TRANSMISSION AND REFLECTION OF AO MODE LAMB WAVE
IN A PLATE OVERLAP
Won-Joon Song1, Joseph L. Rose1, Jose M. Galan2, and Ramon Abascal2
Ultrasonic Lab, Department of Engineering Science and Mechanics,
Penn State University, University Park, PA 16802, USA
2 Escuela Superior de Ingenieros
Camino de los Descubrimientos, s/n.
E41092, Sevilla, Spain
ABSTRACT. Lamb wave propagation in a plate overlap is investigated. Transmission and reflection
coefficients for incident Lamb waves of AO mode across the overlap region are numerically calculated
using a hybrid BE-FE method. Transmission and reflection coefficients of the Lamb wave across the
overlap region are studied as a function of frequency and overlap length. In addition, mode conversion
phenomena from the incident waves within the overlap region are also included in the numerical study.
A few experiments were also conducted for measurements of transmission and reflection coefficients
for incident AO mode wave in overlap-shaped steel plates with two different overlap areas. The
experimental results are in good agreement with the numerical calculations. The numerical and
experimental results can be used to establish guidelines for NDE in overlapped plates and in multilayer structures with various joints by selecting modes and tuning frequency.
INTRODUCTION
Ultrasonic guided wave techniques using have been successfully applied to plate-like
structures, pipes, rods, and multilayer structures due to the characteristics of guided
waves such as long propagation distances, controllability of sensitivity to defects, and
high penetration power [1, 2, 3]. The geometry of industrial structures becomes more
complex than simple structures like pipes, rods, etc. In order to evaluate these complex
structures, ultrasonic waves should be able to travel through branch, weld, and overlap
joints. In many cases, this geometrical complexity in the structure causes difficulties in
their effective nondestructive evaluation because of scattering and the existence of hidden
areas. It has been reported that applications of guided wave NDE already extend to such
structures with complex geometry as lap-jointed plates, rails, curved pipes, rods with an
arbitrary cross-section and overlapped plates [4 -8].
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
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In this study, guided wave scattering in a plate with an overlap is numerically and
experimentally studied in terms of frequency and overlap dimensions. The overlap region
simulates a perfect joint of two semi-infinite plates. A Hybrid Boundary Element (BE)Finite Element (FE) technique is employed to numerically calculate the scattering of
guided waves from an overlap, which combines a Boundary Element method with a
semi-analytical Finite Element method and a normal mode expansion [9 - 14]. Mode
conversion within the overlap region is also considered in the numerical calculation. The
transmission and reflection of AO Lamb wave impinging on an overlap are
experimentally measured for comparison with the numerical results.
NUMERICAL CALCULATION OF TRANSMISSION AND REFLECTION
FACTORS FOR AO MODE INCIDENT LAMB WAVE.
A hybrid Boundary Element (BE)- Finite Element (FE) method was utilized in order
to calculate the transmission and reflection factors in a plate overlap. In this hybrid BE FE technique, the guided wave scattering problem is solved by defining two regions as
shown in Figure 1. The neighborhood of the defects is discretized with a quadratic
Boundary Element method and the elastodynamic solution in each waveguide is
expressed as a normal mode expansion [9, 12, and 13]. Each semi-infinite plate region is
modeled with a FE technique based on a semi-analytical formulation of the wave
propagation in a plate, from which an absorbing boundary condition is derived that will
be later applied on the BE region [14]. The FE and BE regions are coupled by
equilibrium and compatibility on their common interface.
The transmission and reflection factors are numerically calculated as mode conversion
factors. In this paper, they are defined as
__
where C)m) is the mode conversion factor for the scattered mode m under the incidence of
mode /, E(m) is the energy flux averaged over a period carried by the propagating mode
Boundary Element Method
in overlap region
pinite Element Method in
semi-infinite plate region
• • • • • • • • • • • • • • • • i
Finite Element Method in
semi-infinite plate region
Transmitted wave
Incident wave Reflected wave
Absorbing boundary condition
FIGURE 1. Illustration of the boundary configuration for a hybrid Boundary Element-Finite Element
method in a plate overlap.
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m through any cross section of the plate, Einc is the total incident energy, and a(m} is the
participation coefficient of mode m in the normal mode expansion of the scattered field.
