▎ 摘　　要

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 [P. L. de Andres, F. Guinea, and M. I. Katsnelson, Phys. Rev. B 86, 144103 (2012)]. 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. DOI: 10.1103/PhysRevB.86.245409