{"id":"4e4bdad7-13e8-4383-b722-3ece6f100572","arxiv_id":"2411.16393","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A dynamic phase-field model that accounts for material strength as an independent property reproduces crack paths in Kalthoff-Winkler, Brazilian, and soda-lime glass experiments better than classical phase-field models.","lead":"This paper extends a phase-field fracture theory that includes material strength to dynamic loading, and uses it to simulate several dynamic fracture experiments. The simulations suggest that capturing the material's strength surface independently of its stiffness and toughness is essential for predicting crack paths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Dynamic loading is inserted only in momentum balance; ce and δl are validated for uniform stress but applied pointwise to transient, nonuniform fields, and no check ties nucleation events back to the strength surface.","rationale":"The paper does what it claims: it extends a previously validated quasi-static phase-field theory to dynamic loading and shows improved agreement with three dynamic fracture experiments. The simulations are reproducible through RACCOON/MOOSE, the regularization-length insensitivity is checked in Sec. 4.1, and comparisons against two classical phase-field variants are informative. I do not see an internal contradiction in the governing equations. The remaining issue is the one identified by the reader: the dynamic generalization is asserted rather than derived, and the strength surface is only asymptotically recovered for uniform stress. My proposed check tests whether the pointwise local rule used in Eq. (9) actually reproduces the strength-surface nucleation condition in a transient, nonuniform setting. If it does, the concern is resolved; if it does not, the central claim needs qualification. Because the concern is addressable and does not undermine the core evidence, the CONDITIONAL verdict stands.","tokens_in":15714,"tokens_out":8237,"duration_ms":87153,"concrete_test":"Run an uncoupled elastodynamic simulation of the basalt Brazilian test with the same geometry, contact settings, and load history (Eq. (24)), and record for every point the first time the local stress state satisfies Eq. (17) with the NucCe2024 ce and δl. Compare that first-violation time and location with the nucleation site and time in the fully coupled simulation (Fig. 15g-i). If the coupled crack nucleates at a measurably different point or after a significant delay relative to the first violation of Eq. (17), the pointwise use of the uniform-stress strength surface in Eq. (9) is not self-consistent, and the agreement with Yin et al. (2022) cannot be attributed specifically to the strength surface.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.2 introduces inertia by replacing the quasi-static balance of linear momentum with Eq. (8), while the phase-field equation (9), the driving force ce in Eq. (13), and δl in Eq. (14) are carried over unchanged from the quasi-static theory. The only evidence that ce and δl encode the correct strength surface is the uniform-stress comparison in Fig. 2, via the strength surface F^PF in Eq. (17). In the dynamic benchmarks, however, the stress fields are strongly nonuniform and transient; Eq. (9) evaluates ce pointwise from local I1 and J2, thereby assuming that crack nucleation in a transient, gradient-dominated stress field is governed by the same uniform-stress Drucker-Prager surface. This is not derived or separately validated. The comparisons that support the central claim are integrated outcomes—crack paths and branching angles—and two of them (Brazilian u0 in Eq. (24), glass pressure profile in Sec. 5.2) involve load calibration. A mismatch in nucleation location or time would not be visible in the final path comparison. Thus the claim that an independent strength surface is essential could be an artifact of the local pointwise rule used to encode it, rather than evidence for the macroscopic strength surface itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper extends a phase-field theory of fracture that incorporates an independent material strength surface (Drucker-Prager) to the dynamic regime by adding inertia to the momentum balance and retaining the quasi-static phase-field evolution equation. The authors present an adaptive finite-element and implicit time-stepping scheme implemented in RACCOON, and apply it to a single-edge notched specimen, the Kalthoff-Winkler experiment, a pressurized hollow sphere, dynamic Brazilian tests on basalt, and impact experiments on soda-lime glass. The central claim is that accounting for the material strength surface is essential for reproducing experimentally observed crack paths and branching angles in dynamic fracture, whereas classical phase-field models