THE SCATTERING OF ULTRASONIC GUIDED WAVES IN PARTLY
EMBEDDED CYLINDRICAL STRUCTURES
T. Vogt, M. J. S. Lowe, and P. Cawley
Department of Mechanical Engineering, Imperial College, London, SW7 2BX, UK
ABSTRACT. When a waveguide surrounded by vacuum enters an embedding material, guided waves
are scattered at this entry point due to the change in the surface impedance. This paper is concerned
with the numerical modelling of the entry reflection of the fundamental longitudinal mode, L(0,l), in
cylindrical waveguides. An approach based on modal analysis is used to characterize this reflection.
It is validated against Finite Element modeling, very good agreement being obtained. As an example
application, the cure monitoring of epoxy resins is discussed.
INTRODUCTION
In a previous paper given at the QNDE conference the authors reported on waveguide
techniques to monitor the cure process of epoxy resins [1], One of the techniques is based
on the measurement of the reflection of the fundamental longitudinal mode, L(0,l), from the
point where a free wire waveguide enters a surrounding epoxy resin. This entry reflection
is due to the change in surface impedance. As a rule of thumb, the larger the change in
the surface impedance, the larger is the magnitude of the entry reflection. The nature of the
reflection of L(0,l) was examined using Finite Element (FE) modeling. It was found that,
in the range of 0-0.3 MHz mm, the entry reflection is a strong function of the epoxy shear
velocity, and only minimally depends on the epoxy longitudinal velocity. Thus, during cure
of an epoxy resin, where the epoxy shear velocity increases due to the cross-linking of polymer chains, a characteristic increase in the magnitude of the entry reflection can be observed.
However, the computational analysis using FE modeling can be tedious, particularly when
analysing the entry reflection as a function of the material properties of the embedding material. Also, modeling small epoxy shear velocities, representing the early stages of the cure
cycle, is not efficient in FE because a small element size and time step is required. For these
reasons, an alternative approach based on modal analysis was developed. The basic ideas
of this approach are described in the following sections, and the results obtained using this
method are compared to FE modeling. Then, its application to the cure monitoring of epoxy
resins is demonstrated.
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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SCATTERING ANALYSIS
Background
Consider the scattering problem as depicted in Fig. 1. A longitudinal mode is incident
from the free section of the waveguide, and is reflected and scattered due to mode conversion
at the point where the waveguide enters an embedding material. The waveguide modes in the
free part of the waveguide will be referred to as the "vacuum modes", whereas the modes in
the embedded part will be referred to as the "embedded modes".
The particle velocity, v, and stress, T, of a waveguide mode in a cylindrical structure
can be described by the following expressions:
where CO is the circular frequency, kn is the complex wavenumber of mode n in the propagation direction, and v is the angular wavenumber. For longitudinal modes, v = 0. Since by
geometry, the incident longitudinal mode can only be scattered into longitudinal modes, the
circumferential order is omitted in the remainder of this paper. The variation of the modal
fields in the plane perpendicular to the propagation direction z is determined by the radial
distribution functions, v^(r) and T^(r), respectively. They are in general complex functions,
and are referred to as the mode shapes of the waveguide mode.
In modal waveguide analysis, one makes use of the fact that an arbitrary field distribution can be expanded into so-called normal waveguide modes. These normal modes have to
fulfill two main conditions. Firstly, the set of modes has to be complete, which means that
they are sufficient, by superposition, to describe any arbitrary field distribution. Secondly,
the modes must be orthogonal, which means that the expansion of an arbitrary field into
normal modes is unique. Mode orthogonality for layered plate waveguides was treated by
Auld [2], and was generalized to cylindrical waveguides by Ditri [3], The real orthogonality
relationship for two modes n and m in a free waveguide is given by the following expression:
vn-rYm-vm-Tn}zdS = Q
forn^-m,
(3)
u
where S is the cross-section of the waveguide. Here we use the negative sign in "-m" to label
a mode m propagating in the negative z-direction. We refer to modes whose energy propagates
™0
incident mo&
"1
•<
^^*
waveguide
V
scattered modes
FIGURE 1. Schematic of the problem considered in this paper. A propagating longitudinal mode incident from
the free section of the bar is scattered at the interface at z = 0 due a change in surface impedance.
222
in the positive axial direction as "forward" modes, and those whose energy propagates in the
negative axial direction as "backward" modes. Non-propagating and inhomogeneous modes,
however, do not propagate energy in any direction. One can imagine these modes as local
vibrations, which exist only at waveguide discontinuities such as the edge of a plate or the
surface discontinuity considered in this work. In this case, the equivalent to a forward mode
is a non-propagating or inhomogeneous mode which is attenuated in the positive z-direction.
The embedded modes are in general leaky modes, and there exist proofs of orthogonality in the literature for leaky layered waveguides [4, 5]. However, it has to be mentioned
that it is usually assumed that the wavenumber in the propagation direction is either purely
real or purely imaginary, or that the fields vanish far away from the interface. This is not
necessarily true, as leaky guided waves have a complex axial wave number. If they have a
purely real wavenumber, they are essentially non-leaky modes. Nevertheless, orthogonality
of these modes is assumed in this paper (see Ref. [6] for a detailed discussion).
