Microscopic processes controlling 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.

9 pages, 3 figures

References in corpus (23)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Scaling description of the yielding transition in soft amorphous solids at zero temperature
• Subextensive Scaling in the Athermal, Quasistatic Limit of Amorphous Matter in Plastic Shear Flow
• Statistics of Earthquakes in Simple Models of Heterogeneous Faults
• A mode-coupling theory for the pasty rheology of soft glassy materials
• Elastic consequences of a single plastic event : a step towards the microscopic modeling of the flow of yield stress fluids
• Yielding and flow in adhesive and non-adhesive concentrated emulsions
• Statistical Physics of the Yielding Transition in Amorphous Solids
• Slow flows of yield stress fluids: complex spatio-temporal behaviour within a simple elasto-plastic model
• Driving rate dependence of avalanche statistics and shapes at the yielding transition
• Avalanches in Strained Amorphous Solids: Does Inertia Destroy Critical Behavior?
• Avalanches, precursors and finite size fluctuations in a mesoscopic model of amorphous plasticity
• The effect of inertia on sheared disordered solids: Critical scaling of avalanches in two and three dimensions
• Avalanche Size Scaling in Sheared Three-Dimensional Amorphous Solid
• On the density of shear transformation zones in amorphous solids
• Connecting diffusion and dynamical heterogeneities in actively deformed amorphous systems
• Criticality in the approach to failure in amorphous solids
• Inhomogeneous shear flows in soft jammed materials with tunable attractive forces
• On the relevance of disorder in athermal amorphous materials under shear
• Mechanical noise dependent Aging and Shear Banding behavior of a mesoscopic model of amorphous plasticity
• Nontrivial rheological exponents in sheared yield stress fluids
• Avalanche size distributions in mean field plastic yielding models
• Different universality classes at the yielding transition of amorphous systems

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• The yielding of amorphous solids at finite temperatures
• Elastic interfaces on disordered substrates: From mean-field depinning to yielding
• Criticality in sheared, disordered solids. I. Rate effects in stress and diffusion
• Properties of the density of shear transformations in driven amorphous solids
• Motility-induced buckling and glassy dynamics regulate three-dimensional transitions of bacterial monolayers
• Criticality in sheared, disordered solids. II. Correlations in avalanche dynamics
• Thermally activated intermittent flow in amorphous solids
• Ductile and brittle yielding of athermal amorphous solids: a mean-field paradigm beyond the random field Ising model
• Random organization criticality with long-range hydrodynamic interactions
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• Dynamic phase diagram of plastically deformed amorphous solids at finite temperature