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International Journal of Fatigue 188 (2024) 108501
Available online 17 July 2024
0142-1123/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Contents lists available at ScienceDirect
International Journal of Fatigue
journal homepage: www.elsevier.com/locate/ijfatigue
Review
Measurements and prediction of extreme defect distributions for fatigue
assessment in multi-pass weld of 13%Cr–4%Ni alloy for hydraulic turbines
A. D’Andrea a,∗, M. Gagnon b, S. Beretta c,d, P. Bocher a
aÉcole de technologie supérieure, 1100 Notre-Dame St W, Montréal, Québec H3C 1K3, Canada
bInstitut de recherche d’Hydro-Québec (IREQ), 1800 Bd Lionel-Boulet, Varennes, QC J3X 1S1, Canada
cPolitecnico di Milano, Department of Mechanical Engineering, via La Masa 1, 20156 Milano, Italy
dAuburn University, National Center for Additive Manufacturing Excellence, Department of Mechanical Engineering, Auburn, AL 36849, United States of America
ARTICLE INFO
Keywords:
Martensite stainless steel
13%Cr–4%Ni alloy
410NiMo
Defects analysis
Francis runner
Hydraulic turbine
Metallography
X-ray
Kitagawa–Takahashi diagram
Extreme Value Statistics
Fatigue limit
Flux cored arc welding (FCAW)
ABSTRACT
Flux cored arc welding (FCAW) is a manufacturing process commonly used for hydro-turbine application,
especially in the assembly of Francis runners made of 13%Cr–4%Ni stainless steel. The welded connection
between the blade and band/crown is particularly susceptible to high stress levels due to the design of the
runner and the presence of welding defects which act as localized stress concentrations. This research aims to
predict the effect of these defects on the performance of the runner during operating condition. To investigate
the occurrence of the defects, the distributions of the observed discontinuity are measured and analyzed on
T-shape joint specimens using metallographic and X-ray 2D images. The data obtained from these two methods
of acquisition are compared to identify any discrepancy. Extreme Value Statistic (EVS) is used to estimate the
likely occurrence of the largest defects within the welded runner zone. Based on these findings, a probabilistic
Kitagawa–Takahashi diagram with the El-Haddad approximation is adopted to quantify the influence of the
defects size in the fatigue assessment.
Contents
1. Introduction ..................................................................................................................................................................................................... 2
2. Material and methods ....................................................................................................................................................................................... 3
2.1. Material and welding parameters ............................................................................................................................................................ 3
2.2. Method of acquisition ............................................................................................................................................................................. 3
2.2.1. X-ray ...................................................................................................................................................................................... 4
2.2.2. Metallographic macro-graphs ................................................................................................................................................... 4
2.3. Defect analysis strategy ......................................................................................................................................................................... 4
2.3.1. Classification of defects............................................................................................................................................................. 4
2.3.2. Statistical approach ................................................................................................................................................................. 6
2.3.3. Kitagawa-Takahashi diagram..................................................................................................................................................... 8
3. Results & Discussion .......................................................................................................................................................................................... 8
3.1. Extreme values and defect analysis ......................................................................................................................................................... 8
3.2. Fatigue limit estimation .......................................................................................................................................................................... 11
4. Conclusions ....................................................................................................................................................................................................... 13
CRediT authorship contribution statement ........................................................................................................................................................... 14
Declaration of competing interest........................................................................................................................................................................ 14
Data availability ................................................................................................................................................................................................ 14
Acknowledgments .............................................................................................................................................................................................. 14
References......................................................................................................................................................................................................... 14
∗Corresponding author.
E-mail address: antonio.dandrea.1@ens.etsmtl.ca (A. D’Andrea).
https://doi.org/10.1016/j.ijfatigue.2024.108501
Received 26 April 2024; Received in revised form 29 June 2024; Accepted 11 July 2024
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
Nomenclature
Abbreviations
μ-CT Micro-Computed Tomography
AM Additive Manufacturing
AR Aspect Ratio
Circ Circularity
BM Block Maxima
c.d.f Cumulative distribution function
EVS Extreme Values Statistics
FCAW Flux Cored Arc Welding
HSV Highly Stressed Volume
KT Kitagawa–Takahashi
MLE Maximum Likelihood Estimation
p.d.f. Probability distribution function
SIF Stress Intensity Factor
Symbols
δScale parameter
λLocation parameter
ξShape parameter
𝛥Kth,LC Long cracks threshold SIF range
𝛥𝜎w0 Fatigue limit stress range defects-free
𝛥𝜎wFatigue limit stress range
𝜖fElongation at fracture
𝜎UTS Ultimate tensile stress
𝜎y,0.2% Yield strength (offset 0.2%)
area Square root area of defect
Frisk c.d.f. of Competing risk distribution
GEV c.d.f. of GEV distribution
Gc.d.f. of Gumbel distribution
S0Surface area BM
S1Surface area BM (full weld)
V0Investigated volume BM
V1Investigated volume BM (full weld)
V80%
HSV Highly Stressed Volume at blade hot-spot
ξUpper bound shape parameter
ξLower bound shape parameter
F Cumulative frequency
R Load ratio σmin∕σmax
T Return period
x Generic random variable
Y Murakami boundary correction factor
1. Introduction
Martensite stainless steel containing 13%Cr–4%Ni is commonly used
to produce large Francis hydraulic turbine runners because of its ex-
cellent resistance to corrosion and cavitation erosion [1]. In a real
turbine runner, as represented schematically in Fig. 1(b) for medium
head geometry, the most critical region is the welded zone where the
blade-band and blade-crown interface converge, specifically around the
trailing edge, as reported in [2–4]. The design and the existence of
welding defects make the volume of material in these zones susceptible
to high stress levels, especially at the ‘‘hotspots’’ location, highlighted
in red in Fig. 1(c). Given the thickness of the region that needs to
be assembled, flux cored arc welding (FCAW) represents an efficient
and productive method to perform high-volume material deposition.
However, FCAW wire electrodes contain a flux core composed of
Fig. 1. Representation of (a) Francis turbine; (b) medium head Francis runner geometry
and (c) blade geometry detail highlighting the high stress level zones (‘‘hotspots’’
locations).
several metallic and non-metallic compounds (oxides of silicon, tita-
nium, zirconium, etc.), that might get entrapped during the multi-pass
welding procedure, creating inclusions [5]. Moreover, various welding
discontinuities originated from other sources (gas porosity or lack of
fusion) will also influence the material resistance during the cyclic load.
As a consequence, different sources of heterogeneity in the material can
act as crack nucleation sites.
Several studies in the literature have made efforts to identify dis-
continuities within multi-pass FCAW metal. Boukani et al. [5] have
performed an investigation on Not-Destructive Testing (NDT) where
slag welding inclusions were reported as expected. Similarly, Bajgholi
et al. [6] have used Ultrasonic Testing (UT) to prove and generate a
probability of occurrence of similar welding defects.
