{"id":"55c3e78b-31fe-4c27-bbd6-461b3a2e4296","arxiv_id":"2607.29157","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In antiplane shear, a strength-degrading phase-field model yields an ℓ-independent equivalent cohesive law with independent strength and toughness, verified by closed-form solutions and simulations.","lead":"This paper analyzes a phase-field fracture model that degrades material strength rather than stiffness, in antiplane shear, and shows it produces an equivalent cohesive crack law whose parameters are strength and toughness, independent of the regularization length. The result matters because it unifies plastic yielding, cohesive cracks, and brittle fracture in one variational framework, removing a known weakness of standard phase-field models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The equivalent cohesive law (47) rests on a concentration ansatz and a discrete-only Γ-convergence result; without a continuum Γ-limit proof for (9), the ℓ-independence may fail for global minimizers.","rationale":"Good faith reading: the paper makes a specific, strong claim that the strength-degradation model admits an equivalent cohesive law with independent τ_c and G_c, ℓ being a numerical parameter. The construction in §3.2 is algebraically correct and the closed-form results are verified by careful numerical experiments in §6. I give credit for the parameter-free closed-form cohesive law, the mesh-toughening quantification, and the explicit acknowledgment in §1 that the existence theory does not rule out intermediate-dimensional fracture sets. However, the central claim has two layers: (i) the localized solution satisfies the optimality conditions, and (ii) it represents the effective behavior of the model, including the limit ℓ→0. Layer (i) is demonstrated; layer (ii) is not, because the continuum Γ-convergence of (9) to (4) is not proven and the only cited Γ-result is discrete. This does not invalidate the paper, but it makes the unification claim conditional on a missing global lower bound. The reader's weakest_assumption identifies exactly this gap, and I agree. The numerical tests support the constructed branch but cannot rule out other branches; the proposed Γ-analysis (or the 1D special case) would settle the matter. Therefore the reader's CONDITIONAL verdict is appropriate; no change.","tokens_in":41570,"tokens_out":15116,"duration_ms":161897,"concrete_test":"Perform a rigorous Γ-convergence analysis of the continuum antiplane functional (9), following Dal Maso-Orlando-Toader (2016) and Conti-Focardi-Iurlano (2016). Concretely, prove the liminf inequality for all bounded-energy sequences (u_ℓ,p_ℓ,α_ℓ), and construct recovery sequences realizing E_0 of (4). The critical check is that the lower bound integrates to ∫_{J_u} [k(α(α̂))τ_c|⟦u⟧|+G_c α̂] dS with no negative cross-term from p concentrated inside the damage transition profile. A tractable first case is the 1D bar with ζ=1: the Γ-limit should be τ_c|δ|−τ_c^2 δ^2/(4G_c) for |δ|≤2G_c/τ_c and G_c beyond; any deviation (e.g., an extra O(1) term) would invalidate Eq. (47).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2 constructs the localized solution by assuming the singular strain is exactly a codimension-one jump (Eq. 15) and that the damage profile is the first-integral solution (43) with c0=0. These assumptions are not consequences of the minimization problem (10)-(13); they yield a stationary point, not a proof that all relevant minimizers have this form. The paper's introduction explicitly notes that existence theory does not rule out minimizers whose 'fracture' is a Cantor-like set of intermediate dimension. Moreover, the continuum Γ-convergence of the exact functional (9) to the sharp cohesive energy (4) is not established; the cited Maggiorelli et al. (2025) result is for a spatially discrete antiplane model. If, as ℓ→0, a sequence with p_ℓ supported on a set of intermediate dimension (or with a non-vanishing diffuse part) has limit energy strictly below the recovery value inf_α ϕ(δ,α) of (51), then (47) would not be the effective cohesive law, and the claim that ℓ is a purely numerical parameter would be unsupported. The numerical experiments in §6 verify the selected branch (using an initial imperfection in the simple-shear test) but do not explore alternative lower-energy configurations or establish a global lower bound. Because the abstract's unification of cohesive and brittle fracture depends on the sharp-interface limit being the true Γ-limit, this gap is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":41921,"tokens_out":5167,"duration_ms":62223,"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":[{"comment":"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","section":"§3.2, Eqs. (15), (43), and the claim after (47)"},{"comment":"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.","section":"§6.1, Fig. 8(c), and Remark 9"},{"comment":"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.","section":"§5.2, Algorithm 1"}],"minor_comments":[{"comment":"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.","section":"Eq. (45)"},{"comment":"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.","section":"§6.3, text near Fig. 