  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.
