Effective elasticity of a solid suspension of spheres: I. Stress-strain constitutive relation

Physica 125A (1984) 381-397

North-Holland, Amsterdam

EFFECTIVE ELASTICITY OF A SOLID SUSPENSION OF SPHERES I. STRESS-STRAIN CONSTITUTIVE RELATION

R.B. JONES

Department of Physics, Queen Mary College, Mile End Road, London El 4NS, England

R. SCHMITZ

The Rockefeller University, 1230 York Avenue, New York, NY 10021, USA

Received I5 August 1983

We consider the problem of the elastic behaviour of a random medium composed of identical

spherically symmetric inclusions, each made of a linear elastic material with varying shear and bulk

moduli, distributed randomly in a uniform matrix made of a linear elastic material with moduli h,

K,,. We consider the response of a finite sample to an externally imposed incident displacement field

and by eliminating the external field obtain a linear constitutive relation between mean stress and mean

strain in the material. Using the cluster expansion method of Felderhof, Ford and Cohen we show

that in the thermodynamic limit of an infinite system there is a well defined effective elasticity operator

for the medium expressible as a sum of terms which involve successively the problem of 1,2, . , p,

inclusions in an infinite matrix. The elasticity operator is given in detail through second order terms in the number density of inclusions. By passing to a long wavelength limit we show how the constitutive

relation reduces to a local relation expressible in terms of effective moduli peR, Key.

1. Introduction

In the theory of random elastic media one well developed approach to the

calculation of effective macroscopic elastic moduli rests on the use of a variational

principle to obtain upper and lower bounds for these moduli’-3). The advantage

of the variational approach is that it includes both multiphase and polycrystalline

materials and gives results that depend on only a limited statistical knowledge of

the random microscopic distribution of moduli in the medium. If, however, we

restrict the randomness by assuming it to result from a random distribution of

inclusions, each of simple fixed geometry, in a uniform background matrix, it is

possible to adopt a virial expansion approach to the calculation of effective moduli

whereby a systematic expansion in the volume fraction of inclusions is obtained.

For spherically symmetric inclusions this virial expansion method has been studied

0378-4371/84/$03.00 0 Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

382 R.B. JONES AND R. SCHMITZ

extensively in the related problems of random dielectric media4.6) and random

fluid suspensions7m9). For elastic media there are some studies”) to second order

in volume fraction for uniform spherical inclusions, but for general spherically

symmetric inclusions there seems to be no systematic treatment of the virial

expansion to compare with that already given for a fluid suspension of spheres9).

In this and subsequent articles we present a systematic calculation through

second order in volume fraction of the effective shear and bulk moduli for a

medium which consists of a random solid suspension of spherically symmetric (but

not uniform) inclusions in a uniform elastic matrix. Both inclusions and matrix

are assumed to obey the equations of linear elasticity. In this first article we derive

a general expression for a linear non-local elasticity operator relating average

stress and average strain in an infinite medium which on average is homogeneous

and isotropic. In subsequent articles we will present the numerical results of

evaluating our expression in a long wavelength limit which leads to effective shear

and bulk moduli for the medium. In section 2 we establish notation by considering

the problem of one inclusion in a matrix of finite volume V subject to a given

incident displacement field. In section 3 we consider N inclusions in volume V and

solve formally the multiple scattering problem involved which leads to a linear

constitutive relation between average stress and average strain in the medium. In

section 4 we use the cluster expansion method of Felderhof, Ford and CoherP)

to re-express the linear stress-strain relation in a form which enables us to pass

to the infinite volume limit without divergent or ambiguous integrals appearing.

In section 5 we obtain the long wavelength limit of the constitutive equation which

is a local relation defining effective elastic moduli for the averaged medium.

