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

(Kurokawa et al. 1995 Rosell and Persson 2012 Camerini et
al. 2019). Furthermore, fatigue cracks often exhibit complex
microscopic features such as branching, making precise geo-
metric modeling impractical. Since eddy current signals are
generated by disturbances in induced eddy currents, it is more
critical to reproduce a similar distribution of eddy currents
than to replicate the detailed microstructure of the crack itself.
Previous research has shown that assigning electrical conduc-
tivity to a simple domain of constant width—on the order of a
few hundred micrometers—is sufficient to simulate the eddy
current response from real cracks (Tanaka and Tsuboi 2001
Yusa et al. 2007b Yusa and Hashizume 2009).
The permeability assigned to the modeled domain is
intended to simulate the martensitic phase formation induced
by fatigue damage. In reality, the martensitic phase appears not
inside the fatigue crack but near its faces (Uchimoto et al. 2012
Wang et al. 2013). However, assuming a domain with a width
much greater than the actual crack opening, as done in this
study, enables modeling a fatigue crack as a magnetic domain.
2.3.2. PROBABILISTIC SIZING OF A FATIGUE CRACK
This study estimates the possible discontinuity parameters, X,
as a probability density function from the measured signals, V,
using Bayes’ theorem: ​ ​ ​
(1)​ P(X|V) =​ ___________​​​)X​​​|V​​​(P​)X​(P
∫ ​ P​(X)​P(​​​V​|X)​ ​ dX​​​
Specifically, the real and imaginary parts of eddy current
signals measured along a scanning line—containing the
maximum signal and parallel to the discontinuity—were used
for the subsequent analyses and estimation as follows: ​ ​
(2)​ P​(X|​Vre,1​​,​Vim,1​​, ​ ​ ​ Vre,2​​,​Vim,2​​ ​ ​ ⋯,​Vre,​Nm​​​​,​Vim,​Nm​​​​)​ ​ ​ ​ ​
=​ P(​​X)​​​∏q=m​​​ ​ ​ N​ ​ ​ ∏ P​(​​​V​ |X​)​​ ​ ​ ​ ​
____________________​​​q,p​​mi,er=​p1​
∫ ​ P(​​X)​​​∏q=m​​​ ​ ​
1 ​
N​ ​ ​ ∏
p=re,im​​ ​ P​(​​​Vp,q​​​|​​X​)​​ ​ ​ ​ ​ dX​​
where ​ ​
V​re,q​​​ and ​​ ​ im,q​​​ are the real and imaginary parts of eddy
current signals measured at the ​ ​ –th scanning point,
respectively, and ​ ​
N​m​​​ is the total number of scanning points.
The distance between two neighboring scanning points
was 1.0 mm, and signals whose amplitude did not exceed 40%
of the amplitude of the maximum signal were excluded to
mitigate the effect of noise. This limited the maximum number
of scanning points to 41. The prior distribution, ​​ ​ (​​X)​​​​ ​ was set
as a uniform distribution, ​​ ​ (​​​X​min​​,​X​max​​​)​​​​ where ​​ ​ min​​​ and ​​ ​ max​​​
represent the minimum and maximum discontinuity parame-
ters used in the numerical simulations, respectively. The like-
lihood function, ​​ ​ (​​​V​p,q​​​|​​X)​​​​ ​ was obtained by assuming that the
measured signal from a discontinuity with profile ​ ​ ​​ ​ ​ p,q​​​ (​​X)​​​​, ​
could be correlated with the signals obtained through numeri-
cal simulations, ​​ ​ p,q​ ​ ​ (X)​​ as: ​ ​ ​
(3)​ ​ Vp,q​​​(X)​ ​ =N​(​​μ1,p,q​​,​σ​1,p,q​​​)​​Vpi,q​ ​ ​ 2 ​ ​ s m(X)​ ​ ​ +N​(​​μ​2,p,q​​,​σ​2,p,q​​​)​​​​2​
where​
N​(μ,​σ​​2​)​​ denotes a normal distribution with a mean of ​ ​ and
a standard deviation of ​ ​ .
The parameters in Equation 3 were estimated using the
experimental data obtained from the 32 type 316L stainless
steel plates and the corresponding numerical simulations.
Since the discontinuities in these plates were mechanical slits,
and because type 316L stainless steels are generally less mag-
netized by mechanical damage than type 304 stainless steels,
the numerical simulations used to obtain ​​ ​ p,q​ ​ ​ (X)​​ modeled the
discontinuity as an air region. While this means the param-
eters were estimated without considering the effects of the
conductivity and permeability of a discontinuity, this approach
remained reasonable because it is impossible to fabricate
artificial discontinuities with known conductivity and perme-
ability. More details about how to estimate the parameters
in Equation 3 can be found in an earlier study by the authors
(Tomizawa and Yusa 2024).
The posterior distribution was obtained using the
Metropolis–Hastings method. The total number of evaluations
and burn-in samples were 200 000 and 50 000, respectively.
The proposed distribution used in the method was a normal
distribution with a mean of 0. The standard deviations were
specified as follows: 0.1 mm for crack length, 0.1 mm for crack
depth, 0.05% of the base material’s conductivity for conductiv-
ity, and 0.1 for relative permeability. This study assumed that
the distributions were uncorrelated. The length, depth, con-
ductivity, and relative permeability used for the initial values
of the sampling were 10 mm, 5 mm, 0.6% of the base material’s
conductivity, and 1.1, respectively.
It should be noted that the electromagnetic properties of
the discontinuity, namely the conductivity and permeability,
were treated as unknown variables in this study.
2.4. Numerical Simulation
The numerical simulations to obtain ​​ ​ p,q​ ​ ​ (X)​​ in Equation 3
were carried out using commercial finite element model (FEM)
software, COMSOL Multiphysics Version 5.2, and its AC/DC
module. The simulations were performed in a frequency
domain using the following governing equation: ​ ​
(4)​ ​ (jωσ − ​ ω​​2​ε)​A +∇ ×​​ 1
μr​​​μ0​​​​ ​ ​ (∇×A)​ =​ Je​​ ​ ​
where
vectors ​ ​ and ​​ ​ e​​​ represent the magnetic vector potential
and the external current density flowing in the probe,
respectively, ​
j​ is the imaginary unit, ​
ω​ is the angular frequency, ​
σ​ is the electrical conductivity, ​
ε​ is the permittivity, ​ ​
μ​r​​​ is the relative permeability, and ​ ​
μ​0​​​ is the permeability of vacuum.
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