that lack an independent strength surface fail.","tokens_in":16021,"tokens_out":4546,"duration_ms":40411,"significance":"If the central claim holds, the paper provides a useful computational framework and a series of instructive benchmark comparisons that support the notion that material strength is an independent macroscopic property in dynamic fracture. The work is notable for making the RACCOON implementation available, for explicitly comparing the effective strength surfaces of competing phase-field models, for demonstrating regularization-length insensitivity in the single-edge notch problem, and for producing quantitative comparisons with experimental crack paths, branching angles, and crack speeds. The main caveat is that the dynamic extension is assumed rather than derived, and several loads are calibrated to experiment, so the predictive claims should be read with those qualifications in mind.","major_comments":[{"comment":"The dynamic extension is assumed, not derived: inertia enters only in the momentum balance, while the driving force ce (Eq. 13) and coefficient δℓ (Eq. 14) are imported unchanged from the quasi-static theory and are validated only for uniform stress states in Fig. 2 via Eq. (17). The paper applies these pointwise in transient, nonuniform, gradient-dominated stress fields without a derivation or a numerical check. Because the central claim relies on crack paths and nucleation locations produced by this model, the authors should either provide a derivation/justification for the dynamic form of the phase-field equation or supply a specific numerical validation (e.g., a dynamic homogeneous-stress test or a convergence study in ℓ) that ties nucleation events back to the intended strength surface.","section":"Section 5.1, Eq. (24)"},{"comment":"The prescribed displacement amplitude u0 in the Brazilian simulation is calibrated to match the experimentally measured fracture stress using an elastodynamic simulation without fracture (text preceding Eq. 24). As a result, the agreement between the simulated and experimental central crack nucleation is not a fully predictive test of the strength surface. The authors should report the sensitivity of nucleation location and time to u0 and explicitly state which features of the comparison are predictive as opposed to fitted.","section":"Section 5.1, Eq. (24)"},{"comment":"The pressure profile applied to the V-notch in the soda-lime glass simulations is adjusted based on a wave-propagation estimate, and the friction coefficient of 0.35 is assumed rather than measured. Since the reported branching angles and times depend on these choices, the claim that the model captures branching angles requires a sensitivity study with respect to the pressure duration and friction coefficient, or independent justification of those values.","section":"Section 5.2, Fig. 18(b) and Eq. (26)"},{"comment":"The effective strength surface F^PF in Eq. (17) is constructed, through the choices of ce and δℓ, to converge to the input Drucker-Prager surface; hence the agreement in Fig. 2 is by construction and does not independently validate the strength-surface concept. The actual support for the central claim must come from the dynamic benchmark comparisons. The manuscript should state this distinction explicitly and, ideally, provide a test that separates the role of the macroscopic strength surface from the specific pointwise prescription of ce.","section":"Section 2.3, Eq. (17)"}],"minor_comments":[{"comment":"The abstract states that the discretized equations are solved in a staggered manner, but Section 3 describes a fixed-point iteration scheme; please align the two descriptions.","section":"Abstract and Section 3"},{"comment":"The parameter ψc (nucleation energy) is listed for the cohesive model but is not used by the NucCe models; please clarify its role in the table and in the text where it appears.","section":"Tables 2, 3, 5"},{"comment":"The text and figure use 'Coh2019 (V-D)' while Table 1 uses 'Vol./Dev.'; please make the label for the volumetric-deviatoric split consistent throughout.","section":"Figure 7"},{"comment":"The text states that the spurious branch vanishes at ℓ=0.2 mm, but Fig. 10 shows NucCe2020 only; please include results for the same regularization lengths for the NucCe2024 model for a complete comparison.","section":"Section 4.2, Fig. 10"},{"comment":"The dissipated energy D = W − K − U is defined as a residual; because the model contains an explicit external driving force ce, please comment on the accuracy of this energy-balance definition in the presence of the driving force term.","section":"Section 4.1, Eq. (18)"},{"comment":"Reference [41] lists the author as 'Tipper, H.V.'; this appears to be a typo for 'Tippur, H.V.' and should be corrected.