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Wave attenuation and dispersion in randomly cracked solids—II. Penny-shaped cracks
Inr. 1. Engng Sci. Vol. 31, No. 6, pp. 859-872, Printed in Great Britain. All tights reserved
1993
0020-7225/93 $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
WAVE ATTENUATION AND DISPERSION IN RANDOMLY CRACKED SOLIDS-II. PENNY-SHAPED CRACKS CH. ZHANG
and D. GROSS
Institute of Mechanics, TH Darmstadt, W-6100 Darmstadt,
Germany
Abstract-In this paper, attenuation and dispersion of elastic waves in a solid permeated by a random distribution of penny-shaped micro-cracks are analysed. The Foldy’s equation is applied for computing the complex effective wave number of the cracked solid. By taking the real and the imaginary part of the complex effective wave number, the effective wave velocity and the attenuation coefficient are subsequently obtained. Both the aligned and the randomly oriented penny-shaped micro-cracks are investigated. Numerical results for the attenuation coefficient and the effective wave velocity are presented, as functions of the crack density parameter, the crack orientation or the direction of wave incidence, and the wave frequency. Results obtained in this paper for penny-shaped micro-cracks are compared with those for slit micro-cracks given in a previous paper by the authors, to explore the effect of the micro-crack type on the attenuation coefficient and the effective wave velocity.
1. INTRODUCTION Analytical and experimental investigations of attenuation and dispersion of ultrasonic waves in a solid permeated by distributed micro-cracks are of particular importance to quantitative non-destructive evaluation by ultrasonic methods, for detecting and characterizing the damaged state of the cracked solid [l]. In our previous paper [2], wave attenuation and dispersion in a solid with randomly distributed slit micro-cracks in 2-D plane strain have been analysed. Both the theory of Foldy [3] and the causal approach based on Kramers-Kronig relations have been applied to calculate the attenuation coefficient and the effective wave velocity. Under the assumption that the high frequency limit of the effective wave velocity of a cracked solid is identical to the wave velocity of the untracked solid, both approaches yielded qualitatively the same feature for the effective wave velocity. The attenuation coefficient obtained by the two different approaches differs only slightly, as long as the crack density parameter is small. In this paper, the solution procedure used in [2] is extended to three-dimensional configurations. Wave propagation in a solid permeated by a random distribution of pennyshaped micro-cracks is investigated. The Foldy’s equation is applied to calculate the complex effective wave number. The effective wave velocity and the attenuation coefficient are subsequently obtained by taking the real and the imaginary part of the complex effective wave number. In principle, the causal approach based on Kramers-Kronig relations can also be applied for calculating the attenuation coefficient and the effective wave velocity. However, it has been shown in [2] that the nonlocality of the Kramers-Kronig relations requires generally a large amount of computational works to evaluate the infinite integrals arising in the Kramers-Kronig relations, especially in cases where the integrand converges slowly. For this reason and due to the fact that for small crack densities both approaches should give qualitatively the same behaviour for the attenuation coefficient and the effective wave velocity as shown in [2] for aligned slit micro-cracks, only the Foldy’s equation is used in this paper. The basic equations needed for this analysis are briefly summarized in Section 2. As in our previous paper [2], correlations and interactions between individual micro-cracks are not considered, and the analysis is thus only appropriate for small micro-crack densities. Numerical results for aligned penny-shaped micro-cracks are presented in Section 3, while the corresponding results for randomly oriented penny-shaped micro-cracks are given in Section 4. 859
860
CH. ZHANG
and D. GROSS
The use of a single universal crack density parameter enables us to compare the individual results for different micro-crack systems, to explore the effects of the micro-crack orientation or the direction of wave incidence, the micro-crack density, and the wave frequency. The influence of the micro-crack type on the attenuation coefficient and the effective wave velocity is discussed by comparing the results obtained in this paper for penny-shaped micro-cracks with those given in [2] for slit micro-cracks. Conclusions from this study are summarized in Section 5. It should be noted here that elastic wave propagation in a solid containing distributed penny-shaped micro-cracks has been previously analyzed in [4-171. With the exception in [16] and [17], these works were based on static analysis or were limited to low frequencies. The present analysis puts, however, no limitation on frequencies.
2. ATTENUATION
COEFFICIENT
AND EFFECTIVE
WAVE
VELOCITY
We consider first an infinite, homogeneous, isotropic and linearly elastic solid containing a penny-shaped crack in a three-dimensional configuration, as shown in Fig. 1. The deformation of the solid is assumed to be infinitesimally small, and on the faces of the crack, traction-free boundary conditions are assumed. The solid is in time-harmonic motion, but the term eFior is suppressed throughout the analysis, where w is the angular frequency. Without loss of generality, we assume that the direction of wave incidence is in the x1x,-plane. The angle of wave incidence is denoted by 8 as depicted in Fig. 1. The interaction of an incident time-harmonic elastic wave with a crack generates scattered waves. The total wave field, ui and oii, can be written as a sum of the incident wave field and the scattered wave field &=Ujn+$c,
oil = 0;; + o!!,
(2.1)
where z$ and o$ denote the displacement and the stress components of the incident wave in the absence of the crack, while UT and a? represent the corresponding quantities of scattered waves due to the interaction of the incident wave with the crack. In the present analysis, the incident wave is taken to be plane time-harmonic waves of the form up(x) = r/f %+9
g=L,T
(no sum over c),
(2.2)
in which “L” and “T” designate the longitudinal wave (L-wave) and the transverse wave (TV-wave), U,” and I!J~ denote the amplitudes of the incident L- and the incident TV-waves, nj represents the unit wave propagation vector, and kL and kT represent the L- and the TV-wave numbers, respectively. The scattered wave field is unknown and it has to be determined. Far away from the penny-shaped crack, the scattered waves consist of plane time-harmonic waves. By using the following spherical coordinate system with the origin at the center of the penny-shaped crack x,=RsiniPcos+,
x,=RsinYsin@,
Fig. 1. Penny-shaped
crack
x3 = R cos Y,
(2.3)
Wave attenuation and dispersion-II
is can be shown that for large R, the scattered displacement
861
field can be expressed as [B]
ik#
G(R, Y, a) = &
K(Y\v, @I,
R -+M,
(2.4)
R-+@J,
G-5)
ikTR u”,(R,
Y,
Q>)
=
p&
&v(~y,
@>,
ikTR
where 4, Z’& and fir+ are the scattering amplitudes of the scattered plane longitudinal, the scattered plane vertically polarized transverse, and the scattered plane horizontally polarized transverse waves. In particular, they are given by q(Y,
+ (1 - 2~~~) Sis] i e-ikt4y* AU&) &li,
@) = -ikJ2K-2Zif3
(2.7)
(2.9) where K = kT/kL, 6, is the Kronecker symbol, A is the surface of the penny-shaped &heunit vectors 5, i; and & have the following forms
~={~~~~},
+={ZZ;Z}.
I&{::?}.
