# Unfolding engineering metamaterials design: relaxed micromorphic modeling of
large-scale acoustic meta-structures.

F. Demore  Laboratoire de Tribologie et de Dynamique des Système, Ecole
Centrale de Lyon, Ecully 69134, France  G. Rizzi  Institute for Structural
Mechanics and Dynamics, Technical University Dortmund, August-Schmidt-Str. 8,
44227 Dortmund, Germany  M. Collet  Laboratoire de Tribologie et de Dynamique
des Système, Ecole Centrale de Lyon, Ecully 69134, France  P. Neff  Chair for
Nonlinear Analysis and Modeling, Faculty of Mathematics, University of
Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany  A. Madeo  Institute
for Structural Mechanics and Dynamics, Technical University Dortmund, August-
Schmidt-Str. 8, 44227 Dortmund, Germany

###### Abstract

In this paper, we present a unit cell showing a band-gap in the lower acoustic
domain. The corresponding metamaterial is made up of a periodic arrangement of
this unit cell. We rigorously show that the relaxed micromorphic model can be
used for metamaterials’ design at large scales as soon as sufficiently large
specimens are considered. We manufacture the metamaterial via metal etching
procedures applied to a titanium plate so as to show that its production for
realistic applications is viable. Experimental tests are also carried out
confirming that the metamaterials’ response is in good agreement with the
theoretical design. In order to show that our micromorphic model opens
unprecedented possibilities in metastructural design, we conceive a finite-
size structure that is able to focus elastic energy in a confined region, thus
enabling its possible subsequent re-use. Indeed, thanks to the introduction of
a well-posed set of micromorphic boundary conditions, we can combine different
metamaterials and classical Cauchy materials in such a way that the elastic
energy produced by a source of vibrations is focused in specific collection
points. The design of this structure would have not been otherwise possible
(via e.g., direct simulations), due to the large dimensions of the
metastructure, couting hundreds of unit cells.

Keywords: Mechanical metamaterials, micromorphic models, Band gaps, meta-
structures, energy focusing.

##  1 Introduction

Metamaterials are architectured materials whose mechanical properties go
beyond those of classical materials thanks to their heterogeneous
microstructure. This allows them to show exceptional static/dynamic features
such as negative Poisson’s ratio [17], twist in response to being pushed or
pulled [11, 31], band-gaps [19, 41, 29, 6], cloaking [5, 24], focusing [12,
2], channeling [13, 10], negative refraction [46, 14, 43, 18, 3], etc.

In the last two centuries, the advancement of knowledge on finite-size
classical materials modeling has enabled the design of engineering structures
(buildings, bridges, airplanes, cars, etc.) resisting to static and dynamic
loads. Today, while the modeling of infinite-size metamaterials is achievable
via reliable homogenization techniques [7, 44, 8, 45, 4, 38], we must
acknowledge that these techniques are usually unsuitable for finite-size
metamaterials’ modeling since these homogenization methods cannot provide the
right tools to “cut finite-size metamaterials’ LEGO bricks out” of an infinite
block. This gap prevents the exploration of the response of
metamaterials/classical-materials structures and their optimization towards
efficient wave control and energy recovery. In this paper, we show that this
gap can be filled by using the relaxed micromorphic model [27, 26, 32, 33] as
a gold standard to conceive realistic finite-size metamaterials/classical-
materials structures that can control waves and eventually recover energy.

At present, the response of finite-size metamaterials’ structures is mostly
explored via direct Finite Element (FEM) simulations that implement all the
details of the involved microstructures (e.g., [16]). Despite the precise
propagation patterns that these direct numerical simulations can provide, they
suffer from unsustainable computational costs when considering large
metamaterials’ specimens or high frequencies. Therefore, it is very difficult
today to explore large meta-structures combining metamaterials and classical-
materials bricks of different type, size and shape. The awareness of this
limitation triggered all the recent advances on dynamical homogenization
methods [7, 44, 8, 45, 4, 38]. Such methods share the idea that a periodic
infinite-size metamaterial can be replaced by a homogenized continuum,
mimicking its response without accounting for all the microstructures’
details. This leads to an important simplification of metamaterials’
description at the macroscopic scale. Unfortunately, often homogenization
methods cannot describe the response of finite-size metamaterials due to the
difficulty of establishing well-posed boundary conditions at the macro-level.
The unsuitability of classical homogenization methods for finite-size
metamaterials’ modeling in dynamic regime has been very recently acknowledged
by the cutting-edge research groups in dynamical homogenization [40, 39].
Being aware of the homogenization’s limits concerning finite-size
metamaterials’ modeling in dynamics, Sridhar et al. [39] recently proposed an
alternative ad hoc upscaling procedure, only valid for locally resonant
metamaterials, leading to a homogenized equation which the authors recognize
to be of the micromorphic type. In a similar spirit, [40] obtained a
homogenized continuum with extended kinematics, classifying it as
micromorphic, and proposed its use to study a simple 1D boundary value problem
for a periodic metamaterial. It is also common today to find models,
alternative to homogenization, that entail frequency-dependent parameters to
describe dynamic metamaterials’ response at different frequencies (e.g., [20,
7]). Even if these models can give useful information about metamaterials’
response under particular loading conditions, they cannot provide a
comprehensive characterization of metamaterials.

The results presented in this paper show that the relaxed micromorphic model’s
structure, coupled with the introduction of well-posed boundary conditions,
allowed us to unveil both the static and dynamic response of metamaterials
bricks of finite size. Playing LEGO with such bricks enables the design of a
highly performing structure, combining metamaterials and classical-materials
in such a way to focus energy in specific collection points for eventual
subsequent re-use.

##  2 A titanium-based metamaterial for acoustic control: experimental set-up

A consistent branch of research on metamaterials focuses on how to engineer
the unit cell geometry and material properties to optimize their response with
respect to elastic wave manipulation [30, 36, 37, 35, 22, 24, 23, 42, 43, 25,
8, 28, 21, 15]. In particular, optimizing the size, the mass, and the
stiffness distribution within the unit cell can synergize to define the
position of the band-gap. Often, in order to obtain band-gaps in the acoustic
regime, it is necessary to have unit cells whose size are in the range of tens
of centimeters. The unit cell that we present in this paper (Figure 1) has an
optimized geometry that allows us to obtain a band-gap in the low acoustic
frequency range with a cell size of only 2 centimeters.

$a$ | $e_{g}$ | $e_{p}$
---|---|---
[mm] | [mm] | [mm]
20 | 0.35 | 0.25
$\rho_{\text{Ti}}$ | $\lambda_{\text{Ti}}$ | $\mu_{\text{Ti}}$
[kg/m3] | [GPa] | [GPa]
4400 | 88.8 | 41.8
|  |

Figure 1:  (left) unit cell whose periodic repetition in space gives rise to
the metamaterial studied in this paper. (right) geometry and material
parameters characterising the unit cell. The parameters
$\rho_{\tiny\mbox{Ti}}$, $\lambda_{\tiny\mbox{Ti}}$, and
$\mu_{\tiny\mbox{Ti}}$ are the density and the Lamé constants of the titanium
alloy used, respectively.

The four squares in which the unit cell can be divided in act as local
resonators (Figure 1 left) localizing the energy at the microscopic level,
thus creating the band-gap effect. In order to lower the band-gap to the
acoustic frequency range, it is required simultaneously to increase the mass
of the resonators (i.e. maximize the size of such internal squares) and
decrease their stiffness (i.e. minimize the thickness $e_{\rm p}$ of the
stripe on the outline of the cell). To minimize the unavoidable presence of
defects related to the extremely thin holes that must be drilled in the
metallic plate, the Electrical Discharge Machining wire erosion was used for
metal etching. Titanium was chosen as base material to maximize the overall
strength while minimizing losses due to damping. The values of $a$, $e_{\rm
p}$ and $e_{g}$ are given in Figure 1 (right) and the out-of-plane cell
thickness $e$=1[mm] were thus determined by taking into account:

  * •

desired characteristics of the band-gap (acoustic regime);

  * •

manufacturing constraints of the chosen process (limitations for the possible
values of $e_{\rm p}$ and $e_{g}$);

  * •

static resistance of the structure.

Using this unit cell, a 9$\times$11 cells meta-structure has been designed to
experimentally explore the band-gap attenuation: the used set-up is presented
in Figure 2.

Figure 2:  (left) Experimental set-up: glued in the center of the
metamaterial’s plate there is the top piezoelectric patch (another one being
placed on the other side of the plate) that has been used as an actuator for
the external excitation. The tapes placed on the top-right quarter of the
plate can reflect a laser’s beam for speed measurements. (right) Detail of the
upper piezoelectric patch with its electric supply.

Two piezoelectric patches (MEGGIT PZ 21, $\varnothing$16[mm] 2[mm]-thick) are
used as actuators to generate in-plane extension pulse waves in the plate (see
Figure 2 right). Excitation signals are generated by a function generator and
then amplified to power the piezoelectric patches. Sine sweeps are chosen to
impose the external load and the signal’s frequency is swept from 0 to 2500
Hz. Measured speeds are acquired by a 3-D laser (Polytec CLV-3D). An interface
under MATLAB®has also been designed, allowing to easily choose the main
parameters for each test, namely the required frequency range, the resolution
and coordinates of the considered measurement points.

The power supply of the piezoelectric patches is designed to avoid flexural
vibration modes in the plate at the considered frequencies, so that the
applied load is a pure in-plane expansion as shown in Figure 3 (a). In Figures
3 (b) and 3 (c) other possible loads are depicted but not used in the
experiment.

Figure 3:  Scheme of possible power supplies of the piezoelectric patches. The
dashed lines outline the non-powered piezoelectric patches and undeformed
center of the plate while the solid lines depict the deformed powered
piezoelectric patches deforming the microstructured plate. In particular, blue
lines represent the side of the piezoelectric patches connected to the ground
while red lines represent the powered side of the piezoelectric patches. The
same electric potential is applied on both red areas.

##  3 Modeling and simulation

In this section, we give an outlook on the possible modeling tools allowing to
catch the dynamical response of the metamaterial presented in Section 2. In
particular, we simulate the experimental set-up presented in Figure 2 both by
using a detailed finite element model and a novel micromorphic homogenized
approach. The model’s features will be presented in detail except the loading
conditions that will be introduced in section 4 (pulse load) and 5
(piezoelectric load).

###  3.1 Detailed direct element simulations

We present there the detailed finite element simulations of the structure
presented in Section 2, where the material composing the unit cell is modelled
as a classical isotropic Cauchy continuum. The structure and load symmetry
allow us to consider one eighth of the system as presented in Figure 4 (the
plate’s thickness implemented in the simulation is 0.5 mm instead of 1 mm).

