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ADE | Chapter 2
surface of the one-side water-immersed plate, which is excited by a circular
piezoceramic transducer, is dominated by quasi-Scholte waves. The accuracy of the
simulation results has been verified by comparing the phase velocities with the
theoretical dispersion curves as well as the experimental measurements. The
numerical simulations of wave scattering of the quasi-Scholte mode at a circular
blind hole have been compared with the experimental measurements. A good
agreement has been observed between the FE simulations and experimental
measurements. It has been concluded the 3D FE model is able to accurately simulate
quasi-Scholte wave propagation and its scattering characteristics for non-regular
geometries.
Further numerical studies have demonstrated that the scattering directivity
patterns (SDPs) depend on both the diameter and the depth of the circular blind hole.
At a given depth of the damage, the amplitudes of the backward scattered waves
are comparable to the forward scattering amplitudes for small values of RDW. For
larger RDW, the forward scattering amplitudes increase quickly with slight
variation while the backward scattering magnitudes fluctuate following a sinusoidal
pattern with the overall trend being a slow increase. In general, the forward
scattered waves are more suitable to be used for identifying the size of the damage
since they have larger amplitudes and follow a relatively simple scattering pattern.
For the local damage of the same diameter, the forward and backward scattering
amplitudes increase with the depth of the damage. The backward scattering
amplitudes increase faster than the forward scattered waves. Also, for the directions
perpendicular to the incident wave, the scattering amplitudes are weak for damage
whose depth is less than half of the plate thickness but significantly increase for
deeper damage.
Finally, this study has provided a comprehensive investigation of the
scattering phenomena due to low-frequency quasi-Scholte waves interacting with a
circular blind hole. The findings of this study can be used to provide a guide on
selecting appropriate excitation frequencies, guided wave modes, and transducer
locations, and hence, it will help to improve the performance of in-situ damage
detection techniques for structures exposed to the corrosive environment.
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ADE | Chapter 3
Chapter 3. Numerical and experimental investigations on mode
conversion of guided Waves in partially immersed plates
Abstract
This paper numerically and experimentally investigates the guided wave
propagation in a steel plate with one side partly exposed to water. The fundamental
anti-symmetric Lamb wave (A ) is excited on the dry plate section and travels to
0
the water-immersed plate section, where the generated A wave is mode converted
0
to the quasi-Scholte (QS) wave. The results demonstrate that the energy of QS wave
converted by the A wave decreases when the excitation frequency increases. In
0
addition, it is revealed that the guided wave energy can shift in the frequency
domain if the phase velocity of the incident A wave is larger than the sound speed
0
of water. The frequency shift phenomenon should be noticed in practical
applications because the behaviors of guided waves vary with frequency. Finally,
discussions are provided on the frequency selection for exciting guided waves to
detect damage on partially immersed structures and assess liquid properties.
Keywords: Quasi-Scholte waves; Lamb waves; leaky guided waves; mode
conversion; submerged structures.
39 |
ADE | Chapter 3
3.2. Introduction
Ultrasonic guided waves are elastic waves that travel along the boundary of a
structure and have been widely used for identifying damage in structures [1-5],
detecting debonding in adhesively bonded structures [6-9], sensing liquid levels and
properties [10-12], and assessing coatings on the substrate surface [13, 14]. The
advantages of ultrasonic guided waves are that they can propagate for a long
distance, enabling an efficient large-area inspection. Lamb waves are guided waves
in thin-walled structures, such as plates, shells, and pipes. When the plate is
surrounded by air, Lamb waves are composed of multiple symmetric and anti-
symmetric wave modes [15]. When one or both sides of the plate are exposed to
liquid, there is a substantial increase in the energy leakage into the surrounding
liquid medium [16, 17]. Therefore, guided waves in immersed plates are called
leaky Lamb waves and they behave differently from their counterparts in the plates
surrounded by air.
In addition to the symmetric and antisymmetric leaky Lamb wave modes,
there is an interface mode called quasi-Scholte (QS) wave in the plate immersed in
liquid [18]. Recently, studies have been conducted on the QS wave for a wide range
of applications because of its ability to propagate along the plate-fluid interface over
a long distance and high sensitivity to changes in the properties of both the plate
and fluid. Tietze et al. [19] experimentally demonstrated that QS wave propagating
at the interface between electrode and electrolyte is able to remove the diffusion
boundary layer, which can be employed to accelerate the electrochemical process.
Aubert et al. [20] invented a low-cost fluid manipulation device that employed the
generation of QS wave to sort living cells and separate plasma from a blood
microdroplet. Through schlieren imaging, the acoustic fields of the QS wave were
experimentally visualized to be evanescent in the direction normal to the plate
surface, which is a promising characteristic for microfluidic applications. Hayashi
and Fujishima [21] experimentally confirmed that QS wave could be excited by
applying the normal vibration directly on the surface of a plate loaded with water.
The generated QS wave was shown to be sensitive to the change of the physical
conditions on the plate surface. Thus, the QS wave is feasible for non-destructive
testing (NDT) of water-filled storage tanks and pipes.
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There are other studies focused on the application of QS wave, in which the
QS wave is excited by mode conversion from the fundamental anti-symmetric mode
(A ) of Lamb waves. Cegla et al. [18] developed a novel method for sensing fluid
0
property by exciting guided waves on a plate that was partially immersed in the
fluid. A transducer was attached at the end of the dry plate section (outside the fluid)
to excite A wave. When the generated A wave traveled from the dry section of
0 0
the plate to the section immersed in the fluid, part of the wave energy was reflected
backward, and the rest of the wave energy was converted into the leaky A and QS
0
waves. Leaky A wave decayed rapidly and disappeared after a short propagation
0
distance. While QS wave propagated along the immersed plate with low attenuation,
and then reached the end of the immersed section and reflected back to the
measurement location. At the point where the plate was outside the fluid, the QS
wave was converted back to A wave. The time-of-arrival and amplitude of the
0
measured signals changed with the viscosity and bulk longitudinal velocity of the
fluid, and hence, they could be employed to measure the fluid properties. Yu et al.
[22] proposed a Lamb wave-based method for assessing liquid levels in the nuclear
cooling pipe system. The method used a pair of piezoelectric wafer transducers that
were mounted on the wall of a test tank. One of the transducers was used as an
actuator and the other was used as a receiver. The wave signals were measured on
the test tank filled with different amounts of water. It was found that the
fundamental symmetric mode (S ) of Lamb waves was not influenced by the change
0
of water level. In contrast, the presence of water significantly changed both the
amplitude and phase of A wave. The phase change was shown to have a linear
0
relationship with the change of water level. It should be noted that this study did
not take into account the QS and leaky A waves, which also exist in the water-
0
immersed plate [21, 23].
Guo et al. [24] developed two-dimensional (2D) finite element (FE) models
to simulate guided wave propagation along an empty steel vessel and the steel
vessel filled with water, respectively. At the selected excitation frequency, A and
0
QS waves were identified on the water-free vessel and the water-filled vessel,
respectively. The latter was found to propagate more slowly than the former.
Therefore, the traveling time of the guided waves between two transducers could
be also utilized for measuring the liquid level in the steel vessel. This study only
42 |
ADE | Chapter 3
considered a single excitation frequency, at which the leaky A wave mentioned in
0
[18] was not detected in the water-immersed plate-like structure by both the 2D FE
simulations and the experimental measurements [24].
The aforementioned studies employed the mode conversion between QS
and A waves at different excitation frequencies to achieve different applications,
0
where the interactions among various guided wave modes were shown to be
different. To date, there are very limited studies on the variation with frequency of
the mode conversion phenomenon. However, studying the influence of excitation
frequency on guided wave propagation is very important because the behaviors of
guided waves are frequency-dependent. For example, it was reported that the
displacements of the QS wave mainly occur in the liquid, and the majority of studies
on QS wave had been focused on the fluid properties sensing [10, 18]. Only in
recent years, its applications were extended to detect damage for plate structures
submerged in liquid due to the observation that the QS wave at low frequencies has
most of its wave motions conserved in the immersed plate [21, 25]. QS wave is
dispersive at low frequencies and becomes nondispersive at high frequencies. The
wave structure of QS wave at a low frequency significantly differs from that at a
high frequency. Between the high and low frequencies, there is a frequency range,
at which QS wave transitions from dispersive to nondispersive. In this frequency
range, the wave structure of the QS wave changes rapidly with frequency, while
that of the A wave does not change much. Therefore, the mode conversion between
0
QS and A waves should also vary significantly with frequency, which has not been
0
discussed in the literature.
In the present study, the frequency dependence of the mode conversion from
A wave to QS wave is studied numerically and experimentally. The findings of
0
this study complement the current knowledge about guided wave propagation in
partially immersed plates and provide a guide on selecting appropriate excitation
frequencies for NDT of partially immersed structures and assessing liquid
properties and levels. The numerical method using a three-dimensional (3D) FE
model is proposed to portray guided wave propagation in a steel plate, of which one
side is partly exposed to water. A wave is excited at different frequencies on the
0
dry section of the plate and travels to the immersed section. The simulation results
43 |
ADE | Chapter 3
provide a visualization of the interaction of different guided wave modes in both
the plate and water. Then, experiments are conducted on a steel tank that is partially
filled with water. The time-space wave fields are captured by a scanning laser
Doppler vibrometer (SLDV) before and after guided waves travel from the dry
section of the plate into the immersed section. The mode conversion process is
graphically shown with the use of 2D Fourier transform (FT). The experimental
results show a good agreement with the numerical simulations. After that, the
variation of the mode conversion from A wave to QS wave with the excitation
0
frequency is analyzed based on the theoretical dispersion curves and mode shapes
of guided waves. Furthermore, it is observed that the energy of guided waves can
shift in the frequency domain during the mode conversion process if the phase
velocity of the incident A wave is larger than the sound speed of the surrounding
0
liquid medium. The energy shift in frequency can change the behaviors of guided
waves, which should be paid attention to in practical applications.
The paper is organized as below. Section 3.3 compares the theoretical
dispersion curves and mode shapes of guided waves in a plate surrounded by air
and the plate with one side exposed to water. Section 3.4 describes the 3D FE model
and presents the simulated guided wave fields in the partially immersed plate. After
that, Section 3.5 shows the experimental setup and the configuration of
measurement points. Section 3.6 illustrates the signal processing techniques and the
analysis of experimentally measured signals. Then, Section 3.7 summarizes the
frequency dependence of the mode conversion from A wave to QS wave and
0
explains the mechanism of the energy shift in frequency phenomena according to
the theoretical dispersion curves and mode shapes of the guided waves. Based on
the findings, the selection of appropriate excitation frequency is discussed for
reliable testing through the mode conversion from A wave to QS wave. Finally,
0
conclusions are drawn in Section 3.8.
44 |
ADE | Chapter 3
3.3. Guided waves in plates surrounded by air and plates with one
side exposed to water
Guided waves behave differently in plates surrounded by air and plates immersed
in water. When guided waves propagate in a plate in gaseous environments, there
is a very small energy leakage from the plate to the air. The energy leakage to the
air is not modeled in this study because the resistance of air to the displacements of
particles at the plate surface is very small. As shown in Figure 3.1, traction-free
boundary conditions are applied to the plate surface open to the air. In comparison,
when one side of the plate is exposed to water, the out-of-plane displacements and
stresses at the plate-water interface become continuous. The shear stresses are
disconnected because water cannot sustain shear forces [26]. The energy leakage to
the water layer is substantially larger than that to the air.
Figure 3.1. Guided wave propagation models and boundary conditions for (a) a dry
plate surrounded by air, and (b) a plate with one side exposed to water
The properties of guided waves vary with frequency, which can be
theoretically predicted by dispersion curves. The present study employed the global
matrix method to calculate the dispersion curves and the theoretical results were
used to interpret the following numerical and experimental data. Two guided wave
propagation models were constructed using the commercially available software
DISPERSE [27]. They were a 2 mm thick steel plate and the plate with water loaded
45 |
ADE | Chapter 3
on one side, respectively. Table 3.1 gives the material properties of the steel plate
and water. The water layer was defined as a semi-infinite half-space. The boundary
conditions of the plate-air interface and plate-water interface were modeled by the
solid-vacuum and solid-liquid interfaces, respectively. Based on the geometry and
material properties, the stresses and displacements in the plate and water layers
could be determined in terms of the partial waves. Then, a global matrix equation
representing the whole model was assembled by matching the boundary conditions
of each interface. The global matrix equation is a function of frequency,
wavenumber, and attenuation. Solving this global matrix equation gives a series of
combinations of frequency, wave number, and attenuation, at which the partial
waves can combine to a guided wave mode that propagates on the plates in the
directions as shown in Figure 3.1.
Table 3.1. Material properties of the steel and water
Young’s Bulk Bulk wave
Density Poisson’s
Material modulus modulus velocity
(kg m-3) ratio
(GPa) (GPa) (m s-1)
Steel 7800 212.038 0.287 --
Water 1000 -- -- 2.2 1480
Figure 3.2 compares the dispersion curves of guided waves for the 2 mm
thick steel plate surrounded by air and the plate with water loaded on its one side.
At the frequency range up to 500 kHz, only A and S waves exist in the plate
0 0
without water. They are represented by the green and blue dash-dot lines in Figure
3.2, respectively. The black solid lines, red dashed lines, and magenta dotted lines
denote QS, leaky A , and leaky S waves in the one-side water-immersed plate,
0 0
respectively. The phase velocity C and the group velocity C can be related to
p g
the angular frequency w and the real wavenumber k as C w k and
p
C w k. As shown in Figure 3.2(c), the wavenumber dispersion curves of A
0
g
wave and S wave are almost overlapped with those of leaky A wave and leaky S
0 0 0
wave, respectively. It should be noted that leaky A wave appears only after 150
0
kHz where its phase velocity is greater than the sound speed of the surrounding
46 |
ADE | Chapter 3
water [28]. At a frequency lower than 150 kHz, the wavenumber of QS wave in the
one-side water-immersed plate (black solid line) is just slightly larger than that of
A wave in the dry plate (green dash-dot line) and the difference between them
0
increases with frequency.
Figure 3.2. Comparison of dispersion curves of a 2 mm thick steel plate and the
plate with one side exposed to water: (a) phase velocity curves; (b) group velocity
curves; (c) wavenumber curves; (d) attenuation curves (the legends in Figure 3.2(d)
are applied for Figures 3.2(a)-3.2(d)).
Figure 3.2(d) shows the attenuation dispersion curves where significant
deviations can be observed between the dry plate and the one-side water-immersed
plate. Obviously, leaky A wave (red dashed line) has a much higher attenuation
0
than any other wave mode. This is because leaky A wave is dominated by the out-
0
of-plane displacements so that the wave energy can easily and massively radiate
into the surrounding water [29]. The attenuation dispersion curve of leaky A wave
0
declines sharply in the selected frequency range. This means that the low-frequency
leaky A wave has larger attenuation than the high-frequency leaky A wave. The
0 0
levels of attenuation of other wave modes are close for the frequency lower than
47 |
ADE | Chapter 3
200 kHz. Over 200 kHz, the attenuation of QS wave drops to almost zero, while the
attenuations of A , S , and leaky S waves slowly increase with frequency. From
0 0 0
the above observations, it can be concluded that when guided waves propagate from
the dry plate to the water-immersed plate, S wave is converted to leaky S wave
0 0
that has the same wavenumber but slightly higher attenuation. A wave is converted
0
to QS wave at a frequency lower than 150 kHz where both the wavenumber and
attenuation of QS wave in the immersed plate are similar to those of A wave in the
0
dry plate. However, leaky A wave appears when the excitation frequency exceeds
0
150 kHz. A wave can be mode converted to both QS wave and leaky A wave.
0 0
Thus, the mode conversion process can be different with frequency.
The similarity between A wave in the dry plate and QS wave in the one-
0
side water-immersed plate is also studied by their mode shapes. The mode shape of
a guided wave mode shows the distributions of the displacements through the
thickness of the structure and it can be calculated using DISPERSE [27]. The
frequency band of interest is selected from 100 kHz to 200 kHz, in which the
wavenumber dispersion curves of the A and QS waves gradually separate as the
0
frequency increases. Figure 3.3 shows the mode shapes of A wave at 100 kHz, 150
0
kHz, and 200 kHz for the 2 mm thick steel plate that is not in contact with water.
The deformation of the dry plate is dominated by out-of-plane displacements
denoted by the blue solid lines. As the frequency increases, the mode shape
diagrams of A wave do not display much difference.
0
Figure 3.4 shows the mode shapes of QS wave at 100 kHz, 150 kHz, and
200 kHz for the 2 mm thick steel plate with one side exposed to water. The water
layer was defined as a semi-infinite half-space. As shown in Figure 3.4, the
normalized displacement fields of the mode shapes in the water layer monotonically
decrease with the distance away from the plate-water interface. To better compare
the wave structures in the plate with and without water, the mode shape diagrams
only show the 2 mm water regions near the plate-water interface. As shown in
Figures 3.3(a) and 3.4(a), the deformation of QS wave in the immersed plate is
similar to that of A wave in the dry plate when the frequency is lower than 150
0
kHz. However, the deformation in the immersed plate of QS wave decreases rapidly
with frequency. At frequencies above 150 kHz, most of the displacements of QS
48 |
ADE | Chapter 3
3.4. Numerical simulation of guided wave propagation
To portray the guided wave propagation in partially water-immersed plates, a 3D
FE model was developed using the commercial FE software ABAQUS. Table 3.1
gives the material properties used for the FE simulation. A 300 mm × 150 mm × 2
mm steel plate was modeled with symmetry boundary conditions applied to the top
and right edges and absorbing regions attached to the left and bottom edges, as
shown in Figure 3.5. The absorbing regions were 50 mm wide and were divided
into 50 layers. The mass-proportional damping of the material in the absorbing
regions gradually increased layer by layer from zero at the innermost layer to 4×106
at the outmost layer. The absorbing region by increasing damping can reduce
unwanted waves reflected from the plate edges and has been widely used for
ultrasonic guided wave simulation analysis [25, 30-33].
Figure 3.5. Schematic diagram of the 3D FE model for a steel plate with one side
partly exposed to water: (a) front view and (b) side view.
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A wave was generated by applying the out-of-plane displacements to the
0
plate surface covered by a 5 mm diameter quarter-circle located at the top right
corner of the plate [34]. Figure 3.9(b) shows the waveform of the excitation signal,
which is a 10-cycle narrowband tone burst. To define the locations of the
measurement points, a one-dimensional coordinate, scanning distance (SD), was
defined along the right edge of the plate vertically downward as shown in Figure
3.5. The origin (SD = 0 mm) was set at the position of 50 mm below the excitation
center. Then, the out-of-plane displacements were collected at five measurement
points, which were evenly distributed at 50 mm apart from SD = 0 mm to SD = 200
mm. The other side of the plate was partially in contact with water as shown in
Figure 3.5(b). The water level was located at the second measurement point (SD =
50 mm). The thickness of the water layer was chosen to separate the pressure wave
reflections from the incident wave signals. The steel plate and the water layer were
meshed using 3D eight-node solid elements with reduced integration (C3D8R) and
3D eight-node acoustic elements with reduced integration (AC3D8R), respectively.
The fluid and solid interface was simulated by node-surface tie constraints [16, 25].
The element size was set as 0.5 mm, which ensured approximately fifteen elements
exist per wavelength of QS wave at 200 kHz. The simulation results were calculated
using the central-difference integration by ABAQUS/Explicit [35].
Figure 3.6 presents the snapshots of the simulation results with the
excitation frequency of 120 kHz. The color in the water regions denotes the acoustic
pressure. At this excitation frequency, a large proportion of the QS wave energy is
conserved in the one-side water-immersed plate, of which the deformation is similar
to that of A wave in the dry plate as shown in Figures 3.3 and 3.4. When A wave
0 0
travels from the dry section of the plate into the water-immersed section, part of the
wave energy is converted to the pressure waves in the water, and the rest of the
wave energy continues to propagate along the water-immersed plate at a speed
slightly quicker than the pressure waves in the water. After a short propagation
distance in the water-immersed plate, the first wave packet decays slowly and the
acoustic field is tethered to the plate-water interface as shown in Figure 3.6(d).
These are the typical characteristics of QS wave.
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Figure 3.7 shows the snapshots of the simulation results with the excitation
frequency of 170 kHz. The mode shape of QS wave at 170 kHz is dominated by the
in-plane displacements of water and the deformation of QS wave in the water-
immersed plate is no longer similar to that of A wave in the dry plate (see Figures
0
3.3 and 3.4). The first wave packet in Figure 3.7 continuously radiates wave energy
into the surrounding liquid medium, as shown by the skewed acoustic fields
(skewed yellow lines) in the water layer in Figures 3.7(b) and 3.7(d). This wave
packet with continuous wave energy leakage cannot be detected in Figure 3.6.
