Nonlinear dynamics and chaos of functionally graded graphene origami-enabled auxetic metamaterials doubly curved shells with bi-directionally stepped thickness in thermal environment

This paper presents the dynamic and chaotic responses of functionally graded graphene origami (GOri)-enabled auxetic metamaterials (GOEAMs) doubly curved shells featuring stepped thickness profiles in thermal environment. A novel analytical framework is introduced to investigate geometric variations and material distributions, highlighting their crucial roles. Specifically, the investigation delves into structural variations in shell thickness, characterized by abrupt changes in uni- or bi-directional orientations, encompassing both single and double stepped thickness profiles. The five geometric configurations of the stepped structures encompass plates, cylindrical shells, spherical shells, hyperbolic paraboloid shells, and elliptical paraboloid shells. The FG-GOEAM structures, comprising multiple layers with varied GOri distributions across their thickness, are scrutinized using genetic programming-assisted micromechanical models. Central to the present approach is the formulation of nonlinear kinematic relationships using Reddy's third-order shear deformation theory alongside von Kármán's nonlinear geometric assumptions, with equations of motion being solved utilizing Galerkin's technique. Notably, the enhanced model is developed to address non-continuous thickness variation through integral calculus operations, enhancing computational efficiency and obviating the need for complex algorithms. To verify the accuracy of the proposed method, the obtained results are compared with those from published literature. The study thoroughly investigates the influence of material properties, thermal conditions, and geometric parameters on the free vibration and nonlinear behaviors of the structures. Key findings include: The shell stiffness of the functionally graded (FG) shell (X−W_Gr) surpasses that of the homogeneous shell (U−W_Gr). The transitions from periodic to chaotic states of the stepped structures are discerned through the analysis of the time history response, the phase plane illustrating the deflection-velocity relationship, and the Poincaré map. Increasing the thickness ratios and GOri content while reducing the folding degree of GOri results in a significant increase in the fundamental frequency and critical load, along with a simultaneous decrease in vibrational amplitudes.