SCATTERING OF THE SHO MODE FROM GEOMETRICAL
DISCONTINUITIES IN PLATES
A. Demma, P. Cawley and M. J. S. Lowe
Department of Mechanical Engineering, Imperial College, London, SW7 2BX, UK
ABSTRACT. This paper investigates the interaction of the SHO elastic wave mode with
discontinuities in plates and explains some of the physical phenomena involved in this interaction. The
significance of the non-propagating modes in the scattering from thickness steps and notches is
examined. A method for predicting the reflection from notches using knowledge of the reflection from
steps is proposed. The limits of validity of this procedure are also investigated.
INTRODUCTION
Guided waves can be used for rapid NDT of plates and pipes [1,2]. One of the main
issues in practical testing of structures is the choice of the mode which is most suitable for
the given application. SHO can be very useful in practical testing because there is no mode
conversion when there is a long defect aligned in the same direction of the displacement.
The SHO mode is relatively easy to excite and it has the characteristic of being nondispersive. SH waves in plates are also very similar to torsional waves in pipes so the
results obtained for SH waves in plates can usually be extended to torsional waves in pipes.
This is of great interest because the torsional waves are established in use in practical pipe
testing [1].
The subject of this paper is to study the reflection of the SHO mode from thickness
changes and notches in plates. In fact notches are the simplest representation of corrosion
defects in plates and they simulate the crack case when the axial extent of the notch tends
to zero. The interaction of guided elastic waves in plates and pipes with discontinuities
such as thickness steps [3-5] and notches [3, 6-10] has been studied by previous
researchers. The aim of this paper is to investigate the possibility of deriving the reflection
from notches from knowledge of the reflection at thickness steps.
The paper first reviews the properties of the SH guided waves, and then gives the
background for the Finite Element (FE) modelling which was employed in the studies. An
analysis of the scattering of the SHO mode at a step down and at a step up is then presented.
This shows and compares results both for a simplified analytical approach and for accurate
FE predictions. Finally, the scattering behaviour at a notch is predicted by superposition of
the analytical results, and again these are compared with accurate FE predictions.
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
149
SH WAVES IN PLATES
In general two different families of guided waves can exist in a free isotropic plate:
Lamb waves and SH waves. In SH waves the particle displacement is perpendicular to the
plane of propagation. Figure 1 shows the phase velocity dispersion curves for SH waves in
a steel plate in vacuum. The curves scale linearly with frequency and thickness so that the
use of the frequency-thickness abscissa allows these curves to be used for plates of any
thickness. The software Disperse [11] has been used to trace the dispersion curves. SH
waves can be either symmetric or antisymmetric with respect to the mid-plane of the plate,
the modes with even counter variable being symmetric and the modes with odd counter
variable being antisymmetric. The fundamental SH mode, present at zero frequency, is the
SHO (symmetric) mode. This mode is non-dispersive at all frequencies and its phase
velocity is the bulk shear velocity. This study considered SHO incident in all cases and the
frequency thickness range was 0-0.55 MHz-mm so SHO was the only propagating mode.
Figure 2 shows the attenuation dispersion curves for non-propagating modes in a plate
in vacuum. The non-propagating modes describe localized vibration and their attenuation
shows the decay of their amplitudes with distance along the plate. An infinite number of
non-propagating modes exist at any frequency; in Figure 2 only the first five modes are
shown. It is clear from Figure 2 that the attenuation increases with mode number at a
defined frequency-thickness and it decreases with frequency-thickness for a given mode.
The consequence of a stronger attenuation is that the local vibration decays in a shorter
distance.
2
4
6
Frequency-Thickness (MHz-mm)
FIGURE 1. Phase velocity dispersion curves for SH waves in a plate in vacuum.
150
1.5 10
1.25
110 7.510 510
2.5 10 -
0
2
4
6
Frequency-Thickness (MHz-mm)
FIGURE 2. Attenuation dispersion curves for SH non-propagating modes in a plate in vacuum.
(a)
(b)
RBi
FIGURE 3 Schematic diagram of the set-up used for the FE analysis on (a) thickness steps and (b)
rectangular notches.
FINITE ELEMENT ANALYSIS
An FE time marching procedure was used to model the SH wave propagating in a
plate with a geometrical discontinuity (more details on the model used for this specific case
can be found in [12]). It was possible to vary both the axial extent and depth of a defect. In
all cases the defects in the plate were infinitely long in the direction normal to the
151
propagation. These geometrical discontinuities were introduced at some distance from the
end of the system. The SHO wave was excited by prescribing a windowed toneburst
displacement input at one end of the plate. Since the SHO mode shape is constant with
frequency, pure mode excitation was obtained by simply imposing the mode shape at the
center frequency [13].
Two types of discontinuity were modeled:
1. thickness step (see Figure 3a). This was used to study the scattering from thickness
changes in plates. From this model we derived modulus and phase of both reflection and
transmission at a thickness step.
