An Over-Deterministic Method for Mode III SIF Calculation Using Full-Field Experimental Displacement Fields

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript applsci-3457250-v1 (“An Over Deterministic Method for Mode III SIF Calculation Using Full Field Experimental Displacement Fields”) must be reviewed by authors in accordance with the followings comments:

 

i) The “Abstract” must clarify the main contribution of the paper including the present state of the art in which the research will be inserted in the specialized literature.

ii) The list of “Nomenclature” is incomplete, so it must be reviewed by authors.

iii) It is not recommended the inclusion of “Figures” and “Equations” in “Introduction”. All they must be removed from “Section 1” to “Section 2”. In general, the “Section 1” a) must state the motivation for writing the paper and what it will contribute in the specialized literature; b) must include a brief overview of the present state concerning aspects of the adopted methodology; c) must include some more detailed information on the specific topic of the research; and d) must also include a precise description of the exact question or hypothesis that the paper will address in addition with a brief summary of the adopted methodology to solve the investigated problem.

iv) Please, include a “Table” summarising date and measurements uncertainty.

v) Figure 6 must be better introduced and explained.

vi) What are the drawbacks of the present research methodology? And what are the ways to resolve them? More details should be provided by authors.

vii) In “Conclusion”, please, include perspectives for a future research concerning the topic discussed in the present manuscript. The main contribution of the present research must be clarified and justified by authors; which will justify the intended publication in AS.

Comments on the Quality of English Language

English language and style are fine/minor spell check required.

Author Response

We would like to thank Reviewer #1 for the detailed reading and constructive comments. Your insights have made us think of ways to improve the paper, resulting in a much clearer manuscript. Here is a detailed response for the observations raised. We hope the response meets your criteria.

 

Reviewer 1

1. The “Abstract” must clarify the main contribution of the paper including the present state of the art in which the research will be inserted in the specialized literature.

R: Contribution was explicitly stated in the abstract

The main contribution is the method for calculating SIF mode III from full-field experimental displacement data. The approach takes advantage of the over-deterministic Least Squares Method (LSM) to fit displacement data to Williams' series, specifically addressing the calculation of SIFs in scenarios where traditional methods, such as COD, may be inadequate.

Additionally, the paper compares results obtained from the proposed LSM method with those derived from conventional COD equations. Thus, it demonstrates that the LSM method can provide less noisy results and effectively addresses nonlinear variations in SIF calculations across different loading conditions, which ultimately enhances the understanding of crack behavior in presence of mode III.

 

2. The list of “Nomenclature” is incomplete, so it must be reviewed by authors.

R: Thanks for noticing that. Nomenclature was amended.

3. It is not recommended the inclusion of “Figures” and “Equations” in “Introduction”. All they must be removed from “Section 1” to “Section 2”. In general, the “Section 1” a) must state the motivation for writing the paper and what it will contribute in the specialized literature; b) must include a brief overview of the present state concerning aspects of the adopted methodology; c) must include some more detailed information on the specific topic of the research; and d) must also include a precise description of the exact question or hypothesis that the paper will address in addition with a brief summary of the adopted methodology to solve the investigated problem.

R: Introduction was rewritten, and such information was moved to a new section 2.

A justification and brief explanation of current methods and their inconveniences were explicitly given in section 1.

4. iv) Please, include a “Table” summarising date and measurements uncertainty.

R: We believe that date of experiments is irrelevant as the processed data is not time-specific dependent. For uncertainty, along with other experimental details, we would like to refer the reviewer to Refs [7] and [40] in the revised version. We added the camera resolution in L216:

… using two Allied Vision Tech. GT 2450 cameras with 2448x2050 resolution, giving a 3.45 μm x 3.45 μm pixel size.

5. v) Figure 6 must be better introduced and explained.

R: Thanks for pointing that out.