crack, and
(2.10)
The forward scattering amplitudes are defined as the values of &(Y, (9) (5 = L, TV, TH) in the direction of wave incidence, i.e. &(9,0). To determine the scattering amplitudes Z$(Y, a), knowledge of the crack opening displacements Au;(y) are required. In this analysis, the following boundary integral equations are applied for calculating AU,(y) [19]
(2.11) where the integrals in (2.11) are understood as Cauchy principal value integrals, x and y are the position vectors of the observation point and the source point, f: are the traction components of incident waves, &,,kt is the elasticity tensor, n, are the components of the unit normal vector of the crack surface, E,~ is the permutation tensor, p is the mass density, and ~2 and o$ are the displacement and the stress Green’s functions of the full-space. Analytical expressions for us and o:k are given in [18] and they are not repeated here for the sake of brevity. Also, a comma after a quantity denotes partial derivatives of the quantity. For isotropic solids the elasticity tensor is given by &jkl = A 6ij
Sk1
+
p(hk
ajl
+
ail
ajk),
(2.12)
where il and (11are Lame’s elastic constants. The boundary integral equations (2.11) are solved numerically by adopting a boundary element method developed by the authors [19]_ We consider now a homogeneous, isotropic and linearly elastic solid containing a random distribution of penny-shaped micro-cracks. The location of the penny-shaped micro-cracks is ES 31:6-C
862
CH. ZHANG and D. GROSS
assumed to be random, while the orientation of the penny-shaped micro-cracks is either aligned or random. In the case of aligned penny-shaped micro-cracks, whose faces are parallel to the x,x,-plane, the macroscopic response of the cracked solid is transversely isotropic with respect to the x,-axis, whereas with randomly oriented penny-shaped micro-cracks the cracked solid exhibits a macroscopic isotropy. The presence of dispersed penny-shaped micro-cracks gives rise to wave scattering which withdraws the intensity of the incident wave. Consequently, the wave amplitude decreases or attenuates as the wave propagates through the cracked solid. Also, the presence of distributed penny-shaped micro-cracks changes the wave velocity which is frequency-dependent too. Although in a real situation the material microstructures, such as grain boundaries and second phase particles, may also have a strong influence on wave attenuation and dispersion, their contributions are not considered in this analysis. This simplification is less crucial as long as the wave length is sufficiently larger than the characteristic dimensions of the material microstructures. For simplicity, it is further assumed that the cracked solid contains only a single family of micro-cracks, i.e. either aligned penny-shaped micro-cracks or randomly oriented penny-shaped micro-cracks. Within a family of micro-cracks, all cracks are geometrically identical and have the same radius a. As mentioned in [2], the assumption of identical micro-cracks can be removed, if the relevant quantities entered in the analysis are replaced by their corresponding average quantities with respect to the crack sizes, provided that the statistical distribution of the micro-crack sizes is available. To describe elastic wave propagation in a randomly cracked solid, a complex and frequency-dependent effective wave number K(w) is introduced as
K(w) = 0
c(w)
+i&(w),
(2.13)
where Z(w) is the effective wave (phase) velocity and a(w) is the attenuation coefficient. An exact determination of the effective wave velocity I?(W) and the attenuation coefficient a is in general very difficult, due to multiple scattering or interaction effects between individual micro-cracks. For simplicity, it is assumed in this analysis that no correlation between the micro-cracks exists, and the density of the micro-cracks is sufficiently small to ignore the multiple scattering or the interaction effects among them. By using these assumptions, the complex effective wave number K can be determined by using the equation of Foldy [3]. K2=k2+nF,
(2.14)
where k is the wave number of the untracked solid, n is the number density of the micro-cracks, and F = FE(6, 0) (f = L, TV, TH) are the forward scattering amplitudes defined by equations (2.7)-(2.9). Once the complex effective wave number K has been determined via equation (2.14), the effective wave velocity and the attenuation coefficient can be immediately obtained by taking the real and the imaginary part of K, and by considering the definition (2.13) for K. This results in
E(w) =Re[Z(w)]
’
(Y(W)= Im[K(w)].
(2.15) (2.16)
It has been shown in our previous paper [2] that in the case of small crack number density n the attenuation coefficient a can be simplified to n Im(F) (Y=2kt where F = FE(f3, 0) are the forward scattering amplitudes of a single penny-shaped
(2.17) crack. For
Wave attenuation
convenience,
and dispersion-II
863
equation (2.17) is rewritten as
-WF)
c
--
a_2(ka)
a2
[
1
(2.18)
’
in which E = na3 is the crack density parameter, and ka is the dimensionless wave number of the untracked solid. In this paper, equation (2.18) is used instead of equation (2.14) for calculating the attenuation coefficient LY.It has been shown in [2] that equation (2.18) is a fairly good approximation of the Foldy’s equation (2.14), as long as the crack density parameter E is small. Note here that equation (2.18) can also be obtained by appealing to a simple energy consideration.
3. ALIGNED
PENNY-SHAPED
CRACKS
Let us consider a solid containing a random distribution of aligned penny-shaped microcracks and assume that all cracks have the same radius a (Fig. 2). The faces of the cracks are parallel to the x1x2-plane and their normal lies in the x3-direction. The overall average response of the cracked solid is transversely isotropic with respect to the x3-axis. Consequently, the attenuation coefficient and the effective wave velocity depend on the angle of wave incidence 8. The forward scattering amplitudes F,(B, 0) (5 = L, TV) of a single penny-shaped crack are computed numerically by adopting a boundary element method developed by the authors [19]. A total number of 200 constant elements is used and the Poisson’s ratio Y is selected as l/3. For convenience, the following normalized attenuation coefficient & is introduced 2a &=--CL
(3.1)
3dE
Also, the dimensionless TV-wave number kTa is used as frequency parameter, where kT = w/c, and cT is the TV-wave velocity of the untracked solid. Figure 3 presents the normalized attenuation coefficient ii vs the dimensionless wave number kTa, for incidence of plane time-harmonic L-waves. Since & is weakly dependent on E [2] and to keep the &-curves plausible, only the results obtained via equation (2.18) are presented here. The behaviour of & is similar to that for aligned slit cracks given in Fig. 4 of [2], with the exception that for fixed 8 the peaks of IL for aligned penny-shaped cracks are slightly larger than the corresponding values for aligned slit cracks. The &-curves for aligned penny-shaped cracks are shifted to slightly larger values of k-,-a. In general, the normalized attenuation coefficient & decreases with increasing 8. In the low frequency region, the normalized _-_ --_-
-
-
_
-
--
-
----
--_ - - _
---
Fig. 2. Aligned
-
-
-
_
--
-
---_ ---_-
--
penny-shaped
cracks.
864
CH. ZHANG
and D. GROSS
2.4
0.8
Fig. 3. Normalized attenuation coefficient C?vs k,n (L-wave incidence).
attenuation coefficient & increases rapidly with increasing k,a, and it oscillates about its high frequency limit &(m) in the high frequency range. Also in the case of aligned penny-shaped cracks, the high frequency limit ii(a) can be obtained analytically by using Kirchoff approximation [2]. For 8 = O”, 30” and 45”, the maximum normalized attenuation coefficient occurs at k,a = 2, while for 8 = 60” and 0 = 90” the occurrence of the maximum normalized attenuation coefficient is shifted to somewhat larger values of kTa. Numerical results for the normalized effective L-wave velocity CL/cL are shown in Fig. 4(a)-(e). These figures are very similar to those for aligned slit cracks given in Fig. 5(a)-(e) of [2]. The effective L-wave velocity ?r/c,_ decreases first with increasing kTa, after reaching a dip it then increases with further increasing kTa, and it stabilizes in the high frequency region. For fixed 8 and E, the maximum reduction of FL occurs approximately at k,a = 1.4. At low frequencies the reduction of CL decreases with increasing 8, whereas at high frequencies the opposite may be the case. Also, the reduction of C, increases with increasing E. The high frequency limit EL(~) of the cracked solid is larger than its corresponding static limit EL(O), and it approaches cL of the untracked solid. Also here, the effective L-wave velocity CLof a cracked solid is below the L-wave velocity cL of the untracked solid. By comparing Fig. 4(a)-(e) with Fig. 5(a)-(e) of [2] one can conclude that for fixed 8 and E and throughout the entire frequency range considered, the normalized effective wave velocity c,_/cL for aligned penny-shaped cracks is in general smaller than that for aligned slit cracks, and it approaches its high frequency limit slower that in the latter case. It should be noted here that in certain frequency range the variation of EL/cL with kg is very intense, especially for large values of E. Compared to the results for aligned slit micro-cracks, the dips of CL/c, occur at larger values of k,a. In Fig 5(a) and 5(b), the dependence of the normalized effective L-wave velocity EL/c,_ on the crack density parameter E is shown more explicitly for 13= 0” and 8 = 45”. For normally incident L-waves, 8 = 0”, the normalized effective wave velocity cr./c, decreases with increasing E. Depending on k,a the effective L-wave velocity EL may be over or under its static limit EL(O). Also for oblique incidence of L-waves with 8 = 45”, the normalized effective wave velocity EJcL decreases monotonically with increasing E. Figure 5(a) and 5(b) show that EL/c,_ for 8 = 0” is smaller at k,a = 0, 1 and 2, but larger at k,a = 3 than that for 8 = 45” at the corresponding wave numbers. Compared to the results for aligned slit cracks shown in Fig. 6 of [2], no dramatic differences are noted here, except that for fixed E and k,a the reduction of CL, is larger for aligned penny-shaped cracks than for aligned slit cracks. For incident TV-waves, the normalized attenuation coefficient 5 calculated by using equation (2.18) is given in Fig. 6 vs the dimensionless wave number kLa. As for aligned slit cracks the
Wave attenuation and dispersion-II
1.0
1.0
-
0.9
; 0.05
0.8
0.71 \/I -1 -I
\
I
.
