  We investigate the effect of shear and strain in graphene bilayers, under
conditions where the distortion of the lattice gives rise to a smooth
one-dimensional modulation in the stacking sequence of the bilayer. We show
that strain and shear produce characteristic Moir\'e patterns which can have
the same visual appearance on a large scale, but representing graphene bilayers
with quite different electronic properties. The different features in the
low-energy electronic bands can be ascribed to the effect of a fictitious
non-Abelian gauge field mimicking the smooth modulation of the stacking order.
Strained and sheared bilayers show a complementary behavior, which can be
understood from the fact that the non-Abelian gauge field acts as a repulsive
interaction in the former, expelling the electron density away from the
stacking domain walls, while behaving as a confining interaction leading to
localization of the electronic states in the sheared bilayers. In this latter
case, the presence of the effective gauge field explains the development of
almost flat low-energy bands, resembling the form of the zeroth Landau level
characteristic of a Dirac fermion field. The estimate of the gauge field
strength in those systems gives a magnitude of the order of several tens of
Tesla, implying a robust phenomenology that should be susceptible of being
observed in suitably distorted bilayer samples.
