Analysis of Cold Composite Sheet Rolling Considering Anisotropic Effect and Position-Dependent Friction Model

Round 1

Reviewer 1 Report (New Reviewer)

The authors present a combined experimental and theoretical study of cold Ti/Al composite sheet rolling. The developed theoretical background of the model is presented in details and it looks reliable. This is supported by the performed experiments, which revealed a good applicability of the reported theoretical approach. Therefore, I can recommend the manuscript for publication after a very minor revision.

1.      The authors mention in the Introduction the use of finite element method to analyze the deformation of composite sheets, but they missed an example of the discrete element method for stress propagation simulation (https://doi.org/10.1016/j.fpc.2021.09.004). Mention of this paper is much suggested.

2.      Is it possible to apply atomistic modeling with anisotropic compression for simulation of the sheet deformation? A little comment on this issue would be very interesting for the readers.

3.      The authors use term “scholars”, which seems not very appropriate. I suggest using the term “researchers”.

Author Response

Reviewer(s)' Comments to Author:
Reviewer 1:

The authors present a combined experimental and theoretical study of cold Ti/Al composite sheet rolling. The developed theoretical background of the model is presented in details and it looks reliable. This is supported by the performed experiments, which revealed a good applicability of the reported theoretical approach. Therefore, I can recommend the manuscript for publication after a very minor revision.

1) The authors mention in the Introduction the use of finite element method to analyze the deformation of composite sheets, but they missed an example of the discrete element method for stress propagation simulation (https://doi.org/10.1016/j.fpc.2021.09.004). Mention of this paper is much suggested.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice. The article you mentioned is added to the manuscript (as shown in Line 42-44, Page 1 and Line 549-550, Page 23). Thank you again for your helpful advice.

2) Is it possible to apply atomistic modeling with anisotropic compression for simulation of the sheet deformation? A little comment on this issue would be very interesting for the readers.

Our reply: We express our sincere gratefulness to the reviewer for this professional advice. According to your requirement, we added this paragraph to the manuscript (as shown in Line 44-48, Page 1-2):

“It should be mentioned here that multi-scale simulation has become more and more popular in studying the deformation behavior of sheets. Combining the macroscopic finite element model with the atomistic modeling with anisotropic compression is useful for studying the deformation behavior and deformation coordination between layers of composite sheets.”

Thank you again for your helpful advice.

3) The authors use term “scholars”, which seems not very appropriate. I suggest using the term “researchers”.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice. According to your requirement, we change “scholars” to “researchers” in the manuscript (as shown in Line 43, Page 1, Line 49, Page 2, Line 55, Page 2). Thank you again for your helpful advice.

Author Response File: Author Response.pdf

Reviewer 2 Report (New Reviewer)

In this paper, the authors used a cold rolling deformation model that includes an anisotropic effect and a position-dependent friction model to analyze the impact of various process and material variables on the Ti/Al composite sheet.

The article contains an analysis of the problem from the mathematical point of view and research results, including comparisons of the calculated parameters based on the theoretical model with the results of experimental research. The issue of the production of multilayer materials discussed in the article is vital from the point of view of the development of technology and, therefore, worth publishing; however, minor corrections are required before publication.

The introduction is based on the literature from recent years. It contains essential information necessary to determine the topicality of the work, but there needs to be a reference to literature items covering the considered combinations of materials. The article contains a broad description of the mathematical model. The "Results and Discussion" chapter includes tables and graphs to compare the results obtained, but the different scales hamper their analysis of the careful figures. The results are clearly described and adequately analyzed, but the conclusions should be more clearly defined.

SUGGESTED IMPROVEMENTS

1.       The conclusions should clearly state the implications of the resulting theoretical considerations and research.

2.       The results in Figures 8, 9, 10 and 13 should have the same vertical scales for more accessible data analysis.

3.       Line 300, please provide data on the dimensions of the samples.

Author Response

Reviewer(s)' Comments to Author:
Reviewer 2:

In this paper, the authors used a cold rolling deformation model that includes an anisotropic effect and a position-dependent friction model to analyze the impact of various process and material variables on the Ti/Al composite sheet.

