Visco-elastic effects on wave dispersion in three-phase acoustic metamaterials

This paper studies the wave attenuation performance of dissipative solid acoustic metamaterials (AMMs) with local resonators possessing subwavelength band gaps. The metamaterial is composed of dense rubber-coated inclusions of a circular shape embedded periodically in a matrix medium. Visco-elastic material losses present in a matrix and/or resonator coating are introduced by either the Kelvin–Voigt or generalized Maxwell models. Numerical solutions are obtained in the frequency domain by means of $\mathbf{k}\left(\omega \right)$-approach combined with the finite element method. Spatially attenuating waves are described by real frequencies ω and complex-valued wave vectors $\mathbf{k}$. Complete 3D band structure diagrams including complex-valued pass bands are evaluated for the undamped linear elastic and several visco-elastic AMM cases. The changes in the band diagrams due to the visco-elasticity are discussed in detail; the comparison between the two visco-elastic models representing artificial (Kelvin–Voigt model) and experimentally characterized (generalized Maxwell model) damping is performed. The interpretation of the results is facilitated by using attenuation and transmission spectra. Two mechanisms of the energy absorption, i.e. due to the resonance of the inclusions and dissipative effects in the materials, are discussed separately.It is found that the visco-elastic damping of the matrix material decreases the attenuation performance of AMMs within band gaps; however, if the matrix material is slightly damped, it can be modeled as linear elastic without the loss of accuracy given the resonator coating is dissipative. This study also demonstrates that visco-elastic losses properly introduced in the resonator coating improve the attenuation bandwidth of AMMs although the attenuation on the resonance peaks is reduced.