Localized thermal loading effects on nanobeam vibrations resting on Pasternak foundations: Analytical and ANN surrogate approaches

This study investigates the vibration behavior of nanobeams partially subjected to localized thermal fields and resting on Pasternak-type viscoelastic foundations. An analytical framework is developed using the nonlocal elasticity theory to capture nanoscale size effects while accounting for the combined influence of foundation stiffness, shear interaction, and temperature gradients. Closed-form expressions for the natural frequencies are derived for different thermal loading scenarios, including uniform and linear through-thickness temperature distributions with various heated length ratios. The results highlight the strong interplay among thermal softening, scale-dependent stiffness reduction, and foundation-induced stabilization. To complement the analytical model, a surrogate prediction strategy based on artificial neural networks (ANNs) is constructed, trained, and validated using analytical data to ensure consistency between physics-based and data-driven approaches. The trained ANN exhibits excellent accuracy in reproducing the analytical results and provides a fast and reliable tool for predicting vibration frequencies over wide parametric ranges. The proposed analytical–ANN framework thus represents a novel contribution that integrates nonlocal elasticity, localized thermal effects, and viscoelastic foundation modeling into a unified predictive approach, offering valuable insights and practical guidance for the design and optimization of nanoscale thermo-mechanical systems.