# Fatigue Life Prediction of Steam Generator Tubes by Tube Specimens with Circular Holes

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Procedure and Results

#### 2.1. Experimental Procedure

#### 2.2. Results and Analysis

## 3. Evaluation of Stress and Strain near the Hole

#### 3.1. Finite Element Model

#### 3.2. Elastic Analysis

#### 3.3. Elastoplastic Analysis

_{q}and ε

_{q}are local equivalent stresses and strains, K

_{tq}is the equivalent elastic stress concentration factor, S* and e* are plastically corrected nominal stresses and nominal strains, E is the elastic modulus, and S is the elastic nominal stress. The asterisked nominal stress and strain are corrected for nominally inelastic loadings, which are not considered in this work as the nominally applied stress is lower than the yield strength. Figure 10 compares the local equivalent strain as functions of nominal stress estimated with elastoplastic finite element analysis and Neuber’s formula. As shown in Figure 10, the estimation is basically consistent with the finite element calculation result with an error less than 6%. In this work, the Neuber’s method is also used to analyse the local stress-strain of the notch to estimate the fatigue life.

## 4. Fatigue Life Analysis

_{f}:

_{t}is the elastic stress concentration factor, r is the notch radius, and ρ is the material characteristic length. The material characteristic length can be estimated on the ultimate tensile strength σ

_{u}of alloy by the following empirical relation [24,25]:

_{t}= 2.94 for d = 2 mm and K

_{t}= 3.54 for d = 4 mm. Combining Equations (1) and (2), the local equivalent strain near the hole can be obtained by replacing equivalent elastic stress concentration factor K

_{tq}with fatigue notch coefficient K

_{f}. Then, the equivalent strain amplitude ε

_{a}of the smooth tube can be calculated.