The article contains an analysis of the problem from the mathematical point of view and research results, including comparisons of the calculated parameters based on the theoretical model with the results of experimental research. The issue of the production of multilayer materials discussed in the article is vital from the point of view of the development of technology and, therefore, worth publishing; however, minor corrections are required before publication.

The introduction is based on the literature from recent years. It contains essential information necessary to determine the topicality of the work, but there needs to be a reference to literature items covering the considered combinations of materials. The article contains a broad description of the mathematical model. The "Results and Discussion" chapter includes tables and graphs to compare the results obtained, but the different scales hamper their analysis of the careful figures. The results are clearly described and adequately analyzed, but the conclusions should be more clearly defined.

• The conclusions should clearly state the implications of the resulting theoretical considerations and research.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice. According to your requirement, we have added some contents to the conclusion:

“This model enhances the reliability of the slab method in studying the rolling deformation of layered metal composites.”(as shown in Line 496-497, Page 22)

“This study can provide a theoretical reference for the design of layered metal composites with large deformation differences.”(as shown in Line 510-511, Page 23)

Thank you again for your helpful advice.

2) The results in Figures 8, 9, 10 and 13 should have the same vertical scales for more accessible data analysis.

Our reply: We express our sincere gratefulness to the reviewer for this professional advice. According to your requirement, we reprocessed Figure 8, 9, 10 and 13 as follows. Thank you again for your helpful advice.

Fig. 8. Ti/Al thickness ratio values measured by theoretical model and experiment under different aluminum strengths. (a) S1, (b) S2, (c) S3.

Fig. 9. Effect of total reductions on the stress distribution,,  and  in the deformation zone. (a) S1, (b) S2, (c) S3.

Fig. 10. Effect of aluminum strength on the stress distribution, ,  and  in the deformation zone. (a) 35% reduction, (b) 50% reduction.

Fig. 13. Effect of friction coefficient between titanium and aluminum on the stress distribution, ,  and  in the deformation zone. (a) S1, (b) S3.

• Line 300, please provide data on the dimensions of the samples.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice. provide data on the dimensions of the samples: “plate-type tensile specimens (gage length: 25 mm, gage width: 6 mm) were prepared in the rolling and transverse directions”(as shown in Line 300-301, Page 14). Thank you again for your helpful advice.

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

- The abstract should be revised and the important results of this research should be given quantitatively. - The number of references given in the introduction section is small, and the authors should provide more references in this section so that the readers are more familiar with the history of research done in this field. - Authors should provide relevant sources or standards for the assumptions considered. For example, for the assumption "The cold rolling process is plane strain.", they should provide the appropriate reference. - On page 3, line 93, the authors have stated that: "In region I, the titanium layer does not yield, the aluminum layer yields, and there is relative sliding between the titanium and aluminum layers." On what basis is this stated? - For what reason "Because of brittleness and inextensible of the covering layer, the friction coefficients between surfaces depend on the surface extension deformation."? - Why did the authors use Hill's failure criterion for yielding behavior? Why other criteria such as Hosford's failure criteria were not used. - The explanations given for sections 3.2-3.5 are few and not enough. Please provide more complete explanations in this field using relevant sources. - Can the authors also refer to metallurgical points to interpret the results? Can they also use related resources?

Author Response

RESPONSE TO THE REVIEWERS

ID/Title: Manuscript metals-1958935 entitled " Analysis of the cold composite sheet rolling considering anisotropic effect and position-dependent friction model ".

First of all, we would like to express our sincere appreciation to the editor and reviewers for your careful examinations and detailed comments on our work. All the comments are very helpful for us to improve our manuscript. Here, we offer our opinions about the issues raised by the editor and reviewer point-by-point. Our modified content is marked in yellow in the manuscript.

