Bounds and self-consistent estimates for the overall properties of anisotropic composites

J. Mech. Phys. Solids, 1977, Vol. 25, pp. 185 to 202.

PergamonPress. Printedin Great Britain.

BOUNDS AND SELF-CONSISTENT ESTIMATES FOR THE OVERALL PROPERTIES OF ANISOTROPIC COMPOSITES By J. R. WILLIS School of Mathematics,

University of Bath, Claverton Down, Bath BA2 7AY, England

(Received 6th January 1977)

BOUNDS of Hashin-Shtrikman type and self-consistent estimates for the overall properties of composites, which may be anisotropic, are developed. Bodies containing aligned ellipsoidal inclusions are considered particularly, generalizing previously known results. The overall thermal conductivity of a body containing aligned spheroidal inclusions is discussed as an example including, as limiting cases, bodies containing highly-conducting aligned needles and bodies containing aligned pennyshaped cracks.

1.

INTRODUC~ON

WE CONSIDER the determination of the overall properties of a composite material comprising a matrix in which are embedded inclusions at arbitrary concentrations. The main theoretical results are formulae from which bounds on the overall properties of interest (elastic moduli, conductivities, etc.) can be obtained, and from which ‘self-consistent’ estimates of the relevant properties can also be calculated. The work is, in a sense, a generalization of that of WALP~LE (1969), who considered the elastic moduli of a composite in the limiting cases where the inclusions were needleor disc-shaped. Walpole’s results were obtained essentially by exploiting the geometry of the inclusions to reduce an intermediate problem for a single inclusion to a simpler one in which edge effects were disregarded. They cannot, therefore, be specialized, for example, to obtain the overall properties of a body containing cracka, in which the crack edges are of vital significance. The present work is free of this limitation. Its starting point is the variational principle of HASHIN and SHTRIKMAN(1962a,b), a reasonably direct derivation of which is given for completeness. Expressions for bounds are then deduced from it, which contain those of HASHIN and SHTRIKMAN (1963), HASHIN (1965) and WALPOLE (1966a,b, 1969) as special cases. The self-consistent approximation is discussed in relation to the bounds, and a possible slight generalization is outlined. As a simple example, bounds and self-consistent estimates are given for the components of the overall thermal conductivity tensor of a material containing aligned spheroidal inclusions including, as special cases, circular cracks and highlyconducting thin needles. The thermal conductivity of a body containing aligned cracks may have application to the analysis of thermal stress in a nuclear fuel rod, for example; and composites containing highly-conducting needles at low concen185

J. R. WILLIS

186

trations have recently been studied by CHENand ACRIVOS(1976). The corresponding elastic results would be more laborious to obtain, and it is intended to report this work elsewhere. 2.

THE HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLE

The linear constitutive denoted by

relation, for the physical property in question will be a=L’e.

(2.1) This conforms with the symbolic notation employed by WALPOLE(1966a,b, 1969) when Q and e represent stress and strain tensors and L’ represents the fourth-order tensor of elastic moduli. For heat conduction, on the other hand, e would represent the negative of the heat flux vector, e the temperature gradient, and L’ the secondorder tensor of thermal conductivity. Other properties could be interpreted similarly. In each case, the tensor L’ may vary through the material, the symbol L being reserved for the corresponding overall tensor, which we now define. Consider a composite material occupying a volume V, subjected to boundary displacements (or temperature, erc.) over av that are compatible with the uniform strain (or temperature gradient, etc.) Z throughout I/. The problem is to find d throughout V and to deduce its mean value Z. The overall tensor L is then defined by a = Le,

(2.2)

and it can be shown (HILL, 1952) that the mean energy density E is given by E = @ia = @LE.

(2.3)

The tensor L will be positive definite if L’ is positive definite; this property will be assumed in the sequel. For conciseness, the tensor L will be referred to as a tensor of mod&, for which may be read elastic moduli, thermal conductivity, etc., as appropriate. Likewise, d and e will be called stress and strain respectively, until particular examples are discussed. Following HASHIN and SHTRIKMAN(1962a,b) and WALPOLE(1966a,b), a homogeneous comparison material is introduced, with moduli L,, and a polarization z is defined as z = (L’ - L,)e, (2.4) so that B = L,e+z. (2.5) If t were known, then e could be expressed in terms of a Green’s function, so that, symbolically, e = E-IYz, (2.6) where -I? is the linear operator that produces from z a strain that is consistent with zero displacements over av and for which the divergence of @(with c given by (2.5)) is zero; details for both elasticity and thermal conductivity are given in the Appendix. An equation for z is now obtained by combining (2.4) and (2.6), to give (I&L,)_‘r+l-%

= 8,

(2.7)

which, if regarded as an equation for e rather than 2, is the Lippmann-Schwinger

Overall properties of anisotropic composites

187

equation, given by KR~NER (1977). It is shown in the Appendix that, with respect to an inner product (2.8)

(f, 9) = ; i f(X)g(X) dX,

the operator I? is self-adjoint, as also is (L’-LO)-‘, from the usual symmetry of moduli. Hence, immediately, (2.7) is equivalent to the variational principle 6{(r, (L’-LJ’r)+(z,

rz)-2(2,

@)}= 0.

