Construction of a Bending Model of Micropolar Elastic Thin Beams with a Circular Axis and Its Implementation Using the Finite Element Method

• Published in: Journal of applied mechanics and technical physics Vol. 63; no. 7; pp. 1205 - 1217
• Main Authors: Sargsyan, S. H., Khachatryan, M. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Bending; Classical and Continuum Physics; Classical Mechanics; Curved beams; Elastic deformation; Elasticity; Energy conservation law; Finite element analysis; Finite element method; Fluid- and Aerodynamics; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mechanical Engineering; Physics; Physics and Astronomy; Static deformation; Variational principles; beam with a circular axis; finite element method; plane bending; micropolar theory of elasticity; one-dimensional model
• ISSN: 0021-8944; 1573-8620
• DOI: 10.1134/S0021894422070100

The problem of a transition from a system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved region to a one-dimensional system of equations of deformation of a micropolar elastic thin beam with a circular axis (a curved beam with a median surface in the form of a circular arc is meant) is discussed. In carrying out this transition, so-called Timoshenko hypotheses generalized to the micropolar case are used. Based on them, an applied model is constructed that describes the stress–strained state during bending of a micropolar (with independent fields of displacements and rotations) elastic thin beam with a circular axis. It is shown that the model includes the law of conservation of energy, energy theorems, and variational principles. All the main functionals of the constructed model are derived from the functional of the two-dimensional micropolar theory of elasticity that contains only the first-order derivatives of displacements and rotations. To solve boundary problems of statics and dynamics on the basis of an applied bending model of a micropolar elastic thin beam with a circular axis, an appropriate variant of the finite element method (FEM) is being developed. The basic concepts and stages of implementation of the modified FEM are formulated: discretization, selection of the main nodal unknowns, approximation of the desired solution, and construction of the basic FEM equations. Examples of finite element solutions of static deformation problems and problems of natural vibrations of beams with a circular axis in the framework of both the micropolar and classical elasticity theories are given. A comparative analysis of the solutions has been performed, as a result of which some effective properties of beams with a circular axis are established when considering their deformations according to the micropolar theory of elasticity.