The probe and other conditions in the simulations were
the same as those used in the experiment. The conductivity
and relative permeability of a plate in the simulations were
1.35 MS/m and 1, respectively, regardless of whether the
material was actually type 316L or type 304 stainless steel.
The discontinuity was modeled as a rectangular domain.
Its conductivity in the numerical simulations was 0, 0.1, 0.5,
1, 3, and 5% of the base material’s conductivity, ​​σ​0​​​, and its
relative permeability was 1, 3, 5, 7, and 10. The nonlinearity
and hysteresis of the magnetic properties were neglected, as
is typical in numerical simulations of eddy current signals.
While the actual profile of the probe, shown in Figure 3b,
differed somewhat from its design, the numerical simulations
modeled the probe as illustrated in Figure 3a to simplify the
modeling process. The boundary condition ​ × A =0​ was
imposed at the outermost boundary second-order nodal
elements were used to discretize the entire computational
domain.
3. Results and Discussion
Figure 4 shows the estimated dimensions—specifically, the
probability densities of the depth and length—of the three
fatigue cracks. The red marks in the figure represent the actual
dimensions of the fatigue cracks listed in Table 2. The range
of the red marks in the vertical direction corresponds to the
uncertainty in the actual depth due to machining used to
confirm the depth, as explained in Section 2.1.
10
8
6
4
2
0
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20
Length (mm)
10
8
6
4
2
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 5 10 15 20
Length (mm)
10
8
6
4
2
0
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 5 10 15 20
Length (mm)
Figure 4. Estimated lengths and depths of the fatigue cracks for (a) TP1,
(b) TP2, and (c) TP3.
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
1 2 3 4 5 6 7 8 9 10
μr (–)
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0 1 2 3 4 5
σ/σ0 (%)
Figure 5. Marginal distribution of the estimated electromagnetic
properties of the fatigue crack in TP1: (a) conductivity (b) relative
permeability.
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 77
Probability
density
(mm
–2 )
Probability
density
(mm
–2 )
Probability
density
(mm
–2 )
Depth
(mm)
Depth
(mm)
Depth
(mm)
Probability
density
(–)
Probability
density
(–)

The bright area in Figure 4a indicates that the possible
ranges for the length and depth of the crack are 7–9 mm and
1.5–2.5 mm, respectively. The actual length and depth of the
crack introduced into TP1 are 8.6 mm and 2.2 mm, respec-
tively, which fall within this range. By contrast, the bright areas
in Figures 4b and 4c are much more elongated in the depth
direction—indicating increased uncertainty in depth estima-
tion. This is reasonable because induced eddy currents are
predominantly concentrated near the surface of the target
thus, the shallow portion of a discontinuity has a dominant
effect on eddy current signals.
It should be noted that the depths of the fatigue cracks
in TP2 and TP3 are much greater than the standard depth of
penetration. This highlights the difficulty in quantitatively eval-
uating their depths, a limitation clearly reflected in the results
shown. The reason why the red marks in Figures 4b and 4c
fall somewhat outside the bright areas is unclear. In practice,
however, it would be acceptable to overestimate the surface
length by a few millimeters.
The marginal distributions of the estimated electromag-
netic properties of the cracks are presented in Figures 5–7.
While it is difficult to analyze these results quantitatively, the
figures suggest that it is improper to model the fatigue cracks
as an air region, like artificial slits. Notably, the multimodal
characteristics observed in Figures 5a, 5b, and 6b imply non-
unique inverse solutions, where multiple combinations of
discontinuity parameters can reproduce the measured signals.
In other words, this finding challenges the conventional
single-value deterministic approach to discontinuity parameter
evaluation and underscores the need for probabilistic assess-
ment, as implemented in this study.
To confirm the necessity of considering the electromag-
netic properties of a crack, additional sizing was performed
assuming that the conductivity and relative permeability
of the discontinuity were zero and unity, respectively, and
the results are presented in Figure 8. The bright areas in
the figure are much more localized compared with those
in Figure 4. Although this implies that the results are highly
reliable, the estimated dimensions—especially the depths of
the deeper cracks—are largely different from the actual ones.
This result indicates that improper discontinuity modeling
can lead not only to an error in the sizing (Yusa et al. 2003)
but also to a significant misevaluation of the uncertainty of
the results.
1 2 3 4 5 6 7 8 9 10
μr (–)
σ/σ0 (%)
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0 1 2 3 4 5
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Figure 6. Marginal distribution of the estimated electromagnetic
properties of the fatigue crack in TP2: (a) conductivity (b) relative
permeability.
1 2 3 4 5 6 7 8 9 10
μr (–)
σ/σ0 (%)
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0 1 2 3 4 5
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 7. Marginal distribution of the estimated electromagnetic
properties of the fatigue crack in TP3: (a) conductivity (b) relative
permeability.
ME
|
FATIGUECRACKS
78
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
Probability
density
(–)
Probability
density
(–)
Probability
density
(–)
Probability
density
(–)