Electro-mechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory

In this paper, the vibration of the isotropic single-walled piezoelectric conic nanotube was investigated using Love's thin shell model and couple stress theory. In this formulation, to model the size effects in the nanoscale, the size-dependent couple stress theory was employed. This theory has recently been developed for piezoelectric materials. Besides, to model the nanotube in a more precise fashion, the shell model was used instead of the beam model. By measuring strain energy, kinetic energy and the work of external forces, Hamilton's principle was used to develop the couple equations between the mechanical and electoral effects in the piezoelectric nanotube along with the related boundary conditions. To evaluate the equations in a special case, the shell equations together with the related boundary conditions were solved using the two Galerkin and Kantorovich Methods. Finally, size effects, flexoelectric effects, and the effects of geometric dimensions on the vibrations of the piezoelectric nanotube were investigated. The findings demonstrated that this model is capable of eliminating the difference between the results of MD simulation and classical theoretical models.