{"id":"f2687726-5f23-4614-a469-7b275c5629a0","arxiv_id":"2607.13599","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Wave–crack interaction in dynamic phase-field fracture is governed by the regularization-length-to-wavelength ratio, and a strength-degrading cohesive model—unlike stiffness- or stiffness+density-degrading brittle models—recovers sharp-crack response under conditions the paper makes quantitative.","lead":"Dynamic phase-field fracture models distort elastic waves at cracks: this paper shows the distortion is intrinsic, controlled by the ratio of the regularization length to the wavelength, not a numerical artifact. It also extends a strength-degrading cohesive phase-field model to dynamics and derives an analytical crack-opening law for it.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central ℓ/λ thresholds are established with a no-split TMM but applied to a strain-split FEM; the acknowledged qualitative-only bridge leaves the quantitative mechanism unverified.","rationale":"The reader's stated weakest assumption—that the eigenstrain carries no inertia—is a modeling premise rather than an internal flaw; within the proposed cohesive model it is consistently derived from the action functional, and no evidence suggests it fails. The more pressing concern is the evidence bridge for the central quantitative claim. The paper's own acknowledgment that the FEM/TMM comparison is only qualitative means the specific ℓ/λ thresholds and the impedance-based explanation are not directly verified for the strain-split model used in the numerical demonstrations. This does not overturn the qualitative conclusion, but it justifies the reader's conditional verdict, which I recommend keeping unchanged pending a quantitative check.","tokens_in":42231,"tokens_out":18537,"duration_ms":212183,"concrete_test":"First validate the numerical pipeline: run the fixed-phase-field 1D setup of §2.1.5 with the no-split equation (16) for a long quasi-monochromatic tone burst, extract reflected/transmitted power coefficients from the FEM time histories, and check that they collapse onto the TMM curves in Fig. 2a. Then repeat with the same α(x), mesh, and tone burst using the actual strain-split model (20). If the split-model coefficients deviate by more than about 10% in the ℓ/λ range of interest, the stated thresholds are not representative of the model used in the paper's FEM results; if they agree, the impedance mechanism is confirmed for the split model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.1.3 derives the governing parameter ℓ/λ and the TMM reflection/transmission thresholds (Fig. 2a) from the scalar wave equation (16), in which degradation multiplies the full strain. Section 2.1.5 then tests the actual brittle model, whose 1D form uses the tension–compression split (20), so stiffness degradation is applied only where u′>0. The paper explicitly states at the end of §2.1.5 that the FEM/TMM comparison is 'only qualitative'. This gap is load-bearing because the central claim is that the observed oscillations and transmission are governed by the acoustic-impedance profile and by ℓ/λ, with specific thresholds (≲0.08 reflection, ≳0.221 transmission). Once incident and reflected waves superpose, both strain signs coexist in the phase-field support; the effective stiffness profile becomes solution-dependent, and the moving tension/compression interface can itself generate reflections and oscillations. Without a quantitative FEM/TMM comparison for the split model, it is not established that the impedance mechanism—rather than the split-induced nonlinearity—produces the reported FEM response, nor that the stated thresholds apply to the model actually used in the simulations. The qualitative conclusion (not purely numerical artifacts) may survive, but the quantitative mechanism is the weakest link.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes three phase-field formulations for dynamic fracture in 1D and 2D. For brittle stiffness-degradation models, it argues—through WKB/scattering analysis, transfer-matrix computations, and FEM experiments with two time integrators and two mesh resolutions—that high-frequency oscillations and phase-field widening in wave-crack interaction are intrinsic consequences of the regularization, governed by the ratio ℓ/λ of regularization length to wavelength. It shows that adding density degradation restores constant wave speed but violates mass balance and still fails compressive-wave transmission. The paper then extends a recently proposed eigenstrain-based cohesive phase-field model (strength degradation only) to elastodynamics, derives a condition (ℓ/λ, σ̃/σc) for sharp-crack-like response, and derives a closed-form dynamic cohesive opening law governed by c0/ℓch, verified against FEM. A 2D pre-notched plate benchmark compares branching behavior.","tokens_in":42550,"tokens_out":13553,"duration_ms":153582,"significance":"If the claims hold, the paper clarifies a long-debated issue: the oscillations reported in dynamic phase-field fracture are not merely numerical artifacts but an intrinsic feature of stiffness-degrading regularization, and the proposed cohesive formulation offers a principled alternative. The