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