Viscoelastic reduced enhanced isotropic continua as acoustic metamaterials

We consider a linear enhanced viscoelastic continuum of general nature but of specific type. Namely, we consider a reduced elastic continuum, satisfying Lagrange equations, where the strain energy depends on a certain (special) vectorial generalized coordinate, but does not depend on its gradient, and then add linear dissipation to the existing elastic connections. We may also represent this model as a 'bearing continuum', where all the connections are present (described by one vectorial generalized coordinate), enriched in each point by a 'distributed dynamic absorber' (described by 'special' vectorial generalized coordinate). We look for free harmonic waves in this infinite medium and obtain a reduced spectral problem for the vectorial generalized coordinate of the bearing continuum, for an arbitrary number of degrees of freedom. It was shown earlier that under certain symmetry conditions in the elastic case we obtain a single negative acoustic metamaterial, i.e. a medium that has band gaps. Further, we consider isotropic and gyrotropic reduced media, described by two three-dimensional vectorial generalized coordinates. First, we generalize results of previous studies for more complex elastic coupling, discovering a polarized shear wave, which has both bandgaps and zones of anomalous refraction. Then we introduce linear dissipation of different kinds. We find that viscosity yields in existence of travelling harmonic waves for all frequencies, possibly except for some points. Logarithmic decrement, infinite for the elastic material in bandgaps, becomes finite and decreases as the dissipation increases, at least for small viscosity. An important observation is: an infinitesimal dissipation in most cases transforms bandgaps into zones of travelling evanescent waves that partially are zones of anomalous refraction (decreasing parts of dispersion curves), where the medium is a double negative acoustic metamaterial. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.