Following the first wave packet, another wave packet propagates along the
immersed plate at a speed slightly quicker than the pressure waves in water. Unlike
the first wave packet, the second wave packet propagates with most of the energy
confined to the plate-water interface. Based on the propagation speeds and the
acoustic pressure in the water, the first and second wave packets in Figure 3.7 are
identified as leaky A and QS waves, respectively.
0
To better observe how the signals change with the propagation distance,
Figure 3.8 presents the simulated out-of-plane displacements at the five
measurement points that are denoted by the red dots in Figure 3.5. Figures 3.8(a)
and 3.8(b) show the time-domain data for the excitation frequencies of 120 kHz and
170 kHz, respectively. Their corresponding frequency spectrums are given in
Figures 3.8(c) and 3.8(d), respectively. The amplitudes are normalized by the
maximum absolute amplitudes of the signals measured at the first measurement
point (SD = 0 mm). It should be noted that when the excitation frequency is 170
kHz, the time-domain signals measured at SD = 100 mm, 150 mm, and 200 mm are
so small that they are magnified by a factor of four and shown in Figure 3.8(b). The
normalized amplitudes of the signals collected in the immersed section of the plate
(SD > 50 mm) significantly decrease with the excitation frequency. In addition,
when the excitation frequency is 170 kHz, the simulated signals show apparent
frequency shifts in the frequency spectrums. For example, the central frequency of
the signal measured at SD = 100 mm, denoted by the orange dash-dot line in Figure
3.8(d), shifts to a frequency slightly higher than the central excitation frequency of
170 kHz. Subsequently, the wave energy shifts to a lower frequency as shown by
the signals measured at SD = 150 mm (purple solid line) and SD = 200 mm (yellow
solid line) in Figure 3.8(d). The simulation results demonstrate that the guided wave
53 |
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propagation in the partially water-immersed plate varies significantly with the
excitation frequency. The following sections present experimental investigations to
validate the simulation results, and the phenomenon of energy shift in the frequency
domain due to the presence of water is discussed in detail.
Figure 3.8. Simulated time-domain signals for a steel plate with one side partly
exposed to water (the same conditions as Case B presented in the experimental
section) with the excitation frequency of (a) 120 kHz and (b) 170 kHz, respectively;
(c) and (d) are the frequency spectrums of (a) and (b), respectively (the time-domain
signals measured at SD = 100 mm, 150 mm, and 200 mm in Figure 3.8(b) are
magnified by a factor of four).
3.5. Experiment setup
Experiments were conducted on a steel tank, of which the front wall was used as
the test plate. The test plate was 2 mm thick and made of mild steel. A circular
piezoceramic wafer (Ferroperm Pz27, Denmark) was used as the guided wave
actuator and it was bonded on the external surface of the test plate as indicated by
the PZT transducer in Figure 3.9(a). The diameter and the thickness of the
54 |
ADE | Chapter 3
defining measurement points. Then, the out-of-plane displacements were recorded
by the SLDV at a sampling rate of 10.24 MHz. Each measurement was improved
by averaging the signals with 800 acquisitions and applying a low-pass filter with
the cut-off frequency being 1MHz. Figure 3.9(c) shows the schematic diagram of
the experiment setup.
Similar to the 3D FE model, a one-dimensional coordinate, SD, was defined
on the external surface of the test plate. The origin (SD = 0 mm) was set at the
position of 50 mm vertically below the center of the piezoceramic wafer. The
experimental study included two parts, which were five-point scan tests and line
scan tests, respectively. Firstly, the five-point scan tests were conducted on the steel
tank to validate the simulation results. The signals were measured at SD = 0 mm,
50 mm, 100 mm, 150 mm, and 200 mm, which were at the same locations as the
simulations as shown in Figure 5.
Then, line scan tests were carried out to visualize guided wave fields on the
test plate. The objective of the line scan test was to experimentally demonstrate the
interaction of each guided wave mode during the mode conversion process.
According to the simulation results shown in Figures 3.6, 3.7, and 3.8, the mode
conversion process mainly occurs between SD = 50 mm and SD = 150 mm. After
SD = 150 mm, only the QS wave can be detected as shown by the signals measured
at SD = 150mm and SD = 200 mm in Figure 3.8. Therefore, the line scan tests
focused on the region between SD = 0 mm and SD = 150 mm as shown in Figure
3.10(a). The signals were collected at 127 measurement points, which were evenly
distributed along the scan line from SD = 0 mm to SD = 150 mm. With these
measurements, the time-space wave fields could be constructed by plotting the
amplitudes of the measured signals versus time and SD.
56 |
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Figure 3.10. Schematic diagram of the scan line on the test plate (a) front view; (b)
side view of Case A (empty tank); (c) side view of Case B (partially water-filled
tank).
In order to demonstrate the influence of water on the guided wave
propagation, both the five-point scan tests and line scan tests were carried out on
the empty tank (Case A) and the partially water-filled tank (Case B), respectively.
Figures 3.10(b) and 3.10(c) show the side view of the scan line on the test plate for
Case A and Case B, respectively. In Case B, the steel tank was partially filled with
water with the water level set at SD = 50 mm. Therefore, one-third of the scan line
was located in the non-immersed section of the plate (from SD = 0 mm to SD = 50
mm), called “dry plate”, and the rest of the scan line was located in the section of
the one-side water-immersed plate (from SD = 50 mm to SD = 150 mm), which
was denoted as “immersed plate”. Guided waves were generated on the test plate
by the piezoceramic wafer located at 100 mm above the water level. The time-space
wave fields were captured before and after guided waves traveled from the dry plate
into the immersed plate.
3.6. Experimental results and analysis
3.6.1. Validation of numerical simulations
This section presents the experimental measurements to validate the accuracy of the
3D FE model. Figure 3.11 presents the experimental results with the central
excitation frequency of 120 kHz. Figures 3.11(a) and 3.11(b) show the time-space
57 |
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wave fields for the empty tank and the partially water-filled tank, respectively. The
amplitudes are normalized by the maximum absolute amplitudes of the signals
measured at the first scan point (SD = 0 mm). The black dashed line in Figure 3.11(b)
denotes the water level located at SD = 50 mm. The generated guided wave fields
in the dry section of the plate (between SD = 0 mm and SD = 50 mm) are similar
for both the empty tank in Figure 3.11(a) and the partially water-filled tank in Figure
3.11(b). However, the amplitudes of the signals measured in the immersed plate
(SD > 50 mm) of the partially water-filled tank in Figure 3.11(b) are smaller than
their counterparts in the empty tank in Figure 3.11(a). This indicates that part of the
wave energy leaks from the immersed plate into the water as shown in the snapshots
of the simulation results (see Figures 3.6(a) and 3.6(b)).
Figures 3.11(c) and 3.11(d) show typical examples of the signals measured
at SD = 0 mm, 50 mm, 100 mm, 150 mm, and 200 mm for the empty tank and the
partially water-filled tank, respectively. The wave speed in the immersed plate of
the partially water-filled tank, as denoted by the dark blue dash-dot line in Figure
3.11(b), is slightly slower than the wave speed in the dry plate that is marked by the
red dashed line. From the wave speed evaluation, the wave packets measured from
the empty tank in Figure 3.11(a) and the dry section of the partially water-filled
tank in Figure 3.11(b) are identified as A wave [24]. S wave can be not observed
0 0
because it has negligible out-of-plane motions at the selected excitation frequency
[25, 36]. The changes of the wave speed and amplitude in the immersed section of
the test plate demonstrate that A wave is mode converted to QS wave. Figures
0
3.11(e) and 3.11(f) present the frequency spectrums of Figures 3.11 (c) and 3.11(d),
respectively. The amplitudes are normalized by the signal measured at the first scan
point (SD = 0 mm). The peaks of all measured signals are concentrated around the
central frequency of the excitation signal of 120 kHz. There is a good agreement
between the experimental measurements shown in Figures 3.11 (d) and 3.11(f) and
the simulation results (see Figures 3.8(a) and 3.8(c)).
58 |
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Figure 3.11. Experimental results with the excitation frequency of 120 kHz: (a) and
(b) are time-space wave fields for the empty tank and the partially water-filled tank,
respectively; (c) and (d) are typical examples of the time-domain signals for the
empty tank and the partially water-filled tank, respectively; (e) and (f) are the
frequency spectrums of (c) and (d), respectively.
For comparison, Figure 3.12 presents the experimental results with the
central excitation frequency of 170 kHz. Figures 3.12(a) and 3.12(b) present the
time-space wave fields for the empty tank and the partially water-filled tank,
respectively. For the empty tank, the waves propagate at a consistent speed as
represented by the red dashed line in Figure 3.12(a). The same wave fields are
observed in the dry section of the plate (SD < 50 mm) of the partially water-filled
tank as shown in Figure 3.12(b). However, when guided waves propagate into the
water-immersed section (SD > 50 mm), the wave amplitudes decrease rapidly and
59 |
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Figures 3.12(c) and 3.12(d) show typical time-domain signals measured at
SD = 0 mm, 50 mm, 100 mm, 150 mm, and 200 mm for the empty tank and the
partially water-filled tank, respectively. To provide better observation, the signals
measured from the partially water-filled tank at SD = 100 mm, 150 mm, and 200
mm are magnified by a factor of four and shown in Figure 3.12(d). Guided waves
propagate as a single wave packet along the dry section of the plate (between SD =
0 mm and SD = 50 mm). However, after a short propagation distance in the
immersed plate, the signal measured at SD = 100 mm shows two wave packets,
each propagating at different speeds. It should be noted that the mode split
phenomenon is not observed on the water-immersed plate with the central
excitation frequency of 120 kHz (see Figure 3.11(d)). The first wave packet decays
quickly and disappears as shown by the signal measured at SD = 150 mm in Figure
3.12(d), while the second wave packet propagates with low attenuation at a slower
wave speed (dark blue dash-dot line). Figures 3.12(e) and 3.12(f) show the
frequency spectrums of Figures 3.12(c) and 3.12 (d), respectively. The signals
measured at SD = 100 mm, 150 mm, and 200 mm from the partially water-filled
tank in Figures 3.12(f) have much smaller amplitudes than their counterparts from
the empty tank in Figures 3.12(e). In addition, Figure 3.12(f) displays the energy
shift in frequency, which has a good agreement with the simulation results (see
Figure 3.8(d)). It is confirmed that the energy shift in the frequency domain is due
to the presence of water because this phenomenon does not occur in the case of the
empty tank as shown in Figure 3.12(e).
To further investigate the accuracy of the 3D FE model, Figure 3.13
compares the peak amplitudes of the simulated and experimentally measured
signals in the frequency domain for both the empty tank (Case A) and the partially
water-filled tank (Case B), respectively. The magnitudes are normalized by their
corresponding peak amplitudes of the signals measured at SD = 0 mm. Figures
3.13(a) and 3.13 (b) present the results with the excitation frequency of 120 kHz
and 170 kHz, respectively. In the dry section of the plate (SD < 50 mm), the
amplitudes of the simulated and experimentally measured signals are identical for
both cases. When guided waves just pass the water level (from SD = 50 mm to SD
= 100 mm), the measured signals of Case A decrease slowly and smoothly, but the
signal amplitudes of Case B drop substantially. After SD = 100 mm, the guided
61 |
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wave amplitudes in both Case A and Case B decrease slowly with distance. When
the excitation frequency increases from 120 kHz to 170 kHz, the amplitudes of the
signals of Case A do not show obvious changes as shown by the red hexagons and
circles in Figure 3.13. However, the amplitudes of the signals of Case B
significantly decrease when the excitation frequency increases. In general, the
proposed 3D FE model well predicts the frequency shift phenomena and wave
attenuation characteristics and hence, the simulation results are validated to
interpret the experimental data.
Figure 3.13. Normalized amplitudes of the simulated and experimentally measured
signals in the frequency domain for the empty tank (Case A) and the partially water-
filled tank (Case B) with the excitation frequency of (a) 120 kHz and (b) 170 kHz,
respectively.
3.6.2. Segmented frequency wavenumber analysis
Although the time-space analysis displays the variation of wave amplitudes with
the time and propagation distance, it cannot determine the wave mode conversion
characteristics such as mode identities and their corresponding frequency
components. To graphically demonstrate the mode conversion process, 2D FT is
employed to identify the mode information of the experimental data collected along
the scan line on the partially water-filled tank (Case B). The scan line is divided
into three segments, each of which is 50 mm long and comprises 43 measurement
points as shown in Figure 3.10(c). The generated guided waves first propagate
through Segment 1 (Dry plate) and then to Segment 2 (Water-immersed plate) and
62 |
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finally to Segment 3 (Water-immersed plate). The water level is between Segment
1 and 2. The time-space data of each segment is converted to the frequency-
wavenumber spectrum through 2D FT, which is defined as:
uk, f = ux,te-i(2ft-kx)dtdx (3.1)
where uk, f and ux,t are the data in the frequency-wavenumber domain and
time-space domain, respectively. k and f denote the wavenumber and frequency,
respectively. x and t represent the space and time coordinate, respectively.
Figures 3.14(a)-3.14(c) present the experimentally measured data in the
time-space domain for the three segments and their corresponding frequency-
wavenumber spectrums are given in Figures 3.14(d)-3.14(f), respectively. The
excitation frequency is 120 kHz. The color in the frequency-wavenumber spectrums
denotes the wave energy of the experimentally measured signals, which is
calculated by 2D FT. The black solid lines are the theoretical wavenumber
dispersion curves of A , S , and QS waves calculated by DISPERSE. As mentioned
0 0
in Section 3.3, the wavenumber dispersion curves of leaky A and leaky S waves
0 0
overlap with A and S waves. For convenience, leaky A and leaky S waves are
0 0 0 0
labeled as A and S in the figures, respectively. In Segment 1, where the plate is
0 0
not immersed in water, only the energy of A wave is identified in the frequency-
0
wavenumber spectrum as shown in Figure 3.14(d). The energy of S wave is absent
0
because it has negligible out-of-plane displacements [25, 36]. Next, the wave
energy is converted from A wave to QS wave immediately in Segment 2 as shown
0
in Figure 3.14(e). The mode conversion occurs rapidly and the energy of QS wave
decays slowly with distance and dominates the frequency-wavenumber spectrums
of Segments 2 and 3. At this excitation frequency, leaky A wave is not detected
0
and the energy shift in frequency is not observed.
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QS wave. Considering that the deformation in the immersed plate of QS wave at
low frequencies is greater than that at high frequencies (see Figure 3.4), the anti-
symmetrical excitation of leaky A wave is more likely to produce QS wave at lower
0
frequencies. This can be also manifested by the frequency spectrums (see Figures
3.8(d) and 3.12(f)) where the wave energy is progressively transferred from a higher
frequency at SD = 100 mm to a frequency lower than the central excitation
frequency at SD = 150 mm. After that, the wave energy is conserved at the
frequency (lower than the excitation frequency of 170 kHz) and propagates with
small attenuation.
3.6.3. Further study by sweeping the excitation frequency
To further investigate the phenomenon of energy shift in the frequency domain, the
five-point scan tests were conducted on the partially water-filled tank using the
excitation signals with the central frequencies of 110 kHz, 130 kHz, 140 kHz, 150
kHz, 160 kHz, and 180 kHz. The collected signals were transformed to the
frequency domain and shown in Figure 3.16. The amplitudes were normalized by
the signals measured at the first scan point (SD = 0 mm). As it is confirmed from
the segmented frequency wavenumber plots (see Figures 3.14 and 3.15), the signals
measured at SD = 150 mm to 200 mm, denoted by the purple and yellow solid lines
in Figures 3.11(f), 3.12(f), and 3.16, are dominated by the low-attenuation QS wave.
Therefore, it can be concluded that the normalized amplitude of QS wave converted
by A wave decreases when the excitation frequency increases.
0
For excitation frequencies below 140 kHz, the signals measured from both
the dry section and water-immersed section of the plate are concentrated around the
central excitation frequency (see Figure 3.11(f) and Figures 3.16(a)-3.16(c)). The
energy shift in frequency can be observed for excitation frequencies over 150 kHz,
where leaky A wave appears as the phase velocity of the incident A wave becomes
0 0
larger than the sound speed of the surrounding water. Under the central excitation
frequency of 150 kHz and 160 kHz, the signals measured at SD = 100 mm, 150 mm,
and SD = 200 mm have most of their wave energy concentrated at a frequency
lower than the central excitation frequency (see Figures 3.16(d) and 3.16(e)). The
range of the frequency shift also increases when the central excitation frequency
67 |
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increases. However, the signals measured at SD = 100 mm under the central
excitation frequency of 170 kHz and 180 kHz are shown to have more energy
conserved at higher frequencies (see Figures 3.12(f) and 3.16(f)). This is because
the amplitudes of QS wave converted by A wave are so small that leaky A wave
0 0
can be detected clearly at this measurement point. Leaky A wave at higher
0
frequencies decays more slowly than that at lower frequencies as discussed in
Section 3.3, making the central frequency of the signal shift to a relatively higher
frequency. However, leaky A wave completely disappears after a short
0
propagation distance, and only the QS wave can be detected at SD =150 mm and
SD = 200 mm, as shown in Figure 3.15. Since the low-attenuation QS wave at lower
frequencies has a larger deformation fraction in the immersed plate, the measured
signals from the surface of the immersed plate eventually concentrate at a frequency
lower than the central excitation frequency. The next section summarizes the
frequency dependence of the mode conversion from A wave to QS wave and
0
further explains the mechanism of the frequency shift phenomena.
3.7. Discussion and application
3.7.1. The influence of excitation frequency on the mode conversion process
The mode conversion from A wave to QS wave has been numerically and
0
experimentally shown to be dependent on the excitation frequency. The frequency
dependence is summarized in this section and analyzed according to the dispersion
behaviors of the guided waves. The theoretical dispersion curves and mode shapes
of guided waves have been derived from the global matrix theory and are present
in Section 3.3. In the following discussion, the term “high frequency” means the
frequency range, in which the phase velocity of the incident A wave is larger than
0
the sound speed of the surrounding liquid medium and the leaky A wave appears.
0
The term “low frequency” indicates the frequency range, in which the phase
velocity of the incident A wave is smaller than the sound speed of the surrounding
0
liquid medium and the quasi-Scholte wave is dispersive. The transition frequency
between the high and low frequency ranges can be estimated from the phase
velocity dispersion curves. For example, the phase velocity of A wave for a
0
metallic plate monotonically increases with frequency until it reaches the Rayleigh
68 |
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wave speed, which is around 3000 m/s for steel. The sound speed of water is around
1500 m/s, which is constant for all frequencies. Therefore, the transition frequency
is around 150 kHz, at which the phase velocity of A wave traveling along the 2
0
mm thick steel plate is around 1500 m/s, as shown in Figure 3.2(a). The phase
velocity dispersion curves can be calculated by the commercial software
DISPERSE. The input data includes the material properties of the plate, the plate
thickness, and the sound speed of the surrounding liquid medium.
At low frequencies, the difference in wavenumber between A wave and QS
0
wave is small. In addition, the deformation of QS wave in the immersed plate is
similar to that of A wave in the dry plate (see Figures 3.3 and 3.4). Thus, A wave
0 0
can be mode converted to QS wave rapidly with most of the wave energy conserved
in the plate. When the excitation frequency increases, the energy distribution of QS
wave in the immersed plate decreases sharply and the similarity reduces between
A and QS waves (see the dispersion curves in Figure 3.2 and mode shapes
0
diagrams in Figures 3.3 and 3.4). Therefore, the amplitude of QS wave converted
by A wave significantly decreases with frequency.
0
At high frequencies, leaky A wave appears when the phase velocity of the
0
incident A wave becomes larger than the sound speed of the water. A wave is
0 0
mode converted to both QS wave and leaky A wave with more energy transferred
0
to the latter. Leaky A wave that has the flexural mode shape in the plate
0
continuously radiates compressional waves in the liquid and also excites QS wave.
After a short propagation distance, leaky A wave decays quickly and disappears so
0
that only QS wave can be detected.
3.7.2. The mechanism of the energy shift in frequency phenomenon
The energy shift in frequency occurs during the mode conversion process when the
incident A wave is generated at high frequencies. The mechanism of the energy
0
shift in frequency phenomenon can be explained by the dispersion curves and the
mode shapes of guided wave modes as follows.
