Size dependent coupled electromechanical torsional analysis of porous FG flexoelectric micro/nanotubes

In this paper, the governing equations of torsional porous flexoelectric microtubes made of functionally graded (FG) materials are investigated and obtained for the first time. First, the governing electromechanical coupling equations of the problem are extracted, and to achieve this goal, a modified non-classical flexoelectric theory has been used to derive these equations. The independent parameters of polarization in enthalpy relationships and the mechanical and electrical size effect are the important points in this theory. Noted that the independent parameter effects of polarization, mechanical and electrical size that is considered for the first time in this paper, are the capability of this theory, and the effect of these parameters is evaluated on direct and reverse flexoelectric. After that, the static analysis and free vibrations have been reviewed. And in the conclusion section interesting and valuable results have been obtained, that could help the torsional micro-sensors and torsional micro-actuators designers.