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Scattering of ultrasound from clusters of inclusions
Scattering of ultrasound inclusions* I.D. Culverwellt
from clusters of
and J.A. Ogilvy
Theoretical Studies Department, Materials and Manufacturing Technology Division, Industrial Technology, Building 424.4, Harwell Laboratory, Oxon OX1 1 ORA, UK Received
19 February
1991;
revised
AEA
79 July 1991
The scattering of low frequency ultrasound by a cluster of arbitrary ellipsoidal inclusions is studied numerically, using the extended quasi-static approximation. The statistical variations in the scattered intensities are investigated. The relationship between the back-scattered signal and the attenuation of the ultrasound is determined. The implications of these results for the ultrasonic examination of steels containing inclusions are discussed. Keywords:
ultrasound
attenuation;
scattering;
The ultrasonic examination of an engineering structure may be impeded by the presence of small imperfections in the region under inspection. This is because inhomogeneities in a material can attenuate a beam of ultrasound by scattering its energy away from the forward direction. Furthermore, this scattered energy may arrive at the detector and interfere with the signal produced from another defect such as a crack. One example of this behaviour occurs in some steels, when these contain clusters of non-metallic inclusions, originally present as impurities in the melt. It is therefore important to understand the scattering of ultrasound from clusters ofinclusions. This is the question addressed in this paper. The scattering of plane longitudinal waves by spherical obstacles in an isotropically elastic solid was first studied by Ying and True11 in 1956l. Their solution was exact (although not in closed form) and included special cases such as the Born approximation, which is valid when the elastic properties of the scatterer and the surrounding medium are similar, and the quasi-static approximation, in which the stress field in the scatterer is given by the static field produced by the displacement of the incident wave, which is valid in the low frequency regime (for discussions of both see Reference 2). In later work by Gubernatis3 the quasi-static approximation was extended to include the phase of the incident wave in the stress field. This ad hoc modification increased the region of validity of the approximation from ka CC1 to ka < 2, where k is the wavenumber of the incident field and a is the radius &fthe scatterer. At typical ultrasonic frequencies in steel, k < lo4 m-l, so that the extended quasi-static approximation (EQSA) will adequately describe the scattering from inclusions smaller than a few hundred microns in diameter, a size range which covers all of the scatterers of interest in this paper. * Copyright
belongs to the United Kingdom Atomic Energy Authority
(AEA) t Present address: BP International Chertsey Road, Sunbury-on-Thames, 0041-624X/92/010008-07 @ 1992 Butterworth-Heinemann
8
Ultrasonics
Ltd, Sunbury Research Centre, Middlesex TW16 7LN, UK
Ltd
1992 Vol 30 No 1
inclusions
The question of multiple scattering arises. Does each inclusion in the cluster see the same incident field, or is each one influenced by the ultrasound scattered from all of the others? General considerations [see Equations (l)-(3)] indicate that in the far-field the spherically spreading scattered wave from one inclusion should scale as uSCaz uinck2a3exp(ikR)/R, where uinc is the amplitude of the incident field and R is the distance from the centre of the scatterer. Manganese sulphide (MnS) inclusions in steel have been examined microscopically (see Reference 4 and section 0, Reference 5) and found to have sizes a in the range 5-50pm, and mean separations R about five times larger. Thus the largest value of k2a3/R is ~0.05, which is much less than unity, so that multiple scattering can be ignored. Each inclusion is therefore assumed to scatter an incident field of the same magnitude as (but of different phase from) the field seen by its neighbours. Previous analytical work on the scattering of waves by a collection of obstacles assumed that all the scatterers had the same simple size and shape (spheres or planes) 6-8 . They were also assumed to be distributed isotropically or periodically. In this problem we are interested in inclusions which have a wide variety of shapes and sizes, sometimes with a strongly preferred direction (see later section on P-wave scattering from an elongated cluster). Numerical simulation would seem to be the most straightforward way of calculating the scattering from such agglomerations. It has the further advantage that it is easy to calculate the scattering from different clusters, each having the same statistical properties (i.e. mean inclusion-inclusion separation, standard deviation of inclusion sizes, etc.). The statistical variations of the scattered field, which are of great value experimentally, can therefore be found directly. The structure of this paper is as follows. In the next section the basic theory underlying the model is briefly reviewed. The next three sections describe the results of the scattering of compressional and shear waves from a
Scattering
from clusters of inclusions:
spherical cluster of inclusions, and of the scattering of P-waves from an elongated cluster, similar to those sometimes found in rolled steel plate. The attenuation, back-scatter and the relationship between the two are described. The relevance of these results to the nondestructive testing of steel plates is also discussed. Some results are then presented of a generalized model which simulates the scattering of realistic ultrasonic pulses. The final section discusses the conclusions of the paper, together with possible generalizations of the work. Theory Extended scatterer
quasi-static
approximation
for
a
single
The main features of the EQSA, the low frequency scattering theory which forms the basis of our model, have already been described, but, for completeness, the basic equations are given here. All are taken from Reference 3. Consider an ellipsoidal inclusion of density p’, elastic constants CijL1,centred at the origin of a coordinate system, surrounded by a medium of density p and elastic constants C,,. If there is an incident plane wave . r), then in the EQSA the far-field U Inc = u” exp(ik" scattered displacement at position r is given by Pa(r)
= A exp(ik,r)/r
+ B exp(ik,r)/r
(1) where A (the amplitude of the scattered compressional wave) and B (the amplitude of the scattered shear wave) are given by Ai = Pitjfj(k,i) Bi = (Sij - f,Pj)fj(k,i)
(2)
(summing over repeated indices). In Equation (2) the function f is defined by
in which o is the angular frequency, 6p = p’ - p, 6C,, = Cjjkl - Cijkl, &, (the incident strain field) = i/2(uEkf + u,0kz) and Fijkl is a fourth-rank tensor which depends only on the size of the scatterer and the Poisson’s ratio of the surrounding medium. For ellipsoidal inclusions, Fijrr is constant throughout the inclusion because the static stress field inside an ellipsoid is uniform’. S(q) is a shape factor given by the Fourier transform of the shape of the inclusion with respect to q = kP - k”Go, with S(0) equal to the volume of the scatterer. For spherical scatterers we find that ikr u
SCB cc
kza3
5
Y
(4)
i.e. the amplitude of the scattered field has a strong dependence on the wavevector and the size of the scatterer. Scattering
I.D. Culverwell
and J.A. Ogilvy
then chosen to define the positions of the centres of the ellipsoids (these points are sufficiently widely spaced such that overlap between ellipsoids cannot occur). Finally, 3N Euler angles lo define the orientations of the inclusions with respect to the global coordinate system. The scattered field produced at the receiver by each inclusion is then computed using Equations ( l)-( 3). The separate contributions are then summed to produce the total field at the detector. This procedure is repeated many times using different, but statistically identical, clusters of inclusions so that the mean and standard deviation of the field scattered from a cluster of given statistical parameters can be deduced. The back-scattered signal is calculated by positioning the detector in the same location as the emitter. Calculation
of attenuation
If an inclusion cluster of diameter D scatters a total fraction of the energy which is incident upon the cluster 4, then each slab of thickness dz will scatter an amount +dz/D times the energy that is incident upon that slab. This means that the intensity I(z) obeys the law I = Z(O)exp( -+z/D), so that the amplitude uo(z) (which is proportional to I”‘) follows the law uo(z) = u,(O)exp( -&z/20), while propagating through the cluster. On emerging from the cluster we have that z = D, so that the final amplitude is exp( -$/2), relative to the amplitude of the wave on entering the cluster. The quantity 4 is termed the attenuation factor and is found by calculating the scattered field over a wide range of angles and using this to estimate the total scattered energy passing through a large sphere enclosing the cluster. This gives the fraction of the total energy lost from the beam, 4. The above working shows that the usual attenuation coefficient equals l/2 times this attenuation factor. Although this argument is only rigorous for cylindrical clusters and is approximate for clusters of other shapes, such as spherical, the above argument gives a good approximation for estimating the attenuation coefficient from the total scattered energy. The procedure for determining 4 from the fraction of total energy lost still makes sense when this quantity is greater than unity. This does occur, since our model takes no account of the decrease in the amplitude of the beam as it passes through the cluster; as stressed before, each inclusion scatters an incident field of the same magnitude. It is therefore formally possible to find more energy scattered out of the incident beam than is contained within it. Such results are made legitimate by the above prescription for the attenuation coefficient, so that, for example, an attenuation factor of unity (4 = 1) means that the amplitude of the transmitted beam is 4.3 dB less than the amplitude of the beam when incident upon the cluster. If the beam width is greater than the cluster width, then obviously this attenuation figure is relevant only to that part of the beam interacting with the inclusion cluster. A 20dB reduction in amplitude corresponds to 4 = 4.6.
