  Graphene has become in last decades a paradigmatic example of two-dimensional
and so-called van-der-Waals layered materials, showing large anisotropy in
their physical properties. Here we study the elastic properties and mechanical
stability of graphene bilayers in a wide temperature range by molecular
dynamics simulations. We concentrate on in-plane elastic constants and
compression modulus, as well as on the atomic motion in the out-of-plane
direction. Special emphasis is placed upon the influence of anharmonicity of
the vibrational modes on the physical properties of bilayer graphene. We
consider the excess area appearing in the presence of ripples in graphene
sheets at finite temperatures. The in-plane compression modulus of bilayer
graphene is found to decrease for rising temperature, and results to be higher
than for monolayer graphene. We analyze the mechanical instability of the
bilayer caused by an in-plane compressive stress. This defines a spinodal
pressure for the metastability limit of the material, which depends on the
system size. Finite-size effects are described by power laws for the
out-of-plane mean-square fluctuation, compression modulus, and spinodal
pressure. Further insight into the significance of our results for bilayer
graphene is gained from a comparison with data for monolayer graphene and
graphite.
