Crack Propagation on the Surface of the Bottleneck of a Pressure Vessel Considering the Initial Crack Angle

Round 1

Reviewer 1 Report

A. The description of the method should be improved.

B. There is no link with the life prediction model.

C. There is no evidence the presented model is optimal.

1. line 28: which is suitable for external The SIF of the buried pipe with surface defects was evaluated - something is missing here, I suppose

2. line 44: The above model assumes - which one? Or do you mean: 'The above models assume'?

3. line 51: optimize the pressure vessel crack Extend the model. - something is missing here, I suppose

4. line 62: there is no Poisson's ratio in Eqn (1) or (2) - why is it defined here?

5. line 64: Explain better how function F_s is defined.

6. line 75: Eqn (7) is not a combination of Enqs (2) and (5), it defines new variable \Delta G^{\star} using mentioned ones.

7. line 80: Do you assume \Delta G = \Delta G^{\star}, see Eqn. (10)?

8. line 81: Here da is damage, while in line 71 it was crack growth, so?

9. line 100: What is the value of the yield stress for von Mises criterion?

10. line 103: What is 'moment module'?

11. lines 99-108: what type of analysis do you perform? Static, dynamic, heat transfer or coupled?

12. line 109: The curved surface is set to 0.25mm, and the other parts are set to 5mm. - what does it mean? Do you describe the size of FEs?

13. line 119: Neither the Poisson's ratio, nor the k variable is defined in Eqn. (15), so why are they defined there?

14. Sec. 3.1. Describe clearly what output is taken from FE calculations.

15. line 132: what about other dimensions? Are they specified in the paper?

16. line 176: What is a threshold (start) value?

17. line 182: What cycles? Do you mean: increments?

18. line 224: do you perform any fatigue simulations? What is the amplitude definition?

19. line 243: Authors should discuss the ... - this paragraph should be removed.

20. Table 4: Why do you present four cases, when only one is used in comparison?

21. line 260: It is very risky to draw such conclusions based only on one case problem.

22. line 274: This section is not mandatory ... - this paragraph should be removed.

Author Response

Point 1: A. The description of the method should be improved.

1. There is no link with the life prediction model.
2. There is no evidence the presented model is optimal.

Response 1: A. I changed my expression in my paper to make it more logical and readable. B. yes, no link directly, so I delete this from the key words, life prediction model is corresponding to the rest part of it, and it does not reflect in this paper. C. The precise word in title is optimized not optimal. Our work is to optimize the traditional model making it work.

Point 2: 4. line 62: there is no Poisson's ratio in Eqn (1) or (2) - why is it defined here?

6. line 75: Eqn (7) is not a combination of Enqs (2) and (5), it defines new variable \Delta G^{\star} using mentioned ones.
7. line 80: Do you assume \Delta G = \Delta G^{\star}, see Eqn. (10)?
8. line 103: What is 'moment module'?
9. lines 99-108: what type of analysis do you perform? Static, dynamic, heat transfer or coupled?
10. line 109: The curved surface is set to 0.25mm, and the other parts are set to 5mm. - what does it mean? Do you describe the size of FEs?
11. line 119: Neither the Poisson's ratio, nor the k variable is defined in Eqn. (15), so why are they defined there?
12. Sec. 3.1. Describe clearly what output is taken from FE calculations.
13. line 176: What is a threshold (start) value? 30° do you means the start angle?
14. line 224: do you perform any fatigue simulations? What is the amplitude definition?
15. line 260: It is very risky to draw such conclusions based only on one case problem.

Thanks for all the comments.Other comments not mentioned here have been revised in paper.

Response 2: 2. It defined in equation (3).

4. It’s corresponding to the formula (3).
5. Added some expression for it in line62, and the figure of this F_s is as following figure 1.

