Nonlinear isotropic elastic reduced Cosserat continuum as a possible model for geomedium and geomaterials. Spherical prestressed state in the semilinear material

We suggest a nonlinear elastic reduced Cosserat continuum as a possible model for geomedium and geomaterials and also for a soil or rock with heterogeneities. If a medium has a block structure, or if it contains heterogeneities that may have their proper rotational dynamics, the presence of rotational degrees of freedom may influence wave propagation and stability of the medium. In the reduced Cosserat model, translations and rotations are independent, the medium resists to the rotation of each ??particle?? relatively to the background continuum, but it does not resist to the gradient of rotation. We consider a nonlinear spherical stress state in an isotropic elastic reduced Cosserat material. We write down small deviations from this nonlinear equilibrium. They coincide in form with the equations of the linear elastic isotropic reduced Cosserat continuum. Depending on the level of the stress and on the type of elastic energy, the equilibrium can be stable or unstable. In the domain of stability, shear?Crotational wave demonstrates dispersion. There is a resonant frequency corresponding to the independent rotational oscillations. The bulk plane shear?Crotational wave has a forbidden band of frequencies. In this zone, the shear?Crotational wave localises near heterogeneities or external sources. We show that for a semilinear medium (medium with large deformations but linear dependence of stress on the strain tensor), strong compression leads to the material instability caused by shear perturbations, and strong tension for some class of parameters yields in instability caused by rotational perturbations.