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We present an inertia-augmented relaxed micromorphic model that enriches the relaxed micromorphic model previously introduced by the authors via a term Curl P⋅ in the kinetic energy density. This enriched model allows us to obtain a good overall fitting of the dispersion curves while introducing the new possibility of describing modes with negative group velocity that are known to trigger negative refraction effects. The inertia-augmented model also allows for more freedom on the values of the asymptotes corresponding to the cut-offs. In the previous version of the relaxed micromorphic model, the asymptote of one curve (pressure or shear) is always bounded by the cut-off of the following curve of the same type. This constraint does not hold anymore in the enhanced version of the model. While the obtained curves’ fitting is of good quality overall, a perfect quantitative agreement must still be reached for very small wavelengths that are close to the size of the unit cell., Mathematics and mechanics of solids

In this thesis we consider wave propagation phenomena in generalized continuum models. The classical linear continuum theory is inadequate to explain new effects of microstructural motion; for example, in an unbounded elastic medium, the propagation of plane waves is non-dispersive. On the contrary, experiments with real solids show that waves propagation is dispersive, i.e., the wave speed is depending on the frequency. To incorporate better the microstructure of the matter into the classical theory, generalized continuum models may be employed. The higher gradient elasticity theories and micromorprhic models are the most familiar models. The mechanical behaviour of isotropic Eringen-Mindlin micromorphic media is generally defined by means of 18 elastic constants. However, the huge set of parameters cannot unveil the main characteristics of the micromorphic media due to inevitable computational difficulties. To counter this situation, Neff recently presented the relaxed micromorphic model, featuring six parameters and having the capability to characterize the crucial feature of the micromorphic continua comprehensively. It is a well-documented fact that the relaxed micromorphic model includes various classical models as special cases, e.g., the microstretch model, the linear Cosserat model, microvoids model and the classical linear model. Traveling waves can exist within a small depth from a free surface of an elastic continuum, while the bulk of the continuum remains almost at rest. Such waves are called Rayleigh waves, named after the English physicist Lord Rayleigh, who carried out pioneering work in studies of wave propagations in isotropic elastic media. The studies of Rayleigh waves captivated the attention of many scientists owing to its industrial application such as material characterization, nondestructive evaluation, acoustic microscopy and geophysical exploration. These waves are also employed to detect cracks and other defects in the material. Recently, Mielke and Fu have devised a new method to study the surface wave propagation in an anisotropic linear elastic Cauchy material. This method is based on the surface impedance matrix, which is a solution of the algebraic Riccati equation. They presented the proof of the existence and the uniqueness of the surface wave speed of anisotropic elastic material by carrying out an algebraic matrix analysis of the surface impedance matrix in the subsonic range. The studies of Rayleigh waves related to generalized continuum theories are scarce in the literature. In fact, most of the studies focus on the computational aspect rather than establishing a unified theory for the existence and uniqueness of the Rayleigh wave speed. In this thesis we address the existence and uniqueness question for Rayleigh waves in a linear Cosserat material. For this task we apply and generalize the method by Mielke and Fu. After a general introduction (including some necessary mathematical definitions) we revisit first Mielke and Fu���s method as applied to classical linear isotropic elasticity. This serves the didactic purpose but we are already able to derive the specific form of the secular equation. Then we generalize the method to the Cosserat continuum obtaining novel existence and uniqueness results. Detailed comparisons (analytical and numerical) with alternative classical approach are carried out (these approaches do not lead to existence and uniqueness results). The main part of the thesis is accepted for publication [1]. [1] H. Khan, I.