Firstly, the central frequency of the signal can shift to a frequency higher
than the center frequency of the excitation, when the wave fields are dominated by
69 |
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the leaky A wave (see signals at SD = 100 mm in Figures 3.12(f) and 3.16(f)). This
0
is due to the fact that the attenuation dispersion curve of leaky A wave declines
0
sharply in the selected frequency region (see Figure 3.2). The low-frequency
components of leaky A wave decay much quicker than the high-frequency
0
components. Therefore, leaky A wave at high frequencies can travel longer
0
distances, making the central frequency of the signals measured in the immersed
plate near the water level relatively higher than the central excitation frequency.
Secondly, the central frequency of the mode converted QS wave is relatively
lower than the central frequency of the excitation. One reason is that the
deformation in the immersed plate of QS wave at low frequencies has a flexural
mode shape, which is similar to A wave in the dry plate. But the similarity between
0
QS and A waves reduces with frequency. This makes the mode conversion from
0
A wave to QS wave (at the intersection of the dry plate and the water-immersed
0
plate) much easier at low frequencies than that at high frequencies, with more
energy transferred and conserved in the plate. The other reason is that the
deformation of QS wave in the plate is rapidly reduced with frequency (see Figure
3.4). Thus, QS wave at low frequencies can be excited on the plate more easily by
the out-of-plane motions of leaky A wave after guided waves propagate into the
0
water-immersed plate. Since the generated QS wave has low attenuation, the
measured signals in the immersed plate eventually shift to a frequency lower than
the central excitation frequency (see signals at SD = 150 mm and SD = 200 mm in
Figure 3.12(f) and Figures 3.16(d)-3.16(f)).
3.7.3. Implications for practical applications
The findings of the present study suggest that the low-attenuation QS wave can be
easily excited by mode conversion from A wave that is generated at low
0
frequencies on the dry plate section. This phenomenon can be employed to detect
damage for the plate structures that are partially immersed in liquid, such as
partially water-filled tanks and pipelines. These structures generally experience
uniform corrosion and pitting corrosion. The latter is more critical because it
damages the deep structures with little loss of metal [37].
70 |
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Previous studies have characterized corrosion damage using A wave for
0
plates surrounded by air [38-40] and QS wave for plates in contact with liquid [21,
41]. For partially immersed plates, it is also possible to evaluate the defects by
sending A waves on the dry section of the plate and measuring the QS wave signals
0
on the immersed section. This method is very promising for long-range inspection
because QS wave does not radiate energy in the liquid (see Figures 3.6 and 3.7) and
is able to travel along the plate-fluid interface with low attenuation. The
measurement range through using the dispersive QS wave by mode conversion
from A wave can be of the order of several meters based on the low attenuation
0
characteristics as shown in Figure 3.13. However, the actual propagation distance
is dependent on the material properties of the plate and the surrounding liquid
medium as well as the excitation frequency. As discussed in Section 3.7.1, the lower
the excitation frequency, the more wave energy conserved in the plate during the
mode conversion process, and therefore the longer the propagation distance.
Another advantage is that this method has the potential to characterize the structural
defects entirely based on the QS wave, which is appealing for accurate detection
and imaging of the defects [42]. After a short propagation distance, leaky A wave
0
decays quickly and disappears due to high attenuation. The low-attenuation QS
wave can be well separated from the leaky S wave, of which the propagation speed
0
is three times that of QS wave (see Figure 3.2(b)). Although leaky S wave has low
0
attenuation, it is not sensitive enough to identify small and shallow corrosion
damage in the early stage [43-45]. In contrast, the QS wave has the shortest
wavelength at a given frequency and provides better sensitivity than the leaky S
0
wave to shallow hidden corrosion pits in immersed plates [41].
It should be noted that the damage detection algorithm for the partially
immersed structures should consider the change of wave behaviors due to the mode
conversion phenomenon and the presence of liquid. It is recommended to select an
excitation frequency below the transition frequency, at which the proposed
phenomenon of guided wave energy shift in frequency can be avoided. (see Figures
3.11 and 3.14). Otherwise, the effect of the potential wave energy shift in the
frequency domain should also be carefully considered and compensated. For
instance, it is observed in the present study that the wave energy of the mode
converted QS wave in the immersed plate moves to a frequency below the central
71 |
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excitation frequency, making the actual wavenumber of the QS wave smaller than
that at the central excitation frequency (see Figure 3.15(f)). This also indicates a
smaller phase velocity, a higher group velocity, and a larger wavelength (see Figure
3.2). The change of the propagation characteristics of guided waves will affect the
performance of conventional damage detection and imagining algorithms [16, 46].
The mode conversion between QS and A waves in partially immersed
0
plates was widely used for liquid-level assessing [22, 24, 47] and fluid-property
sensing [18, 48, 49]. The behaviors of A wave depend on the geometry and
0
material properties of the plate, while QS wave reflects the properties of both the
plate and the surrounding fluid medium. Generally, the difference between QS and
A waves becomes larger with frequency. Therefore, increasing the excitation
0
frequency can result in larger deviations in the signals measured from the partially
immersed plate in terms of the time of arrival, amplitude, and phase angle, and
hence, it can potentially increase the sensitivity of the signals to the variation of
liquid level and fluid properties. However, the results of the present study show that
the amplitude of the QS wave converted by A wave significantly decreases with
0
frequency. Therefore, the optimal excitation frequency is a trade-off between the
sensitivity and the amplitudes of the measured signals. For simplicity, it is also
recommended to excite the guided waves at a low frequency to ensure that the phase
velocity of the incident A wave is smaller than the sound speed of the surrounding
0
liquid medium. Without the interference of the leaky A wave, the mode conversion
0
process is simple and the frequency shift phenomena can be avoided.
Lastly, QS wave at high excitation frequencies becomes nondispersive and
is promising for fluid manipulation [20] and removing diffusion boundary layer [19,
50], where the focus is directed on the movement of the fluid particles and the
deformation in the plate is not interested. The low-attenuation QS wave with most
of the wave energy concentrated at the plate-fluid interface has the potential to
cover a large area of the fluid near the plate surface. However, attention should be
paid to the potential frequency shift of the measured signals. As shown in Figures
3.15 and 3.16, guided wave energy can shift in the frequency domain during the
mode conversion process, which gives rise to the change of wave behavior such as
group and phase velocities.
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3.8. Conclusion
This paper has provided an insight into the measurement of ultrasonic guided waves
in partially immersed plates. The main contributions are concluded as follows:
(1). Global matrix method is employed to derive the theoretical dispersion
curves and modes shapes of guided waves for a 2 mm thick steel plate and
the plate with one side loaded with water. It is found that the low-
frequencies QS wave in the one-side water-immersed plate and the A wave
0
in the dry plate have similar wavenumbers and deformations. But the
similarity reduces with frequency.
(2). A 3D FE model is developed to simulate the guided wave field in the steel
plate with one side partially immersed in water. The simulation results are
validated by the experimental data. The frequency shift phenomenon and
the guided wave amplitudes with propagation distance can be well predicted.
(3). The experimental studies are conducted on the empty tank and the partially
water-filled tank, respectively. It is confirmed that the frequency shift
phenomenon is due to the presence of water. The further investigation
presents a segmented frequency wavenumber analysis to graphically
demonstrate the mode conversion process, which is divided into three
segments: (i) before guided waves propagate into the water-immersed plate,
(ii) guided waves just propagate into the water-immersed plate, and (iii)
after a short propagation distance in the water-immersed plate. The
experimental data are compared with the theoretical dispersion curves,
through which the mode identities and the corresponding experimentally
measured wave energy can be determined.
(4). The guided wave energy shift in the frequency occurs during the mode
conversion process, which is not caused by the material nonlinearity (micro
cracks) of the plate. The amplitudes of the guided wave signals measured
from the water-immersed plate section are much smaller than those obtained
from the water-free plate section, on which the frequency shift phenomenon
is not observed. Then, the mechanism of the energy shift in frequency
73 |
ADE | Chapter 3
phenomenon is explained by the attenuation dispersion curves of leaky A
0
wave and the mode shapes of QS wave.
(5). Based on the findings, the selection of appropriate excitation frequency is
discussed for damage detection of partially submerged structures, assessing
liquid properties and levels, and fluid manipulations.
In summary, comprehensive investigations have been carried out for the
frequency dependence of the mode conversion from A wave to QS wave in a steel
0
plate with one side partially immersed in water. The findings can provide support
for the further development of guided wave-based techniques for damage detection
on partially immersed structures, liquid-level assessing, and fluid-property sensing.
This paper has only focused on the plate partially immersed in water. Future work
can study the partially immersed structures with different geometries and
investigate the effect when the structure is immersed in other types of liquid. In
addition, the influence of damage such as corrosion pits or stress cracking on the
guided wave propagation and mode conversion can be investigated.
3.9. Acknowledgment
This work was funded by the Australia Research Council (ARC) under grant
numbers DP200102300 and DP210103307. The authors are grateful for this support.
3.10. Reference
[1] J. Moll, C.P. Fritzen, Guided waves for autonomous online identification of
structural defects under ambient temperature variations, Journal of Sound and
Vibration, 331 (2012) 4587-4597.
[2] J. He, C.A.C. Leckey, P.E. Leser, W.P. Leser, Multi-mode reverse time
migration damage imaging using ultrasonic guided waves, Ultrasonics, 94 (2019)
319-331.
[3] A. Aseem, C.T. Ng, Debonding detection in rebar-reinforced concrete structures
using second harmonic generation of longitudinal guided eave, NDT & E
International, (2021) 102496.
[4] S. He, C.-T. Ng, C. Yeung, Time-domain spectral finite element method for
modeling second harmonic generation of guided waves induced by material,
74 |
ADE | Chapter 4
Chapter 4. Early damage detection of metallic plates with one side
exposed to water using the second harmonic generation of
ultrasonic guided waves
Abstract
Metallic plates are the main structural components in a wide range of thin-walled
structures, such as nuclear cooling pipes, pressure vessels, rocket fuel tanks, and
submarine hulls. These structures operate in extreme environments and are
subjected to transient and repetitive loads. Real-time health monitoring of these
structures is indispensable because they are vulnerable to fatigue and corrosion
damage. Second harmonic generation is one of the reliable damage detection
approaches to evaluate microstructural evolution and has been successfully applied
to characterize initial damage on different structures in gaseous environments.
However, there have been very limited studies on the second harmonics generation
on the structures submerged in liquid. This paper experimentally and numerically
investigates the feasibility of using second harmonic generation to evaluate the
material degradation of metallic plates with one side exposed to water. The
fundamental leaky symmetric Lamb wave mode (leaky S ) at low frequencies is
0
selected because it has low-attenuation and weakly dispersive features, enabling
approximate internal resonance. The experimental results show that the second
harmonics of leaky S waves grow linearly with the propagation distance. The
0
growth rate of the relative nonlinearity parameter (') can be related to the material
nonlinearity of the one-side water-immersed plate. In addition, this study proposed
a three-dimensional (3D) finite element (FE) model to simulate the generation of
second harmonics in the one-side water-immersed plate. The material properties of
the plate are modeled by the Murnaghan strain energy function. The Murnaghan
constants of aluminum that describe the material nonlinearity at different fatigue
levels are obtained from a previous experimental study. The simulation results
demonstrate that the values of ' change significantly with the material properties
79 |
ADE | Statement of Authorship
Title of Paper Early damage detection of metallic plates with one side exposed to water using the second
harmonic generation of ultrasonic guided waves
Publication Status Published Accepted for Publication
Unpublished and Unsubmitted work written in
Submitted for Publication manuscript style
Publication Details X. Hu, C.T. Ng, A. Kotousov, (2022). Early damage detection of metallic plates with one side
exposed to water using the second harmonic generation of ultrasonic guided waves. Thin-
Walled Structures, 176, 109284.
Principal Author
Name of Principal Author (Candidate) Xianwen Hu
Contribution to the Paper Conceptualization, Developing and validating numerical models, Conducting experimental
measurements, Signal processing and data analysis, Writing the original draft and editing.
Overall percentage (%) 80%
Certification: This paper reports on original research I conducted during the period of my Higher Degree by
Research candidature and is not subject to any obligations or contractual agreements with a
third party that would constrain its inclusion in this thesis. I am the primary author of this paper.
Signature Date 07/03/2022
Co-Author Contributions
By signing the Statement of Authorship, each author certifies that:
i. the candidate’s stated contribution to the publication is accurate (as detailed above);
ii. permission is granted for the candidate in include the publication in the thesis; and
iii. the sum of all co-author contributions is equal to 100% less the candidate’s stated contribution.
Name of Co-Author Ching-Tai Ng
Contribution to the Paper Supervision, writing – review & editing.
Signature Date 9/3/2022
Name of Co-Author Andrei Kotousov
Contribution to the Paper Supervision, writing – review & editing
Signature Date 07/03/2022
Please cut and paste additional co-author panels here as required. |
ADE | Chapter 4
4.2. Introduction
Thin-walled structures are widely used in energy, petrochemical, aerospace, civil,
and ocean engineering, such as nuclear cooling pipes, pressure vessel, rocket fuel
tanks, storage tanks, and submarine hulls. These structures usually have one side
exposed to liquid and serve in extreme environments. They are subjected to cyclic
loads due to the draining and refilling process, liquid sloshing impacts, and
temperature fluctuations [1, 2]. Over time, fatigue damage takes place even though
the peak values of the cyclic loads are much smaller than the loading capacity of
the structures [3, 4]. In the early damage stage, fatigue is distributed in the material
and appears as microscopic imperfections, such as dislocations, persistent slip
bands, precipitates, and short cracks at the micro-scale. Optical images of the
microstructural features can be found in [5, 6]. As the number of loading cycles
increases, these microscopic defects accumulate, coalesce, and form macroscopic
cracks, which continuously grow to their critical sizes and cause catastrophic
failures [7]. In addition to the cyclic loading, the corrosive operational conditions
of the one-side submerged plate can accelerate the damage-accumulation process
by corrosion, which results in the metal wear through electrochemical reaction [8,
9]. To mitigate the risk of in-service failure of the structures, real-time health
monitoring of the submerged plate structures is indispensable. Existing non-
destructive testing (NDT) for submerged structures has visual inspection by divers
and robots [10], acoustic emission [11, 12], hydro test [13], magnetic flux leakage
test [14] and eddy current test [15]. Most of these approaches require periodic
,
shutdown of the devices and can only inspect a localized area. They are costly,
inconvenient, and inefficient for scanning a large structure.
Guided waves have been extensively studied as a potential alternative to
overcome the aforementioned limitations of existing NDT techniques. They have
the capability of fast propagation over long distances, volumetric inspection of a
relatively large area with a small number of sensors, the ability to scan structures
with coatings and insulations, and the ability to monitor defects in inaccessible
regions [16, 17]. The majority of the studies on guided wave applications were
carried out on the structures in gaseous environments. However, guided waves
behave differently when the structures are exposed to liquid. The liquid coupling
82 |
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can change the dynamic properties of the plate structures [18] and provides a way
for the guided wave energy to leak into the surrounding liquid medium [19, 20].
Due to the energy leakage, guided waves that propagate in submerged plates are
called leaky Lamb waves, which have multi-modal and dispersive features. At any
excitation frequency, multiple leaky symmetric and antisymmetric Lamb wave
modes can exist simultaneously. Each of these wave modes behaves differently and
varies with the excitation frequency. Compared with structures surrounded by air,
guided wave applications on the submerged structures are much more challenging
because most of the leaky Lamb wave modes decay quickly and disappear after a
short propagation distance. Only a limited number of leaky Lamb wave modes at
their corresponding low-attenuation frequency bands can travel a long distance,
which has the potential to enable large-area inspection for the submerged structure
[19]. Therefore, a good understanding of the wave propagation characteristics is
desired for the practical application of leaky Lamb wave-based techniques.
Several researchers studied different leaky Lamb wave modes to evaluate
various defects in the submerged plate structures. Santos and Perdigao [21]
investigated the fundamental leaky symmetric Lamb wave mode (leaky S ) in a
0
pitch and catch configuration to detect and estimate the size of circular hole defects
in bonded aluminum lap joints fully immersed in water. An empirical parameter
that was defined based on the amplitudes of the received signals was shown to have
a linear correlation with the dimensions of the defects. Chen, Su, and Cheng [22]
investigated the propagation characteristics of the fundamental leaky anti-
symmetric Lamb wave mode (leaky A ) in a submerged plate. Circular holes
0
created on the submerged plate mechanically and chemically were accurately
detected by leaky A waves with the appropriate rectification for the medium
0
coupling effect. Rizzo et al. [23] used a pulsed laser focusing on the upper surface
of the plate to excite leaky Lamb waves in an immersed aluminum plate. The signals
received by immersed transducers included leaky S waves, quasi-Scholte waves,
0
and pressure waves. The leaky S mode that has the fastest propagation speed was
0
well separated from the other wave modes. Then, the leaky S waves were extracted
0
from the rest of the signals for further processing with continuous wavelet transform.
Artificial defects, such as notches and circular holes, which were as small as a few
millimeters, were successfully captured. Sharma and Mukherjee [24] studied leaky
83 |
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Lamb waves on an underwater steel plate using two immersed transducers inclined
at specific angles. Three leaky Lamb wave modes were generated at their
corresponding low-attenuation frequencies, which were the leaky S wave mode,
0
the first-order leaky symmetric Lamb wave mode (leaky S ), and the first-order
1
leaky anti-symmetric Lamb wave mode (leaky A ). Each of these wave modes
1
showed different sensitivities in monitoring the progressive notch damage
machined on the underwater steel plate. They concluded that leaky S and leaky A
1 1
waves were more sensitive to surface defects, of which the depth was less than 37.5%
of the plate thickness. In contrast, leaky S waves were more suitable for evaluating
0
the deeper defects since the amplitudes of the transmitted leaky S waves
0
consistently decreased with the notch depth. Sharma and Mukherjee [25] used
similar techniques to monitor corrosion damage in an underwater plate. The initial
surface degradation could be successfully identified by leaky S waves. Further
1
progression of the corrosion was evaluated better by leaky S waves. Takiy et al.
0
[19] conducted experimental measurements on a submerged aluminum plate to
confirm the existence of leaky S , leaky A , leaky S , and the second-order leaky
0 1 1
symmetric Lamb wave mode (leaky S ) at their corresponding low-attenuation
2
frequency bands. Then, leaky S waves at 3.4 MHz-mm were selected for
1
characterizing damage. An image of the submerged plate was obtained to precisely
identify the locations of five drilled holes. Xie, Ni, and Shen [26] proposed an
experimental method to generate pure leaky S waves by applying the pulsed laser
0
radiation laterally at the whole side of an aluminum plate submerged in water.
Through interacting with the damage, the generated leaky S waves can mode
0
convert into leaky A waves, which have mostly out-of-plane wave motions. Hence,
0
the damage can be easily recognized. However, clear edges of the real structures
are not always accessible, making this pure wave mode excitation at the structural
edges very challenging in practice.
These studies demonstrated that it is feasible to use leaky Lamb waves for
identifying damage in plate structures submerged in liquid. The presence of defects
with sizes in the order of millimeters can be detected and quantified based on the
linear features of leaky Lamb waves, such as the change of wave speed and
amplitude. It was reported that the sensitivity of linear ultrasonic guided waves is
limited to damage of a size comparable to the wavelength of the selected guided
84 |
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wave mode [27, 28]. However, they are insensitive to smaller defects in the initial
damage stage. Before macroscopic cracks nucleate, the evolution of microstructural
defects with load accounts for a major part of the total service life of a structure. In
many cases, when the microstructural damage grows into macro scale, the
remaining life of the structures is very short [29]. Therefore, it is better to identify
defects sooner than later. Earlier detection of damage allows more time to
characterize the evolution of the damage and schedule the maintenance actions for
improving safety. Recent studies have proposed several nonlinear guided wave
techniques to capture the microstructure evolution and early-stage material
degradation [30]. Second harmonic generation is one of the most popular nonlinear
techniques and has been successfully applied to different structures in gaseous
environments to evaluate plasticity-induced damage [31, 32], thermal degradation
[33, 34], precipitation [35], small fatigue cracks [36-38], debonding [39] and bolt
loosening [40]. In the early damage stage (before the appearance of macro cracks),
the generation of second harmonics takes advantage of the fact that the
microstructural features in real materials distort the passing sinusoidal ultrasonic
waves. The distortion can generate new wave components at frequencies other than
the excitation frequencies, which provides a way for the evaluation of the
microstructural defects. However, the measurement of second harmonics is
challenging due to the fact that the material nonlinearity is weak [41]. To ensure
measurable generation of second harmonics, the incident waves should have finite
amplitudes so that there is sufficient wave energy to interact with the
microstructural features. In addition, wave mode selection is required to ensure that
the primary waves and second harmonics conform to non-zero power transfer and
phase velocity matching conditions [42-47].
