Investigation of Elastic Properties of the Single-Crystal Nickel-Base Superalloy CMSX-4 in the Temperature Interval between Room Temperature and 1300 °C
, Gert Nolze 2, Claudia Leistner 3, Nikolay Petrushin 4 and Igor Svetlov 4Round 1
Reviewer 1 Report
Summary:
The elastic properties of single-crystal nickel-base superalloy CMSX-4 are investigated by the sonic resonance method between room temperature and 1300 °C. It is stressed that high reliability of the results is achieved by using samples with different geometries and crystallographic orientations.
Vibrational measurements have been done on those samples, precise measurement of the material density and thermal expansion where done since they are required for modeling the resonance frequencies by finite element method.
Combining the results measured in this work and literature data the elastic constants of both phases were predicted. It is stressed that obtained data should be available and furnished and may be be used elsewhere.
General Remarks:
This article is sound and no, or minor modifications are necessary.
The article is well written, one finds all the necessary references to necessary articles for its understanding. Thermal expansion and ?’-phase fraction were thoroughly evaluated in the whole temperature range.
My only remarks and concerns refer to the order of the arguments within the presentation and to the description of the analysis, which, even if it should not be complete, should however stress the points important for the Finite Element Analysis to get insight in the elasticity of this 2-phase crystals compared to a (simple) cubic crystal.
Hence I would be happy if the authors could mention and answer the following questions directly in the text (although answers to my questions can certainly be found in the various references given, or even I did not recognized/over-read some explanations):
With regard to the model used as a basis for the (by Finite Element Analysis (FEA) -) calculations, I have following questions:
1.) Pre-strained specimens were used to get the 3 parameters of the stiffness/elastic tensor of the superalloy. Why? Was the rafted (lamellar) structure deliberately chosen? If so, could the (same?) parameters be obtained for a structure without pre-straining (in which the rafts could develop in all <100> directions during heating (and holding) due to existing internal stresses)? Further could the modeled microstructure (dimensions of rafts/channels) be described (equivalent to the SEM micrographs?), and is it needed (or not) as input for the FEA-calculations.
2.) For the calculus (how) do the authors take into account the internal stresses of the sample (coherence stresses between the phases)?
3.) Different specimen shapes were cut. The (non-)influence of the specimen geometry was never discussed, could this be added? What knowledge is obtained from vibration analysis for different shapes/crystallographic orientations? The elastic moduli in the different directions of a “lamella”-sample are different. Does this difference combined with the sample geometries affect the calculated stiffness parameters in any way? The parameters shown in the different figures are mean values of the values calculated from the vibration analysis (obtained for different geometries/crystal directions)? If so, how big are the deviations/uncertainties?
4.) The results of this work were combined with the results reported by Siebörger et al. [16] to evaluate the elastic constants of the matrix over the entire temperature range. From the actual work the resonance measurements performed at super-solvus temperatures were used only? Or has been found an estimate of the elastic constants in the entire temperature range (RT-1300°C) for the matrix using this work exclusively, and the results combined (compared only) to those of Siebörger et al. later?
5.) Has the effect of porosity within the two-phased material on the effective elastic behaviour been included and investigated/evaluated and what is (would be) its influence?
Some more specific comments:
page 3:
line 100 (section 2.1): You should state that you are working on pre strained (and hence rafted) specimens (if I get you right you first mention it in section 2.4)
page 4:
line 131: put figure caption on page 3 (next to figure 1)
line 148: horizontally enlarge figure 2 to fit page width (the same for many figures: figs 2,3,4,7,8)
page 5:
line 177: horizontally enlarge figure 3
line 194: you state with EPMA to have an error of more than 10%, could you please give an estimate of your error using SEM analysis.
page 6:
line 222: horizontally enlarge figure 4
page 7:
line 229: “The values elastic stiffnesses c11,c12,c44 of superalloy CMSX-4 measured at … “
replace “measured” with “determined (equation 2)”
line 229: Are the determined cij the same for all sample geometries or are the following observations and data the mean values of all cij determined for individual samples?
line 231: Please explain more explicitly how the material parameters (elasticity) were obtained! They are a result of the FEA analysis? Please state the elements used (in FEA) and/or cite a publication here.
line 251: elastic modulus and Poisson's ratio: were these determined via the cij (eq. 2) or vice versa? (even if the relations in the cubic system are simple).
