Self-consistent analysis of waves in a matrix-inclusion composite—III. A matrix containing cracks

J. Mech. Plrr’s. Solih

Printed rn &eat

0022-5096193 $6.00 + 0.00 ,” 1993 Pergamon Press Ltd

Vol. 41. No. 12, pp. 1809-1X24, 1993.

Britain.

SELF-CONSISTENT ANALYSIS OF WAVES IN A MATRIX-INCLUSION COMPOSITE~III. A MATRIX CONTAINING CRACKS V. P. SMYSHLYAEV and J. R. WILLIS School of Mathematical

Scicnccs.

Tlniversity

of Bath. Bath BA? 7AY. U.K

and F. J. SABINA lnstituto

de Investigaciones en Matemiticas Aplicadas y en Sistemas.UnivcrsidadNational Authnoma de MCxico, Apdo. Postal 20-726. Admon. No. 20. DelegaciOn dc Alvaro Obregon. 01000 Mexico D.F.. Mexico

ABWRACT THE SELFCONSISTENT analysis developed in Parts I and II is applied to the study of waves in a body containing cracks, by taking the limits of the formulae already derived BS the aspect ratio of the spheroids tends to zero. A direct formulation for cracks, which leads to the same equations, is briefly summarized. The numbers of equations that require solution for the various cases (empty or fluid-filled cavities. aligned or randomly oriented) are reduced substantially for cracks in comparison with spheroids, because there is only one density (that of the matrix) and cfl‘cctive moduli are only altcrcd from their matrix values by components of stress that interact with the cracks. Sample results are presented. These confirm calculations reported in Parts I and II. when the aspect ratio of the cavities was taken to bc small, and demonstrate that the “crack” limit is approached much more slowly as the aspect ratio tends to zero when the cavities are fluid-filled than when they are empty.

I.

INTR~IIUCTI~N

THIS PAPER FOLLOWS

from two preceding papers (SABINA et ul., 1993; SMYSHLYAEV ct cd., 1993), which are designated henceforth I and II. These papers dealt with waves through a composite containing spheroidal inclusions, with ratio of polar to equatorial diameter 6. Here, the limiting case 6 + 0 is developed explicitly, when the inclusions are cavities, which may be empty or fluid-filled. The inclusions then become circular or “penny-shaped” cracks. A direct formulation, which leads to the same equations, is also briefly outlined. In the static limit, the formulations become equivalent to those of BUDIANSKY and O’CONNELL (1976) for randomly oriented cracks, and HOENIG (I 979) for aligned cracks. Previous work on waves through a matrix containing cracks has been subject to one or both of the restrictions to low frequencies and dilute arrays of cracks. The “Rayleigh limit” of low frequency was addressed, for dilute arrays, by PIAU (l979), and also by CRAMPIN (l984), following work of HUDSON (1981). ZHANG 1809

IXIO

\’

I’.

Shl\rSHI.YA~~ <‘I c/l.

and ACHENRAC.H ( I99 I ) made

approximate allowance for crack interactions in ;I \vay for widely-separated. and hence dilutely distributed. cracks. Their dctailcd calculations were for low-frequency scattering from aligned cracks but these are not inherent limitations of their method. GROSS and ZHANG (1992) and ZHAUC; and GROSS (l993a, b) performed calculations for ;I body containing dilute arrays of cracks. tither aligned or randomly oriented. They estimated attenuation at any frcquency by a method akin to that of FOI.DY (1945) and deduced the associated phase speed by use of the Kramers Kronig relations. KIICUCHI ( 19XI ) earlier applied E‘oldylike theory to calculate dispersion and attenuation of waves in ;I body containing t\vodimensional “slit” cracks. Again. there is ;I restriction to ;I dilute array. WIL.I.IS (19X0) made allowance for a non-dilute array of aligned cracks by invoking the quasicr~stallinc approximation of LAS (I 952). but prcscntcd results only in the Kaylcigh limit. The present work is ~IILIS believed to be the lirst source for formulae relating to multiple scattering from non-dilute arrays of cracks. for waves whose wavelength is comparable to crack diameter. that

was

valid

7 -.