Reviewer(s)' Comments to Author:
Reviewer 1:

The abstract should be revised and the important results of this research should be given quantitatively. - The number of references given in the introduction section is small, and the authors should provide more references in this section so that the readers are more familiar with the history of research done in this field. - Authors should provide relevant sources or standards for the assumptions considered. For example, for the assumption "The cold rolling process is plane strain.", they should provide the appropriate reference. - On page 3, line 93, the authors have stated that: "In region I, the titanium layer does not yield, the aluminum layer yields, and there is relative sliding between the titanium and aluminum layers." On what basis is this stated? - For what reason "Because of brittleness and inextensible of the covering layer, the friction coefficients between surfaces depend on the surface extension deformation."? - Why did the authors use Hill's failure criterion for yielding behavior? Why other criteria such as Hosford's failure criteria were not used. - The explanations given for sections 3.2-3.5 are few and not enough. Please provide more complete explanations in this field using relevant sources. - Can the authors also refer to metallurgical points to interpret the results? Can they also use related resources?

Our reply:

• Firstly, please allow us to express our cordial gratefulness for your valuable advice.I made some modifications according to your requirements.

“The large difference in mechanical properties and plastic deformation ability of each layer will have a great impact on the overall performance of the composite sheet prepared by cold roll bonding. The effect of rolling and material variables on the stress distribution and bonding state in the rolling deformation zone should be studied. In this work, an accurate cold rolling deformation model considering anisotropic effect and position-dependent friction model is established by the slab method. Effects of different process and material variables are analyzed. Related experiments were performed on Ti-Al clads and calculation results from the deformation model were compared with the experimental results. This model can well predict the Ti/Al thickness ratio after rolling, and the smaller the initial aluminum strength, the more accurate the predicted value, the minimum error is within 1%. The deformation coordination between titanium and aluminum layers becomes better with the increase of rolling reduction and initial aluminum strength. At 50% reduction, the deformation ratio of titanium and aluminum increases from 93.8% to 98.1%, which is consistent with the trend of the results calculated by this model.”(as shown in Line 13-25, Page 1)

• We are very grateful for the reviewer’s kind advice.According to your requirement, some contents have been modified and references have been added to the introduction 

“Layered metal composites are more and more widely used because of their excellent comprehensive properties compared with single metal. In addition, layered metal composites can reduce the cost of manufacturing process or product application, so they are widely used in aerospace, national defense and military industry, transportation and equipment manufacturing [1-7].

Rolling is the most economical and efficient method to prepare layered metal composites. The mechanical properties and plastic deformation capacity of each layer of the composite sheet are quite different. In the rolling process, the mutual restriction of each layer makes the composite sheet show different deformation behavior from the single metal as a whole. This deformation characteristic of the composite sheet makes the stress field change significantly. Meanwhile, it will also affect the bonding state of the composite sheet in the deformation zone, which determines the bonding formation. Therefore, the deformation behavior of each layer in the rolling process should be comprehensively analyzed by numerical methods and theoretical models. Many scholars use finite element method to analyze the deformation of composite sheets in the rolling process [7-10]. Other scholars use mathematical models to study the deformation behavior of each layer in the rolling process. These mathematical models include upper bound method [11-13], stream function method [14], slip line method [15-16] and slab method [17-18]. Among them, the stress field cannot be obtained by the upper bound method and stream function method. Moreover, the slab method is one of the best methods to solve mechanical problems. Anisotropy effects are very important in the rolling process, but many scholars seldom consider this when establishing deformation model by slab method. Chaudhari et al. [19] considers the anisotropy of titanium layer when analyzing the cold rolling deformation of multilayer titanium/aluminum composite. Moreover, in the rolling process, different friction states exist at the interfaces of composite sheet with the rolls and at the interface of two layers. In the theoretical and numerical analysis of rolling process, there are usually two friction models: Coulomb friction model and constant shear friction model. Tzou [20] proposed an analytical Coulomb model. Pan et al. [21] established an analytical model considering constant shear for asymmetric cold and hot rolling of composites. Meanwhile, Huang et al. [22] analysed these two friction models. The analytical results have the identical trends for the two models. The relationships between frictional coefficient and frictional factor can be obtained effectively, and the limit frictional coefficient generating sticking friction is found.

