Variable Dilation Angle Models in Rocks, a Review

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article comprehensively reviews 10 expansion angle models developed in the past 20 years, analyzes their equations, parameters, and application limitations, fills the gap in the systematic review of expansion angles in rock mechanics, and has direct application value in tunnel design, mining engineering, and other fields, with particular emphasis on the influence of confining pressure and plastic strain. The research results have certain significance, but there are still some issues, as follows:

(1) Chapter 3 describes the constitutive models one by one, but lacks standardized comparisons, such as the number of parameters, applicable rock types, error analysis, etc. Please supplement relevant content;

(2) There are multiple grammar and spelling errors in the text, such as the alternating use of "dilation angle" and "dilatancy angle" in the text, which affect the professionalism. The author is requested to further check and revise it;

(3) Section 1.2 requires referencing newer research literature, and the reference format in the text is not standardized, with some content missing. The author is requested to check all entries to ensure the completeness of the author, year, and DOI;

(4) Figure 8 compares the difference between the constitutive model and the actual data, but the main text does not delve into the reasons for this difference, such as the fitting deviation of sandstone. Please ask the author to further analyze;

(5) Some of the images in the text are unclear and have inconsistent formatting (font size, font, line width, etc.), which affects analysis and recognition. Please check and replace them.

Author Response

This article comprehensively reviews 10 expansion angle models developed in the past 20 years, analyzes their equations, parameters, and application limitations, fills the gap in the systematic review of expansion angles in rock mechanics, and has direct application value in tunnel design, mining engineering, and other fields, with particular emphasis on the influence of confining pressure and plastic strain. The research results have certain significance, but there are still some issues, as follows:

Thank you for your comprehensive comments, we have considered them to improve our manuscript. Any changes performed in the text to consider your comments are marked in blue.

(1) Chapter 3 describes the constitutive models one by one, but lacks standardized comparisons, such as the number of parameters, applicable rock types, error analysis, etc. Please supplement relevant content;

Thank you for the comment. We tried to summarize the most important features of each model in the summary table (Table 1), including the number of parameters, the initial point of plastic strains and the applicable rock types. We have revised the main features in the table to ensure all this information is available for each model.

On the other hand, as we indicated in the cover letter, we have also performed a comparison of the ten models with our own experimental data, but the final manuscript including this literature review and the model comparison resulted too long, so we decided to separate them in two manuscripts. Second manuscript (which includes the error analysis you indicate) will be sent to a journal after this one is accepted for publication.

(2) There are multiple grammar and spelling errors in the text, such as the alternating use of "dilation angle" and "dilatancy angle" in the text, which affect the professionalism. The author is requested to further check and revise it;

We appreciate your comment. We have revised all the document and used “dilation angle” instead of “dilatancy angle”. In any case, in the field of rock mechanics both terms are used to designate the same concept, so they are synonymous, but we appreciate your comment in the sense that the manuscript maintains a more consistent language.

We have also used the AI-based software Paperpal v. 2.129.3.0 for the purposes of checking English grammar and usage.

(3) Section 1.2 requires referencing newer research literature, and the reference format in the text is not standardized, with some content missing. The author is requested to check all entries to ensure the completeness of the author, year, and DOI;

Thank you for your comment. Only the 35% of the bibliographical references are previous to 2010 (and all of them relevant since they put the basis of the actual development), so the 65% of the cited literature comes from the last 15 years. Considering that the dilatancy is not a common topic of research in rock mechanics, we consider that the references are reasonably up to date. In any case, if you know any relevant reference(s) we missed, but should have considered, do not hesitate to indicate it(them), we will be happy to include it(them) in the manuscript if it (they) is (are) appropriate.

We have revised all the bibliographical references in the text. The format used for in-text references is the number between square brackets ([XX]), as it is indicated by the instructions for authors of the journal, but, in some places, we wanted to explicitly include the authors names to better identify the models. We have also checked that all the required information is available on the references section, regretfully, not all the references feature a DOI (like those of the ARMA symposia or the very old ones).

(4) Figure 8 compares the difference between the constitutive model and the actual data, but the main text does not delve into the reasons for this difference, such as the fitting deviation of sandstone. Please ask the author to further analyze;

As commented in the answer to comment (1), we tried to not perform any analysis on the fitting goodness of each model, because we plan to publish such analysis in another manuscript using our own experimental data. We just wanted to present the models as they are and cite the original reference.

In any case, and answering your particular question, the difference between actual experimental data and W&D model and Z&C model in the case of the weak sandstone (Fig. 8.c) could come from the low number of experimental data points and the high initial scatter of the experimental results (roughly between 30º to 55º in the four initial points) at the lowest confining pressure (0.345 MPa).

