Vibration and dynamic analysis of functionally graded graphene origami-enabled auxetic metamaterial shells on a Kerr foundation using an improved triangular finite element formulation

This study presents a node-based smoothed formulation based on the hybrid-mixed shear projected three-node triangular element concept, referred to as NS-MiSP3, for the free-vibration and transient analyses of functionally graded graphene origami-enabled auxetic metamaterial (FG-GOEAM) shells resting on a Kerr foundation with elastic edge restraints. The formulation is developed within the first-order shear deformation theory (FSDT), and the governing equations are derived from Hamilton’s principle. The Kerr foundation and elastic boundary conditions are included through stiffness terms and penalty parameters, respectively. The formulation is verified through patch tests, convergence studies, and comparisons with published results. Numerical examples are then conducted to examine the effects of graphene-origami (GOri) distribution, reinforcement weight fraction, folding degree, number of layers, foundation stiffness, and boundary flexibility. The results show that the natural frequencies generally increase with reinforcement weight fraction, number of layers, and foundation stiffness, but decrease with folding degree and boundary flexibility. The transient results also indicate that GOri distribution and material parameters affect both the vibration response and the through-thickness stress distribution.