Effects of Initial Small-Scale Material Nonlinearity on the Pre-Yield and Pre-Buckling Response of an Externally Pressurized Ring

Round 1

Reviewer 1 Report

Please see the attached.

Comments for author File: Comments.pdf

The language is good.

Author Response

Authors’ (A) Response to the Reviewer-1’s (R1) Comments

First, we wish to thank Reviewer 1 (R1) for his/her time, a positive comment, “the manuscript provides valuable insights”, and some constructive suggestions, which have helped improve the quality of the presentation of the manuscript. What follows is a point-by-point response to his/her comments:

1. R1: “In lines 21, the use of the keywords "yield" and "buckling" in this section seems a bit redundant.”

A: The respected reviewer has a valid point from semantic point of view. However, from phenomenological point of view, pre-yield describes a different deformation/stress of state than its yield counterpart, and the same is true for buckling/pre-buckling.

2. R1: “The title is recommended to be revised more accurately to represent the content and scope of the paper.”

A: The title is modified to reflect the exact content and scope of the revised manuscript. If the respected Reviewer has any specific suggestion, we will be more than happy to incorporate it.

3. R1: “The formula layout is chaotic and there are problems with the image format and inconsistent font in Figs.2, 3, 4, 5, and 5.’

A: We had the same complaint on one of our earlier publications in another MDPI journal as well as the present one. Some of the Figures, etc. are unrecognizable to us also. For example, the software used by the MDPI cannot handle some of the features, e.g., Figures, symbol font, etc. Hopefully, this will be fixed by the Journal IT experts and the editor.

4. R1: “The literature review is insufficient. Please consider the following studies: Marine Struct, 90, 103424; Thin-Walled Struct, 183, 110370; Eng Struct, 282, 115780; Eng Struct, 252, 113670; Applied Ocean Res, 141, 103799; Marine Struct, 94, 103548.”

A: The literature review is more than adequate from the specific point of view of covering the articles that are needed to explain the specific topical content of the present manuscript, as well as in keeping with the editorial requirement of this Journal. It is not meant to be a review article covering the entire literature survey on shell buckling/post-buckling. There are probably more than 1000 articles on buckling/post-buckling of rings/arches/cylindrical shells. The five articles suggested  by the esteemed Reviewer cover an entirely different, albeit very important topic, and does not fit with the content of the present manuscript (It is like an apple and orange case; both are good, but very different). This said, we appreciate this suggestion. This is an excellent topic for future research, and we hope these suggested articles to inspire us to write a future paper on that topic, entirely different from the present one.

5. R1: “In line 201, it is recommended that the author provide a more detailed explanation for the omitted content.”

A: Omitted content is, in part, due to keeping the size of the manuscript to some manageable length, and partly, due to the Journal’s plagiarism policy. If we quote from my own earlier work to offer some detailed explanations, we will be accused of plagiarism.

6. R1: “In lines 278, is the use of the 16-node layered element model (LLDT) reasonable for describing material nonlinearity? Are there other numerical models used for comparison to validate the accuracy of the results?”

A: The first question is nonsensical in the sense that there is no connection between the shape functions and the constitutive relations of the material. As regards the second question, there are certainly possibilities for future researchers to come up with other and better models. In that case, the present initial small-scale nonlinearity results for rings should serve as benchmark or baseline solutions. But one has to start somewhere.

7. R1: “As shown in Figs.7 and 8, why do the trends of the two exhibit opposite directions?”

A: This is a very good question. The reason is the Poisson effect. This said, the radial strains are so small that this effect does not have practical impact. Please also see the revised manuscript.

8. R1: In lines 349-355, when approaching the buckling pressure, the pressure-strain curve of the ideal ring with initial small-scale material nonlinearity deviates by as much as 16.67%, and the deviation for its linear elastic response curve is 13.64%. What are the reasons for this deviation?