It is convenient to define the quantity Qmn according to
Qmn =
"4
5
and to normalize the fields such that
Qm-m = I(5)
In fact, propagating modes with purely real wavenumbers are power normalized by this expression, since
G m - m = f l » m = l,
(6)
where Pmm represents the power flow of mode m in the axial direction (see, for example, [2]).
Modal solution
We now describe the particle velocity and stress field in the free part of the waveguide,
denoted by "1", as a superposition of the forward and backward vacuum modes, m. If the
amplitudes of the incident modes are denoted by a's, and the scattered mode amplitudes are
denoted by £'s, the fields can be written as
Vi = £tfmv m + £_ m v_ m ,
(7)
m
T, = £amTm + fc_ m T_ m ,
(8)
m
where the sums run over positive integers only. Similarly, it is now assumed that the fields
in the embedded part of the cylinder, denoted by "2", are composed of the embedded modes,
m'
T
2 = £<vV
m'
Note that there is no incident mode from the embedded section. In principle, the sums in
equations (7) - (10) extend over an infinite number of modes. However, for the numerical
calculations, these sums have to be truncated after a certain mode M .
First, the boundary conditions are applied on the interface between the free and the
embedded section:
Vl
= v2
Tj z = T2 z.
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(11)
(12)
Substituting Eqs. (7) - (10) into the boundary conditions yields:
(13)
(14)
m'
Multiplying equation (13) by T_ n and equation (14) by \_n, and subtracting (14) from (13)
yields:
,). (15)
m'
Then, by integrating over the cross-section of the waveguide one makes use of the orthogonality relation (3), and some terms in equation (15) vanish. Noting that the fields are normalized
according to equation (4), it follows that
(16)
m'
This can be written in matrix form:
"V
«2
— -
-aM-
"•
Q-IM1
Q-2M'
M
by
'"
Q-MM'_
-bM>-
2-H'
2-12'
'"
2-21'
2_22'
Q-MV
Q-M21
(17)
In an abbreviated notation, this is
(18)
so
Secondly, multiplying equation (13) by Tn and equation (14) by vn, and subtracting (14) from
(13) yields:
^am (vw • Tn - vn • Tw) + J>_w (v_ m - Tn - vn - T_ m ) m
m
Again, this is integrated over the cross-section of the waveguide, and some terms vanish due
to orthogonality. One is left with the following equation:
(21)
m'
Replacing n by m, this can be abbreviated to give
(22)
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Recalling equation (19), the coefficients b_m are calculated as follows:
The scattered and the incident mode amplitudes can now be collected in column vectors,
and they are related to each other by the scattering matrix, [S]. Equations (19) and (23) are
combined to give the solution to the scattering problem:
_ I
~
\Qmm'\\Q-nnm'
"1
The scattering sub-matrices are M x M matrices, whereas the full scattering matrix [S] is an
M x 2M matrix.
The scattering amplitudes are here calculated using a normalization procedure for the
modal fields according to equation (4). For the interpretation of the results it is convenient
to state the scattering amplitudes in terms of the scattered power. The power normalized
scattering coefficients, denoted by capital letters Am, B_m, and Bm, respectively, are obtained
from the amplitudes calculated here, am, b_m, and bm, respectively, by multiplying by the
square-root of the power of the corresponding mode, for example:
B'm = b'm^/P^,.
(25)
Note that for non-propagating and inhomogeneous modes, Pmm — 0, and therefore no power
is carried away from the scattering region by these modes.
RESULTS
The calculation of the elements of the scattering matrix requires the knowledge of the
radial distribution functions of the modes which are included in the solution. The software
DISPERSE [7], which was developed at Imperial College, has been used to trace the dispersion
curves and to obtain the particle velocity and stress mode shapes of each mode. These mode
shapes are used to calculate the elements of the scattering matrix. Typically, a resolution of
200 points over the radius of the waveguide was found to give a reasonable accuracy for the
calculation of the integrals.
The results from the modal solution for L(0,l) as the incident mode were compared
to those obtained with FE analysis in the range 0-0.3 MHz mm. For a discussion of the
properties of the L(0,l)-mode, and for a description of the FE method, the reader is referred
to Ref. [1]. However, it should be noted here that in this frequency-radius range, L(0,l) is
the only propagating longitudinal mode in the free part of the cylinder. All higher order
longitudinal modes are inhomogeneous. The material properties for the following analysis
are given in Tab. 1, unless otherwise stated.
Figure 2 shows the reflection coefficient for a steel cylinder partly embedded in epoxy
resins with different shear velocities. The agreement between the modal solution and the
FE analysis is very good. To obtain this agreement, it was sufficient to only include the
incident and reflected vacuum L(0,l)-mode and the transmitted embedded L(0,l) mode in the
TABLE 1. Material parameters used for the analysis.