-D. Ghiba, A. Madeo, and P. Neff. ���Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects���. arXiv preprint arXiv:2104.13143 (2021), to appear in Wave Motion (2022)., In dieser Arbeit betrachten wir Wellenausbreitungsph��nomene in verallgemeinerten Kontinuumsmodellen. Die klassische lineare Kontinuumstheorie ist unzureichend, um neue Effekte der mikrostrukturellen Bewegung zu beschreiben; in einem unbegrenzten elastischen Medium beispielsweise ist die Ausbreitung ebener Wellen nicht-dispersiv. Im Gegensatz dazu zeigen Experimente mit realen Festk��rpern, dass die Wellenausbreitung dispersiv ist, d. h. die Wellengeschwindigkeit ist von der Frequenz abh��ngig. Um die Mikrostruktur der Materie besser in die klassische Theorie einzubeziehen, k��nnen verallgemeinerte Kontinuumsmodelle verwendet werden. Die Theorien der h��heren Gradientenelastizit��t und der mikromorphen Modelle sind die bekanntesten Modelle. Das mechanische Verhalten isotroper mikromorpher Eringen-Mindlin-Medien wird im Allgemeinen mit Hilfe von 18 Konstanten der Elastizitat definiert. Aufgrund der unvermeidlichen Rechenschwierigkeiten kann die gro��e Anzahl von Parametern jedoch nicht die wichtigsten Eigenschaften der mikromorphen Medien enth��llen. Um dieser Situation entgegenzuwirken, hat Neff vor kurzem ein relaxiertes mikromorphes Modell mit sechs Parametern vorgestellt, das in der Lage ist, die entscheidenden Merkmale der mikromorphen Kontinua umfassend zu charakterisieren. Es wurde bereits gezeigt, dass das relaxierte mikromorphe Modell verschiedene klassische Modelle als Spezialf��lle enth��lt, z. B. das Mikrodehnungsmodell, das lineare Cosserat-Modell, das Modell der Mikrovoids und das klassische lineare Modell. Fortschreitendewellen k��nnen innerhalb einer geringen Tiefe von einer freien Oberfl��che eines elastischen Kontinuums existieren, w��hrend der Hauptteil des Kontinuums nahezu in Ruhe verbleibt. Solche Wellen werden Rayleigh-Wellen genannt, benannt nach dem englischen Physiker Lord Rayleigh, der Pionierarbeit bei der Untersuchung der Wellenausbreitung in isotropen elastischen Medien geleistet hat. Die Untersuchungen der Rayleigh-Wellen erregten die Aufmerksamkeit vieler Wissenschaftler aufgrund ihrer industriellen Anwendung, z. B. bei der Materialcharakterisierung, den zerst��rungsfreien Pr��fverfahren, der akustischen Mikroskopie und der geophysikalischen Erkundung. Diese Wellen werden auch eingesetzt, um Risse und andere Defekte im Material aufzusp��ren. K��rzlich haben Mielke und Fu eine neue Methode zur Untersuchung der Ausbreitung von Oberfl��chenwellen in einem anisotropen linear elastischen Material entwickelt. Diese Methode basiert auf der Oberfl��chenimpedanzmatrix, die eine L��sung der algebraischen Riccati-Gleichung ist. Sie haben den Beweis f��r die Existenz und die Eindeutigkiet der Oberfl��chenwellengeschwindigkeit eines anisotropen elastischen Materials erbracht, indem sie eine algebraische Matrixanalyse der Oberfl��chenimpedanzmatrix im Unterschallbereich durchgef��hrt haben. Die Studien ��ber Rayleigh-Wellen im Zusammenhang mit verallgemeinerten Kontinuumstheorien sind in der Literatur sp��rlich. In der Tat konzentrieren sich die meisten Studien auf den rechnerischen Aspekt, anstatt eine einheitliche Theorie f��r die Existenz und Eindeutigkeit der Rayleigh-Wellengeschwindigkeit aufzustellen. In dieser Arbeit befassen wir uns mit der Frage der Existenz und Eindeutigkeit von Rayleigh Wellen im linearen Cosserat-Material. F��r diese Aufgabe verwenden und verallgemeinern wir die Methode von Mielke und Fu. Nach einer allgemeinen Einf��hrung (einschlie��lich einiger notwendiger mathematischer Definitionen) gehen wir zun��chst auf die Methode von Mielke und Fu ein, die auf die klassische lineare isotrope Elastizit��t angewendet wird. Dies dient einem didaktischen Zweck, aber wir sind bereits in der Lage, die spezielle Form der s��kularen Gleichung herzuleiten. Dann verallgemeinern wir die Methode auf das Cosserat-Kontinuum und erhalten neue Existenz und Eindeutigkeitsergebnisse. Detaillierte Vergleiche (analytisch und numerisch) mit alternativen klassischen Ans��tzen werden durchgef��hrt (diese Ans��tze f��hren nicht zu Existenz- und Eindeutigkeitsresultaten). Der Hauptteil der Arbeit ist bereits zur Ver��ffentlichung angenommen worden [1]. [1] H. Khan, I.-D. Ghiba, A. Madeo, and P. Neff. ���Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects���. arXiv preprint arXiv:2104.13143 (2021), to appear in Wave Motion (2022).

