Research on Rock Energy Constitutive Model Based on Functional Principle

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper develops a new rock energy constitutive model that considers functional and momentum principles. This approach seeks to advance rock deformation and failure knowledge by incorporating energy characteristics into the constitutive formulations. In general, the organization and content of the paper are well presented, and the authors provide valuable contributions to rock mechanics.

In order to improve the quality of the work, it is recommended to answer and do the following issues.

It is recommended that the abstract briefly state the study's innovative aspects and significance and provide more information about the results' practical applications.

In the introduction, the study's original value should be explained more clearly in 1-2 sentences, and its contribution to the literature should be emphasized. The literature review covers previous studies, but the gaps addressed by the research need to be better emphasized. It should be clearly stated that the proposed model solves existing problems.

Some equations are presented without an adequate explanation of symbols and units. A glossary or appendix listing all variables, their units, and physical meanings should be added. It will help readers follow the mathematical derivations more easily.

A more detailed explanation of the intermediate steps during deriving the main constitutive model equation is needed.

Section 3.1 briefly describes the experimental setup, but critical details such as sample preparation, environmental conditions, and measuring instrument calibration are missing. More information is needed to ensure the results' repeatability. The potential sources of error in the experimental process and how these were mitigated are also not discussed.

The paper should include a more detailed discussion of the proposed model's limitations compared to existing models. Highlighting where the new model outperforms and where it may fail would provide a more balanced assessment.

It would be helpful if readers were provided with comprehensive text explaining all figures. In addition, Figure 5 presents the energy development curves, but the significance of the differences between model and test data under various confining pressures is not adequately explained. The interpretation of such figures needs to be described in detail.

The conclusion should clearly state the practical applications of the findings and summarize potential future research directions based on this study's results.

The study's similarity rate is 36%. Plagiarism should be avoided, and the rate should be reduced to at least 20%.

Comments on the Quality of English Language

The manuscript contains various grammatical errors and sentence structures that affect understanding. The manuscript needs to be revised in terms of grammar.

Author Response

This paper develops a new rock energy constitutive model that considers functional and momentum principles. This approach seeks to advance rock deformation and failure knowledge by incorporating energy characteristics into the constitutive formulations. In general, the organization and content of the paper are well presented, and the authors provide valuable contributions to rock mechanics. In order to improve the quality of the work, it is recommended to answer and do the following issues.

1.It is recommended that the abstract briefly state the study's innovative aspects and significance and provide more information about the results' practical applications.

Response:Amended according to the opinions of experts.

Compared with other models, the model based on energy principle not only reflects the stress-strain curve of rock well. It also well reflects the energy change law of rock.

The essence of rock fracture is the comprehensive result of energy storage, dissipation, surplus and release. From the perspective of energy consumption, the failure of rock materials must be accompanied by energy dissipation. The dissipated energy is the internal driving force of rock damage and progressive failure.

 

2.In the introduction, the study's original value should be explained more clearly in 1-2 sentences, and its contribution to the literature should be emphasized. The literature review covers previous studies, but the gaps addressed by the research need to be better emphasized. It should be clearly stated that the proposed model solves existing problems.

Response:According to the opinions of experts, the original value of this study is supplemented and its contribution is emphasized.

The above constitutive models describe the stress-strain curve of rock well. However, the essence of object destruction is state instability driven by energy. The deformation and failure of the loaded rock occur in the process of energy input, accumulation, dissipation and release. The law of rock deformation and failure is studied from the perspective of energy, which is closer to its failure essence than other angles. It can better analyze and predict the law of rock energy change and stress-strain law. The research results can provide theoretical support and practical means for studying rock mass instability from the perspective of energy.

3.Some equations are presented without an adequate explanation of symbols and units. A glossary or appendix listing all variables, their units, and physical meanings should be added. It will help readers follow the mathematical derivations more easily.

Response:The interpretation of the variables has been supplemented.

 

4.A more detailed explanation of the intermediate steps during deriving the main constitutive model equation is needed.

Response:The solution process is appropriately improved.

According to condition (1), we can get

(28)

According to condition (2), we can get

(29)

(30)

By combining Eqs. (28) - (30), it can be obtained that

 

5.Section 3.1 briefly describes the experimental setup, but critical details such as sample preparation, environmental conditions, and measuring instrument calibration are missing. More information is needed to ensure the results' repeatability. The potential sources of error in the experimental process and how these were mitigated are also not discussed.