2. One-body problem

We first consider the case of one spherical inclusion in order to establish a useful

notation and to recall some definitions from our earlier treatment”) of the

one-body problem. Consider a cubical or otherwise regular volume Vof a uniform

linear elastic material with shear and bulk moduli ,LQ and K~ respectively. A

given displacement field U, in V is determined uniquely by giving its value on the

surface of V. Next we imagine that within a spherical region of radius a centred

at R the original material is replaced by another linear elastic material with

position dependent moduli p(r), rc(r) which are spherically symmetric about R. Now impose the same surface displacement field at the surface of V and ask what

is the new displacement field u inside V. The field u(r) is to be determined by

solving the linear equations of elastic equilibrium which can be written as”,“)

V-a =o, (2.1)

ELASTICITY OF SUSPENSION OF SPHERES 383

with

rr ZB = ~(r)[Q+ + a,u, - $7 . US,,] + Ic(r)V - US,, . (2.2)

The position dependent moduli are constant outside the inclusion and spherically symmetric inside

P(r) = bLg, K(r) = rco, (r - RI > a,

p(r) = Air -RI), K(r)=x(lr-RI), Ir-Rl<a. (2.3)

If we define the functions

iI = P(r) - A y i(r) = K(r) - Kg, (2.4)

which vanish outside the inclusion, and if we introduce the stress s induced by the inclusion as

s&r) = b(r)[a,us + a,u, - ;V . US,,] + k(r)V - I&,, (2.5)

then the equilibrium eq. (2.1) can be written as

~V’U + (po/3 + rco)W * II = -F(r) , (2.6)

F(r) = V . S(r) . (2.7)

In eq. (2.6) we have re-expressed the problem as that of the equilibrium of a uniform medium subject to a force density F(r). The definition of F in (2.7) shows it to vanish outside the inclusion and that it may contain delta-functionlike contributions at the surface of the inclusion. The induced stress s here is analogous to the polarization in the dielectric problem6).

The formal solution to (2.6) may be obtained with the help of the Green tensor for volume V, GV(r, r’), defined as the solution to

poV2G,Y + (k/3 + rc,)a,a,G,YB = -6,,6(r -r’) (2.8)

satisfying the boundary condition that G ‘(r, r’) vanishes for r on the surface of V. Here and below the summation convention holds for repeated tensor indices. For later use we note that in an infinite and unbounded medium G m can be written as

G$(r, r’) = G,“,(r -r’)

1

i

2 + a0 a,, %I =G l+ccOIr-r’l+l+a,

(x, -X3($ - x;> -- - I )r--‘I3 ’ (2.9)

(2.10)

where

1 Ko

ao=3+G.


The solution to (2.6) which satisfies the boundary condition at the surface of V can be written formally as

u(r) = uO(r) + s

G ‘(r, r’) * F(r’) dr’ , (2.11)

Y

or, in a condensed notation,

u =u,,+G’.F. (2.12)

The incident displacement field u, is the analogue of the incident electric field in the dielectric problem6).

From the definitions (2.5) and (2.7) it is clear that F itself is linearly dependent upon the actual displacement field U. We express this formally by writing

F(r) = s

K(r, r’) - u(r’) dr’ ,

F= Km u,

where

(2.13)

K&r, r’) = kJ(jI[f!JJ,, + a&, - ;s,,a,] + r26,,$)6(r - r’) . (2.14)

Combining (2.12) and (2.13) gives

u=u,+G”*K.u (2.15)

with formal solution

u=(Z-GY*K)-l-u,,. (2.16)

By use of (2.16) in (2.13) we can write the induced force density Fin terms of the incident field u,, as

F = R ‘u,, (2.17)

where

R=K.(z-GV.K)-‘=(I-K.GY)-‘.K. (2.18)

In terms of R we can express (2.16) as

u=(Z+G”~R)~u,. (2.19)

We conclude this section by re-expressing the results (2.17) and (2.19) in terms of strain field and induced stress rather than displacement field and force density. For this purpose we note that the operator K can be expressed’) as

K&, r’) = apX,,&r, ,‘)a; , (2.20)


where the fourth rank kernel X is given as

XUpy&rY r’) = {ji(r)[6,%8 + QI,fl - %,&pl + WN&B)~(r - r’) , (2.21)

and a; denotes a/ax;. We note the following symmetry properties of X:

(2.22)