from a cluster of inclusions
The scattering of an ultrasonic wave from a cluster of inclusions is calculated from Equations ( 1)-( 3) as follows. The positions of the ultrasonic source and receiver define a three-dimensional global coordinate system. For a cluster of N inclusions, 3N numbers are chosen to represent the semi-axes of the ellipsoids. N points are
Compressional
wave
scattering
In this section the scattering of compressional ultrasonic waves from an approximately spherical cluster of inclusions is described. The averages of the three inclusion semi-axes have been chosen to’be the same, such that
Ultrasonics
1992
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Scattering
from
clusters
of inclusions:
I.D. Culverwell
and J.A.
Ogilvy
the average inclusion eccentricity is small. This does not, however, preclude the appearance of ellipsoidal inclusions within the cluster. This simulation is performed by choosing the (x, y, z) coordinates of each inclusion at random from Gaussian distributions with the same standard deviations; by selecting each Euler angle at random from a distribution uniform between --7c and n; and by choosing each semi-axis of each inclusion at random from a one-sided Gaussian distribution. A different configuration is described in a later section. Spatial
distribution
of the scattered
fields
There are two distinct scattering regimes: coherent scattering, which occurs when the diameter D of the cluster is much less than the wavelength II; and incoherent scattering, which occurs when the mean separation between inclusions, 5, is much greater than the wavelength. Both regimes are likely to be met experimentally. Figure I shows the amplitudes of the compressional and shear waves scattered from a 400 pm diameter spherical cluster of 50 voids with mean semi-axes of 16 pm, buried 10 mm deep in a slab of mild steel having the Lame elastic constants d = 1.1 x 10”N me2 and p = 8.1 x 10” N m-‘, and a density of 7.9 x lo3 kg rnm3. [We choose to study voids because MnS inclusions are frequently disbonded from their steel host (Reference 5, section Q) and consequently appear vacant to ultrasound.] A compressional wave of frequency 1 MHz (wavelength 5.9 mm) is incident from the left. The smooth features of the coherent scattering pattern are clear. Contrast this with Figure 2 which is the scattering pattern for a cluster of diameter 2.4mm, when the wavelength of the incident wave is 0.8 mm. The forward scattered signal is still coherent (because there is little phase difference between rays going through the centre of the cluster and those that are scattered from the edge of the cluster in the forward direction), but away from this direction the scattered amplitudes are smaller and more rapidly varying in direction. This is incoherently scattered ultrasound. We note that, because of its shorter
‘\ \
i
G \
\
ir
/’
‘1 /’
‘1._ /-y--
_=-
/’ //
/--
) 1/ I /’ -=’ \ I’ ‘Y \ I’\ ‘._0 I I
Figure1 P-wave (---) and S-wave (--) scattered amplrtudes in the coherent scattering regime, for an incident P-wave. The amplitude of the forward-scattered P-wave is 1.7 x 10 -e times the amplitude of the incident wave
10
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1992
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Figure2 P-wave (----) and S-wave (--) scattered amplitudes in the incoherent scattering regime, for an incident P-wave. The amplitude of the forward-scattered P-wave is 8.6 x 1 0e4 times the amplitude of the incident wave
wavelength, the scattered shear wave is already fully incoherent. It would be possible to obtain a wholly incoherently scattered compressional wave by increasing the frequency further, but this circumstance is unlikely to be relevant experimentally. Effect
of coherency
on scattered
field
strengths
It might be expected that in the coherent scattering regime the intensity scattered from a cluster of N inclusions would be N2 larger than the intensity of the scattered signal from one inclusion, because in coherent scattering there is no interference between the scattered waves. In the incoherent regime the scattering from N inclusions might be expected to be N times larger than the scattering from one, because here there is almost totally destructive interference between the waves scattered from different inclusions. These suppositions have been verified numerically. Table I gives the amplitudes of the forwardscattered and back-scattered signals, relative to the incident amplitude, for the clusters whose scattering patterns are shown in Figures I and 2. Using the table we find that the ratio of the forward-scattered amplitudes in the two cases is 51, whereas the ratio of the back-scattered amplitudes is 2. The single particle scattering function [see Equation (4)] would predict that all the amplitudes in Figure 2 would be larger than those of Fiqure I by a factor of the square of the ratio of the two wavenumbers [i.e. by (7.86/1.07)2 = 541, since all other relevant parameters, such as the number of inclusions and the inclusion size distribution, are the same in the two clusters. Thus, the scattering in the forward direction agrees with this k2 scaling, because the 0 = 0 scattering is coherent in both examples. However, the back-scattered amplitude in Figure 2 is 27 times smaller than the frequency scaling would suggest. This is because of destructive interference between the back-scattered amplitudes, i.e. it is a result of incoherent scattering. The different scattering behaviours in the coherent and incoherent regimes will be important when we discuss the attenuation of the beam, the back-scattered signal and the relationship between the two.
