A Solid-Based Approach for Modeling Simple Yield-Stress Fluids: Rheological Transitions, Overshoot and Relaxation

This study proposes a novel solid-based constitutive equation derived from a Zener-type viscoelastic element and nonlinear viscosity models that successfully predicts both steady and transient rheological behaviors of simple yield-stress fluids, including stress overshoot driven by normal stress differences rather than isotropic hardening.

The Gist

The Big Picture: What is this paper about?

Imagine you have a jar of thick, gooey toothpaste or a bucket of mud. If you poke it gently, it acts like a solid—it holds its shape. But if you squeeze it hard enough, it suddenly flows like a liquid. These are called yield-stress fluids.

Scientists have been trying to write a "rulebook" (a mathematical model) to predict exactly how these fluids behave. The problem is that existing rulebooks are good at predicting how they flow after they start moving, but they fail miserably at predicting what happens in the split second before they start, or how they relax after you stop squeezing them.

This paper introduces a new, smarter rulebook. It treats these fluids not just as liquids, but as solids that can flow. By doing this, the authors successfully predict tricky behaviors that other models miss, like a "stress overshoot" (a sudden spike in pressure before the fluid settles down).

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The Core Idea: The "Jammed Crowd" Analogy

To understand the new model, imagine a crowded concert venue.

1. The Microstructure (The Crowd): The fluid (like Carbopol, a common gel used in toothpaste) is made of tiny, squishy particles packed so tightly they are "jammed" together, like a crowd of people standing shoulder-to-shoulder.
2. The Solvent (The Air): Between these people, there is air (or in the fluid's case, water).
3. The Behavior:
   • Gentle Push (Elasticity): If you push the crowd gently, they lean back and spring back when you let go. They act like a solid.
   • Hard Push (Plasticity): If you push hard enough, people start shuffling, stepping over each other, and rearranging. The crowd flows. This is the "yielding" point.

The New Model: A Mechanical Toy

The authors built a mechanical toy to represent this crowd. Instead of just a bucket of soup, they built a machine with three main parts:

1. The Spring (The Crowd's Memory): This represents the particles pushing back when you squeeze them. It stores energy like a rubber band.
2. The Shock Absorber (The Crowd's Shuffle): This represents the particles rearranging themselves. It's like a door closer that resists movement but eventually lets you through.
3. The Water Pipe (The Solvent): This is the "extra" part. In previous models, scientists forgot the water between the particles. This model adds a parallel pipe that represents the water flowing alongside the crowd.

Why is this important?
Previous models treated the fluid like a single, messy blob. This model separates the "solid-like" crowd behavior from the "liquid-like" water behavior. This separation is the secret sauce that allows the model to predict complex real-world behaviors.

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The Three Big Tests (What the Model Predicts)

The authors tested their new rulebook against three real-world scenarios. Here is how the model performed:

1. The "Start-Up" Spike (Stress Overshoot)

The Scenario: You start stirring the fluid very quickly.
The Old Problem: Old models predicted the pressure would just go up and stay up smoothly.
The Reality: In real life, the pressure spikes way higher than expected for a split second, then drops down to a steady level. It's like when you try to push a heavy sofa; it feels incredibly heavy at first, then suddenly "breaks loose" and slides easier.
The New Model: It predicts this spike perfectly!
The Secret: The authors discovered this spike isn't caused by the fluid getting "thinner" over time (as some thought). Instead, it's a geometric trick. As the fluid starts to flow, the internal "crowd" rearranges in a way that temporarily creates extra pressure (normal stress), causing the spike, before settling down.

2. The "Never-Ending" Relaxation (Stress Relaxation)

The Scenario: You stir the fluid, then suddenly stop and hold it still.
The Old Problem: Old models predicted the pressure would drop all the way to zero.
The Reality: The pressure drops, but it stops at a specific, non-zero number. It's like a rubber band that you stretch and hold; even if you stop moving your hand, the rubber band still pulls back with some force.
The New Model: It predicts this "leftover" force perfectly. This proves that even when the fluid looks stopped, it still has a solid-like structure holding it together.

