PREDICTION OF TRANSIENT FLAW SIGNALS OF THE
ULTRASONIC BENCHMARK PROBLEM
Martin Spies
Fraunhofer-Institute for Nondestructive Testing (IZFP)
University of Saarland, Bldg. 37, 66123 Saarbriicken, Germany
ABSTRACT. Using a point source superposition method, the signals from spherical,
cylindrical and crack-like reflectors observed in a pulse-echo measurement are predicted.
Normally incident longitudinal waves as well as obliquely incident transverse and longitudinal waves, generated by circular focused and unfocused transducers, respectively,
of 5 MHz center frequency are assumed in an immersion geometry. Three sets of scattering objects are considered: spherical pores, cylindrical scatterers and circular cracks
of various diameters. The problem addressed in this contribution is an extension of the
2001 UT Benchmark Problem [1].
INTRODUCTION
The simulation of experimental results without time and cost of constructing
specimens and performing measurements requires the validation of respective physical models. As a first step within the process of comparing model-based simulations
to benchmark experiments, a standard type problem has been addressed to allow for
a preliminary comparison of the various models available. In this contribution, the
transient signals from spherical and cylindrical, as well as crack-like reflectors observed
in a pulse-echo measurement are predicted. Normally incident longitudinal waves as
well as transverse and longitudinal waves of oblique incidence, generated by circular focused and unfocused transducers, respectively, of 5 MHz center frequency are assumed
in an immersion geometry. Three sets of scattering objects are considered: spherical,
cylindrical and disk-shaped scatterers of diameters ranging from 0.125 mm to 4 mm
centered at a distance of 25.4 mm into an aluminum block. For oblique incidence, the
transducers are assumed to be inclined at an angle such that waves are excited at refracted angles of 30°, 45°, 60° and 75°. Using a point source superposition technique,
the generation, propagation and refraction of the incident signal is simulated, where
an appropriate spectral function is considered to characterize the reference waveform.
To predict the electric voltage signal generated in the transducer, Auld's reciprocity
formula is used, while the scattering process at the defect is modeled using Kirchhoff's
theory. The calculations are performed in frequency domain with subsequent inverse
Fourier transform into the time domain.
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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ULTRASONIC TESTING PROBLEM
The addressed problem is to predict the signal from the reflectors in a solid,
observed in a pulse-echo measurement. Also, the dependence of the maximum signal
amplitude from the reflector size is of interest for the various reflectors. It is assumed
that normally incident longitudinal waves and obliquely incident transverse and longitudinal waves in an immersion geometry are used, with input parameters as follows.
The material is an aluminum block with a planar surface (density = 2.71 g/cm3,
longitudinal wave speed = 6374 m/s, shear wave speed = 3111 m/s); the block is placed
in a water bath (density = 1 g/cm3, wave speed = 1470 m/s).
Three sets of six scattering objects are considered. The first set consists of
spherical pores, centered at a distance of 25.4 mm (1 inch) into the material. They
have respective diameters of 0.125 mm, 0.25 mm 0.5 mm, 1 mm, 2 mm and 4mm. The
two other sets consist of circular cracks and of cylindrical scatterers, respectively, with
the same depth and diameters as the pores. The diameters have been selected to range
from the regime in which the defect is small with respect to the wavelength to the case
in which it is large with respect to the spot size of the focused beam.
A circular, unfocused transducer of 12.7 mm (half inch) diameter and of 5 MHz
center frequency is considered. The transducer is assumed to act as a piston source,
i.e. all points on its surface move with equal amplitude. As a second transducer, a
circular, focused probe of 25.4 mm (one inch) diameter and of 5 MHz center frequency
is addressed. The transducer is considered to have a curved element with a radius
of curvature of 152.4 mm (six inches) and to act as a piston source. The radius of
curvature is equal to the geometrical focal length, i.e. the point of maximum amplitude
in the absence of diffraction.
The waveform to be considered is described by a three-cycled raised cosine function according to
V(t) = [1 - cos(27r/t/3)] cos(o;t),
0 < t < 3/f .
(1)
Here t is in microseconds and / is the center frequency of the transducer, given in MHz.
