REFLECTION AND TRANSMISSION OF LAMB WAVES AT DISCONTINUITY IN PLATE
Z. Liu
NDT Systems & Services AG, Stutensee, Germany
Abstract: Lamb waves can be used for testing thin plate and pipe because they provide great potential for
improving inspection efficiency and sensitivity. In this paper, numeric analysis implemented by boundary element
methods (BEM) and the normal mode expansion technique is used to study the Lamb wave reflection and
transmission coefficients at discontinuity in plate. It is considered the interaction of Lamb waves with two basic
discontinuities—thickness step and thickness incline. Interaction of individual Lamb wave modes with
discontinuity in a plate is investigated by calculating the reflection and transmission coefficients for varying
incident Lamb wave modes, frequency, and discontinuity shape. A0 and S0 Lamb mode in aluminum plate incident
at frequency to 1.4 MHz, the plate thickness variation from 0.5 mm to 2.0 mm and incline angle from 0°
to 60°
at
the point of discontinuity are considered. The mode conversion phenomena through step and incline discontinuity
in a plate are also given. Results in this paper can be used to improve inspection sensitivity and penetration power
for a variety of practical NDE applications, notably those in which thickness variation is found.
Introduction: Ultrasonic Lamb wave inspection is an elegant and promising technique whereby an ultrasonic pulse
is sent to scan a long path through the structure, such as plate, tube and pipe. Lamb wave propagates in the direction
along the surface of plate or in the axial direction of pipe, and as the material like metal has very little damping, the
energy of the waves is practically not absorbed as they travel, and thus they can travel over long distances.
Therefore a large area of the tested structure can be inspected from a single location. Many work has been
published on the use of Lamb waves for inspection purposes because this approach provides great potential for
improving inspection efficiency and sensitivity compared with the conventional bulk wave techniques.
However, previous studies show that even simple structural configurations can lead to complex response signals for
lamb waves. Hence, it is still needs further study and validation before it can be accepted in the natural industry. On
the other hand, theoretical analysis of Lamb wave reflection and transmission (or scattering) at discontinuity in
structure is extremely difficult because of the complex multi-mode conversion phenomena. Analytical Approach
already has been done for simple geometries with identical thickness and perfect specimens without defects (for
example see [1]]). However, these equations become intractable for more complicated geometries or for a non-
perfect specimen. Many researchers have tried to tackle Lamb wave scattering problems using numerical methods,
e.g. The finite difference method (FDM)[2], the finite element method (FEM)[3] and the boundary element method
(BEM) [4-5]. The FDM was the first numerical method to be applied to investigate the propagation of stress waves.
The FEM is sometimes very cumbrous to use for more complex shaped waveguide problems due to the nature of its
domain type formulation. And the BEM has the advantage that just the surface of the specimen needs to be
discretized; the numerical problem itself is therefore reduced by one dimension.
A hybrid BEM (HBEM) combined with the normal mode expansion technique has been used to study Lamb wave
interaction with surface breaking defects. This technique is here applied to study of Lamb wave reflection and
transmission at step discontinuity and inclined discontinuity in plate. Interaction of individual Lamb wave modes
with discontinuity is investigated by calculating the reflection and transmission coefficients in scattered fields for
varying incident modes, frequency, and the step elevations or the angles of the incline. Lamb wave propagation in
inhomogeneous plates whose thickness varies at definite position is interesting problem but have been reported
rarely[6].
HBEM numerical modeling: The calculation model in this study considers only the plate with spatially varying
thickness without considering the defect in plate (e.g. notch or crack in plate). This paper presents two HBEM
models for step discontinuity (thickness step) and inclined discontinuity of plate respectively. Fig.1a shows the
calculation model of step discontinuity with thickness variations. The calculation model for the inclined
discontinuity is shown in Fig.1b. BoundaryΓ
2 and boundaryΓ
4 are two real free surfaces of the plate, and there is
a typ of discontinuity onΓ
4. Γ
1 and Γ
3 are two created boundary for HBEM calculation. An incident time
harmonic Lamb wave is considered propagating in the positive x direction. The mode incident onto the
discontinuity results in both reflected and transmitted waves of all orders of the propagating.
