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Propagation characteristics of elastic circumferential waves in nuclear fuel cladding tubes
Propagation characteristics of elastic circumferential waves in nuclear fuel cladding tubes M.S. Choi*, H.C. Kim and M.S. Yang* Department of Physics, The Korea Advanced Institute of Science and Technology, PO Box 150, Cheongryang, Seoul 130-650, Korea
Received 21 June 1991; revised 12 December 1991 The backscattering of a plane acoustic wave from an air-filled Zircaloy-4 nuclear fuel cladding tube in water is analysed as a function of kla using the resonance scattering theory. In accordance with the cut-off frequencies, the groups of resonances are identified as the elastic circumferential waves corresponding to the Lamb plate waves. In order to determine accurate propagation characteristics, each resonance is evaluated using the very small calculation step, A ( k l a ) = 0.0001, and the results are presented in the form of phase and group velocities, attenuation constants and angles, and backscattered amplitudes of the elastic circumferential waves as function of the frequency, tube thickness and curvature.
Keywords: circumferential wave; resonance; Lamb wave; Zircaloy-4 t u b e
In the present commercial nuclear power reactors, Zircaloy-4 is one of the more important materials for cladding fissile fuels because of its favourable mechanical, chemical and physical properties such as small absorption cross-section for thermal neutron 1. Any failure in the cladding tube will directly cause radioactive fission gas leakage to the coolant, and thus may result in a serious economic and safety problem. Various non-destructive testing ( N D T ) techniques have been applied to evaluate the quality of the cladding tube. In particular, ultrasonic N D T has been successfully used in the manufacturing process 2, during the operating period in reactor 3 and in the post-irradiation examination 4. However, at present, the types and application techniques of ultrasonic waves used are limited. The pulse-echo or through-transmission techniques using elastic bulk waves have been commonly used. Since the discovery by Barnard and McKinney 5 of multiple echo returns from a single underwater sound pulse incident on the elastic cylinder, a considerable number of experimental investigations 6-9 have been conducted to exploit this phenomenon. It is generally ascribed to the presence of circumferential waves with the behaviour of the surface wave type, i.e. the waves that travel circumferentially along the surface of the scattering body before proceeding to an observer. On the other hand, the sound scattering from an elastic cylinder immersed in a fluid has been investigated theoretically by means of the Watson-Sommerfeld transformation (WST) applied to the classical Rayleigh * Quality Control Department, HWR Fuel Fabrication Division, Korea Atomic Energy Research Institute, PO Box 7, Daeduk-danji, Taejon 305-606, Korea
normal mode series of the scattering solution 1°. Uberall obtained a sum of residue terms which corresponds to circumferential waves divided into two classes: those with velocity close to the elastic bulk wave velocity, i.e. Rayleigh and Whispering Gallery waves, and those with velocity close to the sound velocity in the surrounding fluid, i.e. Stoneley and Franz waves. Frisk et a1.11"12 and Dickey et al. 13 systematically investigated all the types of surface wave by both analytical and numerical methods. They showed that, in the limit of large cylinder radius, the Rayleigh, Stoneley and Whispering Gallery waves tend towards the generalized Rayleigh, Stoneley and lateral waves on a flat surface, respectively, while the Franz waves disappear. Circumferential waves on the elastic cylindrical shell were studied numerically by Ugineius and [Sberall TM and Dickey et al. 15 using the WST method. The solution is more complex than that of the solid cylinder, because of two boundaries, the inside and outside surfaces, and therefore a faster and larger computer is required. Recently, however, many authors 16-z6 have reported that the circumferential wave phenomenon can be more easily analysed using the resonance scattering theory (RST) known as the 'Regge' pole technique in nuclear physics. The backscattering of plane acoustic wave from the air-filled Zircaloy-4 nuclear fuel cladding tube in water is analysed using RST and the results are presented in the form of phase velocities, group velocities, attenuation constants, attenuation angles and backscattered amplitudes of the elastic circumferential waves as a function of frequency, shell thickness and curvature. An easy method for identification of resonances is described and each resonance is evaluated using the very small calculation
0041-624X/92/040213-07 © 1992 Butterworth-Heinemann Ltd
Ultrasonics 1992 Vol 30 No 4
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al.
Propagation characteristics of elastic circumferential waves. M.S. C h o i e t step, A( k ~a ) = 0.0001, to determine accurate propagation characteristics of the elastic circumferential waves. Relationships a m o n g the circumferential waves in a cylindrical shell, those in a solid cylinder and L a m b waves in a plate are also discussed. N u m e r i c a l
analysis
Computation of the scattering form functions When the plane acoustic wave exp[i(k~x-~,Jt)] is incident normally on fluid-filled cylindrical shell immersed in another fluid, as shown in Fi,qure 1, the scattered pressure at the position (r, 05) of the observer located outside the shell is given by the Rayleigh normal mode series ~,t t
P.~(r. 05.t)=exp(-i(.)t) ~
coefficient matrix obtained from the six boundary' conditions. The elements of these determinants, listed in Ref. 14, contain the cylinder functions and their derivatives with arguments kia or kih(i = 1. L,T, 3), where k~ and k s are the acoustic wave numbers in the outside and inside fluids, respectively, and k L and k~ are the wave numbers of bulk longitudinal and transverse elastic waves in the shell, respectively. Employing the Hankel asymptotic form 27 for H,,~t)(ktr), the far-field value of the scattered pressure becomes exp[ilktr
.~t)]
.I;,(05)
(2)
n = O
where
1;,(05) =
c,,
cos(nO )
(3 )
c.i"b"H~lqk~r)cos(n05)
n = o
,t
D, I
(1) where ~;, is called the N e u m a n n factor, % = 1, c,, = 2 for n>~ 1, and b, and D, are 6 x 6 determinants of the
exp[i ( k l x -- ~ot)]
~
~
(r, ~o)
2
The function,l;,(05) represents the angular distribution of the scattered pressure and is called the scattering form function of the nth normal mode. In this study, the scattering form function, .11,(0), was examined in the backscattering direction (05 = ~z) and the backscattered amplitude,f,(n), was calculated on Hewlett Packard 9000 series model 310 computer for the range of n - 0 60, kla 0 200 and h / a = 0.01 0.99. All calculations were for thc air-filled Zircaloy-4 nuclear fuel cladding tube immersed in water, where ,)~ = 1 . 0 g c m s. p, = 6 . 5 5 g c m s, Ps = 0 - 0 0 1 2 g c m 3. The longitudinal and transverse sound velocities were taken to be ('~, 4540 m s i = 3.068 C,, and ('] 2440 m s ~ = 1.649 (',,, respectively', where C,, = C~ is the sound velocity in water. 1480m s ].
