Small perturbations of the spherical prestressed state in a nonlinear isotropic elastic reduced Cosserat medium: Waves and instabilities

Abstract

We consider small deviations from a nonlinear equilibrium in a nonlinear elastic reduced Cosserat medium (a Cosserat continuum which does not resist to the gradient of rotation). We obtain the equations for small deviations for an arbitrary nonlinear equilibrium and any type of elastic energy, which can be explored into the Taylor series near this equilibrium. Then we consider an isotropic material in the spherically symmetric stress state and show that the equations for small deviations coincide with the equations of motion for the reduced linear elastic Cosserat continuum. For a wide class of materials, strong compression leads to the instability of the medium caused by shear perturbations, and strong tension -to the instability with respect to rotational perturbations. 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. In the forbidden zone localisation phenomena near heterogeneities are present.

Band gaps in the relaxed linear micromorphic continuum

“…In the papers of Grekova and co-workers [7,14] the reduced Cosserat medium is analyzed. The reduced Cosserat model can be obtained by our relaxed micromorphic model by setting α c = 0 and μ h → ∞.…”

“…We start from the uncoupled Eqs. (7) and assume that the involved unknown variables take the harmonic form P (23) = Re β (23) e i(kX −ω t) , P [23] = Re β [23]…”

“…Replacing this wave form in Eqs. (7) and simplifying one obtains the following dispersion relations, respectively:…”

“…It is worth noticing that in our relaxed micromorphic model the observed forbidden range of frequency is valid for every plane wave solution (compression and shear). In the papers of Grekova and co-workers [7,14] the reduced Cosserat medium is analyzed. The reduced Cosserat model can be obtained by our relaxed micromorphic model by setting α c = 0 and μ h → ∞.…”

“…We start from the uncoupled Eqs. (7) and assume that the involved unknown variables take the harmonic form…”