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The effects of material attenuation on acoustic resonance scattering from cylindrical tubes
ELSEVIER
Ultrasonics 34 (1996) 737-745
The effects of material
attenuation on acoustic resonance cylindrical tubes
Jong PO Lee a,b,*, Ji Ho Song ‘, Myoung
scattering
from
Seon Choi a
‘Division ofNondestructive Evaluation, Korea Atomic Energy Research Institute (KAERI). PO Box 105, Yusong, Taejon 305-600, Korea bDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-l Kusung-dong, Yusong-ku, Taejon 305-701, Korea ‘Department of Automation and Design Engineering, Korea Advanced lnsfitute of Science and Technology (KAIST). Seoul 130-012, Korea
Received 20 November 1995
Abstract This paper presents the effects of material attenuation on acoustic resonance scattering (ARS) from cylindrical shells for the purpose of applying it to ultrasonic nondestructive evaluation (NDE). Complex wave numbers are introduced to include the effects of material attenuation. Resonance widths versus resonance frequencies with attenuation neglected have been numerically analyzed (up to k,u = 230) for five (S,, A,, S1, SZ, and A,) modes not only to determine resonance frequency regions in which attenuation effects must be investigated intensively, but also to select appropriate frequencies from an NDE point of view. It has been found that frequency regions of interest are those which have relatively narrow resonance widths but not too narrow ones. Material attenuation results in a decrease of resonance peak amplitude as much as an increase of resonance width. The increase of resonance width attributed to material attenuation alone is linearly related to attenuation coefficients and its proportional constants are dependent on the types of modes. The ratio of the resonance width caused by material attenuation alone to that caused by radiation Keywords:
damping
governs
the relationship
Resonance scattering; Attenuation;
between
resonance
Ultrasonic NDE
1. Introduction
In order to evaluate the integrity of cylindrical tubes, ultrasonic NDE, such as pulse-echo or throughtransmission techniques using elastic bulk waves, have been commonly used. However, these techniques have their unique advantages and limitations. As a new approach to the evaluation of tube quality, acoustic scattering or acoustic resonance scattering from submerged shells has been good subject of research. This includes analytical, numerical, and experimental studies. The pertinent literature over four decades has been well summarized by Gaunaurd and Werby [ 11, and Gaunaurd [ 21. Faran [3] obtained solutions for the scattering of plane sound waves from an infinite elastic cylinder immersed in a fluid medium. Later, Doolittle and ijberall [4] obtained the general solutions for the scattered field * Corresponding author. Fax: +82-42-861-1184; e-mail: jplee2Bkaeri.re.kr
width and amplitude.
corresponding to a plane sound wave incident upon an infinite elastic circular-cylindrical concentric shell imbedded in a fluid and enclosing another fluid. There exist detailed methods for calculating the acoustic pressure scattered by an elastic cylinder. The majority of the calculated form functions have been obtained by the direct summation of the Rayleigh normal mode series. The Watson-Sommerfeld Transformation (WST) [S] is often applied to this series to interpret the results obtained. This is accomplished in the complex plane and is not a simple way in the resonance region. Resonance Scattering Theory (RST) [6--91 enables us to easily analyze the circumferential waves and separate them into background terms and resonance peaks. The backgrounds are a contribution due to the scatterer’s shape and the resonances are a contribution due to material composition [ 21. One of the important things in material composition affecting resonances is material attenuation. Furthermore, most industrial tube materials possess, more or less, attenuation of which the attenuation coefficients are dependent on the frequencies of the incident waves.
0041-624X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PII SOO41-624X(96)00075-3
738
Jong PO Lee et al. / Ultrasonics 34 (1996)
Acoustic scattering with absorption taken into account has been studied by many authors [lo-171. Hageman et al. [lo] calculated the scattered wave pressure from irradiation of cylindrical, thin viscoelastic shell by a plane monochromatic sound wave. Vogt et al. [ 111 measured monostatic reflection from an absorbing sphere in water and compared it with the calculation results which include the effects of absorption of shear and compressional waves in lucite. They introduced complex wave numbers in the theory to include absorption considerations. Later, acoustic reflection was studied for absorbing cylinders by Schuetz and Neubauer [12], for absorbing spheres by Hasegawa et al. [ 131, and for layered elastic absorptive cylinders by Flax and Neubauer [14]. Since the advent of the resonance scattering theory, resonant scattering from elastic and/or viscoelastic objects, either spherical or cylindrical shells, has been studied using the Kelvin-Voigt model. Gaunaurd and ijberall [ 151 studied theory of resonant scattering from spherical cavities in elastic and viscoelastic media and Gaunaurd and Kalins [ 161 studied resonances from coated spherical shells. Ayres and Gaunaurd [ 171 drew their conclusions that the locations of the resonances are shifted slightly and their widths are broadened. In addition, the amplitudes of the resonances peaks decreased as viscoelastic losses increased. Recently, Huang [ 181 analyzed energy dissipation in sound scattering by a submerged cylindrical shell. He introduced complex Young’s modulus of shell material to include a loss factor due to structural damping. He showed that dissipation effects are important near the resonance frequencies of the shell. In spite of a number of previous studies on resonance scattering by attenuating cylindrical shells, relationships between attenuations and resonances are not analyzed in detail such that the results can be utilized. The motivation of the present work was provided by this fact and by the necessities of the application of acoustic resonance scattering to ultrasonic NDE. The objective of this study is to investigate the effects of material attenuation on acoustic resonance scattering in detail and to analyze characteristics of resonance behaviour for Zircaloy-4 tubes. Material attenuation effects are taken into account by introducing complex wave numbers whose imaginary parts are given by the attenuation coefficients (damping factors). Relationships between resonance widths and peak amplitudes are numerically analyzed and dicussed for S,, A,, S1, Sz, and A, modes with different attenuations.
2. Theoretical considerations
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an infinite and elastic cylindrical shell of outer radius a and inner radius b as shown in Fig. 1. The fluid media outside and inside the shell are labelled 1 and 3, respectively and the shell material 2. Longitudinal and transverse wave velocities of the elastic shell are cL and cT, respectively. Wave numbers are kj = o/cj (j = 1, L, T, 3). The scattered field P,, at an observer’s point (I; 0) located outside the shell is given by P,,(r, 19,t) = epiW’ f q,i”b,H!,l’(k,r) n=O
cos(n0),
(1)
where E, = 1 for n = 0 and E, = 2 for n 2 1. The coefficients b, are determined from the boundary conditions and have the form of the ratios of two 6 x 6 determinants, b, = BJD,. The 30 nonvanishing elements of these determinants B, and D, have all been given in the Appendix of Ref. [ 193. HL”(k,r) is replaced by its asymptotic form in the far field region. Then, the scattered pressure becomes P,,(r, 8, t) 2:
\i
c ei(klr-@f(O, x,),
(2)
n=O where x1 = kla. The form function f,(0, x1) represents the angular distribution of the scattered pressure for a given x1 and is called the scattering form function of the nth partial wave. A number of previous workers on RST such as Gaunaurd and Werby [20] showed that the scattering form function of partial waves consists of resonance peaks and smooth backgrounds. Resonance peaks can be seen clearly by subtracting an appropriate background which depends on the shell thickness, material properties, and frequency. For metal shells, it has been well known that the scattering amplitude from the rigid cylinder of the same size could be used as a proper background. For a rigid target, there is only one boundary condition of no displacement at the surface and the scattering form function has the following simple
II
2.1. Scattering form function
A plane acoustic wave (angular frequency w) of unit amplitude ei(kl*-“t) interacts at normal incidence with
Fig. 1. The geometry of a plane acoustic wave scattering from an infinite cylindrical shell. Here, Cj and p, denote wave velocity and material density, respectively.
Jong PO Lee et al. / Ultrasonics 34 (1996)
737-745
139
by the complex wave numbers (4) where the superscript (r) refers to rigid background. The modal resonances of elastic partial waves are given by background subtraction %)I = -% 6
If,(R x1) -f?(R
(b” - b2’) cos(ne) .
