3D free vibration of nonlocal piezoelectric rectangular nanoplates under various boundary conditions using a triple Legendre polynomial method

Piezoelectric nanoplates play a crucial role in micro/nanoelectromechanical systems (MEMS/NEMS), yet their vibration behavior is significantly influenced by size effects at the nanoscale. Traditional 2D nanoplate models have limitations in predicting complex stress-strain distributions and higher-order modes due to simplifying assumptions regarding the thickness direction. This paper establishes an electromechanical coupling vibration model for piezoelectric rectangular nanoplates by integrating Eringen’s nonlocal theory with 3D elasticity theory. The complex 3D partial differential equations are transformed into a matrix eigenvalue problem by employing the analytically integrated triple Legendre polynomial expansion method, yielding series solutions for the natural frequencies and modal fields. The validity of the proposed method is verified against existing data in the literature, followed by a discussion on convergence and computational efficiency. The influences of the nonlocal effect, piezoelectric effect, boundary conditions, and geometric properties on the vibration characteristics of piezoelectric nanoplates are investigated in detail. Compared with the finite element method or meshless methods, the proposed method provides analytical solutions based on series expansion, which can reveal the effects of system parameters on vibration behavior more deeply, offering theoretical guidance for the design of NEMS devices.