# Open Review of "How to introduce an initial crack in phase field simulations to accurately predict the linear elastic fracture propagation threshold?"

Flavien Loiseau, Veronique Lazarus, Tushar Kanti Mandal, Vinh Phu Nguyen

► To cite this version:

Flavien Loiseau, Veronique Lazarus, Tushar Kanti Mandal, Vinh Phu Nguyen. Open Review of "How to introduce an initial crack in phase field simulations to accurately predict the linear elastic fracture propagation threshold?". 2025, 10.46298/jtcam.15198. hal-05080459

HAL Id: hal-05080459

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Submitted on 21 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Identifiers

Open Review

OAI hal-05080459

Reviewed Article

DOI 10.46298/jtcam.15198

Licence

CC BY 4.0

©The Authors

# Review of “How to introduce an initial crack in phase field simulations to accurately predict the linear elastic fracture propagation threshold?”

Flavien LOISEAU<fnref n="1" />, Véronique LAZARUS<fnref n="1" />, Tushar KANTI MANDAL<fnref n="2" />,R, and

Vinh Phu NGUYEN<fnref n="3" />, E

## Review of version 1

Permalink: hal-04931758v1

## Authors

We would like to thank both reviewers for their thoughtful and constructive feedback on our manuscript, “How to introduce an initial crack in phase-field simulations to accurately predict the linear elastic fracture propagation threshold?”.

We appreciate the time and effort invested in evaluating our work and providing valuable insights.

We are pleased to note that both reviewers found the topic timely and relevant, and we are committed to addressing the concerns raised to further strengthen our study.

## Reviewer 1 (Tushar KANTI MANDAL)

## Reviewer

The manuscript “How to introduce an initial crack in phase-field simulations to accurately predict the linear elastic fracture propagation threshold?” by Flavien Loiseau and Véronique Lazarus addresses an important and often underexplored aspect of phase-field modelling in fracture mechanics: the influence of initial crack implementation on the accuracy of fracture propagation threshold predictions.

This topic is both timely and relevant. While phase field methods are increasingly used to simulate brittle fracture, how pre-existing cracks are introduced can significantly bias the mechanical response. The authors present a thorough and systematic investigation of several commonly adopted techniques, including geometric and embedded phase field approaches, with variations in crack thickness and boundary conditions. These methods are rigorously benchmarked against a sharp crack model based on Griffith’s theory, using path-following strategies to maintain equilibrium and make comparisons meaningful.

A key strength of the study lies in its demonstration that introducing an infinitely thin crack—whether geometrically or via a narrow phase field—results in artificial toughening due to the need for crack widening during early propagation. This behaviour leads to an overestimation of the critical load. The authors identify two techniques, GEO-T1-WHL and PHA-T1, which consistently yield accurate, unbiased results that match the sharp crack reference solution.

The manuscript is well-written and well-structured. The visual comparisons, force-displacement plots, and discussion of numerical performance provide a strong foundation for the conclusions. The extension of the analysis to unstructured meshes further confirms the robustness of the recommended approaches. The brief but thoughtful discussion on potential extensions to curved

<footnote n="1">¹ IMSIA - Institut des Sciences de la Mécanique et Applications Industrielles</footnote>

<footnote n="2">² Department of Engineering Science, University of Oxford, Oxford</footnote>

<footnote n="3">³ Department of Civil Engineering, Monash University, Clayton, Victoria, Australia</footnote>

<footnote>R Reviewer</footnote>

<footnote>E Editor</footnote>
cracks and history-variable models points towards valuable future work.

In summary, this is a well-executed and practically valuable study that will be of interest to both computational modellers and applied mechanicians working with phase field methods. It offers clear guidance on best practices for crack initialization and provides insights that are likely to improve the reliability of future simulations.

**Authors** We are grateful for the positive feedback and the recognition of the importance and relevance of our study. We are pleased that the reviewer found our investigation thorough and systematic, and that our benchmarking against a sharp crack model was rigorous.

Reviewer 2 (Anonymous)

**Reviewer** The manuscript presents a systematic investigation of different methods for introducing initial cracks in phase-field fracture simulations, focusing on their impact on accurately predicting the linear elastic fracture propagation threshold. By employing path-following techniques, the study highlights how improper crack initialization can lead to artificial toughening effects and provides recommendations to mitigate these issues. The work is insightful and contributes valuable guidance to the field; however, major revisions are required before acceptance.

**Authors** We appreciate the reviewer's acknowledgment of the systematic nature of our investigation and the insightful contributions our work brings to the field of phase-field fracture simulations. We understand the importance of addressing the concerns raised to ensure the manuscript meets the standards required for acceptance.

**Reviewer** The authors state that the geometric crack models (GEO-To-NEU and GEO-T1-NEU) result in unrealistically high fracture loads for the single-edge notch tension test. However, previous phase-field studies have used similar initial crack models without reporting such discrepancies. It would be beneficial for the authors to validate their findings by comparing them with results from existing literature.

