left[1\xi\right],1},\nonumber \\
\nonumber \\
\ddot{P}_{\left(1\xi\right)} & =\frac{\mu_{e}}{\eta}u_{\xi,1}+\frac{1}{2}c_{m}^{2}P_{(1\xi)}{}_{,11}+\frac{1}{2}c_{m}^{2}P_{\left[1\xi\right],11}-\omega_{s}^{2}P_{(1\xi)},\label{Transverse}\\
\nonumber \\
\ddot{P}_{\left[1\xi\right]} & =-\frac{1}{2}\omega_{r}^{2}u_{\xi,1}+\frac{1}{2}c_{m}^{2}P_{(1\xi),11}+\frac{1}{2}c_{m}^{2}P_{\left[1\xi\right]}{}_{,11}-\omega_{r}^{2}P_{\left[1\xi\right]},\nonumber\end{aligned}$$

-   One equation only involving the variable $P_{\left(23\right)}$:

$$\ddot{P}_{\left(23\right)}=-\omega_{s}^{2}P_{\left(23\right)}+c_{m}^{2}P_{\left(23\right),11},\label{Shear}$$

-   One equation only involving the variable $P_{\left[23\right]}$ :

$$\ddot{P}_{\left[23\right]}=-\omega_{r}^{2}P_{\left[23\right]}+c_{m}^{2}P_{\left[23\right],11},\label{Rotations23}$$

-   One equation only involving the variable $P^{V}$:

$$\ddot{P}^{V}=-\omega_{s}^{2}P^{V}+c_{m}^{2}P_{,11}^{V}.\label{VolumeVariation}$$

These 12 scalar differential equations will be used to study wave propagation in our relaxed micromorphic media.

It can be checked that, in order to guarantee positive definiteness of the potential energy (\[DefPos\]) , the characteristic velocities and frequencies introduced in Eq.(\[Definitions\]) cannot be chosen in a completely arbitrary way. In the numerical simulations considered in this paper, the values of the elastic parameters are always chosen in order to guarantee positive definiteness of the potential energy according to Eqs.(\[DefPos-1\]).

The classical micromorphic continuum
------------------------------------

For the classical micromorphic continuum, introducing the new characteristic velocity $$c_{g}=\sqrt{\frac{\alpha_{g}}{\eta}},$$ we get the following simplified one-dimensional equations

-   Longitudinal

$$\begin{aligned}
\ddot{u}_{1} & =c_{p}^{2}u_{1,11}-\frac{2\mu_{e}}{\rho}\, P_{,1}^{D}-\frac{3\lambda_{e}+2\mu_{e}}{\rho}P_{,1}^{S},\nonumber \\
\nonumber \\
\ddot{P}^{D} & =\frac{4}{3}\,\frac{\mu_{e}}{\eta}u_{1,1}+c_{g}^{2}P_{,11}^{D}-\omega_{s}^{2}P^{D},\label{Longitudinal-1}\\
\nonumber \\
\ddot{P}^{S} & =\frac{3\lambda_{e}+2\mu_{e}}{3\eta}u_{1,1}+c_{g}^{2}P_{,11}^{S}-\omega_{p}^{2}P^{S},\nonumber\end{aligned}$$

-   Two sets of three equations only involving transverse quantities in the $k$-th direction, with $\xi=2,3$:

$$\begin{aligned}
\ddot{u}_{\xi} & =c_{s}^{2}u_{\xi,11}-\frac{2\mu_{e}}{\rho}\, P_{\left(1\xi\right),1}+\frac{\eta}{\rho}\omega_{r}^{2}P_{\left[1\xi\right],1},\nonumber \\
\nonumber \\
\ddot{P}_{\left(1\xi\right)} & =\frac{\mu_{e}}{\eta}\, u_{\xi,1}+c_{g}^{2}P_{(1\xi)}{}_{,11}-\omega_{s}^{2}P_{(1\xi)},\label{Transverse-1}\\
\nonumber \\
\ddot{P}_{\left[1\xi\right]} & =-\frac{1}{2}\omega_{r}^{2}u_{\xi,1}+c_{g}^{2}P_{\left[1\xi\right]}{}_{,11}-\omega_{r}^{2}P_{\left[1\xi\right]},\nonumber\end{aligned}$$

-   One equation only involving the variable $P_{\left(23\right)}$:

$$\ddot{P}_{\left(23\right)}=-\omega_{s}^{2}P_{\left(23\right)}+c_{g}^{2}P_{\left(23\right),11},\label{Shear-1}$$

-   One equation only involving the variable $P_{\left[23\right]}$ :

$$\ddot{P}_{\left[23\right]}=-\omega_{r}^{2}P_{\left[23\right]}+c_{g}^{2}P_{\left[23\right],11},\label{Rotations23-1}$$

-   One equation only involving the variable $P^{V}$:

$$\ddot{P}^{V}=-\omega_{s}^{2}P^{V}+c_{g}^{2}P_{,11}^{V}.\label{VolumeVariation-1}$$

Linear waves in micromorphic media
==================================

In this section we study the dispersion relations for the considered relaxed micromorphic continuum, as well as for the Classical micromorphic continuum. We also consider dispersion relations for Cosserat media obtained as a degenerate limit case of our relaxed model. Finally, dispersion relations for a second gradient continuum obtained as a limit case of the classic micromorphic continuum are also presented. It will be pointed out that only our relaxed micromorphic model can disclose the presence of band-gaps.

Micro-oscillations
------------------

We start studying a particular solution of the set of introduced differential equations by setting $$\begin{gathered}
u_{i}=0,\ i=1,2,3,\quad P_{(23)}=Re\left\{ \alpha_{(23)}e^{i\omega t}\right\} ,\quad P_{[23]}=Re\left\{ \alpha_{[23]}e^{i\omega t}\right\} ,\quad P^{V}=Re\left\{ \alpha^{V}e^{i\omega t}\right\} ,\\
\\
P_{(1\xi)}=Re\left\{ \alpha_{(1\xi)}e^{i\omega t}\right\} ,\quad P_{[1\xi]}=Re\left\{ \alpha_{[1\xi]}e^{i\omega t}\right\} ,\ \xi=2,3.\end{gathered}$$ Replacing these expressions for the unknown variables in each of Eqs.(\[Longitudinal\])-(\[VolumeVariation\]), noticing that the space derivatives are vanishing, one gets that the first of Eqs.(\[Longitudinal\]) and (\[Transverse\]) are such that the frequency calculated for the waves $u_{1}$ and $u_{\xi}$ is vanishing, i.e. $$\omega=0.\label{Acoustic}$$ Moreover, from the remaining equations one gets the values of frequency $\text{\ensuremath{\omega}}$ for the waves $P^{D}$, $P^{S}$, $P_{(1\xi)}$, $P_{[1\xi]}$, $P_{(23)}$, $P_{[23]}$ and $P^{V}$ respectively $$\omega^{2}=\omega_{s}^{2},\quad\omega^{2}=\omega_{p}^{2}\quad\omega^{2}=\omega_{s}^{2},\quad\omega^{2}=\omega_{r}^{2},\quad\omega^{2}=\omega_{s}^{2},\quad\omega^{2}=\omega_{r}^{2},\quad\omega^{2}=\omega_{s}^{2}.\label{Optic}$$ It is clear that one gets exactly the same result when considering the complete Mindlin-Eringen model, since the only difference is on the space derivatives of $\mathbf{P}$ which do not intervene when studying micro-oscillations. The characteristic values of the frequencies given in Eq.(\[Optic\]) are fixed once the material parameters of the system are specified (see Eqs.(\[Definitions\])) and they represent the limit values for $k\rightarrow0$ of the eigenvalues $\omega(k)$ associated to propagative waves, as it will be better shown in the next section. This preliminary study of micro-oscillations allows a precise classification of waves which can propagate in a micromorphic medium. More particularly, we can distinguish two types of propagative waves:

