# Two-Stage Model for Fatigue Life Assessment of High Frequency Mechanical Impact (HFMI) Treated Welded Steel Details

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. High-Frequency Mechanical Impact (HFMI) Treatment

#### 2.2. Two-Stage Model (TSM)

#### 2.2.1. General

#### 2.2.2. Crack Initiation Period—Notch Strain Approach (NS)

_{a}is the total strain amplitude, ε

_{a,el}is the elastic strain amplitude, ε

_{a,pl}is the plastic strain amplitude, σ

_{local}is the stress at weld toe, E is the modulus of elasticity, K′ is the cyclic strain hardening coefficient, n′ is the cyclic strain hardening exponent. K′ and n′ are values that can be obtained from the literature or experimental tests. The cyclic load forms the hysteresis loop, shown in Figure 5.

_{y}). After yielding, the stress concentration factor decreases (K

_{tσ}), and the strain concentration factor (K

_{tε}) increases. This means that the stress is lower than the elastic stress, and the deformation is higher than the elastic deformation. To describe the nonlinear analysis by a linear model, Neuber’s expression is used [28]. For each elastic deformation in the σ-ε diagram of the material, Neuber’s expression that gives the energy balance in σ-ε diagram between the elastic stress and strain and the corresponding elastic–plastic stress and strain, as follows:

_{max,local}and ε

_{max,local}are maximal local stresses and strains in the weld toe. K

_{t}is the elastic stress concentration factor. Thus, for given nominal stress and the stress concentration factor K

_{t}, it is possible to calculate the stress and strain in the elastoplastic state. Equation (6) includes residual stress (σ

_{residual}) as the HFMI improvement parameter of fatigue life. The geometry change by HFMI treatment is taken into account by stress concentration factor K

_{t}. To calculate the crack initiation period, it is necessary to calculate stresses and strains in the hysteresis loop (Figure 5). The Ramberg–Osgood Equations (1) and (6) give two equations with two unknowns, from which the maximum local stress in the weld notch σ

_{max,local}is calculated. This procedure provides the maximum stress at the weld toe in the elastic–plastic state:

_{local}:

_{local}showed in Figure 5 can be obtained. Then, the local mean stress is:

_{f}′ is the fatigue strength coefficient, ε

_{f}′ is the fatigue ductility coefficient, b is the fatigue strength exponent, c is the fatigue ductility exponent, σ

_{m}is the mean stress, 2N

_{i}is the number of cycles to crack initiation. The crack initiation period ends when the technical crack is initiated. There are many suggestions in the literature for technical crack size, such as a

_{i}= 0.5–0.8 mm [30] and a

_{i}= 0.25 mm, according to Lawrence et al. [10]. Hou et al. [31] recommend a crack of 0.25 mm.

#### 2.2.3. Crack Propagation Period–Linear Elastic Fracture Mechanics (LEFM)

_{k}is the stress magnification factor due to the stress concentration at the crack tip. A crack starts to propagate if the stress intensity factor range ∆K exceeds the threshold stress intensity factor range ∆K

_{th}. The Paris–Erdogan equation [32], which approximates the rate of crack growth under cyclic planar deformation conditions at the crack tip, is used to estimate the crack propagation period for fatigue loaded welded joints. The crack propagation perpendicular to the load direction is assumed. The Paris–Erdogan equation is:

_{i}to a

_{c}. When the crack size reaches the critical value of a

_{cr}, failure is assumed. The initial crack size is equal to the technical crack size a

_{i}at the end of the crack initiation period. By integrating the Paris–Erdogan equation over the crack size, the service life of the welded detail at fatigue can be calculated [10] as follows:

#### 2.2.4. Flowchart of the Two-Stage Model (TSM)

## 3. Results and Discussion

#### 3.1. Calibration of the TSM

#### 3.1.1. General

#### 3.1.2. Material Parameters for NS Approach

#### 3.1.3. Material and Geometry Parameters for the LEFM Approach

^{−12}. The local stress magnification factor M

_{k}(a) is calculated according to [26]. The parameter Y(a) is a function of crack size, but the average value of a parameter can be considered a constant if Y does not vary strongly [10,26]. For this calculation, it is assumed that Y = 1.12 [43]. The integration was performed by the stepwise integration of crack and cycle increments and by the parallel computing of M

