Computational homogenization of locally resonant acoustic metamaterial panels towards enriched continuum beam/shell structures

Locally resonant acoustic metamaterial (LRAM) panels are excellent candidates for the low-frequency flexural wave attenuation in thin structures. To enable the efficient analysis and design of LRAM beam/shell structural elements for practical applications, a computational homogenization method for modelling wave propagation phenomena in LRAM panels is presented in this work. The approach is based on the notion of a relaxed separation of scales, which tailors the methodology to the phenomena governed by local resonators embedded in a host medium. The macroscopic LRAM panel is modelled as a thin continuum beam/shell described by proper structural kinematics and momentum balance relations. At the microscale, a LRAM unit cell is considered with lamina-wise in-plane boundary conditions and zero out-of-plane tractions, representing the free top and bottom surfaces of the macroscopic LRAM panel. Under the assumption of linear elasticity, in the relaxed separation of scales regime, the microscale unit cell problem can be represented by the superposition of a static and a dynamic problem, hereby enabling a significant model reduction. As a result, the macroscale effective material properties can be computed once and for all off-line for a given unit cell. In addition, a new macroscale evolution equation emerges describing the effect of the microscale internal dynamics on the macroscopic fields through the introduction of new macroscale enrichment variables, which reflect the modal amplitudes of localized microscale modes. The proposed homogenization methodology reveals a high level of accuracy and numerical efficiency compared to the reference direct numerical simulation (DNS) for all relevant analyses: computation of real and complex dispersion spectra for an infinite LRAM panel, as well as steady-state frequency response and transient behaviour for a finite LRAM panel.