An exact analytical solution for a free-supported micropolar rectangle by the method of initial functions

• Published in: Zeitschrift für angewandte Mathematik und Physik 2022-04, Vol.73 (2), Article 74
• Main Author: Matrosov, Alexander V.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Cartesian coordinates; Differential equations; Elasticity; Engineering; Equilibrium equations; Exact solutions; Isotropic material; Mathematical analysis; Mathematical Methods in Physics; Operators (mathematics); Plane strain; Theoretical and Applied Mechanics
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-022-01714-y

In this work, the method of initial functions for a micropolar medium under conditions of plane strain has been developed. In a Cartesian rectangular coordinate system, the solution of the differential equilibrium equations in displacements of the micropolar isotropic theory of elasticity in the form of a linear combination of the stress–strain state (SSS) components defined on the line x = 0 (initial functions) with coefficients in operator form is constructed. The choice of initial functions in the form of trigonometric series makes it possible to solve the boundary problem of deformation of a micropolar rectangle ( h × l ) with arbitrary boundary conditions on the sides x = 0 , h and the type of free support on the sides y = 0 , l . The obtained solution was used to study the SSS of a rectangle made of human bone considered as an isotropic micropolar material. The results of the SSS study, depending on the size of the rectangle, are presented. The limiting dimensions at which the SSS values of the components differ by 5% from the values of the corresponding components calculated according to the classical theory of elasticity are determined.