Fig 1: HBEM modeling statement of Lamb wave reflection and transmission in plate with (a) step discontinuity; (b)
inclined discontinuity
Unlike the conventional BEM, the boundary element displacement or strain on boundary Γ
1 and Γ
3 can not be
expressed by certain definite values. The total displacement and surface traction on Γ
1 and Γ
3 are the
superposition of all of the incident and scattered propagating Lamb wave fields which can be obtained by Lamb
wave expansion technique [5]. The jth mode Lamb wave reflection coefficient
j
R and the mth mode Lamb wave
transmission coefficient
T can be calculated as follows:
m
R Α
I
j
=α
j
T Α
m
m
= β
I
Α
Ι
α
j
β
m
where is the incident Lamb wave amplitude, and denote the amplitude of the jth Lamb wave traveling
in the negative x directions onΓ
1 and the amplitude of the mth Lamb wave traveling in the positive x directions
onΓ
3 respectively. Detailed Description for HBEM can be found in [5] and will not be iterated here.
In this study, the aluminum plate was considered for simulation. The material properties for aluminum were chosen
as: longitudinal wave velocity , the transverse wave velocity C and density
CL /
=6260 s
m
T /
s
m
=3045
/
7
.
2 cm
g
=
ρ
3
.
Results and Discussion: Matlab codes for two-dimensional HBEM were developed for calculating the reflection
and transmission coefficients of Lamb waves on plate with arbitrary shaped internal or surface defects [7]. A simple
case Lamb wave scattering, reflection from a free edge in a semi-infinite plate was already successfully solved by
various experimental and numerical methods [4,8]. As a comparison, the reflection coefficient curves of plate edge
reflection for the case of A0 and A1 incident are show in Fig. 2a and Fig 2b respectively. The plate thickness here
is identical with 2mm (d1=d2=2mm in Fig.1). These curves are similar with previous experimental and numerical
results. Because the HBEM simulation is a closed form solution, it is confirmed that the HBEM codes developed in
this study can be successfully applied to solve Lamb wave scattering problems.
.
1,0
A0
A1
on
ect
i
cient
s
0,5
f
l
R
e
coef
f
i
0,0
0,6 1,1 1,6 2,1
(a) Frequency (MHz)
1,0
A0
A1
on
ect
i
cient
s
0,5
f
l
R
e
coef
f
i
0,0
0,6 1,1 1,6 2,1
(b) Frequency (MHz)
Fig. 2. The Lamb wave reflection curve for a free edge in semi-infinite aluminum plate with thickness 2mm. (a) A0
incident; (b) A1 incident
The HBEM calculated Lamb wave reflection and transmission coefficients versus frequency are plotted in Fig.3
and Fig.4 for the S0 mode and A0 mode incident onto the step discontinuity (d1=1mm and d2=2mm) in an
aluminum plate respectively. As seen in Fig.3 and Fig.4, there are at least two wave modes in plate, and three wave
modes within the range of high frequency in a plate with thickness 2mm. They show that the reflection and
transmission coefficients for any mode under both of S0 mode or A0 mode incident are strongly dependent on
frequency. However, at low frequencies(e.g. below 0.5 MHz), it can be seen that the transmission coefficients of S0
mode under S0 incident remain on high level, but the transmission coefficients of A0 mode are relative lower. Fig.5
and Fig.6 show the reflection and transmission curves for the S0 mode and A0 mode incident onto the step
discontinuity (d1=2mm and d2=1mm). This similar phenomenon is also shown in Fig.5 and Fig.6.
Figs. 7- 10 show the S0 reflection and transmission curves under S0 incident, the A0 reflection and transmission
curves under A0 incident, for several step discontinuity, while Figs. 7-8 show the Lamb waves propagate from thin
side to thick side in plate, and Figs. 7-8 show the Lamb waves propagate from thick side to thin side in plate. Note
that in these four figures, the reflection and transmission coefficients do not increase or decrease monotonically
with the difference between d1 and d2 by most situation, only it is clear that the reflection and transmission
coefficients of A0 at low frequencies have the monotonic trend.
Figs. 11- 14 show the reflection and transmission coefficients under S0 or A0 incident for inclined discontinuity
with s=h=1mm, and d1=1mm or d2=1mm. Compared with the Figs. 3-6, it is seen that the influence of two
type of discontinuity is relatively small. Furthermore, the S0 reflection and transmission curves under S0 incident,
the A0 reflection and transmission curves under A0 incident, for different several inclined discontinuity, are given
in Figs. 15- 18. Here is the inclined degree arctan(s/h=0.57735)=30°; arctan(s/h=1)=45° and
arctan(s/h=0.57735)=60° respectively. It is also found that the influence of different inclined degree (up to 60°) for
reflection and transmission curves is also not very large.
