MODELING ULTRASONIC PROBLEMS FOR THE 2002
BENCHMARK SESSION
Lester W. Schmerr Jr.1'2 and Alexander Sedov3
Center for NDE, Iowa State University, Ames, IA, 50011, USA
2
Dept. of Aerospace Eng. and Eng, Mech., Iowa State University, Ames Iowa, 50011, USA
3
Dept. of Mechanical Eng., Lakehead University, Thunder Bay, Ontario Canada, P7B 5E1
ABSTRACT. In the 2001 RPQNDE conference, a series of ultrasonic benchmark problems were
compared using different model-based approaches [1-5]. Here, an extended set of benchmark problems
are considered. Paraxial beam models are used in conjunction with various measurement models to
demonstrate the effects of various modeling assumptions on the waveforms predicted for these
benchmark problems.
INTRODUCTION
In the 2001 Review of Progress in Quantitative Nondestructive Evaluation conference, a
series of ultrasonic benchmark problems were modeled by various researchers [1-4]. Their
results for the peak-to-peak voltage amplitude responses of ideal pores and cracks were
summarized by R. B Thompson [5]. Unfortunately, the results reported in Table 2 of [5]
from the paper of Schmerr [1] contained several incorrect entries. The correct amplitude (in
dB) for the response of the 0.125 mm diameter pore to the focused probe in Table 2 should
have been listed as - 41.5 dB and the response of the same pore to the planar probe as - 42.5
dB. When those corrections are made all the modeling results of [1] agreed to within 0.3 dB
with those of [3] and generally were within 2 dB of [4].
A second benchmark modeling session was held at the 2002 Review of Progress in
Quantitative Nondestructive Evaluation conference. Two pulse-echo modeling
configurations were considered, as shown in Figures l(a), (b). The first configuration
(Fig.l(a)) consisted of a planar or spherically focused piston transducer radiating at normal
incidence through a plane fluid-solid interface. The setup parameters of Fig.l (a) are
identical to those considered in the previous benchmark study [1]. Here, however the
scatterer is a side-drilled hole (SDH) having six diameters ranging over the same set of
values (0.125, 0.25, 0.5, 1.0, 2.0, and 4.0 mm) previously considered for cracks and pores
[1]. In the second modeling configuration (Fig.l(b)) a pore or SDH, having the same range
of sizes, is placed on the central ray of a probe radiating at oblique incidence to the
interface, where the angle, 9, was specified as having the values 9 = 30,45,60,75° for either
a refracted P- or SV-wave.
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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(b)
(a)
FIGURE 1 (a) Benchmark Problem Configuration I: response of an on-axis side-drilled hole to a planar or
spherically focused piston probe at normal incidence through a fluid-solid interface, (b) Benchmark Problem
Configuration II: response of a side-drilled hole or spherical pore on the central ray of a planar or spherically
focused probe where (0 = 30,45,60,75° J are the refracted angles of either a P- or SV-wave.
Again, the same probe parameters specified in [1] were used in this oblique incidence case.
MEASUREMENT MODELS FOR THE BENCHMARK PROBLEMS
All of the measurement models used in this study are based on a reciprocity-based
relationship, similar to that originally derived by Auld [6], where the frequency components
of the received voltage, VR(ai) (for harmonic disturbances of the formexp(-/##)), are
related to stress and velocity fields on the surface of the scatterer in two problems, labeled a
and b, respectively. The fields in problem a are due to the transducer firing with the flaw
present while in problem b the fields are with the scatterer absent. In terms of these fields
VR(CD} is given by [7]
(1)
where
'"> v / w
are me
stresses and velocity fields for problems m = a,b, respectively, ST
is the area of the transducer and Sf is the surface of the scatterer, n. are components of the
outward normal to the scatterer (pointing into the solid), (pl9cpl) are the density and wave
/ \
speed of the fluid, and VQ ; are the velocities on the face of the transducer (which is
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(a)
(b)
FIGURE 2 (a) Geometry of the side-drilled hole, and (b) the "lit" contour in the Kirchhoff approximation.
assumed to act as a piston) for problems m = a,b. The quantity /?(#>) is the system
"efficiency", which can be obtained through a specification of the measured voltage in a
particular reference calibration setup [7]. Since the details for obtaining fi(co) in our
present benchmark problems are identical to those described in [7], here we will simply
assume fi(co) is a known quantity.