(2.9)

Furthermore, it is shown that the stationary value of the functional in (2.9) is just 2(E, - E), where E is the mean energy density and E,, = -)eL&

(2.10)

Finally, it is shown in the Appendix that the operator (L’-L)-‘+T is positive definite so long as L’-LO is positive definite at each point of V, and is negative definite so long as L’- L, is negative definite at each point of V. Hence, immediately, whenever the former condition holds, the stationary value of the functional is a minimum and whenever the latter holds, it is a maximum. Thus, 2(& - E) < (Z, (L’-LJ’z)+(r, (2)

R)-2(t,

CJ

(2.11)

for any polarization r, whenever Lo is such that L’ -Lo is positive (negative) definite. This result is the variational principle of HASHINand SHTRIKMAN (1962a,b).

3.

GENERALIZEDHASHIN-SHTRIKMAN BOUNDS

We consider now the application of the inequalities (2.11) to a composite comprising a matrix containing n distinct types of inclusions, with moduli L,, at volume concentrations c, (r = 1,2, . . ., n). The matrix has moduli Lnfl and volume concentration c, + i. The limiting case of a polycrystal is included in this framework by taking cn+ 1 = 0 and allowing it to become large. The composite is taken as statistically uniform, in the sense that all possible local configurations of the composite are sampled as x ranges over V. Then, so long as the linear operator I’ can be replaced, for the purpose of taking a volume average, by an operator that is translation invariant, the inner products in (2.11) may be interpreted as expectation values (or ensemble averages), taken at a$xedpoint, and (2.11) becomes 2(E, -E)

< &{Z(X)(L’-L,,)-%(X) +z(x>(~z)(x)-22(x)~). (2)

(3.1)

The validity of (3.1) thus depends upon the properties of r, whose detailed form is respectively. For elasticity, for instance, (A.6) yields the translation-invariant approximation given in (A.6) and (A.7) for elasticity and thermal conductivity

(rt)(x)= Jr yx - q(t(d) - 3)dx’,

(3.2)

for large Y and x remote from 8 V, where i represents the mean value (or expectation value) of t and r;kl(x) = - G’, jdx)/(i, j). (k. w (3.3)

188

J. R.

WILLIS

G” being the infinite-body Green’s function and (i,j) representing symmetrization with respect to the suffixes i, j. With the definition (3.3), the singularity in I?” is to be interpreted in the sense of distributions and, since l?‘(x) = O(IX~-~) as 1x1+ co, the integral converges because (Z - Z) has zero mean-value. In approximating rz by (3.2), boundary terms are neglected which are significant only in a layer adjacent to 8V whose thickness is of the order of microstructural dimensions, so that they do not contribute to the mean value when V is large. The approximation (3.2) was used in a related context by WILLIS and ACTON (1976). HASHIN (1965) used the polarization in the combination (r-2) to secure convergent expressions, but he did not introduce Green’s functions. Generalized forms of the Hashin-Shtrikman bounds are now obtained by substituting into (3.1) a polarization z(x) that is piecewise constant, taking the value t, in the rth phase (r = 1,2, . . . , n+ 1). This gives, upon interchanging the orders of integration and ensembIe averaging, n+l 2(&--E) < c c,z,.(L,--LO)-%,+ (a)‘=’

n+l +

c

n+1 c

II+1

,=l s=l

z,

j roo(x-x’)[~,,(x-x’)-c,cs]

dx’

2,

-

2 c c,zp,

(3.4)

r=1

where the two-point correlation function x,,(x-x’) represents the probability that phases r and s will be found at x and x’ respectively ; that x19depends only upon x - x’ follows from the assumed statistical homogeneity. It may be noted that the integrals in (3.4) converge because if (as we assume) there is no long-range order in the composite, then properties at x’ are independent of those at x when lx- x’l is large, that is, x,~(x-x’) + c,c, as Jx--x’J -+ 00. (3.5) Suppose, now, that the two-point correlation functions x,* are isotropic, and so depend upon ix-x’\ only. In this case, it can be shown that J ra(x-

x’)cx,x] x-+c,cJ

dx’ = P&s(O)-cd,

(3.6)

where the constant tensor P,, has the value J r”(x)dx. (3.7) ixI 0, so that PO =

j

r”(x) dx

= 0

(3.8)

a
for all Q and b, with 0 < a < b. The left side of (3.6) can therefore be replaced by an integral over a small sphere centred at x’ = x, and the right side follows. The result (3.7) is due to ESHELBY(1957), the expression e(x) =

