ULTRASONIC BENCHMARK PROBLEM: APPLICATION OF A
PARAXIAL MODEL TO SIDE-DRILLED HOLES AND OBLIQUE
INCIDENCE
T. A. Gray
Center for Nondestructive Evaluation and Department of Aerospace Engineering and Engineering
Mechanics, Iowa State University, Ames, IA 50011, USA
ABSTRACT. The Thompson-Gray ultrasonic measurement model is applied to two benchmark
problems. The first problem considers the ultrasonic reflection from spherical pores and side-drilled
holes (SDK) for normally incident longitudinal waves. The second problem examines refracted
longitudinal and shear wave reflections from both SDH and spherical pores. Results are obtained for
both planar and spherically focused transducers.
INTRODUCTION
This paper summarizes one of a set of benchmark ultrasonic modeling calculations that
extends earlier results presented at the 2001 Review of Progress in QNDE conference.
Those prior results are found in several papers [1-4] and are summarized in Reference [5].
In short, the previous calculations examined the simulated pulse echo response from
spherical pores and circular cracks as measured using planar and focused immersion
transducers at normal incidence to a fluid-solid interface. In the present study, the previous
calculations are extended to include reflections from side-drilled holes (SDH) at normal
incidence as before, as well as oblique incidence of the transducers to generate longitudinal
or vertically polarized shear waves at refracted angles 30, 45, 60, and 75 degrees. Both
spherical and SDH reflectors are considered for the oblique incidence calculations. The
SDH are assumed to have the same range of diameters (0.125, 0.25, 0.5, 1.0, 2.0 and 4.0
mm) as were considered for the previous pore and crack calculations. The measurement
configurations for the present study are illustrated in Figure 1. Other details of the
benchmark calculations, including probe characteristics, material properties, etc., are the
same as described in Reference [1].
The model used here is the Thompson-Gray measurement model [6]. This model is
based upon Auld's electromechanical reciprocity integral [7], and, in order to simplify
calculations, employs the paraxial approximation in computation of ultrasonic beam
displacements and transmission quantities. The Gaussian-Hermite model [8] was used to
calculate the beam displacements. Scattering from spherical pores was represented using
far-field plane wave scattering amplitudes based on exact solutions for longitudinal [9] and
shear wave [10] backscattering. Scattering from circular cracks was computed using a
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
1800
(a)
(b)
FIGURE 1. Benchmark measurement configurations for normal incidence (a) and oblique incidence (b)
cases. The angle 6 in (b) can assume values of 30, 45, 60, and 75 degrees. The transducer is immersed in
water and the specimen is aluminum.
Kirchhoff approximation [11] integrated over the face of the crack to account for beam
variation [1]. For SDH, a new Kirchhoff model developed by Schmerr and Sedov [12] was
used to compute the flaw scattering. The form of the measurement model in this case is
denoted "Form III" and shown as Equations 7 and 8 in Reference [12].
RESULTS
The first set of benchmark computations is for the normal incidence configuration
shown in Fig. l(a). The responses from spherical pores, circular cracks, and SDH were
calculated for flaw sizes in the range 0.125 mm < diameter < 4.0 mm and using both a
planar (5 MHz, 12.7 mm diameter) and a focused (5 MHz, 25.4 mm diameter, 152.4 mm
curved element) transducer. The results are shown in Figure 2. Note that the vertical
scales on the graphs are expressed in dB computed from the ratio of the peak-to-peak
voltage of the unrectified RF waveform relative to the peak-to-peak voltage of a reference
waveform. The reference waveform, as shown in Figure 1 of Ref. [1], was the reflected
signal from the front surface of the sample taken at a water path of 50.8 mm, for the case of
the planar transducer, or of 152.4 mm, for the focused transducer. In both cases, the peakto-peak voltage of the reference waveform was 3.5 volts. As would be expected, as the
reflector diameter increases, the amplitude for signal from the spherical void is remains less
than that of the SDH while the amplitude of the crack signal starts out lower but eventually
exceeds that of the SDH. Table 1 contains the raw peak-to-peak voltages corresponding to
the dB results shown in Figure 2.
Next, we turn our attention to the oblique incidence calculations. The first of these
results are for refracted longitudinal waves. The various probe tilt angles, <|>, and metal path
lengths, z, as depicted in Figure l(b), are listed in Table 2. Note that for the refracted 75°
longitudinal wave, the probe tilt angle is 12.87°, which is rather near the critical angle of
13.33°. Therefore, we do not expect the paraxial approximation to be accurate in this case.
The refracted longitudinal wave results for the planar and focused transducer are shown in
1801
-10
-20
30
€ -
SDK
-8" -40
1-50
| -60
-70
-80
J____L
0
1
2
3
Diameter, mm
(b)
1
2
3
Diameter, mm
(a)
FIGURE 2. Amplitudes of SDH, crack and spherical reflectors for normal incidence longitudinal waves.
Results are for the planar transducer in (a) and for the focused transducer in (b). Amplitudes in dB relative to
reference waveform.
