Application of the modified Fourier series method and the genetic algorithm for calibration of small‐scale parameters in the nonlocal strain gradient nanobeams

In recent years, nonlocal strain gradient theory (NSGT) has been widely applied by researchers for the analysis of nanostructures. In this theory, the appropriate selection of small‐scale parameters and the type of high‐order boundary conditions is of great importance. In the current paper, free vibrations of carbon nanotubes are studied using a nonlocal strain gradient Timoshenko beam model, and the small‐scale parameters are calibrated based on the molecular dynamics (MD) results. The calibration process is conducted by defining an optimization problem, and the genetic algorithm is applied for obtaining the best values of the small‐scale parameters. The governing equations are derived based on Hamilton's principle and solved analytically using the modified Fourier series method. The variational consistent high‐order boundary conditions are obtained applying the weighted residual approach. The small‐scale parameters are calibrated for the first three natural frequencies considering two different boundary conditions. Moreover, the type of high‐order boundary conditions that is more consistent with the MD results is determined. The outcomes of this paper provide an appropriate reference for researchers who are using the NSGT for the analysis of micro/nanobeams.