title: On the role of micro-inertia in enriched continuum mechanics.
abstract: In this paper, the role of gradient micro-inertia terms [Formula: see text] and free micro-inertia terms [Formula: see text] is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the term [Formula: see text] alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term [Formula: see text] alone describes the full complex behaviour of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia [Formula: see text], in the sense of Cartan-Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia [Formula: see text] on more classical enriched models such as the Mindlin-Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin-Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials.
authors.affiliation: LGCIE SMS-ID, Université de Lyon, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France; IUF, Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France.
authors.forename: Angela
authors.identifier: 0000-0003-1940-9853
authors.initials: A
authors.lastname: Madeo
authors.affiliation: Fakultät für Mathematik , Universität Duisburg-Essen, Mathematik-Carrée , Thea-Leymann-Straße 9, 45127 Essen, Germany.
authors.forename: Patrizio
authors.initials: P
authors.lastname: Neff
authors.affiliation: Aristotle University of Thessaloniki, Thessaloniki 54124, Greece; Michigan Technological University, Houghton, MI 49931, USA; ITMO University, St. Petersburg 197101, Russia.
authors.forename: Elias C
authors.initials: EC
authors.lastname: Aifantis
authors.affiliation: LGCIE SMS-ID, Université de Lyon, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France; LaMCoS-CNRS, INSA-Lyon, Université de Lyon, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France.
authors.forename: Gabriele
authors.initials: G
authors.lastname: Barbagallo
authors.affiliation: LGCIE SMS-ID , Université de Lyon , 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France.
authors.forename: Marco Valerio
authors.initials: MV
authors.lastname: d'Agostino
keywords: complete band-gaps; free micro-inertia; generalized continuum models; gradient micro-inertia; non-local effects; relaxed micromorphic model