_{k}(a). The step increment was Δa = 0.01 mm. The calculations were performed with the initial crack that equals technical crack size a

_{i}= 0.50 mm, implemented at the maximum principal stress location. Various methods can determine the final crack size, but very often it is not necessary. The biggest portion of the fatigue propagation period is related to small cracks. For a steel plate thickness of 8 mm, it is insignificant whether a final crack size is chosen as 5 mm, 6 mm or 7 mm since there is only a negligible difference in the number of cycles. Accordingly, in this research, 5 mm is assumed as a final crack size.

#### 3.1.4. HFMI Parameters for TSM

_{t}= 2.69–3.85 [44]. For the corresponding HFMI-treated detail, the stress concentration factor is about K

_{t}= 1.92–2.17 [44].

#### 3.1.5. Results of the TSM

_{t}= 3.27 was adopted for the AW condition and K

_{t}= 2.05 for the HFMI-treated condition. Residual stresses are assumed to be 340 MPa for the AW condition and −400 MPa for the HFMI-treated condition. The hardness value in the AW condition in a weld toe area is adopted as HB = 230, and for the HFMI-treated condition, it is HB = 240. The stress ratio is assumed to be R = 0.1. The general concept and ability of the TSM to calculate the fatigue life of longitudinal attachment welded steel detail in the AW condition and the HFMI-treated condition is validated in terms of a comparison of the model results, with test results provided in the literature. S800, S355 and 16Mn are different steel grades used in similar welded details, with plate thicknesses of t = 8 mm. The results are presented in Figure 9 and Figure 10.