1,0
S0
A0
on
ect
i
cient
s
f
l
0,5
R
e
coef
f
i
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
S0
A0
A1
Transmission coef
f
i
cient
s
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 3. (a) Reflection and (b) transmission coefficients under S0 incident for aluminum plate with step
discontinuity d1=1mm and d2=2mm
1,0
S0
A0
on
cient
s
ect
i
f
l
0,5
R
e
coef
f
i
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
S0
A0
A1
Transmission coef
f
i
cient
s
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 4. (a) Reflection and (b) transmission coefficients under A0 incident for aluminum plate with step
discontinuity d1=1mm and d2=2mm
1,0
S0
A0
A1
n
tio
cient
s
c
fle
0,5
Re
coef
f
i
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
S0
A0
ts
Transmission co
e
fficie
0,5
n
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 5. (a) Reflection and (b) transmission coefficients under S0 incident for aluminum plate with step
discontinuity d1=2mm and d2=1mm
1,0
S0
A0
A1
n
tio
cient
s
c
fle
0,5
Re
coef
f
i
0,0
0 0,5 1
(a) Frequency (MHz)
1,5
1,0
S0
A0
ts
Transmission co
e
fficie
0,5
n
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 6. (a) Reflection and (b) transmission coefficients under A0 incident for aluminum plate with step
discontinuity d1=2mm and d2=1mm
1,0
1.0 mm
1.5mm
2.0 mm
on
ect
i
cient
s
0,5
R
e
coef
f
i
f
l
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
Transmission coef
f
i
cient
s
0,5
0,0
1.01,5
0 0,5 1
(b) Frequency (MHz)
mm
1.5mm
2.0 mm
Fig. 7. (a) Reflection and (b) transmission coefficients for S0 mode under S0 incident for aluminum plate with step
discontinuity d1=0.5mm and d2=1mm, 1.5mm, 2.0mm
1,0
1.0 mm
1.5mm
2.0 mm
n
tio
cient
s
c
fle
0,5
Re
coef
f
i
0,0
0 0,5 1
(a) Frequency (MHz)
1,5
1,0
1.0 mm
1.5mm
2.0 mm
ts
Transmission co
e
fficie
0,5
n
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 8. (a) Reflection and (b) transmission coefficients for A0 mode under A0 incident for aluminum plate with
step discontinuity d1=0.5mm and d2=1mm, 1.5mm, 2.0mm
1
0 .5mm
1.0 mm
1.5mm
n
tio
cient
s
c
fle
0,5
Re
coef
f
i
0
0 0,5 1
(a) Frequency (MHz)
1,5
1,0
Transmission coef
f
i
cient
s
0,5
0,0
,5
0 0,5 1 1
(b) Frequency (MHz)
0 .5mm
1.0 mm
1.5mm
Fig. 9. (a) Reflection and (b) transmission coefficients for S0 mode under S0 incident for aluminum plate with step
discontinuity d1=2mm and d2=0.5mm, 1.0mm, 1.5mm
1,0
0 .5mm
1.0 mm
1.5mm
on
ect
i
cient
s
f
l
0,5
R
e
coef
f
i
0,0
0 0,5 1
(a) Frequency (MHz)
1,5
1,0
0 .5mm
1.0 mm
1.5mm
Transmission coef
f
i
cient
s
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 10. (a) Reflection and (b) transmission coefficients for A0 mode under A0 incident for aluminum plate with
step discontinuity d1=2mm and d2=0.5mm, 1.0mm, 1.5mm
1,0
S0
A0
on
cient
s
ect
i
f
l
0,5
R
e
coef
f
i
0,0
0 0,5 1
(a) Frequency (MHz)
1,5
1,0
S0
A0
A1
Transmission coef
f
i
cient
s
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 11. (a) Reflection and (b) transmission coefficients under S0 incident for aluminum plate with inclined
discontinuity d1=s=h=1mm and d2=2mm
1,0
S0
A0
n
tio
cient
s
c
fle
0,5
Re
coef
f
i
0,0
0 0,5 1
(a) Frequency (MHz)
1,5
1,0
S0
A0
A1
Transmission coefficie
n
ts
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 12. (a) Reflection and (b) transmission coefficients under A0 incident for aluminum plate with inclined
discontinuity d1=s=h=1mm and d2=2mm
1,0
S0
A0
A1
on
ts
ect
i
fficie
0,5
n
R
e
co
e
f
l
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
S0
A0
ansmission coefficients
0,5
0,0
r
T
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 13. (a) Reflection and (b) transmission coefficients under S0 incident for aluminum plate with inclined
discontinuity d1=2mm and d2=s=1mm; h= -1mm
1,0
S0
A0
A1
on
coefficients
ect
i
0,5
f
l
R
e
0,0
,5
0 0,5 1 1
(a) Frequency (MHz)
1
S0
A0
