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Microscopic processes controlling the Herschel-Bulkley exponent.

Jie Lin1, Matthieu Wyart2

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This study reveals that the mechanical noise in yield stress materials significantly impacts the Herschel-Bulkley exponent (β). Improved models with fat-tailed noise suggest β≈1, challenging previous theories and highlighting finite-dimensional effects.

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Area of Science:

  • Rheology
  • Materials Science
  • Statistical Mechanics

Background:

  • Yield stress materials exhibit singular flow curves as strain rate approaches zero.
  • The Herschel-Bulkley exponent (n=1/β) characterizes this behavior.
  • The Hebraud-Lequeux (HL) mean-field model, assuming Gaussian mechanical noise, predicts β=2, showing good agreement with experimental data.

Purpose of the Study:

  • To investigate the influence of mechanical noise properties on the Herschel-Bulkley exponent (β).
  • To develop an improved mean-field model that accounts for non-Gaussian mechanical noise.
  • To provide a theoretical framework for understanding the finite-dimensional effects on β.

Main Methods:

  • Development of an improved mean-field model incorporating fat-tailed mechanical noise.
  • Theoretical analysis of elastoplastic models and avalanche propagation in plasticity.
  • Derivation of a relationship between β, spatial dimension (d), and avalanche fractal dimension (d_f).

Main Results:

  • The improved mean-field model with fat-tailed noise predicts β=1 with logarithmic corrections.
  • This finding challenges the suitability of the HL model for explaining the observed β values.
  • A theoretical argument suggests β=1+1/(d-d_f), supported by measurements of d_f yielding β≈2.1 (2D) and β≈1.7 (3D).

Conclusions:

  • The Herschel-Bulkley exponent (β) is significantly influenced by finite-dimensional effects, not solely by Gaussian mechanical noise.
  • Fat-tailed noise distributions offer a more accurate theoretical basis for understanding yield stress material flow.
  • The derived relationship β=1+1/(d-d_f) provides a predictive tool for β in different spatial dimensions.