ИСТИНА

The problems of decomposition of initial-boundary value problems in the classical (micropolar) three-dimensional theory of elasticity are considered. In addition, similar problems are considered for the theory of thin bodies with one and two small dimensions when using systems of orthogonal polynomials and shell theories obtained from the corresponding three-dimensional theories. In particular, the first initial-boundary value problem for bodies with an arbitrary boundary is split, and also the second and third (mixed) initial-boundary value problems for bodies with a piecewise-plane boundary for different anisotropy are split. From the decomposed equations of the classical (micropolar) three-dimensional theory, the corresponding decomposed equations of the theory of prismatic bodies with one small dimension of constant thickness in displacements (displacements and rotations) are obtained. From the latter equations, in turn, the equations are derived in the moments of unknown vector-functions with respect to any system of orthogonal polynomials. The systems of the equations of various approximations in the moments with respect to the systems of Legendre and Chebyshev polynomials are obtained. On the basis of the constructed tensor-operator of cofactors for the operator of any of these systems of equations these systems split and for each moment of the unknown vector-function a high order elliptic type equation is obtained (the system order depends on the order of approximation), the characteristic roots of which are easily found. Using the Vekua method, one can obtain their analytical solutions. Acknowledgements: this research was supported by the Shota Rustaveli National Science Foundaiton (project no. DI-2016-41) and the Russian Foundation for Basic Research (project no. 15-01-00848-a) Keywords: thin body; micropolar theory; orthogonal polynomials; tensor-operator of cofactors