Compression of a Polar Orthotropic Wedge between Rotating Plates: Distinguished Features of the Solution

Round 1

Reviewer 1 Report

As attached file.

Comments for author File: Comments.pdf

Author Response

1.      Since this problem is well-posed, The authors should clearly describe how many unknown variables (eg.,stresses, strain rates, velocity vectors,…), equations (equilibrium equations, constitution equations, equations of flow rule,…) and associated boundary conditions are involved and should be dealt with.

A description is provided in Section 2 of the original manuscript? We have summarized the statement of the boundary value problem at the end of this section.

2.      Since the material within two rotating rough semi-infinite plates is anisotropic, the constitution equations as well as the solutions should be related to the angle q. However, why we cannot find this kind of dependence?

It is because the material is polar orthotropic. Please see the title of the paper.

3.      The final solutions should be dependent upon the angular velocity w of two rotating plates. The authors should illustrate the effects of w in the solutions.

The solution is indeed independent of the angular velocity. It is because a rate-independent model of plasticity is adopted. The term rate-independent used in the conventional classification of plasticity theories emphasizes that the solution is independent of the rate of loading.

4.      Some Figures of Tables should be included in this article to illustrate the effects of some important parameters on final results.

Could you please be more specific? The final result is some qualitative mathematical features of the solution. How to illustrate these qualitative features using tables or figures?

Author Response File: Author Response.docx

Reviewer 2 Report

The paper has not enough original outcomes. I think the authors should better clarify the novelties of this work and present also a comparison with some numerical simulations or at least some experiments.

Author Response

The paper has not enough original outcomes. I think the authors should better clarify the novelties of this work and present also a comparison with some numerical simulations or at least some experiments.

Do you know any other paper in which all the qualitative mathematical features of the solution investigated in our solution are discussed for any model of orthotropic plasticity? If so, then please let us know. If you, as an expert, do not know such a paper, then it is the novelty of our paper.

We do not know who can carry out any experiment using infinite plates. No experiment is required to study mathematical properties of models or boundary value problems.

Standard numerical methods are not capable of solving this boundary value problem because of (55) or (56). Also, closed form solutions are used for verifying numerical solutions, not vice versa.

Reviewer 3 Report

I have some difficulties understanding the statement of the problem.

First of all, I never heard of rigid plastic materials and rigid plastic solution. I mean, for what I know a rigid body is one for which a reference frame exists where \(u_i(t) = 0, \forall t \geq 0\). I cannot understand how a rigid material can undergo plastic deformations (or deformations of any type). I suppose that the term _rigid_ is used here in some other way, and I would be interested to know.

Moreover, I cannot understand how the plates can rotate, but the angle \(\alpha\) that gives the positions of the plates could be considered as constant.

Also, the conditions (1), (2), (3) are given without any explanation.

Given these doubts, I cannot procede further in the understanding of the paper. 

Author Response

1. First of all, I never heard of rigid plastic materials and rigid plastic solution. I mean, for what I know a rigid body is one for which a reference frame exists where \(u_i(t) = 0, \forall t \geq 0\). I cannot understand how a rigid material can undergo plastic deformations (or deformations of any type). I suppose that the term _rigid_ is used here in some other way, and I would be interested to know.

The rigid plastic solid is one of the classical models in plasticity (Hill R. The mathematical theory of plasticity, Pergamon Press, 1950, Chapter 6). The term rigid emphasizes that elasticity is neglected.

2. Moreover, I cannot understand how the plates can rotate, but the angle \(\alpha\) that gives the positions of the plates could be considered as constant.

It is not considered to be constant. The solution is instantaneous. When a ball is falling from a tower, you can fix its position at one instant and find its velocity, acceleration, force and so on at this instant. It is the same concept. You may easily find thousands of instantaneous rigid plastic solutions in the literature.

3. Also, the conditions (1), (2), (3) are given without any explanation.

Please see the line above each of these boundary conditions. (1) is because of there no flux through O, (2) and (3) are due to symmetry. You can find these boundary conditions in any textbook on theory of plasticity. The statement of the problem is not new and is very well know. We present a new solution to a known boundary value problem.

Given these doubts, I cannot procede further in the understanding of the paper. 

Round 2

Reviewer 1 Report

Since this problem is well-posed, it is better to clearly describe how many unknown variables (eg., stresses, strain rates, velocity vectors, ...),  equations (eg., equilibrium equations, constitution equations, equations of flow rule, ...) and associated boundary conditions are involved and should be dealt with. 

Since the material within two plates is anisotropic, the constitution equations as well as final solutions should be dependent upon the angle theta. Why there is no such dependence?

The final solution should be dependent upon the angular velocity omega of two rotation plates. The authors had better to describe the effects of omega on the final results.

Some key figures or tables should be supplemented to illustrate the effects of some important parameters on the final results.

Author Response

Please see the file attached

Author Response File: Author Response.docx

Reviewer 2 Report

I appreciate the efforts made by the authors

Author Response

Thank you for your comment

Reviewer 3 Report

I have no particular comments.

Author Response

Thank you for your comment