-   **acoustic waves**, i.e. waves which have vanishing frequency when the wavenumber $k$ is vanishing,

-   **optic waves,** i.e. waves which have non-vanishing frequency when the wavenumber $k$ is vanishing.

Indeed, we will see in the following, that a third type of wave may exist in relaxed micromorphic media under suitable hypothesis on the constitutive parameters:

-   **standing (or evanescent) waves,** i.e. waves which have imaginary wavenumber $k$ corresponding to some frequency ranges. These waves do not propagate, but keep oscillating in a given, limited region of space.

According to the performed study of micro-oscillations we can conclude that

-   For the **uncoupled waves** $P_{(23)}$, $P_{[23]}$ and $P^{V}$, only optic waves are found with cutoff frequencies $\omega_{s}$, $\omega_{r}$ and $\omega_{s}$. Moreover, the structure of the governing equations for $P_{(23)}$ and $P^{V}$ are formally identical (see Eqs. (\[Shear\]) and (\[VolumeVariation\])) so that the associated dispersion relations will give rise to superimposed curves. In the limit $\mu_{c}\rightarrow0$ one has from Eq. (\[Definitions\]) that $\omega_{r}\rightarrow0.$ This means that in this limit case one has one acoustic waves and two superimposed optic waves with cutoff frequency $\omega_{s}$.

-   For the **longitudinal waves** $u_{1}$, $P^{D}$, $P^{S}$, we can identify one acoustic wave and two optic waves with cutoff frequencies $\omega_{s}$ and $\omega_{p}$. In the limit $\mu_{c}\rightarrow0$, which implies that $\omega_{r}\rightarrow0$, the situation for longitudinal waves remains unchanged.

-   For the **transverse waves** $u_{\xi}$, $P_{(1\xi)}$, $P_{[1\xi]}$, ($\xi=1,2$) we identify one acoustic wave and two optic waves with cutoff frequencies $\omega_{s}$ and $\omega_{r}$. In the limit $\mu_{c}\rightarrow0$ which implies that $\omega_{r}\rightarrow0$, we have two acoustic waves and one optic wave with cutoff frequency $\omega_{s}$.

Planar wave propagation in the relaxed micromorphic continuum
-------------------------------------------------------------

We now look for a wave form solution of the previously derived equations of motion. We start from the uncoupled equations (\[Shear\])-(\[VolumeVariation\]) and assume that the involved unknown variables take the harmonic form $$P_{\left(23\right)}=Re\left\{ \beta_{\left(23\right)}e^{i(kX-\omega t)}\right\} ,\quad P_{\left[23\right]}=Re\left\{ \beta_{\left[23\right]}e^{i(kX-\omega t)}\right\} ,\quad P^{V}=Re\left\{ \beta^{V}e^{i(kX-\omega t)}\right\} ,\label{WaveForm1}$$ where $\beta_{\left(23\right)}$, $\beta_{\left[23\right]}$ and $\beta^{V}$ are the amplitudes of the three introduced waves. It can be remarked that the variables $P_{\left(23\right)}$, $P_{\left[23\right]}$ and $P^{V}$ respectively represent transverse (with respect to wave propagation) micro-shear, transverse micro-rotation and transverse micro-deformations at constant volume. Replacing this wave form in Eqs. (\[Shear\])-(\[VolumeVariation\]) and simplifying one obtains the following dispersion relations respectively: $$\omega(k)=\sqrt{\omega_{s}^{2}+k^{2}c_{m}^{2}},\qquad\omega(k)=\sqrt{\omega_{r}^{2}+k^{2}c_{m}^{2}},\qquad\omega(k)=\sqrt{\omega_{s}^{2}+k^{2}c_{m}^{2}}.\label{Eigen1}$$ We notice that for a vanishing wave number ($k=0$) the dispersion relations for the three considered waves give non-vanishing frequencies which correspond to the ones calculated in the previous subsection for the same waves. This is equivalent to say that the waves associated to the three considered variables $P_{\left(23\right)}$, $P_{\left[23\right]}$ and $P^{V}$ are so-called optic waves. Moreover, we also notice that the dispersion relation for the variables $P_{\left(23\right)}$ and $P^{V}$ are the same: this means that the wave-form solutions for these variables coincide modulo a scalar multiplication factor (see Eqs. (\[WaveForm1\])).

