Necessary and sufficient criteria for bistability in electrostatically actuated initially curved pre-stressed micro-plates

The study brings forth extended criteria for electrostatically actuated micro-plates, which take into account the presence of prestress. Following previous studies that analysed the veracity of a reduced order (RO) model, based on Berger’s plate approximation, we present here an analysis of a single degree-of-freedom (DOF) model. The resulting simplified model has enabled a semi-analytical derivation of criteria, for which the plate must adhere to in order to gain bistability in the presence of electrostatic load. The study entails a precursor analysis of a “mechanically” loaded plate, so as to inspect the reliability of Berger’s RO (BRO) model, whilst laying the analytical–numerical foundation for the subsequent electrostatic analysis. The end result is a set of criteria in the space of the plate elevation, thickness-to-gap ratio, and membrane load, splitting it into mono- and bistable domains. Numerical fits, obtained for various cases are also presented. In the process of the study, an upper bound for the condition was also disclosed, representing a necessary condition for bistability. The resulting criteria were compared to numerically obtained ones, extracted from the validated Föppl–von-Kármán’s RO (vKRO) model, as well as from direct finite differences (FD) based solutions (vKFD). The result has shown a remarkable qualitative resemblance, especially in low enough thickness-to-gap ratios, providing a much needed tool in the design of such structures.