Modeling strategies of electrostatically actuated initially curved bistable micro plates

Micro- and nanolectromechanical systems (MEMS/NEMS) incorporating two-dimensional structural elements such as plates and shells attracted significant interest in recent years. These structures demonstrate rich electromechanical behavior and could be advantageous in applications. In this work, we explore implementation of two models describing axisymmetric behavior of initially curved circular micro plates, subjected to a distributed nonlinear electrostatic force. While both models are based on the Kirchoff hypothesis and on the nonlinear Föppl von Kármán (FvK) strain-displacements relations, the second model employs the Berger approximation, which significantly simplifies the formulation and describes the plate by a single governing equation. In both cases, the solution is based on the Galerkin decomposition with buckling modes of an initially flat plate used as the base functions. To track the unstable branches of the equilibrium curve, continuation methods in conjunction with the Riks algorithm are implemented. The validation of the models is conducted for two loading cases, namely “mechanical” deflection-independent load, and electrostatic displacement-dependent load. Results of a finite elements (FE) analysis, as well as of a finite differences (FD) solution of the differential equations, were used as a reference. We estimate the accuracy of the RO models and provide recommendations concerning the number of degrees of freedom (DOF) required to reach a desired accuracy. We show that a simple RO model based on Berger plate theory, can be conveniently used for analysis of electrostatically actuated plates with low initial curvature and small thickness to electrostatic gap ratio.