MODELS FOR CRACK DETECTION IN A CYLINDRICAL HOLE
CONTAINING AN ELASTIC LAYER AND A FLUID-FILLED
CYLINDRICAL HOLE
J. C. Aldrin1, J. D. Achenbach2
Computational Tools, Gurnee, IL 60031, USA
2
Center for Quality Engineering and Failure Prevention,
Northwestern University, Evanston, IL 60208, USA
ABSTRACT. This paper presents a study of ultrasonic NDE models for a cylindrical hole containing an
elastic layer and a fluid-filled hole. For the elastic layer case, both dispersion relations and analytical
solutions were solved to assess the sensitivity of leaky Rayleigh waves to annulus properties. A BEM
model was formulated for the scattering of shear waves by a fluid-filled hole with a notch. Through
simulated studies, a viable inspection technique was derived and validated experimentally.
INTRODUCTION
Prior work has shown the feasibility of a two element ultrasonic transducer approach for
C-141 weep hole inspection [1,2]. Current weep hole inspection requires that the wet wing
be completely purged of fuel before inspection can begin. Due to the time and cost of
emptying and drying out a wing, the capability of weep hole inspection for a wing
containing fuel is of interest to the Air Force. Prior work has been performed to understand
the effect of a fluid-filled cavity on the performance of the two-element weep hole
inspection technique for radial fatigue cracks [3-4]. These papers indicated significant
challenges for directly applying the dual transducer weep hole inspection approach for top
crack detection. However, models addressing the scattering response for an incident shear
wave transducer signal and the case of fluid-filled cavity with radial fatigue crack are
needed to provide additional insight for a potential ultrasonic inspection approach.
Another factor that may affect the performance of the ultrasonic inspection technique is
a polyurathane coating on the inside edge of the C-141 weep holes. This coating is used to
protect the edge of the hole from corrosion. The current procedure requires that the coating
be removed prior to inspection. A significant reduction in time and effort would be gained
by eliminating the need for the removal and the subsequent reapplication of this coating.
The development of a model will generate the needed understanding in order to assess the
viability of using the weep hole inspection procedure for this case.
This paper presents a study of ultrasonic NDE models for both a cylindrical hole
containing an elastic layer and a fluid-filled cylindrical hole. For a cylindrical hole with an
elastic annulus representing the layer, the dispersion relations are first derived and solved
numerically for the phase velocity of the lowest modes. An analytical solution for the
scattering response to an incident plane shear wave by a cylindrical hole with an elastic
annulus is also derived. Parametric studies are performed to assess the sensitivity of leaky
Rayleigh waves to variation in the properties of the annulus. A BEM (boundary element
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
181
method) model is formulated for the scattering response of an incident shear wave by a
fluid-filled hole with a notch. Through simulated measurement results, the measured
signals are explored for potential use in crack detection.
MODELS
Figure l(a) displays an elastic annulus embedded in an infinite elastic medium. The
approach to derive the dispersion relation will be similar to the approach used by Epstein
for the composite cylinder [5]. The general solution for the scalar and vector potentials for
two dimensional plane strain elasticity in terms of cylindrical coordinates is given by:
ft (r, 0, t) = {A[jn fa fcr)]+ B[Yn fafoOjJe-'* sin »# •
cvsnO ,
¥zl (r,0,t) = [C[jn (#*r)]+ D[Yn (frkr)]}^
d)
(2)
where:
co
c
c
CL
c
CT
k7 = -, (%=—, 013,= — .
To satisfy the radiation condition for the infinite elastic medium, the general solutions for
the scalar and vector potentials are written as follows:
<p(r, 0, t) = {E[H™ (afcrjjje-'"* sin n 9 ,
(3)
(4)
These solutions can subsequently be applied to evaluate the displacements, ur and UQ, and
the stresses, Tr and Tr.0for the two regions. Following application of traction free boundary
conditions at r = a, and continuity of displacement and traction at r = b, with some
manipulation, the homogeneous system of equations is obtained. In order to solve the
dispersion relation, solutions in terms of complex angular wave numbers are necessary.
