  Structured metamaterials are at the core of extensive research, promising for
acoustic and thermal engineering. Nevertheless, the computational cost required
for correctly simulating large systems imposes to use a continuous model to
describe the effective behavior without knowing the atomistic details.
Crucially, a correct description needs to describe both the extrinsic
interface-induced and the intrinsic atomic scale-originated phonon scattering,
especially when the component material is made of glass, a highly dissipative
material in which wave attenuation is strongly dependent on frequency as well
as on temperature. In amorphous systems, the effective acoustic attenuation
triggered by multiple mechanisms is now well characterized and exhibits a
nontrivial frequency dependence with a double crossover of power laws. In this
work, we propose a continuum viscoelastic model based on the hierarchical
strategy multi-scale approach, able to reproduce well the phonon attenuation in
a large frequency range, spanning three orders of magnitude from GHz to THz
with a $\omega^2-\omega^4-\omega^2$ dependence, including the influence of
temperature.
