Effects of Relative Density and Grading on the Particle Breakage and Fractal Dimension of Granular Materials

Round 1

Reviewer 1 Report

Some of the conclusions are obvious and do not need any proof. It's apriory clear that the smaller is initial size of particles and higher fractal dimension the smaller is influence of pressure on the system behaviour.

Probably authors mean that it's not necessary to determine the properties of materials and the size is the most important on behavior and final properties ?!

Author Response

Response: Our study focused on the effect of particle breakage on the fractal dimension of aggregates. Particle breakage would generate smaller-sized particles that influenced the fractal dimension of aggregates. Aggregate size is also a physical property of the material. So it is necessary to determine the properties of materials, including aggregate size.

Author Response File: Author Response.docx

Reviewer 2 Report

In this paper, the authors studied experimentally particle breakage effects on granular aggregates by considering relative density and initial grading. The paper seems  interesting, but it is not well written. Hence, I DO NOT recommend its acceptance in the present form. However, I encourage the authors to resubmit their work by considering the cited remarks.

• What is the granular aggregates? What is the adhesion type between primary particles in aggregate? It seems to me granular aggregates for the authors mean single grains. This point should be clarified in the paper.
• As a general remark, in the section of ‘Analysis of test results’, the authors described the curves without any physical explanations. In order to improve the quality of the paper, I propose to add some discussions about why one observes the phenomena described in this section. 
• In Fig. 3, the residual shear stresses don’t become completely stable and the experiences may be continued. It is also interesting to plot q/p as a function of the shear strain in order to compare different results for different confining pressures.
• In page 8, the breakage ratio B_g should be defined.
• Equations (3) and (4) propose two fitting models for B_g. It is not clear why the authors define two relations by knowing that they are equivalent. Indeed, if in Eq. (4), we set Cu=10 (as it is in the case of Gc3), we obtain the same fitting parameters as Eq. (3). 
• In Fig. 9, it is more interesting to render the percentage increment dimensionless to appreciate more the results.

Author Response

1. What is the granular aggregates? What is the adhesion type between primary particles in aggregate? It seems to me granular aggregates for the authors mean single grains. This point should be clarified in the paper.

Response: Thanks for your suggestions. In the manuscript, granular aggregates are composed of coarse and fine particles of different sizes, the largest particles can reach more than 20 mm, the finest can be less than 0.1 mm, the particle size varies a wide range, and the characteristics of coarse and fine particles vary greatly. For ease of understanding, granular aggregates have been replaced by granular materials.

2. As a general remark, in the section of ‘Analysis of test results’, the authors described the curves without any physical explanations. In order to improve the quality of the paper, I propose to add some discussions about why one observes the phenomena described in this section.

Response: Thanks for your suggestions. We have added some discussions about the phenomena described.

This can be attributed to that the greater the relative density, the greater the interlocking between the particles; the greater the loading and the greater the peak strength of the specimen under the same confining pressure. As the load increased, particle breakage occurred, the interlocking force between particles would decrease, and the shear strength would decrease.

…

This can be attributed to the enhanced sliding rather than rotation between aggregates when the compressive pressure increases.

…

Median diameter d₅₀ is an important soil grading curve index. The larger the median diameter, the higher the shear strength and the more obvious the shear expansion effect[31]. Figure 9 shows the relationship between particle breakage ratio with d₅₀ under different confining pressures after monotonic loading. The greater d₅₀, the greater the particle breakage ratio. According to the previous research, the confining pressure effect can be expressed by a power function. A normalized power function is also found to well fit the relationship between breakage ratio, d₅ and confining pressure. The expression is shown as follows:

where k₂ and n₃ are model parameters, determined to be 16.5 and 0.46 (which is the same as the previous value of the coefficient of uniformity.).

3. In Fig. 3, the residual shear stresses don’t become completely stable and the experiences may be continued. It is also interesting to plot q/p as a function of the shear strain in order to compare different results for different confining pressures.

Response: Thanks for your suggestions. Due to the limitations of triaxial apparatus, the sample can no longer be loaded. However, according to the test curve and the existing research results, it can also be approximated that the experiment results has reached a critical state. When apparatus permits, we will continue to study the relationship between critical states and axial strain.