In this contribution, the following computational approach for modeling of transient
signals is used. The harmonic (cw) solution is calculated at many frequencies and then
this data is numerically Fourier transformed into the time domain. The function used
for the frequency spectrum of the transducer input signal is given by
0 ,
otherwise
'
(2)
With this frequency function a probe will have its -6 dB bandwidth, i.e. the bandwidth
measured at the half of the maximum amplitude, given as (uj^ — cji)/2. To match the
waveform described by Eq. (1), a bandwidth of 75 % is used.
The time domain response (i.e. the radio frequency (rf) waveform as a function
of time) of the signal scattered from each of the spheres and cylinders is to be predicted
for all combinations of transducers, wave types and illumination angles (30° to 75°). For
comparison some of the calculations are also performed for inclined cracks, which are
oriented perpendicularly to the incident central ray of the beam. For all calculations,
it is assumed that the transducer is positioned such that its distance to the aluminum
plate - measured along the central ray of the beam - is 50.8 mm (2 inches). This should
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place the flaw in the far field of the radiation of the unfocused probe. For the focused
probe the flaw is near the focal point for normal incidence, while it is remote from the
focal point for the oblique incidence cases.
MODELING PROCEDURE
To predict the time-domain signal for this problem, the various physical processes involved have to be modeled, which are: (i) the radiation of ultrasonic waves by
circular, flat or focused transducers and the propagation thru the respective medium;
(ii) the reflection and refraction process at the water - aluminum interface; (iii) the
scattering of the waves incident on the defect. To account for these processes, a point
source superposition technique is applied assuming that the transducer is acting as a
piston source. The method has been briefly summarized in Ref. 2, while a detailed
description can be found in Ref. 3. The relationships given in Ref. 3 for anisotropic
material symmetry are accordingly exploited for isotropic elastic conditions. To model
the reflection and refraction process at the water - aluminum interface, the continuity of
the normal tractions and the displacements is used to calculate the particle displacement
distribution on the aluminum surface. This distribution is then applied to determine
the propagation of the ultrasonic waves into the aluminum test block. Finally, the resulting displacement on the scatterer is calculated using Kirchhoff's theory as described
in Ref. 4 for the case of anisotropic media. In the calculations, numerical integration is
performed on the basis of equally spaced rectangular grids for planar surfaces, while for
the cylindrical or spherical scatterers respective projections of a planar grid onto the
curved surfaces are applied. While the 2001 Benchmark Problem has been addressed by
assuming cw excitation in the calculation of the particle displacement distribution on
the aluminum surface and rf-impulse propagation in the aluminum block [1], the 2002
calculations have been performed assuming rf-impulse propagation in both cases. The
time-domain signal detected by the transducer is finally determined using Auld's reciprocity theorem for traction-free scatterers which exploits the displacement and traction
at the scatterer's position in presence and absence of the scatterer, respectively [5].
AMPLITUDE VS. REFLECTOR RADIUS - EXPECTED VARIATION
Simple expressions can be formulated for the amplitude dynamic curves measured in a contact pulse-echo inspection experiment for the case where transducer and
reflector are in mutual far field. Using the far field expressions for planar vibrating
sources and assuming that the reflector is - at least in one dimension - smaller than the
beam width, the magnitude of the particle displacement due to wave mode a detected
by the transducer can be expressed according to [6]
• circular disk
|Ua(Br)l/Uo
=
Fcirc-r2T(Rs-E^)-rs(E,T-Rs)-r2circ,
(3)
• spherical pore
/U0
= Fsph • r|(Rs - fir) • rsph ,
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(4)
• cylindrical reflector
(5)
With R/F and Rs designating the spatial coordinates of the transducer and the center
of the scatterer, respectively, the F-quantities are constant factors and TT/S are the
directivity functions characterizing the transducer and the scatterer, respectively (the
transducer directivity function is squared to account for the pulse-echo mode; the directivity functions for the cylinder and the sphere are equal to one). In the derivation
of these approximated expressions the circular reflector has been treated as a secondary
source, while for the sphere and the cylinder a geometrical reflection model has been
applied [6]. Also, the influence of the stress-free material surface on the transducer
radiation characteristics as well as the particular boundary conditions at the scatterer
have been neglected. From these relationships it is expected that a doubling of the
scatterer's diameter leads to an increase of the scattered signal amplitude by a factor
of 12 dB, 6 dB and 3 dB for the circular disk, the sphere and the cylinder, respectively.
