Nonlinear analysis of unimorph piezoelectric nanobeam with a variable cross-section and flexoelectric effect

An analytical method is presented to investigate the nonlinear electromechanical coupling response of an unimorph piezoelectric nanobeam with a variable cross-section, taking into account the flexoelectric effect. Considering the geometric nonlinearity, strongly nonlinear governing equation with variable coefficients is derived using the energy variation method and solved by the differential quadrature method. The nonlinear effects of geometric nonlinearity, electromechanical loadings, flexoelectric effect and width ratio on the electromechanical coupling response are analyzed in detail. The results show that under different loadings, the flexoelectric parameter and width ratio have distinct effects on the electromechanical response of the piezoelectric variable cross-section nanobeam. When the mechanical loads are applied on the nanobeam, a larger flexoelelctric parameter results in a greater bending deflection and higher output of electric field and polarization. The width ratio of variable cross-section can also influence the bending deflection, but its impact on the output of electric field and polarization is limited. When the applied voltage loads are applied to the nanobeam, the bending direction can be controlled by the applied voltage, flexoelectric parameter or width ratio. Due to the strongly nonlinear governing equation, these results in this study differ significantly from those of piezoelectric nanobeam with uniform cross-section. This study can contribute to the understanding of electromechanical coupling response in piezoelectric variable cross-section nanobeam with flexoelectric effect. Additionally, it can be helpful in guiding the structure optimization of piezoelectric nanodevices.