We now want to study harmonic solutions for the differential systems (\[Longitudinal\]) and (\[Transverse\]). To do so, we introduce the unknown vectors $\mathbf{v}_{1}=\left(u_{1},P^{D},P^{S}\right)$ and $\mathbf{v}_{\xi}=\left(u_{\xi},P_{(1\xi)},P_{[1\xi]}\right),\ \xi=2,3$ and look for wave form solutions of equations (\[Longitudinal\]) and (\[Transverse\]) in the form $$\mathbf{v}_{1}=Re\left\{ \boldsymbol{\beta}e^{i(kX-\omega t)}\right\} ,\qquad\mathbf{v}_{\xi}=Re\left\{ \boldsymbol{\gamma}^{\xi}e^{i(kX-\omega t)}\right\} ,\ \xi=2,3\label{WaveForm2}$$ where $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\beta_{3})^{T}$ and $\boldsymbol{\gamma}^{\xi}=(\gamma_{1}^{\xi},\gamma_{2}^{\xi},\gamma_{3}^{\xi})^{T}$ are the unknown amplitudes of considered waves. Replacing this expressions in equations (\[Longitudinal\]) and (\[Transverse\]) one gets respectively $$\mathbf{A}_{1}\cdot\boldsymbol{\beta}=0,\qquad\mathbf{A}_{\xi}\cdot\boldsymbol{\gamma}^{\xi}=0,\qquad\xi=2,3,\label{AlgSys}$$ where $$\begin{gathered}
\mathbf{A}_{1}=\left(\begin{array}{ccc}
-\omega^{2}+c_{p}^{2}\, k^{2} & \, i\: k\:2\mu_{e}/\rho\  & i\: k\:\left(3\lambda_{e}+2\mu_{e}\right)/\rho\\
\\
-i\: k\,\frac{4}{3}\,\mu_{e}/\eta & -\omega^{2}+\frac{1}{3}k^{2}c_{m}^{2}+\omega_{s}^{2} & -\frac{2}{3}\, k^{2}c_{m}^{2}\\
\\
-\frac{1}{3}\, i\, k\:\left(3\lambda_{e}+2\mu_{e}\right)/\eta & -\frac{1}{3}\, k^{2}\, c_{m}^{2} & -\omega^{2}+\frac{2}{3}\, k^{2}\, c_{m}^{2}+\omega_{p}^{2}
\end{array}\right),\\
\\
\\
\mathbf{A}_{2}=\mathbf{A}_{3}=\left(\begin{array}{ccc}
-\omega^{2}+k^{2}c_{s}^{2}\  & \, i\, k\,2\mu_{e}/\rho\  & -i\, k\,\frac{\eta}{\rho}\omega_{r}^{2},\\
\\
-\, i\, k\,2\mu_{e}/\eta, & -2\omega^{2}+k^{2}c_{m}^{2}+2\omega_{s}^{2} & k^{2}c_{m}^{2}\\
\\
i\, k\,\omega_{r}^{2} & k^{2}c_{m}^{2} & -2\omega^{2}+k^{2}c_{m}^{2}+2\omega_{r}^{2}
\end{array}\right).\end{gathered}$$ In order to have non-trivial solutions of the algebraic systems (\[AlgSys\]), one must impose that $$\mathrm{det}\,\mathbf{A}_{1}=0,\qquad\mathrm{det}\,\mathbf{A}_{2}=0,\qquad\mathrm{det}\,\mathbf{A}_{3}=0,\label{Dispersion}$$ which allow us to determine so-called dispersion relations $\omega=\omega\left(k\right)$ for the longitudinal and transverse waves in the relaxed micromorphic conyinuum. As it will be better explained in the next section, the eigenvalues $\omega(k)$ solutions of (\[Dispersion\]) are associated both to optic waves and to acoustic waves.

Planar wave propagation in the classical micromorphic continuum
---------------------------------------------------------------

We want to deduce in this subsection the dispersion relations for the classical micromorphic model, i.e. for the dispersion relations associated to the energy (\[KinPot-1-1\]). If we replace the wave-form solution (\[WaveForm2\]) in the longitudinal and transverse equations (\[Longitudinal-1\]) and (\[Transverse-1\]) we get $$\mathbf{B}_{1}\cdot\boldsymbol{\beta}=0,\qquad\mathbf{B}_{\xi}\cdot\boldsymbol{\gamma}^{\xi}=0,\qquad\xi=2,3,\label{AlgSys-1}$$ where $$\begin{gathered}
\mathbf{B}_{1}=\left(\begin{array}{ccc}
-\omega^{2}+c_{p}^{2}\, k^{2} & \, i\: k\:2\mu_{e}/\rho\  & i\: k\:\left(3\lambda_{e}+2\mu_{e}\right)/\rho\\
\\
-i\: k\,\frac{4}{3}\mu_{e}/\eta & -\omega^{2}+k^{2}c_{g}^{2}+\omega_{s}^{2} & 0\\
\\
-\frac{1}{3}\, i\, k\:\left(3\lambda_{e}+2\mu_{e}\right)/\eta & 0 & -\omega^{2}+k^{2}\, c_{g}^{2}+\omega_{p}^{2}
\end{array}\right),\\
\\
\\
\mathbf{B}_{2}=\mathbf{B}_{3}=\left(\begin{array}{ccc}
-\omega^{2}+k^{2}c_{s}^{2}\  & \, i\, k\,2\mu_{e}/\rho\  & -i\, k\,\frac{\eta}{\rho}\omega_{r}^{2},\\
\\
-\, i\, k\,2\mu_{e}/\eta, & -2\omega^{2}+2\, k^{2}c_{g}^{2}+2\omega_{s}^{2} & 0\\
\\
i\, k\,\omega_{r}^{2} & 0 & -2\omega^{2}+2\, k^{2}c_{g}^{2}+2\omega_{r}^{2}
\end{array}\right).\end{gathered}$$ In order to have non-trivial solutions of the algebraic systems (\[AlgSys-1\]), one must impose that $$\mathrm{det}\,\mathbf{B}_{1}=0,\qquad\mathrm{det}\,\mathbf{B}_{2}=0,\qquad\mathrm{det}\,\mathbf{B}_{3}=0,\label{Dispersion-1}$$ which allow us to determine so-called dispersion relations $\omega=\omega\left(k\right)$ for the longitudinal and transverse waves in the classical micromorphic continuum.

As for the uncoupled waves, the dispersion relations associated to Eqs.(\[Shear-1\]),(\[Rotations23-1\]) and (\[VolumeVariation-1\]) are respectively: $$\omega(k)=\sqrt{\omega_{s}^{2}+k^{2}c_{g}^{2}},\qquad\omega(k)=\sqrt{\omega_{r}^{2}+k^{2}c_{g}^{2}},\qquad\omega(k)=\sqrt{\omega_{s}^{2}+k^{2}c_{g}^{2}}.\label{Eigen1-1}$$

\[NumericalSim\]The relaxed micromorphic model: numerical results 
==================================================================

In this section, following Mindlin [@Mindlin; @EringenBook], we will show the dispersion relations $\omega=\omega(k)$ associated to the considered relaxed micromorphic model. The analysis of dispersion relations in [@Mindlin; @EringenBook] was of qualitative nature, due to the huge number of constitutive parameters (18 elastic coefficients) which are assumed to be non-vanishing and to computational difficulties. On the other hand, thanks to the relaxed constitutive assumption (\[KinPot\]), we are able to clearly show precise dispersion relations for the considered cases and to associate to few constitutive parameters the main mechanisms associated to wave propagation in micromorphic media. As principal obtained result, we will clearly point out that band-gaps can be forecast only when considering a non-vanishing Cosserat couple modulus $\mu_{c}$ in our relaxed micromorphic model. This parameter is associated to the relative rotation of the microstructure with respect to the continuum matrix.

