Thermal Analysis and Thermal–Mechanical Stress Simulation of Polycrystalline Diamond Compact Bits During Rock Breaking Process

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors present simulations and experimental studies on PDC cutters when applied in rock breaking. The thermal and mechanical behaviour -- stress induced and temperature variations during rock breaking -- of the cutters are analyzed and discussed in detail. Overall speaking, the reviewer had a pleasant time reading the manuscript (i.e., the manuscript was well presented, with organized flow and only minor grammatical errors). The manuscript is also informative and the results deserve to be shared among researchers working in the relevant fields.

Below are some comments to further improve the quality of the manuscript:

1. Minor grammatical errors were spotted. Below are some (not all) of the errors spotted:
   • Abstract: add the article "the" before "PDC cutter".
   • Introduction section: add the article "a" before "common practice".
   • Section 4.2: Add "the" before "Process during ..." and "be" before "divided".
   • Section 4.3: "makes the" could be replaced with "results in".
   • Section 5.2: Since the curves in Fig. 15 do not overlap exactly, it may be more appropriate to use "agree closely", instead of "is the same as".
   • Section 5.3: Add "the" before "PDC bit" and "PDC bits".
2. According to the authors, 3 assumptions have been made when performing calculations and analysis. Justifications should be given why these assumptions were made and how these assumptions affect the accuracy of the results.
3. The authors claimed that the Drucker-Prager criterion was employed. They should cite references to show that this criterion is widely adopted.
4. The authors mentioned that the rock-breaking process was analyzed using software. The name of the software should be specified.
5. The schematic diagram in Fig. 14(a) should be called out and described in the text.
6. The abstract should include quantitative results.
7. The variables (i.e., sigma and T) in eq. (1) should be defined.

Comments on the Quality of English Language

Minor grammatical errors were spotted. Below are some (not all) of the errors spotted:

• • Abstract: add the article "the" before "PDC cutter".
  • Introduction section: add the article "a" before "common practice".
  • Section 4.2: Add "the" before "Process during ..." and "be" before "divided".
  • Section 4.3: "makes the" could be replaced with "results in".
  • Section 5.2: Since the curves in Fig. 15 do not overlap exactly, it may be more appropriate to use "agree closely", instead of "is the same as".
  • Section 5.3: Add "the" before "PDC bit" and "PDC bits".

Author Response

Comment 1: Minor grammatical errors were spotted. Below are some (not all) of the errors spotted:

Abstract: add the article "the" before "PDC cutter".

Introduction section: add the article "a" before "common practice".

Section 4.2: Add "the" before "Process during ..." and "be" before "divided".

Section 4.3: "makes the" could be replaced with "results in".

Section 5.2: Since the curves in Fig. 15 do not overlap exactly, it may be more appropriate to use "agree closely", instead of "is the same as".

Section 5.3: Add "the" before "PDC bit" and "PDC bits".

Response: Thank you for your valuable and thoughtful comments. We are sorry for this language mistake. We have corrected the sentence according to your comment. Furthermore, we have had the manuscript polished with a professional assistance in writing.

We have added “the” before “PDC cutter” in the Abstract.

We have added “a” before “common practice” in the Introduction.

The sentence now reads: “The process during 1.0–1.4 s is rock breaking. It can be observed that the temperature rise of the PDC cutters can be divided into three stages”.

We have replaced “makes the” with “results in” for better clarity.

The phrase has been changed to: “…the temperature change trend of the two probes agrees closely with that of the simulation results…”

We have add "the" before "PDC bit" and "PDC bits"

Comment 2: According to the authors, 3 assumptions have been made when performing calculations and analysis. Justifications should be given why these assumptions were made and how these assumptions affect the accuracy of the results.

Response:

These assumptions are commonly adopted in similar thermal-mechanical simulations of PDC cutters to simplify the complex drilling environment, reduce computational cost, and focus on the primary heat generation and stress mechanisms. While neglecting pore media and thermal convection may slightly overestimate temperature gradients, the overall trends in temperature and stress distributions remain valid for comparative analysis under varying formation temperatures and rock strengths.

Comment 3: The authors claimed that the Drucker-Prager criterion was employed. They should cite references to show that this criterion is widely adopted.

Response: Thank you for your valuable and thoughtful comments, we really agree with your viewpoints. The Drucker-Prager failure criterion was employed to model rock behavior, which is widely used in geomechanical simulations to capture pressure-dependent yield of geological materials. According to your comment, we have cited 2 papers, thank you again.

References

[31] Huang, J.; Zeng B.; He, Y., et al, Numerical study of rock-breaking mechanism in hard rock with full PDC bit model in compound impact drilling. Energy Reports 2023, 9, 3896-3909.

[32] Kazi, A.; Antao, D.; Staack, D. , et al, Numerical Investigation of the Effect of Pre‑induced Cracks on Hard Rock Cutting Using Finite Element Analysis. Rock Mechanics and Rock Engineering 2024, 57 , 7997–8011.

Comment 4: The authors mentioned that the rock-breaking process was analyzed using software. The name of the software should be specified.

Response: Thank you for reviewing our manuscript. In Section 1, it is explicitly mentioned to use Abaqus software for rock breaking analysis.

Comment 5: The schematic diagram in Fig. 14(a) should be called out and described in the text.

Response: We have added a callout and description of Fig. 14(a) in Section 5.1.

A schematic diagram of the experimental setup is shown in Fig. 14(a), which illustrates the arrangement of the drilling rig, PDC bit, thermocouple wiring, and data acquisition system.

Comment 6: The abstract should include quantitative results.

Response: We have revised the abstract to include the parameter ranges and key quantitative results.

Change in Manuscript:

Abstract: Polycrystalline diamond compact (PDC) bits are widely used in oil, gas, and geological exploration. During rock breaking, most of the work is converted into cutting heat, leading to cutter temperature rise and potential damage. However, the influence of formation temperature and rock properties on cutting temperature and thermal stress remains insufficiently understood. This study combined numerical simulation and experimental methods to investigate the temperature rise and thermal stress of a single PDC cutter during rock breaking, focusing on the effects of formation temperature (27–250 °C) and rock strength (sandstone, marble, and granite). The results show that the temperature rise of the PDC cutter follows three distinct stages: rapid increase, slow increase, and stabilization. Rock strength significantly affects the temperature rise rate and stress; when breaking granite, the cutter temperature reached approximately 131.4 °C, about 2–3 times higher than for marble and sandstone, while the rate of penetration (ROP) decreased by 70.6–75.6%. As formation temperature increased from 27 °C to 250 °C, the internal temperature difference within the cutter decreased from 72.6 °C to 35.6 °C, and the equivalent stress first increased and then decreased, peaking at 2.84 GPa at 50 °C. The ROP initially increased and then decreased with rising formation temperature. Numerical simulations and experimental findings are in good agreement. This study provides theoretical and technical guidance for optimizing cutter design and improving the rock-breaking efficiency and service life of PDC bits in deep and high-temperature formations.

