ON THE NECKING OF GRAPHENE NANOSTRUCTURES

A 1D continuum model endowed with internal structure is proposed as a coarse descriptor of the necking of graphene nanostructures under tension, in the nonlinear elasticity framework. For the sake of simplicity, the analysis is performed with reference to a graphene sheet but the procedure can be applied, as it stands, to other 2D graphene nanostructures like nanoribbons or nanotubes. A representative elementary volume (REV) which includes two atoms and five C-C bonds is chosen at the nano scale for both the armchair and zigzag carbon atoms' arrangements. The modified Morse potential is employed for describing the interatomic actions. The strain energy density of the REV is first given in terms of nanoscopic strain measures. Then it is expressed as a function of the continuum strain measures by assuming suitable relationships between the discrete and the continuum kinematics. The procedure leads to the definition of a 1D continuum equivalent to the given graphene sheet. The constitutive functions for the 1D stresses are obtained by making the derivative of the strain energy with respect to its arguments. Using the linear approximation of the constitutive functions, the pure tension stress state is examined and the elastic stiffness of the graphene sheet determined. Then the case of a straight beam subjected to an axial end displacement is examined. At first, a trivial solution is determined both for armchair and zigzag carbon atoms' arrangements. A perturbation analysis is performed to obtain the first bifurcation point on the trivial equilibrium path, together with the initial tangent to the bifurcated path. A peculiar postbuckling behavior is detected and commented.