2. rectangular notch (see Figure 3b). This was used to validate the results obtained for the
reconstruction of the reflection from a rectangular notch by superimposing the reflections
from down and up steps.
SCATTERING FROM THICKNESS STEPS
The scattering produced by a thickness step in a structure has already been studied [35]. The effect of the non-propagating modes on the reflection and transmission coefficient
at a step is shown in other work by the authors [12]. There it was confirmed that it is
possible to obtain a good approximation for the reflection and transmission coefficients
using a finite number of non-propagating modes, the accuracy increasing with the number
of modes included in the computation.
If we assume that only the SHO mode gives a contribution to the scattering at the step,
the reflection and transmission coefficient would only be dependent on the impedance
difference at the discontinuity.Under this assumption it is possible to derive the modulus of
reflection and transmission coefficient at the thickness step down (start of the notch in
Figure 3b) and at the thickness step up (end of the notch in Figure 3b) by using the
following expressions:
RAl = ——
((1)
)
in which a = — , where ti is the thickness of the plate before the start of the notch and t2 is
*i
the thickness at the notch (see Figure 3b). The transmission coefficient past the step would
be:
a
(2)
From a step up in plate we would have:
(3)
In this simplified model, the phase of the reflection and transmission coefficient is
zero in all cases except in the step up reflection case where a 180 degrees phase shift is
152
expected due to the increase of impedance (this is taken into account in the sign reversal in
the numerators between Equation (3) and Equation (1)).
Figures 4a and 4b show the Finite element predictions for the modulus and phase of
the reflection at a step down when the step is 50% of the thickness; they also show the
results obtained using the approximation Formula (1). The phase shift was derived by
comparing the phase of the reflection and transmission in the plate with a step with a
reference signal that had traveled along the same path length in a plain plate. The
difference between the approximation and the results obtained using FE is due to the effect
of the non-propagating modes (this is discussed in detail in [12]). When using only the
SHO mode to fulfill the boundary conditions at the step it is not possible to take into
account any frequency effect due to the non-propagating modes and therefore both the
modulus and phase of the reflection coefficient are found not to vary with frequency. The
approximation for the modulus of the reflection using Equation (1) is good enough for
practical application, in fact the difference between the two curves shown in Figure 4a is
just a few percent in the frequency-thickness range 0-0.55 MHz-mm. The phase difference
is more critical. For example the phase of the reflection at 0.3 MHz-mm obtained using FE
is about 0.1 radians (see Figure 4b) and this has a non-negligible effect on the interference
behavior which occurs at a notch (as we will show later in this paper).
(a)
0.345
^ . ••'
fitinp0ui
0.34
.....-••••-""
0.335
*2 0.33
5
35
0.325
0 32
025
f
-
0
0.1
0.2
0.3
0.4
Frequency-Thickness (MHz-mm)
0.5
————————————————————— (trr
0.2 -
I 0.15
M
&
tf
0
0.1
0.05
0 .
-0.05
• .• . • • • • *
...-•••••"'"
0.1
0.2
0.3
0.4
Frequency-Thickness (MHz-mm)
0.5
FIGURE 4. Modulus (a) and phase angle (b) of reflected fundamental SHO mode from a thickness step
down of 50% of total thickness. The plot shows the results obtained with FE (dots) and approximated using
Equation (1) (solid line).
153
SCATTERING FROM NOTCHES
We propose here a new method to approximate the reflection from notches with
varying axial extent using the results obtained at a step down and a step up. The basic idea
of our new method is summarized in Figure 5. The start of a rectangular notch is a decrease
in the thickness of a plate (thickness step down) and the end of the notch is an increase of
the thickness (thickness step up). Therefore it is intuitive to think that the notch case is
equivalent to the superposition of a thickness step down followed by a thickness step up.
The use two step cases (step down and step up) instead of a series of notch cases could
potentially enable a great reduction of computational time.
The total reflection from a notch is given by the sum of the consecutive reflections at
the notch. The first reflection is given by Equation (1). The second reflection is due to the
signal transmitted through a down step (start of notch), reflected at an up step (end of
notch) and transmitted back through an up step (start of notch). The amplitude of the
second reflection is:
R
A2 = TA\ • Rm • Tm
(5)
2L
and its time delay is — where L is the axial extent of the notch and V is the SHO velocity.
The amplitude of the jth reflection is given by:
Rj=TM.(RBl)^.TBl
(6)
with time delay - — . The total reflection is:
If we neglect the phase shift due to the contribution of the non-propagating modes in
the scattering phenomenon, the behavior of the reflection for a 50% depth rectangular
notch with varying axial extent is described by the dashed line in Figure 6. In this
simulation we considered the first four reflections (the amplitude of the fourth reflection
has already decayed to a value of the order of 10~6 of the input amplitude in this case). In
order to verify the accuracy of the prediction we predicted the reflection from rectangular
notches with varying axial extent using Finite Elements. The results for the Finite Element
simulations are also shown in Figure 6 (dots).