Identification of resonances The backscattered amplitude,.ti,(rt), plotted vcrsus k]a consists of the smooth and regular b a c k g r o u n d and a series of resonance spikes, for example as shown in Fi.qure 2a for the shell with h..'a = 0.82. Three resonances labelled as A o, S o and A~ appear and the background agrees closely, except at the very low frequency region, with the backscattered amplitude of the "acoustically rigid' cylinder ~o F i g u r e 1 Geometry of a plane wave scattering from an infinite elastic cylindrical shell. Here, C1 and [)1 denote sound velocity and material density, respectively
t.i~qn )
.....
1.0
[
1
=,
..--,
,:,,
j,,,(kla }
,4)
cos(nrc)
tti,11'(kla)
1.0 a
b
~0 7D
E E <
12
\=lgm/
~0 -(3
0.5
0.5
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Ao
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,
0
L
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L
L
]
10
,
,
,
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,
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J
,
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I
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I
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I
I
I
1
1
1
10
,
I
20
,
,
,
~
I
30
J
~
40
kla
Figure2 Modulus of n = 2 normal mode backscattering amplitude (a) foraZircaloy-4shell ( b / a = 0 . 8 2 , ) and a rigid cylinder ( If2(n)l and If},n (Tt)l, and (b) for the shell with the rigid background subtracted, If2(~) - f~n (~)1, plotted versus k~a
214 Ultrasonics 1992 Vol 30 No 4
,
),
Propagation characteristics of elastic circumferential waves. M.S. Choi et al. where the prime on the cylinder functions denotes the differentiation with respect to their argument. The calculation step size, A(k~a), was 0.05. Figure 2b shows the enhanced appearance of the resonance responses of the elastic shell by subtracting the rigid background, ft,)(g), from the backscattered amplitude,f,(Tt). The rigid background appears as a result of the existence of the Franz-type circumferential waves and the specularly reflected waves from the apex of the rigid cylinder TM. However, the rigid background may be substantially different from the actual background at low frequencies. In fact, this difference causes the resonance-like feature appearing as the broad peaks denoted by the arrow in Figure 2b. Breitenbach et al. 2° described the broad peak as a resonance response suggesting the existence of the Rayleigh surface wave. However, as shown in Figure 3 for the shell with b/a = 0.88, the broad peak can be found only in the neighbourhood of kla = n implying that the peak is related to the circumferential wave whose velocity is close to the sound velocity in water, i.e. the Stoneley or Franz surface wave rather than the Rayleigh surface wave. Thus, we believe that the broad peak is not the Rayleigh-type resonance response of elastic shell. Furthermore, the broad peak does not appear in the backscattered amplitudes of the n = 0 normal mode for any shells, but appears as only one peak for each normal mode n ~> 1 and becomes weak as the normal mode number increases or as the ratio of radii, b/a, decreases. Almost all the resonances reappear in the subsequent normal modes at subsequently higher frequencies. Thus, the resonance peaks can be classified into groups corresponding to the elastic circumferential waves. In accordance with the cut-off frequencies, [fd], the Sz and A~(/>/ 1) circumferential waves correspond to the symmetric and antisymmetric L a m b waves in Zircaloy-4 plate, respectively. The cut-off frequency is determined from the position, kla, of the n = 0 resonance as
fd = k,a (1 - b/a) Cw/2n
So
n=0 ~$1 80'.16
~.~ So
A3 n=1
Sl S2
A3
I
n=2
I S,~.~ 43.41 $1 S2
A1
A2
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,
I
n=3
$1 S2
1
A3
A2,
n=4 "r-
I
~
1
.£
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A2
A3 n=5
v
A1
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A3
D r~
E <
n=6 A1
Sl
J
S2
~.S
A2 l
3 n=7
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A1
$1
S2
.1 J L J L
A2
A3 S3
1 n=8
A1
$1
S2
I _A.jL
A2
A3
1
Ao
n=9
(5) A1
For example, the n = 0 resonance at k l a = 80.16 corresponds to the cut-off frequency [fd] = 2.27 MHz.mm, which is the same as that of the first-order symmetric L a m b plate mode, CL/2. Thus the group of resonances is labelled $1. The cut-off frequency of the group without the n = 0 resonance can be approximated from the position of the next normal mode resonance. For example, the n = 1 resonance at kla = 43.41 corresponds to the cut-off frequency [fd] = 1.227 MHz.mm, which is almost the same as that of the first-order antisymmetric L a m b plate mode, CT/2. Thus, the group of resonances is labelled Aa. It can be inferred from the above result that the other elastic circumferential waves, the So and A 0 waves, correspond to the zeroth order symmetric and antisymmetric L a m b plate waves, respectively. This correspondence will be ensured in the following section. Therefore, it can be concluded that the elastic circumferential waves in the shell make one-to-one correspondence to the L a m b waves in the plate of the same material and thickness. According to RST, the propagation characteristics of the elastic circumferential waves are determined from the position, width and height of the corresponding resonance peaks. The accuracy of the measured quality of resonance peaks depends on the calculation step size. All previous authors ~6-26 used the steps of the same order of magnitude as that of Figures
i
i
Sl i
$2 i
A2 i
,
100
0
i
200
kla Figure 3 Modulus of nth normal mode backscattering amplitude ( n = 0 - 9 ) for a Zircaloy-4 shell (b/a=0.88) with a rigid background subtracted, Ifn(TZ) f(/)(~)l, plotted versus kla
2 and 3. However, the step size is rather large, thus high quality (narrow) resonances are occasionally lost. In order to determine the more accurate propagation characteristics described in the following sections, we recalculated the backscattered amplitudes, f , ( ~ ) and f~,r)(~Z),in the neighbourhood of the resonance peaks using the narrower step size, A(kla ) = 0.0001.
Velocity dispersion The phase velocity, Cp, of circumferential wave is determined from the position, kla, of the corresponding resonance peak 16
Cp = C w kla/n
(6)
The group velocities, Cg, are obtained from the following
Ultrasonics 1992 Vol 30 No 4
215
Propagation characteristics of elastic circumferential waves. M.S, Choi et al. 3.0
2.5
3.0
l
b
..............
