A GUIDED WAVE TECHNIQUE FOR THE CHARACTERIZATION
OF HIGHLY ATTENUATIVE VISCOELASTIC MATERIALS
F. Simonetti and P. Cawley
NDT Group, Department of Mechanical Engineering, Imperial College, Exhibition
Road, London. SW7 2BX, United Kingdom.
ABSTRACT. The measurement of the acoustic properties of highly attenuative materials such as
bitumen is very difficult. One possibility is to use measurements of the extent to which filling a
cylindrical waveguide with the material affects the dispersion relationship of the cylinder. Under the
hypothesis of linear viscoelasticity and on the basis of a monochromatic approach, the bulk shear
velocity and the bulk shear attenuation spectra have been derived from the measured torsional
dispersion curves.
INTRODUCTION
The ultrasonic non-destructive testing of metallic structures in which viscoelastic
components are present, such as pipelines coated with bitumen [1], requires the acoustic
properties of the viscoelastic materials to be known. A number of techniques have been
developed in order to measure material velocities and attenuation. Reflectometry [2] and
transmission [3] methods, for example, are based on the measurement of the reflection and
transmission coefficient of a sound beam incident on the sample to be characterised. These
methods are easy to set up, but the measurements can be affected by several factors,
including beam spreading, transducer coupling variations, non parallelism of sample faces,
and magnification of bias and variance errors in the time trace of the consecutive echoes
emitted from the back of the specimen [4]. Moreover in the case of highly lossy materials,
the transmitted signal could be too weak to be detected. The amplitude spectrum method
[5] and the phase spectrum method [6] are based on the properties of the Fourier transform
of the signal reflected from a sample. These methods exhibit the same limitations as the
former techniques.
The limitations of the previous methods can be overcome by using guided waves.
Torsional waves in rods submerged in a Newtonian liquid of known density have been
employed to measure the viscosity of the liquid [7]. The method is extremely rapid and
ideal for measurement on line and in real time. On the other hand, the technique cannot be
used for viscoelastic materials as the guided wave attenuation itself cannot provide the
unknown material velocity and attenuation unless a frequency dependence for the acoustic
properties is assumed.
The approach presented in this paper is based on the dispersion characteristics of
guided waves propagating in a hollow cylindrical wave-guide filled with the unknown
viscoelastic material. Since the attenuation and the velocity of the guided waves are related
to the acoustic properties of the inner core, by measuring the dispersion of these waves the
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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properties of the viscoelastic material can be determined. The first important feature of the
method is that the guided wave attenuation is only a consequence of the material damping
within the inner core. If the core was perfectly elastic no guided wave attenuation would
occur (assuming the wave-guide to be elastic), in contrast with the case of a rod embedded
in an elastic material. Moreover, excellent geometry control can be achieved, since the
wave-guide works as mould. As laser interferometry can be used for the detection of the
propagating modes, transducer coupling problems can be avoided.
FORMULATION
Under the hypothesis of linear viscoelasticity the Fourier transformed equations of
motion for a homogeneous and isotropic medium are
0,
(1)
where u is the Fourier transformed displacement vector, p is the density and V the threedimensional differential operator (The linearity of eq. (1) follows from the hypothesis of
linear viscoelasticity.) The complex frequency-dependent functions A* and /x* are related to
the relaxation functions of the material A(t) and
X =^+ico(f}ei(0tdt,
"dt,
(2)
(3)
where AO and /i# are the asymptotic values of the relaxation curves.
By using the Helmholtz decomposition the displacement field can be expressed as a
sum of the gradient of a compressional scalar potential, 0, and the curl of a equivoluminal
vector potential, y,
u = V(|) + Vx\|/.
