# Crack Initiation and Propagation Fatigue Life of Ultra High-Strength Steel Butt Joints

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Fatigue Assessment of Welds Considering Crack Initiation and Propagation

_{f}sums up the cycles of both phases N

_{init}and N

_{cp}, respectively. The transition between these two phases is referred to as technical crack initiation at a crack depth a

_{th}; its value varies depending on the selected model [3,4,5,6]. In the literature, various approaches exist for an assessment of the crack initiation phase, such as the fatigue notch factor K

_{f}[7,8,9], strain life approaches [10,11] or the notch stress intensity factor [12,13]. The crack propagation phase, on the other hand, is generally assessed using linear elastic fracture mechanics.

#### 1.2. Detection of the Crack Initiation Point

#### 1.3. Effect of Local Weld Geometry on Fatigue Performance

_{f}in order to take support effects into account [9] (see Equation (1)). In [33], this procedure leads to better matching results than the use of the actual transition radius.

_{ref}= 1 mm independent of the actual geometry. Thereby, a crack-like weld toe exhibiting an actual weld toe radius of ρ = 0 mm is assumed leading to a fictitious radius ρ

_{f}=ρ

_{ref}= 1 mm in Equation (1). This concept is part of the International Institute of Welding (IIW) guideline [38], verified and consistent for structural steels and aluminium alloys. However, it does not consider the actual weld toe geometry. Therefore, it tends to deliver conservative results especially for high-quality high-strength welds with extremely shallow weld toes.

_{f}is calculated from the stress concentration factor K

_{t}using the Peterson formula. The required parameter a*, the critical distance, is related to the ultimate tensile strength of the material.

_{eff}obtained by averaging of the stress distribution in depth σ(x) over the microstructural support length ρ

^{*}at the surface layer instead of the linear elastic stress maximum (see Equation (2)). The fraction of σ

_{eff}and the nominal stress σ

_{n}gives the fatigue relevant effective stress concentration factor K

_{f}(see Equation (3)).

^{*}are provided by Neuber himself for various materials in [8]. For welded structural steels exhibiting a microstructure similar to steel castings a microstructural support length of ρ

^{*}= 0.4 mm is suggested leading to the effective notch stress concept with a fictitious transition radius of ρ

_{ref}= 1 mm, which is mentioned above in Radaj’s concept. In [40], comprehensive investigations on the effect of weld geometry on the fatigue strength under consideration of the microstructural support length are published. Here, a significantly smaller microstructural support length of around ρ = 0.1 mm reveals the lowest scatter band of all fatigue test results. This is traced back to the high hardness of the weld, where Neuber’s initial suggestion for ferritic steel is more appropriate instead of cast steel.

_{init}= 0.05 mm. The work published in [42] presents a holistic study on the effects of the weld geometry applying the IBESS-model [43], a fracture mechanics-based prediction of the fatigue strength of welded joints. Mashiri et al. [44] carried out crack propagation analysis using the boundary element analysis. Their investigations on the effect of weld profile and undercuts on fatigue life shows the best fit at an initial crack length of a

_{init}= 0.1 mm. In [45,46], the Frost diagram is utilized to assess undercuts from a fracture mechanical perspective. The focus hereby lies in the establishment of undercut tolerances for industry.

## 2. Experimental Investigations

#### 2.1. Fatigue Tests

#### 2.2. Determination of Crack Propagation Parameters

_{P}and m

_{P}(see Equation (4)). A non-linear fitting procedure was used to determine these parameters for the present experimental data, whose result is plotted in Figure 4 as well. The consequential values are C

_{P}= 8.3509×10

^{−10}and m

_{P}= 1.721 for units of mm/cycle and MPa√mm, respectively.

## 3. Optical Detection of Crack Initiation and Propagation

#### 3.1. Experimental Setup

_{u}causing possible cracks to open. Before running the test, a reference image is taken at maximum loading as a basis of comparison, which is required by the subsequent post-testing digital image correlation procedure. At first, the possibility of synchronized image acquisition during fatigue testing was checked. However, as the image has to be taken exactly at the maximum load, a very short shutter time is required in order to ensure a clear picture with adverse impact on the image quality. Hence, the run cycle of the fatigue test stops at maximum load for the image acquisition for a few seconds (see Figure 6). Due to the short retention time and standard conditions around the specimen, holding the specimen at maximum load does not affect the crack initiation and propagation behaviour.

_{initial}. This initial cycle number is based on previous tests and strongly depends on the load level of the respective test. The a priori fatigue testing reference image and the image taken at the initial cycle number show no distinct difference, implying early-stage crack propagation. After the initial image acquisition, the fatigue test is continued block by block. The number of load cycles between two images N

_{interval}increases with the load cycle number according to Table 2. Both measures reduce the number of images to be taken without a significant decrease of the procedure’s accuracy. A final image of the broken specimen at N

_{f}provides insight on the failure-tripping crack in case of several initiated cracks.

#### 3.2. Crack Detection and Tracking Procedure

_{n}= 400 MPa, an image acquisition after N

_{initial}= 60,000 load cycles and N

_{f}= 85,039 load cycles until burst fracture.