$\longrightarrow$

Figure 4:  (left) Full microstructured plate with its top piezoelectric patch.
(right) Reduced microstructured symmetrical plate: $\Omega_{\rm p}$ is the
region occupied by the piezoelectric patch, $\Omega_{\rm c}$ is the region
made up of titanium, and $\partial\Omega_{s}$ is the union of the boundaries
on the plane of symmetry. The thickness of the reduced plate is 0.5 mm instead
of 1 mm.

The action functional of the considered reduced system is111Here and in the
sequel, we consider the plane-strain hypothesis which allows to sort out the
thickness $e$ out of the intergration with respect to $x_{3}$.

$\mathcal{A}\left(u,V\right)=\int^{T}_{0}\left[\iiint_{\Omega_{\rm p}}(K_{\rm
p}-W_{\rm
p}-Q)\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}x_{3}+\frac{e}{2}\iint_{\Omega_{\rm
c}}(K_{\rm t}-W_{\rm t})\mathrm{d}x_{1}\mathrm{d}x_{2}\right]\mathrm{d}t$ (1)

where $K_{\rm p}$, $W_{\rm p}$, and $Q$ being respectively the kinetic, strain
and electric potential energy density of the piezoelectric patch while $K_{\rm
t}$ and $W_{\rm t}$ are the kinetic and potential energy density of the
isotropic Cauchy material (titanium) constituting the domain $\Omega_{\rm c}$,
respectively. Finally $e$ is the plate’s thickness and $[0,T]$ is the time
interval during which the system’s response is observed. Given the symmetry of
the system, only one piezoelectric patch can be considered.

The strain and kinetic energy density, and the electric potential expression
are respectively given by

$\displaystyle W_{\rm p}=\frac{1}{2}\langle\mathbb{C}_{\rm
p}\,\text{sym}\,\nabla u+\xi^{T}\,E,\text{sym}\,\nabla u\rangle\,,\qquad
K_{\rm p}=\frac{1}{2}\rho_{\rm p}\,\langle\dot{u},\dot{u}\rangle\,,\qquad
Q=\frac{1}{2}\langle\epsilon_{0}\varepsilon E+\xi\,\text{sym}\,\nabla
u,E\rangle\,,$ (2) $\displaystyle W_{\rm t}=\frac{1}{2}\langle\mathbb{C}_{\rm
t}\,\text{sym}\,\nabla u,\text{sym}\,\nabla u\rangle\,,\qquad K_{\rm
t}=\frac{1}{2}\rho_{\rm t}\,\langle\dot{u},\dot{u}\rangle\,,\,$

where $\mathbb{C}_{\rm p}$ and $\mathbb{C}_{\rm t}$ are 4th order elasticity
tensors, $\text{sym}\,\nabla u$ is the symmetric part of the gradient of the
displacement field, $\xi$ is the 3rd order piezoelectric coupling tensor (in
C/m2), $E$ is the electric vector field (in V/m), $\varepsilon$ is the
relative permittivity tensor, and ${\epsilon_{0}=8.86\times 10^{-12}}$[F/m] is
the vacuum permittivity. Using the approximation of electrostatic and the
Maxwell-Faraday equation, $E$ derives from the potential $V$, i.e.

$E=-\nabla V$ (3)

Requiring the first variation of the total energy with respect $u$ and $V$ to
be zero gives the following equilibrium equations

$\displaystyle\rho_{i}\,\ddot{u}=\text{Div}\,\sigma\quad\text{(Cauchy
equilibrium)}\,,\qquad\qquad\text{Div}\,D=0\quad\text{(Maxwell-Gauss law)}$
(4)

where the 2nd order Cauchy stress tensor $\sigma$ and the electric induction
vector $D\in\mathbb{R}^{6}$ are given by

$\displaystyle\sigma\coloneqq\mathbb{C}_{i}\,\text{sym}\,\nabla
u-\xi^{T}\,E\qquad\qquad\qquad D\coloneqq\varepsilon_{0}\,\varepsilon
E+\xi\,\text{sym}\,\nabla u\,.$ (5)

with $i=\\{{\rm p,t}\\}$. In the domain $\Omega_{\rm c}$, only equation (4)1
is required and the definition (5)1 becomes $\sigma\coloneqq\mathbb{C}_{\rm
t}\,\text{sym}\,\nabla u$. Given the cylindrical symmetry of the piezoelectric
patches, the Voigt representation of elastic tensor $\mathbb{C}_{\rm p}$, the
piezoelectric coupling tensor $\xi$, and the relative permittivity tensor
$\varepsilon$ is

$\displaystyle\mathbb{C}_{\rm p}=\begin{pmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\\
C_{12}&C_{11}&C_{13}&0&0&0\\\ C_{13}&C_{13}&C_{33}&0&0&0\\\
0&0&0&C_{44}&0&0\\\ 0&0&0&0&C_{44}&0\\\
0&0&0&0&0&C_{66}\end{pmatrix},\,\xi=\begin{pmatrix}0&0&0&0&\xi_{15}&0\\\
0&0&0&\xi_{15}&0&0\\\ \xi_{31}&\xi_{31}&\xi_{33}&0&0&0\\\
\end{pmatrix},\,\varepsilon=\begin{pmatrix}\varepsilon_{11}&0&0\\\
0&\varepsilon_{11}&0\\\ 0&0&\varepsilon_{33}\\\ \end{pmatrix}.$ (6)

The matrix $\mathbb{C}_{\rm t}$ has the structure of the classical elasticity
isotropic tensor and its coefficients are reported in Table 1 in terms of Lamé
constants together with the density $\rho_{\rm t}$. The values of the
parameters in $\mathbb{C}_{\rm p}$, $\xi$, and $\varepsilon$, together with
the density $\rho_{\rm p}$ are reported in Table 1.

| $\rho_{\rm p}$ | $C_{11}$ | $C_{12}$ | $C_{13}$ | $C_{33}$ | $C_{44}$ | $C_{66}$ |
---|---|---|---|---|---|---|---|---
| [kg /m3] | [GPa] | [GPa] | [GPa] | [GPa] | [GPa] | [GPa] |
| 7780 | 1140 | 757 | 724 | 1110 | 263 | 403 |
| $\varepsilon_{11}$ | $\varepsilon_{33}$ | $\xi_{31}$ | $\xi_{33}$ | $\xi_{15}$ |  |  |
| [-] | [-] | [C/m2] | [C/m2] | [C/m2] |  |  |
| $3.24\cdot 10^{3}$ | $3.98\cdot 10^{3}$ | -2.92 | 23.4 | 16.2 |  |  |
|  |  |  |  |  |  |  |
Table 1: Mechanical parameters of the piezoelectric patches, coupling and
electrical parameters. Figure 5: Section of the symmetrized microstructured
plate with the piezoelectric patch (one eight on the whole system) and the
boundaries’ denomination $\partial\Omega_{\text{top}}$ and
$\partial\Omega_{\text{bottom}}$ where the electric potential is imposed.

To simulate the experimental setup of Section 2 via the reduced problem of
Figure 4, proper boundary conditions need to be imposed on the boundaries
defined in Figure 5. The first interface conditions represent the imposition
of the electric potential while the second ones are associated to the symmetry
conditions of the reduced problem

$\displaystyle\begin{cases}V=0&\text{on }\partial\Omega_{\text{bottom}}\\\
V=V_{0}&\text{on
}\partial\Omega_{\text{top}}\end{cases},\qquad\begin{cases}\langle
u,n\rangle=0\\\ \langle D,n\rangle=0\end{cases}\text{on }\partial\Omega_{s}$
(7)

where $V_{0}=100$[V]. No condition have to be imposed on the free boundaries
$\partial\Omega_{f}$. The microstructured plate is studied under the plain
strain hypothesis while the piezoelectric region is kept as a full 3D medium:

$u^{\rm c}=\left(u^{\rm c}_{1},u^{\rm c}_{2},0\right)^{\rm
T}\quad\text{in}\quad\Omega_{\rm c}\qquad\qquad\text{and}\qquad\qquad u^{\rm
p}=\left(u^{\rm p}_{1},u^{\rm p}_{2},u^{\rm p}_{3}\right)^{\rm
T}\quad\text{in}\quad\Omega_{\rm p}.$ (8)

Since a thin plate is considered here, a plane stress hypothesis could have
been used instead, but since the difference between the response of the system
in terms of displacement for the two hypothesis is always smaller than $5\%$,
the plane strain hypothesis is kept for the rest of the study for the sake of
simplicity. This plane strain hypothesis in $\Omega_{\rm c}$ together with the
perfect contact condition interface on $\partial\Omega_{\text{bottom}}$
requires the following interface conditions

$\displaystyle\begin{cases}u^{\rm c}_{1}=u^{\rm p}_{1}\\\ u^{\rm c}_{2}=u^{\rm
p}_{2}\\\ u^{\rm
p}_{3}=0\end{cases}\text{on}\quad\partial\Omega_{\text{bottom}}$ (9)

The differential problem in equations (4), (7) and (9), to which we refer in
the following sections as “the microstructured simulation” or “direct
simulation” has been implemented and numerically solved under the finite
element multiphysics software Comsol Multiphysics®using the “Solid Mechanics”
and “Electrostatics” modules. Special attention has been brought to the domain
meshing as presented in Figure 6. The small dimension of the quadrangular and
triangular elements used to mesh the slender portion of the domain of the unit
cell is necessary for the description of the band-gap through the local
resonance, and this significantly increases the number of degrees of freedom
of the finite element problem that has to be solved. To reduce the
computational burden, we chose a coarser mesh on the remaining domain that
allows us to reduce the total number of degrees of freedom while still
guaranteeing the reliability of the results.

Figure 6:  Mesh for a quarter unit cell of the microstructured model: the
portions of domain with coarse mesh allow a reduction of the total number of
degrees of freedom, while a finer mesh is needed in the slender portions of
the domain in order to be able to properly describe the behaviour of the
microstructure in the band-gap frequency range.

Since there is a considerable number of modes in the frequency range that have
been studied (0 to 2500 Hz), numerical damping has been implemented for
numerical stability reasons, modifying the strain energy density of the system
by

$W^{\text{non-conservative}}_{\rm t}=(1+i\times
0.002)W^{\text{conservative}}_{\rm t}$ (10)

where i is the imaginary unit.

###  3.2 Relaxed micromorphic simulations

We introduce here the effective model that we will use to describe the
metamaterial’s response at the macroscopic scale and which is known as relaxed
micromorphic medium. The strain and kinetic energy density expressions for the
relaxed micromorphic model are [32, 33] 222The presence of curvature terms
associated to higher space derivatives of $P$ is not accounted for in the
present paper since their effect is not predominant in the dynamic regime.

