Alexios Aivaliotis
email: alexios.aivaliotis@insa-lyon.fr

Domenico Tallarico
email: domenico.tallarico@empa.ch

Marco Valerio D'agostino
email: marco-valerio.dagostino@insa-lyon.fr

Ali Daouadji
email: ali.daouadji@insa-lyon.fr

Patrizio Neff
email: patrizio.neff@uni-due.de

Angela Madeo
email: angela.madeo@insa-lyon.fr

Marco -Valerio D'agostino

Relaxed micromorphic broadband scattering for finite-size meta-structures -a detailed development

Keywords: enriched continuum mechanics, anisotropic metamaterials, band-gaps, wave-propagation, relaxed micromorphic model, interface, scattering, finite-sized meta-structures AMS 2010 subject classification: 74A30 (nonsimple materials), 74A60 (micromechanical theories), 74B05 (classical linear elasticity), 74J05 (linear waves), 74J10 (bulk waves), 75J15

Introduction

Recent years have seen the rapid development of mechanical metamaterials and phononic crystals showing unorthodox behaviors with respect to elastic wave propagation, including focusing, channeling, cloaking, filtering, etc. [START_REF] Chen | Acoustic cloaking and transformation acoustics[END_REF][START_REF] Craster | Acoustic metamaterials: Negative Refraction, imaging, lensing and cloaking[END_REF][START_REF] Kadic | Metamaterials beyond electromagnetism[END_REF][START_REF] Misseroni | Cymatics for the cloaking of flexural vibrations in a structured plate[END_REF][START_REF] Norris | Acoustic cloacking[END_REF]. The basic idea underlying the design of these metamaterials is that of suitably engineering the architecture of their microstructure in such a way that the resulting macroscopic (homogenized) properties can exhibit the desired exotic characteristics. One of the most impressive features provided by such metamaterials is that of showing band-gaps, i.e. frequency ranges for which wave propagation is inhibited. The most widespread class of metamaterials consists of those which are obtained by a periodic repetition in space of a specific unit cell and which are known as periodic metamaterials. For such metamaterials, renown scientists have provided analytical [START_REF] Willis | Dynamics of composites[END_REF][START_REF] Willis | Exact effective relations for dynamics of a laminated body[END_REF][START_REF] Willis | Effective constitutive relations for waves in composites and metamaterials[END_REF][START_REF] Willis | The construction of effective relations for waves in a composite[END_REF] or numerical [START_REF] Marc | Multi-scale computational homogenization: Trends and challenges[END_REF] homogenization techniques (based on the seminal works of Bloch [START_REF] Bloch | Über die Quantenmechanik der Elektronen in Kristallgittern[END_REF] and Floquet [START_REF] Floquet | Sur les equations differentielles lineaires[END_REF]), allowing to obtain a homogenized model which suitably describes, to a good extent, the dynamical behavior of the bulk periodic metamaterial at the macroscopic scale. Rigorous models for studying the macroscopic mechanical behavior of non-periodic structures become rarer and usually rely on the use of detailed finite element modeling of the considered microstructures (see e.g. [START_REF] Anastasiia | Towards optimal design of locally resonant acoustic metamaterials[END_REF]), thus rendering the implementation of large structures computationally demanding and impractical.

At the current state of knowledge, little effort is made to model large metamaterial structures (which we will call meta-structures), due to the difficulty of imposing suitable boundary conditions in the framework of homogenization theories (see [START_REF] Srivastava | Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach[END_REF]). In this article, we shed new light in this direction by the introduction of an enriched continuum model of the micromorphic type (relaxed micromorphic model), equipped with the proper boundary conditions for the effective description of a 2D finite-sized band-gap meta-structure. The relaxed micromorphic model (see [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF][START_REF] Barbagallo | Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics[END_REF][START_REF] Barbagallo | Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures[END_REF][START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF][START_REF] Madeo | First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model[END_REF][START_REF] Madeo | Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia[END_REF][START_REF] Madeo | On the role of micro-inertia in enriched continuum mechanics[END_REF][START_REF] Madeo | A review on wave propagation modeling in band-gap metamaterials via enriched continuum models[END_REF][START_REF] Madeo | Complete band gaps including non-local effects occur only in the relaxed micromorphic model[END_REF][START_REF] Madeo | Band gaps in the relaxed linear micromorphic continuum[END_REF][START_REF] Madeo | Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps[END_REF][START_REF] Madeo | Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model[END_REF][START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF][START_REF] Neff | A unifying perspective: the relaxed linear micromorphic continuum[END_REF][START_REF] Neff | Real wave propagation in the isotropic-relaxed micromorphic model[END_REF] for preliminary results) has a suitable structure which allows to describe the average properties of (periodic or even non-periodic) anisotropic metamaterials with a limited number of constant material parameters and for an extended frequency range going from the long-wave limit to frequencies which are beyond the first band-gap. The rigorous development of the relaxed micromorphic model for anisotropic metamaterials has been given in [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF], where applications to different classes of symmetry and the particular case of tetragonal periodic metamaterials are also discussed. In [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF], a procedure to univocally determine some of the material parameters for periodic metamaterials with static tests is also provided. It is important to point out that the relaxed micromorphic model is not obtained via a formal homogenization procedure, but is developed generalizing the framework of macroscopic continuum elasticity by introducing enriched kinematics and enhanced constitutive laws for the strain energy density. In this way, extra degrees of freedom are added to the classical macroscopic displacement via the introduction of the micro-distortion tensor and the chosen constitutive form for the anisotropic strain energy density. This allows to introduce a limited number of elastic parameters through fourthorder macro and micro elasticity tensors working on the sym/skew orthogonal decomposition of the introduced deformation measures (see [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF] for details). The need of using an enriched continuum model of the micromorphic type for describing the broadband macroscopic behavior of acoustic metamaterials as emerging from a numerical homogenization technique was recently shown in [START_REF] Sridhar | A general multiscale framework for the emergent effective elastodynamics of metamaterials[END_REF]. Nevertheless, the authors of [START_REF] Sridhar | A general multiscale framework for the emergent effective elastodynamics of metamaterials[END_REF] show that a huge number of elastic parameters (up to 600 for the studied tetragonal two-dimensional structures) is indeed needed to perform an accurate fitting of the dispersion curves issued by the Bloch-Floquet analysis. This extensive number of parameters can also be found in other micromorphic models of the Eringen [START_REF] Cemal | Microcontinuum Field Theories[END_REF] and Mindlin [START_REF] David | Micro-structure in linear elasticity[END_REF] type. The need of this vast number of parameters is related to the fact that the macroscopic class of symmetry of the metamaterial and the correct (i.e. sym/skew-decomposed) deformation measures are usually not accounted for. The fitting of the dispersion curves, which can be obtained by the procedure proposed in [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF], cannot reproduce pointby-point the dispersion curves issued via Bloch-Floquet analysis (which is not the aim of our work), but is general enough to capture the main features of the studied metamaterials' behavior including dispersion, anisotropy, band-gaps for a wide range of frequencies and for wavelengths which can become very small and even comparable to the size of the unit cell. A more refined fitting of the dispersion curves, as well as the possibility of including in the modeling additional symmetries via odd and higher order elasticity tensors, calls for the establishment of a new comprehensive model of the micromorphic type. Given the results presented in this paper for the particular case of tetragonal symmetry, we can claim that the relaxed micromorphic model is an important starting point for the development of such a model.

Stemming from a variational procedure, the relaxed micromorphic model is naturally equipped with the correct macroscopic boundary conditions which have to be applied on the boundaries of the considered metamaterials. This implies that the global refractive properties of metamaterials' boundaries can be described in the simplified framework of enriched continuum mechanics, thus providing important information while keeping simple enough to allow important computational time-saving. Very complex phenomena take place when an incident elastic wave hits a metamaterial's boundary, resulting in reflected and transmitted waves which can be propagative or evanescent depending on the frequency and angle of incidence of the incident wave itself. The primordial importance of evanescent (non-propagative) waves for the correct formulation of boundary value problems for metamaterials has been highlighted in [START_REF] Srivastava | Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach[END_REF][START_REF] Willis | Negative refraction in a laminate[END_REF], where the need of infinite evanescent modes for obtaining continuity of displacement and of tractions at the considered metamaterial's boundary is pointed out. One of the advantages of the enriched continuum model proposed in the present paper is that the generalized boundary conditions (continuity of macroscopic displacement and of generalized traction) associated to the considered boundary value problem are exactly fulfilled with a finite number of modes.

We will show that the proposed framework allows to describe, to a good extent, the overall behavior of the reflection and transmission coefficients (generated by a plane incident wave) of a metamaterial slab of finite width treated as an inclusion between two semi-infinite homogeneous media. We obtain the scattering properties of the slab as a function of the frequency and angle of incidence of the plane incident wave. To the authors' knowledge, a boundary value problem which describes the broadband and dynamical behavior of realistic finite-size metamaterial structures via the introduction of enriched macroscopic broad-angle boundary conditions, is presented here for the first time.

The results show very good agreement (for a wide range of frequencies extending from the lowfrequency Cauchy limit to frequencies beyond the first band-gap and for all the possible angles of incidence) with the direct FEM numerical implementation of the considered system in COMSOL, where the detailed geometry of the unit cell has been implemented in the framework of classical linear elasticity. We observe a tremendous advantage in terms of the computational time needed to perform the numerical simulations (few hours for the relaxed micromorphic model against weeks for the direct FEM simulation).

The paper is structured as follows: In Section 1 we present an introduction together with the notation used throughout the paper. Section 2 briefly recalls the governing equations describing the motion of Cauchy and relaxed micromorphic continua, with specific reference to the definition of the energy flux for both cases. In Section 3 the plane-wave solutions for the Cauchy and relaxed micromorphic continua are obtained as solutions of the corresponding eigenvalue problems. In Section 4 we provide essential information concerning the correct boundary conditions which have to be imposed at the metamaterial's boundaries, in the relaxed micromorphic framework. In Sections 5 and 6 the problems of the scattering from a relaxed micromorphic single interface and relaxed micromorphic slab of finite size, respectively, are rigorously set up and solved. Section 7 presents the detailed implementation of the microstructured metamaterial slab based on classical linear elasticity and implemented in the commercial Finite Element software COMSOL Multiphysics. Section 8 thoroughly presents our results by means of a detailed discussion. Section 9 is devoted to conclusions and perspectives.

Notation

Let R 3×3 be the set of all real 3×3 second order tensors which we denote by capital letters. A simple and a double contraction between tensors of any suitable order is denoted by • and : respectively, while the scalar product of tensors of suitable order is denoted by •, • . 1 The Einstein sum convention is implied throughout this text unless otherwise specified. The standard Euclidean scalar product on R 3×3 is given by X, Y = tr(X • Y T ) and consequently the Frobenius tensor norm is X2 = X, X . The identity tensor on R 3×3 will be denoted by 1; then, tr(X) = X, 1 .

