Effective elastic properties of an anisotropic material with arbitrarily oriented interacting cracks

J.

Med~. P/Iw. So/id\.

Pergamon

Vol. 42. No.

4. pp. 56 I 584. I994

Copyright( 1994 Elsevier Scicncr Ltd Printed inGreatBritain. All rights reserved

0022 5096:94$6.00+0.00

0022-5096(93)E0013-G

EFFECTIVE ELASTIC PROPERTIES OF AN ANTSOTROPIC MATERIAL WITH ARBITRARILY ORIENTED INTERACTING CRACKS C. MAUGE Radionics, Inc., Burlington, MA 01803, U.S.A.

and M. KACHANOV Department of Mechanical Engineering. Tufts University, Medford, MA 02155. U.S.A.

ABSTRACT EFFKTIVE MODULI of a two-dimensional matrix of general anisotropy, containing an arbitrary orientational distribution of cracks, are found. For non-interacting cracks. the results are rigorous and are given in closed form. For interacting cracks, extensive computer experiments-numerical solutions of the interaction problem for a number of sample crack arrays-show that the approximation of non-interacting cracks remains accurate at high crack densities, provided that mutual positions of cracks are random. This is explained by cancellation of the competing interaction eirects of shielding and amplilication similarly to the cast of cracksin the isotropic matrix (Kachanov. 1992, A,IJ~/.Mcclt. Ret.. 4.5, 304- 335).

1.

INTRODUCTION

IN THE TWO-DIMENSIONAL case, the effective moduli of anisotropic materials with cracks were first analysed, probably, by GOTTESMAN et al. (1980). They considered an orthotropic matrix with one family of cracks parallel to one of the principal directions of orthotropy and used the self-consistent scheme for finding the effective moduli. The same problem was considered in more detail by LAWS et al. (1983) in the framework of the self-consistent scheme and by HASHIN (1988) who used both the self-consistent and the differential schemes. [The applicability of the self-consistent and differential schemes to cracked solids in general was discussed by KACHANOV (1992. 1993)]. MAUGE and KACHANOV (1990) analysed an orthotropic material with interacting cracks that are parallel to one, or both, of the orthotropy axes and have arbitrary mutual positions ; they produced a number of numerical solutions for sample crack arrays. DENG and NEMAT-NASSER (1992) analysed periodic arrangements of parallel cracks in an orthotropic sheet. In the thrre-dimensional case, LAWS et al. (1983) considered, in the framework of a self-consistent scheme, a transversely isotropic material with one family of pennyshaped cracks parallel to the plane of isotropy. Note that the solution of this problem (either for non-interacting cracks or in the framework of the self-consistent or the

C.

562

MAUGE andM. KACHAWV

di~er~ntial schemes) can be obtained front the results of HOEMG (1979) ; although he considered parallel cracks in the initially isotropic material, the self-consistent scheme used by him places a representative crack in the effective material which is transversely isotropic. In all the mentioned works, it was assumed that (i) the matrix is orthotropic: (ii) the cracks are parallel to one of the axes of orthotropy. For arhitruri/~ o/,ientetlcracks in an anisotropic solid, KACHANOV(1980) suggested. as a possible approach, to construct the elastic potential in stresses in terms of two parameters : the stress tensor 0 and the crack density tensor a :,f’=,f’(a, a), by entering G and Q through their simultaneous invariants with respect to the group of symmetry of the matrix [as a generalization of his work for the isotropic materials with cracks for which .k’(a,a) is expressed in sinlultall~ous ~~~variallts with respect to the full orthogonal group]. This approach, that has been further developed by TALREJA(see, for example, 1990), has the disadvantage of introducing a large number of adjustable parameters-coefhcients at the mentioned invariants. The present work eliminates the need for such constructions, since the elastic potentials are constructed explicitly. The case of arbitrarily oriented cracks was also discussed by SUMARAC-ct al. (I 992). but the actual results were given only for the isotropic matrix; the authors also suggested that one handle the case of an anisotropic matrix by a “quasi-isotropic approximation” [their formula (3.25)], which seems to be based on contradictory assumptions: for each crack, matrix stiffnesses in the direction normal and parallel to the crack are different, but in the fixed (“laboratory”) coordinate system, the matrix is assumed to be isotropic. Below. we analyse the general 2-D case when the matrix anisotropy is arbitrary and cracks have arbitrary orientational distribution, as well as arbitrary mutual positions. Some preliminary findings are reported by MAIJCXand KAC’HANOV(1992). In the case of non-interacting cracks, the results are given in closed form and are quite simple. For interacting cracks, the moduli are obtained analytically for any particular crack arrangement; based on these results, extensive computer experiments solutions of the interaction problem for a number of sample crack arrays-have been performed. The results of these computer experiments are used to examine the validity of various approximate schemes. As is well known (see a large number of works, starting from the early 197Os), the overall strain E in a solid with N cracks can be represented in the form (written below for the 2-D case) E=S:o=

00)

+ (b)n)‘“‘/‘“’ = (S”+AS)

:G

(1)

where S is the effective compliance (to be found), S” is the compliance of the matrix without cracks, n”), 21’/” and (b)‘“’ are the unit normal vector, length of the kth crack and average displacement discontinuity (COD) vector for the kth crack; A is the averaging (representative) area and a colon denotes contraction over two tensorial indices (in the 3-D case, the representative area A changes to a representative volume, and crack lengths to crack areas). The problem of finding the additional compliance AS due to cracks is thus reduced

563

Anisotropic material with arbitrarily oriented interacting cracks

to finding (b)‘“’ in terms of the average stress 0. [The latter can, under certain conditions, be identified with the externally applied stress; see discussions of HASHIN (1983) and KACHANOV (1992).] We find it important to analyse the problem in terms of the elastic potential in stresses (complementary energy density). The analysis in terms of elastic potential establishes the proper parameters of crack density (see the discussion of Section 3) and provides a unified description for various orientational distribution of cracks. The potential is a sum of two terms :

*

n ts * (b))‘k’l’k’

=,f”(a) + 4f

(2)

where,f’ is the potential without cracks and df is the change off due to cracks. It follows from (2) that the group of elastic symmetry of the effective properties is an intersection of the group of symmetry off0 (elastic symmetry of the matrix without cracks) and the group of symmetry of 46 If the matrix is isotropic, then the overall anisotropy is determined solely by Af [and is the one of orthotropy ; see KACHANOV ( 1980, 1992, 1993) for a detailed discussion]. If, as another example, an anisotropic matrix contains one family of parallel cracks arbitrarily inclined to the anisotropy axes, then the effective properties possess no elements of symmetry (,f” and Af have no common symmetry elements).

2.

