Continuum Mech. Thermodyn.DOI 10.1007/s00161-013-0329-2

ORIGINAL ARTICLE

A. Madeo · P. Neff · I. D. Ghiba · L. Placidi · G. Rosi

Wave propagation in relaxed micromorphic continua:modeling metamaterials with frequency band-gaps

Received: 4 September 2013 / Accepted: 27 November 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, the relaxed micromorphic model proposed in Ghiba et al. (Math Mech Solids, 2013),Neff et al. (Contin Mech Thermodyn, 2013) has been used to study wave propagation in unbounded continuawith microstructure. By studying dispersion relations for the considered relaxed medium, we are able to dis-close precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersionrelations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium.We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couplemodulus μc, which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parame-ter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around agiven threshold drastically changes the observed response of the material with respect to wave propagation. We

Communicated by Andreas Öchsner.

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

A. MadeoUniversité de Lyon-INSA, 20 Av. Albert Einstein, 69100 Villeurbanne Cedex, France

A. Madeo · P. Neff · I. D. Ghiba · L. Placidi · G. Rosi (B)International Center M&MOCS “Mathematics and Mechanics of Complex Systems”,Palazzo Caetani, Cisterna di Latina, ItalyE-mail: giuseppe.rosi@univ-paris-est.fr

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

I. D. GhibaDepartment of Mathematics, University “A.I. Cuza” Blvd. Carol I, no. 11, 700506 Iasi, Romania

I. D. GhibaOctav Mayer Institute of Mathematics, Romanian Academy, 700505 Iasi, Romania

I. D. GhibaInstitute of Solid Mechanics, Romanian Academy, 010141 Bucharest, Romania

L. PlacidiUniversità Telematica Internazionale Uninettuno, Corso V. Emanuele II 39, 00186 Rome, Italy

Present address:G. RosiLaboratoire Modélisation Multi Echelle, MSME UMR 8208 CNRS, Université Paris-Est,61 Avenue du Géneral de Gaulle, 94010 Créteil Cedex, France

A. Madeo et al.

finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosseratand second gradient ones. The potential fields of application of the proposed relaxed model are manifold,above all for what concerns the conception of new engineering materials to be used for vibration control andstealth technology.

Keywords Relaxed micromorphic continuum · Cosserat couple modulus · Wave band-gaps ·Phononic crystals · Lattice structures

1 Introduction

Engineering metamaterials showing exotic behaviors with respect to wave propagation are recently attractinggrowing attention from the scientific community for what concerns both modeling and experiments. Indeed,there are multiple experimental evidences supporting the fact that engineering microstructured materials suchas phononic crystals and lattice structures can inhibit wave propagation in particular frequency ranges (seee.g., [68,69]). Both phononic crystals and lattice structures are artificial materials which are characterized byperiodic microstructures in which strong contrasts of the material properties can be observed. Also, suitablyordered granular assemblies with defects (see e.g., [36,41,42]) and composites [19] have shown the possibilityof exhibiting band-gaps in which wave propagation cannot occur under particular loading conditions. In allthese cases, the basic features of the observed band-gaps are directly related to the presence of an underlyingmicrostructure in which strong contrasts of the elastic properties may occur. Indeed, the rich dynamic behaviorof such materials stems mainly from their hierarchical heterogeneity at the microscopic level which producesthe mixing of the longitudinal and the transverse components of traveling waves. It is clear that the applicationsof such metamaterials would be very appealing for what concerns vibration control in engineering structures.These materials could be used as an alternative to currently used piezo-electric materials which are commonlyemployed for structural vibration control and for this reason are widely studied in the literature (see, amongmany others [4,16,39,40,63,70]).

1.1 Motivation and originality of the present work

Different modeling efforts were recently made trying to account for the observed band-gaps in a reliable man-ner. The most common models are based on homogenization procedures on simple periodic microstructureswith strong contrast giving rise to Cauchy-type equivalent continua. Often, it is remarked that the equiva-lent continua obtained by means of these homogenization techniques are able to predict band-gaps when theequivalent mass of the macroscopic system becomes negative (see e.g., [6,31,72]). Homogenized systemswith negative equivalent mass can be successfully replaced by suitable generalized continua. Indeed, it isknown that homogenization of strongly heterogeneous periodic systems may lead to generalized continua asequivalent macroscopic medium (see e.g., [5]). Some rare papers (see e.g., [32,71,72]) can be found in theliterature in which lattice models with enriched kinematics are used to model band-gaps without the need ofintroducing negative equivalent masses. Nevertheless, to the authors’ knowledge, a systematic treatment ofband-gap modeling based on the spirit of micromorphic continuuum mechanics is still lacking and deservesattention. Indeed, the idea of using generalized continuum theories to describe microstructured materials needsto be fully developed in order to achieve a simplified modeling and more effective conception of engineeringdevices allowing to stop wave propagation in suitable frequency bands. Continuum models are, in fact, suitableto be used as input in finite element calculations, which are for sure one of the main basis of engineering design.

In this paper, we propose to use the relaxed micromorphic continuum model presented in [53] to studywave propagation in microstructured materials exhibiting exotic properties with respect to wave propagation.The used kinematics is enhanced with respect to the one used for classical Cauchy continua by means of sup-plementary kinematical fields with respect to the macroscopic displacement. The introduced supplementarykinematical fields allow to account for the presence of microstructure on the overall mechanical behavior ofconsidered continua. The considered extended kinematics has a similar form as the one used to describe phe-nomena of energy trapping in internal degrees of freedom (see e.g., [8] and references there cited). However,the aforementioned papers deal with damped systems differently from what done in the present paper whereonly conservative systems are considered.

It is well known that the mechanical behavior of isotropic Mindlin–Eringen media is, in general, describedby means of 18 elastic constants (see [21,44,53]). Nevertheless, the set of 18 parameters introduced by Mind-lin in [44] and Eringen in [21] is not suitable to disclose the main features of micromorphic media mainly

Wave propagation in relaxed micromorphic continua

because of unavoidable computational difficulties. We propose here to consider the relaxed micromorphicmodel presented in [53] which only counts 6 parameters and which, nevertheless, is fully able to describe themain characteristic features of micromorphic continua. Indeed, in [53], we showed that the linear isotropicmicrovoid model, the linear Cosserat model, and the linear microstretch model are special cases of our relaxedmicromorphic model in dislocation format. A direct identification of the coefficients gave us that the coefficientof the Cowin-Nunziato theory [14], the Mindlin–Eringen theory [21,44], and the microstretch theory [21,33]can be expressed in terms of our constitutive coefficients (see also [53,61]). Using these identifications, onecould compare all the qualitative results obtained using our relaxed micromorphic model with those alreadyavailable in the literature (see for instance [10–12,21,27–30,44] and references therein) concerning classicaltheories of elastic materials with microstructure.