Response:The sample conditions have been supplemented according to the requirements of experts.

The experimental data used are from Reference [21]. The samples were taken from the surrounding rock of the roadway in Fuxin Hengda Coal Mine. The buried depth of surrounding rock is about 800m-1000m. The rock mass is made into a standard cylinder with h = 100mm and D = 50mm. The samples with obvious defects in appearance are eliminated. Ensure that the diameter deviation of the remaining rock samples is within 0.3mm. The height error is within 0.05 mm and the maximum deviation of the axis is less than 0.25 °. Finally, the rock sample with the same wave velocity is taken as the sample object by the wave velocity tester. The maximum horizontal in-situ stress can reach about 30 MPa. The confining pressure value is 30 MPa. However, in order to study the influence of confining pressure on the mechanical properties of rock, the confining pressure is 20,25,35 and 40 MPa.

 

6.The paper should include a more detailed discussion of the proposed model's limitations compared to existing models. Highlighting where the new model outperforms and where it may fail would provide a more balanced assessment.

Response:Compared with other models, the model based on energy principle not only reflects the stress-strain curve of rock well. It also well reflects the energy change law of rock. The constitutive model shows an excellent fitting effect on rock samples with significant compaction characteristics. The model also describes the stress-strain curve of rock well. But the degree of agreement in the post-peak strain softening stage is slightly low.

 

7.It would be helpful if readers were provided with comprehensive text explaining all figures. In addition, Figure 5 presents the energy development curves, but the significance of the differences between model and test data under various confining pressures is not adequately explained. The interpretation of such figures needs to be described in detail.

Response:The change rule of the curve in Fig.5 has been described according to the expert opinion.

In the pre-peak stage, the elastic energy model curve has a high degree of fitting with the experimental data. However, the elastic energy model cannot describe the variation of the post-peak curve well. On the whole, the elastic energy model curve has a high degree of agreement with the test curve. This shows that the elastic energy model is closer to the actual deformation and failure evolution law of surrounding rock. The established model can better reflect the relationship between elastic energy and strain of rock. However, the total energy and dissipation energy models can better describe the change rule.

 

8.The conclusion should clearly state the practical applications of the findings and summarize potential future research directions based on this study's results.

Response:According to the results of this study, the possible future research directions are summarized.

(4)The results show that the proposed model can accurately predict the mechanical behaviors of rock, such as pre-peak and post-peak strain softening, confining pressure effect and energy evolution. The research results provide theoretical reference and practical means for the subsequent research on the construction of deep underground engineering from the perspective of energy. The model established in this paper has good identification ability for the whole process of deformation and failure of rock in high stress area. However, this paper only proposes a constitutive relationship based on indoor test results. Due to the length and research progress, there is no secondary development and engineering application in the finite element software. In the later stage, our team will develop energy models for practical engineering applications.

The study's similarity rate is 36%. Plagiarism should be avoided, and the rate should be reduced to at least 20%.

Response:The repetition rate has been reduced to journal requirements.

Reviewer 2 Report

Comments and Suggestions for Authors

All my remarks are included in the file that is attached. Paper provides it like a report and does not engage in in depth discussion. The introduction must be strengthened (see my remarks). There are also no pictures or details on the experiment in the methodology section. The discussion section is entirely missing, but the result section is fine. 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Moderate editing of English language required.

Author Response

Reviewer #3:

1.Add one or two lines regarding the importance of rock energy in the industrial or commercial aspect. Reader should know the importance of this study.

Response:The content was supplemented according to the opinions of experts.

The essence of rock fracture is the comprehensive result of energy storage, dissipation, surplus and release. From the perspective of energy consumption, the failure of rock materials must be accompanied by energy dissipation. The dissipated energy is the internal driving force of rock damage and progressive failure.

 

2.specify if only applicable for the sandstone.

Response:The verification of other types of rocks has been supplemented.

In order to verify whether the model is suitable for other types of rocks, a triaxial compression test was carried out using marble. The method of determining the parameters still uses the method of determining the parameters of the sandstone model. The comparison between the stress-strain curve of marble and the model curve is shown in Fig.8.

 

Fig.8 The comparison between the stress-strain curve of marble and the model curve

It can be seen from Fig.8 that the model also describes the stress-strain curve of marble well. The degree of agreement in the post-peak strain softening stage is slightly low. The overall agreement is more than 0.90.It also shows that the model can be applied to the prediction and description of stress-strain curves of various rocks.

