Plane waves in the linear elastic reduced Cosserat medium with a finite axially symmetric coupling between volumetric and rotational strains

Abstract

We consider plane waves in the linear elastic reduced Cosserat continuum, a medium where point-bodies have independent rotational and translational degrees of freedom, but couple stresses are zero, since the medium does not react to the gradient of turn. We consider the case of the simplest anisotropic coupling between volumetric and vortex-rotational strain, determined by a certain constant vector n. All other tensors are isotropic. For almost all directions of wave propagation there exists a frequency band gap below a specific frequency ω ₀, i.e. the medium is a single negative acoustic metamaterial in a certain domain of frequencies. The longitudinal (in the sense of translations) wave, propagating along this vector, has a behaviour similar to the shear-rotational wave in the isotropic case (high dispersion in the domain close to ω ₀ and the band gap below it). Most of waves are mixed (neither longitudinal nor transversal), and the relation between amplitude components is frequency dependent. The wave, propagating in the direction perpendicular to n, has no forbidden band, but it is mixed (the relation of components depends on frequency). We hope that this or a similar model can be applied for understanding wave propagation in granular media.

Keywords

Reduced Cosserat medium elasticity waves anisotropy coupling

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