#### 3.2.1. Local Distortion Field of the Specimen’s Surface

_{y}in the load direction of the specimen. This outcome includes the local displacement as well as a potential global movement of the specimen due to a small amount of sliding in the clamping area. Therefore, the vertical gradient of the vertical displacement s’

_{y}is introduced in order to highlight local changes in vertical displacement and simultaneously exclude possible global movement. The corresponding result of the displacement gradient is depicted in Figure 8b explicitly indicating the present cracks only.

#### 3.2.2. Crack Detection and Tracking

_{y,thres}of two pixels/pixel. If the maximum gradient value of an image exceeds forty pixels/pixel, the threshold value is adapted to five percent of the maximum value. The reason for this lies in the enlargement of the plastic zone at the crack tip with increasing crack length [55], which is considered by this adaptive threshold setting. It has to be noted that the definition and the course of the threshold value is, besides the parameters of the image correlation, the primary influencing factor regarding the finally-measured crack length. Therefore, it plays a decisive role in the procedure’s verification process (see Section 3.3.). Figure 9 shows the black-white image related to Figure 8b with a threshold value of s’

_{y,thres}= 2.39. Figure 10 plots the result of crack identification process in the gradient field of Figure 8b.

#### 3.3. Validation of the Procedure

#### 3.3.1. Beach Marks

#### 3.3.2. Fracture Surfaces

## 4. Fatigue Assessment

#### 4.1. Neuber’s Stress Averaging Method

^{*}in regard to microstructural support. As already stated in Section 1, suggestions for this parameter often deviate from determinations using the lowest scatter-band of the resultant S-N curve as a key factor, especially in the case of welded specimens.

_{eff}(ρ

^{*}) and the corresponding fatigue notch factor K

_{f}(ρ

^{*}) are calculated for each specimen at the respective points of crack initiation. The required stress distribution in depth σ(x) is taken from the numerical analysis of the actual specimen’s weld topography. Hereby, the value of ρ

^{*}is varied in order to obtain the S-N curve with minimum scatter-band, where a minimum value of ρ

^{*}= 0.1 mm must be maintained due to the mesh seed of about 30 µm in depth. Figure 16 shows the development of the consequential S-N curves for specimen fracture with varying ρ

^{*}including the P

_{S}= 97.7% line of the statistical evaluation. Additionally, the curves for nominal stress Δσ

_{n}and notch stress Δσ

_{notch}using the individual stress concentration factors K

_{t}are displayed. The result of the statistical evaluation for crack initiation and total fatigue life for effective notch stress using selected values of ρ

^{*}as well as nominal and notch stress are listed in Table 5; Table 6 provides the corresponding data for each specimen.

_{σ}and the fatigue class FAT of crack initiation fatigue life (a) and total fatigue life (b) with respect to ρ

^{*}. Thereby, the statistical evaluation of the resulting S-N curves according to [48] is performed separately for each selected ρ

^{*}. In the case of technical crack initiation, the least scattering is obtained by the minimum evaluated microstructural support length of ρ

^{*}= 0.1 mm. The least scattering for total fatigue life is reached by ρ

^{*}= 0.13 mm. In both cases, the consequential scatter-band shows very low values of 1:T

_{σ}= 1.081 and 1:T

_{σ}= 1.088. This result corresponds to the findings in [40], where a similar procedure yields in a stress averaging length of ρ

^{*}= 0.05–0.1 mm for the minimum scatter-band. Whereas the slope of the evaluated S-N curve s remains rather constant, the FAT value naturally increases with decreasing averaging length. In Figure 18, the resulting effective S-N curve is pictured showing a significant decrease in scattering compared to the nominal stress results in Figure 2.

#### 4.2. Crack Initiation and Propagation Life

_{th}= 0.5 mm for all investigated specimens [3,4,63]. According to Equation (5), the corresponding surface crack length is 2c = 2.9 mm, giving a rather low crack aspect ratio of a/c = 0.345. This value is quite comprehensible, as the local stress concentration at the weld toe decreases rapidly in the sheet thickness direction and therefore encourages the crack to grow in the surface direction first [64,65]. The fracture surface in Figure 14b exhibits a small crack on the left hand side marked by a yellow arrow exhibiting a very low aspect ratio. A detailed measurement reveals an aspect ratio of a/c = 0.35 validating the result of Equation (5) for the present case. The crack detection procedure delivers the surface crack length in quite coarse steps of load cycle numbers. In order to determine the according threshold load cycle number N

_{init}for a

_{th}, linear interpolation is applied.

_{th}= 0.5 mm in green dye. Table 6 summarizes the effective notch stress fatigue test data. Furthermore, the statistical evaluation including lines of 50% and 97.7% probability of survival are depicted. Some specimens exhibit more than one propagating crack leading to a number of fourteen crack initiation points of ten specimens. In such cases, the effective notch stress values are evaluated at the respective position of the crack initiation points of the particular specimen.

_{S}-value. The result of Equation (6) is depicted in Figure 19 for P

_{S}= 97.7%.

#### 4.3. Assessment of Crack Propagation Life by Fracture Mechanics

- Start of calculation at the previously determined threshold load cycle number N
_{th}with an initial crack length of a_{init}= a_{th}= 0.5 mm. - Calculation from the test start with an initial crack length of a
_{init}= u + 0.1 mm as recommended in [38].