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size effects., Archive of applied mechanics; Vol. 93. 2023, Issue1, pp 5-21

We solve the St.Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical micromorphic model. The results can be used to determine the material parameters of these models. Special attention is given to the possible nonphysical stiffness singularity for a vanishing rod diameter, since slender specimens are in general described as stiffer., 41 pages, 29 figures

While the design of always new metamaterials with exotic static and dynamic properties is attracting deep attention in the last decades, little effort is made to explore their interactions with other materials. This prevents the conception of (meta-)structures that can enhance metamaterials' unorthodox behaviours and that can be employed in real engineering applications. In this paper, we give a first answer to this challenging problem by showing that the relaxed micromorphic model with zero static characteristic length can be usefully applied to describe the refractive properties of simple meta-structures for extended frequency ranges and for any direction of propagation of the incident wave. Thanks to the simplified model's structure, we are able to efficiently explore different configurations and to show that a given meta-structure can drastically change its overall refractive behaviour when varying the elastic properties of specific meta-structural elements. In some cases, changing the stiffness of a homogeneous material which is in contact with a metamaterial's slab, reverses the structure's refractive behaviour by switching it from an acoustic screen (total reflection) into an acoustic absorber (total transmission). The present paper clearly indicates that, while the study and enhancement of the intrinsic metamaterials' properties is certainly of great importance, it is even more challenging to enable the conception of meta-structures that can eventually boost the use of metamaterials in real-case applications., 21 pages, 14 figures

We consider the cylindrical bending problem for an infinite plate as modelled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length scale effects in the sense that thinner specimens are comparatively stiffer. We provide the analytical solution for each case and exhibit the predicted bending stiffness. The relaxed micromorphic continuum shows bounded bending stiffness for arbitrary thin specimens, while classical micromorphic continuum or gradient elasticity as well as Cosserat models [35] exhibit unphysical unbounded bending stiffness for arbitrary thin specimens. This finding highlights the advantage of using the relaxed micromorphic model, which has a definite limit stiffness for small samples and which aids in identifying the relevant material parameters., Comment: 50 pages, 28 figures (38 pictures)

International audience; In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micro-morphic model, also introducing their second order counterpart by using a Voigt-type vector notation. In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit case Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial the meso-and micro-coefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

Mechanical microstructured metamaterials are increasingly gaining attention from the scientific and engineering community. The question of modeling the behavior of metamaterials is of extreme importance. Some choose an approach, which is reminiscent of the classical theory of elasticity: enriched continuum mechanics. We employ the enriched continuum model named relaxed micromorphic model in order to study wave propagation and scattering at interfaces between materials and metamaterials. Dealing with the correct boundary conditions at the macroscopic scale becomes challenging. We show how finite-domain boundary value problems can be set-up in the framework of the relaxed micromorphic model. We set up the full plane wave solution of the scattering from an interface separating a Cauchy medium from a relaxed micromorphic one. Both media are isotropic and semi-infinite. Generalized macroscopic boundary conditions are presented, which allow the effective description of the scattering properties of an interface between a homogeneous solid and a mechanical metamaterial. The associated generalized energy flux is introduced. We show that the contrast of the macroscopic stiffnesses of the two media, together with the type of boundary conditions strongly influence the onset of Stoneley waves at the interface. This allows to tailor the scattering properties of the interface at both low and high frequencies, ranging from zones of complete transmission to zones of zero transmission well beyond the band-gap. We then consider a bulk wave propagation problem and show that the transient waveforms arising from several localised pulses in a micro-structured material can be reproduced. We compare the dynamic response of a bounded micro-structured material to that of bounded continua with special kinematic properties. We show that, while the Cauchy theory is able to describe the overall behavior of the metastructure only at low frequencies, the relaxed micromorphic model goes far beyond by giving a correct description of the pulse propagation in the frequency bandgap and at frequencies intersecting the optical branches. Finally, we present the case of a metamaterial slab of finite width. Its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to numerical simulations encoding all details of the selected microstructure. The reflection coefficient obtained via the two methods is presented as a function of the frequency and the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies. The case of a semi-infinite metamaterial is also presented and is seen to be a reliable measure of the average behavior of the finite metastructure., Les métamatériaux mécaniques microstructurés attirent de plus en plus l'attention de la communauté scientifique et technique. Nous choisissons une approche qui évoque la théorie classique de l'élasticité : celle de la mécanique des milieux continus enrichie. Le but de cette thèse est d'utiliser le nouveau modèle de continuité enrichie appelé modèle micromorphique relaxé détendu afin d'étudier la propagation des ondes et les phénomènes de diffusion aux interfaces entre matériaux et métamatériaux. Lorsqu'il s'agît d'étudier les propriétés de diffusion de structures de taille finie, il devient difficile de traiter les conditions limites correctes. Nous montrons comment les problèmes aux valeurs limites de domaines finis peuvent être mis en place dans le cadre du modèle micromorphique relaxé. Nous mettons en place la solution d'onde plane complète de la diffusion à partir d'une interface séparant un milieu de Cauchy d'un milieu micromorphique relaxé. Les conditions aux limites macroscopiques généralisées sont présentées. Ils permettent de bien décrire les propriétés de diffusion. Le flux d'énergie généralisé associé est introduit. On considère deux cas différents dans lesquels le milieu homogène gauche est soit plus rigide soit plus flexible que le métamatériau droit et le coefficient de transmission est obtenu en fonction de la fréquence et de la direction de propagation. Le contraste des raideurs macroscopiques des deux milieux, influencent l'apparition des ondes Stoneley. On considère par la suite un problème de propagation des ondes de volume et on démontre que les formes d'ondes transitoires résultant de plusieurs impulsions localisées dans un matériau microstructuré peuvent être reproduite. Nous comparons la réponse dynamique d'un matériau microstructuré et lié à celle d'un milieu lié avec des propriétés cinématiques particulières. On démontre que, bien que la théorie de Cauchy soit capable de décrire le comportement global de la métastructure à de basses fréquences, le modèle micromorphique détendu va bien au-delà en donnant une description correcte de la propagation de l'impulsion dans la bande de fréquence et à des fréquences qui croisent les branches optiques. Enfin, on présente le cas d'une dalle de métamatériau de largeur finie. Ses propriétés de diffusion sont étudiées en utilisant une solution semi-analytique du modèle micromorphique relaxé et comparées à des simulations. Le coefficient de réflexion obtenu par les deux méthodes est présenté en fonction de la fréquence et la direction de propagation de l'onde incidente. On trouve un excellent accord pour une large gamme de fréquences, allant de la limite des ondes longues aux fréquences au-delà de la première limite de la bande, et pour des angles d'incidence allant d'une incidence normale à une incidence presque parallèle. Le cas d’un métamatériau semi-infinie est également présenté et est considéré comme une mesure fiable du comportement moyen de la métastructure finie.