Comment 7: The variables (i.e., sigma and T) in eq. (1) should be defined.

Response:

where σ is the uniaxial compressive strength (MPa) and T is the temperature (°C).

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript by Zengzeng Zhang, the authors study the thermodynamics of PDC bits during the breaking of sandstone, marble, and granite. The thermal conductivity and mechanical properties of the rock play a crucial role in the efficiency and failure risk of these devices. The authors provide an interesting study using both experiments and simulations, demonstrating the importance of temperature in the process. The introduction and abstract of the article are well written, clearly conveying the significance of the work to the community and the timeliness of the topic. In addition, the conclusions of the article are solid and supported by the experiments and the simulations. However, there are a few points the authors must address before publication in the journal Coatings:

1. Some parts of the article are very similar to previously published works and flagged as such by iThenticate. Please rewrite the text accordingly to avoid any further issues.
2. Simulations of the rock breaking process end at 0.4 seconds, why didn’t the authors study this process further?
3. The timestamps used in the article are not consistent. In figure 4 the rotation starts at 0.0 seconds, but in figure 5 it starts at 1.0 seconds. This is confusing to the reader, and I would like to ask the authors to correct his issue.
4. In figure 5 the temperature drops around 1.4 seconds, does this mean the temperature stabilizes at a temperature that is lower than the one measured after the first round of drilling?
5. What is the formation temperature? It is not indicated in the text.
6. In figure 10 the temperature map is very inconsistent. In figure 10a one cell of the mesh shows temperature that is more than 2 times the one in the neighboring cells. This issue also appears in other panels, where the area of maximum temperature moves through the disc. Is there any problem with the simulations?
7. The maximum stress values of PDC cutters at different temperatures are indicated in the main text, but I would like to kindly ask the authors to represent them in a separate graph or table.
8. In figure 15 the temperature as a function of the time is represented. The authors show three different regimes spanning from 0 to 100, 200 and 300 seconds. However, the simulations only cover a fraction of this, even showing changes in the temperature that are 100 times faster than those observed in the experiments. For this reason, I would like to ask the authors that in future works they simulate these thermodynamic properties in timescales that are more relevant to this system.

Author Response

Response to Reviewers’ Comments

We sincerely thank the reviewers for their careful reading and constructive suggestions, which have greatly helped us improve the quality of the manuscript. Below we provide point-by-point responses to the comments.

Comment 1:

Some parts of the article are very similar to previously published works and flagged as such by iThenticate. Please rewrite the text accordingly to avoid any further issues.

Response: We appreciate the reviewer’s attention to originality. We have carefully examined the iThenticate report and identified the overlapping sections, mainly in the Introduction and literature review. These parts have now been thoroughly rewritten to present the background and existing research in our own words, with proper citations. The revised text ensures originality while maintaining clarity and academic rigor.

Comment 2:

Simulations of the rock breaking process end at 0.4 seconds, why didn’t the authors study this process further?

Response: The simulation duration of 0.4 s was chosen because it covers one complete rotation cycle of the cutter under the set rotational speed (150 RPM). Within this period, the cutter completes a full engagement with the rock, and the temperature and stress fields reach a representative steady-state distribution. Extending the simulation further would not yield significantly new physical insights but would considerably increase computational cost. However, we acknowledge the reviewer’s point and will consider longer simulation times in future studies to capture possible long-term thermal accumulation effects.

Comment 3:

The timestamps used in the article are not consistent. In figure 4 the rotation starts at 0.0 seconds, but in figure 5 it starts at 1.0 seconds. This is confusing to the reader, and I would like to ask the authors to correct this issue.

Response: Thank you for highlighting this inconsistency. Figure 4 illustrates the rock-breaking and crater formation process during Step 2 of the simulation (cutter rotation), starting at 0 s. Figure 5 plots the temperature evolution beginning from Step 1 (vertical penetration), with the 1.0 s mark indicating the start of rotational cutting. We will revise the captions and text to clearly explain the different simulation stages and time origins to avoid confusion.

Comment 4:

In figure 5 the temperature drops around 1.4 seconds, does this mean the temperature stabilizes at a temperature that is lower than the one measured after the first round of drilling?

Response: The temperature drop around 1.4 s is due to the cutter completing one full rotation and moving into a region of less intense frictional contact as the rock fragment is removed. This leads to a transient decrease in heat generation. Subsequently, as the cutter continues to engage fresh rock, the temperature rises again and eventually stabilizes. The stabilized temperature is indeed slightly lower than the peak at the end of the first rotation, reflecting a dynamic balance between heat generation and conduction/convection losses.

Comment 5: What is the formation temperature? It is not indicated in the text.

Response: “Formation temperature” refers to the initial temperature of the rock before cutting. In our simulations and experiments, formation temperature is set as a variable (27, 50, 100, 150, 200, and 250 °C). This has now been explicitly defined in Section 2 (Rock mechanics parameters) and again in Section 3.3 (Boundary and load) to avoid ambiguity.

Comment 6: In figure 10 the temperature map is very inconsistent. In figure 10a one cell of the mesh shows temperature that is more than 2 times the one in the neighboring cells. This issue also appears in other panels, where the area of maximum temperature moves through the disc. Is there any problem with the simulations?

Response: Thank you for your careful observation regarding Figure 10. We would like to clarify that Figure 10 is a stress nephogram (thermal stress contour) showing the equivalent stress distribution during the rock-breaking process, not a temperature map. The caption of Figure 10 reads: “Stress nephogram of sandstone broken by PDC cutter.” However, your observation about localized high values and the movement of the high-stress zone is valid and physically meaningful even in the context of stress distribution: Localized High Stress Cells: The isolated cells showing stress values much higher than their neighbors correspond to local stress concentrations at asperity contacts or at the instantaneous cutting front. Such concentrations are typical in dynamic cutting simulations and reflect real contact mechanics where load is not uniformly distributed. Movement of High-Stress Zone: As the cutter rotates (from 0.1 s to 0.4 s), the region of maximum stress naturally shifts along the cutter crown, following the changing contact area with the rock. This is consistent with the cyclic loading experienced by the cutter during rotation.

  We apologize for any confusion caused by the misinterpretation of the figure type. The simulation results are physically sound, and the stress concentrations and their movement are expected behaviors in a dynamic cutting process. If it would be helpful, we can provide additional temperature-field figures (which are presented separately in Figures 6 and 9) for further clarity.

Comment 7: The maximum stress values of PDC cutters at different temperatures are indicated in the main text, but I would like to kindly ask the authors to represent them in a separate graph or table.

