A constitutive equation for graphene based on density functional theory

An anisotropic strain energy function is proposed for tensile loading in graphene that provides a nonlinear, hyperelastic constitutive equation. In the proposed function, the energy depends on the principal invariants of the right Cauchy-Green tensor and the strains in the zigzag and armchair directions. The use of the zigzag and armchair strains gives the model the ability to account for anisotropic behavior at moderate deformations. The constitutive law parameters are determined by a least squares fit to the energies predicted by density functional theory (DFT) calculations, and a good match is obtained to the DFT results for zigzag and armchair graphene sheets with various loading combinations. The law is applied in a continuum calculation of nanoindentation of a graphene membrane. The force-deflection predicted with this model show excellent agreement with analogous experimental results, thus providing a strong link between DFT calculations and nanoexperiments.