Mechanical microstructured metamaterials are increasingly gaining attention from the scientific and engineering community. The question of modeling the behavior of metamaterials is of extreme importance. Some choose an approach, which is reminiscent of the classical theory of elasticity: enriched continuum mechanics. We employ the enriched continuum model named relaxed micromorphic model in order to study wave propagation and scattering at interfaces between materials and metamaterials. Dealing with the correct boundary conditions at the macroscopic scale becomes challenging. We show how finite-domain boundary value problems can be set-up in the framework of the relaxed micromorphic model. We set up the full plane wave solution of the scattering from an interface separating a Cauchy medium from a relaxed micromorphic one. Both media are isotropic and semi-infinite. Generalized macroscopic boundary conditions are presented, which allow the effective description of the scattering properties of an interface between a homogeneous solid and a mechanical metamaterial. The associated generalized energy flux is introduced. We show that the contrast of the macroscopic stiffnesses of the two media, together with the type of boundary conditions strongly influence the onset of Stoneley waves at the interface. This allows to tailor the scattering properties of the interface at both low and high frequencies, ranging from zones of complete transmission to zones of zero transmission well beyond the band-gap. We then consider a bulk wave propagation problem and show that the transient waveforms arising from several localised pulses in a micro-structured material can be reproduced. We compare the dynamic response of a bounded micro-structured material to that of bounded continua with special kinematic properties. We show that, while the Cauchy theory is able to describe the overall behavior of the metastructure only at low frequencies, the relaxed micromorphic model goes far beyond by giving a correct description of the pulse propagation in the frequency bandgap and at frequencies intersecting the optical branches. Finally, we present the case of a metamaterial slab of finite width. Its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to numerical simulations encoding all details of the selected microstructure. The reflection coefficient obtained via the two methods is presented as a function of the frequency and the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies. The case of a semi-infinite metamaterial is also presented and is seen to be a reliable measure of the average behavior of the finite metastructure.; Les métamatériaux mécaniques microstructurés attirent de plus en plus l'attention de la communauté scientifique et technique. Nous choisissons une approche qui évoque la théorie classique de l'élasticité : celle de la mécanique des milieux continus enrichie. Le but de cette thèse est d'utiliser le nouveau modèle de continuité enrichie appelé modèle micromorphique relaxé détendu afin d'étudier la propagation des ondes et les phénomènes de diffusion aux interfaces entre matériaux et métamatériaux. Lorsqu'il s'agît d'étudier les propriétés de diffusion de structures de taille finie, il devient difficile de traiter les conditions limites correctes. Nous montrons comment les problèmes aux valeurs limites de domaines finis peuvent être mis en place dans le cadre du modèle micromorphique relaxé. Nous mettons en place la solution d'onde plane complète de la diffusion à partir d'une interface séparant un milieu de Cauchy d'un milieu micromorphique relaxé. Les conditions aux limites macroscopiques généralisées sont présentées. Ils permettent de bien décrire les propriétés de diffusion. Le flux d'énergie généralisé associé est introduit. On considère deux cas différents dans lesquels le milieu homogène gauche est soit plus rigide soit plus flexible que le métamatériau droit et le coefficient de transmission est obtenu en fonction de la fréquence et de la direction de propagation. Le contraste des raideurs macroscopiques des deux milieux, influencent l'apparition des ondes Stoneley. On considère par la suite un problème de propagation des ondes de volume et on démontre que les formes d'ondes transitoires résultant de plusieurs impulsions localisées dans un matériau microstructuré peuvent être reproduite. Nous comparons la réponse dynamique d'un matériau microstructuré et lié à celle d'un milieu lié avec des propriétés cinématiques particulières. On démontre que, bien que la théorie de Cauchy soit capable de décrire le comportement global de la métastructure à de basses fréquences, le modèle micromorphique détendu va bien au-delà en donnant une description correcte de la propagation de l'impulsion dans la bande de fréquence et à des fréquences qui croisent les branches optiques. Enfin, on présente le cas d'une dalle de métamatériau de largeur finie. Ses propriétés de diffusion sont étudiées en utilisant une solution semi-analytique du modèle micromorphique relaxé et comparées à des simulations. Le coefficient de réflexion obtenu par les deux méthodes est présenté en fonction de la fréquence et la direction de propagation de l'onde incidente. On trouve un excellent accord pour une large gamme de fréquences, allant de la limite des ondes longues aux fréquences au-delà de la première limite de la bande, et pour des angles d'incidence allant d'une incidence normale à une incidence presque parallèle. Le cas d’un métamatériau semi-infinie est également présenté et est considéré comme une mesure fiable du comportement moyen de la métastructure finie.