Figure1. the relation between  and

6. 7. \Delta G means the traditional crack growth model formula, \Delta G^{\star} means the optimized crack growth model formula. We have modified the expression for it in line 81. We changed the description for the \delta G, named modified \Delta G^{\star}. The same description applied in \K{\star} in line 65 to differ traditional stress intensity model and modified stress intensity model.
7. moment module is to simulate elastic flexural capacity.
8. static mechanics analysis.
9. Yes, means the mesh density in FEA(finite element analysis), I have added some description in the paper.
10. Something wrong here, we have revised the formula(15) .
11. We add one figure (figure 3) and some description to illustrate the meaning of the FEA calculations result in line 113-118.
12. Yes, it’s means the start angle. The purpose of this paper is to optimize the model of pressure vessel.Our point is to braek the assumption that the start angle of crack is 0°.
13. In formula (4), the cycles Equivalent to the dN.The unit of Dn is cycle. The unit of da/dN is mm/cycle.
14. No, we did not the fatigue simulation. We concentrate on the fracture propagation analysis. The fatigue mainly happens in the crack initiation.
15. Some description wrong here, we have advised in paper in line 258-264.The room temperature can get the most oblivious result, other temperature the modified stress intensity model almost has the same curve with the traditional stress intensity model.

Author Response File: Author Response.docx

Reviewer 2 Report

Dear authors thank You for this interesting paper. In the reviewers opinion the paper should have a chance to be published but it needs major revision. The introduction is missing major information in terms of existing crack propagation models. That means that the literature review needs to be largely extended. The quality of some figures is very poor as there are some tables inside them and they are unreadable. The authors are performing a comparison of their model for different simulations and what the reviewer believes some experimental results obtained from Lanzhou University scientists. It would be wise to perform a procedure check with the use of other experimental results of material crack propagation. The literature is full of experimental results that can be used for that. The reviewer is missing an clear error analysis of the comparison, if there is an error analysis then it is unreadable. The simulations alone are insufficient to support the claims of the authors stated in the conclusions. Please revise the paper in these terms.

Author Response

Point 1: Dear authors thank You for this interesting paper. In the reviewers opinion the paper should have a chance to be published but it needs major revision. The introduction is missing major information in terms of existing crack propagation models. That means that the literature review needs to be largely extended. The quality of some figures is very poor as there are some tables inside them and they are unreadable. The authors are performing a comparison of their model for different simulations and what the reviewer believes some experimental results obtained from Lanzhou University scientists. It would be wise to perform a procedure check with the use of other experimental results of material crack propagation. The literature is full of experimental results that can be used for that. The reviewer is missing an clear error analysis of the comparison, if there is an error analysis then it is unreadable. The simulations alone are insufficient to support the claims of the authors stated in the conclusions. Please revise the paper in these terms.

Response 1:

We revised many descriptions of the formula and simulation process. And we revised the expression of introduction in line 48-55.

We extend the literature for crack propagation of pressure vessel, [3],[4] and [5] are about the simulation and modeling for the crack propagation of pressure, [13] and [14] are about novel approach calculating crack propagation, [15] is about composite plate for pressure vessel modeling.

We modified figure 4(a)(b), 4(a)(b), 6(a)(b) and 6 for clarity.

The comparsion between modified stress intensity factor model and traditional stress intensity factor model is oblivious in figure 10.

The crack propagation experiment is still in progress, and the crack propagation needs much time getting the result, so We use the opening real experiment data for comparsion.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments to the Authors

This manuscript reports a life prediction model of the pressure vessel subjected internal pressure. Unfortunately, the analysis method used in this paper is wrong. It is hoped that the authors will seriously consider the following comments.

1. As the problem of low-cycle fatigue prediction is considered in this paper, the failure of bottleneck of pressure vessel is usually caused by strain fatigue. However, the author used the method of linear elastic fracture mechanics analysis. I think the problem of this paper should be analysed using the method of elastic-plastic fracture mechanics. Therefore, the analysis method used in this paper is incorrect.
2. English needs improvement. The content in manuscript lacks clear expression and a lot of non-conventional terminology were used. For example, the title of the paper is confusing. The propagation of cracks in a cracked structure under alternating external loads is a natural phenomenon. There is no any 'optimized model'. In addition, cracks should be initiated on the surface of the bottleneck of pressure vessel and along the depth and hoop-direction growth. It does not propagation on the surface.
3. The first sentence in the introduction '...when the initial crack angle of the curved surface is not 0°...' is problematic. The crack initiation angle is only related to the coordinate system set up by the authors. If rotating the yoz plane to the angle of the crack initial, can the crack surface change from a curved surface to a straight plane? The fractured section should have a precise definition.
4. Due to the lack of legends in Figures 8 and 9, it is difficult to understand what the contours in these figures represent.
5. In what direction does v in Equation 15 represent the displacement?
6. How is Equation 3 obtained? Can't seem to find it in reference 11.
7. The definition of the symbol K in Equation 1 and the symbol K in Equation 13 should be different but use the same notation.