S P(x-x’)pT(x’)

Ixl
dx’

(3.9)

being precisely the strain at x produced by an isolated spherical inclusion with polarization - p'. ESHELBY(1957) gave P,, explicitly for isotropic LO but also proved

189

Overall properties of anisotropic composites that e(x) was constant for 1x1< a, even when L,, is anisotropic. in the anisotropic case by KNEER (1965). Since, plainly,

P, was first evaluated

we have now n+1

It+1

2&-E)

< c c,z,[(L,-Lo)-‘+P,]z, (a>‘=’

-

n+l

n+l

,z, zr

c,c,t,P&

- 2 c c,z,q

(3.11)

r=l

which is extremized with respect to e, when [(L,-LJ’+P&-P,i-L where

(3.12)

= 0,

ll+1 i = c c,?,. r=l

The

optimal bounds (with respect to z,) now follow as 2(&-E)

with n+1 f=

I

-,~lc,[(Lr-Lo)-l+P,I-‘P,

(3.13)

Q -?iB, (a)

1

-lx

I

(3.14)

c,[(LS-L,)-‘+P&‘8.

By performing routine matrix algebra, the result (3.13) can be expressed in the form (3.15)

E(L-E)@ 2 0, (G) whenever L,-LO is positive (negative) semi-definite for all r, where II+1

II+1

E =,~~c,L,cI+P,(L,-L,)I-’

-l.

S~lclcI+P,(L,-L,)l-’

(3.16)

>

The result embodied in (3.15) and (3.16) constitutes WALPOLE’S(1966a,b) generalization of HASHINand SHTRIKMAN’S (1963) result. It may be noted that ICRGNER(1977) has recently discussed the Hashin-Shtrikman bounds and has concluded that they apply only to material of grades (co, co, 2) in his terminology. That is, the composite should possess n-point correlation functions that are translation invariant and isotropic for all n, and ‘disordered’ two-point correlation functions. The concept of ‘disorder’ is difficult to express in a simple sentence, but it is not quite realized for the composite for which (3.15) was derived; nor did the derivation of (3.15) use isotropy of any correlation functions except for n = 2. Translation invariance of all correlation functions was, however, assumed implicitly, in replacing (2.11) by (3.1). Other bounds of Hashin-Shtrikman type could be obtained from (3.4), by evaluating the integrals for the relevant functions x,,(x-x’), and then extremizing with respect to t,. One simple example which will be considered in the sequel is for a matrix containing aligned, similar, ellipsoidal inclusions, a typical inclusion centred at x0 being described by (x - x’)~A~A(x - x0) < 1,

(3.17)

190

J. R. WILLIS

say. For such a composite, it is possible that x,,(x) may depend upon x only through the combination p = (xTATAx)+. (3.18) ESHELBY(1957) has shown that the constant tensor P, defined by P = j l?(x) P-=0

dx,

(3.19)

is independent of the value of a > 0, and it follows that j ~m(x-x’)C~,,(x-x’)-c,c,l

dx’ = PCL(O)-c,csl,

(3.20)

by the same reasoning that led to (3.6). The bounds (3.15) therefore remain valid for this more general composite, so long as P, is replaced by P in the definition (3.16) of E. Detailed expressions from which P can be evaluated have been given by WILLIS (1970), KINOSHITAand MURA (1971) and KUNIN and SOSNINA(1971).

4.

SELF-CONSISTENT ESTIMATES

The “self-consistent” estimate of the overall moduli of a composite of the type considered in Section 3 is obtained, following WALPOLE(1969), by first defining the average values of stress and strain in the rth phase as or and e,, where 0, = L,e,.

(4.1)

Then, II+1

Cc,e,

ii=

(4.2)

r=l and

II+1 8= CclL,e,

(4.3)

r=1

or, by eliminating c,,+ ,e,+ 1,

5 = L+ 9 +

@lc.(k-4, &,.

(4.4)

The values of e, are now estimated by embedding one isolated inclusion of rth type in a matrix with moduli L, subiected to the uniform strain ZZfar from the inclusion, and averaging over all possible-inclusion shapes, if necessary. There results e, = A,&

(4.5)

say, where A, depends upon L, and L, and (4.4) then implies + i c,(L,--L,+,)A, r=l

z.

(4.6)

>

For consistency of (4.6) with is = LB then, we deduce L=L,+1 as an equation for L.