- S1100 base material determined in Section 2.2.;
- IIW parameters for welds suggested in [38]; and
- Best fit parameters resulting by a least square fit of the fatigue test results.

#### 4.3.1. Crack Propagation Analysis

#### 4.3.2. Weight Functions Approach

_{1a}–M

_{3a}and M

_{1c}–M

_{3c}is covered by many publications for various crack shapes. A sound approach is provided in [69]. However, [71] offers a refined set of formulas enabling a wider range of crack proportions; therefore, these are selected for the present work.

#### 4.3.3. Stress Distribution

_{m}, a correction for the stress distribution away from the stress peak, can be found in the cited work. This set of equations is also referred to and listed in the IIW recommendations [38].

#### 4.3.4. Calculation of Crack Propagation

_{f}to ensure at least fifty steps to a final crack length ${a}_{f}=0.9t$ where burst fracture is assumed.

#### 4.3.5. Determination of “Best Fit” Parameters for the Paris Power Law

_{P}and m

_{P}are varied within a meaningful range of $1\times {10}^{-13}\le {C}_{P}\le 1\times {10}^{-7}$ and $1.5\le {m}_{P}\le 3$.

#### 4.3.6. Crack Propagation Results

_{init}= 0.5 mm, is illustrated in Figure 21. Case 1 one employs the crack propagation parameters of the S1100 base material. The IIW parameters for welded joints in case 2 deliver a conservative approach across all considered specimens. In general, the use of these parameters leads to a very short crack propagation phase. The application of the S1100 base material parameters in case 1 leads to a non-conservative assessment for specimens with burst fracture below 1 × 10

^{5}load cycles. Naturally, the best fit parameters show the best accordance, as expected. Hereby, the exponent m

_{P}is similar, but C

_{P}is about three times smaller compared to the IIW values. Interestingly, all three parameter sets underestimate the crack propagation phase of the three specimens with the burst fracture above one million load cycles, significantly.

^{5}load cycles.

## 5. Summary and Conclusions

_{t}of about 1.9–2.5 for regions without undercut compared to values for K

_{t}from 3.5–5.4 for local undercuts.

^{*}= 0.1 mm for technical crack initiation and ρ

^{*}= 0.13 mm for burst fracture are obtained for minimum scattering regression analysis. These results show good accordance with the published results for similar steel grades.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$a$ | Crack depth (mm) |

${a}_{f}$ | Final crack depth for crack propagation calculation (mm) |

a_{init} | Initial crack depth for crack propagation calculation (mm) |

${a}_{th}$ | Threshold crack depth between crack initiation and propagation (mm) |

$a/c$ | Crack aspect ratio (–) |

$c$ | Half surface crack length, half width of elliptical crack (mm) |

${C}_{P}$ | Crack growth rate coefficient according to Paris law (ΔK in MPa√mm; da/dN in mm/cycle) |

$E$ | Young’s modulus (MPa) |

$FAT$ | Fatigue class according to IIW, stress range Δσ at N = 2·10^{6} load cycles and P_{S} = 97.7% (MPa) |

$K$ | Stress intensity factor (MPa√mm) |

${K}_{f}$ | Fatigue notch factor (–) |

${K}_{t}$ | Stress concentration factor (–) |

$k$ | Inverse slope of S/N-curve (–) |

${M}_{a}$ | Weight function parameters for deepest point of a surface crack |

${m}_{a}\left(x,a\right)$ | Weight function for deepest point of surface crack |

${M}_{c}$ | Weight function parameters for surface point of a surface crack |

${m}_{c}\left(x,a\right)$ | Weight function for surface point of surface crack |

m_{p} | Slope of crack growth rate curve according to Paris law (–) |

$N$ | Load cycle number (–) |

${N}_{f}$ | Load cycle number at specimen burst fracture (–) |

${N}_{initial}$ | Load cycle number before start of image acquisition (–) |

${N}_{interval}$ | Interval load cycle number between two image acquisitions (–) |

${N}_{init}$ | Load cycle number at crack length a_{th}, threshold between crack initiation and propagation (–) |

${N}_{cp}$ | Load cycle number of crack propagation until burst fracture (–) |

${N}_{k}$ | Transition knee point of S/N-curve (–) |

${P}_{S}$ | Probability of survival (–) |

$R$ | Load stress ratio (–) |

${s}_{y}$ | Displacement in y-direction (specimen loading direction) (pixel) |

${{s}^{\prime}}_{y}$ | Gradient of displacement in y-direction (–) |

${{s}^{\prime}}_{y,\mathrm{thres}}$ | Threshold for gradient of displacement in y-direction (–) |

$SSE$ | Sum of squared errors (–) |

$t$ | Sheet thickness (mm) |

$1:{T}_{\sigma}$ | Scatter index of S/N-curve, ratio of stress range Δσ at P_{S} = 10% and P_{S} = 90% (–) |

