{"id":"4b5522b1-ed8a-460c-80e2-0f5fa8ed039f","arxiv_id":"2603.21811","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"Eigenstrain-based cohesive phase-field fracture is recast as a plasticity-style local return map with consistent tangents, enabling standard FE implementation and mesh/length-scale-independent brittle-to-cohesive response.","lead":"A phase-field fracture model with explicit material strength and cohesive behaviour is rewritten as a local return-mapping law at each integration point, needing no extra global unknowns. Existing finite-element codes can therefore adopt cohesive nucleation and mixed-mode cracking without specialized global optimizers.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates residual regularisation as the softest modelling choice while recognising that it does not overturn the algorithmic contribution. The mathematics is standard continuum mechanics, the consistent tangents are derived carefully (including the Schur-complement treatment of the poorly conditioned multi-mode system), and the three benchmarks plus open code supply independent, reproducible support. Because the strongest claim is about the feasibility and correctness of a local return-mapping reformulation rather than about a new physical theory, residual-parameter sensitivity and the low-Gc compressive mode remain minor caveats. No stronger load-bearing concern emerges on re-examination; the ACCEPT verdict with high confidence therefore stands.","tokens_in":19192,"tokens_out":490,"duration_ms":5073,"concrete_test":"Re-run the plate-with-hole tension and compression cases of Sec. 4.1 with κ_t raised from 10^{-9} to 10^{-6} (and, separately, κ from 10^{-3} to 10^{-2}); if peak loads and the brittle-to-cohesive transition with Gc remain within a few percent of the published curves, residual-parameter sensitivity is confirmed to be non-critical for the central claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is algorithmic and implementation-focused: because fracture eigenstrains carry no spatial gradients, their evolution can be treated as a local return-mapping constitutive model (Eqs. 23–30), so the cohesive phase-field needs no extra global degrees of freedom beyond a standard phase-field formulation. That claim is supported by closed-form consistent tangents for both the non-smooth r1 and smooth Drucker-Prager-like surfaces, three standard benchmarks that exhibit mesh- and length-scale-independent load-displacement response, and openly released FEniCSx source. The residual parameters κ and κ_t (Table 1, Eqs. 6 and 23b) are acknowledged regularisation devices whose precise values affect only post-peak residual transfer and conditioning; they do not undermine the local-return-mapping construction itself. The unexplained low-Gc compressive mode (Sec. 4.1.2) is a secondary modelling observation, not a contradiction of the implementation claim. No load-bearing inconsistency or missing derivation was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper reformulates the eigenstrain-based cohesive phase-field energy of Vicentini et al. so that fracture eigenstrains, which carry no spatial gradients, are evolved by a local return-mapping constitutive model at each integration point. Two strength surfaces are treated (non-smooth r1 with independent tensile/shear strengths, and a smooth Drucker-Prager-like surface with compressive strengthening controlled by ε_ref), closed-form consistent tangents are derived (including Schur-complement forms for multi-mode activation), and residual strength/stiffness parameters are introduced for conditioning. The scheme uses only the usual displacement and phase-field global degrees of freedom and is implemented in FEniCSx. Three plane-strain benchmarks (plate with hole under tension/compression, single-edge notched shear, dynamic notched plate) are used to argue mesh- and length-scale-independent load-displacement response, Gc-controlled brittle-to-cohesive transition, and natural dynamic branching. Source code is released.","tokens_in":19498,"tokens_out":1388,"duration_ms":26331,"significance":"If the local return-mapping is equivalent to the parent variational energy and the reported mesh/length-scale independence holds, the contribution is practically important: it removes the main barrier to using eigenstrain-based cohesive phase-field models inside standard finite-element codes without extra global fields or symbolic global optimizers. Explicit consistent tangents for both smooth and non-smooth surfaces, open FEniCSx sources, and standard benchmarks that separate nucleation (strength surface) from propagation (Gc) are concrete strengths. The work is primarily algorithmic and implementation-focused rather than a new fracture theory, but that is a legitimate and useful contribution for computational solid mechanics.","major_comments":[{"comment":"The central claim is that the local return-mapping (Eqs. 23–30) recovers the stationarity conditions of the Vicentini et al. energy without extra global DOFs. The derivation from the same energy is clear, but Section 4 never reports a side-by-side comparison (load-displacement curves, crack paths, or energy dissipation) against a global energy-minimization solution of the same energy on at least one shared benchmark. A short quantitative check would substantially strengthen the equivalence