(5)
2.2. Resonance response with no attenuation Each resonance is characterized by two integer numbers (n, 1). The first one, n represents the number of wave lengths of the surface wave needed for resonance around the circumference of the cylinder. The second one, 1 is considered to be related to the nature of the surface wave [ 211. According to the resonance scattering theory [6-g], the term of form function of resonance without consideration of attenuation can be expressed by;
k,=k,+ia,,
(loa)
k,=k,+icc,,
(lob)
where ctL and ~1~are attenuation coefficients of longitudinal and shear waves, respectively (Np/m). The bar denotes complex numbers. Here, CQ,and c+ are assumed to be a linear function of the incident wave frequency, because the linear term is predominant in most industrial materials. In other words, attenuation coefficient times wave length is constant (/?). %JI, 5 Pi_,
(Ha)
%& = BT,
(lib)
where I, and & are wave length (m) of longitudinal and shear wave, respectively. Then, the complex arguments of Bessel and Neumann functions in b, of the Eq. (3) can be expressed in terms of k,a as follows xj~~a=kja+i(olja)=k,a~
(j = L, T) .l
(6) yj~k,b=kjb+i(~jb)=$(kIa)~
where x,,~- (k, a),, is the resonance frequency, r,, resonance width, and x = x1 = kIa, respectively. The magnitude of backscattered partial wave resonance (0 = x) can be simply derived from the Eq. (6)
Mx)I =
4 &
1 Jl
(n 2 1).
(7)
+ ((x - xJ(W2))2
Peak amplitude, p is simply obtained from the Eq. (7) by putting x = x,~. P=
Ig”tx”dl = Jk
(n 2 1).
(8)
Eq. (8) is the same as the equation derived differently by Veksler [ 221. For a single resonance, the relationship between resonance width r,, and radiation attenuation a,, can be expressed by [23] a,=--.---,
Cl
5
r, 2
(9)
where ~1,is attenuation coefficient (Np), c1 sound velocity in water, cg group velocity of resonances in the shell. 2.3. Resonance response with attenuation
The effects of shell attenuation can be considered by introducing the complex wave numbers. For this purpose, the real wave numbers included in the arguments of the elements in the 6 x 6 determinants are replaced
(j = L, T). I (13)
The imaginary parts of xj and 5 contribute to the effects of attenuation rather than /IL or pT alone. Then, the effects of ultrasonic attenuation can be analyzed in terms of k,a by replacing real arguments of Bessel and Neumann functions with complex arguments of Eqs. ( 12) and (13). If total resonance width r, and the resonance width r, attributed to radiational damping are calculated, resonance width l-‘, attributed to shell attenuation can be obtained.
r,=r,-r,.
(14)
3. Numerical analysis and discussion 3.1. Calculation Numerical analysis is carried out using Mathematics for an air-filled Zircaloy4 cylindrical tube immersed in water. There are several different sizes of Zircaloy4 tubes which are used as nuclear fuel cladding tubes. Here, calculation is made for a typical size of the tubes. The ratio of b/a is 0.888 (2a = 10.7 mm, 2b = 9.50 mm, d = 0.6 mm). The properties of the Zircaloy4 tube, water, aluminium, and air used in calculations are listed in Table 1. It is well known that a Zircaloy4 tube has attenuation coefficients which are in the medium range among metals. Its attenuation coefficients of longitudinal
Jong PO Lee et al. / Ultrasonics 34 (1996)
140 Table 1 The properties
of substance
Substance
Water Air Aluminium Zircaloy-4
1416 _
1000 1.2 2740 6.550
344 _
3100 2440
6380 4540
_
and shear waves, BL2: PT = 0.0028 Np. Analysis is done for four cases, that is, (A) j& = /IT = 0, (B) j3r = & = 0.0014 Np, (C) /IL = /IT = 0.0028 Np, and (D) /Ii, = Pr = 0.00467 Np. Resonances are separated with rigid background subtraction for So, A,, Si, S2, and A, modes. Resonance frequency, peak amplitude, and resonance width for the five modes are investigated with attenuation taken into account. A small calculation step is used such as k,a = 1O-2-1O-6 depending on each resonance width. In order to calculate resonance width accurately with an error less than l%, the small calculation step has been used such that at least 100 calculation points are included within each resonance width. 3.2. Validity of the program The validity of our program is checked. It is difficult to find previous results for direct comparison to ensure whether the results produced are reasonable and valid. The only possibilities of checking are based on comparison with the previous results in the literature for the cases where attenuation is neglected. The first six sets of modal resonances are calculated for a thick (b/a = 0.1) aluminium, air-filled, cylindrical shell in water by setting the attenuation coefficients, /IL and /?r to zero. The present results are shown in Fig. 2 and compared with 1OL
I n=O
0.5 0.0
_
A
L
1.0
40.5 %
0.0
.z 1.0 EO.5 z
0.0
8 1.0 f
0.5
iE 0.0 z" 1.0
l_A_AJ__ 1.0 -
Il=5 A__&_
I 0
10
20
30
40
ha
Fig. 2. First six sets of modal resonances (n = 0, 1, , 5) for a thick (b/a=O.l) aluminium, air-filled, cylindrical shell in water (no attenuation, /& = pT = 0).
737-745
the Fig. 1 of Gaunaurd and Werby [ZO]. Excellent agreement is shown between the present and previous results for backscattering resonances from aluminium shell in water. What is distinct in the present results in Fig. 2 is that some resonances having narrow resonance width are more distinguishable than those of previous results. This is caused by a small calculation step. Other comparisons have also been made. One is the calculation of the first six sets of modal resonances and the other is that of modal backgrounds for a thin, air filled, aluminium, cylindrical shell in water (b/a = 0.9). These two present results, which are not shown here due to limited space, are compared with the Fig. 6 and Fig. 7 of previous results reported by Gaunaurd and Werby [ 201. Good agreement is also obtained. 3.3. Resonances with attenuation neglected Before the analysis of attenuation effects on acoustic resonance scattering, resonance frequency, peak amplitude, and its width for various modes with attenuation neglected must be determined. This is because it is very important to determine the regions of resonance frequency in which we are going to analyze attenuation effects in detail. Choi et al. [23] identified the first several modes of resonances for an air-filled Zircaloy-4 cylindrical tube immersed in water. The first two modes for thin aluminium and Zircaloy4 tubes have been studied by many authors [ 19-241 but it does not seem to be clear. One of the possible reasons appears to be that those frequency regions are in the transition area of soft and rigid background. Therefore, resonances are not clearly separated by using either soft or rigid background. If an ideal background which can encompass these transition regions is known, it would be very helpful to characterize those modes. The studies on the first two modes continues to be an important area of research. A new background using the effects of material attenuation on ARS has been found (submitted to the J. Acoust. Sot. Am.). The results of the study on the first two modes using a new background will be reported later. So, we are here dealing with the next five modes of resonances, which are So, A,, S,, S2, and A, modes. Fig. 3 shows that resonance widths versus resonance frequencies, both k,a and frequency x thickness (fd up to k,a = 230 for the five modes with attenuation neglected. An fd value is obtained using the formula fd = k, a( 1 - b/a)c,/2x. Resonance peak amplitude is not shown here as it can be simply obtained from Eq. (8) without tedious calculations. As can be well explained by Eq. (9) resonance frequencies having large radiation attenuation are not useful from an NDE point of view. The frequency range of interest is around the valley as shown in Fig. 3 because those frequency regions with wide resonance widths have large radiation attenuation.
Jong PO Lee et al. / Ultrasonics 34 (1996) 737-745
Frequency x thickness, 1.32
10.00
2.63
741
fd (MHz mm) 3.95
5.26
6.56 1
I
i
1 - (c)
S,
I,
I,,
&,
n=1
mode
,
,
,
,
,
:
:
(.
El3
,
-
...* Q..
.*.... .. _... mmm.........r.m
"='O
~~-,,.,,,,,,,,,,,,,,,,,,,,, 5
-
t c2-
1
0 3
^
(d)
S,
mode
1
.:
n=61
.‘...
/--‘-“..
.