A: Initial small-scale material nonlinearity, which is the primary goal of the present study.

9. R1: “In lines 449-458, it is observed that the pressure-deflection curves resulting from the initial small-scale material nonlinearity exhibit differences of up to 21.07% and 21.03%, respectively, compared to their linear elastic counterparts. How might these significant differences impact practical engineering applications?”

A: This part is deleted in the revised manuscript. However, even for the isotropic case, this is important for deep (> 12,000 ft., abyss [1]) submergence pressure hulls. Abyss is unforgiving to even small infractions, such as 16.67% under-design.

10. R1: “The format and writing of the manuscript need to be rechecked.”

A: The manuscript has been written in the same style/format as that of the authors’ earlier publications in such top journals as Proc. Royal Society A, London, Phil. Mag., Int. J. Solids Struct., AIAA Journal, Compos. Struct., Int. J. Nonlin. Mech., J. Compos. Mater., Comput. Struct., ASME J. Engineering Mater. Tech., ASCE J. Engineering Mech., Int. J. Num. Meth. Engrg., Engineering Fract. Mech. And so on. PRSA, London and Phil Mag are the two oldest scientific journals, which have published Nobel prize winning research by Rutherford, de Broglie, and many others. If the respected Reviewer has any specific suggestions, we will be more than happy to take that into account.

In conclusion, we thank Reviewer 1 for his/her in-depth critique, positive comments, and for the constructive suggestions, which have helped improve the quality of the manuscript.

Reviewer 2 Report

The review is in the attached file

Comments for author File: Comments.pdf

Author Response

Authors’ (A) Response to the Reviewer-2’s (R2) Comments

First, we wish to thank the Reviewer 2 (R2) for his/her time and some constructive comments, have helped improve the quality of the presentation of the manuscript. What follows is a point-by-point response to his/her comments:

1.R2: “The article is difficult to read because not all formulas and figures are displayed

correctly.”

A: We had the same complaint on one of our earlier publications in another MDPI journal as well as the present one. Some of the Figures, etc. are unrecognizable to us also. For example, the software used by the MDPI cannot handle some of the features, e.g., Figures, symbol font, etc. Hopefully, this will be fixed by the Journal IT experts and the editor.

2. R2: “ Understanding important issues is difficult due to frequent references to external

articles.”

A: This is, in part, due to keeping the size of the manuscript to some manageable length, and partly, due to the Journal’s plagiarism policy. If we quote from my own earlier work to offer some detailed explanations, we will be accused of plagiarism.

3. R2: “All indices in formulas should be unified. In Figure 1, axis is b , in formulas (1)

index is q , in formulas (2) indices are 1, 2, 3, in formulas (8) are x, q , z , in Figure

4, axis z is and so on.”

A: Thanks for the suggestion/correction. This is taken care of in the revised manuscript.

4. R1. “Figures 3 and 4 are badly informative.”

1. We have no idea what the respected Reviewer is alluding to. It is difficult to address such vague comments. If there are any specific concerns, such as point 3 above, we will be more than happy to address them.

5. R2: “What are the values of e f , ee in formulas (17a) and (17c)? How many iterations

did it take to get the results in Examples 1 and 2?”

A: Thanks for the query.  e_f, e_e = 5 x 10^-3. Here,  convergence is rapid (less than 4 iterations were needed for convergence). Please see the revised manuscript for details.

6. R2: “Is there a typo in the word COLSOL line 237?”

A: No. COLSOL stands for Column Solver.

7. R2: “How do the results, taking into account Karman’s theory, differ from solving the

problem in a linear formulation?”

A: We are not sure which results the respected Reviewer is referring to. The present manuscript clearly shows the difference between the linear and nonlinear material properties.

If the question is unrelated to the present manuscript, he can read the Refs. [14, 17, 39-42, 43], where such issues are addressed in much greater detail.