BA(‘I*;ciIu)t;xI)

F~R>IIJI.AI

A gcncral self-consistent schcmc was developed in Part I, which resulted in a set 01 conditions represented by equations (1.23). Part I addrcsscd particularly ;I single population of aligned spheroidal inclusions; solving equations (1.26) was idcntifed as the preferred means of satisfying (1,23). In Part II. when specialized to the case of randomly-oriented spheroids, equations (1,23) reduced to (11,l). To avoid conflicting ~lsage of symbols in this Part. the number density of the cracks will be denoted by N, instead of the 11, LISC~ in Parts I and II. Likewise. v will denote the direction of the axis of symmetry of a spheroid or ;I crack. In either the aligned or randomly oriented case, expressions for cracks follow capon taking the limit ii + 0. while keeping the number density N fixed. T~LIS, limiting forms for the tensors S, and A, arc required. Considering first iz;i,, in either ol the casts under discussion. this is ;I second-order tensor with transversely isotropic symmetry. It has the general form (R,),, = .~1,((5,,~~,,~,,)+.W,,,,,,,,,.

(1)

where M, and M,,, are defined by equations (1-B. 12). In the case of aligned spheroids, the background medium has the same axis of symmetry as the spheroids. and this is taken to be aligned with O.\V~: when the spheroids arc oriented at random. the background medium is taken as isotropic and so has no intluencc on the symmetry. The integrands which appear in (I.B. 12) contain functions F, and F,. which arc given by the first of equations (I,B.6). It is not difficult to conclude that the limit 0 + 0 may bc applied directly to the integrand to yield &I, + 0 as (5 + 0. Since. for ii matrix containing cracks at number density N II the volume fraction of inclusion material C, = 4nN,l/‘ci;3 + 0 as 0 --f 0. it follows that the second of equations (1.26). and likewise the second of (II, 1). both reduce to

Self-consistent

of waves

analysis

1811

Thus, the effective density of a cracked body is its actual density, that of the matrix material. Now it is necessary to consider the tensor of effective moduli, Lo. Explicit equations for L,, follow once the limiting form of the tensor S, has been established. This time, the limit (S+ 0 cannot be applied directly to the integrands, as given in (1,B. I I), because not all of the resulting expressions would be integrable. The offending functions are H,, (0. as defined in equations (I.B.6). The integrals to be evaluated have the general form

(3) where &u) - ~,,(~~)+6’~,(~~)+0(~5’), <(u) = [l + (6’- I )u~] manipulation yields

’ ’ [X

in (I.B. 13)]. and 4,, and

(4)

(b, are

smooth

functions.

Simple

(5) having employed

the elementary

integral

I

‘du

I

(6)

,, i”(U) - s.

Analysis of the integrands in (1.B.l I) shows that c$,)(I) # 0 only for the “transversely isotropic” constants M,~and 2ps. In the former case.

and in the latter,

4(,;(j) +&i(j) 1=y;,). 3

(Of course, in the present

application,

S,(S) =

o.o,o, i

where (s,),

/I, = P,,, = L)?.) Thus, ’ .o I +S(S,), ’ G,, 1 I’ll

+0(d2).

(7)

has the full form (3,) , = (2/F, z /:. ri. 2ni, 2p)

(8)

v.

1812 For

the

sake

of

completeness.

S\lYSHL Yhl v

P.

expressions

l’or

oi

<‘I

the

constants

in

(8)

of

tensor

are

given

in

the

Appendix. Equations When

(1.26)

8 +

0. this

and

(II,])

tensor

require

has

the

[I+S,(L,

the

inversion

the

[I+S,(L,

-

I_,,)].

Ibrm

-L,,)]

=

l:,,+Ci(S,),(L,

PL,,)+o(CS2),

(9)

where

(IO) The

desired

not.

\vhen

empty

inverse L,

crack,

with

Iluid

arc

also

are

not

is dominated

= 0. b,hich nor

with

when

bulk

other

L,

=

(2/i,.

modulus

cases

for

by

the

corresponds

an

again

in the

of 0).

cavity which

lilmit

C:,, is singular.

(! ,/. so long

umpty

K,. I<,, I;,,().

I.L ,.

which

invct-sc to

in

these

are

cxisls.

limit

corresponds

ii --f 0. i.c.

but

as Ihis the

It doc5

ci ---f 0. i.c. to ;I cavil),

a fluid-tilled

crack.