/ ..fl.*
.. ‘. “.Yn....n.~~
...y
n=1
~~1'1~11111/1111,,,,,,,,
- (e)
A2
mode
n=l
t,,.,,.,,,,,,,.,,,,,,,,,,I 0
50
100
150
200
250
Fig. 3. Resonance widths versus resonance frequencies of (a) So mode, (b) A, mode, (c) Sr mode, (d) Sa mode, and (e) A, mode for an air-filled Zircaloy-4 tube (b/a = 0.888) immersed in water when attenuation is neglected.
Hence, it is needed to pay attention to the frequency regions having relatively narrow resonances. However, there are some limitations to the proper selection of appropriate frequencies having relatively narrow resonances because these frequencies are quite dependent on shell attenuation as investigated in the next section. From Fig. 3, the frequency regions for the five modes having relatively narrow resonance smaller than 0.5 are shown in Table 2. What we must pay attention to is the area around these frequency regions. 3.4. The egects of material attenuation on resonance scattering Three backscattered resonances which have relatively wide (S, mode, n = 64), intermediate (A, mode, n = 8), and narrow resonance width (So mode, n = 7) are
Table 2 Resonance frequency regions of interest having narrow resonance width, r, smaller than 0.5 (b/a = 0.888) Mode
SO A, SI SZ SZ A,
From
To
n
kra
fd
?I
kla
fd
2 1 29 1 43 1
6.57 46.61 99.62 93.01 161.07 139.15
0.173 1.226 2.621 2.447 4.238 3.660
17 30 50 3 67 38
47.77 94.13 150.86 94.39 205.65 154.17
1.257 2.477 3.969 2.483 5.411 4.056
compared for four different attenuations as shown in Fig. 4. They are (A) /IL = PT = 0, (B) pL = /%r= 0.0014, (C) jL = pT = 0.0028, and (D) pL = /ST= 0.00467 Np. When attenuations are included in the calculations,
142
Jong PO Lee et al. / Ultrasonics 0.25
a 0.20
O.OO L
b
0.25
b
0.20 P go.15
_&I.,0 ,’ * ‘, ,a; _ . ‘, \s ,’ \\
0.05
2 0.00 ELI 138.4
138.8
139.2
139.6
‘v 0.5
C
0.4
m 6 0.3
‘: .c
0.1
p
0.0 0.2 r-I---l
II.
20.4
I.
>I
I.
20.6
20 8
21.0
ha
Fig. 4. Resonance characteristics with different attenuation coefficients Br=Br of 0 (solid), 0.0014 (dashed), 0.0028 (dashed and dotted), 0.00467 (dotted) Np for (a) S2 (n = 64), (b) A, (n = 8), and (c) So (n = 7) mode.
resonance peak amplitude decreases and resonance width increases due to shell attenuation effects as reported in the previous literature [ 17,181. These effects are more significant in narrow resonances (high Q factor)
Table 3 Resonance
characteristics
with different
34 (1996)
737-745
than wide resonances (low Q factor). Fig. 4 is not enough to see attenuation effects quantitatively in detail. Hence, Table 3 summarizes resonance characteristics which include resonance frequency, peak amplitude, percentage of amplitude relative to those in the case of no attenuation, resonance width, and the value of peak amplitude times resonance width. As shown in Table 3, resonance frequency is slightly changed in which variations are almost negligible and amplitude in the case of no attenuation follows the formula of Eq. (8). In addition, the amplitude drop is very significant depending not only on attenuation coefficients but also on Q factor. Here, attention must be paid to each resonance frequency. Fig. 4c is for low frequency, Fig. 4b for middle frequency, and Fig. 4a for high frequency, respectively. It is expected that frequency effects will be higher at high frequency than at low frequency because attenuation coefficients are given as a linear function of incident wave frequency. This will be shown in the next section. In spite of that, amplitude of S, mode (n = 7) decreases 17.6% more significantly than either 47.2% of S2 mode (n = 64) or 22.8% of A2 mode (n = 8). What is most important is that this amplitude drop makes resonance width broaden due to shell attenuation. Thus, the value of amplitude times resonance width for one single resonance remains the same with different attenuations. We have tried to prove this important fact theoretically, but have not been successful. This proof will be left to others. The reason why pT is not the same for SO mode (n = 7) is because this resonance is not symmetric. The resonance frequency region is near the transition area of soft and rigid background. Thus, background is not fully subtracted with rigid background used. Therefore, additional resonance width seems to be included during the calculation of resonance width in the case of high material attenuation. Single
attenuations
Atten.
Freq.
Amp].
Ampl.
%I
P
W)
Width r
PT
fL=h 0
0.0014 0.0028 0.00467
200.100 200.101 200.102 200.104
0.15954 0.11939 0.09540 0.07523
100 74.8 59.8 47.2
0.266 0.354 0.442 0.560
0.0424 0.0423 0.0422 0.0421
Az mode n=8 A@, a) = 0.0005”
0 0.0014 0.0028 0.00467
139.009 139.008 139.006 139.002
0.06315 0.04369
100 49.6 33.0 22.8
0.0610 0.1230 0.1850 0.2680
0.0117 0.0117 0.0117 0.0117
S, mode n=7 A(k,a) = 0.00005”
0 0.0014 0.0028 0.00467
0.49630 0.19663 0.12352 0.08748
100 39.6 24.9 17.6
0.00595 0.01575 0.02690 0.04525
0.00295 0.00310 0.00332 0.00396
Mode
S2 mode n=64 A&a) = 0.002”
a Calculation
step.
20.6764 20.6773 20.6789 20.6820
Jong PO Lee et al. / Ultrasonics 34 (1996) 737-745
143
resonances are investigated, but it is necessary to know resonance characteristics for all the frequency regions of interest as shown in Table 2. 3.5. Resonance characteristics of So, A,, and S1 modes It has already been shown that the frequency regions of interest are listed in Table 2. Here, resonance characteristics of So, A,, and S1 modes are analyzed in detail for these resonance frequency regions. Figs. 5,6, and 7 show the variations of resonance amplitudes and total resonance widths with three different attenuation coefficients, that is (1) jI,_= BT= 0 Np, (2) fiL = PT = 0.0014 Np, and (3) fiL = fiT = 0.0028 Np for the So, Al, and S1 modes, respectively. Resonance width has its minimum value at n = 6 for So mode, n = 1 for A, mode, and n = 40 for S, mode, and varies with n or resonance frequency. As was mentioned in the previous section, amplitude drops take place significantly in the frequency region having narrow resonance width. Exact calculation of resonance width is difficult for the resonances of which amplitude reaches a value smaller than 0.1 due to shell attenuation because background can be involved in the calculation of resonance width in the cases where background has not been removed perfectly. Therefore, there are no data points in the deep valley area when attenuation is included in the analysis. As shown in Figs. 5, 6, and 7, the resonances having relatively narrow resonance widths in the case of no attenuation are greatly affected by material attenuation.
O.:I k,a”
b
50
60
100
I
80
90
Fig. 6. Same as in Fig. 5, except for A, mode.
110
120
130
140
150
I
140
150
160
k,a
IIT = 0.6 & 9 5 0.4
70 k,a
0.6 _ c L5 % g
0.2
E 5: g
0.5 0.4 0.3 0.2 0.1. 0.0 t
-o.ll.~.~.~.,.~.~ 100
110
120
130
I
k,a
Fig. 7. Same as in Fig. 5, except for S1 mode.
I
I.
5
I.
10
I.
15
%
20
I.
25
I.
30
I
35
.I
40
k,a
Fig. 5. Variations of (a) resonance amplitudes and (b) total resonance widths for So mode with different attenuation coefficients (/IL =/IT); ...M...: 0, ...O...: 0.0014, and . ..A.... 0.0028 for an air-filled Zircaloy4 tube immersed in water (b/a = 0.888).
Resonance width caused by attenuation alone can be calculated using the Eq. ( 14), namely r. = r, - r,. Fig. 8 shows each component of resonance widths for S1 mode. Fig. 8a is for material attenuation /IL = /& = 0.0014 Np and Fig. 8b is for /I,_= j& = 0.0028 Np. r, is linearly increased as k,a increases. From the fact that the value of resonance peak amplitude times total resonance width is constant with attenuations for a single resonance, it
Jong PO Lee et al. / Ultrasonics 34 (1996) 737-745
744
0.0
1
0.7,
'v
b .
