Shear Banding of Soft Glassy Materials in Large Amplitude Oscillatory Shear

We study shear banding in soft glassy materials subject to a large amplitude oscillatory shear flow (LAOS). By numerical simulations of the widely used soft glassy rheology model, supplemented by more general physical arguments, we demonstrate strong banding over an extensive range of amplitudes and frequencies of the imposed shear rate $\dot{\gamma }(t)={\dot{\gamma }}_0\cos (\omega t)$, even in materials that do not permit banding as their steady state response to a steadily imposed shear flow $\dot{\gamma }={\dot{\gamma }}_0=\text{const}$. Highly counterintuitively, banding persists in LAOS even in the limit of zero frequency $\omega \to 0$, where one might a priori have expected a homogeneous flow response in a material that does not display banding under conditions of steadily imposed shear. We explain this finding in terms of an alternating competition within each cycle between glassy aging and flow rejuvenation. Our predictions have far-reaching implications for the flow behavior of aging yield stress fluids, suggesting a generic expectation of shear banding in flows of even arbitrarily slow time variation.

Figure 1

Response of the SGR model to an imposed LAOS deformation with ${\gamma }_0=1.59$ and $\omega =0.01$. Signals (shifted such that $\dot{\gamma }=0$ at $t=0$) show (a) shear stress and $\dot{\gamma }/{\dot{\gamma }}_0$ (inset), (b) inverse effective sample age, and (c) degree of banding over a cycle period $T=2\pi /\omega$. Flow profiles are shown in (d) for the times indicated by the corresponding symbols in (a)–(c). Noise temperature $x=0.3$, initial sample age $t_w=10$, cycle number $N=50$. $m=100$, $n=100$, $w=0.1$.

Figure 2

Top: Dynamic phase diagram showing as a color map the cycle-averaged degree of banding in the SGR model in LAOS. Dashed lines show constant ${\dot{\gamma }}_0$. Bottom: Elastic Lissajous-Bowditch curves for the homogeneous (dashed lines) and heterogeneous (solid lines) SGR model, with the instantaneous degree of banding $\mathrm{\Delta }\dot{\gamma }$ indicated by the color scale, for the grid of ${\gamma }_0,\omega$ values shown by $\times$ in the top panel. (The circle in the top panel shows ${\gamma }_0,\omega$ values of the run in Fig. 1.) Noise temperature $x=0.3$, initial sample age $t_w=10.0$. $m=100$, $n=25$, $w=0.05$.

Figure 3

Maximum discrepancy between stress predicted by a calculation that assumes homogeneous flow and by the full heterogeneous calculation. Parameter values as in Fig. 2.