It follows from (2.18) that R can be expressed analogously as

R&r, r ‘) = Q%pys(r, r’)aG . (2.23)

Although .%? cannot be written explicitly it follows from (2.18) that it has precisely the same symmetry properties as given in (2.22) for X. Both X(r, r’) and W(r, r’)

vanish whenever either argument r, r’ lies outside the inclusion. Introduce the strains fields e,, and e associated respectively with u,, and U,

e,&) = fK&(r) + ~p,(r)l . (2.24)

By use of (2.7), (2.17) and (2.23) we can express the induced stress s as

+dr) = s %bev(r, r’)eo,,(r’) dr’,

s = B - e, . (2.25)

By differentiating eq. (2.11) and then using (2.7) together with an integration by parts we can express e in terms of s by

e&r) = co,,(r) + s

+?&&, r ‘)s& ‘) dr ’ ,

e =e,+%v-s. (2.26)

The Green tensor 9’ is given in terms of G FJ by

%?y = -;[a@;~; + a,a;c& + a,a;q + a,a;c,',] . (2.27)

By combining (2.25) with (2.26) we obtain e in terms of e. as

e =(2+9”.9P)*e,. (2.28)

Eqs. (2.25) and (2.28) are the desired re-expressions of (2.17) and (2.19), respectively.

3. The many-body problem

In this section we consider the situation when there are N identical spherical inclusions in the volume V. Let the ith inclusion be centred at Ri and denote the


configuration of all inclusions by X = {R,, R,, . . . , RN}. We describe the statistical properties of the inclusions by assuming that the configuration X is a random variable with a normalized probability density P,,,(X),

s P,&Y)dX=l. (3.1)

The distribution P,&%‘) is chosen so that no two inclusions overlap or intersect the surface of the volume V. For a macroscopic theory of the elastic properties of such a medium we seek a description in terms of the averaged microscopic stress and strain fields. These average fields are most conveniently computed as an average over an ensemble of realizations described by the density P,,,(X). For a large enough sample (V+ CO) the ensemble average will be equivalent to a volume average by the usual ergodic hypothesis. To compute these average fields we must first generalise eqs. (2.25) and (2.28) to the case of N inclusions when the induced stress s and the strain e are functions of the configuration X.

Just as in the case of one inclusion we first re-write the equilibrium equation for the medium in terms of induced forces. In doing this it is convenient to identify the contribution of each inclusion separately. Thus we define

jii(r) = O(a - jr - &l@(r),

i’(r) = @(a - Ir - R,l)ll(r) , (3.2)

where 0 is a Heaviside step function; both bi and rc”’ vanish outside the ith inclusion. Define induced stresses s’ by

Shs = /i$3& + aflu, - fV - u&] + Is * us,, ) (3.3)

induced force densities by

F’= v ‘S’, (3.4)

and corresponding total force and stress by

s(r, X) = f s’(r, X) , i=l

F(r, X) = t F’(r, X) . ,=I

(3.5)

(3.6)

Here we have explicitly indicated the dependence upon field point r and configuration X; for the sake of compactness we will often not indicate such dependences explicitly but they are to be understood in what follows. The equation of equilibrium for the medium subject to incident field U, is still (2.6) with formal solution (2.11) but F is now the total induced force density (3.6).

As in section 2 the force density F’ is given in terms of the actual displacement


field u by

F’=K’.rr, (3.7)

where K’ is defined as in (2.14) with the replacement b, Iz+fii, Ic”‘. To obtain the analogue of (2.17) we must take account of the fact that for the ith inclusion the incident field is not u,, but u, modified by the presence of the other N - 1 inclusions. We achieve this by a simple multiple scattering calculation just as in the case of a fluid suspension’). We use (3.6) to express (2.12) as

u=uo+Gv. F=u,,+x G”.Fj+G”-Fi. /#i

With the definitions

uj= G”.FJ, (3.8)

u’,=u,+ c uk, (3.9) k#j

we express (2.12) as

u=u;+ul=~;+Gv.Fi. (3.10)

From this result we see that ub serves as the incident displacement field for inclusion i. Using (3.7) and proceeding as in section 2 we derive