RELEVANT RESULTS FOR ONE ARBITRARILY ORIENTED CRACK

2.1. General representations,for

a 2-D anisotropic solid

Stresses and displacements in an anisotropic elastic solid can be represented in terms oftwocomplexstressfunctions4(z,)and$(z,),wherez, = _u+~,~,z~ = x+~~~.The complex parameters p,, p? and their conjugates cl,, pz are roots of the characteristic equation s, I I ,P4-22S,,,2p3+(2S,,22+S

12,2)P2--~2212P+s2222

=

(3)

0

where S,,, (i, j, k, I = 1, 2) are elastic compliances in the coordinate system x, I:. Positive definiteness of the strain energy function implies that ,u,, p2 cannot be real (LEKHNITSKY, 1950). We denote pLk= xk +$, where CQ, PA are real constants and Pk > 0. The stresses and displacements

are expressed

c,., = 2 Re [dcb’@ ,) +&Y(z2)l,

as follows :

cVV= 2 Re L&(2,) + $‘(.Jl,

c.,~ = -2Re[p,@(=J+pL?ICI’(Z2)1 ul(x,,v)

=

~R~~,~GJ+P~$(--~)I,

(4)

~JGY) = 2Re[q,4(z,)+q2@J1

1

wherep, = SI111~~--S111211k+S1122, qk = ~L~‘(S,,?~~~-S?~,?~~+S~~?~). In the case of orthotropy (coordinate of orthotropy), (3) becomes biquadratic

axes are aligned :

with the principal

directions

564

C.

and has purely imaginary

MAWE and M.

KACHANOV

roots

that can be expressed in terms of “engineering constants” (Young’s moduli E,, E,, the shear modulus G, 2 and Poisson’s ratio \I,~) by finding ~1~~1~and ,u, +,u2 from Viete’s theorem :

If the matrix is orthotropic, the parameters 11~for an arbitrarily oriented crack arc obtained from the parameters pf-(purely imaginary) roots of a biquadratic equation corresponding to a crack parallel to one of the orthotropy axes by a simple transformation (LEKHNITSKY, 1950) : ,u:’cos cp-sin A =

47 (8)

cosy,+pFsinq

where cp is the angle between the crack line and the .v,-axis of orthotropy. This leads to considerable simplifications for orthotropic matrices since ,u:’ can then be taken in the principal axes of orthotropy and arc. therefore. given by simple expressions (7).

2.2.

Cruck oppniny displmwnen

ts

Displacements of the faces of a crack embedded in a matrix of arbitrary anisotropy can be derived from the known solutions of anisotropic elasticity (LEKHNITSKY, 1950; SAWN, 1951). They are briefly summarized in formulas (9))(12). In the coordinate system .v._r of the crack (crack occupies 1.~1< 1. J’ = *O). they arc as follows (superscript “0” refers hereafter to the moduli of matrix without cracks). For the uniform normal traction (of unit intensity).

For the uniform

shear traction

24~(s,j9

=

*.s:‘,,,

(p,

21’(.VJ’)

=

is~,,,(c(,pl+c(~~,)/J’l-

(of unit intensity),

+~L)/&.w+[S(:,

, ,(cc,

+x1)

-s:‘,

,21.x (10)

where

x1.? and /I,.2 are real and imaginary

x’/12+[S:‘,,

parts

,(~,~z~p,~~z)-s:‘,21].Y

of ,u,.?. and where the identities

565

Anisotropic material with arbitrarily oriented interacting cracks

= sY222(~1Pz+azPl>/[(c(:+Bf)(a:+8:)3and ~?III(~G-PIPJ -S:‘,22 = s~,,z-S~~?z(a,a2-B,B2)/[(~:+~:)(~~+B:)1 are used.

G,,,(x,B~+QBJ

The UWYKJ~ CODS (normal, for the uniform

normal

traction

(h,)

and shear, (h,))

are, therefore,

(of unit intensity),

(h”) = sL2 (bz) = S:‘,,,(c(,B2+~zlj,)711/2,

(&; I

for the uniform

shear traction

as follows :

1

742; +a2$Bi 2

(11)

(of unit intensity),

where, in accordance with Betti’s reciprocity theorem, (h,) induced by shear loading is equal to (h,) induced by normal loading. As seen in the text to follow, coupling of nmtes I and II, i.e. the fact that normal (shear) traction on a crack produces shear (normal) CODS, has a significant impact on the effective properties. If the matrix is orthotropic and the crack is parallel to the orthotropy axis x,, then modes I and II are uncoupled and the average CODS due to uniform normal (or shear) traction p (or t) can be written in terms of the “engineering constants” of the matrix as follows :

(13)

2.3.

COD tensor of u cuuck

Analysis of the effective properties can be conveniently done in terms of the COD tensor (KACHANOV, 1992, 1993). For a crack of length 21 in an infinite plate loaded by a uniform remote stress 0 the COD tensor B is defined by the relation (b) = n*a*B.

(14)

In the crack coordinate system (t, n) where t is a unit vector tangential to the crack (the directions oft and n are chosen in such a way that t, n form a right hand coordinate system : t = cos cpe, + sin qe2, n = -sin cpe, + cos cpe2; the dyadic products nt or tn remain invariant with respect to the choice of direction of n) B = B,,nn + B,,tt + B,,nt + B&n

(15)

where the off-diagonal components B,,,, B,, characterize coupling of the modes (absent in the case of the isotropic matrix). Betti’s reciprocity theorem implies that B,, = B,,,, so that B is symmetric. Components of B depend on the matrix compliance So and on the orientation of the crack with respect to the matrix anisotropy axes [which enters (16) through CQ,ljkreal and imaginary parts of ~~1. As follows from (1 1))( 13), they are given by formulas

566

C.MAUGE andM.

K.ACHANOV

If the matrix is orthotropic and the crack is parallel to one of the orthotropy axes (no coupling of the modes, B!,,,= 0), then, the components of B are expressed in terms of “engineering constants” as follows :

For a crack arbitrarily oriented in an orthotropic matrix, B can be found in the closed form by utilizing the solution given by LEKHNITSKY (19.50) and SAWN (1951) (the crack line forms an angle (p with the .Y, principal axis of orthotropy.