3. The "Solid vs. Liquid" Switch (Creep)

The Scenario: You hang a weight on the fluid and watch it over time.
The Old Problem: Models struggled to predict when the fluid would stop stretching and when it would start flowing forever.
The Reality:

• If the weight is light, the fluid stretches a little and then stops (Solid behavior).
• If the weight is heavy, the fluid stretches a little and then keeps flowing forever (Liquid behavior).
  The New Model: It finds the exact "tipping point" (yield stress) where the switch happens. It correctly identifies that below a certain weight, the fluid acts like a solid, and above it, it acts like a liquid.

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Why Does This Matter?

You might ask, "Who cares about toothpaste physics?"

This model is a game-changer for industries that use thick, gooey materials:

• Battery Manufacturing: Making lithium-ion batteries involves pumping thick slurries (mixtures) into tiny spaces. If the fluid behaves unexpectedly, the battery fails.
• 3D Printing (Direct Ink Writing): Printing with soft materials requires knowing exactly when the ink will hold its shape and when it will flow.
• Food & Cosmetics: Designing the perfect texture for ketchup, shampoo, or frosting.

The Bottom Line

The authors of this paper realized that to understand these "sticky" fluids, we need to stop treating them like simple liquids and start treating them like solids that can flow.

By building a model that accounts for the "solid crowd" and the "liquid water" separately, and by using advanced math to handle 3D movements, they created a tool that finally explains the weird spikes, the leftover pressure, and the solid-to-liquid switches that scientists have been struggling to predict for decades.

In short: They found the missing piece of the puzzle that explains why yield-stress fluids are so tricky, and they did it by remembering that these fluids are actually just solids in disguise.

Technical Summary

1. Problem Statement

Yield-stress fluids (e.g., Carbopol dispersions, battery electrode slurries) exhibit complex rheological behaviors that cannot be fully captured by steady-state models like the Herschel-Bulkley equation. Key experimental challenges include:

• Stress Overshoot: A transient peak in stress during start-up shear flow, which some existing models attribute to thixotropy (time-dependent structural breakdown) or spatial heterogeneity. However, simple yield-stress fluids with negligible thixotropy also exhibit this.
• Non-Zero Fully Relaxed Stress: Upon cessation of flow, stress does not decay to zero but stabilizes at a finite value, a characteristic of solids, yet these materials flow like fluids under high stress.
• Creep Transition: A transition from solid-like behavior (strain saturation) to fluid-like behavior (continuous strain increase) depending on the applied stress magnitude.
• Model Limitations: Existing constitutive equations often fail to simultaneously predict steady-state behavior, transient overshoots, and relaxation characteristics without relying on isotropic hardening or thixotropic evolution parameters.

2. Methodology

The authors propose a new constitutive framework based on a viscoelastic solid rather than a viscoelastic fluid.

Microstructural Basis

The model is motivated by the microstructure of Carbopol dispersions:

• Gel Phase: Jammed microgel particles. Deformation is decomposed into reversible elastic strain (within particles) and irreversible plastic strain (rearrangement of the jammed structure).
• Solvent Phase: Interstitial solvent providing parallel viscous dissipation.

1D Constitutive Model (Small Deformation)

• Mechanical Analogue: A Zener element (Spring in series with a Kelvin-Voigt unit) representing the gel phase, connected in parallel with a linear dashpot representing the solvent.
• Strain Decomposition: Additive decomposition: $\gamma = \gamma^e + \gamma^p$.
• Stress Components: Total stress $\tau = \tau_{gel} + \tau_{sol}$.
  • $\tau_{gel}$: Governed by elastic strain and back stress ($\tau_B$).
  • $\tau_{sol}$: Proportional to total strain rate (Newtonian solvent).
• Flow Rules:
  • Plastic strain rate follows a generalized flow rule driven by the difference between gel stress and back stress.
  • Back Stress Evolution: Adopted from the Armstrong-Frederick kinematic hardening theory to capture the Bauschinger effect and finite relaxation.
  • Viscosity Models: Two shear-thinning models were tested: Eyring and Carreau-Yasuda, formulated as functions of stress to ensure flexibility.