The resulting imaginary part of the angular wave number corresponds to the attenuation of
the modes necessary for an infinite elastic domain. A program was written in Maple to
search for the complex roots in n for the lowest modes of interest over the range of
frequencies corresponding with the inspection problem. With each solution for the
complex angular wave number, the phase velocity of the circumferential mode was
calculated. For additional details on the solution of this problem, see reference [6].
An extension of this analytical derivation was performed by deriving the transient
scattering response due to a plane shear wave incident on a cavity with an elastic layer.
Pao and Mow previously investigated the case of a cavity in an infinite elastic medium with
an elastic liner for an incident plane longitudinal wave [7]. For the present study, the case
of an incident plane shear wave will be solved. To obtain the transient response, a plane
wave pulse is first defined in the frequency domain. Then, the expansion coefficients are
solved for a selected range of frequencies, and an inverse Fourier transform is applied to
calculate the solution for the locations of interest. For each frequency, the six expansion
coefficients are solved for TV angular wave numbers in order to properly construct the
response. The value of TV is determine through verification of convergence for each
frequency. Hassan and Nagy [4] used a similar approach to study the 'leaky' Rayleigh
wave for the fluid-filled hole case. For additional details on the solution of this particular
problem, see reference [6].
Figure l(b) displays the configuration of a fluid-filled cylindrical hole with a radial
notch emanating from the hole. To address this problem, the boundary element method
182
FIGURE 1. Diagrams of (a) elastic annulus bonded to a cylindrical cavity in an infinite elastic solid, and (b)
fluid-filled cylindrical cavity in an infinite elastic medium with a fluid-filled notch.
(BEM) was used. The frequency domain BEM formulation presented here is based an
extension of the derivation for a fluid inclusion presented by Niwa et al [8]. For this
problem, the notch is finite in width and filled with a fluid. To minimize computational
time, a single boundary was used to define the shape of the cylindrical hole and notch. For
addition details on the solution approach and parameters for the problem, see reference [6].
RESULTS FOR ELASTIC LAYER CASE
Figure 2 displays the dispersion curves where the phase velocity of the circumferential
mode normalized with respect to the shear wave velocity of the elastic layer (c / CT) is
plotted with respect to the normalized frequency (12 = 0) - b / CT .) In this plot, the two
lowest circumferential modes for the cylindrical cavity with an elastic insert are compared
with the dispersion curves for the cylindrical cavity, and for the elastic annulus. For this
study, the material properties of aluminum were used for the infinite elastic medium, and
the properties of polyurethane were applied to the elastic layer. As a rule of thumb, the
wave speeds and density of the polyurethane are approximately one half the values for the
corresponding properties in aluminum. The geometric parameter, h/b, was set to 0.05,
representing the worst case layer thickness found in actual applications. Of particular
interest, the normalized frequency corresponding with an ultrasonic transducer center
frequency of 5.0 MHz for the C-141 weep hole inspection case is 32.
At lower frequencies, the first circumferential mode equates well with the dispersion
curve for the cylindrical hole case. Clearly, when the wavelength of the circumferential
mode is large, the relative thickness of the layer is small, and thus the layer has little impact
on the response. Therefore, very thin layers composed of soft material produce little
change in the 'leaky' Rayleigh wave for the empty cylindrical hole case. As the frequency
is increased, the first circumferential mode decreases in phase velocity and shifts toward
the lowest mode of the elastic annulus case. Likewise for the second circumferential mode,
a decrease in the phase velocity is observed, where the second lowest mode for the elastic
annulus case is approached. Thus, as the wavelength becomes small with respect to the
layer thickness, the mode becomes concentrated within the elastic layer. At the center
frequency of the inspection transducer, the existence of the layer produces a significant
difference in the phase velocity with respect to the empty cylindrical hole case. Clearly, a
measurable change in the 'leaky' Rayleigh wave is expected for the selected layer
parameters. Due to the thickness of the layer being relatively thin, only a few significant
modes are observed within the range of frequencies and phase velocities solved. However,
increases in the thickness of the layer would further impact the nature of the
circumferential modes.
183
To better explore the sensitivity of the 'leaky' Rayleigh wave to variation in the
thickness of the elastic annulus, the transient response for the analytical model was studied.