Considering the extrusion process of the fresh metal at the two faying surfaces, it can be known that the frictions between composite sheet and the rolls and between the two layers are related to the position in the deformation zone. Based on this, in the analysis of cold rolling of Ti/Al composite sheet, the deformation anisotropy effects of titanium and aluminum are introduced, and the position correlation of friction caused by extrusion process of bottom metal is also considered. In this work, the effects of rolling reduction, different initial aluminum strength, friction coefficient between titanium and aluminum and initial thickness of aluminum layer on the stress field distribution and bonding state of Ti/Al composite sheet during cold rolling are analyzed and discussed, which provides a theoretical basis and related process reference for solving the problem of poor deformation coordination and bonding performance of the composite sheet in the cold rolling process.”(as shown in Line 29-74, Page 1-2)

• We are very grateful for the reviewer’s kind advice.According to your requirement, we have provided relevant and appropriate references to the assumptions.

“In order to simplify the formulation involved in developing the analysis for cold roll bonding of composite sheet, the following assumptions are made [18-21]:”(as shown in Line 78-79, Page 3)

In addition, it is also reasonable for us to divide the deformation zone into four regions. We divide the deformation zone into four regions based on the deformation ability of each layer and the essence of the formation of interfacial bonding. We can use a diagram to briefly illustrate. When the rolling pressure acts on the Ti/Al composite sheet, the aluminum layer yields due to its low strength, and the covering layer on the surface of the aluminum layer first breaks (Corresponding to Region I). At this time, the bottom fresh aluminum begins to flow to the crack, and the flow velocity is relatively small. With the deformation progress, the flow velocity of the bottom fresh aluminum is accelerated. When the deformation reaches a certain value, the titanium layer yields, and the corresponding covering layer on the surface of the titanium layer breaks (Corresponding to Region II); then, the bottom fresh titanium starts to flow, and it has a lower flow velocity than aluminum. Therefore, there will be 3 bonding areas on the real contact surface, of which only four bonding areas in form A are called strong bonding. The initial bonding between titanium and aluminum depends on the bonding area of A. When all four strong bonding areas are formed, the entire bonding process is completed, this area corresponds to region III. So, the division of the deformation zone proposed by us has an internal basis. At the same time, I would like to point out that similar assumptions are also used in other articles that analyze the deformation behavior of composite sheets by slab methods (Ref. [18-22]). Thank you again for your helpful advice. 

Fig. a. Extrusion flow of aluminum and titanium in cracks.

• We express our sincere gratefulness to the reviewer for this professional advice.In region I, only the soft layer (Al) deform and it can slip relative to the hard layer (Ti) since they are not bonded. Large deformation of soft layer (Al) and small deformation of hard layer (Ti) will result in relative sliding between them. Thank you again for your helpful advice. 
• We express our sincere gratefulness to the reviewer for this professional advice.Let's take the contact between aluminum layer and roll as an example. When the rolling pressure acts on the Ti/Al composite sheet, because the aluminum layer has greater reduction, the covering layer on its surface first breaks. At this time, the bottom fresh aluminum begins to flow to the crack, and the flow velocity is relatively small. With the deformation progress, the flow velocity of the bottom fresh aluminum is accelerated. When the deformation reaches a certain value, the bottom fresh aluminum is extruded from the crack to contact the roll. We can know that the contact states between the roll and the aluminum surface are different at various surface expansion deformation stages, from the initial contact between the covering layer and the roll to the subsequent contact between the bottom fresh aluminum and the roll, so the friction coefficients are different. Therefore, the friction coefficients between surfaces depend on the surface extension deformation. Thank you again for your helpful advice. 
• We express our sincere gratefulness to the reviewer for this professional advice.First of all, I want to say that our model has universal adaptability. It can be applied to layered composite sheets with different mechanical properties in each layer, such as Ti/Al, Cu/Al, Mg/Al, and so on. Here, In order to verify the accuracy of the model, a Ti/Al cold-rolled experiment was carried out to examine the post-rolled Ti/Al thickness ratio as a target for comparison with the model predicted values. the minimum error is within 1%, the maximum error is not more than 10%. Meanwhie, we conducted an experiment on the deformation of each layer. Comparing Fig. 11 and Fig. 12, the predicted results with our model are in good agreement with the experimental results. We consulted the corresponding literatures, some scholars believe that the Hill anisotropic criterion is inappropriate to describe the yield behaviour of Al alloys, using Barlat model instead of Hill, or, as you said, using Hosford's failure criteria. I want to emphasize that: the focus of this article is to establish cold rolling deformation model considering anisotropic effect and position-dependent friction model. Our another article prepared will discuss the accuracy of the results predicted by different anisotropic criterion including Hosford's failure criteria and Barlat model. Thank you again for your helpful advice. 

