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18 R. ROSCOE

has in general to be taken into account. The present work is restricted, however,to cases in which the frequency of the disturbanee is sufficiently low for inertialeffects to be neglected.

2. THE COMPLEX MODULI AKD COMPLIANCF.S

The system here considered is subjected to a disturbance varying sinusoidallywith time (frequency w/~QT), so it is convenient to use complex Cartesian tensorsU, E, u varying as exp (iwt) to represent displacement, strain and stress respectively.These are t.aken to obey the same linear constitutive equations and kinematicequations as the actual ~~is~la~ern~~lt, train and stress in each phase, and to obeythe same boundary conditions. Thus the actual d~s~~lacer~le~li:, train and stressare the real parts of these tensors.

Since the phases are mechanically isotropic, the constitutive equation for eachyields a relation of the form

u == 2/+* E + 3fcr* EV, (1)

where E, are respectively the deviatoric and isotropic parts of E and the subscriptr refers to the phase. The complex rigidity modulus pr* and complex bulk modulusKy* may be written in terms of their real and imaginary parts :

pr = @ + /&*, Kr = Kr + iKr. (2)

As an a~te~ati~-e to (I), the stress-&rain relation may be written

E = t.j$.* ot f 3 z,* ue (3)

where a, u are respectively the deviatoric and isotropic parts of a. The complexcompliances jr*, l , are the reciprocals of TV+.*, ~*. They may be expressed in termsof their real and imaginary parts :

.lr * = jr - ijrn, &.* = &. - il r > (4)

and si nce pr' , pr , Q , I + are non-negative it follows that j,, jr, lr Ito are alsonon-negative.

3. THE REPKESEXTATIVE VOLCME ELEMEKT

Suppose the con1posit.e system is subjected to oscillatory displa~en~ents ti$over its surface S. Then as its phases are firmly bonded the displacement withinS is continuous. Application of Gausss divergence theorem thus gives

:s

(Ui ??,& Uk Q) dS = 27 s

Eik dV,, (5)

where ~6 is the unit vector drawn normally outwards from the surface element dS,and dV, is a volume element of the rth phase. Since the surface tractions on phase

boundaries are in equilibrium, the theorem also gives the following results forthe surface tractions (force per unit area) Tt on S :

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Bounds for the real and imaginary parts of Lhe dynamic moduli

ITz Gg dS = 2

I(ai9 cti + UZ~J Gi) W,,

r

19

(7)

where Et3 Q represent the complex conjugates of us, EZ~. The second term in eachint,cgral may be neglected since the kinematic equation gives aij,i dV, as equal tothe mass-acceleration of the material within dV,., and here inertial effects areconsidered to be negligible.

The concept of macroscopic homogeneity may be concisely expressed in termsof representative Yolume elements, as defined by HILL (1963), HASIXIN 1964)and Wri~~ord~ (1966). Thus a composite system subjected to prescribed surfacedisplacements may be said to behave on a macroscopic scale as a homogeneousbody if (and only if) its volume can be broken down into such elements. It is notnecessary here to consider the very special conditions (involving the size of thephase particles and their statistical geometry as well as the specified distributionof surface displacements) under which this may be possible.

Now let the surface S considered above be subjected to such surface displace-ments as would be consistent with a uniform strain of homogeneous material withinit. Then it follows from (5) that this uniform strain is equal to (E), where

Further, let conditions be such that the volume V enclosed by S is itself a represen-tative volume element. This implies that the smoothed values of Tg over S are

such as would be consistent with a uniforr~ ~~istribution of stress in a homogeneousmaterial within it. Then it follows from (6) th t this uniform stress is equal to

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20 R. ROSCOE

2/L. E) (?> + 3K E) a ) = _: L?s

2/ *, . E E + 3K; E 2) dV,., (13)7

and a relation of the same form must hold for the imaginary parts of the moduli.

As an alternative to (ll), the relation between (u) and (E) can be put in theform

(E) = *j* (a) + 1* (u), (I++)

where j*, 1 are the macroscopic compliances of the composite material. The com-plex conjugates of the expressions for strain given by (14) and (3) can be insertedin (lo), and a similar argument then gives

3 j (u} (e) + 3 I (a) (5) = Z 1 (4 j,. u O + 4 l,. U 3) dV, (15)7

together with a relation of the same form for the ima.ginary parts of the compliances.

4. BOUNDS FOR THE MODULI AND COMPLIANCES

It is now convenient to introduce a tensor q with deviatoric and isotropicparts defined for a point in the rth phase by

and (17)where C, is the volume fraction of the rth phase. Then it follows from (8) that

.X qdVr=o=Z ijdV,,s s

(18)r 7

where 11 is the complex conjugate of q. Insertion of (16) and (17) in (13) thereforegives

2,A (E) (ii) + 3K El) c ) = 2 E)2)

+ $ Zs

(2~~ I ? + 34. E q 7) dV,. (19)I

Now p;, K,. and the tensor products are non-negative, and so on considering thecase when (6) is deviatoric it is seen that

while consideration of the case when (c) is isotropic gives

(21)

A similar argument based on the equation obtained by changing real to imaginaryparts of the moduli in (13) gives

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22 R. ROSCOE

parallelism between the equations governing infinitesimal elastic deformations andthose governing viscous flow under quasi-static conditions.

Considerable improvements exemplified by the work of WALPOLE 1966),can be made on the Reuss and Voigt bounds in the purely elastic and viscous cases.This suggests that the bounds obtained here for the viscoelastic case by an ele-mentary method are by no means the best that may be found.

HASHIN, 2.

HILL, R.

WALPOLE, L.J.

REFERENCES

1964 Appl. M ech. Rev. 17 1.1965 Proc. 4th I nt. Congr. Rhrol ogy Edited by LEE, E. H. and

COPLEY, A. L.), Part 3, p. 30. Interscience, New York.

1952 Proc. phys. Sot. A 65 349.1963 J. M ech. Phys. Soli ds 11, 357.

1966 J. M ech. Phys. Solids 14, 151.