             Parameter                   Value              Unit
  ------------------------------- -------------------- ---------------
             $\mu_{e}$                   $200$              $MPa$
      $\lambda_{e}=2\mu_{e}$              400               $MPa$
       $\mu_{c}=2.2\mu_{e}$              $ $440             $MPa$
             $\mu_{h}$                    100               $MPa$
           $\lambda_{h}$                 $100$              $MPa$
             $L_{c}\ $                    $3$               $mm$
             $L_{g}\ $                    $3$               $mm$
   $\alpha_{c}=\mu_{e}L_{c}^{2}$   $1.8\times10^{-3}$   $MPa\: m^{2}$
   $\alpha_{g}=\mu_{e}L_{g}^{2}$   $1.8\times10^{-3}$   $MPa\: m^{2}$
              $\rho$                     $2000$          $Kg/m^{3}$
              $\rho$’                    $2500$          $Kg/m^{3}$
                $d$                       $2$               $mm$
        $\eta=d^{2}\,\rho'$           $ $$10^{-2}$         $Kg/m$

  : \[ParametersValues\]Values of the parameters of the relaxed model used in the numerical simulations (left) and corresponding values of the Lamé parameters and of the Young modulus and Poisson ratio (right).

   Parameter   Value    Unit
  ----------- -------- -------
   $\lambda$   $82.5$   $MPa$
     $\mu$     $66.7$   $MPa$
      $E$      $170$    $MPa$
     $\nu$     $0.28$    $-$

  : \[ParametersValues\]Values of the parameters of the relaxed model used in the numerical simulations (left) and corresponding values of the Lamé parameters and of the Young modulus and Poisson ratio (right).

We start by showing in Tab.\[ParametersValues\] (left) the values of the parameters of the relaxed model used in the performed numerical simulations. In order to make the obtained results more exploitable, we also recall that in [@NeffRelaxed] the following homogenized formulas were obtained which relate the parameters of the relaxed model to the macroscopic Lamé parameters $\lambda$ and $\mu$ which are usually measured by means of standard mechanical tests $$\mu_{e}=\frac{\mu_{h}\,\mu}{\mu_{h}-\mu},\qquad2\mu_{e}+3\lambda_{e}=\frac{(2\mu_{h}+3\lambda_{h})\left(2\mu+3\lambda\right)}{(2\mu_{h}+3\lambda_{h})-\left(2\mu+3\lambda\right)}.\label{Homogenized}$$ These relationships imply that the following inequalities are satisfied $$\mu_{h}>\mu,\qquad3\lambda_{h}+2\mu_{h}>3\lambda+2\mu.$$ It is clear that, once the values of the parameters of the relaxed models are known, the standard Lamé parameters can be calculated by means of formulas (\[Homogenized\]), which is what was done in Tab.\[ParametersValues\] (right). To the sake of completeness, we also show in the same table the corresponding Young modulus and Poisson ratio, calculated by means of the standard formulas $$E=\frac{\mu\left(3\lambda+2\mu\right)}{\lambda+\mu},\qquad\text{\ensuremath{\nu}}=\frac{\lambda}{2\left(\lambda+\mu\right)}.$$

The relaxed micromorphic model with $\mu_{c}=0$.
------------------------------------------------

We start by showing the dispersion relations of the algebraic problem (\[Dispersion\]) for the particular case $\mu_{c}=0$. Figure \[DispersionDiag\] shows separately the behaviors of the uncoupled waves $P_{\left(23\right)}$, $P_{\left[23\right]}$ and $P^{V}$ (Fig. \[DispersionDiag\](a)), of the longitudinal waves $u_{1},\, P^{D},\, P^{S}$ (Fig. \[DispersionDiag\](b)) and of the transverse waves $u_{\xi},P_{(1\xi)},P_{[1\xi]}$ (Fig. \[DispersionDiag\](c)).

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
   ![\[DispersionDiag\]Dispersion relations $\omega=\omega(k)$ for the relaxed model with vanishing Cosserat couple modulus ($\mu_{c}=0$): uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRA: transverse rotational acoustic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TO: transverse optic, TA$_{1}$-TA$_{2}$: first and second transverse acoustic.](mc0-uncoupled "fig:")   ![\[DispersionDiag\]Dispersion relations $\omega=\omega(k)$ for the relaxed model with vanishing Cosserat couple modulus ($\mu_{c}=0$): uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRA: transverse rotational acoustic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TO: transverse optic, TA$_{1}$-TA$_{2}$: first and second transverse acoustic.](mc0-long "fig:")   ![\[DispersionDiag\]Dispersion relations $\omega=\omega(k)$ for the relaxed model with vanishing Cosserat couple modulus ($\mu_{c}=0$): uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRA: transverse rotational acoustic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TO: transverse optic, TA$_{1}$-TA$_{2}$: first and second transverse acoustic.](mc0-transv "fig:")     
                                                                                                                                                                                                                                                         (a)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (b)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (c)                                                                                                                                                                                                                                                          
  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --

We can recover from Fig. \[DispersionDiag\](a) one acoustic, rotational wave ($TRA$) and two superimposed optic waves ($TSO$ and $TCVO$) with cutoff frequency $\omega_{s}$. The acoustic wave shows a non-dispersive behavior (the dispersion curve is a straight line). This implies that, when considering relaxed micromorphic media with $\mu_{c}=0$, there is always at least one wave for which the wavenumber is always real independently of the value of frequency. This fact guarantees wave propagation inside the considered medium for all frequency ranges (no global band-gaps).

Figure \[DispersionDiag\](b) shows that longitudinal waves indeed involve one acoustic wave ($LA$) and two optic waves ($LO_{1}$ and $LO_{2}$) with cutoff frequencies $\omega_{p}$ and $\omega_{s}$ respectively. Moreover, it is found that the acoustic wave has an horizontal asymptote at $\omega=\omega_{l}$. As a consequence, it can be seen that a frequency range $\left(\omega_{s},\omega_{l}\right)$ exists in which the wavenumber becomes imaginary for longitudinal waves. According to Eqs. (\[WaveForm2\]), an imaginary wavenumber $k$ gives rise to solutions which are exponentials decaying in space. Waves of this type are so-called standing waves which do not propagate, but keep oscillating in a limited region of space.

Finally, figure \[DispersionDiag\](c) confirms that, for transverse waves, two acoustic waves ($TA{}_{1}$ and $TA{}_{2}$) and one optic wave ($TO$) with cutoff frequency $\omega_{s}$ can be identified. One of the acoustic waves has an horizontal asymptote at $\omega=\omega_{t}$. It is easy to recognize that, due to the existence of the non-dispersive, transverse, acoustic wave $TA_{1}$, there always exists at least one propagative wave for any chosen value of the frequency.