By a normalization procedure which SQtsE(m} equal to unity for all modes m, the mode
conversion factors between the incident and the scattered modes are equal to the
participation coefficients. The numerical results include the factors for mode converted
waves (secondary mode conversion factors) as well as for the same mode as the incident
mode (primary mode conversion factors).
Transmission and reflection factors over an overlap region are calculated as a function
of frequency for AO mode incident Lamb wave. The material properties used in the
numerical study are those of steel: CL = 5.9 mm/jusec, CT= 3.2 mm/fisec, and density 7.8
g/cm3. The problem geometry corresponds to two 6.35 mm thick plates with a 12.7 mm
thick overlap region. First, a case with a fixed overlap length of 6.35mm is considered.
The higher modes generated by mode conversion phenomena within the overlap region
beyond their first cut-off frequency up to 1.5 MHz are also calculated. The calculation
results show that the transmission and reflection factors across the overlap region vary
dramatically along frequency as shown in Figures 2. The peaks or valleys of these
transmission and reflection factors are obtained in narrow frequency ranges. In the
frequency range beyond the first cut-off frequency, a significant amount of energy is
transferred from the incident mode into the higher converted modes in the scattered field.
Furthermore, a parametric study over the overlap length is performed with the overlap
parameter ranging from 0.5 to 13mm. Figure 3 presents the contour map plots of the
primary transmission and reflection factors for AO Lamb mode as a function of frequency
and overlap length. It is noted that the transmission and reflection of AO Lamb wave in a
plate overlap is a complicated phenomena with respect to frequency and overlap length.
In general, a high level of transmission (low in reflection) over the plate overlap occurs in
the frequency range lower than 100 kHz for all the overlap lengths. But, maxima in the
primary mode conversion factors are localized in the frequency-overlap length plane as
shown in Figure 4.
MEASUREMENTS ON TRANSMISSION AND REFLECTION FACTORS OF AO
MODE LAMB WAVE
Experimental Setup
Experiments were conducted in two steel plates with an overlap. A 12.7 mm thick
steel plate was machined into the half-thickness overlapped plates leaving a 12.7 mm
thick overlap region and 6.35 mm thick transmitting and receiving regions. The overlap
lengths, L, are 6.35 and 12.7 mm (Plates 1 and 2 respectively). Two variable angle
transducers are located with a total wave path of 304.8 mm in both through-transmission
and pulse-echo configurations. For reflection factor measurement, the transducers are
located at an angle of 15 degrees with respect to the central line in order to avoid the
beam divergence effect in the transducers. The incident wedge angles of the transmitter
and receiver are identical so that the same scattered mode as the incident mode is
collected (primary conversion factor). The peak-to-peak amplitude of scattered waves
were measured and transmission and reflection factors were calculated as the ratio of the
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FIGURE
of the
the transmission
transmission and
and reflection
reflection factors
factors for
forAO
A0incident
incidentmode
modeininaa6.35
6.35
FIGURE 2.
2. Numerical
Numerical calculation
calculation of
mm
plate with
with aa 6.35mm
6.35mm long
long and
and 12.7
mm thick
thick overlap
overlap region,
region. (a)
(a) Reflection
Reflection factor
factor for
for
mm thick
thick steel
steel plate
12.7 mm
antisymmetric
Transmission factor
factor for
for antisymmetric
antisymmetric modes,
modes, (c)
(c) Reflection
Reflection factor
factor for
for
antisymmetric modes,
modes, (b)
(b) Transmission
symmetric
factors for
for symmetric
symmetric modes.
modes.
symmetric modes,
modes, and
and (d)
(d) Transmission
Transmission factors
amplitude
waves to
to that
that of
of the
the reference
reference waves
waves which
which were
were
amplitude of
of the
the transmitted
transmitted or
or reflected
reflected waves
collected
thick plain
plain steel
steel plate.
plate.
