A Symplectic Method for Analyzing the Nonlocal Modal Behavior of Kirchhoff Plates and Numerical Validation

Eringen’s integral constitutive relation is more general than its differential counterpart for modeling small-scale effects in micro- and nanostructures; however, it leads to integro-differential governing equations that are difficult to solve, which has limited the practical use of integral formulations. To directly address this gap, this paper introduces a novel symplectic-based numerical method that efficiently and accurately analyzes the free vibration of small-scale Kirchhoff plates governed by Eringen’s integral nonlocal model. The method discretizes the nonlocal integral operator by introducing inter-belt elements for long-range interactions and adopting a truncated influence domain, while balancing computational efficiency and accuracy. The effects of the nonlocal parameter, two-phase mixture parameter, mode numbers, kernel types, and geometric parameters on the natural frequencies are systematically investigated. The results indicate stiffness softening. For a simply supported square nanoplate with side length a=10 nm, the first-order frequency parameter decreases by approximately 25% as the nonlocal parameter increases from 0 to 4 nm, and higher-order modes exhibit substantially greater sensitivity to nonlocal effects. Convergence and accuracy are validated against published continuum-level solutions and molecular dynamics simulations; relative deviations are below 2% in most cases, and the local limit (la=0) yields errors on the order of 10−3.