  We investigate theoretically shear banding in large amplitude oscillatory
shear (LAOS) of polymers and wormlike micelles. In LAOStrain we find banding at
low frequencies and sufficiently high strain rate amplitudes in fluids for
which the underlying constitutive curve of shear stress as a function of shear
rate is non-monotonic. This is the direct analogue of quasi steady state
banding seen in slow strain rate sweeps along the flow curve. At higher
frequencies and sufficiently high strain amplitudes we report a different but
related phenomenon, which we call `elastic' shear banding. This is associated
with an overshoot in the elastic (Lissajous-Bowditch) curve of stress as a
function of strain. We suggest that this may arise widely even in fluids that
have a monotonic underlying constitutive curve, and so do not show steady state
banding under a steadily applied shear flow. In LAOStress we report banding in
fluids that shear thin strongly enough to have either a negatively, or weakly
positively, sloping region in the underlying constitutive curve, noting again
that fluids in the latter category do not display steady state banding in a
steadily applied flow. This banding is triggered in each half cycle as the
stress magnitude transits the region of weak slope in an upward direction, such
that the fluid effectively yields. Our numerics are performed in the Rolie-poly
model of polymeric fluids, but we also provide arguments suggesting that our
results should apply more widely. Besides banding in the shear rate profile,
which can be measured by velocimetry, we also predict banding in the shear and
normal stress components, measurable by birefringence. As a backdrop to
understanding the new results on shear banding in LAOS, we also briefly review
earlier work on banding in other time-dependent protocols, focusing in
particular on shear startup and step stress.