By studying dispersion relations in the proposed relaxed model, we show that wave propagation in suchsimplified micromorphic media is affected by the presence of microstructure in a quite controllable manner.More precisely, we show that if only longitudinal waves are allowed to propagate in the considered medium(no displacements and micro-deformations allowed in the direction orthogonal to the direction of wave prop-agation), then the presence of band-gaps can be forecasted only with 5 parameters. The main features of theselongitudinal band-gaps are seen to be directly related to the absolute rotations of the microstructure. Never-theless, the proposed 5-parameters relaxed model was not able to account for the presence of band-gaps whenalso transverse waves are allowed to propagate in the considered material. In order to treat the more generalcase, we hence consider a slightly generalized relaxed model by introducing only one extra elastic parameter,the so-called Cosserat couple modulus μc, which is related to the relative deformation of the microstructurewith respect to the macroscopic matrix. We hence end up with a relaxed micromorphic model with only 6parameters (5+1) which is able to account for the prediction of band-gaps when generic waves can travel inthe considered medium.

Moreover, we consider the so-called classical micromorphic medium, i.e., a micromorphic continuum inwhich the whole gradient of the micro-deformation tensor plays a role (instead of only its curl). We showthat also the classical micromorphic continuum is not suitable to account for the description of band-gaps.Finally, a particular second gradient material is obtained as limit case of the proposed classical micromorphiccontinuum by letting μe → ∞ and μc → ∞. Also in this case, we will show that the existence of band-gapscannot be accounted for.

We can summarize by saying that the proposed relaxed micromorphic model (6 elastic coefficients) accountsfor the description of frequency band-gaps in microstructured media which are “switched on” by the onlyparameter μc related to relative rotations of the microstucture with respect to the solid macroscopic matrix.These band-gaps cannot be predicted by means of classical micromorphic or Cosserat and second gradientmedia, as it will be extensively pointed out in the body of the paper. Moreover, band-gaps are impossibleto observe in the purely one-dimensional situation, since, among others, the curl operator vanishes in thissimplified case. This absence of band-gaps is confirmed by the results found in [7] in which a one-dimensionalmicromorphic model is presented together with a detailed analysis of wave propagation.

We finally remark that there are some similarities between dispersion relations for micromorphic continua(including the proposed relaxed model) and dispersion relations of various types of plates models, see [1,2]where Kirchhoff, Mindlin, and Reissner types of plates including consideration of rotatory inertia are inves-tigated. In particular, for almost all models of plates, there are optical branches of dispersion relations andlinear asymptotes when the wave number tends to infinity. On the other hand, in the theory of plates, there areno band-gaps discovered within the considered models. Also, porous media with strong contrast between thesolid and the fluid phases (see e.g., [66,67]) and biological tissues as etched dentine (see e.g., [38]) may showa marked dispersive behavior, but band-gaps are not observed in such media.

1.2 Notations

Let x, y ∈ R3 be two vectors and X,Y ∈ R

3×3 be two second-order tensors of components xi , yi andXi j , Yi j , i, j = {1, 2, 3}, respectively. We let1 <x, y >R3= xi yi and <X, Y >R3×3= Xi j Yi j denote the scalarproduct on R

3 and R3×3 with associated vector norms ‖x‖2

R3 = <x, x >R3 and ‖X‖2R3×3 = <X, X >R3×3 ,

respectively. In what follows, we omit the subscripts R3 and R

3×3 if no confusion can arise. We define thestandard divergence, gradient, and curl operators for vectors and second-order tensors, respectively as2

1 Here and in the sequel Einstein convention of sum of repeated indices is used if not differently specified.2 Here and in the sequel, a subscript i after a comma indicates partial derivative with respect to the space variable Xi .

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Div x = xi,i , (∇ x )i j = xi, j , (Curl x )i = xa,b εiab,

(Div X )i = Xi j, j , (∇ X )i jk = Xi j,k, (Curl X )i j = Xia,bε jab,

where ε is the standard Levi–Civita tensor.For any second-order tensor X ∈ R

3×3, we introduce its symmetric, skew-symmetric, spheric, and devia-toric part, respectively, as

sym X = 1

2

(X + XT

), skew X = 1

2

(X − XT

), sph X = 1

3tr X1, dev X = X − sph X,

or equivalently, in index notation

(sym X)i j = 1

2

(Xi j + X ji

), (skew X)i j = 1

2

(Xi j − X ji

), (sph X)i j = 1

3Xkk δi j ,

(dev X)i j = Xi j − (sph X)i j ,

where δi j is the Kronecher delta tensor which is equal to 1 when i = j and equal to 0 if i �= j . TheCartan-Lie-algebra decomposition for the tensor X is introduced as

X = dev sym X + skew X + sph X. (1)

2 Equations of motion in strong form

We describe the deformation of the considered continuum by introducing a Lagrangian configuration BL ⊂ R3

and a suitably regular kinematical field χ(X, t) which associates with any material point X ∈ BL its currentposition x at time t . The image of the function χ gives, at any instant t , the current shape of the body BE (t)which is often referred to as Eulerian configuration of the system. Since we will use it in the following, we alsointroduce the displacement field u(X, t) = χ(X, t)− X. The kinematics of the continuum is then enriched byadding a second-order tensor field P(X, t)which accounts for deformations associated with the microstructureof the continuum itself. Hence, the current state of the considered continuum is identified by 12 independentkinematical fields: 3 components of the displacement field and 9 components of the micro-deformation field.Such a theory of a continuum with microstructure has been derived in [44] for the linear-elastic case andre-proposed, e.g., in [22,23,25,26] for the case of nonlinear elasticity.

Once the used kinematics has been made clear, we can introduce the action functional of the consideredmicromorphic system as

A =T∫

0

∫

BL

(T − W ) dX dt (2)

where T and W are the kinetic and potential energies of the considered system, respectively. Denoting byρ and η the macroscopic and microscopic mass densities, respectively, we choose the kinetic energy of thesystem to be

T = 1

2ρ ‖ ut ‖2 + 1

2η ‖ Pt ‖2 , (3)

where we denote by a subscript t partial derivative with respect to time. We remark that since the micro-defor-mation tensor P is dimensionless, the micro-density η has the dimensions of a bulk density (Kg/m3) timesthe square of a length. This means that if ρ′ is the true density (per unit of macro volume) of the materialconstituting the underlying microstructure of the medium, one can think to write the homogenized density ηas (see also [44])

η = d2 ρ′, (4)

where we denoted by d the characteristic length of the microscopic inclusions.3

3 We remark that by considering a scalar microscopic density η, we are limiting ourselves to cases in which the microscopicinclusions only have one characteristic length (e.g., cubes or spheres). On the other hand, expression (3) for the kinetic energycan be easily generalized by replacing the second term with 1/2 ηi j (Pki )t (Pkj )t , where the tensor η can be written as ηi j = d2

i jρ′.

In this way, one can account for as many microscopic characteristic lengths di j as needed.

Wave propagation in relaxed micromorphic continua

On the other hand, we will specify the choice of the strain energy density W in the next subsections by dis-cussing the cases of the relaxed micromorphic continuum introduced in [53] and of the classical micromorphiccontinuum.

