{"id":"64f9450d-e17c-4d9b-99a2-561e6642918f","arxiv_id":"2502.04487","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Under mixed-mode loading, the complete nucleation phase-field model matches benchmarks, the star-convex variational model produces spurious compressive cracks, and the hybrid cohesive-zone model propagates at an incorrect toughness.","lead":"This computational study compares three phase-field models of brittle fracture under shear and mixed-mode loading. It finds that the strength-based complete nucleation model is reliable, while a popular energy-split variant can create false compressive cracks and a hybrid cohesive-zone variant grows cracks at the wrong toughness.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Complete Nucleation model's central claim is not independently established: the calibrated delta_epsilon of Eq. (18) is carried over from prior mode-I work, and every non-mode-I benchmark here kinks to locally mode-I growth, leaving true mode-II/III propagation untested.","rationale":"The reader's weakest_assumption identifies the calibrated delta_epsilon formula, Eq. (18), as the load-bearing support for the Complete Nucleation model's Griffith-consistent propagation. I find the same concern, and the manuscript's own Section 4 limitation statement strengthens it: the paper admits that its non-mode-I global loadings produce locally mode-I crack growth. This means the six benchmarks do not directly test the regime in which a mode-dependent effective toughness or a calibration failure would matter. The CZM-Hybrid incorrect-toughness observation and the star-convex spurious-compressive-crack observation are credible internal comparisons, and the authors are appropriately cautious in calling for further experimental validation and a convergence proof. However, because the practical model-selection conclusion rests on the Complete Nucleation model's transferability of delta_epsilon from prior calibrations, the conditional verdict is justified. No code or mesh-convergence data are provided, which adds uncertainty, but I do not see an internal inconsistency that would require rejection. Since the reader's CONDITIONAL verdict already reflects this concern, no verdict adjustment is needed.","tokens_in":25466,"tokens_out":5158,"duration_ms":61632,"concrete_test":"Run pure mode-II and pure mode-III boundary-layer (K-field) or Arcan/compact-shear specimens with known analytical stress-intensity solutions, using the same graphite parameters and mesh resolution as Section 3.1, and compute the J-integral at onset of crack advance with the Complete Nucleation model. Compare the resulting effective energy release rate with Gc and with the value implied by Eq. (18); if G/Geff deviates from 1 by more than the 5-10% error band claimed for Eq. (18), the calibration is mode-dependent and the central claim fails. As a complementary check on the existing results, compute the local mode mixity along the simulated crack paths in Figs. 4, 7, 9, and 15; if K_II/K_I and K_III/K_I drop to near zero immediately after kinking, those benchmarks cannot distinguish a model that handles true mode-II/III propagation from one that only handles local mode-I growth.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.3 constructs the Complete Nucleation model on Eq. (18), an approximate, numerically calibrated formula for delta_epsilon that restores the effective critical energy release rate to Gc. The paper uses this same delta_epsilon in every mixed-mode benchmark, and the good agreement with Griffith-type behavior in Figs. 2 and 12 is the main support for the claim that this model predicts non-mode-I crack growth. That support is conditional in two connected ways. First, the manuscript offers no independent check of delta_epsilon under pure mode-II or mode-III crack propagation: Sections 3.2, 3.3, 3.5, and 3.6 compare crack paths or load-displacement responses, and Section 4 explicitly concedes that although the global loading is non-mode I, the crack locally tends to propagate in a mode I fashion. The benchmarks therefore do not exercise a regime in which a mode-dependent effective toughness, or a failure of the calibration, would reveal itself. Second, Section 4 also states that a mathematical proof of convergence to Griffith theory as epsilon decreases to zero remains desirable; absent such a proof, the propagation behavior is an emergent property of Eq. (18) plus the PDE system, not a theorem. If delta_epsilon from Eq. (18) is not independent of boundary-value problem and loading mode for shear-dominated cracks, a property asserted from previous numerical observations but not demonstrated here, then the agreement reported in Section 3 is partly built in by the calibration. This is the load-bearing assumption behind the practical conclusion that only the Complete Nucleation model is trustworthy for non-mode-I loading.