Size-dependent nonlinear analysis of piezo-electrostatically actuated porous functionally graded nanobeams incorporating flexoelectricity

Based on the Euler–Bernoulli beam theory and von Kármán nonlinear hypothesis, this paper presents a piezo-electrostatically actuated nonlinear porous functionally graded (FG) nanobeam model with flexoelectricity taken into consideration. The strain gradient elasticity theory and surface elasticity theory are adopted to incorporate size-dependency and surface energy. Employing Hamilton’s principle, the governing equations and associated boundary conditions are obtained. The nonlinear static pull-in voltage and natural frequency are derived from the generalized differential quadrature method (GDQM) and the Newton iteration method. By combining the multiple times scales method and Galerkin’s discretization technique, the nonlinear amplitude-frequency response under superharmonic excitation is obtained. Eventually, comprehensive parametric investigations are presented to illustrate the influences of geometric nonlinearity, flexoelectricity, porosity, dispersion atomic forces (Casimir and van der Waals forces), surface energy, material distribution and size-dependency on the pull-in instability, free vibration and superharmonic resonance.