0.7 e
0.6
/
865
&=O.V
0. I ii
0.2
N 0
L-wave
8=30°
1
2
r ,,,,,,,,,, kTi 4
(b) (,,,r 5
ha
0.8
0.76?,,,,,,,,,,,,,,,.,, 0 1
2
3
,,,,,I 4
(4 5
I,,,
Fig. 4. Normalized effective L-wave velocity vs k,a.
dependence of & on k,a is much more complicated for incident TV-waves than for incident L-waves. The normalized attenuation coefficient & is generally larger for normal incidence than for oblique incidence, except at some small values of k,a. For obliquely incident TV-waves the dependence of E on 8 is very tangled and no general conclusions can be drawn here. Figure 6 shows for instance that at low frequencies the normalized attenuation coefficient & for 8 = 90” is smallest, but this conclusion is no longer valid at high frequencies. For 8 = 0” and in the low
866
CH. ZHANG
and D. GROSS
go.7
0.1 &
0.3 E
Fig. 5. Normalized
effective
L-wave velocity
vs E.
frequency range, the normalized attenuation coefficient Cuincreases with increasing kTa until a peak is reached, and it oscillates thereafter about its high frequency limit L+(m).The normalized attenuation coefficient ii for obliquely incident TV-waves reaches its high frequency limit very slowly. Compared to the results for aligned slit cracks, the maximum normalized attenuation coefficient & for aligned penny-shaped cracks appears generally at somewhat larger values of k,a.
For several values of E and 0, numerical results for the normalized effective TV-wave velocity ET/cT are given in Fig. 7(a)-( e ) vs the dimensionless wave number kTa. These figures are again very similar to those for aligned slit cracks given in Fig. 8(a)-(e) of [2]. By comparing Fig. 7(a)-(e) with Fig. S(a)-(e) of [2] it is concluded that for fixed 8 and E the reduction of ET due to the presence of aligned penny-shaped cracks is larger than that due to the presence of aligned slit cracks. Compared to Er/cT for aligned slit cracks, the minima of ET/cT for aligned penny-shaped cracks are shifted to somewhat larger values of kTa. Also, Er/cT for aligned penny-shaped cracks approaches its high frequency limit more slower than ET/+ for aligned slit cracks does. Similarily to Fig. 8(a)-(e) of [2], the effective TV-wave velocity ET of a cracked solid is smaller than CT of the untracked solid. As for aligned slit cracks, the dependence of ET/+ on 8 is very involved. For fixed E and kTa and at low frequencies the reduction of ET is largest at 8 = 0”. In comparison with the results given in Fig. 4(a)-(e) for CL/cl., CT/cT reaches
Fig. 6. Normalized
attenuation
coefficient
& vs k,.a (TV-wave
incidence).
Wave attenuation
and dispersio-II
867
0.85
0.80
0.80
0.85 TV-wave
3
8=60"
0.80
0.80-
0.85 TV-wave 8=90'
0.80
(e) 0.75,,,,,,,,,,,,,,.,,,,,,~,,,,,,,, 0 I 2
4
5
6
kTi Fig. 7. Normalized
effective
high frequency limit ET(~)/+ = 1 somewhat and E, the reduction of ET is smaller than that grazing incidence, i.e. 8 = 90”, except near the incidence, the normalized effective TV-wave normalized effective L-wave velocity EJcL, frequencies. its
‘IV-wave
velocity vs /~.,.a.
slower. For normal incidence and for fixed 8 of EL, whereas a reverse tendency is noted for dips of EL/c,_ and ET/+. In the case of oblique velocity ET/+ is in general larger than the with the exception for 8 = 60” and at high
CH. ZHANG
868
and D. GROSS
0.80 0.75 -
f3=45”
:
0. I
0.2 &
0.3
0.70
(b) IIJ111111,111111111,/11111111 0.0 0.1 0.2 &
0.3
Fig. 8. Normalized effective TV-wave velocity vs E.
Figure S(a) and 8(b) sh ow the dependence of the normalized effective TV-wave velocity ET/cT on the crack density parameter E. Similarily to the results for aligned slit cracks given in Fig. 9 of [2], these figures show that for fixed k,a and 8, a larger value of E gives rise to a Tlc =. Comparisons of Fig. 8 with Fig. 9 of [2] imply that for k,a considered larger decrease in 1-7 here the reduction of ET due to aligned penny-shaped cracks is larger than that due to aligned slit cracks. Further comparisons of Fig. 8 with Fig. 5 show that for 8 = 0” and at k,a = 0, 1 and 2, the normalized effective wave velocity EL/c,_ is smaller than E-,./c~, whereas this feature is reversed at k,a = 3. For 8 = 45” and for kTa considered, the reduction of C,_is larger than that
4.
RANDOMLY
ORIENTED
PENNY-SHAPED
CRACKS
In this section, a solid permeated by a random distribution of randomly oriented penny-shaped micro-cracks is considered (Fig. 9). Following the same procedure used in section 5 of [2] for randomly oriented slit micro-cracks, the forward scattering amplitudes F,(R 0) (5 = I, TV) are first computed numerically via a boundary element method of [19]. Then, the average forward scattering amplitudes (FE) for randomly oriented penny-shaped micro-cracks are obtained by taking the spherical average of &(8, 0) (4.1)
(~~)=&Jzrr~~4(8.0)sinBdBd~=~~~F,(e,0)sinede. 0 0 0
Fig. 9. Randomly oriented penny-shaped
cracks.
Wave attenuation and dispersion-II
869
I.8
1.0
Its 0.8 0.6
0.4
-
: , ,, , , ,
Random
orientation
Fig. 10. Normalized attenuation coefficient iL vs k,a.
The average normalized attenuation coefficient (&) and the average effective wave velocity (E) are computed by using equations (2.18) and (2.19, and by invoking the following definition for (&)
(ii) =g JrE
((Y).
(4.2)
Here, it should be mentioned again that the average attenuation coefficient (&) and the average effective wave velocity (2) are independent of the angle of wave incidence 8, since the random orientation of the penny-shaped micro-cracks leads to a macroscopic isotropy of the cracked solid. Also in this section, 200 boundary elements are used for solving the boundary integral equations (2.11), and the Poisson’s ratio Y is set to be l/3. For the sake of brevity, the bracket in (&) and (E) is omitted in the following. Figure 10 presents the normalized attenuation coefficient & vs the dimensionless wave number k,a. Also here, only the results obtained from equation (2.18) are given. For incident L-waves and in the low frequency range, the normalized attenuation coefficient & increases rapidly with increasing wave number kTa, and it becomes stabilized in the high frequency range. In this case, & has a typical peak as for aligned penny-shaped cracks (see Fig. 3). The normalized attenuation coefficient & for incident TV-waves is somewhat different: it increases more or less monotonically with increasing k,a and no distinct peak in & is noted, at least in the frequency range considered here. The normalized attenuation coefficient & for incident L-waves is larger at low frequencies, but smaller at high frequencies than that for incident TV-waves. Compared to the results for randomly oriented slit cracks given in Fig. 11 of [2], the normalized attenuation coefficient & of incident L-waves is smaller for randomly oriented penny-shaped cracks. This conclusion applies also for the normalized attenuation coefficient E of incident TV-waves at low frequenties, but it turns over from certain frequency on. By comparing Fig. 10 with Figs 3 and 6 it is concluded that for incident L-waves the &-curve for randomly oriented penny-shaped cracks lies between the corresponding &-curves for aligned penny-shaped cracks with 8 = 45” and 0 = 60”, whereas the situation for incident TV-waves is again very involved. Indeed, the normalized attenuation coefficient & of incident TV-waves is smaller for randomly oriented penny-shaped cracks than for aligned penny-shaped cracks with 13= 0” as expected. Numerical results for the normalized effective wave velocity are given in Fig. 11(a) and 11(b), as functions of the dimensionless wave number k+z. These figures are, up to their
CH. ZHANG
870
and
L-wave Random
orientation
Random
orientation
Fig. 11. Normalized effective wave velocity vs k,a.
