Microscopic processes controlling the Herschel-Bulkley exponent

The flow curve of various yield stress materials is singular as the strain rate vanishes and can be characterized by the so-called Herschel-Bulkley exponent $n=1/\beta$. A mean-field approximation due to Hebraud and Lequeux (HL) assumes mechanical noise to be Gaussian and leads to $\beta =2$ in rather good agreement with observations. Here we prove that the improved mean-field model where the mechanical noise has fat tails instead leads to $\beta =1$ with logarithmic correction. This result supports that HL is not a suitable explanation for the value of $\beta$, which is instead significantly affected by finite-dimensional effects. From considerations on elastoplastic models and on the limitation of speed at which avalanches of plasticity can propagate, we argue that $\beta =1+1/(d-d_f)$, where $d_f$ is the fractal dimension of avalanches and $d$ the spatial dimension. Measurements of $d_f$ then supports that $\beta \approx 2.1$ and $\beta \approx 1.7$ in two and three dimensions, respectively. We discuss theoretical arguments leading to approximations of $\beta$ in finite dimensions.

Figure 1

Distribution $P\left(x\right)$ of local stabilities $x$ at $\sigma >{\sigma }_c$ (blue) and at ${\sigma }_c$ (red). The area in the negative $x$ corresponds to the strain rate $\dot{\gamma }$. The green (red) circle represents a (un)stable site, which implements random long jumps and drifts towards the negative, unstable direction.

Figure 2

[(a)–(c)] $D_{+}$ and $D_{-}$ vs. $\dot{\gamma }$. Here $\mu =1.5$ for (a) and (d), $\mu =1.0$ for (b) and (e), and $\mu =0.5$ for (c) and (f). In all simulations, we take $N={256}^2$, and $A=0.3$. We note that to verify the logarithmic correction at $\mu =1$, we plot $D_{+}/|log\left(\dot{\gamma }\right)|$ vs. $\dot{\gamma }$. [(d)–(f)] Comparison of the numerical flow curves (blue circles) and the theoretical predictions (lines).

Figure 3

[(a)–(c)] Tests of the predicted scaling collapses of ${\mathrm{\Delta }}_2\left(x\right)$ ($x>0$) for $\mu =1.5$ (a), $\mu =1$ (b), and $\mu =0.5$ (c). Colors represent different strain rate, shown in the legend of (a). Dashed lines indicate the slopes predicted by theory. [(d)–(f)] Collapses of ${\mathrm{\Delta }}_2\left(x\right)$ ($x<0$) for $\mu =1.5$ (d), $\mu =1$ (e), and $\mu =0.5$ (f). In the inset of (f), we rescale the $x$ axis to confirm the prediction that the width of the plateau scales as $\dot{\gamma }$ for $\mu <1$.