Flexoelectric Response Of Piezoelectric Nano-structure Based On Generalized Strain Gradient Theory

Flexoelectricity,as a special electromechanical coupling effect different from piezoelectricity,is the coupling of strain gradient and polarization or polarization gradient and strain,exists in all dielectric materials.However,the initial,there almost no researcher study this special electromechanical coupling effect.With the deepening of scholars' research on electromechanical coupling effect and the development of nanotechnology,the flexoelectricity has attracted more and more attention.In recent decades,researchers have not only observed the flexoelectricity in liquid crystal,traditional micro-beam,micro-plate,and other micro-structures,but also observed the flexoelectric phenomenon in biological tissues such as lipid membrane,mammalian cochlear hair and bone crack.On the other hand,researchers obtained some materials flexoelectric coefficient by experiment.In addition,in terms of theoretical research,numerous models of flexoelectricity have been proposed.These theoretical models explain to some extent the experimental phenomena observed in the experiments.At the same time,it also provides a theoretical basis forthe application of flexoelectricity in micro-sensors,micro-actuators,and energy harvesters.Since the flexoelectricity is related to the strain gradient,the high order stain tensors are inevitably introduced in the theoretical model of flexoelectricity.However,the existing theoretical models of flexoelectricity have not been uniformly determined for which high order strain tensors and which high order strain tensors can be ignored.This leads to some existing theoretical models are too complicated to apply in practice,and some are too simple to accurately describe the flexoelectricity.Based on this,this paper introduces the polarization into the generalized strain gradient theory to establish a new theoretical model of flexoelectricity.In this theory,the high order strain tensors only the dilatation gradient tensor,the stretch gradient tensor,and the symmetric rotation gradient tensor are considered.The electric field ignores the polarization gradient terms and only considers the polarization and strain coupling and the coupling of polarization and strain gradient.Based on the new theoretical model,in this paper,the governing equations and boundary conditions of a single material Euler microbeam and a functionally graded material Euler microbeam are given.The main research contents of this paper include:Based on the generalized strain gradient theory,a new theoretical model of flexoelectricity is established.Then,the mechanical and electrical governing equations and boundary conditions of the Euler micro-beam are derived by the Hamiltonian variational principle,and the expression of deflection and polarization of the beam are obtained.Comparing and analyzing the static bending of the cantilever beam,simply supported beam and clamped-clamped beam under different theoretical models by a numerical example.It is found that the flexoelectricity effect on the static bending of the beam is divided into two aspects.On the one hand,reducing the bending stiffness of the beam,on the other hand,coupling with the applied electric load and apply to the end of the beam,that is,affecting the boundary conditions of the beam.At the same time,analyzed the variation of the beam deflection with different structural dimensions.It is found that the flexoelectricity shows obviously size effect,and the classical continuum mechanics theory cannot explain this size effect,but the strain gradient theory can explain,which indicates the importance of strain gradient theory at the micro-nano scales.Based on the static bending of the beam to generate polarization,this paper defines the calculation method of energy conversion efficiency when the beam is polarized and compares the variation of energy efficiency of the beam under different models with structural scales.Through comparative analysis,it can be found that the energy conversion efficiency of the beam exists size effect,that is,the flexoelectricity exists size effect.In addition,energy efficiency has extreme value in the process of changing with the structural scales when considering the strain gradient theory.However,when the strain gradient theory is not considered,the energy efficiency can be infinitely increased as the structure size decreases,which is unreasonable.This validates the importance of strain gradient theory.Changing the material of the Euler beam,based on the new flexoelectric theory model,the governing equations and boundary conditions of the functionally graded material beams are re-derived.Then,the influence of the proportion of material volume in the functionally graded beam on the static bending and energy efficiency of the beam is analyzed by a numerical example.It is found that the higher the proportion of soft materials in the beam,the smaller the bending stiffness of the beam,and the more obvious the bending of the beam under the same load,by comparative analysis.But unlike static bending,the energy efficiency of the beam does not follow this rule.In the process of changing the volume fraction of materials,it is found that existing an optimal ratio of material volume to make the energy efficiency of the beam attain maximum value and the optimal structural size of the beam is also affected by the proportion of materials.Simply,the beam could achieve greater energy efficiency at larger structural sizes with the variation of material volume proportion.In this paper,a new theoretical model of flexoelectricity is established.The size effect of the flexoelectricity and the importance of the strain gradient theory are verified by numerical examples.The micro-beam model established in this paper canprovide a theoretical basis for design flexoelectrical micro-components such as micro-sensors,micro-actuators,and energy harvesters.