**Authors** Unfortunately, those results are not observed in most of the literature as no path-following techniques are employed. As the snap-back is not captured, those studies cannot conclude on this overshoot. Still, Klinsmann et al. (2015) observed that the critical energy release rate at the start of crack growth was larger than the expected one with a crack initialization method similar to our GEO-To-NEU. Moreover, Singh et al. (2016), which employs path-following techniques, also show similar results.

To address this comment, we have modified Section 5 to clarify how path-following techniques enable the capture of snap-back behavior and highlight the artificial overshoot in the force-displacement curve. Furthermore, we have added references to studies that directly or indirectly support these findings.

**Reviewer** To further strengthen the study's contributions, the authors are encouraged to validate their numerical results against experimentally observed load-displacement curves. This comparison would provide deeper insight into the influence of initial crack considerations on fracture predictions and enhance the credibility of the findings.

**Authors** Before proceeding with experimental comparisons, we aim to ensure that the numerical methods used to solve the regularization of Linear Elastic Fracture Mechanics (LEFM) indeed converge towards LEFM. To achieve this, we demonstrate that phase-field fracture, when parameterized similarly to LEFM, accurately reproduces LEFM results. This step is crucial as it allows us to confirm that there are no numerical biases in our approach. This work serves as a milestone towards a comprehensive comparison with experimental results (see previous work in our group: Zhai (2023)).

**Reviewer** In Figures 4(a) and 5(a), the load does not drop to zero even after failure, which differs from the results reported by Miehe et al. (2010). The authors may consider providing a discussion on this observation to clarify potential reasons for the deviation.

**Authors** We appreciate the reviewer's observation and recognize that our initial explanation was unclear. The load in Figures 4(a) and 5(a) does not drop to zero after failure because the crack has not fully
propagated; a small ligament remains. In practice, we stop the simulation when the elastic energy
drops to 0.2 % of its maximum value during crack growth. Since our primary focus is on the peak
load, this does not significantly impact our analysis. Further continuing the simulation would
make the load drop to zero. Additionally, the validity of LEFM as the crack tip approaches the
boundary is a topic that warrants further discussion.

To address this comment, we have clarified in Section 5.2 that we stop the simulation before
full crack propagation.

**Reviewer** The manuscript suggests that the geometric crack model with a crack face phase-field value set to 1 (GEO-T1-WHL) performs better than the simpler geometric crack model (GEO-T1-NEU). Since Neumann boundary conditions typically ensure a stress-free crack face, it would be helpful for the authors to elaborate on why these additional boundary conditions are necessary.

**Authors** Our explanation are unclear in the draft. When discussing the geometric crack model, we were referring to Neumann boundary conditions specifically for the crack phase field, not the displacement field. For the displacement field, stress-free boundary conditions are indeed applied to ensure a stress-free crack face. To account for this comment, we updated Section 4.1 and Figure 1 to make this point clearer.

**Reviewer** The authors are encouraged to extend their study by incorporating additional numerical examples, particularly cases involving more complex crack paths, such as mixed-mode fracture propagation. This would provide a more comprehensive assessment of the impact of different initial crack considerations.

**Authors** Extending this study to more complex crack paths, such as mixed-mode fracture propagation, is indeed an important consideration. While we recognize that such cases would provide a more comprehensive assessment, incorporating mixed-mode fracture would significantly increase the complexity of the study. This extension introduces several additional considerations, including the need for further LEFM developments to predict the crack path for reference results. These considerations are beyond the scope of our current work, which aims to establish a baseline for studying crack initialization techniques. However, we acknowledge the importance of addressing mixed-mode fracture in future research to further validate and refine our findings. Additionally, we are working on a separate publication focusing on path-following techniques that will include mixed-mode cases.

1 Editor's comments (Vinh Phu NGUYEN)

**Editor** This paper addresses a very practical and technically important problem in computational fracture mechanics: how to correctly implement initial cracks in phase field fracture models so that the simulated propagation threshold matches the expected physical behavior, especially in the Griffith framework of Linear Elastic Fracture Mechanics (LEFM). The conclusion is actionable: to avoid artificial toughening, either (i) use a fully damaged one-element-wide band as the initial crack, or (ii) explicitly mesh the crack as a slit and impose fully broken conditions. This guidance is highly useful for researchers and engineers using phase field methods for fracture.

References

Klinsmann, M., D. Rosato, M. Kamlah, and R. M. McMeeking (2015). An assessment of the phase field formulation for crack growth. DOI: 10.1016/j.cma.2015.06.009.

Miehe, C., M. Hofacker, and F. Welschinger (2010). A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. *Computer Methods in Applied Mechanics and Engineering*. [DOI].

Singh, N., C. V. Verhoosel, R. de Borst, and E. H. van Brummelen (2016). A fracture-controlled path-following technique for phase-field modeling of brittle fracture. *Finite Elements in Analysis and Design*. DOI: 10.1016/j.finel.2015.12.005.

Zhai, X. (2023). Crack propagation in elastic media with anisotropic fracture toughness: experiments and phase-field modeling. PhD thesis. ENSTA - Institut Polytechnique de Paris.
OAI: tel-04528865.

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