Even though previous failure investigations have shown that crack
could initiate around material defects and affect the crack propagation,
there is a limited research on the influence of defects on fatigue
behavior for these materials [7]. Nowadays, the influence of defects on
fatigue performance still remains a challenge to mitigate and multiple
studies have emphasized the importance of addressing the impact of de-
fects during the design process [8,9]. Therefore, the characterization of
defects is primarily important when a defective component is subjected
to a cyclic load.
In the world of design and manufacturing, reliability is a paramount
concern. Ensuring the reliability of a component involves the under-
standing of the statistical probability of its fracture event, which is
related to the probability of finding a weak point in the material
microstructure (e.g., porosity or inclusion). The simplest probabilis-
tic model to describe such effect is the concept of the weakest link
(WL), widely adopted in literature to compute probabilistic fatigue
assessment for component containing defects [10]. In the context of
Francis runners, Gagnon et al. [11] have used such probabilistic fatigue
assessment to test the fatigue reliability considering the distribution
of typical defects. Defect analysis for other applications were also
conducted by Romano et al. [12], which used a statistical approach
known as ‘‘Extreme Value Statistics’’ (EVS) to perform a probabilistic
fatigue assessment [13,14]. The concept of EVS applied to defects was
initially introduced by Murakami to study the effects of small inclusions
on fatigue strength. In this study, the fatigue assessment was computed
by experimental formulations based on defect size measurements. The
dimension of the defects was quantified by area parameter repre-
senting the square root of the measured inclusion area [15]. Most of
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
Fig. 2. Image of the T-shape welded joint: (a) geometry and (b) cross-section slices.
these concepts are recently applied to additive manufacturing (AM)
fatigue assessment for structural part highlighting the impact of defects
in fatigue performance [10].
In literature, the Kitagawa–Takahashi (KT) diagram is often adopted
to establish a correlation between the distribution of the identified
defects and the fatigue performance [16]. This diagram combines the
concept of fatigue crack growth threshold and the fatigue endurance
limit into a single plot, relating the size of defects and the endurance
limit. It defines a transition zone from short to long crack propagation
regime. This approach is commonly employed in fracture mechan-
ics design of components and fracture control, such as the fail-safe
principle [17–19].
In this contest, the paper aims at presenting a methodology to
document the impact of defects in multi-pass welded material made
of martensite stainless steel. Defects are measured on large T-shape
joint specimens representative of Francis runners and analyzed with
EVS statistics for fatigue assessment. The distributions of the expected
defects rely on 2D measurements of the welded zone using radiogra-
phy (as in the industry) and metallography images. The defects are
characterized in terms of defect size (area). Additional parameters
such as circularity and aspect ratio (AR) are used to better describe the
defect statistical behavior and estimate the size controlling the fatigue
endurance limit before running any experimental fatigue tests.
2. Material and methods
The blade-band and blade-crown geometry were simplified to a
T-shape welded joint manufactured in agreement with the common
welding process used in Francis runner production to generate equiva-
lent defects. Fig. 2 shows the specimen geometry 2(a) and an image of
the welded part cut in cross-sections for the analysis 2(b).
Table 1
Chemical compositions of 13%Cr–4%Ni alloys: CA6NM (ASTM A743) and 410NiMo (As
measured) (wt%).
Material Cr Ni Mo C Mn Si P S
CA6NM 11.5−14 3.5−4.5 0.4−1<0.06 <0.5 <0.1 <0.04 0.03
410NiMo 11.7 5.32 0.66 0.025 0.34 0.51 0.008 0.01
Table 2
Tensile properties of welded metal 410NiMo after post welded tempering treatment
(PWHT) at 610(±5) ◦C for 8 h.
Material 𝜎y,0.2% [MPa] 𝜎UTS [MPa] 𝜖f[%]
410NiMo 681 ±25 826 ±27 11 ±03
Table 3
Long crack thresholds stress intensity factor (SIF) range for 13%Cr–4%Ni stainless
steels.
R0.7 0.05 −1
𝛥Kth,LC [MPa m] 2.24 [22] 5.37 [22] 11.00 [23,24]
2.1. Material and welding parameters
The material under investigation is a martensite 13%Cr–4%Ni metal
cored wire (410NiMo), welded on a base metal CA6NM (ASTM A743).
The chemical nominal compositions of the base and welded metals are
indicated in Table 1.
The welded region includes about thirty weld beads of 410NiMo
material deposited using a FCAW in semi-automatic configuration. The
welder was manipulating the welding gun while the electrode was au-
tomatically fed to the arc. The welding parameters used to manufacture
the T-shape welded joint represent typical manufacturing conditions
for Francis runners, and they are reported in Table 4. A post welded
tempering treatment (PWHT) at 610(±5)◦C for 8 h was performed,
with air as cooling medium without convection. This solution generates
a significant decrease in residual stress [20] with a well characterized
microstructure [21]. In this study, the residual stresses were low and
thus not included in the analysis below. The heterogeneity in the
microstructure of the weld was not considered, in accordance with
the hypothesis of Linear Elastic Fracture Mechanics (LEFM). In this
condition, the microstructure consists of some tempered martensite and
about 17% of reformed austenite [21].
The mechanical properties of 410NiMo welded metal are reported
in Tables 2 and 3. The tensile properties of welded metal 410NiMo were
computed with a tensile test performed according ASTM E8 whereas the
long crack thresholds stress intensity factor (SIF) range were obtained
from literature [22–24].
The T-joint was cut with an abrasive cutting machine in 34 cross-
section slices (Z–Y plane) with similar thickness (3 mm). Six of them
were selected randomly and used to extract the data necessary for the
analysis of the defects. The region of interest was focused on the welded
material 410NiMo in the Z–Y plane (see Fig. 2(a)).
2.2. Method of acquisition
The acquisition of defects was conducted on 3 mm thick cross-
section slices using X-ray 2D scans and metallographic macro-graphs.
The choice of 3 mm thickness was a compromise to detect defects with
acceptable contrast in X-ray. The shape of the slices have prevented
the use μ-CT scanning, a common method for 3D defect analysis,
widely documented in the literature. The two methods have different
image resolutions: 1 pixel in X-ray radiography is 12.5 μm whereas in
metallography images 1 pixel is 9.5 μm.
A threshold detection limit was set for the minimum defect size
which was chosen as twice the resolution of the corresponding tech-
nique of acquisition, as illustrated in Fig. 3. This criterion was im-
plemented to ensure that observations with a resolution of only one
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
Table 4
Welding parameters of T-shape welded joint.
Process Shielding gas Wire diameter Current Voltage Torch speed Distance Pre-heating Temperature Interpass Temperature
[mm] [A] [V] [cm/min] [mm] [◦C] [◦C]
FCAW 75%Ar–25%CO2 1.6 245 26 30 20 120 170
Fig. 3. Minimum size of the acquired defects: 25 μm for X-ray and 19 μm for
metallography.
pixel (possibly noise) were not mistakenly considered as defects. As
a consequence, the smallest defect size documented by the X-ray and
metallography method was area = 25 μm and area = 19 μm,
respectively.