manuscript has notable strengths: the scattering analysis is self-contained, the transfer-matrix predictions are falsifiable, the analytical opening law is checked against direct FEM solutions of the same PDE system (Fig. 27), the numerical studies include systematic mesh and time-integration checks, and the implementations are publicly available. These features make the paper a potentially valuable contribution to computational fracture mechanics. However, the quantitative threshold claims for the brittle model rest on an unsplit surrogate, and the cohesive model's predictions rest on a rate-independent elimination of the eigenstrain that is stated but not discussed in depth; these caveats need attention.","major_comments":[{"comment":"The TMM thresholds in Fig. 2 (ℓ/λ≈0.08 and 0.221) are computed for the unsplit heterogeneous wave equation (16) with the fixed optimal AT1 profile, whereas the FEM simulations of §2.1.5 use the tension–compression split (20), as the paper itself states ('only qualitative'). This gap is load-bearing for the quantitative claim that the wave-crack interaction is governed by the impedance profile with these specific thresholds. Since incident and reflected waves superpose during the interaction, both signs of u′ coexist in the phase-field support; in the split model the effective stiffness profile becomes solution-dependent, so the impedance profile from the TMM is not the one actually experienced by the FEM solution. Moreover, the pulse (21) is a truncated sinusoid, so its spectral content is not captured by the monochromatic TMM parameter ℓ/λ̃. I therefore do not consider the quantitative","section":"§2.1.3 / §2.1.5, Eq. (16), (20), Fig. 2 & 5"},{"comment":"The dynamic extension of the cohesive model eliminates the eigenstrain η by invoking, in the §3.1 footnote, that 'the eigenstrain carries no inertia' and that pointwise stationarity coincides with quasi-static optimality. This is a strong modeling assumption, and every subsequent cohesive result—the condensed energy (46), the diffuse-jump criterion (59), and the opening law (60)-(62)—inherits it. The manuscript does not discuss the physical scope of this assumption (e.g., what a finite relaxation time for η would change, or why zero inertia is the appropriate limit for a fracture process). Because this is the first dynamic extension of this cohesive model, the authors should state this limitation explicitly in the conclusions and, if possible, support the rate-independent elimination by an asymptotic argument. As written, the 'dynamic cohesive law' is dynamic only in the displacement fie","section":"§3.1 footnote / Appendix D, Eq. (46)"},{"comment":"The no-diffuse-jump condition (56) is converted into the regime diagram of Fig. 13b using the stress envelope (57) and the strength profile (58). Equation (57) is stated without derivation; it is not the incident-wave maximum, since for a free-end reflection of a half-sine pulse the total (incident+reflected) stress at a fixed point has this form. The derivation should be given, including the assumptions (half-sine pulse, no secondary reflections, α frozen at its initial profile before the criterion is violated). The figure caption calls (b) 'numerically obtained limits', but it is obtained by solving the inequality analytically; this is misleading. Since Fig. 13b defines the regime in which the cohesive model is claimed to recover sharp-crack behavior, this is a central element and needs a rigorous derivation or a direct FEM verification along the boundary.","section":"§3.4, Eqs. (57)–(59), Fig. 13"}],"minor_comments":[{"comment":"The relationship between the nominal wavelength λ̃ of the truncated sinusoid and the monochromatic λ of the TMM deserves a word; the pulse's spectral width is probably why FEM at ℓ/λ̃=0.075 still shows visible transmission although the monochromatic TMM would place it in the near-full-reflection regime.","section":"§2.1.5, Eq. (21), Fig. 5"},{"comment":"The footnote could be more precise: the action (6) contains no kinetic term for η by construction; this is a modeling choice, not a consequence of the quasi-static model. Please phrase it that way.","section":"§3.1 footnote"},{"comment":"Replace 'numerically obtained limits' with 'analytically obtained from (59)' to avoid implying a parameter sweep over FEM solutions.","section":"Fig. 13 caption"},{"comment":"The 2D comparison uses ℓ/∆x≈5 only. Given the 1D sensitivity to discretization, a brief mesh-convergence statement for the branching patterns would increase confidence.","section":"§4.2.1, Figs. 20–23"},{"comment":"The notation λ̃ vs λ is used inconsistently between the pulse section and the TMM section; please unify and clarify which quantity is being used in each threshold.","section":"Eq. (57) / §2.1.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be a significant contribution after major revision, provided the quantitative threshold claims are either verified for the split model or reformulated as qualitative statements, and the rate-independent eigenstrain assumption is discussed as a limitation. The 2D comparison is compelling but should not be overinterpreted given the single realization and lack of a resolution study."