(5) Some of the images in the text are unclear and have inconsistent formatting (font size, font, line width, etc.), which affects analysis and recognition. Please check and replace them.

Thank you for your comment. This is an aspect we have already discussed before sending the manuscript. The reason behind your comment is that each figure comes from the original paper of the authors. After analysing the possibility of reformatting each figure, we considered that it was better to maintain the original format since we could miss some detail. We made our best on improving the quality of the figures (Figures 3, 4, 5, 6 and 10 were completely redrawn since they had very low quality) and modified the size of each figure to get a more consistent format and a better readability of the content.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents a comprehensive review of ten variable dilation angle models in rock mechanics developed over the past two decades. The authors systematically analyze how the dilatancy angle depends on plastic strain history and confining stress, examining models ranging from simple approaches with few parameters to complex formulations with multiple coefficients. Please see my comments below:

1. More details needed about the experimental data used to validate different models.
2. As this is a review paper, the references in current version is still limited, and please add more review papers.
3. Some figures (especially Figure 3) could benefit from higher resolution. And Figure captions could be more descriptive.
4. The paper lacks discussion of emerging challenges.

Author Response

This manuscript presents a comprehensive review of ten variable dilation angle models in rock mechanics developed over the past two decades. The authors systematically analyze how the dilatancy angle depends on plastic strain history and confining stress, examining models ranging from simple approaches with few parameters to complex formulations with multiple coefficients. Please see my comments below:

Thank for your comments. We have considered them to improve our manuscript. Any changes performed in the text to consider your comments are marked in green.

(1) More details needed about the experimental data used to validate different models.

Thank you for your comment. Each model has been validated by the original authors with different experimental data, some of them are own data and other models used available data in the literature. We tried to summarize the main characteristics of the models indicating aspects like the type of rock used by each model or the point considered as the beginning of plastic strains in Table 1. Any further detail on each model can be consulted in the original publication, adding more details of each model would make this manuscript too long.

Accordingly, as we indicated in the cover letter, we have also performed a comparison of the ten models with our own experimental data, but the final manuscript including this literature review and the model comparison resulted too long, so we decided to separate them in two manuscripts. Second manuscript (which will include a more detailed section about the used experimental data) will be sent to a journal after current one is accepted for publication.

(2) As this is a review paper, the references in current version is still limited, and please add more review papers.

Thank you for your comment, we also think that a good bibliographical review should begin in a number near a hundred references. The performed literature review revealed many other papers not cited in the manuscript but related to the dilatant behaviour of rocks because they did not add a new model but used existing ones. The number of references may appear low, but although there are a significant number of papers including “dilation angle” and “rocks” as keywords (305 according to scopus), most of them are referred to the effect of using dilation on numerical models. Up to the knowledge of the authors there is not so much research on the development of variable dilation angle models.

In order to increase the number of references, we could explore some topics aside the variable dilation angle models in rocks, like the effect of considering variable dilatancy in numerical models of excavations, or the effect of combining temperature or fluids on the determination of the dilation angle, but in that case we think we would be moving away of the main focus of the paper (presenting the available variable dilation angle models for rocks) and would artificially enlarge the length of the paper.

In any case, we have added 7 more bibliographical references (#60 to #66) derived from your comment #(4), and, if you know some relevant papers we have missed, do not hesitate on indicating them, we would be happy to incorporate them in the manuscript if they are appropriate.

(3) Some figures (especially Figure 3) could benefit from higher resolution. And Figure captions could be more descriptive.

Thank for your comment. We have made our best to improve the quality of the images and clarify the captions. Figures 3, 4, 5, 6 and 10 were completely redrawn and were also included in the manuscript in a vectorial format.

(4) The paper lacks discussion of emerging challenges.

Thank you for your comment. We tried to slightly touch this point in the concluding remarks section but without deepening in it because we are currently working on it. In any case, we added next paragraphs to the Concluding Remarks section:

“These improvements could begin with the standardization of the procedures for the determination of the dilation angle. Until now, the ISRM Suggested Method for the com-plete stress-strain curve for intact rock in uniaxial compression [60] is the document that indicates how to perform uniaxial compression tests reaching the residual state, but there is no other Suggested Method for triaxial tests reaching the residual phase or a Suggested Method for determining the dilation angle. In addition, the existing Suggested Method does not describe how to decompose the total strain into its elastic and plastic compo-nents. It is known that plastic strains begin at a stress known as Crack Initiation (CI), but they are negligible compared to those after the Crack Damage stress (CD). Which stress should be considered as the initial point for the plastic strains? Considering the CI as the initial point for plastic strains is strictly the correct decision, but making so will result in negative dilation angles according to the Vermeer and De Borst [35] equation (Equation 9) because the rock is still contracting between the CI and the CD. Therefore, it seems rea-sonable to consider CD as the initial point of the plastic strain, disregarding those between CI and CD. On the other hand, disregarding the plastic strains between CD and the peak strength may induce errors because it is precisely around this zone where the peak dila-tion angle is attained.

