ИСТИНА

In this paper, we considered some issues of a new parametrization for three-dimensional domain of a thin body and presented some geometric characteristics of this parametrization. Next, we formulated the statements of initial-boundary value problems of three-dimensional linear and some nonlinear classical and linear micropolar theories of viscoelastic bodies and some second-gradient theories of elasticity. From them, one version of the statements of initial-boundary value problems of the three-dimensional nonlinear classical theory of elastic thin bodies was obtained under the new parametrization of the thin body domain. In addition, statements of initial-boundary value problems are given 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 considered thin bodies’ domain. From the last statements, in turn, we have obtained the statements of initial-boundary value problems in moments with respect to systems of orthogonal polynomials and, in particular, with respect to the system of Legendre polynomials. Statements of initialboundary value problems 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. We presented the constitutive relations using the tensors and tensor-block matrices, as well as taking into account the canonical representations of these tensor objects. In addition, the static boundary conditions and equations of motion and equations of equilibrium were represented by differential tensor operators in the case of the classical theory and by differential tensor-block-matrix operators in the case of micropolar and gradient theories. Tensor operators of cofactors for tensor operator of motion and tensor operator of equilibrium and tensor stress operators are constructed, which allow splitting of initialboundary value problems of linear classical and linear micropolar theories of viscoelastic bodies, as well as some gradient theories. It should be noted that all of the above is easily extended to the theories of other rheological bodies, including linear and non-linear 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).