; , - ~ o ~ ~
...................... J~
2.0
E
¢¢~ooo
o %oooooooo~8o%o uO oOc
2.5
//
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E
l v
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1.5
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1.0
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4.0
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0.0
1.0
fd (MHz'mm)
,
',
2.0
:
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Dispersion curves for the phase (a) and group (b) velocities of the A o Lamb plate wave and the A o circumferential wave in Figure 4 , CR = 2.26 Zircaloy-4 cylindrical shell as function of fd and b/a. ~ , b / a = 0.76; O, b / a = 0.82; ~ , b / a = 0.88; +, b / a = 0.93;
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L 4.0
1.0 0.0
1.0
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2.0
3.0
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fd (MHz'mm)
F i g u r e 5 Dispersion curves for the phase (a) and group ( b ) velocities of the So Lamb plate wave and the So circumferential wave in Zircaloy-4 cylindrical shell as function of f d a n d b/a. ~ , b / a = 0 . 7 6 ; ©, b / a = 0 . 8 2 ; D, b / a = 0 . 8 8 ; +, b / a = 0.93; , b/a=0.97; - - - , CR = 2.26
relationship with the phase velocity, C v Cp
Cg-
1
fd 8Cp
(7)
Cp ~9(.fd)
Figures 4 and 5 show the dispersion curves for the phase and g r o u p velocities of the circumferential waves corresponding to the A 0 and So g r o u p of resonances, respectively, together with the velocity dispersion curves of the A o and So L a m b plate waves. The dispersion curves were plotted against fd (expressed as M H z - m m ) instead of kla in order to observe the dependence of shell thickness of the circumferential waves. The results show the transition of the A o and S o circumferential waves to the A o and S o L a m b plate waves, respectively, as b/a increases. In contrast to the S o L a m b plate wave, the S o circumferential wave has a cut-offfrequency caused by the curvature of shell. Breitenbach et al. 2° showed that the S o circumferential wave on the air-filled aluminium shell undergoes the transition to the S o L a m b plate wave in the region offd < 0.5 and described the S o circumferential wave as the lowest transverse mode (T01). Talmant, Fekih
216
Ultrasonics 1992 Vol 30 No 4
and Quentin 24'28'2~ observed experimentally the S O circumferential wave and referred to it as the 'pseudoL a m b ' wave. As Jd comes close to infinity, the lowest L a m b modes, So and A 0, are seen to converge and to coincide with the Rayleigh velocity, CR, on the free elastic plane surface, while all the higher L a m b modes, an example of those is shown in Figure 6 for the A 1 mode, a p p r o a c h to the transverse bulk wave velocity, C-r. The So and A o groups of resonances are thus identified as the Rayleigh-type circumferential waves and the S~ and At(l ~> 1 ) groups as the Whispering Gallery-type circumferential waves. Attenuation characteristics The attenuation constant, ~, of circumferential wave is determined from the width, F -= F W H M (full width half maxima), of the corresponding resonance peak 24 ~ ( N e p e r / r a d ) = FCw/2Cg
(8)
The attenuation of waves that do not exhibit velocity dispersion does not generally depend on the dimensions of the specimen in which they are propagating, and has a fairly smooth dependence on the frequency only.
Propagation characteristics of elastic circumferential waves." M.S. Choi et al. 10.0
3.5
b 8.0
..0. 2 o 2.5
I¢n
;~o-~. . . . . . . . . . . . . . . . . . . . . .
;-. . . . . . . . . . . .
L
°
~=, o °
ooo oo°
6.0
Plate A1
4.0
...........
2.0
1.0
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). . . . . .
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....
,. . . . . . i . . . . . . ;
2.0
3.0
-
-
1.5
~
0.5
4.0
5.0
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f
Plate A1
I
I 2.0
fd (MHz-mm)
i
[ 3.0
,
I 4.0
,
5.0
fd (MHz'mm)
Figure 6 Dispersion curves for the phase (a) and group ( b ) velocities of the A1 Lamb plate wave and the A1 circumferential wave in Zircaloy-4 cylindrical shell as function of fd and b/a. <>, b / a = 0.76; O, b / a = 0.82; D, b / a = 0.88; +, b / a = 0.93; - - - , CT = 2.44
2.0
5.0
a 000
1.5
_
0O0
. Z
1.0
0.5
b
oooO 4.0
O0 00
0o 0 ~. 0 000 ~' 00 0 D.='v='v 000 0 0 0
~
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+
--
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CO c')
3.0
0 0
+ ,,nt,t ~ ~ ~ . . . , . ~ v , ^0 0 00 D.~ - - ~ D r O '~'t~" ~0oUo 0 tl ~ ~00 ^ 0 ~0000000000000~0 0 v
--
1.0
=4000 0 0 0 0 0 0 0 0"'
0.0 I~I~ ~q~ , 0.0
I 1.0
,
I 2.0
,
I 3.0
,
0.0 4.0
i
0.0
I 0.1
fd (MHz'mm) Figure 7
0.2
I 0.3
0.4
fd (MHz-mm)
The attenuation constant (a) and angle ( b ) of the A o circumferential wave in Zircaloy-4 cylindrical shell as function of fd and
b/a. <~, b / a = 0.76; O, b / a = 0.82; D, b / a = 0.88; +, b / a = 0.93 3.0
5.0
a
b
! +
4.0 2.0
4÷
--
+
L
CO 03
÷
o~
3.0
v
z
::
\
J2~
1.0
2.0
r
N 1.0 Ooo
0.0
I
0.0 0.0
1.0
2.0
3.0
4.0
0.0
fd (MHz-mm) Figure
8
0.5
1.0
1.5
, 2.0
fd (MHz,mm)
The attenuation constant (a) and angle ( b ) of the So circumferential wave in Zircaloy-4 cylindrical shell as function of fd and
b/a. <>, b / a = 0.76; O, b / a = 0.82; D, b / a = 0.88; +, b / a = 0.93; - , b / a = 0.97
However, the feature of the circumferential waves is a l t o g e t h e r different. T h e presence of dispersion in the phase a n d g r o u p velocities of the circumferential waves has an i m p o r t a n t effect on the a t t e n u a t i o n characteristics of these waves. F o r examples, the attenuation characteristics
of the A o, So a n d A 1 circumferential waves in Figures 7, 8 a n d 9 show very sensitive frequency, shell thickness and c u r v a t u r e dependence. In the region of fd > 0.5, the a t t e n u a t i o n of the Ao wave (Figure 7a) is large a n d increases with the b/a, but
Ultrasonics 1992 Vol 30 No 4
217
et al.
Propagation characteristics of elastic circumferential waves." M.S. Choi 2.0
:5
3.0
1.5
2.0 (4:) 03
7
1.0
~,~
c~
/ 0.5
-..~..~0~
0.0 Figure0 b/a. ~ , b / a
~
~
Atl~q9 ~~ a ~ ° ? .0
c?~,
4+
2.0
oOOO ~
00
~ 00 00 O0oO ¢
[
,~
-"'-'00
000000000
00
,
3.0 fd (MHz,mm)
I 4.0
0.0
, 5.0
I 1.0
2 .O
,
3.0 fd
4.0
(MHz'mm)
The attenuation constant (a) and (b) angle of theA~ circumferential wave in Zircaloy-4 cylindrical shell as function of f d a n d = 0.76; C), b / a = 0.82; D, b / a - 0.88; +, b / a - 0.93
the A 0 wave propagates well where the Jd is small. In order to emphasize the characteristics in the low attenuation region, as shown in Fi,qure 7h, we introduce a notion of the attenuation angle, q~(degree)_ 180/7z~, which is the travelling angle required for the amplitude of circumferential wave to decrease to 1/e of its initial value. As the h/a increases, the attenuation angle increases and the fi/ region in which the A o wave can encircle the scattering object once or more becomes broad. At a particular value of./d and h/a, the A o wave has a very small attenuation constant and can encircle the scattering object several thousand times. In a scattering experiment, the circumferential wave may not encircle the scattering object several thousand times because of the internal attenuation caused by the grain boundary scattering and thermal relaxation in material. However, the internal attenuation is not considered in this work. The attenuation of the So wave (Fi,qure 8)is minimum when the velocity dispersion (Figure 5) is minimum. The minimum point of the attenuation constant, the maximum point of the attenuation angle, comes close to Id = O, as the h/a increases. The attenuation constant of the S, wave has a maximum in the neighbourhood offd = 2.0 and decreases with.[d. The So wave at large wdue of Jd, Rayleigh surface wave, has small attenuation constant increasing with the h/a. As shown in Figure 3, the resonance peaks corresponding to the A~ circumferential wave have a narrow width in the neighbourhood of the cut-offfrequency I/ill = C~,/2 = 1.22 of the A 1 Lamb plate wave, thus, its attenuation constant is small in spite of the slow group velocity. The attenuation constant of the A~ wave (Fiqure 9) has a minimum in the neighbourhood ofjd - 2.2 and increases with the h/a. If the b/a is larger than 0.93, the A~ wave has difficulty encircling the scattering object only once. These observations lead us to the conclusion that each elastic circumferential wave propagates with a very small attenuation constant, thus, can encircle the scattering body several times for the characteristicjd region which varies with the h/'a.