One of the aspects that may be identified is that, usually, the number of available ex-perimental dilation angle data points is limited (Figures 6, 9, and 10, for instance), espe-cially in the surroundings of the peak dilation angle. This may be because conventional testing equipment may be inappropriate for correctly capturing the failure of rock speci-mens. Fairhurst and Hudson [60] pointed out that the testing setup requires high-stiffness load frames, fast servocontrol, correct choice of feedback signal and strain measurement transducers, specimen preparation techniques, etc., but also indicated that testing some high-strength rocks becomes at best difficult. The unfulfilment of all these requirements may result in the loss of data points, especially immediately after the peak strength, or even recording the whole system (testing equipment and rock specimen) characteristics, which may result in an inaccurate calculation of the dilation angle.

Another assumption generally considered is that the elastic components of the strains are calculated using the pre-peak elastic moduli, irrespective of the analyzed phase (pre- or post-peak). Some studies, like that of Hou and Cai [61] for instance, show that Young’s modulus and Poisson’s ratio are not constant all along the stress-strain curve, but according to the CWFS model [28, 62, 63] they depend on the relation between cohe-sive and frictional components of the strength. If one assumes that the transition from peak to residual strength and the residual phase itself are cohesion loss processes, it seems reasonable to also consider the dependence of the elastic moduli on plastic strain. It is also important to note that the cyclical loading-unloading used to identify the elastic and plastic components of strains (as shown in Figure 1) induces increasing damage to the rock each time a cycle is performed, so the number of cycles performed per test should also be standardized to make test results comparable.

Apart from these practical concerns, there are still some other aspects that have not been addressed. As previously stated, it is known that the dilation angle depends on the plastic strain history and confining stress, and some efforts have been made to identify its dependence on the structure by the research group of the authors [64-66]. However, there have not been any studies on the dilation angle dependence on size or the effects of other variables usually considered in determining the mechanical properties of rocks, such as the temperature or water content.

The standardization of the procedures for determining the dilation angle could be a good starting point to address these dependencies of the dilation angle as well as new ones that will surely arise. More standardized experimental results will allow researchers to set the limits of existing models or to propose new models that better reflect the dilatant behavior of rocks.”

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Editor,

Thank you for the opportunity to review the manuscript titled “Variable dilation angle models in rocks, a review”. Below is my detailed feedback:

This manuscript is a solid and valuable contribution to the geomechanics community. It effectively synthesizes, analyzes, and compares a range of important variable dilatancy models, providing a useful resource for both researchers and practitioners. The logical structure, clarity, and high quality of the figures further enhance its value.

1) Clarity, relevance, and contribution to knowledge

Clarity: The review is well-structured and clear. The introduction establishes the context, Section 2 explains stress-strain behavior and key terms (pre-peak and post-peak), and Section 3 details eight variable dilatancy models. The conclusion (Section 4) summarizes the information and outlines future directions. The paper offers a comprehensive overview of variable dilation angle models in rock mechanics.

Relevance: The subject is highly relevant to the fields of geomechanics and mining engineering. The dilation angle is a crucial parameter for the numerical modeling of rock behavior, particularly in applications like the design of tunnels and mining excavations.

Gap in knowledge: While the paper doesn't identify a specific, unaddressed knowledge gap, it does suggest there is room for improvement in our understanding and modeling of dilatancy behavior. The conclusion states that "there is still room for improvement in our knowledge of dilatancy" and raises important questions for future research, such as the influence of sample size and rock structure.

2) Relevance and interest to the scientific community: The manuscript mentions previous reviews, but focuses on a more detailed and updated analysis of variable dilatancy models. The fact that this review compares multiple recent models (some published as recently as 2021) and synthesizes them into a detailed table (Table 1) makes it highly relevant and useful, avoiding redundant publication. Its value lies not merely in summarizing but in its comparative analysis of different approaches.

