Calibration of Eringen's small length scale coefficient for buckling circular and annular plates via Hencky bar-net model

This paper is focused on the modeling of circular and annular graphene sheets via Hencky bar-net model (HBM¹) and calibrating the Eringen's small length scale coefficient e₀ in Eringen's nonlocal theory. The buckling solutions of circular and annular graphene sheets based on Eringen's nonlocal continuum plate theory are first obtained. On the other hand, HBM is developed to model the same structure from the discrete view. HBM is a grid system comprising rigid bars and arcs connected by frictionless hinges with elastic rotational and torsional springs. By regarding the length of straight segments in HBM equal to the characteristic length of Eringen's nonlocal model (ENM²) and matching their solutions, the Eringen's small length scale coefficient e₀ is calibrated. It is found that for circular graphene sheet, e₀ = 0.258 for clamped edge and e₀ = 0.300 for simply supported edge. For annular graphene sheet, e₀ is dependent on the inner to outer radius ratio χ and boundary conditions. The scale coefficient e₀ takes 0.307–0.367 for clamped edges while 0.219–0.290 for simply supported edges with χ varying from 0.2 to 0.8. Another finding is that the graphene sheet will buckle with a very small load when its dimension is large, regardless of models adopted. However for small dimensions, ENM and HBM predict lower buckling loads than the classical local model because the scale effect is more obvious.