Figure 3 displays the transient in-plane deflection amplitude at r = 3 b, 0 =180°, for a
cylindrical hole with an elastic layer where h/b was varied from 0.00 to 0.20. This choice
of location corresponds to the approximate location of the pulse-echo transducer. First,
with an increase in the width of the elastic layer, the time of flight of the 'leaky' Rayleigh
wave increases. This can be correlated to a decrease in the phase velocity of the 'leaky'
Rayleigh wave as observed in the dispersion relation shown in Figure 2. In addition, as the
layer width is increased, loss into the layer and subsequent re-radiation from the layer
produces the observed spreading of the 'leaky' Rayleigh wave signal. In terms of the
dispersion relation, the higher circumferential modes contribute more to the response as the
layer thickness is increased. Lastly, due to the spreading of the signal, the peak amplitude
of the 'leaky' Rayleigh wave signal is generally reduced. In conclusion, given the low
levels of noise present within the measurement gate, the observed variation in the signal
will have only a limited impact on the peak-to-peak measurement performed for top crack
detection. However, one can likewise conclude that this added variation would greatly
hinder any methodology to characterize the crack size.
——cylindrical hole
——-elastic annulus
cylindrical hole with layer
50.0
60.0
FIGURE 2. Comparison of dispersion curves for circumferential waves propagating around a cylindrical
cavity with an elastic layer (solid gray) with curves for a cylindrical cavity (black), and an elastic annulus.
20.0
FIGURE 3. Total field in-plane response (peak to peak) for an incident shear plane wave at location r = 3 b,
0- 180° for various layer geometric parameter settings (h/b.)
184
FLUID-FILLED HOLE RESULTS WITH EXPERIMENTAL COMPARISON
Figures 4(a) and 4(b) display contour plots of the total field response to an incident inplane shear pulse on a fluid-filled cylindrical cavity for no-notch and with-notch cases
respectively, for eight snapshots in time. The magnitude of the in-plane displacement for
each field location is represented by a gray scale, where the darkest and lightest regions
represent locations of greatest and lowest amplitude, respectively. The properties of water
were used for the fluid. See reference [6] for the transducer and material property values.
In Figure 4(a), initial reflection of longitudinal and transverse waves is first observed.
Also, a longitudinal wave is transmitted into the fluid-filled cavity that propagates as a halo
wave [3]. The incident transverse wave also generates a Rayleigh wave that propagates
along the surface of the weep hole. The Rayleigh wave can clearly be observed leaking
energy into the surrounding solid due to the curvature of the weep hole and thus generating
a significant circumferential shear wave propagating from the hole. However, a significant
loss of the 'leaky' Rayleigh wave in the form of a longitudinal wave into the fluid at the
Rayleigh angle is observed. Additionally, as time marches forward, a halo wave continues
to propagate through the fluid, becomes incident upon the wall of the cavity, and transmits
a portion of the wave back into the elastic media. Thus the reradiated waves from the fluid
dominate the signal which returns to the catch transducer, as was previously shown [3-4].
FIGURE 4. Contour plots of the total displacement field response to an incident in-plane shear pulse on a
1/4" fluid filled weep hole for (a) no crack and (b) with crack cases for eight time steps, t = 1,2,3,4,5,6,7,8.
185
For the notch case in Figure 4(b), observations can be made from the scattering response
For the notch case in Figure 4(b), observations can be made from the scattering response
to examine
potential signals for crack detection. The incident transverse wave again
to examine potential signals for crack detection. The incident transverse wave again
generates a Rayleigh wave that propagates along the surface of the weep hole. With a top
generates a Rayleigh wave that propagates along the surface of the weep hole. With a top
notch
present, a portion of the 'leaky' Rayleigh wave is reflected. The reflected Rayleigh
notch present, a portion of the ‘leaky’ Rayleigh wave is reflected. The reflected Rayleigh
wave leaks energy into the riser and thus generates a small but detectable circumferential
wave leaks energy into the riser and thus generates a small but detectable circumferential
shear wave that propagates from the hole. Time step t = 4 displays this detectable signal.
shear wave that propagates from the hole. Time step t = 4 displays this detectable signal.
