Pull-in voltage of a circular nanoplate with elastically restrained edge subject to van der Waals attractions

This work provides accurate analytic relations for the pull-in voltage of electrostatically actuated circular nanoplates with an elastically restrained edge, modeled according to the classic Kirchhoff thin plate theory, taking into account the van der Waals attractive force. The cases of clamped or simply-supported edges are recovered as limit cases of the more realistic boundary conditions considered here. The electrostatic loading distribution is approximated by considering the deflection of the elastically restrained circular plate under a uniformly distributed transverse loading. The Green's functions of the plate under the elastically restrained boundary conditions are then used for calculating the deflection of the plate center due to the assumed loading distribution, which also depends on the deflection of the plate center. A closed-form nonlinear relation between the applied electrostatic loading and the deflection of the plate center is thus obtained, which displays a maximum at the pull-in voltage. The analytical model can also capture the effects of intermolecular attractions, which becomes relevant when the distance gap between the electrodes becomes of few nanometers. A comparison with the numerical and experimental results available in the literature for the limit case of clamped edge is then provided to assess the accuracy of the approximated analytical model.