CALCULATION AND MEASUREMENT OF ULTRASONIC
ATTENUATION FOR DISTRIBUTED CRACKS IN A SOLID
M. Kitahara and T. Takahashi
Department of Civil Engineering, Tohoku University
Aoba-yama06, Aoba-ku, Sendai, Miyagi 980-8579, Japan
ABSTRACT. A cement-based specimen that contains penny shaped cracks is prepared and experimental
measurements of scattered waveforms are performed. After the processing of measured waveforms, the
ultrasonic attenuations are obtained and compared with the numerically calculated attenuations. Relatively
good agreement is confirmed for the measured and numerically calculated attenuations in the range of
transducer bandwidth.
INTRODUCTION
For the quality assessment of concrete structures, one of the most widely used nondestructive techniques is the ultrasonic pulse velocity method[l, 2]. An ultrasonic attenuation
caused by the scattering also provides useful information on the damaged state of structural
components. Often an increase of the ultrasonic attenuation is an indication of the overall
stiffness degradation[3].
In this paper, ultrasonic attenuations caused by the scattering are studied from numerical and experimental viewpoints for distributed cracks in a cement-based material. First, the
calculation process of the scattering cross sections is summarized for cracks. Then the dilute
assumption for the distribution of cracks is investigated by considering the multiple scattering
effect of cracks. In this process, the distance of cracks for which the multiple scattering effect
can be approximately neglected is estimated from the numerical calculations. This measure
of the dilute distance of cracks is used to prepare the cement-based specimen that contains
penny shaped artificial cracks. Finally the experimental measurement of scattered waveforms
is performed. After the processing of measured waveforms, the measured attenuation is obtained and it is compared with the numerically calculated attenuation.
Theoretical and numerical calculations on attenuations caused by the scatterings from
distributed cracks have been done by Zhang and Achenbach[3], Angel and Bolshakov[4].
Experimental measurements of attenuations from volumetric scatterers have been reported for
porosities by Adler et al.[5]. A comparison of attenuations between theoretical estimations
and experimental results has been shown for inclusions by Sayers and Smith[6]. A method to
use the frequency dependence of the ultrasonic attenuation has been proposed for the porosity
estimation by Nair et al.[l]. A comprehensive review has been given by Beltzer[8] for the
ultrasonic attenuation. An attempt to measure the attenuation from distributed cracks can be
seen in a paper[9] and more precise arrangement of dilute distribution for cracks is considered
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/$20.00
1156
in this paper.
SCATTERING CROSS SECTION AND ACATTERING AMPLITUDE
The scattering cross section plays a key role in the evaluation of attenuation caused by
the scattering. The calculation process of the total scattering cross section is summarized
here for cracks. Fig.l shows the cracks Sc(= Sci + SC2 + • • • + Scm) in the three dimensional
elastic solid. The incident wave u1 is propagating in the a direction and the scattered wave us
is propagating in all directions. We consider the steady state with the time factor exp(-/<jf)Then the total scattering cross section P is defined as[10]:
ps
(1)
where < • > indicates the time average. In this equation, I1 is the intensity of the incident
wave and it has the form
I1 = 0^$
(2)
where cr\. is the stress associated with the incident wave and uf. is the time derivative of the
incident wave u1.. In Equation(l), Ps is the power of the scattered wave and it is expressed as
Ps = fxtfrf
(3)
JA
where x is a unit normal vector, which points to the field point jc, on a spherical surface A
enclosing all cracks and us. is the scattered wave.
In order to obtain the suitable form of the total scattering cross section for the numerical
calculation, it is convenient to consider the differential cross section:
4^P = lim
dLl
*->°o
FIGURE 1. Cracks Sc(= Sci + $C2 + • • • + Scm), incident wave u1 and scattered wave us.
1157
(4)
where dQ = sinOdOdfi is an element of the solid angle, and us.* is the complex conjugate of
uSj. In Equation (4), the time average of the incident intensity for the longitudinal plane wave
(5)
is introduced.