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper makes a real contribution. It shows that the high-frequency oscillations and phase-field widening seen in dynamic brittle phase-field models are not purely numerical artifacts but consequences of the stiffness-degrading regularization, governed by ℓ/λ through the acoustic impedance profile. That is new relative to Steinke, Schlüter, and Durussel, who chalked it up to numerics. The negative result on density degradation—it restores a constant wave speed but leaves a stronger impedance well and violates mass conservation—is a useful corrective to the Chen/Tian/Karma line.\n\nWhat the paper does well: the WKB/TMM analysis is clean and exact within its assumptions, and it separates the roles of wave speed and impedance nicely. The mesh-refinement study (ℓ/Δx≈5 vs. 50, two time integrators) supports the claim that the behavior is intrinsic, not discretization-induced. The dynamic extension of the authors' cohesive model is a genuine step forward: the analytical opening law (ODE (60), solution (61)-(62)) is a real derivation, not a fit, and it matches FEM in Fig. 27. Code is public, and the self-citation to [39,40] is legitimate.\n\nThe soft spots, in proportion: the stress-test note is right that the quantitative thresholds (ℓ/λ≈0.08, 0.221) come from TMM on the unsplit wave equation, while the FEM model uses the tension–compression split. The paper explicitly says the comparison is \"only qualitative.\" That honesty cuts in its favor, but it does mean the stated numbers are indicative rather than exact for the model actually simulated. The impedance mechanism almost certainly carries over, but a quantitative split-model TMM/FEM match would harden the central claim. Not fatal, but the quantitative claim is shakier than the qualitative one.\n\nThe inertia-free elimination of the eigenstrain is an assumption. If η carried inertia or rate-dependent evolution, the cohesive law and ODE would change. The paper states the stationarity argument; that's defensible, but it should be flagged more prominently. The 2D branching results rest on single runs, and the density-degraded model needed h0=10⁻² hand-raised—minor. The regularity violation when shock waves emerge is acknowledged and deferred, which is acceptable as a limitation.\n\nWho this is for: anyone working on dynamic phase-field fracture, especially on wave–crack interaction and on alternatives to stiffness degradation. I'd take it seriously in review. The revisions should address the split-model gap and the eigenstrain assumption head-on, but the core contribution is solid.","headline":"Solid, useful paper: it gives a credible mechanistic explanation for the oscillations and widening in dynamic phase-field fracture, and a new dynamic cohesive model with an analytical opening law; the TMM/FEM split-model gap and the inertia-free eigenstrain assumption are the main soft spots.","tokens_in":43076,"tokens_out":2447,"would_cite":true,"duration_ms":28697,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Phase-field regularization itself, not numerics, causes oscillations and widening in dynamic brittle fracture; a cohesive variant recovers sharp-crack behavior.","keywords":["phase-field fracture","dynamic fracture","cohesive fracture","elastodynamics","wave-crack interaction","acoustic impedance","crack branching","eigenstrain"],"falsifier":"Compute the reflection and transmission power coefficients for a fixed phase-field crack as a function of ℓ/λ, e.g. by transfer-matrix or spectral simulations; if the transition does not fall near ℓ/λ ≈ 0.08–0.221, the impedance-based explanation is wrong.","tokens_in":42003,"feed_emoji":"⚡","tokens_out":7837,"duration_ms":79488,"temperature":0.7,"pith_summary":"The paper examines what happens when an elastic wave strikes a pre-existing crack in phase-field models of dynamic fracture. It argues that for the standard brittle model with stiffness degradation, the high-frequency oscillations and widening of the damaged band are inherent consequences of the regularization, not numerical artifacts, and that they are controlled by the single ratio ℓ/λ of regularization length to elastic wavelength, acting through the acoustic-impedance profile. It also shows that degrading the mass density along with stiffness does not fix the problem and violates mass conservation. The paper then extends a strength-degrading cohesive phase-field model to elastodynamics and shows it preserves the sharp-crack wave response under a condition on ℓ/λ and the stress amplitude ratio σ̃/σc, under which it further derives an analytical dynamic cohesive opening law governed by c0/ℓch. The 2D benchmarks indicate the cohesive model branches like the brittle stiffness-degraded model, while the density-degraded model fails to branch.","feed_headline":"One ratio, ℓ/λ, decides how a phase-field crack scatters waves","feed_subtitle":"Standard brittle models distort dynamic fracture unless ℓ/λ is tiny; a strength-degrading cohesive model fixes it.","key_machinery":"The key object is the acoustic impedance profile Z(α) = sqrt(g(α)E0ρ0) and the WKB adiabaticity parameter δ(x) = |Z'(x)|/Z(x) · c(x)/ω. For the optimal AT1 phase-field profile, δ tends to