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Cohesive fracture emerges from a phase-field model that degrades strength instead of stiffness, with an equivalent cohesive law independent of the regularization length and shear modulus.

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2026-08-03 12:31 UTC pith:3F5NSBFE

load-bearing objection The antiplane specialization is clean and the numerics are careful, but the ℓ-independent cohesive law is proven only on a formal concentration branch, not for all minimizers; worth reviewing with the Γ-limit gap made explicit. the 3 major comments →

arxiv 2607.29157 v1 pith:3F5NSBFE submitted 2026-07-31 cond-mat.mtrl-sci math-phmath.MPphysics.class-ph

Strength-degradation phase-field regularization of cohesive fracture: the antiplane case

Blaise Bourdin , Corrado Maurini This is my paper
classification cond-mat.mtrl-sci math-phmath.MPphysics.class-ph MSC 74R1074C0574G65
keywords phase-field fracturecohesive fracturestrength degradationantiplane shearcrack nucleationconic programmingsoftening plasticitysize effect
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that a phase-field model in which damage degrades the material's strength rather than its elastic stiffness produces, in antiplane shear, localized solutions that obey an exact equivalent cohesive law. The traction and opening are set by the strength and toughness alone, with no dependence on the regularization length or the elastic modulus, so strength, stiffness, and toughness become independent material data. The authors derive closed-form solutions, verify them numerically, and show that the model reproduces perfect plasticity, cohesive fracture, and brittle fracture as regimes of a single variational framework. If true, this would make the regularization length a purely numerical parameter and unify several classical fracture descriptions.

Core claim

In antiplane shear, the localized minimizers of the strength-degradation energy obey the equivalent cohesive law τ = k(α*)τc with opening JuK = −Gc/(cwτc) √(w(α*))/k'(α*), where α* is the maximal damage. The associated surface energy, φ(δ,α*) = k(α*)τcδ + Gc α̂(α*), is precisely the conjectured sharp-interface cohesive energy (5) and depends neither on the regularization length ℓ nor on the shear modulus µ. This gives a Barenblatt-type cohesive surface energy Φ(δ) with initial slope τc, plateau Gc, and a shape controlled by the constitutive pair (k,w).

What carries the argument

The central object is the energy functional Eℓ(u,p,α) = ∫ (µ/2|∇u−p|² + k(α)τc|p|) dA + (Gc/4cw)∫(w(α)/ℓ + ℓ|∇α|²)dA, where p is a plastic-like deformation whose linear-growth term is the support function of the strength disk of radius τc. In a simple shear bar, localized solutions concentrate p on a jump set and the damage profile satisfies the first integral ℓ²(α')² = w(α) − c0 with c0=0; combining this with the damage criterion on the jump set yields the equivalent cohesive law, parametrized by the maximal damage α*.

Load-bearing premise

The closed-form localized solution assumes all singular deformation concentrates on a codimension-one jump set with a damage profile satisfying the first integral with c0=0; if minimizers develop diffuse localization bands or intermediate-dimensional fracture sets, the equivalent cohesive law, its ℓ-independence, and the sharp-interface identification all fail.

What would settle it

Run the simple-shear bar with regularization length ℓ comparable to ℓch and measure the post-nucleation force–displacement curve: the predicted equivalent cohesive law (47) and its independence of ℓ would be contradicted if the traction–opening curve shifts measurably with ℓ. A sharper test: a two-dimensional antiplane simulation without an initial imperfection, observing whether damage localizes to a band of width scaling with ℓ (diffuse) or to a jump line of vanishing width as ℓ→0.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Strength, stiffness, and toughness become independent material data, with the regularization length ℓ acting purely numerically when ℓ ≪ ℓch = µGc/τc².
  • The global response of a bar is governed by the brittleness ratio L/ℓch: short bars fail with progressive cohesive softening, long bars undergo snap-back and nucleate a brittle crack.
  • At a V-notch under monotonic loading, the model replays small-scale yielding, a Barenblatt cohesive crack, and a Griffith brittle crack as successive regimes, without prescribing which regime applies.
  • When the nonlinear deformation p is reversible, the measured effective toughness equals Gc; imposing irreversibility on p leaves a plastic wake and raises the effective toughness to about 1.3Gc.
  • The equivalent cohesive law depends on ℓ and µ only through the no-snap-back condition, which selects the size-effect regime but does not alter the intrinsic surface energy.

