Two scale homogenization in ternary locally resonant metamaterials

In this work, we exploit the two-scale homogenization approach to compute explicitly the band gaps for out-of-plane wave propagation in ternary locally resonant metamaterials (LRM) with two-dimensional periodicity. The homogenization approach, recently developed by the authors for binary LRM, leads to the definition of the dynamic effective mass density, depending on the frequency, that becomes negative near the resonant frequencies of the inclusions. The intervals of negative effective mass give the band gaps. These explicit solutions put in evidence the dependence of the spectral gaps on the geometric parameters of the unit cell and on the mechanical properties of the three constituent materials. The range of frequency where the asymptotic homogenization approach is equivalent to the Bloch-Floquet theory is also established and confirmed by numerical simulations.