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares three phase-field approaches for brittle fracture that incorporate material strength—the classical variational model with the star-convex energy split, the Complete Nucleation model of Kumar et al., and the CZM-Hybrid model—on six benchmark problems including mode II, mode III, and mixed-mode loadings. The authors report that the Complete Nucleation model reproduces Griffith-type propagation and agrees well with the Yosibash–Mittleman experimental crack paths, that the star-convex variational model can nucleate spurious compressive or shear cracks under shear-dominated loading, and that the CZM-Hybrid model propagates cracks at an incorrect effective fracture toughness because its stress-based crack-driving force differs from the strain energy. The paper is framed as the first part of a series on strength-aware phase-field models.","tokens_in":25765,"tokens_out":6366,"duration_ms":65421,"significance":"If the conclusions hold, the paper provides practically useful guidance on which phase-field model is reliable for non-mode-I and mixed-mode fracture problems. The systematic comparison across six benchmarks, the validation against the Yosibash–Mittleman experiments, and the sensitivity checks with respect to residual stiffness, regularization length, Poisson ratio, and compressive strength are notable strengths. The authors are also candid about limitations: Section 4 concedes that all simulated cracks propagate locally in a mode-I fashion and that a convergence proof to Griffith theory as epsilon tends to zero remains desirable. These self-identified limitations are central to the assessment, because they narrow the scope of what the numerical evidence can claim.","major_comments":[{"comment":"The parameter δ_ε is introduced as a numerical calibration whose purpose is to restore the effective critical energy release rate to G_c, and Eq. (18) was obtained in prior mode-I work. The agreement of the Complete Nucleation model with Griffith-type predictions in Section 3 (e.g., Fig. 2(b) and Fig. 12(a–d)) is therefore not an independent validation of the model's propagation law; a significant part of that agreement is built into the model through δ_ε. The authors should either determine δ_ε from a boundary value problem with sustained non-mode-I propagation (a constrained shear or anti-plane shear crack) and show that it coincides with Eq. (18), or explicitly temper the abstract's claim that the model \"effectively predicts crack growth under mode II, mode III\" to acknowledge that the propagation response has so far been calibrated toward Griffith behavior in mode I.","section":"§2.3, Eq. (18)"},{"comment":"The manuscript concedes that \"while the global loading studied in this work is non-mode I, the crack locally tends to propagate in a mode I fashion.\" Consequently, none of the six benchmarks actually exercises a regime in which the crack advances under sustained pure mode-II or mode-III conditions; the global loading is mixed-mode, but the local propagation mode is mode I. The abstract states that the Complete Nucleation approach \"effectively predicts crack growth under mode II, mode III, and mixed-mode loading,\" which is stronger than the evidence presented. The authors should add a benchmark that suppresses out-of-plane kinking (for example, a weak interface or a crack front constrained by geometry) so that pure mode-II or mode-III propagation is tested, or they should rephrase the conclusions to say that the model has been validated for mixed-mode loading with locally mode-I propagation.","section":"§4"},{"comment":"The three models are run with different regularization lengths: ε = 0.1 mm or 0.25 mm for the Complete Nucleation and CZM-Hybrid models, and ε = l_ch for the Classical Variational model in several benchmarks. Because the strength surfaces of all three models depend on ε (Eqs. (13), (19), (30), and Fig. 1), the differences attributed to model formulation—for instance, the underpredicted shear strength in Fig. 12(d) or the spurious compressive crack in Fig. 4(c)—may partly reflect the different ε values rather than intrinsic differences between the models. The authors should include at least one key benchmark with all three models at the same ε (with appropriate mesh-size corrections) or justify why comparing at different ε is the fair and meaningful comparison for the claims made.","section":"§3, Figs. 2, 4, 7, 9, 12, 13"}],"minor_comments":[{"comment":"The correction term in c_e contains the factor (1 − √(I_1²)/I_1), which is discontinuous at I_1 = 0; please clarify how this term is regularized in the numerical implementation.","section":"§2.3, Eq. (20)"},{"comment":"The caption does not specify the regularization lengths used to generate the Classical Variational and CZM-Hybrid strength surfaces; please provide these values so that the comparison in Fig. 1 is reproducible.","section":"Fig. 1 caption"},{"comment":"The tensile strength of marble is adopted as 10 MPa and then as 4 MPa, with the paper noting a reported range of 2–30 MPa. The sensitivity of the secondary-crack predictions to σ_ts is only probed at two values; a short discussion of where within this range the experimental comparison in Fig. 15(c) is robust would strengthen the argument.","section":"§3.6"},{"comment":"The irreversibility constraints in Eq. (9) are said to be enforced with a penalty method, but no penalty parameter, convergence study, or implementation reference is reported; please provide this detail.","section":"§2.2, Remark 3"},{"comment":"The effective fracture toughness corrections for the finite mesh size, G_eff = G_c(1 + 3h/8ε) and G_eff = G_c(1 + h/πε), are stated without derivation; a brief explanation of the origin of these factors would help readers apply the corrections to other problems.","section":"§3.1, Fig. 2(b)"},{"comment":"The manuscript does not include a data availability statement, code, mesh sizes, or convergence tolerances. Since the central evidence is computational, a repository with input files and solver settings would materially improve reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper's central gap is the one the authors themselves identify in Section 4: the benchmarks do not exercise pure mode-II or mode-III propagation, and the δ_ε calibration connecting the Complete Nucleation model to Griffith toughness is carried over from prior mode-I work. A revision that adds a truly non-mode-I propagation benchmark, or carefully recasts the claims, would make this a strong contribution. The paper is also heavily self-referential in its citations, which is understandable given the model lineage, but the independent validation essentially rests on a single experimental dataset (Yosibash–Mittleman). The lack of code or data release is a common issue but worth encouraging here given the breadth of the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is worth reading if you work on phase-field fracture: it is the first systematic three-model comparison under non-mode-I loading, and it delivers two results that actually change how you would pick a model. The star-convex energy split, previously argued to be superior, nucleates a spurious compressive crack under in-plane shear; the authors show it persists across regularization length and residual stiffness, and they trace it to an artificial pressure term amplified by the gamma* parameter. The CZM-hybrid model propagates cracks at an incorrect effective toughness, and the paper explains why: the modified von Mises crack-driving energy differs from the strain energy near the tip. Those are solid, useful findings.\n\nThe Complete Nucleation model is the centerpiece, and here the evidence is good but not airtight. The model uses the calibrated delta_epsilon from Eq. (18), which was constructed to restore the effective critical energy release rate to Gc, so its Griffith-consistent propagation in mode I is partly built in. For non-mode-I loading, the paper is honest in Section 4 that the cracks locally tend to propagate in a mode I fashion, so the benchmarks never really isolate sustained pure mode II or mode III propagation. The stress-test concern is accurate on this point. The absence of a mesh convergence study and the lack of code or data release also make the quantitative claims harder to verify. A few material parameters (compressive strength for the graphite benchmark, tensile strength of marble) are adopted rather than measured, though the authors flag this.\n\nThat said, the central argument is not broken. The comparison is fair, the models are implemented consistently, the V-notch experimental benchmark matches all three models reasonably well, and the new observations about the star-convex split and CZM-hybrid are reproducible in principle. The calibration concern mainly weakens the claim that Complete Nucleation is independently verified for non-mode-I propagation; it does not invalidate the comparison itself.\n\nThis paper deserves a serious referee. A revision should include a mesh convergence study, release or thoroughly document the numerical implementation, and ideally test a configuration that forces a crack to grow under sustained mode II or mode III rather than kinking to mode I. Even without those additions, the existing content is publishable once the limitations are stated as cleanly as the authors already state some of them.\n\nRecommendation: send to peer review, not desk reject.","headline":"A useful, honest numerical comparison with two new findings, but the headline claim for the Complete Nucleation model rests on a calibrated parameter and cracks that mostly kink to mode I.","tokens_in":26317,"tokens_out":2199,"would_cite":true,"duration_ms":23257,"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":"Only the complete nucleation phase-field model predicts crack growth correctly under mode II, mode III, and mixed-mode loading; the classical variational model mispredicts paths, and the cohesive-zone hybrid model grows cracks at the…","keywords":["phase-field fracture","brittle fracture","crack nucleation","strength surface","mixed-mode loading","Griffith criterion","cohesive zone model","variational fracture"],"falsifier":"Run a mode II or mixed-mode crack-growth experiment on a material with independently measured $G_c$ and strength surface, simulate the same geometry with the complete nucleation model using $\\delta_\\varepsilon$ from Eq. (18), and check whether the crack initiates at the Griffith load. A deviation beyond the reported 5--10\\% error would show the calibration is not transferable and would undercut the paper's central comparison.","tokens_in":25228,"feed_emoji":"🧱","tokens_out":11150,"duration_ms":106607,"temperature":0.7,"pith_summary":"Three phase-field models of brittle fracture try to combine material strength (when cracks form) with fracture toughness (how existing cracks grow), and