Thermal postbuckling analysis of functionally graded auxetic metamaterial plates supported by Winkler-Pasternak elastic foundation using artificial neural networks

This study develops an integrated analytical and machine learning framework to investigate the nonlinear thermal postbuckling behavior of functionally graded graphene origami-enabled auxetic metamaterial (FG-GOEAM) plates resting on a two-parameter Winkler-Pasternak elastic foundation. The effective material properties are characterized through a genetic programming-based micromechanical model. The nonlinear governing equations are formulated utilizing Reddy’s higher-order shear deformation theory coupled with von Kármán kinematic nonlinearity and are analytically solved via the Galerkin procedure. To overcome the computational cost associated with iterative nonlinear analytical solvers, two artificial neural network (ANN) architectures trained by Bayesian Regularization (BR) and Levenberg-Marquardt (LM) algorithms are constructed to predict the critical buckling temperature and postbuckling equilibrium paths. Validation results demonstrate that the trained ANNs function as highly efficient surrogate models, achieving a mean squared error of 1 × 10 − 5 with maximum prediction errors consistently below 1%. Comparative evaluations confirm that the BR algorithm provides superior generalization capability and numerical stability across the highly nonlinear domain compared to the LM method. Parametric investigations establish several critical conclusions regarding structural stability. Among the evaluated reinforcement configurations, the X-shaped distribution pattern maximizes the effective bending stiffness, thereby yielding the highest critical thermal buckling threshold. Furthermore, driven by the combination of high elastic modulus and negative thermal expansion coefficient inherent to graphene, increasing the graphene origami content and folding degree significantly improves the thermal postbuckling resistance. Finally, the interaction with the Pasternak shear layer exerts a dominant stabilizing effect against transverse deformation gradients, further enhancing the overall thermal resilience of the plates.