We denote by B L a bounded domain in R 3 , by ∂B L its regular boundary and by Σ any material surface embedded in B L . The outward unit normal to ∂B L will be denoted by ν as will the outward unit normal to a surface Σ embedded in B L . Given a field a defined on the surface Σ, we define the jump of a through the surface Σ as:

[[a]] = a + -a -, with a -:= lim x∈B - L \Σ x→Σ a,
and a + := lim

x∈B + L \Σ x→Σ a, (1.1)
where B - L , B + L are the two subdomains which result from splitting B L by the surface Σ. Classical gradient ∇ and divergence Div operators are used throughout the paper. 2 Moreover, we introduce the Curl operator of the matrix P as (Curl P ) ij = jmn P in,m , where jmn denotes the standard Levi-Civita tensor, which is equal to +1, if (j, m, n) is an even permutation of (1, 2, 3), to -1, if (j, m, n) is an odd permutation of (1, 2, 3), or to 0 if j = m, or m = n, or n = j. The subscript , j indicates derivation with respect to the j-th component of the space variable, while the subscript , t denotes derivation with respect to time.

Given a time interval [0, T ], the classical macroscopic displacement field is denoted by:

u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) T , x ∈ B L , t ∈ [0, T ]. (1.2)
In the framework of enriched continuum models of the micromorphic type, extra degrees of freedom are added through the introduction of the micro-distortion tensor P denoted by:

P (x, t) =      P 11 (x,
     , x ∈ B L , t ∈ [0, T ]. (1.3)
2 Governing equations and energy flux

The classical isotropic Cauchy continuum

The equations of motions in strong form for a classical Cauchy continuum are:

ρ u ,tt = Div σ, ρ u i,tt = σ ij,j , (2.1)
where σ = 2µ sym ∇u + λ tr(sym ∇u)1,

σ ij = µ(u i,j + u j,i ) + λu k,k δ ij , (2.2)
is the classical symmetric Cauchy stress tensor for isotropic materials and µ and λ are the classical Lamé parameters.

The mechanical system we are considering is conservative and, therefore, the energy must be conserved in the sense that the following differential form of a continuity equation must hold:

E ,t + divH = 0, (2.3)
where E is the total energy of the system and H is the energy flux vector, whose explicit expression is given by (see e.g. [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF] for a detailed derivation)

H = -σ • u ,t , H i = -σ ik u k,t . (2.4)

The anisotropic relaxed micromorphic model

The kinetic energy density of the anisotropic relaxed micromorphic model considered here is [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]:

J(u ,
where u is the macroscopic displacement field, P ∈ R 3×3 is the non-symmetric micro-distortion tensor and ρ is the apparent macroscopic density. Moreover, J micro , J c , T e and T c are 4th order inertia tensors, whose precise form will be specified in the following. An existence and uniqueness result for models with these types of generalized inertia terms is given in [START_REF] Owczarek | Nonstandard micro-inertia terms in the relaxed micromorphic model: well-posedness for dynamics[END_REF]. The strain energy density for an anisotropic relaxed micromorphic medium is given by [10, 28]:

where C e , C micro , C c , L e and L c are 4th order elasticity tensors, whose precise form will be given in the following and µ macro is a scalar stiffness weighting the curvature term. Moreover, L c is a characteristic length, which may account for size-effects in the metamaterial. These size-effects are known to be important for metamaterials in the static regime (see e.g. [START_REF] Rokoš | Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields[END_REF]), but are not usually explored in dynamics.

In this paper, we start exploring the role which L c may play in dynamical size-effects. Nevertheless, these results are preliminary and new modeling tools will need to be developed in order to comprehensively unveil non-local effects arising in finite-sized metamaterials.

Let t 0 > 0 be a fixed time and consider a bounded domain B L ⊂ R 3 . The action functional of the system at hand is defined as

A = t0 0 B L (J -W )dXdt, (2.7)
where J is the kinetic and W the strain energy of the system, defined by (2.5) and (2.6), respectively. The minimization of the action functional provides the governing equations for the anisotropic relaxed micromorphic model [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]:

ρ u ,tt -Div σ ,tt = Div σ,

J micro sym P ,tt = σ e -s -sym Curl m, J c skew P ,tt = σ c -skew Curl m, (2.8)
where we set (2.9)

Conservation of energy for an anisotropic relaxed micromorphic continuum is formally written as in equation (2.3). The specific form for the energy flux for an anisotropic relaxed micromorphic continuum is (see Appendix A.1.1 for detailed derivation of this expression):

H = -( σ + σ) T • u ,t -m T • P ,t : , H k = -u i,t ( σ ik + σ ik ) -m ih P ij,t jhk .
(2.10)

The plane-strain tetragonal symmetry case

We are interested in plane-strain solutions of the system (2.8). This means that the macroscopic displacement u and the micro-distortion tensor P introduced in (1.2) and (1.3) are supposed to take the following form:

u = u(x 1 , x 2 ) = (u 1 (x 1 , x 2 ), u 2 (x 1 , x 2 ), 0) T , and u i,3 = 0, i = 1, 2, (2.11)
and

P = P (x 1 , x 2 ) =      P 11 (x 1 , x 2 ) P 12 (x 1 , x 2 ) 0 P 21 (x 1 , x 2 ) P 22 (x 1 , x 2 ) 0 0 0 0     
, and P ij,3 = 0, i = 1, 2, j = 1, 2.

(2.12)

Following [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF], we denote the second order constitutive tensors in Voigt notation corresponding to the fourth order ones appearing in the kinetic and strain energy expressions (2.5) and (2.6) by a tilde. 4For the tetragonal case, the constitutive tensors in Voigt notation are (see [START_REF] Barbagallo | Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics[END_REF][START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]):

Ce =               2 µe + λe λe 0 0 0 λe 2 µe + λe 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ * e               , Cmicro =               2 µmicro + λmicro λmicro 0 0 0 λmicro 2 µmicro + λmicro 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ * micro               , Cc =      0 0 0 0 0 0 4 µc      , Jmicro =               2 η1 + η3 η3 0 0 0 η3 2 η1 + η3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 η * 1               , Jc =      0 0 0 0 0 0 4 η2      , Te =               2 η1 + η3 η3 0 0 0 η3 2 η1 + η3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 η * 1               , Tc =      0 0 0 0 0 0 4 η2      ,
where we denoted by a star those components, which work on out-of-plane macro-and micro-strains and do not play any role in the considered plane-strain problem.

3 Bulk wave propagation in Cauchy and relaxed micromorphic continua

Isotropic Cauchy continuum

We make the plane-wave ansatz for the solution to (2.1):

u(x 1 , x 2 , t) = ψ e i( k,x -ωt) = ψ e i(k1x1+k2 x2-ωt) , ψ ∈ C 2 , (3.1)
where k = (k 1 , k 2 ) T is the wave vector and ω is the angular frequency.5 Plugging (3.1) into equation (2.1) we get a 2 × 2 algebraic system of the form

A • ψ = 0, (3.2)
where

A =   ρ ω 2 -(2µ + λ) k 2 1 -µ k 2 2 -(µ + λ) k 1 k 2 -(µ + λ) k 1 k 2 ρ ω 2 -(2µ + λ) k 2 2 -µ k 2 1   . (3.3)
The algebraic system (3.2) has a solution if and only if det A = 0. This is a bi-quadratic polynomial equation which has the following four solutions:

k L,r 1 = - ρ 2µ + λ ω 2 -(k L,r 2 ) 2 , k S,r 1 = - ρ µ ω 2 -(k S,r 2 ) 2 , k L,t 1 = ρ 2µ + λ ω 2 -(k L,t 2 ) 2 , k S,t 1 = ρ µ ω 2 -(k S,t 2 ) 2 , (3.4)
where we denote by L and S the longitudinal and shear waves and by r and t whether they travel towards -∞ or +∞, respectively. We plug the solutions (3.4) into (3.2) to calculate the corresponding eigenvectors

ψ L,r =   1 - k L,r 2 k L,r 1   , ψ S,r =   1 k S,r 1 k S,r 2   , ψ L,t =   1 k L,t 2 k L,t 1   , ψ S,t =   1 - k S,t 1 k S,t 2   . (3.5)
Normalizing these eigenvectors gives:

ψ L,r := 1 ψ L,r ψ L,r , ψ S,r := 1 ψ S,r ψ S,r , ψ L,t := 1 ψ L,t ψ L,t , ψ S,t := 1 ψ S,t ψ S,t . (3.6)
Then, the general solution to (2.1) can be written as:

u(x1, x2, t) = a L,r ψ L,r e i(k L,r 1 x 1 +k L,r 2
x 2 -ωt) + a S,r ψ S,r e i(k S,r

1 x 1 +k S,r 2 x 2 -ωt) + a L,t ψ L,t e i(k L,t 1 x 1 +k L,t 2
x 2 -ωt) + a S,t ψ S,t e i(k S,t

1 x 1 +k S,t 2 x 2 -ωt) , (3.7)
where a L,r , a S,r , a L,t , a S,t ∈ C are constants to be determined from the boundary conditions, k L,t

1 = -k L,r 1
and k S,t 1 = -k S,r 1 , according to (3.4). Depending on the specific problems which are considered (e.g. semiinfinite media), only some modes may propagate in specific directions. In this case, some of the terms in the sum (3.7) have to be omitted. In particular, if we are considering waves propagating in the x 1 < 0 half-space, then the solution to (2.1) reduces to:

6 u(x1, x2, t) = a L,r ψ L,r e i(k L,r 1 x 1 +k L,r 2 x 2 -ωt) + a S,r ψ S,r e i(k S,r 1 x 1 +k S,r 2 x 2 -ωt) , (3.8)
while if we are considering the x 1 > 0 half-space, then the solution to (2.1) reduces to:

u(x1, x2, t) = a L,t ψ L,t e i(k L,t 1 x 1 +k L,t 2
x 2 -ωt) + a S,t ψ S,t e i(k S,t

1 x 1 +k S,t 2
x 2 -ωt) .

(3.9)

Relaxed micromorphic continuum

We start by collecting the unknown fields for the plane-strain case in a new variable: The plane-wave ansatz for this unknown field reads:

v := (u 1 , u
v = φ e i( k,x -ωt) = φ e i(k1 x1+k2x2-ω t) , (3.11)
where φ ∈ C6 is the vector of amplitudes, k = (k 1 , k 2 ) T ∈ C 2 is the wave-vector 7 and ω is the angular frequency. We plug the wave-form (3.11) into (2.8) and get an algebraic system of the form

A(k 1 , k 2 , ω) • φ = 0, (3.12)
where

A(k 1 , k 2 , ω) ∈ C 6×6 is a matrix depending on k 1 , k 2
, ω and all the material parameters of the plane-strain tetragonal relaxed micromorphic model (see A.2 for an explicit presentation of this matrix).