246+: To what extent is the definition of the bulk modulus meaningful for an anisotropic substance (even though not expecting the bulk modulus to be deleted from the publication)? If necessary, shouldn't the bulk modulus also be determined via the cij (Reuss, Voigt or other means)?
page 8:
line 258: citation: “Therefore, when ν001 approaches 0.5 the values of B, c11 and c12 either increase, as B and c12, or remain nearly constant as c11.” Please modify this sentence to: “the values of B, and c12 increase, while c11 remains nearly constant.”
line 279: Move table 3 to the back to explain first and better (so bring forward lines 287 to 291 (page 9)) how to get the parameters from this table 3 step by step.
Speculating: (actually as explained in lines 377 to 386 on page 12)
- first fit for the composite
- then fit for the ? phase between 1280℃ and 1300℃ , carry the same out for low temperatures where values of Siebörger et al. of the isolated gamma phase were used, ...
- …
page 10:
line 304+: Is your available data valuable to exclusively work the inverse problem of the RVH-rule of mixture? If so, could you explicitly state this?
line 318: you write “The notation <x> represents the volume average of the quantity x in a representative volume”. Is this also the correct phrase for the “mean-eigenvalues” <λi>, or should one not introduce <λi> at the beginning of page 11 (line 342) as by the phase fraction weighted eigenvalues?
page 11:
line 363: I do not understand how the eigenvalues λieff,1, λieff,2 defined in 361, 362 can be equal (λieff,1= λieff,2 ) if λiR < λiV.
line 370: λiR and λiV depend on the "eigenvalues" of both phases, writing λiR( λip) suggests that the dependence on λim disappears (or the latter is assumed constant). I think your idea is to try to resolve for λip, and assume that the fraction λia/λim is nearly constant in equation 37 allowing you to determine the cijp. I would not write λiR( λip) …
Near line 388 : Did you cross-check the cijm values if determined by the same method as presented for the cijp taking the the latter and the effective ones as known?
page 12:
line 394: horizontally enlarge figure 7
page 13:
line 423: in the conclusion you state that: ”the specimen geometry and its exact crystallographic orientation have to be accurately considered”, however you never discussed this (FEA) thoroughly in section 2 or 3 (Results).
page 14:
line 447: horizontally enlarge figure 8
Further no comments (also none on language)
Author Response
Comments and Suggestions for Authors
Summary:
The elastic properties of single-crystal nickel-base superalloy CMSX-4 are investigated by the sonic resonance method between room temperature and 1300 °C. It is stressed that high reliability of the results is achieved by using samples with different geometries and crystallographic orientations.
Vibrational measurements have been done on those samples, precise measurement of the material density and thermal expansion where done since they are required for modeling the resonance frequencies by finite element method.
Combining the results measured in this work and literature data the elastic constants of both phases were predicted. It is stressed that obtained data should be available and furnished and may be used elsewhere.
General Remarks:
This article is sound and no, or minor modifications are necessary.
The article is well written, one finds all the necessary references to necessary articles for its understanding. Thermal expansion and ?’-phase fraction were thoroughly evaluated in the whole temperature range.
My only remarks and concerns refer to the order of the arguments within the presentation and to the description of the analysis, which, even if it should not be complete, should however stress the points important for the Finite Element Analysis to get insight in the elasticity of this 2-phase crystals compared to a (simple) cubic crystal.
Hence I would be happy if the authors could mention and answer the following questions directly in the text (although answers to my questions can certainly be found in the various references given, or even I did not recognized/over-read some explanations):
With regard to the model used as a basis for the (by Finite Element Analysis (FEA) -) calculations, I have following questions:
1.) Pre-strained specimens were used to get the 3 parameters of the stiffness/elastic tensor of the superalloy. Why? Was the rafted (lamellar) structure deliberately chosen? If so, could the (same?) parameters be obtained for a structure without pre-straining (in which the rafts could develop in all <100> directions during heating (and holding) due to existing internal stresses)? Further could the modeled microstructure (dimensions of rafts/channels) be described (equivalent to the SEM micrographs?), and is it needed (or not) as input for the FEA-calculations.
