EXPERIMENTAL OBSERVATION OF LINEAR AND NON-LINEAR
GUIDED WAVE PROPAGATION IN ROLLED ALUMINUM SHEETS
D. A. Scott and D. C. Price
CSIRO Telecommunications and Industrial Physics,
P.O. Box 218,
Lindfield, NSW, 2070
Australia
ABSTRACT. An experimental investigation of guided waves in rolled aluminum sheets has been
undertaken, and a companion paper (following) reports on theoretical work relevant to these
experiments. Transmitting and receiving transducers were coupled to one side of an aluminum sheet,
and the propagation characteristics of linear and non-linear elastic waves were measured. Scans of
frequency and transducer separation showed the effects of mode coupling and anisotropy in the
aluminum sheets, and indicated the presence of cumulative resonant harmonic generation.
INTRODUCTION
The propagation of ultrasonic guided waves in plates has been used for many years as
the basis of a number of NDE techniques, particularly long-range inspection and the
characterization of materials. Non-linear elastic waves are highly sensitive to the
properties of the material through which they propagate, and have been used to detect the
early stages of fatigue, micro-cracking and other microstructural features in a wide range of
applications. Non-linear guided waves offer the potential of long-range inspection for the
detection of these microstructural defect conditions. Central to the interpretation of such
measurements is a knowledge of the characteristics of propagating linear and non-linear
guided waves in plates. This paper presents results of experiments aimed at understanding
such propagation characteristics in rolled aluminum sheet.
GUIDED WAVE MODES IN ALUMINUM PLATES
The dispersion curves for guided wave (Lamb) modes in an isotropic aluminum plate
are shown in Figure 1. These have been calculated using a density of 2.7 gem"3, a
longitudinal velocity of 6447 m s"1 and a transverse velocity of 3132 m s"1. The labels An
and Sn refer to the nth antisymmetric and nth symmetric modes respectively, where the
plane of symmetry is halfway through the thickness of the plate. It is evident that for any
given phase velocity where a mode exists at a frequency/, in general there does not exist a
mode at 2f at that same phase velocity. Closer inspection of the curves reveals that at a
specific phase velocity, in this case at 8357 m s"1, A2 and 82 cross (the modes are
orthogonal) at a frequency/around 2.5 MHz, the A4 and 84 modes cross at 2f, the Ae and
85 modes cross at 3/ and so on. This means that, for example, if A4 and S4 are excited at
this particular phase velocity, conditions for the propagation of energy along the plate in
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
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, A2 S2 A3 S3 A4 S4 A5 S5 A6 Sg A, S7
10
20
frequency (MHz)
FIGURE 1. Dispersion curves for a 2 mm thick aluminum plate, with the symmetric (S, solid lines) and
antisymmetric (A, dashed lines) modes labelled individually up to the eighth order and as pairs up to the
sixteenth order.
the Ag and Sg modes are also satisfied. In this case, therefore, if a signal at a fundamental
frequency is propagating along the plate, any harmonic signal (generated as a result of the
passage of this wave) may also propagate along the plate. There is a similar sequence of
modes at a phase velocity of 6932 m s"1, beginning with A3/S3, A6/S6, A9/S9, and so on. To
observe resonant harmonic generation in a plate one needs to have symmetric or
antisymmetric modes at / and a symmetric mode at 2/ at approximately the same phase
velocity. If, in addition to this, the fundamental and harmonic modes have the same group
velocity, the amplitude of the harmonic signal will increase linearly with propagation
distance for pulsed measurements.
EXPERIMENTS
Experimental Arrangement
A schematic of the experimental set-up is shown in Figure 2. The RITEC SNAP
5000 system is a high-power transmitter and superheterodyne receiver, used here to deliver
tone-burst excitation voltages to the transmitting transducer, and detect and analyse the
received signal to obtain an integrated amplitude and phase with respect to the transmitted
signal. This instrument is computer-controlled and allows narrow bandwidth generation
and detection as well as limited calculation and display of the signals.