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a mechanics-oriented journal and the numerical comparisons are valuable. The main risk is overstatement: the dynamic extension is assumed, the Brazilian load is calibrated, and the glass pressure profile is adjusted, so the claim that the strength surface is 'essential' is not as cleanly established as the abstract suggests. A revision that adds sensitivity studies or a dynamic validation of the strength surface would substantially strengthen the paper. Also note that the comparison with 'classical phase-field models' depends on the specific models chosen; the authors should temper the general statement that such models cannot account for strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is the inertia-inclusive version of the Kumar/Lopez-Pamies phase-field theory, plus a robust numerical implementation and three dynamic benchmark comparisons that classical phase-field models cannot match. In the Kalthoff-Winkler setting, the strength-aware model reproduces the ~70-degree crack angle; in the Brazilian basalt tests it nucleates in the interior rather than at the contact; in the soda-lime glass it brackets the observed branching angles. That is genuine new evidence that strength, as an independent macroscopic property, matters under dynamic loading. Good credit to the authors for showing the effective strength surfaces and the stress trajectories that explain why the competing models fail. The adaptive FE implementation in RACCOON and the energy-based regularization study are also solid, reproducible pieces of work.\n\nNow the soft spots. The dynamic extension is assumed, not derived: inertia is added to the momentum balance, while the phase-field driving force ce and the coefficient delta_l are carried over unchanged from the quasi-static theory and only validated against the uniform-stress strength surface. The stress-test note worries that pointwise use of this driving force in transient, nonuniform fields is an unvalidated assumption. That is a legitimate concern, but not fatal. Using a local stress-based nucleation criterion pointwise is the standard continuum assumption, and the paper does partially check it by plotting stress histories against the strength surfaces in the Kalthoff-Winkler and Brazilian cases. A real weakness is load calibration: the Brazilian displacement u0 is matched to the measured fracture stress and the glass pressure profile is adjusted, and a friction coefficient of 0.35 is assumed. That weakens the predictive power of those two comparisons, though the qualitative failure of the classical models remains meaningful. Also, the paper cites the peridynamics comparison from Mehrmashhadi et al. but never actually shows those results, so the claimed contrast with [32] is not directly documented.\n\nOn balance, the central claim holds up: an independent strength surface appears to be essential for describing dynamic fracture in these brittle materials. The paper is for researchers in computational fracture mechanics, especially phase-field practitioners. It deserves a serious referee. I would recommend acceptance after the authors either derive or more carefully justify the dynamic form of the driving force, and add the missing peridynamics comparison. Not a desk reject.","headline":"A solid dynamic extension of the strength-aware phase-field theory with convincing benchmark evidence that the strength surface governs dynamic crack paths; the main caveats are the assumed dynamic driving force and some calibrated loads.","tokens_in":16498,"tokens_out":1954,"would_cite":true,"duration_ms":22277,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74R10","74S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A phase-field fracture theory that treats material strength as an independent property, extended to include inertia, reproduces the crack angles and nucleation sites measured in dynamic impact experiments on steel, basalt, and glass.","keywords":["dynamic fracture","phase-field fracture","material strength surface","crack nucleation","crack propagation","crack branching","Drucker-Prager strength","Kalthoff-Winkler benchmark"],"falsifier":"Take a brittle material whose tensile and compressive strengths have been measured independently under multiaxial quasi-static loading, run a dynamic Brazilian test with high-speed imaging, and compare where the crack first appears and the angle it grows. If cracks initiate at the compressive loading platens before the independently measured strength surface is exceeded, or if the crack path deviates from the centered horizontal crack the model predicts, the dynamic strength-surface extension is falsified.","tokens_in":15519,"feed_emoji":"💥","tokens_out":12112,"duration_ms":105704,"temperature":0.7,"pith_summary":"The paper extends the Griffith phase-field theory of fracture with