Abstract
The static deformation of micromachined beams under prescribed in-plane compressive stress is studied by analytical and experimental means over the pre-buckling, transition, and post-buckling load ranges. The finite amplitude of the beam in its post-buckled state is predicted by modeling the non-linear dependence of its compressive stress on the out-of-plane deformation. In addition, the model explicitly considers the net effect of slight imperfections, which can include fabrication defects, geometric irregularities, or non-ideal loading, on the beam's behavior in the near-buckling regime. As an application, clamped-clamped silicon dioxide beams are fabricated through conventional bulk micromachining, and their deflected profiles are measured through three-dimensional optical profilometry. The measurements are compared to the post-buckled amplitudes and shapes that are predicted by the model, and by existing simpler models that do not include the effects of both non-linearity and imperfection. As borne out by the data, when imperfections are considered, the beams exhibit continuous growth of the out-of-plane amplitude during transition from the pre-buckled state to a post-buckled one, in contrast to sudden bifurcation at a critical load. By accounting for this behavior, the estimate of residual stress in the thin film from which the beams are fabricated can be improved, and the amplitude of common post-buckled micromachined structures can be predicted.