Nonlinear elastic properties of graphene sheet under finite deformation

The elastic properties of graphene sheet at finite strain and curvature tensors are studied employing atomistic–continuum multiscale modelling approach in Lagrangian framework. The strain energy density function at continuum level is expressed as total interatomic potential per unit area of a unit cell incorporating continuum deformation through Cauchy–Born rule. Two different multibody interatomic potentials namely Tersoff–Brenner potential and second generation REBO potential are used to model the interactions between carbon atoms. The in-plane tangent extensional stiffness, bending stiffness, bending–stretching coupling stiffness matrices are obtained by differentiating the strain energy density function. The effect of different combinations of induced strain/curvature on stiffness coefficients is studied for graphene sheet with zigzag, armchair and chiral configurations. It is found that the graphene sheet possesses a material softening behaviour at finite strains and hardening behaviour at finite curvatures. The nonzero normal-shear coupling and tangent bending–stretching coupling stiffness coefficients are reported at finite strain/curvature for the first time in this work.