The conception of new metamaterials showing unorthodox behaviors with respect to elastic wavepropagation has become possible in recent years thanks to powerful dynamical homogenization techniques. Such methods effectively allow to describe the behavior of an infinite medium generated by periodically architectured base materials. Nevertheless, when it comes to the study of the scattering properties of finite-sized structures, dealing with the correct boundary conditions at the macroscopicscale becomes challenging. In this paper, we show how finite-domain boundary value problems canbe set-up in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized traction when non-local effects areneglected.The case of a metamaterial slab of finite width is presented, its scattering properties are studied viaa semi-analytical solution of the relaxed micromorphic model and compared to numerical simulationsencoding all details of the selected microstructure. The reflection coefficient obtained via the twomethods is presented as a function of the frequency and of the direction of propagation of the incidentwave. We find excellent agreement for a large range of frequencies going from the long-wave limitto frequencies beyond the first band-gap and for angles of incidence ranging from normal to nearparallel incidence. The case of a semi-infinite metamaterial is also presented and is seen to be areliable measure of the average behavior of the finite metastructure. A tremendous gain in termsof computational time is obtained when using the relaxed micromorphic model for the study of theconsidered metastructure.

International audience; In this paper the relaxed micromorphic continuum model with weighted free and gradient microinertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths arriving down to the size of the unit cell. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic structure as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure.

In this paper we study the reflective properties of a 2D interface separating a homogeneous solid from a band-gap metamaterial by modeling it as an interface between a classical Cauchy continuum and a relaxed micromorphic medium. We show that the proposed model is able to predict the onset of Stoneley interface waves at the considered interface both at low and high-frequency regimes. More precisely, critical angles for the incident wave can be identified, beyond which classical Stoneley waves, as well as microstructure-related Stoneley waves appear. We show that this onset of Stoneley waves, both at low and high frequencies, strongly depends on the relative mechanical properties of the two media. We suggest that a suitable tailoring of the relative stiffnesses of the two media can be used to conceive "smart interfaces" giving rise to wide frequency bounds where total reflection or total transmission may occur.