+ f c,(L,-L,+,)A, r=1

(4.7)

Overall properties of anisotropic

composites

191

For ellipsoidal inclusions, A, can be given explicitly, using a construction due to ESHELBY(1957) or, equivalently, from the operator equation (2.7), with L, replaced by L and L’ taken equal to L, within the inclusion and equal to L in the matrix. Using (3.19X then, e, = (L,-L)-‘[(L,-L)-‘+P]-‘g, (4.8) so that A, = [I+P(L,-L)]-’ (4.9) and (4.10)

L= Ln+l + r=1 i c,(L,-L,+,)cI+P(L,-L)l_‘.

For a polycrystal now, for which c,+ r = 0, the above approach has to be modified for, if e, is defined as above, the necessary relation B = C c,e, fails to hold. Instead, we employ a comparison material with moduli L, relative to which the polarization in the rth phase is estimated as (4.11)

r, = [(L,-L)-‘-l-P]-%, and hence, by averaging (2.5) with L, = L,

(4.12)

a = Le + i c,[(L,-L)-‘+P]-‘8. r=1

It follows, since Z = Li!, that

,&rw+-L)-‘+P1-’

= 0,

(4.13)

or, rearranging, -1

L= in which the It is, in fact, by n+ l), by self-consistent

i c,L,CI+P&-Ln-’

1 r=l

&L-I+P(L.--L)-J_’ 11

,

(4.14)

>

right side is identical to (3.16) with cn+ 1 = 0 and Lo replaced, by L. not difficult also to reduce (4.10) to the form (4.14), (with n replaced algebraic manipulation when n = 1. In either case, therefore, the aproximation asserts that (4.15) L = YL),

where L(L) is given by (3.16), with Lo replaced by L and PO by P. The same approximation can now be interpreted in a different way. The exact value of L is obtained by extremizing the functional on the right side of (3.1) over all possible z(x), for any choice of Lo. Correspondingly, estimates for L may be formed by extremizing the functional over the class of piecewise constant fields used in Section 3, though these estimates L will not give bounds unless (L,-L,) is positive (or negative) definite for each r. It is plausible, now, that the corresponding “best” estimate L for L will be obtained when the comparison material L, is identified with L since then, presumably, the “smallest” deviations L,.-L are obtained, in some average sense. This is, at least, an interpretation of (4.15) in terms of the Hashin-Shtrikman variational principle, which leads to the following possible extension for more general composites: evaluate the integrals and extremize the 13

J.R. WILLIS

192

right side of (3.4) to give an estimate of E0 -E. When L, = L, then E.-E = 0; the stationary value of the right side of (3.4) is thus set equal to zero, to give an equation for L, of which (4.15) is a special case. It should be noted, however, that not all approximations of “self-consistent” type conform to this prescription. The recent estimates given by BUDIANSKY and O’CONNELL(1976) for example, for the moduli of a randomly-cracked body could be expressed in terms of a limiting case of (4.7), but could not be put in the form just introduced, because the two-point correlation functions are not known.

5. For

THE THERMAL CONDUCTIVITY OF A BODY CONTAINING SPHEROIDAL I~cLusI0Ns

the thermal

conductivity

problem,

the infinite-body

Green’s function G”

satisfies the equation K,qG,$(x) + 6(x) = 0,

(5. 1)

where the thermal conductivity tensor (called Lo in the general development) has components KP,. An expression for G”(x) is easiest developed by employing the plane-wave expansion (GEL’FAND and SHILOV, 1964)

(5.2) We observe that Gr(g.x) = -(K$~i~,)-‘b(~.x)

(5.3)

satisfies the equation K&Gf,j+6”(&x)

= 0

(5.4)

and deduce, by superposition, (5.5) The analogue of equation (3.3) for thermal conductivity now gives (5.6) and, from (3.19) J’ij=-&

J Ce/(K~~S~53-‘d~~~~6”(g.~)dx. ICI=1

(5.7)

To evaluate the integral with respect to x, it is useful to set y = Ax,

6 = (A-‘)%

(5.8)

so that j BII(g.x)dx=

PC1

jAI-‘,Y,[ls”(G.y)dy=

IAl-* ~~*(,*i!16(1.y-p)dy).

(5.9)

evaluated for p = 0. The last integral in (5.9) represents IQ-’ times the area of the

193

Overall properties of anisotropic composites

circle 6.y = p, IyI -=z1 and so has the value n(l{12-p2)/lQ3. Hence, (5.10) It may be remarked in passing that the corresponding derived similarly, to give

result for elasticity may be

(5.11) where &k(g) =

(5.12)

cijkl tj
for a comparison material with da& moduli ci/kl (WILLIS, 1970). We now specialize (5.10) for spheroidal inclusions aligned parallel to the x,-axis, with length 21 and maximum diameter 2.a = 281, for which we may take (5.13)

p = (x: + x; +&2X:)*.