(4)
By substituting (4) in (1) the equations of motion break down into two uncoupled equations
for the two unknown potentials
a2LV 2<j) + ico</> = 0 ,
Q,
(5)
(6)
where the frequency-dependent complex velocities a^ and as are given by
m
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For an unbounded space the solution to (5,6) can be thought of as a superposition of
longitudinal and shear harmonic waves of the form
0,i|/ = A L y k '* x ,
(9)
where A^s are arbitrary complex constants, x is the position vector and k^s are the complex
wavenumber vectors, which satisfy the secular equation
The harmonic waves propagate in the direction of the real part of kL>s at a phase velocity
CphL,s which depends on the modulus of the real part of kL,s according to
Along the direction of the imaginary part of k^s an exponential decay with the distance
occurs. The acoustic properties of a viscoelastic material are defined considering the case in
which the real and the imaginary parts of k^s are parallel. In this case the phase velocity is
referred to as the bulk velocity, cL>5, while the modulus of the imaginary part is the bulk
attenuation in Nepers per unit length. The complex velocities (7,8) depend on the acoustic
properties according to
OCr
2n
where (XL,S are the bulk attenuations in Nepers per wavelength. From (12) it follows that the
bulk properties of the material are frequency dependent. Moreover, through (12), (7,8) and
(2,3) these properties can be linked to the relaxation functions.
The propagation of stress waves along the filled tube can be studied by considering a
hollow, elastic and isotropic cylinder filled with the unknown viscoelastic material.
Moreover, due the axis symmetry of the system, cylindrical coordinates (r,0,z) are
appropriate (r,9,z, represent the radial, angular and axial positions respectively). For
torsional waves the scalar potential vanishes while the vector potential,^, is parallel to the
cylinder axis and satisfies (6). The generic solution for a cylindrical layer will be of the
form
V = [Vo(V)+ B0YQ(krr)]e-^e^z ,
(13)
where AO and Bo are arbitrary constants, kr and £ + / ^are the projections of ks along r and z
respectively. Moreover, £ represents the guided wave attenuation whereas £ is the
component of the wavenumber which is related to the phase velocity according to
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c ~-- •
(14)
Jo and YO are the Bessel functions of the first and second kind. While for the outer tube both
of the Bessel functions are present, in the inner core only Jo exists due to the singular
behavior of Y0 at the origin.
By imposing the condition (10) on the wavenumber vector ks, and by employing a
matrix technique based on satisfying the continuity of the stresses and displacements at the
interface between the inner core and the external tube, and the stress free condition over the
external surface of the elastic cylinder, the dispersion relationship can be obtained (see for
instance, [8]). Here, the solution to the dispersion equation has been calculated by using the
modelling software Disperse [9].
The guided wave attenuation and the phase velocity depend on a number of
parameters. However, for the purpose of this paper, it is relevant to consider only the
dependence on the acoustic properties of the inner core and on the frequency.
METHOD
For a given frequency, assigned geometry of the tube and known density of the
viscoelastic core, the guided wave attenuation and the phase velocity are functions of the
bulk shear velocity and the bulk attenuation only. Fig. la shows the contour plot of the
attenuation at constant frequency (50kHz) as a function of the bulk properties. Each curve
corresponds to a value of ^and provides all the values of as and cs which, at the prescribed
frequency and geometry, result in the specified value of guided wave attenuation. In other
words, each curve provides as as a function of cs and £
a .s = 6Ql \(c
r i >£}
S/'
1J
\(\^
/
While all the couples (as, cs) provided by (15) result in the same value of £ they do not
result in the same value of the phase velocity. A second relationship can be obtained from
1.4
1.4
110
1.3
^->s
4* 1-3
11.2
& 1.2
1.1
•B 1.1
1.0
g
3
0.9
<u
0.8
0.8
0.7
0.6
0.25
1.0
PQ
0.7
0.6
0.3
0.35
0.4
0.45
0.5
Bulk shear velocity (m/ms)
0.55
0.6
(a)
0.25
0.3
0.35
0.4
0.45
0.5
Bulk shear velocity (m/ms)
0.55
0.6
(b)
FIGURE 1. Contour plots at 50kHz for a 6.8mm inner radius, 0.7mm wall thickness filled copper tube and
0.97 g/cm3 core density: (a) guided wave attenuation (dB/m); (b) phase velocity (m/ms). The arrows indicate
the direction of increasing guided wave attenuation (a) and phase velocity (b).
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the contour plot of the phase velocity (Fig. Ib). It is possible to obtain a function that, for a
given value of the phase velocity, links as to cs and Cph
as=g2(cs,Cph).