TABLE 1. Normal incidence results, peak-to-peak voltage.
Planar
probe
Focused
probe
SDH
sphere
crack
SDH
sphere
crack
0.125mm
0.024873 V
0.0004086
0.000511
0.078893
0.0035812
0.0042146
0.25mm
0.048279
0.002811
0.002042
0.15259
0.023939
0.016768
0.5 mm
0.086064
0.00738
0.008132
0.26805
0.057254
0.065668
1.0mm
0.11952
0.015365
0.031975
0.35784
0.11662
0.24181
2.0mm
0.16574
0.029622
0.12317
0.51147
0.2262
0.72826
4.0mm
0.22712
0.056014
0.37607
0.70333
0.42961
1.1455
Figures 3 and 4, respectively, which display the signal amplitudes in dB relative to the
reference waveform amplitude. The corresponding raw peak-to-peak voltages are shown in
Tables 3 and 4.
The refracted shear wave results are presented next. Plots of those results in dB relative
to their corresponding reference waveform amplitude are shown in Figures 5 and 6,
respectively, for the planar and focused transducer. Raw peak-to-peak voltages are listed in
Tables 5 and 6. Note that the probe tilt angle is 13.67° for the 30° refracted shear wave.
This is very close to the longitudinal wave critical angle of 13.33°. Further, the shear wave
critical angle is 28.20°, which is close to the probe tilt angle of 27.16° used for the refracted
TABLE 2. Probe tilt angles and refracted metal paths for oblique incidence calculations.
Probe tilt angle, <|)
Refracted angle, 6
30°
45°
60°
75°
L-wave
6.62°
9.39°
11.52°
12.87°
S-wave
13.67°
19.52°
24.16°
27.16°
1802
Metal path, z, mm
29.3
35.9
50.8
98.1
1
2
3
Diameter, mm
1
2
3
Diameter, mm
(b)
(a)
FIGURE 3. Oblique incidence longitudinal wave results for the planar transducer. Results for spherical
pores are shown in (a) and for SDH in (b). Amplitudes in dB relative to reference waveform.
-20
-30
-40
8 -50
-60
• -70
-80
-90
-100
i
1
2
3
Diameter, mm
0
1
2
3
Diameter, mm
(b)
(a)
FIGURE 4. Oblique incidence longitudinal wave results for the focused transducer. Results for spherical
cores are shown in (a) and for SDH in (bV Amplitudes in dB relative to reference waveform.
1803
-20
-25
45
3-30
4 -35
1 -40
| -45
-50
-55
1
2
3
Diameter, mm
0
1
2
3
Diameter, mm
(b)
(a)
FIGURE 5. Oblique incidence shear wave results for the planar transducer. Results for spherical pores are
shown in (a) and for SDH in (b). Amplitudes in dB relative to reference waveform.
1
2
3
Diameter, mm
1
2
3
Diameter, mm
(b)
(a)
FIGURE 6. Oblique incidence shear wave results for the focused transducer. Results for spherical pores are
shown in (a) and for SDH in (b). Amplitudes in dB relative to reference waveform.
1804
TABLE 3. Planar probe results for refracted L-waves, peak-to-peak voltage.
sphere
SDH
0.125mm
6 = 30° 0.000365 V
6 = 45° 0.000256
L_ e = 60° 0.000108
6 = 75° 0.000016
6 = 30° 0.023427
0 = 45° 0.017743
6 = 60° 0.008473
0 = 75° 0.001569
0.25mm
0.002484
0.001729
0.000727
0.000105
0.045376
0.034337
0.016386
0.003031
0.5mm
0.006300
0.004285
0.001773
0.000252
0.080332
0.060469
0.028757
0.005311
1.0mm
2.0mm
0.013019 0.025145
0.008792 0.016995
0.003619 L0.007002
0.000513 0.000994
0.109620 0.153300
0.081524 0.114710
0.038492 0.054371
0.007079 0.010015
4.0 mm
0.047587
0.032180
0.013270
0.001885
0.209750
0.156840
0.074364
0.013702
TABLE 4. Focused probe results for refracted L-waves, peak-to-peak voltage.
sphere
SDH
0.125mm
0 = 30° 0.001044V
0 = 45° 9.85E-05
0 = 60° 6.67E-06
0 = 75° 7.62E-07
0 = 30° 0.027029
0 = 45° 0.003808
0 = 60° 0.000735
0 = 75° 9.49E-05
0.25 mm
0.007165
0.000706
4.65E-05
5.28E-06
0.052512
0.007456
0.001431
0.000185
0.5 mm
0.018502
0.002596
0.00018
1.36E-05
0.093827
0.013755
0.002571
0.000333
1.0mm
0.038934
0.005101
0.000326
2.56E-05
0.13104
0.020889
0.003617
0.000476
2.0mm
0.0755
0.00912
0.000661
4.99E-05
0.1851
0.027125
0.005097
0.00067
4.0mm
0.14302
0.016989
0.001265
9.73E-05
0.25455
0.037767
0.006925
0.000909
2.0mm
0.004738
0.022981
0.015791
0.003488
0.039978
0.21131
0.16654
0.045739
4.0mm
0.009881
0.047378
0.032326
0.007207
0.054826
0.28995
0.22855
0.062825
TABLE 5. Planar probe results for refracted S-waves, peak-to-peak voltage.