Both matrix and inclusions will be taken isotropic, for simplicity. The composite is then transversely isotropic, and Kz may consequently be specialized to be diagonal, with KY, = Kz,. In this case, Pij is correspondingly diagonal, with PII = P,,. The integrals, furthermore, are elementary and give

(5.14)

where (5.15)

’ =

Equations (5.14) remain valid when I is complex. Both bounds and self-consistent estimates may now be obtained from (3.16), with P, replaced by P. If the inclusions and matrix have thermal conductivities K, and K,, respectively, then R,

1=

K, -~-- CI____ 1 +P,,(K, -K:,)

C2K2

+ i+P,l(K2-K:11

> ’

Cl x K33

Cl K, =

1+P,3(K1

{

Ct +

~+P,,W,--K~,)

1+P, I(K2 -KY,)

-I,

(5.16)

C2 K2

-K&)

+ i;

P,,(K,

-Kz,)

lx

Cl ]+PdK1

-1

C2

-KY,)

+ t+

P&K,

- K,O,)>

.

(5.17)

RI, and & yield lower bounds to K,, and K,, so long as KY, and K& are both smaller than K, and K,, the greatest lower bounds being obtained by taking = Kg, = min (K,, K2). Dually, the least upper bounds are obtained by taking ;;: = K& = max (K,, K2). Self-consistent estimates, on the other hand, are obtained by setting KE = K,j and identifying Rci with Kii.

J. R. WILLIS

194

6.

SPECIALCASES

Consider first the simplest case of spherical inclusions (E = 1) so that, always, K”11 = K;3’ , since the composite is isotropic. Equations (5.14) become degenerate __ ~ but, by taking limits as A --) co, it is easy to deduce that 1

(6.1)

PI1 = p33 = Elj, and &(2KC:, K:,(K1 ~~~~~+K,)+2c, __ ~~________. K1l = K33 = --- (2Ky,+K,)-c,(K,-Kt) -

- K2) ’

(6.2)

using c3 = 1 -cr. The associated bounds and self-consistent estimate for the overall thermal conductivity were derived respectively by HASHINand SHTRIKMAN(1962~) and BRAILSFORD and MAJOR (1964). They will not therefore be discussed, except for two limiting cases. First, if the inclusions in fact are caoities, so that K, = 0, the lower bound produced by setting K,’ 1 - 0 degenerates to zero, while the upper bound (K:, = Kz) gives

(6.3) and the self-consistent estimate (KY, = K, , + RI *) is (6.4)

KII = U-3WL

Dually, if K, --f co, then the upper bound (KY, = K,) becomes infinite, lower bound

while the

(KY, = K2) gives

u+2c,)K

__._

(l-c,)

._

< 2’

K

‘I’

(6.5)

and the self-consistent estimate is (6.6) K11 = K,/(l - 3c,). There is no novelty in these results, but (6.4) shows that Kr, for a body containing cavities falls to zero at c1 = 2/3 and (6.6) shows that KI1 for a body containing highly-conducting particles becomes singular at c1 = I /3; such features might therefore be expected also for spheroids. We now consider the particular case of long thin inclusions, obtained by letting E + 0. We have, from (5.14), as E + 0,

(6.7)

For a composite for which the ratio K,/K, is neither very small nor very large, the relevant values of Kt will be of a similar order to K,, and the expressions (5.17) for Kij reduce to (c~Kr+czK3)K:,+K,Kz __ R,, Z __________._~ (6.8) (crK~+czK,)+K& ’ (6.9) K33 z c,K,+c2K3,

Overall properties of anisotropic composites

195

to zero-th order in E. Thus, for very long fibres, upper and lower bounds, and the self-consistent estimate for &s, all reduce to the Voigt “law of mixtures” estimate, equation (6.9). Bounds upon K,,, on the other hand, are obtained by setting KY, = KI or K2. When KY, = K,, we obtain

Rll

1+

ZK,

1

2c,(Kt -K,) 2K,+ c,(K, -K,) >

and, when KY, = K2, K,,

2c,(K1 -K,) 2Kz + c,(K, -K,)

>-

(6.11)

The bounds (6.10) and (6.11) for K1 1 agree with those obtained by HASHIN(1965) for the mathematically-identical problem of bounding the elastic modulus cd4 of a composite reinforced by long fibres. The associated “self-consistent” estimate (K:, = K1 1 = RI ,) is obtained by solving the quadratic equation K:, -(cl - cz)(Kr - Kz)K, I -K,

Kz = 0,

(6.12

which agrees with WALPOLE’S(1969, equation (56)) result. Suppose now that K1 B K2. The upper bounds obtained by letting KY1 = K& = K1 become large and their detail is uninteresting; therefore, Kc will be taken as comparable in magnitude with K2. The approximation (6.8) remains valid and simplifies to R,, Z (~1KY, +Kz)lcz, (6.13) but (6.9) no longer applies. Instead, equation (5.17,), when KI 4 K2, yields (6.14) so

that, when E 4 I, K33

~~-~K~3+~~-~~{ln(~(~)i)-1~1.