In order for this system to have a solution other than the trivial one, we impose det A = 0. The equation det A = 0 is a polynomial of order 12 in ω and it involves only even powers of ω. This means that, plotting the roots ω = ω(k) gives 6 dispersion curves in the ω -k plane (see Fig. 3). On the other hand, the same polynomial is of order 8 (and again involves only even powers of k 1 ), if regarded as a polynomial of k 1 (k 2 is supposed to be known when imposing boundary conditions). We can write the roots of the characteristic polynomial as:

k (1) 1 (k2, ω), k (2) 1 (k2, ω), k (3) 1 (k2, ω), k (4) 1 (k2, ω), (3.13)
k (5) 1 (k2, ω) = -k (1) 1 (k2, ω), k (6)
1 (k2, ω) = -k (3) 1 (k2, ω), k (7)
1 (k2, ω) = -k

(3) 1 (k2, ω), k (8)
1 (k2, ω) = -k (4)
1 (k2, ω).

We plug the solutions (3.13) into (3.12) and calculate the eigenvectors of A, which we denote by: φ (1) , φ (2) , φ (3) , φ (4) , φ (5) , φ (6) , φ (7) , φ (8) . We normalize these eigenvectors, thus introducing the normal vectors

φ (1) := 1 | φ (1) | φ (1) , φ (2) := 1 | φ (2) | φ (2) , φ (3) := 1 | φ (3) | φ (3) , φ (4) := 1 | φ (4) | φ (4) , φ (5)
:= 1

| φ (5) | φ (5) , φ (6) := 1

| φ (6) | φ (6) , φ (7) := 1

| φ (7) | φ (7) , φ (8) := 1

| φ (8) | φ (8) .

(3.14)
Considering a micromorphic medium in which all waves travel simultaneously, the solution to (2.8) is:

v(x 1 , x 2 , t) = 8 j=1 α j φ (j) e i k (j) 1 x1+k (j) 2 x2-ωt , (3.15)
where α j ∈ C are unknown constants which will be determined from the boundary conditions. If, on the basis of the particular interface problem one wants to study, only some waves travel in the considered medium, then the extra waves must be omitted from the sum in (3.15). This means that if we are considering waves traveling in the x 1 > 0 direction, then the solution to (2.8) is given by

v(x1, x2, t) = α1 φ (1) e i k (1) 1 x 1 +k (1) 2 x 2 -ωt + α2 φ (2) e i k (2) 1 x 1 +k (2) 2 x 2 -ωt + α3 φ (3) e i k (3) 1 x 1 +k (3) 2 x 2 -ωt + α4 φ (4) e i k (4) 1 x 1 +k (4) 2 x 2 -ωt , (3.16)
while if we consider waves only traveling in the x 1 < 0 direction, the solution to (2.8) is given by

v(x1, x2, t) = α5 φ (5) e i k (5) 1 x 1 +k (5) 2 x 2 -ωt + α6 φ (6) e i k (6) 1 x 1 +k (6) 2 x 2 -ωt + α7 φ (7) e i k (7) 1 x 1 +k (7) 2 x 2 -ωt + α8 φ (8) e i k (8) 1 x 1 +k (8) 2 x 2 -ωt . (3.17)

Boundary Conditions

We will consider two types of interface problems: (i) a single interface separating a Cauchy and a relaxed micromorphic medium, both assumed to be semi-infinite and (ii) a micromorphic slab of finite size embedded between two semi-infinite Cauchy media. In the following, we will simply denote "single interface" and "micromorphic slab" the first and second problem, respectively.

In the single interface problem, two homogeneous infinite half-spaces are occupied by two materials in perfect contact with each other. The material on the left of the interface is an isotropic classical Cauchy medium, while the material on the right is a microstructured tetragonal metamaterial modeled by the tetragonal relaxed micromorphic model (see Fig 1(a)).

In the micromorphic slab problem, two homgeneous infinite half-spaces are separated by a micromorphic slab of finite width h. Three materials are thus in perfect contact with each other: the material on the left of the first interface is a classical linear elastic isotropic Cauchy medium, the material in the middle is an anisotropic relaxed micromorphic medium, while the material on the right of the second interface is again a classical isotropic Cauchy medium (see Fig. 1(b)).

Boundary conditions at an interface between a Cauchy continuum and a relaxed micromorphic continuum

As is well established in previous works [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF][START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF][START_REF] Madeo | Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model[END_REF] there are four boundary conditions which can be imposed at a Cauchy/relaxed micromorphic interface. These are continuity of displacement, continuity of generalized traction and an extra set of boundary conditions which involve only the micro-distortion tensor P . These additional boundary conditions stem from the least action principle, and are called free and fixed microstructure boundary conditions [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF], depending on whether we assign the dual quantity to the micro-distortion, or the micro-distortion itself.

For the displacement, we have:

[[u]] = 0 ⇒ u -= u + , on x 1 = 0, (4.1)
where u -is the macroscopic displacement on the "minus" side (the x 1 < 0 half-plane, occupied by an isotropic Cauchy medium) and u + is the macroscopic displacement on the "plus" side (the x 1 > 0 half-plane, occupied by an anisotropic relaxed micromorphic medium: the first two components of the vector v, defined in (3.10)). As for the jump of generalized traction we have:

t = t, (4.2)
where t is the Cauchy traction on the "minus" side and t is the generalized traction on the "plus" side. We recall that in a Cauchy medium, t = σ • ν, ν being the outward unit normal to the surface and σ being the Cauchy stress tensor given by (2.2). The generalized traction for the relaxed micromorphic medium is given by

t = ( σ + σ) • ν, t i = ( σ ij + σ ij ) • ν j , on x 1 = 0, (4.3)
where σ, σ are defined in (2.9). In order to impose suitable boundary conditions for the micro-distortion tensor P , we need the concept of the double traction τ which is the dual quantity of the micro-distortion tensor P and is defined as [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF][START_REF] Madeo | Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model[END_REF]:

τ = -m • • ν, τ ij = -m ik kjh ν h (4.4)
where m is given in (2.9), is the Levi-Civita tensor and ν is the outward unit normal to the interface.

Free microstructure

In this case, the macroscopic displacement and generalized tractions are continuous while the microstructure of the medium is free to move along the interface [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF][START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF][START_REF] Madeo | First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model[END_REF][START_REF] Madeo | Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model[END_REF]. Leaving the interface free to move means that P is arbitrary, which, on the other hand, implies that the double traction τ must vanish. We have then:

[[u i ]] = 0, t i -t i = 0, τ ij = 0, i = 1, 2, j = 2. (4.5)

Fixed microstructure

This is the case in which we impose that the microstructure on the relaxed micromorphic side does not vibrate at the interface. The boundary conditions in this case are [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF][START_REF] Madeo | Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model[END_REF]:

8 [[u i ]] = 0, t i -t i = 0, P ij = 0, i = 1, 2, j = 2. (4.6)

Continuity of macroscopic displacement and of generalized traction implies conservation of energy at the interface

We have previously shown that conservation of energy for a bulk Cauchy and relaxed micromorphic medium is given by equation (2.3), where the energy flux is defined in (2.4) and (2.10), respectively. It is important to remark that the conservation of energy (2.3) has a "boundary counterpart". This establishes that the jump of the normal part of the flux must be vanishing, or, in other words, the normal part of the flux must be continuous at the considered interface (this comes from the bulk conservation law and the use of the Gauss divergence theorem). In symbols, when considering a surface Σ separating two continuous media, we have

[[H • ν]] = 0, on Σ. (4.7)
We want to focus the reader's attention on the fact that, in the framework of a consistent theory in which the bulk equations and boundary conditions are simultaneously derived by means of a variational principle, the jump conditions imposed on Σ necessarily imply the surface conservation of energy (4.7), as far as a conservative system is considered. We explicitly show here that this is true for an interface Σ separating a Cauchy medium from a relaxed micromorphic one. The same arguments, however, hold for interfaces between two Cauchy or two relaxed micromorphic media.

To that end, considering for simplicity that the interface Σ is located at x 1 = 0 (so that its normal is ν = (1, 0) T ) we get from equation (2.10) that the normal flux computed on the "relaxed micromorphic" side is given by:

(H • ν) + := H + 1 = -ν + • ( σ + σ) T • u + ,t -m T • P ,t : at x 1 = 0. (4.8)
By the same reasoning, the flux at the interface on the "Cauchy" side is computed from (2.4) and gives:

(

H • ν) -:= H - 1 = -ν -• σ • u - ,t . (4.9)
Equation (4.7) can then be rewritten as:

-ν + • ( σ + σ) T • u + ,t -m T • P ,t : = -ν -• σ • u - ,t . (4.10)
It is clear that, given the jump conditions (4.5) or (4.6) the latter relation is automatically verified. As we will show in the remainder of this paper, when modeling a metamaterial's boundary via the relaxed micromorphic model, we only need a finite number of modes in order to exactly verify boundary conditions and, consequently, surface energy conservation. This provides one of the most powerful simplifications of the relaxed micromorphic model with respect to classical homogenization methods, in which infinite modes are needed to satisfy conservation of stress and displacement at the metamaterial's boundary (see [START_REF] Willis | Effective constitutive relations for waves in composites and metamaterials[END_REF]).

Boundary conditions for a micromorphic slab embedded between two Cauchy media

The boundary conditions to be satisfied at the two interfaces separating the slab from the two Cauchy media in Fig. 1(b), are continuity of displacement, continuity of generalized traction and a condition on the micro-distortion tensor P (free or fixed microstructure). This means that we have twelve sets of scalar boundary conditions, six on each interface. The finite slab has width h and we assume that the two interfaces are positioned at x 1 = -h/2 and x 1 = h/2, respectively. The continuity of displacement conditions to be satisfied at the two interfaces of the slab are:9

u - i = v i , on x 1 = - h 2 , v i = u + i , on x 1 = h 2 , i = 1, 2. (4.11)
As for the continuity of generalized traction, we have:

t - i = t i , on x 1 = - h 2 , t i = t + i , on x 1 = h 2 i = 1, 2, (4.12)
where t ± = σ ± • ν ± are classical Cauchy tractions and t is again given by (4.3). Furthermore, we need the additional free or fixed microstructure boundary conditions to be satisfied at both interfaces. These are given by:

τ ij = 0, on x 1 = - h 2 , τ ij = 0, on x 1 = h 2 , i = 1, 2, j = 2, (4.13)
or

P ij = 0, on x 1 = - h 2 , P ij = 0, on x 1 = h 2 , i = 1, 2, j = 2, (4.14)
were the expression for the double traction τ is given in (4.4). We remark once again, that only the tangential parts of the double traction τ or the micro-distortion tensor P must be assigned in order to fulfill conservation of energy.

