Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material

Round 1

Reviewer 1 Report

Please see the attachment

Comments for author File: Comments.pdf

Author Response

Author’s Reply to Reviewer 1

Firstly, we are very grateful to reviewer for the comments. The questions raised by the reviewer are answered one by one below (the revised text in the updated version of the manuscript shall be marked in red):

 

Question 1:

Figure 4. Contours and statistical distribution of the longitudinal strain and stress at tension peak for strain amplitude 205 0.01003: (a) and (b) for longitudinal strain at 3rd and 840th cycle, respectively; (c) and (d) for longitudinal stress at 3rd 206 and 840th cycle, respectively I have two problems with their method Using strain or stress alone as a FIP measure violates the second law of thermodynamics. According to the second law of thermodynamic only entropy can be a degradation measure.

Reply:

Figure 4 shows the distribution changes of normal strain and normal stress in the direction of macro tensile axis in RVE corresponding to the tensile peak. Only the results of the 3rd and 840th cycles under the strain amplitude of 0.01003 are given. Through Figure 4 we want to explain that:

(1) When considering the polycrystalline structure of the material, due to the different plastic slip in each grain, the inhomogeneous deformation will make the difference of the internal strain distribution of the material increase obviously.

(2) The difference of the stress distribution changes is very small, which mainly determined by Hooke's law.

(3) The change of the strain distribution indirectly reflects the change and evolution of the microstructure and deformation state of the material.

In this paper, the standard deviation of strain distribution is taken as FIP instead of the strain itself. The standard deviation of strain distribution increases with the cycle, which reflects the change of material micro geometry, that is, the change of material microstructure. The calculation results show that both the standard deviation and entropy of strain distribution increase irreversibly with the cycle under the same conditions (there is a functional relationship between the both), so it does not violate the second law of thermodynamics.

To avoid misunderstanding, we mentioned this in the new version (275-277).

 

Question 2:

They use Shannon entropy of strain field for fatigue life prediction. Actually, using entropy for fatigue is not a new topic but I don’t see it in their references. I assume they did not do a strong literature survey.

 

Reply:

Thanks to the reviewer's suggestions!

In the introduction of the updated manuscript (55-59), we add some references ([21-26]) to the study of entropy and fatigue damage, including the discussion of using different definitions of entropy to describe damage.

 

Author Response File: Author Response.docx

Reviewer 2 Report

The paper describes a computational approach to predict fatigue life of a metal alloy based on the inhomogeneous strain distribution in the microstructure. Unfortunately there exists too many English language errors combined with long sentences that make the paper very difficult to follow. Please consult a language expert for revision.

In the scientific approach there are certain open points which make the approach difficult to judge. One of the most important is the reproducibility and the dependency of the strain (and stress) distributions is the RVE. Only one RVE is chosen and all calculations are done using that. What happens when the grain orientations/number of grains/size of grains change. The model is of course size independent but that also decreases the reliability since grain size distribution would affect the strain inhomogeneity.

The validation part of the model is not conclusive. In order to validate use either a different alloy or a different RVE by excluding the validation test from the fitting.

1. There have been recent studies of void growth based on crystal plasticity and especially the results show that the GNDs have an effect on the growth rate. As the GNDs are a result of inhomogeneous slip activity it would be nice to see in the introduction references to those studies.
2. The novelty of the study is not very clear. It appears that Zhang et.al. [22] has done a similar study. What exactly is the improvement should  be made clear. Is it only 'tracking the entire strain cycles'?
3. It should be mentioned how the Boundary Conditions are applied. Displacement vs Force, Corners vs Edges etc. It is however clear that they are not periodic. Periodic BCs are known to give superior results compared to regular BCs. The choice of this type of BCs and their expected effect on the results should be discussed.
4. Eq 10 gives a wrong impression of the stress-update routine. In order to update the new stress the Jaumann rate must be used to calculate the change of stress in time and then added to the previous stress. The implementation in this is correct since ABAQUS does that calculation for UMAT but the equation must be corrected.
5. There are 3 elastic constants (cubic) which are impossible to completely and uniquely determine using uniaxial tests. It is mentioned however it is mentioned that they are determined exactly like that (line 180-182). 
6. Please correct the language in Eq 13 and explain it in more detail.
7. The method described in section 3.3.2 is not clear at all. This is partly due to errors in language and partly due to the method of description. Please rewrite clearly how figures 7a and 7b are obtained.
8. The validation part (section 3.3.3) is not very clear in the sense that what is the data used and was that used also in fitting? 
9. It is not also clear from the conclusions what the novelty of the paper is. How do the results compare with Zhang et.al. [22]?

