Bending modulus of the rippled graphene: the role of thickness

Bending modulus is a key parameter to characterize the stiffness of materials. Commonly, it is believed that the bending modulus is closely related to the thickness as described by the thin plate theory. However, the thin plate theory fails in multilayer van der Waals materials like multilayer graphene, suggesting a more complex relationship between the bending modulus and thickness. Here, rippled graphene structures containing non-hexagonal carbon rings with different thicknesses are constructed to study the thickness-dependent bending modulus by the first-principles calculations. It is found that the bending modulus of rippled graphene depends on several factors, such as geometry, bending curvature, and thickness. Particularly, for the egg-tray graphene structures with similar structural pattern and bending curvature, i.e., eliminating the effects of structural pattern and bending curvature, the bending modulus could show a linear relationship to the thickness. Moreover, this linear relationship is very robust even in the case of changing the thickness through heteroatom doping.