Where Pith is reading between the lines

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

  • In the multiaxial case, the same construction should hold only for jump directions compatible with the strength domain; for strength domains bounded along the hydrostatic axis, opening cracks would be forbidden, restricting the unified framework to shear-dominated or suitably shaped strength surfaces.
  • For constitutive choices with w'(0)=0 (e.g. the classical quadratic w), the closed-form first integral with c0=0 is valid only up to corrections of order e^{−L/ℓ}; at small L/ℓ these corrections could re-introduce a weak ℓ-dependence in the effective cohesive law.
  • A direct test of the model's predictive content would be to prescribe a measured multiaxial strength surface and compare the predicted cohesive traction–separation shape (through k and w) with independent interface experiments.
  • The conic-programming numerical scheme is not limited to antiplane problems; it extends to vector-valued elasticity, where the jump-compatibility condition becomes the main new ingredient.

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

3 major / 5 minor

Summary. The paper analyzes a strength-degradation phase-field model, specialized to antiplane shear, as a regularization of cohesive fracture. For a one-dimensional simple-shear bar it constructs homogeneous and localized solutions, and from the localized branch derives an equivalent cohesive law: traction τ = k(α*)τc and opening JuK = −Gc/(cw τc)√(w(α*))/k′(α*), Eq. (47). The associated surface energy (51) is shown to be independent of both the regularization length ℓ and the shear modulus μ, with initial slope τc and plateau Gc. The paper then proposes an alternate-minimization scheme in which both subproblems are recast as second-order cone programs, and uses it to verify the equivalent cohesive law in simple shear, to measure effective toughness in a “surfing” problem, and to show that a V-notch tip passes through small-scale yielding, cohesive, and brittle regimes under monotone loading. The central claim is that strength, stiffness, and toughness are independent material data and that ℓ is purely numerical in the limit ℓ ≪ ℓch, unifying limit analysis, perfect plasticity, cohesive fracture, and brittle fracture in one variational framework.

Significance. If the sharp-interface identification is valid, the paper is a significant step for phase-field fracture: it gives a clean, parameter-free derivation of an equivalent cohesive law from a regularized model, with independent strength, stiffness, and toughness, and it provides reproducible numerical evidence for the ℓ-independence of the localized branch. The closed-form construction in Section 3 is transparent and the numerics reproduce the analytical curves to the stated accuracy, including a quantitative account of mesh-induced toughening. The conic-programming formulation is a practical contribution, avoiding smoothing or penalization of the non-smooth strength term. The main caveat is that the derivation from the global minimization problem is not complete: the localized solution is constructed under a concentration ansatz, and the continuum Γ-convergence of the exact functional to the conjectured cohesive energy is not established. Thus the significance is real but conditional on closing this gap or on reframing the claims as properties of the constructed branch.