X-ray 2D scan provides the capability to project to a plane the entire
volume of the material and likely the various population of defects
(e.g. pores, voids, inclusions, etc.), whereas metallography is limited
to a single plane of observation, limiting the detection of the defects.
In both cases, the plane of observation does not necessarily coincide
with the plane of the largest cross-section of the defect. For instance,
defect sizes determined with X-ray could have smaller size in the Y-Z
plane of observation, but more elongated along the X dimension. This
is a type of underestimation that characterizes data coming from X-ray
acquisitions, as illustrated in Fig. 4(a). Similarly, in the plane of obser-
vation for metallography the maximum defect size is not precisely the
truth maximum size of the defect, as illustrated in Fig. 4(b). Although
widely used for defect analysis, metallography has its limitations, which
have been extensively recognized in the literature, as discussed [25,26].
Despite these, metallographic macro-graphs are widely used in the
industry for root cause analysis due to their simplicity and ability
to provide quick qualitative analysis [27,28]. They offer a unique
perspective, revealing structural details and defects often invisible to
other techniques like ultrasonic testing or radiography. This capability
allows for the identification of grain boundaries, phase distributions,
and microstructural anomalies, providing insights that other methods
cannot match. For this reason, this method was used and compared
with X-ray technique.
2.2.1. X-ray
The X-ray 2D scans of the welded material were achieved by Nikon
XT-H-225. Each cross-section slice was divided in 3 regions to reach the
highest-resolution possible. The X-ray set up parameters are reported in
Table 5 and each acquired image was an average of 64 shootings.
Image processing was carried out using an auto-threshold method
called ‘‘Otsu’’ in Fiji/ImageJ software. Otsu’s method is an image
processing technique used for automatic image thresholding. It works
by finding the optimal intensity threshold that separates an image
into the foreground and background. The method involves calculating
the histogram, probability distribution, and cumulative distribution of
pixel intensities. By maximizing the variance between foreground and
background, Otsu’s method determines the best threshold for creating
a binary image, supporting tasks like image segmentation [29]. There-
fore, this threshold method has facilitated the quantification of several
parameters including defect size, circularity, and aspect ratio.
An example of post-processed image with highlighted detected de-
fects is provided in Fig. 5(a). Most of the large observed discontinuities
Table 5
X-ray set up parameters.
Voltage Current Exposure time Resolution
180 kV 80 μA 708 ms 12.5 μm/pixel
are located at the top connection between welded and base metal, with
the present of several clusters. A detailed view of a cluster of defects is
shown in Fig. 5(c). X-ray results have also shown large internal defects
at the root of the multi-passes welds, as indicated by the arrow in
Fig. 5(a). These defects were not considered in this analysis because
they are not correlated with those produced by the FCAW process, but
rather related to areas of low accessibility.
2.2.2. Metallographic macro-graphs
Metallographic macro-graphs were performed on one side of the six
selected cross-section slices. The chosen side was initially polished for
the image acquisition to perform the defect analysis. The surface was
prepared using an automatic Tegramin machine (Struers) with P320,
P600 and P1200 SiC abrasive grinding papers followed by polishing
with 9, 3, and 1 μm diamond suspension. To achieve a good compromise
between quality and image resolution, a 50x magnification was set
for the acquisition and images were automatically stitched together
to reconstruct the entire cross-section slice. Image processing was also
performed using the ‘‘Otsu’’ thresholding technique in Fiji/ImageJ to
quantify defect size, circularity, and aspect ratio parameters. After the
image acquisition, on the same polished surface a chemical etching
with Vilella’s reagent was used to reveal the martensite microstructure
and welded passes [30]. The revealed microstructure is consistent
with the findings of M.M. Amrei et al. [31], where fine e columnar
microstructure were reported. A detailed view in Fig. 5(d) highlights
the main feature of the microstructure and the defects dimensions.
Fig. 5(b) corresponds to the same slice of Fig. 5(a) and it shows the
metallographic macro-graph obtained by combining surface polishing
and surface etching together. Thanks to this combination, it was possi-
ble to highlight the weld beads and to clearly show the defects across
the region of interest.
2.3. Defect analysis strategy
2.3.1. Classification of defects
The defects of the entire welded 410NiMo region were compared
in terms of defect size, aspect ratio (AR) and circularity. The location
of the defect due to the complex T-shape slice geometry is simplified
to the assumption that any defect not in contact with the free surface
is considered internal. Due to the limited amount of material on the
free surface of the T-shaped joint (3 mm thick slices), surface defects
are few and not statistically treatable. Therefore, all defects are treated
as internal in the analysis. The adopted defect size terminology is the
area parameter and as mentioned in [15] it is usually represented by
the square root of the measured inclusion area. The aspect ratio is a
parameter that measures the elongation of a defect. This parameter is
obtained by quantifying the ratio between the minor and major axes of
the ellipse that is fitted to the original shape of the defect preserving
its area. The expression of this ratio is defined by Eq. (1a) where
minor axis is 𝑏and major axis is 𝑎and its value varies between 0
and 1 with lower values meaning more elongated defects. Circularity
is another measure of the irregularity of a defect and it is calculated as
the ratio between the equivalent and the real perimeter of the defect,
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
Fig. 4. Graphical descriptions showing the expected underestimation of defect size using (a) X-ray and (b) Metallography method of acquisition.
Fig. 5. Location and dimensions of the detected defects: (a) analysis from X-ray imaging; (b) analysis from metallography imaging; (c) detailed view of defect clusters identified
via X-ray imaging; (d) detailed view of the microstructure and defects identified via metallography imaging. The scale bars in (c) and (d) are different. The color represents the
size of the defect according to area parameter.
respectively expressed as Pequiv and Preal in Eq. (1b). In 2D analysis,
circularity varies between 0 and 1 and a value of 1 represents a perfect
circle, whereas values approaching 0 indicate an increasingly irregular
shape [29]. Fig. 6(a) shows the graphical representations of these two
defect irregularity parameters.
AR = 𝑏
𝑎(1a)
Circ = Pequiv
Preal
=2⋅𝜋A
Preal
(1b)
The AR and circularity parameters of detected defects are shown in
Fig. 7. Particularly, in Figs. 7(a) and 7(b) the aspect ratio and circularity
of each defect are compared with the defect size and divided by method
of acquisition: X-ray data in black and white (o) and metallography
ones in red (+). Aspect ratio and circularity parameters are also dis-
played individually in terms of empirical cumulative distribution to
highlight the frequency of the observed data, as illustrated in Figs. 7(c)–
7(d). When acquired through the metallography method, numerous
detected defects exhibit similar values of aspect ratio (∼0.68) and
circularity (∼1), as shown by Figs. 7(c) and 7(d), respectively. This
Fig. 6. Classification of the detected defects: (a) Graphical representations of aspect
ratio and circularity; (b) Schematic diagram to classify defect morphology with
examples of few detected defects.
