ИСТИНА

The inverse tensor-operators to a tensor-operator of the equations of motion in terms of displacements for an isotropic homogeneous material and to a stress-operator are found. They allow decomposing equations and boundary conditions. The inverse matrix differential tensor-operator to the matrix differential tensor-operator of the micropolar theory of elasticity equations of motion in displacements and rotations is built as for isotropic homogeneous materials with a center of symmetry as for materials without one. We obtain the equations as in vector of displacement as in vector of rotation. As a special case, a reduced continuum is considered. Cases in which it is easy to invert the stress-operator and the moment stress-operator are found out. From the decomposed equations of classical (micropolar) theory of elasticity, the corresponding decomposed equations of quasistatic problems of theory of prismatic bodies with constant thickness in displacement (in displacement and rotation) are obtained. From these systems of equations, we derive the equations in moments of unknown vector functions with respect to any system of orthogonal polynomials. We obtain the various approximations systems of equations (from zero to eighth order) in moments with respect to the systems of Legendre and second kind Chebyshev polynomials. The system splits and for each moment of unknown vector-function we obtain an equation of elliptic type of high order (order of the system depends on the order of approximation), the characteristic roots of which are easily found. Using the method of Vekua, we can get their analytical solution. For micropolar theory of thin prismatic bodies with two small sizes having the rectangular crosssection, the split equations in moments of displacement and rotation vectors via an arbitrary the system of polynomials (Legendre, Chebyshev) are obtained. We also deduce similar equations for the reduced continuum involving the classical equation of the continuum. The split systems of equations of eight approximation for micropolar theory of multilayer prismatic bodies of constant thickness in the moments of the displacement and rotation vectors are obtained. Using Vekua method, we can found the analytical solution for this system and for equations for reduce continuum.