2.1 The relaxed micromorphic continuum

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

W = μe ‖ sym (∇u − P) ‖2 + λe

2(tr (∇u − P))2 + μh ‖ sym P ‖2 + λh

2(tr P)2

+μc ‖ skew (∇u − P) ‖2 + αc

2‖ Curl P ‖2 , (5)

where all the introduced elastic coefficients are assumed to be constant. This decomposition of the strain energydensity, valid in the isotropic, linear-elastic case, has been proposed in [27,53]. Indeed, the fact of consideringonly the curl of the microstrain tensor P, instead of the whole gradient as done in classical micromorphicmodels (se e.g., the energy considered in Eq. (13)), allows to account for microstructures inside the consideredcontinuum which form weaker connections at the microscopic level so allowing for the description of band-gaps. More particularly, band-gaps are not possible when considering energies of the type shown in Eq. (13)since the microstructure associated with the whole gradient of P is somehow related to strong connectionsat the microscopic level which always allow for wave propagation. From a mathematical point of view, thequestion is how the control of the substructure interactions is realized. In the classical Mindlin–Eringen model,the moment stresses are triggered by any inhomogeneity in the microdistortion field P: This is the case underthe crucial assumption of a positive definite energy (estimate > ‖∇P‖2). In contrast, in the relaxed model,we only have an estimate of the energy> ‖CurlP‖2 which means that inhomogeneities in the microdistortionfield do not generate moment stresses. This feature is the deeper reason for the possibility of band-gaps: Somemicrodistortion response gets absorbed. We finally mention the fact that the relaxed energy accounting for theterm > ‖CurlP‖2 still gives rise to a well-posed problem as proved in [48].

Since the classical Mindlin–Eringen model is always taken with the no-absorption rule (positive definite-ness of the energy with the whole gradient term), the classical approach is not able to describe band-gaps. Itis clear that this decomposition introduces a limited number of elastic parameters, and we will show how thismay help in the physical interpretation of these latter. Positive definiteness of the potential energy implies thefollowing simple relations on the introduced parameters

μe > 0, μc > 0, 3λe + 2μe > 0, μh > 0, 3λh + 2μh > 0, αc > 0. (6)

One of the most interesting features of the proposed strain energy density is the reduced number of elasticparameters which are needed to fully describe the mechanical behavior of a micromorphic continuum. Indeed,each parameter can be easily related to specific micro- and macro-deformation modes.

2.1.1 Comparison with Mindlin and Eringen models

It can be checked that the proposed strain energy density (5) represents a particular case of the strain energydensity proposed by Mindlin (cf. Eq. (5.5) of [44]). Indeed, considering that our micro-strain tensor is thetransposed of the one introduced by Mindlin (Pi j = ψ j i ) and using the substitutions (cf. also [55]):

μ = μh, λ = λh, b1 = λe+λh, b2 = μe+μh +μc, b3 = μe + μh − μc, g1 = −λh, g2 = −μh,

(7)

a10 = αc, a14 = −αc, a1 = a2 = a3 = a4 = a5 = a8 = a11 = a13 = a15 = 0, (8)

we are able to recover that our relaxed model can be obtained as a particular case of Mindlin’s one. Therelaxed strain energy density (5) is not positive definite in the sense of Mindlin and Eringen, but it gives riseto a well-posed model (see [53]). It is evident that the representation of the strain energy density (5) is moresuitable for applications than Mindlin’s one due to the reduced number of parameters. Indeed, if we considerall the terms with the space derivatives of P to be vanishing (αc = 0 in our model and ai = 0 in Mindlin),we have complete equivalence of the two models when using the parameters identification (7). We can hencestart noticing that we use in this model only 5 parameters instead of Mindlin’s 7 parameters. Things become

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even more interesting when looking at the terms in the energy which involve derivatives of the micro-straintensor P. Indeed, our relaxed model only provides 1 extra parameter, αc, instead of Mindlin’s 11 independentparameters (a1, a2, a3, a4, a5, a8, a10, a11, a13, a14, a15). We claim that the proposed relaxed model wasable to catch the basic features of observable material behaviors of materials with microstructure.

When considering Eringen model for micromorphic continua (see [21] p. 273 for the strain energy density),we can recover that our relaxed model can be obtained from Eringen’s one by setting

μ = μe − μc, λ = λe, τ = λe + λh, ν = −λe, η = μe + μh − μc, σ = μc − μe, k = 2μc,

(9)

τ7 = αc, τ9 = −αc, τ1 = τ2 = τ3 = τ4 = τ5 = τ6 = τ8 = τ10 = τ11 = 0. (10)

Also, in this case, we replace Eringen’s 7 parameters with only 5 parameters, and when considering termswith derivatives of the microstrain tensor P, we have only one additional parameter αc instead of Eringen’s 11parameters.

2.1.2 Governing equations

Imposing the first variation of the action functional to be vanishing (i.e., δA = 0), integrating by parts asuitable number of times, and considering arbitrary variations δχ and δP of the basic kinematical fields, weobtain the strong form of the bulk equations of motion of considered system which read

ρ ut t = Div[2μe sym (∇u − P)+ λetr (∇u − P)1 + 2μc skew (∇u − P)

],

η Pt t = 2μe sym (∇u − P)+ λetr (∇u − P)1 − 2μh sym P − λh tr P 1

+2μc skew (∇u − P)− αc Curl (Curl P) (11)

or equivalently, in index notation4

ρ ui = μe(ui, j j − Pi j, j + u j,i j − Pji, j

) + λe(u j, j i − Pj j,i

)δi j + μc

(ui, j j − Pi j, j − u j,i j + Pji, j

),

η Pi j = μe(ui, j − Pi j + u j,i − Pji

) + λe(uk,k − Pkk

)δi j − μh

(Pi j + Pji

) − λh Pkkδi j

+μc(ui, j − Pi j − u j,i + Pji

) + αc(Pi j,kk − Pik, jk

). (12)

2.2 The classical micromorphic continuum

From here on, we call classical micromorphic continuum a medium the energy of which is given by

W = μe ‖ sym (∇u − P) ‖2 + λe

2(tr (∇u − P))2 + μh ‖ sym P ‖2 + λh

2(tr P)2

+μc ‖ skew (∇u − P) ‖2 + αg

2‖∇ P‖2 . (13)

Our simple one-parameter formulation ‖∇P‖2 is, of course, not equivalent to the fully general 18-param-eters Mindlin–Eringen model. However, our formulation is the prototype of the no-absorption formulation,which does not lead to band-gaps. In this sense, we believe that the full Mindlin–Eringen model can showdifferent quantitative response with respect to our simplified model, but not with respect to band-gaps. Froma mathematical point of view, we believe that the difference is negligible.

2.2.1 Comparison with the Mindlin and Eringen models

Analogously to what done for the relaxed case, we can recover that the classical micromorphic continuum canbe seen as a particular case of Mindlin’s model by means of the parameter identification (7) to which one mustadd:

4 When using index notation for the components of the introduced tensor fields, we will denote partial derivative with respectto time with a superposed dot instead of a t subscript.

Wave propagation in relaxed micromorphic continua

a10 = αg, a1 = a2 = a3 = a4 = a5 = a6 = a7 = a8 = a9 = a11 = a12 = a13 = a14 = a15 = 0.

Analogously, the classical micromorphic continuum can be obtained from Eringen’s model by means of theparameter identification (9) to which one must add the conditions

τ7 = αg, τ1 = τ2 = τ3 = τ4 = τ5 = τ6 = τ8 = τ9 = τ10 = τ11 = 0.

