Analytical closed-form solutions for size-dependent static pull-in behavior in electrostatic micro-actuators via Fredholm integral equation

In this paper, a novel method is proposed for the first time to obtain static pull-in voltages with fringing field effects in electrostatically actuated cantilever and clamped-clamped micro-beams where the mid-plane stretching and the residual axial load are taken into account for clamped-clamped boundary conditions. The non-classical Euler–Bernoulli beam model containing one material length scale parameter is adopted to effectively capture the size effect. In the solution procedure, the governing fourth-order differential equation of variable coefficients is converted into a Fredholm integral equation. By adopting the first natural mode of the cantilever and clamped-clamped micro-beams as a deflection shape function, the resulting equation is solved for the static pull-in voltages. The accuracy of the present analytical closed-form solution is verified through comparing with the experimentally measured and numerical data conducted in the published works. From the experimental data available in the literature, the value of the material length scale parameter for the (poly)silicon is estimated to be in the order of magnitude of 10−1 μm. Then, the effect of the material length scale parameter on the pull-in voltages of the cantilever and clamped-clamped micro-beams is investigated. The results indicate that the tensile residual stress can extend the validity range of the classical continuum mechanics to lower beam thicknesses. It is also found that microcantilever beams are more sensitive to the size effect than their corresponding clamped-clamped micro-beams.