  Ab initio density functional theory has been used to analyze flexural modes,
elastic constants, and atomic corrugations on single and bi-layer graphene.
Frequencies of flexural modes are sensitive to compressive stress; its
variation under stress can be related to the anomalous thermal expansion via a
simple model based in classical Elasticity Theory [Phys. Rev. B 86, 144103].
Under compression, flexural modes are responsible for a long wavelength
rippling with a large amplitude and a marked anharmonic behavior. This is
compared with corrugations created by thermal fluctuations and the adsorption
of a light impurity (hydrogen). Typical values for the later are in the
sub-Angstrom regime, while maximum corrugations associated to bending modes
quickly increase up to a few Angstroms under a compressive stress, due to the
intrinsic instability of flexural modes
