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REVIEW 4 major objections 6 minor 43 references

Revisit of Two-dimensional CEM on Crack Branching: from Single Crack-tip Tracking to Multiple Crack-tips Tracking

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A single energy-based tracking rule now reproduces single cracks, branching, and fragmentation in 2D dynamic fracture.

desk verdict A real but under-specified extension of 2D CEM; the algorithm as written can't produce the branching and fragmentation figures. read the letter →

arxiv 2509.01827 v1 pith:YP3FAMH5 submitted 2025-09-01 cs.CE

Yuxi Xie , Hongyou Cao , Miao Su , Zhipeng Lai , Xiaolong He This is my paper
classification cs.CE MSC 74R1074S05
keywords crackbranchingfragmentationElementModelmultiplecrack-tiptrackingdynamicfractureenergyreleaserateedge-basedsmoothedFEMGPUacceleration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the two-dimensional Crack Element Model from following one crack tip to tracking every crack tip at once. The proposed Multiple Crack-tips Tracking (MCT) algorithm inspects all edge quadrature points on free surfaces, keeps those whose fracture energy release rate exceeds the critical value, and advances the crack at the candidate with the largest rate. With this one set of rules, the method reproduces a Kalthoff-Winkler single crack, crack branching under both traction and displacement loading, and fragmentation of a pressurized cylinder—without the elastic-modulus perturbations other methods use to break symmetry. If the local energy-based ranking is reliable, complicated dynamic fracture patterns no longer require problem-specific branching criteria or added microstructural noise; the same tracking rule drives them all.

What carries the argument

Multiple Crack-tips Tracking (MCT) algorithm: a scanning rule over edge quadrature points in the two-dimensional Crack Element Model. It evaluates the local fracture energy release rate G_{G1-G0} = delta_d * sigma_perp / 2 at all free-surface edge quadratures, applies the critical Gc criterion, then picks the surviving candidate with the largest G, subject to the split-ratio inequality that smooths stress-wave fluctuations. It carries the argument by making crack branching an emergent consequence of ranking local energy release at every possible crack exit, rather than a separately imposed branching criterion.

What would settle it

Run the branching benchmarks on a sequence of refined meshes and compute a path-independent J-integral around each candidate branch; if the J-integral ranking of two competing crack paths ever reverses the G_{G1-G0} ranking that the MCT rule uses, the central claim fails. Alternatively, quantitatively compare simulated PMMA branch angles and crack speeds with high-speed experimental photographs.

Watch

Extended reading notes

Core claim

The central claim is that one algorithmic rule—the Multiple Crack-tips Tracking (MCT) algorithm—can drive the two-dimensional Crack Element Model through single-crack propagation, crack branching, and fragmentation. At each step the algorithm scans every edge quadrature point on all free surfaces, keeps those with fracture energy release rate above the critical value Gc, and advances the crack tip to the candidate with the largest rate, using a split ratio to avoid false choices caused by stress oscillations. The same rule reproduces the Kalthoff-Winkler crack with micro-cracks, branching in a pre-notched plate under both Neumann and Dirichlet loading, and breakup of an internally pressurize

Load-bearing premise

The whole scheme rests on one local formula—the fracture energy release rate at an edge quadrature, developed for a single crack tip—still ranking competing branches correctly when many crack tips interact, and no convergence or J-integral check is given for branched configurations.