Scattering Table 1
from clusters of inclusions:
I.D. Culverwell
and J.A. Ogilvy
Forward- and back-scattered signal amplitudes for the coherent and incoherent scattering patterns of Figures 1 and 2
Figure
Nature of scattering
Forward signale
Back-scattered signala
Wavenumber, k (mm-‘)
1 2
Coherent Incoherent
1.7 x 10-s 8.6 x IO-4
3.0 x lo-5 6.0 x lop5
1.07 7.86
a P-wave amplitude relative to incident wave amplitude
Attenuation
produced
Table 2
by clusters of inclusions
As described in a previous section we determine the attenuation of an ultrasonic beam on passing through a cluster by calculating the amount of energy that is scattered through a large sphere surrounding the cluster. An alternative method, really only useful for spherical scatterers, is to solve Ying and Truell’s equations for the scattering of ultrasound to a high order in ka, and then use Waterman and Truell’s’l formula for the complex propagation constant K to calculate the attenuation, exp [ - Zm( lc)D] (Reference 12). This calculation agrees with the attenuation found by our method (in the incoherent regime) when identical spherical scatterers are considered, but for the reasons discussed above we wish to study the scattering from inclusions of widely varying shapes and sizes, and the method described below is the most efficient way of doing this. The flux of energy carried by a shear wave is proportional to kyu,2, and that carried by a compressional wave is proportional to k,(l + 2p)ui (see, for example, p. 91 of Reference 13). Thus, if a longitudinal wave of magnitude a0 is incident upon a cluster of diameter D, and if ( ui ) and ( u,” ) are the average intensities of the compressronal and shear waves at a distance L from the cluster, then the ratio of the energy scattered out of the beam to the energy incident upon the cluster is 4 = (4LID)’
>/u;
4N = PN(5/A)&
x(=5/1) 3.83 3.83 1.21 3.83 3.83 3.83
2500
-
2000
-
(6)
where 41 represents the mean energy scattered from one inclusion and is proportional to ( 4k2 ( a3 ) /D)*, 5 is equal to the mean inclusion-inclusion separation and the function P, is known to have the limits PJO) = N2 (coherent scattering) and PN( co) = N (incoherent scattering). The form of P, between these limits may be determined from our numerical simulation. For example, when simulations of the scattering from a large number ( z 50) of clusters, each containing 50 inclusions, are made for several different values of x = (/,I, the mean and standard deviations of P,,(x) shown in Table 2 are found. The same results are plotted in Figure 3. The results for 4 were then fitted to a function of the form
x x x x x x
lO-3 lop2 IO-’ IO-’ IO0 IO’
P,o(x)
Q5o(x)
2450 + 20 2110* 150 720 + 240 180+50 72 + 24 79 + 8
50*10 47+10 30*13 6.5 + 4.9 6.7 f 4.9 6.5 f 5.0
1500 -
1000 -
500 -
O-6
(5)
where c, and cP are the shear and compressional wave speeds, respectrvely, and we have used the relationship (k,p)/[ k,(l + 2~)] = c,/cP. For incident shear waves (see section on Shear wave scattering) Equation (5) must be multiplied by cJc,. Equation (5) assumes that after interaction with the cluster the wave is spreading spherically. This will be true for the spherical clusters being considered here. From the discussion in the previous section we can write the energy scattered from N inclusions, e!~~,as
Means and standard deviations of P,, and Q5c
-I.
f
-2
fig1
0
f
I 2
L
I L
. 6' h(C/A)
Fig 2
Figure 3 Values of P50(l/A) given in Tab/e 2, together with least-squares fit P,,(t/E.). Values of t/I, for Figures 1 and 2 are also shown, where 4 is the mean inclusion-inclusion separation within the cluster and 1 is the incident wavelength
The fit was found to have the general (6) with C$= (0.18 + 0.03)[4k2(
form of Equation
a3)/D]‘P,(t/A)
(8)
and
The numerical factor (0.18 k 0.03) arises from the averaging over many clusters and is, in fact, the average of those terms of the order of unity in Equation (3). The error bar of + 0.03 was obtained by calculating the mean fractional error in the data points for 4N and assuming the same percentage error for pN(x). The fit given above is shown in Figure 3 for N = 50, where p,,(x), the empirical least-squares fit to P,,(x), is accurate to within 40% over the entire range of x. To illustrate the sensitivity of the attenuation to inclusion size, Equation (8) is used to calculate the attenuation of a beam propagating through two different clusters. The results are given in Table 3. The average distance between inclusions is taken to be 5 z D/N’13.
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1992 Vol 30 No 1
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Scattering
from clusters of inclusions:
I.D. Culverwell
and J.A. Ogilvy
Table 3 Compression wave attenuation and back-scattered signal strength, from Equations (8) and (lo), differing only in the size of the voids within the clusters. The incident wave freauencv is 5.0 MHz
respectively,
Diameter of cluster (mm)
Radius of voids (pm)
Number of voids
Depth of cluster (mm)
4
Attenuation (d5)
ubsI”O
1 .o 1 .o
20.0 30.0
1000 1000
20.0 20.0
0.14 1.60
0.6 7.0
5.3 x 10-s 1.8 x lo-*
For the first cluster detailed in Table 3, this gives 5 zz 0.1 mm, which is much less than the wavelength. The scattering will be predominantly coherent so we put P, = N* = lo6 in Equation (8) to calculate C#LHowever, with only a 50% increase in void size the attenuation is doubled (i.e. increases by 6 dB), because of the strong dependence of the scattering on the radius of the inclusions. This example shows that the size distributions of the inclusions must be known rather accurately before predictions of the attenuation can be made. The few measurements that have been made (see Reference 5, section 0) suggest a negative exponential distribution of inclusion sizes: p(a) x exp( -a/a,,). Unfortunately this is a very broad distribution, satisfying ( a” ) = n! ( a )“, which means that if the cube of the mean radius is used as an estimate of the mean of the cube, then the right-hand side of Equation (8) will be in error by a factor of 36, leading to large errors in the predicted attenuation. It is therefore important to characterize in detail the inclusion distributions within representative specimens of interest, in order to predict attenuation figures. Back-scattered
signals from clusters of inclusions
When a very similar analysis to the one which led to Equations (9) and (8) is made of the back-scattered compressional wave amplitude ubs we find that ubJuO = (0.48 f 0.25)[k2 (a3
>/LlQd
(10)
where several values of Qso(x) are quoted in Table 2 and plotted in Figure 4, together with the function Q,,(x), an empirical fit to QsO(x) which is accurate to within 50%. Here Q,(x) is defined by 1 + 9.1x2 Q”&) = N
(11)
1 + 9.1 N”‘x’
a function which has the properties ON(O) = N (coherent scattering) and Q,( “o) = Nr” (incoherent scattering). Table 3 gives values of ubJuO for the two clusters considered in the previous section. A 50% increase in void size increases the back-scattered signal by 10.6 dB. For the scattering patterns for 50 inclusions shown in Figures 1 and 2, Equation (10) gives that the ratio of the back-scattered amplitudes in the two cases would be 8.9 k 9.3, which compares satisfactorily with the calculated ratio of 2.0 for two sample realizations (see Table 1). Relationship
between
attenuation
andback-scattering
By combining Equations (8) and (10) we may write the attenuation factor in terms of the back-scattered signal
Figure 4 Values of C&,(5/E.) given in Table 2, together with least-squares fit &,(
the whole range of wavelengths.) Thus, the attenuation of a wave by a cluster of inclusions may be estimated from a measurement of the back-scattered signal strength. We note that by combining the back-scatter and the attenuation in this way the strong dependences of both quantities on the size of the scatterers have cancelled out, leaving the more robust expression (12).
Shear wave
scattering
The results for incident shear waves are very similar to those described in the previous section for incident compressional waves. The analogue of Equation (8) is C/I= (0.030 f 0.006)[4k2
( a3 )/D]‘S,(t/A)
(12)
(The wavelength dependent factor PN/Qi has been omitted since it differs from unity by less than 10% over
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1992
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(13)
where k and ,I now refer to the incident shear wave, and S,(x) is tolerably well (within 50%) represented by (14) Similarly, (polarized given by
the back-scattered in the same direction
shear wave amplitude as the incident wave) is
ubJuO = (0.37 + 0.22)[k2 ( a3 )/L] =
12
for two clusters
where T, is approximated
TN(x) = N
1 + 10.2x2 1 + 10.2N”2x2
TN(5/A)
(15)
by
(16)
Scattering from clusters of inclusions: I.D. Culverwell and J.A. Ogilvy
In view of the similarity of these results to those of the previous section, the implications for the use of backscattered shear waves to predict attenuation are much the same as for compressional waves, and will not be discussed further here.