Fig. 12. Deformation of each layer of three types of Ti/Al composite sheets under different rolling reductions. (a) S1; (b) S2; (c) S3; (d) variation of deformation ratio of titanium and aluminum with total rolling reduction. (Our article accepted for publication)(as shown in Line 430-433, Page 20)

• .Firstly, please allow us to express our cordial gratefulness for your valuable advice.According to your opinion, I have made changes in the corresponding parts in the manuscript, which have been which have been marked yellow.

3.2 (as shown in Line 357-360, 362-364 Page 17)

3.3 (as shown in Line 387-393, Page 18) and (as shown in Line 420-426, Page 19)

3.4 (as shown in Line 435-440, Page 20)

3.5 (as shown in Line 466-470, Page 21)

(8).Firstly, please allow us to express our cordial gratefulness for your valuable advice. The focus of this article is to establish cold rolling deformation model considering anisotropic effect and position-dependent friction model. These two aspects do not appear simultaneously in other models. We hope to use our new model to better predict the deformation behavior of composite sheets. We explain the results calculated using our model by considering the fracture of the covering layer and extrusion process of underlying metal, which is the essence of bonding formation. The change of the three characteristic points mentioned in the article with the parameters is such a consideration. As for the metallurgical points, we need to do corresponding experiments, such as using SEM, TEM and EBSD to observe the interface. In our article accepted to be published (https://doi.org/10.3390/cryst12111665), the distribution of elements on the peeling surfaces was investigated, and the changes of texture near the interface were observed using EBSD. The bonding mechanism was explained by the fracture of the covering layer and extrusion process of underlying metal. Thank you again for your helpful advice.

Author Response File: Author Response.pdf

Reviewer 2 Report

This research investigated the anisotropic effect and position-dependent friction model of cold-rolled Ti/Al composite sheets using the analytical slab method. This is a good work, and its results are interesting for researchers in this field. Accordingly, this manuscript is publishable after some revisions:

1. The literature survey can be improved.

2. All symbols and abbreviations should be collected in a Table at the beginning of the manuscript.

3. The novelty of your work compared to the previous studies should be written clearly.

4. Why did the authors use this analytical (slab) solution compared to the other methods?

5. The following papers should be mentioned in the Introductions:

10.1051/metal/2015044

10.1016/j.jmrt.2020.10.031

10.1016/j.pnsc.2021.08.005

6. The conclusion is too long. Please shorten it.

7. The manuscript should be revised based on the English language point of view.

Author Response

RESPONSE TO THE REVIEWERS

ID/Title: Manuscript metals-1958935 entitled " Analysis of the cold composite sheet rolling considering anisotropic effect and position-dependent friction model ".

First of all, we would like to express our sincere appreciation to the editor and reviewers for your careful examinations and detailed comments on our work. All the comments are very helpful for us to improve our manuscript. Here, we offer our opinions about the issues raised by the editor and reviewer point-by-point. Our modified content is marked in yellow in the manuscript.

Reviewer(s)' Comments to Author:

Reviewer: 2

This research investigated the anisotropic effect and position-dependent friction model of cold-rolled Ti/Al composite sheets using the analytical slab method. This is a good work, and its results are interesting for researchers in this field. Accordingly, this manuscript is publishable after some revisions:
1. The literature survey can be improved.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice. According to your requirement, some contents have been modified and references have been added to the introduction section. Thank you again for your helpful advice.

“  Layered metal composites are more and more widely used because of their excellent comprehensive properties compared with single metal. In addition, layered metal composites can reduce the cost of manufacturing process or product application, so they are widely used in aerospace, national defense and military industry, transportation and equipment manufacturing [1-7].