Author Response

Response to Reviewer 3 Comments

Point 1: As the problem of low-cycle fatigue prediction is considered in this paper, the failure of bottleneck of pressure vessel is usually caused by strain fatigue. However, the author used the method of linear elastic fracture mechanics analysis. I think the problem of this paper should be analysed using the method of elastic-plastic fracture mechanics. Therefore, the analysis method used in this paper is incorrect.

Response 1: Fatigue is divided into high-cycle fatigue and low-cycle fatigue. The research object here is the pressure vessel, and the pressure vessel will have low cycle fatigue. And our research is the low-cycle fatigue.

Low cycle fatigue does require a combination of elastic fracture mechanics and plastic fracture mechanics for analysis, but we research the direction and rate of crack propagation. For the part that has formed a crack (plastic fracture mechanics should be used), the stress of plastic area is a fixed value by the von Mises criterion in simulation. For crack propagation, elastic fracture mechanics analysis is used to judge the direction of crack propagation in the next step for the maximum stress based on the stress distribution, so it’s using the elastic fracture mechanics. In the actual simulation process, our simulation processing is to decompose the crack propagation process into each small unit, and when the energy is accumulated to the energy of the failure of the small unit, the unit is removed to form a new crack.

Point 2: English needs improvement. The content in manuscript lacks clear expression and a lot of non-conventional terminology were used. For example, the title of the paper is confusing. The propagation of cracks in a cracked structure under alternating external loads is a natural phenomenon. There is no any 'optimized model'. In addition, cracks should be initiated on the surface of the bottleneck of pressure vessel and along the depth and hoop-direction growth. It does not propagation on the surface.

Response 2: Some descriptions in the papar have advised, including the description for formulas, for figure and conclusion.

The purpose of this paper is to optimize the crack propagation model.For this point, it could be optimized. Because we use the model not assuming the crack angle, changes the model.And We modeled the pressure vessel model more precisely, beacause we take the neck joint surface body of pressure vessel bottle as the research object.

Point 3: The first sentence in the introduction '...when the initial crack angle of the curved surface is not 0°...' is problematic. The crack initiation angle is only related to the coordinate system set up by the authors. If rotating the yoz plane to the angle of the crack initial, can the crack surface change from a curved surface to a straight plane? The fractured section should have a precise definition.

Response 3: Yes, the crack angle only related to the coordinate system.The angle of 0° means the normal direction of the curved surface.We added an explanation for this in the introduction in line 51 (equivalent to normal direction).

Point 4: Due to the lack of legends in Figures 8 and 9, it is difficult to understand what the contours in these figures represent.

Response 4: These figures are the result of the FEA(finite element analysis). The mesh of these figures are the mesh of FEA.The FEA calculation results’ description have beend added in 3.1 in line 113-118.

Figure 8 and figure 9 are the view of the curved surface’s normal direction. We want to illustrate the influence of the initial angle of crack, but it’s hard to explain it from the positive.And we explain it by comparion of two different initial angle illustrating the crack growth length and crack growth direction are affected by the initial angle.So we demonstrate that relevant research is meaningful.

Point 5: In what direction does v in Equation 15 represent the displacement?

Response 5: Opening displacement of the crack surface.

Point 6: How is Equation 3 obtained? Can't seem to find it in reference 11.

Response 6: We have revised paper for this problem, and added some description for Equation 3 in line 66-70.

Point 7: The definition of the symbol K in Equation 1 and the symbol K in Equation 13 should be different but use the same notation.

Response 7: Yes, should no be the same.we have revised paper for this problem in equation (13).

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

I accept the revised version.

Author Response

.

Reviewer 3 Report

As the authors mentioned in their response their research is limited to the low-cycle fatigue (Maximum number of cycles is about 250). It is correct to use the energy release rate as the crack driving force for solving the elastic-plastic fracture mechanics problem. Unfortunately, Equation 5 is not a right relationship in the elastic-plastic state (Equation 5 only holds under linear elasticity fracture mechanics), so it is wrong to choose Equation 10 as the crack propagation formula. Therefore, the correctness of the analysis results in this paper is questionable. I suggest that the authors either calculate the stress intensity factor in the elastic-plastic state and use Equation 10 for the crack growth analysis, or directly do the crack growth analysis by using the crack growth equation with the energy release rate as the crack driving force.