amplitudes, very similar to those for randomly oriented slit cracks given in Fig. 12 of [2]. For fixed E, the reduction of EL is smaller at low frequencies, but larger at high frequencies for randomly oriented penny-shaped cracks than that for randomly oriented slit cracks. The normalized effective TV-wave velocity ET/cT is smaller for randomly oriented penny-shaped cracks than that for randomly oriented slit cracks. Another comparison can be made with Figs 4(a)-(e) and 7(a)-(e) for aligned penny-shaped cracks. The normalized effective L-wave velocity EL/c,_ for randomly oriented penny-shaped cracks is larger than that for aligned penny-shaped cracks with 8 = O”, 30” and 0 = 45”, but smaller than that for aligned penny-shaped cracks with 8 = 60” and 8 = 90”. In the case of TV-wave incidence, the normalized effective wave velocity ET/cT for randomly oriented penny-shaped cracks is generally larger than that for aligned penny-shaped cracks with 8 = 0”, but smaller than that for aligned penny-shaped cracks with 0 = 45”. For incident TV-waves with 8 = 30”, 60” and 90”, &.I+ for randomly oriented penny-shaped cracks could be smaller or larger than that for aligned penny-shaped cracks, depending on k,a and E. For four values of kTa, the dependence of the normalized effective wave velocity on the crack density parameter E is shown in Fig. 12(a) and 12(b). Here again, the normalized effective wave velocity decreases monotonically with increasing crack density parameter E. For k,a considered here, the reduction of I?,_and ET is larger for randomly oriented penny-shaped 1.00 _ 0.95
1
0.90
1
0.85 1 0.80 0.75 1 L-wave 0.70 1
Random
0.80 orientation
TV-wave Random
orientation
0.75 -
0.65 1 : (a) 0.60,.~,~,,~,~,~.,,,,,,,,,.,..,,,, 0.0 0. I
0.2
0.3
: (b) 0.70 0.0 1111111111111111111111 0.1 0.2
& Fig. 12. Normalized effective wave velocity vs E.
&
,,,,,/I
1 0.3
Wave attenuation and dispersion-II
871
cracks than for randomly oriented slit cracks, except for EL at kTa = 0. By comparing Fig. 12(a) and 12(b) with Figs 5 and 8 it is seen that EL/c,_ and ET/+ are larger for randomly oriented penny-shaped cracks than for aligned penny-shaped cracks with 8 = O”, with an exception in EL/c= at kTa = 3. Also, EL/c,_ for randomly oriented penny-shaped cracks is larger than that for aligned penny-shaped cracks with 8 = 45”, whereas the opposite is concluded for ET/c=, at least for the values of k,a considered. Moreover, Fig. 12(a) and 12(b) show that for randomly oriented penny-shaped cracks and for all four values of kTa considered here, the reduction of E,_is larger than that of ET.
5. CONCLUSIONS
Wave attenuation and dispersion in a solid permeated by randomly distributed penny-shaped micro-cracks have been investigated in this paper. The Foldy’s equation has been applied to compute the complex effective wave number. The effective wave velocity and the attenuation coefficient have been obtained by taking the real and the imaginary part of the complex effective wave number. Numerical results for both the aligned and the randomly oriented penny-shaped micro-cracks have been presented and discussed. By using a single universal crack density parameter the numerical results obtained in this paper for penny-shaped micro-cracks have been compared with those given in [2] for slit micro-cracks. The present analysis puts no limitation on frequencies. Correlations and interactions between individual micro-cracks have been neglected. Thus, the numerial results presented in this paper are only appropriate for a dilute distribution of micro-cracks or a small micro-crack density. To compare the results obtained in this paper with those given in [2], the normalized attenuation coefficient and the effective wave velocity for aligned penny-shaped micro-cracks are denoted by &i(e) and E;(8) (f = L, T), while the corresponding quantities for randomly oriented penny-shaped micro-cracks are designated by ( &g) and (Eg). Similarily, the associated quantities for slit micro-cracks are represented by 5$(e), E:(8), (&g) and (@ as introduced in [2]. Comparisons of the present results for penny-shaped micro-cracks with those for slit micro-cracks given in [2] lead to the following conclusions: (1) The global behaviour of the normalized attenuation coefficients &i(e) and ( &g) for penny-shaped micro-cracks is very similar to &g(e) and (&g) for slit micro-cracks. For incident L-waves and for fixed 8, the peak values of &E(e) for aligned penny-shaped micro-cracks are somewhat larger than their corresponding values of &E(e) for aligned slit micro-cracks, and they appear at larger values of kTa. For incident TV-waves and for fixed 8, 5$(e) for aligned penny-shaped micro-cracks is smaller in the low frequency range than &c(e) for aligned slit micro-cracks, while the situation is very tangled at high frequencies. In general, the dependence of ET(e) on 8 and k,u is much more complex than that of &E(e) on 8 and kTu. For randomly oriented micro-cracks and in the case of L-wave incidence, the normalized attenuation coefficient (&[) for penny-shaped micro-cracks is smaller than (ii:) for slit micro-cracks. In the case of TV-wave incidence, &F is smaller than (&) in the interval 0 G k,u < 10, while an opposite feature is noted for 10 < k,u s 15. The normalized attenuation coefficient (&F) for randomly oriented penny-shaped micro-cracks approaches its high frequency limit even more slower than (&;) for randomly oriented slit micro-cracks does. (2) The variations of the effective wave velocities E!(8) and (5:) with 8, kTu and E are comparable to those of Z$(0) and (ES,) with 8, k,u and E. The effective wave velocities FE(e) and (Ei) of a cracked solid are reduced in comparison to the untracked solid, and they approach cs of the untracked solid in the high frequency limit. For fixed 8 and E and in the frequency range considered, the normalized wave velocities $(e)/cs for aligned penny-shaped cracks is smaller than E~(e)/c, for aligned slit cracks. For randomly oriented micro-cracks and for fixed E, the reduction of (EC) for penny-shaped cracks is larger than that of (E:) for slit
872
CH. ZHANG
and D. GROSS
cracks. This conclusion applies also for (5:) and (EE) at low frequencies, while an opposite feature is noted at high frequencies. Compared to E”,(O) and (ci) for slit micro-cracks, the maximum reduction of Z;(O) and (Eg) occurs at larger values of k,a. Furthermore, E:(O) and (Ei) approach their high frequency limit more slower than Eg(0) and (Eg) do. (3) The present analysis in conjunction with [2] implies that the slit crack model based on 2-D plane strain is able to describe the essential features of elastic wave attenuation and dispersion in randomly cracked solids, provided that a suitable crack density parameter is used. This is advantageous from the analytical and the computational points of view, since the 2-D problem is much easier to handle than the 3-D problem.
REFERENCES In Elastic Waves and Ultrasonic Nondestructive Evaluation (Edited by S. K. DA’ITA, J. D. ACHENBACH and Y. S. RAJAPAKSE). North-Holland, Amsterdam (1990). CH. ZHANG and D. GROSS, Znt. J. Engng Sci. 31, 841-858 (1993). L. L. FOLDY, Z’hys. Rev. 67, 107-119 (1945). H. D. GARBIN and L. KNOPOFF, Q. Appl. Math. 30,453-464 (1973). D. L. ANDERSON, B. MINISTER and D. COLE, J. Geophys. Res. 79,4011-4015 (1974). R. J. O’CONNELL and B. BUDIANSKY, J. Geophys. Res. 79, 5412-5426 (1974). H. D. GARBIN and L. KNOPOFF, Q. Appl. Math. 33,296-300 (1975). H. D. GARBIN and L. KNOPOFF, Q. Appl. Math. 33, 301-303 (1975). M. PIAU, Int. J. Engng Sci. 17, 151-167 (1979). M. PIAU, Int. 1. Engng Sci. 18, 549-568 (1980). A. K. CHATTERJEE, A. K. MAL, L. KNOPOFF and J. A. HUDSON, Math. Proc. Cumb. Phil. Sot. 88, 547-561 (1980). iI21 J. A. HUDSON, Geophys. J. R. Asrr. Sot. 64, 133-150 (1981). iI31 M. F. McCARTHY and M. M. CARROLL, In Wave Propagation in Homogeneous Media and Ulrrasonic Nondesmctiue Evaluation (Edited by G. C. JOHNSON), Vol. 62, pp. 141-153. ASME, New York (1984). 1141 J. A. HUDSON, Geophys. J. R. Asrr. Sot. 87, 265-274 (1986). CH. ZHANG and J. c. ACHENBACH, Int. J. Solids S&l. 27, 751-767 (1991). ;:j D. GROSS and CH. ZHANG. Inl. J. Sol&- Skuct. 29. 1763-1779 (1992). 1171 CH. ZHANG and D. GROSS; In 1st European Solid Mechanics Cdnfereme, Munich, Germany (1991). PSI J. D. ACHENBACH, A. K. GAUTESEN and H. McMAKEN, Ray Methodr for Waves in Elastic Solids, Pitman, Boston (1982). P91 CH. ZHANG and D. GROSS, Compur. Mech. 9, 137-152 (1992).