3D Constitutive Model (Finite Deformation)

To satisfy the principle of material frame invariance and handle general deformations:

• Kinematics: The Kröner–Lee multiplicative decomposition of the deformation gradient ($F = F^e \cdot F^p$) replaces additive strain decomposition.
• Stress Evolution: Uses the upper-convected derivative for spatial tensors to ensure objectivity.
• Flow Rule: The plastic deformation rate ($D^p$) is defined based on the deviatoric stress difference between the gel stress and back stress.
• Viscosity: Defined via the second invariant of the deviatoric stress difference ($\xi$), incorporating the Eyring or Carreau-Yasuda models.

3. Key Contributions

1. Solid-Based Framework: The study successfully models yield-stress fluids as viscoelastic solids with a parallel solvent dissipation pathway, challenging the traditional fluid-centric view.
2. Mechanism of Stress Overshoot: The paper demonstrates that stress overshoot in start-up shear arises from a homogeneous mechanism driven by the tensorial structure of the 3D model. Specifically, normal stress differences enhance the stress invariant, accelerating plastic flow and causing a transient overshoot. This occurs without invoking isotropic hardening or spatial microstructural heterogeneity.
3. Unified Constitutive Equation: The model satisfies four critical experimental requirements simultaneously:
   • Steady-state stress following Herschel-Bulkley behavior.
   • Stress overshoot in start-up shear.
   • Non-zero fully relaxed stress dependent on pre-shear rate.
   • Solid-to-fluid transition in creep tests.
4. Analytical and Numerical Validation: The authors provided rigorous analytical derivations for the 1D case and numerical simulations for the 3D case, comparing results against experimental data for Carbopol 940.

4. Results

• Steady State: Both 1D and 3D models reproduce the Herschel-Bulkley relationship at low shear rates. At high shear rates, the models predict a second Newtonian plateau due to the solvent dashpot, matching experimental observations of Carbopol.
• Stress Overshoot:
  • The 1D model failed to predict overshoot; stress increased monotonically to steady state.
  • The 3D model successfully predicted stress overshoot. The overshoot magnitude increased with shear rate. Analysis revealed that in 3D, normal stress components contribute to the viscosity function, temporarily accelerating plastic flow and causing the stress rate to become negative (overshoot) before stabilizing.
• Stress Relaxation:
  • The model predicts a finite, non-zero fully relaxed stress.
  • Crucially, the 3D model predicts that the magnitude of this relaxed stress decreases as the pre-shearing rate increases, aligning with recent experimental findings (Vinutha et al., 2024). The 1D model predicted a relaxed stress independent of pre-shear rate.
• Creep Behavior:
  • The model captures a distinct transition: below a critical stress (identified with the Herschel-Bulkley yield stress), strain saturates (solid-like); above this threshold, strain increases indefinitely (fluid-like).
  • The 3D simulations confirmed that the Herschel-Bulkley yield stress serves as an accurate threshold for this transition.

5. Significance

• Theoretical Advancement: This work bridges the gap between metal plasticity theories (kinematic hardening) and soft matter rheology, providing a unified framework for yield-stress fluids.
• Elimination of Ad Hoc Assumptions: It proves that stress overshoot is an intrinsic consequence of the viscoelastic solid nature and tensorial stress evolution, removing the need to assume thixotropy or spatial heterogeneity to explain the phenomenon.
• Practical Application: The proposed 3D constitutive equation is suitable for implementation in finite element simulations (FEM) for complex industrial processes involving yield-stress fluids, such as:
  • Lithium-ion battery manufacturing: Modeling electrode slurries.
  • Direct Ink Writing (3D Printing): Predicting flow and shape retention.
  • Geophysical flows: Modeling muddy flows and debris.

In conclusion, the paper establishes a robust, frame-invariant constitutive model that accurately captures the complex transient and steady-state rheology of simple yield-stress fluids, offering a deeper physical understanding of their solid-like and fluid-like transitions.