Critical Resolved Shear Stress and Work Hardening Determination in HCP Metals: Application to Zr Single Crystals
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper addresses very important issue of crystal plasticity concerning determination of the CRSSs and hardening parameters of HCP metals on the example of Zr. I beleive this paper will be of great interest for many researchers dealing with crystal plasticity studies. However, a major revision of the paper is necessary before its acceptance. The following issues must be addressed/explained/revised:
1. I do not recommend using "CRSS" abbreviation in the title. Use "critical resolved shear stress" or "threshold stress" instead.
2. How were the stress tensor components entering Eq(1) calculated ?
3. It seems the authors confusingly mix RSS (TAUr) and CRSS (tau0). The RSS is a material stressed state, it depends on loading parameters. The CRSS is a material property (yield strength), which depends on material microstructure, accumulated slip, etc. In unloading, e.g., RSS can come to 0, while the CRSS remains unchanged. Therefore, Eq. (6) and (8) are given for CRSS which is denoted as "tau0" according to Eq. (3) while RSS are calculated by Eq. (1).
4. Eq. (6): First of two Eqns is redundant as it is exactly a part of the second formula
5. Check Eq. (11). I suppose x1/x2 should be ctg(Teta), not tg(Teta) as it is given in Line 150
6. Line 164: Wrong ref. to Fig. 4
7. Table 4: How were the UTS and YS values determined from the stress-strain curves in Fig. 5? The YS and UTS values for Zr2, Zr8, Zr11 given in the table seem to be overestimated. How was the strain rate determined? Is it an equivalent value or some tensor component? Is it an average value?
8. Captions on the axes are too small. Figure 9 is referred to in the text before Fig. 7 and 8. Discussion of Figure 7 is missing in the text.
9. Line 180: wrong ref. to Fig. 9a
10. Strain rate in Table 8 is given for gamma dotted. How this value was obtained?
11. Figure 12 is referred in the text before Figure 11 (line 208).
13. What does the sign $ in Figure 11's legend means?
14. Line 214: "...two shears" sounds strange. Consider replacing with slip or shear strains
15. Discussion given in Section 5.2 contradicts to its title. It focuses on a number of factors with only minor attention being given to the strain rate effects.
16. Figure 15 is not discussed in the text
17. Line 243: a wrong ref to Fig. 13
Moreover, the text is overwelmed with grammar errors and typos. Some of them are given below:
1. Line 56: Use a capital letter for "...[22]. Twinning..."
2. Check Eq. (2) correctness
3. Line 82: a verb is missing.
4. Line 83: "By" is duplicated
5. Eq.(5): taur=tau0 is included in both conditions
6. Eq.(4): a left-hand-side term should be bold as it is a tensorial quantity
7. Line 89: Use bold font for tensors
8. Line 100: replace "stress shear" by "shear stress"
9. Line 108: "And" should be replaced by "and"
10. Line 110: Ref. to Figure 3 should be replaced by Ref. to Fig. 1
11. Line 111: "...a system of two equations...", however, Eq. (7) and (8) consist of four Eqns
12. Line 113: "coefficient" should be "coefficients"
13. Caption to Fig. 5: "exhibits" should be "exhibit"; start caption with a capital letter
14. Table 4, Section 4.2 title, Fig. 8 caption, Fig. 11: "stade" should be replaced with "stage"
15. Fig. 11, Section 5.2 captions: start with a capital letter;
16. Lines 217, 231, : check grammar
17. Line 236: replace "zr" with "Zr"
18. Lines 245-246: "plan(s)" should be "plane(s)"
19. Lines 246-247: check grammar (a verb is missing)
20: Line 253-254, 257-249, 260, 268: check grammar
21. Line 307: replace <UVW) with <UVW>
Comments on the Quality of English LanguageAuthors are requested to carefully double-check English.