01‘ no physical

an

filled There

intorcst

and

pursued.

The

implications

of

consists

of

;I niatrix

cracks.

Since

the

dixcttsscd

the

prcccding

containing

l’orniulac

arc

arc

~OI-~LI~IC

aligned. sitnplest

hut \vhen

no\v

discussed

when

randomly-positioned. the

cracks

the

composite

penny-shopcd

arc

fluid-fitted.

this

cast

is

first.

When

the

inversion

of

tensor (9)

of

moduti

L,

is chosen

to

correspond

to

it non-viscous

IlLlid.

yields

(II) Regarding order

the

normals

to

lhc

cracks

as aligned

with

2

’ is consistent with the fact that ;I tluid-filled components 0, 1 and o2 1: there is no interaction normal

load

consistent in

the

matrix

is transmitted

equations Iiniit value.

(5 +

(1.26).

0. only

Thus.

across the

the

the

inw33c

component

I.accs

of

O.\Y~. crack

M ith the

the

crack

(I I) is niLtItiplied Z/I,,

01‘ L,,

is altcrcd

the

li~rrm

only

tcrni with

component by

by

ol’thc

interacts the

cl;

Iluid.

In

c’, = 4rr,V,(,‘M from

the

01’

stress

beca~~w lhc and

corresponding

sellso.

1813

Self-consistent analysis of waves

11/z=

111(, =

(12)

,ll?,

in terms of the bulk and shear moduli /iI and iiT of the matrix, assuming that this is equation isotropic. The final constant. p,,, satisfies the single self-consistent 71N,C/7/I’0’(k)h”‘~(-li)p: PO = 112+ Here, the function

(13)

VP 0

l?““(k) is given by (14)

where I; is defined in (I.B. 15) and 0 is the angle between

the wave normal

n and 0.~~.

Consider now the case L, = 0. It is convenient to work in terms of compliances rather than moduli. To this end, the self-consistent equation (I,26), with L, = 0. is by L,, ’ = M,, to yield pre-multiplied by L? ’ = M2 and post-multiplied Iv,, = M2 +

Explicitly,

4y(2’h(kn)h(

-kn)[I-S,L,,]

‘Mo.

(1%

M,, is given as

where A = li,,n,,-/,,I;,. NOW substituting the asymptotic I-S,L,,

form (7) for 3,. it is obtained

= (1,O. -/;,:n,,,o,

that

l,O)-6(s,),L,,+o(~‘)

(17)

and hence that S[Z-S,L,,]

‘M,, = (O.O,O. -fi,o,

-2/,)+0(d),

(18)

where Ii = [17~/?+21?,,(/;,~+/,~1;) +4/&F]

’

(19)

and 2/, = [8/J,;/?] ‘. The self-consistent

equations M,,

=

A/f-

(20)

(I 5) thrls reduce to

4y

/I’“‘(k)l?“J’( -k)(O,

0, 0, ri, 0, 2/g.

Equation (21) shows that only two “transversely isotropic” components of the effective compliance tensor MC, differ from the corresponding matrix values ; inversion yields

1814

v. P.

whose

L,,.

from

components

the invasion

In

the

in

equations

(II,

Part I).

equation.

the efrective

density

the cracks the

notation)

I!.

coincides

with

gi\,en

Now

weighted

is \I?, just

as

will

be

apparent

cavities

tilled

I

reduces

equations the

for

lirst

ol

:

the stress

to the statement

non-viscous

) when

to

;i

of a crack,

(in

-/a).

the tensor

ofthis

that

II.

isotropic”

[I+

whose

average

an

result

3-asis

’ is

- L,,)]

s, (I,,

premultiplicd

The

it is necessary

frame.

of

tensor,

3 of Part

fluid,

“transverscl~

rotated

the evaluation

in Section

of by

of self-consistency

with

(I I,

require

/~(/a)/~(

l’orms

is provided

cracks.

equations

(11.1)

limiting

point

the momentum.

to the faces

over

as

oricn-

(L, ~ I_,).

by

simplilic\

upon

LISC

( I I). P-m~vcs,

the relc\ant

d + 0. only

It follous

directly

nonlinear

equation

that,

relations

since apply

Now

for

coniponcnts

the constant

Equation

corresponding

(11,14)

reduces

of

(II. I) are (11.14) and (II.1 5) to the non-xro

twin

ol‘ordc~

only

;i single

to

would

from L,,‘is

(93)

and

which

isotropic,

(23)

,I!,,

has

k,, = 1~~; thcrc

that to

be determined.