Figure 6(a) shows the strong nonlinearity between mode I loading and unloading paths. One differential aspect of the load is the full load inversion for axial force and torque. As the respected reviewer points out, one might need to crosscheck the results with the load to avoid incorrectly interpreting it as a negative K_I value instead of a a ΔK_I.

6. vi) What are the drawbacks of the present research methodology? And what are the ways to resolve them? More details should be provided by authors.

Thanks for pointing that out. We added the following text in the introduction.

L59. From data acquired through experimental methods, clip gauges are known for reliable measures and low cost, but in the case of kinked cracks, the displacements given by the COD gauge may not adequately describe the stress field on the crack tip. However, at the time of writing this paper, there are no clip gauges that can simultaneously measure the crack's opening and sliding displacements. One would require clip gauges plus a second technique, like strain gauge or extensometers. Nevertheless, one would require separate setups for opening and sliding. The digital image correlation (DIC) technique provides independent full-field displacement measurements in 2D or 3D with the use of two or more cameras. Even with a larger budget, it has become standard equipment in many laboratories. Therefore, such kinds of readings are readily available. So, one can use such experimental measurements and fit them to a stress function. … … Other methods used are J integral decomposition, but one needs to calculate stresses too and need a method to decompose the integral as it gives the total energy used to create a new surface crack regardless of direction. We also pointed out the availability of DIC so independent displacement measurements are more common and one needs a method that can take advantage of the full field.

 

Finally, we pointed out that the best of our knowledge is that there are no clip gauges that can measure at the same time crack opening and sliding to use the COD formulation.

 

In the discussion´s paragraph, starting at L440, we pointed out the drawbacks.

The method relies on knowing the CTL as it places the coordinate system on the CTL. Stresses and deformations must be sufficiently linear to comply with the linear elastic assumption of the Westergaard stress function. The CTL is current problem of investigation even though there are standardized methods, such as Current Drop, but that is out of the scope. For the linear part, one must check the stress field and the plastic zone to discard the near region.

7. vii) In “Conclusion”, please, include perspectives for a future research concerning the topic discussed in the present manuscript. The main contribution of the present research must be clarified and justified by authors; which will justify the intended publication in AS.

The conclusion specifically highlights now the following contributions and areas of future research:

SIF calculation is more robust when using an area comprised of many data points, instead of two points.

A solution to approximate Williams’ solution for curved surfaces was proposed.

The authors identify that future research could focus in optimizing the area of interest (AOI) selection could mitigate the impact of accumulated plasticity and enhance the accuracy of SIF calculations.

 

Comments on the Quality of English Language

English language and style are fine/minor spell check required.

R; English Language was thoroughly checked

Once again, we thank the reviewer for the constructive criticism. We appreciate your keen eye and sharp questioning that have improved the manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

We would like to thank Reviewer #2 for the detailed reading and constructive comments. Your insights have made us think of ways to improve the paper, resulting in a much clearer manuscript. Here is a detailed response for the observations raised. We hope the response meets your criteria.

 

Reviewer 2

1. The topic is original and relevant to the field and could help developed and implemented for other experimental techniques or other numerical simulations. I would recommend more details regarding the digital images (magnitude, scale).

We would like to refer the respected reviewer to Ref [7] on the revised version where extensive experimental details are given.

 

2. An application of this approximation at the microscopic level, if it is possible, for SEM images of the tensile/torsional specimen.

We thank the reviewer for the idea. However, we do not have access to that kind of equipment.

 

3. The references are close to the topic, but their insertion in the text, as well as the absence of some of them, requires more special

We have amended the references adding to MDPI citation style and the citation within the text. In addition, we added the suggested reference

4. Diameters should be expressed in the Figure 3.

Diameter symbol was added

5. I recommend adding the reference: Michael Vormwald et al. Variable mode-mixity during fatigue cycles — crack tip parameters determined from displacement fields measured by digital image correlation, Frattura ed lntegrità Strutturale, 41 (2017) 314-322; DOI: 10.3221/IGF-ESIS.41.42

Thanks for pointing that out. Reference was added

 

6. Fix format T1

Table 1 is formatted according to the style manual

 

7. Add scale F5 and F9

Thanks for noticing that. Guiding scales were added to both figures indicating a known distance, such as the machined slit.