Response: We agree that a separate presentation would be clearer. A new table (Table X) summarizing the maximum equivalent stress of the PDC cutter at each formation temperature (27–250 °C) will be added in Section 4.3.1. The data will also be plotted in a new figure for visual comparison.

Fig. 12. The relationship between temperature and maximum stress.

Comment 8: In figure 15 the temperature as a function of the time is represented. The authors show three different regimes spanning from 0 to 100, 200 and 300 seconds. However, the simulations only cover a fraction of this, even showing changes in the temperature that are 100 times faster than those observed in the experiments. For this reason, I would like to ask the authors that in future works they simulate these thermodynamic properties in timescales that are more relevant to this system.

Response: We acknowledge the discrepancy in time scales between simulation (sub-second) and experiment (hundreds of seconds). The simulation focuses on the transient thermo-mechanical response during a single cutter–rock engagement cycle, which occurs rapidly. The experiment captures longer-term cumulative heating under continuous drilling with fluid circulation. Both approaches are valid for their respective objectives. However, we agree that a coupled long-timescale simulation would be valuable for direct comparison with experiments. In future work, we plan to develop a multi-scale modeling approach that bridges the short-term cutting physics with long-term thermal accumulation.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

While the underlying research topic is interesting, the manuscript, as currently written, suffers from a lack of clarity and essential detail in several sections. Specifically, literature review in the introduction does not adequately summarize necessary details of the methodology and key results, specifically. The conclusions section does not sufficiently and precisely capture the key findings, not indicating the completion/success of the objectives and scope of this work. The following concerns should be addressed before publication.
1.    For enhanced specificity and clarity, the full name of the material (e.g., Polycrystalline Diamond Compact) should be used instead of the abbreviation PDC in the manuscript's title. Also, the title must also clearly state the focus on both thermal and mechanical (stress) properties to better reflect the work presented.
2.    The abstract must be revised to include a statement regarding the range of studied parameters. Furthermore, it is necessary to incorporate quantitative data for the key findings and outcomes, ensuring the summary provides specific, numerical evidence of the research results.
3.    The concluding paragraph of the Introduction needs to be strengthened to precisely define the objective(s) and scope of the investigation, thereby demonstrating how the research gap will be addressed, specifically, in a way that the completion/success of this work can be evaluated. This requires the explicit identification of both the independent/varying parameters (such as material types and time) and the focused measurable properties (such as surface temperature and stress profile).
4.    The Conclusions section must be revised to provide a more specific summary of the key outcomes and findings, including both the concept/observation and important quantitative data.
5.    Several parts in the introduction lack necessary details:
5.1.    In the first paragraph (lines 33-43), please provide background information on the PDC bits, including their shape, size, and the location installed in the cutting system, and how they are fixed with other components.
5.2.    Lines 47-48, please specifically indicate the range for “high temperatures”.
5.3.    Throughout the paragraph in lines 56-81, most of the statements introducing the literature review lack clarity and fail to present the necessary details of the range of studied parameters and key results.
5.3.1.    Lines 57-58, “… it is common practice to study the cutting behavior of a single PDC cutter [15].”: Please elaborate on the measurable physical properties of the mentioned “cutting behavior”.
5.3.2.    Line 60: Please elaborate on the “…temperature increase laws of PDC bits in rock drilling.”.
5.3.3.    Lines 61-63, “Glowka et al. [18–20] established a finite-element model for a single PDC cutter to predict the average cutting temperature of a wear plane…”: Please clarify the size of the system and temperature range, more specifically.
5.3.4.    Lines 63-64, “…The results showed that there was a correlation between the wear plane temperature and bit wear rate.”: What exactly is the correlation between them? How would increasing the wear plane temperature affect the bit wear rate (increase, decrease, u-shaped/bell-shaped, etc.)?
5.3.5.    Lines 65-66, “…when cutting granite with a PDC cutter and obtained that the average wear plane temperature was constant, whereas the wear rate of the bit varied.”: As this statement may be valid only in a certain condition (i.e., not always be true at all conditions), please indicate the size of the PDC, cutting speed, and the constant value and range of temperatures, specifically.
5.3.6.    Lines 66-68, “Che et al. [21] introduced the theory of heat generation in metal cutting and analyzed the temperature change of a PDC cutter during the rock-breaking process.”: Please briefly describe key details of the theory such as assumptions, related variables, and core concept.
5.3.7.    Lines 68-69, “Beste et al. [22] concluded that there is uncertainty in the temperature of the bit…”, please elaborate on the term “uncertainty”.
5.3.8.    Lines 70-72, “Shao et al. [23] studied the contact surface temperature …, and found that the temperature of the cutter showed a positive correlation with the cutting depth and cutting speed.”: Please clarify the reason for this observation. Why do the cutting depth and cutting speed increases the contact surface temperature.
5.3.9.    Lines 72-74, “Kim et al. [24] studied … the effect on temperature was more evident when the contact area between the cutter and rocks was larger.”: Please clarify the reason for this observation.
5.3.10.    Lines 78-81, “Zhang et al. [26] established a full-size PDC bit model using the finite element method to study the temperature and stress fields during the rock crushing process and verified the accuracy of the finite element method through laboratory experiments.”: Please clarify the size of the system and the temperature and stress ranges, specifically.
6.    The following parts need clarification and/or supporting information:
6.1.    Lines 217-219, “The temperature of the cutter increases with an increase in the formation temperature because of the superposition of the cutting temperature produced by the friction between the cutter and the rock, and the formation temperature promotes the corresponding temperature rise in the cutter.”: Please provide references or a similar finding in the literature.
6.2.    Lines 246-248, “It can be observed that with an increase in the rock temperature, the internal temperature difference of the PDC cutter showed a decreasing trend.”: So what? Is this result preferable?
6.3.    Lines 250-266, please provide background information about the strength values of Sandstone, marble, and granite, with credible references.
6.4.    Lines 275-277, “During the rock-breaking process, the stress on the PDC cutter is mainly concentrated near the crown of the cutter and at the junction surface between the diamond layer and tungsten carbide matrix.”: Please improve logical reasoning, describing why the stress on the PDC cutter is mainly concentrated near the crown of the cutter and the junction surface.
6.5.    Line 322, related to Figure 13, “…shows that the stress increases first and then fluctuates.” Please clarify why this trend is observed.
6.6.    Lines 312-313, section 4.3.2, “The maximum stress of PDC cutters when breaking sandstone, marble, and granite is 1.65, 2.304, and 2.44 GPa, respectively.”: The condition for these results is unclear.
7.    I recommend presenting key data of stress profiles in Section 4.3 visually, along with FEM simulation captures:
7.1.    Section 4.3.1 (lines 287-288), I suggest visualizing the data in a plot of Stress vs Temperature (associated with Figure 1) at a certain location and/or the max values.
7.2.    Section 4.3.2 (lines 310-315), I suggest presenting a plot of characteristic values of the stress (e.g., for max, min, and mean values) for various rock types, adding as Figure 12(d).
7.3.    Section 4.3.2 (lines 325-327), the average stresses of the cutter when breaking stones, marble, and granite (associated with Figure 13) are reported. However, each graph exhibits two distinct regimes (e.g., initial low stress phase and subsequent active phase). Reporting only the single overall average value does not seem to adequately capture the characteristic stress profile across these regimes to offer a more complete comprehensive analysis. Therefore, I recommend reporting additional average values for the two regimes (such as before and after a certain point in time, e.g., 1 s). Please also consider adding an accompanying figure (as Figure 13b) that visually plots these three calculated averages (overall, initial regime, and active regime) for various material types.
8.    Comments on table and figure presentation:
8.1.    In Table 1, please include the unit of the temperature (the first column).
8.2.    In Table 2, I suggest presenting the value of the thermal expansion coefficient (the last column) in the unit of “˚C-1” to avoid confusion. For example, it is unclear whether the thermal expansion value of PCD is 2.5×10-6 ˚C-1 or 2.5×106 ˚C-1.
8.3.    Please ensure that Figure 3 (and other with subfigures) indicates (a), (b), … for each subfigure. 
8.4.    In Figure 2, please show the fitting curve based on the correlation in Equation (1), instead of the connecting line, for the uniaxial compressive strength. Also, the unit of thermal conductivity in Figure 2 is incorrect.
8.5.    In Figure 3, please show the color scale.
8.6.    Clarify the unit of TEMP P (Figures 6 and 9) and S MISES (Figures 10-12) in the figure caption.
8.7.    Please ensure the alignment and readable font size, for example, Figure 10.
8.8.    In Figure 14a, please clarify the dimension of important parts of the system shown.
8.9.    I suggest combining Fig. 15-16 as (a)-(b). Also, it is unclear the difference between the results of these two figures. The purpose of presenting these data is unclear.
9.    Other comments
9.1.    Lines 106-107, there seems to be a paragraphing error.
9.2.    I suggest using the full word for “FEM” for the heading of section 3 (line 142).
9.3.    Equation (1) is an empirical correlation, which is valid for a specific unit of the uniaxial compressive strength and temperature. Therefore, it is necessary to note such units when presenting the equation. 
9.4.    The expression format for units should be consistent throughout the manuscript. Both negative exponent (kg·m^-3) and slash notation (W/(m·˚C)) formats are used inconsistently.
9.5.    Use of “Figure” and “Fig.” is inconsistent, for example, Figure 12.
9.6.    Line 345, please include the country of Hai’an City.

