We consider a linear isotropic viscoelastic reduced Cosserat medium, developing reactions to the classical strain and the difference of micro- and macro-rotation and their rates but not reacting to the gradient of micro-rotation or its rate. It was shown before that such an elastic continuum has a bandgap for the transversal wave, being a single negative acoustic metamaterial, and that adding either translational or rotational Kelvin–Voigt-type dissipation for the transversal strain drastically changes its dispersion curves. We perform a dispersional analysis combining both types of linear viscosities and find the asymptotics of dispersion relations for infinitesimal dissipation. The bandgap, present in the elastic case, is transformed into a domain of running decaying waves. The dispersion curve for the real part of the wave number has a descending part, where the medium behaves as a double negative acoustic metamaterial with attenuation. In some frequency domain inside the zone of essential attenuation, the dissipation enhances signal propagation. The asymptotics in the vicinity of characteristic frequencies shows the existence of “cusps” near the lower limit of this zone in the graphs for the wave number. This may lead to an unusual wave behaviour under dynamic loads. The case of the combined dissipation is closer to the case of rotational dissipation than to the translational one. Translational dissipation enlarges the decaying part of the dispersion curve for the real part of the wave number and changes the positions and values of extrema of both dispersion curves. It determines the medium behaviour in the low frequency limit. Rotational dissipation is crucial at the upper boundary of the essential attenuation zone. An illustration of the wave propagation in a linear granular viscoelastic medium simulated by discrete element method (DEM) is presented, qualitatively showing the effect of enhanced wave propagation in some frequency domain by means of dissipation.
Barsukov et al. (Mon,) studied this question.