ИСТИНА

A set of variety thin bodies used in engineering and construction is increasing at the present time. There were chosen some subsets of elements from this set with similar properties for which the corresponding versions of the thin body theories were developed. We note some of them: 1) the thin rods theory; 2) the single-layered thin shells theory; 3) the ribbed shells theory; 4) the single-layer thick shells theory; 5) the theory of multilayer structures and others. We can refer to several versions of each theories that differ both in the original assumptions and in the final equations. Thus, we can conclude that the modern theory of thin bodies is a deeply developed part of the solids mechanics. However, the development the thin body theories are not complete, since new structures are continuously arising in the engineering. But it turns out it is impossible to make calculations of these constructions using existing versions of the theories. In this regard, we should expect the appearance of new versions of refined theories and improved methods for their calculation. Therefore, the construction of the refined theories of thin bodies and the development of effective methods for their calculation are an important and actual. Here we consider some problems of modeling the deformation of micropolar thin bodies with two small sizes. For this purpose, the parametrization of the thin body domain with respect of arbitrary point is considered. In particular, the vector parametric equation of the thin body domain is given. There are introduced the geometric characteristics that appear under this parametrization. Different families of basis (frames) are considered and expressions for the components of the second-rank identity tensor are obtained. The representations of the gradient and the divergence of the tensor under the parametrization of the thin body domain are obtained. Based on the three-dimensional equations of motion, the constitutive relations and the boundary conditions of the micropolar elasticity theory, we got the equations of motion, the constitutive relations and the boundary conditions of the micropolar theory of thin bodies under the offered parametrization. Based on the main recurrence formulas several recurrence formulas for Legendre polynomials playing an important role in the construction of various versions of the thin bodies theories are obtained. Definitions of the (m, n)-th order moment of some quantity with respect to an arbitrary system of orthogonal polynomials and the Legendre system of polynomials are given. The expressions for the moments of partial derivatives and some relations with respect to the system of Legendre polynomials, as well as the boundary conditions and various representations of the system of equations of motion and the constitutive relations of physical content in moments are obtained. It should be noted that using this method of construction the thin bodies theory with two small sizes, we get an infinite system of ordinary differential equations. This system contains quantities which depends on one variable, namely, depends on the parameter of the base line. Thus, decreasing the number of independent variables from three to one we increase the number of equations to infinity, which, of course, has its obvious practical inconveniences. In this regard, the next necessary step is made to simplify the problem. We reduce an infinite system to the finite system. A simplified method of reduction an infinite system of equations to a finite one is presented. The initial-boundary value problems are formulated. To satisfy the boundary conditions on the front surfaces we constructed correcting terms. As a special case, we considered a prismatic body. In particular, the equations with respect to displacement and rotation vectors in moments with respect to arbitrary systems of orthogonal polynomials, including the system of Legendre polynomials, have been derived. Several equations of the first approximation were obtained. We used the tensor calculus to do this research. Acknowledgements: this work was supported by the Shota Rustaveli National Science Foundation (project no. DI-2016-41).