A few notes about viscoplastic rheologies

This paper employs rigorous convex analysis to synthesize and compare general viscoplastic dissipation potentials derived from serial and parallel combinations of linear and nonlinear viscous and plastic elements, specifically contrasting mathematically rigorous serial-viscosity models with empirical harmonic mean approaches commonly used for geomaterials.

The Gist

Imagine you are trying to understand how different materials flow, stretch, or break under pressure. Think of a material like rock, ice, or even honey. Sometimes they act like a solid (they hold their shape), and sometimes they act like a liquid (they flow).

This paper is a mathematical "instruction manual" for building models that predict exactly how these materials behave when they are doing both at the same time. The author, Tomáš Roubíček, uses a branch of math called Convex Analysis (think of it as the geometry of "bowl-shaped" curves) to figure out how to combine different rules of physics into one master rule.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Two Basic Building Blocks

To understand the complex models, we first need two simple "bricks":

• The Perfect Plasticity Brick (The "Bouncer"): Imagine a heavy bouncer at a club. If you push him gently, he doesn't move. But if you push hard enough (past a specific "threshold"), he suddenly starts moving and keeps moving as long as you push. This represents perfect plasticity. It's like a material that is rigid until it breaks, then flows.
• The Viscous Damper Brick (The "Honey"): Imagine pushing a spoon through thick honey. The harder you push, the faster it moves, but it never stops moving as long as you apply force. This represents viscosity. It's a smooth, continuous flow.

2. The Two Ways to Combine Them (The "Circuit")

The paper asks: What happens if we connect these two bricks together? There are two main ways to wire them up, like connecting batteries in a flashlight.

Option A: Parallel (The "Bingham" Model)

• The Setup: Imagine the "Bouncer" and the "Honey" are standing side-by-side. You push on both of them at the same time.
• The Result: You have to push hard enough to wake up the Bouncer plus push hard enough to move the Honey.
• Real-world example: This is like toothpaste or ketchup. It sits still in the tube (the Bouncer holds it), but once you squeeze hard enough, it flows out (the Honey takes over). In physics, this is called a Bingham fluid.

Option B: Series (The "Maxwell" Model)

• The Setup: Imagine the "Bouncer" and the "Honey" are in a line, one after the other. You push the first one, which pushes the second one.
• The Result: The total movement is the sum of both. If the Bouncer is stuck, the whole line is stuck. But if the Bouncer finally moves, the Honey flows through it.
• The Magic Trick: The author shows that when you combine these in a line using strict math, the result is surprisingly smooth. Even though the Bouncer is "jumpy" (stuck then moving), the combination creates a smooth, continuous flow curve. This is great for modeling things like earthquakes or glaciers, where slow creep can suddenly turn into a fast slip.

3. The "Harmonic Mean" vs. The "Real Math"

In the real world, scientists often use a shortcut to combine these rules. They use a formula called the Harmonic Mean (a specific type of average).

• The Analogy: Imagine you have two pipes carrying water. One is narrow, one is wide. The shortcut says, "Let's just average their widths to guess the total flow."
• The Problem: The paper argues that this shortcut is often just a guess (empirical). It works okay for simple cases, but if you have complex materials (like rock deep underground or ice sheets), the shortcut can be wrong.
• The Author's Solution: Instead of guessing, use the rigorous math (Convex Analysis) to build the model from the ground up. This ensures the physics actually makes sense.

4. Adding "Spicy" Ingredients (Nonlinear Viscosities)

The paper also talks about materials that don't just flow like honey; they flow like shear-thinning fluids (like paint or blood).

• The Analogy: Imagine a crowd of people walking. If they walk slowly, they bump into each other and move slowly. But if they start running, they actually move faster because they get out of each other's way.
• The Math: The author shows how to mix the "Bouncer" with these "spicy" flowing rules. He demonstrates that while the math gets very complicated (involving cubic equations and fancy approximations), it is possible to create a single, unified formula that describes the whole system.

5. Why Does This Matter?

Why do we care about combining a bouncer and honey?

• Geophysics: It helps us understand how the Earth's mantle moves (which causes earthquakes and volcanoes) and how glaciers slide over the ground.
• Engineering: It helps design better materials, from polymers to blood flow simulations.
• The Big Picture: The paper proves that even though materials can be messy and complex, we can describe them with a single, elegant mathematical "potential" (a kind of energy map). This map tells us exactly how much force is needed to make the material move, whether it's flowing slowly or slipping fast.

Summary

Think of this paper as a chef's guide to mixing ingredients.

• Ingredients: Rigid blocks (plasticity) and sticky fluids (viscosity).
• Method: Instead of just throwing them together and guessing the taste (the old empirical way), the author uses a precise recipe (convex analysis) to mix them.
• Result: A perfect, smooth sauce (a unified mathematical model) that accurately predicts how complex materials like ice, rock, and magma will behave under stress.

The author essentially says: "Don't just guess how these materials flow. Use the geometry of 'bowl shapes' to build a model that is mathematically perfect, even if the material itself is messy."

Technical Summary

1. Problem Statement

The paper addresses the challenge of rigorously constructing viscoplastic rheological models by combining linear viscosity (creep) and perfect plasticity.

• Context: In engineering and geophysics (e.g., modeling rocks, ice, magma, and polymers), materials often exhibit both viscous flow and plastic yielding.
• The Issue: While various empirical models exist (e.g., Bingham, Maxwell, Norton-Hoff), there is often a lack of mathematical rigor in how these elements are combined, particularly regarding serial vs. parallel arrangements.
• Specific Gap: Empirical models often rely on "harmonic means" of viscosities to combine elements in series. The author questions whether these empirical shortcuts are mathematically equivalent to rigorous combinations derived from convex analysis, especially when nonlinear (power-law) viscosities are involved.