#### 3.2. Parametric Study

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Schijve, J. Fatigue as a Phenomenon in the Material. In Fatigue of Structures and Materials; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp. 7–44. [Google Scholar]
- Maddox, S.J. Fatigue Strength of Welded Structures, 2nd ed.; Abington Publishing: Cambridge, UK, 1991. [Google Scholar]
- Haagensen, P.J.; Maddox, S.J. IIW Recommendations on Methods for Improving the Fatigue Strength of Welded Joints: IIW-2142-110; Woodhead Publishing: Cambridge, UK, 2013; ISBN 978-1-78242-065-1. [Google Scholar]
- Marquis, G.B.; Barsoum, Z. IIW Recommendations for the HFMI Treatment for Improving the Fatigue Strength of Welded Joints; Springer Science & Business Media: Singapore, 2016; ISBN 978-981-10-2503-7. [Google Scholar]
- Mikkola, E. A study on effectiveness limitations of high-frequency mechanical impact. Doctoral Dissertation, Aalto University, Espoo, Finland, 2016. [Google Scholar]
- Fuštar, B.; Lukačević, I.; Dujmović, D. High-Frequency mechanical impact treatment of welded joints. Gradjevinar
**2020**, 72, 421–436. [Google Scholar] [CrossRef] - European Committee for Standardization (CEN). Eurocode 3: Design of steel structures, Part 1–9: Fatigue (EN 1993-1-9:2005); CEN: Brussels, Belgium, 2005.
- European Committee for Standardization—Technical Committee 250 (CEN/TC 250). Eurocode 3: Design of Steel Structures (prEN 1993-1-9 Final Draft: 2020); CEN/TC 250: Brussels, Belgium, 2020.
- Fuštar, B.; Lukačević, I.; Dujmović, D. Review of fatigue assessment methods for welded steel structures Fatigue of welded joints. Adv. Civ. Eng.
**2018**, 2018, 3597356. [Google Scholar] [CrossRef][Green Version] - Radaj, D.; Sonsino, C.M.; Fricke, W. Fatigue Assessment of Welded Joints by Local Approaches, 2nd ed.; Woodhead Publishing and Maney Publishing: Cambridge, UK, 2006. [Google Scholar] [CrossRef]
- Hobbacher, A.F. Erratum to: Recommendations for Fatigue Design of Welded Joints and Components; Springer: Cham, Switzerland, 2019. [Google Scholar] [CrossRef][Green Version]
- Ottersböck, M.J.; Leitner, M.; Stoschka, M.; Maurer, W. Crack Initiation and Propagation Fatigue Life of Ultra High-Strength Steel Butt Joints. Appl. Sci.
**2019**, 9, 4590. [Google Scholar] [CrossRef][Green Version] - Leitner, M.; Simunek, D.; Shah, S.F.; Stoschka, M. Numerical fatigue assessment of welded and HFMI-treated joints by notch stress/strain and fracture mechanical approaches. Adv. Eng. Softw.
**2016**, 120, 96–106. [Google Scholar] [CrossRef] - Röscher, S.; Knobloch, M. Towards a prognosis of fatigue life using a Two-Stage-Model: Application to butt welds. Steel Constr.
**2019**, 12, 198–208. [Google Scholar] [CrossRef] - Baptista, C.; Reis, A.; Nussbaumer, A. Probabilistic S-N curves for constant and variable amplitude. Int. J. Fatigue
**2017**, 101, 312–327. [Google Scholar] [CrossRef] - Yıldırım, H.C. Recent results on fatigue strength improvement of high-strength steel welded joints. Int. J. Fatigue
**2017**, 101, 408–420. [Google Scholar] [CrossRef] - Yıldırım, H.C.; Marquis, G.B. A round robin study of high-frequency mechanical impact (HFMI)-treated welded joints subjected to variable amplitude loading. Weld. World
**2013**, 57, 437–447. [Google Scholar] [CrossRef] - Marquis, G.; Barsoum, Z. Fatigue strength improvement of steel structures by high-frequency mechanical impact: Proposed procedures and quality assurance guidelines. Weld. World
**2014**, 58, 19–28. [Google Scholar] [CrossRef] - Peterson, R.E. Notch sensitivity. In Metal Fatigue; McGraw-Hill: New York, NY, USA, 1959; pp. 293–306. [Google Scholar]
- Neuber, H. Über die Berücksichtigung der Spannungskonzentration bei Festigkeitsberechnungen. Konstruktion
**1968**, 20, 245–251. [Google Scholar] - Radaj, D. Design and Analysis of Fatigue Resistant Welded Structures; Woodhead Publishing: Cambridge, UK, 1990. [Google Scholar]
- Seeger, T. Grundlagen für Betriebsfestigkeitsnachweise (Stahlbau Handbuch); Stahlbau-Verlagsges: Köln, Germany, 1996. [Google Scholar]
- Lihavainen, V.M.; Marquis, G. Estimation of fatigue life improvement for ultrasonic impact treated welded joints. Steel Res. Int.
**2006**, 77, 896–900. [Google Scholar] [CrossRef] - Atzori, B.; Lazzarin, P. Notch sensitivity and defect sensitivity under fatigue loading: Two sides of the same medal. Int. J. Fract.
**2001**, 107, 1–8. [Google Scholar] [CrossRef] - Glinka, G. Energy density approach to calculation of inelastic strain-stress near notches and cracks. Eng. Fract. Mech.
**1985**, 22, 485–508. [Google Scholar] [CrossRef] - Hobbacher, A. The use of fracture mechanics in the fatigue analysis of welded joints. In Fracture Fatigue Welded Joints Struct; Elsevier Ltd.: Amsterdam, The Netherlands, 2011; pp. 91–112. [Google Scholar] [CrossRef]
- Ramberg, W.; Osgood, W.R. Description of Stress–Strain Curves by Three Parameters; National Advisory Comittee for Aeronautics, Rep. 902; University of Washington Libraries: Washington, DC, USA, 1943. [Google Scholar]