Author Response

Response to Reviewer Comments

We sincerely thank the reviewer for the careful reading of our manuscript and for providing constructive and detailed comments. These suggestions are invaluable for improving the quality and clarity of our work. We have revised the manuscript accordingly, and a point-by-point response is provided below. All changes in the manuscript are highlighted in red for ease of identification.

Comment 1: For enhanced specificity and clarity, the full name of the material (e.g., Polycrystalline Diamond Compact) should be used instead of the abbreviation PDC in the manuscript's title. Also, the title must also clearly state the focus on both thermal and mechanical (stress) properties to better reflect the work presented.

Response: Thank you for this suggestion. We have revised the title accordingly.

Change in Manuscript:

Original Title: Thermal analysis and simulation research of PDC bits during rock breaking process

Revised Title: Thermal analysis and thermal-mechanical stress simulation of polycrystalline diamond compact bits during rock breaking process

Comment 2: The abstract must be revised to include a statement regarding the range of studied parameters. Furthermore, it is necessary to incorporate quantitative data for the key findings and outcomes.

Response: We have revised the abstract to include the parameter ranges and key quantitative results.

Change in Manuscript:

Abstract: Polycrystalline diamond compact (PDC) bits are widely used in oil, gas, and geological exploration. During rock breaking, most of the work is converted into cutting heat, leading to cutter temperature rise and potential damage. However, the influence of formation temperature and rock properties on cutting temperature and thermal stress remains insufficiently understood. This study combined numerical simulation and experimental methods to investigate the temperature rise and thermal stress of a single PDC cutter during rock breaking, focusing on the effects of formation temperature (27–250 °C) and rock strength (sandstone, marble, and granite). The results show that the temperature rise of the PDC cutter follows three distinct stages: rapid increase, slow increase, and stabilization. Rock strength significantly affects the temperature rise rate and stress; when breaking granite, the cutter temperature reached approximately 131.4 °C, about 2–3 times higher than for marble and sandstone, while the rate of penetration (ROP) decreased by 70.6–75.6%. As formation temperature increased from 27 °C to 250 °C, the internal temperature difference within the cutter decreased from 72.6 °C to 35.6 °C, and the equivalent stress first increased and then decreased, peaking at 2.84 GPa at 50 °C. The ROP initially increased and then decreased with rising formation temperature. Numerical simulations and experimental findings are in good agreement. This study provides theoretical and technical guidance for optimizing cutter design and improving the rock-breaking efficiency and service life of PDC bits in deep and high-temperature formations.

Comment 3: The concluding paragraph of the Introduction needs to be strengthened to precisely define the objective(s) and scope of the investigation... explicitly identify both the independent/varying parameters and the focused measurable properties.

Response: We have rewritten the final paragraph of the Introduction to clearly state the objectives, scope, independent variables, and dependent measurable properties.

Change in Manuscript:

Therefore, to understand the temperature and thermal stress field distribution rules of the PDC cutter during the rock-breaking process, this study aims to: (1) investigate the influence of formation temperature (varied from 27 °C to 250 °C) and rock strength (represented by sandstone, marble, and granite) on the cutting temperature, temperature distribution, and thermal stress field of a single PDC cutter; (2) analyze the evolution of the stress profile (e.g., equivalent von Mises stress) and rate of penetration (ROP) under these conditions. Using finite element simulation (Abaqus) coupled with experimental validation, we focus on measurable outcomes including maximum/minimum cutter temperature, temperature gradient, stress magnitude at the crown and diamond-matrix interface, and ROP. This work addresses the identified gap by quantifying the thermo-mechanical response of PDC cutters to downhole temperature and formation strength, providing a basis for bit design in deep, high-temperature formations.

Comment 4: The Conclusions section must be revised to provide a more specific summary of the key outcomes and findings, including both the concept/observation and important quantitative data.

Response: We have restructured the Conclusions section to first list key scientific findings with integrated quantitative data, followed by engineering implications and future work.

