Size-dependent torsional wave propagation in FG flexoelectric micro/nanotubes

In the present study, torsional wave propagation in a functionally graded flexoelectric micro/nano tube was investigated. The governing coupled equations of the torsional flexoelectric micro/nanotube have been developed based on strain gradient and micro-inertia. To derive the governing coupling equations, the variation method has been used and the formulation in general with classical and non-classical boundary conditions for functionally graded flexoelectric micro/nanotube has been extracted in this article. Dispersion phenomenon (asymptotic phase velocity-bounded value) which has not been paid attention in classical elastic theories is observed here for flexoelectric micro/nanotubes. However, the main contribution of this paper is the derivation of governing equations for the torsional functionally graded flexoelectric micro/nanotubes with the investigation of the effects of length scale parameters, electromechanical coupling, and micro-inertia on the prediction of actual torsional wave behavior. The analytical solution for phase velocity in this paper is calculated for harmonic decomposition and the magnitude of the phase velocity and its changes based on the wave velocity are plotted in the results section. The results show that the effect of flexoelectricity, micro inertia as well as the effect of size have a great effect on predicting the actual torsional wave propagation behavior in the micro/nanotubes.