Rolling is the most economical and efficient method to prepare layered metal composites. The mechanical properties and plastic deformation capacity of each layer of the composite sheet are quite different. In the rolling process, the mutual restriction of each layer makes the composite sheet show different deformation behavior from the single metal as a whole. This deformation characteristic of the composite sheet makes the stress field change significantly. Meanwhile, it will also affect the bonding state of the composite sheet in the deformation zone, which determines the bonding formation. Therefore, the deformation behavior of each layer in the rolling process should be comprehensively analyzed by numerical methods and theoretical models. Many scholars use finite element method to analyze the deformation of composite sheets in the rolling process [7-10]. Other scholars use mathematical models to study the deformation behavior of each layer in the rolling process. These mathematical models include upper bound method [11-13], stream function method [14], slip line method [15-16] and slab method [17-18]. Among them, the stress field cannot be obtained by the upper bound method and stream function method. Moreover, the slab method is one of the best methods to solve mechanical problems. Anisotropy effects are very important in the rolling process, but many scholars seldom consider this when establishing deformation model by slab method. Chaudhari et al. [19] considers the anisotropy of titanium layer when analyzing the cold rolling deformation of multilayer titanium/aluminum composite. Moreover, in the rolling process, different friction states exist at the interfaces of composite sheet with the rolls and at the interface of two layers. In the theoretical and numerical analysis of rolling process, there are usually two friction models: Coulomb friction model and constant shear friction model. Tzou [20] proposed an analytical Coulomb model. Pan et al. [21] established an analytical model considering constant shear for asymmetric cold and hot rolling of composites. Meanwhile, Huang et al. [22] analysed these two friction models. The analytical results have the identical trends for the two models. The relationships between frictional coefficient and frictional factor can be obtained effectively, and the limit frictional coefficient generating sticking friction is found.

Considering the extrusion process of the fresh metal at the two faying surfaces, it can be known that the frictions between composite sheet and the rolls and between the two layers are related to the position in the deformation zone. Based on this, in the analysis of cold rolling of Ti/Al composite sheet, the deformation anisotropy effects of titanium and aluminum are introduced, and the position correlation of friction caused by extrusion process of bottom metal is also considered. In this work, the effects of rolling reduction, different initial aluminum strength, friction coefficient between titanium and aluminum and initial thickness of aluminum layer on the stress field distribution and bonding state of Ti/Al composite sheet during cold rolling are analyzed and discussed, which provides a theoretical basis and related process reference for solving the problem of poor deformation coordination and bonding performance of the composite sheet in the cold rolling process.”(as shown in Line 29-74, Page 1-2)

2. All symbols and abbreviations should be collected in a Table at the beginning of the manuscript.

Our reply: We are very grateful for the reviewer’s question. According to your requirements, all symbols and abbreviations were collected in a box. Thank you again for your helpful advice.

3. The novelty of your work compared to the previous studies should be written clearly.

Our reply: We are very grateful for the reviewer’s professional advice. The focus of this article is to establish cold rolling deformation model considering anisotropic effect and position-dependent friction model. These two aspects do not appear simultaneously in other models. We hope to use our new model to better predict the deformation behavior of composite sheets. The anisotropic effect and position-dependent friction model are both considered in our slab method is the novelty of our work. The novelty of our work was described in the manuscript (as shown in Line 65-68, Page 2) . Thank you again for your helpful advice.

4. Why did the authors use this analytical (slab) solution compared to the other methods?

Our reply: We are very grateful for the reviewer’s professional advice. As mentioned in ref. [18], the slab method is still one of the best analysis means to solve the mechanical problems. On the other hand, the slab method used by many researchers either does not consider the anisotropic effect of each layer, or uses Coulomb friction or some simple friction models. The anisotropic effect and position-dependent friction model are both considered in our slab method, which can more accurately reflect the actual situation. Thank you again for your helpful advice.

5. The following papers should be mentioned in the Introductions:

10.1051/metal/2015044

10.1016/j.jmrt.2020.10.031

10.1016/j.pnsc.2021.08.005

Our reply: We are very grateful for the reviewer’s question. We have added these three papers to the references, namely, ref. [6-7,13]. Thank you again for your helpful advice.

