Basic equations of Kelvin's medium and analogy with ferromagnets

A nonlinear elastic Cosserat continuum of special kind whose particles have dynamical spins (Kelvin's medium) is considered. The basic equations for this medium are obtained on the basis of fundamental mechanical laws. Various complete sets of strain tensors are discussed. The analogies between Kelvin's medium and other types of media are discussed. First, we show the analogy between constitutive equations for Kelvin's medium and for elastic shells. The main difference is caused by the dimensionality of the media. Then, for a special case of fast proper rotation we establish the analogy between the dynamic equations of Kelvin's medium and elastic ferromagnetic saturated insulators. The most general way to take into account the coupling between magnetic and elastic subsystems is presented. All results can be interpreted in terms of a mechanical medium as well as in terms of ferromagnets.