Change in Manuscript:

This study systematically investigated the temperature and thermal stress evolution of PDC cutters during rock breaking through integrated numerical simulation and experimental validation, with a particular focus on the effects of formation temperature and rock strength. The main findings and their implications are summarized as follows:

(1) The temperature rise of a PDC cutter during rock breaking exhibits a consistent three-stage pattern: rapid increase, slow increase, and stabilization. Rock strength is a dominant factor influencing the rate of temperature rise and the magnitude of cutter temperature. When breaking granite (uniaxial compressive strength ≈134.8 MPa at 27 °C), the cutter temperature reached approximately 131.4 °C, about 2 and 3 times higher than when cutting marble (≈75.2 °C) and sandstone (≈46.3 °C), respectively. Correspondingly, the rate of penetration (ROP) decreased by 70.6% and 75.6% when drilling granite compared to marble and sandstone.

(2) Increasing formation temperature reduces the internal temperature gradient within the cutter, thereby mitigating thermal stress. As the formation temperature rose from 27 °C to 250 °C, the temperature difference between the maximum and minimum points on the cutter decreased from 72.6 °C to 35.6 °C. However, due to material heterogeneity and differential thermal expansion, significant thermal stress still develops, with the maximum equivalent stress (2.84 GPa) occurring at 50 °C formation temperature.

(3) The stress distribution in the PDC cutter is highly concentrated at the crown and at the interface between the diamond layer and the tungsten carbide matrix. Both the magnitude and fluctuation amplitude of stress increase with rock strength. The average stress when breaking granite (2.128 GPa) was 58.2% and 134.8% higher than when breaking marble (1.345 GPa) and sandstone (0.906 GPa), respectively. Stress evolution during cutting shows an initial sharp increase followed by fluctuations, with greater instability observed in harder rocks.

(4) High formation temperatures alter the rock failure mode from brittle to plastic, which affects cutting efficiency. Although rock strength decreases at elevated temperatures (e.g., granite compressive strength dropped from 134.8 MPa at 27 °C to 75.1 MPa at 250 °C), the increased plasticity can lead to a "rubber layer effect," reducing the ROP. The highest ROP (0.06 mm/s) was observed at 150 °C, compared to 0.044 mm/s at 27 °C and 0.052 mm/s at 250 °C.

Comment 5:  Several parts in the introduction lack necessary details:

Response: We have expanded the indicated paragraphs with the requested details.

5.1.    In the first paragraph (lines 33-43), please provide background information on the PDC bits, including their shape, size, and the location installed in the cutting system, and how they are fixed with other components.

Response:

Response: Thank you for the suggestion. We have revised the first paragraph of the Introduction to include the following background information:

“PDC bits are typically composed of a steel or matrix body on which multiple cutters are mounted. Each cutter consists of a polycrystalline diamond layer sintered onto a tungsten carbide substrate. The cutters are usually circular or slightly chamfered in shape, with typical diameters ranging from 13 mm to 19 mm. They are installed on the bit face in a patterned layout (e.g., spiral, blade, or roller-cone arrangements) and are rigidly fixed into pockets via brazing or mechanical pressing. This assembly allows the cutters to effectively engage and break rock under combined axial and rotational loads.”

5.2.    Lines 47-48, please specifically indicate the range for “high temperatures”.

Response: We have clarified the term “high temperatures” by adding a typical temperature range based on literature:

“High temperatures (typically above 300–800 °C) cause changes in the physical and mechanical properties of the PDC cutter…”

5.3. Throughout the paragraph in lines 56-81, most of the statements introducing the literature review lack clarity and fail to present the necessary details of the range of studied parameters and key results.

Response: We agree and have revised the literature review to include more specific information regarding study parameters and key findings. The updated text now provides clearer and more detailed summaries of each cited work.

5.3.1.    Lines 57-58, “… it is common practice to study the cutting behavior of a single PDC cutter [15].”: Please elaborate on the measurable physical properties of the mentioned “cutting behavior”.

Response:We have expanded the sentence to clarify what “cutting behavior” entails:

“…it is common practice to study the cutting behavior of a single PDC cutter [15], which typically includes measurable quantities such as cutting forces (normal and tangential), cutter temperature, wear volume, rate of penetration, and specific energy.”

5.3.2.    Line 60: Please elaborate on the “…temperature increase laws of PDC bits in rock drilling.”.

Response: The phrase has been clarified as follows:

“…relatively few studies have been conducted on the heat generation and temperature increase laws of PDC bits in rock drilling, particularly regarding how temperature evolves with drilling time, depth, and operational parameters.”

5.3.3.    Lines 61-63, “Glowka et al. [18–20] established a finite-element model for a single PDC cutter to predict the average cutting temperature of a wear plane…”: Please clarify the size of the system and temperature range, more specifically.

Response:

We have added details from the cited references:

“Glowka et al. [18–20] established a finite-element model for a single PDC cutter (13.44 mm diameter) to predict the average cutting temperature of a wear plane under steady-state and transient downhole conditions. Their simulated temperatures ranged from 200 °C to over 700 °C depending on the wear flat size and cutting conditions.”

5.3.4.    Lines 63-64, “…The results showed that there was a correlation between the wear plane temperature and bit wear rate.”: What exactly is the correlation between them? How would increasing the wear plane temperature affect the bit wear rate (increase, decrease, u-shaped/bell-shaped, etc.)?

Response: We have specified the nature of the correlation: “The results showed a positive correlation between the wear plane temperature and bit wear rate; specifically, as the wear plane temperature increased, the wear rate of the PDC cutter also increased, primarily due to accelerated thermal degradation and diamond graphitization.”

5.3.5.    Lines 65-66, “…when cutting granite with a PDC cutter and obtained that the average wear plane temperature was constant, whereas the wear rate of the bit varied.”: As this statement may be valid only in a certain condition (i.e., not always be true at all conditions), please indicate the size of the PDC, cutting speed, and the constant value and range of temperatures, specifically.

Response: They have added experimental conditions based on the original study:

“Appl et al. [12] experimentally measured the temperature and wear rate of a 13.44 mm PDC cutter when cutting granite at a cutting speed of 2 m/s. They reported that the average wear plane temperature remained nearly constant at approximately 350 ± 20 °C under steady-state conditions, while the wear rate varied with changes in normal load and rock properties.”

5.3.6.    Lines 66-68, “Che et al. [21] introduced the theory of heat generation in metal cutting and analyzed the temperature change of a PDC cutter during the rock-breaking process.”: Please briefly describe key details of the theory such as assumptions, related variables, and core concept.

Response:We have added a brief description of the theory:

“Che et al. [21] introduced the theory of heat generation in metal cutting, which assumes that nearly all plastic work during cutting is converted into heat. The model relates heat generation to cutting forces, speed, and material properties, and was adapted to analyze temperature rise in PDC cutters during rock breaking. Key variables include shear plane energy, friction work, and partition coefficients for heat flow into the cutter and rock.”

5.3.7.    Lines 68-69, “Beste et al. [22] concluded that there is uncertainty in the temperature of the bit…”, please elaborate on the term “uncertainty”.