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
Fig. 7. Properties of detected defects according to Aspect ratio (AR), circularity and defect size parameter: (a) AR vs defects size; (b) circularity vs defects size; (c) empirical
cumulative distribution for AR parameter; (d) empirical cumulative distribution for circularity parameter.
repetition of the same value is assumed to be attributed to the inher-
ent characteristics of the acquisition method itself. On the contrary,
detected defects acquired through the X-ray method show a smoother
behavior.
Based on the analysis of the AR and circularity data, the defects
were classified into two main categories: defects with AR and cir-
cularity >0.7identified as ‘‘Spherical’’ discontinuity, and the remain-
ing defects treated as ‘‘Elongated’’ discontinuity. The two categories
are respectively displayed in Fig. 6(b) as green and magenta area.
Sanaei et al. [32] have introduced a similar methodology to classify
2D observed discontinuities to characterize their variability in AM
components. In the current study, the distribution of the observed
defects was analyzed both as a unified distribution including spherical
and elongated defects together, and additionally, the defects were also
examined independently.
2.3.2. Statistical approach
The extreme values of the defect size obtained experimentally were
modeled using the ‘‘Generalized Extreme Value’’ (GEV) distribution
and its asymptotic formulations. The cumulative distribution function
(c.d.f.) of GEV is expressed by Eq. (2), whereas c.d.f. of Gumbel
distribution (asymptotic case) is expressed by Eq. (3).
GEV
ξ,
λ,
δ(x) = exp
−1 +
ξx −
λ
δ−1∕
ξ
(2)
G
δ,
λ(x) = exp −exp −x −
λ
δ (3)
where
λ,
δand
ξare the ‘‘location’’, ‘‘scale’’ and ‘‘shape’’ distribution
parameters. The estimated location parameter
λis the modal value
of the distribution shifting the probability density function (p.d.f.) to
the right or to the left,
δcompresses or stretches the p.d.f. of the
distribution, and
ξcontrols the shape of the distribution. Depending
on the value of the estimated shape parameter
ξin Eq. (2), three
possible asymptotic distributions are achievable: Fréchet distribution
(
ξ>0), Weibull distribution (
ξ<0) and Gumbel distribution (
ξ =
0) [33]. The parameters of the distribution are calculated by Maxi-
mum Likelihood Estimation (MLE), which consists of maximizing the
probability that sample data belong to the selected distribution [14].
Additionally, MLE has also given the estimation of the confidence
intervals for these parameters. This becomes crucial as it provides a
range of values within which the true parameters value can reasonably
be expected to fall, offering a measure of the uncertainty associated
with the estimate. The importance of the confidence intervals becomes
evident in determining whether to apply an asymptotic distribution or
not, particularly for the shape parameter
ξ. Indeed, if both confidence
bounds are negative/positive, there is a statistical evidence suggesting
that the estimated parameter will be negative/positive, whereas if the
lower bound
ξis negative and the upper bound
ξis positive, there is no
statistical evidence favoring either a positive or negative value. In such
cases, the value
ξ=0is chosen, simplifying the decision-making process
and aligning with the asymptotic case corresponding to the Gumbel
distribution represented by Eq. (3).
The dimensions of the largest inclusions were measured and ana-
lyzed, according to the block maxima (BM) sampling technique [13].
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
Fig. 8. Representation of sampling blocks in the case of a quarter of weld (a), (c) and
full weld (b), (d).
According to the mentioned block maxima strategy, only the largest de-
fects observed in the investigated volume were used for this statistical
approach. In the analysis, two main block sampling sizes were selected:
one representing a quarter of the welded region and the other including
the entire welded region itself.
Fig. 8 illustrates the X-ray investigated volumes V0and V1and the
corresponding surface area for metallographic macro-graphs S0and
S1, respectively. The investigated volume V0and V1were computed
considering the actual volume analyzed by the radiography and used
for the X-ray analysis. On the other hand, the surface areas S0and S1
were adopted to obtain equivalent volumes necessary to compare met-
allographic results with those obtained from X-ray analysis. Murakami
in [7,34] has proposed a methodology to predict larger inclusion size
based on 2D metallography observations. The predicting method was
based to assign a fictitious thickness h0to the surface area S0. The
fictitious thickness h0was computed considering the average value
among the observed largest defect size in the different surface areas S0.
In this way, it was possible to determine an equivalent volume (S0⋅h0)
necessary to predict the largest inclusion within a target volume. In
the present study, the number of surface areas S0and S1are 24 and
6, respectively. The average size of the largest defects observed on
the surface areas were used as a fictitious thickness to calculate the
equivalent volumes V∗
0and V∗
1. This approach offers the advantage of
conducting a 3D analysis based on 2D measurements, but the associated
volumes could be significantly different from the actual ones.
The use of extreme value statistics solely based on the full weld
sampling blocks (see Fig. 8), providing only six defects in total, may not
yield sufficiently robust results. Therefore, to avoid this, the researchers
opted to perform the analysis on the smaller investigated volume, V0,
represented by a quarter of the full welded 410NiMo region (see Fig. 8).
This adjustment aimed to enhance the reliability and accuracy of the
analysis.
Firstly, the attention was centered on the distribution of all the
defects which were fitted with a GEV and a Gumbel distribution and
displayed on the Gumbel probability paper plot. On this probability
plot, it is possible to illustrate the fitted cumulative distribution func-
tion and also the experimental data in terms of empirical cumulative
probability.
Secondly, the defects were separated in the category mentioned in
Section 2.3.1 and analyzed according to the extreme value statistics.
Maxima defects acquired on V0were considered separately as maxima
spherical discontinuity and as maxima elongated discontinuity. To
examine the impact that each category of defects had in the overall
distribution, spherical and elongated defects were fitted with a GEV
distribution independently. Afterwards, the resulting estimated shape
parameters and its confidence intervals were used to choose the proper
asymptotic distribution [33]. At the end, the superimposition of the
spherical and elongated fitted distributions was treated according to
Fig. 9. Flow chart of competing risk strategy applied to EVS.
a competing risk approach, a strategy adopted by Beretta et al. [35]
for the investigation of multiple particles effects. The entire process
is schematically presented in Fig. 9 and the c.d.f. representing the
competing risk distribution on the volume V0is expressed by the
Eq. (4), assuming that it consists of a combination of a Gumbel and
a GEV distribution.
