Size-dependent vibration and stability of moderately thick functionally graded micro-plates using a differential quadrature-based geometric mapping scheme

In this paper, we propose a novel differential quadrature-based geometric mapping scheme to address the C-1-continuity requirements of the in-plane and out-of-plane kinematic quantities of a couple stress-based Mindlin-Reissner functionally graded (FG) plate model. This mapping scheme is combined with the minimum total potential energy principle to construct a four-node size-dependent plate element being free of shape functions with 20 degrees of freedom (DOFs) per node. To validate the convergence and adaptability of the element formulation derived, a wide range of numerical examples regarding the free vibration and stability problems of FG macro-/micro-plates are presented. Finally, parametric studies are carried out to investigate the influences of material and geometrical parameters on the vibration and stability behavior of moderately thick FG micro-plates. It is shown that the developed element possesses distinct advantages over the corresponding shape function-based element in terms of convergence and computational accuracy, which can lead to a remarkable reduction of the number of meshes employed. Besides, the size-dependence of vibration and buckling mode shapes is exhibited in the form of displacement contour plots for the first time.