https://hal-polytechnique.archives-ouvertes.fr/hal-02146908Comi, ClaudiaClaudiaComiPOLIMI - Politecnico di Milano [Milan]Marigo, Jean-JacquesJean-JacquesMarigoLMS - Laboratoire de mécanique des solides - X - École polytechnique - Mines Paris - PSL (École nationale supérieure des mines de Paris) - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche ScientifiqueHomogenization approach and Bloch-Floquet theory for band-gap prediction in 2D locally resonant metamaterialsHAL CCSD2019[PHYS.MECA.SOLID] Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph]Marigo, Jean-Jacques2019-06-04 11:51:022023-02-08 17:11:072019-06-07 15:25:01enJournal articleshttps://hal-polytechnique.archives-ouvertes.fr/hal-02146908/document10.1007/s10659-019-09743-xapplication/pdf1This paper provides a detailed comparison of the two-scale homogenization method and of the Bloch-Floquet theory for the determination of band gaps in locally resonant metamaterials. A medium composed by a stiff matrix with soft inclusions with 2D periodicity is considered and the equivalent mass density of the homogenized medium is explicitly obtained both for in-plane and out-of-plane wave propagation through two-scale asymptotic expansion. The intervals of frequency where the effective mass is negative identify the band gaps of the material. The Bloch-Floquet problem is then considered and, through an asymptotic analysis, its is shown that it leads to the same prediction of the band gaps. The results are confirmed by some examples and the limits of the asymptotic approach are explicitly given and numerically verified.