Author Response
Reviewer #1: This paper addresses very important issue of crystal plasticity concerning determination of the CRSSs and hardening parameters of HCP metals on the example of Zr. I beleive this paper will be of great interest for many researchers dealing with crystal plasticity studies. However, a major revision of the paper is necessary before its acceptance. The following issues must be addressed/explained/revised:
Thank you for your support and we understand your comments; they are judicious; we had our publication proofread by a colleague who corrected our grammatical errors.
1-I do not recommend using "CRSS" abbreviation in the title. Use "critical resolved shear stress" or "threshold stress" instead.
We have taken your comment into account and have changed the title to:
Critical Resolved Shear Stress and Work hardening determination in HCP metals: Application to Zr single crystals
- How were the stress tensor components entering Eq(1) calculated ?
The mechanical tests consist of in situ tensile tests performed in an SEM, allowing us to determine the macroscopic stress tensor (with a single active component). We have added the following text: for uniaxial tension along the x-axis, the macroscopic stress tensor is expressed as:
Where eand all other components are zero.
The projection onto the active slip systems is made possible through EBSD indexing of the crystal orientations with respect to the macroscopic reference frame.
- It seems the authors confusingly mix RSS (TAUr) and CRSS (tau0). The RSS is a material stressed state, it depends on loading parameters. The CRSS is a material property (yield strength), which depends on material microstructure, accumulated slip, etc. In unloading, e.g., RSS can come to 0, while the CRSS remains unchanged. Therefore, Eq. (6) and (8) are given for CRSS which is denoted as "tau0" according to Eq. (3) while RSS are calculated by Eq. (1).
You are right, we did not clearly explain these equations, which has caused some confusion. We have added the following sentences,
as Equations 6 and 8 pertain to strain hardening and not to critical shear stress.
At the onset of plasticity, corresponds to noted CRSS
We are only considering resolved shear stress (RSS).
- Eq. (6): First of two Eqns is redundant as it is exactly a part of the second formula
We have corrected our mistake.
- Check Eq. (11). I suppose x1/x2 should be ctg(Teta), not tg(Teta) as it is given in Line 150
We have revised and clarified the equations.
- Line 164: Wrong ref. to Fig. 4
Thank you, we have corrected this incorrect reference
- Table 4: How were the UTS and YS values determined from the stress-strain curves in Fig. 5? The YS and UTS values for Zr2, Zr8, Zr11 given in the table seem to be overestimated. How was the strain rate determined? Is it an equivalent value or some tensor component? Is it an average value?
The values for E, UTS, and YS were calculated using a program written in Python. We have redone our calculations and made some adjustments.
The strain rate is calculated by differentiating the strain with respect to time.
- Captions on the axes are too small. Figure 9 is referred to in the text before Fig. 7 and 8. Discussion of Figure 7 is missing in the text.
Thank you, we have enlarged the scales and added the discussion of Figure 7.
- Line 180: wrong ref. to Fig. 9a
- Strain rate in Table 8 is given for gamma dotted. How this value was obtained?
Knowing sigma and epsilon, as well as how the Schmid factor varies at each moment, we were able to calculate tau and gamma at every instant during the measurement. To calculate gamma dot, we differentiated gamma with respect to time
- Figure 12 is referred in the text before Figure 11 (line 208).
We have made the change and placed the sentence referring to Figure 12 after Figure 11
- What does the sign $ in Figure 11's legend means?
It was a typo that we have removed. Thank you for your careful review.
- Line 214: "...two shears" sounds strange. Consider replacing with slip or shear strains
We have changed the sentence taking into account your remark.
In this case, Γ represents the accumulated shear strain: where N is the total number of active systems (see [6]). However, for the purpose of this study, we focus solely on stage I, where only one slip system is active. Thus, Γ = γ
- Discussion given in Section 5.2 contradicts to its title. It focuses on a number of factors with only minor attention being given to the strain rate effects.
we have reworked our sentence to make it more understandable
As can be seen in the figure 12, the effect of deformation rate on the critical resolved shear stress (CRSS) for prismatic ⟨a⟩ slip is minimal, contrary to the findings of Pujol [ 40 ].
