Multiscale Green's function for the deflection of graphene lattice

We show analytically that the continuum limit of the lattice-statics Green's function of a graphene sheet corresponds to the Green's function for an elastically stable Kirchhoff plate but not the Green's function for two-dimensional Christoffel equations. This correspondence demonstrates the mechanical stability of graphene in deflection and is necessary for relating its mechanical parameters to its lattice parameters. An explicit expression is derived for relating the continuum flexural rigidity to the force constants of graphene. This relationship can be used to measure flexural rigidity of graphene directly from experimentally observed phonon dispersion curves. The flexural rigidity is predicted to be 0.797 eV by using the Tersoff-Brenner empirical potential. Numerical examples are presented to show the usefulness of the correspondence in bridging the lattice and continuum length scales in graphene.