Textual and style errors:

• Mathematical fonts in text (and vice versa) should be adjusted.
• Typo line 194
• Style error lines 232-237, 250-252
• ... Please check for more.

Author Response

 

Author’s Reply to Reviewer 2

 

Firstly, we are very grateful to reviewer for the constructive comments! They are of great help to improve the quality of this paper and the future research work. The comments raised by the reviewer are answered one by one below (the revised text in the updated version of the manuscript shall be marked in red):

Comment:

The paper describes a computational approach to predict fatigue life of a metal alloy based on the inhomogeneous strain distribution in the microstructure. Unfortunately there exists too many English language errors combined with long sentences that make the paper very difficult to follow. Please consult a language expert for revision.

Reply:

There are some deficiencies in English of this manuscript, indeed. In the course of revision, we carefully checked and revised the full text, trying to use simple sentence patterns. Thank the reviewer for pointing this out.

Comment:

In the scientific approach there are certain open points which make the approach difficult to judge. One of the most important is the reproducibility and the dependency of the strain (and stress) distributions is the RVE. Only one RVE is chosen and all calculations are done using that. What happens when the grain orientations/number of grains/size of grains change. The model is of course size independent but that also decreases the reliability since grain size distribution would affect the strain inhomogeneity.

Reply:

The questions raised by the reviewer are very important.

One random RVE was used in the study. The correlation between the results and RVE and whether it can be reproduced need to provide evidence. In reference [31] (new version), the RVEs with the same number of grains and units of 8,000, 27,000 and 64,000 were calculated. The results show that although the local stress and strain in RVE vary with the element size, the statistical mean value remains unchanged, and the statistical standard deviation only increases slightly with the decrease of element size.

In order to further discuss this problem, we added an example change the loading direction of RVE (from axis-3 to axis-1) in this paper (430-444). Although the number of grains and the size of the finite element remain unchanged, the grain configuration and orientation distribution of the model are greatly changed due to the crystal anisotropy. The results of calculation are shown in Figure 14. Standard deviation of statistics changes slightly in value, but it can be seen that the curve changes very small. It shows that the change of model polycrystalline configuration and orientation distribution will have no noticeable influence on the statistical results and fatigue judgment.

Above results show that the reproducibility of RVE statistical results can be guaranteed when the unit size and grain size are unchanged or little changed compared with RVE size, and the RVE statistical results are not sensitive to the grain orientation and division in RVE.

As for the influence of size, it involves more and more profound aspects of the mechanical behavior of materials, which will not be discussed in this paper. We mentioned this in manuscript, see Section 2.2.

It should be pointed out that these discussions are focused on the case of equiaxed grains, excluding the case of a large difference in the slenderness ratio of grains.

Comment:

The validation part of the model is not conclusive. In order to validate use either a different alloy or a different RVE by excluding the validation test from the fitting.

Reply:

(1) In the validation part of the model, each prediction curve is based on the critical value of FIP determined by the cyclic test of one strain amplitude. That is, from one point predicts other points, and the prediction curve is not a fitting curve.

(2) In addition, we add an example to load RVE in another direction. Considering the random structure of RVE and the anisotropy of grain, the reversed loading is equivalent to changing an RVE. The results are shown in Figure 14. Considering the page space, only one examples is added in the manuscript.

                                       

The questions raised by the reviewer are answered one by one below:

1. There have been recent studies of void growth based on crystal plasticity and especially the results show that the GNDs have an effect on the growth rate. As the GNDs are a result of inhomogeneous slip activity it would be nice to see in the introduction references to those studies.

Reply:

In the introduction, we added literature introductions (51-54, [18,19]) related to studies of void growth based on crystal plasticity and GNDs effect on the void growth.

 

1. The novelty of the study is not very clear. It appears that Zhang et.al. [22] has done a similar study. What exactly is the improvement should be made clear. Is it only 'tracking the entire strain cycles'?