4. In page 8, the breakage ratio B_g should be defined.

Response: Thanks for your suggestions. The expression of breakage ratio Bg has been added in manuscript.

It can be expressed as the following:

where P_test is percentage by mass of particles after test, P_ini is percentage by mass of particles before test.

5. Equations (3) and (4) propose two fitting models for B_g. It is not clear why the authors define two relations by knowing that they are equivalent. Indeed, if in Eq. (4), we set Cu=10 (as it is in the case of Gc3), we obtain the same fitting parameters as Eq. (3).

Response: Thanks for your suggestions. Two expressions are defined to illustrate the process that affects particle breakage. Essentially, they are the same. However, Equation 3 is to study the effects of relative density. Equation 4 is based on Equation 3 after consideration of the effects of experimental grading index of granular.

6. In Fig. 9, it is more interesting to render the percentage increment dimensionless to appreciate more the results.

Response: Thanks for your suggestions. Particle percentage increment is defined as the percentage of particle mass to the total mass of the specimen. The variable (F) is dimensionless.

Author Response File: Author Response.docx

Reviewer 3 Report

The article discuss the topic of Normalized relation between pressure-dependent particle breakage extent, relative density and grading index of granular. In my opinion article should be improved before potential publication. The following modification should be considered:

1. Introduction part is little short and should be rewritten. Consider to contain more literature related to the essence of this studies (especially morphology of aggregate; including shape of grains).
Moreover, please describe cited literature [lines 39-40].
2. I suggest to add separated point - Research significance - Please describe here the main essence of the research. 
What was the inspiration for such an analysis? Why presented studies are so important?
3. Line 62- what type of rock was analysed? Please determine clearly.
4. Did you use any standard procedure during triaxial shear tests?
5. Please remove text in lines 87-89.
6. Lines 96-100; if you use subscripts in formulas, you should use subscripts in description too.
7. What is "Dr"? Is it relative densities? If yes, please change into "Rd".
8. Please add name of horizontal axis in each top chart (figure 3).
9. Results shown in point 3.2 must be deeper analysed.
10. Final results should be more accurate.
11. It is recommended to add graphical abstract, emphasizing the importance of research.
12. It is recommended to indicate potential application of research results in engineering or another discipline.

Author Response

1. Introduction part is little short and should be rewritten. Consider to contain more literature related to the essence of this studies (especially morphology of aggregate; including shape of grains).

Moreover, please describe cited literature [lines 39-40].

Response: Thanks for your suggestions. We have added the relevant references and described them in the revised text to read: “Especially, the grading curve changed evidently before and after loading^[4]. Such degradation behaviour of granular materials were found to depend not only on stress history^[5,6], but also on grain size distribution curve^[7,8], parent rock type, particle size^[9,10], particle shape^[11,12] and relative density^[13,14]. For example, particle shape has a significant influence on the particle breakage, which increase with the increasing of shape index sphericity, aspect ratio, convexity and overall regularity^[12]. The critical state parameters ( M, ϕ_cs, e_Γ, and λ_c) decrease with increasing aspect ratio, sphericity and convexity^[8]… It was also found to have a great influence on the shear strength and deformation of granular materials. The greater the relative density, the greater the initial elastic mod-ulus and peak friction angle and the smaller the volume strain^[16,17]… The critical state parameters (M and λ_c) are less affected by relative density and particle grading^[7,13]. However, the critical state parameters (e_Γ) decrease with increasing relative density. The relative breakage index decreases with increasing relative density^[14]”.

2. I suggest to add separated point - Research significance - Please describe here the main essence of the research. 

What was the inspiration for such an analysis? Why presented studies are so important?

Response: Thanks for your suggestions. The Grading of a granular material significantly influences its strength and deformation behavior. Fractal dimension is a signature of the grading, which reflects its physical property. In geotechnical engineering with granular materials, particle breakage would occur, which changes the grading and thus the fractal dimension of the granular material, and in turn influences the long-term stability of the geotechnical facilities constructed by granular materials. Therefore, there is a need to investigate the effect of relative density and initial grading on the particle breakage behavior.

