We present a novel analytical prediction for the effective bending rigidity γ eff of multi–layer graphene sheets. Our approach involves using a variational model to determine the folding conformation of multi–layer graphene sheets where the curvature of each graphene layer is taken into account. The Lennard–Jones potential is used to determine the van der Waals interaction energy per unit area and the spacing distance between graphene layers. The mid–line of the folded multi–layer graphene is described by a solution derived in previous work for folded single– and multi–layer graphene. Several curves are obtained for the single–layer solution using different values of the bending rigidity γ, and compared to the mid–line of the folded multi–layer graphene. The total area between these curves and the mid–line is calculated, and the value of γ eff is determined by the single–layer curve for which this area is minimized. While there is some disagreement in the literature regarding the relationship between the bending rigidity and the number of layers, our analysis reveals that the bending rigidity of multi–layer graphene follows an approximate square–power relationship with the number of layers N, where N < 7. This trend is in line with theoretical and experimental studies reported in the literature.

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In this paper, the method of upper and lower solutions is employed to obtain uniqueness of solutions for a boundary value problem at resonance. The shift method is applied to show the existence of solutions. A monotone iteration scheme is developed and sequences of approximate solutions are constructed that converge monotonically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.