100
110
120
130
140
150
ha
Fig. 8. Variations of total resonance widths (r, = r, + r,) for Sr mode with different attenuations, (a) b = 0.0014, (b) fi = 0.0028: .t. 0 ..‘: r,, ...W,..: I-,, and . ..A...: r,.
can be written PJ-, = P*C = P*(C + T*), where the asterisk denotes material attenuation. p* can be rewritten
fed into the transducer. However, the reasons can be explained based on the present results, that is, those resonances with small resonance widths could not be detected due to material attenuation. If attenuation factors are taken into account during theoretical calculation, those sharp resonances are damped out due to material attenuations. Thus, they are not present theoretically and experimentally. In other words, if r, >>r,, then p*/p, 1: l/[ 1 + (r,/r,)] + 0. From the Figs. 5 and 6, r, curve can be drawn for S, and A, as has been done in Fig. 8. Fig. 9 shows r, curve versus k,a with different material attenuations for So, A,, and S, mode. r, is linearly increased with k,a, but the slopes of the curves are different depending on attenuations and types of modes. Table 4 shows the slopes of S,,, A,, and S, mode with two different attenuations. If material attenuation is increased by two times, the slopes of r, curves are also increased by two times for all three modes. In order to confirm the relationships between attenuation (fi) and r,, calculations are made for four different attenuations. Table 5 shows the results. It is confirmed that r. is linearly proportional to the /? and p,T, N p*r,* is maintained as shown in Table 5.
4. Conclusions (15) Hence,
The effects of material attenuation on acoustic resonance scattering from an air-filled cylindrical tube
(16) The most important factor affecting the relationship between resonance width and peak amplitude is the ratio of r,/r,. If the resonances have r, larger than r,, which means r, 2 2r, or those having r, curve underlying r, curve as in Fig. 8, then resonance amplitudes are decreased to more than half of the original resonance amplitudes. Eq. (16) indicates that the ratio of r, to r, governs the percentage of the amplitude decrease. The very narrow and sharp resonances (r, <<1) when attenuation is not considered will be severely attenuated if attenuation is taken into account, because r, will be far greater than r, even though attenuation of shell material is small. Talmant et al. [25] calculated resonance frequencies and their widths without attenuation taken into account for air-filled duraluminium cylindrical shells immersed in water. They also measured resonances experimentally and compared theoretical results with experimental results. According to their results, some resonances which have narrow resonance widths in theoretical calculations were not observed even though they were inside the range of measurements. They explained that the reasons were probably because those resonances were very narrow and very little energy was
m D
10
20
30 ha
40
50
50
60
70
80
90
140
150
L
I
40
ha
P
100
110
120
130 0
Fig. 9. Resonance widths, r, attributed to different attenuations; ...O..‘: 0.0014, ..‘A.-.: 0.0028 for (a) S,, (b) A,, and (c) S, mode. Table 4 The slopes attenuations Attenuation
A.=/% 0.0014 0.0028
of r, curves for S,,, A,, (refer to Fig. 9)
and
S, modes
with
different
Slope of the r, curve S, mode
A, mode
S, mode
0.00042 0.00084
0.00046 0.00092
0.00048 0.00096
Jong PO Lee et al. / Ultrasonics
Table 5 The variations
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745
References of r, value with different
Atten.
So mode, n = 9
BL=BT
p
0.4395 0.3367 0.2731 0.2297 0.1984
0.0384 0.0505 0.0620 0.0745 0.0865
attenuations A, mode, n = 5
r, 0 0.0014 0.0028 0.0042 0.0056
34 (1996)
0 0.0121 0.0236 0.0361 0.0481
P
r,
r,
0.3217 0.2511 0.2059 0.1745 0.1514
0.078 0.100 0.122 0.144 0.166
0 0.022 0.044 0.066 0.088
(Zircaloy4) immersed in water is investigated. Complex wave numbers are introduced to consider the material attenuation of cylindrical shells. The conclusions obtained are summarized as follows. First, from the view point of the application of acoustic resonance scattering to ultrasonic NDE, resonance width (r,) versus resonance frequency @,a) is calculated up to k,a = 230, or fd = 6 (MHzmm) for the five (So, A,, S1, ST, and A2) modes. Frequency regions of interest which have relatively narrow resonance width have been found to be those listed in Table 2. Second, ultrasonic attenuation contributes to an increase of resonance width and brings a decrease of resonance amplitude. The value of peak amplitude times total resonance width (r, = r, + K‘,) remains the same for each resonance with different attenuations. The increase of resonance width (r,) caused by material attenuation is linearly increased with the values of attenuation. The relationship between resonance width and peak amplitude is given by Eq. (16) and the ratio of r, to r, governs amplitude drop. Sharp resonances with attenuation neglected are damped out when material attenuation is taken into account. This is because the resonance width caused by material attenuation is far greater than that caused by radiation damping (I’, >>r,). The slope of r, curve versus k,a is linearly dependent on attenuation coefficients (/I) and varies with types of modes.
and M.F. Werby, Appl. Mech. Rev. 43(8) Cl1 G.C. Gaunaurd (1990) 171. c21 G.C. Gaunaurd, Appl. Mech. Rev. 42(6) (1989) 143. c31 J.J. Faran, Jr., J. Acoust. Sot. Am. 23 (1951) 405. and H. iiberall, J. Acoust. Sot. Am. 39(2) c41 R.D. Doolittle (1966) 272. J. Acoust. c51 J.W. Dickey, D.A. Nixon and J.M. D’Archangelo, Sot. Am. 74( 1) (1983) 284. in: Proc. IUTAM Symp.: Northwestern Univ., C61 H. iiberall, Evanson, IL, Sept 12-15, 1977, Eds. J. Milkowitz and J.D. Achenbach (Wiley Interscience, New York, 1978) p. 239. and H. ijberall, J. Acoust. Sot. Am. 63 (1978) [71 G.C. Gaunaurd 1699 [reviewed in Appl. Mech. Rev. 32 (1979) 12551. and H. nberall, Physical Acoustics C81 L. Flax, G.C. Gaunaurd (Academic Press, 1981) XV. J. Acoust. Sot. Am. 61 (1977) 360. c91 G.C. Gaunaurd, Cl01 D.H. Hageman and V.L. Glass, J. Appl. Phys. 44(90) (1973) 3907. R.H. Vogt, L. Flax, L.R. Dragonette and W.G. Neubauer, [Ill J. Acoust. Sot. Am. 57(3) (1975) 558. J. Acoust. Sot. Am. 62(3) Cl21 L.S. Schuetz and W.G. Neubauer, (1977) 513. Y. Kitagawa and Y. Watanabe, J. Acoust. Sot. [I31 T. Hasegawa, Am. 62(5) (1977) 1298. J. Acoust. Sot. Am. 61(2) Cl41 L. Flax and W.G. Neubauer, (1977) 307. and H. iiberall, J. Acoust. Sot. Am. 63(6) Cl51 G.C. Gaunaurd (1978) 1699. [16] G.C. Gaunaurd and A. Kalins, Int. J. Sol. Structures 18(12) (1982) 1083. [17] V.M. Ayres and G.C. Gaunaurd, J. Acoust. Sot. Am. 81(2) (1987) 301. [ 181 X.Y. Huang, Acustica 77 (1992) 221. [19] G.C. Gaunaurd and D. Brill, J. Acoust. Sot. Am. 75(6) (1984) 1680. [20] G.C. Gaunaurd and M.F. Werby, J. Acoust. Sot. Am. 77(6) (1985) 2081. [21] H. Bartard and G. Quentine, J. Acoust. Sot. Am. 91(2) (1992) 581. [22] N.D. Veksler, Acustica 79 (1993) 22. [23] M.S. Choi, H.C. Kim and M.S. Yang, Ultrasonics 30(4) (1992) 213. [24] M.S. Choi, M.S. Yang and H.C. Kim, Ultrasonics 30(4) (1992) 221. [25] M. Talmant, G. Quentin, J.L. Rousselot, J.V. Subrahmanyam and H. iiberall, J. Acoust. Sot. Am. 84(2) (1988) 681.
