An Analytical Solution for the Mechanical Responses of Graphene Using Semi-Rigid Node Beam Element Theory

This research introduces an analytical model based on a semi-rigid nodal bar system. In this model, carbon–carbon covalent bonds are represented as beam elements, while carbon atoms are treated as semi-rigid nodes connecting these elements. A spring coefficient is incorporated to quantify nodal stiffness. Building upon this construct, spatial stiffness equations for the semi-rigid beam elements are derived, enabling a systematic exploration of how boundary conditions and dimensional factors influence the Young’s modulus and buckling stress in both pristine and defect-laden graphene. The findings reveal that defect-free graphene exhibits remarkable dimensional stability, with its Young’s modulus consistently approximating 1.0 TPa and fluctuating within ±2%. Upon the introduction of defects, the material’s stiffness diminishes significantly, with a maximum reduction of 19% observed when the density of defective elements surpasses a critical threshold. Moreover, the relationship between boundary conditions and buckling stress aligns closely with classical thin plate theory. Under identical dimensional constraints, the buckling stress ratios for fully fixed, fixed-simple, simply supported, and cantilevered boundaries conform to the theoretical ratio of 16:8:4:1.