ИСТИНА

The representations of some differential operators, the system of equations of motion, the heat flow equations, the constitutive relations of the micropolar elasticity theory and the Fourier thermal conductivity law under a new parametrization of the thin body domain are given. The corresponding relations in the moments of the unknown quantities with respect to the Legendre polynomials are obtained. A system of equations in displacements and rotations in moments for any anisotropic material is derived. Interlayer contact conditions are written out under certain connections of adjacent layers of a multilayer body. The initial-boundary value problems are formulated. The tensor-block operator of cofactors to the tensor-block operator of the equations of motion in displacements and rotations for isotropic homogeneous materials as with a center of symmetry as well without a center of symmetry is constructed. It allows us to decompose the equations and, separately, to obtain the equations with respect to the displacement and rotation vectors. Static boundary conditions for bodies with a piecewise-plane boundary are also decomposed. From the decomposed equations of the micropolar theories of elasticity, the corresponding decomposed equations of the static (quasistatic) problem of the theories of constant thickness prismatic bodies in displacements and rotations are obtained. From the latter systems of equations it is derived the equations in the moments of unknown vector functions with respect to any systems of orthogonal polynomials. As a special case, the system of equations of the ninth approximation in moments with respect to the system of Legendre polynomials is obtained, which splits into two systems. One of them is a system with respect to the moments of even orders of an unknown vector function, and the other is a system with respect to moments of odd orders of the same functions. Based on the obtained operator of cofactors to the operator of any of these systems, an elliptic equation of high order (the system order depends on the order of approximation) is obtained for each moment of the unknown vector function. The characteristic roots of these equations can be easily found. Using the Vekua method for solving such equations, we can find their analytical solution. Acknowledgements: this research was supported by the Shota Rustaveli National Science Foundaiton (project no. DI-2016-41).