Editorial extensions

If this is right

  • The same MCT rule reproduces single cracks, branching, and fragmentation without a velocity threshold or user-defined branching criterion.
  • Micro-cracks along the main path appear automatically and consume extra energy, which the paper identifies as the reason its dissipated-energy curves lie above phase-field and cohesive references.
  • Under displacement (Dirichlet) loading the computed crack patterns agree more closely with experiments than under traction (Neumann) loading, attributed to steadier stress fields.
  • The pressurized-cylinder fragmentation benchmark yields 11–13 major fragments on two unstructured meshes with no modulus perturbation, matching phase-field and cohesive reference counts.
  • All simulations run with GPU acceleration, suggesting the method can move into industrial-scale dynamic fracture workflows.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the local energy ranking is valid at branch points, the same free-surface scanning strategy could extend to three-dimensional crack fronts, where a surface quadrature point with maximum G would play the role of each crack-tip candidate; the paper does not test this.
  • The absence of modulus perturbations raises a question the paper leaves open: whether the discretization itself supplies the symmetry-breaking perturbation that real microstructures do, or whether perfectly symmetric structured meshes would spuriously branch.
  • A direct comparison of G_{G1-G0} with a path-independent J-integral at a branched tip would tell whether the ranking survives where interacting stress fields dominate; the paper reports no such comparison.
  • Because the method introduces micro-cracks even in the single-crack Kalthoff case, its dissipated energy should be treated as an upper bound; recalibrating Gc downward may be needed for quantitative energy matching.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes an extension of the two-dimensional Crack Element Model (CEM) from single crack-tip tracking to Multiple Crack-tips Tracking (MCT-2D-CEM). The method considers edge quadrature points on all free surfaces, evaluates a local fracture energy release rate based on edge stretch and normal stress, and selects new crack paths among candidate tips. The authors validate the approach on four dynamic fracture benchmarks: the Kalthoff-Winkler plate, crack branching under Neumann loading, crack branching under Dirichlet loading, and fragmentation of a pressurized cylinder. They report qualitative agreement with experimental and reference numerical crack patterns, present mesh-dependence studies for some benchmarks, and compare dissipated energy and fragment counts with existing methods. GPU acceleration is used in all simulations.

Significance. If the algorithm is correctly specified and reproducible, the work would be a useful contribution: it offers a finite-element-based, crack-tracking alternative to phase-field and peridynamic methods for dynamic branching and fragmentation, with the practical advantage of GPU acceleration. The qualitative match with experiments and with independent phase-field/cohesive results, and the absence of artificial modulus perturbations in the cylinder example, are genuine strengths. The paper also honestly reports mesh-sensitivity and excess micro-crack dissipation. However, the central algorithmic claim is not yet reproducible from the text, and several load-bearing numerical ingredients are either unspecified or unvalidated in branched configurations.

major comments (4)
  1. [Section 3, 'Multiple Crack-tips Tracking'] The algorithm as written cannot produce branching or fragmentation. The text states that MCT 'considers all edge quadrature points along the free surfaces, selects the edge quadrature points satisfying fracture criteria as current crack-tip candidates and picks the edge quadrature point with the largest fracture energy release rate among current crack-tip candidates as the current crack tip.' This is a global selection of a single tip. The example in Figure 4(b) also picks only one of the three candidates. Nowhere is there an operation that turns one active tip into two, a rule for updating a set of active tips, or a rule for nucleating a branch off an existing crack face. Sections 4.2–4.4, however, present branching and fragmentation patterns that require multiple simultaneously advancing tips. This internal contradiction means the figures cannot be inferred from the described algorithm
  2. [Section 2.2 and Section 3, Eq. for G] The driving-force measure G = delta_d * sigma_perp / 2 is carried over from a single-crack-tip formulation, but no evidence is given that this local quantity remains valid at interacting or branching crack tips. In branched configurations, the stress state at a candidate tip is influenced by the other branch, and a simple edge-stretch/sigma_perp product may mis-rank competing branches. The paper provides no convergence study, no J-integral comparison, and no experimental measurement of, say, branch angles or crack speeds to validate this measure in the branched regime. Since the MCT selection rule depends entirely on this quantity, this is a load-bearing point. A quantitative verification, e.g., comparison with J-integral or stress-intensity-factor-based criteria on a static branched geometry, and a mesh-convergence study of branch patterns, would be needed.
  3. [Section 3, split-ratio criteria] The split ratio gamma is never given a numerical value and appears in inequalities that are questionable as written. For instance, the condition G1 > gamma*(G1 - G2) in Section 2.2 is automatically satisfied for many positive G1, G2 when gamma is small; with gamma in (0,1) the inequality is trivially true because the right-hand side is smaller than G1 whenever G2 > 0. The paper does not state gamma's range, its default value, or its sensitivity. Section 3 repeats the same ambiguity. Without a concrete value or an explicit calibration, the branching criterion is not a well-defined function of the computed fields, and the reported patterns cannot be reproduced or assessed for robustness.
  4. [Sections 4.2–4.4, quantitative validation] The validation is mostly qualitative visual comparison. The dissipated-energy comparisons in Figures 18 and 24 show large differences from reference methods, which the paper attributes to extra micro-cracks, but no quantitative convergence of crack path, branch angle, number of branches, or fragment size distribution with mesh refinement is provided. The fragment-count comparison in Table 1 spans a wide range across methods, and the present results agree with one reference at one mesh and another at another mesh, so the fragment count is not a decisive validation. The paper should report quantitative metrics of the crack patterns and demonstrate that the MCT selection rule converges or is at least stable as the mesh is refined.
minor comments (6)
  1. [Abstract] Typo: 'MCR-2D-CEM' should be 'MCT-2D-CEM'.
  2. [Section 2.2] The text refers to 'Griffth' in the introduction; correct to 'Griffith'.
  3. [Section 3, Figure 4] The text says 'including green points, orange points and the red point', but the colors in the figure are not clearly labeled. It would help to mark the candidate numbers directly on the figure.
  4. [Section 4.1] The problem dimensions and notch geometry for the Kalthoff-Winkler plate are not given explicitly in the text; only boundary conditions and material properties are provided. A dimensioned figure or table would aid reproducibility.
  5. [Section 4.4, Table 1] The table lists fragment counts for various methods but does not define 'major fragments' or specify the criterion used to count them. This should be clarified.
  6. [Conclusions] The final paragraph correctly identifies the need for a 'more robust and effective stress regularization scheme'. This limitation should be moved to the front of the validation discussion, because it directly affects the reliability of the branching patterns reported in Sections 4.2–4.4.