$\displaystyle W\left(\nabla u,P\right)$
$\displaystyle=\dfrac{1}{2}\langle\mathbb{C}_{\rm e}\,\mbox{sym}\left(\nabla
u-\,P\right),\mbox{sym}\left(\nabla
u-\,P\right)\rangle+\dfrac{1}{2}\langle\mathbb{C}_{\rm
c}\,\mbox{skew}\left(\nabla u-\,P\right),\mbox{skew}\left(\nabla
u-\,P\right)\rangle$ $\displaystyle\quad+\dfrac{1}{2}\langle\mathbb{C}_{\rm
micro}\,\mbox{sym}\,P,\mbox{sym}\,P\rangle\,,$ (11) $\displaystyle
K\left(\dot{u},\nabla\dot{u},\dot{P}\right)$
$\displaystyle=\dfrac{1}{2}\rho\,\langle\dot{u},\dot{u}\rangle+\dfrac{1}{2}\langle\mathbb{J}_{\rm
m}\,\mbox{sym}\,\dot{P},\mbox{sym}\,\dot{P}\rangle+\dfrac{1}{2}\langle\mathbb{J}_{\rm
c}\,\mbox{skew}\,\dot{P},\mbox{skew}\,\dot{P}\rangle+\dfrac{1}{2}\langle\mathbb{T}_{\rm
e}\,\mbox{sym}\nabla\dot{u},\mbox{sym}\nabla\dot{u}\rangle$
$\displaystyle\quad+\dfrac{1}{2}\langle\mathbb{T}_{\rm
c}\,\mbox{skew}\nabla\dot{u},\mbox{skew}\nabla\dot{u}\rangle\,,$

where $u\in\mathbb{R}^{3}$ is the macroscopic displacement field,
$P\in\mathbb{R}^{3\times 3}$ is the non-symmetric micro-distortion tensor,
$\mathbb{C}_{\rm e}$, $\mathbb{C}_{\rm micro}$ and $\mathbb{C}_{\rm c}$ are 4
order elastic tensors, $\rho$ is the macroscopic apparent density, and
$\mathbb{J}_{\rm m}$, $\mathbb{J}_{\rm c}$, $\mathbb{T}_{\rm e}$ and
$\mathbb{T}_{\rm c}$ are 4 order micro-inertia tensors. The action functional
for the micromorphic medium is defined as :

$\mathcal{A}\left[u,P\right]=\int^{T}_{0}\iint_{\Omega}(K-W)\mathrm{d}\Omega\mathrm{d}t$
(12)

Where $\Omega$ is the domain occupied by the relaxed micromorphic medium.
Requiring the first variation of the action functional with respect to $u$ and
$P$ to be zero gives the following two sets of equilibrium equations, as well
as the associated boundary conditions [32, 33, 9]

$\displaystyle\begin{cases}\rho\,\ddot{u}-\text{Div}\,\widehat{\sigma}=\text{Div}\,\widetilde{\sigma}\,,\\\
\mathbb{J}_{\rm m}\,\text{sym }\ddot{P}+\mathbb{J}_{\rm c}\,\text{skew
}\ddot{P}=\widetilde{\sigma}-s\,,\end{cases}\qquad\text{in
$\Omega$}\,,\qquad\qquad
t_{m}\coloneqq\left(\widetilde{\sigma}+\widehat{\sigma}\right)n=t_{\rm m}^{\rm
ext}\qquad\text{on $\partial\Omega$}\,,$ (13)

where $n$ is the normal to the boundary $\partial\Omega$, $t_{m}$ is the
generalized traction vector, $t_{\rm m}^{\rm ext}$ represent the external
traction load, and

$\displaystyle\widehat{\sigma}=\mathbb{T}_{\rm
e}\,\text{sym}\,\nabla\ddot{u}+\mathbb{T}_{\rm
c}\,\text{skew}\,\nabla\ddot{u}\,,\qquad\widetilde{\sigma}=\mathbb{C}_{\rm
e}\,\text{sym}\,(\nabla u-P)+\mathbb{C}_{\rm c}\,\text{skew}(\nabla
u-P)\,,\qquad s=\mathbb{C}_{\rm m}\,\text{sym}\,P\,.$ (14)

All the previous results hold for a generic class of material symmetry and in
the following the elastic and the micro-inertia tensors will be presented in
the Voigt notation for the tetragonal class of symmetry

$\begin{array}[]{rlrl}&\mathbb{C}_{\rm e}=\begin{pmatrix}\lambda_{\rm
e}+2\mu_{\rm e}&\lambda_{\rm e}&\dots&\bullet\\\ \lambda_{\rm e}&\lambda_{\rm
e}+2\mu_{\rm e}&\dots&\bullet\\\ \vdots&\vdots&\ddots&\\\
\bullet&\bullet&&\mu_{\rm e}^{*}\\\
\end{pmatrix}\,,\quad\qquad\qquad\mathbb{C}_{\rm
micro}=\begin{pmatrix}\lambda_{\rm m}+2\mu_{\rm m}&\lambda_{\rm
m}&\dots&\bullet\\\ \lambda_{\rm m}&\lambda_{\rm m}+2\mu_{\rm
m}&\dots&\bullet\\\ \vdots&\vdots&\ddots&\\\ \bullet&\bullet&&\mu_{\rm
m}^{*}\\\ \end{pmatrix}\,,\\\\[28.45274pt] &\mathbb{J}_{\rm
m}=\begin{pmatrix}\eta_{3}+2\eta_{1}&\eta_{3}&\dots&\bullet\\\
\eta_{3}&\eta_{3}+2\eta_{1}&\dots&\bullet\\\ \vdots&\vdots&\ddots&\bullet\\\
\bullet&\bullet&\bullet&\eta^{*}_{1}\\\
\end{pmatrix}\,,\qquad\qquad\quad\;\;\mathbb{T}_{\rm
e}=\begin{pmatrix}\overline{\eta}_{3}+2\overline{\eta}_{1}&\overline{\eta}_{3}&\dots&\bullet\\\
\overline{\eta}_{3}&\overline{\eta}_{3}+2\overline{\eta}_{1}&\dots&\bullet\\\
\vdots&\vdots&\ddots&\\\
\bullet&\bullet&&\overline{\eta}^{*}_{1}\end{pmatrix}\,,\\\\[28.45274pt]
&\;\mathbb{C}_{\rm c}=\begin{pmatrix}\bullet&&\bullet\\\ &\ddots&\vdots\\\
\bullet&\dots&4\mu_{\rm c}\end{pmatrix}\,,\qquad\qquad\mathbb{J}_{\rm
c}=\begin{pmatrix}\bullet&&\bullet\\\ &\ddots&\vdots\\\
\bullet&\dots&4\eta_{2}\end{pmatrix}\,,\qquad\qquad\mathbb{T}_{\rm
c}=\begin{pmatrix}\bullet&&\bullet\\\ &\ddots&\vdots\\\
\bullet&\dots&4\overline{\eta}_{2}\end{pmatrix}\,,\end{array}$ (15)

Given the plane strain hypothesis, the values of the ”dotted” coefficients are
not necessary to our study. The assumption regarding the class of symmetry is
driven by considerations regarding the symmetry of the unit cell in Figure 1.
In previous papers [32, 33, 1], we showed that a specific calibration
procedure can be applied to compute the values of the relaxed micromorphic
parameters for the metamaterial issued from the unit cell in Figure 1. This
procedure is based on the comparison of the relaxed micromorphic dispersion
curves with those obtained via a classic Bloch-Floquet analysis done on the
unit cell. The comparison of the dispersion curves of the relaxed micromorphic
medium with those obtained via Bloch-Floquet analysis is shown in Figure 7.

Figure 7: (left) Dispersion curves of the microstructured and the relaxed
micromorphic systems along $\Gamma$X (propagation at 0°). (right) Dispersion
curves of the microstructured and the relaxed micromorphic systems along
$\Gamma$M (propagation at 45°).

The values of the relaxed micromorphic parameters relative to the metamaterial
in Figure 1 are presented in Table 2 (left), while in Table 2 (right) are
reported also the coefficients of the Cauchy material resulting from the long-
wave limit of the relaxed micromorphic one.

| $\rho$ | $\mu_{\rm e}$ | $\lambda_{\rm e}$ | $\mu_{\rm e}^{\star}$ |
---|---|---|---|---|---
| [kg/m3] | [Pa] | [Pa] | [Pa] |
| 3841 | 2.53$\crossproduct 10^{9}$ | 1.01$\crossproduct 10^{8}$ | 1.26 $\crossproduct 10^{6}$ |
| $\mu_{\rm m}$ | $\lambda_{\rm m}$ | $\mu_{\rm m}^{\star}$ | $\mu{\rm c}$ |
| [Pa] | [Pa] | [Pa] | [Pa] |
| 4.51 $\crossproduct 10^{9}$ | 1.83 $\crossproduct 10^{8}$ | 2.70 $\crossproduct 10^{8}$ | $10^{5}$ |
| $\eta_{1}$ | $\eta_{2}$ | $\eta_{3}$ | $\eta_{1}^{\star}$ |
| [kg/m] | [kg/m] | [kg/m] | [kg/m] |
| 38.99 | 5.99$\crossproduct 10^{-3}$ | 1.58 | 2.31 |
| $\overline{\eta}_{1}$ | $\overline{\eta}_{2}$ | $\overline{\eta}_{3}$ | $\overline{\eta}^{\star}_{1}$ |
| [kg/m] | [kg/m] | [kg/m] | [kg/m] |
| 8$\crossproduct 10^{-4}$ | 0.02 | 0.008 | 0.09 |
|  |  |  |  |

| $\lambda_{\tiny\rm macro}$ | $\mu_{\tiny\rm macro}$ | $\mu^{*}_{\tiny\rm macro}$ |
---|---|---|---|---
| [Pa] | [Pa] | [Pa] |
| $6.51\times 10^{7}$ | $1.62\times 10^{9}$ | $1.25\times 10^{6}$ |
|  |  |  |

Table 2:  (left) Values of the elastic and micro-inertia relaxed micromorphic
parameters calibrated on the metamaterial whose unit cell is reported in Fig.
1, and (right) the corresponding long-wave limit Cauchy material
$\mathbb{C}_{\tiny\rm macro}$.

In Figure 8 we present a reduced problem also for the relaxed micromorphic
model consisting of an equivalent plate mimicking the considered
microstructured plate under piezoelectric excitation.

