A problem of harmonic oscillations of a rectangle in the theory of micropolar elasticity:the analytical solution

We consider the plane problem of natural harmonic oscillations of a rectangle with mixed boundary conditions in the framework of the linear micropolar theory of elasticity. The micropolar or Cosserat model is used for many modern materials with microstructure, when an elementary particle of a continuous medium has six degrees of freedom. A method for solving the original boundary value problem, when it is divided into separate sequences of consistent scalar boundary value problems, including one for rotational component, is proposed. It was revealed that in a micropolar medium there are two «sorts» of natural oscillations of a rectangle, one of which is bounded from below, while in a classical medium there is only one «sort» of natural oscillations and there are no such restrictions. The proposed method can be developed for the case of other boundary conditions and for the three-dimensional case.