2.2.2 Governing equations

The equations of motion are the same as Eq. (11), except for the gradient term in the second equation:

ρ ut t = Div[2μe sym (∇u − P)+ λetr (∇u − P)1 + 2μc skew (∇u − P)

],

η Pt t = 2μe sym (∇u − P)+ λetr (∇u − P)1 − 2μh sym P − λh tr P 1

+2μc skew (∇u − P)+ αg Div (∇P). (14)

The second equation can hence be rewritten in index notation as

η Pi j = μe(ui, j − Pi j + u j,i − Pji

) + λe(uk,k − Pkk

)δi j − μh

(Pi j + Pji

) − λh Pkkδi j

+μc(ui, j − Pi j − u j,i + Pji

) + αg Pi j,kk . (15)

3 Plane wave propagation in micromorphic media

In our study of wave propagation in considered micromorphic media, we will limit ourselves to the case ofplane waves traveling in an infinite domain. With this end in mind, we can suppose that the space dependenceof all the introduced kinematical fields is limited only to the component X of X which we also suppose to be thedirection of propagation of the considered wave. It is immediate that according to the Cartan–Lie decomposi-tion for the tensor P (see Eq. (1)), the component P11 of the tensor P itself can be rewritten as P11 = P D + P S

where we set

P S := 1

3(P11 + P22 + P33), P D := (dev sym P)11 . (16)

We also denote the components 12 and 13 of the symmetric and skew-symmetric part of the tensor P, respec-tively, as

(sym P)1ξ = P(1ξ), (skew P)1ξ = P[1ξ ], ξ = 1, 2. (17)

We finally introduce the last new variable

PV = P22 − P33. (18)

3.1 The relaxed micromorphic continuum

We want to rewrite the equations of motion (12) in terms of the new variables (16), (17) and, of course, of thedisplacement variables ui . Before doing so, we introduce the quantities5

cm =√αc

η, cs =

√μe + μc

ρ, cp =

√λe + 2μe

ρ,

ωs =√

2 (μe + μh)

η, ωp =

√(3λe + 2μe)+ (3λh + 2μh)

η, ωr =

√2μc

η, (19)

ωl =√λh + 2μh

η, ωt =

√μh

η.

With the proposed new choice of variables and considering definitions (19), we are able to rewrite the governingequations as different uncoupled sets of equations, namely:

5 Due to the chosen values of the parameters, which are supposed to satisfy (6), all the introduced characteristic velocitiesand frequencies are real. Indeed, the condition (3λe + 2μe) > 0, together with the condition μe > 0, imply the condition(λe + 2μe) > 0.

A. Madeo et al.

• A set of three equations only involving longitudinal quantities:

u1 = c2pu1,11 − 2μe

ρP D,1 − 3λe + 2μe

ρP S,1,

P D = 4

3

μe

ηu1,1 + 1

3c2

m P D,11 − 2

3c2

m P S,11 − ω2

s P D, (20)

P S = 3λe + 2μe

3ηu1,1 − 1

3c2

m P D,11 + 2

3c2

m P S,11 − ω2

p P S,

• Two sets of three equations only involving transverse quantities in the kth direction, with ξ = 2, 3:

uξ = c2s uξ,11 − 2μe

ρP(1ξ),1 + η

ρω2

r P[1ξ ],1,

P(1ξ) = μe

ηuξ,1 + 1

2c2

m P(1ξ),11 + 1

2c2

m P[1ξ ],11 − ω2s P(1ξ), (21)

P[1ξ ] = −1

2ω2

r uξ,1 + 1

2c2

m P(1ξ),11 + 1

2c2

m P[1ξ ],11 − ω2r P[1ξ ],

• One equation only involving the variable P(23):

P(23) = −ω2s P(23) + c2

m P(23),11, (22)

• One equation only involving the variable P[23]:

P[23] = −ω2r P[23] + c2

m P[23],11, (23)

• One equation only involving the variable PV :

PV = −ω2s PV + c2

m PV,11. (24)

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

It can be checked that in order to guarantee positive definiteness of the potential energy (6), the character-istic velocities and frequencies introduced in Eq. (19) cannot be chosen in a completely arbitrary way. In thenumerical simulations considered in this paper, the values of the elastic parameters are always chosen in orderto guarantee positive definiteness of the potential energy according to Eq. (6).

3.2 The classical micromorphic continuum

For the classical micromorphic continuum, introducing the new characteristic velocity

cg =√αg

η,

we get the following simplified one-dimensional equations

• Longitudinal

u1 = c2pu1,11 − 2μe

ρP D,1 − 3λe + 2μe

ρP S,1,

P D = 4

3

μe

ηu1,1 + c2

g P D,11 − ω2

s P D, (25)

P S = 3λe + 2μe

3ηu1,1 + c2

g P S,11 − ω2

p P S,

Wave propagation in relaxed micromorphic continua

• Two sets of three equations only involving transverse quantities in the kth direction, with ξ = 2, 3:

uξ = c2s uξ,11 − 2μe

ρP(1ξ),1 + η

ρω2

r P[1ξ ],1,

P(1ξ) = μe

ηuξ,1 + c2

g P(1ξ),11 − ω2s P(1ξ), (26)

P[1ξ ] = −1

2ω2

r uξ,1 + c2g P[1ξ ],11 − ω2

r P[1ξ ],

• One equation only involving the variable P(23):

P(23) = −ω2s P(23) + c2

g P(23),11, (27)

• One equation only involving the variable P[23]:

P[23] = −ω2r P[23] + c2

g P[23],11, (28)

• One equation only involving the variable PV :

PV = −ω2s PV + c2

g PV,11. (29)

4 Linear waves in micromorphic media

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

4.1 Micro-oscillations

We start studying a particular solution of the set of introduced differential equations by setting

ui = 0, i = 1, 2, 3, P(23) = Re{α(23)e

iωt}, P[23] = Re

{α[23]eiωt

}, PV = Re

{αV eiωt

},

P(1ξ) = Re{α(1ξ)e

iωt}, P[1ξ ] = Re

{α[1ξ ]eiωt

}, ξ = 2, 3.

Replacing these expressions for the unknown variables in each of Eqs. (20)–(24), noticing that the space deriv-atives are vanishing, one gets that the first of Eqs. (20) and (21) are such that the frequency calculated for thewaves u1 and uξ is vanishing, i.e.,

ω = 0. (30)

Moreover, from the remaining equations, one gets the values of frequencyω for the waves P D, P S, P(1ξ), P[1ξ ],P(23), P[23] and PV , respectively

ω2 = ω2s , ω2 = ω2

p ω2 = ω2s , ω2 = ω2

r , ω2 = ω2s , ω2 = ω2

r , ω2 = ω2s . (31)

It is clear that one gets exactly the same result when considering the complete Mindlin–Eringen model, sincethe only difference is on the space derivatives of P which do not intervene when studying micro-oscillations.The characteristic values of the frequencies given in Eq. (31) are fixed once the material parameters of thesystem are specified (see Eq. (19)), and they represent the limit values for k → 0 of the eigenvalues ω(k)associated with propagative waves, as it will be better shown in the next section. This preliminary study ofmicro-oscillations allows a precise classification of waves, which can propagate in a micromorphic medium.More particularly, we can distinguish two types of propagative waves:

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

A. Madeo et al.

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

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

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

• For the uncoupled waves P(23), P[23] and PV , only optic waves are found with cutoff frequencies ωs, ωr

and ωs . Moreover, the structure of the governing equations for P(23) and PV are formally identical (seeEqs. (22) and (24)) so that the associated dispersion relations will give rise to superimposed curves. In thelimit μc → 0 one has from Eq. (19) that ωr → 0. This means that in this limit case one has one acousticwaves and two superimposed optic waves with cutoff frequency ωs .

