Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

Round 1

Reviewer 1 Report

This manuscript presented the comparison between Ru’s and He’s formulations of van der Waals interaction coefficient for the analysis of the natural frequencies of DWCNTs. A discrete model for the analysis of DWCNT is proposed as a couple of concentric equivalent continuous cylindrical shells using the Donnell shell theory. Especially, the chirality effects of DWCNTs can be taken into account by using an anisotropic elastic shell model. Considering the detail comparisons presented in this manuscript for the applicability and limitation of the analytical models are useful for researchers to investigate the mechanical behaviors of double-walled carbon nanotubes, this manuscript is worth of publication in this journal subject to the follow optional suggestions.

1.      Vibration mode shape of DWCNTs is sensitive to the vdW interaction, could the authors give a comparison for vibration mode shapes of DWCNTs by using Ru’s vdW coefficient and He’s coefficient? If possible, please indicate the critical order number (m, n) of the mode shape with which the difference by using Ru’ coefficient and He’s coefficient cannot be ignored.

2.      The quality of figures 4-10 could be improved. Please provide more clear curves in the figures, especially the differences between different curves.

Author Response

First of all, the Author would like to thank Reviewer 1 for the interesting and useful comments that, in his opinion, have strongly improved the scientific quality of the manuscript. Below, the Author answers the questions of the Reviewer.

This manuscript presented the comparison between Ru’s and He’s formulations of van der Waals interaction coefficient for the analysis of the natural frequencies of DWCNTs. A discrete model for the analysis of DWCNT is proposed as a couple of concentric equivalent continuous cylindrical shells using the Donnell shell theory. Especially, the chirality effects of DWCNTs can be taken into account by using an anisotropic elastic shell model. Considering the detail comparisons presented in this manuscript for the applicability and limitation of the analytical models are useful for researchers to investigate the mechanical behaviours of double-walled carbon nanotubes, this manuscript is worth of publication in this journal subject to the follow optional suggestions.

Q1. Vibration mode shape of DWCNTs is sensitive to the vdW interactions, could the author give a comparison for the vibration mode shapes of DWCNTs by using Ru’s and He’s vdW coefficient? If possible, please indicate the critical order number (m, n) of the mode shape with which the difference by using Ru’s and He’s vdW coefficient cannot be ignored.

A1. As stated by Reviewer 1, the vibrational mode shapes, and the corresponding natural frequencies, of DWCNTs are sensitive to van der Waals interactions. Actually, from Figure 3 it can be noted that the maximum difference between the values of Ru’s and He’s van der Waals interaction coefficient is present at the inner radius R₁ = 0.5 nm. So, the investigation of the critical order number (m, n) of the mode shape with which the difference between Ru’s and He’s formulations cannot be ignored should be carried out at that value of inner radius, see Figure 4. First of all, from Figure 4 it can be observed that the behaviour is almost the same for every number of longitudinal half-waves m, and therefore the analysis will be focused on the critical number of circumferential waves n. By imposing the limit value of percentage difference of 5%, below which Ru’s formulation can be considered as sufficiently accurate, it is found that Ru’s formulation cannot be applied in the range of number of circumferential waves n = (2,3), which denotes the critical order number of the mode shapes. This explanation was added in the discussion of numerical results in Section 5.3.

Q2. The quality of figures 4-10 could be improved. Please provide more clear curves in the figures, especially the differences between different curves.

A2. The quality of Figures 4-10 was improved, and in particular the differences between different curves were highlighted, eliminating the intervals of number of circumferential waves n in which the curves have the same behaviour, and therefore enlarging the intervals in which the curves show the greatest differences as the number of longitudinal half-waves m varies.

To conclude, the Author would like to thank again Reviewer 1 for the very appreciated work and hopes to have answered the comments comprehensively.

Reviewer 2 Report

The article reviewed about most important literatures recently and constructed new model, the formulations, the calculations, the drafts all are acceptable for publishing at CARBON.

Only suggestion is to shorten the size of some parts, e.g. reduce the words number of conclusion.