 

3.Also add here few lines for the application of this study in the industry.

Response:The content was supplemented according to the opinions of experts.

The research results can fill the blank of energy method in rock deformation and failure. It provides a theoretical basis for deep rock engineering. On the other hand, it can further improve and extend the rock mechanics research system based on energy.

 

4.Seems like all these three theories missing some crucial aspect of rock's properties. Could author make a table mentioning these three categories and also mention flaws in them and how present research fill those missing gaps.

Response:Forms have been established in accordance with the opinions of experts.

The advantages and disadvantages of the above three modeling theories are shown in Table 1.

Table 1 The advantages and disadvantages of the above three modeling theories

Modeling theory method	advantages	disadvantages	Ways to make up for disadvantages
Method (1)	1.The model is simple; 2.few parameters; 3.Engineering can be directly applied.	1.It cannot reflect the stress-strain characteristics of sandstone well and truly; 2.The parameters have no clear physical meaning.	A unified damage constitutive model of rock is constructed from the perspective of energy evolution.
Method (2)	1.The model is simple; 2.There is a fixed method to establish the model.	1.It is necessary to continuously build new model elements according to the existing conditions; 2.Most components can only describe linear mechanical properties.
Method (3)	1.The model can well describe the nonlinear mechanical properties; 2.It can also well reflect the damage evolution law of the internal structure of the material; 3.The physical meaning of parameters is clear.	1.The model parameters are increased. 2.The calculation amount of model derivation is increased.

 

5.Rewrite the statement

Response:It is modified according to the opinions of experts.

Scholars have studied the energy evolution law of rock mass and the factors affecting energy characteristics in detail.

 

6.Also make a table on research on the energy evolution law of rock mass and the factors affecting energy characteristics.

Response:It is modified according to the opinions of experts.

he factors affecting the energy evolution characteristics of rock mainly include stress, osmotic pressure, temperature, loading rate, confining pressure, weathering degree, rock type and so on.

 

7.If possible illustrate this phenomenon through image.

Response:It is modified according to the opinions of experts.

The relationship between the two is shown in Fig. 2.

 

Fig. 2 The relationship between the two

8.Add images of each steps. Also add flow chart mentioning each steps with all details. It would be easier for the readers to understand the whole process.

Response:The specific test operation steps are supplemented.

The specific test steps are shown in Fig.3.

 

Fig.3 The specific test steps

9.Any image of rock ?

Response:A sample image has been added to the flow chart.

 

10.Any specification of the equipment ?model name and parameter used ?

Response:It is modified according to the opinions of experts.

The test equipment is MTS815.02 sandstone test system imported from the United States. The system is equipped with three main systems : axial pressure, confining pressure and seepage. The stiffness of the testing machine reached 1600 kN. The maximum confining pressure can be applied to 70 MPa. The displacement measurement range is ± 4 mm. Fully meet the requirements of this test.

 

11.Add here reference of the image as mentioned it comes from the dissertation.

Response:It is modified according to the opinions of experts.

 

12.Any indications of that certain stage ?

Response:This paragraph is re-described.

After loading to the peak stress, the rock is unloaded. A unloading path curve is obtained.

 

13.Any specific reason of why author used 20-40 MPa pressure ? add scientific implication of parameter used.

Response:The maximum horizontal in-situ stress can reach about 30 MPa. The confining pressure value is 30 MPa. However, in order to study the influence of confining pressure on the mechanical properties of rock, the confining pressure is 20,25,35 and 40 MPa.

 

14.What is the % of error for test and model data ?

Response:The correlation coefficients were all above 0.90.

 

15.Is it the conventional way to validate the data ?

Response:The method of verifying the model is supplemented.

In the previous section, the method of determining the parameters of the damage model is determined by the characteristics of the stress-strain curve and the damage model. Combined with the stress-strain data of rock under different conditions, the damage model parameters under this condition are determined. Substituting the obtained damage model parameters into the model, the model curve can be obtained.

 

16.why data only compared with this reference ?

Response:Thank the experts for their opinions. At that time, a model of a better journal was selected to verify. In the follow-up study, other literature models will be used for comparative analysis.

 

17.Any value ?

Response:The correlation coefficients were all above 0.90.

