Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity

The symmetry breaking of inversion in solid crystals will induce electric polarization in all solid crystals, which is well known as flexoelectricity. At the nanometer scale, due to the large ratio of surface to volume, piezoelectric structures always exhibit distinct mechanical and electrical behaviors compared with their bulk counterparts. In the current work, the effects of surface and flexoelectricity on the buckling and vibration of piezoelectric nanowires is investigated based on a continuum framework and the Euler-Bernoulli beam hypothesis. Analytical solutions of the electric field in the piezoelectric nanobeam subjected to electrical and mechanical loads are obtained with the surface, flexoelectric and nonlocal electric effects. Numeric simulations demonstrate that the Young's modulus and bending rigidity of PZT and BaTiO₃ (BT) nanowires are enhanced by flexoelectricity. In addition, the critical buckling voltage is calculated with consideration of the effects of surface and flexoelectricity, and it is found that the effects of surface piezoelectricity, flexoelectricity and residual surface stress play significant roles in determining the critical buckling voltage. Results obtained for the first resonance frequency also indicate that the effects of surface and flexoelectricity are more significant at a narrow range of beam thickness. The first resonance frequency of PZT and BT nanowires is also influenced by the residual surface stress and external applied voltage. The current work is expected to provide a fundamental study on the buckling and vibration behaviors of piezoelectric nanobeams, and it might also be helpful in devising piezoelectric nanowire-based nanoelectronics.