A semi-analytical approach towards plane wave analysis of local resonance metamaterials using a multiscale enriched continuum description

This work presents a novel multiscale semi-analytical technique for the acoustic plane wave analysis of (negative) dynamic mass density type local resonance metamaterials with complex micro-structural geometry. A two step solution strategy is adopted, in which the unit cell problem at the micro-scale is solved once numerically, whereas the macro-scale problem is solved using an analytical plane wave expansion. The macro-scale description uses an enriched continuum model described by a compact set of differential equations, in which the constitutive material parameters are obtained via homogenization of the discretized reduced order model of the unit cell. The approach presented here aims to simplify the analysis and characterization of the effective macro-scale acoustic dispersion properties and performance of local resonance metamaterials, with rich micro-dynamics resulting from complex metamaterial designs. First, the dispersion eigenvalue problem is obtained, which accurately captures the low frequency behavior including the local resonance bandgaps. Second, a modified transfer matrix method based on the enriched continuum is introduced for performing macro-scale acoustic transmission analyses on local resonance metamaterials. The results obtained at each step are illustrated using representative case studies and validated against direct numerical simulations. The methodology establishes the required scale bridging in multiscale modeling for dispersion and transmission analyses, enabling rapid design and prototyping of local resonance metamaterials.