this paper asks whether they stay reliable when loading is not simple tension. Using mode II, mode III, and mixed-mode benchmarks, including an experimentally documented four-point bending test, it claims that only the complete nucleation model—which adds the strength surface to the phase-field evolution equation as a stress-based driving force—predicts Griffith-type crack growth in every investigated case. The classical variational model with a star-convex energy split can nucleate spurious compressive cracks under in-plane shear and tends to select shear cracks under mode III and compression, because its strength surface is too weak in shear. The cohesive-zone hybrid model follows roughly correct crack paths but propagates cracks at an incorrect effective fracture toughness, because its stress-based crack-driving energy differs from the strain energy near the crack tip. If the paper is right, engineering use of the two flawed approaches in non-tension-dominated problems would mispredict either crack paths or critical loads.","feed_headline":"Only one crack-growth model matches Griffith under shear","feed_subtitle":"Of three phase-field fracture models, the complete nucleation one matches Griffith; rivals mispredict paths or loads.","key_machinery":"The load-bearing objects are the three distinct mechanisms used to embed strength. In the complete nucleation model, an explicit stress-based driving force $c_e(I_1,J_2;\\varepsilon)$ is added to the right-hand side of the phase-field evolution equation; its coefficients are fitted to the chosen strength surface (in these tests, a standard two-parameter Drucker-Prager-type surface), and the coefficient $\\delta_\\varepsilon$ in Eq. (18) restores the effective critical energy release rate to the experimental $G_c$. The classical variational model instead uses a star-convex energy split, degrading only a 'tensile' part of the strain energy and leaving a 'compressive' part intact. The CZM-Hybrid model replaces the variational driving force $g'(v)W(E)$ with $g'(v)\\Gamma(\\sigma)$, where $\\Gamma$ is computed from a modified von Mises or Rankine equivalent stress, so that crack evolution and elasticity are governed by different energy functions.","core_discovery":"The central discovery, stated on the paper's own terms, is that of the three strength-aware phase-field formulations only the complete nucleation model reproduces crack growth from pre-existing cracks under mode II, mode III, and mixed-mode loading. That model augments the Euler-Lagrange equation for the phase field with an extra stress-based driving force built from the material's strength surface and multiplies the fracture energy by a coefficient $\\delta_\\varepsilon$ calibrated so the effective critical energy release rate matches the measured $G_c$. In the benchmarks, this model predicts the same load-displacement response and crack paths as Griffith theory for large cracks, converges to the uniaxial strength as notches shrink, and matches the experimental V-notch mixed-mode paths. The classical variational model with the star-convex split, in contrast, fails to represent the full strength surface: its shear strength is artificially low, leading to a spurious compressive crack under in-plane shear and shear-branch cracks in mode III and compression problems. The CZM-Hybrid model, which drives fracture with an equivalent-stress energy $\\Gamma(\\sigma)$ rather than the strain energy $W(E)$, produces qualitatively similar paths but systematically underpredicts peak loads because $\\Gamma \\neq W$ near the crack tip.","pith_inferences":["If the star-convex split fails under shear because no energy split can represent a general strength surface, then the entire energy-split line of variational models may need a structural change rather than another ad-hoc split; the paper hints at this but does not develop it.","The $\\delta_\\varepsilon$ calibration is the obvious pressure point: testing it on boundary value problems, materials, and mesh sizes outside the calibration set would determine whether the complete nucleation model's Griffith consistency is emergent or fitted.","The $\\Gamma \\neq W$ mismatch that hurts the CZM-Hybrid model is likely shared by any phase-field model that drives fracture with a separate stress-based equivalent quantity, so the quantitative deficiency may extend beyond the specific model tested here.","The numerical tensile-to-shear crack transition seen when compressive strength is reduced suggests a directly testable prediction: materials with a lower compressive-to-tensile strength ratio should show shear-dominated crack growth in nominally mode I-dominated geometries."],"forward_implications":["For shear-dominated or mixed-mode simulations of brittle fracture, only the complete nucleation model's predictions of critical load and crack path can be taken at face value among the three compared models.","The star-convex variational model should not be used for shear-dominated problems: it can nucleate compressive cracks and may select shear cracks over tensile ones because its strength surface is too weak in shear.","CZM-Hybrid results in mixed-mode problems carry a built-in quantitative error: the crack path may look correct, but the peak load is systematically too