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-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 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. 
Angela Madeo and Patrizio Neff and Gabriele Barbagallo and Marco   Valerio d'Agostino and Ionel-Dumitrel Ghiba
math-ph math.MP