As for the empty cylindrical hole case, the pitch transducer can be used to detect this
As for the empty cylindrical hole case, the pitch transducer can be used to detect this
signal. Additional observations can be made for the notch case. Again, a significant loss
signal. Additional observations can be made for the notch case. Again, a significant loss
of the propagating 'leaky' Rayleigh wave into the fluid as a halo wave is observed.
of the propagating ‘leaky’ Rayleigh wave into the fluid as a halo wave is observed.
However, the magnitude of the halo wave transmitted into the fluid for the notch case is not
However, the magnitude of the halo wave transmitted into the fluid for the notch case is not
asasgreat
as for the no-notch case. Since a portion of the 'leaky' Rayleigh wave is reflected
great as for the no-notch case. Since a portion of the ‘leaky’ Rayleigh wave is reflected
before
the
wave in
in the
the
before thesignal
signalisisfully
fullyleaked
leaked into
into the
the fluid
fluid (and
(and solid),
solid), the
the transmitted
transmitted halo
halo wave
fluid
is
reduced.
This
results
in
the
first
reradiated
halo
wave
being
significantly
smaller
fluid is reduced. This results in the first reradiated halo wave being significantly smaller
forforthe
of the
the
thenotch
notchcase.
case. Time
Timestep
stept t ==88clearly
clearlydisplays
displays this
this reduction
reduction in
in the
the magnitude
magnitude of
first
reradiated
halo
wave.
As
for
the
empty
cylindrical
hole
case,
a
second
transducer
first reradiated halo wave. As for the empty cylindrical hole case, a second transducer
located
locatedabove
abovethe
thepitch
pitch transducer
transducer can
can be
be used
used to
to detect
detect this
this signal.
signal. These
These observations
observations
indicate
two
potential
methods
for
detecting
the
existence
of
a
top
notch.
indicate two potential methods for detecting the existence of a top notch.
Figures
signals
Figures 55 and
and 66 display
display the
the comparison
comparison of
of pulse-echo
pulse-echo and
and pitch
pitch catch
catch signals
respectively
with notch,
notch,
respectivelyfor
forfour
fourcases:
cases: (1)
(1) no
nonotch,
notch, no
no fluid,
fluid, (2)
(2) no
no notch,
notch, with
with fluid,
fluid, (3)
(3) with
No fluid,
fluid,No
Notop
topnotch,
notch,Pulse-echo
Pulse-echo
No
specular
reflection
transmitted
transmitted
leaky Rayleigh
wave
Experiment
Experiment
BEMSimulation
Simulation
BEM
Fluid-filled, No
Notop
top notch,
notch,Pulse-echo
Pulse-echo
Fluid-filled,
No
No fluid,
fluid, 0.07"
0.07"top
topnotch,
notch,Pulse-echo
Pulse-echo
w**$j*—*~
Fluid
F luid filled,
filled,0.07"
0.07"top
top notch,
notch,Pulse-echo
Pulse-echo
reflected
leaky Rayleigh
wave
specular
reflection
0.0
2.0
2.0
4.0
4.0
6.0
8.0
6.0
8.0
time (µsec)
first
first
reradiated
reradiated
halo
halo wave
wave
second
second
reradiated
reradiated
halo
halowave
wave
10.0
10.0
12.0
12.0
14.0
14.0
FIGURE5.5. Simulated
Simulated and
and experimental
experimental pulse-echo
FIGURE
pulse-echo signals
signalsfor
foraatransducer
transducersignal
signalincident
incidentonona arib
ribclip
cliphole
hole
withand
and without
without an
an insert
insert and/or
with
and/or aa notch.
notch.
