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.
Keywords: metamaterials, homogenization, effective mass, band gaps, wave propagation
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