A Method to Determine the Geometry-Dependent Bending Stiffness of Multilayer Graphene Sheets

We consider how the bending stiffness of a multilayer graphene sheet relies on its bending geometry, including the in-plane length L and the curvature kappa. We use an interlayer shear model to characterize the periodic interlayer tractions due to the lattice structure. The bending stiffness for the sheet bent along a cylindrical surface is extracted via an energetic consideration. Our discussion mainly focuses on trilayer sheets, particularly the complex geometry-dependency of their interlayer stress transfer behavior and the overall bending stiffness. We find that L and kappa dominate the bending stiffness, respectively, in different stable regions. These results show good quantitative agreement with recent experiments where the stiffness was found to be a non-monotonic function of the bending angle (i.e., L kappa). Besides, for a given in-plane length, the trilayer graphene in the flat state (kappa -> 0) is found to have the maximum bending stiffness. According to our analytical solution to the flat state, the bending stiffness of trilayer graphene sheet can vary by two orders of magnitude. Furthermore, once multilayer graphene sheets are bent along a cylindrical surface with small curvature, the sheets perform similar characteristics. Though the discussion mainly focuses on the trilayer graphene, the theoretical framework presented here can be readily extended for various van der Waals materials beyond graphene of arbitrary layer numbers.