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 elasto-plastic 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.