Ultrasonics 34 (1996) 737-745
The effects of material
attenuation on acoustic resonance cylindrical tubes
Jong PO Lee a,b,*, Ji Ho Song ‘, Myoung
scattering
from
Seon Choi a
‘Division ofNondestructive Evaluation, Korea Atomic Energy Research Institute (KAERI). PO Box 105, Yusong, Taejon 305-600, Korea bDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-l Kusung-dong, Yusong-ku, Taejon 305-701, Korea ‘Department of Automation and Design Engineering, Korea Advanced lnsfitute of Science and Technology (KAIST). Seoul 130-012, Korea
Received 20 November 1995
Abstract This paper presents the effects of material attenuation on acoustic resonance scattering (ARS) from cylindrical shells for the purpose of applying it to ultrasonic nondestructive evaluation (NDE). Complex wave numbers are introduced to include the effects of material attenuation. Resonance widths versus resonance frequencies with attenuation neglected have been numerically analyzed (up to k,u = 230) for five (S,, A,, S1, SZ, and A,) modes not only to determine resonance frequency regions in which attenuation effects must be investigated intensively, but also to select appropriate frequencies from an NDE point of view. It has been found that frequency regions of interest are those which have relatively narrow resonance widths but not too narrow ones. Material attenuation results in a decrease of resonance peak amplitude as much as an increase of resonance width. The increase of resonance width attributed to material attenuation alone is linearly related to attenuation coefficients and its proportional constants are dependent on the types of modes. The ratio of the resonance width caused by material attenuation alone to that caused by radiation Keywords:
damping
governs
the relationship
Resonance scattering; Attenuation;
between
resonance
Ultrasonic NDE
1. Introduction
In order to evaluate the integrity of cylindrical tubes, ultrasonic NDE, such as pulse-echo or throughtransmission techniques using elastic bulk waves, have been commonly used. However, these techniques have their unique advantages and limitations. As a new approach to the evaluation of tube quality, acoustic scattering or acoustic resonance scattering from submerged shells has been good subject of research. This includes analytical, numerical, and experimental studies. The pertinent literature over four decades has been well summarized by Gaunaurd and Werby [ 11, and Gaunaurd [ 21. Faran [3] obtained solutions for the scattering of plane sound waves from an infinite elastic cylinder immersed in a fluid medium. Later, Doolittle and ijberall [4] obtained the general solutions for the scattered field * Corresponding author. Fax: +82-42-861-1184; e-mail: jplee2Bkaeri.re.kr
width and amplitude.
corresponding to a plane sound wave incident upon an infinite elastic circular-cylindrical concentric shell imbedded in a fluid and enclosing another fluid. There exist detailed methods for calculating the acoustic pressure scattered by an elastic cylinder. The majority of the calculated form functions have been obtained by the direct summation of the Rayleigh normal mode series. The Watson-Sommerfeld Transformation (WST) [S] is often applied to this series to interpret the results obtained. This is accomplished in the complex plane and is not a simple way in the resonance region. Resonance Scattering Theory (RST) [6--91 enables us to easily analyze the circumferential waves and separate them into background terms and resonance peaks. The backgrounds are a contribution due to the scatterer’s shape and the resonances are a contribution due to material composition [ 21. One of the important things in material composition affecting resonances is material attenuation. Furthermore, most industrial tube materials possess, more or less, attenuation of which the attenuation coefficients are dependent on the frequencies of the incident waves.
0041-624X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PII SOO41-624X(96)00075-3
738
Jong PO Lee et al. / Ultrasonics 34 (1996)
Acoustic scattering with absorption taken into account has been studied by many authors [lo-171. Hageman et al. [lo] calculated the scattered wave pressure from irradiation of cylindrical, thin viscoelastic shell by a plane monochromatic sound wave. Vogt et al. [ 111 measured monostatic reflection from an absorbing sphere in water and compared it with the calculation results which include the effects of absorption of shear and compressional waves in lucite. They introduced complex wave numbers in the theory to include absorption considerations. Later, acoustic reflection was studied for absorbing cylinders by Schuetz and Neubauer [12], for absorbing spheres by Hasegawa et al. [ 131, and for layered elastic absorptive cylinders by Flax and Neubauer [14]. Since the advent of the resonance scattering theory, resonant scattering from elastic and/or viscoelastic objects, either spherical or cylindrical shells, has been studied using the Kelvin-Voigt model. Gaunaurd and ijberall [ 151 studied theory of resonant scattering from spherical cavities in elastic and viscoelastic media and Gaunaurd and Kalins [ 161 studied resonances from coated spherical shells. Ayres and Gaunaurd [ 171 drew their conclusions that the locations of the resonances are shifted slightly and their widths are broadened. In addition, the amplitudes of the resonances peaks decreased as viscoelastic losses increased. Recently, Huang [ 181 analyzed energy dissipation in sound scattering by a submerged cylindrical shell. He introduced complex Young’s modulus of shell material to include a loss factor due to structural damping. He showed that dissipation effects are important near the resonance frequencies of the shell. In spite of a number of previous studies on resonance scattering by attenuating cylindrical shells, relationships between attenuations and resonances are not analyzed in detail such that the results can be utilized. The motivation of the present work was provided by this fact and by the necessities of the application of acoustic resonance scattering to ultrasonic NDE. The objective of this study is to investigate the effects of material attenuation on acoustic resonance scattering in detail and to analyze characteristics of resonance behaviour for Zircaloy-4 tubes. Material attenuation effects are taken into account by introducing complex wave numbers whose imaginary parts are given by the attenuation coefficients (damping factors). Relationships between resonance widths and peak amplitudes are numerically analyzed and dicussed for S,, A,, S1, Sz, and A, modes with different attenuations.
2. Theoretical considerations
737-745
an infinite and elastic cylindrical shell of outer radius a and inner radius b as shown in Fig. 1. The fluid media outside and inside the shell are labelled 1 and 3, respectively and the shell material 2. Longitudinal and transverse wave velocities of the elastic shell are cL and cT, respectively. Wave numbers are kj = o/cj (j = 1, L, T, 3). The scattered field P,, at an observer’s point (I; 0) located outside the shell is given by P,,(r, 19,t) = epiW’ f q,i”b,H!,l’(k,r) n=O
cos(n0),
(1)
where E, = 1 for n = 0 and E, = 2 for n 2 1. The coefficients b, are determined from the boundary conditions and have the form of the ratios of two 6 x 6 determinants, b, = BJD,. The 30 nonvanishing elements of these determinants B, and D, have all been given in the Appendix of Ref. [ 193. HL”(k,r) is replaced by its asymptotic form in the far field region. Then, the scattered pressure becomes P,,(r, 8, t) 2:
\i
c ei(klr-@f(O, x,),
(2)
n=O where x1 = kla. The form function f,(0, x1) represents the angular distribution of the scattered pressure for a given x1 and is called the scattering form function of the nth partial wave. A number of previous workers on RST such as Gaunaurd and Werby [20] showed that the scattering form function of partial waves consists of resonance peaks and smooth backgrounds. Resonance peaks can be seen clearly by subtracting an appropriate background which depends on the shell thickness, material properties, and frequency. For metal shells, it has been well known that the scattering amplitude from the rigid cylinder of the same size could be used as a proper background. For a rigid target, there is only one boundary condition of no displacement at the surface and the scattering form function has the following simple
II
2.1. Scattering form function
A plane acoustic wave (angular frequency w) of unit amplitude ei(kl*-“t) interacts at normal incidence with
Fig. 1. The geometry of a plane acoustic wave scattering from an infinite cylindrical shell. Here, Cj and p, denote wave velocity and material density, respectively.
Jong PO Lee et al. / Ultrasonics 34 (1996)
737-745
139
by the complex wave numbers (4) where the superscript (r) refers to rigid background. The modal resonances of elastic partial waves are given by background subtraction %)I = -% 6
If,(R x1) -f?(R
(b” - b2’) cos(ne) .
