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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. To introduce material linearity energy is assumed to have aquadratic dependence on the strain tensor, the curvature tensor, the shiftvector, as well as to combinations of them. Hexagonal symmetry then reduces theoverall number of independent material constants to nine. We present ananalysis of simple loading histories such as axial, biaxial tension/compressionand simple shear for a range of problems of increasing difficulty for thegeometrically and materially linear case. We start with the problem of in-planemotions only. By prescribing the displacement, the components of the shiftvector are evaluated. This way the field equations are satisfied trivially.Out-of-plane motions are treated as well; we assume in-planetension/compression that leads to buckling/wrinkling and solve for thecomponents of the shift vector as well as the function present in buckling'smodeling. The assumptions of linearity adopted here simplifies the analysis andfacilitates analytical results.

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