/I(, = K,,+~~L,,:?I.

is, in elrect. Here.

it should

I,, = /~;,,-2ji,,,‘3

and

p,,

be borne =

in

Siniilai

pli.

to L,.

The scheme

preferable

l‘roni

S-waves,

Ihe relevant

employed

then the solution

There

way,

15) gives

(I I,

option

of

explained

’ in (I I) contributes.

mind

oriented.

starting

for aligned

as

form

equations

form first

simple

S;I~K

requirement

with

us

are tread

was

in the limit

and

The the

deals

by the fktor

calculation

Considering (S

which

the normal

of the limiting

and.

in the

arc‘ randomly

required.

expresses

limiting

C’, ul.

, = (l?r;,. rc,. K,. I;,, 0,O). Relative

in (I I).

tations. This

cracks are

which

the second

When

the II

separated

\

( 16).

formula

that

cast

developed

to address

arc not

S\IYSHLYAI

in Part

of(24).

with

be to take/i,, an

II rcquircs

r;,,+4~(,,:3 speed

additional

(11.18). This rcduccs

is

the solution

taking

= JC?throughout.

to tix the ell‘ective is. however,

equation

the This

value

to

oi‘(22).

(23)

already

l’ound.

was not taken

up.

for

P-waves.

and

An alternative

because

ir cccmcd

of P-waves. possibility

: this is to solve,

simultaneously,

the

Self-consistcn1analysis

1815

of waves

two equations that define the active stress components for the two wave types, namely (22) and (24). Then. a .singk effective medium is generated, which supports, “selfboth wave types. This possibility was not discussed for general consistently”, inclusions in Part II. because there would still be a need for two different densities. Results are given later, both for the scheme as employed in Part IT and for this alternative, which (at least as developed here) is specific to cracks.

The limiting form of the first of equations full equation is

(11,l) is now needed.

L,,e,) = (t2-~.IL2(h(kn)lz(-kn)[l-S,L,,]

when L, = 0. The

‘>)e,,.

(2%

The limiting form of the inverse tensor given in (25). relative to rotated coordinates attached to the crack, is easily obtained from the related tensor (I 8). Then. considering P-waves. equations (11.14) and (11-15) reduce to II ,) = 112+

4rru’N, 3

(4~2p,,(4ji-/ITi)1(3., ; (I -G)u-)

,

and I,, = 1, +

~TUI’N, (2/‘1/-‘r,(n-4~)/(;.p 3

For S-waves, equation p<, = /)2 +

; (I -u2)z1?)

(11,19) gives

47m’N, ~~3 p2po(hl(&;

(I -21r’)z4~)+l,l(j.~;4zr~-_3u~+

1)).

(28)

As in the case of fluid-filled cracks, the scheme developed in Part II requires the solution of (26) and (27). followed by the solution of (28), with n,, taking the value already found. Alternatively, a single effective medium can be defined by solving, simultaneously. (26) and (28).

5.

ALTERNATIVE DERIVATION

It is possible to formulate the problem of waves propagating through a medium containing cracks directly. The mean stress in the medium is the mean stress in the matrix, since the cracks support at most finite stress (it is non-zero when they are filled with fluid) and have zero volume. The mean stress is related linearly, through the matrix moduli L?, to the mean strain in the matrix, but this differs from the overall mean strain, which contains a finite contribution from the relative displacements of

v. I’

IXlh

Shl\rSHl

Y II

\

<‘I

C/l

crack. centred at x’. L end with normal

the crack faces. Any individual with it it strain

v’, has associntcd

b)ci(v’*(x--w’)),

;(b@c’+v’&!

aherc ci is the Dirac delta-function and the relative displacement b of the crack litccs is regarded as defined over the entire plane of ~hc crack. but difl‘crcnt from ycro oni! \vithin ;t radius (I ofx’. It follows that the mean strain in the matrix is gi\cn 3s (CL,,,

(20)

= (c> - (C>,,,,J,\.