claim that the paper is built on.","section":null},{"comment":"Table 1 and Eqs. (6), (23b) introduce residual strength κ = 10^{-3} and residual stiffness κ_t = 10^{-9} solely for well-posedness once the surface is fully degraded. These parameters are free regularisation devices; the manuscript does not show how peak load, post-peak residual transfer, or Newton convergence change when they are varied over even one order of magnitude. For a methods paper aimed at drop-in use in existing codes, a brief sensitivity note (or a recommended range) is needed so that practitioners know what is material and what is numerical.","section":null},{"comment":"Section 4.1.2 documents a low-Gc compressive failure mode (horizontal crushing then delayed diagonal localization) that is robust to time-step, length-scale, staggered iterations, and both strength criteria, yet is left as a plausible stress-field explanation without further diagnostics (e.g., driving-force maps or comparison to an analytical hoop-stress estimate). Because this mode changes the qualitative crack path for the same strength surface, the paper should either (i) demonstrate that it is the intended continuum response of the energy or (ii) state clearly that it is a modelling limitation of the chosen potentials under mixed tension-shear around a hole.","section":null}],"minor_comments":[{"comment":"Nomenclature lists both κ (residual strength) and κ_t (residual stiffness); the abstract and introduction sometimes use “residual stiffness” language familiar from standard phase-field. A single clarifying sentence early in Section 2 would avoid confusion with the classical residual-stiffness parameter.","section":null},{"comment":"Figure 3 and related load-displacement plots use ε_yy on the abscissa without stating whether it is the applied far-field strain U_ext/H or a local measure; please define consistently in the captions.","section":null},{"comment":"The multi-pass staggered scheme (max 5 passes) and Newmark parameters (β = 0.5625, γ = 1.0) are stated but not justified beyond numerical damping. A short remark on why five passes suffice for the reported residuals would help reproducibility.","section":null},{"comment":"Eq. (18a) inserts tr(ε) into F_d under compression while η remains the internal variable; a one-line remark that this is intentional (because tr(η) ≥ 0 by construction) would make the non-standard dependence on total strain easier to follow.","section":null},{"comment":"In Section 4.3 the crack-length proxy L_crack = ∫ γ(φ) dΩ is correctly noted to underestimate length when φ < 1; consider also reporting a simple iso-φ contour length for the more cohesive cases so that branching trends remain comparable across Gc.","section":null},{"comment":"Typos / polish: “Theresultingload-displacementbehaviour” and similar missing spaces appear in Section 4.2; “Allcrackspropagate” likewise. A pass for spacing and hyphenation would help.","section":null},{"comment":"Data availability points to a public GitHub repository; please pin a release tag or commit hash corresponding to the submitted manuscript so that the exact results remain reproducible after future code changes.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid methods/implementation paper with open code; the main novelty is algorithmic rather than theoretical. I do not see a load-bearing error. The three major points are fixable with limited additional work (one comparison plot, a short residual-parameter note, and clearer framing of the low-Gc compressive mode). Scope fits a computational-mechanics / CE venue well. No concerns about citation pattern or authorship."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this paper turns the recent eigenstrain cohesive phase-field idea into something you can actually drop into a standard staggered FE code. Because the eigenstrains have no spatial gradients, Hageman treats their evolution as a local return-mapping constitutive model (plasticity-style residuals and consistent tangents, including a Schur-complement treatment of the non-smooth multi-mode surface). No extra global unknowns. That is the real contribution; the energy itself is taken from Vicentini et al. and Bourdin et al.\n\nWhat it does well is the engineering. Closed-form tangents for both the non-smooth r1 surface and a new smooth Drucker-Prager-like criterion are derived carefully, residual stiffness is introduced to keep the tangent well-conditioned once the surface is active, and three standard benchmarks (plate with hole, SEN shear, dynamic branching) show mesh- and length-scale-independent load-displacement curves, Gc controlling the brittle-to-cohesive transition, and natural branching. Full FEniCSx source is released. That combination of derivation + evidence + code is solid.\n\nSoft spots are minor and acknowledged. Residual strength κ and residual stiffness κ_t are hand-tuned regularisers; they affect only post-peak residual transfer and conditioning, not the local-return-map claim itself. The low-Gc compressive mode that starts horizontal before localising into shear bands is unexplained but consistent across refinements and both criteria, so it is a modelling observation rather than a bug. Free parameters (Newmark, damping, ε_ref) are the usual ones for this class of problem.