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Graphene sheet (GS) is a low-dimensional carbon nano material with many outstanding mechanical properties, such as high strength. However, lattice defects in GS are often observed, which induce out-of-plane deformation as an eigen deformation mode, and deformation is process of stress relaxation. As it is known that 4-, 5-, 7-, 8-membered ring carbon atoms exist as lattice defect in GS, which is usually composed only 6-membered rings. In this study, based on Japanese art of paper kirigami and origami, we build initial models oftwo-dimensional carbon nano material GS with isolated positive disclination, isolated negative disclination and disclination dipole. Obvious out-of-plane deformation is caused dueto spontaneous curvature of the disclination, and the GS is changed from two-dimension tothree-dimension. We use conjugate gradient method to perform energy minimization to obtainstable structure of curved surface. Energetic analysis is discussed by molecular dynamicsmethod and the mean curvature of curved surfaces is investigated by differential geometry.We discuss the effect of distance and angle from disclination core on potential energy andmean curvature in details. Further we also analyze the relationship between distance andenergy in disclination dipole. As an important result, we establish a new model ofmathematical mechanics, as the energy features can be expressed geometrically element(mean curvature), and it is necessary for nanoscale order design.

Disclination, Out-of-plane deformation, Spontaneous curvature, Graphene sheet, Carbon nano material