## 5. Conclusions

- (1)
- Under the axial cyclic loading, the fatigue crack initiates on the boundary of the hole and propagates perpendicular to the axial of the tube. The fracture of the sample can be divided into three areas: the crack initiation zone, the fracture zone transition zone, and the long crack extension zone. In the thickness direction, the crack face exhibits different angles as the crack propagates.
- (2)
- The elastic stress concentration factor of the tube specimen with a hole is obtained by elastic finite element analysis. Also, the stress distribution is discussed in this paper. The fatigue damage in the middle of the wall thickness is the highest, which is consistent with the test fracture analysis.
- (3)
- The local stress-strain field of the tube specimen with a hole is obtained by elastoplastic finite element analysis. Compared with the Neuber’s formula estimation results, the error between them is less than 6%. The Neuber’s method can be used to estimate the local stress-strain of the notch.
- (4)
- Based on the local stress-strain field of the tube specimen with a hole, the equivalent fatigue life of the smooth heat transfer tube was obtained and compared with the fatigue life of the raw material. Compared with the mean fatigue life of nickel-based alloy materials, the equivalent fatigue life of smooth heat transfer tubes is higher. Further, the heat transfer tubes involved should have a better fatigue performance.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Auvinen, A.; Jokiniemi, J.K.; Laehde, A.; Lähde, A.; Routamo, T.; Lundström, P.; Tuomisto, H.; Dienstbier, J.; Güntay, S.; Suckow, D.; et al. Steam generator tube rupture (SGTR) scenarios. Nucl. Eng. Des.
**2005**, 235, 457–472. [Google Scholar] [CrossRef] - Diercks, D.R.; Shack, W.J.; Muscara, J. Overview of steam generator tube degradation and integrity issues. Nucl. Eng. Des.
**1999**, 194, 19–30. [Google Scholar] [CrossRef] - Japan Nuclear Energy Safety Organization Nuclear Energy System Safety Division. Environmental Fatigue Evaluation Method for Nuclear Power Plants, JNES-SS-1005; Japan Nuclear Energy Safety Organization: Tokyo, Japan, 2011. [Google Scholar]
- Higuchi, M.; Sakaguchi, K.; Hirano, A.; Nomura, Y. Revised and new proposal of environmental fatigue life correction factor (Fen) for carbon and low-alloy steels and nickel base alloys in LWR water environments. In Proceedings of the PVP2006-ICPVT-11, ASME Pressure Vessels and Piping Division Conference, Vancoucer, BC, Canada, 23–27 July 2006. [Google Scholar]
- Arora, P.; Gupta, S.K.; Bhasin, V.; Singh, R.K.; Sivaprasad, S.; Tarafder, S. Testing and assessment of fatigue life prediction models for Indian PHWRs piping material under multi-axial load cycling. Int. J. Fatigue
**2016**, 85, 98–113. [Google Scholar] [CrossRef] - Neuber, H. Theory of stress concentration for shear strained prismatical bodies with arbitrary non-linear stress-strain law. J. Appl. Mech.
**1961**, 28, 544–550. [Google Scholar] [CrossRef] - Zhang, C.C.; Yao, W.X. Typical fatigue life analysis approaches for notched components. J. Aerosp. Power
**2013**, 28, 1223–1230. [Google Scholar] - Ellyin, F.; Kujawski, D. Generalization of notch analysis and its extension to cyclic loading. Eng. Fract. Mech.
**1989**, 32, 819–826. [Google Scholar] [CrossRef] - Karakaş, Ö. Application of Neuber’s effective stress method for the evaluation of the fatigue behaviour of magnesium welds. Int. J. Fatigue
**2017**, 101, 115–126. [Google Scholar] [CrossRef] - Bentachfine, S.; Pluvinage, G.; Gilgert, J.; Azari, Z.; Bouami, D. Notch effect in low cycle fatigue. Int. J. Fatigue
**1999**, 21, 421–430. [Google Scholar] [CrossRef] - Taylor, D. Geometrical effects in fatigue: A unifying theoretical model. Int. J. Fatigue
**1999**, 21, 413–420. [Google Scholar] [CrossRef] - Sun, D.; Hu, Z.D. Research of the size factor of fatigue strength base on TCD theory. Chin. Quart. Mech.
**2015**, 36, 288–295. [Google Scholar] - Adib-Ramezani, H.; Jeong, J. Advanced volumetric method for fatigue life prediction using stress gradient effects at notch roots. Comp. Mater. Sci.
**2007**, 39, 649–663. [Google Scholar] [CrossRef] - Yao, W.X. Stress field intensity approach for predicting fatigue life. Int. J. Fatigue
**1993**, 15, 243–246. [Google Scholar] - Shen, J.B.; Tang, D.L. Predicting method for fatigue life with stress gradient. Chin. Mech. Eng.
**2017**, 28, 40–44. [Google Scholar] - Branco, R.; Costa, J.; Berto, F.; Antunes, F.V. Fatigue life assessment of notched round bars under multiaxial loading based on the total strain energy density approach. Thoret. Appl. Fract. Mech.
**2018**, 97, 340–348. [Google Scholar] [CrossRef] - Sonsino, C.M.; Bruder, T.; Baumgartner, J. S-N Lines for welded thin joints—Suggested slopes and FAT values for applying the notch stress concept with various reference radii. Weld World
**2010**, 54, 375–392. [Google Scholar] [CrossRef] - Sonsino, C.M.; Fricke, W.; De Bruyne, F.; Hoppe, A.; Ahmadi, A.; Zhang, G. Notch stress concepts for the fatigue assessment of welded joints—Background and applications. Int. J. Fatigue
**2012**, 34, 2–16. [Google Scholar] [CrossRef] - Bertini, L.; Frendo, F.; Marulo, G. Fatigue life assessment of welded joints by two local stress approaches: The notch stress approach and the peak stress method. Int. J. Fatigue
**2018**, 110, 246–253. [Google Scholar] [CrossRef] - He, X.; Chen, J.; Tian, W.; Li, Y.; Jin, W. Low cycle fatigue behavior of steam generator tubes under axial loading. Materials
**2018**, 11, 1944. [Google Scholar] [CrossRef] - Guo, W. Three-dimensional analyses of plastic constraint for through-thickness cracked bodies. Eng. Fract. Mech.
**1999**, 62, 383–407. [Google Scholar] [CrossRef] - Seeger, T.; Heuler, P. Generalized application of Neuber’s rule. J. Test. Eval.
**1980**, 8, 199–204. [Google Scholar] - Gates, N.; Fatemi, A. Notched fatigue behavior and stress analysis under multiaxial states of stress. Int. J. Fatigue
**2014**, 67, 2–14. [Google Scholar] [CrossRef] - Kuhn, P.; Hardrath, H.F. An Engineering Method for Estimating Notch-Size Effect in Fatigue Tests on Steel; NACA Technical Note; NACA: Langley Field, VA, USA, 1952. [Google Scholar]
- Gates, N.; Fatemi, A. Notch deformation and stress gradient effects in multiaxial fatigue. Theor. Appl. Fract. Mech.
**2016**, 84, 3–25. [Google Scholar] [CrossRef] - Langer, B.F. Design of pressure vessels for low-cycle fatigue. Trans. ASME J. Basic Eng.
**1962**, 84, 389–399. [Google Scholar] [CrossRef] - Chopra, O.; Stevens, G. Effect of LWR Coolant Environments on the Fatigue Life of Reactor Materials; NUREG/CR-6909, ANL-12/60; U.S. Nuclear Regulatory Commission: Washington, DC, USA, 2014. [Google Scholar]