The aforementioned studies on the second harmonic generation were carried
out on the structures in gaseous environments. Although the nonlinear
characteristics of guided waves have been demonstrated by a number of studies to
be more sensitive to the microstructural defects in the early stage of damage and
are less influenced by environmental changes, there have been very limited studies
on the use of nonlinear guided waves for damage detection on the submerged
structures. Undoubtedly, the vibration of the plate submerged in liquid behaves
differently from that in the air [48-50]. Compared with structures in gaseous
85 |
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environments, the generation and measurement of second harmonics on the
submerged structure are more challenging because the wave energy can be absorbed
by the surrounding liquid medium. So, the application of nonlinear guided waves
for submerged structures deserves separate and careful studies considering the high
reward for the earlier detection of material degradation.
This paper presents experimental and numerical investigations on the
feasibility of using second harmonic generation to evaluate material degradation in
metallic plates with one side in contact with water. Leaky S mode is selected to
0
generate second harmonic leaky S waves due to the observations from previous
0
studies that leaky S mode at low frequencies has very low attenuation and its phase
0
velocity decreases slowly with frequency [51-53]. These features make the primary
and second harmonic leaky S waves satisfy non-zero power flux and approximate
0
phase velocity matching conditions. Then, experiments are carried out on a metal
tank filled with water. Leaky Lamb waves are generated on the wall of the water-
filled tank by a piezoceramic transducer and measured by a scanning laser
vibrometer. It is demonstrated that second harmonics can be generated by leaky S
0
waves at low excitation frequencies and the corresponding relative nonlinearity
parameters are growing linearly with the propagation distance. The growth rate of
the relative nonlinearity parameters can be used to characterize the material stratus
of the one-side water-submerged plate. After that, a three-dimensional (3D) finite
element (FE) model is developed with the material nonlinearity of the submerged
plate simulated by the Murgnahan strain energy function. The material properties
of aluminum at different levels of fatigue damage are obtained from previous
experimental results [54]. The numerical simulations are validated through the
experimental data. Next, the experimentally validated 3D FE model is employed in
the parametric study to analyze the second harmonic generation in the submerged
plate at different levels of fatigue damage. The results show that leaky S mode at
0
low frequencies can generate measurable second harmonics, which are sensitive to
the change of material properties of the one-side water-immersed plate at the initial
stages of the fatigue damage. This paper is organized as follows. Section 4.3
introduces the second harmonic generation techniques and discusses the selection
of leaky Lamb modes for the generation of second harmonics in the plate with one
side exposed to water. Section 4.4 describes the experimental study. Section 4.5
86 |
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the nonlinear material can be written in terms of the displacement gradient as a
Taylor series expansion truncated at order 2
1
Eu' Eu'2 (4.1)
2
where , u , E , and are the stress, displacement, Young’s modulus of the
medium, and nonlinearity parameters, respectively. u'u x is the displacement
gradient. The particle motions can be described as
'u (4.2)
where represents the mass density; ' x and u2u t2 . Perturbation
theory is employed to solve Eqs. (4.1) and (4.2), leading to the final solution [7]
u A cos(kxwt)A cos(2kx2wt) (4.3)
1 2
where k and w are the wavenumber and angular frequency of the excited primary
1
waves. A is the amplitude of the primary waves at w. A k2A2x represents
1 2 8 1
the amplitude of the second harmonics at 2w. x is the propagation distance.
Two observations can be obtained from Eq.(4.3). Firstly, the primary waves
and the second harmonics should have the same phase velocity, i.e.,
c w k 2w 2k. Secondly, when the structure is excited by waves with a fixed
p
wavenumber value k , the nonlinearity parameter that correlates to the material
nonlinearity can be determined by measuring the magnitudes of the primary waves
and the second harmonics as
8 A
2 (4.4)
k2x A2
1
For practical applications, the change in with its initial value is more
important than its absolute value. Therefore, a relative nonlinearity parameter '
is defined as
A
' 2 x (4.5)
A2
1
88 |
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It can be seen from Eq.(4.5) that ' is proportional to and grows linearly with
the propagation distance x. can be evaluated by the gradient of accumulation of
'. Therefore, any abnormal increase in ' indicates an increase in the material
nonlinearity and progress in material degradation [36]. Although Eq. (4.1)-Eq. (4.4)
are derived for longitudinal waves, Eq.(4.5) has been widely considered to be
applicable for characterizing the second harmonic generation of Lamb waves [37,
44], Rayleigh waves [55], and Edge waves [56]. Therefore, ' is employed in this
study to quantify the change of the second harmonics of the leaky Lamb waves.
4.3.2. Selection of primary leaky Lamb wave modes
Although the generation of second harmonics by guided waves has been studied for
evaluating the incipient damage in various structures that are open to the air, the
feasibility of using leaky Lamb wave modes to generate second harmonics in the
plate with one side exposed to water has not been explored. Considering the multi-
modal and dispersive features, the selection of primary leaky Lamb waves is
important. This section introduces the theoretical derivation of the dispersion curves,
which describe the number of leaky Lamb wave modes and their corresponding
properties with the frequency. Based on the dispersion curves, the leaky Lamb wave
modes that have low attenuation and low dispersion characteristics are identified.
Then, the primary wave mode and excitation frequency are selected to meet the
following three conditions. Firstly, the primary wave mode should have sufficient
wave energy propagating in the submerged structure to interact with the
microstructural features so that the generation of second harmonics best reflects the
material nonlinearity. Secondly, the primary waves should be of the same type of
wave mode as the second harmonics to ensure that the wave energy can be
transferred between the primary waves and the second harmonics [46]. Thirdly, the
primary waves and the second harmonics should have similar phase velocities.
Consider a plate loaded with water on its bottom surface, as shown in Figure
4.2. Traction-free boundary conditions apply to the top surface of the plate as
Plate_0 Plate_0 0 (4.6)
33 31
89 |
ADE | Chapter 4
where Plate_0 and Plate_0 are the normal and shear stress on the top surface of the
33 31
plate, respectively. The bottom surface of the plate is coupled to the water layer.
Under the non-viscosity assumption that water cannot sustain shear forces, the
boundary conditions at the plate-water interface can be described as
uPlate_d uWater_d
33 33
Plate_d Water_d (4.7)
33 33
Plate_d 0
31
where uPlate_d , Plate_d , and Plate_d represent the normal displacement, normal
33 33 31
stress, and shear stress of the plate at the plate-water interface, respectively. uWater_d
33
and Water_d are the normal displacement and normal stress of the water at the plate-
33
water interface, respectively.
Figure 4.2. Schematic diagram of a plate loaded with water on its bottom surface
Previous studies have derived the characteristic equation of the leaky Lamb
waves for the one-side water-immersed plate [57]
q2 k2 q2 k2 2qk 2qk 0
2pk 2pk q2 k2 q2 k2 0
w2
(q2 k2)eipd (q2 k2)eipd 2qkeiqd 2qkeiqd w (4.8)
0
2pkeipd 2pkeipd (q2 k2)eiqd (q2 k2)eiqd 0
w2
peipd peipd keiqd keiqd k2
c2
w
90 |
ADE | Chapter 4
where p w2 c2 k2 , q w2 c2 k2 , c 2 , c .
L S L S
E 112 and E 21 are the first and second Lame
constants of the plate, respectively. E and are Young’s modulus and Poisson’s
ratio of the plate, respectively. and represent the density of the plate and the
w
surrounding water, respectively. c is the speed of the bulk wave in the water.
w
Eq.(4.8) can be solved numerically and the solutions can be presented by a series
of dispersion curves.
Table 4.1. Material properties of the aluminum plate and water
Aluminum density (kg/m3) 2700
Aluminum 1st Lame parameter (GPa) 51.64
Aluminum 2nd Lame parameter (GPa) 26.60
Water density (kg/m3) 1000
w
Water bulk wave velocity c (m/s) 1500
w
Figure 4.3 shows the dispersion curves of a 1.6 mm thick aluminum plate
with one side in contact with water. The material properties are given in Table 4.1.
Within the frequency range up to 1MHz, there are only three wave modes, which
are leaky S , leaky A , and quasi-Scholte waves. Other higher-order wave modes
0 0
appear when the frequency increases. The quasi-Scholte wave that propagates along
the plate-water interface has very low attenuation (close to zero) for the entire
frequency bandwidth as shown in Figure 4.3(c). This wave mode is highly
dispersive in the low-frequency range (below 250 kHz) as denoted by the blue
dotted lines in Figures 4.3(a) and 4.3(b). The phase velocity of the quasi-Scholte
wave increases monotonically with frequency until its value reaches the speed of
the bulk wave in the surrounding water (around 1500 m/s). Further increasing the
frequency, the quasi-Scholte wave becomes nondispersive. Although the quasi-
Scholte wave at the frequency range over 250 kHz has low attenuation and satisfies
phase velocity matching conditions, it is not considered in this study. The reason is
that the quasi-Scholte wave in the nondispersive frequency range has most of its
91 |
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In addition to quasi-Scholte waves, leaky S waves also have low
0
attenuation and low dispersion characteristics within the frequency range up to 600
kHz. The phase velocity of leaky S waves decreases slowly with frequency. For
0
example, the phase velocity of the leaky S mode at 100 kHz is 5435 m/s, and that
0
at 500 kHz is 5383 m/s. The deviation is around 0.96%. A previous study defined
the approximate phase velocity matching condition as the relative phase velocity
deviation less than 1% [59]. Therefore, if the leaky S waves below 250 kHz are
0
selected as the primary waves, the second harmonics at twice the frequency should
satisfy the approximate phase velocity matching condition with the primary waves.
Another key factor is that the low-attenuation frequency band of the leaky S mode
0
is limited to the frequency range up to 600 kHz as shown by the black solid line in
Figure 4.3(c). As the frequency increases beyond 600 kHz, the attenuation of the
leaky S wave increases exponentially. Also, the phase velocity and group velocity
0
decrease quickly with the frequency.
Figure 4.4 presents the mode shapes of leaky S waves for the aluminum
0
plate with one side in contact with water. The mode shape diagrams show the
distributions of the displacements of leaky S waves through the thickness of the
0
plate. The x-axis denotes the magnitudes that are normalized by the maximum
displacement amplitudes. The y-axis denotes the thickness location x that is
3
defined in Figure 4.2. The red dashed lines represent the particle’s displacements
in the direction parallel to the plate surface (in-plane displacements). The blue solid
lines show the displacement in the direction normal to the plate surface (out-of-
plane displacements). It can be seen that the out-of-plane displacements between
the plate and the water areas are continuous. In comparison, the in-plane
displacements are disconnected. When the frequency is below 600 kHz, the mode
shape of leaky S waves is dominated by the in-plane displacements in the plate as
0
indicated by the red dashed lines in Figure 4.4(a). The vast majority of the wave
energy is conserved in the one-side water-immersed plate with minimal loss,
making it ideal for using the leaky S wave to scan the one-side water-immersed
0
plate. However, the out-of-plane displacements in both the plate and water regions
increase with the frequency. This indicates more energy leakage from the structure
into the surrounding liquid medium. From these observations, the excitation
93 |
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frequencies are chosen to be below 250 kHz in the following experimental and
numerical studies.
Figure 4.2. Mode shapes of leaky S wave at (a) 300 kHz, (b) 600 kHz, and (c) 900
0
kHz for a 1.6 mm thick aluminum plate loaded with water on its bottom surface
(the red dashed lines represent the in-plane displacements and the blue solid lines
denote the out-of-plane displacements)
4.4. Experimental study
4.4.1. Experimental setup
This section presents an experimental study on the leaky Lamb wave propagation
in a metal plate with one side in contact with liquid, which aims to simulate a variety
of thin-walled structures operating in extreme conditions, such as nuclear cooling
pipes, pressure vessels, rocket fuel tanks, and submarine hulls. The experiments
were carried out using a metallic tank fully filled with water. The front wall of the
tank was used as the test plate, which was a 1.6 mm thick aluminum plate. The
internal surface of the test plate was in contact with water, while the outer surface
was exposed to air.
Figure 4.5 illustrates the overall experiment setup and the top view of the
water-filled tank. A computer-controlled signal generator (NI PIX-5412) was
employed to generate a six-cycle Hanning window modulated sinusoidal tone burst
pulse. Then, the signal was sent to a power amplifier (Ciprian HVA-800-A) and the
voltage was increased to 160 Vp-p. After that, the amplified signal was sent to a
94 |
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piezoceramic transducer (Ferroperm Pz27) which was bonded to the outer surface
of the test plate. The circular piezoceramic transducer has a diameter of 10 mm and
a thickness of 0.5 mm and can convert the electric signals to mechanic motions,
exciting leaky lamb waves on the test plate. The thin and circular piezoceramic
wafer deforms mainly in the radial direction which is parallel to the plate surface.
As a result, the excitation should be dominated by the in-plane motions of the plate
and leaky S waves could be generated effectively.
0
The response signals were collected on the water-free surface by a non-
contact scanning laser Doppler vibrometer (Polytec PSV-400-M2-20). Taking the
center of the piezoceramic transducer as the origin, a Cartesian coordinate system
was defined as shown in Figure 4.5. The x -axis denotes the in-plane direction
1
parallel to the plate surface, and the x -axis is the out-of-plane direction that is
3
normal to the plate surface. The measurement points are defined along a line parallel
to x -axis. The signals were collected at a sampling rate of 25.6 MHz. To improve
1
the quality of measurements, each signal was averaged by 1000 recordings and
filtered by a low-pass filter with a cut-off frequency of 1MHz.
Figure 4.5. Schematic diagram of the experiment setup
95 |
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4.4.2. Experimental results
Figure 4.6(a) presents an example of the experimentally measured signals. The
excitation frequency was 170kHz, at which there were only leaky S , leaky A , and
0 0
quasi-Scholte waves, as shown by the dispersion curves in Figure 4.3. The
measurement point was 200 mm away from the excitation center. The first wave
packet was identified as the leaky S wave mode, which has the fastest group
0
velocity, as shown in Figure 4.3(b). The phase velocity of the leaky S wave mode
0
at 170 kHz is 5431 m/s, and that at 340 kHz is 5413 m/s. So, the primary leaky S
0
wave is almost phase matched with its second harmonic with a deviation of 0.33%.
It should be noted that the experimental data mainly captured the out-of-plane
motions on the plate surface because the laser beam was perpendicular to the test
plate during the test. Since the out-of-plane displacement of the leaky S wave was
0
small as shown by the blue solid lines in Figure 4.4, the actual magnitudes of the
leaky S wave should be strong enough so that it could be measured by the scanning
0
laser Doppler vibrometer.
The following wave packet should be dominated by the quasi-Scholte wave
mode because the leaky A wave mode decays quickly due to high attenuation [51].
0
At this excitation frequency, the quasi-Scholte wave is highly dispersive. The phase
velocity of the quasi-Scholte mode at 170 kHz is 1317 m/s, and that at 340 kHz is
1488 m/s. The deviation of the phase velocity between the primary waves and the
second harmonics is around 13%. Therefore, the quasi-Scholte waves do not satisfy
the approximate phase velocity matching condition [59].
To further investigate, the first wave packet was cut from the rest of the
signal to exclude the quasi-Schole waves and the unwanted reflections in the data.
Figure 4.6(b) presents the window-cut signal. Then, the chopped signal was
transferred to the frequency domain by FFT and is shown in Figure 4.6(c). There
are two peaks for the primary waves (at 170 kHz) and the second harmonics (at 340
kHz), respectively, as highlighted by the black dotted line in Figure 4.6(c). There is
also a small peak at three times the excitation frequency, which is the third harmonic
[29, 60]. However, the signal-to-noise ratio of the third harmonics is much lower
than that of the second harmonics, and the third harmonics are not the focus of the
paper, so the third harmonics are not discussed in this study. After that, ' were
96 |
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4.5. Finite element simulation
4.5.1. Model description
A 3D FE model was developed to simulate the leaky Lamb wave propagation in the
one-side water-immersed plate. By applying the symmetry boundary conditions to
the left and bottom edges of the plate, only the top right part of the plate was
modeled. The plate modeled in the FE was 430 mm long, 250 mm wide, and 1.6
mm thick. The bottom surface was in contact with a water layer of the same planar
area. The thickness of the water layer was 90 mm, which was chosen to avoid
unwanted reflections from the bottom of the water layer. Figure 4.7 presents the
schematic diagram of the FE model, which was modeled using the commercial
software, ABAQUS. The bottom surface of the plate and the top surface of the
water layer were tied together using the surface-based tie constraint, which
connected the acoustic pressure of the water and the out-of-plane translations of the
plate. Previous studies of ultrasonic guided waves in the solid-liquid coupled
medium experimentally validated that the tie constraint could accurately model the
solid-liquid interactions [22, 51, 58, 61]. The plate and the water layers were
modeled by 3D eight-node reduced integration solid elements and 3D eight-node
reduced integration acoustic elements, respectively. The largest dimension of the
element size was less than 0.5 mm, ensuring that there were at least 20 FE nodes
within the wavelength of the leaky S wave [59, 62].
0
Figure 4.7. Schematic diagram of the 3D FE model.
98 |
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To simulate the material nonlinearity of the one-side water-immersed plate,
the Murnaghan strain energy function was introduced to define the material
properties of the aluminum plate using VUMAT subroutine in ABAQUS.
Murnaghan strain energy function includes the third-order Taylor series expansion
of the strain potential and has been widely used in the analysis of second harmonic
generation for modeling the nonlinear material behaviors [62-64]. This function can
be written as
1 1
WE trE2 tr E2 l2m trE3
2 3
(4.9)
mtrE trE2 tr E2 ndetE
where l, m, and n are the Murnaghan constants which are related to the third-
1
order elastic constants; E FTFI is the Lagrangian strain; F and I denote
2
the deformation gradient and the identity tensor, respectively. Table 4.2 presents
the values of Murnaghm constants of aluminum at different levels of fatigue
damage, which are obtained from previous experimental results [54].
Table 4.2. Murnaghan constants of aluminum at different levels of fatigue damage
[54]
Murnaghan constants
Fatigue life l (GPa) m (GPa) n (GPa)
0% -252.2 -325.0 -351.2
40% -266.8 -332.8 -358.3
80% -271.2 -335.8 -359.8
The leaky S wave was excited by applying nodal displacements at the
0
circumference of a circular transducer represented by the quarter-circle of 10 mm
diameter located at the left-bottom corner of the plate. The excitation signal was a
six-cycle Hanning window modulated sinusoidal tone burst pulse. The explicit
module of ABAQUS was employed to solve the dynamic simulations. Figure 4.8
99 |
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shows a snapshot of the simulation results with the excitation frequency of 170 kHz.
The rainbow color represents the acoustic pressure in the water layer. It can be seen
that the leaky S wave propagates fastest with minimum wave energy leaking into
0
the liquid. Following the leaky S wave, the quasi-Scholte wave propagates at a
0
speed slightly faster than the pressure wave in water. The acoustic pressure of quasi-
Scholte waves is concentrated around the plate-water interface [58]. The leaky A
0
wave dominated by the out-of-plane displacements was not observed from the
surface of the submerged plate. In general, the simulation results have a good
agreement with the experimental data as shown in Figure 4.6(a). The numerically
calculated acoustic wavefields provide additional information to interpret the
experimental data. The simulation results of the 3D FE model are further validated
in the following sections to gain physical insights into the second harmonic
generation on the one-side water-immersed plate.
Figure 4.8. Snapshot of the simulation results at 72 s
4.5.2. Experimental validation of the FE model
In this section, the 3D FE model is validated by comparing the simulation results
with the experimental data. The simulations carried out using the 3D FE model with
the VUMAT subroutine are labeled as nonlinear FE because the material
nonlinearity was modeled by the Murnaghan strain energy function. Firstly, the
linear features of the leaky S waves that were simulated by nonlinear FE and
0
100 |
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measured from experiments were investigated and compared. From both the
experiments and the nonlinear simulations, the out-of-plane displacements were
obtained at 21 measurement points along a line from 200 mm to 300 mm away from
the excitation center. Figure 4.9(a) shows the signal simulated by the nonlinear FE.
The measurement point was 200 mm away from the excitation center. Figure 4.9(b)
compares the waveforms of the window-cut signals obtained from the experiments
and the simulations from the nonlinear FE at the same measurement point. The
amplitudes are normalized by the maximum peak magnitudes of the signals. In
general, the nonlinear FE well predicts the waveform and the time of arrival of leaky
S waves.
0
Figure 4.9. (a) Signal simulated by the nonlinear FE (b) Comparison of the window-
cut signals measured from the experiment and simulated by the nonlinear FE.
