micropolar elasticity, displacement vector, micro-rotation vector, vector potential, vortex part, Helmholtz equation, wave, cylinder, wavenumber, azimuthal number

The coupled system of vector differential equations of the linear theory of micropolar elasticity presented in terms of displacements and micro-rotations in the case of a harmonic dependence of physical fields on time is considered in the three different variants of which the two are due to W. Nowacki and H. Neuber. A new scheme of splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled ones is proposed. The scheme is based on proportionality of the vortex parts of the displacements and micro-rotations to the single vector, which satisfies the screw equation. The problem of determination of the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. A new representation of displacement and micro-rotation vectors is obtained by using two uncoupled metaharmonic vectors. The separation of spatial variables in the Helmholtz metaharmonic equations in a cylindrical coordinate net is described. Solutions of the scalar and vector Helmholtz equations in an infinite cylindrical domain containing a series of arbitrary constants are obtained. Representation of displacement and micro-rotation vectors in a long micropolar cylinder containing eight arbitrary constants are explicitly found. The corresponding solutions are proved to determine the modes of harmonic waves of displacements and micro-rotations propagating along the axis of a long circular cylinder. The obtained modes of the harmonic displacements and micro-rotations waves are valid only for those characterized by a given azimuthal number.

Radayev Yuri Nickolaevich e-mail: radayev@ipmnet.ru, y.radayev@gmail.com, Dr. Sci. Phys. & Math., Professor, Leading Researcher, Institute for Problems in Mechanics of RAS, Moscow, Russia.