111J. D. ACHENBACH,
(Received 2 January
1992; revision received 4 August 1992)
1993
0020-7225/93 $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
WAVE ATTENUATION AND DISPERSION IN RANDOMLY CRACKED SOLIDS-II. PENNY-SHAPED CRACKS CH. ZHANG
and D. GROSS
Institute of Mechanics, TH Darmstadt, W-6100 Darmstadt,
Germany
Abstract-In this paper, attenuation and dispersion of elastic waves in a solid permeated by a random distribution of penny-shaped micro-cracks are analysed. The Foldy’s equation is applied for computing the complex effective wave number of the cracked solid. By taking the real and the imaginary part of the complex effective wave number, the effective wave velocity and the attenuation coefficient are subsequently obtained. Both the aligned and the randomly oriented penny-shaped micro-cracks are investigated. Numerical results for the attenuation coefficient and the effective wave velocity are presented, as functions of the crack density parameter, the crack orientation or the direction of wave incidence, and the wave frequency. Results obtained in this paper for penny-shaped micro-cracks are compared with those for slit micro-cracks given in a previous paper by the authors, to explore the effect of the micro-crack type on the attenuation coefficient and the effective wave velocity.
1. INTRODUCTION Analytical and experimental investigations of attenuation and dispersion of ultrasonic waves in a solid permeated by distributed micro-cracks are of particular importance to quantitative non-destructive evaluation by ultrasonic methods, for detecting and characterizing the damaged state of the cracked solid [l]. In our previous paper [2], wave attenuation and dispersion in a solid with randomly distributed slit micro-cracks in 2-D plane strain have been analysed. Both the theory of Foldy [3] and the causal approach based on Kramers-Kronig relations have been applied to calculate the attenuation coefficient and the effective wave velocity. Under the assumption that the high frequency limit of the effective wave velocity of a cracked solid is identical to the wave velocity of the untracked solid, both approaches yielded qualitatively the same feature for the effective wave velocity. The attenuation coefficient obtained by the two different approaches differs only slightly, as long as the crack density parameter is small. In this paper, the solution procedure used in [2] is extended to three-dimensional configurations. Wave propagation in a solid permeated by a random distribution of pennyshaped micro-cracks is investigated. The Foldy’s equation is applied to calculate the complex effective wave number. The effective wave velocity and the attenuation coefficient are subsequently obtained by taking the real and the imaginary part of the complex effective wave number. In principle, the causal approach based on Kramers-Kronig relations can also be applied for calculating the attenuation coefficient and the effective wave velocity. However, it has been shown in [2] that the nonlocality of the Kramers-Kronig relations requires generally a large amount of computational works to evaluate the infinite integrals arising in the Kramers-Kronig relations, especially in cases where the integrand converges slowly. For this reason and due to the fact that for small crack densities both approaches should give qualitatively the same behaviour for the attenuation coefficient and the effective wave velocity as shown in [2] for aligned slit micro-cracks, only the Foldy’s equation is used in this paper. The basic equations needed for this analysis are briefly summarized in Section 2. As in our previous paper [2], correlations and interactions between individual micro-cracks are not considered, and the analysis is thus only appropriate for small micro-crack densities. Numerical results for aligned penny-shaped micro-cracks are presented in Section 3, while the corresponding results for randomly oriented penny-shaped micro-cracks are given in Section 4. 859
860
CH. ZHANG
and D. GROSS
The use of a single universal crack density parameter enables us to compare the individual results for different micro-crack systems, to explore the effects of the micro-crack orientation or the direction of wave incidence, the micro-crack density, and the wave frequency. The influence of the micro-crack type on the attenuation coefficient and the effective wave velocity is discussed by comparing the results obtained in this paper for penny-shaped micro-cracks with those given in [2] for slit micro-cracks. Conclusions from this study are summarized in Section 5. It should be noted here that elastic wave propagation in a solid containing distributed penny-shaped micro-cracks has been previously analyzed in [4-171. With the exception in [16] and [17], these works were based on static analysis or were limited to low frequencies. The present analysis puts, however, no limitation on frequencies.
2. ATTENUATION
COEFFICIENT
AND EFFECTIVE
WAVE
VELOCITY
We consider first an infinite, homogeneous, isotropic and linearly elastic solid containing a penny-shaped crack in a three-dimensional configuration, as shown in Fig. 1. The deformation of the solid is assumed to be infinitesimally small, and on the faces of the crack, traction-free boundary conditions are assumed. The solid is in time-harmonic motion, but the term eFior is suppressed throughout the analysis, where w is the angular frequency. Without loss of generality, we assume that the direction of wave incidence is in the x1x,-plane. The angle of wave incidence is denoted by 8 as depicted in Fig. 1. The interaction of an incident time-harmonic elastic wave with a crack generates scattered waves. The total wave field, ui and oii, can be written as a sum of the incident wave field and the scattered wave field &=Ujn+$c,
oil = 0;; + o!!,
(2.1)
where z$ and o$ denote the displacement and the stress components of the incident wave in the absence of the crack, while UT and a? represent the corresponding quantities of scattered waves due to the interaction of the incident wave with the crack. In the present analysis, the incident wave is taken to be plane time-harmonic waves of the form up(x) = r/f %+9
g=L,T
(no sum over c),
(2.2)
in which “L” and “T” designate the longitudinal wave (L-wave) and the transverse wave (TV-wave), U,” and I!J~ denote the amplitudes of the incident L- and the incident TV-waves, nj represents the unit wave propagation vector, and kL and kT represent the L- and the TV-wave numbers, respectively. The scattered wave field is unknown and it has to be determined. Far away from the penny-shaped crack, the scattered waves consist of plane time-harmonic waves. By using the following spherical coordinate system with the origin at the center of the penny-shaped crack x,=RsiniPcos+,
x,=RsinYsin@,
Fig. 1. Penny-shaped
crack
x3 = R cos Y,
(2.3)
Wave attenuation and dispersion-II
is can be shown that for large R, the scattered displacement
861
field can be expressed as [B]
ik#
G(R, Y, a) = &
K(Y\v, @I,
R -+M,
(2.4)
R-+@J,
G-5)
ikTR u”,(R,
Y,
Q>)
=
p&
&v(~y,
@>,
ikTR
where 4, Z’& and fir+ are the scattering amplitudes of the scattered plane longitudinal, the scattered plane vertically polarized transverse, and the scattered plane horizontally polarized transverse waves. In particular, they are given by q(Y,
+ (1 - 2~~~) Sis] i e-ikt4y* AU&) &li,
@) = -ikJ2K-2Zif3
(2.7)
(2.9) where K = kT/kL, 6, is the Kronecker symbol, A is the surface of the penny-shaped &heunit vectors 5, i; and & have the following forms
~={~~~~},
+={ZZ;Z}.
I&{::?}.