To further validate the accuracy of the nonlinear FE, the group and phase
velocities were calculated using the simulated signals, and the simulation results
were compared with theoretical values and experiment data. The excitation
frequency was swept from 150 kHz to 390 kHz in steps of 20 kHz. The signals were
collected at the first 10 points for each excitation frequency to calculate the
averaged phase and group velocities. The distance between the two consecutive
measurement points was 5 mm and it was less than half of the wavelength of the
C
selected leaky S wave. The phase velocity was calculated by
0 p
C f 2f x, where f represents the central frequency of the excitation.
p c c c
101 |
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and x are the phase difference and the distance between the two
measurement points, respectively. The group velocity C was calculated by
g
C f x t, where t is the time lag between the two measurement points.
g c
Figures 4.10(a) and 4.10(b) present the phase velocity and group velocity
dispersion curves for the aluminum plate loaded with water on the single side,
respectively. In both figures, there are three wave modes represented by three lines,
which are calculated based on the global matrix theory by the commercial software
DISPERSE [65]. The black solid lines on the top represent the leaky S mode. The
0
values calculated by the experimental data and the nonlinear FE simulations are
denoted by cycles and stars, respectively. It can be seen that the simulation results
have a good agreement with the theoretical derivations and experimental
measurements. The maximum deviation is less than 2%.
Figure 4.10. (a) Phase velocity dispersion curves and (b) group velocity dispersion
curves calculated by the theoretical derivations (black solid line, black dashed line,
and blue dotted line), nonlinear finite element simulations (stars), and experimental
measurements (circles)
Then, the nonlinear features of the simulated and experimentally measured
leaky S waves were analyzed. The simulations were carried out using the 3D FE
0
model with and without the VUMAT subroutine. The simulations, solved using
only the linear elastic material properties as shown in Table 4.1, are labeled as linear
FE because they do not consider the inherent material nonlinearity. The nonlinear
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FE simulates the material nonlinearity of the aluminum plate by introducing the
Murnaghan constants of zero fatigue damage in Table 4.2. Figure 4.11(a) compares
the window-cut signals obtained from the experimental measurements and the
simulated signals from both the linear and nonlinear FE models. The amplitudes are
normalized by their corresponding peak magnitudes. The simulated linear and
nonlinear signals do not show much difference in the time domain, and both have a
good agreement with the experimental data. Figure 4.11(b) shows the
corresponding data in the frequency domain. It can be seen that nonlinear FE
captures the second harmonics at twice the excitation frequency, as highlighted by
the black dotted lines in the figure. In general, the second harmonics simulated by
the nonlinear FE have a good agreement with the experimental data (see the red
dash-dotted line and black solid line in Figure 4.11(b)). However, the linear FE that
uses only the second-order elastic constants could only predict the primary waves
as shown by the blue dashed lines. The second harmonics are not observable in the
linear FE. The comparison between the linear and nonlinear FE further confirms
that the generated second harmonics are due to the material nonlinearity by
considering the Murnaghan strain energy function and Murnaghan constants.
Figure 4.11. (a) Comparison of the time domain signals obtained from the
experimental measurement, linear finite element simulation, and nonlinear finite
element simulation; and (b) their corresponding frequency spectra.
It should be noted that the Murnaghan strain energy function, incorporating
the third-order approximation of the constitutive relation, could be used to predict
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the generation of only up to the second harmonics. To study the third-order
harmonics, the fourth-order expansion of the constitutive relation and fourth-order
elastic material constants should be included in the constitutive model [60]. This
explains why the nonlinear FE is unable to capture the peak located at three times
the excitation frequency. Since this study focuses on the generation of the second
harmonics, the Murnaghan strain energy function is sufficient for analysis.
Therefore, the simulation results are validated.
4.5.3. Second harmonic generation in the submerged plate at different levels of
fatigue damage
The experimentally validated 3D FE model was employed to explore the influence
of evenly distributed fatigue damage on the generation of the second harmonics.
Stobbe [54] experimentally measured the values of Murnaghan constants for
aluminum at different fatigue levels. In his experimental studies, a series of dog-
bone samples made of aluminum were fatigued by repeated uniaxial tensile loads.
One sample was loaded to 52800 load cycles and failed, which was defined as 100%
fatigue damage. The rest samples were then fatigued to different cycles and were
referenced to different percentages of fatigue damage. The numerical simulations
in the present study employed the experimentally measured Murnaghan constants
for the undamaged aluminum and the aluminum at 40% and 80% fatigue damage
[54], as shown in Table 4.2. They have been used in previous studies to simulate
the generation of nonlinear guided waves in aluminum plates [59] and pipes [64] in
gaseous environments. For the first time, this paper presents the numerical
simulations of an aluminum plate with one side loaded with water and the
sensitivity of the second harmonics generated by leaky S waves to fatigue damage
0
is investigated. Three simulations were carried out using the experimentally
validated 3D FE model with three different sets of Murnaghan constants (see Table
4.2), which represent the aluminum plates at 0%, 40%, and 80% fatigue damage.
The other settings remained unchanged.
The simulated out-of-plane displacements were obtained at 21 points from
200 mm to 300 mm away from the excitation center to calculate '. For direct
comparison, the values are normalized by the initial value at 200 mm for the
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aluminum plate at zero fatigue damage. Figure 4.12(a) presents the normalized
nonlinear parameters versus the propagation distance. It can be seen that the
normalized nonlinearity parameters increase linearly with the propagation distance
for the three cases. In addition, the slopes of the best-fit lines increase with the
fatigue damage levels.
Figure 4.12 (a) The normalized nonlinearity parameters versus propagation distance
and (b) normalized slopes of the best-fit lines at different levels of fatigue damage
Figure 4.12(b) compares the slopes of the best fit lines at different levels of
fatigue damage. The values are normalized by the initial value at zero fatigue
damage. As mentioned in Section 4.3.1, the material nonlinearity can be evaluated
by the gradient of accumulation of '. Thus, it shows that material nonlinearity
increases quickly during the initial stage of fatigue damage and the increasing rate
becomes much slower after 40% fatigue damage. This behavior is in agreement
with the preceding studies on fatigue damage evaluation by longitudinal waves [66,
67], Rayleigh waves [68, 69], and Lamb waves [33, 44, 70]. Therefore, the
numerical simulations reveal that second harmonics generated by the low-
attenuation leaky S waves have the potential to characterize the material
0
nonlinearity of the plate when one side of the plate is exposed to water.
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4.5.4. Comparison between the free plate and water-immersed plate
The experimentally validated 3D FE model was also employed to explore the
influence of the surrounding liquid on the generation of the second harmonics. Two
simulations were carried out for the undamaged aluminum plate with and without
the water layer, respectively. The simulation for the plate without the water layer is
labeled as the free plate, while that for the plate with one side exposed to water is
labeled as the water-immersed plate. The other settings remained the same.
Guided waves that propagate in the free plate are called Lamb waves, which
consist of multiple symmetric and antisymmetric Lamb wave modes. Figure 4.13(a)
shows the simulated signals for the free plate. The first wave packet is identified as
the fundamental symmetric Lamb (S ) mode that propagates fastest in the selected
0
excitation frequency. The second wave packet that arrives around 80 s is the
fundamental antisymmetric Lamb (A ) mode. It propagates slightly faster than the
0
quasi-Scholte wave as shown in Figure 4.9(a). Figure 4.13(b) compares the
window-cut signals for the free plate and water-immersed plate, respectively. Their
corresponding frequency spectra are shown in Figure 4.13(c). The amplitudes are
normalized by the corresponding peak magnitudes for comparison. It can be seen
that the leaky S wave has a similar waveform as the S wave in the time domain
0 0
due to the fact that they have similar group and phase velocities [71, 72]. However,
the ratios of the second harmonics to the primary waves are smaller for the leaky S
0
wave compared to that of the S wave. This is because the attenuation of the leaky
0
S wave gradually increases with the frequency as discussed in Section 4.3.2. This
0
phenomenon also has a significant influence on the growing trends of the
nonlinearity parameters as shown in Figure 4.13(d). The slope of the best-fit line of
the free plate is 7.0876, which is nearly double the value of the one-side water-
immersed plate. Thus, the simulation results reveal that the surrounding liquid can
reduce the accumulation rate of the nonlinearity parameters. In addition, the
influence of water is much stronger than that caused by fatigue damage as discussed
in Section 4.5.3. So, the liquid coupling effects should be considered when the
nonlinear guided wave techniques are used for damage detection for the immersed
plate structures.
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Figure 4.13 (a) Signal simulated by the free plate FE; (b) Comparison of the
window-cut signals simulated by the free plate FE and water-immersed plate FE;
(c) frequency spectra of the signals in (b), and (d) comparison of the normalized
nonlinearity parameters versus propagation distance.
4.6. Conclusion
This paper has investigated experimentally and numerically the second harmonic
generation by guided waves in plates immersed in liquid on one side, which has the
potential to characterize the microstructural evolution before the appearance of
macroscale damage and fraction. The findings can provide support for the further
development of NDT techniques for partially submerged structures, such as nuclear
cooling pipes, pressure vessels, rocket fuel tanks, storage tanks, and submarine hulls.
Earlier damage detection of these partially immersed structures allows more time
to schedule the maintenance actions and reduces the risks of in-service failure.
Firstly, the dispersion behavior of guided waves has been analyzed for
metallic plates with one side immersed in water. It has been found that the leaky S
0
mode at low frequencies has low attenuation and low dispersion features. This
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analysis leads to the selection of leaky S to generate second harmonics of the same
0
type of wave mode in the one-side water-immersed plate because the primary and
second harmonic leaky S waves satisfy approximate phase velocity matching and
0
non-zero power flux conditions. Next, experimental studies have been conducted
on the metal tank filled with water. A case study using experimentally measured
signals at the excitation frequency of 170 kHz has been presented. Both the primary
and second harmonic leaky S waves can be identified in the frequency spectrums.
0
In addition, the relative nonlinearity parameter ' has been calculated and shown
to grow linearly with the propagation distance. The experimental results confirm
that leaky S waves can generate measurable second harmonics due to the material
0
nonlinearity of the one-side water-immersed plate. After that, numerical
simulations have been carried out using a 3D FE model and validated through
experimental measurements. The experimentally validated 3D FE model has been
employed in parametric studies to explore the second harmonic generation in the
one-side water-immersed plate at different levels of fatigue damage. The results
have shown that the second harmonic generation techniques are promising for non-
destructively evaluating microstructural defects in plate structures with one side
immersed in liquid.
The present study only demonstrates that the interaction between guided
waves and microstructural features of partially immersed plates can generate
measurable and low-attenuation second harmonics. When the microscopic defects
grow into macro scale, there can be a substantial increase in amplitudes for the
nonlinear guided waves due to the clapping behavior between the surfaces as the
primary guided waves pass through. For the structures in gaseous environments, the
clapping effect of macro cracks is classified as contact-type nonlinearity, which has
been demonstrated to generate second harmonics with much larger amplitudes than
the material nonlinearity [28, 37]. Future studies can investigate the effect of the
size, shape, and location of macro cracks (e.g. stress corrosion cracking) on the
nonlinear guided wave features for the structures immersed in liquid.
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Chapter 5. Structural health monitoring of partially immersed
metallic plates using nonlinear guided wave mixing
Abstract
Metallic plates are important structural components of many liquid containment
structures, such as liquid storage tanks and sewer pipes. Time-dependent loads can
result in fatigue and degradation of the metallic material. Nonlinear guided wave
mixing has been demonstrated to be sensitive to microstructural change at the early
stage of material degradation. Previous studies have been carried out using the
nonlinear guided wave mixing technique on various structures in gaseous
environments. However, its application to structures immersed in liquid has not
been explored. This paper numerically and experimentally investigates the
nonlinear guided wave mixing in an aluminum plate loaded with water on one side.
Experiments are carried out with an empty metal tank and the tank filled with water,
respectively. The results show that cumulative generation of harmonics at the sum
frequency due to the material nonlinearity of the partially immersed plate can be
achieved by mixing the fundamental leaky symmetrical Lamb (leaky S ) waves at
0
two different frequencies. Under the same experimental conditions, the amplitudes
of the guided wave signals and the values of the relative nonlinearity parameters on
the partially immersed plate are different from their counterparts on the plate
without water. Finally, numerical simulations are performed with the material
nonlinearity of the test plate simulated by the Murnaghan constitutive model. The
numerical results reveal that both the second harmonics and the combination
harmonics are sensitive to the material nonlinearity of the plate loaded with water
on one side.
Keywords: Structural health monitoring; Water containment structures; Metallic
plates; Leaky Lamb waves; Nonlinear guided waves; Guided wave mixing; Second
harmonics
115 |
ADE | Statement of Authorship
Title of Paper Structural health monitoring of partially immersed metallic plates using nonlinear guided wave
mixing
Publication Status Published Accepted for Publication
Unpublished and Unsubmitted work written in
Submitted for Publication manuscript style
Publication Details X. Hu, T. Yin, H. Zhu, C.T. Ng, A. Kotousov, (2022). Structural health monitoring of partially
immersed metallic plates using nonlinear guided wave mixing. Construction and Building
Materials (In-print).
Principal Author
Name of Principal Author (Candidate) Xianwen Hu
Contribution to the Paper Conceptualization, Developing and validating numerical models, Conducting experimental
measurements, Signal processing and data analysis, Writing the original draft and editing.
Overall percentage (%) 80%
Certification: This paper reports on original research I conducted during the period of my Higher Degree by
Research candidature and is not subject to any obligations or contractual agreements with a
third party that would constrain its inclusion in this thesis. I am the primary author of this paper.
Signature Date 07/03/2022
Co-Author Contributions
By signing the Statement of Authorship, each author certifies that:
i. the candidate’s stated contribution to the publication is accurate (as detailed above);
ii. permission is granted for the candidate in include the publication in the thesis; and
iii. the sum of all co-author contributions is equal to 100% less the candidate’s stated contribution.
Name of Co-Author Tingyuan Yin
Contribution to the Paper Analytical derivations, Writing – review & editing.
Signature Date 07/03/2022
Name of Co-Author Hankai Zhu
Contribution to the Paper Experimental measurements, Writing – review & editing.
Signature Date 07/03/2022 |
ADE | Chapter 5
5.2. Introduction
Metallic plates are commonly used for constructing undersea tunnels [1], storage
tanks [2], sewer pipes [3], and containment buildings [4]. These structures have one
side immersed in liquid and are subjected to cyclic loads with varying amplitudes.
Material degradation and fatigue are the primary culprits for the failure of these
partially immersed metallic structures [5]. In the early damage stage, dislocations
and slip bands occur in the materials and then micro cracks are formed. With the
increase in loading cycles, the micro cracks continue to accumulate and grow to a
critical point, which can cause catastrophic failures [6, 7]. Continuous evaluation
of material properties of partially immersed metallic plates is crucial to maintain
the structural integrity of high-valued infrastructures.
Guided wave testing is a non-destructive inspection technique that has
attracted extensive research interest. It outperforms other non-destructive testing
methods, such as eddy current testing, acoustic emission, and conventional
ultrasonic testing, because guided waves can travel relatively long distances on
various structures and have a high sensitivity to different kinds of damage [8]. The
structural health can be monitored by both linear and nonlinear features of guided
waves. Conventional guided wave testing is based on linear features. Specifically,
the presence of defects changes the transmitted guided wave signals, typified as
scattering, mode conversion, attenuation, and change in wave velocity. Linear
guided wave testing was successfully applied to immersed structures to characterize
cracks [9], pits [10], notches [11], and corrosion [12, 13]. In these studies, guided
waves in immersed structures were shown to behave differently from their
counterparts in structures without exposure to liquid. In addition, the sizes of the
defects were around a few millimeters, which were comparable to the wavelength
of the selected guided wave modes. However, linear guided waves are insensitive
to smaller defects such as micro cracks and dislocations in the early damage stage.
When the micro cracks evolve into macro cracks and become identifiable through
the linear guided wave features, the metallic structures, in many cases, reach more
than 80% of its total service life [14].
Recent studies have focused on the nonlinear features of guided waves,
which provide much better sensitivity than the linear features in detecting
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microstructural defects that precede the macro-scale damage [15]. The second
harmonic approach is one of the most popular nonlinear guided wave methods.
When the structural material is excited by guided waves with finite amplitudes, the
ultrasonic guided wave energy can be transferred from the excitation frequency to
twice the excitation frequency due to the interaction of the primary guided waves
with the microstructural features of the material. This phenomenon provides a way
to identify and characterize material degradation at its early stage. For example, the
second harmonics generated by guided waves were used to evaluate the evolution
of thermal aging [16, 17] and fatigue [18, 19] for metallic plates in gaseous
environments. A comprehensive review of the second harmonic guided wave
approach can be found in [15]. The major difficulty hindering the applications of
second harmonics is that the instrumentation of the measurement system can also
produce nonlinear signals at the integer multiples of the excitation frequency [20,
21]. It is difficult to distinguish the nonlinearity due to the material from the
nonlinearity caused by the instruments.
To tackle this limitation of the second harmonic approach, a number of
researchers proposed the nonlinear guided wave mixing technique, in which the
structural material is excited by guided waves with two different frequencies.
Hasanian and Lissenden [22] conducted a wave vector analysis for the mutual
interaction of two guided waves with different frequencies. The mutual interaction
can generate combination harmonics at the sum and difference frequencies that are
far from the nonlinear waves produced by the instrumentation. Hasanian and
Lissenden [23] further studied the internal resonance criteria for the non-collinear
guided wave interaction, where the guided waves propagate in different directions
and meet in a localized mixing zone. They concluded that the amplitudes of the
generated combination harmonics are dependent on the size of the wave mixing
areas. Jiao et al [24] demonstrated both experimentally and numerically that the
combination harmonics of guided waves at the sum frequency are sensitive to micro
cracks in metallic plates. Metya et al [25] revealed that the nonlinear guided wave
mixing technique can also evaluate localized deformation of a steel plate during
creep. Shan et al [26] mixed two shear horizontal waves that propagated in the same
direction to generate cumulative combination harmonics at the sum frequency. The
combination harmonics demonstrated a high sensitivity to degradation of the
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aluminum plate during early fatigue stages. Cho et al [27] proposed a novel
technique to detect localized fatigue damage in aluminum plates by the interaction
of two counter-propagating shear horizontal waves. The wave mixing area can be
controlled and moved to different locations on the sample by adjusting the time
delays of the input signals. Thus, the whole area of the plate can be scanned. Li et
al [28] employed the guided wave mixing and mixing frequency peak counting
techniques to assess low-velocity impact damage in CFRP composite laminates.
The value of the mixing frequency peak count could be correlated with the impact
energy in the test. Guan et al [29] developed a three-dimension (3D) finite element
(FE) model to demonstrate that the directions and locations of the micro cracks in
plates can affect the amplitudes of the nonlinear waves generated by nonlinear
guided wave mixing. All the aforementioned studies demonstrated that the guided
wave mixing technique has many advantages over the second harmonic approach.
One of the most important merits is that the combination harmonics generated by
mixing two guided waves are less affected by the higher harmonics produced by
the instrumentations, such as amplifiers and transducers. In addition, mutual
interaction between guided waves propagating in different directions provides more
flexibility for the selection of guided wave modes and their corresponding
excitation frequencies.
The majority of the work on nonlinear guided waves has focused on the
structures in gaseous environments. Before the initiation of macro-scale damage,
the material nonlinearity due to the microstructural features is very weak, making
the generation and measurement of nonlinear guided waves very challenging. Only
a few guided wave modes within limited frequency bandwidths, satisfying the
phase velocity matching and non-zero power flux conditions, have the potential to
generate cumulative and measurable nonlinear guided waves and can be used to
characterize the material nonlinearity [15]. Phase velocity matching refers to that
the primary and the corresponding nonlinear guided waves should have the same
phase velocity. Non-zero power flux means that there must be nonzero power flow
from the primary waves to the nonlinear guided waves. They have been widely
recognized as the criteria for selecting guided wave modes and excitation
frequencies for the nonlinear guided wave methods for the structures in gaseous
environments. When the structures are exposed to liquid, the fluid-solid coupling
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allows the guided wave energy to leak from the structures into the surrounding
liquid medium. On the immersed structures, the generation and measurement of
nonlinear guided waves are more challenging because most of the guided wave
modes have higher attenuation [10]. In addition, the fluid-solid coupling makes the
guided waves in the immersed structures behave differently from their counterparts
in the structures surrounded by air [30-33]. Therefore, a comprehensive
investigation is desired for the nonlinear guided waves in the immersed structures.