low because the crack-driving energy $\\Gamma$ is not the strain energy $W$.","For tension-dominated cases the three models agree, so the practical discrepancies are confined to loadings with significant compressive or shear stress states.","The complete nucleation model also describes the transition from Griffith-dominated to strength-dominated nucleation as notch size decreases, matching analytical limits for uniaxial and biaxial tension."],"supporting_citations":[{"why":"Defines the complete nucleation formulation and its stress-based strength driving force; the central method the paper validates.","marker":"[10]"},{"why":"Introduces the earlier phase-field model that treats the Euler-Lagrange equations rather than the energy functional as primal, motivating the complete nucleation approach.","marker":"[12]"},{"why":"Provides the approximate formula for $\\delta_\\varepsilon$, Eq. (18), which restores the effective Griffith toughness in the complete nucleation model.","marker":"[14]"},{"why":"Introduces the hybrid idea of driving fracture with a separate damage-threshold energy $\\Gamma$ rather than the strain energy $W$.","marker":"[24]"},{"why":"Supplies the star-convex energy split used for the classical variational model, the approach found to mispredict shear-dominated paths.","marker":"[35]"},{"why":"Gives the specific linear-softening phase-field cohesive-zone model and the modified von Mises / Rankine equivalent-stress choices used for the CZM-Hybrid model.","marker":"[37]"},{"why":"Supplies the experimental multiaxial strength data for graphite used to set the strength surface and material parameters.","marker":"[45]"},{"why":"Provides the V-notch four-point-bending benchmark with mode I+II+III loading used to validate crack paths.","marker":"[59]"},{"why":"Reports the experimental crack-initiation angles and material data for the same benchmark, used as the experimental comparison.","marker":"[60]"}],"fun_headline_variants":["Only one phase-field model nails Griffith crack growth","Strength-aware fracture models: only one gets shear right","Two fracture models mispredict cracks; one matches Griffith","Phase-field showdown: one model beats two on strength"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The complete nucleation model's Griffith-consistent behavior depends on the approximate formula for $\\delta_\\varepsilon$ (Eq. 18) being transferable across loading modes and geometries; if that calibration is problem-dependent, the agreement reported here is partly manufactured rather than emergent, and a convergence proof as $\\varepsilon \\to 0$ is still missing.","fun_headline_variants_meta":{"raw":{"variants":["Only one phase-field model nails Griffith crack growth","Strength-aware fracture models: only one gets shear right","Two fracture models mispredict cracks; one matches Griffith","Phase-field showdown: one model beats two on strength"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000495,"raw_usage":{"total_tokens":2487,"prompt_tokens":1063,"completion_tokens":1424,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":1362}},"tokens_in":679,"tokens_out":1424,"duration_ms":11228,"temperature":1.0,"reasoning_tokens":1362,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:32:44.396047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a mode II or mixed-mode crack-growth experiment on a material with independently measured $G_c$ and strength surface, simulate the same geometry with the complete nucleation model using $\\delta_\\varepsilon$ from Eq. (18), and check whether the crack initiates at the Griffith load. A deviation beyond the reported 5--10\\% error would show the calibration is not transferable and would undercut the paper's central comparison.","supporting_citations":[{"cited_title":"Kamarei, A","cited_arxiv_id":null,"evidence_quote":"Provides the approximate formula for $\\delta_\\varepsilon$, Eq. (18), which restores the effective Griffith toughness in the complete nucleation model."},{"cited_title":"Lorentz, A nonlocal damage model for plain concrete c onsistent with cohesive fracture, International Journal o f Fracture 207 (2017) 123–159","cited_arxiv_id":null,"evidence_quote":"Introduces the hybrid idea of driving fracture with a separate damage-threshold energy $\\Gamma$ rather than the strain energy $W$."},{"cited_title":"Vicentini, C","cited_arxiv_id":null,"evidence_quote":"Supplies the star-convex energy split used for the classical variational model, the approach found to mispredict shear-dominated paths."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the specific linear-softening phase-field cohesive-zone model and the modified von Mises / Rankine equivalent-stress choices used for the CZM-Hybrid model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the experimental multiaxial strength data for graphite used to set the strength surface and material parameters."},{"cited_title":"Yosibash, B","cited_arxiv_id":null,"evidence_quote":"Provides the V-notch four-point-bending benchmark with mode I+II+III loading used to validate crack paths."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the experimental crack-initiation angles and material data for the same benchmark, used as the experimental comparison."}],"review_version":1}