Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Abstract

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.