3. Line 62- what type of rock was analysed? Please determine clearly.

Response: Thanks for your suggestions. The expression of test material has been modified in manuscript. “Aggregates were derived from the parent of sandstone rock.”

4. Did you use any standard procedure during triaxial shear tests?

Response: Thanks for your suggestions. During the test, we used Chinese standards and have been supplemented in the manuscript.

“The specimen preparation and loading process are carried out step by step with reference to Specification of soil test (SL237-1999).”

Ministry of Water Resources of the PRC. 1999. Specification of soil test. SL237. Beijing: China Water Conservancy Hydropower Publishing House.

5. Please remove text in lines 87-89.

Response: Thanks for your suggestions. The text has been removed.

6. Lines 96-100; if you use subscripts in formulas, you should use subscripts in description too.

Response: Thanks for your suggestions. The expressions of variables have been checked in all manuscript.

7. What is "Dr"? Is it relative densities? If yes, please change into "Rd".

Response: Thanks for your suggestions. Dr is relative density. We have modified the expression in manuscript with Rd.

8. Please add name of horizontal axis in each top chart (figure 3).

Response: Thanks for your suggestions. The name of horizontal axis has been added in Fig.3. For example, the follow is Fig.3(a). Where red box is modified.

Fig.3 (a)

9. Results shown in point 3.2 must be deeper analysed.

Response: Thanks for your suggestions. According your suggestion, we have added some expressions.

Median diameter d₅₀ is an important soil grading curve index. The larger the median diameter, the higher the shear strength and the more obvious the shear expansion effect^[31]. Figure 9 shows the relationship between particle breakage ratio with d₅₀ under different confining pressures after monotonic loading. The greater d₅₀, the greater the particle breakage ratio. According to the previous research, the confining pressure effect can be expressed by a power function. A normalized power function is also found to well fit the relationship between breakage ratio, d₅ and confining pressure. The expression is shown as follows:

where k₂ and n₃ are model parameters, determined to be 16.5 and 0.46 (which is the same as the previous value of the coefficient of uniformity.).

Figure 9.  Breakage ratio vs median diameter under different confining pressures

10. Final results should be more accurate.

Response: Thanks for your suggestions. The conclusions have been modified.

(2) …The relationship between breakage ratio with median diameter can be described by a linear function.

(3) The conclusions are mainly based on the results after test. In fact, fractal dimension and particle breakage change with axial loading. The future research should focus on the results during shear test. The relationship between fractal dimension and particle breakage with shear modulus and volume strain should be created.

11. It is recommended to add graphical abstract, emphasizing the importance of research.

Response: Thanks for the suggestion. However, the option for graphical abstract is not provided by the journal.

12. It is recommended to indicate potential application of research results in engineering or another discipline.

Response: Thanks for your suggestions. We have added some expressions about potential application of research results in manuscript.

“The research results play an important role in understanding and mastering the gradation change law and strength characteristics of granular materials before and after loading, and have an important reference for the stable design parameters of the structure.”

Author Response File: Author Response.docx

Reviewer 4 Report

Normalized relation between pressure-dependent particle 2

breakage extent, relative density and grading index of granular

Authors: Gui Yang, Zhuanzhuan Chen, Yifei Sun and Yang Jiang

The paper is focused on the effects of relative density and initial grading on the particle breakage behaviour of granular aggregates under different confining pressures, underlining that the particle breakage increases as the confining pressure or relative density increases, whereas it decreases with the increasing coefficient of uniformity. A relation between the particle breakage extent and confining pressure by considering relative density and grading index was found.

Although the subject is very interesting some remarks are to be made:

Reviewer comments:

- English needs to be carefully revised and some type-writing errors must be carefully revised; 

-Please revise the title; I think it is too long – not entirely showed.

-In my opinion more explanations about drained triaxal test are necessary;

-Chapter 3: “This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn” – please erase the paragraph

-Figure 4: Grading curves – can you provide standard deviations for fractal dimension? How is fractal dimension computed – some words about fractal dimension determination method would be if great interest. What are self-similarity limits? Please be more explicit.

