  Yield stress fluids display a rich rheological phenomenology. Beyond the
defining existence of a yield stress in the steady state flow curve, this
includes in many materials rather flat viscoelastic spectra over many decades
of frequency in small amplitude oscillatory shear; slow stress relaxation
following the sudden imposition of a small shear strain; stress overshoot in
shear startup; logarithmic or sublinear power law creep following the
imposition of a shear stress below the yield stress; creep followed by yielding
after the imposition of a shear stress above the yield stress; richly featured
Lissajous-Bowditch curves in large amplitude oscillatory shear; a Bauschinger
effect, in which a material's effective yield strain is lowered under straining
in one direction, following a preceding strain in the opposite direction;
hysteresis in up-down shear rate sweeps; and (in some materials) thixotropy
and/or rheological ageing. A key challenge is to develop a constitutive model
that contains enough underlying mesoscopic physics to have meaningful
predictive power for the full gamut of rheological behaviour just described,
with only a small number of model parameters, and yet is simple enough for use
in computational fluid dynamics to predict flows in complicated geometries, or
complicated flows that arise due to spontaneous symmetry breaking instabilities
even in simple geometries. Here we introduce such a model, motivated by the
widely used soft glassy rheology model, and show that it captures all the above
rheological features.
