Mathematical Modeling of Elastic Thin Bodies with one Small Size

Some questions on parametrization with an arbitrary base surface of a thin-body domain with one small size are considered. This parametrization is convenient to use in those cases when the domain of the thin body does not have symmetry with respect to any surface. In addition, it is more convenient to find the moments of mechanical quantities than the classical. Various families of bases (frames) and the corresponding families of parameterizations generated by them are considered. Expressions for the components of the second rank unit tensor are obtained. Representations of some differential operators, the system of motion equations, and the constitutive relation (CR) of the micropolar theory of elasticity are given for the considered parametrization of a thin body domain. The main recurrence formulas of system of orthogonal Legendre polynomials are written out and some additional recurrence relations are obtained, which play an important role in the construction of various variants of thin bodies. The definitions of the moment of the kth order of a certain value with respect to an arbitrary system of orthogonal polynomials and system of Legendre polynomials are given. Expressions are obtained for the moments of the kth order of partial derivatives and some expressions for the system of Legendre polynomials. Various representations of the system of motion equations and CR in the moments for the theory of thin bodies are given. Boundary conditions are derived. The CR of the classical and micropolar theory of the zero approximation and approximation of order r in the moments are obtained. The boundary conditions of physical and thermal contents on the front surfaces are given. The statements of dynamic problems in moments of the approximation (r, M) of a micropolar thermomechanics of a deformable thin body, as well as a non-stationary temperature problem in moments are given. It should be noted that using the considered method of constructing a theory of thin bodies with one small size, we obtain an infinite system of equations, which has the advantage that it contains quantities depending on two variables, the base surface Gaussian coordinates x1, x2. So, to reduce the number of independent variables by one we need to increase the number of equations to infinity, which of course has its obvious practical inconveniences. In this connection, the reduction of the infinite system to the finite is made.