This paper presents a series of experimental and numerical investigations
on the nonlinear guided wave mixing in partially immersed metallic plates. In the
experiments, the fundamental leaky symmetric Lamb (leaky S ) waves are excited
0
at two different frequencies on an aluminum plate, of which one side is exposed to
water. The response signals display the combination harmonics at the sum
frequency. Next, the amplitudes of the combination harmonics are investigated with
varying excitation magnitudes and propagation distances. The effect of the liquid-
structure coupling is also explored by comparing the guided wave signals measured
from the test plate with and without water. Then, numerical simulations using a 3D
FE model are implemented to further investigate the sensitivity of the combination
harmonics to the material nonlinearity of the metallic plate partially immersed in
water. The findings of this study can provide support for the development of
structural health monitoring techniques for metallic liquid containment structures.
The remainder of the paper is organized as follows. Section 5.3 introduces
the theoretical background of the mutual interaction of ultrasonic waves in materials
with weak nonlinearity. Section 5.4 presents the experimental study, including the
overall experimental setup, preliminary tests to select the guided wave modes and
excitation frequencies, and the results of the experimental investigations. The
numerical study is illustrated in Section 5.5. The 3D FE model is described and
validated through experimental measurements. This section also includes a
parametric study, in which the experimentally validated 3D FE model is employed
to investigate the characteristics of the nonlinear guided wave mixing in materials
with different levels of fatigue. Finally, conclusions are drawn in Section 5.6.
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5.3. The theoretical background of the wave mixing technique
This section describes the theoretical derivations for the mutual interaction of
ultrasonic waves with two different frequencies in nonlinearly elastic materials. The
material nonlinearity is small and attributed to the microstructural features such as
dislocations, microvoids, and micro cracks. For one-dimensional problems, the
stress-strain relationship of the nonlinear elastic material can be expressed as [15]
Eu u2 (5.1)
2
where represent the stress. uu x with u and x representing the
displacement and the position, respectively. E and denote the linear elastic
modulus and the second order nonlinear parameters, respectively. The quadratic
term accounts for the weak nonlinearity of the material, which is ignored in the
linear theory.
Considering that two waves with different frequencies ( f and f , f f )
1 2 2 1
travel in the material, the equation of motions can be described as
u (5.2)
where is the mass density; x and u2u t2 ; t denotes the time.
Substituting Eq (5.1) into Eq (5.2) gives
E
uuuu (5.3)
where u2u x2. Eq (5.3) can be solved using the perturbation approach with
the assumption that the solution form of the total displacements is the sum of the
primary waves and the nonlinear waves
uu u (5.4)
P N
where u and u represent the primary and nonlinear components of the total
P N
displacements, respectively. The primary waves are also called linear waves
because they have the same frequencies as the input signals. The nonlinear waves
are generated by the interaction of the primary waves with the material nonlinearity.
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The amplitude of the nonlinear waves is much smaller than that of the primary
waves ( u u ). Then, the governing equation can be obtained by substituting
P N
Eq (5.4) into Eq (5.3) as follows
E
u u u u u u u u (5.5)
P N P N P N P N
Since the amplitudes of the nonlinear waves are very small, the derivatives with
respect to u can be neglected [15]. Then the governing equation can be further
N
divided into two differential equations as follows.
E
u u 0
P P
(5.6)
E E
u u uu
N N P P
Finally, the solution of the primary waves is
u A cos(wtk x)A cos(wtk x) (5.7)
P f 1 1 1 f 2 2 2
1 2
The primary waves combine the two excitation frequencies f and f . A and
1 2 f
1
A are the amplitudes of the primary waves at f and f , respectively. w, k , and,
f 1 2
2
represent the angular frequency, wavenumber, and phase shift, respectively, with
the subscripts denoting the first and second frequency components.
The solution of the nonlinear waves at the frequencies other than the
excitation frequencies can be written as
u A cos(2wt2k x) A cos(2w t2k x)
N 2f 1 1 1 2f 2 2 2
1 2
A cos(w w tk k x) (5.8)
f f 1 2 1 2 1 2
1 2
A cos(w w tk k x)
f f 1 2 1 2 1 2
1 2
where A A2k2x 8 and A A2k2x 8 are the amplitudes of the second
2f f 1 2f f 2
1 1 2 2
harmonics at 2f and 2f , respectively. A A A kk x 4 is the amplitude
1 2 f f f f 1 2
1 2 1 2
of the combination harmonics (sum harmonics) at the sum frequency ( f f ) and
1 2
A A A kk x 4 is the amplitude of the combination harmonics (difference
f f f f 1 2
1 2 1 2
harmonics) at the difference frequency ( f f ). Therefore, when the structural
1 2
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material is excited by waves with two different frequencies, the nonlinear
parameters can be estimated as follows
8 A 1
2f 1 for the second harmonics at2f
2f 1 k2 A2 x 1
1 f
1
8 A 1
2f 2 for the second harmonics at2f
2f 2 k2 A2 x 2
2 f
2 (5.9)
4 A 1
f 1f 2 for the sum harmonics at f f
f 1f 2 k k A A x 1 2
1 2 f f
1 2
4 A 1
f 2f 1 for the difference harmonics at f - f
f 2f 1 k k A A x 2 1
1 2 f f
1 2
It can be seen that when the propagation characteristics of the primary
waves do not change (e.g. fixed wavenumber values k and k ), the material
1 2
nonlinearity at any location can be correlated to the amplitudes of the primary
waves and the nonlinear waves. For simplicity, the material properties can be
characterized by relative nonlinearity parameters and they are defined as follows
A
2f i for the second harmonics at2f
2f i A2 i
f
i
A
f 1f 2 for the sum harmonics at f f (5.10)
f 1f 2 A A 1 2
f f
1 2
A
f 2f 1 for the difference harmonics at f - f
f 2f 1 A A 2 1
f f
1 2
where the relative nonlinearity parameter is proportional to the second order
nonlinear parameter and the propagation distance x.
The above derivations consider only the simplest case that ultrasonic waves
travel in the isotropic material in one direction. However, the relative nonlinearity
parameters defined by Eq. (5.10) have been widely recognized to be applicable for
characterizing the material nonlinearity of various structures using different guided
wave modes. In the literature, there are two popular approaches for evaluating
nonlinear elastic properties. The first approach is to excite structures with the input
signals of various magnitudes and measure the response signals at a single location
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[34-37]. If the measured nonlinear guided waves are generated due to the material
nonlinearity, the amplitudes of the second harmonics (A ) should increase linearly
2f
i
with the square of the amplitudes of the corresponding primary waves ( A2 ). In
f
i
contrast, the amplitudes of the combination harmonics, including the sum
harmonics (A ) and difference harmonics (A ), should have a positive linear
ff f f
1 2 2 1
relationship with the product of the primary waves at the two excitation frequencies
(A A ). The increasing rate of the nonlinear guided wave magnitudes to the
f f
1 2
corresponding primary waves can be correlated to the material nonlinearity of
structures.
The second approach estimates the material nonlinearity by measuring
signals at several locations with different propagation distances [26, 38-41]. Based
on Eqs (5.9) and (5.10), the relative nonlinearity parameters () will grow linearly
with the propagation distance, provided that the primary waves have sufficiently
large motion magnitudes to interact with the microstructural features of the material
[40, 42, 43]. The material's nonlinear elastic properties can be characterized by the
accumulation gradient of with the propagation distances. Within the same
sample and identical experimental setup, any abnormal increase of the nonlinear
guided waves and the relative nonlinearity parameters indicate the growth of the
material nonlinearity and degradation. In the present study, the relative nonlinearity
parameters are calculated by Eq. (5.10) to characterize the nonlinear guided wave
features on the partially immersed plate. The growing trends of the nonlinear guided
wave features with varying excitation magnitudes and propagation distance are
investigated, respectively.
5.4. Experimental study
5.4.1. Experimental setup for actuating and sensing guided waves
Experiments were conducted on a 1.6 mm thick aluminum plate, which was fixed
to the front of a metal tank with bolts. To investigate the effect of the liquid coupling
on the nonlinear guided wave mixing, experimental measurements were collected
on the external surface of the aluminum plate when the tank was empty and when
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it was filled with water, respectively. Due to the isotropic features of the metal
materials, the findings of this study could be also applicable to the isotropic plates
made of other metal materials such as steel and alloy. Figure 5.1 shows a photo of
the experimental setup. A computer-controlled signals generator (NI PIX-5412)
was employed to generate the excitation signals. The waveforms of the excitation
signals and the frequency selection are discussed in detail in Section 5.4.2. Then,
the voltage of the input signals was magnified by a voltage amplifier (Ciprian HVA-
800-A). After that, the amplified excitation signals were sent to the piezoceramic
transducer (Ferroperm Pz27, 10 mm diameter and 0.5 mm thick) that was bonded
to the outer surface of the test plate. The piezoceramic transducer could convert the
electric signals to mechanical motions and generate guided waves on the test plate.
The response signals on the plate surface were measured by a scanning laser
vibrometer (Polytec PSV-400-M2-20) and then further processed using the
software MATLAB. Each measurement was collected at 25.6 MHz and averaged
by 1000 acquisitions. The signal-to-noise ratios were improved by applying a low-
pass filter, of which the cut-off frequency was set to 1MHz.
Figure 5.1. Experimental setup
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5.4.2. Mode tuning and frequency selection
This section describes the selection of guided wave modes and excitation
frequencies for the nonlinear guided wave mixing in partially immersed plates. To
begin with, the dispersion features of guided waves were studied using DISPERSE.
Based on the global matrix method, the dispersion curves were derived for the 1.6
mm thick aluminum plate surrounded by air (when the metallic tank was empty)
and the same plate with one side exposed to water (when the metallic tank was filled
with water), respectively [44]. The material properties of the plate and the water are
shown in Table 5.1. The water layer was defined as a non-viscous semi-infinite
acoustic medium. The air properties were not considered in modeling because the
influence of the air on the guided wave propagation was very small. The air-coupled
plate surfaces were assumed to be traction-free. The water-coupled plate surface
was defined by the structural-liquid boundary conditions, which connected the
normal stresses and displacements at the plate-water interface [44]. Through the
out-of-plane motions, the guided waves in the liquid-coupled structures could
continuously radiate wave energy into the surrounding liquid medium.
Table 5.1: Material properties for the aluminum plate and the water layer
Density Young’s Poisson’s Longitudinal
(kg/m3) modulus ratio velocity (m/s)
(GPa)
Aluminum 2704 70.76 0.33 --
Water 1000 -- -- 1500
Figure 5.2 presents the dispersion curves for the 1.6 mm thick aluminum
plate without water and loaded with water on one side, respectively. The frequency
range was selected to be below the cut-off frequency of higher-order guided wave
modes. Therefore, only the fundamental guided wave modes could be excited by
the piezoceramic transducer. When the tank is empty, the test plate is surrounded
by air on both surfaces. There are only the fundamental symmetric (S ) and
0
antisymmetric (A ) wave modes on the air-coupled metallic plate as shown in
0
Figures 5.2(a)-5.2(c). Previous studies demonstrated that S wave on air-coupled
0
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plates satisfies approximate phase velocity matching and nonzero power flow
conditions [19, 45]. Specifically, the phase velocity of S mode decreases very
0
slowly with frequency. When the phase velocity of S mode at the excitation
0
frequency matches that of S mode at twice the excitation frequency with a relative
0
deviation of less than 1%, the interaction of the primary S wave at the excitation
0
frequency with the material nonlinearity of a metallic plate can generate measurable
second harmonic S waves at twice the excitation frequency. In addition, the
0
calculated relative nonlinearity parameters can grow linearly with the propagation
distance and can be used to characterize the microstructural change of the material
before the initiation of macro-scale damage [19, 45, 46].
In contrast, the phase velocity of the A wave increases rapidly with
0
frequency, making the phase velocity of the A wave at the excitation frequency
0
significantly different from that of A wave at other frequencies. To date, there are
0
very limited studies using A wave to evaluate the microstructural changes of
0
material in the early damage stage. Chillara and Lissenden [47] numerically
demonstrated that the interaction of A wave with the material nonlinearity of a
0
metallic plate can only generate second harmonic S waves, which propagate
0
independently and separate from the primary A wave. The generated second
0
harmonic S waves are so small that it is difficult to measure in practical
0
applications. However, A wave has been extensively employed to identify contact-
0
type damage, such as open fatigue cracks [18], delamination [48], and the bonding
effects of bolts [49]. The clapping behaviors between the surfaces as guided waves
pass through can generate measurable nonlinear guided waves. Since the contact-
type nonlinearity is much larger than the material nonlinearity, the evaluation of the
contact-type defects does not require phase velocity matching and nonzero power
flow conditions.
When the tank is filled with water, one side of the test plate is in contact
with water, and the other side is exposed to air. Guided waves in the partially
immersed plate include the leaky S wave and the fundamental leaky antisymmetric
0
(leaky A ) wave as well as the quasi-Scholte wave as shown in Figures 5.2(d)-5.2(f).
0
The leaky A wave has high attenuation and is unable to propagate a long distance
0
in the immersed plates, as shown in Figure 5.2(f). The quasi-Scholte wave has low
128 |
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black solid lines in Figures 5.2(d)-5.2(f), the phase and group velocities of the leaky
S wave decrease very slowly with frequency, which is similar to those of the S
0 0
wave in the dry plate. In addition, the attenuation of the leaky S wave is close to
0
zero at the frequency range below 600 kHz. These features enable the leaky S
0
waves to have similar propagation characteristics across a relatively wide frequency
range, which provides good flexibility for the selection of excitation frequencies for
guided wave mixing.
For further investigation, mode shapes were extracted by DISPERSE for the
S mode and the leaky S mode, respectively, as shown in Figure 5.3. The mode
0 0
shape diagrams display the relative displacements of guided wave modes across the
thickness of the structure. The red dashed lines and the blue solid lines represent
the particle displacements in the in-plane direction (parallel to the wave propagation)
and the out-of-plane direction (normal to the plate surface), respectively. The
amplitudes are normalized by the maximum absolute magnitudes. Figures 5.3(a)
and 5.3(b) show the mode shapes for the S mode at 100 kHz and 500 kHz,
0
respectively. Within the selected frequency range, the S mode is dominated by the
0
in-plane displacement that is uniformly distributed across the plate thickness. The
out-of-plane displacement component is small at low frequency and gradually
increases with frequency. For comparison, Figures 5.3(c) and 5.3(d) show the mode
shapes for the leaky S mode at 100 kHz and 500 kHz, respectively. Generally, the
0
wave structure of leaky S mode in the immersed plate is similar to that of S mode
0 0
in the plate without water. The leaky S mode has predominately wave motions
0
conserved in the plate structure, which ensures sufficient wave motions to interact
with the material microstructures. The in-plane displacement between the plate and
water is disconnected, while the out-of-plane displacement is continuous. Therefore,
as the frequency increases, more wave energy can leak into the liquid medium
through the out-of-plane wave motions. For these observations, the frequency range
of interest was chosen to be below 600 kHz to ensure that the generated nonlinear
guided waves have low attenuation and are measurable.
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Figure 5.3. Mode shapes for S mode in a 1.6 mm thick aluminum plate at (a) 100
0
kHz and (b) 500kHz and mode shapes for leaky S mode for the plate loaded water
0
on one side at (c) 100 kHz and (d) 500kHz. (the red dashed lines represent the in-
plane displacements and the blue solid lines denote the out-of-plane displacements)
Preliminary tests were implemented with single-frequency excitation
signals on the empty tank and the water-filled tank, respectively, to evaluate the
excitability of the piezoceramic transducers. The preliminary tests aimed to select
two excitation frequencies, at which the selected guided wave modes have
comparable wave motions. The single-frequency excitation signals were 6-cycle
narrow-band tone burst pulses modulated by Hanning window [15]. The excitation
frequency swept from 90 kHz to 410 kHz in steps of 20 kHz. Figure 5.4(a) shows
typical examples of the guided wave signals measured at 250 mm away from the
excitation center. The excitation frequency was 230 kHz. The red dash-dot and
black solid lines denote the signals obtained from the empty tank and water-filled
tank, respectively. From the wave speed evaluation, the first wave packets between
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45 s and 80 s were identified as the S wave for the empty tank and the leaky S
0 0
wave for the water-filled tank, respectively. It should be noted that the scanning
laser vibrometer measures the normal displacements on the plate surface [32].
Although S and leaky S waves are relatively small in the received signals, their
0 0
actual wave motions in the plate should be strong because both wave modes have
mostly in-plane motions as shown in Figure 5.3. Following the first wave packets,
the second wave in the empty tank should be the A wave, while the second wave
0
in the water-filled tank is the quasi-Scholte wave. The latter travels much slower
than the former, which is in good agreement with the theoretical predictions by the
group velocity dispersion curves as shown in Figures 5.2(b) and 5.2(e).
Figure 5.4. (a) Comparison of the time-domain signals experimentally collected
from the empty tank and water-filled tank and (b) the peak amplitudes of the
extracted signals across various frequencies.
For further signal processing, the first wave packets were extracted from the
rest of the signals. Figure 5.4(b) shows the peak amplitudes of the first wave packet
for different excitation frequencies. In general, the magnitudes of the signals
measured from the empty tank and water-filled tank change with the excitation
frequency, following a similar pattern. When the frequency increases from 90 kHz
to 250 kHz, the amplitudes of the signals increase with frequency. For the frequency
range over 250 kHz, the amplitudes of the signals decrease with frequency. The
decreasing rates of the signals measured from the water-filled tank are much
quicker than those obtained from the empty tank. From these observations, f and
1
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f were selected to be 170 kHz and 270 kHz, respectively, for the following three
2
reasons. Firstly, the difference between f and f was chosen to be 100 kHz,
1 2
which enables good separation between the second harmonics and the combination
harmonics. Secondly, the signals collected at both excitation frequencies had
relatively high signal-to-noise ratios. Thirdly, the received signals show the normal
displacement components on the plate surface. For the S and leaky S waves, the
0 0
out-of-plane displacements increase with frequency as shown in Figure 5.3.
Therefore, the signal amplitude at f should be slightly lower than that at f to
1 2
ensure the actual wave motions at the two selected excitation frequencies have
comparable magnitudes on the plate.
Figure 5.5. Merging two single-frequency tone burst signals to generate a mixed
frequency signal.
Next, the mixed frequency excitation signals were generated by merging
two single-frequency signals, which were a 6-cycle Hanning window-modulated
tone burst at a central frequency of 170 kHz and a 9-cycle Hanning window-
modulated tone burst at a central frequency of 270 kHz. The higher frequency signal
had more cycles, which was to ensure that the two frequency components had
similar duration in the time domain and comparable energy contents in the
133 |
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frequency spectrums. A mix ratio of 1:1 (where 1:1 mixing means 1 part 170 kHz
and 1 part 270 kHz) is promising to generate larger combination harmonics than
other mix ratios [52]. Figure 5.5 shows the waveforms of the single-frequency and
mixed frequency signals and their corresponding frequency spectrums.
5.4.3. Experimental results
5.4.3.1. Guided wave mixing in partially immersed metal plates
A demonstration of the guided wave mixing phenomenon was presented for the
partially immersed plate. Firstly, experiments were carried out with the water-filled
tank using the two single-frequency excitation signals as shown in Figure 5.5. The
measurements were collected at a fixed location that was 250 mm away from the
actuator center. The voltage of the input signals was increased to 160 V. Figures
5.6(a) and 5.6(b) show the time-domain signals for 170 kHz and 270 kHz,
respectively. The first wave packets in the two figures are identified as the leaky S
0
waves that propagate faster than any other wave mode as shown by the group
velocity dispersion curves in Figure 5.2(e). Secondly, experiments with the same
settings were performed using the mixed frequency signal. The response signals
measured at the same location are shown in Figure 5.6(c). It can be seen that the
waveform of the first wave packet is similar to the input of the mixed frequency
signal as shown in Figure 5.5. This is because leaky S waves have similar phase
0
and group velocities and low attenuation within the selected frequency range. The
leaky S waves at 170 kHz and 270 kHz can propagate together and the wave-
0
mixing zone is maximized.
Following the first wave packet, there are other wave components in Figures
5.6(a)-5.6(c). Since leaky A waves have high attenuation at the selected
0
frequencies, the remaining wave components should be dominated by the low-
attenuated quasi-Scholte waves. It can be seen that the waveform of the quasi-
Scholte wave in Figure 5.6(c) is similar to that at 170 kHz as shown in Figure 5.6(a).
This is because the deformation fraction in the immersed plate of the quasi-Scholte
wave decreases rapidly with frequency [53]. As a result, when the structure is
excited by the mixed frequency signal, the quasi-Scholte waves measured on the
plate surface are dominated by the low-frequency components. The high-frequency
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components of the quasi-Scholte wave are too weak to interact with the low-
frequency components. Therefore, the quasi-Scholte wave can be filtered out for
the study of nonlinear guided wave mixing.