8. R2: “Condition (19c) does not agree with the signature in Figure 4.”

A: “What does the respected Reviewer mean by the term, “signature”? Condition (19c)” does not exist either in the original or the revised manuscript.

9. R2: “What should be the boundary conditions regarding the function w when solving the

given problems?”

A: It was earlier pictorially depicted in Fig. 4. Please see Fig. 4 of the revised manuscript. Thanks for the suggestion anyway.

10. R2: “Which formula in the article gives the conditions for matching layers?”

A: The revised manuscript deals with a single-layer shell only.

11. R2: “in lines 321-322 line 318 is repeated.”

A: Thanks for the correction. This is taken care of in the revised manuscript.

12. R2: “Why was point D chosen to demonstrate the numerical results in example 1 and,

accordingly, points D and C in example 2? Show results at other points in the sector

A: The deformation of the perfect ring prior to buckling is axi-symmetric.

In conclusion, we thank Reviewer 2 for his/her in-depth critique, positive comments, and for the constructive suggestions, which have helped improve the quality of the manuscript.

Reviewer 3 Report

The paper deals with the buckling response of shell structural components, specifically ring elements, under pressurised conditions, affected by initial defects, namely material non-linearities.

The topic, widely investigated in the literature with various applications, is worth of investigation and is tackled in the article with a computational approach.

Despite the scientific interest, the innovative contributions fostered by the paper appears rather limited, and constrained to the specific application case and computational procedure.

In order to gather a larger interest for the readers, it is suggested to:

- complement the results with parametric summarising tables and/or charts, which may generalise the obtained results and allow for guidelines or validity rules and further applications;

- revise the conclusions to highlight the innovative contributions and toward structural considerations to be extended also from a theoretical viewpoint.

A minor editing of English language is suggested to carefully remove misprints. Consistently, a detailed revision is expected on style: e.g., missing symbol at line 125, axis labels in Figure 2, missing terms at line 162, etc.

Author Response

Authors’ (A) Response to the Reviewer-3’s (R3) Comments

First, we wish to thank Reviewer 3 (R3) for his/her time and positive comment, “The topic … is worth of investigation”. What follows is a point-by-point response to his/her comments:

-R3: “Despite the scientific interest, the innovative contributions fostered by the paper appears rather limited, and constrained to the specific application case and computational procedure.”

- The entire premise of this reviewer’s comment, ‘…limited interest…’ is flawed to start with. Does the respected Reviewer expect the manuscript to cover the recent solar eclipse in North America, or treatment of lung cancer to make it of interest to a larger group? Sarcasm aside, we have a specific objective, and that objective is met successfully by employing an appropriate methodology, resulting in valid conclusions.

-A: This manuscript is a part of a much larger investigation dating back all the way to 1990’s, which is summarized in the Introduction backed by citing relevant references. It appears that the respected Reviewer has made the comment without reading and/or comprehending that aspect.

-R3: “revise the conclusions to highlight the innovative contributions and toward structural considerations to be extended also from a theoretical viewpoint.”

-A: The respected reviewer appears to have made this vague comment without reading and/or comprehending the manuscript.

-R3: “Comments on the Quality of English Language：A minor editing of English language is suggested to carefully remove misprints. Consistently, a detailed revision is expected on style: e.g., missing symbol at line 125, axis labels in Figure 2, missing terms at line 162, etc.”

A: We had the same complaint on one of our earlier publications in another MDPI journal as well as the present one. Some of the Figures, etc. are unrecognizable to us also. For example, the software used by the MDPI cannot handle some of the features, e.g., Figures, symbol font, etc. Hopefully, this will be fixed by the Journal IT experts and the editor.

In conclusion, Reviewer 3 (R3) has made mostly vague, worthless and/or nonsensical comments without reading and/or comprehending the manuscript.

Round 2

Reviewer 1 Report

The present version is fine.

Reviewer 2 Report

The text of the article has been significantly improved. I think that the article can be published in this form.