(5)
2.2. Resonance response with no attenuation Each resonance is characterized by two integer numbers (n, 1). The first one, n represents the number of wave lengths of the surface wave needed for resonance around the circumference of the cylinder. The second one, 1 is considered to be related to the nature of the surface wave [ 211. According to the resonance scattering theory [6-g], the term of form function of resonance without consideration of attenuation can be expressed by;
k,=k,+ia,,
(loa)
k,=k,+icc,,
(lob)
where ctL and ~1~are attenuation coefficients of longitudinal and shear waves, respectively (Np/m). The bar denotes complex numbers. Here, CQ,and c+ are assumed to be a linear function of the incident wave frequency, because the linear term is predominant in most industrial materials. In other words, attenuation coefficient times wave length is constant (/?). %JI, 5 Pi_,
(Ha)
%& = BT,
(lib)
where I, and & are wave length (m) of longitudinal and shear wave, respectively. Then, the complex arguments of Bessel and Neumann functions in b, of the Eq. (3) can be expressed in terms of k,a as follows xj~~a=kja+i(olja)=k,a~
(j = L, T) .l
(6) yj~k,b=kjb+i(~jb)=$(kIa)~
where x,,~- (k, a),, is the resonance frequency, r,, resonance width, and x = x1 = kIa, respectively. The magnitude of backscattered partial wave resonance (0 = x) can be simply derived from the Eq. (6)
Mx)I =
4 &
1 Jl
(n 2 1).
(7)
+ ((x - xJ(W2))2
Peak amplitude, p is simply obtained from the Eq. (7) by putting x = x,~. P=
Ig”tx”dl = Jk
(n 2 1).
(8)
Eq. (8) is the same as the equation derived differently by Veksler [ 221. For a single resonance, the relationship between resonance width r,, and radiation attenuation a,, can be expressed by [23] a,=--.---,
Cl
5
r, 2
(9)
where ~1,is attenuation coefficient (Np), c1 sound velocity in water, cg group velocity of resonances in the shell. 2.3. Resonance response with attenuation
The effects of shell attenuation can be considered by introducing the complex wave numbers. For this purpose, the real wave numbers included in the arguments of the elements in the 6 x 6 determinants are replaced
(j = L, T). I (13)
The imaginary parts of xj and 5 contribute to the effects of attenuation rather than /IL or pT alone. Then, the effects of ultrasonic attenuation can be analyzed in terms of k,a by replacing real arguments of Bessel and Neumann functions with complex arguments of Eqs. ( 12) and (13). If total resonance width r, and the resonance width r, attributed to radiational damping are calculated, resonance width l-‘, attributed to shell attenuation can be obtained.
r,=r,-r,.
(14)
3. Numerical analysis and discussion 3.1. Calculation Numerical analysis is carried out using Mathematics for an air-filled Zircaloy4 cylindrical tube immersed in water. There are several different sizes of Zircaloy4 tubes which are used as nuclear fuel cladding tubes. Here, calculation is made for a typical size of the tubes. The ratio of b/a is 0.888 (2a = 10.7 mm, 2b = 9.50 mm, d = 0.6 mm). The properties of the Zircaloy4 tube, water, aluminium, and air used in calculations are listed in Table 1. It is well known that a Zircaloy4 tube has attenuation coefficients which are in the medium range among metals. Its attenuation coefficients of longitudinal
Jong PO Lee et al. / Ultrasonics 34 (1996)
140 Table 1 The properties
of substance
Substance
Water Air Aluminium Zircaloy-4
1416 _
1000 1.2 2740 6.550
344 _
3100 2440
6380 4540
_
and shear waves, BL2: PT = 0.0028 Np. Analysis is done for four cases, that is, (A) j& = /IT = 0, (B) j3r = & = 0.0014 Np, (C) /IL = /IT = 0.0028 Np, and (D) /Ii, = Pr = 0.00467 Np. Resonances are separated with rigid background subtraction for So, A,, Si, S2, and A, modes. Resonance frequency, peak amplitude, and resonance width for the five modes are investigated with attenuation taken into account. A small calculation step is used such as k,a = 1O-2-1O-6 depending on each resonance width. In order to calculate resonance width accurately with an error less than l%, the small calculation step has been used such that at least 100 calculation points are included within each resonance width. 3.2. Validity of the program The validity of our program is checked. It is difficult to find previous results for direct comparison to ensure whether the results produced are reasonable and valid. The only possibilities of checking are based on comparison with the previous results in the literature for the cases where attenuation is neglected. The first six sets of modal resonances are calculated for a thick (b/a = 0.1) aluminium, air-filled, cylindrical shell in water by setting the attenuation coefficients, /IL and /?r to zero. The present results are shown in Fig. 2 and compared with 1OL
I n=O
0.5 0.0
_
A
L
1.0
40.5 %
0.0
.z 1.0 EO.5 z
0.0
8 1.0 f
0.5
iE 0.0 z" 1.0
l_A_AJ__ 1.0 -
Il=5 A__&_
I 0
10
20
30
40
ha
Fig. 2. First six sets of modal resonances (n = 0, 1, , 5) for a thick (b/a=O.l) aluminium, air-filled, cylindrical shell in water (no attenuation, /& = pT = 0).
737-745
the Fig. 1 of Gaunaurd and Werby [ZO]. Excellent agreement is shown between the present and previous results for backscattering resonances from aluminium shell in water. What is distinct in the present results in Fig. 2 is that some resonances having narrow resonance width are more distinguishable than those of previous results. This is caused by a small calculation step. Other comparisons have also been made. One is the calculation of the first six sets of modal resonances and the other is that of modal backgrounds for a thin, air filled, aluminium, cylindrical shell in water (b/a = 0.9). These two present results, which are not shown here due to limited space, are compared with the Fig. 6 and Fig. 7 of previous results reported by Gaunaurd and Werby [ 201. Good agreement is also obtained. 3.3. Resonances with attenuation neglected Before the analysis of attenuation effects on acoustic resonance scattering, resonance frequency, peak amplitude, and its width for various modes with attenuation neglected must be determined. This is because it is very important to determine the regions of resonance frequency in which we are going to analyze attenuation effects in detail. Choi et al. [23] identified the first several modes of resonances for an air-filled Zircaloy-4 cylindrical tube immersed in water. The first two modes for thin aluminium and Zircaloy4 tubes have been studied by many authors [ 19-241 but it does not seem to be clear. One of the possible reasons appears to be that those frequency regions are in the transition area of soft and rigid background. Therefore, resonances are not clearly separated by using either soft or rigid background. If an ideal background which can encompass these transition regions is known, it would be very helpful to characterize those modes. The studies on the first two modes continues to be an important area of research. A new background using the effects of material attenuation on ARS has been found (submitted to the J. Acoust. Sot. Am.). The results of the study on the first two modes using a new background will be reported later. So, we are here dealing with the next five modes of resonances, which are So, A,, S,, S2, and A, modes. Fig. 3 shows that resonance widths versus resonance frequencies, both k,a and frequency x thickness (fd up to k,a = 230 for the five modes with attenuation neglected. An fd value is obtained using the formula fd = k, a( 1 - b/a)c,/2x. Resonance peak amplitude is not shown here as it can be simply obtained from Eq. (8) without tedious calculations. As can be well explained by Eq. (9) resonance frequencies having large radiation attenuation are not useful from an NDE point of view. The frequency range of interest is around the valley as shown in Fig. 3 because those frequency regions with wide resonance widths have large radiation attenuation.
Jong PO Lee et al. / Ultrasonics 34 (1996) 737-745
Frequency x thickness, 1.32
10.00
2.63
741
fd (MHz mm) 3.95
5.26
6.56 1
I
i
1 - (c)
S,
I,
I,,
&,
n=1
mode
,
,
,
,
,
:
:
(.
El3
,
-
...* Q..
.*.... .. _... mmm.........r.m
"='O
~~-,,.,,,,,,,,,,,,,,,,,,,,, 5
-
t c2-
1
0 3
^
(d)
S,
mode
1
.:
n=61
.‘...
/--‘-“..
.
