Ratcheting of Steel Samples Undergoing Asymmetric Loading Cycles at Elevated Operating Temperatures: Analytical and Numerical Assessments

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The present study intends to assess the ratcheting response of SA508 and SA333 steel alloys subjected to asymmetric loading cycles at various operating temperatures through a hardening framework developed by Ahmadzadeh-Varvaniand the finite element analysis structured by the Chaboche hardening model (CH) in the ANSYS software. The simulated ratcheting responses for brick and tetrahedron solid elements were compared to those predicted analytically by the A-V hardening rule and experimentally measured values. The predicted and simulated ratcheting data were found in good agreement with the measured data.

The topic sounds interesting. The background of the work is solid. The topic also falls within the scope of the target journal. The results sound credible. The work can be considered for publication after some careful modifications.

1. Most of the equations are written in terms of tensor. If is so, they should be in the bold font. Also after each equation, there should be “,” or “.”. There should be no blank place before the following “where” as it is a whole paragraph.
2. There is no “R” in Eq. (6).
3. In L175, is “G” the shear modulus?
4. Fig. 4 is wrong. The strain has no unit.
5. In Table 1, the unit should not be in the italic format.
6. The authors can check the related work consider damage with temperature effect. Please see: Zhang et al., A modified continuum damage model considering cryogenic temperaturesand its application in corrugated plates of the LNG membrane storage tank. International Journal of Pressure Vessels and Piping, 2025.

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Summary

The phenomenon of cyclic creep (ratcheting) of SA508 and SA333 steels under cyclic asymmetric loading and different operating temperatures is considered. The Chaboche hardening model (CH) and the Ahmadzadeh-Varvani hardening structure (A-V) were used for modeling, the parameters of which were obtained experimentally. An increase in the strength of the material at high temperatures due to dynamic strain aging (DSA) was found. A sensitivity analysis of the FE mesh was performed. It was found that the analytical and FE models based on CH and A-V correspond to the experimental results.

The manuscript is relevant, contains new scientific results that are important for understanding the behavior of some steels during cyclic loading and high temperatures. However, the manuscript has many inaccuracies, including methodological ones, that do not allow for a qualitative assessment of these results, make the text incomprehensible to the reader and do not allow other researchers to reproduce the results. I believe that the manuscript needs to be corrected.

Comments on the research methodology:

1.Figure 5. Don't the authors believe that for steels SA508 and SA333 the FE meshes should be identical? That is, a = c , b = d . This will allow for a better comparison of the simulation results.

2."The predicted and simulated ratcheting data were found in good agreement with the measured data". I recommend using a numerical criterion for comparing theoretical, experimental and FE dependencies, for example the coefficient of determination R².

3."...dislocation motion is hindered due to solute atom interactions, resulting in an abrupt increase in material strength and a pronounced hardening response" ( line 242) - this statement requires substantiation or citing of the source.

4.The authors should explain the reason for the discrepancy between the FEA and theoretical data in Fig. 9. Perhaps the slight deviation is explained by the multiaxial stress-strain state of the FE model?

5.It is not clear from the text whether the experimental samples correspond to the FEA model. Briefly describe the experiment.

6.Describe the FEA model in more detail. In particular, indicate how many mesh elements the model contains, according to which the results were obtained (Figure 8, 9). I recommend publishing the model code together with the article.

7.In the conclusions or discussion, I recommend briefly describing the area of application of the model. What are the limitations of the model? Analyze the results from a practical point of view of the operation of these steels. Is DSA good or bad? Make brief recommendations on the optimal use of steels, for example, optimal operating temperatures.

 

Other comments:

8.Many inaccuracies in the description of the formulas in section "2. Mathematical formulation ". For example, in the formula 2 𝜎.I must be replaced by 𝜎∙I. This also applies to the multiplication sign in other formulas (3, 5, 10).

9.In line 129 "I" should be italicized. Also check the italicized symbols in other formulas.

10.𝜀 ̇_{𝑣 p} - a dot above a symbol usually means a derivative. So is this the strain rate? Not strain but strain rate ??? Indicate the sources from which the authors took formulas 1-3.