RESULTS AND DISCUSSION
The reference signal is taken to be the signal observed during the normal incidence reflection from the front surface of the aluminum block. The signals calculated
for the planar and the focused transducer are shown in Figure 1. In the latter case,
the transducer to front-surface distance is taken to be 152.4 mm (6 inches), which is
equivalent to placing the geometrical focal plane on the surface. For the planar probe,
the transducer to front-surface distance is taken to be 50.8 mm (2 inches), which places
the surface into the transducer nearfield.
The beam fields generated by the two transducers under concern have been
calculated in water, as well as in the aluminum block assuming a water path of 50.8
Focused transducer
4000 -j
3000200010000
-1000-2000-3000-4000-5000207.00
207.25
207.50
207.75
208.00
Planar transducer
70.00
69.00
FIGURE 1. Reference signal calculated for the circular, unfocused (bottom) and focused
(top) transducer of 5 MHz center frequency.
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Planar transducer / L-wave
Planar transducer / T-wave
-is
~|S
Focused transducer/ L-wave
-fi
-it
-25
-11:
Focused transducer / T-wave
"$&
^3
<y&
™&,3
iw
"13
**
•">?
"*%
3
x[mm]
FIGURE 2. Longitudinal (left) and transverse (right) wave beam fields in aluminum generated by the circular, unfocused (top) and focused (bottom) transducer. The beam fields are
shown in logarithmic scaling, ranging down to -36 dB.
mm. The normal incidence results have been shown in Ref. [1]. The results for
oblique transverse and longitudinal wave incidence, respectively, are exemplarily shown
in Figure 2 for the 45°-case for both transducers. The probe inclination angles have
been calculated using Snell's law. While the intended refraction angles in the aluminum
block can be achieved with the planar transducer, this is not the case for the focused
probe. Here, the angle of incidence deviates more or less from the nominal value, e.g.
in the case of the 45° L-wave an angle of incidence of about 35° results. To obtain
the intended refraction angles, a respective correction to the probe inclination angles is
required.
For the unfocused transducer, Figure 3 displays the maximum flaw signal amplitudes as a function of the reflector diameter for both normally and obliquely incident
longitudinal waves. For each configuration, the signal amplitude for the 4 mm reflector
is maximal and has accordingly been set to 0 dB. The results obtained for transverse
waves and for the focused transducer in principle show the same behavior. However,
since for the focused transducer the nominal refraction angles are generally less than
intended, the reflectors are not properly positioned on the central ray of the incident
beam field. Thus, in some cases deviations from the expected amplitude behavior have
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Planar transducer/normal L-wave
0
-12
•
-24
•
1
2
3
4
Planar transducer/inclined circular crack/L-roave
-36 •
-48
-60
0
1
2
3
4
Planar transducer/cylindrical reflector/L-wave
1
2
3
4
Planar transducer/spherical pore/L-wave
Diameter (ram)
FIGURE 3. Flaw signal amplitude plotted versus reflector diameter for normal (top) and
oblique (bottom) L-wave incidence (unfocused transducer). The expected amplitude behavior
according to Eqns. (3-5) is indicated by the solid lines. Similar diagrams result for the focused
transducer as well as for incident transverse waves.
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Planar transducer/normal L-wave
-30
•
0.0
0.5
1.0
1.5
2.0
Diameter (mm)
FIGURE 4. Enlarged view of the signal amplitude plotted versus diameter for the cylindrical
and the spherical reflectors (unfocused transducer, normal L-wave incidence at 5 and 10 MHz,
respectively). The expected amplitude behavior according to Eqn. (4) and (5) is indicated
by the solid lines.
occured in the calculations. Adjusting the probe inclination angles is expected to decrease these deviations.
The results obtained for the circular cracks reveal that the amplitude is principally proportional to the square of the reflector radius, as the approximation according
to Eq. (3) also suggests. This holds for both normal incidence and oblique incidence,
where the circular crack has been assumed to be perpendicularly oriented with respect
to the incident central ray. Thus, dividing the scatterer's diameter by a factor of 2 leads
to a decrease of the scattered signal amplitude by a factor of 12 dB. However, referring
to the larger scatterers (1,2 and 4 mm diameter) the decrease in amplitude is less than
12 dB. This is due to the beam field amplitude variation across the defect surface [1],
which is not taken into account in the derivation of Eqs. (3) to (5).
