Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches

Arch-shaped microelectromechanical systems (MEMS) have been used as mechanical memories, micro-relays, micro-valves, optical switches and digital micro-mirrors. A bi-stable structure, such as an arch, is characterized by a multivalued load deflection curve. Here we study the symmetry breaking, the snap-through instability and the pull-in instability of a bi-stable arch-shaped MEMS under static and dynamic electric loads. Unlike a mechanical load, the electric load is a nonlinear function of the a priori unknown deformed shape of the arch. The nonlinear partial differential equation governing transient deformations of the arch is solved numerically using the Galerkin method and a time integration scheme that adaptively adjusts the time step to compute the solution within the prescribed tolerance. For the static problem, the displacement control and the pseudo-arc-length continuation methods are used to obtain the bifurcation curve of the arch's displacement versus a load parameter. The displacement control method fails to compute the arch's asymmetric deformations that are found by the pseudo-arc-length continuation method. For the dynamic problem, two distinct mechanisms of the snap-through instability are found. It is shown that critical loads and geometric parameters for instabilities of an arch under an electric load with and without consideration of mechanical inertia effects are quite different. A phase diagram between a critical load parameter and the arch height is constructed to delineate different regions of instabilities. We compare results from the present model with those from a continuum mechanics based approach, and with results of other models and experiments available in the literature.