11.dS, S, a̅ - explain the symbols after formula 3. a̅ is explained only after formula 5.

12.Also explain f, σ _y (formula 5).

13."𝑅 is the increment of isotropic hardening" (line 158) . But in formula 6 there is no R.

14.Explain the quantities T_l , 𝜀 ̇_𝑣𝑝o , and the power n (formula 7).

15." 5×10 ^-3 G" should probably be "5×10 ^-3 GPa" ? (line 175).

16.Explain the quantities dda , Ci, da_i , 𝛾_𝑖 ′ , 𝛼 ̅ _i (formula 8).

17.𝛾₂ not in formulas 8 (line 189).

Check formula 10: 𝑑𝜀𝑃 . 𝑑𝜀𝑃 ?

18.C_1,3 , 𝛾 ′_1,3 are not in formulas 11.

19.Explain the value of M in formula 11.

20.Figure 2 shows experimental data. If this is not the authors' result, add the source below the figure. If this is the authors' result, describe the experiment in more detail.

21.Same for Figure 4.

22.Figure 6 - Add vertical axis title everywhere. No temperature label.

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents an analytical and numerical investigation about the behaviour of steel samples under asymmetric loads and elevated temperatures. The paper is interesting, well written and presented, and in my opinion should be considered for publication. Few aspects should be improved before final acceptance:

• Introduction is well written. I suggest only to insert a couple of sentences about the possible real-life implications of the proposed study
• Probably a figure reporting the framework of the work and that summarize the analytical assumption could give value to the work
• Authors should add some images about the experiments take into consideration for validating the proposed approach 
• About the numerical modelling approach, some additional information is required. See for example description in 10.1007/s10518-023-01751-6 
• Conclusions should include future developments and highlight also the limitations of the study

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The authors investigated the ratchet behavior of SA508 and SA333 alloys at elevated temperatures. It was studied under uniaxial asymmetric loading cycles using the A-V hardening model and the FE method. The Chabosch hardening rule was used in Ansys to study the effect of the mesh structure on the cyclic behavior of steel alloys at elevated temperatures. At elevated temperatures, due to thermal softening of steels, the samples demonstrated lower resistance to cyclic plastic deformation. However, in a certain temperature range, the DSA phenomenon occurred, which led to an increase in the strength of the material. The obtained results are important for science and practice, but there are several questions:

1. Numerous developed criteria for cyclic strength mostly operate with stress or strain ranges. Using stress or strain as a parameter characterizing fatigue damage leads to the formulation of conditional failure criteria. Please write down the failure criterion for the studied case.

2. A promising method for studying fatigue is based on the principles of thermodynamics of irreversible processes, which, without studying in detail the subtle mechanisms of the fracture process, at the same time allows us to make conclusions as reliable as the fundamental laws underlying thermodynamics. At the same time, it is very important to maintain physical correctness. When choosing the element sizes for FEM, did the authors take the grain size as a base value?

3. The ratchet behavior of SA508 and SA333 alloys under elevated temperatures under cyclic loading was studied, but how many cycles were studied? What loading modes were studied? High-cycle fatigue? or low-cycle fatigue?

4. Was the process of fatigue damage accumulation in SA508 and SA333 alloys studied under a plane stress state? Or under plane strain?

5. What methods for calculating the damage and service life of SA508 and SA333 alloys does the author propose? Is the applicability of the linear damage summation hypothesis confirmed under the studied conditions?

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

This paper investigates the ratcheting behavior of SA508 and SA333 steels under high-temperature asymmetric cyclic loading, incorporating an Armstrong–Frederick (A-V) hardening model and finite element analysis (FEA) based on the Chaboche framework. The study addresses a topic with clear engineering relevance and employs a combined experimental–numerical approach. However, several critical issues undermine the paper’s scientific rigor and clarity:Lack of precision: Inconsistent use of terminology, unclear figure labels, and contradictory statements (e.g., DSA temperature range). Methodological gaps: Weak validation of the finite element model, with no quantitative comparison to experimental data. Loose structure: The introduction lacks focus on innovation; conclusions reiterate results without deeper scientific insights. Academic norms: Reference formatting and figure attribution are inconsistent. Final Recommendation: Major Revision.