3) Timeliness and relevance of references: The reference list is extensive and includes recent publications (2020, 2021, 2022). However, it also contains essential older references, which are critical for tracing the evolution of these concepts from the early 1970s. This combination of historical and recent references provides substance and a proper perspective on the field's development. All references are directly relevant to the topic of dilation angle and rock mechanics, and each model discussed is supported by the corresponding citation.

4) Coherence of claims and conclusions: Each model is well-described, with its equations and characteristics linked to the appropriate references. For instance, Table 1 serves as an excellent synthesis of the information presented in the text, with clear attributions to the authors of each model. The conclusions accurately summarize the information from the preceding sections and logically identify future research areas.

5) Figures and tables: The figures and tables are appropriate and play an important role in illustrating complex concepts. They are well-executed and easy to understand. The legends are clear and effectively explain what each graph or diagram represents.

6) General comments and recommendations

Methodology clarification: I recommend adding an explicit "Methodology" section. The paper currently lacks key details about the literature review process. The scientific transparency of the work would be significantly enhanced by including the following information:

• Details about the databases used (e.g., Scopus, Web of Science, Google Scholar) and the search strategy.
• Inclusion and exclusion criteria used to justify the selection of the models presented.
• The time frame covered by the literature analysis.
• A study selection flow diagram (e.g., a PRISMA-style diagram) to visually illustrate how the selection was made from an initial number of studies to the final set of analyzed models.

Areas for improvement: While the paper is good and useful, there are a few aspects that could be deepened. These are not factual errors, but rather untapped opportunities for further exploration.

a) More detailed and critical comparative analysis: Although the paper compares the models and summarizes them in Table 1, the comparative analysis could be more in-depth.

Detailed discussion of limitations: The review mentions the limitations of each model, but a separate section dedicated to a critical discussion of what works and what doesn't for each model would be beneficial.

"Hands-on experience": An exceptional review could have included a discussion of the authors' practical experience in implementing or using these models in numerical modeling software, discussing the practical difficulties or necessary trade-offs.

b) Lack of in-depth discussion of future research directions: The paper mentions in its conclusions that "there is room for improvement" but does not offer a concrete roadmap. It could suggest specific research directions, such as developing models that better integrate anisotropy, thermal or chemical effects, or that combine micro- and macroscopic mechanisms.

7) English Language and Style

The manuscript is well-written. The English is fine.

In summary, this is a valuable and well-done review. The review is good and useful, but it could be improved by including a more critical analysis, a more detailed discussion of the methodology (as previously discussed), and a clearer roadmap for future research.

Author Response

Reviewer 3

This manuscript is a solid and valuable contribution to the geomechanics community. It effectively synthesizes, analyzes, and compares a range of important variable dilatancy models, providing a useful resource for both researchers and practitioners. The logical structure, clarity, and high quality of the figures further enhance its value.

Thank you for your deep analysis of the manuscript. We have considered it to improve our manuscript. Any changes performed in the text to consider your comments are marked in red.

1) Clarity, relevance, and contribution to knowledge

Clarity: The review is well-structured and clear. The introduction establishes the context, Section 2 explains stress-strain behavior and key terms (pre-peak and post-peak), and Section 3 details eight variable dilatancy models. The conclusion (Section 4) summarizes the information and outlines future directions. The paper offers a comprehensive overview of variable dilation angle models in rock mechanics.

Relevance: The subject is highly relevant to the fields of geomechanics and mining engineering. The dilation angle is a crucial parameter for the numerical modeling of rock behavior, particularly in applications like the design of tunnels and mining excavations.

Gap in knowledge: While the paper doesn't identify a specific, unaddressed knowledge gap, it does suggest there is room for improvement in our understanding and modeling of dilatancy behavior. The conclusion states that "there is still room for improvement in our knowledge of dilatancy" and raises important questions for future research, such as the influence of sample size and rock structure.

2) Relevance and interest to the scientific community: The manuscript mentions previous reviews, but focuses on a more detailed and updated analysis of variable dilatancy models. The fact that this review compares multiple recent models (some published as recently as 2021) and synthesizes them into a detailed table (Table 1) makes it highly relevant and useful, avoiding redundant publication. Its value lies not merely in summarizing but in its comparative analysis of different approaches.

3) Timeliness and relevance of references: The reference list is extensive and includes recent publications (2020, 2021, 2022). However, it also contains essential older references, which are critical for tracing the evolution of these concepts from the early 1970s. This combination of historical and recent references provides substance and a proper perspective on the field's development. All references are directly relevant to the topic of dilation angle and rock mechanics, and each model discussed is supported by the corresponding citation.

