Dynamical Models of Plasticity with Nonmonotonic Deformation Curves for Nanomaterials

Round 1

Reviewer 1 Report

In the manuscript, constitutive models for dynamic plastic deformation have been proposed phenomenologically.  Various experimental results  have been fitted by the proposed models for verification. There are some points to be addressed

1)If how to achieve the fitting is described briefly, it will be more clear since several of the equations are not so explicit, e.g., to calculate Y. 

2) In Eq.(4), b_3 should be beta? In Eq.(5), why does 2G (G the shear modulus) appear when relating the true stress to strain? Should it be 3G or G?

3) During giving the prediction in Fig.4, negative value of K has been adopted. Is it physically sound? 

4) It is noteworthy that the parameteres involved are not constant even for the same material with different grain size. Under such situations, is it possible to give the dependence of the paramters on the grain size? Or alternatively, what are the advantages of the new models?

Author Response

Reviewer 1

 

Dear Rev.1,

Thank you for opportunity to present a revised version of our manuscript for reconsideration. We are thankful to Rev. 1 for useful comments that allow us to improve our presentation; bellow all of them are cited and answered. All changes are marked out by red in the revised manuscript. The text has been checked by MDPI uses experienced, native English speaking editors.

Truly yours,

Authors

In the manuscript, constitutive models for dynamic plastic deformation have been proposed phenomenologically.  Various experimental results  have been fitted by the proposed models for verification. There are some points to be addressed

1) If how to achieve the fitting is described briefly, it will be more clear since several of the equations are not so explicit, e.g., to calculate Y. 

Thanks for your valuable comment. Model parameters excluded from independent variables in model equations (1)–(5), (7), (8). Added explanatory text with a reference to the work, which schematically shows the evaluation of parameters.

The two parameters of α and τ are determined from the dependence of the yield stress on the strain rate obtained from a set of deformation curves over a wide range of strain rates. In general concepts of any two deformation curves with different strain rates, there are three measurable quantities: the yield stress, the distance between the curves at the hardening stage, and the hardening angle. A unique set of three parameters of α, τ, and β is optimal (minimal) since for any two deformation curves with different strain rates, three reactions of a material to a load are observed: amplitude (characterized by α), rate (characterized by τ) and hardening (characterized by β). Three limiting cases α=1, τ=0 and β=0 for each of the parameters describe three cases of insensitivity to amplitude, velocity and hardening, respectively. Technical aspects of the selection of α, τ, and β parameters for plotting strain diagrams discussed in detail [31].

 

2) In Eq.(4), b_3 should be beta? In Eq.(5), why does 2G (G the shear modulus) appear when relating the true stress to strain? Should it be 3G or G?

Thanks for your valuable comment. Model parameters excluded from independent variables in model equations (1)–(5), (7), (8). G correct in Eqs. (4), (5) and (8).

3) During giving the prediction in Fig.4, negative value of K has been adopted. Is it physically sound? 

Thanks for your valuable comment. Physically negative K corresponds to softening processes in the material (Eq. (6)).

4) It is noteworthy that the parameteres involved are not constant even for the same material with different grain size. Under such situations, is it possible to give the dependence of the paramters on the grain size? Or alternatively, what are the advantages of the new models?

 

Thanks for your valuable comment. Text added.

The characteristic relaxation time parameter is a structural-temporal parameter. On the one hand, like all model parameters, it does not depend on the strain rate. On the other hand, for materials composed of the same chemical substances, but having different structures, the values of the characteristic relaxation times are different. The relationship between the obtained characteristic relaxation times and the relaxation times associated with the dominant plasticity mechanisms can be established from the expressions in Appendix A. Unlike the proposed model, the relaxation times in Appendix A can also be applied only for certain values of strain rates.

Since the proposed model is phenomenological, no matter what mechanism of plasticity occurs in the material, the characteristic relaxation time can be calculated. In other words, if the material is dominated by not one, but two or more plasticity mechanisms, then the determination of the characteristic relaxation times using the model proposed in this work becomes more preferable. This is due to the fact that the characteristic relaxation time is responsible for the rate sensitivity of the material to the load, thereby taking into account the unstable nature of the deformation dependence.

In this paper, we only point out the different order of characteristic relaxation times of nanocrystalline (nanostructured) and nanosized materials. This Despite the variation in the grain size of nanomaterials in Table 2 and Table 3, the characteristic time can be of the order of ns and μs. This complicates the construction of the dependence of the characteristic relaxation times on the grain size, and is more tied to the method of obtaining a nanostructure.

