(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,pmi,er=p1
∫ P(X)∏q=m
1
N ∏
p=re,im P(Vp,q|X) dX
where
Vre,q and im,q are the real and imaginary parts of eddy
current signals measured at the –th scanning point,
respectively, and
Nm 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, (Xmin,Xmax) where min and max
represent the minimum and maximum discontinuity parame-
ters used in the numerical simulations, respectively. The like-
lihood function, (Vp,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,σ1,p,q)Vpi,q 2 s m(X) +N(μ2,p,q,σ2,p,q)2
where
N(μ,σ2) 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 − ω2ε)A +∇ × 1
μrμ0 (∇×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,
μr is the relative permeability, and
μ0 is the permeability of vacuum.
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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
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,pmi,er=p1
∫ P(X)∏q=m
1
N ∏
p=re,im P(Vp,q|X) dX
where
Vre,q and im,q are the real and imaginary parts of eddy
current signals measured at the –th scanning point,
respectively, and
Nm 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, (Xmin,Xmax) where min and max
represent the minimum and maximum discontinuity parame-
ters used in the numerical simulations, respectively. The like-
lihood function, (Vp,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,σ1,p,q)Vpi,q 2 s m(X) +N(μ2,p,q,σ2,p,q)2
where
N(μ,σ2) 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 − ω2ε)A +∇ × 1
μrμ0 (∇×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,
μr is the relative permeability, and
μ0 is the permeability of vacuum.
ME
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FATIGUECRACKS
76
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