/ ..fl.*
.. ‘. “.Yn....n.~~
...y
n=1
~~1'1~11111/1111,,,,,,,,
- (e)
A2
mode
n=l
t,,.,,.,,,,,,,.,,,,,,,,,,I 0
50
100
150
200
250
Fig. 3. Resonance widths versus resonance frequencies of (a) So mode, (b) A, mode, (c) Sr mode, (d) Sa mode, and (e) A, mode for an air-filled Zircaloy-4 tube (b/a = 0.888) immersed in water when attenuation is neglected.
Hence, it is needed to pay attention to the frequency regions having relatively narrow resonances. However, there are some limitations to the proper selection of appropriate frequencies having relatively narrow resonances because these frequencies are quite dependent on shell attenuation as investigated in the next section. From Fig. 3, the frequency regions for the five modes having relatively narrow resonance smaller than 0.5 are shown in Table 2. What we must pay attention to is the area around these frequency regions. 3.4. The egects of material attenuation on resonance scattering Three backscattered resonances which have relatively wide (S, mode, n = 64), intermediate (A, mode, n = 8), and narrow resonance width (So mode, n = 7) are
Table 2 Resonance frequency regions of interest having narrow resonance width, r, smaller than 0.5 (b/a = 0.888) Mode
SO A, SI SZ SZ A,
From
To
n
kra
fd
?I
kla
fd
2 1 29 1 43 1
6.57 46.61 99.62 93.01 161.07 139.15
0.173 1.226 2.621 2.447 4.238 3.660
17 30 50 3 67 38
47.77 94.13 150.86 94.39 205.65 154.17
1.257 2.477 3.969 2.483 5.411 4.056
compared for four different attenuations as shown in Fig. 4. They are (A) /IL = PT = 0, (B) pL = /%r= 0.0014, (C) jL = pT = 0.0028, and (D) pL = /ST= 0.00467 Np. When attenuations are included in the calculations,
142
Jong PO Lee et al. / Ultrasonics 0.25
a 0.20
O.OO L
b
0.25
b
0.20 P go.15
_&I.,0 ,’ * ‘, ,a; _ . ‘, \s ,’ \\
0.05
2 0.00 ELI 138.4
138.8
139.2
139.6
‘v 0.5
C
0.4
m 6 0.3
‘: .c
0.1
p
0.0 0.2 r-I---l
II.
20.4
I.
>I
I.
20.6
20 8
21.0
ha
Fig. 4. Resonance characteristics with different attenuation coefficients Br=Br of 0 (solid), 0.0014 (dashed), 0.0028 (dashed and dotted), 0.00467 (dotted) Np for (a) S2 (n = 64), (b) A, (n = 8), and (c) So (n = 7) mode.
resonance peak amplitude decreases and resonance width increases due to shell attenuation effects as reported in the previous literature [ 17,181. These effects are more significant in narrow resonances (high Q factor)
Table 3 Resonance
characteristics
with different
34 (1996)
737-745
than wide resonances (low Q factor). Fig. 4 is not enough to see attenuation effects quantitatively in detail. Hence, Table 3 summarizes resonance characteristics which include resonance frequency, peak amplitude, percentage of amplitude relative to those in the case of no attenuation, resonance width, and the value of peak amplitude times resonance width. As shown in Table 3, resonance frequency is slightly changed in which variations are almost negligible and amplitude in the case of no attenuation follows the formula of Eq. (8). In addition, the amplitude drop is very significant depending not only on attenuation coefficients but also on Q factor. Here, attention must be paid to each resonance frequency. Fig. 4c is for low frequency, Fig. 4b for middle frequency, and Fig. 4a for high frequency, respectively. It is expected that frequency effects will be higher at high frequency than at low frequency because attenuation coefficients are given as a linear function of incident wave frequency. This will be shown in the next section. In spite of that, amplitude of S, mode (n = 7) decreases 17.6% more significantly than either 47.2% of S2 mode (n = 64) or 22.8% of A2 mode (n = 8). What is most important is that this amplitude drop makes resonance width broaden due to shell attenuation. Thus, the value of amplitude times resonance width for one single resonance remains the same with different attenuations. We have tried to prove this important fact theoretically, but have not been successful. This proof will be left to others. The reason why pT is not the same for SO mode (n = 7) is because this resonance is not symmetric. The resonance frequency region is near the transition area of soft and rigid background. Thus, background is not fully subtracted with rigid background used. Therefore, additional resonance width seems to be included during the calculation of resonance width in the case of high material attenuation. Single
attenuations
Atten.
Freq.
Amp].
Ampl.
%I
P
W)
Width r
PT
fL=h 0
0.0014 0.0028 0.00467
200.100 200.101 200.102 200.104
0.15954 0.11939 0.09540 0.07523
100 74.8 59.8 47.2
0.266 0.354 0.442 0.560
0.0424 0.0423 0.0422 0.0421
Az mode n=8 A@, a) = 0.0005”
0 0.0014 0.0028 0.00467
139.009 139.008 139.006 139.002
0.06315 0.04369
100 49.6 33.0 22.8
0.0610 0.1230 0.1850 0.2680
0.0117 0.0117 0.0117 0.0117
S, mode n=7 A(k,a) = 0.00005”
0 0.0014 0.0028 0.00467
0.49630 0.19663 0.12352 0.08748
100 39.6 24.9 17.6
0.00595 0.01575 0.02690 0.04525
0.00295 0.00310 0.00332 0.00396
Mode
S2 mode n=64 A&a) = 0.002”
a Calculation
step.
20.6764 20.6773 20.6789 20.6820
Jong PO Lee et al. / Ultrasonics 34 (1996) 737-745
143
resonances are investigated, but it is necessary to know resonance characteristics for all the frequency regions of interest as shown in Table 2. 3.5. Resonance characteristics of So, A,, and S1 modes It has already been shown that the frequency regions of interest are listed in Table 2. Here, resonance characteristics of So, A,, and S1 modes are analyzed in detail for these resonance frequency regions. Figs. 5,6, and 7 show the variations of resonance amplitudes and total resonance widths with three different attenuation coefficients, that is (1) jI,_= BT= 0 Np, (2) fiL = PT = 0.0014 Np, and (3) fiL = fiT = 0.0028 Np for the So, Al, and S1 modes, respectively. Resonance width has its minimum value at n = 6 for So mode, n = 1 for A, mode, and n = 40 for S, mode, and varies with n or resonance frequency. As was mentioned in the previous section, amplitude drops take place significantly in the frequency region having narrow resonance width. Exact calculation of resonance width is difficult for the resonances of which amplitude reaches a value smaller than 0.1 due to shell attenuation because background can be involved in the calculation of resonance width in the cases where background has not been removed perfectly. Therefore, there are no data points in the deep valley area when attenuation is included in the analysis. As shown in Figs. 5, 6, and 7, the resonances having relatively narrow resonance widths in the case of no attenuation are greatly affected by material attenuation.
O.:I k,a”
b
50
60
100
I
80
90
Fig. 6. Same as in Fig. 5, except for A, mode.
110
120
130
140
150
I
140
150
160
k,a
IIT = 0.6 & 9 5 0.4
70 k,a
0.6 _ c L5 % g
0.2
E 5: g
0.5 0.4 0.3 0.2 0.1. 0.0 t
-o.ll.~.~.~.,.~.~ 100
110
120
130
I
k,a
Fig. 7. Same as in Fig. 5, except for S1 mode.
I
I.
5
I.
10
I.
15
%
20
I.
25
I.
30
I
35
.I
40
k,a
Fig. 5. Variations of (a) resonance amplitudes and (b) total resonance widths for So mode with different attenuation coefficients (/IL =/IT); ...M...: 0, ...O...: 0.0014, and . ..A.... 0.0028 for an air-filled Zircaloy4 tube immersed in water (b/a = 0.888).
Resonance width caused by attenuation alone can be calculated using the Eq. ( 14), namely r. = r, - r,. Fig. 8 shows each component of resonance widths for S1 mode. Fig. 8a is for material attenuation /IL = /& = 0.0014 Np and Fig. 8b is for /I,_= j& = 0.0028 Np. r, is linearly increased as k,a increases. From the fact that the value of resonance peak amplitude times total resonance width is constant with attenuations for a single resonance, it
Jong PO Lee et al. / Ultrasonics 34 (1996) 737-745
744
0.0
1
0.7,
'v
b .