2. Methodology

The author employs the rigorous tools of convex analysis to synthesize single dissipation potentials ($\zeta_{vp}$) from the potentials of individual viscous and plastic elements.

• Core Framework: The dissipative stress $\sigma$ is defined as the derivative (or subdifferential) of a convex dissipation potential $\zeta_{vp}$ with respect to the strain rate $\varepsilon$:
  $$\sigma \in \partial \zeta_{vp}(\varepsilon)$$
• Key Mathematical Tools:
  • Convex Conjugate ($f^*$): Used to switch between stress and strain rate representations.
  • Infimal Convolution ($\square$): The primary operation used to combine potentials in serial arrangements. If elements are in series, the total potential is the infimal convolution of individual potentials: $\zeta_{total} = \zeta_1 \square \zeta_2$.
  • Addition of Potentials: Used for parallel arrangements: $\zeta_{total} = \zeta_1 + \zeta_2$.
• Approach: The paper systematically derives the effective viscosity and stress-strain relationships for:
  1. Linear viscosity + Perfect plasticity (Parallel vs. Serial).
  2. Three-element models (Bi-viscosity).
  3. Nonlinear power-law viscosities (Norton-Hoff model).

3. Key Contributions and Results

A. Rigorous Derivation of Basic Models (Sections 2 & 3)

• Parallel Combination (Bingham/Kelvin):
  • Combines a plastic potential ($\sigma_a |\cdot|$) and a quadratic viscous potential ($\frac{1}{2}D|\cdot|^2$) by addition.
  • Result: Yields the standard Bingham fluid model. The effective viscosity is $\mu_{eff} = D + \sigma_a/|\varepsilon|$.
• Serial Combination (Visco-perfect-plastic):
  • Combines the same elements via infimal convolution ($\zeta_1 \square \zeta_2$).
  • Result: The resulting potential is the Huber function (a smooth approximation of the plastic potential).
  • Significance: This model allows for smooth creep at low stresses and fast slip at high stresses. The effective viscosity is $\mu_{eff} = \min(D, \sigma_a/|\varepsilon|)$.
  • Smoothness: Unlike the parallel model, the serial model yields a smooth (continuously differentiable) dissipation potential, avoiding set-valued subdifferentials at zero strain rate.

B. Equivalence of Three-Element Models (Section 3)

• The paper analyzes two variants of combining two viscous dampers with one plastic element (Bi-viscosity models).
• Finding: Two seemingly different topological arrangements (one adding a damper in parallel to the serial pair, the other adding it in series to the parallel pair) are mathematically equivalent if the parameters are rescaled correctly (Equation 7).
• Contrast with Empiricism: The paper demonstrates that empirical harmonic mean formulas (commonly used in geophysics) applied to these same topologies yield non-equivalent results. The rigorous convex analysis approach is necessary to ensure physical consistency.

C. Nonlinear Viscosities and Power-Law Fluids (Section 4)

• The author extends the analysis to Norton-Hoff (power-law) fluids where $\sigma = D|\varepsilon|^{1/n-1}\varepsilon$.
• Infimal Convolution Complexity: Combining linear viscosity (diffusive creep) and power-law viscosity (dislocation creep) in series requires solving the infimal convolution.
  • For general $n$, an explicit closed-form solution for the stress-strain relationship is often impossible (requires solving high-degree algebraic equations).
  • Example: For $n=2$ and $n=3$, the author derives explicit solutions using quadratic and Cardano's cubic formulas, respectively.
• Comparison of Models:
  • The paper compares the rigorous infimal convolution (Eq. 14) against the empirical harmonic mean (Eq. 15), where partial strain rates are replaced by the total strain rate.
  • Result: These two approaches yield different effective viscosities and stress responses. The empirical approach is shown to be an approximation that fails to capture the true physics of serial combinations in nonlinear regimes.

D. Generalized Maxwell Rheologies (Section 4, Remark 6)

• The paper proposes a formulation for large-strain visco-elastodynamics where the serial arrangement of viscoplastic elements is handled via the sum of conjugate potentials ($\sum \zeta_i^*$) rather than the difficult infimal convolution of primal potentials.
• This avoids the ambiguity of non-commutative multiplicative decompositions in large-strain plasticity.

4. Significance and Implications

• Mathematical Rigor: The paper establishes that "harmonic mean" viscosity rules, widely used in geophysical modeling (e.g., mantle convection, ice sheet dynamics), are empirical approximations that do not strictly follow from the serial combination of convex potentials.
• Model Selection: It clarifies when to use the Bingham model (parallel) vs. the Huber/Visco-perfect-plastic model (serial) based on the physical mechanism (e.g., whether creep and slip occur simultaneously or sequentially).
• Geophysical Application: The results are critical for modeling:
  • Mantle Dynamics: Where shear-thinning (power-law) behavior is dominant.
  • Glaciology: Modeling ice creep (Glen's law) combined with basal sliding.
  • Earthquake Mechanics: Distinguishing between aseismic creep and catastrophic slip.
• Computational Advantage: By utilizing the convex conjugate and summing conjugate potentials (as in the Generalized Maxwell formulation), the paper offers a pathway to numerically implement complex serial rheologies without explicitly solving non-trivial infimal convolutions.

Conclusion

Roubíček demonstrates that while simple linear models can be combined intuitively, the rigorous synthesis of viscoplastic rheologies—especially those involving nonlinear power laws—requires the formal machinery of convex analysis. The paper serves as a corrective to empirical practices in geophysics, showing that rigorous serial combinations produce distinct, often smoother, and physically more consistent behaviors than standard harmonic mean approximations.