• Quantify Key Findings in Abstract: The statement “simulation results agreed well with experimental data” is vague.
• Section 1 lists existing work but fails to specify how this study advances prior research.
• In Equation (1), the viscoplastic strain rate εvp conflicts with total strain ε. Parameters χ\chiχ and S are redundantly defined in Eq. (7).
• The stress–strain curves in Fig. 1 lack details about their origin (e.g., strain rate, referenced figures).
• Model validation in Section 3.5 relies only on mesh convergence, not comparison with experimental strain fields.
• Section 1 states DSA occurs between 473–873K; Section 4 claims it disappears above 623K.
• In Fig. 2, axis labels like U/E are undefined; Fig. 7 includes unnecessary UI screenshots.
• Section 3.6 merely states back stress decreases with temperature, without explaining material-dependent mechanisms.
• The conclusion restates observations without broader implications.
• Reference formatting is inconsistent (e.g., mixed styles, missing years); figure/table captions lack source info.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 6 Report

Comments and Suggestions for Authors

Review of the article «Ratcheting of Steel Samples undergoing Asymmetric Loading  Cycles at Elevated Operating Temperatures: Analytical and Numerical Assessments» (Authors: M. Karimi, A. Varvani-Farahani)

The study of the behavior of a structure under the action of asymmetric cyclic loads is a rather urgent problem of modern applied mechanics. This problem is the subject of the reviewed article. Based on a literature review, which presents the results of using DSA (dynamic strain aging) models, the authors considered a calculation method for studying steels based on these model representations. The reviewer believes that works on such a formulation of the problem are important both for modern mechanical engineering and for engineering in general.

At the same time, a number of comments on the article should be noted that require their resolution.

1. The reviewer did not see any scientific novelty in this publication. DSA models have been known for a long time and this work does not reveal anything new in them.
2. Mathematical formulation of the problem. Formulas (1)-(4) are written with errors and poorly described.
3. The formulation of the finite element method is not presented in the work at all (transition from a discrete to a continuous model). Although, as the authors claim, a numerical assessment of the behavior of samples under asymmetric loads is the basis of this article, it is impossible.
4. Section 3.7 of the article states: "Figure 9 compares the measured, predicted and simulated ratcheting behavior of SA508 and SA333 steel samples at various temperatures using different element types." The reviewer did not see in the article a section devoted to a physical experiment that shows the results of measuring steel samples.

In general, the above comments make it impossible to publish the article. In this regard, the reviewer recommends rejecting this article as such, which lacks scientific novelty and contains numerous errors and shortcomings.

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 7 Report

Comments and Suggestions for Authors

The study investigates the ratchet behavior of SA508 and SA333 steel alloys under asymmetric loading cycles at different operating temperatures (298, 573 and 623 K). The research uses a strain hardening concept developed by Ahmadzadeh-Varvani (A-V) and integrates it with the Chaboche strain hardening model for finite element analysis. The dynamic recovery conditions in the A-V and Chaboche models include temperature-dependent parameters that account for the DSA phenomenon. The study presents both analytical and numerical assessments of the ratcheting behavior and uses finite element analysis (FEA) to evaluate the evolution of the recovery stress. The study compares the performance of two common nonlinear element types (brick and tetrahedron) in the finite element simulations. At this stage, the reviewer believes that the work needs MAJOR REVISION. Please carefully answer each question “Q” suggested below to change the final decision.

General comments

Q1) What gaps in knowledge does the work close on the basis of the literature review?

Q2) What is the novelty and uniqueness of the proposed research?

Q3) Why were the specific temperature ranges of 298, 573 and 623 K chosen for the experiments? Can the authors provide a rationale for this choice based on previous studies or material behavior at different temperatures? What previous studies influenced temperature selection?

Q4) Can the authors explain the specific parameters and functions used in the Ahmadzadeh-Varvani (A-V) and Chaboche hardening models? How were these parameters determined and what experimental data are they based on?

Q5) What specific criteria were used to select the non-linear brick and tetrahedron element types for the finite element analysis? Were preliminary tests conducted to determine their suitability for this study?