Figure 8:  (left, top) One eighth of the microstructured plate with its top
piezoelectric patch. (right, top) Equivalent relaxed micromorphic plate with
its top piezoelectric patch: $\Omega_{\rm m}$ is the domain of the relaxed
micromorphic medium, $\partial\tilde{\Omega}_{\rm f}$ is the generalized
traction-free border, $\partial\Omega_{\rm c}$ is the interface between the
Cauchy and relaxed micromorphic domains, and $\partial\Omega_{\rm s}$ is the
union of the boundaries on the plane of symmetry. We refer to this
configuration as “reduced micromorphic problem”. (bottom) Front view of the
reduced relaxed micromorphic plate.

The piezoelectric patch and the central isotropic Cauchy medium description
presented in Section 3.1 remain unchanged, while it is worth to focus on the
plain strain hypothesis for the relaxed micromorphic model, which implies :

$u^{\rm c}=\begin{pmatrix}u^{\rm c}_{1}\\\ u^{\rm c}_{2}\\\ u^{\rm
c}_{3}\end{pmatrix}\quad\text{in}\quad\Omega_{\rm p},\quad u^{\rm
c}=\begin{pmatrix}u^{\rm c}_{1}\\\ u^{\rm c}_{2}\\\
0\end{pmatrix}\quad\text{in}\quad\Omega_{\rm c},\quad u^{\rm
m}=\begin{pmatrix}u^{\rm m}_{1}\\\ u^{\rm m}_{2}\\\
0\end{pmatrix}\,\,\,\text{and}\,\,\,P=\begin{pmatrix}P_{11}&P_{12}&0\\\
P_{21}&P_{22}&0\\\ 0&0&0\end{pmatrix}\quad\text{in}\quad\Omega_{\rm m}.$ (16)

The perfect contact conditions between the Cauchy material and the relaxed
micromorphic material at the interfaces $\partial\Omega_{\rm c}$ and the
traction-free conditions on $\partial\tilde{\Omega}_{\rm f}$ are

$\begin{cases}u^{c}=u^{m}\\\ (\widehat{\sigma}+\widetilde{\sigma})\cdot
n=\sigma\cdot n\end{cases}\quad\text{on}\quad\partial\Omega_{\rm
c},\qquad(\widehat{\sigma}+\widetilde{\sigma})\cdot
n=0\quad\text{on}\quad\partial\tilde{\Omega}_{\rm f}$ (17)

On the boundary of symmetry $\partial\Omega_{\rm s}$ we have to impose the
following boundary conditions333See Appendix A for a derivation of these
conditions.

$\displaystyle\begin{cases}u_{i}n_{i}=0\\\
(\delta_{ki}-n_{k}n_{i})(P_{ij}n_{j})=0\end{cases}$ (18)

where, $n_{i}$ are the components of the unit normal to each surface and
$\delta_{ij}$ is the Kronecker delta operator. The relaxed micromorphic model
not being implemented under Comsol by default, the “Weak form PDE” module has
been used, requiring to write explicitly energy densities and imposing
manually boundary conditions presented above. The associated Lagrangian for
the plate consisting of Cauchy and relaxed micromorphic media, under the
plane-strain hypothesis, is

$\mathcal{A}\left[u,P\right]=\frac{e}{2}\int^{T}_{0}\left[\iint_{\Omega_{\rm
c}}(K_{\rm c}-W_{\rm
c})\mathrm{d}x_{1}\mathrm{d}x_{2}+\iint_{\Omega_{m}}(K_{m}-W_{m})\mathrm{d}x_{1}\mathrm{d}x_{2}\right]\mathrm{d}t$
(19)

Using a unique field $u$ which is equal to $u^{\rm c}$ when restricted to
$\Omega_{\rm c}$ and to $u^{\rm m}$ when restricted to $\Omega_{m}$ allows to
simplify the numerical implementation of the problem, automatically
guaranteeing the continuity of displacement at the interface
$\partial\Omega_{\rm c}$. The mesh used for the relaxed micrmorphic relaxed
problem is given in Figure 9.

Figure 9: Chosen mesh for the reduced relaxed micromophic plate with its
central Cauchy medium inclusion.

Quadratic Lagrange elements are used for the discretization of $u$, which
allows us to recover the continuity of $\nabla u$ and therefore of generalized
tractions. No gradient being applied upon $P$, we discretize $P$ through form
functions an order below (here, linear Lagrange elements), in terms of
regularity, of the one used for $u$. Despite the introduction of an additional
2nd order tensor to describe the response of the plate and the tight mesh
around the excitation domain required to assure the slow convergence of $P$,
the relaxed micromorphic model allows a considerable reduction in the number
of degrees of freedom with respect to the microstructured model, as shown in
Table 3.

| Degrees of freedom | Microstructured plate | Relaxed micromorphic plate |
---|---|---|---|---
|  $9\times 11$ plate | 493,674 | 67,048 |
|  $49\times 51$ plate | 9,726,194 | 1,300,302 |
|  |  |  |
Table 3: Numbers of degrees of freedom for the symmetrized microstructured and
the relaxed micromorphic plates.

As justified in section 3.1, the considered structure requires to be damped
for numerical stability. As before, a hysteretic damping is introduced in the
plate, modifying the strain energy density in the bidimensionnal plate as

$\displaystyle\begin{cases}W^{\rm c}_{\text{non-conservative}}=(1+i\eta)W^{\rm
c}_{\text{conservative}}\\\ W^{\rm m}_{\text{non-
conservative}}=(1+i\eta)W^{\rm m}_{\text{conservative}}\end{cases}\text{ where
}\eta=0.002$ (20)

##  4 Metamaterial’s plate under a pulse excitation : microstructured versus
relaxed micromorphic simulations

In this section, we start simulating the experimental set-up presented in
section 2 both via the microstructured simulations and the relaxed
micromorphic ones when considering a pulse excitation to give a first
description of the piezoelectric load.

More precisely, to mimick the application of the piezoelectric load, we start
implementing a simplified framework in which the piezoelectric region
$\Omega_{\rm p}$ is modeled as a void region $\Omega_{v}$. The applied load is
given as an imposed displacement on the boundary $\partial\Omega_{v}$ (see
Figure 10) in the form :

$u=\psi\,n\text{ (expansion load)}$ (21)

where $n$ is the unit normal to the each surface and $\psi=10^{-3}$ [m]. The
other interface conditions are given in equations (17) and (18).

Figure 10: (left) Top view of the full microstructured plate and
identification of points A and B. (right) Top view of the full relaxed
micromorphic plate and identification of the corresponding volumes
$\Omega_{A}$ and $\Omega_{B}$. Given the strong directionality of the plate,
we do not consider other points outside the dashed domain to show the
simulation’s results. Points $C_{1}$, $C_{2}$, $C_{3}$ and $C_{4}$ are also
located since they are used to present further results.

Since the relaxed micromorphic model is a macroscopic model it is not always
worth comparing the solution displacement field pointwise with the one issued
via the microstructured simulations. A consistent difference between these
pointwise fields may be expected. To provide a more stable comparison an
average displacement field over a representative portion of the unit cell can
be considered. To this aim we start identifying the points $A$ and $B$ in the
considered structure as (see also Figure 10):

$\displaystyle\begin{cases}A&=\Big{(}3a/2,0\Big{)}^{T}\\\
B&=\Big{(}ma/2,0\Big{)}^{T}\end{cases}\text{ and surfaces
}\begin{cases}\Omega_{A}&=\Big{[}a,3a/2\Big{]}\times\Big{[}0,a/2\Big{]}\\\
\Omega_{B}&=\Big{[}(m-1)a/2,ma/2\Big{]}\times\Big{[}0,a/2\Big{]}\end{cases}$
(22)

where $m$ is the number of cells of the plate on its main axis (see Figure
10). We then introduce a pointwise measure of displacement $p$ and a mean
measure of displacement $m$ as :

$\displaystyle p_{X}=\frac{1}{\psi}\sqrt{\langle u(X),\bar{u}(X)\rangle}\text{
and }m_{X}=\frac{4}{\psi a^{2}}\iint_{\Omega_{X}}\sqrt{\langle
u(x_{1},x_{2}),\bar{u}(x_{1},x_{2})\rangle}\mathrm{d}x_{1}\mathrm{d}x_{2}\text{
for }X=\\{A,B\\}$ (23)

where a superposed bar indicates the complex conjugate operation and $\psi$
the amplitude of the expansion load introduced before. The hermitian norm used
here, not necessary for the static response of the system, where the
displacement stays real despite the hysteretic damping, finds its use
computing the dynamic response of the plate. For the subsequent purposes of
comparison with the experiment, we also introduce four points $C_{i}$
$i\in\\{1,4\\}$ far from the excitation. As these points are only considered
in comparison with the experimental setup, we directly give their coordinates
for $m=11$, $n=9$ and $c=1$ in Table 4.

| Point | $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$ |
---|---|---|---|---|---|---
|  $x_{1}$ coordinate [mm] | 105 | 105 | 5 | 15 |
|  $x_{2}$ coordinate [mm] | 5 | 15 | 85 | 85 |
|  |  |  |  |  |
Table 4: Coordinates of the measurement points.

###  4.1 The long-wave limit : statics

The main challenge for metamaterials’ modelling consists in the description of
their broadband mechanical response. More particularly, a suitable model must
be able to describe metamaterials’ response for the larger possible set of
frequencies. We will show in the following sections that the RMM is able to
correctly describe the metamaterial’s response for a very wide side set of
frequencies going well beyond the first band-gap.

Nevertheless, specific attention must be payed to the so-called long-wave or
static limit which can be recovered from the dynamic model when considering
very small frequencies, in the limit $\omega\rightarrow 0$.

In this section, we explicitely point out this static limit both for the
microstructured and the relaxed micromorphic model. We show that, since
internal lengths are neglected, the relaxed micromorphic static limit
coincides with an equivalent Cauchy medium. We remark that for the
experimental metamaterial’s specimen’s size (9 $\times$ 11 cells), this
equivalent Cauchy medium slightly deviates from the static response of the
full microstructured metamaterial. However, this difference remains smaller
than 10 $\%$ (see Figure 11) and becomes even smaller as soon as higher
frequencies are considered.

To improve the relaxed micromorphic response of these small samples for the
static limit, internal lengths should be introduced. This would lead, on the
other hand, to a more complex identification procedure for the dynamic regime.
We thus limit ourselves to the case of negligible internal lengths, knowing
that this leads to a controlled inaccuracy in the static limit for small
specimens.