Reflection and transmission at the single Cauchy/relaxed micromorphic interface

In this section, we study the two-dimensional, plane-strain, time-harmonic scattering problem from an anisotropic micromorphic half-space (see equations (2.8)), occupying the region x 1 > 0 of Fig. 1(a). With reference to Fig. 1(a), the half-space x 1 < 0 is filled with a linear elastic Cauchy continuum, governed by equation (2.1). Considering that reflected waves only travel in the x 1 < 0 Cauchy halfplane, only negative solutions for the k 1 's must be kept in equation (3.4), so that the total solution in the left half-space is u -(x1, x2, t) = a L/S,i ψ L/S,i e i( x,k L/S,i -ωt) + a L,r ψ L,r e i( x,k L,r -ωt) + a S,r ψ S,r e i( x,k S,r -ωt)

=: u L/S,i + u L,r + u S,r , (

where we write L or S in the incident wave depending on whether the wave is longitudinal or shear and i and r in the exponents stand for "incident" and "reflected". Analogously, the solution on the right half-space, which is occupied by a relaxed micromorphic medium, is10 v(x1, x2, t) = α1 φ (1) e i k

(1)

1 x 1 +k (1) 2 x 2 -ωt + α2 φ (2) e i k (2) 1 x 1 +k (2) 2 x 2 -ωt + α3 φ (3) e i k (3) 1 x 1 +k (3) 2 x 2 -ωt + α4 φ (4) e i k (4) 1 x 1 +k (4)
2 x 2 -ωt + α5 φ (5) e i k

(5)

1 x 1 +k (5) 2 x 2 -ωt , (5.2)
where we have kept only terms with positive k 1 's in the solution (3.13), since transmitted waves are supposed to propagate in the x 1 > 0 half-plane.

Since the incident wave is always propagative, the polarization and wave-vectors are given by:

ψ L,i = (sin θ L , -cos θ L ) T , k L = |k L |(sin θ L , -cos θ L ) T , (5.3)
ψ S,i = (cos θ S , sin θ S ) T , k S = |k S |(sin θ S , -cos θ S ) T , (5.4)
where, according to (3.4), |k L | = ω c L and |k S | = ω c S , with c L = (2µ + λ)/ρ and c S = µ/ρ the longitudinal and shear speeds of propagation and θ L and θ S the angles of incidence when the wave is longitudinal or shear, respectively (see Fig. 1 and [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF] for a more detailed exposition).

The continuity of displacement condition (4.1) provides us with the generalized Snell's law for the case of a Cauchy/relaxed micromorphic interface (see [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF] for a detailed derivation):

k L/S,i 2 = k L,r 2 = k S,r 2 = k (1) 2 = k (2) 2 = k (3) 2 = k (4) 2 = k (5)
2 . Generalized Snell's Law (5.5) As for the flux, the unit outward normal vector to the surface (the x 2 axis) is ν = (-1, 0). This means that in expressions (2.4) and (2.10) for the fluxes, we need only take into account the first component. According to our definitions (2.4) and (2.10), we have:

H - 1 = -u i,t σ i1 , H + 1 = -v i,t ( σ i1 + σ i1 ) -m ih P ij,t jh1 , i = 1, 2, j, k = 2, 3. (5.6)
Having calculated the "transmitted" flux, we can now look at the reflection and transmission coefficients for the case of a Cauchy/relaxed micromorphic interface. We define11

J i = 1 T T 0 H i (x, t)dt, J r = 1 T T 0 H r (x, t)dt, J t = 1 T T 0 H t (x, t)dt, (5.7)
where T is the time period of the considered harmonic waves,

H i = H - 1 (u L/S,i ), H r = H - 1 (u L,r + u S,r
) and H t = H + 1 (v), with H + 1 and H - 1 defined in (5.6). Then the reflection and transmission coefficients are

R = J r J i , T = J t J i .
(5.8)

Since the system is conservative, we must have that R + T = 1.

Reflection and transmission at a relaxed micromorphic slab

As pointed out, in this case there are three media: the first Cauchy half-space, the anisotropic relaxed micromorphic slab and the second Cauchy half-space. The two Cauchy half-spaces are denoted byand +, while the quantities considered in the slab have their own notation. The solution on the first Cauchy half space is given, as in the case of a single interface, by u -(x 1 , x 2 , t) = a L/S,i ψ L/S,i e i( x,k L/S,i -ωt) + a L,r ψ L,,r e i( x,k L,,r -ωt) + a S,r ψ S,r e i( x,k S,r -ωt) . (6.1)

In the case of a relaxed micromorphic slab, when solving the eigenvalue problem we must select and keep all the roots for k 1 , as given in (3.13), both positive and negative. This is due to the fact that there are waves which transmit in the micromorphic part from the first interface (x 1 = -h/2), upon which the incident wave hits and waves which reflect on the second interface (x 1 = h/2). This means that the solution of the PDEs in the slab is v(x1, x2, t) = α1 φ (1) e i( x,k (1) -ωt) + α2 φ (2) e i( x,k (2) -ωt) + α3 φ (3) e i( x,k (3) -ωt) + α4 φ (4) e i( x,k (4) -ωt) + α5 φ (5) e i( x,k (5) -ωt) + α6 φ (6) e i( x,k (6) -ωt) + α7 φ (7) e i( x,k (7) -ωt) + α8 φ (8) e i( x,k (8) -ωt) .

(6.2)

Finally, the solution on the right Cauchy half-space is u + (x 1 , x 2 , t) = a L,t ψ L,t e i( x,k L,t -ωt) + a S,t ψ S,t e i( x,k S,t -ωt) . (6.

3)

The continuity of displacement conditions (4.11) again imply a generalized form of Snell's law for the case of the micromorphic slab (see [START_REF] Aivaliotis | Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxedmicromorphic interface[END_REF]):

k L/S,i 2 = k L,r 2 = k S,r 2 = k (1) 2 = k (2) 2 = k (3) 2 = k (4) 2 = k (5) 2 = k (6) 2 = k (7) 2 = k (8) 2 = k L,t 2 = k S,t
2 . Generalized Snell's law in a micromorphic slab (6.4) In order to define the reflection and transmission coefficients in the case of the anisotropic slab, we follow the same reasoning as for the single interface. However, in this case, the transmitted flux is defined as the flux on the right of the second interface, which is occupied by an isotropic Cauchy medium.

In this case, the reflected flux is evaluated at x 1 = -h/2 and the transmitted flux at x 1 = h/2. Both the reflected and the transmitted fields propagate in isotropic Cauchy media, so that the quantities J i , J r , J t defined in (5.7) are given by (see [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF])

J i slab = 1 2 Re (2µ -+ λ -) ψ L/S,i 1 2 k L/S,i 1 + λ -ψ L/S,i 1 * ψ L/S,i 2 k L/S,i 2 + µ -ψ i 1 ψ L/S,i 2 * k L/S,i 2 + ψ L/S,i 2 2 k L/S,i 1 a L/S,i 2 ω , (6.5)
J r slab = j∈J 1 2 Re (2µ -+ λ -) ψ j,r 1 2 k j,r 1 + λ -ψ j,r 1 * ψ j,r 2 k j,i 2 + µ -ψ j,r 1 ψ j,(2) 2 * k j,i 2 ψ j,r 2 2
k j,r 1 a j,r 2 ω , (6.6) where, J = {L, S}, a j,i , a j,r , a j,t ∈ C and ψ j,i , ψ j, (2) , ψ j,t ∈ C 2 with j ∈ J , are the amplitudes and polarization vectors for incident, reflected and transmitted waves, respectively (see also equation (3.6)), µ -, λ -and µ + , λ + are the Lamé parameters of the left and right Cauchy half-spaces, respectively. The reflection and transmission coefficients for the slab can then be defined as:

J t slab = j∈J 1 2 Re (2µ + + λ + ) ψ j,t 1 2 k j,t 1 + λ + ψ j,t 1 * ψ j,t 2 k j,i 2 + µ + ψ j,t 1 ψ j,t 2 * k j,i 2 + ψ j,t
R slab = J r slab J i slab , T slab = J t slab J i slab . (6.8)
Since the system is conservative, we must have R slab + T slab = 1.

Reflective properties of the detailed micro-structured slab

In this section, we consider the scattering of in-plane elastic waves from a slab containing cross like holes drilled in an isotropic linear elastic material (see Fig. 2(a)). The holes in the micro-structured slab are arranged according to a truncated square lattice, i.e. a finite number N of cells in the x 1 direction and an infinite number of cells in the x 2 direction. The details of the unit cell are given in Fig. 2 We consider an incident time-harmonic plane wave:

u j,i (x, t) = ūj,i (x)e -iωt = d j e i k j ,x -iωt , (7.1)
where the index j ∈ {L, S} denotes longitudinal and shear waves, respectively. Accordingly, k j = ω/c j (sin θ, -cos θ, 0) T , with j ∈ {L, S} and θ the angle of incidence, c L = (λ Al + 2µ Al )/ρ Al and c S = µ Al /ρ Al are the longitudinal and shear wave speeds for aluminum. The Lamé parameters λ Al , µ Al uniquely define the fourth order stiffness tensor C Al , whose Voigt representation is

CAl =               2µ Al + λ Al λ Al 0 0 0 λ Al 2µ Al + λ Al 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ Al               , (7.2)
where the stars denote the components which do not intervene in the plane-strain case. In equation ( 7.1) we have introduced the polarization vectors d j , j ∈ {L, S} of amplitude d 0 for longitudinal and shear waves, defined as d L = d 0 (sin θ, -cos θ, 0) T and d S = d 0 (cos θ, sin θ, 0) T , respectively. The scattering problem in terms of the displacement field u j ≡ u j (x, t), j ∈ {L, S}, in the micro-structured material, according isotropic to linear elasticity, can be written as:

12      ρ Al u j ,tt = Div C Al sym ∇u j , x ∈ Ω 0 (n 1 , n 2 ) f (u j ) = 0, x ∈ ∂ (-) Ω(n 1 , n 2 ), ∀n 1 ∈ {1, • • • , N } and ∀n 2 ∈ Z, (7.3)
where we have introduced the traction vectors

f (u j ) = (C Al sym ∇u j ) • ν, j ∈ {L, S}, (7.4)
ν being the normal unit vector (see black arrow line in Fig. 2(b)) to the cross-like holes boundaries ∂ (-) Ω(n 1 , n 2 ) and where we denote by Ω 0 the part of the domain Ω, which is non-empty (occupied by aluminum). The elastic field in (7.3) can be written according to the scattering time-harmonic ansatz13 u j (x, t) = ūj,i (x) + ūsc (x) e -iωt , j ∈ {L, S}, (7.5) where ūj,i has been introduced in equation ( 7.1) and ūsc is the so-called scattered solution, which typically contains coupled L and S scattered contributions. By linearity of the traction vector (7.4), and using equation (7.5), we obtain

f (u j (x, t)) = f (ū j,i (x)) + f (ū sc (x)) e -iωt = 0, j ∈ {L, S}. (7.6)
Using the fact that u j,i (x, t) is a solution of the PDE in equation ( 7.3), together with equation (7.6), the PDEs system (7.3) can be rewritten in a time-harmonic form with respect to the field u sc (x), as:

     -ω 2 ρ Al ūsc = Div (C Al sym ∇ū sc ) , x ∈ Ω0 f (ū sc ) ≡ (C Al sym ∇ū sc ) • ν = -f (ū j,i ), x ∈ ∂ (-) Ω(n1, n2), ∀n1 ∈ {1, • • • , N } and ∀n2 ∈ Z, (7.7)
with j ∈ {L, S} and where we have canceled out time-harmonic factors. The analytical expressions for the boundary conditions for the scattered field (see right-hand side of the boundary conditions in equation (7.7)) are given by:

f (ū sc ) = f j v (x) := (C Al sym ∇ū j,i ) • e1 = iωρ cj c 2 L sin θ d j,i 1 -(c 2 L -2c 2 S ) cos θ d j,i 2 e1 + c 2 S -cos θ d j,i 1 + sin θ d j,i
2 e2 e i k j,i ,x , (7.8)

with j ∈ {L, S} for vertical boundaries of ∂ (-) Ω(n 1 , n 2 ) with normal vector parallel to e 1 (see Fig. 2(b)).

