Microscopic processes controlling the Herschel-Bulkley exponent

TL;DR

This paper investigates the microscopic processes influencing the Herschel-Bulkley exponent in yield stress materials, challenging previous models and proposing a new relation involving avalanche fractal dimensions to better explain experimental observations.

Contribution

It introduces an improved mean-field model with fat-tailed mechanical noise and derives a relation between the Herschel-Bulkley exponent and avalanche fractal dimensions in finite dimensions.

Findings

Mean-field model with fat-tailed noise predicts β=1 with logarithmic correction.

Measurements of avalanche fractal dimension support β≈2.1 in 2D and 1.7 in 3D.

Finite dimensional effects significantly influence the Herschel-Bulkley exponent.

Abstract

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.