6. The conclusion is too long. Please shorten it.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice. We have modified the conclusion accordingly. Thank you again for your helpful advice.

“    In this work, a Ti/Al cold-rolled deformation model is established by the slab method, which takes into account the anisotropic effect and position-dependent friction model. The Ti/Al thickness ratio calculated by the model is consistent with the experimental results, and the maximum error is not more than 10%. The theoretical model results show that with the increase of rolling reduction, the bonding point gradually moves away from the inlet, which promotes more aluminum to flow to the outlet. Meanwhile, the greater vertical stress after the formation of bonding has a catalytic effect on the enhancement of the interfacial bonding properties. The higher the initial aluminum strength, the more coordinated deformation of the composite sheet and the more stable the properties. Meanwhile, at 50% reduction, the higher the initial aluminum strength, the closer the corresponding bonding point is to the inlet, causing the longer the relative distance experienced by the composite sheet under higher rolling pressure after the joint formation, which can promote the improvement of the bonding property, this is consistent with the experimental results. The greater the friction between the titanium and aluminum layers, the more conducive to the improvement of the bonding property. ”(as shown in Line 488-503, Page 22)

Author Response File: Author Response.pdf

Reviewer 3 Report

1.    1. The paper employs the slab method for investigating the rolling process of two-layer sheets under plane strain conditions. This method itself is oversimplified. Besides, in the case under consideration, additional assumptions are necessary. In particular, the method cannot determine the shape of the bi-material interfaces. Therefore, this shape is prescribed without any rational arguments.

2.      2. The solution is cumbersome, and its description is unclear. It is unreasonable to repeat the derivation of the simplified equilibrium equations for each region. It is sufficient to derive the equations for a generic region using arbitrary geometric parameters. Then, these parameters should be specified in each region.

3.  3.      It is unclear why the paper discusses a crack. This crack is not shown in the figures.

4.      4. The transition conditions between regions 2 and 3 are unclear. The thickness ratio is the same in both. Then, the other non-zero strain should also be the same. How is it possible that there is sliding between the layer in region 2 but sticking in region 3?

5.     5.  The description of anisotropic properties is unsatisfactory. Under the assumptions made in the paper, any anisotropic yield criterion is Sigma1 – Sigma3 = constant (J. Mech. Phys. Solids 21, 63-74 (1973)).

Author Response

RESPONSE TO THE REVIEWERS

ID/Title: Manuscript metals-1958935 entitled " Analysis of the cold composite sheet rolling considering anisotropic effect and position-dependent friction model ".

First of all, we would like to express our sincere appreciation to the editor and reviewers for your careful examinations and detailed comments on our work. All the comments are very helpful for us to improve our manuscript. Here, we offer our opinions about the issues raised by the editor and reviewer point-by-point. Our modified content is marked in yellow in the manuscript.

Reviewer(s)' Comments to Author:

Reviewer: 3
1. The paper employs the slab method for investigating the rolling process of two-layer sheets under plane strain conditions. This method itself is oversimplified. Besides, in the case under consideration, additional assumptions are necessary. In particular, the method cannot determine the shape of the bi-material interfaces. Therefore, this shape is prescribed without any rational arguments.

Our reply: Firstly, please allow us to express our cordial gratefulness for your valuable advice.

Firstly, please allow us to express our cordial gratefulness for your valuable advice. In deed, the slab method is simpler than upper bound method, stream function method and slip line method. The focus of this article is to establish cold rolling deformation model considering anisotropic effect and position-dependent friction model. These two aspects do not appear simultaneously in other models. We hope to use our new model to better predict the deformation behavior of composite sheets. We divide the deformation zone into four regions, which consider the deformation ability of each layer and the essence of the formation of interfacial bonding. In region III, the underlying metal is extruded from the crack and bond formation is completed. We can use a diagram to briefly illustrate. When the rolling pressure acts on the Ti/Al composite sheet, the covering layer on the surface of the aluminum layer first breaks. At this time, the bottom fresh aluminum begins to flow to the crack, and the flow velocity is relatively small. With the deformation progress, the flow velocity of the bottom fresh aluminum is accelerated. When the deformation reaches a certain value, the corresponding covering layer on the surface of the titanium layer breaks; then, the bottom fresh titanium starts to flow, and it has a lower flow velocity than aluminum. Therefore, there will be 3 bonding areas on the real contact surface, of which only four bonding areas in form A are called strong bonding. The initial bonding between titanium and aluminum depends on the bonding area of A. When all four strong bonding areas are formed, the entire bonding process is completed, this area corresponds to region III. So, the division of the deformation zone proposed by us has an internal basis. Thank you again for your helpful advice.

