To mathematical modeling of deformation of micropolar thin bodies with two small sizes

Abstract

We consider some problems of modeling the deformation of micropolar thin bodies with two small sizes. Using the three-dimensional equations of motion, the constitutive relations and the boundary conditions of the micropolar elasticity theory [1, 2], we got the equations of motion, the constitutive relations and the boundary conditions of the micropolar theory of thin bodies under the parametrization of the thin body domain with two small sizes [3]. The boundary conditions and various representations of the system of equations of motion and the constitutive relations of physical content in moments with respect to the Legendre polynomials are obtained. Note 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, we reduce an infinite system to the finite system. The initial-boundary value problems are formulated. To satisfy the boundary conditions on the front surfaces we constructed correcting terms [4]. As a special case, we considered a prismatic body. We used the tensor calculus to do this research [5–8].