  We consider a discrete model of a graphene sheet with atomic interactions
governed by a harmonic approximation of the 2nd-generation Brenner potential
that depends on bond lengths, bond angles, and two types of dihedral angles. A
continuum limit is then deduced that fully describes the bending behavior. In
particular, we deduce for the first time an analytical expression of the
Gaussian stiffness, a scarcely investigated parameter ruling the rippling of
graphene, for which contradictory values have been proposed in the literature.
We disclose the atomic-scale sources of both bending and Gaussian stiffnesses
and provide for them quantitative evaluations.