Answer: The pre-rafted specimens were used only to measure the temperature dependence of ?’-volume fraction. For measurement of the elastic constants the undeformed fully heat treated specimens with usual cuboidal ?/?’-microstructure were used. To avoid this misunderstanding at the beginning of Chapter 2.2 we changed the phrase “The elastic constants of heat treated CMSX-4 have been determined” which is marked by yellow.
2.) For the calculus (how) do the authors take into account the internal stresses of the sample (coherence stresses between the phases)?
Answer: Internal stresses, like coherency stresses have no influence on the effective elastic constants on the compound. This is a consequence of the principle of superposition in linear elasticity. The total stress field being the sum of the coherency stress and the external stress fields . The coherency stress field remains unaffected by an external load and plays no role (see e.g. T. Mura, Micromechanics of defects in solids, Springer Netherlands, 1982). The text has been supplemented in accordance.
3.) Different specimen shapes were cut. The (non-)influence of the specimen geometry was never discussed, could this be added? What knowledge is obtained from vibration analysis for different shapes/crystallographic orientations? The elastic moduli in the different directions of a “lamella”-sample are different. Does this difference combined with the sample geometries affect the calculated stiffness parameters in any way? The parameters shown in the different figures are mean values of the values calculated from the vibration analysis (obtained for different geometries/crystal directions)? If so, how big are the deviations/uncertainties?
Answer: We are very sorry here. Due to the corona-pandemic the communication within our large group was not perfect which resulted in this confusion. It was initially planned to use both cylindrical rods and rectangular, plate-like beams but finally only the plate-like specimens have been used, which is sufficient for a unique determination of the constants. So, all statements about different geometries and cylindrical samples are removed. The obtained elastic constants take into account the resonance measurements from 3 rectangular specimens. There is thus only one set of constants for CMSX-4.
4.) The results of this work were combined with the results reported by Siebörger et al. [16] to evaluate the elastic constants of the matrix over the entire temperature range. From the actual work the resonance measurements performed at super-solvus temperatures were used only? Or has been found an estimate of the elastic constants in the entire temperature range (RT-1300°C) for the matrix using this work exclusively, and the results combined (compared only) to those of Siebörger et al. later?
Answer: From the actual work only the resonance measurements performed at super-solvus temperatures were used.
5.) Has the effect of porosity within the two-phased material on the effective elastic behaviour been included and investigated/evaluated and what is (would be) its influence?
Answer: The volume fraction of microporosity in our samples is low, of about 0.2 vol.%. So, we do not expect a significant effect of such a microporosity on the elastic properties.
Some more specific comments:
page 3: line 100 (section 2.1): You should state that you are working on pre strained (and hence rafted) specimens (if I get you right you first mention it in section 2.4)
Answer: As mentioned above: The pre-rafted specimens were used only to measure the temperature dependence of ?’-volume fraction. For measurement of the elastic constants the undeformed fully heat treated specimens with usual cuboidal ?/?’-microstructure were used.
page 4: line 131: put figure caption on page 3 (next to figure 1)
Answer: The caption is moved as requested.
line 148: horizontally enlarge figure 2 to fit page width (the same for many figures: figs 2,3,4,7,8)
Answer: All these images are now horizontally enlarged.
page 5: line 177: horizontally enlarge figure 3
Answer: This image is now horizontally enlarged.
line 194: you state with EPMA to have an error of more than 10%, could you please give an estimate of your error using SEM analysis.
Answer: For 12 SEM images taken from every specimen the calculated standard error varied within 1-2 vol.% depending on temperature. This value is now given in the text.
page 6: line 222: horizontally enlarge figure 4
Answer: This figure is now horizontally enlarged.
page 7: line 229: “The values elastic stiffnesses c11,c12,c44 of superalloy CMSX-4 measured at … “
replace “measured” with “determined (equation 2)”
Answer: The recommended change is done.
line 229: Are the determined cij the same for all sample geometries or are the following observations and data the mean values of all cij determined for individual samples?
Answer: As was mentioned above: Only the plate-like beam specimens were actually investigated.
