Comparative Study of 2D Mindlin Plate and 3D Elasticity Theory for Free Vibration Analysis of Functionally Graded Piezoelectric Nanoplates Using Legendre Polynomial Method

A new comparative study about two-dimensional (2D) nonlocal Mindlin plate and three-dimensional (3D) nonlocal elasticity theory coupled with Legendre polynomial method is developed for free vibration analysis of functionally graded piezoelectric (FGP) nanoplates. In this paper, the small-scale effects are obtained through a simple modification of the local equations based on the nonlocal theory of Eringen. At the same time, the displacements are expanded into Legendre polynomial series and added to the vibration control equations to solve resonant frequency and harmonic response. In order to verify the convergence and accuracy of the presented method, some examples are carried out for isotopic, piezoelectric, and FGP nanoplates under different boundary conditions, and the results are compared with other reported results and which shows good agreement. Then, the effect of various geometric parameters, nonlocal parameter and material gradient index on frequency parameters is analyzed and discussed. In the end, mode shapes and electric potential are also compared and analyzed by the 2D method and the 3D method.