Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere

Round 1

Reviewer 1 Report

Dear authors,

the paper is well organized and clearly written. However, the following questions should be addressed:

·     In section 2.2. the contact deformation of the sphere is determined. Why is it necessary to repeat the complete derivation of all equations of the theory of elasticity considering axisymmetry? In equations (18) and (21) you even give the field quantities inside the half-space. The listing of equations (11) to (22) are, in my opinion, superfluous. Instead, one can refer to any textbook on contact mechanics, e.g. Johnson [1]. Even the calculation of the displacements in equations (23) and (24) is not mandatory. According to the Hertzian contact theory, it is well-known that a pressure distribution according to equation (9) applied on an elastic half-space leads to a quadratic displacement of the surface in the loading area.

·     However, the most critical point of the whole work concerns the assumed pressure distribution in the contact area between the sphere and the plate according to equation (9), which was taken from Hertz's theory. In general, the pressure distribution is a priori unknown. If you a priori assume a pressure distribution in the form of the Hertzian contact pressure, then you must clearly specify the conditions under which this approach is justified. For example, suppose that the sphere has a much larger Young's modulus than the plate. In the limiting case, let the sphere be rigid. This case has been studied by several authors [2-4] and one important result is that the position of peak contact pressure depends on the ratio of the contact radius and the plate thickness. The peak appears at the center of the contact zone only for small ratios, while the peak contact pressure appears near the edge of the contact zone for larger ones. In the example you showed (same elasticity of both bodies), the plate thickness is about 4 times larger than the contact radius and you get the pressure distribution according to Figure 12 from FEM, which is indeed of the Hertzian type. But what happens if, for example, the contact radius becomes larger than the plate thickness, i.e. if we increase the load? It seems that your theory can not correctly represent this case! Since you are doing FE analysis, it should be easy for you to check it.

·     Regardless of the critical point of the assumed pressure distribution, this work lacks a parameter study. In particular, the influence of different Young's moduli of plate and sphere on the solution should be investigated. In the limiting case of a rigid sphere can be compared with the solutions from references [2-4]. In addition, some geometrical parameters such as the plate thickness, plate radius, radius of curvature of the sphere or their ratio should also be varied. Of particular interest is the influence of the ratio of the contact radius to plate thickness on the solutions.

One more point can be found in the attached pdf-file!

Comments for author File: Comments.pdf

Author Response

Dear Reviewer:

Thank you for your comments concerning our manuscript entitled “Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere” (ID: applsci-1970297). Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. The responds to your comments are as following.

Thanks very much for your suggestion. The Hertz theory is not applicable to the elastic sphere in contact with the thin plate with large deformation. This is due to the fact that the elastic strain energy is determined by the membrane stresses which result from the large out-of-plane deflections of the thin plate. Consequently, geometrical nonlinearity has to be considered and detailed analysis need to be performed. In this paper, a circular thin plate clamped peripherally and frictionless contacting an elastic sphere is studied. Firstly, a set of equations are established to predict the relationship between the contact force and the relative penetration. On this basis, analysis of the contact stress and surface displacements of the circular thin plate and the elastic sphere is carried out. Then, comparison of the results with the Hertz theory is presented. Finally, the validity of the methodology developed is examined by comparing the theoretical results with finite element method.

We are grateful for your kind advice of parameter study. In our future research on this topic, we would like to continue this work to include the influence of different Young's moduli of plate and sphere, the plate thickness, plate radius, radius of curvature of the sphere or their ratio in detail.

 

Special thanks to you for your good comments.

 

