Curvature dependent surface energy for free standing monolayer graphene: Geometrical and material linearization with closed form solutions

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. To introduce material linearity energy is assumed to have a quadratic dependence on the strain tensor, the curvature tensor, the shift vector, as well as to combinations of them. Hexagonal symmetry then reduces the overall number of independent material constants to nine. 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 for the geometrically and materially linear case. We start with the problem of in-plane motions only. By prescribing the displacement, the components of the shift vector are evaluated. This way the field equations are satisfied trivially. Out-of-plane motions are treated as well; we assume in-plane tension/compression that leads to buckling/wrinkling and solve for the components of the shift vector as well as the function present in buckling’s modeling. The assumptions of linearity adopted here simplifies the analysis and facilitates analytical results.