Flexoelectricity, the coupling of strain gradient to polarization, enhances the properties of piezoelectric response desirable for advanced MEMS dramatically even in centrosymmetric dielectrics. In this paper, the general formulations of the flexoelectric couple stress theory presented by Hadjesfandiari in orthogonal curvilinear coordinate system are derived, and are then specified for the case of cylindrical coordinates. A size-dependent flexoelectric model of circular plate is established based on the current formulations in cylindrical coordinates. The governing equations, boundary conditions and initial conditions are derived by applying Hamilton's principle. The static bending and free vibration problems of a simply supported axisymmetric circular plate are carried out to illustrate the applicability of the present model. Numerical results reveal that a homogeneous electric field between the up and down surfaces of the circular plate is induced indeed. The generated deflection, induced voltage and natural frequency show obvious size effect, but the size effect is almost diminishing as the thickness of the plate is far greater than the material length scale parameter. As the increase of the flexoelectric coefficient, the induced voltage increases evidently and the generated deflection and the natural frequency increase weakly.

In this paper, a Levy-type solution based on the modified couple stress theory isdeveloped to study the buckling behaviors of micro-plates. Based on this theory, lengthscale parameter is considered to capture the size effect of rectangular micro-plates.Minimum potential energy and adjacent-equilibrium criteria are exploited to obtain thestability equations and corresponding boundary conditions. Different boundary conditionswith two opposite edges simply supported and arbitrary boundary conditions along the otheredges are considered. To illustrate the new model, both uniaxial and biaxial loads areapplied and the critical buckling loads are defined for over a wide range of thickness,different length scale parameters and various boundary conditions. To show the accuracy ofthe formulations, present results are compared with available results in literature forspecific cases and a very good agreement is observed. Results reveal that the criticalbuckling load increases as the length scale parameter increases especially when thethickness of the micro-plates becomes in order of length scale parameter and this effectis more significant for free boundary condition.