A family of variational principles of minima for the plasticity, the friction contact and the fracture mechanics

The Gist

Imagine you are trying to predict how a complex system behaves over time—like a metal beam bending under heat, two rough surfaces rubbing together, or a crack spreading through glass. Usually, scientists solve these problems step-by-step, like climbing a mountain one foot at a time, calculating the next position based on where you are right now.

This paper proposes a different, more "all-at-once" way of thinking. Instead of climbing step-by-step, it suggests looking at the entire journey from start to finish as a single, unified path and finding the "best" one among all possible paths.

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

1. The Big Idea: The "Movie" vs. The "Snapshot"

Most engineering calculations are like taking a series of snapshots. You calculate the state at 1 second, then 2 seconds, then 3 seconds.
The author, G. de Saxcé, suggests a "movie" approach. He proposes a Variational Principle. Think of this as a rule that says: "Out of every possible movie you could film of this system's history, nature only chooses the one that minimizes a specific 'cost'."

If you can find the path that makes this "cost" zero, you have found the true physical behavior of the system.

2. The Toolkit: Two Geometries

To build this "movie" rule, the author mixes two different types of geometry:

• The Reversible Part (Symplectic Geometry): This handles the "perfect" parts of physics, like a pendulum swinging back and forth without friction. It's like a frictionless ice rink where energy is conserved.
• The Irreversible Part (Convex Analysis): This handles the "messy" parts where energy is lost, like friction, plastic bending (where metal stays bent), or cracking. This is where things get "sticky" or "rough."

The paper's main trick is to combine these two. It treats the system as having a "reversible engine" (like a spring) and a "dissipative brake" (like friction), and it finds a mathematical formula that balances them perfectly over the whole timeline.

3. The "BEN" Principle: Finding the Perfect Path

The core of the paper is an extension of a famous idea called the Brezis-Ekeland-Nayroles (BEN) principle.

• The Analogy: Imagine you are trying to find the smoothest path for a ball to roll from point A to point B while dragging a heavy sack of sand (friction) behind it.
• The Paper's Claim: There is a specific mathematical formula (a "functional") that calculates the "roughness" of any path you imagine.
  • If you guess a path that nature wouldn't take, the formula gives you a positive number (a penalty).
  • If you guess the actual path nature takes, the formula gives you zero.
  • Therefore, to solve the problem, you just need to find the path that makes this formula equal to zero.

4. What Does This Solve?

The author shows this "movie" approach works for three tricky areas where standard math often struggles:

• Plasticity (Bending Metal): When you bend a paperclip, it doesn't spring back. The paper shows how to calculate the whole bending process at once, rather than step-by-step, using the "zero cost" rule.
• Frictional Contact (Rubbing Surfaces): When two rough surfaces touch, they stick or slide in complex ways. The paper uses a tool called a "Bipotential" (think of it as a two-sided map) to describe this sticking/sliding behavior without needing to force it into a simple "smooth" shape.
• Fracture (Cracking Glass): This is the most dramatic example. When a crack grows, it usually jumps in a specific direction.
  • The Problem: Old methods often predicted the crack would go in the wrong direction because they used a "step-by-step" (explicit) calculation that was too sensitive to small errors.
  • The Paper's Solution: By using the "movie" approach with a specific "implicit" calculation (looking at the whole step at once), the author's model predicts the crack's path much more accurately. It matches real-world experiments where cracks "kink" or turn at specific angles.

5. The "Symplectic" Twist

The author introduces a fancy term: Symplectic.

• Simple Explanation: In physics, "symplectic" is a way of organizing information about position and momentum (how fast and where) together.
• The Paper's Contribution: The author takes this "symplectic" organization and applies it to systems that lose energy (dissipative systems). Usually, symplectic math is only for perfect, energy-conserving systems. The author builds a bridge to use this powerful math for messy, real-world systems like friction and cracking.

6. The "Bipotential" for Non-Standard Rules

Some physical laws (like Coulomb friction) don't follow the standard "smooth" rules of math. They are "non-associated," meaning the direction of movement isn't perfectly aligned with the force pushing it.

• The Analogy: Imagine pushing a heavy box. Usually, you push it, and it moves in the direction you push. But with friction, the box might stick until you push hard enough, then slide sideways.
• The Paper's Tool: The author uses a Bipotential. Think of this as a special "translator" that can handle these weird, non-smooth rules. It allows the "movie" principle to work even when the physics is messy and doesn't follow a simple straight line.

Summary

The paper doesn't invent a new physical law; it invents a new way of solving existing laws.
Instead of calculating a system's future one second at a time, it proposes a method to calculate the entire history of the system at once. It uses a "cost function" that should be zero for the correct path. By combining the geometry of perfect motion (symplectic) with the geometry of messy loss (convex analysis), the author creates a unified framework that accurately predicts how metals bend, surfaces rub, and cracks grow, often outperforming traditional step-by-step methods.

