  In this study, a reduced micromorphic model for multiscale materials is
developed. In the context of this model, multiscale materials are modeled with
deformable microstructures. The deformation energy is formed depending on
microstrain and macroscopic strain residual fields. The constitutive equations
according to the reduced micromorphic model only depend on eight material
coefficients for linear elastic materials. These material coefficients are
related to the material micro/macro-stifnesses and the material microstructural
features. The wave dispersions in multiscale materials are then derived
according to the reduced micromorphic model. It is revealed that this model can
reflect nine dispersion curves (three acoustic modes and six optics) for a
two-scale material. To demonstrate the effectiveness of the proposed model, the
wave propagation characteristics, the band structure, and the absolute bandgap
features of phononic materials are investigated. It is demonstrated that the
reduced micromorphic model can effectively reflect the increase in the bandgap
width with the increase in the filling factor in a composite phononic material
with square lattices.