Given that the fatigue life evaluated in this paper is only 250 cycles, it is generally understood that the high cyclic stress at the bottleneck region of the pressure vessel (the remote stress at the local regime should exceed the yield limit) created fatigue failure. For this reason, the authors should provide the constitutive equation of the materials they used, the adopted plastic strengthening model and fracture toughness, etc.

The so-called crack growth model usually refers to the use of mathematical or physical methods to describe the crack growth. Its rationality and correctness and whether it can reflect the actual situation need to be verified by experimental results. Therefore, there is no 'optimized' crack growth model. This is completely different from the concept of 'optimized' in structural optimization design. Therefore, the title should be rewritten. The word 'Optimized' must be removed. When it comes to optimizing a model, it is often necessary to choose an objective function. What is the objective function for this paper? Is it looking for longer durability for a given dimensions of the pressure vessel or for a lighter structure with a certain fatigue life or what…?

The mesh provided in Figures 2, 3 and 8 seem to be very coarse. There is no local mesh refinement in the region of brake and wheel contact. What type element has been used in this analysis? It looks 4 nodes element. Does this mesh can accurately model the real situation? My concern is the accuracy of mesh for fracture analysis. Have authors done mesh density convergence check for their FEM model and how about the results?

The author did not answer my fourth question. What I would like to know is that legends in Figure 9 should be added, i.e. the red in the Figure 9 represents how many MPa the stress is, the blue represents how many MPa the stress is, ..., etc.

Author Response

1． As the authors mentioned in their response their research is limited to the low-cycle fatigue (Maximum number of cycles is about 250). It is correct to use the energy release rate as the crack driving force for solving the elastic-plastic fracture mechanics problem. Unfortunately, Equation 5 is not a right relationship in the elastic-plastic state (Equation 5 only holds under linear elasticity fracture mechanics), so it is wrong to choose Equation 10 as the crack propagation formula. Therefore, the correctness of the analysis results in this paper is questionable. I suggest that the authors either calculate the stress intensity factor in the elastic-plastic state and use Equation 10 for the crack growth analysis, or directly do the crack growth analysis by using the crack growth equation with the energy release rate as the crack driving force.

 (1) The commonly used pressure vessel prediction crack initiation and propagation model is modified on the basis of the Erdogan model. For the stress analysis of pressure vessels, the elastic model is usually used for analysis, and the main load is the internal pressure load. When external force is applied to the wall of the pressure vessel for a long time, stress concentration points will be generated due to the material properties, and the stress concentration points will be unevenly subjected to the elastic limit for a long time and enter the plastic yield stage. At this time, the material transforms from the elastic change area to the plastic change area, forming a crack. In this paper, the crack propagation of pressure vessels is studied based on the maximum circumferential stress criterion.

For the maximum circumferential stress criterion, it is widely used because of its simple form and convenient application, but the criterion ignores the influence of the tip plastic zone, although there is a certain deviation, and this deviation is not large. Based on this criterion, this paper studies the crack propagation of pressure vessels, considering the effect of initial crack initiation angle on crack propagation, and obtains the results under this analysis method.

(2) The fracture mechanics model established in ABAQUS combines elastic mechanics and plastic mechanics. The crack propagation process is decomposed into each small unit, the energy is accumulated to the energy of the failure of the small unit, and the unit is removed to form a crack. Therefore, the crack propagation can be studied according to the maximum circumferential stress criterion in the ABAQUS simulation.

2. Given that the fatigue life evaluated in this paper is only 250 cycles, it is generally understood that the high cyclic stress at the bottleneck region of the pressure vessel (the remote stress at the local regime should exceed the yield limit) created fatigue failure. For this reason, the authors should provide the constitutive equation of the materials they used, the adopted plastic strengthening model and fracture toughness, etc.

the constitutive equation of the materials is as following equation:

The adopted plastic strengthening model used in the ABAQUES simulation in this paper is the Von Mises yield criterion.

The fracture toughness is as shown in the table below.

