Band Gaps in Metamaterial Plates: Asymptotic Homogenization and Bloch-Floquet Approaches

In this work, we study the transversal vibration of thin periodic elastic plates through asymptotic homogenization. In particular, we consider soft inclusions and rigid inclusions with soft coatings embedded in a stiff matrix. The method provides a general expression for the dynamic surface density of the plate, which we compute analytically for circular inclusions or numerically for two-way ribbed plates. Through asymptotic homogenization, we find that band gaps related to in-plane propagating transversal waves occur for frequency intervals in which the effective surface density is negative. The same result is obtained via an asymptotic analysis of the Bloch-Floquet problem on a unit cell, showing the equivalence of the two approaches. Finally, we validate the method by comparing in several examples the predicted band gaps with those obtained from numerical Bloch-Floquet analyses on the real unit cell.