We can conclude that, in general, when considering the relaxed micromorphic medium as a whole (all the 12 waves), there always exist waves which propagate inside the considered medium independently of the value of frequency. On the other hand, if one considers a particular case (obtained by imposing suitable kinematical constraints) in which only longitudinal waves can propagate, then in the frequency range $\left(\omega_{s},\omega_{l}\right)$ only standing wave exist which do not allow for wave propagation. In this very particular case, the frequency range $\left(\omega_{s},\omega_{l}\right)$ can be considered as a band-gap for longitudinal waves. According to definitions given in Eq. (\[Definitions\]) the depth of this frequency band is controlled by the three parameters $\mu_{e}$, $\mu_{h}$ and $\lambda_{h}$.

The relaxed micromorphic model with $\mu_{c}>0$.
------------------------------------------------

In this subsection we discuss the behavior with respect to wave propagation of relaxed micromorphic continua in the general case in which the Cosserat couple modulus $\mu_{c}$ is assumed to be non-vanishing. We start by showing the dispersion relations in figure \[separate\].

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
   ![\[separate\]Dispersion relations $\omega=\omega(k)$ for the relaxed model with non-vanishing Cosserat couple modulus ($\mu_{c}>0$): uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO: transverse rotational optic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TA: transverse acoustic, TO$_{1}$-TO$_{2}$: first and second transverse optic.](Uncoupled "fig:")   ![\[separate\]Dispersion relations $\omega=\omega(k)$ for the relaxed model with non-vanishing Cosserat couple modulus ($\mu_{c}>0$): uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO: transverse rotational optic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TA: transverse acoustic, TO$_{1}$-TO$_{2}$: first and second transverse optic.](Longitudinal "fig:")   ![\[separate\]Dispersion relations $\omega=\omega(k)$ for the relaxed model with non-vanishing Cosserat couple modulus ($\mu_{c}>0$): uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO: transverse rotational optic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TA: transverse acoustic, TO$_{1}$-TO$_{2}$: first and second transverse optic.](Transverse "fig:")     
                                                                                                                                                                                                                                                     (a)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (b)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (c)                                                                                                                                                                                                                                                        
  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --

As for the uncoupled waves we have, in this case, only optic waves: two superimposed ($TSO$ and $TCVO$) with cutoff frequency $\omega_{s}$ and one ($TRO$) with cutoff frequency $\omega_{r}$ (see Fig. \[separate\](a)). When considering longitudinal waves (see Fig. \[separate\](b)) the situation is unchanged with respect to the previous case ($\mu_{c}=0$) and one can observe one acoustic wave ($LA$) and two optic waves ($LO{}_{1}$ and $LO_{2}$) with cutoff frequencies $\omega_{p}$ and $\omega_{s}$ respectively. If we finally consider the transverse waves (Fig. \[separate\](c)), we can remark that there exists one acoustic wave ($TA$ ) and two optic waves ($TO{}_{1}$ and $TO_{2}$) with cutoff frequencies $\omega_{s}$ and $\omega_{r}$ respectively. It can be easily noticed that, in all three cases, there exist frequency ranges in which no propagative wave can be found. This means that the wavenumber becomes imaginary and only standing waves exist. For the situation depicted in Fig. \[separate\], the frequency ranges for which only standing waves exist are $\left(0,\omega_{s}\right)$ for uncoupled waves, $\left(\omega_{l},\omega_{s}\right)$ for longitudinal waves and $\left(\omega_{t},\omega_{s}\right)$ for transverse waves. The intersection of these three intervals being non-empty, we can conclude that a frequency band-gap $\left(\omega_{l},\omega_{s}\right)$ exists in the considered relaxed micromorphic medium corresponding to which any type of wave can propagate independently of the value of $k$. More precisely, if one imagines to excite the considered medium with a signal the frequency of which falls in the range $(\omega_{l},\omega_{s})$, this signal cannot propagate inside the considered medium but it keep oscillating close to the point of application of the initial condition. It is clear that this feature is a fundamental tool to conceive micro-structured materials which can be used as pass- and stop- bands. Investigations in this sense would bring new insights towards high-tech solutions in the field of vibration control and will be the object of forthcoming studies. It is evident (see Eqs. (\[Definitions\])) that, in general, the relative positions of the horizontal asymptotes $\omega_{l}$ and $\omega_{t}$ as well as of the cutoff frequencies $\omega_{s}$, $\omega_{r}$ and $\omega_{p}$ can vary depending on the values of the constitutive parameters. In particular, we can observe that the relative position of the characteristic frequencies defined in (\[Definitions\]) can vary depending on the values of the constitutive parameters. It can be checked that, in the case in which $\lambda_{e}>0$ and $\lambda_{h}>0$ one always has $\omega_{p}>\omega_{s}>\omega_{t}$ and $\omega_{l}>\omega_{t}$. The relative position of $\omega_{l}$ and of $\omega_{s}$ can vary depending on the values of the parameters $\lambda_{h}$ and $\mu_{h}$.

It is easy to verify that one can have band-gaps for longitudinal waves only if the horizontal asymptote $\omega_{l}$ is such that $\omega_{s}>\omega_{l}$ (which implies $\lambda_{h}<2\mu_{e}$). As for transverse waves, it can be checked that band-gaps can exist only if $\omega_{r}>\omega_{t}$ (which implies $\mu_{c}>\mu_{h}/2$). A band gap for the uncoupled waves always exist (independently of the values of the constitutive parameters) for frequencies between $0$ and the smaller among $\omega_{r}$ and $\omega_{s}$. It is clear that some stronger conditions are needed in order to have a global band gap which do not allow for any kind of waves (transverse, longitudinal and uncoupled) to propagate inside the considered microstructured material. More particularly, it can be checked that, in order to have a global band gap, the following conditions must be simultaneously satisfied $$\omega_{s}>\omega_{l}\quad\textrm{and}\quad\omega_{r}>\omega_{l}.$$ In terms of the constitutive parameters of the relaxed model, we can say that global band-gaps can exist, in the case in which one considers positive values for the parameters $\lambda_{e}$ and $\lambda_{h}$ , if and only if we have simultaneously $$\begin{gathered}
\nonumber \\
0<\mu_{e}<+\infty,\qquad0<\lambda_{h}<2\mu_{e},\qquad\mu_{c}>\frac{\lambda_{h}+2\text{\ensuremath{\mu}}_{h}}{2}.\label{Band Gaps}\\
\nonumber\end{gathered}$$ As far as negative values for $\lambda_{e}$ and $\lambda_{h}$ are allowed, the conditions for band gaps are not so straightforward as (\[Band Gaps\]). We do not consider this possibility in this paper, leaving this point open for further considerations.

