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/β. A mean-field approximation due to Hebraud and Lequeux (HL) assumes mechanical noise to be Gaussian and leads to β=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 β=1 with logarithmic correction. This result supports that HL is not a suitable explanation for the value of β, 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 β=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 β≈2.1 and β≈1.7 in two and three dimensions, respectively. We discuss theoretical arguments leading to approximations of β in finite dimensions.