Similarly, for vertical boundaries with normal vector anti-parallel to e 1 we have f (ū j,i ) = -f j v (x). In addition, for horizontal boundaries in ∂ (-) Ω(n 1 , n 2 ) whose normal vector is parallel to e 2 we have:

f (ū sc ) = f j h (x) := (C Al sym ∇ū j,i ) • e2 = iωρ cj (c 2 L -2c 2 S ) sin θ 15d j,i 1 -c 2 L cos θ d j,i 2 e2 + c 2 S -cos θ d j,i 1 + sin θ d j,i
2 e1 e i k j,i ,x , (7.9)

with j ∈ {L, S}. Similarly, for vertical boundaries with normal vector anti-parallel to e 2 we have f (ū sc ) = -f j h (x).

Bloch-Floquet conditions

We recall that the primitive vectors of a square lattice are defined as:

a 1 = a e 1 , and
a 2 = a e 2 , (7.10)
where a is the side of the unit-cell (see Fig. 2(b)). Since the scatterers (i.e. the cross-like holes) are periodic in the x 2 -direction, the displacement field in equation (7.5) satisfies Bloch-Floquet boundary conditions ūsc (x + n 2 a 2 ) = e in2k2a ūsc (x), for x ∈ γ, and n 2 ∈ Z, (7.11) where k 2 is the component along the x 2 -direction of the wave vector k. The value of k 2 is known and should be equal, given the considered geometry represented in Fig. 2(a), to the second component of the wave vector of the incident wave. This requirement is essential in order to construct a solution which satisfies the prescribed boundary conditions within a micro-structured medium which is periodic in one dimension, i.e. in a layered micro-structured medium (see e.g. [START_REF] Steven | Two-dimensional phononic crystals and scattering of elastic waves by an array of voids[END_REF]). The conservation of the value of k 2 provides the mathematical justification of Snell's law governing the refraction of waves at the interface between two half-spaces with different material parameters (see the book by Leckner [START_REF] Leckner | Theory of Reflection: Reflection and Transmission of Electromagnetic, Particle and Acoustic Waves[END_REF] for a mathematical introduction encompassing several physical scenarios). As it is customary in Floquet theory of PDEs with periodic coefficients, we can obtain the solution of the PDEs system (7.7), by solving the problem in its period (i.e. the red strip in Fig. 2(a) here denoted as γ) provided that the Bloch-Floquet condition (7.11) on the scattered field is satisfied.

Although the x 1 extension of the homogeneous part of the domain γ is infinite in our model problem, in the finite-element implementation of the boundary value problem we are of course restricted to finite computational domains. In order to annihilate the reflection from the sides of γ with constant x 1 , we use perfectly matched layers [START_REF] Basu | Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation[END_REF] away from the microstructure.

Reflectance

The time-averaged Poynting vector associated with a 2D time-harmonic displacement field u(x, t) = ū(x) e -iωt is defined as [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF][START_REF] Bertram | Acoustic Fields and Waves in Solids[END_REF]

F = - ω 2 Re (iσ • u * ) , (7.12)
where " * " denotes complex conjugation and σ = C Al sym ∇u is the Cauchy stress tensor associated with the elastic field u. From equation (7.12), and using equation ( 7.1), it follows that the energy flux associated with the incident displacement field (7.1) is

F j,i = 1 2 ρ c j |d j,i | 2 k j,i |k j,i | , j ∈ {L, S}. (7.13)
Similarly, we define the flux F sc of the scattered field to be as in equation ( 7.12) with u(x, t) = ūsc (x)e -iωt , where ūsc is the solution of the PDEs system (7.7)). The reflectance, i.e. the ratio of reflected energy and incident energy passing through a vertical line of length a, is

R j = 1 F j,i , e 1 1 a a/2 -a/2 F sc | x1 -aN/2 , e 1 dx 2 , = 2 ρ c j |d j,i | 2 sin(θ) 1 a a/2 -a/2 F sc | x1 -aN/2 , e 1 dx 2 , j ∈ {L, S}. (7.14)
The reflectance associated with incident shear waves (R S ) differs from that associated with incident longitudinal waves (R L ). The Poynting vector is evaluated at a given x 1 -aN/2 away from the slab to avoid the contribution from the near elastic field close to the microstructure. Provided the condition x 1

-aN/2 is satisfied, we have verified that the reflectance (7.14) does not depend on the exact value of x 1 .

Results and discussion

In this section we present the comparison between the refractive behavior of the finite microstructured metamaterial's slab (see Fig. 2) and the relaxed micromorphic model (see Fig. 1). We also provide the results concerning the relaxed micromorphic single interface, which will be seen as an average behavior with respect to the micromorphic slab of finite size. To this end, we chose the material parameters of the relaxed micromorphic model as in Table 2.

λ e µ e µ * e [GPa] [GPa] [GPa]
2.33 10.92 0.67

λ micro µ micro µ * micro [GPa] [GPa] [GPa]
5.27 12.8 8.33

µ c [GPa] 2.28 • 10 -3 L c [m] 4.41 • 10 -6 λ macro µ macro µ * macro [GPa] [GPa] [GPa]
1.74 5.89 0.62

ρ η 1 η 2 η 3 η * 1 η 1 η 2 η 3 η * 1 kg/m 3 [kg/m] [kg/m] [kg/m] [kg/m] [kg/m] [kg/m] [kg/m] [kg/m] 1485 8.6 • 10 -5 10 -7 -2.2 • 10 -5 3.3 • 10 -5 5.6 • 10 -5 7.3 • 10 -4 1.7 • 10 -4 9 • 10 -7
Table 2: Summary of the numerical values for the elastic (top) and inertia (bottom) parameters of the tetragonal relaxed micromorphic model in the 2D plane-strain case. The macroscopic parameters of the resulting homogenized Cauchy material (see [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF][START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]) are also provided (top right).

We will show that the performed fitting allows the description of the average reflectance of the metamaterial slab even if the Bloch-Floquet patterns are not exactly reproduced point by point (see Fig. 3). The micromorphic modeling framework presented here will need further generalizations in order to exactly match these patterns. Nevertheless, the astonishingly pertinent results presented in section 8.1 clearly indicate that the present paper must be used as a starting stepping stone for a new mindset in the modeling of metamaterials, which will enable the design of complex large-scale meta-structures.

The choice of the metamaterial parameters is made according to the procedure presented in [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF], which allows to determine the parameters of the relaxed micromorphic model on a specific metamaterial by an inverse approach. This fitting procedure is based on the determination of the elastic parameters of the relaxed micromorphic model via numerical static tests on the unit cell and of the remaining inertia parameters via a simple inverse fitting of the dispersion curves on the analogous dispersion patterns as obtained by Bloch-Floquet analysis (see [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF] for details). The fitting of the bulk dispersion curves for the periodic metamaterial, whose unit cell is shown in Fig. 2(b), is presented in Fig. 3. We checked that a sort of distinction between modes which are activated by an L or S incident wave can be made for an incident wave which is orthogonal to the interface. This is exactly true for the relaxed micromorphic model, but an analogous trend can be found for Bloch-Floquet modes at least for the lower frequency modes (before the band-gap). The uncoupling between L and S activated modes is present only for θ = 0 (and, by symmetry, θ = π/2), but is lost for any other direction of propagation. In general, for any given frequency, all modes which are pertinent at that frequency may be simultaneously activated by an L or S incident wave (excluding the particular case of normal incidence). Nevertheless, we will show that this uncoupling hypothesis can be retained with little error for angles of incidence which are close to normal incidence.

Scattering at a relaxed micromorphic slab

We start by comparing the reflection coefficient of the relaxed micromorphic slab as a function of the frequency for two fixed directions of propagation of the incident wave (θ = π 2 and θ = π 4 ) and for both longitudinal and shear incident waves with the fully resolved linear elastic microstructured model.

Figures 4 and5 show the behavior of the reflection coefficient for the considered microstructured slab in the case of a longitudinal incident wave for the two boundary conditions (free and fixed microstructure) at normal incidence (Fig. 4(a), 5(a)) as well as for incidence at 45 • (Fig. 4(b), 5(b)). The black dashed line is the solution issued via the microstructured model obtained by coding all the details of the unit cell presented in section 7. The red continuous line is obtained by solving the relaxed micromorphic problem presented in section 6, using the software Mathematica.

It is immediately evident that the relaxed micromorphic model is able to capture the overall behavior of the reflection coefficient for a very wide range of frequencies. More particularly, for lower frequencies and up to the band-gap region, oscillations of the reflection coefficient due to the finite size of the slab are observed in both models. The band-gap region is also correctly described and corresponds to the frequency interval for which complete reflection (R = 1) is observed. Few differences can be found between the two types of imposed boundary conditions (free and fixed microstructure). This indicates that the characteristic length L c has little effect in the dynamic regime and it remains a quantity that mainly influences the static behavior of the metamaterial by allowing to obtain the homogenization formulas for the identification of static micromorphic stiffnesses [START_REF] Valerio D'agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF].

After the band-gap region, a characteristic frequency can be identified corresponding to which almost complete transmission occurs. This phenomenon is related to internal resonances at the level of the microstructure. It is easy to see that the relaxed micromorphic model is able to correctly describe also the internal resonance phenomenon. The internal resonance is clearly visible in both the discrete and the relaxed micromorphic model at normal incidence. It is, however, lost in the discrete simulation at 45 • notwithstanding the presence of a zero group velocity mode in both models.

Similar arguments can be carried out for an S incident wave with reference to Figs. 6 and7. In addition to the comments carried out for an L incident wave, we remark in Fig. 6(a) the presence of a small frequency interval, in which some transmission takes place correspondingly to the lower part of the band-gap. This small transmission interval is related to the S-incident activation of the first optic mode that can be associated to micro-motions (mainly rotations) at the level of the unit cell.