Frisk ,V0(x) = G
δ,
λ(x) ⋅GEV
ξ,
δ,
λ(x) (4)
Once that the competing risk distribution is identified for the vol-
ume V0, the statistics can be generalized to volumes larger than the
investigated one using the concept of the ‘‘return period’’. For example,
the expected distribution of the maximum defect on the volume V1can
be obtained as:
Frisk ,V1(x) = Frisk ,V0(x)T(5)
where T = V1∕V0is the return period for the largest defects in V1when
block maxima sampling is adopted [13,14].
The discussed volumes shown in Fig. 8 are simplified cases and not
fully representative of the real application. For this reason, another
volume was introduced to simulate the stressed volume at the hotspots
of the Francis runner. Kuguel et al. [36] discussed that for any volume
geometry, there is a portion of volume called Highly Stressed Volume
(HSV) that is particularly critical when assessing the potential fatigue
failure. After the introduction of this concept, the HSV was used by sev-
eral authors [37,38] to determine the highly stressed volume in terms
of a specific percentage of the maximum stress. Romano et al. [10]
and Beretta et al. [39] have demonstrated that a portion of material
subjected to a percentage between 95% and 80% of the maximum stress
is significantly critical in terms of fatigue assessment, especially in
presence of defects. In the present analysis, the highly stressed volume
is represented by the volume subjected at least to 80% of the maximum
principal stress at the hotspot and indicated as V80%
HSV. Finite element
analysis is often used to identify crucial hotspots and their associated
maximum principal stress, which is influenced by both the geometry
of the Francis runner and the operating conditions of the turbine. In
the case under study, V80%
HSV was estimated for a medium head Francis
turbine operating at maximum power with runner geometry similar to
the one represented in Fig. 1(b). For instance, according to this case, the
value of V80%
HSV is 1700 mm3and the hotspot is located at the blade-band
interface, as illustrated in Fig. 10 [40].
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Fig. 10. Finite element analysis showing the highly stress volume for a medium head
Francis runner operating at maximum power: (a) location of the hotspot; (b) normalized
maximum principal stress to identify the V80%
HSV (regions in orange and red).
2.3.3. Kitagawa-Takahashi diagram
The Kitagawa–Takahashi (KT) diagram is commonly used in liter-
ature to illustrate how the fatigue limit decreases with the increasing
size of the defect [16]. This diagram represents a tool widely employed
for fatigue assessment in various studies [17,19,41] and, in this study,
it was used to estimate how the defect size (area) affects the fatigue
limit stress range (𝛥𝜎w) representing the fatigue strength at 10 million
cycles for infinite life. To simplify the investigation, the statistical
approach was only implemented for infinite life avoiding any detailed
analysis for finite life. Among different approaches, the modified El-
Haddad formulation is particularly interesting because it describes a
smooth transition from short to long cracks and it is controlled by the
El-Haddad parameter (area0) expressed by Eq. (6a) [42].
area0=1
𝜋𝛥Kth,LC
𝑌⋅𝛥𝜎w0 2
(6a)
𝛥𝜎w(area) = 𝛥𝜎w0 ⋅
area0
area + area0
(6b)
In Eq. (6a) 𝛥Kth,LC represents the long crack threshold SIF range; 𝛥𝜎w0
is the fatigue limit stress range of the defect-free material [7] and Y
is the Murakami’s boundary correction factor [7,43]. In this study, Y
is equal to 0.5 because internal defects are considered. Then Eq. (6b)
corresponds to the analytical expression for the smooth short-long crack
transition and it was used to determine the value of the fatigue limit
stress range 𝛥𝜎wfor a specific defect size area. In the present study,
the variables used in the El-Haddad model (𝛥Kth,LC ,𝛥𝜎w0) are assumed
to be perfectly correlated as stated in [44] to have a straightforward
link between them.
The El-Haddad parameter is graphically represented in Fig. 11 and
it corresponds to the intersection between the dashed red line defining
the fatigue limit stress range of the defect-free material 𝛥𝜎w=𝛥𝜎w0
and the solid red line computed by the long crack threshold SIF range
𝛥Kth,LC 𝛥𝜎w=𝛥Kt h,LC∕Y𝜋ar ea.
Fig. 11. Schematic Kitagawa–Takahashi diagram and its dependency on R.
The KT diagram varies with the load ratio R because the El-Haddad
parameter depends on 𝛥𝜎w0(R) and 𝛥Kt h,LC(R), both a function of load
ratio R. This modifies the asymptotic lines for each load ratio, leading
to different 𝛥𝜎w(R) curves as illustrated in Fig. 11. For this reason, for
each load ratio R the material parameters 𝛥𝜎w0(R) and 𝛥Kt h,LC(R) are
computed respectively by the Goodman equation as reported in Eq. (7)
and by the long crack threshold SIF range data from experimental tests
or NASGRO fitting [45].
𝛥𝜎w0(R) = 1
1+R
1−R
⋅1
𝜎UTS
+1
𝛥𝜎R=−1
w0
(7)
3. Results & Discussion
In accordance with the detection defect procedure mentioned in
Section 2.2, the number of observed defects and the absolute maximum
defect size for the X-ray and metallography method of acquisition are
reported in Table 6. For the same region of interest, the number of
discontinuities identified by X-ray is found to be lower compared to
those determined by metallographic macro-graphs, but the associated
size is significantly larger.
The difference in the image resolutions of the two methods signifi-
cantly affects the defect detection, as shown by Figs. 3 and 4. For this
reason, X-ray data were analyzed in detail, as this method of acquisition
provides a complete overview of the defects throughout the material.
This is especially true when the focus is on the population of the
largest discontinuities. The divergences between the two methods are
examined at the end of Section 3.2, focusing on the estimation of the
fatigue limit stress range (𝛥𝜎w).
Table 6
Features of the detected defects in the 410NiMo welded material analyzing six T-shape
cross-section slices.
Number of defects Max defect size [μm]
Method of aquisition Spherical Elongated Spherical Elongated
X-ray 327 103 207 444
Metallography 337 131 135 257
3.1. Extreme values and defect analysis
The overall defects (spherical and elongated together), observed on
the investigated volumes V0, were analyzed according to GEV extreme
statistics. The maximum likelihood estimation of the GEV distribution
parameters have provided the estimates of the shape parameter
ξand
its confidence intervals. The confidence interval values of the shape
parameter (
ξ,
ξ) did not exclude the possibility to fit the X-ray data with
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Fig. 12. Results of X-ray data on V0: (a) Gumbel fitting of the overall observed defects; (b) GEV fitting of the overall observed defects; (c) extreme statistics applied to spherical
and elongated category; (d) competing risk applied to the overall defects (Spherical+Elongated).
the Gumbel distribution (
ξ=0). Therefore, Fig. 12(a) illustrates the
Gumbel fitting on a Gumbel probability plot with the experimental ob-
served defects and 90% confidence intervals. The Gumbel distribution
fails at including all the largest defects, as the observed defect at 444 μm
is out from the 90% confidence intervals. The GEV fitting with the
associated 90% confidence intervals (Fig. 12(b)) shows an improvement
as all the largest defects fall within the confidence intervals; however,
it does not yet precisely represent the distribution of the largest sizes
of defects. The inaccuracy of the fittings for the welded discontinuities
might be attributed to the present of more than one population of
defects.