Backscattered amplitude The amplitude of circumferential waves is determined from the height of the corresponding resonance peaks. Fi~lure lOa shows the backscattered amplitude of the A o
218
?
1.0
00060000^^^
Ultrasonics 1992 Vol 30 No 4
0.6
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0.4
+~0
0
~\oo
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J
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-=
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O
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{ ~ %
°°
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I
1.0
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l
2.0
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3.0
4.0
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0.4
02 1 0.0
/
1.0
~
t
,
2.0
I
i
3.0 fd
I
4.0
I
/
5.0
(MHz'mm)
Figure 10 The backscattered amplitude of the Ao (a), So (b), and A~ (c) circumferential wave in Zircaloy-4 cylindrical shell as function of f d and b / a . ~ , b / a 0.76; O, b / a - 0 . 8 2 ; ~, b / a - 0.88; + , b / a - 0.93; , b/a 0.97
Propagation characteristics of elastic circumferential waves: M.S. Choi et al. circumferential wave as function offd for various values of b/a. The amplitude decreases exponentially with the fd, the larger b/a, the smaller the amplitude. The amplitude of the S o and A1 circumferential waves is shown in Figures lOb and lOc, respectively. The dependence of the amplitude on the variablesfd and b/a is the same as in the case of the A 0 wave. Superimposing three figures, we notice that the amplitude of all elastic circumferential waves for each value of b/a falls on the same exponential decay curve. This implies that the amplitude of elastic circumferential waves can be determined by a single characteristic curve which depends onfd and b/a, regardless of the types of circumferential wave. As discussed above, each elastic circumferential wave has its own characteristic fd region of very small attenuation constants. For example, the A o, So and A t circumferential waves in the shell with b/a = 0.88 have the lowest attenuation constant at fd = 002, 050 and 1.23, respectively. The multiple echo returns from a shell in a backscattering of a single underwater sound pulse are ascribed to the presence of a circumferential wave with a very small attenuation. Therefore the amplitudes of the multiple echo returns corresponding to the A o mode in the shell with b/a = 0.88 are larger than those corresponding to the So or A 1 modes.
Conclusion The backscattering of plane acoustic wave from the air-filled Zircaloy-4 nuclear fuel cladding tube immersed in water was analysed using RST. The backscattered amplitude of Rayleigh normal mode was shown to be the superposition of a smooth and regular background term and a series of resonances of the elastic cylindrical shell. The appearance of the resonances was enhanced by the subtraction of the rigid background from the backscattered amplitude. In accordance with the cut-off frequencies, the groups of resonances were identified as the elastic circumferential waves corresponding to the Lamb plate waves. Each resonance could be evaluated accurately using the calculation step, A(kla ) = 0.0001. The circumferential waves underwent a smooth transition to the corresponding Lamb waves, as the b/a increased. The S o circumferential wave, in contrast to the So Lamb plate wave, had a cut-off frequency caused by the curvature of the shell. Each elastic circumferential wave had the characteristic fd region of very small attenuation constants varying with the b/a. The amplitude of elastic circumferential waves could be determined by a single characteristic curve which depends on fd and b/a, regardless of the types of circumferential wave.
References 1 2
Maxwell, R.B. Zirconium alloys, in: Engineering Manual Chalk River Nuclear Lab., Chalk River, Ontario, Canada (1969, 1970) Sec 5, Subsec 3, Part 1-2 Recommended procedure for ultrasonic testing of zirconium and zirconium alloy tubing for nuclear service A S T M B353-77a 248 257
D'Annucci, F. and Schar~nberg, R. Operational experience of ultrasonic testing on fuel assemblies with FFRDS Nuclear Europe (1985) 5 23 24 4 Patel, H.B., Ditsehun, A. and Hunton, A.E. Ultrasonic testing of irradiated fuel sheathing Mat Eval (1975) 33 49-55 5 Barnard, G.R. and McKinney, C.M. Scattering of acoustic energy by solid and air-filled cylinders in water J Acoust Soc Am ( 1961 ) 33 226-238 6 Hampton, L.D. and MeKinney, C.M. Experimental study of the scattering of acoustic emergy from solid metal spheres in water J Acoust Soc Am (1961) 33 644-678 7 Horton, C.W., King, W.R. and Diereks, K.J. Theoretical analysis of the scattering of short acoustic pulses by a thin-walled metallic cylinder in water J Acoust Soc Am (1962) 34 1929-1932 8 Diercks, K.J., Goldsberry, T.G. and Horton, C.W. Circumferential waves in thin-walled air-filled cylinders in water J Acoust Soc Am (1963) 35 59 64 9 Goodman, P.R., Bunney, R.E. and Marshall, S.W. Observation of circumferential waves on solid aluminium cylinders J Acoust Soc Am (1967) 42 523-524 10 ~berall, H. Surface waves in acoustics, in: Physical Acoustics Vol 10 (Eds Marson, W.P. and Thurston, R.N.) Academic Press, New York (1973) 1 60 11 Frisk, G.V., Dickey, J.W. and l[3berall, H. Surface wave modes on elastic cylinders J Acoust Soc Am (1975) 42 996-1008 12 Frisk, G.V. and 1Dberall, H. Creeping waves and lateral waves in acoustic scattering by large elastic cylinders J Acoust Soc Am (1976) 59 46-54 13 Dickey, J.W., Frisk, G.V. and l~berall, H. Whispering Gallery wave modes on elastic cylinders J Acoust Soc Am (1976) 59 1339 1346 14 Ugin~zius, P. and 1D~rall, H. Creeping-wave analysis of acoustic scattering by elastic cylindrical shells J Acoust Soc Am (1968) 43 1025-1034 15 Dickey, J.W., Nixon, D.A. and D'Aehangelo, J.M. Acoustic high-frequency scattering by elastic cylindrical shells J Acoust Soc Am (1983) 74 294 304 16 1Dberall, H., Dragonette, L.R. and Flax, L. Relation between creeping waves and normal modes of vibration of a curved body J Acoust Soc Am (1977) 61 711-715 17 Dickey, J.W. and l~berall, H. Surface wave resonances in sound scattering from elastic cylinders J Acoust Soc Am (1978) 63 319-320 18 Flax, L., Dragonette, L.R. and 