ИСТИНА

Some issues on parametrization with an arbitrary base surface of a thin-body domain with one small size similarly to [3, 4] 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 moments of mechanical quantities than classical [1]. Various families of bases 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 viscoelasticity are given for the considered parametrization of a thin body domain. The main recurrence formulas of the system of orthogonal Legendre polynomials are written out and some additional recurrence relations are obtained [3], which play an important role in the construction of various variants of thin body theories. The definitions of the moment of the kth order of a certain quantity 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 of tensor fields and some expressions with respect to the system of Legendre polynomials. Various representations of the system of motion equations and CR in the moments with respect to the system of Legendre polynomials for the theory of viscoelastic thin bodies are given. Boundary conditions in the moments are derived. The CR of the classical and micropolar elastic [2, 5, 6] (see also [1, 3]) and viscoelastic [5] theories of the zero approximation and approximation of order r in the moments are obtained. The boundary conditions of physical content on the front surfaces are given. The statements of dynamic problems in moments of the approximation (r, M) of a micropolar theory of viscoelastic thin bodies 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 x^1,x^2. 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 system is made. Prismatic bodies are considered as a special case. Keywords: micropolar thin body theory, Legendre polynomials, constitutive relations Acknowledgment: This work was supported by the Russian Foundation for Basic Research, grants no. 18–29–10085–mk References 1.Vekua, I.N.: Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985 2.Купрадзе В. Д., Гегелиа Т. Г.: Башелейшвили М. О., Бурчуладзе Т. В. Трехмерные задачи математической теории упругости и термоупругости. М.: Наука, 1976. 3.Никабадзе М. У.: Развитие метода ортогональных полиномов в механике микрополярных и классических упругих тонких тел. М.: Изд-во Попечительского совета мех.-мат. ф-та МГУ, 2014. 515 c. https://istina.msu.ru/publications/book/6738800/ 4. Nikabadze M. and Ulukhanyan A.: Mathematical Modeling of Elastic Thin Bodies with one Small Size. In: Altenbach H., Müller W., Abali B. (eds) Higher Gradient Materials and Related Generalized Continua. Advanced Structured Materials. Springer, Cham Switzerland. Vol. 120, 2019. DOI: 10.1007/978-3-030-30406-5_9 5.Eringen A. C.: Microcontinuum field theories. 1. Foundation and solids. N.Y.: Springer, 1999. 6. Mindlin R. D.: Microstructure in linear elasticity. Arch. Rat. Mech. Anal. 1964. No 1, p. 51–78.