Standard immersion, longitudinal-wave transducers (KB-Aerotech, 12.7 mm
diameter, center frequencies either 5 MHz or 10 MHz) were used as the transmitter and
receiver. These were coupled to the aluminum samples (2.0 mm thick AA5005 or 1.6 mm
thick AA2024) using Perspex (polymethylmethacrylate) wedges and a thin film of standard
gel (Echotrace) couplant between both the transducer and the wedge, and between the
wedge and the sample. The wedges were accurately machined to hold the transducers at a
specific angle to the aluminum sheet. The angles (19.18°, 19.69°, 20.00° and 20.50°) were
chosen, using Snell's Law, to give phase velocities in the sheets of 8430, 8220, 8100 and
7910 m s"1 respectively, using our experimentally measured velocity of longitudinal sound
waves in Perspex of 2770 m s"1. These wedges were used to probe the sheets with slightly
different phase velocities. Pairs of wedges at the same angle were used for the
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Monitoring
oscilbscope
Control computer
Switchable attenuators and matching circuits
Sample
FIGURE 2. Experimental arrangement for measuring guided waves.
transmitting and receiving transducers. Reproducibility of the results was ensured through
careful experimental technique, with particular emphasis on the couplant thickness, and the
alignment of the transducers and wedges using a specially designed jig. Experiments with
a range of couplants of widely varying viscosity were undertaken, but results from these are
not reported here: all of the experiments yielded results with features similar to the ones
presented herein. The wedges were also designed to hold the transducers close to the
aluminum sheet; the closest point of the transducer to the sheet was 1 mm for all the angles
used. This ensured as short a path length in the Perspex as possible, minimizing the
generation of harmonics in this medium. The SNAP system was set to transmit tone bursts
of duration 40 |is over a range of fundamental frequencies, typically 3 - 8 MHz.
Switchable, matched attenuators in the transmission line facilitated the changing of driving
voltages to check the system linearity and response to different signal levels. A 10 MHz
high-pass filter in the receiver line was used only during the measurement of the amplitude
of the harmonic signal. The measurement procedure is described in detail below. An
oscilloscope was used to monitor signals generated during scans, such as the integration
gate width of the detector, amplitudes and tone burst shapes of the transmitted and received
signals, and their relative timing.
Measurement of Tone Bursts
Dispersion curves such as those shown in Figure 1 are derived from theory (see
following paper) describing the steady state of continuous waves propagating along a plate
of infinite extent. In order to be able to compare theory with experiment with some
validity, long tone bursts were used here as the excitation method. By measuring the
amplitudes of these long tone bursts (several hundred cycles) well after any initial transient
behaviour has decayed, a very close approximation to the steady state value will be
obtained.
A schematic representation of idealized waveforms observed in the experiments is
shown in Figure 3. Figure 3 (a) represents the tone burst incident on the aluminum plate. If
the conditions are such that only one mode propagates along the plate, a single tone burst is
received some time tA later, shown in Figure 3(b). If, however, two modes are excited, in
general they will travel along the plate at different group velocities and therefore arrive at
the receiver at different times, denoted by te and tc in Figure 3(c). In the time interval
during which these two modes overlap (denoted by D in the figure) the waves superpose,
so when either the frequency of excitation is changed or the distance between transmitter
and receiver is changed, the phase relationship of the two wave packets will change. This
results in a large oscillation in the amplitude during time D, the two extremes of which are
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Transmitted tone burst
node2
\\
iiirst shape
mode crossing
•MU
"
\^
frequency
FIGURE 4. Schematic representation of
scanning the transmitting and receiving
frequency.