We can conclude by saying that, the fact of switching on a unique parameter, namely the Cosserat couple modulus $\mu_{c}$, allows for the description of frequency band-gaps in which no propagation can occur. This parameter can hence be seen as a discreteness quantifier which starts accounting for lattice discreteness as far as it reaches the threshold value specified in Eq.(\[Band Gaps\]). The fact of being able to predict band-gaps by means of a micromorphic model is a novel feature of the introduced relaxed model. Indeed, as it will be shown in the remainder of this paper, neither the classical micromorphic continuum model nor the Cosserat and the second gradient ones are able to predict such band-gaps.

The Cosserat model as limit case of the relaxed micromorphic model 
-------------------------------------------------------------------

In this subsection we analyze wave propagation in Cosserat-type media which can be obtained from the proposed relaxed model by letting the parameter $\mu_{h}$ tend to infinity. Indeed, since the strain energy density defined in (\[KinPot\]) must remain finite, when letting $\mu_{h}\rightarrow\infty$ one must have $\mathrm{Sym}\,\mathbf{P}\rightarrow0$, hence $\mathbf{P}=\mathrm{skew}\,\mathbf{P}$. Therefore, the strain energy density becomes the Cosserat energy (see [@Neff_JeongMMS08; @Jeong_Neff_ZAMM08]) $$W_{\mathrm{Coss}}\left(\nabla\mathbf{u},\mathrm{skew}\:\mathbf{P}\right)=\mu_{e}\left\Vert \mathrm{sym}\:\nabla\mathbf{u}\right\Vert ^{2}+\frac{\lambda_{e}}{2}\left(\mathrm{tr}\:\nabla\mathbf{u}\right)^{2}+\mu_{c}\left\Vert \mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right\Vert ^{2}+\frac{\alpha}{2}\left\Vert \mathrm{\mathrm{\, Cur}l}\left(\mathrm{skew}\:\mathbf{P}\right)\right\Vert ^{2}.$$

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
   ![\[separate-1\]Dispersion relations $\omega=\omega(k)$ for the Cosserat model obtained by letting $\mu_{h}\rightarrow\infty$ in the relaxed model: uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO: transverse rotational optic, LA: longitudinal acoustic, TO: transverse optic, TA: transverse acoustic.](UncoupledCosserat "fig:")   ![\[separate-1\]Dispersion relations $\omega=\omega(k)$ for the Cosserat model obtained by letting $\mu_{h}\rightarrow\infty$ in the relaxed model: uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO: transverse rotational optic, LA: longitudinal acoustic, TO: transverse optic, TA: transverse acoustic.](LongCosserat "fig:")   ![\[separate-1\]Dispersion relations $\omega=\omega(k)$ for the Cosserat model obtained by letting $\mu_{h}\rightarrow\infty$ in the relaxed model: uncoupled waves (a), longitudinal waves (b) and transverse waves (c). TRO: transverse rotational optic, LA: longitudinal acoustic, TO: transverse optic, TA: transverse acoustic.](TransvCosserat "fig:")     
                                                                                                                                                                                 (a)                                                                                                                                                                                                                                                                                                                                                              (b)                                                                                                                                                                                                                                                                                                                                                            (c)                                                                                                                                                                                  
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The dispersion relations obtained for this particular limit case are depicted in Fig.\[separate-1\]. It can be immediatly noticed that all the optic waves with cutoff frequency $\omega_{s}$ do not propagate anymore since indeed $\omega_{s}\rightarrow\infty$ (see definition in Eq.(\[Definitions\])). The same is for the optic longitudinal wave with cutoff frequency $\omega_{p}$. As for the acoustic waves, they exist in the limit Cosserat medium, but they do not have horizontal asymptotes anymore since $\omega_{l}$ and $\omega_{t}$ tend to infinity as well. We finally end up with a medium in which we can observe one optic wave ($TRO$) associate to the variable $P_{[23]}$ (micro-rotation) with cutoff frequency $\omega_{r}$, one acoustic non-dispersive longitudinal wave ($LA$), one acoustic slightly dispersive transverse wave ($TA$) and one optic transverse wave ($TO$) with cutoff frequency $\omega_{r}$. It is easy to remark that no band-gaps can be described in the framework of the considered Cosserat medium.

The dispersive behavior of Cosserat media shown in Fig.\[separate-1\] fits with known results in the literature. Indeed, a direct comparison with the micropolar medium studied in [@EringenBook] (p. 150) can be made, by simply considering the parameters identification (\[Identification1-1\]), (\[Identification2-1\]). Moreover, Lakes states in [@Lakes_Cosserat] that “Dilatational waves propagate non-dispersively, i.e. with velocity independent of frequency, in an unbounded isotropic Cosserat elastic medium as in the classical case. Shear waves propagate dispersively in a Cosserat solid (Eringen, 1968). A new kind of wave associated with the micro-rotation is predicted to occur in Cosserat solids”. We also remark that the behavior for high frequencies shown in Fig.\[separate-1\] coincides with that of acceleration waves in micropolar media, see [@Victor02; @Victor2].

Indeed, the linear Cosserat model is undoubtedly the most studied generalized continuum model. This does not mean, however, that its status as a useful model for the description of material behaviour is unchallenged. Quite to the contrary, it appears that after 40 years of intensive research, not one material has been conclusively established as a Cosserat material. We refer to the discussion in [@Neff_Cosserat_plasticity05; @Neff_Gamm04; @Neff_ZAMM05; @Jeong_Neff_ZAMM08; @Neff_Jeong_bounded_stiffness09; @Neff_JeongMMS08; @Neff_Jeong_Conformal_ZAMM08].

By looking at the dispersion behaviour of the linear Cosserat model and comparing it with the more general micromorphic (and relaxed micromorphic model) we get a glimpse on why the status of the linear Cosserat model is really challenging. The equations appear as a formal limit in which $\mu_{h}\to\infty$, while $\ensuremath{0<\mu_{c}<\infty}$. The process $\mu_{h}\to\infty$ corresponds conceptually to assume that the substructure cannot deform elastically, nevertheless the substructures may mutually interact through the curvature energy, which itself is only involving the $\mathrm{Curl}$-operator, since the micro-distortion is constrained to be skew-symmetric.

The usefulness of a geometrically nonlinear Cosserat model, however, is not in general questioned. Indeed, it has been shown in [@Merkel2] that a Cosserat-type continuum theory can be of use to describe the experimental behavior of granular phononic crystals. One may avoid the deficiencies of the linear model and the linear coupling, see e.g. [@Neff_Muench_simple_shear09; @Muench07_diss; @Neff_Muench_transverse_cosserat08; @Neff_Muench_magnetic08; @Neff_Biot07].

The classical micromorphic model: numerical results
===================================================

In this section, we show the dispersion relations for a classical micromorphic continuum.