P-wavescattering
from an elongated
cluster
In this section we simulate the compressional wave scattering from an elongated cluster of inclusions, as is sometimes produced when steel containing MnS inclusions is hot rolled (Reference 4 and section P, Reference 5). This modification is incorporated into the model described above in the following way. Two of the semi-axes of each inclusion are chosen at random from a Gaussian distribution, with the third chosen from a distribution with a larger mean - in this example, five times larger than that of the other two axes. The Euler angles of each ellipsoid are then restricted so that the fibre-like inclusions are roughly aligned, and not orientated randomly. Finally, the coordinates of the inclusions are chosen randomly from distributions with mean values in the ratio 1: 1:5, as for the sizes of the inclusions. In this way we mimic the rolling-out of a spherical cluster of inclusions in a steel plate. The results of calculations of the attenuation and back-scatter of compressional waves from this prolate spheroid cluster of aspect ratio 5: 1 are similar in form to those of P- or S-waves scattering off a spherical cluster. They can be summarized as follows C$= (0.35 f 0.04)[4kZ ( a3 )/D]2v,(r/n)
(17)
where V, is well represented by
(18) For waves incident perpendicular to the long axis of the ellipsoid, the back-scattered P-wave amplitude is given by ubs/uO = (0.67 +_0.39)[k2 ( a3 )/L] W,( t/n) where W, is approximated
m,(x) = N
(19)
by
1 + 27.7x2 1 + 27.7N”‘x’
(20)
We assume the time-dependence of the initial pulse to be given by g(t) = (t - t,)“coso,(t
-
t,)exp[
-cc(t - to)]
(21)
which is known to give a reasonable fit to genuine pulse shapes (see p. 194 of Reference 13). We further assume that the space-dependence of the incident pulse is given by the well known Bessel function model (p. 193 of Reference 13)
A(r) = z
c
2J,(kc sin 0) kc sin 0
1
(22)
where c is the crystal radius of the transducer and 0 is the angle between i and the axis of the probe. Figure 5 shows the back-scattered compressional wave signal from the cluster whose monochromatic plane wave scattering pattern is shown in Figure I (and described in the section on the Spatial distribution of the scattered fields). The incident compressional wave pulse, defined by the parameters n = 2, o0 = 27~x 1 MHz, t, = 0 and CI= 271hs in Equation (21), and c = 5 mm in Equation (22), is also plotted. The delay before the reception of the signal corresponds closely to the 3.4 ~LStaken for the waves to travel from the source to the cluster and then back to the source. Because the transit time across the cluster is only ~0.06 ps the received signal has about the same duration as the incident pulse. The received pulse has a well defined frequency, equal to the central frequency of the incident pulse. The ratio of the back-scattered to incident peak-peak amplitudes is 2.3 x 10m6,to be compared with the ratio of 29.5 x 10e6 for the monochromatic wave scattering shown in Figure I. Most of this discrepancy can be attributed to the decrease in beam amplitude given by the c/4nr term in Equation (22) ( ~0.04 for the values used here). Further differences arise because of the change in pulse shape, caused by the frequency-dependent scattering coefficient [Equation (3)]. Figure 5 should be contrasted with Figure 6, the back-scattering of a higher frequency wave from the larger cluster whose plane wave scattering pattern is shown in Figure 2. The incident pulse in Figure 6 has a frequency o0 = 271x 7.5 MHz, with n, t, and o!remaining the same as before. The ratio of the back-scattered to incident peak-peak amplitudes is 3.1 x 10m6, compared with 56.9 x 10m6 for Figure 2. Again, c/4m z 0.04, which
These results are similar to those of the previous two sections in form, although the magnitude of the scattering is somewhat larger.
Time-dependent
scattering
In this section we briefly report some results from a generalization of the basic model presented in the theory section. The modification consists of replacing the incident monochromatic plane wave with a pulse having a realistic variation in time and space. This pulse is Fourier decomposed into a set of harmonic waves. The incident field at any inclusion is approximated by the plane wave to which the true field is locally equal. The scattering at each frequency is calculated according to the theory described previously. The scattered amplitudes are then Fourier recombined to give the final scattered pulse.
Figure 5 Incident (-) and back-scattered (--) P-wave pulses for coherent scattering. The scattered pulse is multiplied by 4.4 x 10s
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1992
Vol 30 No 1
13
Scattering
from
clusters
of inclusions:
I.D. Culverwell
t
Figure 6 Incident pulses for incoherent
/ps
(--) and back-scattered (--) P-wave scattering. The scattered pulse is multiplied
bv 3.2 x lo5
and J.A.
Ogilvy
We have produced analogous results for the scattering of shear waves from a spherical cluster, and for compressional waves from an elongated cluster similar to those sometimes found in rolled steel plate. The results are of a similar form to those above, although it is interesting to note that the P-wave scattering from a rolled-out cluster is rather stronger than that from a spherical cluster. Nevertheless, the relationship between the attenuation and the back-scatter is similar in all three examples. We have also reported some calculations of the scattering of realistic pulses from clusters of inclusions. Preliminary results show qualitative similarity to the monochromatic calculations which form the bulk of this paper. Possible generalizations of this work include the following. A simple modification would be the study of non-vacuous inclusions. We expect the mean scattering amplitudes to be reduced by something like the mean value of 6p/p, 6L/A and 8p/p [see Equation (3)]. More interestingly, it might also be possible to use this method to calculate the scattering of ultrasound from very long, banded groupings of inclusions by exaggerating the elongated cluster model described. This would be a valuable investigation because such structures occasionally arise in steel plates (see Reference 5, sections 0, P and Q). However, it is possible that fibre-like inclusions may be longer than the wavelength of the ultrasound, causing the low frequency EQSA to break down in some scattering directions. We hope that some sort of long wavelength-short wavelength hybrid theory can be developed to handle this.
accounts for most of this difference. The change in shape and slight increase in the length of the scattered pulse is a consequence of the incoherent scattering. Figures 5 and 6 show that although the main features of the scattering of ultrasonic pulses from clusters of inclusions can be obtained using a monochromatic plane wave model, a detailed calculation of the scattering of a pulse, particularly in the incoherent regime, requires a time-dependent model, such as the one used in this section.
Acknowledgement
Conclusions
This work has been supported by the Health and Safety Executive through its Nuclear Safety Research Programme.
and discussion
In this paper we have studied the scattering of low frequency ultrasound by clusters of ellipsoidal inclusions. We have used the EQSA to calculate the amplitude of the compressional and shear waves scattered by each inclusion separately, and have summed the contributions from each inclusion to produce the final scattered amplitudes. This procedure was repeated for different but statistically identical clusters, so that the statistical variations in the scattered field could be determined. Since we have found that there is a large variance in the scattered signals (see Table 2) this feature of our approach is important: the detailed scattering pattern of a single cluster of inclusions is of very little value. It has been shown that coherent and incoherent scattering can occur in practice. The transition from one regime to another has been incorporated by means of simple approximations to the numerical results [Equations (9))(20)]. We have produced expressions for the attenuation and back-scatter of compressional waves incident upon a spherical cluster of roughly spherical vacuous inclusions [Equations (8) and ( lo), respectively]. These have been combined to relate the measured back-scattered signal to the likely level of attenuation of the beam [Equation
(1211.
14
Ultrasonics
1992
Vol 30 No 1
Correspondence Correspondence concerning this article should be addressed to Dr J. A. Ogilvy at the address shown.