- Figure 15 is not discussed in the text
We added the following sentence:” TEM observations (see Fig. 16) revealed cross-slip from the prismatic to the basal plane at room temperature, involving straight screw dislocations moving via a kink-pair mechanism.”
- Line 243: a wrong ref to Fig. 13
Thank you, we have corrected
Moreover, the text is overwelmed with grammar errors and typos. Some of them are given below:
Thank you for your careful reading and we are sorry to have made so many grammatical errors, all typographical corrections have been written in red
- Line 56: Use a capital letter for "...[22]. Twinning..."
- Check Eq. (2) correctness
- Line 82: a verb is missing.
- Line 83: "By" is duplicated
- Eq.(5): taur=tau0 is included in both conditions
- Eq.(4): a left-hand-side term should be bold as it is a tensorial quantity
- Line 89: Use bold font for tensors
- Line 100: replace "stress shear" by "shear stress"
- Line 108: "And" should be replaced by "and"
- Line 110: Ref. to Figure 3 should be replaced by Ref. to Fig. 1
- Line 111: "...a system of two equations...", however, Eq. (7) and (8) consist of four Eqns
- Line 113: "coefficient" should be "coefficients"
- Caption to Fig. 5: "exhibits" should be "exhibit"; start caption with a capital letter
- Table 4, Section 4.2 title, Fig. 8 caption, Fig. 11: "stade" should be replaced with "stage"
- Fig. 11, Section 5.2 captions: start with a capital letter;
- Lines 217, 231, : check grammar
- Line 236: replace "zr" with "Zr"
- Lines 245-246: "plan(s)" should be "plane(s)"
- Lines 246-247: check grammar (a verb is missing)
20: Line 253-254, 257-249, 260, 268: check grammar
- Line 307: replace <UVW) with <UVW>
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsSee the attachment
Comments for author File: Comments.pdf
Author Response
Reviewer #2: The paper from Jean-Sebastien Lecomte et al. describes an original approach to obtain CRSS and some coefficients of hardening matrix applied in a commercially pure zirconium (α-Zr) at room temperature. Actually, the topic of the entire paper fits pretty well the scope of Metals journal. Anyway, there are some observations
First the bibliography, I would suggest adding some references more recent to justify the study underlying the importance of the developed method.
Thank you for your pertinent remark: current articles on zirconium focus on the effect of hydrogen on mechanical behavior and fatigue resistance. However, we found two recent articles (2020 and 2024) that seem relevant to our publication.
Li, Y.; Po, G.; Cui, Y.; Ghoniem, N. Prismatic-to-basal plastic slip transition in zirconium. Acta Materialia 2023, 242, 118451. 394, https://doi.org/10.1016/j.actamat.2022.118451.
An, X.; Zhang, H.; Ni, S.; Ou, X.; Liao, X.; Song, M. Effects of temperature and alloying content on the phase transformation and {101-1} twinning in Zr during rolling. Journal of Materials Science & Technology 2020, 41, 76–80.
Furthermore, the conclusions should be more incisive, explaining the originality of the work and the proof that a certain oxygen content is needed to detect changes.
Thank you, we decided to add a more incisive sentence, because we believe that the values obtained can serve the community :
“This is the first study to focus on the strain-hardening coefficient. Previously, we shared some of our results with a french research team, and they successfully modeled our experimental data [ 43 ] (see Fig. 15aax and c). The developed method is applicable to all hexagonal metals (such as titanium, zinc, and magnesium) and, more broadly, to all metals, as long as no more than two active slip systems are considered at the same time.”
At the end I also suggest reading one more time the text to avoid typing errors.
Yes, you are right, a more careful reading would have allowed us to make fewer mistakes. We apologize and have taken into account the reviewers' corrections. We also had the text proofread by a colleague
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsIn this manuscript, the authors report on the quantitative evaluation of the CRSS and work hardening behavior in HCP crystal, such as an example of Zr single crystal by in-situ works. The works are contrived, and interesting results seem to be obtained.