\n\nThis is for people who already run phase-field fracture and want a practical cohesive model with an explicit strength surface. The central algorithmic claim is supported by the equations and the numerics. I would send it to peer review without hesitation; a serious referee will improve the residual-parameter discussion and the low-Gc mode note, but the work is already usable.","headline":"Clean, usable local return-map that finally puts the Vicentini/Bourdin eigenstrain cohesive model into ordinary FE codes; the math and benchmarks hold up.","tokens_in":20074,"tokens_out":500,"would_cite":true,"duration_ms":5460,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Fracture eigenstrains can be returned locally at each integration point, so cohesive phase-field fracture needs no extra global degrees of freedom.","keywords":["cohesive fracture","phase-field","eigenstrains","crack nucleation","finite element method","return mapping","strength surface"],"falsifier":"Run the plate-with-hole benchmarks while systematically varying residual strength and residual stiffness over several orders of magnitude; if peak load, post-peak residual force, or Newton convergence change appreciably, the claim that the residuals are merely numerical safeguards is false.","tokens_in":20078,"feed_emoji":"🔩","tokens_out":616,"duration_ms":6242,"temperature":0.7,"pith_summary":"Standard phase-field fracture models inherit two limitations from brittle fracture theory: they do not prescribe a material strength for crack nucleation, and they struggle to produce true cohesive unloading. Recent formulations fix both problems by introducing fracture eigenstrains that decouple strength from fracture energy, but those formulations have so far required global energy-minimization solvers and extra field variables. This paper shows that the eigenstrains need no spatial gradients, so their evolution can be treated exactly like a plasticity return map evaluated at each integration point. The resulting cohesive model uses only the ordinary displacement and phase-field degrees of freedom, supplies consistent tangents for both a non-smooth tensile-shear surface and a smooth pressure-sensitive surface, and produces mesh-independent, length-scale-independent load-displacement curves in which fracture energy alone controls the brittle-to-cohesive transition and dynamic branching appears without extra criteria.","feed_headline":"Cohesive phase-field cracks need no extra global unknowns","feed_subtitle":"Eigenstrains return-map at each integration point, decoupling strength from energy","key_machinery":"The local eigenstrain return map: residual equations that keep the stress on the chosen strength surface (non-smooth r1 or smooth Drucker-Prager-like) together with closed-form consistent tangents obtained via the Schur complement, all evaluated pointwise and inserted into the ordinary staggered phase-field Newton scheme.","core_discovery":"Because fracture eigenstrains carry no spatial gradients, their evolution can be resolved entirely at the integration-point level by a local return-mapping scheme analogous to plasticity. Consequently a cohesive phase-field model that already carries an explicit strength surface can be implemented inside any standard finite-element code without additional global unknowns, while still decoupling nucleation strength from fracture energy and recovering mesh- and length-scale-independent structural response.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Eigenstrain return-mapping yields cohesive phase-field without extra unknowns","Local plasticity-like scheme embeds strength surfaces in phase-field fracture","Cohesive cracks via integration-point eigenstrain evolution alone","Standard FE codes host cohesive phase-field through local return maps","Decoupled strength and energy via local eigenstrain return-mapping"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Two small residual parameters (strength floor and post-fracture stiffness) must be inserted by hand to keep the damaged strength surface and tangent matrix well-conditioned; their values are not fixed by material data.","fun_headline_variants_meta":{"raw":{"variants":["Eigenstrain return-mapping yields cohesive phase-field without extra unknowns","Local plasticity-like scheme embeds strength surfaces in phase-field fracture","Cohesive cracks via integration-point eigenstrain evolution alone","Standard FE codes host cohesive phase-field through local return maps","Decoupled strength and energy via local eigenstrain return-mapping"]},"model":"grok-4.5","effort":"low","cost_usd":0.006942,"raw_usage":{"total_tokens":1723,"prompt_tokens":849,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":69420000,"prompt_tokens_details":{"text_tokens":849,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":803,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":849,"tokens_out":71,"duration_ms":7152,"temperature":1.0,"reasoning_tokens":803,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T20:36:31.910759+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Run the plate-with-hole benchmarks while systematically varying residual strength and residual stiffness over several orders of magnitude; if peak load, post-peak residual force, or Newton convergence change appreciably, the claim that the residuals are merely numerical safeguards is false.","supporting_citations":[],"review_version":1}