References I
2 3 4 5 6 7 8 9
I0 11 12 13
Ying, C.F. and Truell, R. J Appl Phys (1956) 27 1086 Gubernatis, J.E., Domany, E., Krumhansl, J.A. and Huberman, M. J Appl Phys (1977) 48 2812 Gubernatis, J.E. .I 4ppl Phys (1979) 50 4046 Pickering, F.B. inclusions Institute of Metallurgists Monograph No. 3, London, UK (1979) Ch 3 Kiessling, R. Non-Metallic Inclusions rn Steel Parts I-IV, 2nd Edn, Iron and Steel Institute, London. UK (1978) Achenbacb, J.D. and Kitahara, M. J Acousf Sot Am( 1986) 80 1209 Mikata, Y. and Achenbach, J.D. J Acoust Sot Am (1988) 83 38 Sotiropoulos, D.A. and Achenbach, J.D. J NDE (1988) 7 123 Eshelbv. J.D. Proc Rev Sot (London) (1957) A271 376 Mathews, J. and Walkkr, R.i. Math~nkcai Methods qf Ph>tks 2nd Edn, Adison Wesley, London, UK ( 1970) Ch 15 Waterman, P.C. and Truell, R. J Math Phys (1961) 2 512 Sayers, C.M. and Smith, R.L. J Phys D: Appl Phys (1983) 16 1189 Harker, A.H. Elustic, Waws in &lids with Applicutions TO Nondestructive Testing of Pipelines Adam Hilger/British Gas, Bristol, UK (1988)
from clusters of
and J.A. Ogilvy
Theoretical Studies Department, Materials and Manufacturing Technology Division, Industrial Technology, Building 424.4, Harwell Laboratory, Oxon OX1 1 ORA, UK Received
19 February
1991;
revised
AEA
79 July 1991
The scattering of low frequency ultrasound by a cluster of arbitrary ellipsoidal inclusions is studied numerically, using the extended quasi-static approximation. The statistical variations in the scattered intensities are investigated. The relationship between the back-scattered signal and the attenuation of the ultrasound is determined. The implications of these results for the ultrasonic examination of steels containing inclusions are discussed. Keywords:
ultrasound
attenuation;
scattering;
The ultrasonic examination of an engineering structure may be impeded by the presence of small imperfections in the region under inspection. This is because inhomogeneities in a material can attenuate a beam of ultrasound by scattering its energy away from the forward direction. Furthermore, this scattered energy may arrive at the detector and interfere with the signal produced from another defect such as a crack. One example of this behaviour occurs in some steels, when these contain clusters of non-metallic inclusions, originally present as impurities in the melt. It is therefore important to understand the scattering of ultrasound from clusters ofinclusions. This is the question addressed in this paper. The scattering of plane longitudinal waves by spherical obstacles in an isotropically elastic solid was first studied by Ying and True11 in 1956l. Their solution was exact (although not in closed form) and included special cases such as the Born approximation, which is valid when the elastic properties of the scatterer and the surrounding medium are similar, and the quasi-static approximation, in which the stress field in the scatterer is given by the static field produced by the displacement of the incident wave, which is valid in the low frequency regime (for discussions of both see Reference 2). In later work by Gubernatis3 the quasi-static approximation was extended to include the phase of the incident wave in the stress field. This ad hoc modification increased the region of validity of the approximation from ka CC1 to ka < 2, where k is the wavenumber of the incident field and a is the radius &fthe scatterer. At typical ultrasonic frequencies in steel, k < lo4 m-l, so that the extended quasi-static approximation (EQSA) will adequately describe the scattering from inclusions smaller than a few hundred microns in diameter, a size range which covers all of the scatterers of interest in this paper. * Copyright
belongs to the United Kingdom Atomic Energy Authority
(AEA) t Present address: BP International Chertsey Road, Sunbury-on-Thames, 0041-624X/92/010008-07 @ 1992 Butterworth-Heinemann
8
Ultrasonics
Ltd, Sunbury Research Centre, Middlesex TW16 7LN, UK
Ltd
1992 Vol 30 No 1
inclusions
The question of multiple scattering arises. Does each inclusion in the cluster see the same incident field, or is each one influenced by the ultrasound scattered from all of the others? General considerations [see Equations (l)-(3)] indicate that in the far-field the spherically spreading scattered wave from one inclusion should scale as uSCaz uinck2a3exp(ikR)/R, where uinc is the amplitude of the incident field and R is the distance from the centre of the scatterer. Manganese sulphide (MnS) inclusions in steel have been examined microscopically (see Reference 4 and section 0, Reference 5) and found to have sizes a in the range 5-50pm, and mean separations R about five times larger. Thus the largest value of k2a3/R is ~0.05, which is much less than unity, so that multiple scattering can be ignored. Each inclusion is therefore assumed to scatter an incident field of the same magnitude as (but of different phase from) the field seen by its neighbours. Previous analytical work on the scattering of waves by a collection of obstacles assumed that all the scatterers had the same simple size and shape (spheres or planes) 6-8 . They were also assumed to be distributed isotropically or periodically. In this problem we are interested in inclusions which have a wide variety of shapes and sizes, sometimes with a strongly preferred direction (see later section on P-wave scattering from an elongated cluster). Numerical simulation would seem to be the most straightforward way of calculating the scattering from such agglomerations. It has the further advantage that it is easy to calculate the scattering from different clusters, each having the same statistical properties (i.e. mean inclusion-inclusion separation, standard deviation of inclusion sizes, etc.). The statistical variations of the scattered field, which are of great value experimentally, can therefore be found directly. The structure of this paper is as follows. In the next section the basic theory underlying the model is briefly reviewed. The next three sections describe the results of the scattering of compressional and shear waves from a
Scattering
from clusters of inclusions:
spherical cluster of inclusions, and of the scattering of P-waves from an elongated cluster, similar to those sometimes found in rolled steel plate. The attenuation, back-scatter and the relationship between the two are described. The relevance of these results to the nondestructive testing of steel plates is also discussed. Some results are then presented of a generalized model which simulates the scattering of realistic ultrasonic pulses. The final section discusses the conclusions of the paper, together with possible generalizations of the work. Theory Extended scatterer
quasi-static
approximation
for
a
single
The main features of the EQSA, the low frequency scattering theory which forms the basis of our model, have already been described, but, for completeness, the basic equations are given here. All are taken from Reference 3. Consider an ellipsoidal inclusion of density p’, elastic constants CijL1,centred at the origin of a coordinate system, surrounded by a medium of density p and elastic constants C,,. If there is an incident plane wave . r), then in the EQSA the far-field U Inc = u” exp(ik" scattered displacement at position r is given by Pa(r)
= A exp(ik,r)/r
+ B exp(ik,r)/r
(1) where A (the amplitude of the scattered compressional wave) and B (the amplitude of the scattered shear wave) are given by Ai = Pitjfj(k,i) Bi = (Sij - f,Pj)fj(k,i)
(2)
(summing over repeated indices). In Equation (2) the function f is defined by
in which o is the angular frequency, 6p = p’ - p, 6C,, = Cjjkl - Cijkl, &, (the incident strain field) = i/2(uEkf + u,0kz) and Fijkl is a fourth-rank tensor which depends only on the size of the scatterer and the Poisson’s ratio of the surrounding medium. For ellipsoidal inclusions, Fijrr is constant throughout the inclusion because the static stress field inside an ellipsoid is uniform’. S(q) is a shape factor given by the Fourier transform of the shape of the inclusion with respect to q = kP - k”Go, with S(0) equal to the volume of the scatterer. For spherical scatterers we find that ikr u
SCB cc
kza3
5
Y
(4)
i.e. the amplitude of the scattered field has a strong dependence on the wavevector and the size of the scatterer. Scattering
I.D. Culverwell
and J.A. Ogilvy
then chosen to define the positions of the centres of the ellipsoids (these points are sufficiently widely spaced such that overlap between ellipsoids cannot occur). Finally, 3N Euler angles lo define the orientations of the inclusions with respect to the global coordinate system. The scattered field produced at the receiver by each inclusion is then computed using Equations ( l)-( 3). The separate contributions are then summed to produce the total field at the detector. This procedure is repeated many times using different, but statistically identical, clusters of inclusions so that the mean and standard deviation of the field scattered from a cluster of given statistical parameters can be deduced. The back-scattered signal is calculated by positioning the detector in the same location as the emitter. Calculation
of attenuation
If an inclusion cluster of diameter D scatters a total fraction of the energy which is incident upon the cluster 4, then each slab of thickness dz will scatter an amount +dz/D times the energy that is incident upon that slab. This means that the intensity I(z) obeys the law I = Z(O)exp( -+z/D), so that the amplitude uo(z) (which is proportional to I”‘) follows the law uo(z) = u,(O)exp( -&z/20), while propagating through the cluster. On emerging from the cluster we have that z = D, so that the final amplitude is exp( -$/2), relative to the amplitude of the wave on entering the cluster. The quantity 4 is termed the attenuation factor and is found by calculating the scattered field over a wide range of angles and using this to estimate the total scattered energy passing through a large sphere enclosing the cluster. This gives the fraction of the total energy lost from the beam, 4. The above working shows that the usual attenuation coefficient equals l/2 times this attenuation factor. Although this argument is only rigorous for cylindrical clusters and is approximate for clusters of other shapes, such as spherical, the above argument gives a good approximation for estimating the attenuation coefficient from the total scattered energy. The procedure for determining 4 from the fraction of total energy lost still makes sense when this quantity is greater than unity. This does occur, since our model takes no account of the decrease in the amplitude of the beam as it passes through the cluster; as stressed before, each inclusion scatters an incident field of the same magnitude. It is therefore formally possible to find more energy scattered out of the incident beam than is contained within it. Such results are made legitimate by the above prescription for the attenuation coefficient, so that, for example, an attenuation factor of unity (4 = 1) means that the amplitude of the transmitted beam is 4.3 dB less than the amplitude of the beam when incident upon the cluster. If the beam width is greater than the cluster width, then obviously this attenuation figure is relevant only to that part of the beam interacting with the inclusion cluster. A 20dB reduction in amplitude corresponds to 4 = 4.6.