Sincerely Yours,

Wei Han

Reviewer 2 Report

This research reports on an analytical investigation into mechanical deformation response of a circular thin plate under the effect of an elastic Sphere. This is an interesting topic which provides useful information in the deformation response of engineering structures under diverse loading conditions. However, there are important issues that need to be addressed by the authors through Major Revision in the interest of improving this work before I can recommend this manuscript for publication.   1) First and foremost, the abstract is not informative. A good abstract in a research project encompasses a summary that identifies the purpose, problem, methods, results, and conclusion of your work. Your abstract is too short and lacks a brief discussion regarding the results and conclusions of your work.    2) The authors need to do a better job of discussing how to construct the free body diagram. How did you come up with those loading conditions?   3) Please explain what would happen if the contact area were equal to the diameter of the sphere? Is your case being a limited case-study in which the circular contact area, resulting in a semi-elliptic pressure distribution, is smaller than the sphere diameter?   4) I don't see the need to illustrate the stress components in cylindrical coordinates. This is something that can be found in every Solid Mechanics book. You can simply skip Figure.4 to avoid a lengthy manuscript and reference it within the text.     5) How did you derive the results for the displacements and stresses? They must be provided.    6) Another important issue is the Finite Element (FE) mesh refinement/convergence study. Thus, the current description of the commercial numerical apparatus used for this purpose is extremely poor. To rectify the current situation, the following additions are necessary:   (a) Please conduct mesh sensitivity analysis for qualitative verification of your FE simulation results. Mesh sensitivity analysis is conducted for convergence studies and/or adaptive analysis to determine and control the numerical errors, respectively.   (b) Give a complete characteristic of the applied commercial elements in respect to the applied loading, the number of element nodes and the number and kind of degrees of freedom per each node.   (c) For each numerical calculation, the number of elements and degrees of freedom must be given.   (d) You must discuss the issue of potential sources of modeling and discretization errors in your numerical analysis; also, the methods of determination and controlling such errors should be mentioned.  7) The literature review isn't adequate. The authors are apparently not much familiar with some of the leading research regarding the analytical and FEA simulation of various engineering structures under different loading conditions. The introduction is currently not concise and informative. The author must discuss the following references in the introduction portion of the revised manuscript:   "Localized failure analysis of internally pressurized laminated ellipsoidal woven GFRP composite domes: Analytical, numerical, and experimental studies", Archives of Civil and Mechanical Engineering, Vol:19, pp:1235-1250.     "Static and dynamic deformation response of smart laminated composite plates induced by inclined piezoelectric actuators", Journal of Composite Materials, Vol: 56, (2022).     8) The manuscript requires significant language revision. I strongly recommend the authors have their paper reviewed by a native English speaker to improve its written expression.
  I believe responding to the above-mentioned comments and revising the manuscript accordingly are essential before considering the paper for publication.      

Author Response

Dear Reviewer:

 

Thank you for your comments concerning our manuscript entitled “Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere” (ID: applsci-1970297). Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. The responds to your comments are as following.

Thanks very much for your suggestion. The Hertz theory is not applicable to the elastic sphere in contact with the thin plate with large deformation. This is due to the fact that the elastic strain energy is determined by the membrane stresses which result from the large out-of-plane deflections of the thin plate. Consequently, geometrical nonlinearity has to be considered and detailed analysis need to be performed. In this paper, a circular thin plate clamped peripherally and frictionless contacting an elastic sphere is studied. Firstly, a set of equations are established to predict the relationship between the contact force and the relative penetration. On this basis, analysis of the contact stress and surface displacements of the circular thin plate and the elastic sphere is carried out. Then, comparison of the results with the Hertz theory is presented. Finally, the validity of the methodology developed is examined by comparing the theoretical results with finite element method.

We are grateful for your kind advice. We have revised the abstract adding a brief discussion regarding the results and conclusions of our work. You can find them in revised manuscript.

We are truly grateful to your thoughtful suggestions. In our future research on this topic, we would like to continue this work to include the Finite Element mesh refinement/convergence study in detail.

Thanks for the valuable references you provided. After reading them thoroughly, we have decided to mention them in our revised manuscript to help readers better understand our work.

Thanks to the suggestion. We have carefully revised the paper. After rechecking the grammatical error and English usage by ourselves, we asked for some colleagues to go through our manuscript. We think the English presentation has been improved in the revised manuscript.

 

Special thanks to you for your good comments.

 

Sincerely Yours,

Wei Han

Reviewer 3 Report

1-In this paper, an analytical deformation analysis of a thin plate in contact with elastic sphere has been carried and comparison is done using Ansys software. In figures 11 and 12, very good agreement between the two results are observed. Here, normal stress variations, and deflection aspects are reported. The present work is although quite significant in terms of results, but require some major revisions,

2- Various assumptions involved in the modelling approach should be categorically mentioned somewhere in Section 2. Further, it should be explained how the two contacting bodies are replaced by elastic half-space.

3- Various parameters involved in the validation as well as used for generating the present results should be mentioned in separate tables.

4-The introduction section and literature survey needs improvement. The Introduction should be improved by adding more literature references related to various studies on computational schemes used for solving various fluid flow and thermal transport problems. Following references can be discussed: An inverse analysis of a transient 2-D conduction–radiation problem using the lattice Boltzmann method and the finite volume method coupled with the genetic algorithm; Application of Adomian decomposition method and inverse solution for a fin with variable thermal conductivity and heat generation.

5- At the end of the introduction section, please clearly state the main contributions of this paper to the existing body of knowledge. Objectives need to be clearly defined. Also, various aspects associated with the developed system should be mentioned clearly before the actual analysis. For example, various boundary conditions selected for this problem should be mathematically expressed as well as justified. Also, various expressions used in the work should be appropriately referenced.