• For the longitudinal waves u1, P D, P S , we can identify one acoustic wave and two optic waves withcutoff frequenciesωs andωp. In the limitμc → 0, which implies thatωr → 0, the situation for longitudinalwaves remains unchanged.

• For the transverse waves uξ , P(1ξ), P[1ξ ], (ξ = 1, 2) we identify one acoustic wave and two optic waveswith cutoff frequencies ωs and ωr . In the limit μc → 0 which implies that ωr → 0, we have two acousticwaves and one optic wave with cutoff frequency ωs .

4.2 Planar wave propagation in the relaxed micromorphic continuum

We now look for a wave form solution of the previously derived equations of motion. We start from theuncoupled equations (22)–(24) and assume that the involved unknown variables take the harmonic form

P(23) = Re{β(23)e

i(k X−ωt)}, P[23] = Re

{β[23]e

i(k X−ωt)}, PV = Re

{βV ei(k X−ωt)

}, (32)

where β(23), β[23], and βV are the amplitudes of the three introduced waves. It can be remarked that the vari-ables P(23), P[23], and PV , respectively, represent transverse (with respect to wave propagation) micro-shear,transverse micro-rotation, and transverse micro-deformations at constant volume. Replacing this wave formin Eqs. (22)–(24) and simplifying, one obtains the following dispersion relations, respectively:

ω(k) =√ω2

s + k2c2m, ω(k) =

√ω2

r + k2c2m, ω(k) =

√ω2

s + k2c2m . (33)

We notice that for a vanishing wave number (k = 0), the dispersion relations for the three considered wavesgive nonvanishing frequencies which correspond to the ones calculated in the previous subsection for the samewaves. This is equivalent to say that the waves associated with the three considered variables P(23), P[23] andPV are the so-called optic waves. Moreover, we also notice that the dispersion relation for the variables P(23)

and PV is the same: This means that the wave form solutions for these variables coincide modulo a scalarmultiplication factor (see Eq. (32)).

We now want to study harmonic solutions for the differential systems (20) and (21). To do so, we introducethe unknown vectors v1 = (

u1, P D, P S)

and vξ = (uξ , P(1ξ), P[1ξ ]

), ξ = 2, 3 and look for wave form

solutions of equations (20) and (21) in the form

v1 = Re{βei(k X−ωt)

}, vξ = Re

{γ ξ ei(k X−ωt)

}, ξ = 2, 3 (34)

where β = (β1, β2, β3)T and γ ξ = (γ

ξ1 , γ

ξ2 , γ

ξ3 )

T are the unknown amplitudes of considered waves. Replacingthis expressions in equations (20) and (21) one gets, respectively,

A1 · β = 0, Aξ · γ ξ = 0, ξ = 2, 3, (35)

where

A1 =

⎛⎜⎜⎜⎝

−ω2 + c2p k2 i k 2μe/ρ i k (3λe + 2μe) /ρ

−i k 43 μe/η −ω2 + 1

3 k2c2m + ω2

s − 23 k2c2

m

− 13 i k (3λe + 2μe) /η − 1

3 k2 c2m −ω2 + 2

3 k2 c2m + ω2

p

⎞⎟⎟⎟⎠,

Wave propagation in relaxed micromorphic continua

A2 = A3 =

⎛⎜⎜⎝

−ω2 + k2c2s i k 2μe/ρ −i k η

ρω2

r ,

− i k 2μe/η, −2ω2 + k2c2m + 2ω2

s k2c2m

i k ω2r k2c2

m −2ω2 + k2c2m + 2ω2

r

⎞⎟⎟⎠.

In order to have nontrivial solutions of the algebraic systems (35), one must impose that

det A1 = 0, det A2 = 0, det A3 = 0, (36)

which allow us to determine the so-called dispersion relations ω = ω (k) for the longitudinal and transversewaves in the relaxed micromorphic conyinuum. As it will be better explained in the next section, the eigenvaluesω(k) solutions of (36) are associated both with optic waves and with acoustic waves.

4.3 Planar wave propagation in the classical micromorphic continuum

We want to deduce in this subsection the dispersion relations for the classical micromorphic model, i.e., forthe dispersion relations associated with the energy (13). If we replace the wave form solution (34) in thelongitudinal and transverse equations (25) and (26), we get

B1 · β = 0, Bξ · γ ξ = 0, ξ = 2, 3, (37)

where

B1 =

⎛⎜⎜⎝

−ω2 + c2p k2 i k 2μe/ρ i k (3λe + 2μe) /ρ

−i k 43μe/η −ω2 + k2c2

g + ω2s 0

− 13 i k (3λe + 2μe) /η 0 −ω2 + k2 c2

g + ω2p

⎞⎟⎟⎠ ,

B2 = B3 =

⎛⎜⎜⎝

−ω2 + k2c2s i k 2μe/ρ −i k η

ρω2

r ,

− i k 2μe/η, −2ω2 + 2 k2c2g + 2ω2

s 0

i k ω2r 0 −2ω2 + 2 k2c2

g + 2ω2r

⎞⎟⎟⎠ .

In order to have nontrivial solutions of the algebraic systems (37), one must impose that

det B1 = 0, det B2 = 0, det B3 = 0, (38)

which allow us to determine the so-called dispersion relations ω = ω (k) for the longitudinal and transversewaves in the classical micromorphic continuum.

As for the uncoupled waves, the dispersion relations associated with Eqs. (27), (28), and (29) are, respec-tively,

ω(k) =√ω2

s + k2c2g, ω(k) =

√ω2

r + k2c2g, ω(k) =

√ω2

s + k2c2g. (39)

5 The relaxed micromorphic model: numerical results

In this section, following Mindlin [21,44], we will show the dispersion relations ω = ω(k) associated withthe considered relaxed micromorphic model. The analysis of dispersion relations in [21,44] was of qualitativenature, due to the huge number of constitutive parameters (18 elastic coefficients) which are assumed to benonvanishing and to computational difficulties. On the other hand, thanks to the relaxed constitutive assump-tion (5), we are able to clearly show precise dispersion relations for the considered cases and to associate withfew constitutive parameters, the main mechanisms associated with wave propagation in micromorphic media.As principal obtained result, we will clearly point out that band-gaps can be forecast only when considering anonvanishing Cosserat couple modulus μc in our relaxed micromorphic model. This parameter is associatedwith the relative rotation of the microstructure with respect to the continuum matrix.

A. Madeo et al.

Table 1 Values of the parameters of the relaxed model used in the numerical simulations (column 1–13) and corresponding valuesof the Lamé parameters and of the Young modulus and Poisson ratio (column 14–17)

Parameter Value Unit

μe 200 MPaλe = 2μe 400 MPaμc = 2.2μe 440 MPaμh 100 MPaλh 100 MPaLc 3 mmLg 3 mmαc = μe L2

c 1.8 × 10−3 MPa m2

αg = μe L2g 1.8 × 10−3 MPa m2

ρ 2,000 Kg/m3

ρ’ 2,500 Kg/m3

d 2 mmη = d2 ρ′ 10−2 Kg/mλ 82.5 MPaμ 66.7 MPaE 170 MPaν 0.28 –

We start by showing in Table 1 (column 1–13) the values of the parameters of the relaxed model used in theperformed numerical simulations. In order to make the obtained results more exploitable, we also recall thatin [53], the following homogenized formulas were obtained which relate the parameters of the relaxed modelto the macroscopic Lamé parameters λ and μ which are usually measured by means of standard mechanicaltests

μe = μh μ

μh − μ, 2μe + 3λe = (2μh + 3λh) (2μ+ 3λ)

(2μh + 3λh)− (2μ+ 3λ). (40)

These relationships imply that the following inequalities are satisfied

μh > μ, 3λh + 2μh > 3λ+ 2μ.