4) Coherence of claims and conclusions: Each model is well-described, with its equations and characteristics linked to the appropriate references. For instance, Table 1 serves as an excellent synthesis of the information presented in the text, with clear attributions to the authors of each model. The conclusions accurately summarize the information from the preceding sections and logically identify future research areas.

5) Figures and tables: The figures and tables are appropriate and play an important role in illustrating complex concepts. They are well-executed and easy to understand. The legends are clear and effectively explain what each graph or diagram represents.

6) General comments and recommendations

Methodology clarification: I recommend adding an explicit "Methodology" section. The paper currently lacks key details about the literature review process. The scientific transparency of the work would be significantly enhanced by including the following information:

Details about the databases used (e.g., Scopus, Web of Science, Google Scholar) and the search strategy.

Inclusion and exclusion criteria used to justify the selection of the models presented.

The time frame covered by the literature analysis.

A study selection flow diagram (e.g., a PRISMA-style diagram) to visually illustrate how the selection was made from an initial number of studies to the final set of analyzed models.

Thank you for your valuable comments, we have added a summarized description of the methodology used to perform the bibliographical review at the end of the Introduction section, the new paragraphs are as follows:

“The methodology used for the development of this work consisted of a systematic search for information through electronic resources, mainly in the SCOPUS and Web of Science (WoS) databases, selected due to their large number of internationally indexed and cited scientific publications.

The search strategy was based on the use of keywords related to rock dilatancy. To define the framework, filters were applied considering only publications after 2010, and the rel-evance of the articles was prioritized based on the number of citations relative to the year of publication. Subsequently, the abstracts were reviewed, evaluating the objective, the methodology used, and the contributions related to the knowledge of dilatancy, discard-ing those that did not meet the established criteria. The selected works were organized into categories according to the focus of the model and the variables involved.

Inclusion criteria:

• Publications in indexed journals, book chapters, and conference proceedings.
• Studies that explicitly address dilatancy in rocks or models of dilation angle.
• References that include equations related to the dilation angle.
• Publications from the year 2015 onwards

Exclusion criteria:

• Studies applied to other materials (for example, concrete, soils, or others).
• Articles on constitutive models without relevant contributions to dilatancy.
• Articles in languages other than English.”

Areas for improvement: While the paper is good and useful, there are a few aspects that could be deepened. These are not factual errors, but rather untapped opportunities for further exploration.

1. a) More detailed and critical comparative analysis: Although the paper compares the models and summarizes them in Table 1, the comparative analysis could be more in-depth.

Thank you for your comment. As we indicated in the cover letter, we have also performed a comparison of the ten models with our own experimental data, but the final manuscript including this literature review and the model comparison resulted too long, so we decided to separate them in two manuscripts. Second manuscript (which will include a more detailed comparison section with error analysis) will be sent to a journal after this one is accepted for publication.

Detailed discussion of limitations: The review mentions the limitations of each model, but a separate section dedicated to a critical discussion of what works and what doesn't for each model would be beneficial.

Thank you for the comment. As in the previous answer, next publication will include such discussion, because it will benefit of the practical fitting of the different models to the same actual experimental data.

"Hands-on experience": An exceptional review could have included a discussion of the authors' practical experience in implementing or using these models in numerical modeling software, discussing the practical difficulties or necessary trade-offs.

Thank you for your comment. We focused on describing each model, avoiding to artificially increase the length of the paper with aside topics. We sincerely feel that adding a section like that one you mention may fall in these “aside topics”. Nevertheless, next manuscript will also consider the numerical implementation of the different variable dilation angle models in numerical modelling to show the differences between them.

1. b) Lack of in-depth discussion of future research directions: The paper mentions in its conclusions that "there is room for improvement" but does not offer a concrete roadmap. It could suggest specific research directions, such as developing models that better integrate anisotropy, thermal or chemical effects, or that combine micro- and macroscopic mechanisms.

Thank you for your comment. We tried to slightly touch this point in the concluding remarks section but without deepening in it because we are currently working on it. Inm any case, We added the next paragraphs to the Concluding Remarks section:

“These improvements could begin with the standardization of the procedures for the determination of the dilation angle. Until now, the ISRM Suggested Method for the com-plete stress-strain curve for intact rock in uniaxial compression [60] is the document that indicates how to perform uniaxial compression tests reaching the residual state, but there is no other Suggested Method for triaxial tests reaching the residual phase or a Suggested Method for determining the dilation angle. In addition, the existing Suggested Method does not describe how to decompose the total strain into its elastic and plastic compo-nents. It is known that plastic strains begin at a stress known as Crack Initiation (CI), but they are negligible compared to those after the Crack Damage stress (CD). Which stress should be considered as the initial point for the plastic strains? Considering the CI as the initial point for plastic strains is strictly the correct decision, but making so will result in negative dilation angles according to the Vermeer and De Borst [35] equation (Equation 9) because the rock is still contracting between the CI and the CD. Therefore, it seems rea-sonable to consider CD as the initial point of the plastic strain, disregarding those between CI and CD. On the other hand, disregarding the plastic strains between CD and the peak strength may induce errors because it is precisely around this zone where the peak dila-tion angle is attained.

