Dynamic analysis of a nonlinear nanobeam with flexoelectric actuation

Over the past few years, several researchers have been increasingly attracted to flexoelectric transduction because of its potential application for sensing and actuation in NanoElectroMechanical Systems. The flexoelectric effect refers to coupling between polarization and strain gradient in centrosymmetric and non-centrosymmetric dielectrics. Consequently, not only piezoelectric dielectrics with an initial polarization are of interest, but also a larger range of dielectric materials. In contrast to piezoelectricity, the flexoelectric effect is scale-dependent and can exhibit large electromechanical coefficients only at small scales. This paper focuses on the effects of geometric nonlinearity, resulting from relatively large displacements and restrictive boundary conditions, on the static and dynamic responses of piezoelectric flexoelectric nanobeams. The derived equations of motion for the transverse displacement and variation of the internal electric potential are discretized using a Galerkin procedure. A closed-form solution for the nonlinear static response is proposed. The results are compared and validated with those found in the literature. For the dynamic response, a perturbation technique is used to solve analytically the nonlinear equations of motion for the primary and parametric resonances of the first mode. The analytical perturbation solution is validated using a numerical technique. The results show that a general hardening-type behavior is obtained and, therefore, several jumps are observed for the dynamic solution. High sensitivity of the solution to an applied AC voltage is also demonstrated for the principal resonance of the first mode.