However, the explanation of each figure is out of order that the authors organized, so it is difficult to grasp the story of this manuscript. Besides, some figures are not explained (no sentences describing some figures are included in the text). If the authors make figures for their purposes, all of the figures have to be explained. The reviewer recommends that the authors reorganize the story of this paper and organize the figures to suit the revised manuscript.
Also, each figure seems to be a typical example of a specific sample. Based on the analytical techniques proposed by the authors, such as work hardening behavior shown in Fig.7 and the distribution of the strain heterogeneity in Fig.9, the reviewer suggests that the authors discuss the different deformation manner of specimens with slip deformation and twinning.
The small comments,
1. As shown in Fig.3, the strong orientation dependence of the slip deformation and twinning, even in Zr, is obtained in the same way as the other metals. The authors summarize the SF in Table 3. To visualize the results of this dependency, is it possible to make a contour map of the SF for each P<a> slip and Twinning in Fig.3? It will be acceptable with different figures such as Fig.3(a), contour map for P<a> in (b), and twinning in (c).
2. In Fig.5, the reviewer suggests that the authors discuss the reason for different elastic behavior depending on the specimen (orientation) since the elastic curves are highlighted with the dashed line.
3. In Fig.7(c), what is the considerable stress drop detected at the strain at around 0.6?
4. In the bright field image of Fig.8, dislocation contrast exists orthogonally. If the hexagonal illustration inserted on this image are the unit cell of the specimen, is the arrangement of the P<a> correct? The reviewer understands that the P<a> is the prism slip, so the P<a> must not exist with 90 degrees because of the hexagonallity. The reviewer also suggests that the incident beam direction of this image.
5. 3 indices of <uvw>{hkl} and 4 indices of <uvtw>{hkil} are mixed in this manuscript. The authors should unify the index. The reviewer recommends to fix them with the four indices.
6. For what did the authors add the “Appendix A. Demonstration”? The readers need the XXX of “Demonstration of XXX.”
7. The authors must check and revise the manuscript thoroughly; there are typos and incorrect explanations.
The reviewer lists a part of incorrect explanations as examples,
The “work hardening matrix” on L. 100 is revised to the normal style.
Is “zr” on L.236 “Zr”?
“(HKL)<UVW)” on L.307, is it incorrect of (HKL)<UVW>?
Is “(E T H)” on L.318 a correct expression? All symbols will be written in italics. And Is “T” bold?
In the Appendix, “equ.” or “equa.” should be revised as “eq.” of normal abbreviation. (Several abbreviations are seen.)
In references, the names of journals are revised as abbreviations.
The figures showing the electron micrograph should be revised to make the scale bar, indicators, etc more readable.
(Fig.5) Is “Stade” in Fig.5 and Table 5 in French? If correct, the authors should be revised as “Stage.”
(Fig.7) The Number of horizontal and vertical axes should be revised with a larger font size so the readers can recognize them easily.
The reviewer suggests that the authors need to check and revise the manuscript thoroughly because there are typos and incorrect explanations mentioned above.
Author Response
Reviewer #3 :In this manuscript, the authors report on the quantitative evaluation of the CRSS and work hardening behavior in HCP crystal, such as an example of Zr single crystal by in-situ works. The works are contrived, and interesting results seem to be obtained.
However, the explanation of each figure is out of order that the authors organized, so it is difficult to grasp the story of this manuscript. Besides, some figures are not explained (no sentences describing some figures are included in the text). If the authors make figures for their purposes, all of the figures have to be explained. The reviewer recommends that the authors reorganize the story of this paper and organize the figures to suit the revised manuscript.
First of all we wanted to thank you. We have corrected our text and tried to redo the story so that it is more easily readable. We hope you will appreciate our effort.
Also, each figure seems to be a typical example of a specific sample. Based on the analytical techniques proposed by the authors, such as work hardening behavior shown in Fig.7 and the distribution of the strain heterogeneity in Fig.9, the reviewer suggests that the authors discuss the different deformation manner of specimens with slip deformation and twinning.