The fracture toughness of 35CrMo

3. The so-called crack growth model usually refers to the use of mathematical or physical methods to describe the crack growth. Its rationality and correctness and whether it can reflect the actual situation need to be verified by experimental results. Therefore, there is no 'optimized' crack growth model. This is completely different from the concept of 'optimized' in structural optimization design. Therefore, the title should be rewritten. The word 'Optimized' must be removed. When it comes to optimizing a model, it is often necessary to choose an objective function. What is the objective function for this paper? Is it looking for longer durability for a given dimensions of the pressure vessel or for a lighter structure with a certain fatigue life or what…?

We have updated the title of this paper, and the new title is:

Crack Propagation on the Surface of the Bottleneck of Pressure Vessel Considering the Initial Crack Angle

4. The mesh provided in Figures 2, 3 and 8 seem to be very coarse. There is no local mesh refinement in the region of brake and wheel contact. What type element has been used in this analysis? It looks 4 nodes element. Does this mesh can accurately model the real situation? My concern is the accuracy of mesh for fracture analysis. Have authors done mesh density convergence check for their FEM model and how about the results?

Yes, it give good results, and with a smaller mesh the results are exactly the same with 0.25mm mesh size. Therefore, it is suitable to use 0.25mm for the neck part, and there is no need to refine the circumference of the crack.

5. The author did not answer my fourth question. What I would like to know is that legends in Figure 9 should be added, i.e. the red in the Figure 9 represents how many MPa the stress is, the blue represents how many MPa the stress is, ..., etc.

The legends in figure 9 is hard to see, so we do not insert it to the paper.

Author Response File: Author Response.docx

Round 3

Reviewer 3 Report

The first point of the author's response is incorrect as a long-term static loads on the structure will not cause cracking, so the authors believe that "When external force is applied to the wall of the pressure vessel for a long time, stress concentration points will be generated due to the material properties, and the stress concentration points will be unevenly to the elastic limit for a long time and enter the plastic yield stage..." is incorrect. No cracks will occur and growth even after 100 years even more of exposure. This is the basic principle of fracture mechanics. Effects of prolonged loads on structures only involves problems of creep or relaxation, not fatigue. Cracks only initiate and grow under the action of alternating stress.

In addition, the meaning of the other 250 cycles is unclear. What is the load cycle representative in this manuscript? How long? If the frequency is assumed to be 1 Hz, does it mean that the service life is 250 seconds which is approximately 0.0464 hours? The authors should insert the explanation of my question into the manuscript so that the reader can understand what the authors want to express.

The content of point 2 of the response needs to be incorporated into the manuscript, especially the constitutive equations and fracture toughness used in the elastic-plastic analysis done in this paper should be told to the reader.

I am happy with points 3 to 5 of your response.

In the line of 117, the word of ‘greater’ should change to ‘higher’. Stress can only be expressed as high or low.

Comments for author File: Comments.docx

Author Response

1. The first point of the author's response is incorrect as a long-term static loads on the structure will not cause cracking, so the authors believe that "When external force is applied to the wall of the pressure vessel for a long time, stress concentration points will be generated due to the material properties, and the stress concentration points will be unevenly to the elastic limit for a long time and enter the plastic yield stage..." is incorrect. No cracks will occur and growth even after 100 years even more of exposure. This is the basic principle of fracture mechanics. Effects of prolonged loads on structures only involves problems of creep or relaxation, not fatigue. Cracks only initiate and grow under the action of alternating stress.

Sorry for that our previous statement is problematic. The external force is not static but dynamic, the cyclic load curve of the pressure vessel is as figure 1. The x-axis represents the time/s, and the y-axis represents the load/KN.

Figure 1.  cyclic load curve.

2. In addition, the meaning of the other 250 cycles is unclear. What is the load cycle representative in this manuscript? How long? If the frequency is assumed to be 1 Hz, does it mean that the service life is 250 seconds which is approximately 0.0464 hours? The authors should insert the explanation of my question into the manuscript so that the reader can understand what the authors want to express.

The content of point 2 of the response needs to be incorporated into the manuscript, especially the constitutive equations and fracture toughness used in the elastic-plastic analysis done in this paper should be told to the reader.

I am happy with points 3 to 5 of your response.

In the line of 117, the word of ‘greater’ should change to ‘higher’. Stress can only be expressed as high or low.

Other revision you mentioned we’ve updated in the paper highlight red. The location of this paper revision in line 102-104, line 118 and line 187-188. Thanks for your nice comments on our article.

Author Response File: Author Response.docx

Round 4

Reviewer 3 Report

The revised version is barely ready for publication.