Effects of surface and flexoelectricity on a piezoelectric nanobeam

The effects of surface and flexoelectricity have been found in the presence of strong size dependence and should be technically taken into account for nano-scaled dielectric structures. This paper proposes a Bernoulli-Euler beam model to investigate the electromechanical coupling response of piezoelectric nanostructures, in which the effects of surface elasticity, dielectricity and piezoelectricity as well as bulk flexoelectricity are all taken into consideration. The governing equations with non-classical boundary conditions are naturally derived from a variational principle. Then the present beam model is directly applied to solve the static bending problems of cantilever beams. Without considering the residual surface stresses, the bending rigidity can be defined the same as that in classical piezoelectricity theory. The bending rigidity is found to increase for silicon nanowires and decrease for silver nanowires. Also the flexoelectric effect in piezoelectric nanowires has a momentous influence on the bending rigidity. The residual surface stresses which are usually neglected are found to be more important than the surface elasticity for the bending of nanowires. However, this has no influence on the effective electromechanical coupling coefficient. The deflections reveal the significance of the residual surface stresses and the bulk flexoelectric effects. The effective electromechanical coupling coefficient for piezoelectric nanowires is dramatically enhanced, which demonstrates the significant effects of the bulk flexoelectricity and surface piezoelectricity. The effects of surface and flexoelectricity decrease with the increase of the beam thickness, and therefore these effects can be ignored for large-scale structures. This work is very helpful in designing cantilever-beam-based nano-electro-devices.