# Shear Banding in Time-Dependent Flows of Complex Fluids

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## Abstract

We explore theoretically the time-dependent rheological response of complex fluids to step stress, shear startup and strain ramp deformations. We study soft glassy materials and entangled polymeric materials above and below the glass transition using the scalar fluidity, soft glassy rheology (SGR), rolie-poly (RP), Giesekus and glassy polymer (GP) models. For each deformation we investigate fluid-universal, deformation specific criteria for the onset of linear instability to shear banding. In step stress, the shear rate response of the RP and Giesekus models is qualitatively similar to experiment, but only in the RP model does significant transient shear banding arise. Motivated by experiments, we explore `creep' and `fluidisation' in the glass phase of the SGR model. Finally, we show the GP model has similar behaviour in step shear stress as it does in extensional loading; we also explain why strain hardening reduces the magnitude of transient shear banding. In shear startup, we explore `elastic' and `viscous' contributions to linear instability to shear banding. We use this to explain: the range of shear rates for which time-dependent shear banding arises in the RP model; why no significant time-dependent shear banding arises associated with the stress overshoot in the Giesekus model; the occurrence of age-dependent transient shear banding in the scalar fluidity model; and also why strain hardening again decreases the magnitude of transient shear banding in the GP model. Finally, we investigate stress relaxation in the RP model after strain ramps with rates that probe the chain stretch relaxation rate. We show that after `slow' ramps (relative to this rate) linear instability to heterogeneity arises for strain amplitudes greater than 1.7, and that `fast' ramps of the same amplitude result in a delayed onset of linear instability, provided convective constraint release is sufficiently inactive.

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