A C1 Finite Element Formulation for Mindlin-Reissner Microplate Model

In this paper, a C1 finite element (FE) formulation of Mindlin-Reissner microplate based on strain gradient elasticity theory is developed. The general form of the stiffness matrix and force vector of the microplate element is firstly extracted, and then specialized on a four-node quadrilateral element with 36 degrees of freedom. Deformation of rectangular microplates with simply-supported edges, clamped edges, and three edges simply-supported and the fourth edge free, and under uniform external pressure is then studied. For the case of microplate with simply-supported boundaries, comparison between the FE and the corresponding exact solution is made, which shows extremely close results. For the next two examples, a convergent solution by means of mesh refinement is obtained. Moreover, for the case of thin plates and for large values of the thickness-to-material length ratio, the results of gradient-based FE analysis are coincident with those of the Kirchhoff plate model based on classical elasticity. Numerical simulations show that the introduced element is able to capture the size effect phenomenon at micron scale. When the plate thickness is in the order of the material length parameter, the value of deflection is lower than that predicted by the models based on classical elasticity.