Isotropic Linear Viscoelastic Reduced Cosserat Medium: an Acoustic Metamaterial and a First Step to Model Geomedium

The reduced Cosserat medium is a continuum whose body points possess rotational degrees of freedom, and there is a reaction to the rotation of a body point relatively to the background of centres of mass, but no stresses are caused by the gradient of micro-rotation. This theory is useful for modelling rocks and soils containing heterogeneities, a geomedium with blocky structure, certain composites with inclusions as well as seismic metamaterials. In this work we consider the influence of viscosity in the linear isotropic reduced Cosserat medium on the propagation of shear waves. We find that viscosity may change drastically the wave propagation. In some cases, the material behaves as a double negative acoustic metamaterial for shear waves, i.e. there is a decreasing part of the dispersion curve for a certain band of frequencies. We also observe that the attenuation in such a continuum does not necessarily increases with frequency, as it happens in the classical viscoelastic medium. It may have one maximum at a certain frequency, or have maximum and minimum. Similar phenomena are observed in the range of seismic frequencies for a geomedium (Sato et al, 2012). The theory considered in this work is only the first step to model the geomedium since it does not take into account previous stress state and existing couplings between pressure and shear-rotational waves due to rock anisotropy caused by gravity.