Response: We have clarified the meaning of “uncertainty” in this context:

“Beste et al. [22] concluded that there is considerable uncertainty in predicting the exact temperature of the bit during drilling due to variations in contact conditions, cooling effects, and heterogeneous rock properties. Their experimental measurements indicated that the average cutter temperature could range between 300 and 500 °C under typical drilling conditions.”

5.3.8.    Lines 70-72, “Shao et al. [23] studied the contact surface temperature …, and found that the temperature of the cutter showed a positive correlation with the cutting depth and cutting speed.”: Please clarify the reason for this observation. Why do the cutting depth and cutting speed increases the contact surface temperature.

Response: We have explained the underlying mechanism:

“Shao et al. [23] studied the contact surface temperature between the cutter and rock when cutting sandstone, and found that the temperature of the cutter showed a positive correlation with the cutting depth and cutting speed. This is because larger cutting depths and higher speeds increase the volume of rock deformed per unit time and the frictional work at the cutter–rock interface, leading to greater heat generation and higher temperatures.”

5.3.9.    Lines 72-74, “Kim et al. [24] studied … the effect on temperature was more evident when the contact area between the cutter and rocks was larger.”: Please clarify the reason for this observation.

Response: We have added a physical explanation:

“Kim et al. [24] studied the influence of tooth shape on the temperature of a cutter and found that the effect on temperature was more evident when the contact area between the cutter and rocks was larger. This occurs because a larger contact area increases frictional resistance and reduces heat dissipation per unit area, resulting in higher interface temperatures.”

5.3.10.    Lines 78-81, “Zhang et al. [26] established a full-size PDC bit model using the finite element method to study the temperature and stress fields during the rock crushing process and verified the accuracy of the finite element method through laboratory experiments.”: Please clarify the size of the system and the temperature and stress ranges, specifically.

Response:  We have included specific data from the cited study:

“Zhang et al. [26] established a full-size PDC bit model using the finite element method to study the temperature and stress fields during the rock crushing process. Their simulation and experiments showed that the maximum cutter temperature reached 250 °C under typical drilling conditions. The study verified the accuracy of the finite element approach through laboratory measurements. ”

Comment 6: The following parts need clarification and/or supporting information:

6.1 Lines 217-219, “The temperature of the cutter increases with an increase in the formation temperature because of the superposition of the cutting temperature produced by the friction between the cutter and the rock, and the formation temperature promotes the corresponding temperature rise in the cutter.”: Please provide references or a similar finding in the literature.

Response: Thank you for raising this point. The statement is based on the principle of thermal superposition in thermo-mechanical systems, where the total temperature rise in a cutting tool is the sum of the ambient (formation) temperature and the temperature rise due to frictional heating. Similar phenomena have been reported in studies on PDC cutter temperature under downhole conditions. For instance:

Glowka & Stone (1985) noted that downhole temperature significantly influences the thermal response of PDC cutters, with higher ambient temperatures leading to elevated cutter temperatures under identical cutting conditions [18].

Che et al. (2012) also observed that both ambient temperature and frictional heat contribute to the overall temperature of PDC cutters during rock cutting [21].

6.2 Lines 246-248, “It can be observed that with an increase in the rock temperature, the internal temperature difference of the PDC cutter showed a decreasing trend.”: So what? Is this result preferable?

Response: Thank you for this important question. The reduction in internal temperature difference (gradient) with rising formation temperature is indeed a significant finding because a smaller temperature gradient generally leads to lower thermal stress within the cutter, which is beneficial for reducing thermal cracking and extending cutter life. However, it should be noted that while the gradient decreases, the absolute temperature of the cutter increases, which may still pose risks such as diamond graphitization or interfacial debonding. Therefore, the result suggests that in high-temperature formations, although the cutter experiences higher overall temperatures, the thermal stress concentration may be somewhat alleviated due to a more uniform temperature distribution. This insight is valuable for designing cutters with better thermal management in deep, high-temperature wells.

6.3 Lines 250-266, please provide background information about the strength values of Sandstone, marble, and granite, with credible references.

Response: The strength values of sandstone, marble, and granite are shown in Table 2. These data were obtained from experiments.

6.4 Lines 275-277, “During the rock-breaking process, the stress on the PDC cutter is mainly concentrated near the crown of the cutter and at the junction surface between the diamond layer and tungsten carbide matrix.”: Please improve logical reasoning, describing why the stress on the PDC cutter is mainly concentrated near the crown of the cutter and the junction surface.

Response: We appreciate this suggestion. The stress concentration in these regions can be attributed to the following reasons:

    Crown region: This is the primary contact area with the rock during cutting. High contact forces, combined with frictional heating, lead to significant mechanical and thermal stress concentration.

   Diamond–matrix interface: This is a bi-material interface with a mismatch in thermal expansion coefficients (PCD ≈ 2.5×10⁻⁶/°C, WC-Co ≈ 5.2×10⁻⁶/°C) and elastic moduli. Under thermal and mechanical loading, this mismatch induces high interfacial shear and normal stresses, making it a common site for delamination or cracking.

6.5 Line 322, related to Figure 13, “…shows that the stress increases first and then fluctuates.” Please clarify why this trend is observed.

Response: Thank you for this question. The observed trend can be explained as follows:

   Initial sharp increase: Corresponds to the cutter’s initial penetration into the rock, where contact stress builds up rapidly as the cutter engages with the intact rock surface.

   Subsequent fluctuations: Reflect the cyclic nature of rock fragmentation—brittle fracture events release stress temporarily, followed by re-engagement and stress buildup. This is particularly pronounced in hard, brittle rocks like granite, where fracture events are more sudden and energetic.

6.6 Lines 312-313, section 4.3.2, “The maximum stress of PDC cutters when breaking sandstone, marble, and granite is 1.65, 2.304, and 2.44 GPa, respectively.”: The condition for these results is unclear.

Response: We apologize for the lack of clarity. These results were obtained at room temperature, with a weight-on-bit (WOB) of 2kN and a rotational speed of 150rpm. We have now explicitly stated these conditions in Section 4.3.2 for clarity.

Comment 7: I recommend presenting key data of stress profiles in Section 4.3 visually, along with FEM simulation captures.

Response: We agree that visual summaries of stress data would be valuable. We will add the following figures to the revised manuscript:

7.1 Section 4.3.1 (lines 287-288), I suggest visualizing the data in a plot of Stress vs Temperature (associated with Figure 1) at a certain location and/or the max values.