major comments (3)
  1. [§3.2, Eqs. (15), (43), and the claim after (47)] The equivalent cohesive law rests on assumptions that are not consequences of the minimization problem (10)–(13): the singular strain is assumed to be exactly a codimension-one jump, Eq. (15), and the damage profile is taken to satisfy the first integral (43) with c0 = 0. The paper itself notes in the introduction that existence theory does not rule out Cantor-like fracture sets. The cited Γ-convergence result of Maggiorelli et al. (2025) is for a spatially discrete antiplane model, not for the continuum functional (9). As written, Eq. (47) and the ℓ-independence are properties of one stationary branch, not of the sharp-interface limit of all relevant minimizers. This is load-bearing because the abstract's claim that ℓ is a purely numerical parameter and the unification of cohesive and brittle fracture depend on the limit identification. I recommend either proving a continuum Γ-limit (or
  2. [§6.1, Fig. 8(c), and Remark 9] The numerical verification exercises the selected localized branch but does not establish that this branch is the global energy minimizer. In the simple-shear tests an initial imperfection α0 cos²(πx/L) selects the localization point and the bifurcation load depends on α0 (Fig. 8(c)), as the authors themselves show in Remark 9. The simulations therefore confirm the constructed solution, not the absence of lower-energy diffuse or intermediate-dimensional configurations. To make the sharp-interface claim quantitative, the paper should report energy comparisons between the computed state and the homogeneous/localized analytical energies (50)–(51) over a sweep of ℓ/ℓch and L/ℓch, or otherwise provide a lower-bound check.
  3. [§5.2, Algorithm 1] The alternate-minimization scheme is globally convergent only for the two convex subproblems at fixed α and fixed (u,p); the overall energy is not jointly convex. The stopping criterion monitors only the damage increment, and the algorithm may converge to different stationary points depending on initialization. This is not a flaw of the method, but the manuscript should state explicitly that the numerical results are local-minimizer paths, and that the branch selection is controlled by the initial imperfection and load increments. The current wording, especially in the conclusions, sometimes reads as if the numerical experiments validate global minimization of the original functional.
minor comments (5)
  1. [Eq. (45)] The symbol “q” in Eq. (45) is not defined; it appears to denote the jump Jα′yK of the normal derivative of α across the localization point. Use Jα′K consistently and define it.
  2. [§6.3, text near Fig. 18] The phrase “consistent with our one-dimensional analysis and the tearing simulations” references a “tearing” simulation that does not appear in the paper. This is presumably a typo for the simple-shear or surfing simulations; please correct it.
  3. [Abstract and §1] The abstract advertises “arbitrary convex strength surface,” but the paper treats only the isotropic antiplane disk. The multiaxial strength surface is delegated to the companion paper. I suggest a qualifying phrase such as “for the antiplane specialization” in the abstract to avoid overstatement.
  4. [§5.2 and Appendix A] The description “with maximal tolerances set to 10⁻⁸” is slightly ambiguous: it refers to the interior-point solver tolerances for the conic subproblems, not to the alternate-minimization tolerance tolAM. Please distinguish the two.
  5. [Figure 13] P and Q are defined in Figure 12 but used again in Figure 13; the caption should remind the reader of their definitions, since the two figures may be read independently.

Circularity Check

0 steps flagged

No circular reduction: the antiplane cohesive law is derived from optimality conditions, not from fitted inputs; companion-paper self-citations are contextual and not load-bearing.

full rationale

The central result, the equivalent cohesive law (47) and surface energy (51), is derived within this paper from the first-order optimality conditions: mechanical equilibrium (21b), the damage criterion on the jump set (45), and the first integral of the damage profile (43)-(46). No parameter is fitted to produce the law: traction τ=k(α*)τc and opening JuK=−Gc/(cwτc)√w(α*)/k′(α*) are obtained by eliminating the profile slope, and the ℓ-independence follows algebraically. The identification of (51) with the conjectured sharp surface energy (5) is a recognition that the same definition of α̂ has been used, not a reduction of the result to an assumed outcome. The numerical simulations verify the closed-form localized branch and quantify mesh-induced toughening; the only fits (e.g., Fig. 14 and Fig. 18) are diagnostic or visualization fits and do not set constitutive parameters. The paper does rely on the companion paper (Bourdin, Marigo, et al., 2025) for the origin of the model and for the three-dimensional context, and it cites that paper for the general conjecture; this self-citation is present but not load-bearing for the antiplane derivation, which is self-contained here. The paper itself flags the genuine mathematical gaps: the Γ-convergence result cited is spatially discrete (Maggiorelli et al., 2025), the continuum Γ-limit is not proven, and the localized solutions rely on an explicit concentration ansatz (Eqs. 14-15 and c0=0) rather than a proof that all relevant minimizers are of this form. Those are correctness/open-problem concerns, not circularity, because the derived law is a conditional stationary-point construction rather than a disguised restatement of the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central antiplane results are derived from energy (9) with no hidden fitted constants. The only hand-chosen knobs are the constitutive family M1's ζ, the residual strength kres, and numerical seeds; these do not encode the target cohesive law. The main unproven burden is the codimension-one concentration ansatz and reliance on a discrete-only Γ-convergence result, both of which are flagged in the text.

