Advances in Applied Mechanics, Volume 21

Advances in Applied Mechanics Volume 21

Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERM” HILL RODNEY L. HOWARTH T. Y. Wu

Contributors to Volume 21 B. L. N. KENNETT L. J . WALPOLE J . R. WILLIS

AD VANCES I N

APPLIED MECHANICS Edited by Chia-Shun Yih DEPARTMENT OF MECHANICAL ENGINEERING AND APPLIED MECHANICS THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

VOLUME 21

1981

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

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9 8 7 6 5 4 3 2 1

Contents vii

LISTOF CONTRIBUTORS

Variational and Related Methods for the Overall Properties of Composites J . R . Willis I. Introduction 11. 111. IV. V. VI. VII. VIII.

Preliminary Definitions Methods Related to the Classical Variational Principles Methods Related to the Hashin-Shtrikman Variational Principle Self-consistent Estimates Generalizations Problems Which Lack Convergence Wave Propagation IX . Recent Developments References

2 3 13 23 42 47 55

64 74 74

Elastic Wave Propagation in Stratified Media B. L. N . Kennett I. Introduction 11. Elastic Waves in Stratified Regions 111. Reflection and Transmission of Elastic Waves

IV. Half-Space Response in Terms of Reflection Matrices V. Inversion of the Transforms VI. Conclusion References

80 83 100

127 152 163 164

Elastic Behavior of Composite Materials: Theoretical Foundations L . J . Walpole 1. Introduction 11. Preliminary Analysis of Tensors and Elastic Behavior V

169 171

vi 111. The Elastic Field of an Inclusion IV. The Elastic Field of a Composite Body V. The Overall Elastic Behavior of a Composite Body References

Contents 188 202 208 236

AUTHORINDEX

243

SUBJECT INDEX

241

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

B. L. N. KENNETT,Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, England (79) L. J. WALPOLE, School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ, England (169) J. R. WILUS,School of Mathematics, University of Bath, Bath BA2 7AY,

England (1)

vii

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ADVANCES IN APPLIED MECHANICS. VOLUME

21

Variational and Related Methods for the Overall Properties of Composites J . R . WILLIS School of Mathematics University of Bath Bath. England

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Types of Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Definitions of Overall Properties . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3 3 1

I11. Methods Related to the Classical Variational Principles . . . . . . . . . . . . . . . A. Small Variations in Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Methods Related to the Hashin-Shtrikman Variational Principle . . . . . . . . .

13 20

A. The Integral Equation and Variational Principle . . . . . . . . . . . . . . . . . B. Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Variational Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 25 33

13

23

V . Self-Consistent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

VI . Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Problems Which Lack Convergence . . . . . . . . . . . . . . . . . . . . . . . . . .

47

A. A Model Problem: Diffusion to a Random Array of Voids . . . . . . . . . . . B. An Integral Equation and Perturbation Theory . . . . . . . . . . . . . . . . . . C . Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 50 55 56 58 60

VIII . Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

A . Polarization Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 72

IX . Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 14

1 Copyright 0 1981 by Academic Press. Inc . All rights of reproduction in any form reserved. ISBN 0-12-002021-1

2

J. R. Willis

I. Introduction A wide range of phenomena that are observable macroscopically are governed by partial differential equations that are linear and self-adjoint. This article is concerned with such phenomena for materials, such as fiberreinforced composites or polycrystals, whose properties vary in a complicated fashion from point to point over a small, “microscopic” length scale, while they appear “on average” (that is, relative to the larger, macroscopic scale) to be uniform or perhaps, more generally, their properties appear to vary smoothly. The determination of these “overall”properties from the properties and geometrical arrangement of the constituent phases generally requires the solution of boundary value problems for randomly inhomogeneous media. The development of approximations is necessary for two reasons. First, even if the geometry of the medium were known exactly, the solution of any particular boundary value problem would be hopelessly complicated. Second, only certain features of the geometry (typically, concentrations of phases and low-order correlation functions) will be known in practice. In this situation, the best approximations are likely to be those that are derivable from a variational principle: this provides the unifying theme of this article. Problems are formulated in terms of operator equations that are self-adjoint and so generate associated variational principles. The equations are then solved, either by formal perturbation theory or by use of some ad hoc simplifying approximation, but usually the approximations are subsequently shown to have some precise variational status. Most of this article is concerned with static problems, for which many of the variational estimates of overall properties actually provide bounds. This article is concerned more with methods and principles than with results for particular media, though illustrative examples for which the algebra is not too complicated are given at a number of points. Whether or not a variational approach is adopted, a problem appears when any attempt is made to replace a finite body by one that is infinite: direct replacement of the Green’s function for the finite body by the infinite-body Green’s function leads to integrals that may converge conditionally in some problems and diverge in others and so cannot be allowed. The approach that is recommended is to start from a precise formulation for a finite body and to apply to this some mathematical analysis; this contrasts with an alternative approach, termed “renormalization,” which attempts to correct the offending terms without detailed reference to the finite-body problem. Much of this article treats the elastic behavior of composites, but it is emphasized that a number of other properties (conductivity, viscosity of a suspension, etc.) are described by the same equations. Extensions to visco-