186
No
No fluid,
fluid, No
No top
top notch,
notch, Pitch-catch
Pitch-catch
transmitted
transmitted
leaky
leaky Rayleigh
Rayleigh
wave
wave
specular
reflection
Experiment
Experiment
BEM
BEM Simulation
Simulation
Fluid-filled, No
No top
top notch,
notch, Pitch-catch
Pitch-catch
Fluid-filled,
No fluid,
fluid, 0.07"
0.07" top
top notch,
notch, Pitch-catch
Pitch-catch
No
Fluid filled,
filled, 0.07"
0.07" top
top notch,
notch, Pitch-catch
Pitch-catch
Fluid
reflected
reflected
leaky Rayleigh
Rayleigh
leaky
wave
wave
specular
reflection
0.0
0.0
2.0
2.0
4.0
4.0
6.0
6.0
8.0
8.0
second
first
second
reradiated reradiated
reradiated
reradiated
halowavg
halo wave
halo
wave halo
10.0
10.0
12.0
12.0
14.0
14.0
time ((jjsec)
time
µsec)
FIGURE 6. Simulated and experimental pitch-catch signals for a transducer signal incident on a rib clip hole
with and without an insert and/or aa notch.
no fluid,
fluid, and (4) with notch, with fluid.
fluid. The
The magnitude
magnitude of
of the
the BEM
BEM simulations was
was
adjusted
adjusted to match the average specular refection
refection for the
the experimental
experimental data.
data. Overall,
Overall, there
there isis
good agreement between the simulated and experimental
experimental data.
data. For
For the
the pulse-echo
pulse-echo response
response
in Figure 5, the notch case is shown to generate
generate aa significant
significant reflected
reflected 'leaky'
‘leaky’ Rayleigh
Rayleigh
wave signal for both the empty and the fluid-filled
fluid-filled hole
hole cases.
cases. Although
Although the
the magnitude
magnitude of
of
the reflected
'leaky'
Rayleigh
wave
signal
for
the
fluid-filled
hole
case
(4)
is
about
reflected ‘leaky’
the fluid-filled hole case (4) is about 30%
30% of
of
the
the magnitude
magnitude for
for the
the empty
empty hole
hole case
case (3),
(3), this
this signal
signal is
is still
still significantly
significantly greater
greater than
than the
the
noise level
level for
for the
noise
the no-notch
no-notch case
case (2).
(2). ItIt can
can be
be observed
observed that
that accurate
accurate placement
placement of
of gating
gating
can improve
improve the
the signal
signal to
to noise
noise ratio
ratio significantly.
significantly. For
can
For the
the pitch-catch
pitch-catch response
response in
in Figure
Figure 6,
6,
the transmitted
circumferential Rayleigh
Rayleigh wave
the
transmitted circumferential
wave is
is reduce
reduce considerably
considerably by
by the
the addition
addition of
of
fluid to
to the
the insert.
insert. In
In addition,
addition, two
two significant
significant reradiated
fluid
reradiated halo
halo waves
waves are
are also
also shown
shown for
for the
the
fluid
filled
cases.
As
observed
in
the
contour
plots,
there
is
a
significant
reduction
fluid filled cases. As observed in the contour plots, there is a significant reduction in
in the
the
magnitude of
of the
the first
first reradiated
reradiated halo
halo wave
wave due
due to
to the
the existence
existence of
Simulated
magnitude
of aa top
top notch.
notch. Simulated
results produce
produce aa reduction
reduction of
of about
about 50%
50% for
for larger
larger notch
notch cases.
cases.
results
Figure 7(a)
7(a) displays
displays the
the peak
peak to
to peak
peak response
response for
Figure
for the
the BEM
BEM simulation
simulation of
of the
the reflected
reflected
'leaky'
Rayleigh
wave
signal
in
the
pulse-echo
mode
and
the
first
reradiated
‘leaky’ Rayleigh wave signal in the pulse-echo mode and the first reradiated halo
halo wave
wave
signal in
in the
the pitch-catch
pitch-catch mode
mode for
for various
various notch
signal
notch lengths.
lengths. The
The symbols
symbols indicate
indicate the
the results
results
of the
the BEM
BEM calculations,
calculations, while
while the
the lines
lines indicate
indicate the
of
the general
general trend
trend of
of the
the peak
peak to
to peak
peak
187
notch length (in.)
notch length (in.)