 

8. Check nomenclature

Thanks for noticing that. Nomenclature was amended.

 

Once again, we thank the reviewer for the constructive criticism. We appreciate the time spent reading and critiquing the manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

Review for paper applsci-3457250

An Over deterministic Method for Mode III SIF Calculation using Full Field Experimental Displacement Fields

 

This paper proposes a new method for calculating the stress intensity factor (SIF) of cracks in mode III through the analysis of full-field experimental displacement field data. This method aims to solve the deficiencies of existing methods in calculating mode III. There have been a variety of experimental methods and numerical simulation methods for cracks in the literature, but relatively few systematic studies on mode III. In this context, this paper proposes a new calculation method aiming to fill this gap. Nonetheless, the manuscript could be improved if the authors could address the comments and recommendations I listed below.

 

1. Please improve the image resolution (quality) of your Figures. All your plots are very blurry.

2. Please add scale bars in Figures 5 & 9.

3. Under load, the appearance of negative (K_I) values is not in line with physical significance and may lead to misunderstandings of crack behavior.

4. Line 54: So many "Error! Reference source not found" appeared across your article. Please review it.

5. Please indicate how you got the Data source images in Figure 5.

6. The conclusion part should be more concise.

 

Author Response

We would like to thank Reviewer #3 for the detailed reading and constructive comments. Your insights have made us think of ways to improve the paper, resulting in a much clearer manuscript. Here is a detailed response for the observations raised. We hope the response meets your criteria.

 

Reviewer 3

This paper proposes a new method for calculating the stress intensity factor (SIF) of cracks in mode III through the analysis of full-field experimental displacement field data. This method aims to solve the deficiencies of existing methods in calculating mode III. There have been a variety of experimental methods and numerical simulation methods for cracks in the literature, but relatively few systematic studies on mode III. In this context, this paper proposes a new calculation method aiming to fill this gap. Nonetheless, the manuscript could be improved if the authors could address the comments and recommendations I listed below.

 

1. Please improve the image resolution (quality) of your Figures. All your plots are very blurry.

All figures were improved and placed as *SVG in the revised document.

2. Please add scale bars in Figures 5 & 9.

Guiding scales were added to both figures indicating known distances

3. Under load, the appearance of negative (K_I) values is not in line with physical significance and may lead to misunderstandings of crack behavior.

Thanks for pointing that out.

We fully agree with the reviewer. Negative K_I cannot exist. See the paragraph beginning section 4.3

Figure 8(a) illustrates how K_I increases for positive loads, but the approaches differ slightly in their negative slopes. The negative part of K_I is a superposition of plasticity reversion and elastic compression created by the negative portion of the axial load, which results in positive residual stresses upon unloading [9]. This could be explained by the fact that some of the positive plastic strains created when the crack was fully open are now reversed [9]. Nonetheless, when the crack is closed, the singularity required by the Westergaard stress function does not occur.

 

4. Line 54: So many "Error! Reference source not found" appeared across your article. Please review it.

Thank you. Error was fixed

 

5. Please indicate how you got the Data source images in Figure 5.

 

We added the following text starting in L265          

Furthermore, in an effort to reduce the accumulated plasticity ahead of the crack tip, displacement data was collected behind the crack tip, mimicking a COD clip gauge. That short distance minimizes the effects of accumulated plasticity, the uncertainty derived from the low correlation criteria given around the crack lips (which may provide unrealistic strains), and keeping the distance between AOI and CTL as short as possible to minimize the curvature effect as well as not getting into the near zone which would capture plastic strains. Furthermore, the fields must be aligned with the crack growth direction to comply with Williams' premises.