collected in
in aa 6.35mm
6.35mm thick
Experimental
Experimental Results
Results
Figure 4 shows the measured transmission
transmission and
and reflection
reflection factors
factors of
of AO
A0 Lamb
Lambmode
modeinin
comparison with numerical results. In general, the
comparison
the overall
overall experimental
experimentalmeasurements
measurementsare
are
in good
good agreement with numerical calculations. The
in
The experimental
experimental results
results indicate
indicate that
thatthe
the
transmission and reflection of Lamb waves impinging
transmission
impinging on
on an
an overlap
overlap region
region can
can be
be
maximized or minimized for each incident
maximized
incident mode
mode in
in aa specific
specific frequency
frequency range
range asas
predicted in numerical calculations. The maximum
predicted
maximum transmission
transmission occurs
occurs at
at 250
250 kHz
kHz and
and
530 ~~ 550
550 kHz for 6.35 mm long overlap and at
530
at 185 and
and 500
500 kHz
kHz for
for 12.7
12.7 mm
mm long
long
overlap, while the maximum reflection
overlap,
reflection occurs at 200
200 kHz
kHz for
for 6.35
6.35mm
mmlong
longoverlap
overlapand
and
at 150,
150, 250 , and 410 kHz for 12.7
at
12.7 mm
mm long
long overlap.
overlap.
These experimental results demonstrate the possibility
These
possibility of
of controlling
controlling the
the Lamb
Lamb wave
wave
transmission and reflection from an overlap region by
transmission
by selecting
selecting proper
proper incident
incident mode
mode and
and
frequency. In view of practical application, these results
frequency.
results can
can provide
provide aauseful
usefulguideline
guidelinetoto
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calculationof
ofprimary
primarymode
mode conversion
conversion factors
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Lamb mode
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as aa function
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of
FIGURE
frequencyand
andoverlap
overlaplength,
length.(a)
(a)Transmission
Transmissionfactor,
factor,(b)
(b)Reflection
Reflectionfactor.
factor.
frequency
1 .0
1 .0
Ex p
Nu m
0 .8
Ex p
Nu m
0 .8
0 .6
0 .6
R
T
0 .4
0 .4
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0 .0
0
0 .2
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Fr e q u e0.4
n c y ( M Hz ) 0.6
F r e q u e n c y (MHz)
1 .0
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0
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Fr e q u e 0.4
n c y ( MHz )
0 .6
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Nu m
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Nu m
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0
F r e q u e n c y (MHz)
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0 .4
0 .6
Fr e q u e0.4
n c y ( M Hz ) 0.6
F r e q u e n c y (MHz)
0 .8
(b)
(b)
FIGURE 4. Comparison of experimental measurements with numerical calculation on transmission and
FIGURE 4. Comparison of experimental measurements with numerical calculation on transmission and
reflection factors for A0 Lamb incident mode. (a) Transmission and reflection factors with the overlap
reflection factors for AO Lamb incident mode, (a) Transmission and reflection factors with the overlap
length of 6.35 mm, (b) Transmission and reflection factors in the overlap length of 12.7 mm.
length of 6.35 mm, (b) Transmission and reflection factors in the overlap length of 12.7 mm.
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effective guided wave NDE in the area of a plate before or beyond the overlap region and
inside the overlap region itself.
CONCLUSIONS
The scattering of guided waves in a plate overlap has been studied numerically and
experimentally. The numerical studies are performed using a hybrid BE-FE technique.
First, a plate overlap of fixed geometry is considered, for which the primary and
secondary mode conversion factors under AO Lamb incidence are calculated. It is found
that the scattering of Lamb waves by the overlap region has a complicated pattern along
the frequency. High values of transmission and reflection factors are obtained within
narrow frequency ranges. The mode conversion phenomena from the overlap region
above the first cut-off frequency produces higher modes in the scattered field, and the
interference among the modes results in a complicated scattering pattern along the
frequency. Next, a parametric study is performed for varying geometries, which reveals
that the primary mode conversion factors highly depend on the incident wave mode,
overlap dimensions and frequency. The maximum transmission and reflection for each
incident mode are obtained within localized frequency-overlap length ranges.
The experiments on measurement of transmission and reflection factors for AO Lamb
wave agree well with the numerical calculation. Therefore, it can be concluded that best
reflection and transmission points can be obtained by tuning phase velocity and
frequency and observing maximum amplitude signals to find its best test points. In
addition, the numerical studies can provide quantitative prediction of Lamb wave
scattering over an overlap.
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