 

Reply:

"Tracking the whole strain cycle" is just one of the characteristics of this method. The deformation of materials is usually described by strain tensor, and the problem of how to select the strain variable is involved in the discussion of non-uniform deformation. In reference [30] (new version), only the normal axial tensile strain and the first principal strain are discussed. If we want to further study the complex stress state or multiaxial loading fatigue problem, we are confronted with the problem of parameter selection. In this paper, it is found that if the standard deviation or continuous entropy is used as the index parameter, the variation laws of any component of strain tensor, the first principal strain, the maximum principal shear strain and the equivalent strain with the number of cycles are similar. This means that these parameters are almost unaffected by the choice of coordinating system.

Thanks to the reviewers for pointing out this problem. The author revised the state of this study in the introduction.

 

1. It should be mentioned how the Boundary Conditions are applied. Displacement vs Force, Corners vs Edges etc. It is however clear that they are not periodic. Periodic BCs are known to give superior results compared to regular BCs. The choice of this type of BCs and their expected effect on the results should be discussed.

 

Reply:

Thank the reviewer for the suggestions. We have revised them according to the suggestions (132-151, new version). The detailed description of boundary conditions is added, and the statistical standard deviation of RVE calculated according to periodic boundary conditions under the cyclic strain amplitude of 0.01003 is given. Further explanation is given according to the comparison of results (445-452).

 

1. Eq 10 gives a wrong impression of the stress-update routine. In order to update the new stress the Jaumann rate must be used to calculate the change of stress in time and then added to the previous stress. The implementation in this is correct since ABAQUS does that calculation for UMAT but the equation must be corrected.

 

Reply:

It has been modified (see 194, 195 and Equation 12, new version) according to the reviewer's suggestion to avoid misunderstanding.

 

1. There are 3 elastic constants (cubic) which are impossible to completely and uniquely determine using uniaxial tests. It is mentioned however it is mentioned that they are determined exactly like that (line 180-182). 

 

Reply:

The elastic constant is calibrated by referring to the data of 650℃ of the same material in reference [31] (new version) and the test data in reference [33] (new version). The relevant text has been revised to avoid misunderstanding (204, 205).

 

1. Please correct the language in Eq 13 and explain it in more detail.

 

Reply:

Thank you for your comments. We have carefully checked and revised the formula expression and explanation to avoid misunderstanding (see 266-271, and Equation 15, new version).

 

1. The method described in section 3.3.2 is not clear at all. This is partly due to errors in language and partly due to the method of description. Please rewrite clearly how figures 7a and 7b are obtained.

 

Reply:

Thanks for the comments of the reviewer. For this section, we have revised the original expression, and the new expression has explained the key link of fatigue curve prediction as much as possible. The explanatory text of Figure 7 has been rewritten. I hope the readability of the figure is better (see new version, 320-370).

 

1. The validation part (section 3.3.3) is not very clear in the sense that what is the data used and was that used also in fitting? 

 

Reply:

Data fitting is not used in fatigue life prediction.

Thank you for your comments. We have revised the text in Section 3.3.3 and reinterpreted Figure 13 to avoid misunderstanding. (407-417)

 

1. It is not also clear from the conclusions what the novelty of the paper is. How do the results compare with Zhang et.al. [22]?

 

Reply:

Compared with reference [30] (new version), this paper makes a more in-depth study: is the deformation nonuniformity obtained by different strain variables in the cyclic process similar? Can they be used to measure the degree of fatigue damage and predict the fatigue life curve? Is the forecast reasonable? The purpose of these studies is to lay a foundation for further study of complex loading fatigue. It found out that

(1) The variation of the standard deviation of each strain is similar, which means that any strain component can be used. The selected parameters are almost unrelated to the coordinate system, which may bring convenience for further analysis of torsion or tension torsion combined loading.

(2) The variation of the standard deviation of the equivalent strain, the first principal strain and the maximum principal shear strain is very similar, which has not been discussed in previous studies.

(3) It is confirmed that different strain variables can be used as FIP to give a more reasonable fatigue life prediction. This has not been studied and discussed before.

Textual and style errors:

• Mathematical fonts in text (and vice versa) should be adjusted.

Reply:

We made font adjustments

• Style error lines 232-237, 250-252

Reply:

We changed the font style. We have checked and revised the full text.

• ... Please check for more.

Reply:

We have checked and revised the full text.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Authors included my comments 

Reviewer 2 Report

Most of the previous comments of the reviewer are applied. The only minor issue is that there are still small typos. Once final English check is recommended.