 

 

6. The conclusion part should be more concise.

Sentences were added to the conclusion, highlighting the specific contributions and further areas for research:

SIF calculation is more robust when using an area comprised of many data points, instead of two points.

A solution to approximate Williams’ solution for curved surfaces was proposed.

The authors identify that future research could focus in optimizing the area of interest (AOI) selection could mitigate the impact of accumulated plasticity and enhance the accuracy of SIF calculations.

 

Once again, we thank the reviewer for the constructive criticism. We greatly appreciate the time spent reading and critiquing the manuscript.

Reviewer 4 Report

Comments and Suggestions for Authors

Authors proposed a least squares method (LSM) for calculating the Mode III stress intensity factor (SIF) and validates the method using full-field displacement data. The paper compares the LSM method with the COD method and demonstrates its effectiveness in calculating Mode III SIF, particularly under non-proportional loading conditions. However, the current form of this study can be acceptable after minor revision. Some aspects as listed below:

1. Figures in lines 54, 56, 106, 113, 179, and 274 have cross-reference errors, showing "Error! Reference source not found." which need to be re-imported.
2. It is suggested to redraw the Figure 1, as it does not clearly show the out-of-plane shear mode crack coordinate system and displacement. Adding annotations to the figure would help in understanding.
3. It mentions the use of Digital Image Correlation (DIC) technology, but additional details on the DIC operation and data processing methods are needed. It should include camera resolution, calibration process, data processing methods, error estimation, and especially how accurate displacement data around the crack tip is captured.
4. The AOI is chosen based on two 1 x 1 mm squares located 1 mm behind the crack tip. However, is this AOI selection effective when considering plasticity effects? Could there be a more reasonable AOI choice, or should the AOI selection be further validated?
5. In the methods section, especially when introducing the theoretical background, more emphasis should be placed on connecting the theoretical methods with the experimental approach. For instance, it would be helpful to explain how the Williams series is applied to actual experimental data and why least squares fitting was chosen.
6. Has the LSM method been sufficiently validated, particularly in comparison to traditional methods (such as COD)? While the paper compares the LSM and COD results, the uncertainty in these results is not discussed.
7. Negative K_I values do not make physical sense, but is it completely impossible for negative K_I to occur? Is this phenomenon only limited to experimental errors or specific conditions, or are there other influencing factors such as strain hardening or plastic deformation?
8. SIF changes influence crack propagation, but it does not explore the quantitative relationship between crack growth rate (FCG) and SIF in depth. Does SIF have a clear mathematical relationship with crack growth rate?
9. The least squares method (LSM) is used to fit the Williams series but does not clearly explain how the number of terms in the Williams series is chosen. Is the number of terms in the series adjusted for different loading conditions? Additionally, has sensitivity analysis been conducted to verify the dependence of results on the number of terms in the series?

Comments on the Quality of English Language

Some sentences are quite long and may affect the readability.

For example, the sentence "However, a negative K_I does not make physical sense, as the crack lips would need to penetrate each other to produce negative relative displacements, which clearly is physically impossible." could be rewritten as "However, negative K_I values do not make physical sense. For negative relative displacements to occur, the crack lips would need to penetrate each other, which is physically impossible."

Author Response

We would like to thank Reviewer #4 for the detailed reading and constructive comments. Your insights have made us think of ways to improve the paper, resulting in a much clearer manuscript. Here is a detailed response for the observations raised. We hope the response meets your criteria.

 

 

Reviewer 4

Authors proposed a least squares method (LSM) for calculating the Mode III stress intensity factor (SIF) and validates the method using full-field displacement data. The paper compares the LSM method with the COD method and demonstrates its effectiveness in calculating Mode III SIF, particularly under non-proportional loading conditions. However, the current form of this study can be acceptable after minor revision. Some aspects as listed below:

1. Figures in lines 54, 56, 106, 113, 179, and 274 have cross-reference errors, showing "Error! Reference source not found." which need to be re-imported.

