  In this paper the relaxed micromorphic model proposed in [Patrizio Neff,
Ionel-Dumitrel Ghiba, Angela Madeo, Luca Placidi, Giuseppe Rosi. A unifying
perspective: the relaxed linear micromorphic continuum, submitted, 2013,
arXiv:1308.3219; and Ionel-Dumitrel Ghiba, Patrizio Neff, Angela Madeo, Luca
Placidi, Giuseppe Rosi. The relaxed linear micromorphic continuum: existence,
uniqueness and continuous dependence in dynamics, submitted, 2013,
arXiv:1308.3762] has been used to study wave propagation in unbounded continua
with microstructure. By studying dispersion relations for the considered
relaxed medium, we are able to disclose precise frequency ranges (band-gaps)
for which propagation of waves cannot occur. These dispersion relations are
strongly nonlinear so giving rise to a macroscopic dispersive behavior of the
considered medium. We prove that the presence of band-gaps is related to a
unique elastic coefficient, the so-called Cosserat couple modulus $\mu_{c}$,
which is also responsible for the loss of symmetry of the Cauchy force stress
tensor. This parameter can be seen as the trigger of a bifurcation phenomenon
since the fact of slightly changing its value around a given threshold
drastically changes the observed response of the material with respect to wave
propagation. We finally show that band-gaps cannot be accounted for by
classical micromorphic models as well as by Cosserat and second gradient ones.
The potential fields of application of the proposed relaxed model are manifold,
above all for what concerns the conception of new engineering materials to be
used for vibration control and stealth technology.
