  In this paper the role of gradient micro-inertia terms
$\bar{\eta}\lVert\nabla u_{,t}\rVert^{2}$ and free micro-inertia terms
$\eta\lVert p_{,t}\rVert^{2}$ is investigated to unveil their respective effect
on the dynamical behavior of band-gap metamaterials. We show that the term
$\bar{\eta}\lVert\nabla u_{,t}\rVert^{2}$ alone is only able to disclose
relatively simplified dispersive behaviors. On the other hand, the term
$\eta\lVert p_{,t}\rVert^{2}$ is in charge of the description of the full
complex behavior of band-gap metamaterials. A suitable mixing of the two
micro-inertia terms allows 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
$\bar{\eta}\lVert\nabla u_{,t}\rVert^{2}$, in the sense of Cartan-Lie
decomposition of matrices, allows to flatten separately longitudinal and
transverse optic branches thus giving the possibility of a second band-gap.
Finally, we investigate the effect of the gradient inertia
$\bar{\eta}\lVert\nabla u_{,t}\rVert^{2}$ on more classical enriched models as
the Mindlin-Eringen and the internal variable ones. We find that the addition
of such gradient micro-inertia allows for the onset of one band-gap in the
Mindlin-Eringen model and of 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
most performing one, among the considered enriched model, for the description
of non-local band-gap metamaterials.