In the relaxed micromorphic model, the first optic mode is flat if we set L c = 0, which means that there would be no propagation in the frequency interval of interest. On the other hand, setting L c > 0 gives a non-zero group velocity to the first optic mode, thus allowing to catch some transmission in this frequency interval (see Figures 6(a interval has a pattern which is evidently affected by the choice of the free or fixed microstructure boundary conditions. In order to improve the description of the transmission pattern in this single frequency interval, the relaxed micromorphic model needs to be enhanced with space-time non localities to include the possibility of describing negative group velocity patterns. Apart from the differences observed in the aforementioned small-frequency interval, changing the type of boundary conditions does not sensibly affect the transmission pattern for other frequencies. For this reason, we will present te following results only for the free-microstructure boundary conditions.

We depict in Figure 8 the transmission coefficient (T = 1 -R) as a function of both the angle and frequency of the longitudinal incident wave for both the relaxed micromorphic and the microstructured models. We observe an excellent agreement between the continuous and discrete simulation for frequencies lower than the band-gap. For those frequencies, transmission is principally allowed by the blue acoustic mode for all directions of propagation. Even if there exists some coupling at non-orthogonal incidence with the other lower frequency modes, the blue acoustic mode is the one which is predominantly activated by an L incident wave and it is, to a large extent, responsible for the transmission across the metamaterial slab before the band-gap. The band-gap region is also correctly described. The validation of the fitting performed on higher frequencies is less precise due to the difficulty encountered for properly controlling the convergence of the detailed microstructured simulations.

Figure 9 depicts the analogous results for an S incident wave. For frequencies lower than the bandgap, we observe once again an excellent agreement between the discrete and continuous simulations for all angles of incidence. We also remark additional interesting phenomena. Firstly, except for the small transmission frequency interval, we see that the band-gap region extends to lower frequencies when considering angles close to normal incidence. This is related to the previously discussed acoustic mode uncoupling, which is observed for angles close to normal incidence. A shear incident wave mostly activates the green acoustic mode (see Fig. 3) and the first (green) optic one and they are almost entirely responsible for the propagation pattern before the band-gap region. Since the blue acoustic mode is not activated for angles close to normal incidence, the bottom band-gap limit is consequently lower compared to the case of an L incident wave.

A first threshold value of the angle of incidence can be identified (around 5π/24), for which the two (green and blue) acoustic modes start to couple and energy starts being transmitted. More remarkably, a second threshold value of the incident angle exists (around π/3), for which the amount of transmission suddenly increases, approaching a total transmission pattern. This is due to a stronger coupling between the two acoustic modes which are activated by an S incident wave for incident angles beyond the second threshold. This impressive pattern is clearly associated to the tetragonal symmetry of the metamaterial: the need for introducing "generalized classes of symmetry" in an enriched continuum environment becomes thus evident. We deduce that the agreement is very satisfactory for all the considered angles (going from normal incidence to incidence almost parallel to interface) and for the considered range of frequencies. This fact corroborates the hypothesis, which has been made according to Neumann's principle and which states that the class of symmetry of the metamaterial at the macroscopic scale is the same as the symmetry of the unit cell (tetragonal symmetry in this case).

We conclude this section by pointing out that the simulations performed to obtain Figures 8 and 9 ), while the angle of incidence decreases towards the right until it reaches the value θ = 0, which corresponds to the limit case where the incidence is parallel to the interface. The band-gap region is highlighted by two dashed horizontal lines, where, as expected, we observe no transmission. The dark blue zone shows that no transmission takes place, while the gradual change from dark blue to red shows the increase of transmission, red being total transmission.

Scattering at a single relaxed micromorphic interface

In this subsection we show the results for the reflection coefficient obtained by using the single interface boundary conditions for the relaxed micromorphic continuum as described in section 4.1.

Figure 10 shows the reflection coefficient as a function of the frequency for two different angles of incidence (θ = π/2 and θ = π/4), when considering an L incident wave for the "single interface" boundary conditions. Figure 11 shows the analogous results for an incident S wave. As expected, the solution obtained using the "single interface" boundary conditions, provide a sort of average behavior for the oscillations at lower frequencies. This is sensible, since when considering a semi-infinite metamaterial, multiple reflections on the two boundaries of the slab are not accounted for. The difference between "single" and "double interface" boundary conditions in the relaxed micromorphic model becomes less pronounced for higher frequencies, since the wavelength of the considered waves is expected to be much lower than the characteristic-size of the slab. wave. The origin coincides with normal incidence (θ = π/2), while the angle of incidence decreases towards the right until it reaches the value θ = 0, which corresponds to the limit case where the incidence is parallel to the interface. The band-gap region is highlighted by two dashed horizontal lines, where, as expected, we observe no transmission. The dark blue zone shows that no transmission takes place, while the gradual change from dark blue to red shows the increase of transmission, red being total transmission. We conclude this subsection by showing the transmission coefficients at the "single interface", as a function of the frequency and the angle of incidence for L and S incident waves (Fig. 12(a) and 12(b), respectively). Comparing Fig. 8(b) to 12(a) and 9(b) to 12(b), we can visualize the extent to which the single interface can be considered to represent a meta-structure of finite size. Although some basic averaged information is contained in Figures 12(a) and 12(b) (band-gap, critical angles), the detailed scattering behavior of the finite slab cannot be inferred from it. This provides additional evidence for the real need to propose a framework in which macroscopic boundary conditions can be introduced in a simplified way. Semi-infinite problems for metamaterials are solved in the context of homogenization methods in [START_REF] Srivastava | Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach[END_REF][START_REF] Willis | Exact effective relations for dynamics of a laminated body[END_REF], but to the author's knowledge, a systematic method for the solution of scattering problems for metamaterials of finite size is not available in the literature.

In this paper we presented for the first time the scattering solution for a metamaterial slab of finite size, modeled via a rigorous micromorphic-type boundary value problem describing its homogenized behavior.

The correct macroscopic boundary conditions (continuity of macroscopic displacement, of generalized tractions and of micro-distortion/double-traction) are presented and are intrinsically compatible with the used macroscopic bulk PDEs. The scattering properties of the considered finite-size meta-structures, as obtained via the relaxed micromorphic model, are compared to a direct microstructured simulation. This latter simulation is obtained by assuming that the metamaterial's unit cell is periodic and linear-elastic. Excellent agreement is found for all angles of incidence and for frequencies going from the long-wave limit to the first band-gap and beyond. Further work will be devoted to applying the relaxed micromorphic model validated in this paper to the scattering of metamaterial obstacles that have finite size in both the x 1 and x 2 directions.

The results presented in this paper clearly indicate the need of a new mindset with respect to classical homogenization to open new possibilities for large-scale meta-structural design. These results will serve as a starting stepping stone for the development of new modeling tools for the mechanics of finite-sized metamaterials enabling the design of large-scale meta-structures.

σe=

  Ce sym(∇u -P ), σc = Cc skew(∇u -P ), σ = σe + σc, σ = Te sym ∇u + Tc skew ∇u, s = Cmicro sym P, m := µmacro L 2 c Curl P.

Figure 1 :

 1 Figure 1: Panel (a): single interface separating a Cauchy medium from a relaxed micromorphic medium (both semi-infinite in the x 1 direction). Panel (b): A micromorphic slab of width h between two semiinfinite elastic Cauchy media. Both configurations (a) and (b) are semi-infinite in the x 2 direction.

2 2k j,t 1 a

 21 j,t 2 ω , (6.7)

Figure 2 :

 2 Figure 2: Panel (a) is a schematic representation of a slab of cross like-holes which is finite in the x 1 -direction (N = 10 number of unit cells) and periodic in the x 2 -direction. The red shadows represent the finite element domain γ where the scattering problem is set up and solved. The domain γ contains two perfectly matched layers (darker red regions at the sides of the rectangular domain γ). Panel (b) is a schematic representation of a unit cell Ω(n 1 , n 2 ), for a generic pair of integers (n 1 , n 2 ). The inner boundary ∂ (-) Ω(n 1 , n 2 ) (dashed black line), the outer boundary ∂ (+) Ω(n 1 , n 2 ) (dot-dashed black line), and the normal and traction vectors along ∂ (-) Ω(n 1 , n 2 ) (black and red arrow lines, respectively) are also shown.

Figure 3 :+ k 2 2

 32 Figure 3: Dispersion diagrams for normal (a) and 45 degrees (b) incidence. The solid curves are obtained via the tetragonal anisotropic relaxed micromorphic model, while the dashed curves are issued by Bloch-Floquet analysis. In panel (a), green color stands for modes which are mostly activated by a shear incident wave, while blue color indicates modes which are mostly activated by a longitudinal incident wave. This uncoupling between L and S activated modes at normal incidence is analytically checked in the relaxed micromorphic model and only approximate for Bloch-Floquet modes. In panel (b), we keep the same coloring, but all curves are coupled together, which means that L and S incident waves may simultaneously activate all modes at a given frequency.

Figure 4 :Figure 5 :

 45 Figure 4: Reflection coefficient at the relaxed micromorphic slab for an incident L wave and for two directions of propagation θ = π/2 (normal incidence) (a) and θ = π/4 (b) and for the free microstructure boundary condition. The red curve is generated by the analytical tetragonal relaxed micromorphic model and the black dashed line indicates the fully resolved linear elastic microstructured model.

Figure 6 :Figure 7 :

 67 Figure 6: Reflection coefficient at the relaxed micromorphic slab for an incident S wave and for two directions of propagation θ = π/2 (normal incidence) (a) and θ = π/4 (b) and for the free microstructure boundary condition. The red curve is generated by the analytical tetragonal relaxed micromorphic model and the black dashed line indicates the fully resolved linear elastic microstructured model.

Figure 8 :Figure 9 :

 89 Figure 8: Transmission coefficient of the metamaterial slab as a function of the angle of incidence θ and of the wave-frequency ω for an incident L wave and the free microstructure boundary condition. Panel (a) depicts the microstructured simulations, while panel (b) the analytical relaxed micromorphic model. The origin coincides with normal incidence (θ = π/2), while the angle of incidence decreases towards the right until it reaches the value θ = 0, which corresponds to the limit case where the incidence is parallel to the interface. The band-gap region is highlighted by two dashed horizontal lines, where, as expected, we observe no transmission. The dark blue zone shows that no transmission takes place, while the gradual change from dark blue to red shows the increase of transmission, red being total transmission.

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Reflection coefficient at the single interface for an incident L wave and for two directions of propagation θ = π/2 (normal incidence) (a) and θ = π/4 (b) and for the free microstructure boundary condition. The blue curve is generated by the analytical tetragonal relaxed micromorphic model and the black dashed line indicates the fully resolved linear elastic microstructured model.

