Vibration Characteristics of Functionally Graded Graphene Origami-enabled Auxetic Metamaterial Shell Structures: Polynomials Fitting for Various Boundary Conditions

Various metamaterials with unusual behaviors are attracting tremendous attention, while the vibration characteristics of auxetic metamaterial shell structures with the negative Poisson’s ratio remain poorly understood. This work is concerned with the vibration characteristics of the functionally graded graphene origami (FG GOri)-enabled auxetic metamaterial shell structures including conical shells, cylindrical shells and annular plates, based on a novel method. Attention is paid to interpret how the varying Young’s modulus and Poisson’s ratio caused by the variation of GOri weight fraction and folding degree affect the natural frequencies of the structures. Five types of GOri distribution patterns are considered where the maximum values of volume fractions are introduced to ensure equal mass of the FG structure. Based on the polynomials fitting, explicit expressions of the modified Chebyshev polynomials for eight types of boundary conditions of the structures are given. The unified first-order shear deformation theory (FSDT) is adopted to simulate moderately thick structures, where the rotation inertia of the structures is considered. The results reveal that the variations of GOri weight fractions and folding degrees change the effective Young’s modulus and Poisson’s ratio together, both of which affect the stiffness of the structures, and then the varying stiffness and different distribution patterns make the natural frequencies exhibit different variation trends. The novelty of this work is that the semi-analytical solutions for vibration characteristics of FG GOri-enabled auxetic metamaterial shell structures under various boundary conditions are given. The mechanism of the negative Poisson’s ratio affecting the natural frequencies of shell structures is analyzed in detail with the help of the FG GOri-enabled auxetic metamaterials.