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 to a broad range of values of the frequency ω and amplitude of the imposed oscillation. Crucially, and highly counterintuitively, in the glass phase, this persists even to the lowest frequencies accessible numerically (in a manner that furthermore seems consistent with its persisting even to the limit of zero frequency ω0), even though the soft glassy rheology model in which we perform our calculations has a purely monotonic underlying constitutive curve of shear stress as a function of shear rate, and is therefore unable to support shear banding as its true steady state response to a steadily imposed shear of constant 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 model has a flow curve of power law fluid form. In this way, 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 in any calculation that imposes 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”.

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1/τ=i=1i=nj=1j=mexp((Eijklij2)/x)/(mn), where Eij and lij are energy trap depth and strain corresponding to an SGR element ij. k = 1 in our units and is assumed to be same for all elements.

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