One of the aspects that may be identified is that, usually, the number of available ex-perimental dilation angle data points is limited (Figures 6, 9, and 10, for instance), espe-cially in the surroundings of the peak dilation angle. This may be because conventional testing equipment may be inappropriate for correctly capturing the failure of rock speci-mens. Fairhurst and Hudson [60] pointed out that the testing setup requires high-stiffness load frames, fast servocontrol, correct choice of feedback signal and strain measurement transducers, specimen preparation techniques, etc., but also indicated that testing some high-strength rocks becomes at best difficult. The unfulfilment of all these requirements may result in the loss of data points, especially immediately after the peak strength, or even recording the whole system (testing equipment and rock specimen) characteristics, which may result in an inaccurate calculation of the dilation angle.

Another assumption generally considered is that the elastic components of the strains are calculated using the pre-peak elastic moduli, irrespective of the analyzed phase (pre- or post-peak). Some studies, like that of Hou and Cai [61] for instance, show that Young’s modulus and Poisson’s ratio are not constant all along the stress-strain curve, but according to the CWFS model [28, 62, 63] they depend on the relation between cohe-sive and frictional components of the strength. If one assumes that the transition from peak to residual strength and the residual phase itself are cohesion loss processes, it seems reasonable to also consider the dependence of the elastic moduli on plastic strain. It is also important to note that the cyclical loading-unloading used to identify the elastic and plastic components of strains (as shown in Figure 1) induces increasing damage to the rock each time a cycle is performed, so the number of cycles performed per test should also be standardized to make test results comparable.

Apart from these practical concerns, there are still some other aspects that have not been addressed. As previously stated, it is known that the dilation angle depends on the plastic strain history and confining stress, and some efforts have been made to identify its dependence on the structure by the research group of the authors [64-66]. However, there have not been any studies on the dilation angle dependence on size or the effects of other variables usually considered in determining the mechanical properties of rocks, such as the temperature or water content.

The standardization of the procedures for determining the dilation angle could be a good starting point to address these dependencies of the dilation angle as well as new ones that will surely arise.”

 

7) English Language and Style

The manuscript is well-written. The English is fine.

In summary, this is a valuable and well-done review. The review is good and useful, but it could be improved by including a more critical analysis, a more detailed discussion of the methodology (as previously discussed), and a clearer roadmap for future research.

Thank you for your kind comments.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

A review article is presented, devoted to the comparative analysis of dilatancy angle models in rocks (Alejano & Alonso, Zhao & Cai, Pourhosseini & Shabanimashcool, Walton & Diederichs, Chen et al., Rahjoo & Eberhardt, Wang et al., Jin et al., Zhao & Li, Cai et al.). Despite the thorough analysis carried out by the authors and the generally clear presentation of the obtained data, it is recommended to improve the article according to the following comments:

• The large scope of the analysis is not consistent with the very short conclusions in Section 4. Moreover, the first three paragraphs there present widely known statements. This section should be expanded by adding a more detailed comparison of the different models as outlined in the main text.

• It is also suggested to expand Table 1 and add a column with recommendations on the minimum set of tests required to obtain the model parameters. This would also be useful for comparing the models in terms of their practical applicability.

• The article is aimed at a comparative analysis of models for practical application, but there is no real comparison of each model with actual engineering problems. The discussion is limited to laboratory tests. It may be advisable to address this issue in the article or supplement it with examples.

The list of questions may be extended after a second review of the article.

Author Response

Reviewer 4

A review article is presented, devoted to the comparative analysis of dilatancy angle models in rocks (Alejano & Alonso, Zhao & Cai, Pourhosseini & Shabanimashcool, Walton & Diederichs, Chen et al., Rahjoo & Eberhardt, Wang et al., Jin et al., Zhao & Li, Cai et al.). Despite the thorough analysis carried out by the authors and the generally clear presentation of the obtained data, it is recommended to improve the article according to the following comments:

Thank you for your comments. We apologize for the misunderstanding. The last two words of the introduction section “and compared” have been removed. These words remained from a previous version of the manuscript in which we, in addition to the current bibliographical review, performed a comparison of the different models using our own experimental data, but it resulted in a too long document, so we decided to separate them. This manuscript pretends to be the first part of this previous large document, a description of the existing models, emphasizing the different approaches used to develop the models and the lack of standardization. Second manuscript will be sent to a journal after the current one is accepted for publication, we are finishing a set of numerical models using the different variable dilatancy models.

