Three-dimensional mathematical solution of static and free vibration of graphene origami-enabled auxetic metamaterial thick plates

Abstract

This study comprehensively investigates the static bending and free vibration characteristics of functionally graded graphene origami-enabled auxetic metamaterial (FG GOEAM) plates with a copper matrix. By leveraging three-dimensional linear elasticity theory, constitutive relations, and equilibrium/motion equations are derived and reformulated through two pairs of transformations into two independent, order-reduced sets of spatial state-space equations, distinguishing in-plane and out-of-plane motions. This novel formulation establishes a unified framework applicable to plates of arbitrary thickness (from thin to highly thick) while enhancing numerical efficiency. The mechanical properties of the metamaterial are predicted via an AI-assisted model. The governing equations are solved using the transfer matrix method under three boundary condition types: elastic simple supports, rigidly slipping edges, and their combinations. The parametric study systematically evaluates nine distinct dispersion pattern combinations of graphene origami (GOri) platelets across the plate's thickness, alongside variations in folding degree, weight fraction, and temperature. Key findings reveal that GOri dispersion patterns critically influence structural performance: the $X_G-O_H$ configuration maximizes stiffness, yielding the highest natural frequencies (37–39 % higher) and minimal central deflection (90 % lower), whereas $U_G-U_H$ induces the softest response, with non-monotonic stiffness behavior observed at intermediate GOri weight fractions.