$\rho $ | Weld toe radius (mm) |

${\rho}^{\ast}$ | Microstructural support length (mm) |

$\theta $ | Weld flank angle (°) |

$\sigma $ | Stress (MPa) |

Recurring indices | |

$\mathrm{eff}$ | Effective value |

$n$ | Nominal value |

$notch$ | Notch value |

$\u2206$ | Range, difference of upper and lower value |

## References

- Radaj, D. Review of fatigue strength assessment of nonwelded and welded structures based on local parameters. Int. J. Fatigue
**1996**, 18, 153–170. [Google Scholar] [CrossRef] - Lassen, T.; Recho, N. Fatigue Life Analyses of Welded Structures; ISTE: London, UK; Newport Beach, CA, USA, 2006. [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 weldment fatigue crack initiation life. Int. J. Fatigue
**1997**, 19, 537–541. [Google Scholar] [CrossRef] - Remes, H. Strain-based Approach to Fatigue Strength Assessment of Laser-welded Joints. Ph.D. Thesis, Helsinki University of Technology, Espoo, Finland, 2008. [Google Scholar]
- Lihavainen, V.-M. A Novel Approach for Assessing the Fatigue Strenght of Ultrasonic Impact Treated Welded Structures. Ph.D. Thesis, Lappeenranta University of Technology, Lappeenranta, Finland, 2006. [Google Scholar]
- Peterson, R.E. Notch sensitivity. In Metal Fatigue; Sines, G., Waisman, J.L., Dolan, T.J., Eds.; 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; Abington: Cambridge, UK, 1990. [Google Scholar]
- Lawrence, F.V.; Ho, N.J.; Mazumdar, P.K. Predicting the fatigue resistance of welds. Ann. Rev. Mater. Sci.
**1981**, 11, 401–425. [Google Scholar] [CrossRef] - Seeger, T. Grundlagen für Betriebsfestigkeitsnachweise. Stahlbau-Handbuch, 3rd ed.; Stahlbau-Verl.-Ges: Köln, Germany, 1996; pp. 5–123. [Google Scholar]
- 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] - Haibach, E. Betriebsfestigkeit. Verfahren und Daten zur Bauteilberechnung; Springer: Berlin, Germany, 2006. [Google Scholar]
- Krebs, J.; Hübner, P.; Kassner, M. Eigenspannungseinfluss auf Schwingfestigkeit und Bewertung in geschweißten Bauteilen; DVS-Verlag: Düsseldorf, Germany, 2004. [Google Scholar]
- Lieurade, H.-P.; Huther, I.; Maddox, S.J. Recommendations on the Fatigue Testing of Welded Components; LETS Global: Rotterdam, The Netherlands, 2006. [Google Scholar]
- Stoschka, M.; Leitner, M.; Fössl, T.; Posch, G. Effect of high-strength filler metals on fatigue. Weld. World
**2012**, 56, 20–29. [Google Scholar] [CrossRef] - Maddox, S.J. The effect of mean stress on fatigue crack propagation a literature review. Int. J. Fract.
**1975**, 11, 389–408. [Google Scholar] - Verreman, Y.; Nie, B. Short-crack growth and coalescence along the toe of a manual fillet weld. Fatigue Frac. Eng. Mat. Struct.
**1991**, 14, 337–349. [Google Scholar] [CrossRef] - Fricke, W. Fatigue analysis of welded joints: State of development. Mar. Struct.
**2003**, 16, 185–200. [Google Scholar] [CrossRef] - Baumgartner, J.; Bruder, T. Influence of weld geometry and residual stresses on the fatigue strength of longitudinal stiffeners. Weld. World
**2013**, 57, 841–855. [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] - Baumgartner, J.; Waterkotte, R. Crack initiation and propagation analysis at welds—Assessing the total fatigue life of complex structures. Mat. Wiss. Werkstofftech.
**2015**, 46, 123–135. [Google Scholar] [CrossRef] - Simunek, D.; Leitner, M.; Grün, F. In-situ crack propagation measurement of high-strength steels including overload effects. Proc. Eng.
**2018**, 213, 335–345. [Google Scholar] [CrossRef] - Simunek, D.; Leitner, M.; Maierhofer, J.; Gänser, H.-P. Crack growth under constant amplitude loading and overload effects in 1:3 scale specimens. Proc. Struct. Integr.
**2017**, 4, 27–34. [Google Scholar] [CrossRef] - Todorov, E.I.; Mohr, W.C.; Lozev, M.G.; Thompson, D.O.; Chimenti, D.E. Detection and Sizing of Fatigue Cracks in Steel Welds with Advanced Eddycurrent techniques. In Proceedings of the AIP Conference—34th Annual Review of Progress in Quantitative Nondestructive Evaluation, Golden, CO, USA, 22–27 July 2007; pp. 1058–1065. [Google Scholar]
- Lamtenzan, D.; Glenn, W.; Lozev, M.G. Detection and Sizing of Cracks in Structural Steel Using the Eddy Current Method; US Department of Transportation Federal Highway Administration FHWA-RD-00-018; Turner-Fairbank Highway Research Center: McLean, VA, USA, 2000.
- Roux, S.; Réthoré, J.; Hild, F. Digital image correlation and fracture: An advanced technique for estimating stress intensity factors of 2D and 3D cracks. J. Phys. D Appl. Phys.
