Size-dependent thermoelastic vibration of nanobeams under localized heating: a hybrid analytical–ANN approach

Abstract

This study develops an efficient modeling framework to predict the size-dependent thermoelastic vibration behavior of nanobeams subjected to partially localized thermal loading and supported by Pasternak elastic foundations. The governing equations are formulated within a nonlocal elasticity framework to account for nanoscale interactions under prescribed thermomechanical fields. The expressions for natural frequencies are derived for various heating patterns, including uniform, linear, nonlinear, and sinusoidal temperature distributions with different heated length ratios. The semi-analytical solutions are validated through comparisons with available benchmark results, demonstrating excellent accuracy and reliability with a maximum relative error of only 0.0013%. Comprehensive parametric investigations elucidate the coupled influences of nonlocal effects, thermal softening, and foundation stiffness on the vibration characteristics. In addition, a feed-forward artificial neural network trained on the semi-analytical dataset is employed as a surrogate model to enable rapid and accurate frequency prediction across an extended parametric space. A key scientific contribution of this study lies in the consistent formulation of the thermally induced axial force based on equilibrium and axial strain compatibility conditions. The present methodology ensures mechanical consistency and can be readily extended to more complex structural configurations, including multi-directional functionally graded materials, non-uniform geometries, and nonlinear temperature distributions.