crack, and
(2.10)
The forward scattering amplitudes are defined as the values of &(Y, (9) (5 = L, TV, TH) in the direction of wave incidence, i.e. &(9,0). To determine the scattering amplitudes Z$(Y, a), knowledge of the crack opening displacements Au;(y) are required. In this analysis, the following boundary integral equations are applied for calculating AU,(y) [19]
(2.11) where the integrals in (2.11) are understood as Cauchy principal value integrals, x and y are the position vectors of the observation point and the source point, f: are the traction components of incident waves, &,,kt is the elasticity tensor, n, are the components of the unit normal vector of the crack surface, E,~ is the permutation tensor, p is the mass density, and ~2 and o$ are the displacement and the stress Green’s functions of the full-space. Analytical expressions for us and o:k are given in [18] and they are not repeated here for the sake of brevity. Also, a comma after a quantity denotes partial derivatives of the quantity. For isotropic solids the elasticity tensor is given by &jkl = A 6ij
Sk1
+
p(hk
ajl
+
ail
ajk),
(2.12)
where il and (11are Lame’s elastic constants. The boundary integral equations (2.11) are solved numerically by adopting a boundary element method developed by the authors [19]_ We consider now a homogeneous, isotropic and linearly elastic solid containing a random distribution of penny-shaped micro-cracks. The location of the penny-shaped micro-cracks is ES 31:6-C
862
CH. ZHANG and D. GROSS
assumed to be random, while the orientation of the penny-shaped micro-cracks is either aligned or random. In the case of aligned penny-shaped micro-cracks, whose faces are parallel to the x,x,-plane, the macroscopic response of the cracked solid is transversely isotropic with respect to the x,-axis, whereas with randomly oriented penny-shaped micro-cracks the cracked solid exhibits a macroscopic isotropy. The presence of dispersed penny-shaped micro-cracks gives rise to wave scattering which withdraws the intensity of the incident wave. Consequently, the wave amplitude decreases or attenuates as the wave propagates through the cracked solid. Also, the presence of distributed penny-shaped micro-cracks changes the wave velocity which is frequency-dependent too. Although in a real situation the material microstructures, such as grain boundaries and second phase particles, may also have a strong influence on wave attenuation and dispersion, their contributions are not considered in this analysis. This simplification is less crucial as long as the wave length is sufficiently larger than the characteristic dimensions of the material microstructures. For simplicity, it is further assumed that the cracked solid contains only a single family of micro-cracks, i.e. either aligned penny-shaped micro-cracks or randomly oriented penny-shaped micro-cracks. Within a family of micro-cracks, all cracks are geometrically identical and have the same radius a. As mentioned in [2], the assumption of identical micro-cracks can be removed, if the relevant quantities entered in the analysis are replaced by their corresponding average quantities with respect to the crack sizes, provided that the statistical distribution of the micro-crack sizes is available. To describe elastic wave propagation in a randomly cracked solid, a complex and frequency-dependent effective wave number K(w) is introduced as
K(w) = 0
c(w)
+i&(w),
(2.13)
where Z(w) is the effective wave (phase) velocity and a(w) is the attenuation coefficient. An exact determination of the effective wave velocity I?(W) and the attenuation coefficient a is in general very difficult, due to multiple scattering or interaction effects between individual micro-cracks. For simplicity, it is assumed in this analysis that no correlation between the micro-cracks exists, and the density of the micro-cracks is sufficiently small to ignore the multiple scattering or the interaction effects among them. By using these assumptions, the complex effective wave number K can be determined by using the equation of Foldy [3]. K2=k2+nF,
(2.14)
where k is the wave number of the untracked solid, n is the number density of the micro-cracks, and F = FE(6, 0) (f = L, TV, TH) are the forward scattering amplitudes defined by equations (2.7)-(2.9). Once the complex effective wave number K has been determined via equation (2.14), the effective wave velocity and the attenuation coefficient can be immediately obtained by taking the real and the imaginary part of K, and by considering the definition (2.13) for K. This results in
E(w) =Re[Z(w)]
’
(Y(W)= Im[K(w)].
(2.15) (2.16)
It has been shown in our previous paper [2] that in the case of small crack number density n the attenuation coefficient a can be simplified to n Im(F) (Y=2kt where F = FE(f3, 0) are the forward scattering amplitudes of a single penny-shaped
(2.17) crack. For
Wave attenuation
convenience,
and dispersion-II
863
equation (2.17) is rewritten as
-WF)
c
--
a_2(ka)
a2
[
1
(2.18)
’
in which E = na3 is the crack density parameter, and ka is the dimensionless wave number of the untracked solid. In this paper, equation (2.18) is used instead of equation (2.14) for calculating the attenuation coefficient LY.It has been shown in [2] that equation (2.18) is a fairly good approximation of the Foldy’s equation (2.14), as long as the crack density parameter E is small. Note here that equation (2.18) can also be obtained by appealing to a simple energy consideration.
3. ALIGNED
PENNY-SHAPED
CRACKS
Let us consider a solid containing a random distribution of aligned penny-shaped microcracks and assume that all cracks have the same radius a (Fig. 2). The faces of the cracks are parallel to the x1x2-plane and their normal lies in the x3-direction. The overall average response of the cracked solid is transversely isotropic with respect to the x3-axis. Consequently, the attenuation coefficient and the effective wave velocity depend on the angle of wave incidence 8. The forward scattering amplitudes F,(B, 0) (5 = L, TV) of a single penny-shaped crack are computed numerically by adopting a boundary element method developed by the authors [19]. A total number of 200 constant elements is used and the Poisson’s ratio Y is selected as l/3. For convenience, the following normalized attenuation coefficient & is introduced 2a &=--CL
(3.1)
3dE
Also, the dimensionless TV-wave number kTa is used as frequency parameter, where kT = w/c, and cT is the TV-wave velocity of the untracked solid. Figure 3 presents the normalized attenuation coefficient ii vs the dimensionless wave number kTa, for incidence of plane time-harmonic L-waves. Since & is weakly dependent on E [2] and to keep the &-curves plausible, only the results obtained via equation (2.18) are presented here. The behaviour of & is similar to that for aligned slit cracks given in Fig. 4 of [2], with the exception that for fixed 8 the peaks of IL for aligned penny-shaped cracks are slightly larger than the corresponding values for aligned slit cracks. The &-curves for aligned penny-shaped cracks are shifted to slightly larger values of k-,-a. In general, the normalized attenuation coefficient & decreases with increasing 8. In the low frequency region, the normalized _-_ --_-
-
-
_
-
--
-
----
--_ - - _
---
Fig. 2. Aligned
-
-
-
_
--
-
---_ ---_-
--
penny-shaped
cracks.
864
CH. ZHANG
and D. GROSS
2.4
0.8
Fig. 3. Normalized attenuation coefficient C?vs k,n (L-wave incidence).
attenuation coefficient & increases rapidly with increasing k,a, and it oscillates about its high frequency limit &(m) in the high frequency range. Also in the case of aligned penny-shaped cracks, the high frequency limit ii(a) can be obtained analytically by using Kirchoff approximation [2]. For 8 = O”, 30” and 45”, the maximum normalized attenuation coefficient occurs at k,a = 2, while for 8 = 60” and 0 = 90” the occurrence of the maximum normalized attenuation coefficient is shifted to somewhat larger values of kTa. Numerical results for the normalized effective L-wave velocity CL/cL are shown in Fig. 4(a)-(e). These figures are very similar to those for aligned slit cracks given in Fig. 5(a)-(e) of [2]. The effective L-wave velocity ?r/c,_ decreases first with increasing kTa, after reaching a dip it then increases with further increasing kTa, and it stabilizes in the high frequency region. For fixed 8 and E, the maximum reduction of FL occurs approximately at k,a = 1.4. At low frequencies the reduction of CL decreases with increasing 8, whereas at high frequencies the opposite may be the case. Also, the reduction of C, increases with increasing E. The high frequency limit EL(~) of the cracked solid is larger than its corresponding static limit EL(O), and it approaches cL of the untracked solid. Also here, the effective L-wave velocity CLof a cracked solid is below the L-wave velocity cL of the untracked solid. By comparing Fig. 4(a)-(e) with Fig. 5(a)-(e) of [2] one can conclude that for fixed 8 and E and throughout the entire frequency range considered, the normalized effective wave velocity c,_/cL for aligned penny-shaped cracks is in general smaller than that for aligned slit cracks, and it approaches its high frequency limit slower that in the latter case. It should be noted here that in certain frequency range the variation of EL/cL with kg is very intense, especially for large values of E. Compared to the results for aligned slit micro-cracks, the dips of CL/c, occur at larger values of k,a. In Fig 5(a) and 5(b), the dependence of the normalized effective L-wave velocity EL/c,_ on the crack density parameter E is shown more explicitly for 13= 0” and 8 = 45”. For normally incident L-waves, 8 = 0”, the normalized effective wave velocity cr./c, decreases with increasing E. Depending on k,a the effective L-wave velocity EL may be over or under its static limit EL(O). Also for oblique incidence of L-waves with 8 = 45”, the normalized effective wave velocity EJcL decreases monotonically with increasing E. Figure 5(a) and 5(b) show that EL/c,_ for 8 = 0” is smaller at k,a = 0, 1 and 2, but larger at k,a = 3 than that for 8 = 45” at the corresponding wave numbers. Compared to the results for aligned slit cracks shown in Fig. 6 of [2], no dramatic differences are noted here, except that for fixed E and k,a the reduction of CL, is larger for aligned penny-shaped cracks than for aligned slit cracks. For incident TV-waves, the normalized attenuation coefficient 5 calculated by using equation (2.18) is given in Fig. 6 vs the dimensionless wave number kLa. As for aligned slit cracks the
Wave attenuation and dispersion-II
1.0
1.0
-
0.9
; 0.05
0.8
0.71 \/I -1 -I
\
I
.
