Prediction Of Local Resonance Band Gaps In 2d Elastic Metamaterials Via Bloch Mode Identification

This paper investigates methods of band gap formation in two-dimensional locally resonant elastic metamaterials and proposes a technique for computing such gaps as algebraic functions of integrated Bloch modes and metamaterial parameters. Mode shapes associated with the deformation of unit cells consisting of a hard matrix, soft filler, and hard resonator are investigated near the lower and upper bounds of the first local resonance band gap. Similarities reveal that Bloch modes at these bounds have the same non-dimensional geometries and locations within the irreducible Brillouin Zone irrespective of material properties, matrix and resonator thicknesses, and unit cell size. A theoretical model for the unit cell incorporating Mindlin plate theory is employed to compute the frequency of these modes in transverse vibration using conservation of energy. A least squares system identification algorithm is then described to generate surface equations for non-dimensional Bloch mode shapes computed by finite element analysis, employing a computation-saving Kronecker factorization and yielding continuous surface equations for the non-dimensional Bloch modes when rectangular cell components are utilized. Nine sample metamaterials are analyzed with this approach generating reliable predictions of the first band gap bounds. The presented framework offers insights into band gap formation, opens avenues in inverse unit cell design, and provides versatile algebraic relationships between band gap frequencies and unit cell parameters in such resonant metamaterials. (C) 2021 Elsevier B.V. All rights reserved.