Analytical modeling of damped locally-resonant metamaterials

Locally-resonant metamaterials (LRMMs) are architected materials that can be designed to manipulate mechanical wave propagation by tuning their band gaps. Discrete lumped-mass models and discrete distributed-mass finite element models are both generally used to analyze LRMMs. While the former is accurate only near the fundamental resonance frequency of resonators, the latter’s accuracy is tightly coupled to the computational cost. In this study, an analytical procedure based on the spectral element method (SEM) is proposed to analyze both undamped and damped LRMMs as continuous systems. We compare LRMMs’ band structures to those obtained by discrete models and show that the proposed procedure is capable of capturing the wave dynamics of these materials very accurately and with negligible computational cost. The behavior of a finite LRMM waveguide is also studied through displacement transmissibility. In addition to the attenuation provided by band gaps, we investigate the effects of constant viscous damping and frequency-dependent viscoelastic damping, which proved to be a straightforward extension of the undamped spectral element model.