Overall Properties of Composites

3

elastic and thermoelastic behavior are presented, for both of which the variational characterization given is believed to be new. Problems such as the resistance to flow of viscous fluid through a fixed bed of particles are mentioned and a model problem that involves diffusion is presented in some detail. This displays the same difficulty in relation to divergence of an integral and is the one problem of this type that has so far been approached variationally. In the concluding sections wave propagation is considered, using a formulation that is a natural extension of that used for static problems. This is under development at present and only approximate solutions of the operator equations have so far been obtained. However, this article is concluded with the derivation of a variational principle against which it should be possible to judge the approximate dynamical solutions. It is likely (but this cannot be guaranteed because it has not yet been done) that all of the procedures developed for static problems have their dynamic analogs so that, in the future, a unified dynamical treatment may be possible, which includes the static limit as a special case. 11. Preliminary Definitions

A. TYPESOF COMPOSITES For the purpose of this article, a composite is a particular type of inhomogeneous body, whose properties vary from point to point on a length scale (the microscale)which is small in comparison with overall dimensions. Generally, it will contain n distinct phases and perfect bonding will be assumed across phase interfaces; imperfections in bonding may, of course, be important in practice, but relate rather to the failure of a composite, a subject beyond the scope of this review. Examples of composites include glass fiber reinforced plastics, for which n = 2, the phases being the fibers and the resin, and polycrystals, for which n -+ co,each crystal orientation being associated with a different phase. It is reasonable for these two examples to treat the individual phases as continua and attention will be confined to materials (and phenomena) for which this approach is allowed. Thus, so far as the present work is concerned, a crystal such as MgO would be regarded as a homogeneous continuum. The problems that will be considered for composites may be described broadly as those in which the phases are not discernible individually, though they do, of course, determine overall properties. In the simplest case, the composite may appear macroscopically as a body whose properties are uniform, though more generally its overall properties may

4

J . R. Willis

vary: this would be the case, for example, if the mean concentration of reinforcing fibers varied over a specimen. The determination of these overall properties from those of the constituent phases is the subject of this article; they are defined more formally in the sections that follow. In preparation for this, a brief description is now given of the relevant geometrical properties of the microstructure. The simplest idealization of the microstructure of a composite would be to regard it as periodic, with period 6 much smaller than specimen dimension; for example, the composite may comprise a homogeneous matrix containing a regular array of inclusions. The properties of such a material would either be known with precision at each point or else might be subject to uncertainty only in relation to fixing the position of the inclusions relative to the boundary of the specimen: the uncertainty would be resolved if the position of one inclusion (and perhaps also the orientation of the lattice on which the inclusions are arranged) were given. Manufacturing processes are usually less obliging, however, and produce materials about which only statements “on average” can be made. It is desirable, therefore, to treat a composite as a random medium. For this purpose, any particular specimen is regarded as one with a label a which defines its configuration precisely. The parameter a belongs to a sample space 9’over which a probability density p(a) is defined. Then, any particular property f of the composite (such as the mass density at point x, or the product of mass densities at x and x’)is a function of a and has mean value, or ensemble average,