Fi=Ri.& (3.11)

where

Ri=(j_Ki.GY)-i.Ki. (3.12)

Next pass from force density and displacement to stress and strain writing

F’= V .s’, (3.13)

and

si = W1 - eb. (3.14)

The strain field 66 is formed from u; while the kernel W’(r, r’) is related to R’ as in (2.23). We note that W’(r, r’) vanishes if either field point r, r’ lies outside inclusion i, that is if Ir - Ril > a or lr’ - Ril > a. If we now use (3.13) and (3.8) in (3.9) then integrate by parts and differentiate, we obtain

eb=e,+ 1 S”-sj. jti

(3.15)

The coupled equations (3.14) and (3.15) can be solved for s’ in terms of eo. The


result is

sic @se,,

where aB’ is the solution to the coupled integral equations

(3.16)

p=@i+ C wi.~V.@. (3.17) J*’

For the total induced stress s as given in (3.5) we have the result

s = R * e, ,

where

R= 5 A?. i=l

(3.18)

(3.19)

From (3.10) and (3.15) we can express the strain field e corresponding to u as

e=(1+9v*W).e,. (3.20)

The results (3.18) and (3.20) are the desired generalization of (2.25) and (2.28) to the case of many inclusions.

The fields s(r, X) and e(r, X) depend upon the configuration X through the operator R(r, r’; X). We can now average the induced stress and strain over the ensemble of realizations in the following way

(s(r)) = s

p&V(r, X) dX,

s

(3.21)

(e(r)) = P,+,(X)e(r, X) dX.

Using (3.18) and (3.20) gives for these averages

(s) = (R>*e,, (3.22)

(e)=(Z+~v~(W))~e,. (3.23)

The incident field e,, can be eliminated now to give a macroscopic constitutive relation between the average strain and average induced stress in the medium,

(s)=(R)~(l+Sv~(R))-*s(e). (3.24)

Defining a new kernel Wiind(r, r’) by

~i”d=(aB).(I+~v.(aB))-‘, (3.25)

we have a non-local elasticity operator relating average strain and induced stress

(S(r)) = s

wiind(r, r’) * (e(r’)) dr’ . (3.26)


The operator wind is the analogue of the electric susceptibility operator in the

dielectric problem of Felderhof, Ford and Cohen’). The eq. (3.24) is analogous also to a result of Bedeaux, Kapral and Mazur’) in the fluid suspension problem.

We conclude this section with the definition of an effective elasticity operator for the averaged medium by using the equilibrium eq. (2.6) together with (2.7). First express (2.6) in the form

V*V,*e = -F= -V-s, (3.27)

where the unperturbed elasticity operator V,, for the matrix material is defined as

W,,(r, r’) =(2pOP + 3rc,Q)d(r -r’) (3.28)

with constant tensors

P u&y = S&Y + &,&,) - f&,~, 9 (3.29)

Qor~py = f&& . (3.30)

The tensors P, Q are invariant tensors under rotations of coordinates and are orthogonal projectors in the space of symmetric second rank tensors. If we average eq. (3.27) and use (3.26) we obtain

V~%o~(e)=-V~(s)=-V~Vi”d~(e)

or

V*(V,+W”d)*(e)=O. (3.31)

This is the averaged equilibrium equation for the medium where we can identify the averaged total stress (a) as being related to the averaged strain (e) by a non-local elasticity operator %?“(r, r’)

(u)=Wfl.(e), (3.32)

qp = $go + qp”d. (3.33)

4. Cluster expansion

The results of section 3 hold for N inclusions in a finite volume V. For a statistical theory based on ensemble averaging to be valid we must now pass to the thermodynamic limit of an infinite system with I/+ cc, N+co but the number density of inclusions, N/V = n,, finite. In infinite volume, however, the Green tensor g@-r? behaves as (r-f’1 -3 at large distances leading to ambiguous conditionally convergent integrals in any naive low density expansion of the results of section 3. This convergence difficulty has received much attention in the


analogous problems of a dielectric medium or a fluid suspensiotr-‘). We have found the most transparent and useful method for handling this difficulty to be the cluster expansion method of Felderhof, Ford and Cohen6) which has been used successfully in the dielectric problem6) and in the fluid suspension problem’). These authors show by their treatment of the dielectric problem that by first cluster expanding wind as given in (3.25) for the finite system and then rearranging terms one can obtain a series expression for wind which involves only absolutely convergent integrals in the infinite volume limit. In fact this method of proof shows that %Ze”(r - r’) falls off at least as rapidly as Ir - r’lm6 giving a short ranged effective elasticity kernel.