It is well known that a given metamaterial can be modeled as a Cauchy
continuum in the static regime as soon as a specimen of suitable large size is
considered. By direct inspection of Figure 11 we can infer that the
metamaterial considered in this paper behaves as a Cauchy continuum in the
static regime when considering specimens that are greater than 30$\times$30
cells, for the pulse excitation shown in Figure 10.

The convergence of the considered metamaterial towards a proper Cauchy
material in the static regime can be also achieved on smaller specimens by
suitably changing the applied load (e.g., increasing the size of the region
where the load is applied). This can be seen from Figures 12 and 13 ; when
considering a load applied on a square whose side is 3 unit cells, the
obtained static response can be modeled as a Cauchy continuum even for small
specimen sizes (9$\times$11 cells).

In summary, we have shown that our hypothesis of neglecting static internal
lengths may produce a small and controlled inaccuracy for the static limiting
case when considering a specimen of the size considered in our experiment
(9$\times$11 cells) and an external load applied on a unique unit cell
($c=1$)444The parameter $c$ indicates, as presented in Figure 10, the
extension of the region where the applied load is applied : $c=1$ represents a
$1\times 1$ cells central region, $c=2$ a $2\times 2$ cells central region,
etc. We will show in section 4.2 that the extension of this region is
necessary to the correct description of the microstructured plate by the
relaxed micromorphic model..

Since, on the other hand, this hypothesis drastically simplifies the
characterization procedure for the dynamic regime, we decide to keep it in the
remainder of the paper.

Figure 11: Pointwise (left) and mean (right) displacement for the static
responses of the microstructured, the homogenized and the relaxed micromorphic
models at point B for $c=1$. Similar results are valid for point A. Figure 12:
Pointwise (left) and mean (right) displacement for the static responses of the
microstructured, the homogenized and the relaxed micromorphic models at point
B for ($m=11$,$n=9$). Similar results are valid for point A. Figure 13:
Pointwise (left) and mean (right) displacement for the static responses of the
microstructured, the homogenized and the relaxed micromorphic models at point
B for ($m=51$,$n=49$). Similar results are valid for point A.

###  4.2 Broadband dynamics of the metamaterial’s plate

In this section, we present the broadband response for both the
microstructured and relaxed micromorphic plate when considering a plate size
of 51$\times$49 cells and a load applied on a square whose side is 11 unit
cells. Based on the results of the previous section this choice allows to
precisely recover the long-wave limit and shows the capability of the relaxed
micromorphic model to correctly reproduce the dynamical response of the
considered metamaterial for a wide set of frequencies (from zero to beyond the
first band-gap).

Figure 14: (left) Pointwise displacement $p_{B}$ of the microstructured and
relaxed micromorphic models at point $B$ with the theoretical band-gap
(dashed) for $m=51$, $n=49$ and $c=11$. (right) Mean displacement $m_{B}$ of
the microstructured and relaxed micromorphic models with the theoretical band-
gap (dashed) for $m=51$, $n=49$ and $c=11$.

Figure 14 shows this broadband response for the considered metamaterial plate
: it is apparent that the relaxed micromorphic model describes well the
plate’s behavior for the whole considered frequency range. The size of the
considered plate (49$\times$51) was still allowing a direct comparison of the
relaxed micromorphic model with the microstructured simulations. However, the
computational time was considerably higher for the microstructured plate.
Explicit comparison for larger plates would be out of reach with standard
computational tools. This calls for the importance of our model in view of its
use for the design of large-scale engineering metastructures.

To give an outlook of the efficacity of the relaxed micromorphic model, we
plot in Figs. 15-18 the solution for the displacement field at frequencies
$M_{1}$, $M_{2}$, $M_{3}$ as defined in Figure 14 (right). For each point
$M_{i}$, we actually consider two adjacent values of the frequency to compute
the solution (see Figure 14 right). It can be clearly inferred that the
relaxed micromorphic model encodes all the main features of the metamaterial’s
reponse at a fraction of the computationnal cost.

Figure 15: $|u|/\psi$ at frequency $M_{1}$ for the microstructured model and
the relaxed micromorphic model, the first two figures correspond to $M_{1}$
(left) and the last two figures to $M_{1}$ (right) (see Figure 14 for the
definition of these points). Figure 16: $|u|/\psi$ at frequency $M_{2}$ for
the microstructured model and the relaxed micromorphic model, the first two
figures correspond to $M_{2}$ (left) and the last two figures to $M_{2}$
(right) (see Figure 14 for the definition of these points). Figure 17:
$|u|/\psi$ at frequency $M_{3}$ for the microstructured model and the relaxed
micromorphic model, the first two figures correspond to $M_{3}$ (left) and the
last two figures to $M_{3}$ (right) (see Figure 14 for the definition of these
points). Figure 18: $|u|/\psi$ at frequency $M_{4}$ for the microstructured
model and the relaxed micromorphic model, the first two figures correspond to
$M_{4}$ (left) and the last two figures to $M_{4}$ (right) (see Figure 14 for
the definition of these points).

##  5 Comparison of the relaxed micromorphic and microstructured simulations
with the experimental results

To make a direct comparison with the experimental results obtained with the
set-up of section 2, we introduce here a more precise modeling of the applied
piezoelectric load. The details and micromorphic simulations presented in
sections 3.1 and 3.2 are implemented and solved in Comsol. We show that both
approaches give similar results which can be directly superposed to the
experimental ones as soon as manufacturing defects are taken into account.
While a direct comparison of the results obtained via the geometry detailed
finite element simulation and the relaxed micromorphic approach has been
possible given the reduced size of the metamaterial’s sample, finite element
simulations become computationally too expensive for samples of larger size.
On the other hand, the relaxed micromorphic simulations provide a powerful
tool for investigating the behaviour of larger samples with a modest increase
of the computational costs. As we will show in section 6, this feature of the
relaxed micromorphic model is of major importance to simulate realistic,
larger-scale meta-structures that can have a true impact in engineering
science. Figure 19 shows a broadband comparison of the relaxed micromorphic
and microstructured response with the experimental one.

Figure 19: Amplitude of $|\dot{u}|$ at $C_{1}$ (see Figure 10 for the
definition of this point) for the experiment, the microstructured and relaxed
micromorphic models (the dashed black lines locate the theoretical position of
the band-gap for an infinite specimen).

It can be seen that both models reproduce well, on average, the experimentally
observed response, except for a frequency shift. This frequency shift can be
related to defects in the experimental procedure, more particularly to defects
that have occurred during the manufacturing process. To account for the
presence of these defects, a recalibration procedure is presented in section
5.1. As we will discuss more in section 5.1.3, the relaxed micromorphic
material slightly underestimates the resonance peak before the band-gap. This
is due, to a big extent, to the fact that Figs. 19 and 22 are relative to
pointwise displacements (not to mean displacements) because of the need of a
direct comparison with the experiment.

###  5.1 Recalibration procedure accounting for manufacturing defects

In this subsection we present a recalibration procedure for both the
microstructured and the relaxed micromorphic model allowing to account for the
presence of defects in the fabrication process or, to a smaller extent, to
measurement inaccuracy. Given the change of nature of the excitation of the
system, we redefine $p_{X}$ and $m_{X}$ by :

$\displaystyle p_{X}=\sqrt{\langle u(X),\bar{u}(X)\rangle}\text{ and
}m_{X}=\frac{4}{a^{2}}\iint_{\Omega_{X}}\sqrt{\langle
u(x_{1},x_{2}),\bar{u}(x_{1},x_{2})\rangle}\mathrm{d}x_{1}\mathrm{d}x_{2}.$
(24)

####  5.1.1 Recalibration for the microstructured system

To account for the presence of defects and get closer to the experimental
results, the system can be modified, taking into account several potential
differences between the analytical model and experimental system. To simulate
the likely presence of defects in the plate, we chose not to alter geometry
and to modify the mechanical parameters. In formulas :

$\begin{cases}\lambda^{\text{recalibration}}_{\text{Ti}}=(1+\kappa_{E})\lambda_{\text{Ti}}\\\
\mu^{\text{recalibration}}_{\text{Ti}}=(1+\kappa_{E})\mu_{\text{Ti}}\end{cases}\text{and
}\rho^{\text{recalibration}}_{\text{Ti}}=(1+\kappa_{\rho})\rho_{\text{Ti}}$

Leading, for the local energy densities, to

$W_{\text{recalibration}}=(1+\kappa_{E})W_{\text{Ti}}\text{ and
}K_{\text{recalibration}}=(1+\kappa_{\rho})K_{\text{Ti}}$

Confronting the theoretical results to the experiments eventually led to

$\displaystyle\kappa_{\rho}=+0.05\,,\qquad\qquad\qquad\kappa_{E}=-0.10875\,,$
(25)

which, considering the uncertainties due to, among other, the manufacturing
process, the experimental boundary conditions, the gluing of the piezoelectric
patch to the microstructured plate and more particularly the plain-strain
hypothesis in the plate, is quite acceptable.

####  5.1.2 Recalibration for the micromorphic system

In the same way, the kinetic and potential energy density of the micromorphic
medium are modified as

$K_{\text{recalibration}}=(1+\kappa_{K})K_{0}\text{ and
}W_{\text{recalibration}}=(1+\kappa_{W})W_{0}$

To recalibrate the relaxed micromorphic model on the microstructured model, we
set

$\kappa_{K}=\kappa_{\rho}\text{ and }\kappa_{W}=\kappa_{E}$

The relaxed micromorphic model, less expensive in computing time, was
recalibrated on the experiment and the parameters of the microstructured model
were then updated by the following rule :

$\displaystyle\begin{cases}\eta_{i}^{\text{recalibration}}=(1+\alpha)\eta_{i}\text{,
}i\in\\{1,3\\}\\\ \eta_{1}^{*\text{
recalibration}}=(1+\alpha)\overline{\eta}_{1}^{*}\\\
\overline{\eta}_{i}^{\text{recalibration}}=(1+\alpha)\overline{\eta}_{i}\text{,
}i\in\\{1,3\\}\\\ \overline{\eta}_{1}^{*\text{
recalibration}}=(1+\alpha)\overline{\eta}_{1}^{*}\\\
\lambda_{i}^{\text{recalibration}}=(1+\beta)\lambda_{i}\text{,
}i\in\\{e,m\\}\\\ \mu_{i}^{\text{recalibration}}=(1+\beta)\mu_{i}\text{,
}i\in\\{e,m\\}\\\ \mu_{i}^{*\text{ recalibration}}=(1+\beta)\mu_{i}^{*}\text{,
}i\in\\{e,m\\}\\\ \mu_{\rm c}^{\text{ recalibration}}=(1+\beta)\mu_{\rm
c}\end{cases}\longleftrightarrow\begin{cases}\rho^{\text{recalibration}}=(1+\alpha)\rho\\\
\mu^{\text{recalibration}}=(1+\beta)\mu\\\
\lambda^{\text{recalibration}}=(1+\beta)\lambda\\\ \end{cases}\text{where
}\begin{cases}\alpha&=0.05\\\ \beta&=-0.10875\end{cases}.$

Figures 20 and 21 present $p_{B}$ and $m_{B}$ for the microstructured and the
relaxed micromorphic models with the recalibrated parameters. We remark that,
once calibrated, both the pointwise and the mean displacement describe well
the local resonance occuring at the lower band-gap limit. As expected, the
mean displacements for the relaxed micromorphic and microstructured model show
better agreement than the pointwise displacement.

