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

• Published in: Journal of elasticity Vol. 148; no. 1; pp. 55 - 79
• Main Authors: Faraci, David, Comi, Claudia, Marigo, Jean-Jacques
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.01.2022; Springer Nature B.V
• Subjects: Asymptotic properties; Biomechanics; Classical and Continuum Physics; Classical Mechanics; Density; Elastic plates; Energy gap; Engineering; Homogenization; Inclusions; Materials Science; Mathematical Applications in the Physical Sciences; Metamaterials; Theoretical and Applied Mechanics; Unit cell; Wave propagation; Band gaps; 74H10; Homogenization; Elasticity; Metamaterials; Local resonance; 74Q10; Plates
• ISSN: 0374-3535; 1573-2681
• DOI: 10.1007/s10659-022-09879-3

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.