18"},{"comment":"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.","section":"Abstract and §1"},{"comment":"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.","section":"§5.2 and Appendix A"},{"comment":"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.","section":"Figure 13"}],"recommendation":"major_revision","confidential_remarks":"The central concern raised by the stress-test note is valid and lands on a load-bearing point: the equivalent cohesive law and the ℓ-independence are derived for a formal concentration ansatz, while the continuum Γ-limit for (9) is not proven and the cited Γ-convergence result is discrete. This may be fixable within the manuscript's scope by adding a Γ-convergence or lower-bound argument, or by carefully restricting the claims to the localized branch; the numerics are otherwise strong and the closed-form analysis is clean. I would not reject, but the abstract and conclusions currently claim more than the paper establishes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper gives a careful antiplane analysis of the strength-degradation phase-field model from Bourdin, Marigo et al. 2025, and the equivalent cohesive law (47) is real and well verified. The catch is that it is derived on a localized branch that is stationary but not shown to be the global minimizer, and the paper cites only a spatially discrete Γ-convergence result, not a continuum one. So the unification narrative is a conjecture supported by strong evidence, not a theorem.\n\nWhat's actually new: the closed-form homogeneous and localized solutions, the explicit traction-separation law with initial slope τc and plateau Gc, independent of ℓ and µ; the SOCP alternating-minimization scheme that avoids smoothing; and the numerical verification in the simple-shear, surfing, and V-notch problems. The numerics match the analytical curves to the stated accuracy, the mesh-induced toughening is quantified and corrected, and the size effect in L/ℓch is reproduced. The rigid-limit test that measures the cohesive law directly is a nice touch.\n\nThe soft spots are the ones the paper itself flags. The localized solution assumes all singular deformation concentrates on a codimension-one jump (Eqs. 14–15) and uses the first integral with c0=0. The introduction explicitly notes the existence theory doesn't rule out Cantor-like fractures. That means the Γ-limit of the exact functional is open; the cited Maggiorelli et al. result is for a spatially discrete antiplane model. The stress-test note is right that this gap is load-bearing if you read the paper as claiming a full sharp-interface limit. The authors are more careful than their abstract—they say 'conjectured limit' in Eq. (4)—so I'd call it a scoping problem rather than an error. Also minor: no code or data shipped, and the V-notch nucleation overshoot is acknowledged but not fully resolved.\n\nBottom line: this is a solid paper with a worthwhile contribution. The antiplane equivalent cohesive law will be cited. It deserves peer review, with a request to release the numerical artifacts and to state explicitly what is proven versus conjectured about the continuum Γ-convergence. I'd bring it to the reading group to get a second opinion on whether the formal ansatz is a real threat to the ℓ-independence claim.","headline":"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.","tokens_in":42383,"tokens_out":2461,"would_cite":true,"duration_ms":27075,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74R10","74C05","74G65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["phase-field fracture","cohesive fracture","strength degradation","antiplane shear","crack nucleation","conic programming","softening plasticity","size effect"],"falsifier":"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.","tokens_in":41423,"feed_emoji":"","tokens_out":3736,"duration_ms":40072,"temperature":0.7,"pith_summary":"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.","feed_headline":"One fracture model unifies plastic, cohesive, and brittle regimes","feed_subtitle":"Antiplane analysis gives an equivalent cohesive law independent of regularization length and shear modulus.","key_machinery":"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*.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Strength-degradation model unifies all fracture regimes","Equivalent cohesive law independent of regularization length","Same variational framework: plastic, cohesive, brittle","Antiplane shear bridges small-scale and brittle fracture","One model, one energy: cohesion from damage gradient"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Strength-degradation model unifies all fracture regimes","Equivalent cohesive law independent of regularization length","Same variational framework: plastic, cohesive, brittle","Antiplane shear bridges small-scale and brittle fracture","One model, one energy: cohesion from damage gradient"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1310,"prompt_tokens":813,"completion_tokens":497,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":440}},"tokens_in":557,"tokens_out":497,"duration_ms":5814,"temperature":1.0,"reasoning_tokens":440,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:31:56.926750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}