Waves in Elastic Reduced Cosserat Medium with Anisotropy in the Term Coupling Rotational and Translational Strains or in the Dynamic Term

Abstract

We consider wave propagation in a special kind of elastic micropolar continuum: reduced Cosserat medium possessing an anisotropic term coupling rotational and translational strains or anisotropy in the tensor of inertia of point bodies. This medium consists of point bodies which are infinitesimal rigid bodies that can perform independent translations and turns. Reduced Cosserat medium is a Cosserat medium that does not react on the gradient of turn of point bodies. We consider linear theory. In the isotropic case the P-wave is the same as in the classical medium, and shear-rotational waves are dispersive and one branch has a cut-off frequency. Anisotropy in elastic properties allows to introduce a coupling term between these waves. We consider some classes of anisotropy and special directions of wave propagation, and also waves in the vicinity of the characteristic frequency of the medium. We find that the shear-rotational wave splits, and that in some cases the P-wave also becomes dispersional and has a band gap, in another case there exist shear-rotational and mixed wave, both having these properties. Then we consider a reduced Cosserat medium with isotropic elastic properties but with non-spherical tensor of inertia (equal in its initial position for all point bodies). The pressure wave in such a medium is the same as in the classical medium (non-dispersive). The shear-rotational wave changes essentially. For axial symmetry its dispersion graph has two boundary frequencies and two cut-off frequencies.