free parameters (2)
  • ζ (M1 shape parameter) = varied values: 0, 1/4, 2/3, 1, 0.75
    Constitutive shape parameter in k(α)=1−α, w(α)=(1−ζ)α+ζα². Chosen by hand; controls the shape of the cohesive law and the presence of a constant-stress plateau, but is not fitted to data.
  • kres (residual strength) = 10^-6 for simple shear, 10^-4 for surfing and V-notch
    Numerical residual added to k(α) to keep the SOCP well-posed as α→1; small and not part of the continuum model, but a numerical choice that could affect results near full damage.
axioms (3)
  • domain assumption Constitutive functions k,w satisfy Hypothesis 1, including monotonicity and (SH)/(SS) conditions (Eqs. 25–26).
    These monotonicity conditions are needed for the homogeneous and localized branches to be single-valued and are used throughout Sections 3 and 4. They are postulated model inputs, not derived.
  • ad hoc to paper Singular strain decomposition: ∇^S u = p^S = ⟦u⟧ n δ_Ju on a codimension-one jump set (Eqs. 14–15).
    The localized closed-form solutions and the equivalent cohesive law assume all singular deformation concentrates on a codimension-one set. The paper presents this as formal; it is not proven for minimizers of this exact functional in the continuum setting.
  • domain assumption Γ-convergence of the discrete antiplane model to the sharp cohesive energy (Maggiorelli et al., 2025).
    The identification of the ℓ→0 limit with the sharp cohesive model rests on this cited discrete result, which is not reproven here. The paper's own derivation is formal via optimality conditions.

pith-pipeline@v1.3.0-daily-deepseek · 41095 in / 13446 out tokens · 134346 ms · 2026-08-03T12:31:56.926750+00:00 · methodology

0 comments
read the original abstract

Phase-field approaches to fracture, initially designed as regularization of the Griffith model of brittle fracture, are now commonly viewed as gradient-damage models whose regularization length becomes a material property driving crack nucleation. One weakness of this approach is that the strength surface cannot be arbitrary: its shape is dictated by the elastic energy, and its magnitude by the regularization length. We focus on the antiplane version of the model introduced by Bourdin, Marigo, Maurini and Zolesi (arXiv:2506.22558), which handles crack propagation along unknown paths and nucleation governed by an arbitrary convex strength surface by degrading the strength instead of the stiffness. It can be interpreted as a regularization of softening plasticity in which localization bands obey an equivalent cohesive law set by the strength domain and the toughness, while the role of the regularization length, when small compared to the elasto-cohesive length, is purely numerical. Strength, stiffness, and toughness thus become independent material data, and limit analysis, perfect plasticity, cohesive fracture, and brittle fracture merge into a single variational framework. We derive closed-form solutions for a simple shear problem, propose a numerical scheme combining alternate minimization and conic programming, and numerically verify the equivalent cohesive law, its independence of the regularization, and the size effect governed by the elasto-cohesive length. A "surfing" simulation highlights the structure of the propagating crack while a re-entrant V-notch is used to show how the model bridges small-scale yielding, cohesive fracture, and brittle fracture without a priori hypotheses.

Figures

Figures reproduced from arXiv: 2607.29157 by Blaise Bourdin, Corrado Maurini.