sphere
SDH
0.125mm
0 = 30° 0.000181 V
0 = 45° 0.000947
0 = 60° 0.000693
0 = 75° 0.000161
0 = 30° 0.008268
0 = 45° 0.044012
0 = 60° 0.035087
0 = 75° 0.009491
0.25mm
0.000708
0.003226
0.002099
0.000474
0.014777
0.077926
0.061587
0.016608
1805
0.5 mm
0.000958
0.004561
0.003082
0.000671
0.020648
0.10719
0.083421
0.022693
1.0mm
0.001994
0.009711
0.006707
0.001488
0.027495
0.14554
0.11475
0.031459
TABLE 6. Focused probe results for refracted S-waves, peak-to-peak voltage.
sphere
SDK
0.125mm
9 = 30° 0.000989 V
6 = 45° 0.003721
6 = 60° 0.000752
6 = 75° 1.18E-05
6 = 30° 0.014478
6 = 45° 0.056759
0 = 60° 0.011899
6 = 75° 0.000499
0.25mm
0.0035
0.012495
0.002262
5.04E-05
0.02569
0.10094
0.021143
0.000881
0.5 mm
0.004839
0.017734
0.003236
8.3E-05
0.036293
0.138
0.029784
0.001219
1.0mm
0.010268
0.037807
0.007051
0.000148
0.049658
0.18437
0.040606
0.00163
2.0mm
0.024297
0.089418
0.016542
0.000358
0.072444
0.2673
0.058984
0.00236
4.0mm
0.050395
0.18437
0.034969
0.000745
0.099606
0.36644
0.080984
0.003232
75° shear wave. Therefore, in both of these cases, we expect significant error in the
paraxial model predictions.
SUMMARY
As part of a series of the results of ultrasonic modeling benchmark calculations, this
paper presents a series of simulated results for normal and oblique incidence measurements
made on aluminum samples containing circular cracks, spherical pores and side-drilled
holes. The calculations were made using the Thompson-Gray measurement model, which
employed the Gaussian-Hermite ultrasonic beam model and incorporated exact scattering
amplitude formulations for spherical pores and Kirchhoff approximations for the cracks
and SDH. Because of the paraxial approximations used in this modeling approach,
computational results are not expected to be accurate in cases where the transmitted
ultrasonic beam is near the critical angle for longitudinal or shear waves.
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation Industry/University
Cooperative Research Program at the Center for Nondestructive Evaluation, Iowa State
University.
REFERENCES
1. Gray, T. A. and R. B. Thompson, "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, Plenum
Press, New York, vol. 21,2002, pp. 1925-1932.
2. Schmerr, L. W., "Ultrasonic Modeling of Benchmark Problems," ibid., pp. 1933-1940.
3. Song, S.-J., H.-J. Kim, and C.-H. Kim, "Prediction of Flaw Signals of the Ultrasonic
Benchmark Problems by Sungkyunkwan University," in Review of Progress in
Quantitative Nondestructive Evaluation, edited by D. O. Thompson and D. E.
Chimenti, Plenum Press, New York, vol. 21,2002, pp. 1941-1948.
4. 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, Plenum
Press, New York, vol. 21,2002, pp. 1949-1955.
1806
5. Thompson, R. B., "An Ultrasonic Benchmark Problem: Overview and Discussion of
Results," in Review of Progress in Quantitative Nondestructive Evaluation, edited by
D. O. Thompson and D. E. Chimenti, Plenum Press, New York, vol. 21, 2002, pp.
1917-1924.
6. Thompson, R. B. and T. A. Gray, J. Acoust. Soc. Am. 74, 1279-1290 (1983).
7. Auld, B. A., Wave Motion 1, 3-10 (1979).
8. Thompson, R. B., T. A. Gray, J. H. Rose, V. G. Kogan, and E. F. Lopes, J. Acoust. Soc.
Am. 82, 1818-1828(1987).
9. Ying, C. F. and R. Truell, J. AppL Phys. 27,1086-1097 (1956).
10. Gubernatis, J. E., E. Domany, J. A. Krumhansl, and M. Huberman, "The Fundamental
Theory of Elastic Wave Scattering by Defects in Elastic Materials - Integral Equation
Methods for Application to Ultrasonic Flaw Detection," ERDA Technical Report
#COO-3161-42, Cornell University, Ithaca, New York, (1976).
11. Adler, L. and J. D. Achenbach, J. Nondestr. Eval 1, 87-99 (1980).
12. Schmerr, L. W. and A. Sedov, 'Modeling Ultrasonic Problems for the 2002 Benchmark
Session," (these proceedings)
1807