ABSTR ACT
This study aimed to probabilistically evaluate the
size of a fatigue crack on a type 304 austenitic
stainless steel flat plate using eddy current signals.
Three fatigue cracks, with depths ranging from
2 mm to 5 mm, were introduced into the plates
through a four-point bending test. After the starter
notches for the test were removed, eddy current
testing was conducted using a differential-type plus
point probe at a frequency of 200 kHz to collect
signals caused by the cracks. The fatigue crack was
modeled as a rectangular continuous domain with
a constant width and uniform electromagnetic
properties. Since mechanical damage transforms the
austenitic phase into the martensitic phase, both
the conductivity and permeability of the domain
were explicitly considered. The depth and length of
the cracks were evaluated using a Bayesian-based
inverse algorithm, assuming the electromagnetic
properties of the crack were either known (equal to
air) or unknown. When the crack was modeled as
air, the evaluated crack sizes deviated considerably
from the actual sizes. In contrast, assuming the
electromagnetic properties to be unknown provided
better evaluations with quantified uncertainty.
KEYWORDS: electromagnetic testing, cracking, finite element
simulation, uncertainty, profile likelihood
1. Introduction
Eddy current testing (ECT) is a conventional nondestruc-
tive testing method that utilizes eddy currents induced by
time-dependent electromagnetic fields. It has several attractive
characteristics, such as being contactless and highly sensi-
tive to surface discontinuities thus, it is especially effective in
detecting discontinuities on the surface of metallic compo-
nents. However, sizing a detected discontinuity from measured
eddy current signals is challenging because the signals do not
provide direct information on the discontinuity’s profile—
especially its depth—unlike ultrasonic testing. Therefore,
many studies have been performed to develop computational
inversion algorithms to evaluate discontinuity profiles from
measured eddy current signals (Bowler et al. 1994 Auld and
Moulder 1999 Chen et al. 2004 Chen et al. 2009).
One of the major challenges in evaluating the profile of
a discontinuity from eddy current signals using a computa-
tional inversion algorithm is the ill-posedness of the problem.
Usually, the algorithm postulates that discontinuities with
similar profiles produce similar signals. However, in reality,
two discontinuities whose profiles are largely different can
sometimes produce very similar signals (Yusa et al. 2006).
Consequently, the presence of even small noise, which is inevi-
table in actual inspections, could lead to a large error in profile
evaluations (Yusa et al. 2007a Yusa and Hashizume 2017).
This highlights the necessity of taking into consideration the
reliability of discontinuity profile evaluations, specifically the
potential errors in the estimated profiles.
To address this issue, recent studies have proposed more
sophisticated algorithms to evaluate discontinuity profiles
probabilistically—that is, by estimating discontinuity parame-
ters not as single values (i.e., point estimation) but as a prob-
ability density function (Cai et al. 2018 Tomizawa and Yusa
2024). When the probability density function provided by the
algorithm is locally distributed, the results would be reliable.
Conversely, a function with a large distribution indicates less
reliable results. A previous study by the authors demonstrated
that the algorithm could reasonably quantify the reliability
uncertainty in evaluating the length and depth of a rectangular
slit machined into an austenitic stainless steel plate.
This article reports further development of the algorithm
in dealing with a more practical problem: estimating the
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PROBABILISTIC SIZING OF A FATIGUE CRACK
ON TYPE 304 AUSTENITIC STAINLESS STEEL
FROM EDDY CURRENT SIGNALS
TAKUMA TOMIZAWA† AND NORITAKA YUSA†*
† Department of Quantum Science and Energy Engineering, Graduate School of
Engineering, Tohoku University, Sendai, Japan
*Corresponding author: noritaka.yusa.d5@tohoku.ac.jp
Materials Evaluation 83 (8): 73–80
https://doi.org/10.32548/2025.me-04519
©2025 American Society for Nondestructive Testing
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 73

depth and length of a fatigue crack artificially introduced into
a type 304 stainless steel plate. Prior studies have shown that
it is not always proper to model a real crack as an insulating
wall, as induced eddy currents sometimes flow across a crack
because of the electrical contact between the crack surfaces
(Yusa et al. 2007b). Therefore, real cracks generally need to
be modeled as conductive domains. Additionally, mechan-
ical damage can alter the magnetic properties of steel (Jiles
1988 Thompson 1996), and fatigue damage transforms the
austenitic phase of type 304 stainless steel into the martensitic
phase (Chen et al. 2002 Shimamoto et al. 2008 Kinoshita et
al. 2014 Xie et al. 2018 Kinoshita 2020 Lan et al. 2022). Thus,
it is sometimes necessary to consider a magnetic domain
when evaluating eddy current signals due to fatigue cracks
in type 304 steel (Wang et al. 2013). For these reasons, this
study attempts to evaluate the length and maximum depth of
a fatigue crack under the condition that the electromagnetic
properties of the crack are unknown. The results demonstrate
that improper discontinuity modeling could lead to a signifi-
cant overestimation or underestimation of the reliability of the
results.
2. Materials and Methods
This section describes the fatigue cracks targeted for sizing in
this study, as well as the procedures employed for their mea-
surement and sizing.
2.1. Sample Preparation
This study prepared 32 plates made of type 316L stainless steel,
each containing a rectangular artificial slit, and three type 304
stainless steel plates with artificial fatigue cracks.
The purpose of using the type 316L stainless steel plates
was to estimate the likelihood of the Bayesian estimation
explained in Section 2.3. The dimensions of the artificial slits in
the type 316L stainless steel plates are summarized in Table 1.
The slits were machined using electrical discharge machining
and had a width of 0.5 mm. The plate thicknesses ranged from
5 mm to 25 mm, as the samples were made in several earlier
studies by the authors. This study assumed that the effect of
plate thickness on the measured signals was negligible. This
assumption was based on two factors: (1) the maximum slit
depth did not exceed 60% of the plate thickness, and (2) the
standard depths of penetration were much smaller than the
plate thickness. These samples were the same as those used in
a previous study by the authors (Tomizawa and Yusa 2024).
The fatigue cracks in the type 304 stainless steel plates,
which this study aimed to size, were introduced using cyclic
four-point bending tests, as illustrated in Figure 1. The original
dimensions of the plates were 200 mm in length, 72 mm in
width, and 14 mm in thickness. The terminal distances of the
tests were 40 mm and 100 mm. To initiate a fatigue crack,
a semi-elliptic notch with a length of 5 mm and a depth of
0.5 mm was machined into each plate prior to the bending
tests, and the notch was removed after the fatigue test. The
final thickness of the plates was ~13 mm, which was much
thicker than the standard depth of penetration. The results of
the visual inspection of the surfaces of the plates are presented
in Figure 2. Table 2 summarizes the conditions of the fatigue
tests as well as the surface lengths and maximum depths of the
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14 mm
100 mm
Notch
40 mm
200 mm
Figure 1. Schematic of the four-point bending test.
10 mm
10 mm
10 mm
Figure 2. Surface of the type 304 stainless steel plates: (a) TP1 (b) TP2
(c) TP3. The scratches running vertically are caused by machining to
remove the notches.
TA B L E 1
Dimensions of the artificial slits in the type 316L
stainless steel plates
Length (mm) Depth (mm)
5 0.1, 0.2, 0.3, 0.5, 1.0, 1.5, 2.5, 3.0, 5.0
10 0.1, 0.2, 0.3, 0.4, 0.5, 1.0, 1.0, 1.5, 2.5, 3.0, 5.0
20 0.1, 0.2, 0.3, 0.5, 1.0, 1.5, 2.5, 3.0, 5.0, 5.0, 10.0
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