Figure 20: (left) Pointwise displacement $p_{B}$ for the original and
recalibrated relaxed micromorphic models at point $B$. (right) Mean
displacement $m_{B}$ for the original and recalibrated relaxed micromorphic
models. Figure 21: (left) Pointwise displacement $p_{B}$ for the original and
recalibrated microstructured models at point $B$. (right) Mean displacement
$m_{B}$ for the original and recalibrated microstructured models.

####  5.1.3 Discussion of results

Figure 22 shows the comparison of the velocity spectrum at point $B$ for the
experiment, the microstructured and the relaxed micromorphic model. A
generally good agreement is found for both models, the relaxed micromorphic
model showing slightly reduced performances due to the reduced size of the
plate. This result is to be considered satisfying, since all the main response
characteristics are well described.

When increasing the size of the plate the relaxed micromorphic model becomes
more and more accurate with no significant increase of the computational cost
(see Figure 14). On the other hand the microstructured simulations become very
costly in terms of computational time. This opens the way to the effective
design of large-scale metastructures that would not be possible without using
the relaxed micromorphic model. To prove this statement, we present in the
next section the design of a new large-scale engineering meta-structure that
would not have been otherwise possible.

Figure 22: Amplitude of $|\dot{u}|$ at $C_{1}$ (see Figure 10 for the
definition of this point) for the experiment, the microstructured and relaxed
micromorphic models.

##  6 Enabling meta-structural engineering design

As we pointed out so far, the relaxed micromorphic model can be considered as
an appropriate homogenized limit for mechanical metamaterials as soon as
sufficiently large specimens are considered. In particular, for the specific
unit cell presented in this paper (see Fig. 1), we showed that the relaxed
micromorphic model starts giving very good results for specimens that are
larger than 30x30 unit cells. We also showed that the results ulteriorly
improve when considering a central pulse excitation distributed over more than
one unit cell. For example, we showed in Fig. 14 that the metamaterial’s
behavior is perfectly reproduced for the whole frequency range when
considering a specimen of $51\times 49$ unit cells and a pulse excitation
distributed over a region whose side is 11 cells. The results presented in
this paper open unprecedented opportunities of exploring metamaterial
structures at large scales, thus electing the relaxed micromorphic model as a
gold standard for engineering design. To support our claiming, we present in
this section the design of a complex metastructure that is able to concentrate
energy in specific points, so that the eventual use of converters can be eased
for the subsequent conversion of elastic energy into, e.g., heat or
electricity. We consider a structure whose geometry is given in Fig. 23: the
central domain $\Omega_{m}^{1}$ is made up of the metamaterial studied in this
paper (Fig. 1), while the outer domain $\Omega_{m}^{2}$ is made of a
metamaterial with the same geometry whose unit cell is doubled with respect to
the one presented in Fig. 1 (see Appendix C for the details concerning this
larger unit cell). Both metamaterials’ domains are very large ($101\times 51$
unit cells in $\Omega_{m}^{1}$) and $51\times 51$ unit cells in
$\Omega_{m}^{2}$), so that the use of the homogenized model is certainly
pertinent. This large-scale structure points towards realistic structural
engineering design (think, for example, that the domain $\Omega_{m}^{1}$ is
located around a railway truck and that the domain $\Omega_{m}^{2}$ are the
lateral banks). The two metamaterials’ domains are separated by a classical
Cauchy material occupying the annular domain $\Omega_{c}$. The elastic
properties of such soft Cauchy material are given in Table 5. The
metastructure setup is given in Figure 23. The functioning mechanism of this
structure can be summarized as follows: we send a pulse signal in the center
of the metamaterial plate $\Omega_{m}^{1}$ at a frequency that falls in the
band-gap of the outer metamaterial (see Fig. 27 for the relative position of
the dispersion curves for the two metamaterials). When reaching the outer
domain the wave cannot propagate due to the presence of the outer
metamaterial. The annular Cauchy material is chosen so that a diode effect is
triggered (the wave coming from $\Omega_{m}^{1}$ can pass, but cannot go back)
[34]. Thanks to this design, the proposed meta-structure can focus an
important part of the elastic energy in the annular Cauchy region (see Tab.
6). It can be noted that the energy concentration in the annular Cauchy
material is evident, especially considering the very narrow area in which it
occurs (see Fig. 25). A structure of this type could be used to locate energy
converters in the annular Cauchy region for subsequent energy conversion and
re-use. It will be the object of forthcoming works to effectively optimize and
realize a structure of this type.

| $\rho_{2}$ | $\lambda_{2}$ | $\mu_{2}$ |
---|---|---|---|---
| [kg.m-3] | [Pa] | [Pa] |
| $3000$ | $9.74\cdot 10^{8}$ | $5.88\cdot 10^{5}$ |
|  |  |  |
Table 5: Mechanical parameters of the second isotropic Cauchy medium between
the two relaxed micromorphic mediums (red annular region in Fig. 23 and 24).

Figure 23: (left) Top view of the full microstructured plate with the two
different cells. (right) Top view of the equivalent micromorphic plate with
the boundaries and medium denominations.

Figure 24: (left) Top view of the symmetrized microstructured structure with the two different cells. (right) Top view of the symmetrized equivalent micromorphic structure with the boundaries and domains denominations. Only the micromorphic meta-structure has been implemented in the FEM environnement. | 857.5 Hz | $\Omega^{1}_{c}$ | $\Omega^{1}_{m}$ | $\Omega^{2}_{c}$ | $\Omega^{2}_{m}$ |
---|---|---|---|---|---|---
| [J.m-3] | [J.m-3] | [J.m-3] | [J.m-3] |
| W | $55.1\cdot 10^{3}$ | $45.5\cdot 10^{3}$ | $29.5\cdot 10^{3}$ | $7.74\cdot 10^{3}$ |
| K | $3.42\cdot 10^{6}$ | $43.6\cdot 10^{3}$ | $26.3\cdot 10^{3}$ | $7.84\cdot 10^{3}$ |
|  |  |  |  |  |
Table 6: Values of the average total energy for each domain of the structure
at 857.5 Hz. Figure 25: Displacement field for the considered metastructure at
857.5 Hz.

##  7 Conclusions and perspectives

In this paper we showed for the first time how the relaxed micromorphic model
can be operatively used to design realistic large-scale structures that can
control elastic waves and eventually recover energy. Our model was calibrated
on an acoustic metamaterial and the theoretical results were compared to the
experimental ones performed on a specimen manufactured via metal etching
techniques. We presented a metastructure combining metamaterials’ bricks of
different type/shape and bricks of classical Cauchy materials in such a way
that the elastic energy produced by a source of vibrations is concentrated in
specific collection points, thus easing eventual subsequent re-use. The
structure conceived in this paper will be optimized in forthcoming works and a
prototype will be eventually realized to show the new possibilities that a
micromorphic-type modeling of metamaterials opens for practical applications.

##  8 Acknowledgments

Angela Madeo and Gianluca Rizzi acknowledge support from the European
Commission through the funding of the ERC Consolidator Grant META-LEGO, N◦
101001759. Angela Madeo and Gianluca Rizzi acknowledge funding from the French
Research Agency ANR, “METASMART” (ANR-17CE08-0006). Patrizio Neff acknowledges
support in the framework of the DFG-Priority Programme 2256 “Variational
Methods for Predicting Complex Phenomena in Engineering Structures and
Materials”, Neff 902/10-1, Project-No. 440935806.