(9)
0
time'
FIGURE 3. Schematic representation of tone bursts.
shown in Figures 3(c) and (d). If the integration gate (the time interval over which the
amplitude is measured) is set to be long enough to capture all of the received signals as
shown in Figure 3(e), the resulting integrated amplitude needs to be interpreted carefully as
it has contributions from many possible modes. In the case shown in the figure, as the
frequency of the transmitted burst is changed, the measured received amplitude will show a
slowly varying background level with a smaller, more rapidly varying component due to
the interference effects of the overlapping modes. If the characteristics of a particular mode
need to be measured, a narrow integration gate may be set, and usually towards the end of a
tone burst in order to measure, as well as possible, the steady state amplitude of the mode.
Examples of these gates for modes B, A and C are shown as 1, 2 and 3 respectively, in
Figure 3(f). In practice, tone burst shapes are not perfect 'top-hat' functions, but are
subject to transient effects, inter alia, and a more realistic tone burst shape is depicted in
Figure 3(g).
Frequency Scans
In the present experiments, when a tone burst is transmitted into the aluminum sheet
there will be a spread in both the incident angle, and hence phase velocity, and the
frequency. The range of incident angles is largely determined by the effects of diffraction
from the (finite-sized) aperture of the transmitting transducer. The initial angle of the
longitudinal incident waves determines the phase velocity of the partial waves of shear and
longitudinal oscillations within the plate. These are produced by refraction and mode
conversion at the wedge/plate interface, and add to form the propagating mode. Over short
propagation distances a range of phase velocities will approximately satisfy the conditions
for propagation along the plate. Over longer distances only those waves that more closely
satisfy the conditions (a narrower range of angle and frequency) will propagate. The plate
acts as a spatial filter. Also, early in the tone burst a wide range of frequencies are present
as the steady state of the oscillation is reached. Further into the tone burst the frequency
range will be considerably narrower. The combination of these two ranges in phase
velocity and frequency is shown schematically in Figure 4 as a rectangle, and to illustrate
the effect of scanning the frequency of the transmitted burst several such rectangles are
shown sweeping across two intersecting dispersion curves. As the frequency is increased,
the first signal would represent the detection of mode 1, then of both modes at
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0.40
6
7
10
frequency (MHz)
FIGURE 5. Amplitude of received fundamental (solid line) and harmonic (dashed line) signals as a function
of fundamental frequency, for a sample of 2 mm thick AA5005 sheet.
the crossing point, then of both modes beyond the crossing. The range of the phase
velocity varies with propagation distance and the range of frequency varies with time from
the start of the pulse.
RESULTS
Frequency Dependence
Figure 5 shows the amplitude of the received signals at the fundamental frequency
(of the transmitted tone burst) and at the harmonic frequency for 2 mm AA5005 sheet with
19.18° wedges, corresponding to a phase velocity of 8430 m s"1, at a wedge separation of
Ax = 140 mm. A comparison of the results with the calculated dispersion curves yielded
the assignments of the symmetric and antisymmetric modes as shown. In all the results
presented here the amplitude of the harmonic signal has been multiplied by a factor of 400
to plot it on the same scale as the fundamental signal.
Looking first at the fundamental amplitude results, large oscillations in the signals
between SB/AS, A^/Ss and Sy/Ay modes are due to the detection, with the long integration
gate, of two modes overlapping in time. Further, near 5.0 MHz and 7.6 MHz (the A4/S4
and A6/S6 regions respectively) there is a sharp minimum in the signal. In the harmonic
spectrum the existence of the S7/A7, A9/S9, and Sn/An modes indicates the presence of
either residual harmonic in the driving signal, or non-resonant harmonic generation in the
system somewhere near the transmitter (in the transducer itself, in the couplant layers, or in
the Perspex wedge): there are no linear (fundamental) modes in these frequency ranges so
these signals represent signals at twice the fundamental frequency that have been injected
into the plate rather than developed within it, and transmitted along the plate linearly. The
Ag/Sg and Ai2/Si2 spectra show both a background signal level and also a sharp, almost
'bipolar' feature. As modes at both/and 2f exist at these frequencies, the background
signal may come from non-resonant harmonic generation anywhere in the system. The
peaks in the harmonic signals and the minima in the fundamental signals occur at the same
frequency, and are discussed below.
