Flexoelectric Effect On The Electromechanical Coupling Response Of Piezoelectric Nanobeam Under The Different Boundary Conditions

The flexoelectric effect is the coupling relationship between the polarization(electric field)and the strain gradient,which has important influence on the new micro-nano sensors,actuators and energy harvestings.Based on the linear piezoelectric theory,the coupling effect between the strain gradient and the electric field is considered into the Gibbs free energy density function of the flexoelectric-piezoelectric material.The constitutive equations,governing equations and boundary conditions are obtained by the variational method.Based Bernoulli-Euler theory,the governing equations and the coupling boundary conditions of nano beams are deduced.Piezoelectric beams under different electromechanical boundary conditions subjected to the mechanical loading are discussed in detail.For the cantilever piezoelectric beams,we discuss the flexoelectric effect of piezoelectric beams under three electrical boundary conditions:open circuit without surface electrodes(OC),closed circuit with a fixed external electric potential(CCF)and open circuit with surface electrodes and an induced electric potential by mechanical deformation(OCI).Through the analysis of cantilever nanobeam,the analytical solution of the deflection of the beam and the analytic expression of the induced potential due to the flexoelectric effect under the OCI boundary are obtained firstly.The relationship between the induced electric potential and the flexoelectric coefficient/structure sizes of beams is also obtained.The results show that the normalized deflections of beams due to the flexoelectric effect are always smaller than that without the flexoelectricity in the OC and OCI conditions,and the beam exhibits a stiffer elastic behavior.The normalized deflection of beams in the CCF condition is smaller or larger dependent on the direction of the applied electric voltage.A nearly uniform electric potential distribution is generated by mechanical deformation due to the flexoelectric effect in the OCI condition.The induced electric potential causes a reversed moment,which opposes those mechanically induced and decreases the deflection of the beams.Under the OCI condition,we further study the electromechanical coupling response of piezoelectric beams with the different mechanical boundary conditions(clamped-clamped structure and simply supported structure).The results show that the flexoelectricity has no effect on the response of piezoelectric beams of clamped-clamped structure.For the simple support structure,the induced potential and normalized deflection are influenced by the axial force,which is more significant on the nano scale beams.The influence of the induced potential on the elastic behavior of the simply supported beam is the same as that of the cantilever beam.But the influence of induced potential on the electrical property of the cantilever piezoelectric beam(such as peak potential,polarization)is more obvious than that of the simply supported structure.This work provides a helpful guidance for the design and development of flexoelectricity micro-nano devices.