 

18.Add one more section i.e section 5 (Discussion) and section 6 would be conlusion. In section 5 discuss all results obtain and integrate them accordingly. May discuss results of previous studies and how present studies improve our understanding about rock energy.

Response:The discussion section has been added according to expert opinions.

5 Discussion

For the loaded rock system, its energy conversion is roughly divided into energy input, energy accumulation, energy dissipation, four processes of energy release. There are two main mechanisms for energy-driven rock deformation and failure. On the one hand, the external energy input causes energy dissipation behaviors such as damage and plastic deformation inside the rock. This reduces the ability of rock to resist damage. On the other hand, the increase of the elastic energy accumulated in the rock increases the energy source of the overall failure of the rock. This enhances the ability to drive rock failure.

The energy conversion is driven by strain hardening mechanism and strain softening mechanism. The strain hardening mechanism loses the outside world. The energy into the rock system is transformed into the strain energy of the rock system. The strain softening mechanism converts the strain energy in the rock into damage energy, heat energy and other forms of energy.

From the energy point of view, the deformation and failure of rock under external load can be divided into three stages. The first stage is the energy accumulation stage. It roughly corresponds to the compaction stage, the linear elastic stage and the stable fracture development stage. This stage is mainly based on the transformation of external load work and rock elastic energy. The second stage is the energy dissipation stage. It roughly corresponds to the unstable rupture stage. This process is dominated by the transformation of elastic energy and damage dissipation energy. The third stage is the energy release stage. Corresponding to the post-peak softening stage, a large amount of elastic energy is released in this process. Convert into surface energy and kinetic energy of fragments. It should be noted that the strain hardening mechanism and the strain softening mechanism coexist in the whole deformation and failure stage of the rock. paragraph. The strain hardening mechanism in the pre-peak stage is greater than the strain softening mechanism. Therefore, the macro shows strain hardening and energy accumulation. The latter is greater than the former in the post-peak stage.

Rock energy evolution theory is a new field of rock mechanics. It has a wide application prospect. This article is only a preliminary exploration. There are many aspects that have not been well studied. This is also the next step that needs to be made. The future research directions of rock energy evolution theory include the correlation between microscopic mechanism and macroscopic mechanical behavior, energy evolution constitutive theory, engineering rock mass stability, and multi-energy field coupling.

 

19.Add numbers (correlation values etc)

Response:It is modified according to the opinions of experts.

The correlation coefficients of the models in this paper are all greater than 0.90. The correlation coefficient between the literature model and the test curve is less than 0.90.

20.Mention parameters (20 MPa, 25 MPa etc)

Response:It is modified according to the opinions of experts.

Therefore, the damage constitutive model based on the functional principle can better reflect the stress-strain variation of rock under different confining pressures (20, 25, 30, 35 and 40MPa).

 

21. May add future recommendation in few lines.

Response:It is modified according to the opinions of experts.

(4)The results show that the proposed model can accurately predict the mechanical behaviors of rock, such as pre-peak and post-peak strain softening, confining pressure effect and energy evolution. The research results provide theoretical reference and practical means for the subsequent research on the construction of deep underground engineering from the perspective of energy. The model established in this paper has good identification ability for the whole process of deformation and failure of rock in high stress area. However, this paper only proposes a constitutive relationship based on indoor test results. Due to the length and research progress, there is no secondary development and engineering application in the finite element software. In the later stage, our team will develop energy models for practical engineering applications.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors present their new energy constitutive model based on the functional and momentum principles. A triaxial compression test was performed on the sandstone to analyze the main stress-strain characteristics  of the rock during the failure. A sensitivity analysis of a number of parameters introduced into the model was carried out based on the comparison results obtained. The physical meaning of the introduced parameters was also clarified.

My comments are as follows.

1.In Abstract, the authors state that the achieved results: “ show that the energy model can better reflect the nonlinear mechanical behavior of the rock during loading. “, but better than what? Or in what case?

2.Correct the empty line in Abstract.

3.It would be helpful to have a diagram of the stresses, strains, forces acting on the micro-element inside the rock – Section 2.

4.In Section 2, it would be possible to show symmetry in the mechanical properties of the element or loads - because the publication is supposed to be in Symmertry (journal)

5.Why is the energy increase resulting from the temperature change caused by internal forces in the element and loading by external forces not taken into account? What is the meaning of the parameter l?