**2009**, 42, 214004. [Google Scholar] [CrossRef] - Mathieu, F.; Hild, F.; Roux, S. Identification of a crack propagation law by digital image correlation. Int. J. Fatigue
**2012**, 36, 146–154. [Google Scholar] [CrossRef] [Green Version] - Ozelo, R.R.M.; Sollero, P.; Sato, M.; Barros, R.S.V. Monitoring crack propagation using digital image correlation and cod technique. In Proceedings of the COBEM 2009 20th International Congress of Mechanical Engineering, ABCM, Gramado, Brazil, 15–20 November 2009. [Google Scholar]
- Alam, M.M.; Barsoum, Z.; Jonsén, P.; Kaplan, A.F.H.; Häggblad, H.Å. The influence of surface geometry and topography on the fatigue cracking behaviour of laser hybrid welded eccentric fillet joints. Appl. Surf. Sci.
**2010**, 256, 1936–1945. [Google Scholar] [CrossRef] - Caccese, V.; Blomquist, P.A.; Berube, K.A.; Webber, S.R.; Orozco, N.J. Effect of weld geometric profile on fatigue life of cruciform welds made by laser/GMAW processes. Mar. Struct.
**2006**, 19, 1–22. [Google Scholar] [CrossRef] - Lillemäe, I.; Remes, H.; Liinalampi, S.; Itävuo, A. Influence of weld quality on the fatigue strength of thin normal and high strength steel butt joints. Weld. World
**2016**, 60, 731–740. [Google Scholar] [CrossRef] - Rennert, R.; Kullig, E.; Vormwald, M.; Esderts, A.; Siegele, D. Analytical Strength Assessment of Components Made of Steel, Cast Iron and Aluminium Materials in Mechanical Engineering: FKM Guideline, 6th ed.; VDMA-Verl: Frankfurt, Germany, 2012. [Google Scholar]
- Olivier, R.; Köttgen, V.B.; Seeger, T. Welded Joints I—Fatigue Strength Assessment Method for Welded Joints Based on Local Stresses; Forschungshefte No. 143; Forschungskuratorium Maschinenbau: Frankfurt, Germany, 1989. [Google Scholar]
- Olivier, R.; Köttgen, V.B.; Seeger, T. Welded Joints II—Investigation of Inclusion into Codes of a Novel Fatigue Strength Assessment Method for Welded Joints in Steel; No. 180; Forschungskuratorium Maschinenbau: Frankfurt, Germany, 1994. [Google Scholar]
- Morgenstern, C.; Sonsino, C.M.; Hobbacher, A.; Sorbo, F. Fatigue design of aluminium welded joints by the local stress concept with the fictitious notch radius of rf = 1 mm. Int. J. Fatigue
**2006**, 28, 881–890. [Google Scholar] [CrossRef] - Hobbacher, A. Recommendations for Fatigue Design of Welded Joints and Components, 2nd ed.; Springer: New York, NY, USA, 2016. [Google Scholar]
- Nykänen, T.; Björk, T.; Laitinen, R. Fatigue strength prediction of ultra high strength steel butt-welded joints. Fatigue Frac. Eng. Mater. Struct.
**2013**, 36, 469–482. [Google Scholar] [CrossRef] - Liinalampi, S.; Remes, H.; Lehto, P.; Lillemäe, I.; Romanoff, J.; Porter, D. Fatigue strength analysis of laser-hybrid welds in thin plate considering weld geometry in microscale. Int. J. Fatigue
**2016**, 87, 143–152. [Google Scholar] [CrossRef] - Nykänen, T.; Marquis, G.; Björk, T. Effect of weld geometry on the fatigue strength of fillet welded cruciform joints. In Proceedings of the International Symposium on Integrated Design and Manufacturing of Welded Structures; Lappeenranta University of Technology: Lappeenranta, Finland, 2007. [Google Scholar]
- Schork, B.; Kucharczyk, P.; Madia, M.; Zerbst, U.; Hensel, J.; Bernhard, J.; Tchuindjang, D.; Kaffenberger, M.; Oechsner, M.; Zerbst, U. The effect of the local and global weld geometry as well as material defects on crack initiation and fatigue strength. Eng. Fract. Mech.
**2018**, 198, 103–122. [Google Scholar] [CrossRef] - Madia, M.; Zerbst, U.; Th. Beier, H.; Schork, B. The IBESS model—Elements, realisation and validation. Eng. Fract. Mech.
**2018**, 198, 171–208. [Google Scholar] [CrossRef] - Mashiri, F.R.; Zhao, X.L.; Grundy, P. Effects of weld profile and undercut on fatigue crack propagation life of thin-walled cruciform joint. Thin Walled Struct.
**2001**, 39, 261–285. [Google Scholar] [CrossRef] - Steimbreger, C.; Chapetti, M.D. Fatigue strength assessment of butt-welded joints with undercuts. Int. J. Fatigue
**2017**, 105, 296–304. [Google Scholar] [CrossRef] - Steimbreger, C.; Chapetti, M.; Hénaff, G. Undercut tolerances in industry from a fracture mechanic perspective. MATEC Web Conf.
**2018**, 165, 21009. [Google Scholar] [CrossRef] - Ottersböck, M.J.; Leitner, M.; Stoschka, M.; Maurer, W. Effect of Weld Defects on the Fatigue Strength of Ultra High-strength Steels. Proc. Eng.
**2016**, 160, 214–222. [Google Scholar] [CrossRef] [Green Version] - ASTM. Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (-N) Fatigue Data; E739; ASTM International: West Conshohocken, PA, USA, 2015. [Google Scholar]
- Dengel, D.; Harig, H. Estimation of the fatigue limit by progressively-increasing load tests. Fatigue Frac. Eng. Mat. Struct.