Circularity Check

0 steps flagged · score 1.0 of 10

No definitional circularity: the MCT rule is explicitly defined, and the branching/fragmentation claims are validated against external experiments and independent numerical methods, not fitted or derived from the algorithm's own outputs.

full rationale

The paper's central derivation chain is the definition of the Multiple Crack-tips Tracking (MCT) rule on top of the local fracture-energy-release-rate formula G = (delta_d * sigma_perp)/2 from the authors' prior CEM work. That formula is re-stated in Section 2.2 and used as a local crack-driving measure; it is not defined in terms of the final crack patterns it is used to predict. The MCT rule itself is explicitly described in Section 3: evaluate G at all free-surface quadrature points, keep those exceeding G_c as candidates, and pick the largest-G candidate as the current crack tip. This is a selection rule, not a construction that already contains branching or fragmentation as an output. The benchmark results are compared against experimental observations (e.g., Ozbolt et al. for the Dirichlet compact-tension tests) and against independent numerical methods (phase-field, cohesive, cracking-node, local-damage models), so the validation is external and not fitted by construction. There are self-citations to the authors' 2D and 3D CEM preprints, but the paper reproduces the essential formulation rather than deferring the entire derivation to those citations, so the self-citations are not load-bearing in a circular sense. The main weaknesses are non-circular: the split ratio gamma is never given a numerical value or sensitivity study, and the MCT text says it 'picks' a single current crack tip while the figures show multiple simultaneously advancing tips, indicating an omitted algorithmic detail. These are reproducibility and completeness concerns, not evidence that the predictions reduce to the model's inputs. Overall, the derivation is self-contained against external benchmarks, so circularity is minimal.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central method rests on the ES-FEM discretization, the local energy release rate criterion, and the split-ratio heuristic. The only freely adjustable numerical parameter reported is gamma, and its value is not given. Material constants are taken from the cited benchmark literature and are treated as inputs.