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- --
   ![\[Mindlin\]Dispersion relations $\omega=\omega(k)$ for the classical model: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a classical micromorphic continuum. TRO: transverse rotational optic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TA: transverse acoustic, TO$_{1}$-TO$_{2}$: first and second transverse optic.](UncoupledGradP "fig:")   ![\[Mindlin\]Dispersion relations $\omega=\omega(k)$ for the classical model: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a classical micromorphic continuum. TRO: transverse rotational optic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TA: transverse acoustic, TO$_{1}$-TO$_{2}$: first and second transverse optic.](LongGradP "fig:")   ![\[Mindlin\]Dispersion relations $\omega=\omega(k)$ for the classical model: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a classical micromorphic continuum. TRO: transverse rotational optic, TSO: transverse shear optic, TCVO: transverse constant-volume optic, LA: longitudinal acoustic, LO$_{1}$-LO$_{2}$: first and second longitudinal optic, TA: transverse acoustic, TO$_{1}$-TO$_{2}$: first and second transverse optic.](TransvGradP "fig:")     
                                                                                                                                                                                                                                               (a)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (b)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (c)                                                                                                                                                                                                                                                
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These dispersion relations are plotted in Fig.\[Mindlin\]: we recover a similar behavior with respect to the one qualitatively sketched in [@Mindlin] and [@EringenBook]. Indeed, we claim that the classical micromorphic model associated to the strain energy density (\[KinPot-1-1\]) is qualitatively equivalent to the full 18-parameters Mindlin and Eringen micromorphic model. It is clear from Fig.\[Mindlin\] that, even if the behavior of such medium is the same as the one observed in Fig.\[separate\] for the relaxed micromorphic medium when considering small wavenumbers (large wavelengths), the situation is completely different when considering small wavelengths. Indeed, the first macroscopic feature that can be observed is that no band-gaps can be forecasted by a Mindlin-Eringen model due to the fact that no horizontal asymptotes exist for acoustic waves.

A second gradient model as limit case of the classical micromorphic model 
--------------------------------------------------------------------------

In this subsection we analyze wave propagation in second gradient media which can be obtained as a suitable limit case of the classical micromorphic continua. More particularly, if we let simultaneously $$\mu_{e}\rightarrow\infty,\qquad\mu_{c}\rightarrow\infty,$$ this implies that $$\mathrm{sym}\:\mathbf{P}\rightarrow\mathrm{sym}\:\nabla\mathbf{u}\qquad\mathrm{skew}\:\mathbf{P}\rightarrow\mathrm{skew}\:\nabla\mathbf{u}\quad\text{{and\:\ hence}\quad\ensuremath{\mathbf{P}\rightarrow\nabla\mathbf{u}}}.$$ This means that the energy (\[KinPot-1-1\]) reduces to

$$\begin{gathered}
W_{2G}\left(\nabla\mathbf{u},\nabla\nabla\mathbf{u}\right)=\mu_{h}\left\Vert \,\mathrm{sym}\:\nabla\mathbf{u}\,\right\Vert ^{2}+\frac{\lambda_{h}}{2}\left(\mathrm{tr}\,\nabla\mathbf{u}\right)^{2}+\frac{\alpha_{g}}{2}\left\Vert \nabla\nabla\mathbf{u}\right\Vert ^{2}\label{Second Gradient}\end{gathered}$$

which is indeed a second gradient energy for linear-elastic, isotropic media (see e.g. [@2Grad1; @2Grad2]). Indeed, governing equations for second (and higher) gradient continua can be also obtained by using a simplified kinematics with respect to the one introduced in this paper and by adopting variational principles (see e.g. [@SecGrad1; @2Grad2; @SecGrad3]). On the other hand, the fact of obtaining a second gradient model as the limit case of a micromorphic one can have many advantages either with respect to numerical efficiency and to physical interpretation of the boundary conditions (see e.g. [@AngelaMicromorphic; @Microm1; @Microm2]).

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
   ![\[2Grad\]Dispersion relations $\omega=\omega(k)$ for the second gradient model obtained as limit case of the classic model by letting $\mu_{e}\rightarrow\infty$ and $\mu_{c}\rightarrow\infty$: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a second gradient continuum. LA: longitudinal acoustic, TA: transverse acoustic.](Uncoupled2Grad "fig:")   ![\[2Grad\]Dispersion relations $\omega=\omega(k)$ for the second gradient model obtained as limit case of the classic model by letting $\mu_{e}\rightarrow\infty$ and $\mu_{c}\rightarrow\infty$: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a second gradient continuum. LA: longitudinal acoustic, TA: transverse acoustic.](Long2Grad "fig:")   ![\[2Grad\]Dispersion relations $\omega=\omega(k)$ for the second gradient model obtained as limit case of the classic model by letting $\mu_{e}\rightarrow\infty$ and $\mu_{c}\rightarrow\infty$: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a second gradient continuum. LA: longitudinal acoustic, TA: transverse acoustic.](Transv2Grad "fig:")     
                                                                                                                                                                                           (a)                                                                                                                                                                                                                                                                                                                                                                                  (b)                                                                                                                                                                                                                                                                                                                                                                                (c)                                                                                                                                                                                            
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It is immediate to notice from Fig.\[2Grad\] that no coupled waves can exist in this limit case, since the only independent kinematical variable turns to be the displacement $\mathbf{u}$. As for, the longitudinal and transverse waves, they can only be of acoustic type ($LA$ and $TA$). This is in agreement with what is known from the literature (see e.g. [@Waves2Grad1; @Waves2Grad2]): second gradient media are such that only acoustic dispersive waves can propagate inside the medium. In [@Waves2Grad1], Eq. (13), dispersion relations for planar waves in isotropic second gradient media are presented associated to a strain energy density of the type (\[Second Gradient\]) but with a kinetic energy which is simpler than the one used in this paper and which can be obtained from (\[Kinetic\]) by setting $\eta=0$. It can be checked by means of the simple application of a variational principle that the equations of motion associated to the strain energy density (\[Second Gradient\]) and to the kinetic energy $T=\frac{1}{2}\,\rho\left\Vert \,\mathbf{u}_{t}\,\right\Vert ^{2}+\frac{1}{2}\,\eta\left\Vert \,\left(\nabla\mathbf{u}\right)_{t}\,\right\Vert ^{2},$ can be written as $$\rho\,\ddot{u}_{i}-\eta\,\ddot{u}_{i,jj}=\mu_{h}\left(u_{i,jj}+u_{j,ij}\right)+\lambda_{h}u_{j,ji}-\alpha_{g}u_{i,jkkj}.\label{Motion}$$ It can also be cheked that the dispersion relations for longitudinal and transverse planar waves (obtained by studying the wave-form solution of Eq. (\[Motion\]) when setting $i=1$ and $i=2,3$ respectively) are respectively given by $$\omega=\sqrt{\frac{\alpha_{g}\, k^{4}+(2\,\mu_{h}+\lambda_{h})\, k^{2}}{\rho+\eta\, k^{2}}},\qquad\qquad\omega=\sqrt{\frac{\alpha_{g}\, k^{4}+\mu_{h}\, k^{2}}{\rho+\eta\, k^{2}}},$$ which indeed correspond to the two acoustic waves depicted in Fig.\[2Grad\]. It is immediate to check that for vanishing wavenumber $k$ the frequency is always vanishing, which means that only acoustic waves can propagate in such kind of media, optic waves are hence forbidden *a priori* in this kind of models. Moreover, it is possible to remark that the frequency goes to infinity when the wavenumber tends to infinity (no possibility of horizontal asymptotes). Finally, we notice that standing waves can also be present in second gradient media since purely imaginary wavenumbers may exist which give rise to real frequencies. Indeed, when replacing an imaginary wavenumber in the wave-form $e^{i(kX-\omega t)}$ it is immediate to see that the solution is an exponential decaying in space. We can summarize by saying that, when considering second gradient media, one has that, for any value of the frequency, propagative acoustic waves and standing waves exist. It can also be remarked that the inertial term $\frac{1}{2}\,\eta\left\Vert \,\left(\nabla\mathbf{u}\right)_{t}\,\right\Vert ^{2},$ plays a role on the possibility of changing the concavity of the dispersion relations. Indeed, if the microscopic density $\eta$ tends to zero, then it is clear that the concavity of the dispersion curves cannot change. On the other hand, this change of concavity is possible when $\eta\neq0$.