1Dberall, H. Theory of elastic resonance excitation by sound scattering J Acoust Soc Am (1978) 63 723-731 19 Murphy, J.D., Breitenbach, E.D. and Dberall, H. Resonance scattering of acoustic waves from cylindrical shells J Acoust Soc Am (1978) 64 677 683 20 Breitenbach, E.D., 1Dberall, H. and Yoo, K.B. Resonant acoustic scattering from elastic cylindrical shells J Acoust Soc Am (19831 74 1267-1273 21 Solomon, S.G., IUberall, H. and Yoo, K.B. Mode conversion and resonance scattering of elastic waves from a cylindrical fluid-filled cavity Acustica (1984) 55 147 159 22 Gaunaurd, G.C. and Brill, D. Acoustic spectrogram and complexfrequency poles of a resonantly excited elastic tube J Acoust Soc Am (1984) 75 1680- 1693 23 Rousselot, J.L. Comportement acoustique d'une tube cylindrique mince en basse frequence Acustica (1985) 58 291 297 24 Talmant, M., Quentin, G., Rousselot, J.L., Subrahmanyam, J.V. and I~berall, H. Acoustic resonance of thin cylindrical shells and the resonance scattering theory J Acoust Soc Am (1988) 84 681 688 25 Veksler, N.D. The analysis of peripheral waves in the problem of plane acoustic pressure wave scattering by a circular cylindrical shell Acustica (1989) 69 63-72 26 Veksler, N.D. Transverse whispering gallery waves in scattering by elastic cylinders Ultrasonics (1990) 28 67-76 27 Watson, G.N. A Treatise on Theory q/ Bessel Functions 2nd Edn, Cambridge University Press, London (19661 28 Fekih, M. and Quentin, G. Scattering of short ultrasonic pulses by thin cylindrical shells: generation of guided waves inside the shell Phys Lett(1983) 96 379-384 29 Talmant, M. and Quentin, G. Backscattering of a short ultrasonic pulse from thin cylindrical shells d Appl Phys (1988) 63 1857 1863 3
Ultrasonics 1992 Vol 30 No 4
219
Received 21 June 1991; revised 12 December 1991 The backscattering of a plane acoustic wave from an air-filled Zircaloy-4 nuclear fuel cladding tube in water is analysed as a function of kla using the resonance scattering theory. In accordance with the cut-off frequencies, the groups of resonances are identified as the elastic circumferential waves corresponding to the Lamb plate waves. In order to determine accurate propagation characteristics, each resonance is evaluated using the very small calculation step, A ( k l a ) = 0.0001, and the results are presented in the form of phase and group velocities, attenuation constants and angles, and backscattered amplitudes of the elastic circumferential waves as function of the frequency, tube thickness and curvature.
Keywords: circumferential wave; resonance; Lamb wave; Zircaloy-4 t u b e
In the present commercial nuclear power reactors, Zircaloy-4 is one of the more important materials for cladding fissile fuels because of its favourable mechanical, chemical and physical properties such as small absorption cross-section for thermal neutron 1. Any failure in the cladding tube will directly cause radioactive fission gas leakage to the coolant, and thus may result in a serious economic and safety problem. Various non-destructive testing ( N D T ) techniques have been applied to evaluate the quality of the cladding tube. In particular, ultrasonic N D T has been successfully used in the manufacturing process 2, during the operating period in reactor 3 and in the post-irradiation examination 4. However, at present, the types and application techniques of ultrasonic waves used are limited. The pulse-echo or through-transmission techniques using elastic bulk waves have been commonly used. Since the discovery by Barnard and McKinney 5 of multiple echo returns from a single underwater sound pulse incident on the elastic cylinder, a considerable number of experimental investigations 6-9 have been conducted to exploit this phenomenon. It is generally ascribed to the presence of circumferential waves with the behaviour of the surface wave type, i.e. the waves that travel circumferentially along the surface of the scattering body before proceeding to an observer. On the other hand, the sound scattering from an elastic cylinder immersed in a fluid has been investigated theoretically by means of the Watson-Sommerfeld transformation (WST) applied to the classical Rayleigh * Quality Control Department, HWR Fuel Fabrication Division, Korea Atomic Energy Research Institute, PO Box 7, Daeduk-danji, Taejon 305-606, Korea
normal mode series of the scattering solution 1°. Uberall obtained a sum of residue terms which corresponds to circumferential waves divided into two classes: those with velocity close to the elastic bulk wave velocity, i.e. Rayleigh and Whispering Gallery waves, and those with velocity close to the sound velocity in the surrounding fluid, i.e. Stoneley and Franz waves. Frisk et a1.11"12 and Dickey et al. 13 systematically investigated all the types of surface wave by both analytical and numerical methods. They showed that, in the limit of large cylinder radius, the Rayleigh, Stoneley and Whispering Gallery waves tend towards the generalized Rayleigh, Stoneley and lateral waves on a flat surface, respectively, while the Franz waves disappear. Circumferential waves on the elastic cylindrical shell were studied numerically by Ugineius and [Sberall TM and Dickey et al. 15 using the WST method. The solution is more complex than that of the solid cylinder, because of two boundaries, the inside and outside surfaces, and therefore a faster and larger computer is required. Recently, however, many authors 16-z6 have reported that the circumferential wave phenomenon can be more easily analysed using the resonance scattering theory (RST) known as the 'Regge' pole technique in nuclear physics. The backscattering of plane acoustic wave from the air-filled Zircaloy-4 nuclear fuel cladding tube in water is analysed using RST and the results are presented in the form of phase velocities, group velocities, attenuation constants, attenuation angles and backscattered amplitudes of the elastic circumferential waves as a function of frequency, shell thickness and curvature. An easy method for identification of resonances is described and each resonance is evaluated using the very small calculation
0041-624X/92/040213-07 © 1992 Butterworth-Heinemann Ltd
Ultrasonics 1992 Vol 30 No 4
213
al.
