The bending and vibration responses of functionally graded piezoelectric nanobeams with dynamic flexoelectric effect

Abstract

Dynamic flexoelectric effect refers to the coupling between polarization and acceleration of a material, which is usually ignored in previous works. However, when the thickness of the beam decreases to nanometer scale, the strain gradient increases sharply, it is not accurate to only consider the static flexoelectric effect. The aim of this manuscript is to study the static bending and vibration frequency of the functional gradient piezoelectric nanobeam with the dynamic flexoelectric effect. The governing equations and boundary conditions are derived from variational principle. Different from the classical electric Gauss equations, electric field equilibrium equation has be amended in this manuscript. Then, parameter studies are given to examine the influence of dynamic flexoelectric effect coefficient, applied voltage and gradient index on the static deformations and free vibration frequencies of FGM piezoelectricity nanobeam. The numerical results show that the dynamic flexoelectric effect are significant in investigation on higher-order vibration modes of FGM nanobeam, while gradient index has a vital influence on dimensionless frequency of each mode. In addition, it is observed that the effective stiffness is obviously affected by the gradient index, piezoelectricity and flexoelectricity. The model is helpful to supply some useful guidelines for designing energy harvesting devices at nanometer scale.

Keywords

• Flexoelectricity
• Dynamic flexoelectricity
• Functional graded nanobeam
• Static deformation
• Free vibration