To create a new surface in a solid or a liquid material, energy needs to be spent. On the atomistic level dangling bonds are formed, which tend to be reconstructed accompanied by an excess surface energy.
For thin structures like graphene this energy can change the global shape of the structure drastically. In this work the effect of a free edge on the global behaviour of a circular graphene patch is studied.
The second chapter is devoted to the molecular mechanics approach. Small unit cells are used to compute the edge energy and edge stress of armchair and zigzag edges respectively. Graphene is modelled by classical multibody potentials called AIREBO and REBO, which are reasonable for describing hydrocarbon structures. As a result of the changed bonding configuration there is a compressive stress localised at the edge, independent of the underlying geometry of the edge. With an empirical force field formulation the effect of the compressive stress on the global configuration of a circular graphene patch is studied. The free edge shows a wavy out of plane displacement in the circumferential direction. In radial direction the displacement decays away from the edge. To investigate the influence of a substrate on the buckled configuration, an additional fixed graphene sheet interacting via the Van der Waals force is considered. Formulating this problem in the framework of continuum mechanics, offers the possibility of stating an appropriate stability problem. The aim of such an approach is to obtain the stability boundary in the plane of suitable parameters. In the third chapter the plate equations for a non-Euclidean metric are derived. The question of measuring lengths on a surface is intrinsically tied to the metric tensor. The expanding edge is modelled as a perturbation of the metric tensor of the elastic plate, which is called target metric. The strain tensor results from the difference between the actual and the target metric tensor, and is the starting point of deriving the nonlinear Föppl-von-Karman plate equations for a non-Euclidean annular plate. There is no external load, but the attempt of the edge to increase its circumferential line due to the changed metric term.
Performing a linear stability analysis of the flat unbuckled configuration leads to the stability boundary in the dimensionless parameter plane, namely the critical metric coefficient as a function of the foundation stiffness. Curves of different modes of instability cross each other in the parameter plane. At such a point a nonlinear analysis is performed, to answer the question if the two modes may interact at this point. It turns out, that both solutions bifurcate subcritical at the corresponding stability boundaries. To study the post buckling behaviour beyond the stability boundary a discrete model is used. The continuous plate is discretised by means of a triangular spring network model. The expanding edge can be defined very intuitively by increasing the equilibrium lengths of corresponding springs. With this discrete formulation of the plate the postbuckling configurations of the circular plate are computed.