from a cluster of inclusions
The scattering of an ultrasonic wave from a cluster of inclusions is calculated from Equations ( 1)-( 3) as follows. The positions of the ultrasonic source and receiver define a three-dimensional global coordinate system. For a cluster of N inclusions, 3N numbers are chosen to represent the semi-axes of the ellipsoids. N points are
Compressional
wave
scattering
In this section the scattering of compressional ultrasonic waves from an approximately spherical cluster of inclusions is described. The averages of the three inclusion semi-axes have been chosen to’be the same, such that
Ultrasonics
1992
Vol 30 No 1
9
Scattering
from
clusters
of inclusions:
I.D. Culverwell
and J.A.
Ogilvy
the average inclusion eccentricity is small. This does not, however, preclude the appearance of ellipsoidal inclusions within the cluster. This simulation is performed by choosing the (x, y, z) coordinates of each inclusion at random from Gaussian distributions with the same standard deviations; by selecting each Euler angle at random from a distribution uniform between --7c and n; and by choosing each semi-axis of each inclusion at random from a one-sided Gaussian distribution. A different configuration is described in a later section. Spatial
distribution
of the scattered
fields
There are two distinct scattering regimes: coherent scattering, which occurs when the diameter D of the cluster is much less than the wavelength II; and incoherent scattering, which occurs when the mean separation between inclusions, 5, is much greater than the wavelength. Both regimes are likely to be met experimentally. Figure I shows the amplitudes of the compressional and shear waves scattered from a 400 pm diameter spherical cluster of 50 voids with mean semi-axes of 16 pm, buried 10 mm deep in a slab of mild steel having the Lame elastic constants d = 1.1 x 10”N me2 and p = 8.1 x 10” N m-‘, and a density of 7.9 x lo3 kg rnm3. [We choose to study voids because MnS inclusions are frequently disbonded from their steel host (Reference 5, section Q) and consequently appear vacant to ultrasound.] A compressional wave of frequency 1 MHz (wavelength 5.9 mm) is incident from the left. The smooth features of the coherent scattering pattern are clear. Contrast this with Figure 2 which is the scattering pattern for a cluster of diameter 2.4mm, when the wavelength of the incident wave is 0.8 mm. The forward scattered signal is still coherent (because there is little phase difference between rays going through the centre of the cluster and those that are scattered from the edge of the cluster in the forward direction), but away from this direction the scattered amplitudes are smaller and more rapidly varying in direction. This is incoherently scattered ultrasound. We note that, because of its shorter
‘\ \
i
G \
\
ir
/’
‘1 /’
‘1._ /-y--
_=-
/’ //
/--
) 1/ I /’ -=’ \ I’ ‘Y \ I’\ ‘._0 I I
Figure1 P-wave (---) and S-wave (--) scattered amplrtudes in the coherent scattering regime, for an incident P-wave. The amplitude of the forward-scattered P-wave is 1.7 x 10 -e times the amplitude of the incident wave
10
Ultrasonics
1992
Vol 30 No 1
Figure2 P-wave (----) and S-wave (--) scattered amplitudes in the incoherent scattering regime, for an incident P-wave. The amplitude of the forward-scattered P-wave is 8.6 x 1 0e4 times the amplitude of the incident wave
wavelength, the scattered shear wave is already fully incoherent. It would be possible to obtain a wholly incoherently scattered compressional wave by increasing the frequency further, but this circumstance is unlikely to be relevant experimentally. Effect
of coherency
on scattered
field
strengths
It might be expected that in the coherent scattering regime the intensity scattered from a cluster of N inclusions would be N2 larger than the intensity of the scattered signal from one inclusion, because in coherent scattering there is no interference between the scattered waves. In the incoherent regime the scattering from N inclusions might be expected to be N times larger than the scattering from one, because here there is almost totally destructive interference between the waves scattered from different inclusions. These suppositions have been verified numerically. Table I gives the amplitudes of the forwardscattered and back-scattered signals, relative to the incident amplitude, for the clusters whose scattering patterns are shown in Figures I and 2. Using the table we find that the ratio of the forward-scattered amplitudes in the two cases is 51, whereas the ratio of the back-scattered amplitudes is 2. The single particle scattering function [see Equation (4)] would predict that all the amplitudes in Figure 2 would be larger than those of Fiqure I by a factor of the square of the ratio of the two wavenumbers [i.e. by (7.86/1.07)2 = 541, since all other relevant parameters, such as the number of inclusions and the inclusion size distribution, are the same in the two clusters. Thus, the scattering in the forward direction agrees with this k2 scaling, because the 0 = 0 scattering is coherent in both examples. However, the back-scattered amplitude in Figure 2 is 27 times smaller than the frequency scaling would suggest. This is because of destructive interference between the back-scattered amplitudes, i.e. it is a result of incoherent scattering. The different scattering behaviours in the coherent and incoherent regimes will be important when we discuss the attenuation of the beam, the back-scattered signal and the relationship between the two.