Propagation characteristics of elastic circumferential waves. M.S. C h o i e t step, A( k ~a ) = 0.0001, to determine accurate propagation characteristics of the elastic circumferential waves. Relationships a m o n g the circumferential waves in a cylindrical shell, those in a solid cylinder and L a m b waves in a plate are also discussed. N u m e r i c a l
analysis
Computation of the scattering form functions When the plane acoustic wave exp[i(k~x-~,Jt)] is incident normally on fluid-filled cylindrical shell immersed in another fluid, as shown in Fi,qure 1, the scattered pressure at the position (r, 05) of the observer located outside the shell is given by the Rayleigh normal mode series ~,t t
P.~(r. 05.t)=exp(-i(.)t) ~
coefficient matrix obtained from the six boundary' conditions. The elements of these determinants, listed in Ref. 14, contain the cylinder functions and their derivatives with arguments kia or kih(i = 1. L,T, 3), where k~ and k s are the acoustic wave numbers in the outside and inside fluids, respectively, and k L and k~ are the wave numbers of bulk longitudinal and transverse elastic waves in the shell, respectively. Employing the Hankel asymptotic form 27 for H,,~t)(ktr), the far-field value of the scattered pressure becomes exp[ilktr
.~t)]
.I;,(05)
(2)
n = O
where
1;,(05) =
c,,
cos(nO )
(3 )
c.i"b"H~lqk~r)cos(n05)
n = o
,t
D, I
(1) where ~;, is called the N e u m a n n factor, % = 1, c,, = 2 for n>~ 1, and b, and D, are 6 x 6 determinants of the
exp[i ( k l x -- ~ot)]
~
~
(r, ~o)
2
The function,l;,(05) represents the angular distribution of the scattered pressure and is called the scattering form function of the nth normal mode. In this study, the scattering form function, .11,(0), was examined in the backscattering direction (05 = ~z) and the backscattered amplitude,f,(n), was calculated on Hewlett Packard 9000 series model 310 computer for the range of n - 0 60, kla 0 200 and h / a = 0.01 0.99. All calculations were for thc air-filled Zircaloy-4 nuclear fuel cladding tube immersed in water, where ,)~ = 1 . 0 g c m s. p, = 6 . 5 5 g c m s, Ps = 0 - 0 0 1 2 g c m 3. The longitudinal and transverse sound velocities were taken to be ('~, 4540 m s i = 3.068 C,, and ('] 2440 m s ~ = 1.649 (',,, respectively', where C,, = C~ is the sound velocity in water. 1480m s ].
Identification of resonances The backscattered amplitude,.ti,(rt), plotted vcrsus k]a consists of the smooth and regular b a c k g r o u n d and a series of resonance spikes, for example as shown in Fi.qure 2a for the shell with h..'a = 0.82. Three resonances labelled as A o, S o and A~ appear and the background agrees closely, except at the very low frequency region, with the backscattered amplitude of the "acoustically rigid' cylinder ~o F i g u r e 1 Geometry of a plane wave scattering from an infinite elastic cylindrical shell. Here, C1 and [)1 denote sound velocity and material density, respectively
t.i~qn )
.....
1.0
[
1
=,
..--,
,:,,
j,,,(kla }
,4)
cos(nrc)
tti,11'(kla)
1.0 a
b
~0 7D
E E <
12
\=lgm/
~0 -(3
0.5
0.5
So
E ,<
fi
A1 A1
Ao
0.0
,
0
L
Ao ,
L
L
]
10
,
,
,
L
I
L
,
20 k,a
J
,
1
30
J
,
L
0.0
I
40
I
0
I
I
I
1
1
1
10
,
I
20
,
,
,
~
I
30
J
~
40
kla
Figure2 Modulus of n = 2 normal mode backscattering amplitude (a) foraZircaloy-4shell ( b / a = 0 . 8 2 , ) and a rigid cylinder ( If2(n)l and If},n (Tt)l, and (b) for the shell with the rigid background subtracted, If2(~) - f~n (~)1, plotted versus k~a
214 Ultrasonics 1992 Vol 30 No 4
,
),
Propagation characteristics of elastic circumferential waves. M.S. Choi et al. where the prime on the cylinder functions denotes the differentiation with respect to their argument. The calculation step size, A(k~a), was 0.05. Figure 2b shows the enhanced appearance of the resonance responses of the elastic shell by subtracting the rigid background, ft,)(g), from the backscattered amplitude,f,(Tt). The rigid background appears as a result of the existence of the Franz-type circumferential waves and the specularly reflected waves from the apex of the rigid cylinder TM. However, the rigid background may be substantially different from the actual background at low frequencies. In fact, this difference causes the resonance-like feature appearing as the broad peaks denoted by the arrow in Figure 2b. Breitenbach et al. 2° described the broad peak as a resonance response suggesting the existence of the Rayleigh surface wave. However, as shown in Figure 3 for the shell with b/a = 0.88, the broad peak can be found only in the neighbourhood of kla = n implying that the peak is related to the circumferential wave whose velocity is close to the sound velocity in water, i.e. the Stoneley or Franz surface wave rather than the Rayleigh surface wave. Thus, we believe that the broad peak is not the Rayleigh-type resonance response of elastic shell. Furthermore, the broad peak does not appear in the backscattered amplitudes of the n = 0 normal mode for any shells, but appears as only one peak for each normal mode n ~> 1 and becomes weak as the normal mode number increases or as the ratio of radii, b/a, decreases. Almost all the resonances reappear in the subsequent normal modes at subsequently higher frequencies. Thus, the resonance peaks can be classified into groups corresponding to the elastic circumferential waves. In accordance with the cut-off frequencies, [fd], the Sz and A~(/>/ 1) circumferential waves correspond to the symmetric and antisymmetric L a m b waves in Zircaloy-4 plate, respectively. The cut-off frequency is determined from the position, kla, of the n = 0 resonance as
fd = k,a (1 - b/a) Cw/2n
So
n=0 ~$1 80'.16
~.~ So
A3 n=1
Sl S2
A3
I
n=2
I S,~.~ 43.41 $1 S2
A1
A2
A3 $3
,
I
n=3
$1 S2
1
A3
A2,
n=4 "r-
I
~
1
.£
$1 S2
A2
A3 n=5
v
A1
Sl S2
A2
A3
D r~
E <
n=6 A1
Sl
J
S2
~.S
A2 l
3 n=7
Ao
A1
$1
S2
.1 J L J L
A2
A3 S3
1 n=8
A1
$1
S2
I _A.jL
A2
A3
1
Ao
n=9
(5) A1
For example, the n = 0 resonance at k l a = 80.16 corresponds to the cut-off frequency [fd] = 2.27 MHz.mm, which is the same as that of the first-order symmetric L a m b plate mode, CL/2. Thus the group of resonances is labelled $1. The cut-off frequency of the group without the n = 0 resonance can be approximated from the position of the next normal mode resonance. For example, the n = 1 resonance at kla = 43.41 corresponds to the cut-off frequency [fd] = 1.227 MHz.mm, which is almost the same as that of the first-order antisymmetric L a m b plate mode, CT/2. Thus, the group of resonances is labelled Aa. It can be inferred from the above result that the other elastic circumferential waves, the So and A 0 waves, correspond to the zeroth order symmetric and antisymmetric L a m b plate waves, respectively. This correspondence will be ensured in the following section. Therefore, it can be concluded that the elastic circumferential waves in the shell make one-to-one correspondence to the L a m b waves in the plate of the same material and thickness. According to RST, the propagation characteristics of the elastic circumferential waves are determined from the position, width and height of the corresponding resonance peaks. The accuracy of the measured quality of resonance peaks depends on the calculation step size. All previous authors ~6-26 used the steps of the same order of magnitude as that of Figures
i
i
Sl i
$2 i
A2 i
,
100
0
i
200
kla Figure 3 Modulus of nth normal mode backscattering amplitude ( n = 0 - 9 ) for a Zircaloy-4 shell (b/a=0.88) with a rigid background subtracted, Ifn(TZ) f(/)(~)l, plotted versus kla
2 and 3. However, the step size is rather large, thus high quality (narrow) resonances are occasionally lost. In order to determine the more accurate propagation characteristics described in the following sections, we recalculated the backscattered amplitudes, f , ( ~ ) and f~,r)(~Z),in the neighbourhood of the resonance peaks using the narrower step size, A(kla ) = 0.0001.
