Nonlinear isotropic elastic reduced and full Cosserat media: waves and instabilities

Abstract

We consider two types of the nonlinear elastic Cosserat media: a full one, in which the strain energy depends both on the Cosserat deformation tensor and wryness tensor, and the reduced one, in which the strain energy depends only on the Cosserat deformation tensor, thus excluding its dependence on the gradient of micro-rotation. For these media, we obtain the equations for small deviations for an arbitrary nonlinear equilibrium and any type of elastic energy, which can be expanded into the Taylor series near this equilibrium. Then we consider an isotropic material under isotropic tension or compression and show that the equations for small deviations coincide with the equations of motion for the corresponding linear elastic continuum. For a wide class of materials with convex energy, strong compression leads to the instability of the medium caused by shear perturbations, and strong tension—to the instability with respect to shear–rotational perturbations. The last mechanism of instability is absent in the classical elastic continuum. For the reduced medium in the stable domain, shear–rotational wave has a forbidden band of frequencies, demonstrates strong dispersion near this zone, and there is a resonant frequency corresponding to the independent rotational oscillations. Characteristic frequencies depend on the stress state. Thus, it is a single negative acoustic metamaterial with a tunable band gap, depending on the nonlinear stress state, with respect to the shear waves. In the forbidden zone, localization phenomena near heterogeneities are present.