**1980**, 3, 113–128. [Google Scholar] [CrossRef] - Sonsino, C.M. Course of SN-curves especially in the high-cycle fatigue regime with regard to component design and safety. Int. J. Fatigue
**2007**, 29, 2246–2258. [Google Scholar] [CrossRef] - Ottersböck, M.J.; Leitner, M.; Stoschka, M. Characterisation of actual weld geometry and stress concentration of butt welds exhibiting local undercuts. Eng. Fail. Anal.
**2019**. in review. [Google Scholar] - ASTM. Standard Test Method for Measurement of Fatigue Crack Growth Rates; E647; ASTM International: West Conshohocken, PA, USA, 2000. [Google Scholar]
- Tada, H.; Paris, P.C.; Irwin, G.R. The Stress Analysis of Cracks Handbook, 3rd ed.; ASME: New York, NY, USA, 2000. [Google Scholar]
- Beden, S.M.; Abdullah, S.; Ariffin, A.K. Review of fatigue crack propagation models for metallic components. Eur. J. Sci. Res.
**2009**, 28, 364–397. [Google Scholar] - Richard, H.A.; Sander, M. Ermüdungsrisse. In Erkennen, Sicher Beurteilen, Vermeiden, 3rd ed.; Springer Vieweg: Wiesbaden, Germany, 2012. [Google Scholar]
- Paris, P.; Erdogan, F. A critical analysis of crack propagation laws. J. Basic Eng.
**1963**, 85, 528. [Google Scholar] [CrossRef] - Jones, E.M.C. Improved Digital Image Correlation; Version 4; University of Illinois: Champaign, IL, USA, 2015. [Google Scholar]
- Reu, P.L.; Toussaint, E.; Jones, E.M.C.; Bruck, H.A.; Iadicola, M.; Balcaen, R.; Turner, D.Z.; Siebert, T.; Lava, P.; Simonsen, M. DIC Challenge: Developing images and guidelines for evaluating accuracy and resolution of 2D analyses. Exp. Mech.
**2018**, 58, 1067–1099. [Google Scholar] [CrossRef] - Jones, E.M.C. Documentation for Matlab-Based DIC Code; Version 4; University of Illinois: Champaign, IL, USA, 2015. [Google Scholar]
- Ottersböck, M.J. Einfluss von Imperfektionen auf die Schwingfestigkeit hochfester Stahlschweißverbindungen. Ph.D. Thesis, Montanuniversität Leoben, Leoben, Austria, 2019. [Google Scholar]
- Simunek, D.; Leitner, M.; Maierhofer, J.; Gänser, H.-P. Fatigue crack growth under constant and variable amplitude loading at semi-elliptical and V-notched steel specimens. Proc. Eng.
**2015**, 133, 348–361. [Google Scholar] [CrossRef] - Engesvik, K.M. Analysis of Uncertainties in the Fatigue Capacity of Welded Joints. Ph.D. Thesis, University of Trondheim, Trondheim, Norway, 1981. [Google Scholar]
- Baumgartner, J. Enhancement of the fatigue strength assessment of welded components by consideration of mean and residual stresses in the crack initiation and propagation phases. Weld. World
**2016**, 60, 547–558. [Google Scholar] [CrossRef] - Radaj, D.; Sonsino, C.M.; Fricke, W. Fatigue Assessment of Welded Joints by Local Approaches, 2nd ed.; Woodhead Publishing: Sawston, UK, 2006. [Google Scholar]
- Goyal, R.; Glinka, G. Fracture mechanics-based estimation of fatigue lives of welded joints. Weld. World
**2013**, 57, 625–634. [Google Scholar] [CrossRef] - Newman, J.C.; Raju, I.S. Stress-intensity factor equations for cracks in three-dimensional finite bodies, ASTM STP 791. In Fracture Mechanics: Fourteenth Symposium—Volume I: Theory and Analysis; Lewis, J.C., Sines, G., Eds.; ASTM International: West Conshohocken, PA, USA, 1983; pp. I238–I265. [Google Scholar]
- Murakami, Y.; Murakami, Y. Stress Intensity Factors Handbook; Pergamon Press: Oxford, UK, 1990. [Google Scholar]
- Bueckner, H.F. Novel principle for the computation of stress intensity factors. J. Appl. Math. Mech.
**1970**, 50. [Google Scholar] - Shen, G.; Glinka, G. Weight functions for a surface semi-elliptical crack in a finite thickness plate. Theor. Appl. Fract. Mech.
**1991**, 15, 247–255. [Google Scholar] [CrossRef] - Glinka, G.; Shen, G. Universal features of weight functions for cracks in mode I. Eng. Fract. Mech.
**1991**, 40, 1135–1146. [Google Scholar] [CrossRef] - Wang, X.; Lambert, S.B. Stress intensity factors for low aspect ratio semi-elliptical surface cracks in finite-thickness plates subjected to nonuniform stresses. Eng. Fract. Mech.
**1995**, 51, 517–532. [Google Scholar] [CrossRef] - Hall, M.S.; Topp, D.A.; Dover, W.D. Parametric Equations for Stress Intensity Factors in Weldments; Project Report TSC/MSH/0244; Technical Software Consultants Ltd.: Milton Keynes, UK, 1990. [Google Scholar]
- Monahan, C.C. Early Fatigue Crack Growth at Welds; Computational Mechanics: Southampton, UK, 1995. [Google Scholar]