The morphology of the defects is chosen as a criterion to create two
populations of defects (spherical vs elongated category) and each cate-
gory is analyzed and fitted with the best respective extreme statistics.
Spherical defects are best fitted with a Gumbel distribution, while elon-
gated defects follows a Fréchet distribution, as shown in Fig. 12(c). The
separated fitting clearly illustrates that the largest defects are described
by the elongated distribution better than the previous case when spher-
ical and elongated discontinuities were fitted together. The maximum
likelihood estimations of the mentioned distributions parameters are
collected in Table 7 for control volume V0.
In Fig. 12(c), it is clear that for small values of Gumbel cumulative
frequency F (equivalent to small return period), spherical defects are
the largest discontinuities, and they will be the ones controlling the
fatigue performance of the material. On the other hand, when large
volumes are considered (large values of F), elongated defects are the
most critical ones. These results illustrate the importance of considering
the right size of samples to perform the characterization of the defects:
Table 7
Maximum likelihood estimation of distribution parameters for control volume V0(X-ray).
Fitting parameters
Type of distribution
ξ{
ξ,
ξ} [–]
λ[μm]
δ[μm]
Gumbel (Spher.+Elo.) – 121.51 58.42
GEV (Spher.+Elo.) 0.25 {−0.06, 0.56} 121.91 37.75
Gumbel (Spherical) – 108.71 27.92
Fréchet (Elongated) 0.38 {0.01, 0.94} 82.33 50.31
Competing risk 0.38 108.71 & 82.33 27.92 & 50.31
if an excessively small volume (sample size) is considered to investigate
the material, the population of defects that would be characterized
might not be the one that will drive the material fatigue performance
in larger volumes, i.e., in the real part.
The separated investigation of the two populations of defects re-
sulted into a 90% confidence intervals wider than the case when
considering the defects as a single population, as seen by comparing
Figs. 12(b) and 12(c). With this approach, the necessity to use two
separated statistical distributions to predict the fatigue limit makes
the process less straightforward, justifying the development of the
competing risk approach. The resulting competing risk distribution,
generated by merging the two computed distribution from spherical
and elongated defects, is presented in Fig. 12(d) together with its 90%
confidence intervals. For defect sizes smaller than 230 μm, both the
GEV and competing risk distributions accurately predict the sizes of
the defects, as shown by the accurate fitting between the solid lines
(predicted defect size) and the observed discontinuities in Figs. 12(b)
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Fig. 13. Normalized residuals as a function of the defect sizes for the fittings achieved with the X-ray data on V0: (a) Gumbel of the overall observed defects; (b) GEV of the
overall observed defects; (c) Gumbel and GEV of spherical and elongated defects; (d) competing risk of the overall defects. The probability distribution functions (p.d.f.) are also
provided for each fitting.
and 12(d). However, the competing risk distribution provides an im-
provement compared to the previous results, particularly for the largest
defect size. A significant difference between the two distributions is
clearly visible when switching from the Gumbel/GEV distribution to
the competing risk one. To estimate this difference, the predicted defect
size is calculated using the same cumulative frequency F of the largest
observed defect, corresponding to a value of −log(−log(F))=3.2. For
the same value of cumulative frequency F, the predicted defect size
increases by 31% from 309 μm to 405 μm, while the observed defects
size is at 444 μm.
The residuals of the fittings are computed and associated to the
estimated p.d.f. and standard deviations in Fig. 13. These residuals
correspond to the difference between the observed defect size and the
predicted one at the same probability of occurrence normalized by
the predicted defect size. If the data are properly fitted, one would
expect the residuals to be normally distributed with centered and
symmetrical values. The residuals obtained from the Gumbel distribu-
tion of the overall defects highlight the significant inaccuracy of this
fitting because a large standard deviation is found (0.16), as shown in
Fig. 13(a). The adoption of GEV fitting of the same defect distribution
in Fig. 13(b) have a narrow range of residuals, resulting in a lower
standard deviation at 0.10 but still a significantly high residual value
for the largest defect at approximately 0.5. In Fig. 13(c), the spherical
distribution is properly fitted by the proposed Gumbel distribution with
a standard deviation of 0.06. Conversely, the fitting on the elongated
defects shows a dis-symmetric distribution resulting in a quite large
standard deviation at 0.14, despite the fact that the residual associated
with the largest defect was reduced by half at 0.25. Only the use of the
competing risk distribution function in Fig. 13(d) leads to a centered
and symmetric distribution with a small standard deviation of 0.06.
A good prediction of the largest defect size residual is found with a
value of 0.12, i.e., +12% of the predicted defect size. The competing
risk analysis is the most effective way to describe the distribution of
defect sizes, especially in the range of large defects, and it is used to
assess the fatigue performance of the material.
In order to verify that the analysis with different populations of
defects was not affected by the selection of the investigated volume,
the largest defects from the six volumes V1were compared with the
data obtained and analyzed from volumes V0. In each of the six
investigated volumes V1, the largest defects were acquired as spherical
and elongated discontinuity, providing two separated populations: six
maxima spherical defects and six maxima elongated defects. By adopt-
ing the ‘‘return period’’ concept as expressed by Eq. (5) the population
of defect obtained on V0were scaled up to the larger volume V1.
Fig. 14(a) shows that the defects with the corresponding fittings on V0
exhibited similar statistics when they are scaled up and compared to the
separated population of maxima defects observed in the investigated
volume V1. Particularly, the adoption of a smaller volume, such as
V0, has enabled to provide more statistics and it is preferable when
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Fig. 14. Influence of the investigated volumes V0and V1for X-ray data using extreme statistics: (a) effect on spherical and elongated data; (b) competing risk for volume V0
obtained by the related experimental defects and calculated competing risk for V1compared to the six overall largest defects (Spherical+Elongated) in V1.
there is a limited availability of the acquired data, as in the case of the
investigated volume V1.
Similarly, the competing risk distribution obtained from the in-
vestigated volumes V0was scaled up to the larger volume V1and
compared to the six overall largest defects (Spherical+Elongated) mea-
sured on V1. In Fig. 14(b), the competing risk on V1(black thick
line) was obtained by shifting the competing risk distribution on V0
(transparent thin line) according to the return period. The transition
between the two competing risk distribution is governed by Eq. (5). The
obtained distribution representing the welded volume (V1) describes
well the six overall largest defects (Spherical+Elongated) detected on
the investigated volume V1, as shown by Fig. 14(b).
3.2. Fatigue limit estimation
During different operating conditions, such as a start-up, speed no
load, maximum opening, and stop, the runner experiences wide range
of loading conditions, influencing the load ratio R [11]. For this reason,
the fatigue limit was evaluated considering three representative load
ratios: R=−1, 0.1 and 0.7.