We explicitly remark that, in the considered second gradient medium no band-gaps can be forecasted. Nevertheless, it is known that if surfaces of discontinuity of the material properties are considered, then reflected and transmitted energy can be strongly influenced by the value of the second gradient parameter depending on the considered jump conditions imposed at the interface itself (see e.g. [@Waves2Grad1; @Waves2Grad2]).

Conclusions
===========

In this paper we used the relaxed micromorphic model proposed in [@NeffRelaxed; @Ghiba] to study wave propagation in unbounded continua with microstructure. The quoted relaxed model only counts 6 elastic parameters against the 18 parameters appearing in Mindlin-Eringen micromorphic theory (cf. [@Mindlin; @EringenBook]). Despite the reduced number of parameters, we claim that the proposed relaxed model is fully able to account for the description of the mechanical behavior of micromorphic media. More precisely, we have shown that only the relaxed micromorphic continuum model with non-vanishing Cosserat couple modulus $\mu_{c}$ is able to predict frequency band-gaps corresponding to which no wave propagation can occur. The main findings of the present study can be summarized as follows:

-   The relaxed micromorphic model (6 parameters and only the term $\left\Vert \,\mathrm{Curl}\:\mathbf{P}\,\right\Vert ^{2}$ appearing in the strain energy) is fully able to describe the main features of the mechanical behaviour of micromorphic continua, first of all for what concerns the possibility of describing the presence of band gaps.

-   A reduced relaxed model (5 parameters) can be obtained from the previous one by setting the Cosserat couple modulus $\mu_{c}\rightarrow0$. This simplified model produces a *symmetric Cauchy force stress tensor* and is still able to describe the mechanical behavior of a big class of microstructured continua. Nevertheless, this model excludes *a priori* the presence of band-gaps. This means that it is not suitable to fully describe the behavior of sophisticated microstructured engineering materials such as phononic crystals and lattice structures. However, practically all known engineering materials do not show band gaps. For these materials the reduced model is our alternative of choice in the family of micromorphic models.

-   The Cosserat model is obtained as a degenerate limit case of the proposed relaxed model (the full relaxed model with 6 parameters) when letting $\mu_{h}\rightarrow\infty$ . The Cosserat model is not able to describe band-gaps but it introduces a *non-symmetric Cauchy stress tensor*.

-   The classical continuum model (6 parameters but the full $\left\Vert \,\nabla\mathbf{P}\,\right\Vert ^{2}$ appearing in the strain energy) is qualitatively equivalent to the full 18-parameters Mindlin-Eringen micromorphic continuum model. This is true since the classical model is controlling all the kinematical fields of the full model i.e. it is uniformly pointwise definite. The classical continuum model is not able to forecast band-gaps.

-   Second gradient theories can be obtained as a limit case of the classical micromorphic continuum model by letting $\mu_{e}\rightarrow\infty$, $\mu_{c}\rightarrow\infty$. These theories are not able to account for the presence of band-gaps.

We can conclude that only the 6-parameters relaxed model proposed in [@NeffRelaxed; @Ghiba] and used in this paper to study wave propagation in microstructured media is able to describe the presence of band-gaps. These band-gaps are seen to be “switched on” by a unique constitutive parameter, namely the Cosserat couple modulus $\mu_{c}$. The proposed relaxed micromorphic model is hence suitable to be used for the conception and optimization of metamaterials to be used for vibration control.

Acknowledgements {#acknowledgements .unnumbered}
================

I.D. Ghiba acknowledges support from the Romanian National Authority for Scien- tific Research (CNCS-UEFISCDI), Project No. PN-II-ID-PCE-2011-3-0521. I.D. A. Madeo thanks INSA-Lyon for the financial support assigned to the project BQR 2013-0054 Matériaux Méso et Micro-Héterogènes: Optimisation par Modèles de Second Gradient et Applications en Ingenierie.

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[^1]: Corresponding author: angela.madeo@insa-lyon.fr

[^2]: Here and in the sequel Einstein convention of sum of repeated indices is used if not differently specified.

[^3]: Here and in the sequel a subscript $i$ after a comma indicates partial derivative with respect to the space variable $X_{i}$.

[^4]: We remark that by considering a scalar microscopic density $\eta$, we are limiting ourselves to cases in which the microscopic inclusions only have one characteristic length (e.g. cubes or spheres). On the other hand, expression (\[Kinetic\]) for the kinetic energy can be easily generalized by replacing the second term with $1/2\,\eta_{ij}(P_{ki})_{t}(P_{kj})_{t}$, where the tensor $\boldsymbol{\eta}$ can be written as $\eta_{ij}=d_{ij}^{2}\rho'$. In this way one can account for as many microscopic characteristic lengths $d_{ij}$ as needed.

[^5]: When using index notation for the components of the introduced tensor fields, we will denote partial derivative with respect to time with a superposed dot instead of a $t$ subscript.

[^6]: Due to the chosen values of the parameters, which are supposed to satisfy (\[DefPos-1\]), all the introduced characteristic velocities and frequencies are real. Indeed it can be checked that the condition $\left(2\lambda_{e}+\mu_{e}\right)>0$ together with the condition $\mu_{e}>0$, imply both the conditions $\left(3\lambda_{e}+2\mu_{e}\right)>0$ and $\left(\lambda_{e}+2\mu_{e}\right)>0$.