material strength — the idea that a material's strength surface is an independent macroscopic property, separate from elasticity and toughness — to dynamic loading by adding inertia to the balance of linear momentum. The central question is whether the strength surface still governs crack nucleation and propagation when loads change rapidly and stress waves matter. Using a Drucker-Prager strength surface, the model reproduces the Kalthoff-Winkler crack angle, the interior crack nucleation observed in dynamic Brazilian tests on basalt, and the branching angles measured in soda-lime glass, while classical phase-field and cohesive models develop spurious branches or nucleate at the wrong location. The practical stakes are that dynamic fracture simulations which omit an independent strength surface can get both where cracks start and which way they grow wrong.","feed_headline":"Material strength, not just toughness, sets dynamic crack paths","feed_subtitle":"A phase-field model that keeps strength independent of elasticity and toughness reproduces impact crack angles classical models miss.","key_machinery":"The central object is the material strength surface, $F(\\sigma_1,\\sigma_2,\\sigma_3)=0$, the set of critical stress states at which the material fractures under spatially uniform stress; here it is a Drucker-Prager surface with uniaxial tensile strength $\\sigma_{ts}$ and hydrostatic strength $\\sigma_{hs}$ as independent inputs. The theory encodes this surface in the phase-field evolution through a driving force $c_e(X,t)$ and a coefficient $\\delta_\\ell$ chosen so that, in the sharp-interface limit, the phase-field model's own strength surface reduces to the prescribed one. The coupled system consists of a hyperbolic linear-momentum balance with inertia $\\rho \\ddot{u}$ and an elliptic phase-field evolution with irreversibility constraints, discretized in space by adaptive finite elements and in time by an implicit scheme. The strength surface does double duty: under uniform stress it delays fracture until the surface is crossed, and near stress concentrators it suppresses crack growth into compressive regions that would otherwise produce spurious branches.","core_discovery":"The paper's central claim is that in the dynamic regime the material strength surface remains an independent macroscopic material property on par with elasticity and toughness, and that inertia enters the theory simply by adding mass density to the momentum balance of the earlier quasi-static formulation. In concrete terms, the proposed model — a hyperbolic momentum equation coupled to an elliptic phase-field evolution — reproduces the roughly 70 degree crack path in Kalthoff-Winkler impact experiments, nucleates the dynamic Brazilian fracture of basalt in the specimen interior rather than at the compressive contacts, and produces soda-lime glass branching at 54 and 44 degrees, bracketing the experimental 47 to 55 degree range. Classical variational phase-field and cohesive models fail at least one of these benchmarks because their effective strength surfaces are locked to elasticity and toughness and cannot, for example, set compressive strength independently of tensile strength. That the correct crack paths emerge only when the strength surface is represented accurately is taken as evidence that strength governs propagation as well as nucleation under dynamic conditions.","pith_inferences":["This suggests the same construction could be used to import arbitrary experimentally measured strength surfaces, not just Drucker-Prager, into dynamic fracture simulations.","If the dynamic extension holds, part of the historical scatter among cohesive, phase-field, and peridynamic predictions of dynamic fracture may reflect each model's implicit strength surface rather than genuine differences in material behavior.","A natural use would be inverse identification: dynamic Brazilian or impact tests, matched by this model, could extract the compressive branch of the strength surface for materials that are difficult to test under uniform multiaxial compression."],"forward_implications":["The strength surface must be treated as an independently measured input in dynamic fracture simulations, alongside elastic moduli, toughness, and mass density.","Crack nucleation sites under impact are set by the strength surface: in dynamic Brazilian tests the crack forms in the specimen interior rather than at the compressive contacts when the surface is represented accurately.","Spurious lower branches in Kalthoff-Winkler simulations — a known artifact in some classical models — disappear once the effective strength surface matches the material's actual one.","The model brackets the observed soda-lime glass branching angles, with branching time matching experiments when a tangential load is included at the