For details of the cluster expansion method we refer to ref. 6 where formal proofs of results are given. In this section we will summarize the relevant results as applied to the elasticity problem. The cluster expansion itself is carried out directly on the quantity (R) in eq. (3.25). The kernel R(r, r’; X) depends upon the field points r, r’ as well as the configuration X. In what follows we will not explicitly write the field point dependence but will emphasize the configuration dependence by writing

R(r, r’; X) = R( 1,2, . . . , N) = i @( 1,2, . . . , N) , i=l

(4.1)

where 1, 2,..., N is short for R,, R2,. . . , R,. For identical inclusions R is a symmetric function of the labels 1, 2, . . . , N. If no inclusion is present R vanishes and for one inclusion it reduces to W as given in eq. (2.23). We indicate this by writing

R(O)=O, W(l)=S’, (4.2)

where 8 denotes the empty set of labels. Cluster functions T associated with R’ are defined recursively by setting

R(9) = 1 T(9) 9 (4.3) 9cB

where the sum is over all sets of labels 9 which are subsets of the set of labels 9. The inverse relation expressing Tin terms of W is

T(M) = c (- l)“-“R(A) (4.4) Yc”V

with n, m the number of labels in the sets JV, .M respectively. For small sets of labels we have explicitly

U@) = 0 ,

T(l) = R(l) >

J-(1,2) = R(l,2) - R(l) - R(2).

(4.5)


The average (W) may now be decomposed as a sum of cluster contributions. Using the symmetry in inclusion labels of both R’(X) and PN(X) for identical inclusions we write

(R(l,2,... , N)) = s

PdX)R(l, . . . , N) dX

dR, . ..dR,n(l,2 ,..., s)T(1,2 ,..., s)

with partial distribution functions n(1, . . . , s) defined as

N! n(l,2,...,s)=--- ... (N _ S)!

s s dR,,, . . . dR#AX) .

(4.6)

(4.7)

If we use (4.6) in (3.26) and also expand (I + ‘S ’ * (R))-’ in geometric series we obtain an infinite sum of contributions to wind, each composed of averages of the cluster functions Tconnected by the Green operator ‘S “. In the infinite volume limit the explicit evaluation of these terms would require the explicit solution of the problem of p inclusions in an infinite medium for p = 1, 2, . . . . . . . We have given elsewhere a general explicit solution for one inclusion”). From this it is possible by the method of reflections13) to solve the two inclusion problem. Thus we can hope to evaluate those terms in the cluser expansion which involve only a one or two inclusion problem. This suffices to compute %P to second order in volume fraction of inclusions, at least for long wavelength stress and strain fields. These low order contributions can be written simply. For (R) we require

and

dRln(l)T(l)+; dR,dR,n(l,2)J-(1,2)+.*.,

from (2 + % ’ * (Ii?))-’ we require only

(4.8)

(Z+~V~(R))-l=(Z++V~ dR,n(l)T(l)+...)-’ s

=l-BV. s dR,n(l)r(l) + * * * . (4.9)

Combining these gives for wind

wind = s

dR,n(l) T(l) + + iJJ

dR, dR,n(l, 2)T(l, 2)

-2[~dR,.(l)~(l)]+‘~[~dR~~(2)~(2)]}+~~~. (4.10)

Using (4.5), (4.1) and the symmetry with respect to inclusion labels we can express


this as

wind = dR’n(l)W’(l) + s

dR, dR,n(l, 2)(W’(l, 2) - R’(1))