0.7 e
0.6
/
865
&=O.V
0. I ii
0.2
N 0
L-wave
8=30°
1
2
r ,,,,,,,,,, kTi 4
(b) (,,,r 5
ha
0.8
0.76?,,,,,,,,,,,,,,,.,, 0 1
2
3
,,,,,I 4
(4 5
I,,,
Fig. 4. Normalized effective L-wave velocity vs k,a.
dependence of & on k,a is much more complicated for incident TV-waves than for incident L-waves. The normalized attenuation coefficient & is generally larger for normal incidence than for oblique incidence, except at some small values of k,a. For obliquely incident TV-waves the dependence of E on 8 is very tangled and no general conclusions can be drawn here. Figure 6 shows for instance that at low frequencies the normalized attenuation coefficient & for 8 = 90” is smallest, but this conclusion is no longer valid at high frequencies. For 8 = 0” and in the low
866
CH. ZHANG
and D. GROSS
go.7
0.1 &
0.3 E
Fig. 5. Normalized
effective
L-wave velocity
vs E.
frequency range, the normalized attenuation coefficient Cuincreases with increasing kTa until a peak is reached, and it oscillates thereafter about its high frequency limit L+(m).The normalized attenuation coefficient ii for obliquely incident TV-waves reaches its high frequency limit very slowly. Compared to the results for aligned slit cracks, the maximum normalized attenuation coefficient & for aligned penny-shaped cracks appears generally at somewhat larger values of k,a.
For several values of E and 0, numerical results for the normalized effective TV-wave velocity ET/cT are given in Fig. 7(a)-( e ) vs the dimensionless wave number kTa. These figures are again very similar to those for aligned slit cracks given in Fig. 8(a)-(e) of [2]. By comparing Fig. 7(a)-(e) with Fig. S(a)-(e) of [2] it is concluded that for fixed 8 and E the reduction of ET due to the presence of aligned penny-shaped cracks is larger than that due to the presence of aligned slit cracks. Compared to Er/cT for aligned slit cracks, the minima of ET/cT for aligned penny-shaped cracks are shifted to somewhat larger values of kTa. Also, Er/cT for aligned penny-shaped cracks approaches its high frequency limit more slower than ET/+ for aligned slit cracks does. Similarily to Fig. 8(a)-(e) of [2], the effective TV-wave velocity ET of a cracked solid is smaller than CT of the untracked solid. As for aligned slit cracks, the dependence of ET/+ on 8 is very involved. For fixed E and kTa and at low frequencies the reduction of ET is largest at 8 = 0”. In comparison with the results given in Fig. 4(a)-(e) for CL/cl., CT/cT reaches
Fig. 6. Normalized
attenuation
coefficient
& vs k,.a (TV-wave
incidence).
Wave attenuation
and dispersio-II
867
0.85
0.80
0.80
0.85 TV-wave
3
8=60"
0.80
0.80-
0.85 TV-wave 8=90'
0.80
(e) 0.75,,,,,,,,,,,,,,.,,,,,,~,,,,,,,, 0 I 2
4
5
6
kTi Fig. 7. Normalized
effective
high frequency limit ET(~)/+ = 1 somewhat and E, the reduction of ET is smaller than that grazing incidence, i.e. 8 = 90”, except near the incidence, the normalized effective TV-wave normalized effective L-wave velocity EJcL, frequencies. its
‘IV-wave
velocity vs /~.,.a.
slower. For normal incidence and for fixed 8 of EL, whereas a reverse tendency is noted for dips of EL/c,_ and ET/+. In the case of oblique velocity ET/+ is in general larger than the with the exception for 8 = 60” and at high
CH. ZHANG
868
and D. GROSS
0.80 0.75 -
f3=45”
:
0. I
0.2 &
0.3
0.70
(b) IIJ111111,111111111,/11111111 0.0 0.1 0.2 &
0.3
Fig. 8. Normalized effective TV-wave velocity vs E.
Figure S(a) and 8(b) sh ow the dependence of the normalized effective TV-wave velocity ET/cT on the crack density parameter E. Similarily to the results for aligned slit cracks given in Fig. 9 of [2], these figures show that for fixed k,a and 8, a larger value of E gives rise to a Tlc =. Comparisons of Fig. 8 with Fig. 9 of [2] imply that for k,a considered larger decrease in 1-7 here the reduction of ET due to aligned penny-shaped cracks is larger than that due to aligned slit cracks. Further comparisons of Fig. 8 with Fig. 5 show that for 8 = 0” and at k,a = 0, 1 and 2, the normalized effective wave velocity EL/c,_ is smaller than E-,./c~, whereas this feature is reversed at k,a = 3. For 8 = 45” and for kTa considered, the reduction of C,_is larger than that
4.
RANDOMLY
ORIENTED
PENNY-SHAPED
CRACKS
In this section, a solid permeated by a random distribution of randomly oriented penny-shaped micro-cracks is considered (Fig. 9). Following the same procedure used in section 5 of [2] for randomly oriented slit micro-cracks, the forward scattering amplitudes F,(R 0) (5 = I, TV) are first computed numerically via a boundary element method of [19]. Then, the average forward scattering amplitudes (FE) for randomly oriented penny-shaped micro-cracks are obtained by taking the spherical average of &(8, 0) (4.1)
(~~)=&Jzrr~~4(8.0)sinBdBd~=~~~F,(e,0)sinede. 0 0 0
Fig. 9. Randomly oriented penny-shaped
cracks.
Wave attenuation and dispersion-II
869
I.8
1.0
Its 0.8 0.6
0.4
-
: , ,, , , ,
Random
orientation
Fig. 10. Normalized attenuation coefficient iL vs k,a.
The average normalized attenuation coefficient (&) and the average effective wave velocity (E) are computed by using equations (2.18) and (2.19, and by invoking the following definition for (&)
(ii) =g JrE
((Y).
(4.2)
Here, it should be mentioned again that the average attenuation coefficient (&) and the average effective wave velocity (2) are independent of the angle of wave incidence 8, since the random orientation of the penny-shaped micro-cracks leads to a macroscopic isotropy of the cracked solid. Also in this section, 200 boundary elements are used for solving the boundary integral equations (2.11), and the Poisson’s ratio Y is set to be l/3. For the sake of brevity, the bracket in (&) and (E) is omitted in the following. Figure 10 presents the normalized attenuation coefficient & vs the dimensionless wave number k,a. Also here, only the results obtained from equation (2.18) are given. For incident L-waves and in the low frequency range, the normalized attenuation coefficient & increases rapidly with increasing wave number kTa, and it becomes stabilized in the high frequency range. In this case, & has a typical peak as for aligned penny-shaped cracks (see Fig. 3). The normalized attenuation coefficient & for incident TV-waves is somewhat different: it increases more or less monotonically with increasing k,a and no distinct peak in & is noted, at least in the frequency range considered here. The normalized attenuation coefficient & for incident L-waves is larger at low frequencies, but smaller at high frequencies than that for incident TV-waves. Compared to the results for randomly oriented slit cracks given in Fig. 11 of [2], the normalized attenuation coefficient & of incident L-waves is smaller for randomly oriented penny-shaped cracks. This conclusion applies also for the normalized attenuation coefficient E of incident TV-waves at low frequenties, but it turns over from certain frequency on. By comparing Fig. 10 with Figs 3 and 6 it is concluded that for incident L-waves the &-curve for randomly oriented penny-shaped cracks lies between the corresponding &-curves for aligned penny-shaped cracks with 8 = 45” and 0 = 60”, whereas the situation for incident TV-waves is again very involved. Indeed, the normalized attenuation coefficient & of incident TV-waves is smaller for randomly oriented penny-shaped cracks than for aligned penny-shaped cracks with 13= 0” as expected. Numerical results for the normalized effective wave velocity are given in Fig. 11(a) and 11(b), as functions of the dimensionless wave number k+z. These figures are, up to their
CH. ZHANG
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and
L-wave Random
orientation
Random
orientation
Fig. 11. Normalized effective wave velocity vs k,a.
