  We study theoretically shear banding in soft glassy materials subject to
large amplitude time-periodic shear flows, considering separately the protocols
of large amplitude oscillatory shear strain, large amplitude square or
triangular or sawtooth strain rate, and large amplitude oscillatory shear
stress. In each case, we find shear banding to be an important part of the
material's flow response for a broad range of values of the frequency $\omega$
and amplitude of the imposed oscillation. Crucially, and highly
counterintuitively, in the glass phase this persists even to the lowest
frequencies accessible numerically (that seems consistent with it persisting
even when $\omega\to 0$), although the soft glassy rheology model we use has an
underlying monotonic constitutive curve of shear stress as a function of shear
rate, and is therefore unable to support shear banding as its steady state
response at constant shear rate. We attribute this to the repeated competition,
within each flow cycle, of glassy aging and flow rejuvenation. Besides
reporting significant banding in the glass phase, where the flow curve has a
yield stress, we also observe it at noise temperatures just above the glass
point, where the flow curve is of power law fluid form. Thus, our results
suggest a predisposition to shear banding in flows of even extremely slow
time-variation, for both aging yield stress fluids, and for power law fluids
with sluggish relaxation timescales. We show that shear banding can have a
pronounced effect on the shape of the Lissajous-Bowditch curves that are
commonly used to fingerprint complex fluids rheologically. We therefore counsel
caution in seeking to compute such curves calculations that impose upfront a
homogeneous shear flow, discarding the possibility of banding. We also analyze
the stress response to the imposed strain waveforms in terms of a `sequence of
physical processes'.