- dR,n(l)W’(l) .SV* 1 [S dR,n(2)@(2) II + . . . . (4.11)

It is convenient to express this result as a two step average, first a conditional average with the position of inclusion 1 held fixed and then a final average over R,. We write

%Yiind(r, r’) = s

dR,n(l)T”(r, r’; 1) (4.12)

with the kernel TV given as

n(l)~“(l)=n(l)aB’(1)+ dR,n(l,2)[W’(l,2)- B?‘(l)- aB’(l)*%“*R”(2)] s

+ dR,[n(l,2)-n(l)n(2)]R’(l)~~V~R2(2)+~~~. s

(4.13)

The kernel T” has the property that it vanishes if the field point r lies outside inclusion 1.

In passing from (4.11) to (4.13) we have rearranged the series by adding and substracting terms to the integrands. It is this step which allows us now to take the thermodynamic limit (V+ co) of (4.13) giving absolutely convergent integrals. The separate terms of (4.11) would give only conditionally convergent integrals. Let us assume henceforth that our material is statistically homogeneous and isotropic in the infinite volume limit. For such a material we will have

41) = no 3

n(lt 2) = &(lR, - &I), (4.14)

where g(lR, - R2() is the pair distribution function for the inclusions. In the thermodynamic limit the result (4.13) becomes

n,P(l) = n,a8’(l)+n; s

dR~(~R,-Rz~){~‘(l,2)-aB’(l)-R’(l)~~em~z(2))

+ n; s

dR,[g(IR, - R21) - l]@(l) * ~3~ * R2(2) + . . . . (4.15)


The Green tensor 3 m(r - r’) follows from (2.9) and (2.27). Since the Green tensor G” (r - r') behaves as jr - r’l-’ for small Ir - r’l it is clear that %a will possess a delta function singularity when Ir - r’l vanishes. We express this in the following manner. If&,(r) is a well behaved test function in the sense of generalised functions14), then

+ ljz s

g&& - r’)&(r’) dr’ , (4.16)

where in the second term we omit from the r’ integration a sphere of infinitesimal radius t about r and then take L 40 after integration. The invariant tensors P, 0

are given in (3.29) and (3.30). The kernel gm(R) is given by

Q&m = 1

32nv0( 1 + a0)R3 4[“l&&P - &J, - &,&3yl

- $ [6,,R,R, + G,,R,RB] + 6(1;2d(o) [&,R,R, + d,,R,R,

+ &,R,R, -I- 6,R,R,] -I- $ R,R,R,R, . (4.17)

In the following ,article we will use the result (4.15) to give quantitative expressions for the effective elastic moduli to order no’. There are many higher order terms, however, which also involve only the solution of a one or two inclusion problem. Because of the possibility of summing subsets of these terms to all orders’) it is helpful to have a formal expression for wind to all orders. This can be achieved by simply following the method of Felderhof, Ford and CoherP) in their treatment of the electric susceptibility kernel in the dielectric problem. We close this section by summarizing the formulae which give wind to all orders. The notation is as close as possible to that of ref. 6 so that details may be checked there and also in the corresponding discussion of the fluid suspension problem9). The result of the cluster expansion of (R) together with the geometric series expansion of (1 + 3 ‘. (R))-’ can be written to all orders as

(4.18)

where the sth term involves s inclusion labels 1, 2, . . . , s. Each term UF” can be expressed as a sum over ordered partitions (B) = (~~1~~1 . . .IB~) of the s inclusion


labels

$$F”= . . . dR,. s f

. . dR, c ( - l)k-‘,(B,) T#,) (B)

.s”.n(B,)~((B,).9V.....~e”.n(B,)T(B,). (4.19)

In the ordered partition (B) the Ri are mutually disjoint subsets of the s labels with the restriction that label 1 occurs in B,. The number of subsets in the partition is denoted by k and the ordering refers to the order of the B,, B2,. . . , Bk. Within each Bi the labels are not ordered. Thus for s = 3, (1 12, 3) and (113,2) are the same ordered partition but (1 12 13) and (1 1 3 12) are distinct. Just as in going from (4.10) to (4.11) above so in obtaining (4.19) we have used the symmetry with respect to inclusion labels to bring label 1 into the first subset B, and to reduce the first cluster operator to il;(B,) which is defined by

T,(r, r’; B,) = 13(a - (r - R,]) T(r, r’; B,) . (4.20)

The operator r, depends on r only through the combination r - R, and vanishes if r is outside inclusion 1.