B,, = C( I -D

cos 2p)f

B,,,, = C( I + D cos3pjf B,,, = CD (sin 2~)/

(18)

:

where

We note that the dependence of H on the crack orientation cp is remarkably simple (particularly if it is compared with the general orientational dependence of the elastic moduli of the orthotropic solid, see Appendix}. Another interesting observation is that the compliance of an arbitrarily oriented crack is determined by only two combinations, C and n, of the (four) matrix moduli. [The underlying reason being that the COD is determined, aside from the crack orientation q, by only two parameters /Zi,,-roots of the biquadratic equation (5).] An important property of the B-tensor is that the strength of the coupling effect of (the ratio of B,, to B,,, B,,) depends on the ratio EYJE; only and is independent the other matrix moduli. The dependence of B on the matrix compliance S” is non-linear ; it becomes linear only if the matrix is either isotropic or has cubic symmetry [see formulas (21) and

G91. Note the impact of coupling

term &, on the overall compliance

: for a crack of any

Anisotropic material with arbitrarily oriented interacting cracks

567

orientation cp, the coupling produces a stiffening (softening) overall effect under uniaxial loading parallel to the stiffer (softer) direction of the matrix. The coupling can be neglected in the case of weak to moderate anisotropy of the matrix; if the ratio Ey/E! is of the order of 1.5 or less, then ]I&,] is one order of magnitude smaller than 1B,,I and IfInnI. As any symmetric tensor. B can also be represented in the principal form B = C(l +D)l&,&,+C(l

-D)IQ&

(20)

where C( 1 ill) are the eigenvalues of B. As follows from (I 8) and (19), these eigenvalues do not depend on crack orientation cp. The principal directions e,, Gz are the directions of no coupling, i.e. the only additional strain due to a crack produced by a uniaxial loading in the direction of i, (or iZ2) is tit (or ~22). These directions coincide with t, n in the case of the isotropic matrix. For an anisotropic matrix. the cp is remarkably simple: the angle dependence of 6 ,, 6, on the crack orientation between 6, and the crack line t (counted from t to iZ, counterclockwise) is exactly equal to cp. In the case of the isotropic matrix, B = (d/EO)l

(21)

[for plane stress ; E” is to be understood as E”/( 1 -vi) for the case of plane strain] so that (b) is collinear to the vector of applied traction n *CJ[as discussed by KACHAN~V (1980. 1992), this collinearity leads to orthotropy of the effective properties for an arbitrary orientational distribution of cracks]. In the case of cubic symmetry (Ey = Ei, but Gyz remains an independent constant), formula (18) yields that B does not depend on the crack orientation cp and, similarly to the case of the isotropic matrix, B is proportional to I :

4, = &

= 4

(24

B,,, = 0. This is a remarkable simplification; is, to within a constant coefficient, orientation in the isotropic material.

3. 3.1.

Structure

extra strain due to an arbitrarily oriented crack the same as the one for a crack of the same

THE APPROXIMATION OF NON-INTERACTING CRACKS of the elastic potential

This approximation is analysed in detail, since, as shown in Section 5, it remains accurate at high crack densities (provided mutual positions of cracks are random) ; besides, it constitutes the basic building block of various effective media theories. In this approximation, each crack is embedded into the homogeneous stress field b. Substituting (b) = n-o * B into (2) yields the change in elastic potential due to cracks

568

C. MAUGEand M. KACHANOV

(23)

[Equivalently, the change of elastic compliance due to cracks is AS = l/A c (InBn)‘“‘, where the appropriate tensorial symmetrization is assumed.] Since the B-tensor for an arbitrarily oriented crack is constructed above in (16)-~ (I 8), formula (23). together with (18), explicitly express the elastic potential (and. thus, the effective moduli) for all orientational distributions of cracks in a unified way. The general result (23) is specialized to three particular orientational distributions of cracks in subsections 3.3 3.5.

These two approximations arc often considered as equivalent, and their names are used as synonyms. In fact, they are not generally equivalent. In the approximation of non-interacting cracks, each crack is placed in the undamaged matrix; therefore, the potential 47’ is inversely propor;ional to the matrix stiffnesses. see (23), (I 8) and (19). As a result, equating ,fo+Af to the potential,f’of the effective medium yields the following structure for any of the effective stifinesses A4 :

where p is the appropriate crack density parameter and the constant C depends on the particular modulus A4 considered and on the orientational distribution of cracks. The approximation of small density of cracks (“dilute limit”) is based on linearization of (24) M

M,,

= 1-

cp.

It appears that there is no need to linearize (24)-the only result achieved by such linearization is a reduction of the range of applicability of the approximation of noninteracting cracks. Indeed, when cracks have random mutual positions, predictions of the approximation of non-interacting cracks remain accurate at high crack densities (due to cancellation of the competing effects of shielding and amplification, see Section 5). Linearization would render this approximation inapplicable at relatively small crack densities. On the other hand, iflocations ofcrack centers are non-random, interactions can be strong (and the approximation of non-interacting cracks inapplicable) at vanishingly small densities (see KACHANOV. 1992). Thus, the “dilute limit” appears to be an unnecessary construction. In the text to follow, we are using the term “non-interacting cracks” (and refrain from using the terms “small crack density” or “dilute limit”).

Anis~tropic

3.3.

material

with arbitrarily

oriented

~~t~l~t~~~pi~ ~l~t~i.~ with t~~~~,~~~lil~~.~ qfcrwks

interacting

cracks

569

~~r~~l~~ to the (~~thot~~~p~ LI.WS

In the case of two families of cracks normal to the principal directions x,, ,x1 of matrix orthotropy, the tensor B (h) for a crack normal to xk is given by (17). Normal and shear modes on cracks are uncoupled, B,,, = 0, and the potential takes the form

4f =

“‘o~,+~?~Bj,~‘o~,+(l/2)[p,Bj,“f/,2B1,2’]((~:2+(~~,) P , B,,,,

(26)

where p ,, p3 are partial crack densities for cracks with n = e, or e2. As follows from (26), the change of elastic compliance due to cracks AS = p ,e ,B’ “e, +p 7e4B”)eL and the effective “engineering constants” are

GI2

G2 ---=[

l+zG:l

(

G!I2

The results (27) were given by MAUGE and KACHANOV (1990). This case was also considered earlier by G~TTESMANet ul. (1980) (whose expression for GIZ contains a n~isprint) and by HASHIN (1988) in the framework of the self-consistent and the differential schemes. (As discussed in Section 5, these schemes substantially overestimate the effective compliance.) Their results can be transformed to (27) if the mentioned schemes are changed to the approximation of non-interacting cracks. Note that, of the four effective moduli, three [given by (27)] are affected by cracks and the fourth one is a constant of the matrix : v, JE, = vyz/Ey [similarly to the case of cracks in the isotropic matrix (KAC’HANOV,1992)]. Cracks normal to the stiffer direction of the matrix produce a higher impact on stiffness reduction than cracks normal to the softer direction. This dependence of the crack’s “influence” on its orientation is strongly “asymmetric” : enhancement of the crack’s impact (as compared to the case of the isotropic matrix) for the orientations normal to the stiffer direction is sLlbstantiaIly stronger than its reduction for the orientations normal to the softer direction (Fig. I). This “asymmetry” has important physical consequences : randomly oriented cracking in an anisotropic matrix leads to a gradual disappearance of anisotropy (Subsection 3.4). Consider a numerical example (relevant for glass fiber reinforced plastics): EI’/E’?’= 4.1, E’;/2G:> = 5, Y;, = 0.068, v:‘, = 0.277. For parallel cracks normal to the softer direction, E:/Et = (1 + 5.77p,)-- ‘-a small change in the coefficient at crack density {I. as compared to E/E” = (I +2np) ’ for the Lsotropic matrix with parallel cracks. For cracks normal to the stiffer direction, the change in sensitivity to p is much more pronounced : El/E:’ = (1 + 1I ..5p,) ‘. Thus, anisotropic matrices are more “sensitive” to cracks normal to the stiffer direction. Note a somewhat similar conclusion on the impact of matrix anisotropy on crack