Transducer Separation Scans
Results of frequency scans for nine different transducer separations are shown in
Figure 6, for the frequency range 4.8 MHz to 5.6 MHz, for a 2 mm thick sheet of AA5005.
The small oscillations in the fundamental amplitude are again due to the A4 and 84 modes
overlapping under the integration gate; similarly for the A8/S8 harmonic results.
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0.30:
Ax = 20 mm
0.25-
Ax = 40 mm
... Ax = 60 mm
nAx= 100mm
j\Ax= 120mm
0.200.150.100.05-
A Ax = 80 mm
0.100.05
0.25
f\Ax= 140 mm
= 170mm
0.20-
t
Ax = 200 mm
0.15
0.100.05
0.00
4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.64.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.64.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6
frequency (MHz)
FIGURE 6. Amplitudes of the fundamental (solid lines) and harmonic (dashed lines) signals in the region
4.8 MHz to 5.6 MHz for different transducer separations, Ax, for a sample of 2 mm thick AA5005 sheet.
The background amplitudes of both the/and non-resonant 2f signals decrease with
distance as would be expected to result from diffraction and attenuation losses in the
aluminum sheet. The non-resonant component of the 2f signal may originate from two
separate sources: it may be injected into the plate (as discussed above), or it may be
generated along the plate by the propagating linear (/) mode, but does not build up in
amplitude. More important to note is the dependence of the sharp minimum in the
fundamental (/) amplitude and the sharp maximum in the harmonic (2f) amplitude at the
cross-over frequency of the respective modes. At the crossing point of the A4 and S4
modes the surface particle motion of the 84 mode is entirely in-plane, whereas the A4 mode
has an out-of-plane particle motion. It is expected that the couplant used between the
aluminum sheet and the coupling wedge will significantly attenuate the in-plane motion,
resulting in only weak generation of the 84 mode and a low sensitivity of the receiving
system to such in-plane motion. The minimum in the signal at the cross-over frequency of
the A4 and 84 modes is believed to be due to coupling between the A4 and 84 modes,
possibly due to anisotropy in the sheet (see following paper). The deepening of the
minimum in the signal at / with increasing transducer separation is consistent with the
filtering effect referred to above - the proportion of the received wave motion concentrated
in the coupled region is higher.
It is postulated that the sharp peak in the harmonic signal is due to cumulative
resonant harmonic generation. The magnitude of this peak increases linearly with
transducer separation for Ax = 60 mm to Ax = 140 mm, before declining due to attenuation
and diffraction. This is consistent with resonant, cumulative harmonic generation. This is
believed to be the first reported observation of accumulating harmonic generation of Lamb
waves in a plate. It will be discussed further in the following companion paper.
Material Dependencies
The spectral features described in the previous section show different characteristics
in different materials and samples. Figure 7 shows results for different sheets of nominally
the same material (AA5005), and scans at different places on the same sheet. The two left
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- Fundamental, Sheet I, Ax=200mm, 19.18° wedges
-- Harmonic, Sheet 1,Ax=200mm, 19.18° wedges
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
-Fundamental, Sheet II, Ax=200mm, 19.18° wedges
-- Harmonic, Sheet II, Ax=200mm, 19.18° wedges
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
frequency (MHz)
Fundamental, Sheet II, Ax=200mm, 19.18° wedges
--—-top of sheet
—— middle of sheet
........... bottom of sf
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
Fundamental, Sheet II, middle of sheet, 19.18° wedges
——— Ax=200mm
....... Ax=300mm
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
frequency (MHz)
FIGURE 7. Fundamental and harmonic amplitudes as functions of frequency for different sheets of AA5005
(left) and different positions on the same sheet (right).
panels of the Figure 7 show results from two nominally identical sheets of 2 mm AA5005
under the same conditions. The data in the upper left panel is the same data as appears in
the last panel of Figure 6, for a sheet referred to as 'Sheet I'. The data in the lower left
panel is from another sheet of the same material. Not only is the sharp minium in the
fundamental signal absent in this data, but the 'bipolar' spike in the harmonic spectrum is
reversed in polarity. (At larger Ax a minimum in the/signal from Sheet II was observed; it
was not absent, merely much weaker than that found in Sheet I.)