**Figure 2.**Picture of the test device: (

**a**) Test rig, and (

**b**) diagrammatic sketch of clamping technique.

**Figure 3.**Fracture appearance for tube with holes: (

**a**) Macroscopic view of the crack, (

**b**) fracture appearance, (

**c**) diagrammatic sketch, and (

**d**) microscopic morphology.

**Figure 6.**Distribution of Mises stress and max principal stress around the hole circumference (path P1).

**Figure 8.**Distribution of Mises stress and max principal stress radically away from hole edge on specimen outer surface (path P3).

**Figure 9.**Stress contours from elastic finite element analysis: (

**a**) Axial stress, (

**b**) hoop stress, (

**c**) maximum principal stress, and (

**d**) maximum shear stress.

**Figure 10.**Neuber’s rule estimations compared to finite element results for nominal stress vs. local equivalent strain.

Ni | Cr | Fe | C | Mn | Si | Cu | S |
---|---|---|---|---|---|---|---|

58.0 min. | 27.0–31.0 | 7.0–11.0 | 0.05 max. | 0.50 max. | 0.50 max. | 0.5 max. | 0.015 max. |

Item | Young’s Modulus (MPa) | Yield Strength (MPa) | Tensile Strength (MPa) |
---|---|---|---|

Experimental results | 2.12 × 105 | 286 | 702 |

Standard requirement | - | 240 | 586 |

Diameter of Holes | Control Method | Load Level | Test Life |
---|---|---|---|

4 mm | Strain control | ±0.2% | 478, 886 |

2 mm | Stress control | ±250 MPa | 11,722, 13,300 |

2 mm | Stress control | ±200 MPa | 44,200, 58,800 |

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**MDPI and ACS Style**

Wang, Q.; Chen, J.; Chen, X.; Gao, Z.; Li, Y.
Fatigue Life Prediction of Steam Generator Tubes by Tube Specimens with Circular Holes. *Metals* **2019**, *9*, 322.
https://doi.org/10.3390/met9030322

**AMA Style**

Wang Q, Chen J, Chen X, Gao Z, Li Y.
Fatigue Life Prediction of Steam Generator Tubes by Tube Specimens with Circular Holes. *Metals*. 2019; 9(3):322.
https://doi.org/10.3390/met9030322

**Chicago/Turabian Style**

Wang, Qiwei, Junfeng Chen, Xiao Chen, Zengliang Gao, and Yuebing Li.
2019. "Fatigue Life Prediction of Steam Generator Tubes by Tube Specimens with Circular Holes" *Metals* 9, no. 3: 322.
https://doi.org/10.3390/met9030322