Technical Summary

Technical Summary: A Family of Variational Principles of Minima for Plasticity, Friction Contact, and Fracture Mechanics

Problem Statement
The paper addresses the challenge of formulating unified, non-incremental space-time variational principles for dynamic dissipative systems. Traditional variational approaches often struggle with non-smooth constitutive laws (such as plasticity, frictional contact, and fracture) and non-associated flow rules, where the dissipation potential is not convex or the flow rule is not normal to the yield surface. Furthermore, existing unified frameworks for dissipative systems (e.g., GENERIC, rate-independent systems) often rely on restrictive hypotheses, such as the 1-homogeneity of the dissipation potential, which limits their applicability to viscoplasticity or general dynamic cases. The author seeks to extend the Brezis-Ekeland-Nayroles (BEN) principle to a symplectic framework capable of handling these complexities within a consistent geometrical setting.

Methodology
The methodology is grounded in the synthesis of convex analysis and symplectic geometry. The core approach involves the following steps:

1. Symplectic Decomposition: The time rate of the state vector $z$ (comprising degrees of freedom and momenta) is decomposed additively into a reversible part $\dot{z}_R$ (the symplectic gradient of the Hamiltonian $H$) and an irreversible/dissipative part $\dot{z}_I$.
2. Symplectic Convex Analysis: The author introduces the symplectic subdifferential $\partial_\omega \phi$ and the symplectic Fenchel polar $\phi^*_\omega$ of a dissipation potential $\phi$. This generalizes the standard Fenchel inequality to the symplectic context, allowing for the modeling of systems where the dissipation potential is not necessarily 1-homogeneous.
3. The SBEN Principle: A Symplectic Brezis-Ekeland-Nayroles (SBEN) principle is constructed. It posits that the natural evolution path of a dissipative system minimizes a specific functional $\Pi(z)$ over all admissible paths. The functional combines the dissipation potential, its symplectic polar, and the symplectic form, yielding a minimum value of zero for the true physical path.
4. Extension to Non-Associated Laws: To handle non-associated constitutive laws (e.g., Coulomb friction), the paper introduces the concept of bipotentials (functions $b(x,y)$ that are bi-convex and satisfy specific extremality conditions). This is further extended to symplectic bipotentials ($\hat{b}$) to accommodate dynamic non-associated systems.
5. Application to Specific Mechanics: The framework is applied to:
   • Standard and viscoplasticity (recovering Hill's inequality).
   • Signorini unilateral contact (linking to Linear Complementary Problems).
   • Finite strain plasticity (using both additive and multiplicative decompositions).
   • Fluid dynamics (Navier-Stokes and Bingham fluids in Eulerian specification).
   • Crack propagation with friction, modeling the crack extension as a flow on the cracked surface.

Key Contributions

• Unified Symplectic Framework: The paper proposes a general SBEN principle that unifies the treatment of reversible (Hamiltonian) and irreversible (dissipative) dynamics without requiring the restrictive 1-homogeneity of the dissipation potential.
• Symplectic Bipotentials: A novel mathematical tool, the symplectic bipotential, is defined to extend variational principles to non-associated constitutive laws, a class of problems previously difficult to treat variationally.
• Non-Incremental Formulation: The principles are "non-incremental," meaning they consider the entire time interval $[0, T]$ simultaneously rather than solving step-by-step, offering a global perspective on the evolution path.
• Fracture Mechanics Application: The paper applies the SBEN principle to brittle fracture with friction. It models crack extension via a flow field and utilizes an implicit scheme to solve the resulting variational problem.

Results

• Plasticity and Contact: The SBEN principle successfully recovers the classical constitutive laws for plasticity and Signorini contact as extremality conditions. For contact, the formulation reduces to a Linear Complementary Problem (LCP).
• Fluid Dynamics: The principle is shown to apply to Navier-Stokes and Bingham fluids, recovering Bernoulli's principle in the inviscid limit.
• Crack Propagation Validation: In the application to crack kinking under mixed-mode loading, the paper compares the SBEN predictions (using an implicit scheme and the normality law) against experimental data (PMMA specimens) and the empirical criterion by Richard.
  • The explicit Euler scheme is shown to yield "disastrous" predictions compared to experiments.
  • The implicit scheme, combined with the normality law, yields predictions in "very good agreement" with experimental data regarding the kink angle and stress intensity factors.
  • The derived ratio of Mode II to Mode I fracture toughness ($K_{IIc}/K_{Ic} = \sqrt{3}/2 \approx 0.86$) aligns better with experimental ranges (0.88–0.95) than previous theoretical proposals.

Significance and Claims
The paper claims that the SBEN principle offers a robust variational approach suitable for a broad spectrum of applications, spanning smooth and non-smooth mechanics, solids and fluids, and both Lagrangian and Eulerian specifications.

The author emphasizes that this approach is not merely a method for deriving weak formulations but serves as a "source of ideas" for modeling new constitutive laws. The significance lies in the ability to handle non-associated flow rules and dynamic dissipative systems within a single, consistent geometrical framework using tools of convex analysis and symplectic geometry. The paper modestly notes that while the normality law and the Principle of Local Symmetry yield good results for the specific case of kinked cracks, the normality law is recommended as the general rule, and the implicit scheme is preferred over explicit schemes for fracture mechanics problems.

The work is presented as a synthesis of previous research (including works by Brezis, Ekeland, Nayroles, and the author's own prior publications) rather than a presentation of entirely new experimental data, though it validates the theoretical framework against existing experimental benchmarks.