  Transmission single, L incidence, free microstr. Wave frequency [10 7 rad/s] Angle of incidence [rad] Transmission single, S incidence, free microstr. Wave frequency [10 7 rad/s] Angle of incidence [rad]

  µc -µ * e ) -i k1(µc + µ * e ) 0 (η2 -η * 1 ) ω 2 -µc + µ * e + µ * micro -(η2 + η * 1 ) ω 2 + µc + µ * e + µ * micro + k 2 2 L 2

  2 , P 11 , P 12 , P 21 , P 22 ) T . (3.10)

Table 1 :

 1

	(b) and

Geometric parameters of the unit-cell (left), and its constitutive material (right).

For example, (A • v) i = A ij v j , (A • B) ik = A ij B jk , A : B = A ij B ji , (C • B) ijk = C ijp B pk , (C : B) i = C ijp B pj , v, w = v • w = v i w i , A, B = A ij B ij , etc.

The operators ∇, curl and Div are the classical gradient, curl and divergence operators. In symbols, for a field u of any order, (∇u) i = u ,i , for a vector field v, (curl v) i = ijk v k,j and for a field w of order k > 1, (Div w) i 1 i 2 ...i k-1 = w i 1 i 2 ...i k ,i k

We could consider more complex expressions for the curvature term, which would also include anisotropy and in fact, such expressions will be explored in further works. Here we want to show that curvature terms provide small corrections to the overall behavior of the metamaterial and so, we limit ourselves to a simplified isotropic expression.

For example, the 4th order tensor Ce is written as Ce in Voigt notation.

As we will show in the following, k 2 , which is the second component of the wave-number, is always supposed to be known and is given by Snell's law when imposing boundary conditions on a given boundary.

This choice for the sign of k 1 always gives rise to a solution which verifies conservation of energy at the interface. This particular choice, which at a first instance is rather intuitive, is not always the correct one when dealing with a relaxed micromorphic medium.

Here again, as in the case of a Cauchy medium, k 2 will be fixed and given by Snell's law when imposing boundary conditions.

We remark that, in the relaxed micromorphic model, only the tangent part of the double traction in (4.5) or of the micro-distortion tensor in (4.6) must be assigned, see[START_REF] Neff | The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations[END_REF][START_REF] Neff | A unifying perspective: the relaxed linear micromorphic continuum[END_REF] for more details.

We denote by v the first two components of the micromorphic field v defined in equation(3.11).

We suppose here that, for the transmitted wave we have to consider only the positive solutions of (3.13).

In order to easily compute these coefficients in the numerical implementation of the code, we employ Lemma 1 given in Appendix B.

The index j ∈ {L, S} for the elastic field u j indicates the fact that u j is the solution field generated by an L or S incident wave, respectively. Nevertheless, u j typically contains in itself coupled L and S waves as a result of scattering.

We limit ourselves to the plane-strain case, so that we set u j

i,3 = 0, j ∈ {L, s}, i ∈ {1, 2, 3}.

Acknowledgements

Angela Madeo and Domenico Tallarico acknowledge funding from the French Research Agency ANR, "METAS MART" (ANR-17CE08-0006). Angela Madeo acknowledges support from IDEXLYON in the framework of the "Programme Investissement d'Avenir" ANR-16-IDEX-0005. All the authors acknowledge funding from the "Région Auvergne-Rhône-Alpes" for the "SCUSI" project for international mobility France/Germany.

A Appendix

A.1 Energy flux for the anisotropic relaxed micromorphic model

A.1.1 Derivation of expression (2.10)

The total energy is given by: E = J(u,t, ∇u,t, P,t) + W (∇u, P, Curl P ), (A. [START_REF] Aivaliotis | Lowand high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface[END_REF] where J(u,t, ∇u,t, P,t) and W (∇u, P, Curl P ) are defined in (2.5) and (2.6). Differentiating (A.1) with respect to time 14 we have:

Ce sym(∇u -P ), sym(∇u -P ),t + Cc skew(∇u -P ), skew(∇u -P ),t = Ce sym(∇u -P ), sym(∇u -P ),t + skew(∇u -P ),t + Cc skew(∇u -P ), sym(∇u -P ),t + skew(∇u -P ),t = Ce sym(∇u -P ) + Cc skew(∇u -P ), (∇u -P ),t = σ, (∇u -P ),t = σ, ∇u,t -σ, Pt = Div(u,t • σ) -u,t, Div σ -σ, Pt , Cmicro sym P, sym P,t = Cmicro sym P, sym P,t + skew P,t = Cmicro sym P, P,t = s, P,t ,

So, by adding all the above and simplifying (A.5) becomes:

from which we can define the energy flux for the general anisotropic relaxed micromorphic model:

14 Let ψ be a vector field and A a second order tensor field. Then

Taking ψ = u,t and A = σ we have ∇u,t, σ = Div(u,t • σ) -u,t, Div σ . (A.3) Furthermore, we have the following identity m, Curl P,t = Div (m T • P,t) :

+ Curl m, P,t , (A.4) which follows from the identity div(v × w) = w • curl v -v • curl w, where v, w are suitable vector fields, × is the usual vector product and : is the double contraction between tensors.

A.1.2 Analytical expression of the flux for the relaxed micromorphic model when L c = 0

The flux H of the relaxed micromorphic model when Lc = 0 can be written as (using Lemma 1)

1 φ

(1)

1 -φ

(1)

6 ω ,

1 φ

(2)

6 ω ,

4

5 ,

(1) 2

Then, the matrix A is

B Lemma 1

We have the following well-known result.

Lemma 1. Let u(x, t) = A(x)e i( k,x -ωt) , σ(x, t) = B(x)e i( k,x -ωt) be two functions with A, B : R 3 → C. Then the following holds

where T = 2π ω is the period of the functions u, σ, * denotes the complex conjugate, x = (x1, x2) T and k = (k1, k2) T .

Proof. We have:

Be i( k,x -ωt) + B * e -i( k,x -ωt) 2 dt where we used the facts that the integral of the periodic function e 2iωt over its period is zero and that for any complex number z ∈ C: Re(z) = z+z * 2 .


Denis Roegel

A brief survey of 20th century logical notations

We give here the logical notations used in a certain number of works of logic. Most of 20th century's work of historical importance is covered here. Although the list contains secondary work, the overview given here is not necessarily representative of the whole of contemporary logic.

An empty cell means that either there is no symbol corresponding to a concept, or that I didn't find any by skimming through the book. (Some symbols are rarely used and I may have missed them.) One must not confuse an empty cell with the mention "none." The latter means that the concept exists but that no symbol is associated to it. This is sometimes the case with "and." Some authors write pq for p ∧ q (like one often writes ab for a × b).

Note also that the symbol representations in this document are, in general, approximations of the symbols actually used. The latter can be of a different size, larger in width or in height, bolder or lighter, etc.

Many symbols stem from Peano who inspired himself from Gabelsberger's shorthand alphabet.

• the first date between parentheses represents a likely but unchecked date; this can be the case for a first edition of a book of which I only consulted a later edition; it is therefore possible that the notations changed between the editions;

• the second date between square brackets corresponds to the date where the notation has been observed; for instance, " Carnap (1934Carnap ( ) [1951]]" means that I observed the notation in a 1951 edition (actually the third printing of "The logical syntax of language") but the original German publication was in 1934.

The table gives an idea of the introduction dates of the various symbols. However, one should be careful not to draw too hasty conclusions. For instance, it can seem that Schönfinkel introduced "&", "→" and "(Ex)" in 1924, but actually the author claims to borrow the symbols used by Hilbert in his classes. However, it might be the first published use of these symbols.