(16)
In this case all the couples (as, cs) give the same phase velocity but different guided wave
attenuation.
At a given frequency, /#, the phase velocity, Cpho, and the guided wave attenuation,
£0, can be measured experimentally. As a consequence, the curve gi(as, £o) of the guided
wave contour plot and the curve g2(as, CPHO) the phase velocity contour plot are known. By
overlapping the two curves, the intersection provides the bulk acoustic properties as and cs
of the viscoelastic material at the prescribed frequency, /#. This comes from the fact that at
the intersection, as and cs result in the measured value of the guided wave attenuation, £0, as
the intersection point belongs to gi(as, &), and in the measured value CPHO of the phase
velocity as the intersection point belongs to g2(osst Cpho). Since the previous procedure can
be applied to any frequency, it follows that the dispersion curves of the viscoelastic material
(ccs(f) and cs(f)) can be determined over the frequency range of interest.
EXPERIMENTS
The experiments were performed on a 1m length copper tube (internal radius
6.8mm, wall thickness 0.7mm) filled with bitumen TML 24515 45/60. The torsional mode
was excited by means of two piezoelectric transducers clamped to the external surface of
the tube at one end as shown in Fig. 2. The transducers consisted of shear elements
mounted on a steel bulking mass [10] and oriented as shown in Fig. 2 to induce torsion.
The transducers were excited by a Banning windowed toneburst generated by a bespoke
waveform generator-power amplifier. The torsional mode was detected by a laser
interferometer (Sensor Head: Polytec OFV 512, Controller: Polytec OFV 3001) operating in
differential mode. The tangential displacements were measured by focusing the two beams
at ±30° with respect to the radial direction and by orienting the beams in the plane
perpendicular to the axis of the tube. The displacements were sampled along the tube axis
and stored in a PC after 100 averages.
Displacement and
velocity decoder
Digital
oscilloscope
IE
Waveform
generator
FIGURE 2. Experimental setup and measurement equipment.
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PC
Since the main goal of this paper is the measurement of the properties of the
unknown material over a wide range of frequencies, a relatively wide band excitation signal
with a number of cycles ranging from 3, for low frequencies, up to 5 for higher frequencies
was used. More cycles are needed at higher frequencies since the attenuation tends to be
higher and the use of more cycles improves the signal noise ratio and reduces the effects of
dispersion.
In order to work out the dispersion curves, one can observe that for a propagating
mode the Fourier transform of the signal at an arbitrary position, z, can be expressed as a
function of the Fourier transform at the origin, z = 0, multiplied by a suitable complex
exponential
)e-*zeCl* ,
(17)
where the argument of the exponential is the projection of the complex wavenumber along
the propagation direction. Furthermore, if the ratio R is defined as
,co
(18)
it follows that
(19)
(20)
where || || is the norm operator in the complex domain. Since the ratio R can be measured
experimentally, the guided wave attenuation can be determined through a linear
interpolation of the experimental, logarithmic distribution of \\R\\ versus the axial position.
The slope of the interpolation line gives the guided wave attenuation. The phase velocity is
worked out by employing a cosine interpolation of the measured real part of the ratio
/2/||/?||. The argument of the cosine function, co / Cph, is obtained by minimizing the square
root of the error between the experimental distribution and the cosine function.
BITUMEN DISPERSION CURVES
Here the general procedure previously described is applied to the characterization of
the bitumen TML 24515 45/60 acoustic properties. For this bitumen type the measured
density was 0.97 g/cm3. Fig. 3a shows the measured guided wave attenuation as a function
of frequency. The curve is obtained as a superposition of measurements performed at
different center frequencies and successively postprocessed according to the method
described in the previous section. Fig. 3b shows the experimental phase velocity for both
the empty and the filled tube. As an example, let us consider the frequency where the first
attenuation peak occurs (i.e. 40kHz, Fig. 3a). At this frequency the guided wave attenuation
is 127dB/m while the phase velocity is 2.093 m/ms (Fig. 3b). Fig. 4 shows the overlapping
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40
60
40
80
Frequency (kHz)
60
80
100
Frequency (kHz)
FIGURE 3. Experimental dispersion curves for a 6.8mm inner radius, 0.7mm wall thickness copper tube
filled with bitumen: (a) guided wave attenuation; (b) phase velocity.
