The problem of development of a theory for calculating the stress-strain state of thin multilayer elastic plates in the moment (micropolar) theory, is considered. The solution of this problem is built using an asymptotic analysis of the general equations for a 3-dimensional quasi-static problem of the moment theory of elasticity. The asymptotic analysis is carried out with respect to a small parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local problems of the moment theory of elasticity are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged system of equations for multilayer plates is presented. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 9 components of the stress tensor and the moment stress tensor (in general) in the plate. As a special case, the problem of calculating the stress-strain state of a centrally symmetrical hingedly fixed plate when bending under the action of a uniformly distributed pressure. A complete analytical solution of this problem for all non-zero components of the stress tensor and the moment stress tensor is obtained. A numerical analysis of the solution of the problem for a single layer plate for the stress tensor is carried out, basing on the obtained expressions. A comparative analysis of the obtained results with similar calculations for the classical theory of elasticity is carried out, with revealing of similarities and differences for all components of the stress tensor.
Димитриенко Ю.И., Бойко С.В. Асимптотическая теория многослойных тонких микрополярных упругих пластин. Математическое моделирование и численные методы, 2023, № 2, с. 33–66.