interactions. Tnteractions between cracks normal to the stiffer (softer) direction of the matrix are enhanced (weakened) by the matrix anisotropy [see Mnuc;~ {1990), and MAUCE and KACHANOV f 1390)], The effective elastic properties are orthotropic, but, unlike the case ofcracks in the isotropic matrix, the potential &” cannot be represented in terms of the (purely geometrical) crack density tensor a, that is. ~%f’+ ef(~,~). The underlying reason is that cracks of different orientations contribute equally to a, whereas their actual impact on the eflective moduli depends on their orientation with respect to the matrix anisotropy axes. In the case of cubic symmetry (E:’ = Ei), the matrix material is characterized by three independent mod&, E:l = Ei, 11:‘~= vs, and G:‘,. For a crack of arbitrary orientation, coupling between modes vanishes, &;,, = 0 and Bj,,‘,’= RI,:’ = 8),” = El:’ = E,,,,. Then, similarly to the case of the isotropic matrix, Aj’ is expressed in terms of the crack density tensor &‘= B,;,o. -c : a. This case differs from the one of matrix isotropy in that the potential .I‘” of the matrix without cracks has cubic symmetry. Therefore, the group of the overall symmetry in this case is the intersection of the group of cubic symmetry of the matrix and of the group of bob-thotropi~) symmetry of the symmetric second rank crack density tensor cx.The overall compliance has no elements of symmetry (unless cracks are parallel to one, or both, of the axes of matrix symmetry). We note that, using the identity nn+tt = I, one can always represent B in the form B = B,,I-t-(B ,,,,-B,,)nn+B,,(nt+tn).

08)

For the orthotropic matrix with cracks parallel to one of the orthotropy axes? B,,, = 0 and the third term of (2X) vanishes: for an arbitrarily oriented crack in a matrix of cubic symmetry, in addition, B,, = B,, and both the second and the third terms vanish.

Anisotropic

3.4.

Orthotropic

matrix

material

with arbitrarily

oriented

interacting

571

cracks

with one ~rb~trurily oriented.~~m~~y of ~~r~~lei cracks

This case is of fundamental importance, since results for any orientational distribution of non-interacting cracks can be obtained by integration over orientations of the results for this case. Since t, n are the same for all cracks, Aj’ = pa : n{ B,,tt + B,,nn + B,,(nt + tn)} n : (r

(29)

where p = (1 /A ) c I’“” is the conventional scalar crack density parameter. Substituting B given by (18) into (29), we obtain the elastic potential, and therefore, all the moduli. Thus, in the principal axes x,, _xzof the matrix orthotropy, 1 ~ ..~~ Ey - t+2psinZrp(B,,cos~,sin’~-E,,sin24?)E:’

E,

I

& E;=--

G,,

GY2 VI2

1+2pcos”

-

cp(B,, sin’ cp+B,,cos’

&B,,

sin2q)Ey 1

1 .~ I+p(B,,sin22q-B,,cos22q-B,,,sin4q)G’;, 0

(301

VI?

E, =E: (at CJJ= n/2 the results of Section 3.3 are recovered). The last equation of (30) shows that, under uniaxial loading, arbitrarily oriented cracks produce no additional lateral strain. The results (30) are plotted in Fig. 1 (at crack density p = 0.2). Note a strong asymmetry in the impact of matrix anisotropy on the effective moduli ; the impact of cracks normal to the softer direction of the matrix is close to the one for the isotropic matrix (dashed line), whereas the impact of cracks normal to the stiffer direction is much stronger. Young’s modulus E(8) in an arbitrary direction inclined at an angle B to the X, axis is E(O) E”(fl)

1 l+2psin’(~-~)[B,,cos’(tl-rp)+B,,sin2(~-cp)-B,,,sin2(8-cp)]Eo(B)

(31) where E’(ft) is Young’s modulus of the untracked matrix in the same direction 8. Figure 2 shows the impact of cracks on the effective stiffness in an arbitrary direction 0 for qn = o”, 45” and 90‘ (9 is the angle between the crack line and the “stiffer” direction xi of the matrix). Note that, since B,,,, B,,, B,, entering (30) and (31) are functions of cp, the overall dependence of the moduli on y is quite complex, as seen from Figs 1 and 2. We observe that, with the exception of the special case v, = 0 or 7~12, Af has no elements of elastic symmetry. 3.5. Randomly oriented cracks in the orthotropic crnisotropy due to cracking

matrix.

Gradual disappearance

of

Elastic potential Af is obtained by integration of (23) over orientations. If the orientational distribution of cracks is isotropic (random) and the distribution of crack

c‘. MAWI: and M. KACHAF‘;O~

sizes is uncorrelated with orientations, then, accounting for the normalization relation z(P) = p. we obtain 41’ in terms of the conventional scalar crack density parameter p. The effective *“engineering constants” E,. EZS G’, 2 and I’, z are

As discussed above. cracks normal to the stiffer direction of the matrix arc more “influential” : they produce a larger relative reduction of the eR’ectivc stiffness than cracks normal to the softer direction. As a result, the effective anisotropy gradui~lly disappears as the density of randomly oriented cracks increases. This is illustrated in Fig. 3 [a similar plot was given earlier by MANX and KACHANOV (1992) ; their Fig. 3 is scaled as nfj, rather than p. along the horizontal axis]. 3.6.

On proprr uack

ckwsit~~pur.unwtcr,fbr cmdx

ih anisotropic r~~rrtericrls

The B-tensor of a given crack depends on the crack orientation with respect to the anisotropy axes of the matrix. reflecting the fact that cracks normal to the stitfa direction of the matrix produce a stronger impact on the effective compliance than the ones normal to the softer direction. Therefore. conventional crack density parameters, in which cracks of different orientations contribute equally, such as scal;r~ density fj, or its tensorial ~ener~~li~~tion-the crack density tensor r = (I :‘,A)c (hn)‘”

Anisotropic

material

with arbitrarily

oriented

interacting

573

cracks

P FIG. 3. Randomly

oriented

cracks

in the orthotropic density

matrix. p

Gradual

disappearance

ofanisotropy

as crack

increases.