It is clear that once the values of the parameters of the relaxed models are known, the standard Lamé parameterscan be calculated by means of formulas (40), which is what was done in Table 1 (column 14–17). For thesake of completeness, we also show in the same table the corresponding Young modulus and Poisson ratio,calculated by means of the standard formulas

E = μ (3λ+ 2μ)

λ+ μ, ν = λ

2 (λ+ μ). (41)

5.1 The relaxed micromorphic model with μc = 0

We start by showing the dispersion relations of the algebraic problem (36) for the particular case μc = 0.Figure 1 shows separately the behaviors of the uncoupled waves P(23), P[23], and PV (Fig. 1a), of the longi-tudinal waves u1, P D, P S (Fig. 1b) and of the transverse waves uξ , P(1ξ), P[1ξ ] (Fig. 1c).

We can recover from Fig. 1a one acoustic, rotational wave (T R A), and two superimposed optic waves(TSO and TCVO) with cut-off frequencyωs . The acoustic wave shows a nondispersive behavior (the dispersioncurve is a straight line). This implies that when considering relaxed micromorphic media with μc = 0, thereis always at least one wave for which the wave number is always real independently of the value of frequency.This fact guarantees wave propagation inside of the considered medium for all frequency ranges (no globalband-gaps).

Figure 1b shows that longitudinal waves indeed involve one acoustic wave (LA) and two optic waves (LO1and LO2) with cut-off frequencies ωp and ωs , respectively. Moreover, it is found that the acoustic wave hasan horizontal asymptote at ω = ωl . As a consequence, it can be seen that a frequency range (ωs, ωl) existsin which the wave number becomes imaginary for longitudinal waves. According to Eq. (34), an imaginary

Wave propagation in relaxed micromorphic continua

0 0.5 1 1.5 20

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c pk

cmk

csk

TSO TCVO

TRA

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ω p

ω s

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c pk

cmk

cskLO1

LO2

LA

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

ω t

ω s

1

Wavenumber k 1 mm

c pk

cmk

csk

TO

TA1

TA2

(c)(b)(a)

ω

Fig. 1 Dispersion relations ω = ω(k) for the relaxed model with vanishing Cosserat couple modulus (μc = 0): uncoupled waves(a), longitudinal waves (b), and transverse waves (c). TRA transverse rotational acoustic, TSO transverse shear optic, TCVOtransverse constant-volume optic, LA longitudinal acoustic, LO1−LO2 first and second longitudinal optic, TO transverse optic,TA1−TA2 first and second transverse acoustic

wave number k gives rise to solutions which are exponentials decaying in space. Waves of this type are theso-called standing waves which do not propagate, but keep oscillating in a limited region of space.

Finally, Fig. 1c confirms that for transverse waves, two acoustic waves (TA1 and TA2) and one optic wave(TO) with cut-off frequency ωs can be identified. One of the acoustic waves has an horizontal asymptote atω = ωt . It is easy to recognize that due to the existence of the nondispersive, transverse, acoustic wave TA1,there always exists at least one propagative wave for any chosen value of the frequency.

We can conclude that in general, when considering the relaxed micromorphic medium as a whole (all the12 waves), there always exist waves which propagate inside the considered medium independently of the valueof frequency. On the other hand, if one considers a particular case (obtained by imposing suitable kinemat-ical constraints) in which only longitudinal waves can propagate, then in the frequency range (ωs, ωl), onlystanding wave exist which do not allow for wave propagation. In this very particular case, the frequency range(ωs, ωl) can be considered as a band-gap for longitudinal waves. According to definitions given in Eq. (19),the depth of this frequency band is controlled by the three parameters μe, μh and λh .

5.2 The relaxed micromorphic model with μc > 0

In this subsection, we discuss the behavior with respect to wave propagation of relaxed micromorphic con-tinua in the general case in which the Cosserat couple modulus μc is assumed to be nonvanishing. We start byshowing the dispersion relations in Fig. 2.

As for the uncoupled waves, we have, in this case, only optic waves: two superimposed (TSO and TCVO)with cut-off frequency ωs and one (TRO) with cut-off frequency ωr (see Fig. 2a). When considering longi-tudinal waves (see Fig. 2b), the situation is unchanged with respect to the previous case (μc = 0) and onecan observe one acoustic wave (LA) and two optic waves (LO1 and LO2) with cut-off frequencies ωp andωs , respectively. If we finally consider the transverse waves (Fig. 2c), we can remark that there exists oneacoustic wave (TA) and two optic waves (TO1 and TO2) with cut-off frequencies ωs and ωr , respectively.It can be easily noticed that in all three cases, there exist frequency ranges in which no propagative wavecan be found. This means that the wave number becomes imaginary and only standing waves exist. For thesituation depicted in Fig. 2, the frequency ranges for which only standing waves exist are (0, ωs) for uncoupledwaves, (ωl , ωs) for longitudinal waves, and (ωt , ωs) for transverse waves. By the intersection of these threeintervals being nonempty, we can conclude that a frequency band-gap (ωl , ωs) exists in the considered relaxedmicromorphic medium corresponding to which any type of wave can propagate independently of the value ofk. More precisely, if one imagines to excite the considered medium with a signal the frequency of which fallsin the range (ωl , ωs), this signal cannot propagate inside the considered medium, but it keep oscillating closeto the point of application of the initial condition. It is clear that this feature is a fundamental tool to conceive

A. Madeo et al.

0 0.5 1 1.5 20

0.2

0.4

0.6

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ωr

ωs

1

Wavenumber k 1 mm

Freq

uenc

yω

106

rad

s

c pk

cmk

c sk

TRO

TSO TCVO

0 0.5 1 1.5 20

0.2

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ω p

ωs

1

Wavenumber k 1 mm

c pk

cmk

c sk

LO1

LO2

LA

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

ω t

ωr

ωs

1

Wavenumber k 1 mm

c pk

cmk

c sk

TO1

TO2

TA

(c)(b)(a)

Fig. 2 Dispersion relations ω = ω(k) for the relaxed model with nonvanishing Cosserat couple modulus (μc > 0): uncoupledwaves (a), longitudinal waves (b) and transverse waves (c). TRO transverse rotational optic, TSO transverse shear optic, TCVOtransverse constant-volume optic, LA longitudinal acoustic, LO1−LO2 first and second longitudinal optic, TA transverse acoustic,TO1−TO2 first and second transverse optic

micro-structured materials which can be used as pass- and stop-bands. Investigations in this sense would bringnew insights toward high-tech solutions in the field of vibration control and will be the object of forthcomingstudies. It is evident (see Eq. (19)) that, in general, the relative positions of the horizontal asymptotes ωl andωt as well as of the cut-off frequencies ωs, ωr , and ωp can vary depending on the values of the constitutiveparameters. In particular, we can observe that the relative position of the characteristic frequencies definedin (19) can vary depending on the values of the constitutive parameters. It can be checked that in the case inwhich λe > 0 and λh > 0 one always has ωp > ωs > ωt and ωl > ωt . The relative position of ωl and of ωscan vary depending on the values of the parameters λh and μh .

