Static Response of Auxetic Beams: Investigating the Combined Effects of Graphene and Hydrogen Distribution

Auxetic metamaterials are a class of advanced materials with unconventional mechanical properties. This paper examines the static behavior of functionally graded (FG) graphene origami (GOri)-enabled auxetic metallic metamaterial (GOEAM) beams. Methodologically, the mechanical properties of each material layer are derived using a genetic programming (GP)-assisted micromechanical model. The static response is then analyzed using the first-order shear deformation beam theory, with governing equilibrium equations formulated via the principle of minimum potential energy and solved numerically. The beams consist of multiple layers of GOEAM, utilizing varying GOri content in layer-wise patterns to create a graded distribution of Poisson’s ratio and stiffness. This work expands on previous studies by exploring four distinct GOri distribution patterns combined with four hydrogen (H) coverage layouts. Numerical simulations consider the combined effects of GOri content, folding degree, thermal conditions, and boundary conditions. Quantitative findings reveal that increasing graphene content up to approximately 2.0 wt% consistently reduces deflections, while folding degrees exceeding 60% transition the matrix into an auxetic regime characterized by a negative Poisson’s ratio. Among all studied cases, the combination of an X–WGr GOri distribution and an O–HGr coverage pattern emerges as the optimal design. This specific configuration achieves a 50.32% reduction in maximum deflection compared to a pure copper beam under hinged-hinged supports, and its mechanical superiority persists across a thermal range of 300 to 400 K. These results establish that functionally graded GOri architectures offer a powerful synergy of stiffness enhancement and thermal resilience, making them strong candidates for aerospace, MEMS, and advanced mechanical systems.