Material Long. Velocity [m/s] Shear Velocity [m/s] Density [kg/m3]
"Steel
5960
3260
7932 ==
Epoxy_______2610_____
1000 _______1170
225
Shear velocity
1.2m/ms
l.Om/ms
0.8 m/ms
Frequency-Radius [MHz mm]
FIGURE 2. Modulus of the amplitude reflection coefficient of L(0,l) in a steel bar for different values of epoxy
shear velocity. All other material properties of the epoxy are according to Tab. 1.
modal solution. In general, even though potentially an infinite number of non-propagating
and inhomogeneous modes is involved in the scattering process, it was found that only a
minimal number is needed to correctly model the entry reflection coefficient. It should also
be noted that the reflection coefficient depends on the product of frequency and the radius of
the waveguide [6].
APPLICATION TO CURE MONITORING
As mentioned before, the shear bulk velocity of an epoxy resin increases during cure
until the process is complete. At a fixed frequency, the mode shapes can be determined as
a function of the epoxy shear velocity, or indeed any other variable of interest. Thus, the
reflection coefficient can be determined as a function of the epoxy shear velocity in a single
calculation. This is a significant advantage over the FE method, since several models with
different epoxy shear velocities have to be run, possibly with different element sizes and time
steps.
As an example, Fig. 3 shows the dependence of the L(0,l) reflection coefficient at
0.05 MHz mm on the shear velocity of an embedding epoxy resin as calculated using modal
analysis. As expected, with increasing epoxy shear velocity, the reflection coefficient increases. The modal analysis is also capable of determining the reflection coefficient at low
shear velocities where the FE method is inefficient.
In order to demonstrate the validity of the results, experiments were conducted monitoring the reflection coefficient of L(0,l) in a 0.5mm radius steel wire during cure of the
commercial epoxy resin Araldite 2013 (Ciba Specialty Chemicals). For the experimental
procedure, the reader is referred to Ref. [1]. The epoxy was cured at room-temperature in a
bulk sample of 50 ml volume, and measurements were taken at intervals of 5 minutes.
The cure monitoring curves obtained from these experiments at different frequencies are plotted in Fig. 4. As expected, the reflection coefficient is higher at lower frequencies. The reflection coefficient of L(0,l) at the end of cure at 0.1 MHz, which corre-
226
0.6
s
Epoxy Shear Velocity [m/ms]
1.5
FIGURE 3. Predicted entry reflection coefficient of L(0,l) as a function of the epoxy shear velocity at a
frequency-radius product of 0.05 MHz mm (modal solution). The material properties are given in Tab. 1.
Curing Time [min]
500
FIGURE 4. Cure monitoring curves obtained in Araldite 2013 with the reflection coefficient method at different
frequencies.
227
spends to a frequency-radius product of 0.05 MHz mm, was R = 0.451. With a density of
p — 1170kg/m3, this corresponds to an epoxy shear velocity of cs — 1.06 m/ms. This agrees
well with the shear velocity data presented by Freemantle et al in Ref. [8].
CONCLUSIONS
A numerical method based on modal analysis to calculate scattering coefficients in
partly embedded cylindrical structures has been developed. This method is an attractive alternative to Finite Element modelling for two reasons. Firstly, it can model materials with
low shear velocities more efficiently, and secondly, it potentially reduces the computation
times when considering a number of different material properties. In particular, the reflection
of L(0,l) in steel cylinders partly embedded in epoxy resins was discussed and successfully
applied to the cure monitoring of epoxy resins.
REFERENCES
1. Vogt, T., Lowe, M. J. S. and Cawley, P., in Review of Progress in Quantitative NDE,
Vol. 20, edited by D. O. Thompson and D. E. Chimenti, American Institute of Physics,
New York, 2001, pp. 1642-1649.
2. Auld, B. A., Acoustic Fields and Waves in Solids, Vol. 2, Krieger Publishing Company,
Malabar, Florida, 1990.
3. Ditri, J. J., J. Acoust. Soc. Am. 96, 3769-3775 (1994).
4. Murphy, J. E., Li, G. and Chin-Bing, S. A., J. Acoust. Soc. Am. 96, 2313-2317 (1994).
5. Briers, R., Leroy, O., Shkerdin, G. N. and Gulyaev, Yu. V, /. Acoust. Soc. Am. 95,19531966 (1994).
6. Vogt, T., Lowe, M. J. S. and Cawley, P., "The scattering of guided waves in partly embedded cylindrical structures", sumitted for publication to J. Acoust. Soc. Am., May 2002.
7. Pavlakovic, B. N., Lowe, M. J. S., Alleyne, D. N. and Cawley, P., in Review of Progress
in Quantitative NDE, Vol. 16, edited by D. O. Thompson and D. E. Chimenti, Plenum
Press, New York, 1997, pp. 185-192.
8. Freemantle, R. J, and Challis, R. E., Meas. Sci. Tech. 9, 1291-1302 (1998).
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