Size dependent torsional electro-mechanical analysis of flexoelectric micro/nanotubes

The free vibration and static torsion of a coupling electromechanical flexoelectric micro/nanotube have been investigated, in the present study. Using the non-classical theory of continuum mechanics which is based on a strain gradient, the coupling governing equations of torsional flexoelectric micro/nanotubes have been developed. To derive the coupling governing equations the variational method has been used and the formulation with the classical and non-classical boundary conditions is generally derived for torsional flexoelectric micro/nanotubes. For the first time, the formulation of micro/nanotube torsional flexoelectric coupling with independent polarization effect is presented in this paper which can be used for specific applications. At last, the static analysis and free vibrations results are presented for specific mechanical and electrical boundary conditions. The parameter of mechanical and electrical size effect changes has a great effect on the results. However, the effect of the mechanical and electrical size effect parameter on direct flexoelectric torsion is also opposite, but the effect of mechanical and electrical size effect parameter on reverse flexoelectric torsion is similar. Polarization is also introduced as a new variable in the equations and its effect is investigated as an independent variable of the electric field in the results, in this paper. As a significant parameter, the results show that polarization plays an important role to simulate the torsional electromechanical structures.