For the cylindrical and the spherical reflectors, in principle the expected amplitude behavior is also obtained. Here, additional calculations have been performed for a
reflector diameter of 3 mm. Dividing the defect diameter by a factor of two leads to a
decrease of the scattered signal amplitudes by a factor of 3 dB and 6 dB, respectively,
for the 2, 3 and 4 mm reflectors, while for the smaller diameters, the decrease of the
calculated amplitudes is larger. As the results obtained for the spherical pores for a
frequency of 10 MHz reveal, the deviation from the expected behavior depends on the
respective ratio of wavelength to reflector diameter. At 10 MHz, the deviations from
the expected amplitude behavior occur at a reflector diameter of 1 mm, while in the 5
MHz-case, the deviation of calculated from the expected amplitude drop already occurs
at a diameter of 2 mm (Fig. 4).
The scattering process has been considered by using Kirchhoff 's approximation,
which assumes that each point on the scatterer's surface behaves like a point on an
infinitely long reflecting plane. Although the k - r values (wavenumber times scatterer's
radius) are small for some of the defects considered, the obtained results suggest that
here Kirchhoff's approximation is still applicable for the case of planar scatterers, which
is consistent with measurements recently performed on flat-bottomed holes using far
field conditions [7]. However, from the results obtained for the cylindrical and the
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spherical reflectors it can be inferred, that the application of Kirchhoff's theory in the
employed point source superposition technique leads to deviations from the expected
amplitude behavior for curved scatterers in the regime where A; • r is equal to 5 and less.
The results obtained for the circular cracks and the spherical pores are consistent with
those obtained by Gray and Thompson [8], who have applied both approximate and
exact models. In principle, the latter have shown that the amplitude decrease for the
small scatterers is larger than predicted by Kirchhoff models.
Absolute values of the signal amplitudes obtained e.g. for the 4 mm reflectors are
not shown here. Since the applied point source superposition technique provides relative
results, the calculated amplitudes have to be normalized to a respective reference signal,
such as the signal observed during the normal incidence reflection from the (assumed)
backwall of the aluminum block.
ACKNOWLEDGMENT
The author would like to thank Dr. Wolfgang Gebhardt of IZFP's Physical
Basics Department for many valuable discussions. This work was partially sponsored
by the Deutsche Forschungsgemeinschaft DFG, which is gratefully acknowledged.
REFERENCES
1. Spies, M., "Simulating a Standard Type Problem of Ultrasonic Testing - A
Contribution to the Comparison of Models," in Review of Progress in Quantitative
Nondestructive Evaluation, edited by D.O. Thompson and D.E. Chimenti, AIP,
New York, Vol. 21, 2001, 1949-1955
2. Spies, M., "Modeling Transient Radiation of Ultrasonic Transducers in Anisotropic
Materials Including Wave Attenuation," in Review of Progress in Quantitative
Nondestructive Evaluation, edited by D.O. Thompson and D.E. Chimenti, AIP,
New York, Vol. 21, 2001, 807-814
3. Spies, M., J. Acoust. Soc. Am. 110, 68-79 (2001)
4. Spies, M., J. Acoust. Soc. Am. 107, 2755-2759 (2000)
5. Auld, B.A., Wave Motion 1, 3-10 (1979)
6. Werneyer, R., Walte, F. and Miiller, W., Technical Report No. 780852-TW
(in German), IZFP Saarbriicken, Germany (1978)
7. Schmitz, V. and Spies, M., "Measurements on an Aluminum Block with
Flat-Bottomed Holes Using Pulse-Echo Contact Technique," unpublished
8. Gray, T. and Thompson, R.B., "Solution of an Ultrasonic Benchmark Problem
within the Paraxial Approximation ," in Review of Progress in Quantitative
Nondestructive Evaluation, edited by D.O. Thompson and D.E. Chimenti, AIP,
New York, Vol. 21, 2001, 1925-1932
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