The fatigue limit was estimated relatively to the defect-free fa-
tigue limit 𝛥𝜎w0 which is generally set as the stabilized fatigue stress
associated with a strain-controlled value of 0.05% at R=−1 [42,46]
denoted as 𝛥𝜎R=−1
w0 . In the absence of such experimental value, 𝛥𝜎w0
is usually approximated considering the standard tensile properties of
the material. In the present case, as the material undergoes dynamic
strain softening, the 𝛥𝜎w0 value was set at 80% of the 𝜎UTS, i.e., 691
MPa [47]. The dependence of 𝛥𝜎w0 on R was obtained by using the
Goodman equation reported in Eq. (7) [47].
On the other hand, long crack threshold SIF ranges (𝛥Kth,LC ) were
already documented in literature for various load ratio R (Table 3)
and these values were fitted with NASGRO equation [45] to obtain a
𝛥Kth,LC (R) function, as illustrated in Fig. 15.
In order to estimate the fatigue limit, the El-Haddad curves were
computed for each of the mentioned load ratio R and used to assess
the reduction of the fatigue limit stress range (𝛥𝜎w) from its defect-free
condition (𝛥𝜎w0), represented by 𝛥𝜎wat ar ea = 1 μm. The El-Haddad
curves alongside with the p.d.f. of the competing risk distributions are
illustrated in Fig. 16 for the three investigated volumes: V0,V1, and
V80%
HSV. These distributions highlight the probability of the defect sizes
to fall within a particular range of values. The defect size adopted as
area parameter was in accordance with the definition discussed in
Fig. 15. NASGRO fitting considering 𝛥Kth,LC presented in Table 3.
Section 2.3.1. As the volume increases, the competing risk distribution
shifts to the right with a decrease in the p.d.f. intensity of the mode,
illustrating how for larger volumes a wider dispersion is obtained.
Characteristic defect sizes for V0,V1, and V80%
HSV volumes were computed
based on the 90% percentile of their distributions, which correspond to
p.d.f. values of approximately 0.0009 for V0and 0.0005 for both V1,
and V80%
HSV. The 90% percentile was chosen because it focuses on the
largest defect sizes, representing the tail end of the distribution. Never-
theless, the provided distribution parameters (Table 7) and the return
period T allows plotting the c.d.f. of the competing risk distribution
and calculating the defect size corresponding to the desired percentile.
The resulting sizes of the defects were 266 μm,477 μm, and 441 μm for
V0,V1, and V80%
HSV, respectively. The computed defect sizes were then
introduced into Eq. (6b) to estimate the associated fatigue limits. A
parallel research activity, focused on the defects that control the failure
in this material, have shown that similar sizes of defect were critical
under cyclic loading. The experimental data are illustrated in Fig. 16
using a boxplot layout, and they refer to a volume of approximately
1350 mm3, representing the reduced section of a standard axial fatigue
specimen.
The fatigue limit stress range (𝛥𝜎w) is strongly influenced by load
ratio R, as illustrated in Fig. 16. For V0the estimation of 𝛥𝜎wat R=0.1
shows a decrease of approximately 45% compared to R=−1 and a 75%
decrease at R=0.7. Similar values are found for V1and V80%
HSV, with a
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Fig. 16. X-ray results: El-Haddad approximation presented alongside the 90% percentile of the competing risk distribution for load ratios R=−1, 0.1 and 0.7. (a) KT diagram for
control volume V0and V1; (b) KT diagram for control volume V80%
HSV. The boxplot at bottom of the KT diagram in (a) and (b) represents the experimental defect size controlling
the failure for a volume of approximately 1350 mm3(reduced section of axial fatigue specimen).
48% decrease of 𝛥𝜎wat R=0.1 compared to R=−1 and a 77% decrease
at R=0.7. At constant load ratio R in Fig. 16(a),𝛥𝜎wdecreases as the
volume increases from V0to V1with a drop of 15%, 18%, and 18%
for load ratio R of −1, 0.1, and 0.7, respectively. The 𝛥𝜎westimates for
V80%
HSV in Fig. 16(b) are marginally higher (about 3%) to those calculated
for V1as the associated characteristic defect of V80%
HSV is slightly smaller
than V1(about 8%).
A parallel procedure was implemented to compute the fatigue per-
formance using the metallographic observations, mirroring the method-
ology detailed for X-ray data. The competing risk approach was able to
fit rather well the defects detected on the 24 available surface areas of
175 mm2. As discussed in Section 2.3.2, the Murakami approach [7,34]
and the equivalent volumes (V∗
0,V∗
1) were determined by multiplying
the respective surface area by the average size of the largest defect size
observed on the analyzed surface areas. The average defect size calcu-
lation has led to fictitious thicknesses h0=0.087 mm and h1=0.148 mm.
This calculation has provided volumes of 15 mm3and 104 mm3for V∗
0
and V∗
1, respectively. On the other hand, V80%
HSV was kept the same at
1700 mm3as it represents the volume of interest for design purposes.
To quantify better the difference between the two methods, a dis-
crepancy of fatigue limits in terms of percentage was calculated and
reported in Fig. 17: the metallographic fatigue limit estimation was
subtracted to the X-ray fatigue limit estimation and this difference was
divided by the X-ray one. The difference between the two methods
of acquisition significantly varies from +10% to −79%. Particularly,
for smaller volume (V∗
0,V0) the prediction of the fatigue limit stress
range (𝛥𝜎w) through metallographic macro-graphs leads to estimates
less conservative than X-ray ones with a discrepancy limited to an
overestimation of +10% difference. Conversely, for large volumes (V∗
1,
V1)𝛥𝜎wmetallographic estimates are more conservative than X-ray
ones with a discrepancy limited to an underestimation of −27%. This
difference becomes overly conservative when considering the investi-
gated volume V80%
HSV with a discrepancy reaching an underestimation of
−79% difference, highlighting a fundamental divergence between the
two approaches. The discrepancy can be related with the return period
associated with the estimation method.
In the case of metallography, the obtained p.d.f. of the competing
risk distributions together with the El-Haddad curves is illustrated in
Fig. 17. Fatigue limit estimates 𝛥𝜎wusing X-ray and metallography at R=−1,0.1 and
0.7, highlighting discrepancies percentage and defect-free fatigue limit range 𝛥𝜎w0.
