We introduce a torsional force field for sp$^2$ carbon to augment an in-plane atomistic potential of a previous work (Kalosakas et al, J. Appl. Phys. {bf 113}, 134307 (2013)) so that it is applicable to out-of-plane deformations of graphene and related carbon materials. The introduced force field is fit to reproduce DFT calculation data of appropriately chosen structures. The aim is to create a force field that is as simple as possible so it can be efficient for large scale atomistic simulations of various sp$^2$ carbon structures without significant loss of accuracy. We show that the complete proposed potential reproduces characteristic properties of fullerenes and carbon nanotubes. In addition, it reproduces very accurately the out-of-plane ZA and ZO modes of graphenes phonon dispersion as well as all phonons with frequencies up to 1000~cm$^{-1}$.
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