<b>Supplementary material for "</b><b>Buckling behavior of functionally graded nanoplates on Pasternak foundations: A modified nonlocal strain gradient-based analytical solution and artificial neural networks</b><b>"</b>

This study presents analytical solutions and artificial neural network model for buckling analysis of functionally graded nanoplates on Pasternak foundations using novel modified nonlocal strain gradient theory. The main goal of this study is to present theory that integrates both nonlocal elasticity and strain gradient effects within a unified theoretical framework, allowing the capacity to accurately capture size-dependent effect at micro and nanoscale structures. The governing equations of motion are derived using higher-order shear deformation theory and Hamilton’s principle. Besides, a key novelty of this work is the integration of an artificial neural network model with analytical solutions, allowing efficient prediction of the critical buckling loads of functionally graded nanoplates under various parameters. The artificial neural network model is trained using a dataset generated from analytical methodology. A parametric study is conducted to analyze the influence of the nonlocal parameter, material length-scale, power-law index, foundation stiffness, and geometric configurations on the buckling behaviors of functionally graded nanoplates. The outcomes illustrate the effectiveness of the artificial neural network approach and highlight the robustness of the modified theory in modeling the stability of functionally graded nanostructures supported by elastic foundations.