Vibration of piezoelectric nanobeams with an internal residual stress and a nonlinear strain

This Letter reports the effect of an internal residual stress and the local geometric nonlinearity on the vibration of piezoelectric nanowires (NWs). A dynamic equation is derived based on Hamilton's principle, which enables one to capture the above-mentioned effects and the influence of all lateral surfaces of a rectangular NW. Vibration frequencies are obtained for the NWs under an electrical voltage and compared with those given by the existing Young-Laplace model where zero internal stress, a linear strain and the effects of top and bottom surfaces of rectangular NWs are considered. It is found that the internal residual stress can extinguish the effect of the surface-induced residual stress and substantially down shift the frequency or qualitatively alter the size-dependence of the frequency. In addition, with a nonlinear strain the piezoelectric effect is found to be able to exert a direct impact on the bending stiffness of piezoelectric NWs.