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The potential energy surface (PES) of interlayer interaction of infinitetwisted bilayer graphene is calculated for a set of commensurate moir\'epatterns using the registry-dependent Kolmogorov-Crespi empirical potential.The calculated PESs have the same shape for all considered moir\'e patternswith the unit cell size of the PES which is inversely related to the unit cellsize of the moir\'e pattern. The amplitude of PES corrugations is found todecrease exponentially upon increasing the size of the moir\'e pattern unitcell. An analytical expression for such a PES including the first Fourierharmonics compatible with the symmetries of both layers is derived. It is shownthat the calculated PESs can be approximated by the derived expression with theaccuracy within 1%. This means that different physical properties associatedwith relative in-plane motion of graphene layers are interrelated and can beexpressed analytically as functions of the amplitude of PES corrugations. Inthis way, we obtain the shear mode frequency, shear modulus, shear strength andbarrier for relative rotation of the commensurate twisted layers to a fullyincommensurate state for the considered moir\'e patterns. This barrier maypossibly lead to the macroscopic robust superlubricity for twisted graphenebilayer with a commensurate moir\'e pattern. The conclusions made should bevalid for diverse 2D systems of twisted commensurate layers.

Interlayer interaction, shear vibrational mode, and tribological properties of two-dimensional bilayers with a commensurate moir\'e pattern