[introduced by VAKULENK~ and KACHANCW (1971) ; see KACHAN~V (1980, 1992) for a detailed discussion and references] becomes inadequate for cracks in anisotropic materials. The proper crack density parameter that adjusts “relative weight” of a given crack according to its orientation with respect to the matrix is the fourth rank tensor (33) the mentioned weight adjustment being characterized by B. We emphasize that this parameter is not introduced arbitrarily, but emerges naturally, as a term in the elastic potential (23). In the case of the isotropic matrix, the potential Af reduces to (x/E’)a ‘6: a and description in terms of the second rank crack density tensor g is recovered. The importance of the proper crack density parameter lies in the fact that the ell’ectivc moduli arc expressed i/z a u/z$ifiedII‘U_I~ for all orientational distributions in terms of this parameter ; see (23). The moduli for any specific orientational distribution then follow from this general representation [results (27) and (30) being examples]. If one attempts to express the moduli in terms of other parameters (for example, the convcntio~lal scalar crack density parameter p) then such expressions would be valid only for the specific orientational distributiol~ for which they were derived. An interesting observation is that this situation is somewhat similar to inadequacy of the conventional crack density parameters (p or a) in the case when the matrix is isotropic but cracks have non-random mutual positions. Then, cracks located in the stress amplifying positions are more “influential” than the ones in the stress shielding positions, which leads to emergence of the fourth rank tensorial density parameter cu that is sensitive to mutual positions (KACHANOV, 1992, 1993). In the present context, the inadequacy of &I or tl is due to the dependence of the crack’s influence on its orientation with respect to the matrix. The potential 4f can be reduced to a simultaneous invariant of G and a (so that x becomes an adequate crack density parameter) in the following special cases :

574

C.Mizuw

zu& M. KACHANOV

(I) The matrix is isotropic. In this case, B = (z//E”)t and {23) reduces to (z/E”)a.a: a. The material is orthotropic, with the orthotropy axes coinciding with the principal axes of a. Moreover, the orthotropy is of a substantially simplified form (KACHANOV, 1980,1992). (2) The matrix has cubic symmetry. B is proportional to I and the potential 4f’ reduces to a form similar to the one for the isotropic matrix: &‘= L$,p~o: a. Unlike the case of the isotropic matrix, however, the part,f” (no cracks) of the total potential is anisotropic, so that the overall compliance S generally has no elements of symmetry.

Ge~erul rrrmrks. Effective moduh of a 3-D anisotropic solid with arbitrarily oriented cracks are difficult to find in the closed form since solutions for one such crack are not available (to the best of our knowledge). Certain insight into the structure of the elastic potential can, however. be gained by examining the structure of the COD tensor B. For an arbitrarily oriented flat crack, the B-tensor can be represented in the form B = B,,,,nn+ B,,tt + &ss+

R,,,(tn+nt)

+ B,,(ts+st)

+ B,,,(ns+sn)

(34)

where t and s are any two mutually orthogonal directions in the crack plane; offdiagonal terms B,,,, & and B,,, characterize coupling of the modes. (Symmetry of B follows from Betti’s reciprocity theorem.) Loading normal to the crack produces, generally, both normal (h,), and shear (h,), components of the average COD. We denote by z a unit vector in the direction of (h,) (this direction depends on the crack orientation and, in the case of a noncircular crack, on the crack shape). Then, as follows from Betti’s theorem, shear loading in the direction normal to z generates no normal component of(b). Then, if t is chosen as z, the B-tensor takes the form B = B,,,,nn+B,,tt+H,,,ss+B,,,(tn+nt)+B,,(ts+st) which contains only two coupling terms. These representations may be useful if a solution for an arbitrarily oriented crack is constructed by some means (analytically or numerically). Then, representing this solution in the form (35) and inserting it into the potential (23) would yietd the elTective moduli for an arbitrary orientational distriblltion of non-interacting cracks. 3-D T~~II~~sL~~~,~~I~ is~~t~opi~ ~~t~t~~.~with ~~o~-it~t~~~~t~~l~ ~~~l~~~~-.~~~iipt~~i cracks ~~~i~~~~~l to the plunr qf‘ isotrop_y. The effective rnoduli for this case can be derived in a straightforward way, since the solutions for one crack (parallel to the plane of isotropy) are known [SHIELD, 195 1 ; CHEN, 1966 ; see also LAWS (1985) for more explicit expressions for the CODS]. The average CODS (normal and shear) are given by the following formulas :

Anisotropic

materiai

with arbitrarily

oriented

interacting

cracks

575

Using these results, one obtains the following expressions for the effective Young’s modulus in the direction xi normal to cracks and the effective shear modulus in the isotropy plane (.y ,, x2) : &J,Y,(Y, +“JX)(S:lll 3s III,

16(Y,+~,)(S~,1,-S

GI? -(I (3’2

‘+

-S:21*P

3{S,,,r(2S,~,3)“7+(?:,+-1/2)(S-~I~~S,~,2)(S

,,,I -S,Z,~)‘.‘~$ (37)

where rf. 112 are the roots ‘i, =?I= 1): (Sf,,,

of the quadratic

equation

(in the case of isotropy,

,313+2S,,13(5,111-~1212)11/2+S1111S3133-Sf133 =o.

-S:2L?)Y4-_[S1111S

(38) The results (37) are shown on Figs 4 and 5. Note that the impact of matrix anisotropy is strongly asymmetric (similarly to the 2-D case) ; the difference with the case of isotropy is more pronounced in the case when the matrix stiffness E: in the x,-direction is higher than in the x ,x1 plane, as compared to the case when Ei < E:. Computer experiments on parallel cracks in the 2-D anisotropic material (discussed in Section 5) show that the approximation of non-interacting crack remains accurate at high crack density (provided that mutual position of cracks are random). The explanation is that the presence of cracks does not raise the average stress environment in the matrix (if the boundary conditions are in tractions), so that, for randomly located cracks, the competing effect of shielding and amplification, on average, cancel each other. The same argument holds in the 3-D case ; therefore, the results (37) are expected to remain accurate at high crack densities,

1.0 3-D 0.8

~

0

P FIG. 4. Young’s modulus of a transversely isotropic matrix with penny-shaped cracks parallel to the plane of isotropy (approximation of non-interacting cracks). Dotted line gives the approximation of noninteracting cracks for the j.s~~~~~~~~~ matrix, as a reference.