free parameters (1)
  • split ratio gamma
    Used in Section 2.2 and Section 3 crack-path selection, for example G2 > gamma*(G2-G1). No value or tuning rule is reported, yet the branching decision depends on it.
assumptions (4)
  • standard math ES-FEM strain smoothing formulation is a valid discretization for dynamic fracture
    The paper builds on Liu et al. (2009) ES-FEM; accepted background, but its use for post-crack stress fields is not independently examined here.
  • domain assumption Fracture growth is governed by a local critical energy release rate criterion G > G_c
    Section 2.2 uses G_c as the sole fracture criterion; rate dependence, process zone, and cohesive effects are ignored.
  • ad hoc to paper The edge-stretch based energy release rate delta_d * sigma_perp / 2 is valid at interacting and branching crack tips
    This formula is inherited from single-tip CEM and applied to multiple tips in Section 3 without independent validation.
  • ad hoc to paper The split ratio gamma suppresses stress fluctuations without altering physical branch selection
    No value or validation is given; it is central to the candidate-selection inequalities in Section 3.

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Cite this review

Pith. "Pith review of Revisit of Two-dimensional CEM on Crack Branching: from Single Crack-tip Tracking to Multiple Crack-tips Tracking." pith.science (2026). https://pith.science/paper/YP3FAMH5

@misc{pith2026250901827,
  author       = {Pith},
  title        = {Pith review of: Revisit of Two-dimensional CEM on Crack Branching: from Single Crack-tip Tracking to Multiple Crack-tips Tracking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YP3FAMH5}},
  note         = {Machine review of arXiv:2509.01827}
}
read the original abstract

In this work, a Multiple Crack-tips Tracking algorithm in two-dimensional Crack Element Model (MCT-2D-CEM) is developed, aiming at modeling and predicting advanced and complicated crack patterns in two-dimensional dynamic fracturing problems, such as crack branching and fragmentation. Based on the developed fracture energy release rate formulation of split elementary topology, the Multiple Crack-tips Tracking algorithm is proposed and a series of benchmark examples are provided to validate effectiveness and efficiency in modeling crack branching and fragmentation. Besides, the proposed MCR-2D-CEM can still model single crack propagation but extra micro-cracks are introduced. GPU acceleration is employed in all two-dimensional simulations, providing high computational efficiency, consistency, and accuracy.

Figures

Figures reproduced from arXiv: 2509.01827 by the authors.