Figure 5.6. Experimental signals from the water-filled tank with the excitation (a)
at f = 170 kHz, (b) at f = 270 kHz, (c) at mixed frequencies; and (d) the frequency
1 2
spectrum of their corresponding window-cut data.
Then, the first wave packets were extracted from the rest of the signals and
transferred into the frequency domain by fast Fourier transfer (FFT) as shown in
Figure 5.6(d). When the one-side water-immersed plate is excited separately by the
single-frequency signals, the 170 kHz and 270 kHz leaky S waves can generate
0
second harmonics, as manifested by the peaks at double the excitation frequencies
(2f 340kHzand 2f 540kHz). For comparison, when the partially immersed
1 2
plate is excited by the mixed frequency signal, the received signal has an additional
peak at the sum frequency ( f f 440kHz ), as shown by the black solid line in
1 2
Figure 5.6(d). The combination harmonics at the difference frequency
( f f 100kHz) cannot be observed clearly for the mixed frequency excitation.
1 2
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This is because the difference frequency is too close to the lower excitation
frequency ( f 170kHz) with only 70 kHz spacing. So, the difference harmonics
1
can be overwhelmed by the side lobes [46, 54]. The same reason also applies to the
second harmonics at (2f 340kHz) that are very close to the higher excitation
1
frequency ( f 270kHz ). Nevertheless, the experimental results indicate that
2
mixing leaky S waves with two different frequencies can generate low-attenuated
0
sum harmonics on the partially immersed plate.
Figure 5.7. The actual excitation signals from the piezoceramic transducer (a) in the
time domain, and (b) in the frequency domain.
Next, the nonlinearity due to the instrumentations was investigated. Figure
5.7(a) shows the actual signal from the piezoceramic transducer measured by the
scanning laser vibrometer. Figure 5.7(b) presents the corresponding frequency
spectrum. The amplitudes are normalized by the maximum absolute magnitudes.
As shown in Figure 5.7(b), there are no apparent peaks at the sum frequency
( f f 440kHz ) and difference frequency ( f f 100kHz) in the frequency
1 2 1 2
spectrum of the actual signal from the piezoceramic transducer. Therefore, the sum
harmonics observed in Figure 5.6(d) should be generated mainly by the interaction
of the guided waves with the material nonlinearity. However, the actuation system
can produce second harmonics as manifested by the peak at (2f 540kHz) in
2
Figure 5.7(b).
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Figure 5.8. Experimental signals from the water-filled tank excited by mixed
frequency signal (a) in the time domain, and (b) in the frequency domain; (c) the
amplitudes of the sum harmonics versus the product of the primary wave
amplitudes; and (d) the amplitudes of the second harmonics at 2f versus the
2
square of the corresponding primary wave amplitudes at f .
2
After that, the nonlinear response of the guided wave mixing was further
investigated by varying the excitation voltages. The voltage of the input signal was
increased to 40V, 80V, 120V, and 160V, respectively. For each voltage, the
response signals were measured five times at 250 mm away from the excitation
center. Figures 5.8(a) and 5.8(b) show the experimentally measured signals in the
time domain and the frequency domain, respectively. Figure 5.8(c) shows the
amplitudes of the sum harmonics versus the product of the primary waves at the
two excitation frequencies. Figure 5.8(d) shows the amplitudes of the second
harmonics at 2f versus the square of the amplitudes of the corresponding primary
2
waves at f . The black solid lines in Figures 5.8(c) and 5.8(d) represent the best-fit
2
lines by linear regressions with the error bars denoting the standard deviations of
five measurements. As discussed in Section 2, the amplitudes of the second
137 |
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harmonics due to the material nonlinearity should increase linearly with the square
of the amplitudes of the corresponding primary waves, while the amplitudes of the
combination harmonics should grow linearly with the product of the primary waves
at the two excitation frequencies. Overall, this analysis demonstrates that both the
combination harmonics and the second harmonics could be generated due to the
material nonlinearity of the specimen. However, the growing trends of the second
harmonics in Figure 5.8(d) show relatively larger deviations from the best-fit line.
This phenomenon indicates that the second harmonics are more susceptible to the
nonlinearity generated by the instrumentations, which agrees well with Figure
5.7(b).
5.4.3.2. The effect of the surrounding liquid on the guided wave propagation
The section compares the phenomenon of guided wave mixing in the test plate with
and without water. Guided wave signals were collected from the empty tank and
the water-filled tank, respectively, under the same experimental conditions. Figure
5.9(a) presents the time-domain signals measured at 250 mm away from the
excitation center. The red dash-dot and black solid lines denote the signals obtained
from the empty tank and water-filled tank, respectively. The first wave packets are
identified as the mixed frequency S wave for the empty tank and the mixed
0
frequency leaky S wave for the water-filled tank, respectively. Both wave modes
0
have similar amplitudes and waveforms in the time domain. However, an obvious
difference can be observed in the frequency domain. Figure 5.9(b) shows the
frequency spectrums of the window-cut signals extracted from Figure 5.9(a). The
combination harmonics at the sum frequency can be observed in both the empty
tank and the water-filled tank. The amplitudes of the primary waves at fundamental
excitation frequencies (170 kHz and 270 kHz) are similar for the empty tank and
water-filled tank. Over 270 kHz, the signal obtained from the empty tank has larger
amplitudes than that obtained from the water-filled tank.
Further research investigated the growing trends of combination harmonics
at the sum frequency with increasing excitation voltage and propagation distance,
respectively. Figure 5.9(c) shows the amplitudes of the sum harmonics versus the
product of the primary wave magnitudes at the fundamental frequencies. The
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response signals were collected at 250 mm away from the excitation center and the
voltage of the input signal was increased to 40V, 80V, 120V, and 160V. Five
measurements were collected for each voltage on both the empty tank and the
water-filled tank, respectively. It can be seen that the amplitudes of both the primary
waves (A A ) and the sum harmonics (A ) on the empty tank are relatively
f f ff
1 2 1 2
larger than those on the water-filled tank. Also, when the primary waves (A A )
f f
1 2
increase, the sum harmonics (A ) on the empty tank increase much faster than
ff
1 2
those on the water-filled tank.
Figure 5.9. Comparison of experimental signals from the empty tank and water-
filled tank excited by mixed frequency signal (a) in the time domain, and (b) in the
frequency domain; (c) the amplitudes of the sum harmonics versus the product of
the primary waves; and (d) the amplitudes of the nonlinearity parameters for the
sum harmonics versus propagation distance.
To investigate the relationship between the combination harmonics at the
sum frequency and the propagation distance, 21 measurement points were defined
on the external surface of the test plate and equally spaced between 200 mm and
139 |
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300 mm away from the excitation center. The voltage of the input signals was
increased to 160 V. Five measurements using the same settings were performed on
the empty tank and water-filled tank, respectively. Then, the relative nonlinearity
parameters were calculated by Eq. (5.10) for the sum harmonics and plotted against
the propagation distance as shown in Figure 5.9(d). For both the empty tank and the
water-filled tank, the relative nonlinearity parameters grow linearly with the
propagation distance from 200 mm to 270 mm. The cumulative propagation
distances are limited to 270 mm, which may be caused by the small phase velocity
difference between the primary waves and the sum harmonics. As shown in Figure
5.2, the phase velocities of S wave in the dry plate and leaky S wave in the partially
0 0
immersed plate have similar values and decrease slowly with frequency. The phase
velocities of the primary S waves at 170 kHz and 270 kHz are 5430 m/s and 5422
0
m/s, respectively. They are very close to the phase velocity of the sum harmonic S
0
waves at 440 kHz, which is around 5397 m/s. The deviation is less than 1%.
Previous studies have analytically and experimentally demonstrated that the
cumulative propagation distances of second harmonic generation due to material
nonlinearity decrease with the phase velocity difference between the primary S
0
waves and the second harmonic S waves [19, 45]. In the present study, the
0
cumulative propagation distances of sum harmonics determined by experimental
measurements have a similar order of magnitudes to the theoretical predictions for
the second harmonic generation by S waves due to material nonlinearity [45]. Thus,
0
it can be validated that the sum harmonics of the signals measured from 200 mm to
270 mm are generated due to the nonlinearity of the material.
In the linearly cumulative range, the growth rate of the relative nonlinearity
parameters with the propagation distance does not show an apparent difference
between the plate surrounded by air and the plate partially immersed in water.
However, the absolute values of the relative nonlinearity parameters on the empty
tank are much larger than those on the water-filled tank. Therefore, these results
indicate that the cumulative generation of combination harmonics due to the
material nonlinearity of partially immersed plates can be achieved with a mixed
frequency leaky S wave. Under the same experimental conditions, the amplitudes
0
of the guided wave signals and the values of the relative nonlinearity parameters on
140 |
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the partially immersed plate are different from their counterparts on the plate in
gaseous environments.
5.5. Numerical study
5.5.1. Modeling material nonlinearity
Numerical simulations were carried out with ABAQUS to further investigate the
sensitivity of the combination harmonics to the material nonlinearity of the partially
immersed metallic plate. The numerical methods have the advantage of eliminating
unwanted effects of the noises from the measurement system. The nonlinearity of
the material was simulated by incorporating a VUMAT subroutine that introduced
a constitutive model proposed by Murnaghan [55]. This section presents the
constitutive equations. To begin with, X and xare defined as the coordinates in the
reference and current configurations, respectively. The deformation gradient can be
expressed as [56]
x
F IH (5.11)
X
where I is the identity tensor; Hu X is the displacement gradient and
u xX is the displacement vector. The Lagrangian strain tensor can be written
as
1 1
E FTF HHT HTH (5.12)
2 2
For a hyperelastic and homogeneous isotropic solid material, the strain
energy function is [55]
1 1
W(E)
trE2 tr E2
l2m trE3
2 3
(5.13)
mtrE trE2 tr E2 ndetE
where and are the lame constants; l , m, and n are Murnaghan constants
that describe the second order material nonlinearity. Previous studies
experimentally measured the Murnaghan constants from dog bone samples made
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of aluminum [57]. The samples were subjected to repeated uniaxial tensile loads
with different cycles. One sample was loaded to a total of 52800 cycles and failed.
The rest samples were loaded to various cycles and referenced to the percent of
fatigue level. Table 5.2 summarizes the material properties of aluminum with 0,
40%, and 80% fatigue levels [57].
Table 5.2. Lame constants and Murnaghan constants for aluminum [57]
Fatigue 𝜌 𝜆 𝜇 L M N
level (kg m-3) (GPa) (GPa) (GPa) (GPa) (GPa)
(%)
0 2704 51.6 26.6 -252.2 -325.0 -351.2
40 2704 51.6 26.6 -266.8 -332.8 -358.3
80 2704 51.6 26.6 -271.2 -335.8 -359.8
The second Piola-Kirchhoff stress tensor can be obtained by
WE
T (5.14)
PK2 E
The Piola-Kirchhoff stresses are used to describe the reference
configuration and correlated with the Cauchy stress tensor σ as
T detFF1σ F1T (5.15)
PK2
5.5.2. 3D FE model
ABAQUS/CAE was employed to build and mesh the 3D FE model as shown in
Figure 5.10. The model consisted of a test plate that was 250 mm wide and 430 mm
long and had the same thickness as the experimental specimen. The bottom surface
of the test plate was exposed to water. The red quarter circle at the bottom left corner
of the plate represented a quarter of the piezoceramic transducer, which was
perfectly bonded to the top surface of the plate. Symmetric boundary conditions
were defined for the left and bottom edges. Firstly, the numerical simulations were
implemented by considering only the linear elastic material properties of the test
plate and water as shown in Table 5.1. The linear finite element (FE) simulations
consider the metallic plate as a linear elastic material. Then, nonlinear FE
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Leaky S waves were generated by applying nodal displacements to the
0
circumference of the simulated piezoceramic transducer [18]. The displacements
were assigned in the radial direction as shown in Figure 5.10(a). The excitation
signals were the mixed frequency signals as shown in Figure 5.5. The magnitude of
the displacement was 3 m. The simulated guided wave signals were calculated by
the central-difference integration through ABAQUS/Explicit. In all simulations, the
increment time step was automatically controlled by ABAQUS. The maximum time
increment step is less than the ratio of the minimum element size to the dilatational
wave speed [43]. The accuracy of the nonlinear constitutive model was validated
by comparing the results with the outcomes of the linear FE and experimental
measurements.
5.5.3. Experimental validation
Figure 5.11(a) compares the experimental measurements and simulation results in
the time domain. The time-domain signals, simulated by the nonlinear FE model
incorporating the Murnaghan constants for the intact aluminum (zero fatigue in
Table 2), are consistent with those simulated by the linear FE model. Both have a
good agreement with the experimental signals. Next, the signals were windowed
between 45 s and 80 s and transferred into the frequency domain as shown in
Figure 5.11(b). The linear and nonlinear FE models well predict the primary guided
wave components of the experimental signals at the fundamental excitation
frequencies, as highlighted by the dotted lines at 170 kHz and 270 kHz in Figure
5.11(b). The linear FE signals have no nonlinear guided wave components at
frequencies other than the excitation frequencies. But the nonlinear FE well predicts
the location of the peaks for the combination harmonics at the sum frequency
( f f ) and the second harmonics at 2f . The amplitudes of the simulated
1 2 2
nonlinear guided waves are comparable to the experimentally measured data.
Although there is a discrepancy between the experimental signals and simulated
signals at other frequency components, the nonlinear FE simulation well predicts
the generation of the combination harmonics due to the material nonlinearity.
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Figure 5.11: Experimental validations by comparing the experimental and
simulated signals (a) in the time domain and (b) in the frequency domain
5.5.4. Parametric study
The 3D FE model validated by experiments was employed to further investigate the
sensitivity of the nonlinear guided waves mixing to the material nonlinearity of the
partially immersed metallic plate. Nonlinear FE simulations were implemented
using the same 3D FE models, in which the material properties of the test plate were
defined as aluminum at the three different fatigue levels as shown in Table 5.2,
respectively. The out-of-plane displacements were collected at 11 measurement
points that were equally distributed on the top surface of the test plate from 200 mm
to 250 mm away from the excitation center. Then, the nonlinearity parameters were
calculated using Eq. (5.10) for the combination harmonics at the sum frequency
( f f ) and the second harmonics at 2f , respectively. To better observe the
1 2 2
influence of material nonlinearity evolution, the relative nonlinearity parameters
were normalized by their corresponding minimum value at zero fatigue level.
Figures 5.12(a) and 5.12(b) show the normalized nonlinearity parameters for the
combination harmonics and the second harmonics, respectively. Within the selected
measurement range, the normalized nonlinearity parameters grow linearly with
propagation distance. The growth rate (slope values of the best-fitted line) increases
as the material suffers more fatigue damage.
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5.6. Conclusion
This paper presents experimental and numerical investigations on the nonlinear
guided waves mixing in the partially immersed plates. The main contributions are
summarized as follows:
(1). According to the dispersion curves, leaky S waves at low excitation
0
frequencies have low attenuation and low dispersion effects, which provide
good flexibility for the selection of excitation frequencies for guided wave
mixing.
(2). Experiments have been conducted on an aluminum plate loaded with water
on one side using the single-frequency excitation and mixed frequency
excitation, respectively. Leaky S waves with two different frequencies can
0
generate combination harmonics at the sum of the excitation frequencies
that cannot be achieved by single-frequency excitations (see Figure 5.6).
(3). The combination harmonics at sum frequency ( A ) grow linearly with
f +f
1 2
the product of the primary guided waves at the fundamental excitation
frequencies ( A A ) and are less affected by the nonlinearity due to
f f
1 2
instrumentations (see Figures 5.7 and 5.8).
(4). Under the same experimental conditions, the liquid-structure coupling of
the partially immersed plate makes the amplitudes of the guided wave
signals and the relative nonlinearity parameters different from those of the
test plate without liquid coupling (see Figure 5.9).
(5). Numerical studies have been carried out with the material nonlinearity of
the test plate simulated by the Murnaghan constitutive model. The
numerical results reveal that both the second harmonics and the combination
harmonics are sensitive to the material nonlinearity of the partially
immersed plate (see Figures 5.12 and 5.13).
In conclusion, the current study has demonstrated that nonlinear guided wave
mixing has the potential to evaluate the material nonlinearity of metallic plates with
one side exposed to water. This new possibility can be significant considering the
high rewards for earlier detection of the damage to maintain the structural integrity
of high-valued infrastructures. To maximize the wave mixing zone, the selected
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guided waves propagate together in the same direction. Future studies are required
to explore the feasibility of non-collinear guided wave mixing, where the selected
guided waves propagate in different directions and meet in a localized mixing zone.
The non-collinear guided wave mixing has the potential to identify the area of
localized material degradation but the generation and measurement of the nonlinear
waves are more challenging because the nonlinear wave amplitudes can be affected
by the reduced wave mixing zone. In addition, more experimental studies need to
be carried out to investigate the correlation between the nonlinear guided wave
signals and the degree of damage.
5.7. Acknowledgment
This work was funded by the Australia Research Council (ARC) under grant
numbers DP200102300 and DP210103307. The supports are greatly appreciated.
5.8. Reference
[1] R. Liu, S. Li, G. Zhang, W. Jin, Depth detection of void defect in sandwich-
structured immersed tunnel using elastic wave and decision tree, Construction and
Building Materials, 305 (2021) 124756.
[2] R. Ignatowicz, E. Hotala, Failure of cylindrical steel storage tank due to
foundation settlements, Engineering Failure Analysis, 115 (2020) 104628.
[3] N. Balekelayi, S. Tesfamariam, Statistical inference of sewer pipe deterioration
using Bayesian geoadditive regression model, Journal of Infrastructure Systems, 25
(2019) 04019021.
[4] A. Heifetz, D. Shribak, X. Huang, B. Wang, J. Saniie, J. Young, S. Bakhtiari,
R.B. Vilim, Transmission of images with ultrasonic elastic shear waves on a
metallic pipe using amplitude shift keying protocol, IEEE Transactions on
Ultrasonics, Ferroelectrics, and Frequency Control, 67 (2020) 1192-1200.
[5] M. Abbas, M. Shafiee, An overview of maintenance management strategies for
corroded steel structures in extreme marine environments, Marine Structures, 71
(2020) 102718.
[6] M. Hong, Z. Su, Q. Wang, L. Cheng, X. Qing, Modeling nonlinearities of
ultrasonic waves for fatigue damage characterization: Theory, simulation, and
experimental validation, Ultrasonics, 54 (2014) 770-778.
148 |
ADE | Chapter 6
Chapter 6. Conclusions and remarks
6.1. Summary of contributions
This thesis has presented a comprehensive study on the ultrasonic guided waves in
thin-walled structures immersed in liquid on one side. The multimodal and
dispersive features of guided waves have been analyzed and the influence of the
surrounding liquid medium on the guided wave propagation characteristics has
been demonstrated both numerically and experimentally. The quasi-Scholte wave
and the fundamental leaky symmetric Lamb (leaky S ) wave have been intensively
0
studied for characterizing corrosion pits and microstructural evolution, respectively.
Both linear and nonlinear guided wave features have been discussed. The findings
complement the current knowledge about guided waves in submerged structures
and provide support for safety inspections of high-valued and critical infrastructures,
such as liquid storage tanks, vessels and pipelines, and submarine hulls.
Chapter 2 (Paper 1) has investigated the interaction of guided waves with
corrosion pits in a steel plate loaded with water on one side. Among many other
guided wave modes, the quasi-Scholte mode has been selected to characterize the
dimensions of circular blind holes that are the simplest representations of
progressive corrosion pits. The results have indicated that the quasi-Scholte mode
has a high sensitivity to the physical conditions of the plate-water interface and is
promising for evaluating corrosion damage in submerged structures.
In the literature, the quasi-Scholte wave was rarely used for damage
detection. This chapter has highlighted that applying the quasi-Scholte wave for
detecting damage in submerged structures is limited to the low frequency range, at
which the quasi-Scholte wave is dispersive. A case study has been presented for
using the quasi-Scholte wave at 100 kHz to characterize the blind holes in a 2 mm
thick steel plate with one side immersed in water. The advantages of the quasi-
Scholte wave at low frequencies have been demonstrated, which include easy
excitation, low attenuation, strong signal-to-noise ratios, and shorter wavelength
compared to other guided wave modes (higher sensitivity to smaller defects).
However, the quasi-Scholte mode becomes nondispersive at a high frequency and
153 |
ADE | Chapter 6
has most of its wave motions conserved in the liquid medium. The scattered waves
caused by the damage can be hardly measured on the plate surface.