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

The time and space scale effect of plastic deformation is one of the key points to describe the dynamic response of metallic materials. It’s well known that the macro mechanical behavior of metal materials is closely related to microstructure evolution, and thus various at different temporal and spatial scales. How to uniformly describe and understand complex metallic plastic behaviors at different temporal and spatial scales is a long-standing puzzle, which motivates the reviewed research. In this paper, the relaxation model of plasticity for prediction of deformation curves at different strain rates is used. It allows us to study comprehensively the effects of strain hardening in a wide range of deformation conditions for coarse grained materials and nanomaterials. In this paper, the stress evolution of nanomaterials and bulk metal materials at different strain rates is described by relaxation model of plasticity. This demonstrates that the currently developed model can describe the plastic deformation of metallic materials at different temporal and spatial scales. All these points make the paper significant enough to be published in Metals. On the other hand, were is a number of methodological issues to be clarified before such recommendation.

 

1.  This reviewer strongly suggest the authors check and improve the expression of manuscript thoroughly, here report a limited set of examples:

a): The units of stress in Fig. 1 are incorrect.

b): Legend of Fig. 3 is somewhat incomplete. It’s not clear that the symbols of experimental data correspond to which strain rate.

c): The parameters in the model should not be included as independent variables in Equations. As they don't change with physical processes.

For example, Eq. 1 may be written to be more understandable in this form:

,

2.  It is important to clarify the origin of each parameter in the model, is it a known material property or a fit to experimental results? Can all model parameters be determined without directly fitting the experimental results? in other words, how to reflect the predictive ability of the model.

3.  One key point of this paper is the difference in characteristics relaxation times at different spatial scales (nanomaterials and bulk materials). However, the physical significance of this difference is not adequately discussed.

Similarly, the physical mechanisms of plastic strengthening and softening behavior at different strain rates are also not adequately discussed.

Author Response

Reviewer 2

 

Dear Rev.2,

Thank you for opportunity to present a revised version of our manuscript for reconsideration. We are thankful to Rev. 2 for useful comments that allow us to improve our presentation; bellow all of them are cited and answered. All changes are marked out by red in the revised manuscript. The text has been checked by MDPI uses experienced, native English speaking editors.

Truly yours,

Authors

There is a detailed answer to the review in the attached file.

1. This reviewer strongly suggest the authors check and improve the expression of manuscript thoroughly, here report a limited set of examples:

a): The units of stress in Fig. 1 are incorrect.

Thanks for your valuable comment. Error along the y-axis in units corrected for GPa.

b): Legend of Fig. 3 is somewhat incomplete. It’s not clear that the symbols of experimental data correspond to which strain rate.

Thanks for your valuable comment. In Fig. 3, the theoretical strain dependencies have been color-coded depending on the strain rate and the model used. All designations of experimental and theoretical dependencies were added to the legend in Fig. 3.

c): The parameters in the model should not be included as independent variables in Equations. As they don't change with physical processes.

For example, Eq. 1 may be written to be more understandable in this form:

Thanks for your valuable comment. Model parameters excluded from independent variables in model equations (1)–(5), (7), (8).

2. It is important to clarify the origin of each parameter in the model, is it a known material property or a fit to experimental results? Can all model parameters be determined without directly fitting the experimental results? in other words, how to reflect the predictive ability of the model.

Thanks for your valuable comment. Explanatory text with a reference to the work added, which schematically shows the evaluation of parameters.

3. One key point of this paper is the difference in characteristics relaxation times at different spatial scales (nanomaterials and bulk materials). However, the physical significance of this difference is not adequately discussed. Similarly, the physical mechanisms of plastic strengthening and softening behavior at different strain rates are also not adequately discussed.

Thanks for your valuable comment. Text added.

Author Response File: Author Response.pdf

Reviewer 3 Report

The paper deals with the modelling of plastic flow curves of nano-structured materials subjected to quasi-static and high-strain rate loading conditions.

The English form of the paper needs a thorough revision; it is sometimes difficult to understand the meaning of the sentences.

No experiments were conducted, and all the experimental data are borrowed from the literature.

I believe the paper cannot be accepted for publication, not only because of the English form, but mostly because the model is not new, as it appears to be already published in ref [29] with more details, while here the modelling description is insufficient and almost incomprehensible; one must read ref [29] to understand the model. In this paper, only the results of the model on bulk steel and Au-Ag alloy nanowires are presented.

