Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells

Abstract

In the present paper asymptotic solution of boundary-value problem of three-dimensional micropolar theory of elasticity with free fields of displacements and rotations is constructed in thin domain of the shell. This boundary-value problem is singularly perturbed with small geometric parameter. Internal iteration process and boundary layers are constructed, problem of their jointing is studied and boundary conditions for each of them are obtained. On the basis of the results of the internal boundary-value problem the asymptotic two-dimensional model of micropolar elastic thin shells is constructed. Further, the qualitative aspects of the asymptotic solution are accepted as hypotheses and on the basis of them general applied theory of micropolar elastic thin shells is constructed. It is shown that both the constructed general applied theory of micropolar elastic thin shells and the classical theory of elastic thin shells with consideration of transverse shear deformations are asymptotically confirmed theories.

Application of micropolar theory to the description of the skin effect due to hydrogen saturation

“…The solution of the problem, accounting for classical and nonclassical boundary conditions, can, in general, be considered as a sum of a momentless solution and boundary-layer type corrections [66][67][68]. The momentless solution does not satisfy nonclassical boundary conditions.…”

Applied model of elastic thin plates made from micropolar material with constrained rotation and application of the finite element method.

“…В работах [13][14][15][16][17][18] на основе асимптотических свойств решений трёхмерной микрополярной теории упругости со свободным вращением в тонких областях [19] сформулированы достаточно общие гипотезы и построены прикладные теории микрополярных упругих тонких стержней, пластин и оболочек с независимыми полями перемещений и вращений. В работах [20,21] развивается метод конечных элементов решения краевых задач статики и динамики микрополярных упругих тонких стержней и пластин со свободным вращением.…”