Applicability of scale parameters in nonlocal shear deformation circular copper plates for axisymmetric vibrations

Nonlocal shear plate theories, such as nonlocal classical plate theory (Non-CLPT), nonlocal first-order shear deformation theory (Non-FSDT) and third-order shear deformation theory (Non-TSDT), are widely utilized to analyze the mechanical properties of nano-scale. However, the applicability of nonlocal parameters for different theories has always been a challenging problem. The conventional Legendre polynomial method with good orthogonality is widely applied to investigate mechanical problems, but it exhibits significant limitations in addressing complex boundary problems such as partial derivatives and polynomial summations of displacement and stress. In this paper, the constraint representing the boundary is incorporated into the cycle of Legendre polynomials as an independent equation, which can effectively overcome these limitations. The improved method is employed to calculate the natural frequencies of axisymmetric vibrations in nonlocal shear deformation circular plates. The natural frequencies of the circular copper (Cu) plates with a face-centered cubic (FCC) lattice are determined by using fast Fourier transform (FFT) in the molecular dynamics (MD) simulation. The MD results are used to invert the nonlocal parameters for the clamped boundary circular Cu plate. The nonlocal plate results of the calibration parameters are consistent with the MD results. The results indicate that nonlocal parameters based on Non-CLPT are larger than those based on Non-FSDT and Non-TSDT, and this discrepancy increases with the increasing height-to-radius ratio. The nonlocal parameters increase as the radius and height increase, and the fitting functions follow linear and nonlinear relationships, respectively.