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

Abstract : 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.
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Submitted on : Thursday, July 16, 2020 - 10:29:11 AM
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Alexios Aivaliotis. Propagation et diffusion des ondes au niveau macroscopique des métamatériaux limites via le modèle micromorphique relaxé. Matériaux. Université de Lyon, 2019. Français. ⟨NNT : 2019LYSEI073⟩. ⟨tel-02900456⟩

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