As mentioned above: The obtained elastic constants take into account the resonance measurements from 3 rectangular specimens. There is thus only one set of elastic constants for CMSX-4.
line 231: Please explain more explicitly how the material parameters (elasticity) were obtained! They are a result of the FEA analysis? Please state the elements used (in FEA) and/or cite a publication here.
Answer: Additional and detailed explanations have been included in the text.
line 251: elastic modulus and Poisson's ratio: were these determined via the cij (eq. 2) or vice versa? (even if the relations in the cubic system are simple).
Answer: Elastic modulus and Poisson's ratio: were these determined via the cij.
246+: To what extent is the definition of the bulk modulus meaningful for an anisotropic substance (even though not expecting the bulk modulus to be deleted from the publication)? If necessary, shouldn't the bulk modulus also be determined via the cij (Reuss, Voigt or other means)?
Answer: A point of interest of our work was how the elastic properties of superalloy single-crystal change with increasing temperature close to the point. Therefore, a temperature change of the bulk modulus B has been discussed too. Its value has been determined via the cij but formula (5), as well as formulas (6, 7), are used only for the interpretation of temperature dependencies of B, c11 and c12.
Furthermore, structural analysis of components as well as microstructure simulations by phase field require the full stiffness tensor and thus also the bulk modulus.
page 8: line 258: citation: “Therefore, when ν001 approaches 0.5 the values of B, c11 and c12 either increase, as B and c12, or remain nearly constant as c11.” Please modify this sentence to: “the values of B, and c12 increase, while c11 remains nearly constant.”
Answer: This sentence is modified as recommended.
line 279: Move table 3 to the back to explain first and better (so bring forward lines 287 to 291 (page 9)) how to get the parameters from this table 3 step by step.
Speculating: (actually as explained in lines 377 to 386 on page 12)
- first fit for the composite
- then fit for the ? phase between 1280℃ and 1300℃ , carry the same out for low temperatures where values of Siebörger et al. of the isolated gamma phase were used, ...
Answer: To improve the manuscript structure Table 3 (composite+matrix) was split into Table 3 (composite) and Table 4 (matrix), and Fig. 6 (composite+matrix) into Fig. 6 (composite) and Fig. 7 (matrix). Table 4 and Fig. 7 were moved back to the corresponding text (about matrix).
page 10: ine 304+: Is your available data valuable to exclusively work the inverse problem of the RVH-rule of mixture? If so, could you explicitly state this?
Answer: Sorry, we do not understand this question. If you mean a statement about the solution of the inverse problem of the RVH-rule of mixture, it is stated at the beginning of Chapter 3.2, marked by yellow.
line 318: you write “The notation <x> represents the volume average of the quantity x in a representative volume”. Is this also the correct phrase for the “mean-eigenvalues” <λi>, or should one not introduce <λi> at the beginning of page 11 (line 342) as by the phase fraction weighted eigenvalues?
Answer: The notation <x> conserves the same meaning throughout the whole paper. So, for example <λi> is the volume average of the eigenvalues λi in a representative volume. The text has been amended to make this clear.
page 11: line 363: I do not understand how the eigenvalues λieff,1, λieff,2 defined in 361, 362 can be equal (λieff,1= λieff,2 ) if λiR < λiV.
Answer: You are right, it should be an inequality symbol ≠. Sorry, it is a typing mistake. Now it is corrected.
line 370: λiR and λiV depend on the "eigenvalues" of both phases, writing λiR( λip) suggests that the dependence on λim disappears (or the latter is assumed constant). I think your idea is to try to resolve for λip, and assume that the fraction λia/λim is nearly constant in equation 37 allowing you to determine the cijp. I would not write λiR( λip) …
Answer: This is correct. λiR and λiV depend on both λip,and λim . The abbreviated notation λiR( λip) was intended to highlight the fact that the unknowns are λip . The full notations have been now included in the text.
Near line 388 : Did you cross-check the cijm values if determined by the same method as presented for the cijp taking the the latter and the effective ones as known?
