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We present a simple analytical approach to study anharmonic effects in singlelayer, bilayer, and multilayer graphene. The coupling between in plane and outof plane modes leads to negative Gr\"uneisen coefficients and negative thermalexpansion. The value of the thermal expansion coefficient depends on thecoupling to the substrate. The bending rigidity in bilayer graphene shows acrossover between a long wavelength regime where its value is determined by thein plane elastic properties and a short wavelength regime where its valueapproaches twice that of a single layer.

Bending modes, anharmonic effects and thermal expansion coefficient in single layer and multilayer graphene