The reflection coefficient at a rectangular notch can be seen to have a cyclical
behavior with axial extent and it has a maximum of reflection at about a quarter of a
wavelength and a minimum at about half wavelength as seen in previous work [6-7,10].
From the practical point of view it is important to obtain accurate results at relatively low
axial extent because a notch with small axial extent closely represents typical localized
corrosion. In fact the approximation that we obtain using Equation (7) has the limitation of
not taking into account the frequency dependent phase shift at the steps. The results for the
reflection from a notch when using the phase shift obtained at the down and up steps are
also shown in Figure 6 (solid line).
It is clear from these results that the simulation computed using the phase information
gives a very good approximation of the FE predictions. Therefore it is possible to use this
approximation function to study the effect of axial extent on the reflection from notches.
154
IT
4
FIGURE 5. Explanation of combination of step cases to obtain notch case.
Another interesting feature is the result that we obtain for the crack case. The FE
prediction for a zero axial extent notch is not very well approximated using our
superposition method. This is because, when combining the signals coming from the steps,
we assumed that only SHO was present. In fact when the opposite sides of the notch are
very close this assumption is no longer valid and the non-propagating modes localized at
the two discontinuities (step down and step up) interact with each other causing a change in
the interference pattern. In principle, the same observation is valid for any low axial extent
cases. However, in practice, as is clear from Figure 6, the approximation function at low
axial extent (e.g. the point at 1.25 % of the wavelength) closely matches the prediction
from the Finite Element analysis. This is due to the fact that the non-propagating modes
rapidly decay away from the discontinuity, therefore the approximation using the phase
shift is not valid for the crack case but it can be used to approximate a notch characterized
by low axial extent.
o
S
b
i
i
20
40
60
80
100
Axial extent (% of wavelength)
FIGURE 6. Variation of reflection ratio with axial extent of the notch. Results are for a plate with SHO at
0.55 MHz-mm incident on a notch with 50% thickness depth. The dots indicate the FE results obtained for
the rectangular notch case. The dashed line predicts the notch behavior using the simplified theory (no phase
shift) and the solid line reproduces the notch reflection behavior using the complete theory with the phase
shift information.
155
CONCLUSIONS
In this paper we studied the effect of thickness steps and notches in plates. Both the
modulus and the phase of the reflection at a thickness step when SHO is incident are
frequency dependent functions.
We here demonstrated that the reflection from rectangular notches can be derived by
superimposing the reflection at a thickness step down (modeling the start of the notch)
followed by a thickness step up (modeling the end of the notch). Accurate prediction of the
phase of the reflection from steps is vital to obtain good accuracy of the notch reflection
coefficient. The superposition method is valid down to low axial extents but it loses
accuracy for the crack case.
REFERENCES
1. Alleyne D.N., Pavlakovic B., Lowe M. and Cawley P., Insight 43, 93 (2001).
2. Wilcox P., Lowe M. and Cawley P., "An EMAT array for the rapid inspection of large
structures using guided waves.", in this proceedings.
3. Koshiba M., Hasegawa K., and Suzuki M., IEEE transactions on ultrasonics,
ferroelectrics and frequency control 34, 461 (1987).
4. Ditri J., J. Acoust. Soc. Am. 100, 3078 (1996).
5. Engan H., J. Acoust. Soc. Am. 104, 2015 (1998).
6. Lowe M. and Diligent O., J. Acoust. Soc. Am., Ill, 64 (2002).
7. Lowe M., Cawley P., Kao J.Y., and Diligent O., "Low frequency reflection
characteristics of the AO Lamb wave from a rectangular notch in a plate.", in press in J.
Acoust. Soc. Am.
8. Rose J. and Zhao X., Mater. Eval. 59, 1234 (2001).
9. Alleyne D.N., Lowe M. and Cawley P., J. Appl. Mech. 65, 635 (1998).
10. Demma A., Cawley P., Lowe M, and Roosenbrand A., "The reflection of the
fundamental torsional mode from cracks and notches in pipes." Submitted to J. Acoust.
Soc. Am. (2002).
11. Pavlakovic B., Lowe M., Alleyne D. and Cawley P., in Review of Progress in QNDE,
Vol.16, eds. D.O. Thompson and D.E. Chimenti (Plenum Press, New York, 1997),
p.185.
12. Demma A., Cawley P. and Lowe M., "Scattering of the fundamental SH mode from
steps and notches in plates." Submitted to J. Acoust. Soc. Am. (2002).
13. Pavlakovic B., "Leaky guided ultrasonic waves in NDT", Ph.D. thesis, University of
London, 1998.
156