Free vibration and buckling investigation of piezoelectric nano-plate in elastic medium considering nonlocal effects

Abstract

In this paper, buckling and free vibration of piezoelectric nano-plate on elastic foundation are investigated. The assumptions of nonlocal elasticity and classical plate theory are considered. The simply supported boundary conditions are considered and plate is subjected to external electric voltage. The final equations of the motion are derived using Hamilton function and solved by Navier method to find the natural frequency and critical buckling force of the nano-plate. The effect of various parameters on the plate behavior, such as nonlocal parameter, electrical voltage, and Winkler and shear modulus, is investigated and results are compared with those reported in the literature. The results show that by increasing the small-scale effects, the natural frequency and buckling load decrease.

Appendix

$$ \begin{aligned} A = & - D_{11} \alpha^{4} - 2D_{12} \alpha^{2} \beta^{2} - D_{11} \beta^{4} - 4D_{66} \alpha^{2} \beta^{2} - k_{w} \left( {1 + \mu \left( {\alpha^{2} + \beta^{2} } \right)} \right) \\ & \; - k_{\text{g}} \left( {\alpha^{2} + \beta^{2} + \mu \left( {\alpha^{4} + 2\alpha^{2} \beta^{2} + \beta^{4} } \right)} \right) + \omega^{2} \rho h\left( {1 + \mu \left( {\alpha^{2} + \beta^{2} } \right)} \right) + \left( { - \bar{N}_{xx} + 2e_{31} V_{0} } \right)\left( {\alpha^{2} + \mu \left( {\alpha^{4} + \alpha^{2} \beta^{2} } \right)} \right) \\ & \; + \left( { - \bar{N}_{yy} + 2e_{31} V_{0} } \right)\left( {\beta^{2} + \mu \left( {\beta^{4} + \alpha^{2} \beta^{2} } \right)} \right). \\ \end{aligned} $$