- Neuber, H. Kerbspannungslehre: Grundlagen für Genaue Spannungsrechnung; Springer: Berlin/Heidelberg, Germany, 1937. [Google Scholar] [CrossRef]
- Masing, G. Eigenspannungen und Verfestigung beim Messing. In Proceedings of the 2nd International Congress for Applied Mechanics, Zurich, Switzerland, 12–17 September 1926; pp. 332–335. [Google Scholar]
- Chattopadhyay, A.; Glinka, G.; El-Zein, M.; Qian, J.; Formas, R. Stress analysis and fatigue of welded structures. Weld. World
**2011**, 55, 2–21. [Google Scholar] [CrossRef] - Hou, C.Y.; Charng, J.J. Models for the Estimation of Weldemnt Fatigue Crack Initiation Life. Int. J. Fatigue
**1997**, 19, 537–541. [Google Scholar] [CrossRef] - Paris, P.; Erdogan, F. A critical analysis of crack propagation laws. J. Fluids. Eng. Trans. Am. Soc. Mech. Eng.
**1963**, 85, 528–533. [Google Scholar] [CrossRef] - Wang, T.; Wang, D.; Huo, L.; Zhang, Y. Discussion on fatigue design of welded joints enhanced by ultrasonic peening treatment (UPT). Int. J. Fatigue
**2009**, 31, 644–650. [Google Scholar] [CrossRef] - Lihavainen, V.M.; Marquis, G.; Statnikov, E.S. Fatigue strength of a longitudinal attachment improved by ultrasonic impact treatment. Weld. World
**2004**, 48, 67–73. [Google Scholar] [CrossRef] - Huo, L.; Wang, D.; Zhang, Y. Investigation of the fatigue behaviour of the welded joints treated by TIG dressing and ultrasonic peening under variable-amplitude load. Int. J. Fatigue
**2005**, 27, 95–101. [Google Scholar] [CrossRef] - Haagensen, P.J.; Alnes, Ø. Progress Report on IIW WG2 Round Robin Fatigue Testing Program on 700 MPa and 350 MPa YS Steels. IIW Doc, XIII-2081-05; International Institute of Welding: Paris, France, 2005. [Google Scholar]
- Leitner, M.; Ottersböck, M.; Pußwald, S.; Remes, H. Fatigue strength of welded and high frequency mechanical impact (HFMI) post-treated steel joints under constant and variable amplitude loading. Eng Struct
**2018**, 163, 215–223. [Google Scholar] [CrossRef] - Leitner, M.; Barsoum, Z.; Schäfers, F. Crack propagation analysis and rehabilitation by HFMI of pre-fatigued welded structures. Weld. World
**2016**, 60, 581–592. [Google Scholar] [CrossRef][Green Version] - Kim, K.S.; Chen, X.; Han, C.; Lee, H.W. Estimation methods for fatigue properties of steels under axial and torsional loading. Int. J. Fatigue
**2002**, 24, 783–793. [Google Scholar] [CrossRef] - Bäumel, A.; Seeger, T.; Boller, C. Materials Data for Cyclic Loading. Supplement 1. Materials science monographs; Elsevier: Amsterdam, The Netherlands, 1990; Volume 3. [Google Scholar]
- Korkmaz, S. Extension of the Uniform Material Law for High Strength Steels. Master’s Thesis, Bauhaus University Graduate School of Structural Engineering, Weimar, Germany, 2008. [Google Scholar]
- Roessle, M.L.; Fatemi, A. Strain-controlled fatigue properties of steels and some simple approximations. Int. J. Fatigue
**2000**, 22, 495–511. [Google Scholar] [CrossRef] - Dowling, N.E. Notched Member Fatigue Life Predictions Combining Crack Initiation and Propagation. Fatigue Fract. Eng. Mater. Struct.
**1979**, 2, 129–138. [Google Scholar] [CrossRef] - Yıldırım, H.C.; Marquis, G.; Sonsino, C.M. Lightweight design with welded high-frequency mechanical impact (HFMI) treated high-strength steel joints from S700 under constant and variable amplitude loadings. Int. J. Fatigue
**2016**, 91, 466–474. [Google Scholar] [CrossRef] - Suominen, L.; Khurshid, M.; Parantainen, J. Residual stresses in welded components following post-weld treatment methods. Procedia Eng.
**2013**, 66, 181–191. [Google Scholar] [CrossRef][Green Version] - Schubnell, J.; Eichheimer, C.; Ernould, C.; Maciolek, A.; Rebelo-Kornmeier, J.; Farajian, M. The influence of coverage for high frequency mechanical impact treatment of different steel grades. J. Mater. Process. Technol.
**2020**, 277, 116437. [Google Scholar] [CrossRef] - Schubnell, J.; Pontner, P.; Wimpory, R.C.; Farajian, M.; Schulze, V. The influence of work hardening and residual stresses on the fatigue behavior of high frequency mechanical impact treated surface layers. Int. J. Fatigue
**2020**, 134, 105450. [Google Scholar] [CrossRef] - METINVEST. Available online: https://metinvestholding.com/en/products/steel-grades/s690q (accessed on 12 May 2021).
- Wohlfahrt, H. Auswirkungen mechanischer Oberflächenbehandlungen auf das Dauerschwingverhalten unter Einschluss von Rissbildung und Rissausbreitung. In Mechanische Oberflächenbehandlungen; Wohlfahrt, H., Krull, P., Eds.; Wiley-VCH: Weinheim, Germany, 2000; pp. 56–85. [Google Scholar]