The large scope of the analysis is not consistent with the very short conclusions in Section 4. Moreover, the first three paragraphs there present widely known statements. This section should be expanded by adding a more detailed comparison of the different models as outlined in the main text.

Thank you for your comment. Apart from the non-comparative intention of the manuscript, we agree with your comment in that first paragraphs of the Concluding Remarks section are widely known statements. These first paragraphs tried to summarize the importance of characterizing dilatant behaviour of rocks and to present the known dependencies. Although widely known, we considered that it was important to mention this knowledge on the Concluding Remarks section.

It is also suggested to expand Table 1 and add a column with recommendations on the minimum set of tests required to obtain the model parameters. This would also be useful for comparing the models in terms of their practical applicability.

Thank you for your valuable comment. We apologize again for the confusion; we did not want to compare the models in this manuscript. We just wanted to describe the different existing models.

Answering your particular comment (minimum set of tests required), in our next manuscript we will address such detail, but the fast answer is that we can resort to the ISRM Suggested Methods (ISRM, 2007), that recommends a minimum of five compression tests per set of external conditions. In fact, the interesting topic is not the number of tests, since it is stablished by the ISRM, but the amount of data points that can be recovered from each test. If you look at the figures of the different models, you will notice that most of them are based in a limited number of data points, especially on the peak dilation angle estimation zone.

Regarding your comment, as well as based on other reviewers’ comments, we have added next paragraphs in the concluding remarks section talking about emerging challenges.

“These improvements could begin with the standardization of the procedures for the determination of the dilation angle. Until now, the ISRM Suggested Method for the com-plete stress-strain curve for intact rock in uniaxial compression [60] is the document that indicates how to perform uniaxial compression tests reaching the residual state, but there is no other Suggested Method for triaxial tests reaching the residual phase or a Suggested Method for determining the dilation angle. In addition, the existing Suggested Method does not describe how to decompose the total strain into its elastic and plastic compo-nents. It is known that plastic strains begin at a stress known as Crack Initiation (CI), but they are negligible compared to those after the Crack Damage stress (CD). Which stress should be considered as the initial point for the plastic strains? Considering the CI as the initial point for plastic strains is strictly the correct decision, but making so will result in negative dilation angles according to the Vermeer and De Borst [35] equation (Equation 9) because the rock is still contracting between the CI and the CD. Therefore, it seems rea-sonable to consider CD as the initial point of the plastic strain, disregarding those between CI and CD. On the other hand, disregarding the plastic strains between CD and the peak strength may induce errors because it is precisely around this zone where the peak dila-tion angle is attained.

One of the aspects that may be identified is that, usually, the number of available ex-perimental dilation angle data points is limited (Figures 6, 9, and 10, for instance), espe-cially in the surroundings of the peak dilation angle. This may be because conventional testing equipment may be inappropriate for correctly capturing the failure of rock speci-mens. Fairhurst and Hudson [60] pointed out that the testing setup requires high-stiffness load frames, fast servocontrol, correct choice of feedback signal and strain measurement transducers, specimen preparation techniques, etc., but also indicated that testing some high-strength rocks becomes at best difficult. The unfulfilment of all these requirements may result in the loss of data points, especially immediately after the peak strength, or even recording the whole system (testing equipment and rock specimen) characteristics, which may result in an inaccurate calculation of the dilation angle.

Another assumption generally considered is that the elastic components of the strains are calculated using the pre-peak elastic moduli, irrespective of the analyzed phase (pre- or post-peak). Some studies, like that of Hou and Cai [61] for instance, show that Young’s modulus and Poisson’s ratio are not constant all along the stress-strain curve, but according to the CWFS model [28, 62, 63] they depend on the relation between cohe-sive and frictional components of the strength. If one assumes that the transition from peak to residual strength and the residual phase itself are cohesion loss processes, it seems reasonable to also consider the dependence of the elastic moduli on plastic strain. It is also important to note that the cyclical loading-unloading used to identify the elastic and plastic components of strains (as shown in Figure 1) induces increasing damage to the rock each time a cycle is performed, so the number of cycles performed per test should also be standardized to make test results comparable.

