Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory

In this paper, the free vibrations of a short cylindrical nanotube made of piezoelectric material are studied based on the consistent couple stress theory and using the shear deformable cylindrical theory. This new model has only one length scale parameter and can consider the size effects of nanostructures in nanoscale. To model size effects in nanoscale, and considering the nanotube material which is piezoelectric, the consistent couple stress theory is used. First, using Hamilton\'s principle, the equations of motion and boundary condition of the piezoelectric cylindrical nanoshell are developed. Afterwards, using Navier approach and extended Kantorovich method (EKM), the governing equations of the system with simple-simple (S-S) and clamped-clamped (C-C) supports are solved. Afterwards, the effects of size parameter, geometric parameters (nanoshell length and thickness), and mechanical and electric properties (piezoelectric effect) on nanoshell vibrations are investigated. Results demonstrate that the natural frequency on nanoshell in nanoscale is extremely dependent on nanoshell size. Increase in size parameter, thickness and flexoelectric effect of the material leads to increase in frequency of vibrations. Moreover, increased nanoshell length and diameter leads to decreased vibration frequency.