Age-Dependent Transient Shear Banding in Soft Glasses

We study numerically the formation of long-lived transient shear bands during shear startup within two models of soft glasses (a simple fluidity model and an adapted “soft glassy rheology” model). The degree and duration of banding depends strongly on the applied shear rate, and on sample age before shearing. In both models the ultimate steady flow state is homogeneous at all shear rates, consistent with the underlying constitutive curve being monotonic. However, particularly in the soft glassy rheology case, the transient bands can be extremely long lived. The banding instability is neither “purely viscous” nor “purely elastic” in origin, but is closely associated with stress overshoot in startup flow.

Figure 1

(a) Homogeneous startup transient in fluidity model at applied shear rate $\overline{\dot{\gamma }}=0.1$ for waiting times $t_w={10}^0$, ${10}^2$, ${10}^4$, ${10}^6$, ${10}^8$ ($t_w$ increasing with increasing size of overshoot) showing stable region (solid), unstable (dashed). Dotted curve: actual startup curve arising from inhomogeneous flow with $l_0=0.01$ and $\delta =0.01$. (b) Counterpart in modified SGR model for $x=0.3$, $\overline{\dot{\gamma }}=0.1$, $w=0.05$, $n=50$, $m=300$, and waiting times $t_w={10}^0$, ${10}^2$, ${10}^4$, ${10}^6$. Solid lines: banding disallowed. Dotted: banding allowed. Inset: approach to steady state at later times in the banding case. Note that our strain variable is rescaled; in practice we expect strains $O(1)$ at peak.

Figure 2

(a) For chosen $t_w={10}^6$, plots of the instantaneous homogeneous constitutive curves during start up for $t-t_w=40$, 80, 120, 160, 200 with regions of instability dashed. (b) Strength of banding (${\dot{\gamma }}_{\max }-{\dot{\gamma }}_{\min }$) against $t-t_w$ for $\overline{\dot{\gamma }}=0.1$, and $t_w={10}^4$, ${10}^6$, ${10}^8$. (c) For the same $\overline{\dot{\gamma }}=0.1$, and $t_w={10}^4$ (solid), ${10}^6$ (dotted), ${10}^8$ (dashed), plots in the $t-t_w$, $\dot{\gamma }$ plane showing the upper (thick lines) and lower (thin lines) boundary curves of the window in which banding is significant (threshold set by $({\dot{\gamma }}_{\max }-{\dot{\gamma }}_{\min })/\overline{\dot{\gamma }}>0.01$). (d) Snapshots of banded profiles for $\dot{\gamma }=0.1$, $t_w={10}^6$, and $t-t_w=5$, 120, 130, 135, 225 showing onset, development and eventual collapse of the bands. $l_0=0.01$, $\delta =0.01$ in (b)–(d).

Figure 3

Shear-rate profiles corresponding to slowly decaying stress signal at $t_w={10}^6$ in Fig. 1b averaged over time windows 250–500; 500–1000; 1000–1500; 1500–2000; 2000–2500; 2500–3000 (top to bottom at $y=0.5$).