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
ME
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TECHPAPER
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
74
M AT E R I A L S E V A L U AT I O N • A U G U S T 2 0 2 5

fatigue cracks as determined after the subsequent eddy current
testing described in Section 2.2.
Note that the maximum crack depths were evaluated by
gradually thinning the plates until the cracks disappeared.
Therefore, it is probable that the actual crack depths are
~0.1 mm shallower than the values shown in the table.
2.2. Eddy Current Testing
Eddy current signals from the slits and fatigue cracks were col-
lected using a commercial ECT instrument. A differential-type
plus point probe, shown in Figure 3, was used with an exci-
tation frequency of 200 kHz. The probe consisted of two rect-
angular coils that worked as both an exciter and a detector.
The coils were connected in series, so the phase and amplitude
of the currents flowing through them were identical. The probe
outputs a signal corresponding to the difference between the
impedances of the two coils.
The actual profile of the probe, revealed by radiographic
testing and shown in Figure 3b, deviated from the design illus-
trated in Figure 3a. While this deviation could lead to error
in discontinuity evaluation, this study assumes that only the
design profile is known, as the actual probe profile is not
always available in practical nondestructive evaluations. The
number of the turn of the coils is 102.
The probe was attached to an X-Y stage and scanned the
surface of the plate two-dimensionally with a pitch of 0.5 mm
and a liftoff of 1.0 mm. All measured signals were calibrated
such that the amplitude and phase of the maximum signal
from an artificial slit (20 mm in length and 10 mm in depth)
machined in a type 316L stainless steel plate were set to 1.0 V
and 0°, respectively, to enable a direct comparison with signals
obtained using numerical simulations. Note that the standard
depth of penetration is ~1 mm, indicating that the plates
prepared in this study were sufficiently thick.
2.3. Probabilistic Sizing of a Fatigue Crack
This subsection describes the modeling of the fatigue cracks
and the specific sizing procedure used in this study, with consid-
eration given to the uncertainty inherent in the sizing process.
2.3.1. MODELING A FATIGUE CRACK
This study modeled a fatigue crack as a rectangular continuous
domain with a constant width of 0.5 mm and uniform electro-
magnetic properties, specifically conductivity and permeability,
throughout. Since the study uses signals on a scanning line
along a discontinuity, a discontinuity using this model is rep-
resented by a vector X with five scalar elements: length, depth,
center position, conductivity, and permeability.
Machining performed to confirm the depth of the fatigue
crack—by gradually thinning the plate—revealed that the
surface length of the crack also decreased. This observation
suggests that the actual boundary profiles of the fatigue cracks
more closely resemble a semi-elliptical shape rather than a
rectangular one. In this study, however, the rectangular model
was adopted due to its simplicity in numerical modeling. While
this implies that employing a semi-elliptical model would
provide more accurate results, it should be emphasized that
the primary objective of this study is not to precisely deter-
mine the shape of a fatigue crack, but to develop a numerical
method for evaluating possible errors in crack sizing.
The modeled crack width of 0.5 mm is substantially
greater than the actual opening of a fatigue crack—that is, the
distance between two crack faces however, it is important to
note that, in the context of numerical simulations for ECT, the
width of the domain representing a crack serves a different
purpose than the physical crack opening. In general, when
the modeled discontinuity width is smaller than the spatial
resolution of the probe, its effect on eddy current signals is
minimal compared to the influence of the electrical contact
between crack surfaces, as demonstrated in previous studies
TA B L E 2
Conditions of the fatigue test and the dimensions of the
fatigue cracks
ID Load (min/max)
(kN) Cycles Length
(mm)
Maximum
depth (mm)
TP1 1.0/60.0 101 577 8.6 2.2
TP2 1.0/60.0 172 569 11.8 4.0
TP3 1.0/60.0 134 277 15.4 4.9
10 mm
3.6 mm
3.6 mm
2.8 mm
10 mm
4.5 mm
Figure 3. The plus point probe used in this study: (a) design
specification (b) radiographic image.
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 75