In applying Eq.(l) we will make several assumptions that will greatly facilitate the
modeling of the benchmark problems. First, we will assume that the transducer wave fields
incident on the scatterer in problems a and b (which are equal to each other for the pulseecho case) can be written in a quasi-plane wave form (paraxial approximation) [7]. For the
fields in problem b, for example, we have v(6) =Vav^da exp(ika2ea -x) where Vais a
normalized velocity amplitude, and (e a ,d a ) are unit vectors describing the incident wave
direction and polarization, respectively, for a wave of type a (a = /?, s), and ka2 = CD/ ca2 is
the corresponding wave number. With this assumption Eq.(l) becomes
(2)
P\C p\
where r is the radius of the transducer, p2 is the density of the solid, and ca2 (a = p,s] are
the wave speeds of compressional and shear waves in the solid. The quantity Ga is given by
_
G ot =
-
(3)
where (^>vf) are the stresses and velocity components on the surface of the scatterer due
to an incident plane wave of type a (a — /?,s)and unit displacement amplitude.
Equation (3) will be the basis for three types of measurement models that we will
consider here. The first of these models, which we will use for the benchmark problems
involving a pore, is based on the additional assumption that we can neglect the variations of
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-
\2
the velocity amplitude term, (va\ over the surface of the scatterer. In this case Eq.(3)
reduces to what we will call form I:
Form I:
(4)
=/?H(J>)VH
C
Pllpl
_
where Aa(co)is the pulse-echo far field plane wave scattering amplitude [7]. Form I is the
measurement model originally derived by Thompson and Gray [8]. A very important feature
of this model is that the scatterer response is separated from the other parts of the measured
response and thus can be calculated independently.
Form I is not suitable for modeling the response of a side-drilled hole since the incident
fields can extend a significant distance along the length of the hole. However, if one
assumes that the scattered wave field of the SDH can be obtained from the Kirchhoff
approximation [7] then it can be shown that in this approximation we have
en,
(5)
over the "lit" surface of the cylinder (and zero elsewhere), a quantity that is independent of
the y-coordinate along the cylinder (Fig. 2(a)). Thus, in the Kirchhoff approximation,
Equation (3) reduces to what we will call form II:
Form II:
nr
J(c a -n)exp(2it a2 c a .x)
f
dc
length,L
(6)
where Clit is a line integral over the lit portion of the cylinder surface (see Fig. 2(b)).
If we can neglect the variations of the velocity fields in Eq. (6) over Clit, then that
equation reduces to what we call form III:
Form III:
A" (co)
VR(ai) = p(co)
*-" length,L
-ika2r plCpl
(7)
where Aa (co) is the 3-D far field scattering amplitude (in the Kirchhoff approximation)
for the cylinder. We see that form III is very similar to form I with the squared velocity field
term in form I being replaced by the average of this term over the length of the SDH. Form
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FIGURE 3 Peak-to-peak voltage pulse-echo response versus diameter of a side-drilled hole for a focused
probe using form II (dashed line) and form III (solid line).