In the present paper, the material parameters of the isotropic relaxed micromorphic model derived for a specific metamaterial in a previous contribution are used to model its transmission properties. Specifically, the reflection and transmission coefficients at an interface between a homogeneous solid and the chosen metamaterial are analyzed by using both the relaxed micromorphic model and a direct FEM implementation of the detailed microstructure. The obtained results show an excellent agreement between the transmission spectra derived via our enriched continuum model and those issued by the direct FEM simulation. Such excellent agreement validates the indirect measure of the material parameters and opens the way towards an efficient meta-structural design., Comment: The paper has been already accepted in Mathematics and Mechanics of Solids as it is

International audience; In this paper, we show that the transient waveforms arising from several localised pulses in a micro-structured material can be reproduced by a corresponding generalised continuum of the relaxed micromorphic type. Specifically, we compare the dynamic response of a bounded micro-structured material to that of bounded continua with special kinematic properties: (i) the relaxed micromorphic continuum and (ii) an equivalent Cauchy linear elastic continuum. We show that, while the Cauchy theory is able to describe the overall behaviour of the metastructure only at low frequencies, the relaxed micromorphic model goes far beyond by giving a correct description of the pulse propagation in the frequency band-gap and at frequencies intersecting the optical branches. In addition, we observe a computational time reduction associated with the use of the relaxed micromorphic continuum, compared to the sensible computational time needed to perform a transient computation in a micro-structured domain.

The systematic use of a so-called Cauchy theory often leads to too much simplification of reality. Indeed, some features of the microstructure are implicitly neglected in these approaches. Materials have microstructures on a fairly large scale (micron, millimeter, centimeter), the effect of which affects the macroscopic behavior. The Cauchy model is insufficient to describe their specific global behavior, linked for example to the concentration of forces or deformations, or to deformation modes characterized by strong local gradients inducing behaviors related to what happens on smaller scales.One of the most promising fields of application for enriched continuous-media theories concerns woven composite reinforcements. This class of materials is formed by the weaving of yarns, whose rigidities are very different in tension and shear: the yarns are very stiff in tension but the angle between them can vary very easily. This very marked contrast of the mechanical properties of the material's mesostructure makes it possible to describe its homogenized properties as part of a second gradient theory. The macroscopic manifestation of the mesostructure can indeed play a major role in shaping reinforcements composites.Furthermore, Cauchy's models are not adapted to the description of the dynamic response of some microstructured materials showing dispersive behaviors or band-gaps. Enriched continuum theories are good candidates for modeling the effects of the presence of a microstructure. They may also have very particular properties with respect to wave propagation, which gives the resulting structures solutions of choice as a screen or wave absorber which may be innovative in the field of vibration control or in the field of stealth., L’utilisation systématique d’une théorie dite de Cauchy conduit souvent à des simplifications trop fortes de la réalité. En effet, certaines caractéristiques de la microstructure sont implicitement négligées dans ces approches. Des matériaux possèdent des microstructures à une échelle assez grande (micron, millimètre, centimètre), dont l’effet se répercute sur le comportement macroscopique. Le modèle de Cauchy est insuffisant pour décrire leur comportement global spécifique, lié par exemple à la concentration d’efforts ou de déformations, ou à des modes de déformations caractérisés par de forts gradients locaux induisant des comportements liés à ce qui se passe à des échelles plus petites.Un des domaines d’application les plus prometteurs des théories de milieux continus enrichis concerne les renforts tissés de composites. Cette classe de matériaux est constituée par le tissage de mèche, dont les rigidités sont très différentes en traction et en cisaillement : les mèches sont très raides en traction mais l’angle entre deux mèches peut varier très facilement. Ce contraste très marqué des propriétés mécaniques de la mesostructure du matériau permet de décrire ses propriétés homogénéisées dans le cadre d’une théorie de deuxième gradient. La manifestation macroscopique de la mesostructure peut en effet jouer un rôle majeur lors de la mise en forme des renforts de composites. Les modèles de Cauchy ne sont pas adaptés à la description de la réponse dynamique de certains matériaux microstructurés montrant des comportements dispersifs ou des band-gaps. Les théories de milieux continus enrichis sont de bonnes candidates pour modéliser les effets de la présence d’une microstructure. Elles peuvent également posséder des propriétés très particulières vis à vis de la propagation d’ondes, ce qui confère aux structures résultantes des solutions de choix comme écran ou absorbeur d’ondes qui peuvent innovantes dans le domaine du contrôle des vibrations ou dans le domaine de la furtivité.