Figure 1
Figure 1. Figure 1: 0 k(α) τc µ kεk 0 k(α) τc kτ k µkεk p = 0 p 6= 0 (a) 0 k(α) τc µ kεk − k(α) 2 τ 2 c 2µ 0 k(α) 2 τ 2 c ψ 2µ (ε, α) µ 2 kεk 2 slope k(α) τc (b) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Snap-back phase diagrams for the family M1 in the plane (ζ, ℓ/ℓch) for the homogeneous response (a) and (ζ, L/ℓch) for the localized response (b). The solid lines are the critical values ℓ/ℓch = ζ/(2cw) of (65) and Lc/ℓch = (1 + ζ)/(2cw) of (70), with cw = cw(ζ); at ζ = 1 they take the values ℓ/ℓch = 1 and Lc/ℓch = 2. The shaded regions correspond to snap-back of the corresponding branch under displacement… view at source ↗
Figure 3
Figure 3. Figure 3: Homogeneous (material-point) response for the family M1, for several values of ζ and length￾scale ratio ℓch/ℓ = 5: (a) normalized total energy density Whom/(τcεe) of (67) and (b) normalized stress τ /τc, versus the applied strain ε in units of the elastic-limit strain εe = τc/µ. The constant￾stress plateau εe ≤ ε < εc is present only for ζ < 1, and the softening branch snaps back for ζ = 0 and ζ = 1/4. Tur… view at source ↗
Figure 4
Figure 4. Figure 4: Localized solution for the family M1, for the same values of the shape parameter ζ as in [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simple shear, reference case: L/ℓch = 1, ζ = 1, ℓ/ℓch = 0.05, α0 = 10−3 , N = 80 load steps, for uniform meshes h/ℓ ∈ {0.1, 0.2, 0.5}. (a) Stress τ /τc versus end load t/εe; (b) total, elastic and dissipated energies normalized by GcH, against the exact solution (50). The damage and nonlinear deformation fields α and ∥p∥h after failure are shown in [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Damage α (top) and nonlinear deformation scaled by the mesh size, ∥p∥h (bottom), at the end of loading for the test of [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reference test of Figure 5 at [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Stress–strain response of the simple shear test: influence of (a) the brittleness [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Simple shear in the rigid limit µ → ∞ (ζ = 1, L τc/Gc = 1, ℓ/L = 0.05, h/ℓ = 0.05, α0 = 0, N = 80 load steps), for which the imposed load is the opening, t L = JuK: (a) equivalent cohesive law (68), (b) dissipated energy normalized by GcH, (c) displacement u along the bar; color scale: the load JuK τc/Gc. W H/2 L0 usurf = 2K µ r r 2π sin θ 2 on the outer boundary R u = t µλ (2πρ) λ−1ρ sin(λθ) on r = R (λ =… view at source ↗
Figure 10
Figure 10. Figure 10: Geometry and loading of the two antiplane test cases. Half-domain for the (left) Surfing, [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Crack-tip fields at the finest mesh (ℓ/ℓch = 1/6, h/ℓ = 0.1): damage α (top), plastic slip ∥p∥h (middle), normalized stress ∥τ ∥/τc (bottom), for the reversible (left) and irreversible (right) run. Note the residual plastic wake behind the irreversible tip. and a residual plastic wake is left behind, and dragged along with, the advancing tip. The two settings thus realize two distinct fracture phenomenolo… view at source ↗
Figure 12
Figure 12. Figure 12: 1-D field profiles along y = 0 (left) and the tip normal x = xP (right). Top: damage α, plastic slip ∥p∥h (log); bottom: displacement u, stress ∥τ ∥/τc. Solid: reversible, dashed: irreversible. we see that in both cases, the crack edges are stress-free while the stress attains its maximum near Q. We thus think of P as the “brittle crack tip” and of Q as the “cohesive crack tip”. The length of the cohesive… view at source ↗
Figure 13
Figure 13. Figure 13: Energy and crack-tip evolution vs. crack advance [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Effective toughness G eff c = dD/da versus h/ℓ (steady-state slope, one series per ℓ/ℓch; domain length W = 4 ℓch, mesh refined in a band of half-width ℓch/2 around the crack path) for reversible (left) and irreversible (right) plasticity. Dashed: fits G0 (1 + h/(4cwℓ)) with cw = 1 2 , the value of M1 at ζ = 1; red: the same law with G0 = Gc, i.e. the classical mesh-induced toughening. Bands and error bar… view at source ↗
Figure 15
Figure 15. Figure 15: Near-crack overview, ω = 179◦ (ℓ = 0.4, ℓch = 1, R = 10; tc = 1.616, t ∗ = 1.637). Top: (a) force F/(τcℓch) versus the GSIF t = KV , with tc = K∗ V (dashed), the Griffith load Kc (gray) and the snapshots A–C; (b) extents a, b of the strength zone against the SSY a0 = b0 (78), up to nucleation. Bottom: tip-zoom maps of α, ∥p∥ h/ℓch (log) and ∥τ ∥/τc at A–D (rows), D the first localized state. 32 [PITH_FUL… view at source ↗
Figure 16
Figure 16. Figure 16: Nucleated state at ω = 179π 180 (179◦ ; t = t ∗ = 1.637; crack length Lcrack = 1.83, tip process￾zone length ℓtip = 1.34). Top: tip-zoom maps as in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Blunt-wedge overview, ω = 5π 6 (150◦ ; ℓ = 0.4, ℓch = 1, R = 10; tc = 1.456, t ∗ = 1.475). Same panels, fields and colormaps as [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Critical load at the onset of the brittle crack, [PITH_FULL_IMAGE:figures/full_fig_p035_18.png] view at source ↗

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