(6.15)

If E tends to zero while c1 remains finite, then the last term in the right side of (6.15) inevitably dominates. It is of interest, however, to let E tend to zero while keeping the number density n of inclusions fixed. The volume of each inclusion (whose length is 21) is $cs2Z3 and so ci = &rc2n13.

(6.16)

Then, (6.17)

K33

as E -+ 0, and a lower bound for K33 is obtained by setting KY, = Kt3 = K,; thus, K, {l+3d3

[In (:)

- I]-‘}

GK,,.

(6.18)

The left side of (6.18) coincides precisely with the “dilute suspension” estimate for this composite, obtained by ROCHAand Acnrvos (1973). This is to be expected since, as E 3 0 with n fixed, the suspension does ultimately become “dilute”, even

J, R. WILLIS

196

though the inclusions may be very close in comparison with the length-scale 1. The expression (6.15) with K,r’ = K& = K2, furnishes a lower bound for KJ3, even when cr is not small. The corresponding self-consistent estimates of KI1, KJ3 are obtained by setting K: = Kii = Rij in (6.13) and (6.14). From (6.13), (6.19)

KII = K,/(l - 2c,), and then (6.15) gives

(2:) = I+$ (&--+n

(~(l-zc,)-*

(2)‘)

- 1)-r

(6.20)

or, when E + 0 with n tied, K,, = K2 and Ka3 satisfies (6.2 1) The opposite limiting case, E --) co, will now be considered, corresponding to a composite containing parallel flat discs. In this limit, equations (5.14) reduce to (6.22) (6.23) Then, if K,/K,

is neither very large nor very small, then (5.16) and (5.17) reduce to (6.24)

RI, z cl K,+c,K,, L

= Cc,/K,

+cz/&l-‘3

(6.25)

which are, respectively, the Voigt and Reuss estimates of K,, and KsJ; they agree with WALPOLE’S (1969) estimates of c66 and c,, for a “sandwich” composite. The problem becomes more interesting if we consider the cases of highlyconducting and non-conducting discs. As in the case of needles, Kz will be taken comparable with K2, because the bounds obtained by setting KY, = Ksz = K1 are either infinite or zero. First, if K, $ K,, then equation (6.25) reduces to (6.26) Kss z &I(1 -cl) but (6.24) becomes singular. However, equation (5.16) gives, asymptotically, K1, z K1+c,

[k

-K&-j,

(6.27)

and so, from (6.22) and (6.27), (6.28) Setting KY, = Kz3 = Kz in (6.27) now yields the lower bound

Kz

[l+cl(;-l)] GK,,.

(6.29)

197

Overall properties of anisotropic composites

If n is defined as the number density of the discs, then cl

=

$nna3/E,

(6.30)

and (6.29) becomes, asymptotically, K,[l +yna3] as E + co with n fixed. The corresponding (6.26) for K33, and, from (6.28),

6 Kll,

(6.31)

self-consistent estimates are given by

K

--“=l+c,

(6.32)

@JR&J+-‘I’

K,

which can be rearranged as a quadratic equation for (K,,/K,)*. Now consider the limiting case K1 + 0, corresponding to a matrix containing a distribution of aligned cracks. This time, (6.24) reduces to (6.33)

K*, ,” (1 -c&K,, while (6.25) becomes singular. Equation (5.17) reduces, however, to the form c,Cl +P33(&

- K:3)1

>’ ~,C~+P~~(K~-K~~)I+C~C~-P~~K~~I

(6.34)

which, when combined with (6.23), yields (6.35)

having also expressed cl in terms of n, using (6.29). When KY, = Ki3 = K,, the upper bound K33 G

K2 1 + 8na3/3

results for K33. The corresponding self-consistent estimate for K,, is obtained by setting KY, = (1 -c,)K, (from (6.33)) and K33 = K33 = Ki3 in (6.35); this yields the quadratic equation

(2) - +na” (j$J’-

(l -Cd+

(1 -Q

= 0

(6.37)

or, if E + co with n fixed, (2)-+na’(z)t-l=O.

(6.38)

Some sample results are shown in Figs 1 to 4. Figure 1 shows upper and lower bounds for K,, and K33, together with the associated self-consistent estimates, for a composite containing relatively-fat “needles”, with E = 0.1, for which K,/K, = 8; these were obtained from the full equations (5.14) to (5.17). The expected transition is shown, with the self-consistent estimates ranging from K2 at c1 = 0 to Kl at cl = 1, being close to the lower bounds (KY, = KS3 = KJ at low values of c1 and close to the upper bounds (KY, = KS3 = KJ at high values of ci. The bounds

198 (b)

II

( 1.0

0

0

1.0

Cl

Cl

FIG. 1. Plots of upper and lower bounds and self-consistent estimates of (a) KI1 and (b) KS3 versus cl, for a composite containing fairly-thin spheroidal inclusions (E = 0.1) with KJKLE= 8.