C. MALKX and

0.2

0.0

M. KACHANOV

0.4

0.6

0.8

1.0

P

FIG. 5. Shear modulus of a transversely isotropic matrix with penny-shaped cracks parallel to the plane 01 isotropy (approximation of non-interacting cracks). Dotted line gives the approximation of non-intcractin~r cracks for the i.cotro/~i~~matrix. as a reference.

LAWS and DVORAK (1987) considered this problem (cracks parallel to the isotropy plane) in the framework of the self-consistent and differential schemes. As discussed in the following sections, these schemes appear to substantially overestimate the overall compliance. Their results can be transformed to (37), if the mentioned schemes are changed to the approximation of non-interacting cracks.

4. 4.1

The CNS~~ of’thr isotropic

INTERACTING CRACKS

tmtris

Based on the method of analysis of crack interactions developed by KACHAYOV (19X7), we can find vectors (b’) entering the basic equations (I) and (2) for the effective moduli, as follows. Although (b) on a given crack is not exactly proportional to the average traction (t) on this crack (otherwise any traction distribution with zero average would have generated a crack opening with zero avcrage~-generally. a wrong result), it appears that in the 7-D case of the isotropic medium, this proportionality holds with a high degree of accuracy ( KACHANOV. 1987)

[for plane stress ; E” is to be changed to E”/( 1 - ~5) for plain strain]. Then, inserting (t’), as found by the mentioned method in terms of the transmission A-factors (characterizing transmission of average stresses from one crack onto another), into (39) leads to a relatively simple closed form expression for the efl’ective moduli of a solid with any particular deterministic arrangement of cracks :

Anisotropic

material

with arbitrarily

oriented

interacting

577

cracks

where R”“’ = (26,kI-A”“) ‘. We note that the proportionality relation (39) can be written in the vectorial form due to the fact that the normal and shear compliances of a crack are equal. A number of computer simulations, in which the interaction problem for arrays of 30 cracks was solved by both the described method and by a more precise but much more computer intensive alternating technique, show that the method produces highly accurate results for the effective moduli.

4.2.

Anisotropic

rnutrix

The method of analysis of crack interactions developed for the isotropic matrix can be extended to the case of an anisotropic matrix in a straightforward way. The transmission A-factors inter-relating the average tractions on cracks will depend, in this case, not only on the crack array geometry, but, also, on the matrix moduli. Two complicating factors arise in the case of an anisotropic matrix. (I ) Normal and shear compliances of a crack are not equal, so that the vectorial equation (39) should be replaced by separate proportionality relations for the normal p and shear z modes of loading : (41) where B,,,,, B,,, and B,, are the components of the COD tensor. (2) The accuracy of these proportionality relations should be re-examined. Such an examination can be done as follows (similarly to the case of the isotropic matrix). If one assumes that the traction distribution on a crack can be reasonably well (for the purpose of calculating (b)) approximated by a cubic polynomial with appropriately chosen coefficients

(42) (where t stands for either normal

or shear traction),

then, according

to the polynomial

conservation theorem. the COD has the form of an ellipse ,/I -.u’jl’ multiplied by a polynomial of the third degree. Calculating the average CODS produced by the normal p(s) and shear r(.~) tractions and relating it to the average traction (t) = to( I+ q/3) yields

@‘)= (P>

B

‘+11/4. (‘7) = “‘!1 fI7/3 ’

(z)

B

‘+d4. ” I +q/3’

(_‘I,)= (r)

<_e,> =B

l+!jf

HI1 +y/3’

(43)

Deviation from proportionality of (h,), (h?) to (p), (t), is characterized by the ratio (1+~/4)/( 1+~/3) which depends on the coefficient at the quadratic term in traction distribution (42). This ratio is, typically, quite close to 1. Replacing it by

57x

C. MAIJGI: and M. KACHATWV

unity leads to universal relations (43).

(independent

5.

of the traction

RESULTS OF COMPUTER

distribution)

proportionality

EXPERIMENTS

Sample crack arrays contained 25-35 cracks each. This sample size appears to be sufficient since the moduli were found to stabilize for crack arrays containing 15520 cracks; further increase of the number of cracks did not change the results. These arrays were generated, as realizations of certain crack statistics, with the help of a random number generator. For each sample array, the interaction problem was solved using the method outlined in Section 4. Two orientation statistics were considered : randomly oriented cracks and parallel cracks. For each of them, five crack densities were assumed : p = 0.10,0.15,0.20,0.25 and 0.30 (in 2-D, the densities of 0.2550.30 can be considered as quite high). Between 10 and 1.5sample arrays were generated for each of the crack densities. Mutual positions of cracks within the representative area were random (uncorrelated). For parallel cracks, generation of such arrays was straightforward. For randomly oriented cracks, crack intersections had to be avoided; this was achieved by generating cracks successively and discarding a newly generated crack if it intersects the already existing ones, and generating it again. Although such a procedure, strictly speaking. violates the condition that crack locations are uncorrelated, we assume that it does not create errors of a systematic sign. In order to avoid noticeable errors (or the necessity to use a computer intensive alternating or polynomial expansion techniques), the spacings between cracks were not allowed to be overly small. Namely, they were, generally, not smaller than 0.02 of the crack length, and for pairs of parallel cracks that had a significant “overlap”. they were not smaller than 0.2 of the crack length. Figures 66X show some of the sample arrays of randomly oriented and parallel cracks (crack density 0 = 0.25). The representative area A is assumed to constitute a part of a statistically homogeneous field of cracks (extending all the way to the external boundary) and stresses at the boundary f of A are assumed constant and equal to the remotely applied ones. This assumption is rigorously correct for non-interacting cracks. when cracks inside A do not experience any influence of those outside A. For interacting cracks. stresses fluctuate along f. If the area A is sufficiently large to be representative. then substitution of the actual stress on f by its uniform average will affect only a “boundary layer“ having thickness of the order of the fluctuation wavelength. This substitution can be expected to contribute to scatter the results from one sample of the crack statistics to another (as seen from Figs 6--S. the scatter is reasonably small). but is not expected to produce errors of a systematic sign.

5.2.