Author Response

First of all, the Author would like to thank Reviewer 2 for the useful comment that, in his opinion, has improved the quality of the manuscript. Below, the Author answers the question of the Reviewer.

The article reviewed about most important literatures recently and constructed new model, the formulations, the calculations, the drafts all are acceptable for publishing at CARBON.

Q1. Only suggestion is to shorten the size of some parts, e.g. reduce the words number of Conclusions.

A1. Conclusions were strongly reduced by eliminating some minor sentences and bringing together the most significant results obtained in the present paper.

To conclude, the Author would like to thank again Reviewer 2 for the very appreciated work and hopes to have answered the comment comprehensively.

Reviewer 3 Report

The work is devoted to the comparison of two approaches in modeling the interaction of van der Waals in the problems of vibrations of double-walled carbon nanotubes. As the main approach in the modeling of nanotubes, the representation of the tube in the form of a continuous thin two-layer shell was chosen, which was based on the Donnel’s shell theory.

The introduction gives a fairly clear idea of both the essence of the problem under study and the existing methods and approaches to its solution, which have been obtained by other researchers so far. The author analyzed the shortcomings of existing approaches and methods, and clearly presented the relevance of this work, which consists in comparing and establishing the limits of applicability of two formulations of the van der Waals interaction. It may be worthwhile to slightly expand the review of the literature, supplementing it with a slightly larger number of works over the past 3-5 years.

In the problem statement section, everything is stated quite clearly and logically. A full explanation is given about the construction of the model, the boundary conditions, the process of obtaining dimensionless variables and the methods used to solve the problem.

In the results section, large-scale numerical studies were carried out, which substantiate the author's point of view that the He’s formulation gives more accurate results within the given limits.

The author's conclusions are confirmed by the results of calculations, a comparative analysis with other authors, as well as a parametric analysis of the influence of the inner radius R_1.

In general, the article is written quite well, clearly and logically.

However, there were a few misspellings, such as on line 109 “date”.

Author Response

First of all, the Author would like to thank Reviewer 3 for the interesting and useful comments that, in his opinion, have strongly improved the scientific quality of the manuscript. Below, the Author answers the questions of the Reviewer.

The work is devoted to the comparison of two approaches in modelling the interaction of van der Waals in the problems of vibrations of double-walled carbon nanotubes. As the main approach in the modelling of nanotubes, the representation of the tube in the form of a continuous thin two-layer shell was chosen, which was based on the Donnell’s shell theory.

Q1. The introduction gives a fairly clear idea of both the essence of the problem under study and the existing methods and approaches to its solution, which have been obtained by other researchers so far. The author analysed the shortcomings of existing approaches and methods, and clearly presented the relevance of this work, which consists in comparing and establishing the limits of applicability of two formulations of the van der Waals interaction. It may be worthwhile to slightly expand the review of the literature, supplementing it with a slightly larger number of works over the past 3-5 years.

A1. The Author expanded the review of the literature by adding 5 new works published over the past 5 years, see Refs. [51-55], related to vibrations and stability of DWCNTs based on anisotropic model.

In the problem statement section, everything is stated quite clearly and logically. A full explanation is given about the construction of the model, the boundary conditions, the process of obtaining dimensionless variables and the methods used to solve the problem.

In the results section, large-scale numerical studies were carried out, which substantiate the author's point of view that the He’s formulation gives more accurate results within the given limits.

The author's conclusions are confirmed by the results of calculations, a comparative analysis with other authors, as well as a parametric analysis of the influence of the inner radius R₁.

Q2. In general, the article is written quite well, clearly and logically. However, there were a few misspellings, such as on line 109 “date”.

A2. The Author very carefully checked the manuscript and tried to eliminate all the English language errors present.

To conclude, the Author would like to thank again Reviewer 3 for the very appreciated work and hopes to have answered the comments comprehensively.

Reviewer 4 Report

The article was enhanced very well to be suitable to be published. However, there are minor comments needed to be carried as:

1- The paragraphs in the introduction need to be merged as it is not a highlights

2- The novelty and the difference between the current work in compared to others need to be more clear for the reader 

Author Response

See the attached document.

Author Response File: Author Response.pdf