0

Cl

IO

0 Cl

IO

FIG. 2. Plots of upper bounds and self-consistent estimates of (a) KU and (b) Ka3 versus cl, for a composite containing moderately-flat cavities (E = 10).

for KS3 are very close together, compared with those for K,,, and show that Ka3 must vary almost linearly with cr, conforming with (6.9). Figure 2 shows corresponding plots of self-consistent estimates and upper bounds for a composite containing fairly-flat cavities, with E = 10. The self-consistent estimate for K, 1 becomes zero at c1 w O-53, following the same pattern as the result (6.4) for spherical cavities, and the associated KJ3 was not calculated beyond this value. Again, the self-consistent estimate of KS3 (for cr -C O-53) is quite close to the upper bound, which agrees almost exactly with (6.25). Figure 3 shows the lower bound and the containing infinitely-conducting self-consistent estimate of K33, for a composite needles, with E = 0.02, calculated from (6.18) and (6.21); the value nZ3 = 5 still corresponds to a very small concentration by volume, namely c, = 0.0084. Finally, Fig. 4 shows the upper bound and self-consistent estimate for Ks3, for a composite

Overall properties of anisotropic composites

0

nl

FIG. 3.

1

199

5

Lower hound and self-consistent estimate of Ks~ for a composite containing highlyconducting needles (E = 0.02), each of length U, at number density n.

FSG. 4. Upper bound and self-consistent estimate of Ka9 for a composite containing circular cracks of radius a, at number density n.

containing aligned penny-shaped cracks, calculated from (636) and (6 38). It may be noted that na3 could in principle be taken arbitrarily large, since the cracks have no volume (s = co). It may be remarked, also, that in contrast to the self-consistent estimate (6.4) for spherical cavities and the corresponding result shown in Fig. 2(a), the self-consistent estimate of KS3 for aligned cracks does not vanish at any finite value of na3, although, as Fig. 4 shows, it falls rapidly as na3 increases. The corresponding self-consistent estimate (calculated separately using (4.7)) for a body containing cracks at all orientations vanishes at nu3 = l-125.

200

J. R. WILLIS

ACKNOWLEDGMENT Thanks are due to Professor E. Krijner (Institut fur Theoretische und Angewandte Physik, Universit%t Stuttgart) for making available to the writer his work (KR~~NER,1977) in advance of its publication. REFERENCES BRAILSPORD,A. D. and MAJOR, K. G. BUDIANSKY,B. and O’CONNELL,R. J. CHEN, H. S. and ACRIVOS,A. ESHELBY,J. D. GEL’FAND,I. M. and SHILOV,G. E.

1964

Br. J. appl. Phys 15, 313

1976

Znt. J. Solids Struct. 12, 81.

1976 1957 1964

Proc. R. Sot. A 349, 261. Ibid. A 241, 376. Generalized Functions, Vol. 1: Properties and Operations. Academic Press, New York. J. Mech. Phys. Solids 13, 119. Ibid. 10, 335. Ibid. 10, 343. J. appl. Phys. 33, 3125. J. Mech. Phys. Solids 11, 127. Proc. phys. Sot. A 65, 349. Phys. Status Solidi a 5, 759. Ibid. 9, 825. J. Mech. Phys. Solids 25, 137. Dokl. Akad. Nauk SSSR 199, 571. Q. Jl Mech. appl. Math. 26, 217. J. Mech. Phys. Solids 14, 15 1. Ibid. 14,289. Ibid. 17, 235. Asymmetric Problems of Elasticity, Adams Prize Essay, Cambridge University. Q. JI Mech. appl. Math. 29, 163.

WILLIS, J. R.

1965 1962a 1962b 1962~ 1963 1952 1971 1965 1977 1971 1973 1966a 1966b 1969 1970

WILLIS, J. R. and ACTON, J. R.

1976

HASHIN, Z. HASHIN, Z. and SHTRIKMAN,S.

HILL, R. KINOSHITA,N. and MURA, T. KNEER, G. KRBNER, E. KUNIN, I. A. and SOSNINA,E. G. ROCHA, A. and ACRIVOS.A. WALPOLE,L. J.

APPENDIX Properties For elasticity,

equation

of the operator I?

(2.5) assumes the form bij =

C$lu,l+rij,

(A.1)

of the displacewhere the tensor L,, has components c$, and Ui are the components ment field. The problem from which the operator l? is obtained, is to solve the equilibrium equations bij

subject to the boundary

j

E

C$tlUk,

lj

+

denotes

j =

O9

x E

K

(A.3

conditions ur = 0,

If G;(x-x’)

rij,

the infinite-body

Xcav.