Fig. a. Extrusion flow of aluminum and titanium in cracks.

2. The solution is cumbersome, and its description is unclear. It is unreasonable to repeat the derivation of the simplified equilibrium equations for each region. It is sufficient to derive the equations for a generic region using arbitrary geometric parameters. Then, these parameters should be specified in each region.

Our reply: .Firstly, please allow us to express our cordial gratefulness for your valuable advice. As mentioned earlier, we divide the deformation zone into four regions, which consider the deformation ability of each layer and the essence of the formation of interfacial bonding. There is no generic region because the force on each region is different, for example, in region I and region II, we need to conduct separate stress analysis on titanium and aluminum layers, however, in region III and region IV, we only need to conduct stress analysis on the whole composite sheet. So we need to make a detailed analysis for each region. And there are similar practices in other literatures (Ref. [18] and ref. [19] in this work). Thank you again for your helpful advice.

3. It is unclear why the paper discusses a crack. This crack is not shown in the figures.

Our reply: We are very grateful for the reviewer’s question. In addition to anisotropy, the position-dependent friction model is also considered in our slab method .We explain the results calculated using our model by considering the fracture of the covering layer and extrusion process of underlying metal, which is the essence of bonding formation. The change of the three characteristic points mentioned in the article with the parameters is such a consideration. We also gave the process of bonding formation in the previous question and explained it with the help of the Fig. a.Thank you again for your helpful advice.

4. 4. The transition conditions between regions 2 and 3 are unclear. The thickness ratio is the same in both. Then, the other non-zero strain should also be the same. How is it possible that there is sliding between the layer in region 2 but sticking in region 3?

Our reply: We are very grateful for the reviewer’s kind advice. We divide the deformation zone into four regions according to the yielding and bonding state of component layers. At (Boundary point between regions II and III), the extruded metals from component layers contact each other and the diffusion bonding of the extruded metals starts to occur. “The Ti/Al thickness ratio remains unchanged” means Ti/Al thickness ratio remains unchanged after the titanium and aluminum layers are deformed as a single unit. We did not say that the thickness ratio in regions II and III is the same. In fact, the thickness ratio in region II is variable. From the aspect of the yielding and bonding state of component layers, in region II, both titanium and aluminum layers yield, but the covering layer fragments are small or the flow velocity of the extruded metal flowing to the crack is relatively small, the extruded metals from component layers start to contact each other at . In region III, the joining occurs because of the formation of abundant bonding points. The joint formation is the reason of sticking in region III. Because of the different deformation of titanium and aluminum layers, there is sliding between the layers in region II. Thank you again for your helpful advice.

5. 5. The description of anisotropic properties is unsatisfactory. Under the assumptions made in the paper, any anisotropic yield criterion is Sigma1 – Sigma3 = constant (J. Mech. Phys. Solids 21, 63-74 (1973)).

Our reply: We express our sincere gratefulness to the reviewer for this valuable advice. When materials satisfy the von Mises’ yield criterion, . When materials satisfy the modified von Mises’criterion considering anisotropy, here , . For a specific material,  is definite, which can be determined by . This is not inconsistent with Sigma1 – Sigma3 = constant (J. Mech. Phys. Solids 21, 63-74 (1973). Thank you again for your helpful advice.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The manuscript can be accepted in the current form.

Reviewer 2 Report

The authors have addressed all corrections. Now, the manuscript is acceptable.

Reviewer 3 Report

I have not seen any improvement. It is impossible. The theoretical part of the paper has no value. The authors may want to rewrite it as a purely experimental paper.