Integrating analytical and machine learning methods for investigating nonlinear bending and post-buckling behavior of 3D-printed auxetic tubes

This study focuses on two primary objectives regarding 3D-printed tubular metastructures. Firstly, it investigates the nonlinear mechanical bending and post-buckling characteristics of re-entrant perfect and imperfect auxetic tubes analytically. To achieve this, von-Kármán nonlinear assumption and Timoshenko theory are employed to establish the governing equations for the auxetic tubes with geometrical imperfection under transverse and axial mechanical loads. The tensile test and the digital image correlation (DIC) method are used to experimentally extract the mechanical properties of polylactic acid (PLA) biopolymer, and scanning electron microscopy (SEM) images are taken to examine factors influencing differences in Young’s modulus and ultimate tensile strength in dogbone specimens. The nonlinear equations are discretized using the Ritz method with Chebyshev polynomials. The validation study is carried out for the auxetic tube and the unit cell’s representative volume element (RVE), utilizing findings from the literature and simulations in the finite element (FE) software Abaqus. In the second part, machine learning-assisted predictions concerning the behavior of this structure under mechanical loads are explored. Five machine learning algorithms, including Categorical Boosting (CatBoost), Random Forest (RF), Gradient Boosting Tree (GBT), Decision Tree (DT), Extreme Gradient Boosting (XGBoost), and an Artificial Neural Network (ANN) model, are developed to predict the nonlinear mechanical bending and post-buckling behavior of the re-entrant auxetic tube based on analytical data. The anticipated outcomes from both machine learning (ML) and ANN models for the auxetic tube show strong alignment with the Ritz method. Additionally, the effects of various parameters on the mechanical post-buckling and nonlinear bending properties of the re-entrant tubular metastructure are thoroughly investigated, providing insights into design optimization for auxetic metastructures. For instance, the buckling modes under the clamped-clamped boundary condition decrease by approximately 37% when the absolute value of the cell angle is increased from \(15^{\circ }\) ${15}^{\circ }$ to \(30^{\circ }\) ${30}^{\circ }$ . Moreover, the statistical analysis confirmed that for the prediction of post-buckling responses, the Random Forest model provided the most accurate predictions, achieving a mean squared error (MSE) of 13.31 and an \(R^2\) $R^2$ value of 99.9%.