Fig. 18 for the three investigated volumes (V∗
0,V∗
1, and V80%
HSV). The
90% percentile of the competing risk distribution for V∗
0has provided
a characteristic defects size which is 195 μm corresponding to a p.d.f.
value of approximately 0.0007. The resulting characteristic defect size
was 27% smaller than the one found for X-ray observation, highlighting
the tendency of metallography to detect smaller sizes. The return period
between V∗
1,V∗
0is equal to T∗
1=6.9, about 70% higher than the one
observed in the X-ray analysis. The increase in the return period was
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Fig. 18. Metallography results: El-Haddad approximation presented alongside the 90% percentile of the competing risk distributions for the investigated volumes V∗
0,V∗
1, and V80%
HSV
at R=−1, 0.1 and 0.7.
caused by the associated fictitious thicknesses (h1∕h0=1.7) as the
surface areas ratio (S1∕S0=4) matches the volumes ratio in the X-ray
analysis (V1∕V0=4). This large return period moves the V∗
0distribution
to the right, resulting in V∗
1competing risk distribution with a value
of maximum p.d.f. and mode location close to the V1distribution,
computed in the X-ray analysis. The approach seems to work if the
defect size corresponding to the mode is chosen; however, in terms
of extreme values, the increase in the return period results in much
larger defect size prediction and significantly lower 𝛥𝜎westimation.
Particularly, the competing risk distribution for V∗
1corresponding to
90% percentile has led to a characteristic defects size of 1025 μm with
p.d.f. value of approximately 0.0001. The determined characteristic
defect size was about 2.14 times larger than the one obtained in the
X-ray analysis. The small number of large defects in the material leads
to rather small values for the fictitious thicknesses (h0,h1), resulting
in equivalent volumes too small compared to the ones associated with
X-ray. The fact that large defects were too rare to be captured with
metallography in their entirety, has limited the correct estimation of the
fictitious thicknesses. The issue becomes even more problematic when
computing the competing risk distribution for the volume V80%
HSV; the
equivalent volume V∗
0is significantly smaller than V80%
HSV (1700 mm3),
resulting in a return period as large as 113, approximately 34 times
the one observed for the X-ray analysis (THSV =3.3). Competing risk
distribution for V80%
HSV corresponding to 90% percentile has provided a
characteristic defect size extremely large at 13250 μm with a very low
value of p.d.f. at approximately 5⋅10−5. The determined characteristic
defect size was incredibly larger than the one obtained in the X-ray
analysis, showing again that the return period cannot be estimated with
the mentioned approach.
In order to achieve consistent 𝛥𝜎westimations between X-ray and
metallography, the characteristic defect size representing the extreme
defect and identified by the 90% percentile of the competing risk dis-
tribution should have similar value for both methods. In other words,
it is possible to define a proper return period so that the metallography
results would estimate properly the extreme defect size in the volume of
interest (V80%
HSV). The value of the return period necessary for this match
is equal to THSV =2.65 which defines the expected fictitious thickness
to be equal to h0=3.66 mm as the surface area S0is known. The
computed fictitious thickness h0was 42 times the one considered for the
metallography analysis (h0=0.087 mm), a value that was not possible to
estimate from the metallographic data. Therefore, the proposed method
based on metallography data cannot be used to properly estimate the
fatigue limit.
It should be highlighted that the hypothesis that the estimated
defects from X-ray can predict fatigue performance will have to be
validated by a comprehensive fatigue experimental test campaign. In
this campaign, fatigue specimens will be scanned using X-ray and then
subjected to fatigue testing. This process will provide the insights about
the main features of the defects which lead to the fatal failure of the
material. The authors will present the results of such fatigue campaign
in an upcoming article.
4. Conclusions
The present study proposes methodology to implement Extreme
Value Statistics (EVS) to investigate and predict the distribution of
defects and the largest characteristic defect size in target volumes for a
13%Cr–4%Ni multi-pass welded material in order to estimate the fatigue
properties. Defects in T-shape joint specimens produced by multi-pass
welding are documented through both metallographic and 2D X-ray
analyses. The dimension of the defects measured in terms of area
parameter was used to estimate the reduction in fatigue strength using
the Kitagawa–Takahashi (KT) diagram with the El-Haddad formulation.
Based on the results obtained, the following conclusion can be drawn:
•The competing risk approach improves the prediction of the de-
fect distribution compared to using a Generalized Extreme Value
(GEV) and Gumbel distributions on all data set. It is recommended
to classify the defect population in terms of geometry to avoid
inconsistent analyses;
•The distribution of defects in 13%Cr–4%Ni multi-pass welded
material was divided in two populations based on their geometry
(spherical and elongated discontinuities);
•The competing risk approach improves the prediction of the de-
fect distribution compared to using a Generalized Extreme Value
(GEV) and Gumbel distributions on all data set. It is then recom-
mended to classify the defect population in terms of geometry to
improve the statistical analysis;
•By the separated fitting of spherical and elongated discontinu-
ities, it was shown that the wrong population of defects can
be characterized if the size of the analyzed volume is not large
enough;
•Metallography offers greater precision in terms of defect mor-
phology but provides limited information as it cannot capture
defects below the observation plane and no realistic volume can
be associated with the observed data for further statistical pre-
diction. Conversely, X-ray analysis provides more comprehensive
International Journal of Fatigue 188 (2024) 108501
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A. D’Andrea et al.
information as it can penetrate through the material and it can
be associate without ambiguity to a specific volume;
•Defects acquired with X-ray and metallographic measurements
have evidenced a significant difference in terms of fatigue limit
estimations, highlighting the necessity to adopt a discrepancy
factor between the two methods of acquisition;
•The results have shown that metallographic statistical analysis
might generate inaccurate fatigue assessment due to large return
periods resulting into wider dispersion of the defect size;
•The estimated fatigue limit stress ranges 𝛥𝜎wwere presented
for different load ratios, providing typical stress values to per-
form a fatigue test campaign and to design components made of
multi-pass welds with specific material and welding conditions.
CRediT authorship contribution statement
A. D’Andrea: Writing – review & editing, Writing – original draft,
Methodology, Data curation, Conceptualization. M. Gagnon: Writing
– review & editing, Visualization, Supervision. S. Beretta: Validation,
Supervision. P. Bocher: Writing – review & editing, Visualization,
Validation, Supervision, Conceptualization.
Declaration of competing interest
The authors declare the following financial interests/personal rela-
tionships which may be considered as potential competing interests:
Antonio D’Andrea reports financial support was provided by Hydro-
Quebec’s Research Institute. Antonio D’Andrea reports financial sup-
port was provided by Natural Sciences and Engineering Research Coun-
cil of Canada. Antonio D’Andrea reports a relationship with Sacmi
Imola SC that includes: funding grants. If there are other authors,
they declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work
reported in this paper.
Data availability
The data that has been used are confidential.
Acknowledgments
This research activity was part of an international research project
called FatCo supported financially by Institut de recherche d’Hydro-
Québec (IREQ), Natural Sciences and Engineering Research Council of
Canada (NSERC) (CRSNG CRD 530064-18), SACMI, and Consortium de
Recherche et d’Innovation en Transformation Métallique (CRITM). The
authors are grateful to Alexandre Lapointe for his help and knowledge
on specimens preparation and Jean-François Morissette for the finite
element analysis.
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