Multiscale modeling of embedded graphene sheets based on the higher-order Cauchy-Born rule: Nonlinear static analysis

The nonlinear flexural response of single-layer graphene sheets (SLGSs) resting on elastic matrix is studied using an atomistic-based second gradient continuum model. The higher-order Cauchy-Born rule is used to link the interatomic potential to the strain energy induced in the continuum without any parameter fitting. The graphene is modeled by a hyperelastic membrane whose elastic potential energy is exclusively written in terms of the interatomic potential. This results in a constitutive model independent of any additional phenomenological input and thickness. Moreover, through this linkage, both the material and geometrical nonlinearities are exactly reflected in the constitutive model. To solve the continuum boundary value problem, the differential quadrature (DQ) approach is employed in the context of a variational formulation, and the discretized weak form of the equilibrium equation is obtained. The static response of SLGSs under a uniformly distributed load is evaluated. It is found that the present multiscale model can reproduce the results of other coupled atomistic-continuum and full atomistic approaches with a small number of discrete points. Also, the effect of the second-order deformation gradient is found to be significant on the bending deflection of SLGS specifically on the one with high flexural stiffness.