**Figure 1.**Specimen with weld toe exhibiting a local undercut [47].

**Figure 2.**Fatigue test results according to the nominal stress evaluation [47].

**Figure 3.**Fracture surface clearly showing the crack initiation point at the undercut (red arrow) [47] (Δσ

_{n}= 200 MPa, N

_{f}= 2.87×10

^{6}, R = 0.1).

**Figure 7.**Reference and last image of an exemplary specimen, Δσ

_{n}= 400 MPa, N

_{f}= 85,039: (

**a**) Reference image before fatigue test; (

**b**) image at N = 84,000 with crack at upper weld toe; (

**c**) magnification of the crack region of (

**b**).

**Figure 8.**Result of image correlation at N = 84,000: (

**a**) Displacement field s

_{y}in vertical (y)-direction; (

**b**) vertical gradient of vertical displacement field s’

_{y}.

**Figure 9.**Binarized matrix of Figure 8 indicating crack regions.

**Figure 10.**Gradient field s’

_{y}of Figure 8 with marked results of the crack identification process.

**Figure 12.**Position and extension of determined cracks marked at specimen surface and comparison of the crack identification procedures result with the final fracture surface: (

**a**) N = 60,000; (

**b**) N = 84,000; (

**c**) fracture surface at N

_{f}= 85,039 with three clearly visible crack initiation points.

**Figure 13.**Comparison of beach marks length and assessed crack length: (

**a**) Fracture surface with the surface length measurement of visible bench marks; (

**b**) development of the crack length over the load cycle number.

**Figure 14.**Fracture surfaces used for validation of the calculated crack length: (

**a**) Specimen 1; Δσ

_{n}= 400 MPa, N

_{f}= 85,039; (

**b**) specimen 2; Δσ

_{n}= 600 MPa, N

_{f}= 30,871.

**Figure 15.**Measurement of not-through crack size by fitting a semi-ellipse: (

**a**) Specimen 1; (

**b**) specimen 2.

**Figure 16.**S-N curves for specimen fracture depending upon the microstructural support length ρ

^{*}.

**Figure 17.**Development of slope, scatter index and FAT value with microstructural support length; (

**a**) Crack initiation; (

**b**) burst fracture.

**Figure 18.**S-N curve including crack initiation life and cycles to final rupture of all investigated specimens.

**Figure 19.**The fraction of the crack initiation life in total fatigue life over the effective notch stress range.

Material | Yield Strength σ_{y} (MPa) | Tensile Strength σ_{u} (MPa) | Elongation A (%) | Impact Work ISO-V KV (J) |
---|---|---|---|---|

Base, S1100 | ≥1100 | ≥1140 | ≥8 | ≥ 27 @ −20 °C |

Filler, T89 | ≥890 | ≥940 | ≥15 | ≥ 47 @ −40 °C |

Current Load Cycle Number N (–) | Interval between Image Acquisition N_{interval} (–) | ||
---|---|---|---|

0≤ | N | <100,000 | 2000 |

100,000≤ | N | <400,000 | 5000 |

400,000≤ | N | <800,000 | 10,000 |

800,000≤ | N | <1500,000 | 25,000 |

1500,000 ≤ | N | 50,000 |

Beach Mark | Measured Crack Length 2c (mm) | Assessed Crack Length 2c (mm) | Deviation (%) | |
---|---|---|---|---|

No. | N (–) | |||

1 | 740,000 | 17.94 | 18.23 | 1,62 |

2 | 700,000 | 11.98 | 11.97 | −0,08 |

3 | 660,000 | 9.52 | 9.52 | 0 |

4 | 620,000 | 8.00 | 7.92 | −1.00 |

5 | 580,000 | 6.84 | 6.72 | −1.75 |

6 | 540,000 | 6.00 | 6.00 | 0 |

**Table 4.**Crack propagation of selected not-through cracks and comparison to fracture surface measurement.

Specimen 1 | Specimen 2 | ||||
---|---|---|---|---|---|

Number of Load Cycles N (–) | Assessed Crack Length 2c (mm) | Measured Crack Length 2c after Final Rupture (mm) | Number of Load Cycles N (–) | Assessed Crack Length 2c (mm) | Measured Crack Length 2c after Final Rupture (mm) |

60,000 | 1.41 | 7.76 | 20,000 | - | 3.70 |

62,000 | 2.31 | 22,000 | - | ||

64,000 | 2.56 | 24,000 | 1.19 | ||

66,000 | 2.70 | 26,000 | 1.70 | ||

68,000 | 2.95 | 28,000 | 2.39 | ||

70,000 | 3.34 | 30,000 | 3.24 | ||

72,000 | 3.59 | 30,871 | 3.67 * | ||

74,000 | 3.98 | ||||

76,000 | 4.24 | ||||

78,000 | 4.49 | ||||

80,000 | 5.01 | ||||

82,000 | 5.65 | ||||

84,000 | 6.93 | ||||

85,039 | 7.66 * |

Evaluation Method | Crack Initiation | Burst Fracture | ||||
---|---|---|---|---|---|---|