Velocity dispersion The phase velocity, Cp, of circumferential wave is determined from the position, kla, of the corresponding resonance peak 16
Cp = C w kla/n
(6)
The group velocities, Cg, are obtained from the following
Ultrasonics 1992 Vol 30 No 4
215
Propagation characteristics of elastic circumferential waves. M.S, Choi et al. 3.0
2.5
3.0
l
b
..............
; , - ~ o ~ ~
...................... J~
2.0
E
¢¢~ooo
o %oooooooo~8o%o uO oOc
2.5
//
2.0
Plate Ao
E
l v
1.5
1.5
1.0
1.0
0.5
0.5 0.0
1.0
2.0
3.0
4.0
I
0.0
1.0
fd (MHz'mm)
,
',
2.0
:
I
,
3.0
4.0
fd (MHz'mm)
Dispersion curves for the phase (a) and group (b) velocities of the A o Lamb plate wave and the A o circumferential wave in Figure 4 , CR = 2.26 Zircaloy-4 cylindrical shell as function of fd and b/a. ~ , b / a = 0.76; O, b / a = 0.82; ~ , b / a = 0.88; +, b / a = 0.93;
6.0
5.0
b
a
5.0
4.0
o
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E
4.0
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3.0
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L
0.0
I 1.0
,
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,
I 3.0
L 4.0
1.0 0.0
1.0
fd (MHz'mm)
2.0
3.0
4.0
fd (MHz'mm)
F i g u r e 5 Dispersion curves for the phase (a) and group ( b ) velocities of the So Lamb plate wave and the So circumferential wave in Zircaloy-4 cylindrical shell as function of f d a n d b/a. ~ , b / a = 0 . 7 6 ; ©, b / a = 0 . 8 2 ; D, b / a = 0 . 8 8 ; +, b / a = 0.93; , b/a=0.97; - - - , CR = 2.26
relationship with the phase velocity, C v Cp
Cg-
1
fd 8Cp
(7)
Cp ~9(.fd)
Figures 4 and 5 show the dispersion curves for the phase and g r o u p velocities of the circumferential waves corresponding to the A 0 and So g r o u p of resonances, respectively, together with the velocity dispersion curves of the A o and So L a m b plate waves. The dispersion curves were plotted against fd (expressed as M H z - m m ) instead of kla in order to observe the dependence of shell thickness of the circumferential waves. The results show the transition of the A o and S o circumferential waves to the A o and S o L a m b plate waves, respectively, as b/a increases. In contrast to the S o L a m b plate wave, the S o circumferential wave has a cut-offfrequency caused by the curvature of shell. Breitenbach et al. 2° showed that the S o circumferential wave on the air-filled aluminium shell undergoes the transition to the S o L a m b plate wave in the region offd < 0.5 and described the S o circumferential wave as the lowest transverse mode (T01). Talmant, Fekih
216
Ultrasonics 1992 Vol 30 No 4
and Quentin 24'28'2~ observed experimentally the S O circumferential wave and referred to it as the 'pseudoL a m b ' wave. As Jd comes close to infinity, the lowest L a m b modes, So and A 0, are seen to converge and to coincide with the Rayleigh velocity, CR, on the free elastic plane surface, while all the higher L a m b modes, an example of those is shown in Figure 6 for the A 1 mode, a p p r o a c h to the transverse bulk wave velocity, C-r. The So and A o groups of resonances are thus identified as the Rayleigh-type circumferential waves and the S~ and At(l ~> 1 ) groups as the Whispering Gallery-type circumferential waves. Attenuation characteristics The attenuation constant, ~, of circumferential wave is determined from the width, F -= F W H M (full width half maxima), of the corresponding resonance peak 24 ~ ( N e p e r / r a d ) = FCw/2Cg
(8)
The attenuation of waves that do not exhibit velocity dispersion does not generally depend on the dimensions of the specimen in which they are propagating, and has a fairly smooth dependence on the frequency only.
Propagation characteristics of elastic circumferential waves." M.S. Choi et al. 10.0
3.5
b 8.0
..0. 2 o 2.5
I¢n
;~o-~. . . . . . . . . . . . . . . . . . . . . .
;-. . . . . . . . . . . .
L
°
~=, o °
ooo oo°
6.0
Plate A1
4.0
...........
2.0
1.0
~
). . . . . .
/
....
,. . . . . . i . . . . . . ;
2.0
3.0
-
-
1.5
~
0.5
4.0
5.0
.0
f
Plate A1
I
I 2.0
fd (MHz-mm)
i
[ 3.0
,
I 4.0
,
5.0
fd (MHz'mm)
Figure 6 Dispersion curves for the phase (a) and group ( b ) velocities of the A1 Lamb plate wave and the A1 circumferential wave in Zircaloy-4 cylindrical shell as function of fd and b/a. <>, b / a = 0.76; O, b / a = 0.82; D, b / a = 0.88; +, b / a = 0.93; - - - , CT = 2.44
2.0
5.0
a 000
1.5
_
0O0
. Z
1.0
0.5
b
oooO 4.0
O0 00
0o 0 ~. 0 000 ~' 00 0 D.='v='v 000 0 0 0
~
j,='~'~ ~00 0 ~,.¢j~ 0~ 0
+
--
O0 ¢
CO c')
3.0
0 0
+ ,,nt,t ~ ~ ~ . . . , . ~ v , ^0 0 00 D.~ - - ~ D r O '~'t~" ~0oUo 0 tl ~ ~00 ^ 0 ~0000000000000~0 0 v
--
1.0
=4000 0 0 0 0 0 0 0 0"'
0.0 I~I~ ~q~ , 0.0
I 1.0
,
I 2.0
,
I 3.0
,
0.0 4.0
i
0.0
I 0.1
fd (MHz'mm) Figure 7
0.2
I 0.3
0.4
fd (MHz-mm)
The attenuation constant (a) and angle ( b ) of the A o circumferential wave in Zircaloy-4 cylindrical shell as function of fd and
b/a. <~, b / a = 0.76; O, b / a = 0.82; D, b / a = 0.88; +, b / a = 0.93 3.0
5.0
a
b
! +
4.0 2.0
4÷
--
+
L
CO 03
÷
o~
3.0
v
z
::
\
J2~
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2.0
r
N 1.0 Ooo
0.0
I
0.0 0.0
1.0
2.0
3.0
4.0
0.0
fd (MHz-mm) Figure
8
0.5
1.0
1.5
, 2.0
fd (MHz,mm)
The attenuation constant (a) and angle ( b ) of the So circumferential wave in Zircaloy-4 cylindrical shell as function of fd and
b/a. <>, b / a = 0.76; O, b / a = 0.82; D, b / a = 0.88; +, b / a = 0.93; - , b / a = 0.97
However, the feature of the circumferential waves is a l t o g e t h e r different. T h e presence of dispersion in the phase a n d g r o u p velocities of the circumferential waves has an i m p o r t a n t effect on the a t t e n u a t i o n characteristics of these waves. F o r examples, the attenuation characteristics
of the A o, So a n d A 1 circumferential waves in Figures 7, 8 a n d 9 show very sensitive frequency, shell thickness and c u r v a t u r e dependence. In the region of fd > 0.5, the a t t e n u a t i o n of the Ao wave (Figure 7a) is large a n d increases with the b/a, but
Ultrasonics 1992 Vol 30 No 4
217
et al.