Ill can be written in a more explicit form for the SDH since in the Kirchhoff approximation
Aa (ai) can be calculated exactly as
(8)
where b is the radius of the SDH and Sl and Jl are Struve and Bessel functions,
respectively. In contrast, for a spherical pore of radius b, the Kirchhoff approximation gives,
using Eq. (5),
(9)
BENCHMARK PROBLEM CONFIGURATION I
For the configuration of Fig. l(a) we calculated the peak-to-peak voltage response of
side-drilled holes (having the range of diameters described previously) for a 12.7 mm
diameter, 5 MHz planar piston transducer and a 25.4 mm diameter 5 MHz focused probe
(focal length = 152.4 mm). The procedures followed were identical to those described in [1],
for the previous benchmark problems. The normalized velocity field, Va , was calculated
using a multi-Gaussian beam model [9]. Results using both form II and form III were
compared to see if variations of the incident transducer field over the cross-section of the
SDH affected these voltage responses. As Fig. 3 shows for the focused probe case, little
difference was seen even for the largest diameter hole. Similar results (not shown) were
found for the response of the planar probe. Based on these results, all our subsequent SDH
calculations were done with form III.
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1.4
SDH
1
2
3
1
2
3
(b) size (mm)
(a) size (mm)
FIGURE 4 Peak-to-peak voltage versus diameter for a focused probe.(a) Comparison of side-drilled hole
response and crack response, (b) Comparison of side-drilled hole response and spherical pore response.
0.35
0.25
45
30°
45
o
1
2
1
3
(a) size, mm
2
3
(b) size, mm
FIGURE 5 Peak-to-peak voltage versus diameter for a planar probe at oblique incidence, (a) Response of a
side-drilled hole on the axis of a refracted P-wave for the refracted angles indicated, (b) Response of a sidedrilled hole on the axis of a refracted SV-wave for the refracted angles indicated.
The peak-to-peak voltage responses of on-axis cracks and pores in the same configuration of
Fig. l(a) were calculated previously [1]. Figures 4(a), (b) compare the voltage versus flaw
size curves for those cases with that of the SDH for the focused probe. As can be seen from
that figure, the SDH response is always higher than that of the pore, for all sizes considered,
while the SDH response is larger than the crack response only for the smaller diameters.
Similar results (not shown) were found for responses with the planar transducer.
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0.05
0.05
45°
1
2
3
1
(a) size, mm
2
3
(b) size, mm
FIGURE 6 Peak-to-peak voltage versus diameter for a planar probe at oblique incidence, (a) Response of a
spherical pore on the central axis of a refracted P-wave for the refracted angles indicated, (b) Response of a
spherical pore on the central axis of a refracted SV-wave for the refracted angles indicated.
BENCHMARK PROBLEM CONFIGURATION II
For the configuration of Fig. l(b) we considered the responses of both pores and sidedrilled holes with the same planar and focused probes. Again, the velocity field was
calculated with a multi-Gaussian beam model. Since that beam model relies on the paraxial
approximation, it can lose accuracy in the case where one is near a critical angle in oblique
incidence testing because in the paraxial approximation it is assumed that the plane wave
transmission coefficient for the interface is slowly varying. It was found that for the
refracted P-wave case at highest refracted angle (75°) this assumption may be violated while
for the SV-wave case significant changes in the transmission coefficient are present for
9 = 30° and possibly for 0 = 75°. Thus, the predicted voltage responses in those cases may
be in error. With that caveat, the voltage responses versus flaw size for a planar probe are
shown in Figs. 5(a), (b) for a SDH on the axis of a refracted P-wave or SV-wave,
respectively. Similar results (not shown) were found for the response of the SDH to the
focused probe. Figures 6(a), (b) show the comparable planar probe results for the spherical
pore where the response was calculated with the Kirchhoff approximation (Eq.(9)). Again,
the focused probe results with the pore showed similar behavior.
SUMMARY
We have outlined three measurement models suitable for predicting the ultrasonic
responses of spherical pores and side-drilled holes and applied these models to a number of
normal incidence and oblique incidence inspection cases. Comparison of these model results
with those of others will help to define the validity of the various modeling approximations
used so that the models can be reliably used in conjunction with experiments to accurately
predict measured ultrasonic responses.
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ACKNOWLEDGEMENTS
For L.W. Schmerr this work was supported by the National Science Foundation
Industry/University Cooperative Research Program at the Center for NDE, Iowa State
University. A. Sedov was supported by the Natural Sciences and Engineering Research
Council of Canada.
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