It is easy to verify that one can have band-gaps for longitudinal waves only if the horizontal asymp-tote ωl is such that ωs > ωl (which implies λh < 2μe). As for transverse waves, it can be checkedthat band-gaps can exist only if ωr > ωt (which implies μc > μh/2). A band-gap for the uncou-pled waves always exist (independently of the values of the constitutive parameters) for frequenciesbetween 0 and the smaller among ωr and ωs . It is clear that some stronger conditions are needed inorder to have a global band-gap which do not allow for any kind of waves (transverse, longitudinaland uncoupled) to propagate inside the considered microstructured material. More particularly, it canbe checked that in order to have a global band-gap, the following conditions must be simultaneouslysatisfied

ωs > ωl and ωr > ωl .

In terms of the constitutive parameters of the relaxed model, we can say that global band-gaps can exist, in thecase in which one considers positive values for the parameters λe and λh , if and only if we have simultaneously

0 < μe < +∞, 0 < λh < 2μe, μc >λh + 2μh

2. (42)

As far as negative values for λe and λh are allowed, the conditions for band-gaps are not so straightforward as(42). We do not consider this possibility in this paper, leaving this point open for further considerations.

We can conclude by saying that the fact of switching on a unique parameter, namely the Cosserat cou-ple modulus μc, allows for the description of frequency band-gaps in which no propagation can occur. Thisparameter can hence be seen as a discreteness quantifier which starts accounting for lattice discreteness as faras it reaches the threshold value specified in Eq. (42). The fact of being able to predict band-gaps by means ofa micromorphic model is a novel feature of the introduced relaxed model. Indeed, as it will be shown in theremainder of this paper, neither the classical micromorphic continuum model nor the Cosserat and the secondgradient ones are able to predict such band-gaps.

Wave propagation in relaxed micromorphic continua

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2k

c sk

TRO

0 0.5 1 1.5 20

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c pk

c gk

c g

2k

c sk

LA

0 0.5 1 1.5 20

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0.8

ω r

1

Wavenumber k 1 mm

c pk

c gk

c g

2k

c sk

TO

TA

(c)(b)(a)

ω

Fig. 3 Dispersion relationsω = ω(k) for the Cosserat model obtained by lettingμh → ∞ in the relaxed model: uncoupled waves(a), longitudinal waves (b) and transverse waves (c). TRO transverse rotational optic, LA longitudinal acoustic, TO transverseoptic, TA transverse acoustic

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

In this subsection, we analyze wave propagation in Cosserat-type media which can be obtained from the pro-posed relaxed model by letting the parameterμh tend to infinity. Indeed, since the strain energy density definedin (5) must remain finite, when letting μh → ∞ one must have Sym P → 0, hence P = skew P. Therefore,the strain energy density becomes the Cosserat energy (see [34,35])

WCoss (∇u, skew P) = μe ‖sym ∇u‖2 + λe

2(tr ∇u)2 + μc ‖skew (∇u − P)‖2 + α

2‖ Curl (skew P)‖2 .

The dispersion relations obtained for this particular limit case are depicted in Fig. 3. It can be immediatelynoticed that all the optic waves with cut-off frequency ωs do not propagate anymore since indeed ωs → ∞(see definition in Eq. (19)). The same is for the optic longitudinal wave with cut-off frequency ωp. As for theacoustic waves, they exist in the limit Cosserat medium, but they do not have horizontal asymptotes anymoresince ωl and ωt tend to infinity as well. We finally end up with a medium in which we can observe oneoptic wave (TRO) associate with the variable P[23] (micro-rotation) with cut-off frequency ωr , one acousticnondispersive longitudinal wave (LA), one acoustic slightly dispersive transverse wave (TA), and one optictransverse wave (TO) with cut-off frequency ωr . It is easy to remark that no band-gaps can be described in theframework of the considered Cosserat medium.

The dispersive behavior of Cosserat media shown in Fig. 3 fits with known results in the literature. Indeed,a direct comparison with the micropolar medium studied in [21] (p. 150) can be made, by simply consideringthe parameters identification (9), (10). Moreover, Lakes states in [37] that “Dilatational waves propagate non-dispersively, i.e., with velocity independent of frequency, in an unbounded isotropic Cosserat elastic mediumas in the classical case. Shear waves propagate dispersively in a Cosserat solid (Eringen, 1968). A new kindof wave associated with the micro-rotation is predicted to occur in Cosserat solids”. We also remark that thebehavior for high frequencies shown in Fig. 3 coincides with that of acceleration waves in micropolar media,see [3,20].

Indeed, the linear Cosserat model is undoubtedly the most studied generalized continuum model. This doesnot mean, however, that its status as a useful model for the description of material behavior is unchallenged.Quite to the contrary, it appears that after 40 years of intensive research, not one material has been conclusivelyestablished as a Cosserat material. We refer to the discussion in [34,35,54,56–60].

By looking at the dispersion behavior of the linear Cosserat model and comparing it with the more generalmicromorphic (and relaxed micromorphic model), we get a glimpse on why the status of the linear Cosseratmodel is really challenging. The equations appear as a formal limit in which μh → ∞, while 0 < μc < ∞.The processμh → ∞ corresponds conceptually to assume that the substructure cannot deform elastically; nev-ertheless, the substructures may mutually interact through the curvature energy, which itself is only involvingthe Curl-operator, since the micro-distortion is constrained to be skew-symmetric.

A. Madeo et al.

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LO2

LA

0 0.5 1 1.5 20

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ω t

ωr

ωs

1

Wavenumber k 1 mm

c pk

c gk

c sk

TO1

TO2

TA

(c)(b)(a)

ω

Fig. 4 Dispersion relations ω = ω(k) for the classical model: uncoupled waves (a), longitudinal waves (b) and transverse waves(c) for a classical micromorphic continuum. TRO transverse rotational optic, TSO transverse shear optic, TCVO transverse con-stant-volume optic, LA longitudinal acoustic, LO1−LO2 first and second longitudinal optic, TA transverse acoustic, TO1−TO2:first and second transverse optic

The usefulness of a geometrically nonlinear Cosserat model and, however, is not in general questioned.Indeed, it has been shown in [43] that a Cosserat-type continuum theory can be of use to describe the exper-imental behavior of granular phononic crystals. Indeed, it is known that grain packings appear as examplesof materials in which particle micro-rotations and micro-deformations are possible as shown by experimentalmeasurements in [45] and discussed in micro-macro derivations in [9,46]. One may avoid the deficiencies ofthe linear model and the linear coupling, see e.g., [47–50,52].