Response: Thank you for this valuable suggestion. We agree that a graphical representation of the maximum equivalent stress as a function of formation temperature would provide a clearer and more immediate understanding of the observed trend (initial increase followed by a decrease). In the revised manuscript, we will add a new figure to plot the maximum von Mises stress (extracted from the cutter crown region) against the formation temperature (27, 50, 100, 150, 200, and 250 °C). This plot will visually confirm the peak stress occurring at 50 °C (2.84 GPa) and the subsequent decline at higher temperatures, complementing the descriptive text in Section 4.3.1.

Fig. 12. The relationship between temperature and maximum stress.

7.2 Section 4.3.2 (lines 310-315), I suggest presenting a plot of characteristic values of the stress (e.g., for max, min, and mean values) for various rock types, adding as Figure 12(d).

Response: This is an excellent suggestion. Figure 13 has summarized the maximum and average von Mises stresses for sandstone, marble, and granite, highlighting the strong dependence of stress magnitude on rock strength, thereby reinforcing the findings described in the paper. I will incorporate this in future research. Thank you again.

7.3: Section 4.3.2 (lines 325-327), the average stresses of the cutter when breaking stones, marble, and granite (associated with Figure 13) are reported. However, each graph exhibits two distinct regimes (e.g., initial low stress phase and subsequent active phase). Reporting only the single overall average value does not seem to adequately capture the characteristic stress profile across these regimes to offer a more complete comprehensive analysis. Therefore, I recommend reporting additional average values for the two regimes (such as before and after a certain point in time, e.g., 1 s). Please also consider adding an accompanying figure (as Figure 13b) that visually plots these three calculated averages (overall, initial regime, and active regime) for various material types.

Response: We thank the reviewer for this astute observation. The reviewer is correct that the single overall average stress obscures the distinct mechanical behavior during different cutting stages. As seen in Figure 13 (current manuscript), the stress evolution indeed shows two primary regimes:

Initial Penetration/Low-Stress Regime (~1.0–1.1 s): Characterized by a sharp stress rise as the cutter engages the intact rock surface.

    Active Cutting/High-Fluctuation Regime (~1.1–1.4 s): Characterized by stress fluctuations due to cyclic rock fragmentation.

Comment 8: Comments on table and figure presentation:

8.1 In Table 1, please include the unit of the temperature (the first column).

Response: Thank you for pointing this out. We will add the unit "(°C)" to the first column "Temperature" in Table 1, revising it to "Temperature (°C)".

Table 1. Physical and thermodynamic parameters for finite element analysis.

8.2 In Table 2, I suggest presenting the value of the thermal expansion coefficient (the last column) in the unit of “˚C-1” to avoid confusion. For example, it is unclear whether the thermal expansion value of PCD is 2.5×10-6 ˚C-1 or 2.5×106 ˚C-1.

Response: We accept the suggestion. We will revise the header of the last column in Table 2 to "Thermal expansion coefficient (×10⁻⁶ °C⁻¹)" and clearly present all values as, for example, "2.5", "5.2", etc., to unambiguously indicate the 10⁻⁶ scale.

Table 2. The main parameters of PDC cutter and rock used in finite element analysis.

8.3 Please ensure that Figure 3 (and other with subfigures) indicates (a), (b), … for each subfigure.

Response: Thank you for the reminder. We will review all figures containing subfigures (including Figures 3, 4, 6, 10, 11, 12, 14, etc.) and ensure that each subfigure is clearly labeled with identifiers like (a), (b) either within the image or in the caption.

8.4 In Figure 2, please show the fitting curve based on the correlation in Equation (1), instead of the connecting line, for the uniaxial compressive strength. Also, the unit of thermal conductivity in Figure 2 is incorrect.

Response: We appreciate this important correction. While "W/(m·°C)" is thermodynamically equivalent to "W/(m·K)", we will revise it to the more standard "W/(m·K)" in the revised manuscript to avoid any potential confusion and align with common conventions.

Fig. 2. Changes in thermal conductivity and uniaxial compressive strength of rocks with temperature.

8.5   In Figure 3, please show the color scale.

Response: Thank you for pointing this out. Figure 3(a) and (b) represent the geometric model and mesh partitioning diagram, respectively, which typically do not involve color scales. We understand that the reviewer may be referring to the temperature field or stress field contour plots discussed later in the text.

8.6 Clarify the unit of TEMP P (Figures 6 and 9) and S MISES (Figures 10-12) in the figure caption.

Response: Thank you. We will add clarifications in the captions of the respective figures:

"TEMP P" represents temperature in degrees Celsius (°C).

"S MISES" represents the von Mises equivalent stress in Pascals (Pa) or Gigapascals (GPa), which will be kept consistent within the context.

8.7 Please ensure the alignment and readable font size, for example, Figure 10.

Response: Noted. We will re-layout and optimize all figures, especially multi-panel figures like Figure 10, to ensure proper alignment between subfigures and that all labels, legends, and axis text have appropriate, clearly readable font sizes.

8.8 In Figure 14a, please clarify the dimension of important parts of the system shown.

Response: Accepted. We will add dimension labels or descriptions for key components (e.g., the bit, sample size, heating chamber) in the schematic diagram of the experimental setup (Figure 14a).

8.9 I suggest combining Fig. 15-16 as (a)-(b). Also, it is unclear the difference between the results of these two figures. The purpose of presenting these data is unclear.

Response: We thank the reviewer for this constructive suggestion.

Fig. 15: Aims to demonstrate the three-stage temperature evolution over time for a fixed rock type (granite) under constant WOB and RPM, validating the temperature change pattern observed in simulations.

Fig. 16: Aims to compare the significant differences in final steady-state temperature and heating rate when the PDC bit interacts with three rocks of different strengths (sandstone, marble, granite) under the same drilling parameters, thereby highlighting the critical influence of rock strength on cutting temperature.

Comment 9:  Other Comments

9.1.    Lines 106-107, there seems to be a paragraphing error.

Response: Thank you for pointing this out. We have reviewed the text around lines 106–107 and corrected the paragraph formatting to ensure proper structure and logical flow.

9.2.    I suggest using the full word for “FEM” for the heading of section 3 (line 142).

Response: Thank you for the suggestion. We have revised the heading of Section 3 from “FEM of rock breaking process” to “Finite Element Modeling of rock breaking process” for clarity.

9.3.    Equation (1) is an empirical correlation, which is valid for a specific unit of the uniaxial compressive strength and temperature. Therefore, it is necessary to note such units when presenting the equation.

Response: We agree and have now added a note below Equation (1) specifying the units:

where σ is the uniaxial compressive strength (MPa) and T is the temperature (°C).

9.4.    The expression format for units should be consistent throughout the manuscript. Both negative exponent (kg·m-3) and slash notation (W/(m·˚C)) formats are used inconsistently.

Response: Thank you for highlighting this inconsistency. We have unified the unit notation to the slash format throughout the manuscript (e.g., kg/m³, W/(m·°C)) for consistency, except in table headers where the compact form is retained for space reasons, with clarity maintained.

