A non-classical Mindlin plate finite element based on a modified couple stress theory

This paper presents a novel Mindlin plate element based on the framework of a modified couple stress theory for analyzing the static bending, free vibration and buckling behaviors of size-dependent Mindlin micro-plates. The element proposed is a four-node rectangular element which has 15-DOF (degrees of freedom) at C0 each node with considering both bending and stretching deformations, and 9-DOF with only considering bending deformation. Unlike the classical Mindlin plate element, this element satisfies the continuity and C1 weak continuity and contains a material length scale parameter. It can be directly used to predict the size effect on the Mindlin micro-plates. Finite element formulations are derived by applying the corresponding weak form equations. To illustrate the applicability and accuracy of the developed Mindlin plate element, the static bending, free vibration and buckling problems for rectangular Mindlin micro-plates with various boundary conditions are investigated. Convergence and comparison studies are carried out to examine the reliability of the numerical solutions. It is shown that the typical numerical results are in good agreement with those available results reported in previous literature. In addition, the numerical results illustrate that the size effect on Mindlin micro-plates can be effectively predicted by using the proposed plate element. These predicted trends agree with those observed in micro-torsion test of thin copper wires and micro-bending test of epoxy polymeric beams. Some results are believed to be the first known in the open literature and can be used as benchmark for further studies.