Of particular interest is the indicator function f,(x)that takes the value 1 if P,(x) of finding phase

x lies in phase r and zero otherwise. The probability r at a chosen point x is then expressible in the form

and the probability Prs(x,x’)of finding simultaneouslyphase r at x and phase s at x’is

Probabilities involving more points follow the same pattern. Mean values of other properties of the composite are expressible in terms of the essentially geometrical functions P,,P,,, etc. For example, if phase r has mass density p , and the density of the composite at x is denoted by p(x), then, in the case

Overall Properties of Composites

5

of n discrete phases,

Similarly, n

(P(x)P(x’)>=

n

C C PrPsPrs(X,X’)* r = l s=l

(2.6)

Correlation functions, of which (2.6) is an example, will be featured prominently in the analysis to follow. Reduced forms for them have been discussed by Miller (1969a,b) for a particular geometric model in which the space occupied by the composite is divided into cells, to each of which a value of r is assigned. However, the work contains more detail than is needed here, particularly for correlation functions of high order, and also restrictions (particularly isotropy) are imposed that will not always apply in the sequel. The form that probabilities such as Pr, take when phases are distributed isotropically has most recently been discussed by Fokin (1979). It should perhaps be mentioned that definitions like (2.2) to (2.5) rely only upon the notion of the random field f,(x) which need not be associated with a cell model, even though many composites may conveniently be viewed in that way. In the limiting case of a continuous rather than discrete variable r, the probabilities P r , Prs, etc. will be interpreted as densities and the sums in (2.4), (2.5) replaced by integrals. A more general approach based upon measure (both here and for the sample space 9’) is a refinement that is unlikely to be needed in practice; it is definitely not required in the present work. The foregoing discussion applies to any composite. However, for a composite which comprises a matrix containing a distribution of inclusions, there is some advantage in developing a description that allows for this feature. Elaborate notation is avoided by considering for illustration a matrix that contains a single set of inclusions, of identical properties, shape, and orientation. If the individual inclusions are labeled A = l, 2, . . . , the configuration of the composite is defined by the positions { x a ; A = 1,2,. . . } of any conveniently chosen point of each inclusion. For ease of description, the point x A will be referred to as the center of inclusion A, regardless of how it is chosen. In place of (2.2), it is now desirable to define a probability density P A for finding an inclusion centered at x A . This is achieved by evaluating the mean number of inclusions centered in any subset U of the region V

J . R. Willis

6

occupied by the composite, to give

where Y u ( k ) is the subset of Y for which there are exactly k inclusions centered in U . Similarly, the joint probability density PA, for finding distinct inclusions centered at x A , X , is defined so that, for disjoint subsets U and W of V,the expected value of the product of the number of inclusionscentered in U with the number centered in W is expressible as

where Y,,,(k, 1) is the subset of Y for which there are exactly k inclusions centered in U and 1in W .In the particular case where the composite is known to contain exactly N inclusions, Y v ( k ) = Y if k = N and is empty otherwise. Then, Eq. (2.7) implies Jv

dxA = N ,

(2.9)

as it should. Also, if U is so small that it can contain at most one inclusion, and if W = V - U , then Y,,,(k, 1) = Y,(k) if 1 = N - k and k = 0 or 1 and is empty otherwise. Equation (2.8) then implies

The conditional density PBIA is defined by the relation PAB= P B ~ A P A ,

(2.11)

with PBIA = 0 whenever inclusions A and B would overlap. It follows from (2.10) that (2.12) again as expected. The discussion so far covers random media more general than composites, in that no particular microscale has been assumed. Suppose now that the dimensions of the region I/ occupied by the composite are large compared with a typical microstructural dimension such as grain size or mean inclusion spacing. The notion of a statistically uniform material may be defined by the requirement that probabilities such as P,(x),Prs(x,x’), P A ,PA, should be insensitive to translations, so that P,, P A become constants, while P,,, PA, become functions of (x’ - x), (xB- xA),respectively. Of course, for any particular finite V , these requirements cannot be met exactly, but it is

