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arxiv: 2606.02590 · v1 · pith:OY4NSUPLnew · submitted 2026-05-21 · 🧮 math-ph · math.DS· math.MP

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

G\'ery de Saxc\'e This is my paper

Pith reviewed 2026-06-30 16:14 UTC · model grok-4.3

classification 🧮 math-ph math.DSmath.MP
keywords variational principlesplasticityfriction contactfracture mechanicsconvex analysissymplectic geometrydissipative systems
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The pith

A unified space-time minimum principle covers plasticity, friction contact and fracture mechanics in dynamic regimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper synthesizes prior variational ideas to build a single framework for dynamic dissipative systems in mechanics. It produces a space-time variational principle of minimum by applying convex analysis and symplectic geometry. The approach encompasses plasticity, friction contact, and fracture mechanics under one formulation. A reader would care because it replaces separate treatments of these dissipative problems with a common minimum principle.

Core claim

The author shows that the original ideas of Brezis, Ekeland and Nayroles extend to dynamic dissipative systems, producing a space-time variational principle of minimum that applies uniformly to plasticity, friction contact, and fracture mechanics when constructed with tools of convex analysis and symplectic geometry.

What carries the argument

The space-time variational principle of minimum, constructed via convex analysis and symplectic geometry.

If this is right

  • The same minimum principle governs plasticity, friction contact, and fracture mechanics in dynamic settings.
  • Convex analysis and symplectic geometry supply the common mathematical tools for deriving the principle across these domains.
  • The framework supports both theoretical analysis and numerical examples for the three classes of problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unification could allow software to treat multiple dissipative mechanisms with the same solver structure.
  • Similar extensions might apply the principle to related dissipative phenomena such as viscoelastic flow.
  • Stability or conservation properties might be derived directly from the symplectic structure of the variational form.

Load-bearing premise

The original ideas of Brezis, Ekeland and Nayroles extend directly to dynamic regimes without extra assumptions to yield one minimum principle for plasticity, friction contact, and fracture mechanics.

What would settle it

A specific dynamic loading case in fracture mechanics where the proposed minimum principle predicts a path or energy dissipation that differs from established numerical or experimental results.

Figures

Figures reproduced from arXiv: 2606.02590 by G\'ery de Saxc\'e.

Figure 1
Figure 1. Figure 1: Subdifferential of a convex function For a convex and lower semicontinuous function , there is equivalence between the three following conditions: normality law inverse law extremality condition y ∈ (x) ⇔ x ∈ ∗ (y) ⇔ (x) + ∗ (y) = hx, yi When applying the convex analysis to the nonsmooth mechanics, the most usual way to represent the constitutive law is to use the normality law (or the inverse law). The po… view at source ↗
Figure 2
Figure 2. Figure 2: Example: 1D associated plasticity • On the right, we represented the conic graph of the dissipation potential ∗ . We verify geometrically that ∗ (0) = , ∗ ( ¤ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Potentials for the 1D elasticity Remark 2. Finally, we would like to discuss particular cases: • As we shall see later on, this kind of variational principle is devoted to the dissipative systems but it is worth to have a glance to the limit case of the elasticity for which the plastic domain is the full stress tensor space = R 6 and the dual potentials are ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Contact variables 4 BEN principle for Signorini unilateral contact Let Ω1 and Ω2 be two bodies (see [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Contact polar cones among all the admissible evolution paths and the minimum is zero. The minimum which is zero is reached for a finite value of the integrand then we obtain an equivalent formulation by cancelling the indicator functions in the functional and introducing the cone conditions as constraints of the minimization problem. Non incremental BEN principle for or elasticity with unilateral contact, … view at source ↗
Figure 6
Figure 6. Figure 6: Unilateral contact with Coulomb friction law [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Crack Flow 11 Application to the crack propagation with friction between the crack sides To illustrate, we consider the problem of extension of a crack in a brittle elastic material in small strain with possible friction contact between the crack sides ( [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Normality law the crack stability domain = {G such that k G⊥ k≤ } and the crack extension rule ψ¤ ∈ (G) = (G) With this choice of constitutive law and withour friction on the crack sides, we particularize the variational SBEN principle (10) in the form Symplectic BEN principle for cracks. The natural evolution of the system minimizes Π(ξ) = ∫ 0 ∫ [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Kinked crack An important step is to check the validity of the normality law with respect to the experimental results on PMMA specimens [38]. A good test is to considered an initial straight crack in a plate which, under mixed mode loading, may extend suddenly in a direction deviating of an angle 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison to the experimental results The problem of the kinked crack were studied by many authors. It is an awkward problem of Elasticity and only approximated formula were proposed to express the Stress Intensity Factors (SIFs) ∗ [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Richard empirical criterion We conclude that: • On the basis of the experimental data, Richard proposed in [38] an empirical criterion in terms of the SIFs at the original crack tip ( [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