## References

  * [1] Alexios Aivaliotis, Domenico Tallarico, Marco-Valerio d’Agostino, Ali Daouadji, Patrizio Neff and Angela Madeo  “Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model”  In _Archive of Applied Mechanics_ 90.5, 2020, pp. 1073–1096  DOI: 10.1007/s00419-019-01651-9
  * [2] A. Bacigalupo and L. Gambarotta  “Second-gradient homogenized model for wave propagation in heterogeneous periodic media”  In _International Journal of Solids and Structures_ 51.5, 2014, pp. 1052–1065  DOI: 10.1016/j.ijsolstr.2013.12.001
  * [3] Giovanni Bordiga, Luigi Cabras, Andrea Piccolroaz and Davide Bigoni  “Prestress tuning of negative refraction and wave channeling from flexural sources”  In _Applied Physics Letters_ 114.4 AIP Publishing LLC, 2019, pp. 041901
  * [4] Claude Boutin, Antoine Rallu and Stéphane Hans  “Large scale modulation of high frequency waves in periodic elastic composites”  In _Journal of the Mechanics and Physics of Solids_ 70 Elsevier, 2014, pp. 362–381
  * [5] Tiemo Bückmann, Muamer Kadic, Robert Schittny and Martin Wegener  “Mechanical cloak design by direct lattice transformation”  In _Proceedings of the National Academy of Sciences_ 112.16, 2015, pp. 4930–4934  DOI: 10.1073/pnas.1501240112
  * [6] Paolo Celli, Behrooz Yousefzadeh, Chiara Daraio and Stefano Gonella  “Bandgap widening by disorder in rainbow metamaterials”  In _Applied Physics Letters_ 114.9, 2019, pp. 091903  DOI: 10.1063/1.5081916
  * [7] Wen Chen and Jacob Fish  “A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales”  In _J. Appl. Mech._ 68.2, 2001, pp. 153–161
  * [8] R.. Craster, J. Kaplunov and A.. Pichugin  “High-frequency homogenization for periodic media”  In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 466.2120, 2010, pp. 2341–2362  DOI: 10.1098/rspa.2009.0612
  * [9] Marco Valerio d’Agostino, Gabriele Barbagallo, Ionel-Dumitrel Ghiba, Bernhard Eidel, Patrizio Neff and Angela Madeo  “Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model”  In _Journal of Elasticity_ 139.2, 2020, pp. 299–329  DOI: 10.1007/s10659-019-09753-9
  * [10] Tallarico Domenico and A.B. A.  “Edge waves and localization in lattices containing tilted resonators.”  In _Frontiers in Materials_ 4(16) URL: DOI:%2010.3389/fmats.2017.00016
  * [11] Tobias Frenzel, Muamer Kadic and Martin Wegener  “Three-dimensional mechanical metamaterials with a twist”  In _Science_ 358.6366 American Association for the Advancement of Science, 2017, pp. 1072–1074
  * [12] Sébastien Guenneau, Alexander Movchan, Gunnar Pétursson and S Anantha Ramakrishna  “Acoustic metamaterials for sound focusing and confinement”  In _New Journal of Physics_ 9.11, 2007, pp. 399–399  DOI: 10.1088/1367-2630/9/11/399
  * [13] Nadège Kaina, Alexandre Causier, Yoan Bourlier, Mathias Fink, Thomas Berthelot and Geoffroy Lerosey  “Slow waves in locally resonant metamaterials line defect waveguides”  In _Scientific Reports_ 7.1, 2017, pp. 15105  DOI: 10.1038/s41598-017-15403-8
  * [14] Nadège Kaina, Fabrice Lemoult, Mathias Fink and Geoffroy Lerosey  “Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials”  In _Nature_ 525.7567, 2015, pp. 77–81  DOI: 10.1038/nature14678
  * [15] Anastasiia O Krushynska, Ada Amendola, Federico Bosia, Chiara Daraio, Nicola M Pugno and Fernando Fraternali  “Accordion-like metamaterials with tunable ultra-wide low-frequency band gaps”  In _New Journal of Physics_ 20.7 IOP Publishing, 2018, pp. 073051
  * [16] Anastasiia O Krushynska, Marco Miniaci, Federico Bosia and Nicola M Pugno  “Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials”  In _Extreme Mechanics Letters_ 12 Elsevier, 2017, pp. 30–36
  * [17] Roderic Lakes  “Foam structures with a negative Poisson’s ratio”  In _Science_ 235.4792 American Association for the Advancement of Science, 1987, pp. 1038–1040
  * [18] Jensen Li and Che Ting Chan  “Double-negative acoustic metamaterial”  In _Physical Review E_ 70.5 APS, 2004, pp. 055602
  * [19] Chenchen Liu and Celia Reina  “Broadband locally resonant metamaterials with graded hierarchical architecture”  In _Journal of Applied Physics_ 123.9, 2018, pp. 095108  DOI: 10.1063/1.5003264
  * [20] A.N. M.R Haberman  “Acoustic metamaterials.”  In _Acoustics Today_ 12:3, 2016, pp. 31–39
  * [21] Marco Miniaci, Anastasiia Krushynska, Federico Bosia and Nicola M Pugno  “Large scale mechanical metamaterials as seismic shields”  In _New Journal of Physics_ 18.8 IOP Publishing, 2016, pp. 083041
  * [22] Marco Miniaci, Anastasiia Krushynska, Alexander B Movchan, Federico Bosia and Nicola M Pugno  “Spider web-inspired acoustic metamaterials”  In _Applied Physics Letters_ 109.7 AIP Publishing LLC, 2016, pp. 071905
  * [23] D Misseroni, AB Movchan and D Bigoni  “Omnidirectional flexural invisibility of multiple interacting voids in vibrating elastic plates”  In _Proceedings of the Royal Society A_ 475.2229 The Royal Society Publishing, 2019, pp. 20190283
  * [24] D. Misseroni, D.. Colquitt, A.. Movchan, N.. Movchan and I.. Jones  “Cymatics for the cloaking of flexural vibrations in a structured plate”  In _Scientific Reports_ 6.1, 2016, pp. 23929  DOI: 10.1038/srep23929
  * [25] H. Nassar, Q.-C. He and N. Auffray  “Willis elastodynamic homogenization theory revisited for periodic media”  In _Journal of the Mechanics and Physics of Solids_ 77, 2015, pp. 158–178  DOI: 10.1016/j.jmps.2014.12.011
  * [26] P. Neff, I.. Ghiba, M. Lazar and A. Madeo  “The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations”  In _The Quarterly Journal of Mechanics and Applied Mathematics_ 68.1, 2015, pp. 53–84  DOI: 10.1093/qjmam/hbu027
  * [27] Patrizio Neff, Ionel-Dumitrel Ghiba, Angela Madeo, Luca Placidi and Giuseppe Rosi  “A unifying perspective: the relaxed linear micromorphic continuum”  In _Continuum Mechanics and Thermodynamics_ 26.5, 2014, pp. 639–681  DOI: 10.1007/s00161-013-0322-9
  * [28] E. Nolde, R.. Craster and J. Kaplunov  “High frequency homogenization for structural mechanics”  In _Journal of the Mechanics and Physics of Solids_ 59.3, 2011, pp. 651–671
  * [29] C. O.R.  “Architected lattices for simultaneous broadband attenuation of airborne sound and mechanical vibrations in all directions.”  In _Physical Review Applied_ 10.5:054060, 2018
  * [30] K. Pham, V.G. Kouznetsova and M.G.D. Geers  “Transient computational homogenization for heterogeneous materials under dynamic excitation”  In _Journal of the Mechanics and Physics of Solids_ 61.11, 2013, pp. 2125–2146  DOI: 10.1016/j.jmps.2013.07.005
  * [31] G Rizzi, F Dal Corso, D Veber and D Bigoni  “Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation”  In _International Journal of Solids and Structures_ 176 Elsevier, 2019, pp. 19–35
  * [32] Gianluca Rizzi, Manuel Collet, Félix Demore, Bernhard Eidel, Patrizio Neff and Angela Madeo  “Exploring metamaterials’ structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber”  In _Frontiers in Materials_ 7, 2021, pp. 589701  DOI: 10.3389/fmats.2020.589701
  * [33] Gianluca Rizzi, Marco Valerio d’Agostino, Patrizio Neff and Angela Madeo  “Boundary and interface conditions in the relaxed micromorphic model: Exploring finite-size metastructures for elastic wave control”  In _Mathematics and Mechanics of Solids_ SAGE Publications Sage UK: London, England, 2021, pp. 10812865211048923
  * [34] Gianluca Rizzi, Domenico Tallarico, Patrizio Neff and Angela Madeo  “Towards the conception of complex engineering meta-structures: relaxed-micromorphic modelling of mechanical diodes”  In _(arXiv:2012.11192) Accepted in Wave Motion_ , 2021
  * [35] A. Sridhar, V.. Kouznetsova and M… Geers  “Frequency domain boundary value problem analyses of acoustic metamaterials described by an emergent generalized continuum”  In _Computational Mechanics_ 65.3, 2020, pp. 789–805  DOI: 10.1007/s00466-019-01795-z
  * [36] A. Sridhar, V.. Kouznetsova and M… Geers  “Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum”  In _Computational Mechanics_ 57.3, 2016, pp. 423–435  DOI: 10.1007/s00466-015-1254-y
  * [37] A. Sridhar, V.G. Kouznetsova and M.G.D. Geers  “A general multiscale framework for the emergent effective elastodynamics of metamaterials”  In _Journal of the Mechanics and Physics of Solids_ 111, 2018, pp. 414–433  DOI: 10.1016/j.jmps.2017.11.017
  * [38] Ashwin Sridhar, Varvara G Kouznetsova and Marc GD Geers  “A general multiscale framework for the emergent effective elastodynamics of metamaterials”  In _Journal of the Mechanics and Physics of Solids_ 111 Elsevier, 2018, pp. 414–433
  * [39] Ashwin Sridhar, Varvara G Kouznetsova and Marc GD Geers  “Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum”  In _Computational mechanics_ 57.3 Springer, 2016, pp. 423–435
  * [40] Ankit Srivastava and John R Willis  “Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach”  In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 473.2200 The Royal Society Publishing, 2017, pp. 20160765
  * [41] Pai Wang, Filippo Casadei, Sicong Shan, James C. Weaver and Katia Bertoldi  “Harnessing buckling to design tunable locally resonant acoustic metamaterials”  In _Physical Review Letters_ 113.1, 2014, pp. 014301  DOI: 10.1103/PhysRevLett.113.014301
  * [42] J.. Willis  “Effective constitutive relations for waves in composites and metamaterials”  In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 467.2131, 2011, pp. 1865–1879  DOI: 10.1098/rspa.2010.0620
  * [43] J.. Willis  “Negative refraction in a laminate”  In _Journal of the Mechanics and Physics of Solids_ 97, 2016, pp. 10–18  DOI: 10.1016/j.jmps.2015.11.004
  * [44] John R Willis  “Effective constitutive relations for waves in composites and metamaterials”  In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 467.2131 The Royal Society Publishing, 2011, pp. 1865–1879
  * [45] John R Willis  “The construction of effective relations for waves in a composite”  In _Comptes Rendus Mécanique_ 340.4-5 Elsevier, 2012, pp. 181–192
  * [46] R. Zhu, X.. Liu, G.. Hu, C.. Sun and G.. Huang  “Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial”  In _Nature Communications_ 5.1, 2014, pp. 5510  DOI: 10.1038/ncomms6510

##  Appendix A Boundary conditions on a symmetry plane for a relaxed
micromorphic medium using Curie’s Symmetry Principle

$u$ and $P$ not depending on the orientation of space, supposing that our
problem to have a symmetry with respect to the plane $\mathcal{N}$ of normal
$n$, one can apply Curie’s Symmetry Principle, which gives

$\displaystyle\begin{cases}u(x^{\star})=u^{\star}(x)\\\
P(x^{\star})=P^{\star}(x)\end{cases}$ (26)

$\chi^{\star}$ being the symmetric of $\chi$ with respect to $\mathcal{N}$.
Let’s define $t_{1}$ and $t_{2}$ so that $(t_{1},t_{2},n)$ forms an
orthonormal basis, thus

$t_{1}^{\star}=t_{1}\text{ , }t_{2}^{\star}=t_{2}\text{ and }n^{\star}=-n$

And

$(t_{1}\otimes t_{1})^{\star}=t_{1}\otimes t_{1}\text{ , }(t_{1}\otimes
t_{2})^{\star}=t_{1}\otimes t_{2}\text{ , }(t_{1}\otimes
n)^{\star}=-t_{1}\otimes n$ $(t_{2}\otimes t_{1})^{\star}=t_{2}\otimes
t_{1}\text{ , }(t_{2}\otimes t_{2})^{\star}=t_{2}\otimes t_{2}\text{ ,
}(t_{2}\otimes n)^{\star}=-t_{2}\otimes n$ $(n\otimes t_{1})^{\star}=-n\otimes
t_{1}\text{ , }(n\otimes t_{2})^{\star}=-n\otimes t_{2}\text{ and }(n\otimes
n)^{\star}=n\otimes n$

