Size effects on electromechanical coupling fields of a bending piezoelectric nanoplate due to surface effects and flexoelectricity

Due to large surface to volume ratio and manifest strain gradients typically present in nanostructures, it is essential to incorporate both surface effects and flexoelectricity in studying the size-dependent electromechanical coupling behaviors of piezoelectric materials at the nano-scale. In the current work, a modified Kirchhoff plate model with the consideration of residual surface stress, surface elasticity, surface piezoelectricity, and flexoelectricity is developed to investigate the electroelastic responses and vibrational behaviors of a bending piezoelectric nanoplate (PNP). The governing equations and the corresponding boundary conditions accounting for both the surface effects and the flexoelectricity are derived by the variational principle. Ritz approximate solutions of the static bending and the free vibration indicate that these nano-scale features are more prominent for thinner plates with smaller thickness. The simulation results also reveal that the influence of the flexoelectricity and the surface effects upon the bending behaviors of the PNP depends on the applied electrical loading and the plate dimensions. Moreover, it is also observed that the frequency tuning of PNP-based nanoresonators by adjusting applied electrical load can be modified by both the flexoelectricity and the surface effects. The current work is expected to provide increased understanding on the theoretical basis for the design and applications of PNP-based nanodevices.