(a)
(b)0.6as r
0.10
pp(H1PC)
pp(CPE)
0.08
0.5
Ratio - pp(CPE)/pp(H1PC)
Normalized Peak to Peak Amplitude
0.09
(b)
0.07
0.06
0.05
0.04
0.03
0.02
0.4
0.3
0.2
0.1
0.01
0.00
0.0
0.000 0.002 0.004 0.006 0.008 0.010 0.020 0.030 0.040 0.070 0.105
0.000 0.002 0.004 0.006 0.008 0.010 0.020 0.030 0.040 0.070 0.105
crack length (in.)
crack length (in.)
0.000 0.002 0.004 0.006 0.008 0.010 0.020 0.030 0.040 0.070 0.105
0.000 0.002 0.004 0.006 0.008 0.010 0.020 0.030 0.040 0.070 0.105
crack length (in.)
crack length (in.)
FIGURE 7. Plots of simulated normalized peak to peak amplitude of (a) reflected 'leaky' Rayleigh wave
FIGURE 7. Plots of simulated normalized peak to peak amplitude of (a) reflected ‘leaky’ Rayleigh wave
signal (CPE) and first reradiated halo wave signal (HIPC), and (b) the corresponding ratio for various notch
signal (CPE) and first reradiated halo wave signal (H1PC), and (b) the corresponding ratio for various notch
response versus
versus notch
notch length.
length. AA proposed
proposed measure
measure consists
consists of
of the
the ratio
ratio of
of the
the amplitudes
amplitudes
response
of
the
reflected
'leaky'
Rayleigh
wave
by
the
pulse-echo
transducer
over
the
first
reradiated
of the reflected ‘leaky’ Rayleigh wave by the pulse-echo transducer over the first reradiated
halo
wave
by
the
catch
transducer.
Figure
7(b)
displays
the
results
for
this
ratio
for
various
halo wave by the catch transducer. Figure 7(b) displays the results for this ratio for various
numerically
simulated
notch
length
cases,
clearly
indicating
that
notches
of
0.010
in. and
and
numerically simulated notch length cases, clearly indicating that notches of 0.010 in.
larger can be properly classified using this approach.
larger can be properly classified using this approach.
Seven no-notch and seven notch weep hole samples were investigated experimentally.
Seven no-notch and seven notch weep hole samples were investigated experimentally.
The sawcut notch lengths varied between 0.05 and 0.08 in. Gates were used to measure the
The sawcut notch lengths varied between 0.05 and 0.08 in. Gates were used to measure the
peak-to-peak signals from the reflected 'leaky' Rayleigh wave and the first reradiated halo
peak-to-peak signals from the reflected ‘leaky’ Rayleigh wave and the first reradiated halo
wave. Proper classification was made of all no-notch and notch samples. Ratio values
wave. Proper classification was made of all no-notch and notch samples. Ratio values
ranged from 0.04 to 0.08 for the no-notch cases and from 0.29 to 0.43 for the notch cases.
ranged from 0.04 to 0.08 for the no-notch cases and from 0.29 to 0.43 for the notch cases.
Any discrepancies with the simulation may be due to variation in experimental top-notch
Any discrepancies with the simulation may be due to variation in experimental top-notch
locations (0 = 45° for the BEM simulation), saturation of 8 bit A/D, transducer model
locations (θnn = 45° for the BEM simulation), saturation of 8 bit A/D, transducer model
approximations, and coherent noise. In conclusion, the results of this paper support the
approximations, and coherent noise. In conclusion, the results of this paper support the
claim that an NDE technique for crack detection of fluid-filled cylindrical holes is feasible.
claim that an NDE technique for crack detection of fluid-filled cylindrical holes is feasible.
ACKNOWLEDGEMENTS
ACKNOWLEDGEMENTS
The authors would like to acknowledge SAIC Ultra Image Int. for acquisition of
The authors would like to acknowledge SAIC Ultra Image Int. for acquisition of
experimental data and support of this work. Funding for the majority of this effort was
experimental data and support of this work. Funding for the majority of this effort was
provided by Warner Robins Air Logistic Center. Additional support for modeling in NDE
provided by Warner Robins Air Logistic Center. Additional support for modeling in NDE
was provided by AFRL-MLLP, Wright Patterson AFB OH.
was provided by AFRL-MLLP, Wright Patterson AFB OH.
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188