For the evaluation of the differential cross section in Equation(4), it is necessary to get
the explicit expression of the scattered wave us. at the far field (see, Fig.2). The scattered far
field can be obtained from the far field approximation of the integral representation for the
scattered wave and it reduces to
u.
a=L,T
4nx
(6)
where x = \x\. Furthermore Af and A^ are scattering amplitudes of the longitudinal and
transverse waves and they are expressed in the following form for the scattering problem of
the stress free crack:
A?(jt,^) = -ikaB"ikJ (x) \ nj(y)e-ik^y[uk(y,oj)]dS(y).
Jsc
(7)
In this equation, Sc(= SCi + Sc2 + * • * + ^cm) is the surface of all cracks and
[uk(y,a))] = u+k(y,a)) - u~k(y,a)}
(8)
is the crack opening displacement and B?jk has the following forms
BTijk(x)
= -{SikXj + Sijxk-2xixjxk}
where K = kL/kT and x = x/\x\.
FIGURE 2. Source point y on cracks Sc(= Sci + $C2 + • • • + ^c/n) and field point x.
1158
(10)
Introducing Equation(6) into Equation(4), we get the differential cross section expressed
by the scattering amplitudes
(11)
The total scattering cross section P is obtained from the integration of the differential cross
section dP/dQ,
(12)
d£l
In the numerical analysis, the crack opening displacement [uk] contained in the scattering amplitude in Equation(7) is calculated by the boundary element method.
ON A DILUTE DISTRIBUTION OF CRACKS
In the evaluation of attenuation, we consider the dilute distribution of cracks in the next
section. From this dilute assumption, an independent scatterer approximation is adopted and
the multiple scattering effect of cracks is neglected. The dilute assumption is useful for the
evaluation of attenuation. However, the distance of cracks for which the single scattering
approximation can be accepted is not so clear. Here we consider the multiple scattering effect
for two cracks and try to estimate the order of distance of cracks for which the multiple
scattering effect can be approximately neglected.
For multiple scattering calculations, we consider two types of arrangements for two
cracks as shown in Fig.3. One is the horizontal two cracks and the other is the vertical two
cracks. Here, each crack is assumed to be circular crack with radius a. The distance of
two cracks is d. Two cracks are arranged to be parallel in both cases. The incident wave is
propagating along the ^-direction and the incident wave is assumed to be longitudinal plane
waves.
The scattering cross sections for these two cracks are calculated from Equations (11) and
(12). Fig.4 and Fig.5 show the scattering cross sections for horizontal two cracks and vertical
two cracks, respectively. Fig,4 (a) is the results for the distance d/a = 0.5 and Fig,4 (b) is
the results for d/a = 1.0. The abscissa is the nondimensional longitudinal wave number akL
and the ordinate is the nondimensional total scattering cross section P/(na2). The square (D)
shows the total scattering cross section for two cracks and it includes the multiple scattering
effect. For comparison, the value of two times total scattering cross section for a single crack
is shown by the dot (•) and therefore this value does not contain the multiple scattering effect.
X2
FIGURE 3. Horizontal two cracks (left) and vertical two cracks (right).
1159
D:PM
O:PL
D:P( W )
A:Pr • : 2xPsingle crack
O:PL
Jo
1
°\
P
•
4'
*
B
n
O BC W,
1
°B! BE8
23
Tia -
Lo"°o^,
^
gA°
0
> A A AA
IBBBB '
o° °( )no
1
0<
"U°( )C00
O
SiAAAA,
P
0.5
1
B
n
4-
g'
8| 'gc61
'
WH
I
HH
[
BCT
BOO ^
U
2 3
Tia -
a A(
A
f°°n
°£U
iAAAA
ft
0- «M$i
0.0
yu
•
0
SO^AA,
A:Pr •: 2xPsingle crack
,0000, ^n
0
'%A
;
°ooo( $6AA. ^AAAAj
°( )ooo°
,AAAA
_ _ H j l^
1.0
1.5
2.0
2.5
3.0
3.5
4.0
00.0
0.5
1.0
1.5
2.0
2.5
akL
akL
(a) d/a=0.5
(b)d/a=1.0
3.0
3.5
4.0
FIGURE 4. Scattering cross sections for horizontal two cracks: (a) d/a = 0.5, (b) d/a = 1.0.