ts
Transmission co
e
fficie
0,5
n
0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 14. (a) Reflection and (b) transmission coefficients under A0 incident for aluminum plate with inclined
discontinuity d1=2mm and d2=s=1mm; h= -1mm
1,0
30 °
45°
60 °
on
ect
i
cient
s
0,5
R
e
coef
f
i
f
l
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
30 °
45°
60 °
Transmission coef
f
i
cient
s
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 15. (a) Reflection and (b) transmission coefficients for S0 mode under S0 incident for aluminum plate with
inclined discontinuity d1= h=1mm; d2= 2mm; and s=0.57735mm, 1.0mm, 1.73205mm
1,0
30 °
45°
60 °
tio
n
n
ts
fle
fficie
c
0,5
co
Re
e
0,0
,5
0 0,5 1 1
(a) Frequency (MHz)
1,0
30 °
45°
60 °
Transmission coef
f
i
cient
s
0,5
0,0
0 0,5 1
(b) Frequency (MHz)
1,5
Fig. 16. (a) Reflection and (b) transmission coefficients for A0 mode under A0 incident for aluminum plate with
inclined discontinuity d1= h=1mm; d2= 2mm; and s=0.57735mm, 1.0mm, 1.73205mm
1,0
30 °
45°
60 °
n
tio
cient
s
c
fle
0,5
Re
coef
f
i
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
Transmission coef
f
i
cient
s
0,5
30 °
45°
60 °
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 17. (a) Reflection and (b) transmission coefficients for S0 mode under S0 incident for aluminum plate with
inclined discontinuity d1=2mm; h= -1mm; d2= 1mm; and s=0.57735mm, 1.0mm, 1.73205mm
1,0
30 °
45°
60 °
on
ect
i
cient
s
0,5
R
e
coef
f
i
f
l
0,0
0 0,5 1 1
(a) Frequency (MHz)
,5
1,0
30 °
45°
60 °
Transmission coefficients
0,5
0,0
0 0,5 1 1
(b) Frequency (MHz)
,5
Fig. 18. (a) Reflection and (b) transmission coefficients for A0 mode under A0 incident for aluminum plate with
inclined discontinuity d1=2mm; h= -1mm; d2= 1mm; and s=0.57735mm, 1.0mm, 1.73205mm
Conclusions: The HBEM can be used to treat the Lamb wave scattering onto discontinuity in plate. Serial
numerical results show the complex for Lamb wave reflection and transmission at discontinuity in plate. According
to the calculation, both of S0 and A0 mode Lamb wave have a good penetration through step discontinuity and
inclined discontinuity at low frequency, and the sensitivity of inclined degree is also not large (up to 60°). But they
have different range of frequency. On other hand, the reflection and transmission coefficients for any mode under
both of S0 mode or A0 mode incident are strongly dependent on frequency, particularly at high frequency. The
present HBEM is capable of optimizing the Lamb waves and the range of frequency for NDE of plate or pipe with
different discontinuity.
References:
[1] K.F. Graff, Wave motion in elastic solids, Dover Publications, New York(1991)
[2] A.H. Harker, Numerical modeling of the scattering of elastic waves in plates, J. Nondestructive Eval.1984, 4(2):
89-106
[3] D.N.Alleyne, P.Cawley, The interaction of Lamb waves with defects. IEEE Transactions on Ultrasonics,
Ferroelectrics and Frequency Control, 1992,39(3): 381-397.
[4] Y.Cho, J.L.Rose, A boundary element solution for a mode conversion study on the edge reflection. The Journal
of the Acoustical Society of America, 1996, 99: 2097-
2109.
[5] Y.Cho, J.L.Rose, An elastodynamic hybrid boundary element study for elastic guided wave interactions with a
surface breaking defect. International Journal of Solids and Structures, 2000, 37: 4103-
4124.
[6] A.Demma, The Interaction of guided waves with discontinuities in structures, PhD thesis, University of
London, 2003
[7] R.Huang, Z.Liu, Lamb wave scattering by defects in plate, in 8th International Workshop on Modern Acoustics
- NDE, Taylor & Francis, London and Washington(2000)
[8] Y.Zhang, J.Shen, C.Ying, The reflection of the Lamb Wave by a free plate edge: Visualization and theory,
Materials Evaluation, 1988, 46: 638-641.
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