amplitudes, very similar to those for randomly oriented slit cracks given in Fig. 12 of [2]. For fixed E, the reduction of EL is smaller at low frequencies, but larger at high frequencies for randomly oriented penny-shaped cracks than that for randomly oriented slit cracks. The normalized effective TV-wave velocity ET/cT is smaller for randomly oriented penny-shaped cracks than that for randomly oriented slit cracks. Another comparison can be made with Figs 4(a)-(e) and 7(a)-(e) for aligned penny-shaped cracks. The normalized effective L-wave velocity EL/c,_ for randomly oriented penny-shaped cracks is larger than that for aligned penny-shaped cracks with 8 = O”, 30” and 0 = 45”, but smaller than that for aligned penny-shaped cracks with 8 = 60” and 8 = 90”. In the case of TV-wave incidence, the normalized effective wave velocity ET/cT for randomly oriented penny-shaped cracks is generally larger than that for aligned penny-shaped cracks with 8 = 0”, but smaller than that for aligned penny-shaped cracks with 0 = 45”. For incident TV-waves with 8 = 30”, 60” and 90”, &.I+ for randomly oriented penny-shaped cracks could be smaller or larger than that for aligned penny-shaped cracks, depending on k,a and E. For four values of kTa, the dependence of the normalized effective wave velocity on the crack density parameter E is shown in Fig. 12(a) and 12(b). Here again, the normalized effective wave velocity decreases monotonically with increasing crack density parameter E. For k,a considered here, the reduction of I?,_and ET is larger for randomly oriented penny-shaped 1.00 _ 0.95
1
0.90
1
0.85 1 0.80 0.75 1 L-wave 0.70 1
Random
0.80 orientation
TV-wave Random
orientation
0.75 -
0.65 1 : (a) 0.60,.~,~,,~,~,~.,,,,,,,,,.,..,,,, 0.0 0. I
0.2
0.3
: (b) 0.70 0.0 1111111111111111111111 0.1 0.2
& Fig. 12. Normalized effective wave velocity vs E.
&
,,,,,/I
1 0.3
Wave attenuation and dispersion-II
871
cracks than for randomly oriented slit cracks, except for EL at kTa = 0. By comparing Fig. 12(a) and 12(b) with Figs 5 and 8 it is seen that EL/c,_ and ET/+ are larger for randomly oriented penny-shaped cracks than for aligned penny-shaped cracks with 8 = O”, with an exception in EL/c= at kTa = 3. Also, EL/c,_ for randomly oriented penny-shaped cracks is larger than that for aligned penny-shaped cracks with 8 = 45”, whereas the opposite is concluded for ET/c=, at least for the values of k,a considered. Moreover, Fig. 12(a) and 12(b) show that for randomly oriented penny-shaped cracks and for all four values of kTa considered here, the reduction of E,_is larger than that of ET.
5. CONCLUSIONS
Wave attenuation and dispersion in a solid permeated by randomly distributed penny-shaped micro-cracks have been investigated in this paper. The Foldy’s equation has been applied to compute the complex effective wave number. The effective wave velocity and the attenuation coefficient have been obtained by taking the real and the imaginary part of the complex effective wave number. Numerical results for both the aligned and the randomly oriented penny-shaped micro-cracks have been presented and discussed. By using a single universal crack density parameter the numerical results obtained in this paper for penny-shaped micro-cracks have been compared with those given in [2] for slit micro-cracks. The present analysis puts no limitation on frequencies. Correlations and interactions between individual micro-cracks have been neglected. Thus, the numerial results presented in this paper are only appropriate for a dilute distribution of micro-cracks or a small micro-crack density. To compare the results obtained in this paper with those given in [2], the normalized attenuation coefficient and the effective wave velocity for aligned penny-shaped micro-cracks are denoted by &i(e) and E;(8) (f = L, T), while the corresponding quantities for randomly oriented penny-shaped micro-cracks are designated by ( &g) and (Eg). Similarily, the associated quantities for slit micro-cracks are represented by 5$(e), E:(8), (&g) and (@ as introduced in [2]. Comparisons of the present results for penny-shaped micro-cracks with those for slit micro-cracks given in [2] lead to the following conclusions: (1) The global behaviour of the normalized attenuation coefficients &i(e) and ( &g) for penny-shaped micro-cracks is very similar to &g(e) and (&g) for slit micro-cracks. For incident L-waves and for fixed 8, the peak values of &E(e) for aligned penny-shaped micro-cracks are somewhat larger than their corresponding values of &E(e) for aligned slit micro-cracks, and they appear at larger values of kTa. For incident TV-waves and for fixed 8, 5$(e) for aligned penny-shaped micro-cracks is smaller in the low frequency range than &c(e) for aligned slit micro-cracks, while the situation is very tangled at high frequencies. In general, the dependence of ET(e) on 8 and k,u is much more complex than that of &E(e) on 8 and kTu. For randomly oriented micro-cracks and in the case of L-wave incidence, the normalized attenuation coefficient (&[) for penny-shaped micro-cracks is smaller than (ii:) for slit micro-cracks. In the case of TV-wave incidence, &F is smaller than (&) in the interval 0 G k,u < 10, while an opposite feature is noted for 10 < k,u s 15. The normalized attenuation coefficient (&F) for randomly oriented penny-shaped micro-cracks approaches its high frequency limit even more slower than (&;) for randomly oriented slit micro-cracks does. (2) The variations of the effective wave velocities E!(8) and (5:) with 8, kTu and E are comparable to those of Z$(0) and (ES,) with 8, k,u and E. The effective wave velocities FE(e) and (Ei) of a cracked solid are reduced in comparison to the untracked solid, and they approach cs of the untracked solid in the high frequency limit. For fixed 8 and E and in the frequency range considered, the normalized wave velocities $(e)/cs for aligned penny-shaped cracks is smaller than E~(e)/c, for aligned slit cracks. For randomly oriented micro-cracks and for fixed E, the reduction of (EC) for penny-shaped cracks is larger than that of (E:) for slit
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CH. ZHANG
and D. GROSS
cracks. This conclusion applies also for (5:) and (EE) at low frequencies, while an opposite feature is noted at high frequencies. Compared to E”,(O) and (ci) for slit micro-cracks, the maximum reduction of Z;(O) and (Eg) occurs at larger values of k,a. Furthermore, E:(O) and (Ei) approach their high frequency limit more slower than Eg(0) and (Eg) do. (3) The present analysis in conjunction with [2] implies that the slit crack model based on 2-D plane strain is able to describe the essential features of elastic wave attenuation and dispersion in randomly cracked solids, provided that a suitable crack density parameter is used. This is advantageous from the analytical and the computational points of view, since the 2-D problem is much easier to handle than the 3-D problem.
REFERENCES In Elastic Waves and Ultrasonic Nondestructive Evaluation (Edited by S. K. DA’ITA, J. D. ACHENBACH and Y. S. RAJAPAKSE). North-Holland, Amsterdam (1990). CH. ZHANG and D. GROSS, Znt. J. Engng Sci. 31, 841-858 (1993). L. L. FOLDY, Z’hys. Rev. 67, 107-119 (1945). H. D. GARBIN and L. KNOPOFF, Q. Appl. Math. 30,453-464 (1973). D. L. ANDERSON, B. MINISTER and D. COLE, J. Geophys. Res. 79,4011-4015 (1974). R. J. O’CONNELL and B. BUDIANSKY, J. Geophys. Res. 79, 5412-5426 (1974). H. D. GARBIN and L. KNOPOFF, Q. Appl. Math. 33,296-300 (1975). H. D. GARBIN and L. KNOPOFF, Q. Appl. Math. 33, 301-303 (1975). M. PIAU, Int. J. Engng Sci. 17, 151-167 (1979). M. PIAU, Int. 1. Engng Sci. 18, 549-568 (1980). A. K. CHATTERJEE, A. K. MAL, L. KNOPOFF and J. A. HUDSON, Math. Proc. Cumb. Phil. Sot. 88, 547-561 (1980). iI21 J. A. HUDSON, Geophys. J. R. Asrr. Sot. 64, 133-150 (1981). iI31 M. F. McCARTHY and M. M. CARROLL, In Wave Propagation in Homogeneous Media and Ulrrasonic Nondesmctiue Evaluation (Edited by G. C. JOHNSON), Vol. 62, pp. 141-153. ASME, New York (1984). 1141 J. A. HUDSON, Geophys. J. R. Asrr. Sot. 87, 265-274 (1986). CH. ZHANG and J. c. ACHENBACH, Int. J. Solids S&l. 27, 751-767 (1991). ;:j D. GROSS and CH. ZHANG. Inl. J. Sol&- Skuct. 29. 1763-1779 (1992). 1171 CH. ZHANG and D. GROSS; In 1st European Solid Mechanics Cdnfereme, Munich, Germany (1991). PSI J. D. ACHENBACH, A. K. GAUTESEN and H. McMAKEN, Ray Methodr for Waves in Elastic Solids, Pitman, Boston (1982). P91 CH. ZHANG and D. GROSS, Compur. Mech. 9, 137-152 (1992).
111J. D. ACHENBACH,
(Received 2 January
1992; revision received 4 August 1992)