The formula for the contribution to the mean strain from the cracks is given bcio~. The condition for self-consistency ih no\\ (0)

= L,(e)

= L,(e)

(30)

11,. ,,,I\.

The relative displacement of the crack faces has to be calculated. This is done. approxitnatcly. by solvin g the problem of ;I single crack. in ;I matrix with tensor 01‘ nioduli L,, and density J):. scattering an incident M;t\‘e u,,. In this approximation. the scattered field can be expressed in terms of b via an integral inl;olving the Green’s function. To a\,oid complicated notation. the I‘ormulne are given for the cast that the crack is centred at the origin and lies in the plane .Y~= 0; the general case can bc deduced by rcgardinp this description to be rcluti\c to ;I set of local coordinates. ‘l‘hc stress associaled \vith this scattcrcd field is then dj,, d.t,, S,(x-y)L,(b(y)

0 v’)

(31)

and this must gcncrate It-actions on the crack faces that cancel those associated the incident wave. Thus. b satisfies the integral equation

lim v’*L,,

,,+*n

Its solution a spheroid,

dj., d.t*?S,(x-yy)L,(b(y)

(33)

jj (I, 0)

is approximated by employing, as was done in the case of scattering a form which is exact in the static limit: b(x) = Acxp(iL~*x’)(l

and choosing

0 v’) = -v’+L,,e,,.

with

-_Ix--s’~‘~c~~)’

from

(33)

‘.

A so that (31) is satisfied in the mean sense v’* L,,(,?,),L,,

*v’A = -~trl”.L,,e,,/l”“(/\).

(34)

where (S,),

= ;:, jjd.\-,d.\-ll(;;,

j!‘,

,, cl.,,, d.t,,S,(x-y)(I

-~xl’+)’

‘(I -ly12/r/‘)’

‘.

Sell-consisten

1817

analysis of waves

the integrals extending over the crack surface, with II’“‘(~) given as in (14). The notation employed here is deliberate : some analysis is required but it can be verified that (s,), is precisely the tensor introduced in (7). Once the equation for A is solved. the self-consistent condition (30) yields an explicit equation for &. The only formula

FREQUENCY

L

4

/

i

1

I

0.06 t

_01 0

0.5

I

1.5

2

2.5

3

FREQUENCY FIG I.Normalized plots of the variations of wave speeds and attenuations against liequency (X,,,tr. ~hcrc k,,, is the P-wavenumher appropriate to untracked matrix material). for quasi-P waves [(a) and (h)] and quasi-SR waves [(c) and (d)]. Solid curves arc for empty cracks. and dashed curbcs for fluid-tilled cracks. The crack density is such that N,tr3 = 0.1. The cracks are aligned and the direction of propagation of the waves makes an angle 45 with the crack normals. Corresponding curves are shown for spheroidal ca\itics with ;~spcct ratio 6 = 0. I. for comparison. Dash dot curves relate to empty cavities. uhile the dotted curves arc for cavities filled with fluid. Continued orrrh~/.

0.96

0.94 --

0

,’

II.5

I

2

1.5

2.5

FREQUENCY

(d)

0.0x-------

y

---

?

.~~-,

_-. .’

‘. ‘,

,’

0 07 -

‘, ‘,

n.Gfi -

0.02 -

1

0.5

1.5

2

FREQUENCY F‘KI.

I. ~‘oI/li/r/fld

not shown so far is that for the contribution from the cracks to the mean value of the strain. In the approximation being employed. this is

(c>LI.Kh\ where the mean order tensol

value

=

2n(zNi

(/7""(/i)h""(

--k)T

is taken over crack orientations

and

‘)C!,,, T

~xpresents

(36) the

socond-

Self-consistent

analysis

of waves

T = v’~L,,(~,),L,,~v’.

1819

(37)

In the case of fluid-filled cracks, the normal component of b is set to zero, (32), (34) are required to hold only for their shear components and T reduces to a 2 x 2 matrix. These formulae reproduce exactly those given in the preceding two sections.

