Finite element method for calculation of bending of micropolar elastic thin plates

Abstract

A general applied theory of bending deformation of micropolar elastic thin plates subjected to transverse shear deformation is presented in differential and variation formulations. Equilibrium equations, elasticity and geometric relations, natural boundary conditions of bending of micropolar thin plates follow from this variation principle. The theory has been derived from the corresponding three-dimensional theory using the hypothesis method, which adequately expresses the properties of the asymptotic solutions in the case of thin plate. In the present paper, a finite element method is applied to solve the boundary problems of micropolar elastic thin plates under bending loads. On the basis of free displacement and rotation laws, the effective quadrangular finite elements are designed. Application of the constructed stiffness matrix enables us to perform the procedure of formation of the system of algebraic linear equations. A particular problem of bending of a square micropolar elastic plate under evenly distributed power load is considered for the plate with hinged-supported edges. The plate is investigated in the framework of the theory of micropolar elasticity. Numerical implementation of the system of linear algebraic equations of the finite element method is carried out. In parallel, numerical results for the corresponding classical elastic case of the plate material (with consideration of transverse shear deformation) are obtained by the the finite element method, while other values of problem parameters are equal. Analysis of the numerical results shows the advantage of the micropolar approach over the classical one for describing the stiffness and strength of the examined plate.