6 The classical micromorphic model: numerical results

In this section, we show the dispersion relations for a classical micromorphic continuum.These dispersion relations are plotted in Fig. 4: We recover a similar behavior with respect to the one qual-

itatively sketched in [44] and [21]. Indeed, we claim that the classical micromorphic model associated with thestrain energy density (13) is qualitatively equivalent to the full 18-parameters Mindlin and Eringen micromor-phic model. It is clear from Fig. 4 that even if the behavior of such medium is the same as the one observed inFig. 2 for the relaxed micromorphic medium when considering small wave numbers (large wavelengths), thesituation is completely different when considering small wavelengths. Indeed, the first macroscopic featurethat can be observed is that no band-gaps can be forecasted by a Mindlin–Eringen model due to the fact thatno horizontal asymptotes exist for acoustic waves.

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

In this subsection, we analyze wave propagation in second gradient media which can be obtained as a suitablelimit case of the classical micromorphic continua. More particularly, if we let simultaneously

μe → ∞, μc → ∞,

this implies that

sym P → sym ∇u skew P → skew ∇u and hence P → ∇u.

This means that the energy (13) reduces to

W2G (∇u,∇∇u) = μh ‖ sym ∇u ‖2 + λh

2(tr ∇u)2 + αg

2‖∇∇u‖2 (43)

Wave propagation in relaxed micromorphic continua

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0 0.5 1 1.5 20

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c gk

TA

(c)(b)(a)

ω

Fig. 5 Dispersion relations ω = ω(k) for the second gradient model obtained as limit case of the classic model by lettingμe → ∞ and μc → ∞: uncoupled waves (a), longitudinal waves (b) and transverse waves (c) for a second gradient continuum.LA longitudinal acoustic, TA transverse acoustic

which is indeed a second gradient energy for linear-elastic, isotropic media (see e.g., [17,64]). Indeed, govern-ing equations for second (and higher) gradient continua can be also obtained by using a simplified kinematicswith respect to the one introduced in this paper and by adopting variational principles (see e.g., [18,64,65]). Onthe other hand, the fact of obtaining a second gradient model as the limit case of a micromorphic one can havemany advantages either with respect to numerical efficiency and to physical interpretation of the boundaryconditions (see e.g., [13,24,51]).

It is immediate to notice from Fig. 5 that no coupled waves can exist in this limit case, since the onlyindependent kinematical variable turns to be the displacement u. As for the longitudinal and transverse waves,they can only be of acoustic type (LA and TA). This is in agreement with what is known from the literature(see e.g., [15,62]): Second gradient media are such that only acoustic dispersive waves can propagate insidethe medium. In [15], Eq. (13), dispersion relations for planar waves in isotropic second gradient media arepresented associated with a strain energy density of the type (43), but with a kinetic energy which is simplerthan the one used in this paper and which can be obtained from (3) by setting η = 0. It can be checked bymeans of the simple application of a variational principle that the equations of motion associated with thestrain energy density (43) and with the kinetic energy T = 1

2 ρ ‖ ut ‖2 + 12 η

∥∥ (∇u)t∥∥2, can be written as

ρ ui − η ui, j j = μh(ui, j j + u j,i j

) + λhu j, j i − αgui, jkk j . (44)

It can also be cheked that the dispersion relations for longitudinal and transverse planar waves (obtained bystudying the wave form solution of Eq. (44) when setting i = 1 and i = 2, 3, respectively) are, respectively,given by

ω =√αg k4 + (2μh + λh) k2

ρ + η k2 , ω =√αg k4 + μh k2

ρ + η k2 ,

which indeed correspond to the two acoustic waves depicted in Fig. 5. It is immediate to check that for vanish-ing wave number k, the frequency is always vanishing, which means that only acoustic waves can propagatein such kind of media, and optic waves are hence forbidden a priori in this kind of models. Moreover, it ispossible to remark that the frequency goes to infinity when the wave number tends to infinity (no possibilityof horizontal asymptotes). Finally, we notice that standing waves can also be present in second gradient mediasince purely imaginary wave numbers may exist which give rise to real frequencies. Indeed, when replacing animaginary wave number in the wave form ei(k X−ωt), it is immediate to see that the solution is an exponentialdecaying in space. We can summarize by saying that when considering second gradient media, one has thatfor any value of the frequency, propagative acoustic waves and standing waves exist. It can also be remarkedthat the inertial term 1

2 η∥∥ (∇u)t

∥∥2, plays a role on the possibility of changing the concavity of the dispersion

relations. Indeed, if the microscopic density η tends to zero, then it is clear that the concavity of the dispersioncurves cannot change. On the other hand, this change of concavity is possible when η �= 0.

A. Madeo et al.

We explicitly remark that in the considered second gradient medium, no band-gaps can be forecasted. Nev-ertheless, it is known that if surfaces of discontinuity of the material properties are considered, then reflectedand transmitted energy can be strongly influenced by the value of the second gradient parameter depending onthe considered jump conditions imposed at the interface itself (see e.g., [15,62]).

7 Conclusions

In this paper, we used the relaxed micromorphic model proposed in [27,53] to study wave propagation inunbounded continua with microstructure. The quoted relaxed model only counts 6 elastic parameters againstthe 18 parameters appearing in Mindlin–Eringen micromorphic theory (cf. [21,44]). Despite the reduced num-ber of parameters, we claim that the proposed relaxed model is fully able to account for the description ofthe mechanical behavior of micromorphic media. More precisely, we have shown that only the relaxed mi-cromorphic continuum model with nonvanishing Cosserat couple modulus μc is able to predict frequencyband-gaps corresponding to which no wave propagation can occur. The main findings of the present study canbe summarized as follows:

• The relaxed micromorphic model (6 parameters and only the term ‖ Curl P ‖2 appearing in the strain energy)is fully able to describe the main features of the mechanical behavior of micromorphic continua, first of allfor what concerns the possibility of describing the presence of band-gaps.

• A reduced relaxed model (5 parameters) can be obtained from the previous one by setting the Cosseratcouple modulus μc → 0. This simplified model produces a symmetric Cauchy force stress tensor and isstill able to describe the mechanical behavior of a big class of microstructured continua. Nevertheless, thismodel excludes a priori the presence of band-gaps. This means that it is not suitable to fully describe thebehavior of sophisticated microstructured engineering materials such as phononic crystals and lattice struc-tures. However, practically all known engineering materials do not show band-gaps. For these materials,the reduced model is our alternative of choice in the family of micromorphic models.

• The Cosserat model is obtained as a degenerate limit case of the proposed relaxed model (the full relaxedmodel with 6 parameters) when letting μh → ∞. The Cosserat model is not able to describe band-gapsbut it introduces a non-symmetric Cauchy stress tensor.

• The classical continuum model (6 parameters but the full ‖∇P ‖2 appearing in the strain energy) is qual-itatively equivalent to the full 18-parameters Mindlin–Eringen micromorphic continuum model. This istrue since the classical model is controlling all the kinematical fields of the full model, i.e., it is uniformlypoint-wise definite. The classical continuum model is not able to forecast band-gaps.

• Second gradient theories can be obtained as a limit case of the classical micromorphic continuum modelby letting μe → ∞, μc → ∞. These theories are not able to account for the presence of band-gaps.

We can conclude that only the 6-parameters relaxed model proposed in [27,53] and used in this paper to studywave propagation in microstructured media was able to describe the presence of band-gaps. These band-gapsare seen to be “switched on” by a unique constitutive parameter, namely the Cosserat couple modulus μc.The proposed relaxed micromorphic model is hence suitable to be used for the conception and optimizationof metamaterials to be used for vibration control.

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

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