D:P(U)
O:PL
ig,
cP
P
»a •
si
4:
D
AI D
A
n i
Tia23:
»
,
0-
0.0
•I
0.5
1.0
•
:P(u)
O:PL
A:Pr •:2xPsingle crack
•"*•
•
*•.,
*•
'••••
' °nn< n
^D
oUn
AA
a%
A:Pr • :2xPSmgle crack
i ' DDDl
D
>ooo°
^
) ODD
[
°o
AA
, 'AiiA,
1.5
'nnnr,[
'°000<
^
2.0
2.5
A
A
W^
3.0
,AAAA'
3.5
4.0
0.0
0.5
1.0
1.5
2.0
2.5
ak L
akL
(a)d/a=1.0
(b) d/a=4.0
3.0
3.5
4.0
FIGURE 5. Scattering cross sections for vertical two cracks: (a) d/a = 1.0, (b) d/a = 4.0.
In the figures, the longitudinal component of the scattering cross section is shown by the circle
(o) and the transverse component is shown by the triangle (A). In Figs.4 (a) and (b), the values
of the total scattering cross section for two cracks (D) and that for two times single crack (•)
are not exactly the same, but we can see they are very close for horizontal two cracks in the
range of calculated nondimensional wavenumbers. In this sense, multiple scattering effects
are not so large at the distance of d/a =1.0 for horizontal two cracks. Fig.5 (a) is the results
for the distance d/a = 1.0 and Fig.5 (b) is the results for d/a = 4.0 in the case of vertical two
cracks. For vertical cracks, we can see from Fig.5 (a) that multiple scattering effects for the
total scattering cross section are very large at the distance of d/a = 1.0. However, as shown
in Fig.4 (b), multiple scattering effects become small in the range of small nondimensional
wavenumbers for 0 < akL < 2.0 at the crack distance of d/a — 4.0. It is to be careful that the
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multiple scattering effect becomes large in the range of large nondimensional wavenumbers
as can be seen in Fig.4 (b) even in the case of d/a = 4.0.
ATTENUATION CAUSED BY SCATTERING
We consider an elastic solid that contains dilutely distributed circular cracks as shown
in Fig.6. The radius of all cracks is a and all crack planes are parallel in the Xi-x2 plane,
where the distribution of cracks is random. The waves are assumed to be propagating along
the ;c3-direction. Then the plane longitudinal wave can be written as
u(x) = u(
(13)
where CL(CO) is the phase velocity of the longitudinal wave and a(a)) is the attenuation. The
average intensity < / > for the waves in Equation (13) reduces to
< / >=< /o > e-2ax*
(14)
from the definition of intensity in Equation (2), where < /o > is the intensity at the reference
position *3 = 0. By considering the change of intensity in the propagation direction, the
differential equation for the intensity < / > can be obtained
dx3
+ NP(co) < I >= 0.
(15)
In the above equation, the dilute assumption for cracks is adopted and the relation < Ps >=
P(a)) < I > in Equation (1) is used. In Equation (15), N is the number of cracks in the
unit volume and P(OJ) is the total scattering cross section for a single crack. The solution of
Equation(15) is easily obtained as
(16)
From Equations (14) and (16), the attenuation a(cS) is obtained in the following form
(17)
For circular cracks with radius a, it is convenient to introduce a nondimensional crack
density parameter 6 = Na3. Then the nondimensional attenuation coefficient a(co)a can be
written as
(18)
a(a>)a =
2 no
Propagation
direction
Cracks
FIGURE 6. Dilutely distributed circular cracks (Crack planes are parallel in the x\-X2 plane).