of the guided wave attenuation and phase velocity contour plots at 40kHz. The intersection
of the curve at constant attenuation (£ = 127dB/m) and the curve at constant phase velocity
(Cph = 2093m/s) provides the values of the shear velocity, 0.43m/ms, and the bulk
attenuation, 1.35np/wl, of bitumen at 40kHz. It is interesting to note that the two curves
intersect each other almost perpendicularly which implies that the method is numerically
stable. The bitumen bulk shear velocity versus frequency is shown in Fig. 5 (a), and the
bulk shear attenuation against frequency is shown in Fig. 5 (b). The curves have been
derived by applying the former procedure to several frequencies (black squares). The trend
lines (solid lines) have been obtained by best fitting second order polynomials.
I««
2.00
1.75
1.50
1.25
I
1.00
§
1
(a)
10
Frequency (kHz)
*
0.75
0.50
0.3
0.35
0.4
0.45
0.5
0.6
0.55
(b)
Bulk shear velocity (m/ms)
OQ
FIGURE 4. Guided wave attenuation contour plot
(solid lines dB/m) and phase velocity contour plot
(dashed lines m/s) at 40kHz for a 6.8mm inner radius,
0.7mm wall thickness copper tube filled with bitumen
(p = 0.97 g/cm3).
10
Frequency (kHz)
80
FIGURE 5. Dispersion curves for bitumen TML
24515 45/60: (a) bulk shear velocity against
frequency; (b) bulk shear attenuation against
frequency.
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CONCLUSIONS
A novel technique for the characterization of highly attenuative viscoelastic
materials has been presented. It has been shown that by measuring the dispersion curves of
a tube filled with the unknown material the shear acoustic properties can be obtained. The
method is based on the hypothesis of linear viscoelasticity and no assumption are made
about the frequency dependence of the acoustic properties as a monochromatic approach is
followed. The technique is very attractive since many of the limitations common to
traditional method including beam spreading, sample manufacturing, transducer coupling,
etc. are overcome. On the other hand, the need for the material to be molded into the tube
represents a limitation in the case of solid materials, since melting the material would
dramatically change the properties. By contrast, the method is particularly suitable for
materials that will flow as the tube can then easily be filled.
REFERENCES
1.
2.
3.
4.
Alleyne, D. N., Pavlakovic, B., Lowe, M. J. S. and Cawley, P., Insight 43, 93-96 (2001).
Mak. D. K., British Journal ofNDT 35, 443-449 (1993).
Castaings, M., Hosten, B. and Kundu, T., NDT and E International 33, 377-392 (2000).
Challis, R. E., Wilkinson, G. P. and Freemantle, R. J., Measurement Science Technology
9, 692-700 (1998).
5. Pialucha, T., Guyott, C. C. H. and Cawley, P., Ultrasonics 27, 270-279 (1989).
6. Sachse, W., Pao, Y. H., Journal of applied Physics 64, 1645-1651 (1987).
7. Vogt, T., Lowe, M. J. S., and Cawley, P., "Ultrasonic Waveguide Techniques for the
Measurement of Materials Properties," in Review of Progress in QNDE, Vol. 21, eds.
D. O. Thompson and D. E. Chimenti, American Insitute of Physics, New York, 2002,
pp. 1742-1749.
8. Lowe, M. J. S., IEEE Trans. Ulrason. Ferroelectr. Freq. Control 42, 525-542 (1995).
9. Pavlakovic, B. N., Lowe, M. J. S., Alleyne, D. N. and Cawley, P., "Disperse: A General
Purpose Program for creating dispersion curves," in Review of Progress in QNDE, Vol.
16, eds. D. O. Thompson and D. E. Chimenti, Plenum, New York, (1997), pp. 185-192.
10. Evans, M. J., Webster, J. and Cawley, P., Ultrasonic 37, 589-594 (2000).
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