impact surface.","The same framework extends to three-dimensional fragmentation, as demonstrated by the pressurized hollow-sphere simulation."],"supporting_citations":[{"why":"Establishes the quasi-static Griffith phase-field fracture theory with material strength that this paper extends to inertia.","marker":"[21, 23, 20]"},{"why":"Supplies the specific driving-force and coefficient prescriptions used to reproduce the Drucker-Prager strength surface.","marker":"[18]"},{"why":"Provides the variational phase-field formulation with material strength whose balance equations are extended by adding inertia.","marker":"[27]"},{"why":"Reports the Kalthoff-Winkler impact experiments that set the roughly 70-degree crack-angle benchmark.","marker":"[17]"},{"why":"Reports the dynamic Brazilian basalt experiments whose centered crack nucleation is reproduced by the model.","marker":"[42]"},{"why":"Documents the soda-lime glass impact and branching experiments used to compare branching angles and times.","marker":"[40]"},{"why":"Reviews why classical variational phase-field models cannot account for material strength as an independent property, motivating the comparisons.","marker":"[29, 19]"},{"why":"Provides the peridynamic and phase-field simulations of soda-lime glass contrasted with the present results.","marker":"[32]"}],"fun_headline_variants":["Strength, not toughness, steers dynamic crack paths","Dynamic crack angles expose strength's independent role","Inertia enters, but strength still rules fracture paths","Phase-field model: strength separates from toughness in dynamics","Crack paths revealed: strength is an independent dynamic property"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model assumes that the set of stresses at which a material breaks in slow, uniform laboratory tests is the same set that governs fracture in the fast, nonuniform stress fields of an impact, so that the only dynamic change needed is adding inertia to the momentum balance.","fun_headline_variants_meta":{"raw":{"variants":["Strength, not toughness, steers dynamic crack paths","Dynamic crack angles expose strength's independent role","Inertia enters, but strength still rules fracture paths","Phase-field model: strength separates from toughness in dynamics","Crack paths revealed: strength is an independent dynamic property"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00095,"raw_usage":{"total_tokens":4104,"prompt_tokens":1043,"completion_tokens":3061,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":2985}},"tokens_in":659,"tokens_out":3061,"duration_ms":22557,"temperature":1.0,"reasoning_tokens":2985,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:10:11.802973+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a brittle material whose tensile and compressive strengths have been measured independently under multiaxial quasi-static loading, run a dynamic Brazilian test with high-speed imaging, and compare where the crack first appears and the angle it grows. If cracks initiate at the compressive loading platens before the independently measured strength surface is exceeded, or if the crack path deviates from the centered horizontal crack the model predicts, the dynamic strength-surface extension is falsified.","supporting_citations":[{"cited_title":"The poker-chip experiments of synthetic elastomers explained","cited_arxiv_id":null,"evidence_quote":"Supplies the specific driving-force and coefficient prescriptions used to reproduce the Drucker-Prager strength surface."},{"cited_title":"A variational formulation of Griffith phase-field fracture with material strength","cited_arxiv_id":null,"evidence_quote":"Provides the variational phase-field formulation with material strength whose balance equations are extended by adding inertia."},{"cited_title":"Failure mode transition at high rates of shear loading","cited_arxiv_id":null,"evidence_quote":"Reports the Kalthoff-Winkler impact experiments that set the roughly 70-degree crack-angle benchmark."},{"cited_title":"Determination of Dynamic Tensile Strength of Microwave-Induced Basalt Using Brazilian Test","cited_arxiv_id":null,"evidence_quote":"Reports the dynamic Brazilian basalt experiments whose centered crack nucleation is reproduced by the model."},{"cited_title":"Dynamic fracture of soda-lime glass: A full-field optical investigation of crack initiation, propagation and branching","cited_arxiv_id":null,"evidence_quote":"Documents the soda-lime glass impact and branching experiments used to compare branching angles and times."},{"cited_title":"On validating peridynamic models and a phase-field model for dynamic brittle fracture in glass","cited_arxiv_id":null,"evidence_quote":"Provides the peridynamic and phase-field simulations of soda-lime glass contrasted with the present results."}],"review_version":1}