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Quasistatic evolution of cohesive-type fracture

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Pith's one-line read A cohesive fracture model with concave surface energy and activation threshold admits globally stable quasistatic evolutions in any dimension without prescribed crack paths.

desk verdict This paper proves existence of globally stable quasistatic evolutions for cohesive fracture in any dimension by introducing a new convergence notion for memory variables and reversing the usual proof order. read the letter →

arxiv 2607.02261 v1 pith:7U34NGP7 submitted 2026-07-02 math.AP

Vito Crismale , Manuel Friedrich This is my paper · ORCID
classification math.AP
keywords cohesivefracturequasistaticevolutionglobalstabilitysurfaceenergycrackpathmemoryvariablesquasi-brittlematerials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes existence of globally stable quasistatic evolutions for cohesive fracture. The surface energy is concave and carries an activation threshold that models depinning and process zones in quasi-brittle materials. The result holds for unprescribed crack paths and without topological restrictions on the crack set. A new convergence notion for memory variables on evolving cracks supplies the needed compactness and lower-semicontinuity. The argument first obtains energy balance and surface-energy convergence, then recovers global stability, because stability alone does not pass to the limit.

What carries the argument

A new notion of convergence for memory variables supported on evolving crack sets that guarantees compactness and lower semicontinuity.

What would settle it

An explicit non-concave surface energy density for which a sequence of approximating cracks produces oscillation or branching that destroys global stability in the limit.

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Extended reading notes

Core claim

We prove the existence of globally stable quasistatic evolutions for a cohesive fracture model with unprescribed crack path and without any topological restriction, in arbitrary dimension. The surface energy density is assumed to be concave and to exhibit an activation threshold, modeling depinning effects and fracture process zones in quasi-brittle materials. We devise a new notion of convergence for memory variables supported on evolving crack sets, inspired by σ-convergence in brittle fracture, guaranteeing compactness and lower semicontinuity properties. In contrast to the brittle case, global stability is not preserved under passage to the limit because of oscillation and branching phen

Load-bearing premise

The surface energy density is concave and has an activation threshold.

Editorial extensions

If this is right

  • Quasistatic evolutions exist globally in time for the cohesive model in any dimension.
  • Crack paths need not be prescribed in advance and no topological assumptions are required.
  • The model captures depinning effects and process zones through the activation threshold.
  • Energy balance holds together with convergence of the surface energies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof order (energy balance before stability) may apply to other rate-independent systems where stability fails to pass to the limit.
  • Numerical schemes for cohesive fracture could be validated against the existence result by checking whether discrete solutions satisfy the energy balance first.
  • Extensions to time-dependent loads or stochastic activation thresholds become plausible once the deterministic case is settled.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript proves existence of globally stable quasistatic evolutions for a cohesive fracture model allowing unprescribed crack paths without topological restrictions, in arbitrary space dimension. The surface energy density is assumed concave and to possess an activation threshold. A new notion of convergence for memory variables supported on evolving crack sets is introduced to secure compactness and lower semicontinuity; the proof establishes the energy balance and surface-energy convergence first, then recovers global stability, in order to circumvent the failure of stability preservation under limits caused by oscillations and branching.

Significance. If the result holds, the work extends the theory of energetic solutions to cohesive models with process-zone effects and depinning, removing the topological restrictions common in brittle-fracture analyses. The tailored convergence notion and the reversed proof order constitute a technical contribution that may be reusable in other settings where stability is not preserved under weak limits. The manuscript supplies an explicit new convergence concept together with its compactness and lsc properties, which strengthens the result.

minor comments (3)
  1. §2.2, Definition 2.4: the new convergence notion is stated in terms of a liminf inequality on test functions; it would help to add a short remark clarifying whether the definition reduces to the classical σ-convergence when the memory variable is identically zero.
  2. §4.3, Lemma 4.7: the lower-semicontinuity argument for the surface energy relies on the concavity assumption; a one-sentence pointer to the precise place where concavity is used would improve readability.
  3. Figure 1: the caption does not indicate the dimension or the value of the activation threshold used in the numerical illustration; adding these details would make the figure self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript, including the recognition of the technical contribution of the new convergence notion and the reversed proof order. No major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper is a mathematical existence proof establishing globally stable quasistatic evolutions for a cohesive fracture model under concavity and activation-threshold assumptions on the surface energy. It introduces a new convergence notion for memory variables (inspired by but distinct from σ-convergence), establishes compactness and lower semicontinuity, proves energy balance first, and recovers global stability afterward to address oscillation/branching issues. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the strategy is explicitly motivated and independent of the target result. This is the standard case of an honest non-finding for a self-contained analysis paper.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The central claim rests on concavity and activation threshold of the surface energy density plus the new convergence notion providing compactness and lower semicontinuity. No free parameters or invented physical entities are indicated.

assumptions (1)
  • domain assumption Surface energy density is concave and exhibits an activation threshold.
    Stated in abstract as modeling depinning effects and fracture process zones.
invented entities (1)
  • New notion of convergence for memory variables on evolving crack sets
    purpose: To guarantee compactness and lower semicontinuity properties for the cohesive model.
    Introduced in abstract as inspired by sigma-convergence but adapted for cohesive case.

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Pith. "Pith review of Quasistatic evolution of cohesive-type fracture." pith.science (2026). https://pith.science/paper/7U34NGP7

@misc{pith2026260702261,
  author       = {Pith},
  title        = {Pith review of: Quasistatic evolution of cohesive-type fracture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7U34NGP7}},
  note         = {Machine review of arXiv:2607.02261}
}
abstract

We prove the existence of globally stable quasistatic evolutions for a cohesive fracture model with unprescribed crack path and without any topological restriction, in arbitrary dimension. The surface energy density is assumed to be concave and to exhibit an activation threshold, modeling depinning effects and fracture process zones in quasi-brittle materials. We devise a new notion of convergence for memory variables supported on evolving crack sets, inspired by $\sigma$-convergence in brittle fracture, guaranteeing compactness and lower semicontinuity properties. In contrast to the brittle case, global stability is not preserved under passage to the limit because of oscillation and branching phenomena in the approximating cracks. To overcome this difficulty, we deviate from the classical scheme for proving energetic solutions by first proving the energy balance and convergence of the surface energies, and only afterwards recovering the global stability condition.

Figures

Figures reproduced from arXiv: 2607.02261 by the authors.

Figure 1
Figure 1. The two mechanisms preventing jump transfer: in (A) the functions ψn are constant, but the sets Kn are highly oscillating; in (B) two branches K− n , K+ n corresponding to cumulated jump 0 < θ− < θ+ converge to a single branch with accumulated jump θ := θ+ + θ−. A combination of (A) and (B) is also possible. (α) In both cases, the energy cost in the limit for a competitor with crack K and jump height θ + ¯θ is given… view at source ↗

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Works this paper leans on

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