Results

and their

For randomly of non-interacting

interpwtatiotr

oriented cracks, the results are presented cracks remains surprisingly accurate,

in Fig. 8. The approximation well into the domain of high

Anisotropic

material

with arbitrarily

0.0

oriented

0.2

interacting

0.4

cracks

579

0.6

P FIG. 6. Results of computer cxpcriments (vertical bars indicate scatter of the results) on arrays of parallel (randomly located) cracks in an orthotropic matrix ; Ey/Ey = IO. The results arc close to the approximation of non-interacting cracks. (Dotted line gives the approximation of non-interacting cracks for the isotropic matrix, as a reference.) The predictions of the self-consistent and differential schemes are given for comparison ; they substantially underestimate the stiffness.

crack densities and strong interactions (where this approximation is usually considered inapplicable). For parallel cracks, the approximation of non-interacting cracks still provides good results (Figs 6 and 7), although there is a slight but distinguishable tendency to the stiffening overall effect of interactions, indicating a slight dominance of the shielding mode of interactions. Some of these results were reported by MAUGE and KACHANOV (1992), Figs 5 and 6 (where the horizontal axis was scaled as 7cp, rather than y). These conclusions are similar to the ones for the isotropic matrix (KACHANOV,

1992,

1993).

The underlying reason for accuracy of the approximation of non-interacting cracks is that the competing effects of stress shielding and stress amplification, on average, cancel each other. This is a direct consequence of the fact that, as follows from the divergence theorem, introduction of cracks---cavities of zero volumePdoes not change the average stress in the matrix (provided the boundary conditions are in

\ E,‘/E,”

0.0

= 0.10

0.2

1

10.6

0.4

FIG. 7. Same. as Fig. 6; E:iEy

= 0. IO.

/B.

.O

:

0.4

0.2

.--_i

0.6

P

of

FIG. 8. Results computerexpcrinxnts on ;rrrqs of randomly oriented cracks in iin orthotropic matrix. Vertical bars indicate scatter of the results. The results follow the :tpproximntion of non-interacting cracks. solid lines. (Dotted lint gives the approxinwrion of non-interacting cracks for the Iwrropic matrix. II\ it refercncc.)

tractions). Therefore, if a certain number of ‘new” cracks is introduced into the environment of the prc-existing ones in a random fashion, these new cracks will, on average. experience no effect of the preexisting cracks. This statement becomes particularly clear in the language of stress superpositions (representation of a problem with N cracks loaded by d-a as a superposition of N sub-probienls with one crack each, loaded by n’ *CTplus additional interaction tractions x,Atii). The interactjon tractions At” will be of both amplifying and shielding nature on different cracks ; on average, they reflect the average stress “environment” in the matrix which is unaffected by cracks; thus amplifying and shielding effects balance each other. These considerations also explain why MORI and TANAKA‘S (1973) scheme (that places a representative defect into the average stress in the matrix), being applied to a cracked material, coincides with the approximation of non-interacting cracks [but not with the “dilute limit” (25) which is linearized with respect to crack density!]. The reported results are for the 2-D configurations. The approximation of noninteracting cracks can be expected to remain accurate in 3-D as well, due to the same mechanism ~~fc~~ncellation of shielding and amplification effects. Moreover. we expect the scatter of the results to be smaller. since inter~lctions are, generally, weaker in 3-D ( KM,HAIZOV and LAURES, 1989 : KACHANW. 19931.

5.3.

R~w~dc mrr mzdorww.~s of’crwk

incations

‘The conclusion on accuracy of the approximation of non-interacting cracks is based on the assumption of randomness of mutual positions of cracks. Otherwise (in “ordered” periodic arrays, for example) the effective moduli may he very difrercnt from the ones predicted by the approximation of non-interacting cracks (with the ditferencc both in the directions of stiffening or softening). We note that imposition of various constraints on crack locations may violate the randomness. For example, as shown by KANAUX (1980. 1983). the prohjbition for the crack centers to enter

Anisotropicmaterial with arbitrarilyorientedinteractingcracks

5x1

circles drawn around the neighboring cracks lowers noticeably the effective stiffness, [this softening effect of noni.e. creates a “bias” against shielding arrangements randomness was interpreted by VAVAKIN and SALGANIK (1975) as a confirmation of the differential scheme]. Slight stiffening effect observed in our computer experiments on parallel cracks (Figs 6 and 7) is another example: it may be due to the fact that the prohibition for the crack tips to be overly close to each other created a slight “bias” against the amplifying configurations. We also observe that naturally occurring crack systems may not always be fully random. A crack may enhance nucleation of new cracks in its amplification zone and suppress the nucleation in the shielding zone, thus creating an amplification “bias”. This may be relevant for the stages of microcracking when localization is about to occur. As an opposite example, the observations of HAN and SURESH (1989) on microcracking in ceramics seem to indicate (although not fully conclusively) some shielding “bias” in most (but not all) microcracking geometries. In such cases, the approximation ofnon-interacting cracks should be used with caution, and the statistics of mutual positions of cracks becomes important.

6.

A COMMENT ON THE EFFECTIVE MEDIA THEORIES FOR CRACKED MATERIALS

As shown in Section 5, the approximation of non-interacting cracks remains accurate at high crack densities, due to cancellation of the competing interaction effects of stress amplification and stress shielding. This is in contrast with the predictions of the “effective matrix” schemes (self-consistent, differential) which place a representative crack into the effective material of reduced stiffness. These schemes substantially overestimate the effective compliance (in particular, the self-consistent scheme). The underlying reason is that the impact of interactions on a given crack is modeled by a reduced stiffness of the surrounding material ; therefore, interactions are predicted to always reduce stiffness. Such modeling distorts the actual mechanics of interactions (best illustrated by stress superpositions mentioned above). In particular. the account is not taken of the shielding mode of interactions that produces a stiffening effect and neutralizes the impact of the amplifying mode. [If, however, one does wish to undertake such “effective matrix” constructions. then the results of the present work provide a convenient starting point. For example, the result (32) provides a starting point in the case of randomly oriented crack in an orthotropic matrix ; for the self-consistent scheme. for instance, the matrix constants in (32) are to be substituted by the effective ones. and the sum,f‘“+ af’should be equated to the potential of the effective orthotropic medium.] On the other hand, our results are in full agreement with the e$f~ctire,/irlcl theory (see KANAUN, 1980, 1983) which places a representative crack into an effective stress (rather than an effective matrix) environment, and with the simplest version of this method-the scheme of MORI and TANAKA (I 973) which assumes that the effective stress coincides with the average stress. [For the isotropic matrix with cracks, these issues are discussed by KACHAN~V (1992, 1993) and for the isotropic matrix with cavities, see KACHAN~V rt al. (I 994).]

C. MAUGFand M. KACHANOV

582

7.