9.5.    Use of “Figure” and “Fig.” is inconsistent, for example, Figure 12.

Response: We have standardized the abbreviation to “Fig.” throughout the text and captions, including for Figure 12, to maintain consistency.

9.6.    Line 345, please include the country of Hai’an City.

Response: We have revised the sentence to read:

“The device used in this experiment is a high-temperature thermal cycle drilling rig specially manufactured by a company in Hai’an City, Jiangsu Province, China.”

We have diligently addressed all the reviewer's comments through textual revisions, additions of necessary details and justifications, improvement of figures and tables, and formatting corrections. We believe these changes have significantly enhanced the manuscript's clarity, specificity, and scientific rigor. We are grateful for the reviewer's time and insightful feedback.

Thank you for considering our revised manuscript.

Sincerely,

All authors

2025.12.17

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript has been improved significantly. However, the following concerns about figure presentation should be addressed before publication.
1. The authors clarified in the response that the unit of TEMP P representing temperature in degrees Celsius (°C) (Figures 6 and 9) and S MISES representing the von Mises equivalent stress in Pascals (Pa) or Gigapascals (GPa) (Figures 10, 11, and 13) has been described in the figure caption. However, I do not see any description associated with those mentioned units. Please ensure that the units are clarified in the associated figure caption.
2. Please ensure that the numbers (a) and (b) of subfigures of Figure 3 are indicated properly.
3. I have a serious concern about the curve for uniaxial compressive strength in Figure 2. The connecting line does not represent the correlation in Equation (1). The modeling curve based on Equation (1) should be presented instead of the connecting lines.
4. In Figure 12, the curve (solid line) does not represent the data well. Instead, I suggest showing a connecting line between data points rather than a smooth curve. For example, the peak at a temperature of around 70 °C could mislead the representation.
5. The authors clarified in the response that the dimension of key components (e.g., the bit, sample size, and heating chamber) in the schematic diagram of the experimental setup (Figure 15a). However, I could not find any change indicating such dimensions. Please ensure that the manuscript provides the necessary information on the system dimensions for the reader.
6. Please check errors associated with incorrect figure references, e.g., Fig. 14(a) (line 382) should be Fig. 15(a).

Author Response

Responds to the reviewer ’s comments:

We sincerely appreciate the reviewer's thorough evaluation of our manuscript and the valuable suggestions provided. We have carefully considered each point raised and have addressed them accordingly. Below, we provide a detailed response to each concern:

Comment 1: The authors clarified in the response that the unit of TEMP P representing temperature in degrees Celsius (°C) (Figures 6 and 9) and S MISES representing the von Mises equivalent stress in Pascals (Pa) or Gigapascals (GPa) (Figures 10, 11, and 13) has been described in the figure caption. However, I do not see any description associated with those mentioned units. Please ensure that the units are clarified in the associated figure caption.

Response:

Thank you for your careful review and valuable comments. We sincerely appreciate your attention to the details regarding the units in the figures.

In response to your comment, we will revise the captions of Figures 6, 9, 10, 11, and 13 to explicitly include the units of the displayed quantities. Specifically:

For Figures 6 and 9 (temperature contours), we will add “(Unit: °C)” in the caption.

For Figures 10, 11, and 13 (stress contours), we will add “(Unit: Pa)” in the caption.

Change in Manuscript:

Fig. 6. Temperature gradient distribution of cutter (Units: °C).

Fig. 9. Temperature field of cutter for rock breaking (Units: °C).

Fig. 10. Stress nephogram of sandstone broken by PDC cutter (Units: Pa).

Fig. 11. Temperature field of cutter for breaking rocks of different temperatures (Units: Pa).

Fig. 13. Stress field of cutter for breaking three types of rocks (Units: Pa).

Comment 2: Please ensure that the numbers (a) and (b) of subfigures of Figure 3 are indicated properly.

Response:

Thank you for pointing this out. We have revised Figure 3 to clearly label each subfigure as “(a)” and “(b)” in the image itself, and the caption now reads:

      “Fig. 3. PDC cutter and rock model: (a) Cutter-rock model; (b) Mesh.”

Comment 3: I have a serious concern about the curve for uniaxial compressive strength in Figure 2. The connecting line does not represent the correlation in Equation (1). The modeling curve based on Equation (1) should be presented instead of the connecting lines.

Response:

Thank you for your insightful comment regarding Figure 2 and the representation of the uniaxial compressive strength curve. We agree with your observation that the simple connecting line between the data points does not adequately reflect the fitted relationship expressed in Equation (1).

In response to your suggestion, we will revise Figure 2 by replacing the current connecting line with a curve based on the fitting equation (Equation (1)) provided in the text. This will clearly illustrate the trend described by the model and better represent the correlation between temperature and uniaxial compressive strength.

Fig. 2. The variation of uniaxial compressive strength of rocks with temperature.

Comment 4:  In Figure 12, the curve (solid line) does not represent the data well. Instead, I suggest showing a connecting line between data points rather than a smooth curve. For example, the peak at a temperature of around 70 °C could mislead the representation.

Response:

We appreciate this suggestion. We have modified Figure 12 by replacing the smooth curve with a connecting line between data points. This change provides a clearer and more accurate representation of the stress variation with temperature.

Fig. 12. The relationship between temperature and maximum stress.

Comment 5: The authors clarified in the response that the dimension of key components (e.g., the bit, sample size, and heating chamber) in the schematic diagram of the experimental setup (Figure 15a). However, I could not find any change indicating such dimensions. Please ensure that the manuscript provides the necessary information on the system dimensions for the reader.

Response:

Thank you for your attentive review and valuable suggestion regarding the dimensions of the experimental setup. We appreciate your comment and have accordingly revised the manuscript to include the key dimensions of the system components.

    As the schematic diagram (Figure 15a) is already detailed and does not have sufficient space for clear dimension labels, we have added the following specifications in the text of Section 5.1 (“Experimental devices”), where the experimental setup is described. This ensures that readers have access to the necessary dimensional information while maintaining the clarity of the figure.

Change in Manuscript:

     The PDC bit used had an outer diameter of 49 mm, and each cutter had a diameter of 13.44 mm. The rock sample was prepared with dimensions of 150 mm × 100 mm × 240 mm. The heating chamber, which housed the sample during high‑temperature tests, measured 450 mm × 400 mm × 390 mm.

Comment 6: Please check errors associated with incorrect figure references, e.g., Fig. 14(a) (line 382) should be Fig. 15(a).

Response:

Thank you for catching this error. We have corrected the figure reference in line 382 from “Fig. 14(a)” to “Fig. 15(a)”. We have also carefully checked the entire manuscript to ensure that all figure references are accurate and consistent.

Thank you for considering our revised manuscript.

Sincerely,

All authors

2025.12.23

Author Response File: Author Response.pdf