Closed solutions for the electromechanical bending and vibration of thick piezoelectric nanobeams with surface effects

In this paper, a more accurate model is established to study the influences of surface effects (SEs) including the surface elasticity, residual surface stress and surface piezoelectricity, on the electromechanical bending and vibration of the piezoelectric nanobeam (PNB) in the presence of shear deformation and rotary inertia. Analytical solutions are obtained for the electromechanical bending deflection, resonant frequency and mode shape of the PNB for three typical boundary conditions, demonstrating the significance of incorporation of shear deformation, rotary inertia and the surface parameters at different aspect ratios. The analytical solutions are found to be in good agreement with both molecular dynamics results and experimental data. The numerical results reveal that the surface elasticity plays a less significant role on the electromechanical bending and vibration than that of surface piezoelectricity and residual surface stress, and can be neglected for PNB with small aspect ratio and stiffer boundary conditions. In addition, the shear deformation and rotary inertia are found to play a larger impact effect than SEs for a stubby beam at higher vibration mode (i.e., aspect ratio less than 14 for doubly clamped beam). Moreover, the SEs are found to be increasing notably as aspect ratio increases. The continuum model established in this study will be useful for characterizing the mechanical properties of size-dependent piezoelectric structures and the design, calibration and application of PNB-based devices.