To obtain absolute convergence of integrals in the thermodynamic limit the terms of the series (4.19) must be re-arranged as was done in passing from (4.11) to (4.13). Felderhof, Ford and Cohen”) have found an elegant way to express this re-arrangement to all orders. Consider the partitions (B) of s labels written down in rows as follows. In row one write the partition consisting of one subset, in row two write the partitions consisting of two subsets, and in rowj write the partitions consisting ofj subsets. Introduce a partial ordering amongst the (B) by defining (B’)>(B) if (B’) is the same as (B) or if (B’) can be obtained from (B) by removing one or more vertical slashes from its representation (B) = (B, I Bz I . . . I B,J. By the same method of proof as in ref. 6 one can show that the integrand in (4.19) can be written as

;(- l)“-‘n(B,)~~(B,).~Y.....s”.n(B,)~(B,)

=~b(B,/B,I...lB,)c(B,IB,I...IB,), (4.21)

where both sides involve summation over all partitions of s labels and the block distribution function b(B, I B2 I . . . ) B,J and the chain operator c(B, I B2 1. . . I Bk)

are defined as follows:

b(B, I.. . 1 Bk) = 1 (- l)'-'n(B;)n(B;) . . . n(B;,), (4.22) W)~W

c(B,I.../B,)= 1 (-l)k-~~,(B;).beV.~(B;).gV.....,,V.~(B;,). m G cm

(4.23)


According to the definition of the partial ordering, the summation in (4.22) is a sum over partitions (B’) including (B’) = (B) together with all partitions (B’) that can be obtained from (B) by removing slashes while in (4.23) the sum is over partitions (B’) from which (B) can be obtained by removing slashes including (B’) = (B). The final expression for U:” is then

(4.24)

with

%Fd(R) = . . . s s

dR, . . . dR,b(B)c(B). (4.25)

The Green tensor s"(R) has the same R -3 long distance behaviour as the Green function for the dielectric problem. Making the weak assumption that the partial distribution functions (4.7) have the cluster decomposition property, we carry over to the elasticity problem the key result of Felderhof, Ford and Cohen6) that at large separations of inclusion labels Ri either the block distribution function b(B) or the chain operator c(B) vanishes fast enough to ensure absolute convergence of the integral (4.25) in the infinite volume limit. It is easy to check that the s = 2 term of (4.18) gives the same result as already given in (4.12) and (4.13).

5. Long wavelength limit

In (3.32) we have given the constitutive relation between mean stress and strain for a finite system. The results of section 4 enable us to pass to the thermodynamic limit obtaining a well defined short ranged but non-local kernel Ueff. Since we have assumed the medium to be statistically isotropic and homogeneous it follows by translation invariance in the thermodynamic limit that we’ is a difference kernel,

%P’(r, r’) = %P(r - r’) (5.1)

and that we can simplify the constitutive relation (3.32) by Fourier trans- formation. We write Fourier expansions

(e(r)> = s eiq"(&q)) dq,

1 (4q)) = (27~ I eeiq.‘(e(r)) dr ,

and use the convolution theorem to obtain (3.32) in the form

(6(q)) = (2~)3@eff(q) - (wz)) 3

(5.2)

(5.3)


where

(2n)3@e*(q) = s

df e-i~‘%“ff(f) . (5.4)

The q dependence of (e(q)), (u(q)) d escribes the spatial variation of the mean macroscopic fields. For a useful macroscopic description of the medium these average fields must vary extremely slowly over length scales of order a, the radius of an inclusion. This condition can be ensured by applying only slowly varying incident external fields e,. For fields whose spatial variation is of long wavelength only (qa G 1) we need the constitutive relation (5.3) only near q = 0, which corresponds to uniform fields. Henceforth we will consider this problem in the uniform field (or long wavelength) limit at q = 0. We set