Answer: Yes, this has been checked.
page 12: line 394: horizontally enlarge figure 7
This figure is now horizontally enlarged.
page 13: line 423: in the conclusion you state that: ”the specimen geometry and its exact crystallographic orientation have to be accurately considered”, however you never discussed this (FEA) thoroughly in section 2 or 3 (Results).
Answer: As was mentioned above only the plate-like beam specimens were actually investigated.
So this question is not relevant. The influence of deviations of the specimen orientations has been analyzed in a separate investigation. A sentence summarizing the results has been added in the text.
page 14: line 447: horizontally enlarge figure 8
Figure 8 (now 9) is horizontally enlarged.
Further no comments (also none on language)
Reviewer 2 Report
The article presented the elastic properties of single crystal nickel-base superalloy CMSX-4 over a wide range of temperature. The authors investigated the alloy’s elastic constants very carefully and systematically. However, despite their efforts, the same evaluation in elastic properties of the same alloy has already been studied in the previous paper [D. Siebörger, H. Knake, U.Glatzel, Temperature dependence of the elastic moduli of the nickel-base superalloy CMSX-4 and its isolated phases, Mater. Sci. Eng. A 298 (2001) 26-33.]. That is, the present work has a serious problem in its novelty and the reviewer could not recommend this article to be published.
Author Response
Reviewer 2
Comments and Suggestions for Authors
The article presented the elastic properties of single crystal nickel-base superalloy CMSX-4 over a wide range of temperature. The authors investigated the alloy’s elastic constants very carefully and systematically. However, despite their efforts, the same evaluation in elastic properties of the same alloy has already been studied in the previous paper [D. Siebörger, H. Knake, U.Glatzel, Temperature dependence of the elastic moduli of the nickel-base superalloy CMSX-4 and its isolated phases, Mater. Sci. Eng. A 298 (2001) 26-33.]. That is, the present work has a serious problem in its novelty and the reviewer could not recommend this article to be published.
Answer to Reviewer
The reviewer writes that “the same evaluation in elastic properties of the same alloy has already been studied in the previous paper of Siebörger et al.” and for this reason “he could not recommend this article to be published”.
So, to answer to this statement one needs to compare the results of the submitted paper with the results of the paper of Siebörger et al.
- In the paper of Siebörger et al. the elastic constants of superalloy CMSX-4 were measured in the temperature range between the room temperature and 1000 °C (see Figs 7, 8 and 9 of this paper). This maximal temperature of 1000°C is well below the maximal service temperature of single-crystal turbine blades which can be up to 1150 °C. So, for modeling the mechanical behavior of the blade material under service conditions in the full temperature range the results of this paper are not sufficient.
In the submitted paper the elastic constants of superalloy were investigated in the temperature range between the room temperature and 1300 °C. “This wide temperature interval covers all areas where the elastic constants of CMSX-4 are needed for component analysis, namely: service conditions of the blade material, technical accidents of gas turbines (overheating) as well as manufacturing of turbine blades (hot isostatic pressing)”. - as mentioned in the abstract, introduction and conclusions.
Actually we started this work because in our international project we model hot isostatic pressing of CMSX-4 at 1288 °C, but this needs the elastic constants of CMSX-4 at this temperature which were not available up to now.
- In the paper of Siebörger et al., the elastic constants of the samples compositionally close to the matrix- and gamma’-phases of superalloy CMSX-4 were measured, but only in the temperature interval below 800 °C (see Figs 7, 8 and 9 of this paper), which is not relevant for practice. To the best of our knowledge, there are no available results in the open literature for the constants of the matrix- and gamma’-phases at temperatures higher than 800 °C. However, these data are crucially needed for simulations of microstructure evolution by the now well-established phase field method. This concerns for example simulations of rafting that only occurs at higher temperatures over 800°C as well as simulation of heat treatment, which is also performed at temperatures much higher than 800°C. Therefore, there is a need for estimates of the elastic properties of the matrix- and gamma’-phases at higher temperatures than 800 °C. The aim of this paper is just to make these constants available for anyone.
Reviewer 3 Report
The work has its merit and worth being published, although the method is not really robust/deterministic, and relies on many estimations and also FE analyses. A few points are given below for further clarity/improvement.