-Figure 6: as in Figure 4, can you provide standard deviation, determination coefficient ?

-What does it means “DEM simulation”. 

-Whereas conclusions presented the important findings of the paper, the importance of these findings together with some future research directions would be of great interest.

-Please provide more literature about the subject.

 I recommend: accept after minor revision.

Comments for author File: Comments.docx

Author Response

1. English needs to be carefully revised and some type-writing errors must be carefully revised;

Response: we have checked the manuscript carefully and checked by a native English-speaking person.

2. Please revise the title; I think it is too long – not entirely showed.

Response: Thanks. We have changed it to “Effects of relative density and pressure on the particle breakage and fractal dimension of granular materials”

3. In my opinion more explanations about drained triaxal test are necessary;

Response: Thanks for your suggestions. We have added some content about the drained triaxal test in manuscript.

“The specimen preparation and loading process are carried out step by step with reference to Specification of soil test (SL237-1999).

……

A load cell and pore-pressure sensor were used to measure the deviator load and drainage volume, respectively, through the electronic display system. All the tests were conducted up to a maximum axial strain of 25%.”

4. Chapter 3: “This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn” – please erase the paragraph

Response: Thanks for your suggestions. Chapter 3 has been divided into two sub-parts: “3.1. Particle breakage under different relative density, 3.2. Particle breakage under different grading curve”. Detailed descriptions of the experimental results and their interpretations, as well as the experimental conclusions have been also provided.

5. Figure 4: Grading curves – can you provide standard deviations for fractal dimension? How is fractal dimension computed – some words about fractal dimension determination method would be if great interest. What are self-similarity limits? Please be more explicit.

Response: Thanks for your suggestions. We have added the RMSE and determination coefficient in Figure 4, as shown follow.

Fig.4 (a)

Fractal dimension is calculated according to the grain size distribution of granular materials after triaxial test.

In manuscript, we did not discuss the self-similarity. According to the research results, the particle breakage and fractal dimension increase with the increase of the external loading, however the particle breakage and fractal dimension will not increase when external loading more than a certain value. The maximum of fractal dimension is self-similarity limit.

6. Figure 6: as in Figure 4, can you provide standard deviation, determination coefficient ?

Response: Thanks for your suggestions. Due to the legend size, we have added the standard deviation and determination coefficient in Figure 4 – 6, as follow.

Fig.6 (a)

7. What does it means “DEM simulation”.

Response: Thanks for your suggestions. “DEM simulation” is an abbreviation for “discrete element model simulation” which often used in numerical simulation papers. We have revised it.

8. Whereas conclusions presented the important findings of the paper, the importance of these findings together with some future research directions would be of great interest.

Response: Thanks for your suggestions. We will continue our research in this area. “The conclusions are mainly based on the results after test. In fact, fractal dimension and particle breakage change with axial loading. The future research should focus on the results during shear test. The relationship between fractal dimension and particle breakage with shear modulus and volume strain should be created. ”

9. Please provide more literature about the subject.

Response: Thanks for your suggestions. We have added the relevant references and described them in the revised text to read: “Especially, the grading curve changed evidently before and after loading^[4]. Such degradation behaviour of granular materials were found to depend not only on stress history^[5,6], but also on grain size distribution curve^[7,8], parent rock type, particle size^[9,10], particle shape^[11,12] and relative density^[13,14]. For example, particle shape has a significant influence on the particle breakage, which increase with the increasing of shape index sphericity, aspect ratio, convexity and overall regularity^[12]. The critical state parameters ( M, ϕ_cs, e_Γ, and λ_c) decrease with increasing aspect ratio, sphericity and convexity^[8]… It was also found to have a great influence on the shear strength and deformation of granular materials. The greater the relative density, the greater the initial elastic mod-ulus and peak friction angle and the smaller the volume strain^[16,17]… The critical state parameters (M and λ_c) are less affected by relative density and particle grading^[7,13]. However, the critical state parameters (e_Γ) decrease with increasing relative density. The relative breakage index decreases with increasing relative density^[14]”.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Accept

Reviewer 3 Report

All remarks have been considered by authors. Errors have been eliminated.

Currently, the article presents a higher scientific quality.