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Continuum modeling of a free-standing graphene monolayer, viewed as a twodimensional 2-lattice, requires specifications of the components of the shiftvector that act as an auxiliary variable. The field equations are then theequations ruling the shift vector, together with momentum and moment ofmomentum equations. We present an analysis of simple loading histories such asaxial, biaxial tension/compression and simple shear for a range of problems ofincreasing difficulty. We start by laying down the equations of a simplifiedmodel which can still capture bending effects. Initially, we ignore out ofplane deformations. For this case, we solve analytically the equations rulingthe auxiliary variables in terms of the shift vector; these equations arealgebraic when the loading is specified. As a next step, still working on thesimplified model, out-of-plane deformations are taken into account and theequations complicate dramatically. We describe how wrinkling/buckling can beintroduced into the model and apply the Cauchy-Kowalevski theorem to getexistence and uniqueness in terms of the shift vector for some characteristiccases. Finally, for the treatment of the most general problem, we classify theequations of momentum and give conditions for the Cauchy-Kowalevski theorem toapply.

Curvature dependent surface energy for a free standing monolayer graphene: some closed form solutions of the nonlinear theory