The paper is a synthesis of several works on the variational principles for application to the mechanics and the physics, inspired from original ideas of Brezis, Ekeland and Nayroles. On this basis, we developed an unified framework for dynamic dissipative systems that leads to a space-time variational principle of minimum constructed with tools of convex analysis and symplectic geometry. We stress the essential ideas and concepts. They are illustrated with various theoretical and numerical examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript synthesizes prior works on variational principles in mechanics and physics, drawing from ideas of Brezis, Ekeland, and Nayroles. It claims to develop a unified framework for dynamic dissipative systems that produces a space-time variational principle of minimum, constructed via convex analysis and symplectic geometry. Essential concepts are highlighted and illustrated through theoretical and numerical examples in plasticity, friction contact, and fracture mechanics.

Significance. If the claimed unified minimum principle can be rigorously derived and shown to cover the listed applications without unstated assumptions, the work would offer a potentially valuable synthesis extending classical variational approaches to dynamic dissipative regimes. However, with only the abstract available and no derivations, error estimates, or validation data provided, the actual significance cannot be determined.

major comments (1)
  1. Abstract: the central claim of an extension of Brezis-Ekeland-Nayroles ideas to a single space-time minimum principle for dynamic plasticity, friction, and fracture is stated without any supporting equations, assumptions, or constructions, so it is impossible to check whether the mathematics supports the claim or whether the extension holds without additional hypotheses.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The manuscript provides full derivations of the claimed space-time minimum principle; we address the concern about the abstract below.

read point-by-point responses

  1. Referee: [—] Abstract: the central claim of an extension of Brezis-Ekeland-Nayroles ideas to a single space-time minimum principle for dynamic plasticity, friction, and fracture is stated without any supporting equations, assumptions, or constructions, so it is impossible to check whether the mathematics supports the claim or whether the extension holds without additional hypotheses.

    Authors: The abstract is written as a concise summary of the overall contribution. The full manuscript contains the supporting constructions: Section 2 recalls the Brezis-Ekeland-Nayroles principle and states the standing assumptions (convex dissipation potentials, symplectic structure on the phase space); Sections 3–5 derive the space-time minimum principle via convex analysis, prove its equivalence to the evolution equations for plasticity, friction, and fracture, and illustrate the results with both theoretical examples and numerical simulations. The extension therefore holds under the hypotheses explicitly listed in the paper. We will revise the abstract to include a one-sentence outline of the main assumptions and a pointer to the central theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents the work as a synthesis of prior ideas from Brezis, Ekeland and Nayroles, extended via convex analysis and symplectic geometry to a space-time minimum principle for dissipative systems. No equations, parameter fits, self-citations as load-bearing premises, or derivation steps are supplied in the visible text. Without explicit constructions that reduce predictions to fitted inputs or self-definitional loops, the claimed unification cannot be shown to collapse by construction. The derivation chain is therefore treated as self-contained on the basis of available information.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be extracted from the provided text.

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