Slope k (–) | FAT Value (MPa) | Scatter Index 1:T_{σ} (–) | Slope k (–) | FAT Value (MPa) | Scatter Index 1:T_{σ} (–) | |

Nominal stress | 4.43 | 160 | 1.30 | 4.20 | 176 | 1.29 |

Eff. notch stress (ρ^{*} = 0.4 mm) | 4.68 | 307 | 1.16 | 4.46 | 337 | 1.15 |

Eff. notch stress (ρ^{*} = 0.3 mm) | 4.68 | 340 | 1.14 | 4.45 | 372 | 1.13 |

Eff. notch stress (ρ^{*} = 0.2 mm) | 4.66 | 392 | 1.11 | 4.44 | 427 | 1.10 |

Eff. notch stress (ρ^{*} = 0.13 mm) | 4.64 | 452 | 1.09 | 4.43 | 489 | 1.09 |

Eff. notch stress (ρ^{*} = 0.1 mm) | 4.62 | 489 | 1.08 | 4.43 | 523 | 1.10 |

Notch stress (K_{t}) | 4.37 | 626 | 1.41 | 4.38 | 650 | 1.51 |

**Table 6.**Overview on local stress concentration and fatigue test data for each crack initiation point.

Specimen No. | K_{t} (–) | K_{f} (–) | Δσ_{eff} (MPa) | N_{th} (–) | N_{f} (–) | N_{th} /N_{f} (%) | |
---|---|---|---|---|---|---|---|

ρ = 0.10 mm | ρ = 0.13 mm | ||||||

1 | 3.93 | 2.60 | 2.42 | 1453.5 | 10,000 | 23,497 | 43 |

4.50 | 2.69 | 2.49 | 1492.2 | 10,520 | - | - | |

2 | 3.37 | 2.36 | 2.21 | 884.5 | 142,840 | 199,292 | 71 |

3 | 3.97 | 2.60 | 2.40 | 479.8 | 1601,800 | 2281,981 | 70 |

4 | 5.61 | 2.89 | 2.63 | 1052.6 | 52,650 | 85,039 | 53 |

4.75 | 2.83 | 2.60 | 1039.5 | 58,530 | - | - | |

4.45 | 2.71 | 2.50 | 1001.2 | 67,580 | - | - | |

5 | 5.82 | 2.80 | 2.57 | 513.6 | 1596,300 | 2873,617 | 52 |

6 | 3.50 | 2.33 | 2.20 | 660.0 | 635,190 | 728,340 | 87 |

7 | 3.92 | 2.36 | 2.21 | 441.5 | 2610,560 | 3958,013 | 67 |

8 | 3.33 | 2.24 | 2.10 | 1259.4 | 24,000 | 37,839 | 66 |

4.60 | 2.40 | 2.20 | 1322.7 | 19,330 | - | - | |

9 | 3.81 | 2.49 | 2.30 | 1378.7 | 13,790 | 25,060 | 52 |

10 | 4.61 | 3.06 | 2.84 | 852.9 | 155,070 | 232,717 | 67 |

**Table 7.**Summary of the investigated crack propagation cases including the statistical evaluation of the consequential S-N curves.

Start Crack Length a_{init} (mm) | Start Load Cycle Number N_{start} (–) | Parameters for Paris Law | SSE (–) | Statistical Evaluation of S-N Curve | |||||
---|---|---|---|---|---|---|---|---|---|

Material | C_{P} | m_{P} | Slope k (–) | FAT Value (MPa) | Scatter Index 1:T_{σ} (–) | ||||

Case 1 | 0.5 | N_{th} | S1100 | 8.35×10^{−10} | 1.72 | 0.141 | 3.80 | 453 | 1.077 |

Case 2 | 0.5 | N_{th} | IIW | 5.21×10^{−13} | 3.00 | 0.259 | 4.47 | 458 | 1.076 |

Case 3 | 0.5 | N_{th} | Best fit | 1.78×10^{−13} | 2.90 | 0.065 | 4.17 | 467 | 1.084 |

Case 4 | u + 0.1 | 0 | S1100 | 8.35×10^{−10} | 1.72 | 6.710 | 1.73 | 92 | 1.239 |

Case 5 | u + 0.1 | 0 | IIW | 5.21×10^{−13} | 3.00 | 14.251 | 3.03 | 147 | 1.243 |

Case 6 | u + 0.1 | 0 | Best fit | 1.00×10^{−13} | 2.85 | 0.985 | 2.92 | 339 | 1.255 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/app9214590

**AMA Style**

Ottersböck MJ, Leitner M, Stoschka M, Maurer W.
Crack Initiation and Propagation Fatigue Life of Ultra High-Strength Steel Butt Joints. *Applied Sciences*. 2019; 9(21):4590.
https://doi.org/10.3390/app9214590

**Chicago/Turabian Style**

Ottersböck, Markus J., Martin Leitner, Michael Stoschka, and Wilhelm Maurer.
2019. "Crack Initiation and Propagation Fatigue Life of Ultra High-Strength Steel Butt Joints" *Applied Sciences* 9, no. 21: 4590.
https://doi.org/10.3390/app9214590