Propagation characteristics of elastic circumferential waves." M.S. Choi 2.0
:5
3.0
1.5
2.0 (4:) 03
7
1.0
~,~
c~
/ 0.5
-..~..~0~
0.0 Figure0 b/a. ~ , b / a
~
~
Atl~q9 ~~ a ~ ° ? .0
c?~,
4+
2.0
oOOO ~
00
~ 00 00 O0oO ¢
[
,~
-"'-'00
000000000
00
,
3.0 fd (MHz,mm)
I 4.0
0.0
, 5.0
I 1.0
2 .O
,
3.0 fd
4.0
(MHz'mm)
The attenuation constant (a) and (b) angle of theA~ circumferential wave in Zircaloy-4 cylindrical shell as function of f d a n d = 0.76; C), b / a = 0.82; D, b / a - 0.88; +, b / a - 0.93
the A 0 wave propagates well where the Jd is small. In order to emphasize the characteristics in the low attenuation region, as shown in Fi,qure 7h, we introduce a notion of the attenuation angle, q~(degree)_ 180/7z~, which is the travelling angle required for the amplitude of circumferential wave to decrease to 1/e of its initial value. As the h/a increases, the attenuation angle increases and the fi/ region in which the A o wave can encircle the scattering object once or more becomes broad. At a particular value of./d and h/a, the A o wave has a very small attenuation constant and can encircle the scattering object several thousand times. In a scattering experiment, the circumferential wave may not encircle the scattering object several thousand times because of the internal attenuation caused by the grain boundary scattering and thermal relaxation in material. However, the internal attenuation is not considered in this work. The attenuation of the So wave (Fi,qure 8)is minimum when the velocity dispersion (Figure 5) is minimum. The minimum point of the attenuation constant, the maximum point of the attenuation angle, comes close to Id = O, as the h/a increases. The attenuation constant of the S, wave has a maximum in the neighbourhood offd = 2.0 and decreases with.[d. The So wave at large wdue of Jd, Rayleigh surface wave, has small attenuation constant increasing with the h/a. As shown in Figure 3, the resonance peaks corresponding to the A~ circumferential wave have a narrow width in the neighbourhood of the cut-offfrequency I/ill = C~,/2 = 1.22 of the A 1 Lamb plate wave, thus, its attenuation constant is small in spite of the slow group velocity. The attenuation constant of the A~ wave (Fiqure 9) has a minimum in the neighbourhood ofjd - 2.2 and increases with the h/a. If the b/a is larger than 0.93, the A~ wave has difficulty encircling the scattering object only once. These observations lead us to the conclusion that each elastic circumferential wave propagates with a very small attenuation constant, thus, can encircle the scattering body several times for the characteristicjd region which varies with the h/'a.
Backscattered amplitude The amplitude of circumferential waves is determined from the height of the corresponding resonance peaks. Fi~lure lOa shows the backscattered amplitude of the A o
218
?
1.0
00060000^^^
Ultrasonics 1992 Vol 30 No 4
0.6
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,
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,
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2.0
I
3.0
4.0
C
0.4
02 1 0.0
/
1.0
~
t
,
2.0
I
i
3.0 fd
I
4.0
I
/
5.0
(MHz'mm)
Figure 10 The backscattered amplitude of the Ao (a), So (b), and A~ (c) circumferential wave in Zircaloy-4 cylindrical shell as function of f d and b / a . ~ , b / a 0.76; O, b / a - 0 . 8 2 ; ~, b / a - 0.88; + , b / a - 0.93; , b/a 0.97
Propagation characteristics of elastic circumferential waves: M.S. Choi et al. circumferential wave as function offd for various values of b/a. The amplitude decreases exponentially with the fd, the larger b/a, the smaller the amplitude. The amplitude of the S o and A1 circumferential waves is shown in Figures lOb and lOc, respectively. The dependence of the amplitude on the variablesfd and b/a is the same as in the case of the A 0 wave. Superimposing three figures, we notice that the amplitude of all elastic circumferential waves for each value of b/a falls on the same exponential decay curve. This implies that the amplitude of elastic circumferential waves can be determined by a single characteristic curve which depends onfd and b/a, regardless of the types of circumferential wave. As discussed above, each elastic circumferential wave has its own characteristic fd region of very small attenuation constants. For example, the A o, So and A t circumferential waves in the shell with b/a = 0.88 have the lowest attenuation constant at fd = 002, 050 and 1.23, respectively. The multiple echo returns from a shell in a backscattering of a single underwater sound pulse are ascribed to the presence of a circumferential wave with a very small attenuation. Therefore the amplitudes of the multiple echo returns corresponding to the A o mode in the shell with b/a = 0.88 are larger than those corresponding to the So or A 1 modes.
Conclusion The backscattering of plane acoustic wave from the air-filled Zircaloy-4 nuclear fuel cladding tube immersed in water was analysed using RST. The backscattered amplitude of Rayleigh normal mode was shown to be the superposition of a smooth and regular background term and a series of resonances of the elastic cylindrical shell. The appearance of the resonances was enhanced by the subtraction of the rigid background from the backscattered amplitude. In accordance with the cut-off frequencies, the groups of resonances were identified as the elastic circumferential waves corresponding to the Lamb plate waves. Each resonance could be evaluated accurately using the calculation step, A(kla ) = 0.0001. The circumferential waves underwent a smooth transition to the corresponding Lamb waves, as the b/a increased. The S o circumferential wave, in contrast to the So Lamb plate wave, had a cut-off frequency caused by the curvature of the shell. Each elastic circumferential wave had the characteristic fd region of very small attenuation constants varying with the b/a. The amplitude of elastic circumferential waves could be determined by a single characteristic curve which depends on fd and b/a, regardless of the types of circumferential wave.
References 1 2
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