Figure 1
Figure 1. The studied domain is discretized by red points which represent gaussian quadrature points. Notation 𝐴𝑖 represents element ID and 𝑔𝑖 represents edge quadrature ID: (a). ES-FEM formulation of Constant Strain Triangle element, totally 8 related elements and 16 edge quadrature points; (b). ES-FEM formulation of bilinear Quadrilateral element, totally 9 related elements and 24 edge quadrature points. The blue and green … view at source ↗
Figure 2
Figure 2. CST element with single crack tip tracking in two-dimensional Crack Element Model: (a). computation of fracture energy release rate  in CST element, in which 𝐺𝑖 is edge quadrature point ID, 𝐸𝑖 is element ID, 𝑁𝑖 is node ID, 𝝈 ⟂ is normal projection of Cauchy stress 𝝈 at quadrature location; (b). single crack propagation in CST element, in which red star represents current crack tip, blue stars represent crack tip ca… view at source ↗
Figure 3
Figure 3. Quadrilateral element with single crack tip tracking in two-dimensional Crack Element Mode, two possible crack patterns: crack pattern I illustrates a crack path forms from quadrature 𝐺0 to 𝐺1 or 𝐺2 at the adjacent edges; crack pattern II illustrates a crack path forms from quadrature 𝐺0 to 𝐺3 at opposite edge. such as loading rate, material brittleness, and structural geometry. On the theoretical side, branching is… view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: (a). Single Crack-tip Tracking algorithm in two-dimensional CEM: red solid circle denotes current crack tip while black solid circles denote past crack tips, the solid line connecting solid circles denotes single crack path; (b). Multiple Crack-tips Tracking algorithm …
Figure 5
Figure 5. Figure 5: (a). The whole geometry, boundary condition of Kalthoff-Winkler plate experiment; (b). Upper half of the Kalthoff-Winkler plate. To comprehensively test the MCT-2D-CEM algorithm in capturing single crack pattern, three different meshes are selected to implement simulat…
Figure 6
Figure 6. Figure 6: Three representative triangle meshes are illustrated: (a). irregular mesh with 7212 nodes and 14158 elements; (b). irregular mesh with 7173 nodes and 14086 elements; (c). regular mesh with 6727 nodes and 13199 elements. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p0…
Figure 7
Figure 7. Figure 7: The final crack patterns of the three representative meshes are illustrated: (a). irregular mesh with 7212 nodes and 14158 elements; (b). irregular mesh with 7173 nodes and 14086 elements; (c). regular mesh with 6727 nodes and 13199 elements. For example, in Figure.8(c…
Figure 8
Figure 8. Figure 8: The final crack patterns comparison between mutiple crack tips tracking algorithm and single crack tip tracking algorithm in the three representative meshes are illustrated, in which the red solid line is from multiple crack tips tracking algorithm and the green solid …
Figure 9
Figure 9. Figure 9: Stress fields and crack propagation patterns for the grid of 7212 nodes and 14158 elements at times at (a). 𝑡 = 25.5 𝜇𝑠, (b). 𝑡 = 45 𝜇𝑠, (c). 𝑡 = 64 𝜇𝑠, (d). 𝑡 = 90 𝜇𝑠. though the external loading and stress distribution are symmetric, the crack tip becomes unstable as…
Figure 10
Figure 10. Figure 10: Stress fields and crack propagation patterns for the grid of 7173 nodes and 14086 elements at times at (a). 𝑡 = 25.5 𝜇𝑠, (b). 𝑡 = 45 𝜇𝑠, (c). 𝑡 = 64 𝜇𝑠, (d). 𝑡 = 90 𝜇𝑠. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Stress fields and crack propagation patterns for the grid of 6727 nodes and 13199 elements at times at (a). 𝑡 = 25.5 𝜇𝑠, (b). 𝑡 = 45 𝜇𝑠, (c). 𝑡 = 64 𝜇𝑠, (d). 𝑡 = 90 𝜇𝑠 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Geometric dimensions of the two-dimensional pre-notched plate under traction-applied boundary condition. (unit: mm) micro-cracks are captured, which demonstrates the proposed multiple crack tips tracking algorithm is tracking all possible crack tips rather than single…
Figure 13
Figure 13. Figure 13: Three different two-dimensional meshes of the plate with Constant Strain Triangle element are illustrated: (a). 37199 elements and 18981 nodes; (b). 9287 elements and 4835 nodes; (c). 2343 elements and 1268 nodes. snaps are shown in Figure.17. By comparing crack patte…