100
110
120
130
140
150
ha
Fig. 8. Variations of total resonance widths (r, = r, + r,) for Sr mode with different attenuations, (a) b = 0.0014, (b) fi = 0.0028: .t. 0 ..‘: r,, ...W,..: I-,, and . ..A...: r,.
can be written PJ-, = P*C = P*(C + T*), where the asterisk denotes material attenuation. p* can be rewritten
fed into the transducer. However, the reasons can be explained based on the present results, that is, those resonances with small resonance widths could not be detected due to material attenuation. If attenuation factors are taken into account during theoretical calculation, those sharp resonances are damped out due to material attenuations. Thus, they are not present theoretically and experimentally. In other words, if r, >>r,, then p*/p, 1: l/[ 1 + (r,/r,)] + 0. From the Figs. 5 and 6, r, curve can be drawn for S, and A, as has been done in Fig. 8. Fig. 9 shows r, curve versus k,a with different material attenuations for So, A,, and S, mode. r, is linearly increased with k,a, but the slopes of the curves are different depending on attenuations and types of modes. Table 4 shows the slopes of S,,, A,, and S, mode with two different attenuations. If material attenuation is increased by two times, the slopes of r, curves are also increased by two times for all three modes. In order to confirm the relationships between attenuation (fi) and r,, calculations are made for four different attenuations. Table 5 shows the results. It is confirmed that r. is linearly proportional to the /? and p,T, N p*r,* is maintained as shown in Table 5.
4. Conclusions (15) Hence,
The effects of material attenuation on acoustic resonance scattering from an air-filled cylindrical tube
(16) The most important factor affecting the relationship between resonance width and peak amplitude is the ratio of r,/r,. If the resonances have r, larger than r,, which means r, 2 2r, or those having r, curve underlying r, curve as in Fig. 8, then resonance amplitudes are decreased to more than half of the original resonance amplitudes. Eq. (16) indicates that the ratio of r, to r, governs the percentage of the amplitude decrease. The very narrow and sharp resonances (r, <<1) when attenuation is not considered will be severely attenuated if attenuation is taken into account, because r, will be far greater than r, even though attenuation of shell material is small. Talmant et al. [25] calculated resonance frequencies and their widths without attenuation taken into account for air-filled duraluminium cylindrical shells immersed in water. They also measured resonances experimentally and compared theoretical results with experimental results. According to their results, some resonances which have narrow resonance widths in theoretical calculations were not observed even though they were inside the range of measurements. They explained that the reasons were probably because those resonances were very narrow and very little energy was
m D
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130 0
Fig. 9. Resonance widths, r, attributed to different attenuations; ...O..‘: 0.0014, ..‘A.-.: 0.0028 for (a) S,, (b) A,, and (c) S, mode. Table 4 The slopes attenuations Attenuation
A.=/% 0.0014 0.0028
of r, curves for S,,, A,, (refer to Fig. 9)
and
S, modes
with
different
Slope of the r, curve S, mode
A, mode
S, mode
0.00042 0.00084
0.00046 0.00092
0.00048 0.00096
Jong PO Lee et al. / Ultrasonics
Table 5 The variations
737-745
745
References of r, value with different
Atten.
So mode, n = 9
BL=BT
p
0.4395 0.3367 0.2731 0.2297 0.1984
0.0384 0.0505 0.0620 0.0745 0.0865
attenuations A, mode, n = 5
r, 0 0.0014 0.0028 0.0042 0.0056
34 (1996)
0 0.0121 0.0236 0.0361 0.0481
P
r,
r,
0.3217 0.2511 0.2059 0.1745 0.1514
0.078 0.100 0.122 0.144 0.166
0 0.022 0.044 0.066 0.088
(Zircaloy4) immersed in water is investigated. Complex wave numbers are introduced to consider the material attenuation of cylindrical shells. The conclusions obtained are summarized as follows. First, from the view point of the application of acoustic resonance scattering to ultrasonic NDE, resonance width (r,) versus resonance frequency @,a) is calculated up to k,a = 230, or fd = 6 (MHzmm) for the five (So, A,, S1, ST, and A2) modes. Frequency regions of interest which have relatively narrow resonance width have been found to be those listed in Table 2. Second, ultrasonic attenuation contributes to an increase of resonance width and brings a decrease of resonance amplitude. The value of peak amplitude times total resonance width (r, = r, + K‘,) remains the same for each resonance with different attenuations. The increase of resonance width (r,) caused by material attenuation is linearly increased with the values of attenuation. The relationship between resonance width and peak amplitude is given by Eq. (16) and the ratio of r, to r, governs amplitude drop. Sharp resonances with attenuation neglected are damped out when material attenuation is taken into account. This is because the resonance width caused by material attenuation is far greater than that caused by radiation damping (I’, >>r,). The slope of r, curve versus k,a is linearly dependent on attenuation coefficients (/I) and varies with types of modes.
and M.F. Werby, Appl. Mech. Rev. 43(8) Cl1 G.C. Gaunaurd (1990) 171. c21 G.C. Gaunaurd, Appl. Mech. Rev. 42(6) (1989) 143. c31 J.J. Faran, Jr., J. Acoust. Sot. Am. 23 (1951) 405. and H. iiberall, J. Acoust. Sot. Am. 39(2) c41 R.D. Doolittle (1966) 272. J. Acoust. c51 J.W. Dickey, D.A. Nixon and J.M. D’Archangelo, Sot. Am. 74( 1) (1983) 284. in: Proc. IUTAM Symp.: Northwestern Univ., C61 H. iiberall, Evanson, IL, Sept 12-15, 1977, Eds. J. Milkowitz and J.D. Achenbach (Wiley Interscience, New York, 1978) p. 239. and H. ijberall, J. Acoust. Sot. Am. 63 (1978) [71 G.C. Gaunaurd 1699 [reviewed in Appl. Mech. Rev. 32 (1979) 12551. and H. nberall, Physical Acoustics C81 L. Flax, G.C. Gaunaurd (Academic Press, 1981) XV. J. Acoust. Sot. Am. 61 (1977) 360. c91 G.C. Gaunaurd, Cl01 D.H. Hageman and V.L. Glass, J. Appl. Phys. 44(90) (1973) 3907. R.H. Vogt, L. Flax, L.R. Dragonette and W.G. Neubauer, [Ill J. Acoust. Sot. Am. 57(3) (1975) 558. J. Acoust. Sot. Am. 62(3) Cl21 L.S. Schuetz and W.G. Neubauer, (1977) 513. Y. Kitagawa and Y. Watanabe, J. Acoust. Sot. [I31 T. Hasegawa, Am. 62(5) (1977) 1298. J. Acoust. Sot. Am. 61(2) Cl41 L. Flax and W.G. Neubauer, (1977) 307. and H. iiberall, J. Acoust. Sot. Am. 63(6) Cl51 G.C. Gaunaurd (1978) 1699. [16] G.C. Gaunaurd and A. Kalins, Int. J. Sol. Structures 18(12) (1982) 1083. [17] V.M. Ayres and G.C. Gaunaurd, J. Acoust. Sot. Am. 81(2) (1987) 301. [ 181 X.Y. Huang, Acustica 77 (1992) 221. [19] G.C. Gaunaurd and D. Brill, J. Acoust. Sot. Am. 75(6) (1984) 1680. [20] G.C. Gaunaurd and M.F. Werby, J. Acoust. Sot. Am. 77(6) (1985) 2081. [21] H. Bartard and G. Quentine, J. Acoust. Sot. Am. 91(2) (1992) 581. [22] N.D. Veksler, Acustica 79 (1993) 22. [23] M.S. Choi, H.C. Kim and M.S. Yang, Ultrasonics 30(4) (1992) 213. [24] M.S. Choi, M.S. Yang and H.C. Kim, Ultrasonics 30(4) (1992) 221. [25] M. Talmant, G. Quentin, J.L. Rousselot, J.V. Subrahmanyam and H. iiberall, J. Acoust. Sot. Am. 84(2) (1988) 681.