A Review on Wave Propagation Modeling in Band-Gap Metamaterials via Enriched Continuum Models
In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-m
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Abstract In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-motions without considering a micro-macro coupling. Moreover, we show that when adding a gradient inertia term which indeed allows for the description of the coupling of the vibrations of the microstructure to the macroscopic motion of the unit cell, other enriched continuum models of the micromorphic type A. Madeo (B) · M.V. d’Agostino LGCIE, INSA-Lyon, Université de Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France e-mail: [email protected] M.V. d’Agostino e-mail: [email protected] A. Madeo IUF, Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France P. Neff Head of Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Mathematik-Carrée, Thea-Leymann-Straße 9, 45127 Essen, Germany e-mail: [email protected] G. Barbagallo LaMCoS-CNRS \& LGCIE, INSA-Lyon, Universitité de Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France e-mail: [email protected] I.-D. Ghiba Lehrstuhl Für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany e-mail: [email protected], [email protected] I.-D. Ghiba Department of Mathematics, Alexandru Ioan Cuza University of Ia¸si, Blvd. Carol I, no. 11, 700506 Ia¸si, Romania I.-D. Ghiba Octav Mayer Institute of Mathematics of the Romanian Academy, Ia¸si Branch, 700505 Ia¸si, Romania © Springer Nature Singapore Pte Ltd. 2017 F. dell’Isola et al. (eds.), Mathematical Modelling in Solid Mechanics, Advanced Structured Materials 69, DOI 10.1007/978-981-10-3764-1_6
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may allow the description of the onset of band-gaps. Nevertheless, the relaxed micromorphic model proves to be yet the most effective enriched continuum model which is able to describe multiple band-gaps in non-local metamaterials.
1 Introduction In the last years, a lot of interest has been raised from a class of microscopically heterogeneous materials which show exotic behaviors such as that of “stopping” the propagation of elastic waves. In some cases, the waves lose some of the energy due to micro-diffusion phenomena (Bragg scattering) or even local resonance of the microstructure (Mie resonance). These effects can be exploited to design innovative materials whose dynamical behavior differs completely from the classical materials usually employed in engineering sciences. As a matter of fact, classical Cauchy continuum theories, are not always well adapted to cover the wealth of experimental evidences on the dynamical behavior of real materials. As a first point, in fact, real materials commonly show dispersive behaviors, which means that the speed of propagation of the traveling wave changes when considering smaller wave
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