(e(r)) = e”, @(q)) = @d(q) (5.5)

and obtain from (5.3) a uniform stress uU related locally to e” by an effective elasticity tensor,

(i(q)) = a”6(q) 2 (5.6)

b” = (2x)3C&K(q)(,,, * e” . (5.7)

From (5.4) we have that

(2n)3@eff(q)lp=0 = s s

dlweff(l) = d(r - r’)%“(r - r’) . (5.8)

By the assumed isotropy of the medium the integral in (5.8) must be an invariant tensor under rotations of coordinates. It follows then from the tensor symmetry properties of we’ that the integral must be a linear combination of the invariant tensors P, 0 introduced in section 3,

s dl GV(f) = 2pefiP + 3~‘“0 , (5.9)

thus defining effective shear and bulk moduli for the averaged material. Recalling the results (3.33), (3.28), (4.12) and (4.13) we have

s dl%eff(I)=2@+3rc,C2 +~~Sd(r-I’)SdR,T’(r,r.;l). (5.10)

The kernel r”‘(r, r’; 1) is not translation invariant since in it we have fixed the position of inclusion 1 at R, while averaging over the position of all other inclusions. However r” depends upon r only through the combination r - R,,

therefore the double integral in the last term of (5.10) can also be expressed as

Id(r -r’)[dR,rm(r,r’; l)=I/drdr’rm(r,r’; 1). (5.11)


Our final expression for the effective moduli which describe the local mean stress-mean strain relation in the random medium at long wavelengths is

2peflP + 3rc”W = 2p,,P + 3rc,Q + n, ss

dr dr’f O”(r, r’; 1). (5.12)

The kernel f m is given to second order in (4.15) and can be obtained to all orders from the expressions (4.12) (4.18) (4.24) and (4.25). In the following articles we will show how to evaluate pLeff, rce’ to order rri in terms of the scattering coefficients introduced earlier”) to describe the general solution of the one-inclusion problem.

References

1) Z. Hashin, App. Mech. Rev. 17 (1964) 1. 2) P.H. Dederichs and R. Zeller, Z. Physik 259 (1973) 103. 3) E. Kroner, J. Mech. Phys. Solids 25 (1977) 137. 4) V.M. Finkel’berg, Sov. Phys. JETP 19 (1964) 494; Sov. Phys. Doklady 8 (1964) 907. 5) D.J. Jeffrey, Proc. R. Sot. Lond. A335 (1973) 355; A338 (1974) 503; in Proceedings of the Second

International Symposium on Continuum Models of Discrete Systems, J.W. Provan and H.H.E. Leipholz, eds. (Univ. of Waterloo Press, Waterloo, Canada, 1978).

6) B.U. Felderhof, G.W. Ford and E.G.D. Cohen, J. Stat. Phys. 28 (1982) 135. 7) G.K. Batchelor and J.T. Green, J. Fluid Mech. 56 (1972) 401. 8) D. Bedeaux, R. Kapral and P. Mazur, Physica 88A (1977) 88. 9) R. Schmitz, Dissertation R.W.T.H., Aachen, 1981.

10) L.J. Walpole, Quart, Joum. Mech. and Applied. Math XXV (1972) 153. J.R. Willis and J.R. Acton, Quart, Joum. Mech. and Applied Math. XXIX part 2 (1976) 165. H-S Chen and A. Acrivos, Int. J. Solids Structures 14 (1978) 331, 349.

11) R.B. Jones and R. Schmitz, Physica 122A (1983) 105, 114. 12) L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Pergamon, London, 1959). 13) R.B. Jones, Physica 92A (1978) 545. 14) I.M. Gelfand and G.E. Shilov, Generalized Functions Vol 1. (Trans. Saletan) (Academic Press,

New York, 1964).

Documents

Effective elasticity of a solid suspension of spheres: I. Stress-strain constitutive relation