4

0.970.9650.960.955-

FREQUENCY

(b) 0.04

0.03

2 0.0250

FREQUENCY FIG. 2. Plots of variations of wave speeds [Fig. 2(a)] and attenuation [Fig. 2(b)] of P-wave for randomly oriented fluid-tilled cracks. Normalizations and the number density are as for Fig. I, Figures 2(c) and 2(d) give the corresponding information for S-waves. The solid curves arc obtained from the first prescription. in which different tensors L,, are employed for P- and S-waves; the other prescription, which employs a single tensor L,, for both P- and S-waves leads to the dashed curves. The dotted curves are as obtained in Part II, for randomly-oriented fluid-filled spheroids with aspect ratio S = 0.01. Continuedowrfeuf.

0

0.5

1.5

1

2

2.5

2

2.5

FREQUENCY

(1) 0

0.5

1.5

3

FREQUENCY t’lc,.

2

c ‘iullf/ir/l~~l.

As ;I first example. Fig. I shows some plots of normalired wave speeds and attcntiations. comparing when the cracks WC empty and fluid-filled. The cracks are takcll

aligned, and the direction of propagation the direction of the crack normals. Figures

of the waves makes an angle of-45 with I (a) and I(b) show. respectively, quasi-f’ w\e speeds (normalized to the speed of the corresponding wax in Llncrackcd matrix material). and attenuation (in the form Im i/if/j). plotted against /c,,,u. where li,,, is the

as

S&consistent

P-wavenumber properties of is chosen so empty cracks, l(c) and I(d) some curves spheroids at

analysis

1821

of waves

for the untracked matrix and u is the radius of the cracks. The the matrix are taken to be appropriate for rock, and the number density that N,a3 = 0.1, as in Figs 5 and 6 of Part I. The solid curves are for while the dashed curves relate to when the cracks are fluid-filled. Figures provide similar information, for quasi-SR waves. The figures also show which are repeated from Part I. The dash dot curves relate to empty the same number density. and the dotted curves are for fluid-filled

0.98 0.96 -

0 0.92 -

0.82

0

0.5

1

1.5

2

2.5

3

FREQUENCY

(b)

o’25 r

FREQUENCY FIG.

3.

As for Fig. 2 but for empty cracks. The dotted curves, shown for comparison, spheroidal cavities with aspect ratio 6 = 0.1. Continued orerleqf.

relate to empty

0

0.5

I

2

1.5

2.5

3

FREQUENCY

Cd)

O.lhr

-7

-.~-

7-m

-7

--,

spheroids. The aspect ratio ii of the spheroids is 0.1. III the case of quasi-P WIVL‘L th-c is ;I marked difrerencc. both for WIVC speeds and attenuations. between Iluidfilled and empty cracks. Also. while the results for empty spheroids closely approximate those f’or empty cracks. those for fluid-filled spheroids show significant dill‘ercnces. Convergence of the results for fluid-filled spheroids to the crack limit can be demonstrated. but smaller values of 6, of the order 0.001. arc required. The dilYercnccs are Icss pronounced for quasi-SR waves. Figure 2 treats randomly-oriented. fluid-filled cracks in rock, again at number

IX23

Self-consistent analysis of wilvcs

density such that N,a’ = 0.1. Normalizations are analogous to those for Fig. I. Figures 2(a) and 2(b) show, respectively. wave speed and attenuation versus frequency for P-waves, and Figs 2(c) and 2(d) give the corresponding information for S-waves. Two alternative prescriptions were mentioned in Section 4. The solid curves were obtained from the first prescription. in which different tensors Lo are employed for P- and S-waves; this was also employed in Part II for randomly-oriented spheroids. The other prescription, which employs a single tensor L,, for both P- and S-waves, led to the dashed curves. Their close agreement indicates a lack of sensitivity of the results to the assignment of values to “non-active” stress components. The dotted curves were as obtained in Part II, for randomly-oriented fluid-filled spheroids with aspect ratio 6 = 0.01. Figures 3(a)-(d) give corresponding results for empty cracks. The relate to empty cavities with aspect ratio dotted curves, shown for comparison, d = 0. I. These reinforce the observation from Fig. I that empty spheroids approach the crack limit more rapidly as (S--t 0 than those filled with fluid. Results for cracks obtained from the two alternative prescriptions difl‘er more than in the case of fluidfilled cracks, but the difference remains insignificant. in relation to other approximations in the modelling. Qualitatively. the results for cracks show the same features as have already been noted for spheroids in Parts I and II. The most important are the large fluctuations, both for phase speeds and attenuations, when the half-wavelengths are of the same order as the crack diameter. When it is known in advance that the case of cracks is the one of interest, the formulae presented here permit the setting up and solving of fewer equations than are required for spheroids. In addition. when the cracks are oriented randomly, there is the possibility (embodied in the “second” scheme) of generating just one “effective tensor” Lo which supports wave of both P and S type.

