ИСТИНА

Some issues of a new parametrization for the three-dimensional domain of a thin body are considered, and also some geometric characteristics of this parametrization are given [1, 2]. The formulations of initial-boundary value problems are given for three-dimensional linear and some nonlinear classical and linear micropolar theories of viscoelastic bodies [4-6], as well as some second-gradient theories of elasticity [7-10]. Proceeding from the formulations of initial-boundary value problems mentioned above, the corresponding formulations of initial-boundary value problems for three-dimensional theories of thin bodies are derived for the considered parameterizations of the domains of these bodies. In particular, the formulations of initial-boundary value problems for one variant of the three-dimensional nonlinear classical theory of elastic thin bodies under the new parametrization of the domain of a thin body are obtained, and the formulations of initial-boundary value problems for the three-dimensional linear micropolar theory of viscoelastic thin bodies and some second gradient theories of elastic thin bodies under the new parametrization of the domains of the considered thin bodies are derived. From three-dimensional formulations of initial-boundary value problems for theories of thin bodies, in turn, the formulations of initial-boundary value problems at moments with respect to arbitrary systems of orthogonal polynomials, including the system of Legendre polynomials, are obtained. The formulations of initial-boundary value problems in moments are also considered in the case of the classical linear theory and gradient theories of elasticity with respect to the displacement vector, and in the case of the linear micropolar theory with respect to the displacement and rotation vectors. Several first approximations of these formulations of initial-boundary value problems are considered as special cases. The constitutive relations of the bodies are written using tensors and tensor-block matrices, as well as taking into account the canonical representations [2] of these tensor objects. In addition, in the case of the classical theory, the static boundary conditions and the equations of motion and equilibrium are represented by differential tensor-operators, and in the case of the micropolar and gradient theories, by differential tensor-block matrix operators. The tensor operators of cofactors for tensor-operators of motion and equilibrium and stress tensor-operators are constructed, which, similarly to [1, 3], allow splitting of initial-boundary value problems of linear classical and micropolar theories of viscoelastic bodies, as well as some gradient theories. It should be noted that all of the above is easily extended to theories of other rheological bodies, including high-order gradient theories. Acknowledgements: this work was supported by the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No 075-15-2019-1621 and of the Shota Rustaveli National Science Foundation (project No FR-21-3926). 1. Nikabadze MU (2014) Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies. Moscow Univ. 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