Thanks for pointing that out. They were corrected

1. It is suggested to redraw the Figure 1, as it does not clearly show the out-of-plane shear mode crack coordinate system and displacement. Adding annotations to the figure would help in understanding.

R: Fig 1 was enhanced, and the caption was modified to reflect the displacements in the three-coordinate axis.

1. It mentions the use of Digital Image Correlation (DIC) technology, but additional details on the DIC operation and data processing methods are needed. It should include camera resolution, calibration process, data processing methods, error estimation, and especially how accurate displacement data around the crack tip is captured.

R: Thanks for the suggestion. We would like to direct the respected reviewer to REF [7] on the new version where such experimental details are included. We opted to reference them, rather than put them on the manuscript to avoid a lengthy paper and to avoid duplication.

1. The AOI is chosen based on two 1 x 1 mm squares located 1 mm behind the crack tip. However, is this AOI selection effective when considering plasticity effects? Could there be a more reasonable AOI choice, or should the AOI selection be further validated?

We added the following text starting in L265          

Furthermore, in an effort to reduce the accumulated plasticity ahead of the crack tip, displacement data was collected behind the crack tip, mimicking a COD clip gauge. That short distance minimizes the effects of accumulated plasticity, the uncertainty derived from the low correlation criteria given around the crack lips (which may provide unrealistic strains), and keeping the distance between AOI and CTL as short as possible to minimize the curvature effect as well as not getting into the near zone which would capture plastic strains. Furthermore, the fields must be aligned with the crack growth direction to comply with Williams' premises.

 

1. In the methods section, especially when introducing the theoretical background, more emphasis should be placed on connecting the theoretical methods with the experimental approach. For instance, it would be helpful to explain how the Williams series is applied to actual experimental data and why least squares fitting was chosen.

Thanks, we agree with the reviewer. To the best of our knowledge, the LSM has been used since 1980 when Sanford et al. applied the method to photoelastic data. We added the following text starting at L251.

Williams' series has been used in curved samples with cracks smaller than the curvature. On the other hand, the LSM is a tool for solving overdetermined systems. It finds the solution that minimizes the sum of the squared errors between the measured and the predicted responses.

 

We also added the text in L309, right before the image, to highlight cases where a clip gauge may not work.

 

Furthermore, a COD clip gauge must be placed on the slit, and due to the nature fo the crack growth, it may not give displacements that represent the stress field for the cases shown in Figures 5(a) and (b).

 

 

1. Has the LSM method been sufficiently validated, particularly in comparison to traditional methods (such as COD)? While the paper compares the LSM and COD results, the uncertainty in these results is not discussed.

The use of LSM has been sufficiently proven in literature. From the cited references [27, 39, 40] – checked the effectiveness of using different numbers of terms in the Williams series for SIF calculation, indicating some validation of results with respect to varying series truncations, implying the application of LSM and further supporting its validation through comparison across different approaches. Gonzáles (Gonzáles, 2017) compared K_I obtained from COD, J integral and LSM finding close results among the methods.

Gonzáles, G. et al. (2017). Determining SIFs Using DIC Considering Crack Closure and Blunting. In: Zhu, Y., Zehnder, A. (eds) Experimental and Applied Mechanics, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-42028-8_4

1. Negative K_I values do not make physical sense, but is it completely impossible for negative K_I to occur? Is this phenomenon only limited to experimental errors or specific conditions, or are there other influencing factors such as strain hardening or plastic deformation?

We fully agree with the reviewer. Negative K_I cannot exist. See the paragraph beginning section 4.3

Figure 8(a) illustrates how K_I increases for positive loads, but the approaches differ slightly in their negative slopes. The negative part of K_I is a superposition of plasticity reversion and elastic compression created by the negative portion of the axial load, which results in positive residual stresses upon unloading [9]. This could be explained by the fact that some of the positive plastic strains created when the crack was fully open are now reversed [9]. Nonetheless, when the crack is closed, the singularity required by the Westergaard stress function does not occur.