Overall Properties of Composites

7

reasonable to assume their validity except in some “boundary layer” region close to the surface 8V of V . Another assumption that also requires V to be large will be made from time to time. This is that a composite possesses no long-range order. The mathematical implication for the joint probability Prs(x,x’) is that (2.13) when x and x‘ are far apart. This notion is independent of statistical homogeneity. An example of a composite for which (2.13) would not be realized is provided by a periodic structure, for which the only uncertainty may be in the precise location of phase r. Similar concepts may be applied to higher order probabilities and also to the densities P A , P A B ,etc., for a composite containing inclusions. Finally, for statistically uniform media, it is usual to make an ergodic assumption that local configurations occur over any one specimen with the frequency with which they occur over a single neighborhood in an ensemble of specimens. Subject to this assumption, the probability P , (which is now independent of x) may be estimated as the volume average of the function f,(x) for one specimen, P,, (which now depends upon x’ - x) is estimated as the volume average of f,(x x”)f,(x’ x”) (the integration being over x”), and so on. This assumption will be invoked at several points in the arguments that follow. It is as well to admit now, however, that precise conditions under which statistical uniformity and ergodicity(in the sense given)are approached asymptotically for large V are unknown at the present time.

+

+

B. DEFINITIONS OF OVERALL PROPERTIES The basic problem under consideration is to determine the response of a specimen or a structure when any kind of generalized loads (which could take the form of imposed forces, or displacements, surface temperature, electric field, etc.) are applied to it. To be specific, this initial discussion will be phrased in terms of the elastic stress analysis of a composite, since this particular aspect of composite behavior is considered more extensively than any other in the present work, but many of the remarks to follow will apply more generally. For the stress problem, then, for any particular specimen, stress components oijare related to strain components eijthrough the tensor of elastic moduli whose components are L i j k l , so that (2.14)

the strain components being related to displacement components uiby eij =

+ ujJ =

(2.15)

8

J . R. Willis

The summation convention for repeated suffixesapplies in (2.14). In (2.15) the suffixj denotes partial differentiation with respect to x j and the brackets applied to suffixes imply symmetrization. To avoid complicated formulas, a condensed notation will be used wherever possible so that, for example, (2.14) becomes u = Le.

(2.16)

In keeping with this shorthand, u, e, and L are referred to as stress tensor, strain tensor, and tensor of moduli, respectively. The tensor of moduli L varies over the region V occupied by the composite, taking the value L, in phase Y. If boundary conditions (tractions, displacements,or some mixture of these) are applied over dV and body forces fare specified over V ,the problem is to determine the stress, strain, and displacement throughout the composite. This is accomplished in principle by solving the equilibrium equation diva + f = 0

(2.17)

together with the constitutive relation (2.16) and the given boundary conditions. The moduli L vary over such a small scale, however, that the solution would provide an excess of detail even if it could be found and it would be desirable to extract from it some “locally averaged” information. This would consist, perhaps, of the values of stress, strain, and displacement, averaged over some representative volume, small on the scale of V , but large on the scale of the microstructure; such averages might be measured, in practice, with strain gauges or transducers. For some applications, it might also be appropriate to have some information on local fluctuations about the averages. For composites with periodic microstructure, these notions can be made precise by using the method of multiple scales (Cole, 1968), in which stress, strain, and displacement fields are expressed as functions of the macroscopic variable x/D and the microscopic variable x/6, where D, 6 represent length scales for the whole specimen and the microstructure, respectively. The fields are then represented as periodic functions of x/6, whose mean values and amplitudes vary with x/D. The approach was initiated by Sanchez-Palencia (1974) and forms the basis of the method of homogenization, comprehensively reviewed by Bensoussan et al. (1978). The method of multiple scales gives equations for the mean values that are consistent with an “overall” constitutive relation u

= Le,

(2.18)

where E is uniform and defined by a prescription that will be given later, together with equations defining the local fluctuation. Furthermore, the multiple scale solution has been proved to be asymptotic to the exact solution in the limit 6/D -+ 0.

