plots of error in estimated crack depth versus known depth
for aluminum and titanium are shown in Figures 10a and 10b.
(Steel was not included due to poorer results.) The analysis
focuses on the tail of the data that crosses the probability of
0.05 and how well a normal distribution represents the data.
For both cases, the inversion algorithm overestimates the crack
depth, implying that the model is producing conservative
results. While the error trend for cracks in aluminum is right-
skewed, the trend for the left-side tail appears quite normal as
the probability approaches 0.05. Here, the 95% safety LUS bound
is estimated to be –0.26 mm, a significant result considering the
fairly wide variability present for the crack depth sizing data. In
practice, adding 0.26 mm to estimated depth would ensure that
nearly all cracks are mitigated during hole resizing. It is common
practice to perform a final BHEC inspection after hole resizing to
verify that no crack indication remains.
For the titanium error data, the tails of the distribution
were especially wide and indicated that assuming a normal
0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Aluminum
Steell
Titanium
0.2 0.4 0.6 0.8 1.0
Known crack length (mm)
1.2 1.4 0
0
0.220
0.440
0.660
0.880
1.00
1.2
1.44
0.2 0.4 0.6 0.8 1.0
Known crack depth (mm)
1.2 1.4
Aluminummuminu
Steelle
Titaniummuniait 2
1
0 2 0 0 0 8 0 1 2
Al
S
T
0 2
0 4
0 6
0 8
0
1
Alu
ita
Figure 9. Plots of (a) estimated versus known crack length and (b) estimated versus known crack depth, grouped by material.
–0.4
0.003
0.01
0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.95
0.98
0.99
0.997
–0.2 0 0.2 0.4 0.6 0.8
Error – crack depth estimate (mm)
0.003
0.01
0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.95
0.98
0.99
0.997
–0.5 0 0.5 1
Error – crack depth estimate (mm)
3
7
Figure 10. Normal probability plots of error in the estimate of crack depths versus the known values for small (a) aluminum corner cracks
and (b) titanium corner cracks, with a 95% lower bound (aluminum: –0.255 mm [normal], –0.244 mm [tail] titanium: –0.228 mm [normal],
–0.416 mm [tail]).
TA B L E 5
Results for smoothed empirical likelihood quantiles
Quantile Material Estimate (mm) Lower 95% bound (mm) Upper 95% bound (mm)
5% Aluminum –0.255 –0.309 –0.186
5% Titanium –0.417 –0.449 –0.373
ME
|
CRACKSIZING
54
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
(V)
crack
depth
EstEstimated
)(mm)
AmAmplitude
Probability Probability

distribution in the 95% safety LUS bound is incorrect. The tail
of the data crosses the 0.05 probability line at –0.42, which
is much lower than where the normal line crosses around
–0.23. These tails indicate there may be some poorly under-
stood aspects of the inversion results (which drive the larger
amount of error in these results) that require further investi-
gation. Smoothed empirical likelihood quantiles can be used
to estimate error quantiles with confidence bounds and do
not assume a specific distribution (Chen and Hall 1993). The
results are presented in Table 5. These 5% quantile estimates
agree with results from the normal probability plots but addi-
tionally provide 95% confidence intervals for the estimates.
5. Conclusions
This paper summarized recent work to improve and validate
the capability to characterize fatigue cracks in metallic mul-
tilayer fastener sites using bolt-hole eddy current (BHEC)
techniques with model-based inversion. Key enhancements
have been made on data preprocessing, model calibration,
liftoff compensation, and the inversion scheme. The updated
approach now addresses crack sizing in titanium, aluminum,
and stainless steel, across multiple frequencies and varying
hole diameters. A novel method was developed to efficiently
build liftoff compensation relationships using limited experi-
ments combined with statistical model fits.
A comprehensive set of fast surrogate models was created
using an improved numerical solver capability, enabling the
simulation of split-D differential eddy current 2D responses for
discontinuities in bolt holes, including corners. Over 2.4 million
measurement simulations were completed in under 200 days.
A comprehensive crack sizing evaluation study was performed,
including 133 unique cracks with verified length and depth
measurements via microscopy, and nearly 1000 BHEC scans
covering a wide range of test conditions.
The results demonstrate improved sizing performance
over the practice of using peak amplitude alone. The crack
length estimates exhibited less error than the crack depth
estimates, although depth is the more critical parameter for
determining appropriate maintenance actions. The 95% safety
limit against undersizing (LUS) bound for corner cracks in
aluminum—within the range where crack removal is feasible—
was estimated to be –0.26 mm. While the average inversion
error for corner cracks in titanium and steel was acceptable,
the tails of the error distributions were too wide to produce
a practical 95% safety LUS bound. Future work is planned to
investigate the root cause for these poor sizing estimates. This
model-based inversion capability has been integrated and
demonstrated with a commercial BHEC instrument.
While significant progress has been made in producing
quality crack dimension estimates using model-based inver-
sion with current BHEC inspection procedure, fundamentally
the BHEC technique was designed to detect the presence of
cracks, not provide the means to size them. One recommenda-
tion is to develop a separate follow-up eddy current procedure
optimized for crack sizing, minimizing error while providing
the maximum utility for the end user.
First, it is difficult to have sensitivity to increasing crack
depth well beyond the depth of penetration of the probe. The
use of lower and multiple frequencies should provide more
capability and improved accuracy over the current single
high-frequency approach. Second, there would be a large
benefit in using a smaller-profile coil to better resolve the ends
of the crack and for assessing much smaller corner cracks. This
is expected to also improve depth estimation. Third, limitations
were discovered in using only a single-point calibration with
one notch size. The calibrated model did not track amplitude
well with increasing crack depth and length. Essentially, if
the application requires accurate sizing beyond 1.2 mm deep
corner cracks, a larger discontinuity size is needed to be part of
the calibration panel and process. In addition, it would also be
advantageous to have notches with different aspect ratios and
to have widths approaching 0.03 mm, which would improve
representation of a crack-like response. These features are not
available in current calibration standards.
Lastly, some model discrepancies were noted, particularly
for stainless steel (17-7PH TH1050), which has a high per-
meability. A surrogate model with greater flexibility to adapt
to a limited set of empirical samples would likely improve
performance—especially for the special case when cracks
are adjacent to a stainless-steel layer. Further development
is needed to design a surrogate model that provides the best
compromise between speed, flexibility, and accuracy.
ACKNOWLEDGMENTS
The authors would like to acknowledge support for this work by the US
Air Force under the contract FA8650-19-C-5218, as well as support for
the manufacture of crack specimens by Dr. Thomas Mills of Analytical
Processes/Engineered Solutions (AP/ES) software development by Sean
Corley (formerly of TRI Austin) sample manufacture, sample verification,
and data acquisition by John Nagel (formerly of TRI Austin), Cody Morrow
(formerly of TRI Austin), Jameson Pitcheralle (TRI Austin) and general
support in the eddy current technique from ASNT NDT Level III (ET) Mark
Keiser (TRI Austin). The UniWest EVi eddy current inspection system with
an ECS-5 BHEC scanner was used to acquire all experimental data for the
program.
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