 

1. SIF changes influence crack propagation, but it does not explore the quantitative relationship between crack growth rate (FCG) and SIF in depth. Does SIF have a clear mathematical relationship with crack growth rate?

We strongly agree with the reviewer. Although, if SIFs change, but the equivalent SIF does not, the ratio of modes II and III will do, which undoubtedly impact crack path, as recently discussed by Gomez et al. [31]. Beyond that discussion, the relationship between FCG and SIF is beyond the scope of this manuscript. Our contribution is a method to calculate SIG mode III so it could be used to calculate FCG rate and Crack Path.

 

1. The least squares method (LSM) is used to fit the Williams series but does not clearly explain how the number of terms in the Williams series is chosen. Is the number of terms in the series adjusted for different loading conditions? Additionally, has sensitivity analysis been conducted to verify the dependence of results on the number of terms in the series?

Thanks for the observation. We did not perform a strict sensitivity analysis with the number of terms. This approach has been used before [7, 26, 39, 40, 41, 42]. The following text was added at L334

Strohmann et al. [29] discussed how the number of terms can influence fracture mechanics parameters and FCG stability, but there is limited understanding of their physical meaning. Recently, Kolditz et al. [42] used a similar approach but used the stability of the energy release rate in a phase-field simulation as a benchmark. ….  ….  So, to perform an analysis, one typically starts with one term in the Williams' series and gradually raises the number until SIF results stabilize. The attached algorithm includes the number of terms as a variable in the supplemental material.

 

Comments on the Quality of English Language

Some sentences are quite long and may affect the readability.

For example, the sentence "However, a negative K_I  does not make physical sense, as the crack lips would need to penetrate each other to produce negative relative displacements, which clearly is physically impossible." could be rewritten as "However, negative K_I values do not make physical sense. For negative relative displacements to occur, the crack lips would need to penetrate each other, which is physically impossible."

R: Thanks for the suggestion. It was corrected as suggested

 

Once again, we thank the reviewer for the constructive criticism. We appreciate the time spent reading and critiquing the manuscript.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript applsci-3457250-v2 has been sufficiently improved, which warrants its publication in Applied Sciences.

Author Response

Thanks for your time.

Reviewer 2 Report

Comments and Suggestions for Authors

Improvements have been made.

I recommend publishing the article.

Author Response

Thanks for your time and insights.

Reviewer 4 Report

Comments and Suggestions for Authors

Authors proposed a least squares method (LSM) for calculating the Mode III stress intensity factor (SIF) and validates the method using full-field displacement data. The paper have revised according to the comments. It can be accepted after minor revision. The image quality in Fig.2-7 should be improved.

Author Response

Dear revieweer #2. Firts of all, thanks for your time and insights.

Attached you will find the revised version of the paper “An Over Deterministic Method for Mode III SIF Calculation Using Full Field Experimental Displacement Fields”. 

Obs 1. The image quality in Fig.2-7 should be improved.

 

R: All images have been improved and saved as Portable Network Graphics (*.PNG) files with the following resolution:

Figure number	Resolution (pixels)
Figure 1	2661 x 1819
Figure 2	4400 x 2097
Figure 3	1393 x 403
Figure 4	1728 x 821
Figure 5	4042 x 1140
Figure 6	3933 x 2476
Figure 7	4100 x 2423
Figure 8	4036 x 2468
Figure 9	3118 x 2283

 

Additionally, minor details were fixed such as using the same font (Palatino Linotype) in the figures as in the text, and a few spelling mistakes in manuscript were corrected and marked. For clarity, we have attached a revised copy with the tool changes from MS-Word named applsci-3457250-marked V 5.0.docx and for easiness, a revised version with the same changes but already implemented, named applsci-3457250-clean V 5.0.docx in a ZIP file. We also attached the figures as a ZIP file, should the editorial office need them.

 

We appreciate your consideration!