On the size-dependent nonlinear thermo-electro-mechanical free vibration analysis of functionally graded flexoelectric nano-plate

Abstract

In the present paper, a continuous-based electromechanical model has been developed by the Kirchhoff plate's theory and the modified flexoelectric theory in order to study the size-dependent nonlinear free vibration of functionally graded flexoelectric nano-plate. Nonlinear governing coupled differential equations of motion of the nano-plate and their associated different boundary conditions for the four edges have been extracted for the first time by using Hamilton's principle and variational method. The obtained governing equations are solved by using Galerkin's and perturbation methods. The electromechanical coupling (electromechanical stress) in the internal energy function causes nonlinearity in the governing equations.The functionally graded material of the nano-plate is defined by power-law along the plate's thickness. Additionally, the plate is under the applied electric voltage and temperature rise. The natural frequencies and natural mode-shapes have been determined and shown in two cases as direct and inverse flexoelectriceffects. The results show that the size-dependency of the material properties in the presence of flexoelectric effect has significant importance in the nano-scale and with regarding to application of this type of nano-plate in oscillators, considering the flexoelectric effect. The electric potential can play an important role in adjusting and controlling the frequency.