Geometrically Nonlinear Free Vibration of Functionally Graded Auxetic Metamaterial Arches Inclusive of Tunable Graphene Origami

This research deals with the nonlinear free vibrations of laminate shallow functionally graded (FG) arches in a thermal environment. The structure layers are composed of Cu matrix and graphene origami (GOri) which exhibit auxetic metamaterial behavior with a negative Poisson’s ratio (NPR). The properties of these metamaterials are tunable through the GOri weight fraction and its folding degree. The modified Halpin–Tsai and Voigt rules are used to obtain temperature-dependent effective thermomechanical characteristics. The modifier multipliers for each characteristic have been calculated by employing genetic programming. A higher-order shear deformation theory based on the hyperbolic functions along the arch thickness determines the system’s displacements. The nonlinear motion equations are extracted by Hamilton’s principle based on the nonlinear strain equations and the uncoupled thermoelasticity constitutive law. The motion equations are transformed into algebraic form by implementing the Chebyshev collocation method. Next, with the aid of a bilateral displacement control method, the nonlinear natural frequencies of the FG-GOri-enabled auxetic metamaterials (FG-GOEAM) arch are calculated. Some comparison studies validate the formulation and methods. The novel results demonstrate the influence of various parameters in tuning the geometrically nonlinear free vibration response of FG-GOEAM arches.