Supplementary material for "Static bending and free vibration analysis of functionally graded nanoplates on Pasternak foundations using modified nonlocal strain gradient theory and artificial neural network approaches"

This study presents a novel modified nonlocal strain gradient theory for analyzing the static and dynamic responses of functionally graded nanoplates with thin-walled characteristics on Pasternak elastic foundations. Employing higher-order shear deformation theory and Hamilton’s principle, the governing equations are derived and solved analytically using the Navier method. The novelty of this study is that the proposed model integrates nonlocal elasticity and strain gradient effects within a unified theoretical framework to capture size-dependent behavior at the nanoscale. Additionally, an artificial neural network model is developed to enhance computational efficiency and predictive capability. A thorough parametric study is then presented to illustrate the influence of foundation stiffness parameters, nonlocal and length-scale coefficients, geometric ratios, and material gradation on the bending and vibration behaviors. The results demonstrate the accuracy and applicability of the proposed approach in characterizing the mechanical performance of thin-walled nanostructures, such as 2D materials and nanoscale functionally graded shells and plates. This work contributes to the advanced modeling of thin-walled nanosystems and supports the development of lightweight, high-performance structural components.