Chapter 3 (Paper 2) has further investigated and compared the behaviors of
guided waves in a dry plate surrounded by air and the plate with one side immersed
in water. The foregoing studies demonstrated that mode conversion occurs when
guided waves propagate from the dry plate to the immersed plate. However, the
variation with the excitation frequency of the mode conversion phenomenon has
not been discussed before. In this chapter, the fundamental anti-symmetric Lamb
wave (A ) is excited on the dry plate section and travels to the water-immersed plate
0
section, where the generated A wave is mode converted to the quasi-Scholte wave.
0
The frequency dependence of the mode conversion from A wave to QS wave has
0
been studied numerically and experimentally. It has been discovered that the guided
wave energy can shift in the frequency domain when the phase velocity of the
incident A wave is larger than the sound speed of the surrounding liquid medium.
0
Due to the dispersive features of guided waves, the change in frequency
components can make the guided wave propagation properties different. Therefore,
the findings of this study are important for practical applications (e.g. using guided
waves for damage detection of partially immersed structures and assessing liquid
properties and levels).
Chapters 2 and 3 have also indicated that the defects (e.g. corrosion pits) on
partially immersed plates can be evaluated by sending A wave on the dry section
0
of the plate and measuring the quasi-Scholte wave on the immersed section. This
method is very promising for long-range inspection because the quasi-Scholte wave
has very low attenuation and does not radiate energy in the liquid. However, future
study is required to develop an effective damage detection algorithm for the
partially immersed structures to consider the change of wave behaviors due to the
mode conversion phenomenon and the presence of liquid.
The targeted defects in the first two papers (Chapters 2 and 3) are thickness
thinning in local areas and have a size of around a few millimeters. These
macroscopic defects can change the propagation speeds and amplitudes of the
transmitted guided wave signals, of which the wavelength is comparable to the
dimension of the defects. Recent studies on guided wave applications have focused
154 |
ADE | Chapter 6
on nonlinear guided wave features which provide the potential to identify and track
the evolution of damage in the microscale that precedes macroscopic defects.
Although nonlinear guided wave features have better sensitivity to microstructural
defects, their generation and measurements are very challenging because the
microscopic defects in the material are quite small. In the literature, several
conditions were proposed for the structures surrounded by air to ensure that the
nonlinear guided waves generated due to the material nonlinearity are cumulative
and measurable. However, these conditions have not been validated for the
structures submerged in liquid.
Chapter 4 (Paper 3) has explored the feasibility of the second harmonics
generation by guided waves in metallic plates with one side exposed to water. The
dispersive behaviors of multiple guided wave modes have been analyzed and three
criteria have been proposed in regard to selecting appropriate guided wave modes
and excitation frequencies. Firstly, the selected guided wave modes at the excitation
frequency (primary waves) should have low attenuation and have most wave
motions conserved in the submerged structures. Secondly, there should be nonzero
power flux between the primary waves and the corresponding second harmonics.
Thirdly, the phase velocity deviation between the primary waves and second
harmonics should be less than 1%. These criteria can be also applied to different
structures exposed to various liquids. Then, a case study has been presented using
experimental signals with an excitation frequency of 170 kHz. The results have
indicated that cumulative generation of second harmonics can be achieved by leaky
S wave. Next, a three-dimensional (3D) finite element (FE) model has been
0
developed to simulate the nonlinear guided wave generation. The material
nonlinearity of the immersed plate has been simulated by a VUMAT subroutine
that incorporates Murnaghan’s strain energy function. The numerical simulations
have been validated through experimental measurements. After that, a series of
parametric studies have been carried out and the results have shown that the second
harmonics are sensitive to the early-stage damage in plates with one side immersed
in water. However, the shortcoming of the second harmonic approach is that the
instrumentations for sensing and actuating signals can also produce higher
harmonics at the integer multiples of the excitation signals. It is very difficult to
155 |
ADE | Chapter 6
extract the second harmonics due to the material nonlinearity from the instrument
nonlinearity.
To cope with this limitation, Chapter 5 (Paper 4) has numerically and
experimentally investigated the nonlinear guided wave mixing in an aluminum
plate loaded with water on one side. Leaky S waves are excited at two different
0
frequencies on the wall of a metal tank filled with water. The results have shown
that the nonlinear interaction between leaky S waves at two different frequencies
0
can produce cumulative combination harmonics at the sum frequency. Compared
to the second harmonics studied in Chapter 4, the combination harmonics are less
affected by the higher harmonics produced by the instrumentations, such as
amplifiers and transducers. In addition, mixing guided waves with different
frequencies provides more flexibility for the selection of guided wave modes and
excitation frequencies. Finally, Chapter 5 has also presented parametric studies
using the experimentally validated nonlinear FE model. The combination
harmonics have shown a better sensitivity to the early stage of fatigue damage than
the second harmonics.
6.2. Future work and recommendations
The current research only considers the fundamental guided wave modes at low
excitation frequencies for the metallic plates partially immersed in water (non-
viscous liquid). However, the current research has built a foundation for the
development of guided wave-based techniques for safety inspection of submerged
plates or thin-walled structures, based on the excellent ability of guided waves to
travel at fast speed over long distances and the high sensitivity to the evolution of
microscopic defects. Below are the possible research directions for future studies.
1. The propagation characteristics of guided waves in plates immersed in
viscous liquids (e.g. honey, oil, and gas). The guided wave energy can leak
into viscous liquids through both shear and longitudinal wave motions. This
is different from the structures immersed in non-viscous liquids. Shear
waves do not exist in non-viscous liquids because shear forces cannot be
156 |
ADE | Chapter 6
sustained. Therefore guided waves in the plates immersed in viscous liquids
can be significantly different from that in the non-viscous liquids.
2. The propagation characteristics of guided waves in submerged structures
with different geometries (e.g. curved plates and T-section joints) and
various materials (e.g. plastics and composites). Guided waves behave
differently on different structures and materials. This future work is very
important because real-world structures are more complex.
3. The feasibility of using guided waves to characterize multiple corrosion pits
with irregular shapes. The damage in real-world structures can vary
significantly in size, shape, and locations. Therefore, the effect of damage
with increasing numbers and dimensions should be considered.
4. The acoustoelastic effect of submerged plates due to the hydrostatic
pressure. Submerged structures containing liquids are usually subjected to
hydrostatic pressure with varying aplitudes that can potentially induce and
incrase microstructural defects. The acoustoelastic effect can change the
propagation properties of guided waves and the generation of nonlinear
guided wave features.
5. The interaction between guided waves and stress corrosion cracking. Stress
corrosion cracking is one of the major concerns for the immersed metallic
plate structures that are subjected to time-dependent loads with varying
amplitudes. The present study only demonstrated that the micro cracks in
the early stage of damage can generate measurable and low-attenuation
second harmonics. The effect of the size, shape, and location of stress
corrosion cracking in macroscale on the guided wave propagation can be
future work.
6. The development of damage detection tools using guided waves and
machine learning approaches. Current algorithms cannot be applied to
structures with one side partly immersed in water as mentioned in Chapter
3 (Paper 2). A data-derived approach such as machine learning can help
consider the change of wave behaviors due to the mode conversion
phenomenon and the variation of liquid levels.
157 |
ADE | Abstract
This thesis aims to investigate novel approaches in the field of Machine learning and advanced
data analytics that can handle large data volumes and open new doors in the field of reservoir
characterization.
To begin, a new approach for rock typing is introduced using fractal theory where conventional
resistivity logs are the only required data. Fractal analysis of resistivity logs showed that the
fractal dimension of these logs which is a measure of the variability of the signal, is related to
the complexity of the rock fabric. the fractal dimension of multiple deep resistivity logs in the
Cooper Basin, Australia was measured and compared with the fabric structure of cores from
same intervals. The results showed that the fractal dimension of resistivity logs increases from
1.14 to 1.29 Ohm-meter for clean to shaly sands respectively, indicating that the fractal
dimension increases with complexity of rock texture.
The thesis continues with a machine learning application to augment/automate facies
classification using resistivity image logs. Given the complexity of the application, a supervised
learning strategy in combination with transfer learning was used to train a deep convolutional
neural network on available data. The results show that in the absence of other
information/logs, the trained network can detect image facies with a testing accuracy of 82%
form electric image logs and a proposed post-processing method increases the final
categorization accuracy even further.
An important step in reservoir characterization is understanding and quantification of
uncertainty in reservoir models. In the next section a novel Generative Adversarial Network
(GAN) architecture is introduced which can generate realistic geological models while
iv |
ADE | maintaining the variability of the generated dataset. The concept of mode collapse and its
adverse effect on variability is addressed in detail. The new architecture is applied to a binary
channelized permeability distribution and the results compared with those generated by Deep
Convolutional GAN (DCGAN) and Wasserstein GAN with gradient penalty (WGAN-GP). The
results show that the proposed architecture significantly enhances variability and reduces the
spatial bias induced by mode collapse, outperforming both DCGAN and WGAN-GP in the
application of generating subsurface property distributions.
Finally, an advanced analytics technique for efficient history matching is proposed in the
appendix. In this part of the thesis, an ensemble of surrogates (proxies) with generation-based
model-management embedded in CMA-ES is proposed to reduce the number of simulation
calls efficiently, while maintaining the history marching accuracy. History matching for a real
field problem with 59 variables and PUNQ-S3 with eight variables was conducted via a standard
CMA-ES and the proposed surrogate-assisted CMA-ES. The results showed that up to 65% and
50% less simulation calls for case#1 and case#2 were required.
v |
ADE | Declaration
I certify that this work contains no material which has been accepted for the award of any
other degree or diploma in my name in any university or other tertiary institution and, to the
best of my knowledge and belief, contains no material previously published or written by
another person, except where due reference has been made in the text. In addition, I certify
that no part of this work will, in the future, be used in a submission in my name for any other
degree or diploma in any university or other tertiary institution without the prior approval of
the University of Adelaide and where applicable, any partner institution responsible for the
joint award of this degree.
The author acknowledges that copyright of published works contained within this thesis
resides with the copyright holder(s) of those works.
I give permission for the digital version of my thesis to be made available on the web, via the
University's digital research repository, the Library Search and also through web search engines,
unless permission has been granted by the University to restrict access for a period of time.
I acknowledge the support I have received for my research through the provision of an
Australian Government Research Training Program Scholarship.
Roozbeh Koochak 02/04/2023
vi |
ADE | Acknowledgement
I am grateful to all those whose belief in me, encouragement, and support have been the
driving force behind my achievements.
Sahba, my loving wife, your endless patience, understanding, and belief in me have been the
bedrock of my success. Your constant support and sacrifices made it possible for me to focus
on my research and overcome challenges. Your presence by my side has been a source of
strength, motivation, and joy, and I am forever grateful for your unwavering love.
To my dear mother, Farideh, and siblings, Atousa, Reza, and Parisa, your consistent support,
encouragement, and understanding have been invaluable. Your belief in my abilities and
unwavering support, both emotionally and practically, have been instrumental in my
accomplishments. Your presence in my life has filled me with gratitude and motivation to excel.
I would like to extend my deepest appreciation to my esteemed supervisors, Dr. Manoucheher
Haghighi, Dr. Mohammad Sayyafzadeh, and Dr. Mark Buch. Your expert guidance, knowledge,
and dedication have shaped my research, broadened my horizons, and enriched my
understanding of the subject matter. Your tireless efforts to provide me with valuable feedback,
constructive criticism, and mentorship have played a crucial role in my academic growth and
the successful completion of my Ph.D. I am truly grateful for the opportunities you have given
me to explore and contribute to the field.
Lastly, I would like to thank my dear friends, Dr. Ali Nadian, Colin Jordan, and Dr. Martin Roberts
for their unwavering support, interest in my work, and willingness to lend a helping hand
whenever needed. Your intellectual discussions, feedback, and collaborative spirit have been
instrumental in shaping my ideas and sharpening my research focus.
vii |
ADE | Thesis by publication
This is a thesis by publication and is composed of multiple pieces of work. This includes 4
publications in total. 2 published in peer-reviewed journals, 1 submitted for publication to a
peer-reviewed journal and 1 conference paper. These are the details below:
Published Peer-reviewed Journal Papers:
Koochak, R., Haghighi, M., Sayyafzadeh, M. and Bunch, M., 2018. Rock typing and facies
identification using fractal theory and conventional petrophysical logs. The APPEA Journal,
58(1), pp.102-111.
Koochak, R., Sayyafzadeh, M., Nadian, A., Bunch, M. and Haghighi, M., 2022. A variability
aware GAN for improving spatial representativeness of discrete geobodies. Computers &
Geosciences, 166, p.105188.
Submitted to Peer-reviewed journal for publication:
Roozbeh Koochak, Ali Nadian Ghomsheh, Manouchehr Haghighi, Mark Bunch, Mohammad
Sayyafzadeh, A transfer learning approach for facies prediction using resistivity image well
logs. Submitted to Geoscience Frontiers for publication.
Published Conference paper:
Sayyafzadeh, M., Koochak, R. and Barley, M., 2018, September. Accelerating cma-es in history
matching problems using an ensemble of surrogates with generation-based management. In
ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery (Vol. 2018, No. 1,
pp. 1-15). EAGE Publications BV.
viii |
ADE | 1. Introduction
1.1 Problem statement
Reservoir characterization includes all techniques and methods that enhance our
understanding of the geologic, Geo-chemical and petrophysical controls of fluid flow. it is a
constant process that Continuously evolves from field discovery to development and
production down to abandonment phase. Throughout these stages, knowledge of the
subsurface continually enhances as new data is received. This makes characterization of the
reservoir a dynamic process. The challenge is that the dynamic properties of the reservoir such
as Pressure, Saturation or Porosity continually change as the reservoir is produced. This
together with the fact that most of the data measurements are indirect along with error of
measurement, causes uncertainty in the reservoir characterization. Adequate characterization
of a reservoir is increasingly important for optimising field development, reservoir evaluation
and production. Also, in recent years with novel subsurface applications such as carbon
capture and sequestration or hydrogen storage, detailed characterization is imperative to
successful operations throughout the life a field. An integrated approach for characterization
and/or modelling of the reservoir can tear down traditional disciplinary divides and lead to
better understanding and handling of uncertainties in the reservoir. With this statement in
mind, the methodologies presented in this thesis have multidisciplinary applications where the
disciplines of Petrophysics, Reservoir Engineering and Geology have been brought together.
Reservoir characterization generally involves estimating reservoir parameters at different
locations by correlating collected data from a wide variety of sources. Sources of data for
subsurface reservoirs include cores, well logs, Seismic surveys, production data and outcrop
analogues. These sources of data all have different scales and vary in dimensionality. Logs have
1 |
ADE | high resolution but on a small scale, generally, 1D or 2D in dimension. Well logs provide a
vertically high-resolution model at the well locations. However, the distribution of well
locations is sparse and biased towards proven section of the field. Seismic data on the other
hand is large scale and covers extensive areas but with low resolution. Dimensionality of this
data type is 2D, 3D and sometime 4D. Combining well logs with geophysical and geological data
will provide the necessary constrains required to extrapolate the high resolution well data
beyond where they are measured. Core data, as 3-dimension datatype, is classed as hard data
and presents the most accurate information with highest resolution however their availability
is limited as the acquisition of cores, is expensive and requires a lot of effort. The most effective
utilisation of these data sources is, therefore, combining these different data sources. This
approach results in the best and most complete description of reservoir which is generally
referred to as integration modelling. Integration of different data sources enhances
understanding of the reservoir, reduces uncertainties and mitigates risks. The goal of
characterization is to develop different models that can be used in analytical or numerical
evaluation methods.
The oil and gas industry are experiencing a surge in the amount of data they are receiving from
their fields. Field data, in recent years, has expanded in volume, velocity and complexity. This
includes significant increase in the number of sensors in the field and in the pipelines,
connected through Internet of Things (IoT). Resolution and sampling rate of wireline data has
increased and complex data such as 4D seismic are becoming more common and more
frequently recorded. Advent of technologies such as fibre optics has made access to these data
virtually instant. The industry has always been overwhelmed with large quantities of data but
was never able to make efficient and productive use of this data. Traditional methods in
subsurface energy generally need to compromise between data size and complexity on one
2 |
ADE | hand and fidelity of the model (they are used to construct) on the other. In recent years Data
analytics and Artificial Intelligence (AI) - Machine Learning (ML) more specific in this thesis -
have introduced new methodologies that not only handles large and complex data types but
is able to process the data much faster than traditional methods. This in part is also driven by
advances in commodity hardware. Furthermore, ability of these methods to identify and learn
features and patterns in the data makes them a great tool to unlock new insights, extract more
information, develop new usage and leverage and optimize untapped data.
In this work, the aim is to develop and implement advanced analytics and machine learning
techniques with a focus on reservoir characterization using multi-scale, multi-dimensional data.
Firstly, a novel application of fractal dimension of resistivity logs is presented. In this one-
dimensional, small-scale application porosity and permeability derived from cores was rock
typed into categories and the fractal dimension of the corresponding deep resistivity log was
calculated. The correlation between the rock fabric and fractal dimension of the resistivity logs
was investigated and a new method to make direct use of these logs in rock typing is proposed.
This research investigates the effect of pore structure on the variability of resistivity logs. It
takes a step further than just interpreting resistivity logs based on change in average value and
signature and reveals information at pore scale. In this study only the effect of pore structure
on the variability and fractal dimension of logs was investigated. Further, research is required
to determine the effect of other factors like fluid in the rock or different shale types. Bearing
in mind that fractal dimension is rather a quality parameter and a flag for change in pore
structure. Then in a 2-dimensional small-scale application, a machine learning algorithm
complete with practical data pipeline is introduced to augment/automate the tedious process
of resistivity image log interpretation. In this technique, a convolutional neural network (CNN)
is trained to learn interpreted facies categories in one well, then the trained network is used
3 |
ADE | to detect the learned facies categories in a newly drilled well. The ability of CNNs to learn and
identify features in geological facies images is thoroughly investigated by studying the
confusion matrix. Multiple CNN architectures are compared and challenges in the application
are identified, and solutions presented. Next, we apply machine learning to a 2-dimensional
large-scale application. In this study, Generative Adversarial Networks (GANs) are used to
quantify uncertainty in a field or basin wide scale. The concept of Training Image (TI) is
reviewed. Methodology to generate network training data from a single TI is presented and
compared with other traditional geo-statistics methods. The variability of generated
realizations (which is detrimental in geological uncertainty quantification) using GANs is
thoroughly investigated including the concept of mode collapse and the effect of input training
data. A novel architecture specialized for maintaining the variability of geological realizations
at the same level as the input training data is presented.
Working with large data sets and complex algorithm, computation efficiency and cost becomes
an important factor in popularity of a characterization method. We enhance the computational
efficiency of a Covariance Matric Adaptation Evolutionary Strategy (CMA-ES) by proposing an
online learning scheme to update an ensemble of proxies. The effective ness of the technique
was evaluated on two different history matching cases and other techniques such as
generation-based model management and evolution control were examined.
1.2 Thesis Structure
This is a thesis by publication. Chapter 1 begins with an introduction to reservoir
characterization and its challenges. It describes how novel advanced analytics techniques and
machine learning algorithms can enhance the process of reservoir characterization and
describes the aims of this thesis. The contribution of each publication to this thesis, is also
4 |
ADE | analytics and state-of-the-art techniques such as machine learning are utilized to better
understand uncertainty, develop new methods, and potentially extract more information and
propose new usage of the traditional data, while considering, different data sources with
different scales and dimensionality. Furthermore, computational efficiency and automation is
addressed where these techniques are adaptable.
This work begins by introducing a new application for conventional resistivity logs which are
one dimensional and on a centimetre scale. In the paper titled "Rock typing and facies
identification using fractal theory and conventional petrophysical logs" a technique to use
fractal dimension of resistivity logs for rock typing and flow unit classification is proposed. Rock
typing is an integral part of reservoir characterization. In this process the reservoir is subdivided
into layers based on similar properties and flow points. In other words, rock fabric of each layer,
that is, pore throat dimensions, geometry, size, distribution, and capillary pressures must be
similar. This enhances flow behaviour modelling and, significantly reduces uncertainty and risk
of predicting production and/or injection in the field. In this study, porosity and permeability
measured from cores were correlated with fractal dimension of corresponding deep resistivity
logs. A methodology to determine fractal dimension from 1D data is proposed and the
propagation of ions and electric current in the rock fabric along with its relationship and effect
to fractal dimension of the resistivity logs is investigated. Traditionally, only the change in
average of resistivity log over an interval is utilized when interpreting resistivity logs. The
results of this investigation show that further information can be derived from these logs. For
example, presence of layered beds with thicknesses less than the resolution of the tool can be
flagged using is effect of these beds on fractal dimension of the resistivity log. While this is not
possible using conventional interpretation methods. This study is presented in detail in chapter
3.
6 |
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