The right panels of Figure 7 show data taken from the same Sheet II as in the left
panels, only at different positions on the sheet; only the fundamental amplitude is shown
here. In the upper panel are data taken for the same Ax, 200 mm, at the top, middle and
lower sections of the sheet. While there are slight differences in the maximum values of
the overall spectrum (possibly due to slight differences in the couplant thickness), the
frequency at which the sharp minimum occurs and the relative depth of that minimum
change markedly. The lower panel shows results taken from the middle of the same sheet,
but at different separations. The overall signal amplitude decreases with increasing Ax, as
seen before, but the minimum occurs at different but distinct frequencies, sometimes giving
rise to two separate minima in the same frequency scan.
These data clearly indicate that there are fairly large regions, of the order of
centimetres, in these sheets that have markedly different properties. The reasons for such
inhomogeneities are not clear. A variation in the thickness of the sheets, due to the normal
cold-rolling processes used in their manufacture, would not explain such features. Step
changes in thickness would be necessary to be consistent with the data. A slowly varying
thickness would only serve to broaden the observed minimum, not cause distinct minima.
Finally, frequency scans of the fundamental and harmonic signals in a different
aluminum alloy, AA2024, yielded essentially the same results as the AA5005 alloy. The
local minimum at the crossover frequency was still apparent in the fundamental signals,
and the bipolar signals in the harmonic spectra were also clear, although slightly less
pronounced. This alloy is age hardened (precipitation heat-treated) which might be
expected to lessen the anisotropy caused by cold rolling compared to the AA5005 alloy.
This may be a partial explanation for the AA2024 results differing slightly from those of
the AA5005.
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A10/S10
5.5
6.0
S5
6.5
frequency (MHz)
FIGURE 8. Amplitudes of the received signals at / (solid lines) and 2/ (dashed lines) as a function of
frequency for different angles between the propagation direction and the rolling direction in a sheet of
AA5005, 'Sheet IF.
Rolling Direction
All of the results above were obtained from experiments in which the direction of
propagation of the guided waves was parallel to the rolling direction of the aluminum sheet.
Experiments were also performed in which the transmitter and receiver were placed so as
to have the direction of propagation at an angle to the rolling direction. Results of these
experiments are shown in Figure 8. The frequency range of 4.8 MHz to 6.8 MHz was
chosen to observe the effects of rolling direction on the modes we have studied thus far, the
A4/S4 and A8/S8, as well as non-resonant 2/modes (A9/S9), and modes that do not cross at
this phase velocity (As/Ss at/and Aio/Sio at 2/). The results show that the only significant
change with rolling direction occurs for the A4/S4 and A8/S8 modes. The amplitude of the
fundamental signal shows a small local minimum at low angles, but this develops into two
separate minima at 45° and back to a single minimum at higher angles, but at a different
frequency to the low-angle minimum. The 2/amplitude undergoes a more marked change,
the 'bipolar' spike changing into two distinct minima at 45° and 60°, and just a single
minimum at 90° at the same frequency as the observed minimum in the/signal. These
results indicate the effects of both anisotropy and inhomogeneities in this sheet of AA5005.
CONCLUSIONS
The characteristics of guided wave modes propagating in aluminum sheets have been
presented. The effects of different types of aluminum alloy, probing the sheets at different
positions, and of rolling direction have all been studied. It is believed the results represent
the first reported observation of cumulative resonant harmonic generation in aluminum
plates. They also indicate that significant anisotropy and inhomogeneities are present in
cold-rolled aluminum sheet.
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