Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures

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

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

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

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

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

Wave propagation and scattering at metamaterials’ macroscopic boundaries via the relaxed micromorphic model

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

Propagation et diffusion des ondes au niveau macroscopique des métamatériaux limites via le modèle micromorphique relaxé

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

Low-and high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface

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

A Framework for CO₂ Emission Reduction in Manufacturing Industries: A Steel Industry Case

Rising carbon emissions are linked to the increase in global temperature, because of increasing human activities and increasing greenhouse gas emissions. Since manufacturing is one of the most carbon intensive sectors, it is vital to suggest solutions that lead to carbon emission reduction in all sectors, among which is the steelmaking manufacturing sector. The present study focuses on presenting a framework for energy intensive industries introducing digitalization and energy efficient equipment in the production line. The current framework proposes different metrics, both from carbon emissions and cost viewpoints. A secondary steelmaking industry was used as a case study, showcasing the impact of digitalization and energy efficient equipment towards the reduction of carbon emissions. In addition, different metrics were calculated with energy efficient scenarios providing the lowest energy consumption metric, but their high purchasing costs make these scenarios less attractive. However, if carbon emission reduction per cost is the metric, a combination of digital tools and energy efficient equipment is the answer to the company’s needs. Applying the concepts of innovation absorption and digitalization, the number of alternatives is kept low, however, the impact on the line is quite large. The introduction of new technologies is also supported by training of the workforce, aligning the framework with current industrial trends

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

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