Scattering Table 1
from clusters of inclusions:
I.D. Culverwell
and J.A. Ogilvy
Forward- and back-scattered signal amplitudes for the coherent and incoherent scattering patterns of Figures 1 and 2
Figure
Nature of scattering
Forward signale
Back-scattered signala
Wavenumber, k (mm-‘)
1 2
Coherent Incoherent
1.7 x 10-s 8.6 x IO-4
3.0 x lo-5 6.0 x lop5
1.07 7.86
a P-wave amplitude relative to incident wave amplitude
Attenuation
produced
Table 2
by clusters of inclusions
As described in a previous section we determine the attenuation of an ultrasonic beam on passing through a cluster by calculating the amount of energy that is scattered through a large sphere surrounding the cluster. An alternative method, really only useful for spherical scatterers, is to solve Ying and Truell’s equations for the scattering of ultrasound to a high order in ka, and then use Waterman and Truell’s’l formula for the complex propagation constant K to calculate the attenuation, exp [ - Zm( lc)D] (Reference 12). This calculation agrees with the attenuation found by our method (in the incoherent regime) when identical spherical scatterers are considered, but for the reasons discussed above we wish to study the scattering from inclusions of widely varying shapes and sizes, and the method described below is the most efficient way of doing this. The flux of energy carried by a shear wave is proportional to kyu,2, and that carried by a compressional wave is proportional to k,(l + 2p)ui (see, for example, p. 91 of Reference 13). Thus, if a longitudinal wave of magnitude a0 is incident upon a cluster of diameter D, and if ( ui ) and ( u,” ) are the average intensities of the compressronal and shear waves at a distance L from the cluster, then the ratio of the energy scattered out of the beam to the energy incident upon the cluster is 4 = (4LID)’
>/u;
4N = PN(5/A)&
x(=5/1) 3.83 3.83 1.21 3.83 3.83 3.83
2500
-
2000
-
(6)
where 41 represents the mean energy scattered from one inclusion and is proportional to ( 4k2 ( a3 ) /D)*, 5 is equal to the mean inclusion-inclusion separation and the function P, is known to have the limits PJO) = N2 (coherent scattering) and PN( co) = N (incoherent scattering). The form of P, between these limits may be determined from our numerical simulation. For example, when simulations of the scattering from a large number ( z 50) of clusters, each containing 50 inclusions, are made for several different values of x = (/,I, the mean and standard deviations of P,,(x) shown in Table 2 are found. The same results are plotted in Figure 3. The results for 4 were then fitted to a function of the form
x x x x x x
lO-3 lop2 IO-’ IO-’ IO0 IO’
P,o(x)
Q5o(x)
2450 + 20 2110* 150 720 + 240 180+50 72 + 24 79 + 8
50*10 47+10 30*13 6.5 + 4.9 6.7 f 4.9 6.5 f 5.0
1500 -
1000 -
500 -
O-6
(5)
where c, and cP are the shear and compressional wave speeds, respectrvely, and we have used the relationship (k,p)/[ k,(l + 2~)] = c,/cP. For incident shear waves (see section on Shear wave scattering) Equation (5) must be multiplied by cJc,. Equation (5) assumes that after interaction with the cluster the wave is spreading spherically. This will be true for the spherical clusters being considered here. From the discussion in the previous section we can write the energy scattered from N inclusions, e!~~,as
Means and standard deviations of P,, and Q5c
-I.
f
-2
fig1
0
f
I 2
L
I L
. 6' h(C/A)
Fig 2
Figure 3 Values of P50(l/A) given in Tab/e 2, together with least-squares fit P,,(t/E.). Values of t/I, for Figures 1 and 2 are also shown, where 4 is the mean inclusion-inclusion separation within the cluster and 1 is the incident wavelength
The fit was found to have the general (6) with C$= (0.18 + 0.03)[4k2(
form of Equation
a3)/D]‘P,(t/A)
(8)
and
The numerical factor (0.18 k 0.03) arises from the averaging over many clusters and is, in fact, the average of those terms of the order of unity in Equation (3). The error bar of + 0.03 was obtained by calculating the mean fractional error in the data points for 4N and assuming the same percentage error for pN(x). The fit given above is shown in Figure 3 for N = 50, where p,,(x), the empirical least-squares fit to P,,(x), is accurate to within 40% over the entire range of x. To illustrate the sensitivity of the attenuation to inclusion size, Equation (8) is used to calculate the attenuation of a beam propagating through two different clusters. The results are given in Table 3. The average distance between inclusions is taken to be 5 z D/N’13.
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1992 Vol 30 No 1
11
Scattering
from clusters of inclusions:
I.D. Culverwell
and J.A. Ogilvy
Table 3 Compression wave attenuation and back-scattered signal strength, from Equations (8) and (lo), differing only in the size of the voids within the clusters. The incident wave freauencv is 5.0 MHz
respectively,
Diameter of cluster (mm)
Radius of voids (pm)
Number of voids
Depth of cluster (mm)
4
Attenuation (d5)
ubsI”O
1 .o 1 .o
20.0 30.0
1000 1000
20.0 20.0
0.14 1.60
0.6 7.0
5.3 x 10-s 1.8 x lo-*
For the first cluster detailed in Table 3, this gives 5 zz 0.1 mm, which is much less than the wavelength. The scattering will be predominantly coherent so we put P, = N* = lo6 in Equation (8) to calculate C#LHowever, with only a 50% increase in void size the attenuation is doubled (i.e. increases by 6 dB), because of the strong dependence of the scattering on the radius of the inclusions. This example shows that the size distributions of the inclusions must be known rather accurately before predictions of the attenuation can be made. The few measurements that have been made (see Reference 5, section 0) suggest a negative exponential distribution of inclusion sizes: p(a) x exp( -a/a,,). Unfortunately this is a very broad distribution, satisfying ( a” ) = n! ( a )“, which means that if the cube of the mean radius is used as an estimate of the mean of the cube, then the right-hand side of Equation (8) will be in error by a factor of 36, leading to large errors in the predicted attenuation. It is therefore important to characterize in detail the inclusion distributions within representative specimens of interest, in order to predict attenuation figures. Back-scattered
signals from clusters of inclusions
When a very similar analysis to the one which led to Equations (9) and (8) is made of the back-scattered compressional wave amplitude ubs we find that ubJuO = (0.48 f 0.25)[k2 (a3
>/LlQd
(10)
where several values of Qso(x) are quoted in Table 2 and plotted in Figure 4, together with the function Q,,(x), an empirical fit to QsO(x) which is accurate to within 50%. Here Q,(x) is defined by 1 + 9.1x2 Q”&) = N
(11)
1 + 9.1 N”‘x’
a function which has the properties ON(O) = N (coherent scattering) and Q,( “o) = Nr” (incoherent scattering). Table 3 gives values of ubJuO for the two clusters considered in the previous section. A 50% increase in void size increases the back-scattered signal by 10.6 dB. For the scattering patterns for 50 inclusions shown in Figures 1 and 2, Equation (10) gives that the ratio of the back-scattered amplitudes in the two cases would be 8.9 k 9.3, which compares satisfactorily with the calculated ratio of 2.0 for two sample realizations (see Table 1). Relationship
between
attenuation
andback-scattering
By combining Equations (8) and (10) we may write the attenuation factor in terms of the back-scattered signal
Figure 4 Values of C&,(5/E.) given in Table 2, together with least-squares fit &,(
the whole range of wavelengths.) Thus, the attenuation of a wave by a cluster of inclusions may be estimated from a measurement of the back-scattered signal strength. We note that by combining the back-scatter and the attenuation in this way the strong dependences of both quantities on the size of the scatterers have cancelled out, leaving the more robust expression (12).
Shear wave
scattering
The results for incident shear waves are very similar to those described in the previous section for incident compressional waves. The analogue of Equation (8) is C/I= (0.030 f 0.006)[4k2
( a3 )/D]‘S,(t/A)
(12)
(The wavelength dependent factor PN/Qi has been omitted since it differs from unity by less than 10% over
Ultrasonics
1992
Vol 30 No 1
(13)
where k and ,I now refer to the incident shear wave, and S,(x) is tolerably well (within 50%) represented by (14) Similarly, (polarized given by
the back-scattered in the same direction
shear wave amplitude as the incident wave) is
ubJuO = (0.37 + 0.22)[k2 ( a3 )/L] =
12
for two clusters
where T, is approximated
TN(x) = N
1 + 10.2x2 1 + 10.2N”2x2
TN(5/A)
(15)
by
(16)
Scattering from clusters of inclusions: I.D. Culverwell and J.A. Ogilvy
In view of the similarity of these results to those of the previous section, the implications for the use of backscattered shear waves to predict attenuation are much the same as for compressional waves, and will not be discussed further here.
