Wave propagation in two-dimensional viscoelastic metamaterials

Propagation of elastic waves in acoustic metamaterials based on locally resonant viscoelastic phononic crystals is discussed. A variational formulation of the complex band structure for in-plane polarized waves is proposed and used to formulate a finite element model. Two different types of locally resonant band gaps are found for quasilongitudinal and quasishear waves, with distinct features in terms of complex bands and transmission bandwidth. The influence of viscosity on the complex band structure, transmission properties, and effective dynamic mass density of two-dimensional locally resonant metamaterials is further investigated. It is found that bands that were degenerate in the elastic case are separated when viscosity is introduced, and that sharp corners at high symmetry points become rounded. Transmission is generally worsened in passing bands, while it is enhanced inside locally resonant band gaps, contrary to what was observed previously for Bragg band gaps. All changes in the complex band structure and transmission spectra are solely due to the dispersive and dissipative effects of viscosity. It is also found that the negative mass density property may also disappear when viscosity is introduced. These results are relevant to practical applications of elastic and viscoelastic metamaterials.