Figure 14
Figure 14. Figure 14: The final crack patterns of three representative Constant Strain Triangle grids are illustrated: (a). 37199 elements and 18981 nodes; (b). 9287 elements and 4835 nodes; (c). 2343 elements and 1268 nodes. et al. (2012)), local damage model by Bui et al. (Bui et al. (20…
Figure 15
Figure 15. Figure 15: The crack pattern evolution of the mesh with 36922 elements and 188425 nodes are provided: (a). time 𝑡 = 14.4 𝜇𝑠; (b). time 𝑡 = 33.6 𝜇𝑠; (c). time 𝑡 = 56 𝜇𝑠; (d). time 𝑡 = 80 𝜇𝑠. note that different from three-dimensional CEM that has no any explicit criteria of crack…
Figure 16
Figure 16. Figure 16: The crack pattern evolution of medium mesh with 9287 elements and 4835 nodes are provided: (a). time 𝑡 = 14.4 𝜇𝑠; (b). time 𝑡 = 33.6 𝜇𝑠; (c). time 𝑡 = 56 𝜇𝑠; (d). time 𝑡 = 80 𝜇𝑠. In the dynamic fracture study of concrete compact tension specimens, the transition from …
Figure 17
Figure 17. Figure 17: The crack pattern evolution of coarse mesh with 2343 elements and 1268 nodes are provided: (a). time 𝑡 = 14.4 𝜇𝑠; (b). time 𝑡 = 33.6 𝜇𝑠; (c). time 𝑡 = 56 𝜇𝑠; (d). time 𝑡 = 80 𝜇𝑠. In contrast, when the loading rate is increased significantly, such as 3.318 𝑚∕𝑠 or 3.993…
Figure 18
Figure 18. Figure 18: Comparison of dissipated energy among three representative grids and three reference results(Bui et al. (2022), Borden et al. (2012), Hirmand and Papoulia (2019)). material to dissipate the surplus strain energy more broadly, reducing the energy density at any single …
Figure 19
Figure 19. Figure 19: (a). The model geometric dimensions (unit: mm) and boundary conditions of compact tension experiment. (b). The mesh with 11939 Nodes and 23413 Elements; (c). The mesh with 7643 Nodes and 14914 Elements; (d). The mesh with 3563 Nodes and 6874 Elements. shown under thre…
Figure 20
Figure 20. Figure 20: (a-c). experimental crack patterns under applied velocity of 1.375 𝑚∕𝑠, 3.318 𝑚∕𝑠 and 3.993 𝑚∕𝑠 (Ožbolt et al. (2013)); (d-f). numerical crack patterns of 𝑁 = 11939, 𝐸 = 23413 model under applied velocity of 1.375 𝑚∕𝑠, 3.318 𝑚∕𝑠 and 3.993 𝑚∕𝑠; (g-i). numerical crack p…
Figure 21
Figure 21. Figure 21: Max. principle stress contour and crack patterns evolution of fine mesh 𝑁 = 11939, 𝐸 = 23413 with 𝑣0 = 3.993 𝑚∕𝑠 at times at (a). 𝑡 = 60 𝜇𝑠, (b). 𝑡 = 84 𝜇𝑠, (c). 𝑡 = 144 𝜇𝑠, (d). 𝑡 = 300 𝜇𝑠. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p022_21.png]
Figure 22
Figure 22. Figure 22: Max. principle stress contour and crack patterns evolution of medium mesh 𝑁 = 7643, 𝐸 = 14914 with 𝑣0 = 3.993 𝑚∕𝑠 at times at (a). 𝑡 = 60 𝜇𝑠, (b). 𝑡 = 84 𝜇𝑠, (c). 𝑡 = 144 𝜇𝑠, (d). 𝑡 = 300 𝜇𝑠. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p022_22.png]
Figure 23
Figure 23. Figure 23: Max. principle stress contour and crack patterns evolution of coarse mesh 𝑁 = 3563, 𝐸 = 6874 with 𝑣0 = 3.993 𝑚∕𝑠 at times at (a). 𝑡 = 54 𝜇𝑠, (b). 𝑡 = 78 𝜇𝑠, (c). 𝑡 = 144 𝜇𝑠, (d). 𝑡 = 300 𝜇𝑠. Through the comparative evaluation of the proposed Multiple Crack-tips Tracki…
Figure 24
Figure 24. Figure 24: Comparison of dissipated energy 𝑈𝑑 among three representative meshes under three different applied velocities. 4.4. Fragmentation in pressured cylinder Crack fragmentation is the process by which a material under high loading rates or dynamic conditions develops multi…
Figure 25
Figure 25. Figure 25: (a). The whole geometry, boundary condition of pressure-loaded cylinder; (b). internal impact pressure variation with respect to time. (a) (b) (c) (d) (e) (f) [PITH_FULL_IMAGE:figures/full_fig_p024_25.png]
Figure 26
Figure 26. Figure 26: The discretization and final crack patterns of two representative Constant Strain Triangle meshes are illustrated: (a-b). 29598 elements and 15161 nodes; (d-e). 52916 elements and 26942 nodes. And (c,f). reference crack patterns of two meshes (Geelen et al. (2019)). Y…
Figure 27
Figure 27. Figure 27: Development of maximum principal stress through crack propagation process in the mesh of 52916 elements and 26942 nodes, at times: (a). 30 𝜇𝑠; (b). 40 𝜇𝑠; (c). 54 𝜇𝑠; (d). 60 𝜇𝑠 [PITH_FULL_IMAGE:figures/full_fig_p025_27.png]

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