Analytical and ANN-based approaches for free vibration and nonlinear transient analysis of FG-GOEAM toroidal shell segments

Abstract

Objectives

This study investigates the free vibration and nonlinear transient response of functionally graded graphene origami (GOri)-enabled auxetic metamaterials (GOEAMs) toroidal shell segments under thermal conditions. The impact of the Winkler-Pasternak foundation, distributed in two configurations: centered and at both ends of the shell, is thoroughly examined.

Methods

The material properties with GOri distributions through the shell thickness are scrutinized using genetic programming-assisted micromechanical models. Nonlinear kinematic relationships are derived via Reddy's third-order shear deformation theory and von Kármán's geometric assumptions. The equations of motion are solved using Galerkin method. An Artificial Neural Network (ANN), trained with Bayesian regularization backpropagation algorithm, is developed to predict natural frequencies, using comprehensive training data validated against analytical results.

Results

The ANN achieves a target mean squared error (MSE) of $1\times {10}^{-7}$, with error histograms showing minimal and evenly distributed errors. Regression plots confirm perfect correlations (R = 1) between predicted and actual values, indicating robust predictive accuracy. Additionally, increased GOri folding amplifies the negative Poisson's ratio, reduces Young's modulus in GOri/Cu composites, and consequently decreases shell stiffness, lowers natural frequencies, and increases vibration amplitudes. A center-concentrated foundation distribution yields higher natural frequencies and reduced vibration amplitudes compared to end-distributed configurations.

Conclusions

The proposed approaches demonstrate high accuracy and generalization capability in predicting the dynamic responses of FG-GOEAM shells under thermal effects. The findings emphasize the critical role of GOri folding patterns and foundation distributions in tuning vibration characteristics, offering valuable insights for the design and optimization of advanced metamaterial structures.