Symmetry Breaking Criteria in Electrostatically Loaded Bistable Curved/Prebuckled Micro Beams

The symmetric and asymmetric buckling of micro beams subjected to distributed electrostatic force is studied. The analysis is carried out for two separate cases: a case of a stress-free beam, which is initially curved by fabrication and a case of a pre-stressed beam buckled due to an axial force. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with vibrational or buckling modes of a straight beam used as the base functions. The criteria of symmetric, limit point, buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beams. While the necessary symmetry breaking criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve, the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric equilibrium path. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through. A comparison between the results provided by the reduced order model, and those obtained by other numerical analyses confirms the accuracy of the symmetry breaking criteria and their applicability for the analysis and design of micro beams.