6.Unnecessary spaces, for example Lines: 131, 144, etc.

7.Why is the text in Line 217 marked (1)?

8.Page 7: the sentence: In order to make the theoretical value and experimental value as close as possible, m groups of test data and theoretical value of the difference between the sum of squares to take the minimum value[25]. – should be improved.

9.In Eqt. 30, there is the term C not C1 as it is below the equation.

10.Figures 3 and 10. Why is the total energy curve different in (a) and others (b) to (e)? (also differently coloured?) - similarly elastic energy?

11. In the abstract, it should be added how much the improved accuracy of the new model is important for technical sciences.

Comments on the Quality of English Language

A little correction is necessary.

Author Response

Reviewer #2:

The authors present their new energy constitutive model based on the functional and momentum principles. A triaxial compression test was performed on the sandstone to analyze the main stress-strain characteristics  of the rock during the failure. A sensitivity analysis of a number of parameters introduced into the model was carried out based on the comparison results obtained. The physical meaning of the introduced parameters was also clarified.

My comments are as follows.

1.In Abstract, the authors state that the achieved results: “ show that the energy model can better reflect the nonlinear mechanical behavior of the rock during loading. “, but better than what? Or in what case?

Response:According to expert opinion has been amended.

The results show that the energy model can better reflect the nonlinear mechanical behavior of sandstone under high stress conditions during loading.

2.Correct the empty line in Abstract.

Response:According to expert opinion has been amended.

 

3.It would be helpful to have a diagram of the stresses, strains, forces acting on the micro-element inside the rock – Section 2.

Response:According to expert opinion has been amended.

The stress characteristics of rock micro-element are shown in Fig.1.

 

Fig. 1 Stress characteristics of rock micro-element

 

4.In Section 2, it would be possible to show symmetry in the mechanical properties of the element or loads - because the publication is supposed to be in Symmertry (journal)

Response:The symmetry description of axial stress and radial stress is added at the appropriate position in the text.

It can be seen from Fig.1 that the rock micro-element is subjected to axial stress and radial stress. The axial stress acts on the upper and lower ends of the rock. Radial stress acts on the surface around the rock. The axial stress and radial stress are symmetrical.

 

5.Why is the energy increase resulting from the temperature change caused by internal forces in the element and loading by external forces not taken into account? What is the meaning of the parameter l?

Response:Thank the experts for their suggestions. This part of our team has not been considered. The analysis of energy evolution law is based on the literature. In the later stage, we will further consider the energy change caused by this part of stress.

 

6.Unnecessary spaces, for example Lines: 131, 144, etc.

Response:It has been modified according to the opinions of experts.

 

7.Why is the text in Line 217 marked (1)?

Response:The serial number has been deleted.

 

8.Page 7: the sentence: In order to make the theoretical value and experimental value as close as possible, m groups of test data and theoretical value of the difference between the sum of squares to take the minimum value[25]. – should be improved.

Response:It has been improved according to the requirements.

 

9.In Eqt. 30, there is the term C not C1 as it is below the equation.

Response:Thanks to the opinions of the experts, after checking this part should be to solve the parameter C1.

 

10.Figures 3 and 10. Why is the total energy curve different in (a) and others (b) to (e)? (also differently coloured?) - similarly elastic energy?

Response:The method of determining the total energy has been given. It is mainly obtained by the stress-strain curve. Therefore, the total energy evolution curves under different confining pressures are also different.

Through the software Origin, the axial stress-strain and circumferential data stress-strain are respectively integrated by area to obtain the new data after integration. The total energy in the rock loading process is obtained by adding the integrated data. Firstly, the elastic modulus and Poisson's ratio of rock under different confining pressures are determined. The axial stress-strain, circumferential stress-strain data, elastic modulus and Poisson's ratio under different confining pressure conditions can be substituted into Eq. (26) to determine the elastic energy data of rock during loading.

 

11. In the abstract, it should be added how much the improved accuracy of the new model is important for technical sciences.

Response:According to the opinions of experts, this content is added at the corresponding position in the text.

The fitting curve is basically consistent with the change trend of the test curve. The correlation coefficients were all above 0.90.

Author Response File: Author Response.doc

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The requested revisions have been made.

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript is greatly improved by the authors, and I am satisfied with the amendments.

Comments on the Quality of English Language

Minor editing of English language required.