- Optimisation Eq (2) is used to determine c11, c22 and c44 So how Poison ratio v[001] was identified? Please comment on the accuracy of the results as it hugely relies on the FE analysis.
- The property of gamma phase is taken form literature for temperature below the solvus temperature. So, the method is not really robust. Again, please comment the accuracy of the results of gamma matrix for the whole range of temperature.
- Also comment on the accuracy of the results of gamma-prime phase for the whole range of temperature, as many estimations were involved in the process such as precipitate volume fraction, results of gamma phase as well as the RVH rule etc.
Author Response
Reviewer 3
Comments and Suggestions for Authors
The work has its merit and worth being published, although the method is not really robust/deterministic, and relies on many estimations and also FE analyses. A few points are given below for further clarity/improvement.
- Optimisation Eq (2) is used to determine c11, c22 and c44 So how Poison ratio v[001] was identified? Please comment on the accuracy of the results as it hugely relies on the FE analysis.
- The property of gamma phase is taken form literature for temperature below the solvus temperature. So, the method is not really robust. Again, please comment the accuracy of the results of gamma matrix for the whole range of temperature.
- Also comment on the accuracy of the results of gamma-prime phase for the whole range of temperature, as many estimations were involved in the process such as precipitate volume fraction, results of gamma phase as well as the RVH rule etc.
Answer to reviewer 3.
- Concerning the first part of this question: by minimizing the least squares sum in Eq (2) first the elastic stiffnesses c11, c22 and c44 were determined, and then using the c11, c22 and c44 values, the other elastic moduli were calculated, including Poisson’s ratio.
It is very difficult to estimate the accuracy of the determined constants for the CMSX-4 since the final results depend on many factors, including the accuracy and the correct identification of the resonance peaks, the accuracy of the testing temperature, the accuracy of the input parameters in the model (material density, specimen geometry, orientation of the specimens) and the accuracy of the FE analysis itself. A full analysis of the uncertainty is thus beyond the scope of this paper. However, efforts were already made in this sense, which are not completed up to now. The current investigation state can be summarized below:
- A mesh refinement analysis has been conducted. It was found that the maximal relative difference in terms of eigenfrequencies between the used mesh in the paper and a reference mesh (four times refined, i.e. element size divided by 4) is lower than 5x10-5 up to the 10th mode.
- The influence of geometrical imperfections has been investigated. For this purpose, the parameter determination has been repeated with imperfect specimens, that is, such the thickness and the width linearly vary by 2/100 (corresponding to maximal deviations of the dimensions by 1/100). It was found that such perturbations induce a relative error between 3x10-4 for c44 and 2x10-3 for c12.
- The influence of errors concerning the specimen orientations has been analyzed by applying a perturbation to the inverse problem, i.e. an additional rotation of 2° around a random axis to the ideal orientations. It was found that such perturbations induce an average relative error between 5x10-3 for c44 and 4x10-2 for c12.
- The elastic constants of alloy CMSX-4 were measured and the accuracy of these measurements was evaluated, as mentioned above. The elastic constants of gamma-matrix were partially measured (at 20-800 °C by Siebörger et al. and at 1280-300 °C in our work) and partially predicted, that is interpolated between 800 and 1280 °. The elastic constants of gamma’-phase were predicted in the entire temperature range. The reliability of this prediction is supported by the obtained values of the elastic misfit (m) which fit with literature data, see Fig. 9 (before it was Fig. 8). The direct experimental verification of predicted results is problematic because it needs the single-crystals of alloys with the compositions of gamma- and gamma’-phases at high temperatures. However, our experience shows that alloys cast with such compositions again decompose into two phases: gamma and gamma’.
- The use of the RVH averaging rule has been shown in the past to deliver accurate results when the difference between the constants of the different phases is not too large, e.g. see [31]. In fact, the paper presents an improvement of the initial method due to the use of a spectral decomposition of the stiffness tensor. The advantage of the presented method is that it is amenable to a relatively simple analytical solution. A systematic investigation of the difference between this scheme and more advanced schemes, like periodic homogenization would require a large effort. Such an investigation is planned, and the results will published in the future. Concerning the accuracy of the gamma’-fraction measurement, it is within 1-2 vol.%, as now given in the revised manuscript.
