Supplementary material for "Modified nonlocal strain gradient framework and ANN surrogate for size-dependent buckling of functionally graded nanoplates on Pasternak foundations"

This study develops an analytical framework to investigate the buckling response of functionally graded nanoplates resting on Pasternak-type foundations. The formulation is constructed within a modified nonlocal strain gradient theory, allowing the simultaneous consideration of nonlocal stress softening and strain gradient stiffening effects. To represent the kinematics, a higher-order shear deformation model is adopted, and the governing equations are obtained from the principle of minimal potential energy and solved in closed form for simply supported boundaries. Since repeated analytical evaluations are costly, a neural-network surrogate is trained from solutions created by the analytical formulation, enabling rapid prediction of critical buckling loads across a wide parameter range. The impacts of the nonlocal coefficient, intrinsic material length, gradation profile, geometric ratios, and foundation stiffness are examined through a detailed parametric study. The outcomes demonstrate the consistency of the theoretical model and illustrate how a physics-based formulation, when combined with data-driven approximation, can support efficient analysis and design of nanoscale graded structures.