Q6) How did the authors account for the effects of dynamic strain aging (DSA) in their models? Were there specific experimental observations that guided the incorporation of DSA into the hardening frameworks?

Q7) What methods were employed to ensure the accuracy and reliability of the experimental data used for model validation? Were there any control experiments or repeat tests conducted to confirm the findings?

Q8) What specific convergence criteria were applied during the finite element simulations? How did the authors ensure that the results were independent of the number of elements used in the analysis?

Specific comments

Q9) Figure 2: Please place SI units in all axes of the diagrams

Q10) Figure 7: Improvement of the legend and placement of the units of the results shown

Q11) Figure 5: Can the author present and show the mesh analysis of the FEA model?

Q12) How do the authors explain the differences in ratcheting behavior observed between SA508 and SA333 steel samples at various temperatures? What specific factors contribute to the higher strain amplitude seen in SA333 compared to SA508 at 298 K?

Q13) Can the authors provide more insight into how the increase in temperature from 298 K to 473 K affects the ratcheting rates? What mechanisms are responsible for the noticeable decrease in yield strength and the subsequent increase in ratcheting rate at elevated temperatures?

Q14) What specific methods were used to validate the predicted ratcheting responses against experimental data? Were there any discrepancies noted between the predicted and measured values, and how were these addressed?

Q15) How do the Chaboche coefficients influence the predicted ratcheting behavior in the simulations? Can the authors discuss the significance of these coefficients in relation to the observed experimental results?

Q16) Perhaps the analysis can also be applied to other temperatures, such as those described in the following research, which can be included in the proposed study as a suggestion for future research: DOI10.1016/j.applthermaleng.2024.123086

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have mostly taken into account all the critical comments and the manuscript can be published. However, I recommend paying attention to the following:

1. The text does not describe the cause of different geometry of specimen SA508 and specimen SA333.
2. However, the R₂ criterion could be a better estimate than the maximum deviation (line 449), because it assesses similarity "as a whole".

Some inaccuracies in the formulas have not been corrected:

The arguments of the function f( S,a, σ _y ) should be written with a comma, not a period (5). The value of S (lowercase or uppercase?) in this formula is not explained . There is no S on the right side of the equation.

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Accept.

Author Response

Thanks to the reviewer for accepting the manuscript for publication in the journal.

Reviewer 5 Report

Comments and Suggestions for Authors

My comments on the initial version of the manuscript have been sufficiently addressed by the authors in this revised version. I have no further comments on the technical aspects. The manuscript may be considered for publication after a proofreading.

Author Response

Thanks to the reviewer for accepting the manuscript for publication in the journal.

Reviewer 6 Report

Comments and Suggestions for Authors

The authors of the article only partially took into account the reviewer's
comments. It is necessary to note a number of shortcomings of the new version of
the article.

1. The authors did not fully take into account the following preliminary
comment from the reviewer: “The formulation of the finite element method is not
presented in the work at all (transition from a discrete to a continuous model).
Although, as the authors claim, a numerical assessment of the behavior of samples
under asymmetric loads is the basis of this article, it is impossible.”.
2. Formulas (1) – (4) are written with errors. At the same time, the authors
refer to the work [41]. Here it is necessary to note the following: the work of 1963
is not sufficiently modern, and most importantly – the reviewer did not find
formulas (1) – (4), which are presented in this article, in the work [41].
3. A similar remark is made about formulas (6a) – (6b). The reviewer did not
see these relationships in the literary source [42].
4. The reviewer considers it not entirely correct to have references to works of
1963, 1978, 1991 when conducting modern research. Such references are possible
if these are reference books, or the article itself is a review work on research into
the historical aspects of studying this issue.
     These remarks make publication in the presented form impossible.
In this regard, the reviewer recommends reject the article until it is revised in
accordance with the above remarks.

Author Response

The authors wish to extend their sincere gratitude to the respected reviewer for the constructive comments. We did our best to address these valuable comments.

Author Response File: Author Response.pdf

Reviewer 7 Report

Comments and Suggestions for Authors

No more comments.

Author Response

Thanks to the reviewer for accepting the manuscript for publication in the journal.