  The potential energy surface (PES) of interlayer interaction of infinite
twisted bilayer graphene is calculated for a set of commensurate moir\'e
patterns using the registry-dependent Kolmogorov-Crespi empirical potential.
The calculated PESs have the same shape for all considered moir\'e patterns
with the unit cell size of the PES which is inversely related to the unit cell
size of the moir\'e pattern. The amplitude of PES corrugations is found to
decrease exponentially upon increasing the size of the moir\'e pattern unit
cell. An analytical expression for such a PES including the first Fourier
harmonics compatible with the symmetries of both layers is derived. It is shown
that the calculated PESs can be approximated by the derived expression with the
accuracy within 1%. This means that different physical properties associated
with relative in-plane motion of graphene layers are interrelated and can be
expressed analytically as functions of the amplitude of PES corrugations. In
this way, we obtain the shear mode frequency, shear modulus, shear strength and
barrier for relative rotation of the commensurate twisted layers to a fully
incommensurate state for the considered moir\'e patterns. This barrier may
possibly lead to the macroscopic robust superlubricity for twisted graphene
bilayer with a commensurate moir\'e pattern. The conclusions made should be
valid for diverse 2D systems of twisted commensurate layers.