ACKNOWLEDGEMENTS Thanks arc due to Mr P. Bussink (Shell Laboratory. Rjjswj;k) for pointing out a scaling error in an early version of some of the figures. identified from his indcpcndent implementation of the scheme presented in this paper. Support from the Commission of the European Communities. contract number CIl*-CT9I-0883(SSMA) and PAPIID.DGAPA,UNAM. project number IN 103691 is gratefully acknowlcdgcd.

REFERENCES BUIIIANSKY, B. and O’CONNELL. R. J. (1976) Elastic moduli of a cracked solid. ltzt. J. Solirl.~ St~uc’l. 12, 8 l-97. CRAMPIN. S. (1984) Effective anisotropic elastic constants for wave propagation through cracked solids. G’ropl~~x. J. R. Astro. Sot. 76, 135-145. FOLEY, L. L. (I 945) The mutiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Plr~s. Rw. 67, 107?119. GROSS. D. and ZHANG, CH. (1992) Wave propagation in damaged solids. hf. J. Solids Sttxc/.

29, 1763 1779. A. (1979) Elastic moduli of a non-randomly 137-154.

HOENIG.

cracked

body.

htt.

J. Solids

Srtwct.

IS,

1824

v. I’. SraSllI.l.AI.\

<‘I rrl.

HCJI)SO~~.J. A. (19x1) Wave speeds and attenuation of’ clast~c waves in material containing cracks. Gcopl/r.c. .I. R. Asrro. Sock. 64, I33 150. KI~~UCIII. M. (IYXI) Dispersion and attenuation or elastic waves due to multiple scattering from cracks. Ph~x Em-t/t Plrnwror~~ hriwio~r 27, IO0 105. LAX. M. (lYS2) Multiple scattering of’ waves. II. Ett‘cctivc field in dense systctns. /‘/tj..s. Kc-. 85, 62 l--629. PIA~:. M. (1979) Attenuation of’ ;I plane comprcssional w;t\e by a trandom distribution 01‘ circular cracks. //I/. J. b’/tg~~q SC.;. 17, IS I 167. SAHINA, F. J., SMYSHLYAEV. V. P. and WILLIS. J. R. (lYY3) Sell-consistent analysis ol‘wavcs in a matrix-inclusion composite-l. Aligned spheroidal inclusions. .I. Medt. P/t!..\. Solids 41, 1573m 1588. SMYSHLYAEV. V. P.. WILLIS. J. R. and SABINA, F. J. (lYY3) Sell-consistent analysis ol‘wavcs in a matrix-inclusion composite I I. Randomly oriented spheroidal inclusions. J. !Mcdt. 1’1t.1~~. Solids 41, I 58% 1598. WILLIS. J. R. (IYXO) A polarization approach to the scattering of elastic w;tvcs: II. Multiple scattering from inclusions. .J. :lilcc,h. ~%JY. .+dit/.s 28, 307 327. ZIMNG. CH. and ACFUNIACH. J. D. (IYYI) Ef??ctivc wvc vclocily and attenuation in ;I mabial with distributed penny-shaped cracks. /r/i. J. So/iti.c S/r.~tt.f. 27, 75 I 767. ZIIANG. CH. and GKOSS, D. (lYY3u) Wnvc altenuation and dispersion in randomly crackcd solids 1. Slit cracks. //I[. J. E~/17q Sci. 31, X41 X5X. ZHAM;, CH. and Gtwss. D. (lYY3b) Wave attenuation and ciispcrsion in trnndomly cracked solids--- II. Penny-shaped cracks. I/I/. J. G7qr7g SC?. 31, X5Y X72.

This Appendix lists the cxprcssions which define constan& this part, &~~ILISC they rclatc spccilically to cracks.

introduced

for lhc lirst time tn

i;=l: ri=

(A.1) where