Apart from these practical concerns, there are still some other aspects that have not been addressed. As previously stated, it is known that the dilation angle depends on the plastic strain history and confining stress, and some efforts have been made to identify its dependence on the structure by the research group of the authors [64-66]. However, there have not been any studies on the dilation angle dependence on size or the effects of other variables usually considered in determining the mechanical properties of rocks, such as the temperature or water content.

The standardization of the procedures for determining the dilation angle could be a good starting point to address these dependencies of the dilation angle as well as new ones that will surely arise.”

The article is aimed at a comparative analysis of models for practical application, but there is no real comparison of each model with actual engineering problems. The discussion is limited to laboratory tests. It may be advisable to address this issue in the article or supplement it with examples.

Thank you for your comment. We apologize again for the misunderstanding. As stated in the cover letter, and indicated in the previous answers to your comments, we are working on a comparison of the different models with actual experimental data, but including this comparison in this manuscript resulted in a too long document, so we preferred to separate both parts. Under this consideration, we tried to maintain a more descriptive vision of the models in the current manuscript, without giving quantitative or qualitative results.

On the other hand, we limited the study to laboratory tests results because up to the knowledge of the authors there are no recent experimental studies on dilation angle of rock masses.

The list of questions may be extended after a second review of the article.

We thank you for your interesting comments and apologize for the induced confusion.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

For the revision, the references is still in infancy for a review paper, and more relevant references should be critical reviewed.

Author Response

For the revision, the references is still in infancy for a review paper, and more relevant references should be critical reviewed.

Dear Reviewer, thank you for your comment, the manuscript strengthened with your recommendation. The changes performed on the manuscript are marked in green.

We have re-performed the bibliographical review and paid attention to any sentence without references. Another variable dilation angle model was added because it was recently published (and after we performed the bibliographical compilation). We increased the number of references from 66 to 119 according to your recommendation.

In any case, we must insist in the same point as on the previous review, if you think we missed some relevant reference/s, do not hesitate to indicate it/them explicitly, we will check it/them and add to the manuscript if appropriate.

Best regards.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

With all the positive impressions from the presented review, responses to comments and a small addition to the manuscript, the following criticisms remain:
1. In the introduction, the purpose of the article should be clearly written, indicating that this material represents another more extensive work with a comparative analysis of various models. Perhaps it should be pointed out that the paper does not provide a comparative analysis, but draws conclusions about shortcomings in other studies.
2. The manuscript must explicitly present the scientific novelty of the work and its connection with . So far, this has been done very disjointed in the final part.

Author Response

Reviewer 4:

With all the positive impressions from the presented review, responses to comments and a small addition to the manuscript, the following criticisms remain:

Thank you for your kind comments. Your suggestions helped in a good manner to improve the manuscript. The changes performed in the manuscript according to your comments are marked in orange also in this second revision.

1. In the introduction, the purpose of the article should be clearly written, indicating that this material represents another more extensive work with a comparative analysis of various models. Perhaps it should be pointed out that the paper does not provide a comparative analysis, but draws conclusions about shortcomings in other studies.
2. The manuscript must explicitly present the scientific novelty of the work and its connection with . So far, this has been done very disjointed in the final part.

Thank you for both your comments. We are answering them together because we have revised the introduction to clarify the purpose and the scientific novelty of the study. The text now explicitly states that the work focuses on presenting the main characteristics and limitations of eleven (we added a recently published one) existing variable dilation angle models. It also emphasizes that the article does not provide a comprehensive comparative analysis but rather highlights shortcomings and knowledge gaps in previous and actual studies and, finally explicitly presents, the scientific novelty of the study.

Original paragraph:

This study aims to compile ten existing variable dilation angle models, explaining their characteristics as well as their pros and cons.

New paragraph:

Despite significant advances, knowledge on rock dilatancy remains fragmented, and the relationship between the dilation angle, plastic strain, and confinement has not been standardized. This lack of standardization complicates the comparison between studies and the selection of appropriate models for different rock types. Recognizing these gaps, the present study focuses on compiling and examining eleven existing variable dilation angle models, presenting their main characteristics as well as their advantages and limitations. By systematically reviewing these models, the intention of this bibliographical review is not to provide a comprehensive comparative analysis, but rather to highlight common shortcomings and unresolved questions in previous and current research, thereby providing a clearer understanding of rock dilatancy. Ultimately, this study seeks to bridge the gap between existing knowledge and practical application, offering a foundation for the standardization of procedures to determine the dilation angle and guiding the development of improved models in future research.

Author Response File: Author Response.pdf

Round 3

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript lacks scientific novelty. I hope the authors will supplement this before publishing.