Electromechanical Coupling Analysis Of The Flexo-Piezoelectric Functionally Graded Cantilever Nanobeams

The flexoelectric effect has an important influence on the mechanical-electric coupling performance of micro-nano devices.Therefore,improving its role in sensors,actuators and energy capture devices is an important research filed at present.This paper takes functionally graded flexo-piezoelectric materials as the research object,and mainly analyzes the mechanical-electrical coupling performance of functionally graded flexo-piezoelectric beam structures under different electrical boundary conditions.Using the variational principle and linear piezoelectric elastic theory,we derive and obtain the constitutive equations,governing equations and boundary conditions of the functionally graded flexo-piezoelectric cantilever beam.The influence of flexoelectricity,piezoelectricity,functional gradient parameters,structural parameters and boundary conditions on the mechanical and electrical coupling performance of the system have been analyzed and discussed in detail.On the basis of a functionally graded flexo-piezoelectric cantilever beam,the influence of flexoelectric effect on the mechanical-electric coupling performance of the system under two electrical boundary conditions is discussed.Different electrical boundary conditions are closed circuit with a fixed external electric potential(CCF)and open circuit with surface electrodes and an induced electric potential by mechanical deformation(OCI).The exponential distribution function describing the change of material properties along the thickness direction is introduced,and the corresponding governing equations and boundary conditions under the two electrical boundary conditions are obtained through Bernoulli-Euler beam theory and generalized variational principle.The analytical expression of the induced potential of a functionally graded piezoelectric beam is obtained for the OCI condition.Numerical simulation results show that the flexoelectric effect,piezoelectric effect and gradient index have a significant influence on the mechanical-electrical coupling performance of functionally graded beams.Under CCF conditions,the equivalent deflection of a functionally graded beam has better controllability than the homogeneous beam.Under different external loads,the coupling effect of flexo-piezoelectricity on the mechanical deformation and stress distribution of the beam is different.Due to the gradient of piezoelectricity,the difference in polarization direction can change(enhance or weaken)the bending resistance of the beam.Under the OCI condition,due to the flexoelectric effect and the gradient piezoelectric effect,the functionally graded flexo-piezoelectric beam structure will generate an induced potential.Owing to the inverse flexoelectric effect,the induced potential will generate the opposite moment of the mechanical load,which would reduce the bending deformation of the beam,thereby increase the bending stiffness of the gradient beam.This paper further studies the changes in the mechanical and electrical properties of functionally graded piezoelectric beams under OCI conditions with exponentially distributed gradient exponents.Numerical simulation results show that the existence of piezoelectric gradient will cause flexoelectric effect to generate induced potential,and have a greater effect when the thickness of the beam increases.Hence,the flexoelectricity and graded piezoelectricity have a significant effect on the electromechanical responses of nano structures.This paper will provide an important theoretical basis for the design of graded materials under the micro-nano scale.