P-wavescattering
from an elongated
cluster
In this section we simulate the compressional wave scattering from an elongated cluster of inclusions, as is sometimes produced when steel containing MnS inclusions is hot rolled (Reference 4 and section P, Reference 5). This modification is incorporated into the model described above in the following way. Two of the semi-axes of each inclusion are chosen at random from a Gaussian distribution, with the third chosen from a distribution with a larger mean - in this example, five times larger than that of the other two axes. The Euler angles of each ellipsoid are then restricted so that the fibre-like inclusions are roughly aligned, and not orientated randomly. Finally, the coordinates of the inclusions are chosen randomly from distributions with mean values in the ratio 1: 1:5, as for the sizes of the inclusions. In this way we mimic the rolling-out of a spherical cluster of inclusions in a steel plate. The results of calculations of the attenuation and back-scatter of compressional waves from this prolate spheroid cluster of aspect ratio 5: 1 are similar in form to those of P- or S-waves scattering off a spherical cluster. They can be summarized as follows C$= (0.35 f 0.04)[4kZ ( a3 )/D]2v,(r/n)
(17)
where V, is well represented by
(18) For waves incident perpendicular to the long axis of the ellipsoid, the back-scattered P-wave amplitude is given by ubs/uO = (0.67 +_0.39)[k2 ( a3 )/L] W,( t/n) where W, is approximated
m,(x) = N
(19)
by
1 + 27.7x2 1 + 27.7N”‘x’
(20)
We assume the time-dependence of the initial pulse to be given by g(t) = (t - t,)“coso,(t
-
t,)exp[
-cc(t - to)]
(21)
which is known to give a reasonable fit to genuine pulse shapes (see p. 194 of Reference 13). We further assume that the space-dependence of the incident pulse is given by the well known Bessel function model (p. 193 of Reference 13)
A(r) = z
c
2J,(kc sin 0) kc sin 0
1
(22)
where c is the crystal radius of the transducer and 0 is the angle between i and the axis of the probe. Figure 5 shows the back-scattered compressional wave signal from the cluster whose monochromatic plane wave scattering pattern is shown in Figure I (and described in the section on the Spatial distribution of the scattered fields). The incident compressional wave pulse, defined by the parameters n = 2, o0 = 27~x 1 MHz, t, = 0 and CI= 271hs in Equation (21), and c = 5 mm in Equation (22), is also plotted. The delay before the reception of the signal corresponds closely to the 3.4 ~LStaken for the waves to travel from the source to the cluster and then back to the source. Because the transit time across the cluster is only ~0.06 ps the received signal has about the same duration as the incident pulse. The received pulse has a well defined frequency, equal to the central frequency of the incident pulse. The ratio of the back-scattered to incident peak-peak amplitudes is 2.3 x 10m6,to be compared with the ratio of 29.5 x 10e6 for the monochromatic wave scattering shown in Figure I. Most of this discrepancy can be attributed to the decrease in beam amplitude given by the c/4nr term in Equation (22) ( ~0.04 for the values used here). Further differences arise because of the change in pulse shape, caused by the frequency-dependent scattering coefficient [Equation (3)]. Figure 5 should be contrasted with Figure 6, the back-scattering of a higher frequency wave from the larger cluster whose plane wave scattering pattern is shown in Figure 2. The incident pulse in Figure 6 has a frequency o0 = 271x 7.5 MHz, with n, t, and o!remaining the same as before. The ratio of the back-scattered to incident peak-peak amplitudes is 3.1 x 10m6, compared with 56.9 x 10m6 for Figure 2. Again, c/4m z 0.04, which
These results are similar to those of the previous two sections in form, although the magnitude of the scattering is somewhat larger.
Time-dependent
scattering
In this section we briefly report some results from a generalization of the basic model presented in the theory section. The modification consists of replacing the incident monochromatic plane wave with a pulse having a realistic variation in time and space. This pulse is Fourier decomposed into a set of harmonic waves. The incident field at any inclusion is approximated by the plane wave to which the true field is locally equal. The scattering at each frequency is calculated according to the theory described previously. The scattered amplitudes are then Fourier recombined to give the final scattered pulse.
Figure 5 Incident (-) and back-scattered (--) P-wave pulses for coherent scattering. The scattered pulse is multiplied by 4.4 x 10s
Ultrasonics
1992
Vol 30 No 1
13
Scattering
from
clusters
of inclusions:
I.D. Culverwell
t
Figure 6 Incident pulses for incoherent
/ps
(--) and back-scattered (--) P-wave scattering. The scattered pulse is multiplied
bv 3.2 x lo5
and J.A.
Ogilvy
We have produced analogous results for the scattering of shear waves from a spherical cluster, and for compressional waves from an elongated cluster similar to those sometimes found in rolled steel plate. The results are of a similar form to those above, although it is interesting to note that the P-wave scattering from a rolled-out cluster is rather stronger than that from a spherical cluster. Nevertheless, the relationship between the attenuation and the back-scatter is similar in all three examples. We have also reported some calculations of the scattering of realistic pulses from clusters of inclusions. Preliminary results show qualitative similarity to the monochromatic calculations which form the bulk of this paper. Possible generalizations of this work include the following. A simple modification would be the study of non-vacuous inclusions. We expect the mean scattering amplitudes to be reduced by something like the mean value of 6p/p, 6L/A and 8p/p [see Equation (3)]. More interestingly, it might also be possible to use this method to calculate the scattering of ultrasound from very long, banded groupings of inclusions by exaggerating the elongated cluster model described. This would be a valuable investigation because such structures occasionally arise in steel plates (see Reference 5, sections 0, P and Q). However, it is possible that fibre-like inclusions may be longer than the wavelength of the ultrasound, causing the low frequency EQSA to break down in some scattering directions. We hope that some sort of long wavelength-short wavelength hybrid theory can be developed to handle this.
accounts for most of this difference. The change in shape and slight increase in the length of the scattered pulse is a consequence of the incoherent scattering. Figures 5 and 6 show that although the main features of the scattering of ultrasonic pulses from clusters of inclusions can be obtained using a monochromatic plane wave model, a detailed calculation of the scattering of a pulse, particularly in the incoherent regime, requires a time-dependent model, such as the one used in this section.
Acknowledgement
Conclusions
This work has been supported by the Health and Safety Executive through its Nuclear Safety Research Programme.
and discussion
In this paper we have studied the scattering of low frequency ultrasound by clusters of ellipsoidal inclusions. We have used the EQSA to calculate the amplitude of the compressional and shear waves scattered by each inclusion separately, and have summed the contributions from each inclusion to produce the final scattered amplitudes. This procedure was repeated for different but statistically identical clusters, so that the statistical variations in the scattered field could be determined. Since we have found that there is a large variance in the scattered signals (see Table 2) this feature of our approach is important: the detailed scattering pattern of a single cluster of inclusions is of very little value. It has been shown that coherent and incoherent scattering can occur in practice. The transition from one regime to another has been incorporated by means of simple approximations to the numerical results [Equations (9))(20)]. We have produced expressions for the attenuation and back-scatter of compressional waves incident upon a spherical cluster of roughly spherical vacuous inclusions [Equations (8) and ( lo), respectively]. These have been combined to relate the measured back-scattered signal to the likely level of attenuation of the beam [Equation
(1211.
14
Ultrasonics
1992
Vol 30 No 1
Correspondence Correspondence concerning this article should be addressed to Dr J. A. Ogilvy at the address shown.
References I
2 3 4 5 6 7 8 9
I0 11 12 13
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