Bending stiffening of graphene and other 2D materials via controlled rippling

In this paper, the effects of rippling on the bending stiffness of a monolayer graphene are studied. The initial rippling of the surface is modeled by cosine functions with a hierarchical topology. Considering both large displacement and small scale effect, the governing equilibrium equations are determined and solved. Then an equivalent bending stiffness is calculated for a rippled graphene and the effects of rippling, material discreteness, and structural dimension on its stiffness are discussed in details. The results quantify how the rippling strongly increases the effective bending stiffness of graphene and interacts with the discrete nature of the material not only because of increase in the moment of inertia. This approach can be applied to ripples design of 2D materials in order to achieve stiffening in bending as required in specific applications.