1161
MEASURED ATTENUATION FROM ARTIFICIAL CRACKS
A cement-based specimen was prepared for the measurement of attenuation caused by
distributed cracks as shown in Fig.7. In the left half of the specimen, penny shaped artificial
cracks with radius a=2mm and thickness f=0.8mm exist and planes of artificial cracks are
arranged to be parallel to the surface of the specimen. The artificial crack density e = Ncr* =
0.008 is chosen to satisfy the dilute assumption described in the results of Figs.4 and 5. There
are no artificial cracks in the right half of the specimen. The specimen is immersed in water
and the forward scattering waveforms are measured.
In the right half of the prepared specimen, the reference waveform UQ is obtained and it
can be written as
u0(x) = u(xl,x2)e~aQ(^X3ei[(a}/CLo(^}X3~ajt].
(19)
The forward scattering waveform MI is measured in the left half of the specimen with artificial
cracks and it has the form
,. f^.\ _ f i f v
v ^SJ-GI^XI
U\\X) — U{X\,X2)£
6
i[(a}/CLi(cj))x3-a}t]
.
/"")fY\
\^J)
Taking the absolute values of Eqs.(19) and (20) and then taking the natural logarithm, we get
the expression for the measured attenuation:
f \ - aQ(u>)}a
r \\ = -In
i o
f \ = i[ai(a))
a(a))a
h
HI (a;)
(21)
where the thickness of the specimen h is used as the propagation distance of waves in the
prepared specimen.
In the experiment, transducers with center frequency of 0.5 MHz are used for the measurement of forward waveforms in Equations (19) and (20). At the center frequency, the
wavelength is about 8mm in the cement-based specimen.
Fig.8 shows the result of measured attenuation. In this figure, the circles are measured
attenuations obtained from the data processing in Equation (21). For comparison, the attenuation evaluated from the numerical calculation for cracks with the same crack density
6 = No3 = 0.008 as in the cement-based material is shown by the solid line. It can be
seen from the results in Fig.8 that the agreement of the measured and calculated attenuations
is fairly good in the range of the effective bandwidth (from 0.25MHz to 0.85MHz) of the
transducer.
200tfi!n
200mm
100mm
lOQmrn
FIGURE 7. Prepared specimen (Artificial cracks with density e = Na3 - 0.008 exist in the left half and there
is no cracks in the right half of the specimen).
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0.0!
~®~ Measurement
— Numerical analysis
0.04
§ o.o:
§ o.o;
o.oi
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 . 9 1.
Frequency f (MHz)
FIGURE 8. Measured attenuation and numerically calculated attenuation for cement-based specimen with
crack density of e = Na3 = 0.008.
CONCLUSIONS
The dilute assumption for the distribution of cracks was investigated by considering the
multiple scattering effect of cracks. The experimental measurement of scattered waveforms
was performed and the experimentally evaluated attenuation was compared with the numerically calculated attenuation. Relatively good agreement was confirmed for the measured and
numerically calculated attenuations in the range of transducer bandwidth.
REFERENCES
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2. Popovics, S., Rose, J.L. and Popovics, J.S., Cement and Concrete Research 20, 259
(1990).
3. Zhang, Ch. and Achenbach, J.D., Int. J. Solids Structures 27, 751 (1991).
4. Angel, Y.C. and Bolshakov, A., Wave Motion 31, 297 (2000).
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6.
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8.
9.
Adler, L., Rose, J.H. and Mobley, C., J. AppL Phys. 59, 336 (1986).
Sayers, C.M. and Smith, R.L., /. Phys. D: AppL Phys. 16, 1189 (1983).
Nair, S.M., Hsu, D.K. and Rose, J.H., J. Nondestr. Eval. 8, 13 (1989).
Beltzer, A.I., Wave Motion 11, 211 (1989).
Kitahara, M., Kishibe, D., Takahashi, T. and Kishi, N., in Nondestructive Characterization of Materials X, eds. R.E. Green, T. Kishi, T. Saito, N. Takeda and B.B. Djordjevic,
Elsevier Science, Oxford, 2001, p.231.
10. Gubernatis, I.E., Domany, E. and Krumhansl, J.A., / AppL Phys. 48, 2804 (1977).
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