Size-dependent nonlinear bending and vibration of flexoelectric nanobeam based on strain gradient theory

Abstract

In this study, the nonlinear bending and free vibration of Timoshenko piezoelectric nanobeam incorporating flexoelectricity and surface effect are investigated for the first time. To take size effect into account, a size-dependent Timoshenko nanobeam model containing additional material length scale parameters is developed based on strain gradient theory. The governing equations and corresponding boundary conditions are obtained by electric enthalpy variation and Hamilton’s principle. By adjusting the values of material length scale parameters, the current Timoshenko piezoelectric nanobeam formulations can be transformed to those based on modified couple stress theory. The generalized differential quadrature method (GDQM) is employed to discretize governing differential equations and boundary conditions into a series of nonlinear algebraic equations. Then the algebraic equations are solved by using the Newton iteration method. It is found that strain gradient elastic effect, flexoelectricity, surface effect and applied electric voltage have significant influences on the nonlinear mechanical behaviors of nanobeam. Simulation results indicate that both the strain gradient effect and flexoelectric effect have considerable impacts on the electric field distribution in nanobeams. Moreover, the results also indicate that the influence of flexoelectricity is weakened to some extent due to the consideration of surface effect. The numerical analysis reveals that the present model can be considered reliable to quantitatively investigate size-dependent nonlinear bending and nonlinear free vibration of the piezoelectric nanobeam incorporating flexoelectric and surface effects.