6- Abstract section is very short. Please revise it carefully and eliminate any generic information. Also, some quantified results need to be included.

7- Authors are encouraged to provide a greater depth of discussion about each Figure. In connection with figure 10, its importance should be mentioned. Again in Figure 8, nonlinear variation of the deflection mechanism should be explained clearly.

8- A Nomenclature section should be included to list all symbols, notations, subscripts, etc. used in the paper. Appropriate SI units should be written wherever applicable.

9- Conclusion is not written impressively. Change the text from a summary of reported results to a narrative providing insights from analyzed and interpreted information. Make specific recommendations and highlight new knowledge gained from this study. Some scopes for further research should be discussed.

Author Response

Dear Reviewer:

 

Thank you for your comments concerning our manuscript entitled “Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere” (ID: applsci-1970297). Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. The responds to your comments are as following.

Thanks very much for your suggestion. The Hertz theory is not applicable to the elastic sphere in contact with the thin plate with large deformation. This is due to the fact that the elastic strain energy is determined by the membrane stresses which result from the large out-of-plane deflections of the thin plate. Consequently, geometrical nonlinearity has to be considered and detailed analysis need to be performed. In this paper, a circular thin plate clamped peripherally and frictionless contacting an elastic sphere is studied. Firstly, a set of equations are established to predict the relationship between the contact force and the relative penetration. On this basis, analysis of the contact stress and surface displacements of the circular thin plate and the elastic sphere is carried out. Then, comparison of the results with the Hertz theory is presented. Finally, the validity of the methodology developed is examined by comparing the theoretical results with finite element method.

We are grateful for your kind advice. We have revised the abstract and conclusion adding a brief discussion regarding the results and conclusions of our work. You can find them in revised manuscript.

Thanks for the valuable references you provided. After reading them thoroughly, we have decided to mention them in our revised manuscript to help readers better understand our work.

 

Special thanks to you for your good comments.

 

Sincerely Yours,

Wei Han

Round 2

Reviewer 1 Report

 

Dear Authors,

I am surprised that in your revised manuscript you have almost not addressed the points critically mentioned in the review reports. Instead, only 3 phrases of little significance (and some references) were added in the entire manuscript. I can understand that you would like to avoid the effort of additional parameter studies. However, you have picked only one special case here (same materials and fixed geometric specifications).

You emphasize several times in the manuscript that the Hertzian contact theory is not applicable. I agree! Nevertheless, as already noted in the review, the section 2.2 of your manuscript serves to determine the displacement of the half-space surface according to equation (24) from the pressure distribution according to equation (9). But what you calculate here are exactly the solutions of the Hertz theory. On the one hand the repetition of the equations is completely superfluous (You should clarify what comes from you and what is taken from textbooks!) and on the other hand the approach of the pressure distribution according to equation (9) is the main problem of your work. As already noted, the pressure distribution between the sphere and the plate is generally a priori unknown. However, you assume a pressure distribution of Hertzian type a priori and use it later for the calculation of the plate displacement (equations (30) and following). The comparison with your FE-calculations gives you right for the one special case you consider (same elasticity of both bodies, plate thickness about 4 times larger than the contact radius), but for other material pairings and geometrical specifications a clearly different pressure distribution can/will occur! It is essential to draw attention to this fact in the manuscript. In this context, the results of the mentioned pioneering work on the contact of a rigid sphere and an elastic plate should definitely be mentioned (and included in the introduction):

   [1]       Keer, L. M., & Miller, G. R. (1983). Contact between an elastically supported circular plate and a rigid indenter. International Journal of Engineering Science, 21(6), 681-690.

   [2]       Thredgold, J. M., Lucas, S. K., & Howlett, P. G. (2006). On the contact of a rigid sphere and a thin plate. Mathematical and computer modelling, 43(1-2), 119-131.

   [3]       Li, M., Ru, C. Q., & Gao, C. F. (2018). Axisymmetric indentation of an elastic thin plate by a rigid sphere revisited. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 98(8), 1436-1446.

The problem you are studying, i.e. the contact between an elastic sphere and an elastic plate (under large deformations) is definitely very interesting. However, I think the above-mentioned limitation/weakness of your study/approach should definitely be discussed in the manuscript!

Author Response

Dear Reviewer:

 

Thank you for your comments concerning our manuscript entitled “Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere” (ID: applsci-1970297). Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. The responds to your comments are as following.

Author Response File: Author Response.doc

Reviewer 2 Report

The authors have revised the manuscript accordingly. The revised manuscript can be published in the present form. 

Author Response

Thank you very much.

Reviewer 3 Report

Revisions are satisfactory.

Author Response

Thank you very much.