Let’s write $u$ and $P$ in this base :

$\exists!(u_{1},u_{2},u_{3})\in{\mathcal{C}}^{0}(\mathbb{R}^{3},\mathbb{R})^{3}\text{
: }u=u_{1}t_{1}+u_{2}t_{2}+u_{3}n$
$\exists!(P_{11},P_{22},P_{33},P_{12},P_{13},P_{21},P_{31},P_{32},P_{23})\in{\mathcal{C}}^{0}(\mathbb{R}^{3},\mathbb{R})^{9}\text{
: }$ $P=P_{11}t_{1}\otimes t_{1}+P_{12}t_{1}\otimes t_{2}+P_{13}t_{1}\otimes
n+\\\ P_{21}t_{2}\otimes t_{1}+P_{22}t_{2}\otimes t_{2}+P_{23}t_{2}\otimes
n+\\\ P_{31}n\otimes t_{1}+P_{32}n\otimes t_{2}+P_{33}n\otimes n$

Let’s also write $x$ as

$\exists!(x_{0},\epsilon)\in\mathcal{N}\times\mathbb{R}:x=x_{0}+\epsilon n$

Then

$x^{\star}=x_{0}-\epsilon n$

For $u$, substituting in 26, one can get

$u_{1}(x_{0}-\epsilon n)t_{1}+u_{2}(x_{0}-\epsilon
n)t_{2}+u_{3}(x_{0}-\epsilon n)n=u_{1}(x_{0}+\epsilon
n)t_{1}+u_{2}(x_{0}+\epsilon n)t_{2}-u_{3}(x_{0}+\epsilon n)n$

By identification

$\forall
x_{0}\in\mathcal{N},\forall\epsilon\in\mathbb{R}\left\\{\begin{array}[]{rcl}u_{1}(x_{0}-\epsilon
n)&=&u_{1}(x_{0}+\epsilon n)\\\ u_{2}(x_{0}-\epsilon n)&=&u_{2}(x_{0}+\epsilon
n)\\\ u_{3}(x_{0}-\epsilon n)&=&-u_{3}(x_{0}+\epsilon n)\end{array}\right.$

In the same way, we have for $P$

$\forall
x_{0}\in\mathcal{N},\forall\epsilon\in\mathbb{R}\left\\{\begin{array}[]{rcl}P_{11}(x_{0}-\epsilon
n)&=&P_{11}(x_{0}+\epsilon n)\\\ P_{12}(x_{0}-\epsilon
n)&=&P_{12}(x_{0}+\epsilon n)\\\ P_{13}(x_{0}-\epsilon
n)&=&-P_{13}(x_{0}+\epsilon n)\\\ P_{21}(x_{0}-\epsilon
n)&=&P_{21}(x_{0}+\epsilon n)\\\ P_{22}(x_{0}-\epsilon
n)&=&P_{22}(x_{0}+\epsilon n)\\\ P_{23}(x_{0}-\epsilon
n)&=&-P_{23}(x_{0}+\epsilon n)\\\ P_{31}(x_{0}-\epsilon
n)&=&-P_{31}(x_{0}+\epsilon n)\\\ P_{32}(x_{0}-\epsilon
n)&=&-P_{32}(x_{0}+\epsilon n)\\\ P_{33}(x_{0}-\epsilon
n)&=&P_{33}(x_{0}+\epsilon n)\end{array}\right.$

These conditions allow to reconstruct the displacement and microdistorsion
fields in the whole plate, when knowing them in one fourth of the plate. For
consistency reasons, when considering the symmetry planes, these conditions
imply :

$\forall x\in\mathcal{N}\text{, }\left\\{\begin{array}[]{ccl}\langle
u,n\rangle&=&0\\\ \langle P,n\otimes t_{1}\rangle&=&0\\\ \langle P,n\otimes
t_{2}\rangle&=&0\\\ \langle P,t_{1}\otimes n\rangle&=&0\\\ \langle
P,t_{2}\otimes n\rangle&=&0\end{array}\right.\textit{ i.e. }\forall
x\in\mathcal{N},u_{[t_{1},t_{2},n]}(x)=\begin{pmatrix}\star\\\ \star\\\
0\end{pmatrix}\text{ and
}P_{[t_{1},t_{2},n]}(x)=\begin{pmatrix}\star&\star&0\\\ \star&\star&0\\\
0&0&\star\end{pmatrix}$

This can be written, using Einstein’s convention, as

$\displaystyle\forall x\in\mathcal{N},\begin{cases}u_{i}n_{i}=0\\\
(\delta_{ki}-n_{k}n_{i})(P_{ij}n_{j})=0\text{ }\forall
k=\\{1,2,3\\}\end{cases}$

##  Appendix B 2D Stress-free surface boundary conditions

We have, in the frequency domain

$(\tilde{\sigma}+\hat{\sigma})_{\star
1}=\begin{pmatrix}2\mu_{e}(-P_{11}+u_{1,1})+\lambda_{e}(-P_{11}-P_{22}+u_{x}+v_{y})-\omega^{2}(2\bar{\eta}_{1}u_{1,1}+\bar{\eta}_{3}(u_{1,1}+u_{2,2}))\\\
\mu_{\rm
c}(-p_{12}+p_{21}+u_{1,2}-u_{2,1})+\mu_{e}^{*}(-P_{12}-P_{21}+u_{1,2}+u_{2,1})-\omega^{2}(\bar{\eta}_{2}(u_{1,2}-u_{2,1})+\bar{\eta}^{*}_{1}(u_{1,2}+u_{2,1}))\\\
\star\end{pmatrix}$ $(\tilde{\sigma}+\hat{\sigma})_{\star
2}=\begin{pmatrix}\mu_{\rm
c}(P_{12}-P_{21}-u_{1,2}+u_{2,1})+\mu_{e}^{*}(-P_{12}-P_{21}+u_{1,2}+u_{2,1})+\omega^{2}((\bar{\eta}_{2}-\bar{\eta}_{1}^{*})u_{1,2}-(\bar{\eta}_{2}+\bar{\eta}_{1}^{*})u_{2,1})\\\
2\mu_{e}(-P_{22}+u_{2,2})+\lambda_{e}(-P_{11}-P_{22}+u_{1,1}+u_{2,2})-\omega^{2}(2\bar{\eta}_{1}u_{2,2}+\bar{\eta}_{3}(u_{1,1}+u_{2,2}))\\\
\star\end{pmatrix}$

For a vertical stress-free border, the relaxed micromorphic medium verifies

$\displaystyle\begin{cases}2\mu_{e}(-P_{11}+u_{1,1})+\lambda_{e}(-P_{11}-P_{22}+u_{x}+v_{y})-\omega^{2}(2\bar{\eta}_{1}u_{1,1}+\bar{\eta}_{3}(u_{1,1}+u_{2,2}))&=0\\\
\mu_{\rm
c}(-p_{12}+p_{21}+u_{1,2}-u_{2,1})+\mu_{e}^{*}(-P_{12}-P_{21}+u_{1,2}+u_{2,1})-\omega^{2}(\bar{\eta}_{2}(u_{1,2}-u_{2,1})+\bar{\eta}^{*}_{1}(u_{1,2}+u_{2,1}))&=0\end{cases}$

For a horizontal stress-free border, the relaxed micromorphic medium verifies

$\displaystyle\begin{cases}\mu_{\rm
c}(P_{12}-P_{21}-u_{1,2}+u_{2,1})+\mu_{e}^{*}(-P_{12}-P_{21}+u_{1,2}+u_{2,1})+\omega^{2}((\bar{\eta}_{2}-\bar{\eta}_{1}^{*})u_{1,2}-(\bar{\eta}_{2}+\bar{\eta}_{1}^{*})u_{2,1})&=0\\\
2\mu_{e}(-P_{22}+u_{2,2})+\lambda_{e}(-P_{11}-P_{22}+u_{1,1}+u_{2,2})-\omega^{2}(2\bar{\eta}_{1}u_{2,2}+\bar{\eta}_{3}(u_{1,1}+u_{2,2}))&=0\end{cases}$

The continuity of the traction forces at a vertical border is

$\displaystyle\begin{cases}2\mu_{e}(-P_{11}+u_{1,1})+\lambda_{e}(-P_{11}-P_{22}+u_{x}+v_{y})-\omega^{2}(2\bar{\eta}_{1}u_{1,1}+\bar{\eta}_{3}(u_{1,1}+u_{2,2}))=2\mu
u_{1,1}+\lambda(u_{1,1}+u_{2,2})\\\ \mu_{\rm
c}(-p_{12}+p_{21}+u_{1,2}-u_{2,1})+\mu_{e}^{*}(-P_{12}-P_{21}+u_{1,2}+u_{2,1})-\omega^{2}(\bar{\eta}_{2}(u_{1,2}-u_{2,1})+\bar{\eta}^{*}_{1}(u_{1,2}+u_{2,1}))=\mu(u_{1,2}+u_{2,1})\end{cases}$

The continuity of the traction forces at a horizontal border is

$\displaystyle\begin{cases}\mu_{\rm
c}(P_{12}-P_{21}-u_{1,2}+u_{2,1})+\mu_{e}^{*}(-P_{12}-P_{21}+u_{1,2}+u_{2,1})+\omega^{2}((\bar{\eta}_{2}-\bar{\eta}_{1}^{*})u_{1,2}-(\bar{\eta}_{2}+\bar{\eta}_{1}^{*})u_{2,1})=\mu(u_{1,2}+u_{2,1})\\\
2\mu_{e}(-P_{22}+u_{2,2})+\lambda_{e}(-P_{11}-P_{22}+u_{1,1}+u_{2,2})-\omega^{2}(2\bar{\eta}_{1}u_{2,2}+\bar{\eta}_{3}(u_{1,1}+u_{2,2}))=2\mu
u_{2,2}+\lambda(u_{1,1}+u_{2,2})\end{cases}$

##  Appendix C Characteristics of the metamaterials used in the design of the
structure in Fig. 26

We present here the fitting (Fig. 24) for the larger unit cell used in the
metastructure’s design, as well as the relative position of the dispersion
curves for the smaller and larger unit cells (Fig. 25).

Figure 26: (left) Dispersion curves of the microstructured and the relaxed
micromorphic “double cell” along $\Gamma$X (propagation at 0°). (right)
Dispersion curves of the microstructured and the relaxed micromorphic “double
cell” along $\Gamma$M (propagation at 45°). Figure 27: (left) Dispersion
curves for the 4cm unit cell (red) and the 2 cm unit cell (blue) along
$\Gamma$X (propagation at 0°) and (right) along $\Gamma$M (propagation at
45°).