Only object language symbols are shown. Some authors also define other symbols for the metalanguage. For instance, Skolem in 1928 has → for the consequence relation between formulae and → ← for their equivalence.

````````À Frege (1879) 166: 1-8, translated as "On the consistency of arithmetic" in van Heijenoort's "From Frege to Gödel", where the correct symbols are given, which is not the case in the "Logical Writings" (1971, edited by Goldfarb) which use more standard symbols)

A B A x Φ(x) Peirce (1885) x --< Π i Σ j Peano (1889) - ∪ ∩ C C x Schröder (1890) (to check) a + • = C Π i Σ i Peano (ca. 1895) [1901] - ∪ ∩ C C x ∃x Hilbert (1904) a o. u. | Russell (1908) ∼ ∨ . ⊃ (x) ∃x ≡ Russell &Whitehead (1910-1913) ∼ ∨ . ⊃ (x) (∃x) ≡ Löwenheim (1915) a + • = C Π i Σ i Hilbert (1917/18) X ×/none + → (x) = Skolem (1920) a + • Π i Σ i Post (1921) ∼ ∨ . ⊃ ≡ Tarski (1923) ∼ ∨ ⊃ Schönfinkel (1924) a ∨ & → (x) (Ex) ∼ Ramsey (1925) ∼ ∨ . ⊃ (x) (∃x) Hilbert (1925) A ∨ & → (x) (Ex) Kolmogorov (1925) A → (x) (Ex) Hilbert (1927) A ∨ & → (x) (Ex) Skolem (1928) a + • Π i Σ i Herbrand (1928) ∼ ∨ & ⊃ (x) (∃x) ≡ Hilbert & Ackermann (1928) [1950] A ∨ & → (x) (Ex) ∼ Lukasiewicz (1929) Np Apq Kpq Cpq Px Sx Epq Heyting (1929) ¬ ∨ ∧ ⊃ (x) (Ex) ⊃⊂ Gödel (1930) p ∨ & → (x) (Ex) ∼ Heyting (1931) ¬p ∨ Gödel (1931) p ∨ & → (x) (Ex) ≡ (also ∼) (also ⊃) (also xΠ) Herbrand (1931) ∼ ∨ × → (x) (Ex) ≡ Gentzen (1934) ∨ & ⊃ A x E x ⊃⊂ Hilbert & Bernays (1934-39) A ∨ & → (x) (Ex) ∼ Carnap (1934) [1951] ∼ ∨ • ⊃ (x) (∃x) ≡ Tarski (1940) ∼ ∨ ∧ → A x E x ↔ Quine (ML, 1940) ∼ ∨ . ⊃ (x) (∃x) ≡ Henkin (1949) ∼ ⊃ (x) (∃x) Henkin (1950) ∼ ∨ none ⊃ (x) (∃x) Kleene (IM, 1952) ¬ ∨ & ⊃ ∀x ∃x ∼ Carnap (1954) ∼ ∨ . ⊃ (x) (∃x) ≡ Church (1956) ∼ ∨ none ⊃ (x) (∃x) ≡ Suppes (IL, 1957) -p ∨ & → (x) (∃x) ↔ Davis (C&U, 1958) [1982] -A & ⊃ (x) (∃x) Bernays (AST, 1958) [1968] p ∨ & → (x) (Ex) ↔ Suppes (AST, 1960) -p ∨ & → ∀x ∃x ↔ Quine (STL, 1963) ∼ ⊃ (x) (∃x) ≡ Mendelson (1964) [1968] ∼ ∨ ∧ ⊃ (x) (Ex) ≡ Kleene (LM, 1967) [1971] ¬ ∨ & ⊃ ∀x ∃x ∼ Smullyan (1968) ∼ ∨ ∧ ⊃ (∀x) (∃x) ↔ Blanché (1968) ∼ ∨ . ⊃ (x) (∃x) ≡ Putnam (1971) ∼ ∨ ∧ ⊃ (x) (∃x) ≡ Boolos & Jeffrey (1974) [1996] -p ∨ & → ∀x ∃x ↔ Copi (SL, 1979) ∼ ∨ • ⊃ (x) (∃x) ≡ Quine (1982) -p ∨ none → ∀x ∃x ↔ Vax (1982) ∼ ∨ ∧ ⇒ ∀x ∃x ⇔ Rivenc (1989) ¬ ∨ ∧ → ∀x ∃x ↔ Gochet & Gribomont (1990) ¬ ∨ ∧ ⊃ ∀x ∃x ≡ Lalement (1990) ¬ ∨ ∧ ⇒ ∀x ∃x ⇔ Ruyer (1990) ¬ ∨ ∧ → ∀x ∃x ↔ Smullyan (1992) ∼ ∨ ∧ ⊃ ∀x ∃x ≡ Largeault (1992) ¬ ∨ & → ∀x ∃x ↔ Cori & Lascar (1993) ¬ ∨ ∧ ⇒ ∀x ∃x ⇔ Largeault (1993) ∨ & → ∀x ∃x ↔ Chazal (1996) ¬ ∨ ∧ ⊃ ∀x ∃x ⊂⊃ Gauthier (1997) ∼ ∧ ⊃ ∀x ∃x Leroux (1998) ¬ ∨ ∧ ⊃ ∀x ∃x ≡ Encyclop. Britannica (1999) ∼ ∨ none/• ⊃ (∀x) (
• Gentzen (1934): Untersuchungen über das logische Schließen (Mathematische Zeitschrift 39: 176-210, 405-431) (in the French translation "Recherches sur la déduction logique", 1955, the symbols are Bernays (1934Bernays ( -1939)) Collections mentionned:

Γ , ∨, &, U , A x, E x, U U ) • Hilbert &
• Jean van Heijenoort: "From Frege to Gödel", Cambridge, Mass.: Harvard University Press, 1967 • Jacques Herbrand: "Logical Writings", Cambridge, Mass.: Harvard University Press, 1971 • F. P. Ramsey: "Philosophical Papers", Cambridge University Press, 1990• Alfred Tarski: "Logic, Semantics, Metamathematics", Oxford University Press, 1956

  ∃x) ≡Sources• Frege (1879): Begriffsschrift (Halle: Louis Nebert, translated in "From Frege to Gödel") • Peirce (1885): On the Algebra of Logic; A Contribution to the Philosophy of Notation (American Journal of Mathematics 7: 180-202). • Peano (1889): Arithmetices principia, nova methodo exposita (Augustae Taurinorum, Ed. Fratres Bocca. -XVI, 20 p, translated in "From Frege to Gödel" as "The principles of arithmetic") • Schröder (1890): Vorlesungen über die Algebra der Logik (exakte Logik) (volume 1, Leipzig: Teubner) • Peano (ca. 1895): Formulaire de mathématiques (according to Cajori, vol. 2, page 300, Peano used ∃ in volume II, number 1, of his Formulaire de mathématiques, which was published in 1897) • Hilbert (1904): Über die Grundlagen der Logik und der Arithmetik (Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, 174-185) (in van Heijenoort, "u." is replaced by "a.") (translated as "On the foundations of logic and arithmetic" in "From Frege to Gödel") • Russell (1908): Mathematical logic as based on the theory of types (American Journal of Mathematics 30: 222-262, reproduced in "From Frege to Gödel") • Russell & Whitehead (1910-1913): Principia Mathematica (Cambridge: At the University Press) • Löwenheim (1915): Über Möglichkeiten im Relativkalkül (Mathematische Annalen 76: 447-470, translated as "On possibilities in the calculus of relatives" in "From Frege to Gödel") • Hilbert (1917/1918): Prinzipien der Mathematik (Lecture notes by Paul Bernays, Winter-Semester 1917-1918, Unpublished typescript. Bibliothek, Mathematisches Institut, Universität Göttingen) (personal communication by Richard Zach) • Skolem (1920): Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem Theoreme über dichte Mengen. (Videnskabsakademiet i Kristiania, Skrifter I, No. 4, 1919, 1-36, translated as "Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by L. Löwenheim and generalizations of the theorem" in "From Frege to Gödel") • Post (1921): Introduction to a general theory of elementary propositions (American Journal of Mathematics 43: 163-185, also in "From Frege to Gödel") • Tarski (1923): On the Primitive Term of Logistic (in Tarski: "Logic, Semantics, Metamathematics", Oxford University Press, 1956) • Schönfinkel (1924): Über die Bausteine der mathematischen Logik (Mathematische Annalen 92: 305-316, translated as "On the building blocks of mathematical logic" in "From Frege to Gödel") • Ramsey (1925): The Foundations of Mathematics (Proceedings of the London Mathematical Society, series 2, 25: 338-384, also in F. P. Ramsey: "Philosophical Papers", Cambridge University Press, 1990) • Hilbert (1925): Über das Unendliche (Mathematische Annalen (1926) 95: 161-190, translated as "On the infinite" in "From Frege to Gödel") • Kolmogorov (1925): O principe tertium non datur (Matematiqeski sbornik, 32: 646-667, translated as "On the principle of excluded middle" in "From Frege to Gödel") • Hilbert (1927): Die Grundlagen der Mathematik (Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6: 65-85, translated as "The foundations of mathematics" in "From Frege to Gödel") • Skolem (1928): Über die mathematische Logik (Norsk matematisk tidsskrift 10: 125-142, translated as "On mathematical logic" in "From Frege to Gödel") • Herbrand (1928): Sur la théorie de la démonstration (Comptes rendus hebdomadaires des séances de l'Académie des sciences, Paris, 186 : 1274-1276, translated as "On proof theory" in "Logical Writings") • Hilbert & Ackermann (1928): Grundzüge der Theoretischen Logik (translated as "Principles of Mathematical Logic") (Berlin: Springer-Verlag) • Lukasiewicz (1929): Elementy logiki matematycznej (Warsaw) (translated as "Elements of mathematical logic", New York: Macmillan, 1963) • Heyting (1929): Die formalen Regeln der intuitionistischen Logik (Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin, Physikalisch-mathematische Klasse, 42-56, 57-71, 158-169, 1930) • Gödel (1930): Die Vollständigkeit der Axiome des logischen Funktionenkalküls (Monatshefte für Mathematik und Physik 37: 349-360, translated as "The completeness of the axioms of the functional calculus of logic" in "From Frege to Gödel") • Heyting (1931): Die intuitionistische Grundlegung der Mathematik (Erkenntnis, 2: 106-115) • Gödel (1931): Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I (Monatshefte für Mathematik und Physik 38: 173-198, translated as "On formally undecidable propositions of Principia Mathematica and related systems I" in "From Frege to Gödel") • Herbrand (1931): Sur la non-contradiction de l'arithmétique (Journal für die reine und angewandte Mathematik

:

  Grundlagen der Mathematik (Berlin: Springer-Verlag, translated in French as "Fondements des mathématiques," Paris: L'Harmattan, 2001) • Carnap (1934): Logische Syntax der Sprache (Wien: Julius Springer, translated as "The Logical Syntax of Language", 1937) • Tarski (1940): Introduction to logic and to the methodology of deductive sciences (New York: Oxford university press) • Quine (1940): Mathematical Logic (New York: W. W. Norton & company, inc.) • Henkin (1949): The completeness in the first order functional calculus (Journal of Symbolic Logic, 14: 150-166) • Henkin (1950): Completeness in the theory of types (Journal of Symbolic Logic, 15(2): 81-91) • Kleene (1952): Introduction to Metamathematics (Princeton, NJ: D. van Nostrand) • Carnap (1954): Einführung in die symbolische Logik (translated as "Introduction to Symbolic Logic and its Applications", 1958) (Wien: Springer) • Church (1956): Introduction to Mathematical Logic (Princeton: Princeton University Press) • Suppes (1957): Introduction to Logic (New York: Van Nostrand Reinhold Company) • Davis (1958): Computability and Unsolvability (New York: McGraw Hill) • Bernays (1958): Axiomatic Set Theory (Amsterdam: North-Holland) • Suppes (1960): Axiomatic Set Theory (Princeton, N.J.: Van Nostrand) • Quine (1963): Set Theory and its Logic (Cambridge: Belknap Press of Harvard University Press) • Mendelson (1964): Introduction to Mathematical Logic (Princeton, NJ: Van Nostrand) • Kleene (1967): Mathematical Logic (New York: John Wiley) • Smullyan (1968): First-order Logic (New York: Springer-Verlag) • Blanché (1968): Introduction à la logique contemporaine (Paris : A. Colin) • Putnam (1971): Philosophy of Logic (New York: Harper and Row) • Boolos & Jeffrey (1974): Computability and Logic (Cambridge University Press) • Copi (1979): Symbolic Logic (New York: Macmillan) • Quine (1982): Methods of logic (previous editions had Russell and Whitehead's symbols) (Cambridge, Mass.: Harvard University Press) • Vax (1982): Lexique logique (Paris : Presses Universitaires de France) • Rivenc (1989): Introduction à la logique (Paris : Payot) • Gochet & Gribomont (1990): Logique, volume 1 : méthodes pour l'informatique fondamentale (Paris : Hermès) • Lalement (1990): Logique, réduction, résolution ( Études et recherches en informatique, Paris : Masson) (translated as "Computation as logic", Englewood Cliffs, NJ: Prentice-Hall, 1993) • Ruyer (1990): Logique (Paris : Presses Universitaires de France) • Smullyan (1992): Gödel's Incompleteness Theorems (New York: Oxford University Press) • Largeault (1992): L'intuitionisme (Que sais-je ? 2684, Paris : Presses Universitaires de France) • Cori & Lascar (1993): Logique mathématique. Cours et exercices (Paris : Masson, translated in English as "Mathematical logic: A Course with Exercises", 2 volumes, New York: Oxford University Press, 2000 and 2001) • Largeault (1993): La logique (Que sais-je ? 225, Paris : Presses Universitaires de France) • Chazal (1996): Éléments de logique formelle (Paris : Hermès) • Gauthier (1997): Logique et fondements des mathématiques (Paris : Diderot Éditeur) • Leroux (1998): Introduction à la logique (Bibliothèque des sciences, Paris : Diderot Éditeur) • Encyclop. Britannica (1999):