λ/(2πℓ) near the crack center, making ℓ/λ the single governing parameter that separates reflecting (sharp-crack-like) from transmitting (homogeneous-like) regimes. For the cohesive model, the key machinery is the eigenstrain field η and the strength-degradation function a(α)=1−α, which leave the bulk wave equation unchanged; the condition (59) ensures the strength criterion is met only at the crack, and the complementary relations yield the linear ODE (60) whose solution is the analytical dynamic cohesive l","core_discovery":"On the paper's own terms: for a stiffness-degrading brittle phase-field model, a crack is not a free surface but a smooth impedance well; a harmonic wave is reflected only when the adiabatic parameter δ ≈ λ/(2πℓ) is large, and transmitted when ℓ/λ is large, with transfer-matrix results giving the sharp-crack-like reflection regime for ℓ/λ ⪅ 0.08 and transmission for ℓ/λ ⪆ 0.221. The same mechanism, combined with the local strength profile of the AT1 model, explains the widening of the damaged band. Degrading the density in addition to stiffness removes the wave-speed dip but deepens the impedance well, yielding total reflection for all ℓ/λ in tension and also altering compression. The newly","pith_inferences":["Beyond the paper, the finding that the bulk wave equation is untouched in the cohesive model implies that any spurious wave–crack interactions in existing simulations using brittle models can be quantified a priori by computing ℓ/λ; a practical quality metric for dynamic phase-field simulations would be to report it.","Beyond the paper, the condition (59) is derived for the incident-wave stress envelope of a half-sine pulse; for broadband or multiply reflected waves, the boundary in Fig. 13b may shift, so a more general criterion would be needed before using it as a universal safeguard.","Beyond the paper, the analytical opening law suggests that the dimensionless number c0/ℓch is the natural time-scale for dynamic cohesive crack opening, which could be used to design experiments that distinguish cohesive from brittle dynamic response in a material.","Beyond the paper, the similarity of the branching patterns between the brittle and cohesive models—despite very different mechanisms—suggests that branching itself may be insensitive to the regularization details provided the elastic domain and dissipation are matched; testing this hypothesis would require a systematic parameter sweep."],"forward_implications":["The reported 'numerical' artifacts in dynamic brittle phase-field simulations will persist under mesh refinement and even with numerical dissipation, because they stem from the regularization itself as long as ℓ/λ is not small.","For brittle stiffness-degrading models, the sharp-crack response is recovered only as ℓ/λ→0, which conflicts with using ℓ as a material parameter to set the nucleation stress.","The stiffness+density degradation variant, despite constant wave speed in tension, is not a remedy: it produces full reflection for all ℓ/λ, fails on compressive waves, and breaks mass conservation.","For the new cohesive model, the regularization length ℓ is decoupled from the material strength and can be chosen purely from numerical considerations, with the dynamic response governed by c0/ℓch, which is testable.","In the 2D benchmark, crack branching is captured by both the stiffness-degraded brittle model and the cohesive model, while the density-degraded model produces straight cracks; conclusions about branching from the latter should be reconsidered."],"fun_headline_variants":["One ratio, ℓ/λ, decides phase-field crack wave scattering","ℓ/λ: the single parameter that controls dynamic fracture wave interaction","New dynamic cohesive phase-field model fixes wave scattering","Brittle phase-field cracks reflect waves only when ℓ/λ is tiny","Phase-field wave scattering collapses to a single length ratio"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The cohesive model's dynamic law rests on the assumption that the eigenstrain carries no inertia and its pointwise evolution is identical to the quasi-static stationarity condition at every instant.","fun_headline_variants_meta":{"raw":{"variants":["One ratio, ℓ/λ, decides phase-field crack wave scattering","ℓ/λ: the single parameter that controls dynamic fracture wave interaction","New dynamic cohesive phase-field model fixes wave scattering","Brittle phase-field cracks reflect waves only when ℓ/λ is tiny","Phase-field wave scattering collapses to a single length ratio"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00043,"raw_usage":{"total_tokens":2022,"prompt_tokens":719,"completion_tokens":1303,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":1215}},"tokens_in":463,"tokens_out":1303,"duration_ms":11838,"temperature":1.0,"reasoning_tokens":1215,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:56:18.362686+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the reflection and transmission power coefficients for a fixed phase-field crack as a function of ℓ/λ, e.g. by transfer-matrix or spectral simulations; if the transition does not fall near ℓ/λ ≈ 0.08–0.221, the impedance-based explanation is wrong.","supporting_citations":[],"review_version":2}