**Figure 1.**Untreated weld toe—AW (

**left**) and treated weld toe—HFMI (

**right**) on the specimen with longitudinal attachment.

Parameter | UML [10] | Extended UML [41] | HM [42] |
---|---|---|---|

K′ | 1.61 f_{u} | σ_{f}’/(ε_{f}’)n′ | 1.65 f_{u} |

n′ | 0.15 | b/c | 0.15 |

σ_{f}’ | 1.5 f_{u} | f_{u} (1 + ψ) | 4.25 HB + 225 |

ε_{f}’ | 0.59 ψ | 0.58 ψ + 0.01 | 1/E(0.32 (HB)^{2} − 487 (HB) + 191,000 |

b | −0.087 | −log (σ_{f}’/σ_{E})/6 | −0.057 to −0.14 |

σ_{E} * | 0.45 f_{u} | f_{u}(0.32 + ψ)/6 | - |

c | −0.58 | −0.58 | −0.39 to 1.04 |

ψ ** | 1.0 for (f_{u}/E) ≤ 3 × 10^{−3}1.375 − 125 (f _{u}/E) for (f_{u}/E) > 3 × 10^{−3}and ≥0 | 0.5(cos(π(f_{u} − 400)/2200) + 1) | - |

_{E}—technical endurance limit in terms of stress. ** ψ—dimensionless factor related to ultimate strength f

_{u}and elastic modulus E.

Steel Grade | S355 | S690 | S960 | ||||
---|---|---|---|---|---|---|---|

Parameter | Condition | min | max | min | max | min | max |

Stress concentration | AW | 2.6 | 4.0 | 2.6 | 4.0 | 2.6 | 4.0 |

HFMI | 1.8 | 2.2 | 1.8 | 2.2 | 1.8 | 2.2 | |

Residual stress (at surface) (MPa) | AW | 25 | 250 | 25 | 400 | 25 | 600 |

HFMI | −350 | −200 | −650 | −400 | −800 | −600 | |

Hardness (at surface) (HB) | AW | 146 | 187 | 230 | 240 | 320 | 320 |

HFMI | 240 | 290 | 240 | 270 | 320 | 340 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fuštar, B.; Lukačević, I.; Skejić, D.; Lukić, M. Two-Stage Model for Fatigue Life Assessment of High Frequency Mechanical Impact (HFMI) Treated Welded Steel Details. *Metals* **2021**, *11*, 1318.
https://doi.org/10.3390/met11081318

**AMA Style**

Fuštar B, Lukačević I, Skejić D, Lukić M. Two-Stage Model for Fatigue Life Assessment of High Frequency Mechanical Impact (HFMI) Treated Welded Steel Details. *Metals*. 2021; 11(8):1318.
https://doi.org/10.3390/met11081318

**Chicago/Turabian Style**

Fuštar, Boris, Ivan Lukačević, Davor Skejić, and Mladen Lukić. 2021. "Two-Stage Model for Fatigue Life Assessment of High Frequency Mechanical Impact (HFMI) Treated Welded Steel Details" *Metals* 11, no. 8: 1318.
https://doi.org/10.3390/met11081318