 

Equations (1) and (2) provide only implicit definitions of the models, and I doubt that a reader could replicate the same computations. Most of the parameters appearing in equations (1-4) are not explained. Not even the Y function, which is the most important, is clearly explained. What are L, b*, b1, b2…? So, section 2.1 is not self-contained, not to say incomprehensible.

At the beginning of section 2.2 equations (6) does not match with the following σy0 explanation; in addition, there are no stresses under the integral sign, neither in equation (3) nor in any other equation.

Regarding the results shown in figure 3, the legend should help distinguish the markers used for the quasi-static and dynamic results; moreover, the dynamic experimental data shown in figure 3 appear to be similar but not the same as the DP600 reported in the paper [37]. Consider that these kinds of peaks in Hopkinson bar tests are often considered to be due either to ringing effects or to inaccurate measurements at the beginning of the deformation.

Author Response

Reviewer 3

 

Dear Rev.3,

Thank you for opportunity to present a revised version of our manuscript for reconsideration. We are thankful to Rev. 3 for useful comments that allow us to improve our presentation; bellow all of them are cited and answered. All changes are marked out by red in the revised manuscript. The text has been checked by MDPI uses experienced, native English speaking editors.

Truly yours,

Authors

There is a detailed answer to the review in the attached file.

The English form of the paper needs a thorough revision; it is sometimes difficult to understand the meaning of the sentences.No experiments were conducted, and all the experimental data are borrowed from the literature.

Thanks for your valuable comment. The text has been checked by MDPI uses experienced, native English speaking editors.

 

I believe the paper cannot be accepted for publication, not only because of the English form, but mostly because the model is not new, as it appears to be already published in ref [29] with more details, while here the modelling description is insufficient and almost incomprehensible; one must read ref [29] to understand the model. In this paper, only the results of the model on bulk steel and Au-Ag alloy nanowires are presented.

Thanks for your valuable comment. In this work, the dynamic effects of plastic deformation are predicted not only for bulk steel (Fig. 3) and Au-Ag nanowires (Fig. 4), but also nano-sized gold whisker crystals (Fig. 1), tungsten single crystal pillars (Fig. 2), nanocrystalline tantalum and nanocrystalline Cu-Al (Table 3).

Equations (1) and (2) provide only implicit definitions of the models, and I doubt that a reader could replicate the same computations. Most of the parameters appearing in equations (1-4) are not explained. Not even the Y function, which is the most important, is clearly explained. What are L, b*, b1, b2…? So, section 2.1 is not self-contained, not to say incomprehensible.

Thanks for your valuable comment. Model parameters excluded from independent variables in model equations (1)–(5), (7), (8). Additional notation cannot confuse the reader, since their introduction allows us to make a compact representation of the model.

At the beginning of section 2.2 equations (6) does not match with the following σy0 explanation; in addition, there are no stresses under the integral sign, neither in equation (3) nor in any other equation.

Thanks for your valuable comment. Equation (3) contains an integral, which is denoted by the function Y(t,Σ(t)) (See Eq. (1)). For reference, the notation Y(t,Σ(t))  is used in the equations (1)-(3), (7) explicitly and implicitly through the γ function (4),(5) and (8). Recording bug fixed σya on σy0 in Eq. (6).

Regarding the results shown in figure 3, the legend should help distinguish the markers used for the quasi-static and dynamic results; moreover, the dynamic experimental data shown in figure 3 appear to be similar but not the same as the DP600 reported in the paper [37]. Consider that these kinds of peaks in Hopkinson bar tests are often considered to be due either to ringing effects or to inaccurate measurements at the beginning of the deformation.

Thanks for your valuable comment. In Fig. 3, the theoretical strain dependencies have been color-coded depending on the strain rate and the model used. All designations of experimental and theoretical dependencies were added to the legend in Fig. 3.

In this work, the main emphasis is on the application of the relaxation model of plasticity on nanomaterials shown in Fig. 1, Fig. 2 and Fig. 4. We do not deny that a number of researchers believe that such peaks in Hopkinson rod tests are often considered to be due to ringing effects. However, there is another point of view that this effect is a consequence of the reaction of the material to the applied load. These two points of view are a topic of great discussion, which can be stated in the framework of a new large article, but not within the framework of the study under consideration.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have already addressed the questions raised and I think it is acceptable.

Reviewer 2 Report

The revised manuscript has fully referred to the peer review comments, and the quality of the manuscript has been significantly improved. It is recommended to publish in Metals.

Reviewer 3 Report

The authors have significantly improved the paper and reduced the critical issues