Thermal effects on large-amplitude vibrations of doubly curved auxetic shells supported by a visco–Pasternak foundation

Graphene Origami, with its foldable structure and high strength-to-weight ratio, enhances structural performance. Nonlinear free vibration analysis of such reinforced systems is essential for capturing their true dynamic characteristics and ensuring accurate performance prediction. For this purpose, this research addresses the nonlinear free vibration characteristics of a doubly curved shell composed of auxetic metal metamaterials reinforced with graphene origami (GOri), subjected to a thermal environment and resting on an elastic substrate. The integration of GOri reinforcements markedly improves the structural mechanical response; thus, a micromechanical framework is adopted to evaluate these enhancements. Five distinct through-thickness distribution schemes of GOri are explored. The formulation of the shell’s governing equations is based on the first-order shear deformation theory (FSDT) combined with Hamilton’s variational principle, while geometrical nonlinearities are incorporated via the von Kármán strain assumptions. The resulting nonlinear partial differential equations, under simply supported boundaries, are reduced using the Galerkin technique to a single nonlinear equation in the transverse displacement component. This equation is then solved to obtain the nonlinear natural frequencies through perturbation analysis and a modified Poincaré-Lindstedt approach, and also by employing the fifth-order Runge-Kutta method. Unlike the linear case, in nonlinear free vibrations the natural frequencies are dependent on the system’s initial conditions. A series of parameter-based analyses is carried out to determine the impact of non-dimensional initial amplitude, temperature variation, GOri volume fraction, distribution pattern, folding angle, geometric configuration, elastic foundation stiffness and damping coefficient on the frequency. Verification comparisons are undertaken to assess the robustness of the method, the solution process used, and the correctness of the results. Findings reveal that the initial amplitude has a pronounced impact on the dynamic response, and that elevated temperatures cause a substantial reduction in nonlinear frequencies.