Review Reports

Round 1

Reviewer 1 Report

The authors have reported a progress in which the ideal flow condition is incorporated with the double shearing model and the double slip and rotation model, respectively. It seems that the identical work has already been published by the authors before, see reference [22]. Even though this manuscript provides more details than [22], the main conclusion remains unchanged.

Author Response

We believe that it is a misunderstanding. The solution in [22] is a special solution based on a solution for the flow through an infinite wedge-shaped channel with no friction. This solution is not even a particular case of the new solution based on the general theory.

Reviewer 2 Report

The referred paper presents a study extending the ideal plastic theory for stationary planar flow to the double slip and rotation models. Based on this theory, the optimal extrusion and minimum-length dies are calculated. The development of the proposed theory is well described and discussed. The manuscript is structured in a tidy manner. The English writing is nice. I think the paper can be accepted for publication.

 

 

 

The followings are some pieces of advice that may be helpful to improve the manuscript. 

 

 

 

 

 

1.  On line 102, “follows” may be replaced by “following.”

 

2.  On line 172, “combine” may be changed to passive voice. 

 

3.  From equation (3), the angle between line eta and line beta should be (Pi/4+phi/2). It is not clear why it is (Pi/4-phi/2) in Figure 2.

 

4.  On line 156, it is not clear why equation (5) can give principal stress. Equation (5) is simply a summation of principal stress.

 

5.   On lines 177 and 178, it is not clear why psi is independent of alpha, as psi is defined with respect to xi line.

 

6.   On lines 319 and 338, “consider” may be changed to “considering.”

 

7.   On line 358, “summing” may change to “summing them.”

 

8.  The Riemman’s method may be explicitly introduced and referred in the section on calculating the minimum-length dies. 

 

 

 

The wording flow is quite fluent in this paper. However, I noticed a few inappropriate phrases in the manuscript. 

Author Response

1. On line 102, “follows” may be replaced by “following.”

Response.

We have incorporated this correction.

2. On line 172, “combine” may be changed to passive voice. 

Response.

We have incorporated this correction.

3. From equation (3), the angle between line eta and line beta should be (Pi/4+phi/2). It is not clear why it is (Pi/4-phi/2) in Figure 2.

Response.

It was a typographical error in (3). We are sorry for this inconvenience.

4. On line 156, it is not clear why equation (5) can give principal stress. Equation (5) is simply a summation of principal stress.

Response.

This line reads ‘equations (2) and (5).’ These two equations constitute a system for determining the principal stresses if p is known. Line 153 emphasizes that p should be calculated before solving (2) and (5) for the principal stresses.

5. On lines 177 and 178, it is not clear why psi is independent of alpha, as psi is defined with respect to xi line.

Response.

This section treats the case that alpha-lines are straight. Therefore, the angle between any alpha line in this region and any straight line is constant. The angle between any alpha line and xi lines is constant (equation (3)). Therefore, the angle between any xi line in the region and any straight line is independent of alpha. We have clarified it in the revised manuscript.

6. On lines 319 and 338, “consider” may be changed to “considering.”

Response.

We have incorporated this correction.

7. On line 358, “summing” may change to “summing them.”

Response.

We have incorporated this correction.

8. The Riemman’s method may be explicitly introduced and referred in the section on calculating the minimum-length dies.

Response.

We have corrected Section 4.

Reviewer 3 Report

The article presents the mathematical development of a plastic flow theory for pressure-dependent materials. This theories are necessary for designing tooling for cold and hot working materials.

The mathematical concepts were easy to follow and the presented results make sense, although I would like to see an actual application of the theory (i.e., a case study).

Overall, the article is interesting and consider it would have an impact in future tooling developments. Therefore, I recommend its publication.

Author Response

We think that Section 4 provides an application. This section calculates specific die profiles.