Curvature-dependent surface energy for free-standing monolayer graphene

The purpose of the present work is to give a continuum model that can capture bending effects for free-standing graphene monolayers taking material's symmetry properly into account. Starting from the discrete picture of graphene modelled as a hexagonal 2-lattice, we give the arithmetic symmetries. Confined to weak transformation neighbourhoods one is able to work with the geometric symmetry group. Use of the Cauchy-Born rule allows the transition from the discrete case to the continuum case. At the continuum level we use a surface energy that depends on an in-plane strain measure, the curvature tensor and the shift vector. Dependence of the energy on the curvature tensor allows for incorporating bending effects into the model. Dependence on the shift vector is motivated by the fact that discretely graphene is a 2-lattice. We lay down the complete and irreducible set of invariants for this surface energy amenable to available representation theory. This way we obtain the expression for the surface stress as well as the surface couple stress tensor, the first being responsible for the in-plane deformations and the second for the out-of-plane motions. Forms for the elasticities of the material are given accompanied by the field equations. The model, in its simplest form, predicts 13 independent scalar variables in the constitutive relations to be observed in experiments. The framework presented is valid for both materially and geometrically nonlinear theories. We also present the case where symmetry changes at the continuum level, without taking into account how energy behaves at the transition regime.