CONCLUSIONS

Closed form expressions for the effective elastic moduli of an anisotropic matrix with non-interacting cracks of arbitrary orientational distribution are constructed for the case of arbitrary matrix anisotropy. The cases of random (isotropic) crack orientations and of one arbitrarily oriented family of parallel cracks are discussed in detail. For interacting cracks, approximate and quite accurate analytical solutions arc constructed for any particular deterministic arrangement of cracks. The main results are as follows. (1) The approximation of non-interacting cracks (but not the linearized “dilute limit”) remains accurate at high crack densities. This result, obtained in computer experiments on a large number of sample crack arraysPrealizations of certain crack statistics-is similar to the one for the isotropic matrix (KACHANOV, 1992, 1993). It is due to the fact that competing effects of stress shielding and stress amplification, on average. balance each other in an array of interacting cracks (provided mutual positions ofcracks are random). (2) Cracks that are normal to the stiffer direction are more “influential” in their impact on the effective properties than cracks normal to the softer direction. As a result, the conventional scalar crack density parameter p [or its tensorial gcncralization-the crack density tensor a = (l/A) c hn] in which the relative “weight” of a given crack is independent of its orientation, becomes inadequate for cracks in anisotropic materials. Instead. a fourth rank tensor (l/A) c (InBn)‘/“ should be used (whet-c B is the second rank COD tensor, characterizing the average COD in terms of the applied traction). Then the effective moduli are cxpresscd in a unified way for all orientational distributions of cracks. (3) Introduction of randomly oriented cracks into an anisotropic solid leads to the gradual disappearance of anisotropy (due to the fact that cracks normal to the stiffer direction produce a stronger impact on the effective compliance than the ones normal to the softer direction). ACKNOWLEIIGEMENTS This research was supported

Scientific

Research

through

by the U.S. Department

oi” Energy and the Air Force Office of

grants to Tufts University.

REFERENCES CHliN.

W. T. (1966) Some aspects of a flat elliptical crack under shear stress. ./. ,Mn//7. Plr.1~. 45,213 223. DENG, H. and NEMAJ’-NASSER,S. (1992) Microcrack arrays in isotropic solids. Mech. Maw. 13, 15-36. GOTTESMAN, T.. HASHIN, Z. and BRULL, M. A. (1980) Effcctivc elastic moduli of cracked fiber composites. A&unces in Conzposit~~ Materiuls (cd. BUNSELL rt cd.), pp. 749- 758. Pergamon Press, Oxford. HAN, H. X. and SUR~SH, S. (1989)High temperature failure of an alumina-silicon carbide composite under cyclic load. J. Amer. Cwam. Ser. 72, 123331238. a survey. J. Appl. Mech. 50,481 505. HASHIN, Z. (1983) Analysis of composite materials

Anisotropic

material

with

arbitrarily

oriented

interacting

cracks

583

HASHIN, Z. (1988) The differential

scheme and its application to cracked materials. J. Me&. Phys. Solids 36, 7 19-734. HOENIC;, A. (1979) Elastic moduli of a non-randomly cracked body. ht. J. Solids Struct. 15, 137-154. KACHANOV, M. (1980) Continuum model of medium with cracks. J. Eqq Mech. Div. 106, 1039~1051. KACHANOV, M. (1987) Elastic solids with many cracks: a simple method of analysis. Inr. J. Solids Struct. 23, 2344. KACHANOV, M. (1992) Effective elastic properties of cracked solids : critical review of some basic concepts. Appl. Mc&. Rev. 45(8), 304335. KACHANOV, M. (1993) Elastic solids with many cracks and related problems. Adwncrs in Applied MechL/nics (cd. J. HUTCHINSON and T. WU). Vol. 30. pp. 259-445. Academic Press, New York. KA~HANW, M. and LAURES, J. P. (1989) Three-dimensional problems of strongly interacting arbitrarily located penny-shaped cracks. ht. J. Fract. 41, 289-313. KACHANOV, M.. TSUKRW. I. and SHAFIRO, B. (1994) Effective moduli of solids with cavities of various shapes. Appl. Mech. Rrr. 47( 1). 15 I-1 74. KANAUN. S. (I 980) Poisson’s field of cracks in elastic medium. J. Apppl. Math. Me&. 44, 80% 815. KANAUN, S. (1983) Elastic medium with random field of inhomogeneities. Chapter VII in Elustic Mediu with Microstructure (ed. I. A. KUNIN), pp. 165-228. Springer, Berlin. LAWS, N. and DVORAK, G. J. (1987) The effect of fiber breaks and aligned penny-shaped cracks on the stiffness and energy release rates in unidirectional composites. Int. J. Solids Struct. 23, 1269-l 283. LAWS, N.. DVORAK. G. J. and HEJAZI, M. (1983) Stiffness changes in unidirectional composites caused by crack systems. Mech. Mater. 2, 123-l 37. LEKHNITSKY, S. G. (1950) Theory of Elasticity of‘ an Anisotropic Elastic Body. OGIZ (in Russian). English Transl. : Mir Publishers, 1961. MAUGE, C. (1990) M.S. Thesis, Tufts University. MAUGP, C. and KA~HANOV, M. (1990) On interaction of cracks in anisotropic solids. Microcracking Induced Dumuge in Composites (ed. G. DVORAK and D. LAGOUDAS), pp. 95-99. ASME, New York. MAWE, C. and KACHANOV. M. (1992) Interacting arbitrarily oriented cracks in anisotropic matrix. Stress intensity factors and effective moduli. ht. J. Fract. 58, R69+R74. MORI, T. and TANAKA, K. (1973) Average stress in matrix and average elastic energy. Actu Met. 21, 571-m574. SAWN, G. N. (1951) Stress Concentration around Holes. Gostekhteoretizdat (in Russian). English Transl. : Pergamon Press, 1960. SHIFI~D, R. T. (1951) Notes on problems in hexagonal aelotropic materials. Pw~. Cam. Ph_~~l. Sot. 47, 40 l-409. SUMARA~, D., KRAJCINOVIC, D. and MALLICK, K. (1992) Elastic parameters of brittle, elastic solids containing slits-mean field theory. Int. J. Dumuge Me&. 1, 320-346. TALREJA, R. (1990) Internal variable damage mechanics of composite materials. Yielding, Damage, nnd Failure of’Anisotropic Solid.7 (Vol. EGFS), pp. 509-533. Mech. Eng. Pub. VAKULENKO, A. and KACHANOV, M. (1971) Continuum theory of medium with cracks. Mech. Solids 6, 145- 151. VAVAKIN, A. S. and SALC;ANIK, R. L. (1975) Effective characteristics of nonhomogeneous media with isolated inhomogeneities. Mech. Solids 10, 65-75.

APPENDIX Elastic compliances of a 2-D orthotropic solid in the coordinate system x’,, x; are expressed in terms of the compliances in the system x ,, x2 rotated by the angle cp with respect to .u’,. u:. as follows [see, for example, LEKHNITSKY(I 950)] :

C. MAIJGCand M. KACHANOV

(A.])

cosl

sin’ cp

cp

E’2’

-

E’;

,