Green’s

function,

cP/~~~G~,~,(x-x’)+~~~~(x-x’)

(A.3) so that = 0,

(A.4)

201

Overall properties of anisotropic composites

then an application of Gauss’ divergence theorem gives U&X’)= - j GG, ,(x -

X’)[Z,,
-

“ii]

dX -Jr

+ 5 G~(x_x’)C7t,l~l_fi~+ av

C&UL, t(x)Ini dS

(A-5)

for any value Of the constant Equation (A.5) yields, in fact, an integral equation for c&..u~,r(x)nj, since (A.3) requires that u&x’) = 0 on 8J’. If, now fij is taken as the mean value of z,~(x), then the integrands involving 7,j(X)- Tij oscillate about zero, It is plausible that, on av, Cijkl u k, l n i will likewise oscillate about zero and hence, by Saint-Venant’s principle, that the integral over dV in (A.5) is significant only in a “boundary layer” close to 8K The operator r is now obtained by differentiating (A.5): i,j.

e&X’) = -(Tr),(X’)

= [j Gg,j&X-X’)[Tt,..X)-Sri] ’

dx -

-~G~.,(X-X’)[7~XX)-i~,+C~~~U~,~(X)lnjd~I~,.,), (A.6)

in which the singularity at x = x’ is interpreted in the sense of distributions and the value of c&u,, r(x)nj on av, which, fortunately, is not needed explicitly, is defined as the solution of the integral equation obtained by subjecting (A.5) to the boundary condition (A.3). For thermal conductivity, a similar development yields e&x’) = - (I’z),(x’) = d G,=&(x- x )[ri(x) - ?,] dx -p

O”X-X’)[Sj(X)-?j

+ Kiq,e,(x)]n,

dS,

(A.7)

where the temperature gradient has components e,, the conductivity tensor has components K$ and G”(x-x’) satisfies K; G,;Ax - x’) +&(x-x’)

= 0.

(A.8)

Again, it is plausible that the boundary integral in (A.7) is significant only in a boundary layer adjacent to 8K We now derive the relevant properties of r, reverting to symbolic notation. Use will be made of the ‘virtual work’ equality (or, ez) = 0,

(A-9)

for any ‘stress’ bl that is divergence-free and any ‘strain’ e2 that is derived from a ‘displacement’ that is zero over at’. First, to verify self-adjointness, we have (rr, rtJ

1 = - --- j tI e, dx = (Lee, -or, vv

e,),

(A. 10)

where a1 and e, are the ‘stress’ and ‘strain’ associated with rl through (A.2) and (A.3), and e2 is similarly associated with TV. From (A.9), therefore, (tI, rT,) = (Lo+ 4,

(A.1 1)

which is symmetric in the subscripts 1 and 2, from the symmetry of Lo. Next, setting +r =I = z2 in (A. 11) shows that r is positive definite, and hence that (L’- Lo)-'

202

J. R. WILLIS

is positive definite, so long as (L’ -LO) is positive definite at each point of V. If (L’-L,) is negative definite, however, it is convenient to introduce, following HILL (1963), the ‘strain polarization’ q by r = Loq,

(A.12)

q = Mar,

where M, is the inverse of La, so that q = M,a - e.

(A.13)

It now follows, using (A.9) where necessary, that (A.14)

(r’, W = (q, L,q) - (a, Me+ But, (r, (L-LJ’r)

= (Lclq, (L’-LJ’L,q),

(A.15)

= Mo(M,--M’)-’

(A.16)

and, by elementary manipulation, (L’-La)-‘L,

where M’ is the inverse of L’ and I is the identity. (r, (L’-Lo)-‘@

-1,

Hence,

= (q, (MO--M’)-‘q)-(q,

Loq).

(A.17)

Therefore, by addition of (A.14) and (A.17), (z, (L’-LO)-‘r)+(r,

I?) = (q, (M,-M’)-‘q)-(a,

M,u),

(A.18)

which is negative definite so long as (M, -M’) or, equivalently, (L’-L,), is negative definite at each point of V. Finally, to find the stationary value of the functional in (2.9), let r be the solution of (2.7) and let B and e now be the corresponding stress and strain, satisfying (2.5) and (2.6). The stationary value of the functional is then -(r, a), and it follows that - (r,

a) = (Lae - fr, E) = (5, LOS)- (a, a) = 2(E, - E),

(A.19)

from (2.5) and the definitions of iZand 8. REFERENCE HILL,R.

1963

Progress in Applied Mechanics-The Prager Anniversary Volume, p. 99. Macmillan, New York.