This is a great suggestion. We have revised our examples to reflect your feedback.
The small comments,
1.As shown in Fig.3, the strong orientation dependence of the slip deformation and twinning, even in Zr, is obtained in the same way as the other metals. The authors summarize the SF in Table 3. To visualize the results of this dependency, is it possible to make a contour map of the SF for each P<a> slip and Twinning in Fig.3? It will be acceptable with different figures such as Fig.3(a), contour map for P<a> in (b), and twinning in (c).
we have taken your remark into account and we have redone the calculations of schmid factor for the prismatic and for the T1 twinning
we have gathered all the figures in one (as following your advice)
- In Fig.5, the reviewer suggests that the authors discuss the reason for different elastic behavior depending on the specimen (orientation) since the elastic curves are highlighted with the dashed line.
This is excellent advice. So, we added the elastic modulus calculations (with Fisher & Renken constants) and compared them to our results. We also redid a theoretical elastic modulus calculation that we put into an inverse pole figure
- In Fig.7(c), what is the considerable stress drop detected at the strain at around 0.6?
The stress drop in Figure 7c is due to the stress release during the EBSD measurement time and also the time-consuming grid photos. During this time, the single crystal was unfortunately undergoing softening
- In the bright field image of Fig.8, dislocation contrast exists orthogonally. If the hexagonal illustration inserted on this image are the unit cell of the specimen, is the arrangement of the P<a> correct? The reviewer understands that the P<a> is the prism slip, so the P<a> must not exist with 90 degrees because of the hexagonallity. The reviewer also suggests that the incident beam direction of this image.
it was an illustration error. We have changed
Thanks again for looking at our manuscript accurately
- 3 indices of <uvw>{hkl} and 4 indices of <uvtw>{hkil} are mixed in this manuscript. The authors should unify the index. The reviewer recommends to fix them with the four indices.
we have put the notations (hkil) in a homogeneous way
- For what did the authors add the “Appendix A. Demonstration”? The readers need the XXX of “Demonstration of XXX.”
we changed and put "Calculation of slopes A, B, C, D and E"
- The authors must check and revise the manuscript thoroughly; there are typos and incorrect explanations.
Thank you for your comments. We have corrected all our mistakes and you will find them in red in the text.
The reviewer lists a part of incorrect explanations as examples,
The “work hardening matrix” on L. 100 is revised to the normal style.
Is “zr” on L.236 “Zr”?
“(HKL)<UVW)” on L.307, is it incorrect of (HKL)<UVW>?
Is “(E T H)” on L.318 a correct expression? All symbols will be written in italics. And Is “T” bold?
In the Appendix, “equ.” or “equa.” should be revised as “eq.” of normal abbreviation. (Several abbreviations are seen.)
In references, the names of journals are revised as abbreviations.
The figures showing the electron micrograph should be revised to make the scale bar, indicators, etc more readable.
(Fig.5) Is “Stade” in Fig.5 and Table 5 in French? If correct, the authors should be revised as “Stage.”
(Fig.7) The Number of horizontal and vertical axes should be revised with a larger font size so the readers can recognize them easily.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe authors addressed my comments properly.
Author Response
Thank you for patiently reading this manuscript
We have tried to take into account all your comments
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors adequately reply to the reviewer's comment, and their revision makes a significant improvement in this manuscript.
I have a small comment on the revised manuscript, if the authors took my comment into account, I would like to recommend accepting it for publication in Metals.
1. Regarding "IPF" in "IPF of Schmid Factor..." in the caption of Figure 3(b) and (c), Isn't “IPF” needed. It would be acceptable to start from the "Schmid factor." Otherwise, the reviewer suggests that it is more suitable to use the "Standard stereographic triangle" instead of "IPF."
Author Response
Thank you for patiently reading this manuscript
We have tried to take into account all your comments