Overall Properties of Composites

9

For the random composites with which this article is concerned, no corresponding precision has yet been achieved. There are, nevertheless, physically plausible prescriptions which are consistent with what is known for periodic composites. These are now outlined. In contrast to the periodic case, however, the prescriptions generate stochastic equations for which approximate solutions must be sought. 1. Dejnitions Based on Volume Averages

If it is assumed that a specimen behaves “on average” as though it were homogeneous, with constitutive relation (2.18), the overall moduli may be determined practically by a suitable set of tests in which stress and strain are uniform, in some nominal sense. To be specific, the boundary aT/ of the specimen could be subjected to the displacement u.1 = e..x. (2.19) x E dV, 1J 1’ which would generate the strain field 5 throughout V if the specimen were actually uniform. In fact, e varies over V , but its average over V may be expressed in the form

n

=

I/V J ; z i j d V = zij

(2.20)

by application of Gauss’ theorem, together with (2.19) on 8V. In (2.20) V is also used to represent the volume of the region V , to avoid introducing more symbols. Thus, when (2.19)is applied, the average strain over V is E precisely. Overall moduli E may now be defined by the relation

a = L5,

(2.21)

where denotes the average value of o over V ,when the boundary condition (2.19) is applied. It is interesting to note also that the average value 0 of the energy density over V may be calculated as (2.22) since aij,j= 0. Substitution of (2.19) into the last of these expressions and reversing the argument now gives (2.23) Thus, the moduli L may be defined either from the mean stress or from the mean energy density, if boundary conditions (2.19) are imposed.

J. R . Willis

10

Dually, the specimen could be subjected to boundary tractions compatible with a uniform stress field, so that -

aijnj = aijnj,

(2.24)

x E aV.

The equality 1/V Jv

(Tik

dx = 1/1/

Lv

aijxknj

dS

(2.25)

guarantees that cr indeed has average value 5 over V and overall moduli L may again be defined by (2.20) where, this time, the average value F of the strain is found from the solution of the boundary value problem. It follows also that (2.26) by substituting (2.24)into (2.22). Thus, again, overall moduli may be defined either from the average stress and strain or from the average energy density. These observations, made by Hill (1963a), provide rigorous definitions for overall moduli, for one particular specimen. They relate to tests that can actually be performed and apply whether or not the overall constitutive relation (2.18) is useful for predicting composite behavior when other boundary conditions are applied. Indeed, the two separate boundary value problems defined by (2.19)and (2.24)would yield different values for L, if applied to an arbitrary inhomogeneousmedium. It is plausible, rather than rigorously established, that Eq. (2.18) describes “average” composite behavior when arbitrary boundary conditions are applied and the extent to which the boundary conditions (2.19) and (2.24) generate estimates of L that are in agreement might be regarded as a partial check on the validity of the “overall modulus” concept. It may be remarked, finally, that these two definitions of L agree for the periodic composite in which, away from a boundary layer close to aV, they both generate periodic stress and strain fields with mean values 8, F, which Sanchez-Palencia(1974)used to define E over a single cell. The importance of Eq. (2.23) for the mean value of the energy is that it allows the construction of bounds for the components of L for any particular specimen. The principle of minimum energy states that, when displacements are prescribed over aV, the actual displacement field is that which minimizes the energy functional U* = (2V)-’ Jv e$LijkfeZdx

= (2V)-I

Jv e*Le* dx,

(2.27)

where e* is the strain associated with any displacement field u* over I/ that takes the prescribed boundary values. The simplest displacement u* compat-

Overall Properties of Composites

11

ible with (2.19)is

u* = z..x. V J

(2.28)

1

throughout V ;this leads to the inequality

0 I (2V)-'

s,,

ELEdx = (2V)-'E

sv

L,f,(x)dxE,

(2.29)

r= 1

where f,(x) is the indicator function defined in the preceding section. Thus, if the volume concentration of phase r is cr,Eq. (2.29) shows that Lv - L is positive semidefinite, where the estimate L,, for the overall moduli is the Voigt average (Voigt, 1889)

c crLr. n

Lv =

(2.30)

r=l

Dually, if traction boundary conditions are applied over aV, the complementary energy principle states that the actual stress field is that which minimizes the functional

U,* = (2V)-'

f,,o$!4ijkto~ld~ = (2V)-' J,,@*MU* dx,

(2.31)

where M is the tensor of compliances, inverse to L and @* is any selfequilibrated stress field that satisfies the given boundary conditions. The simplest choice is n* = 8, which yields the inequality U

where M is the overall compliance tensor, inverse to L and Mr is inverse to L,. Equation (2.32) shows that M, - is positive semidefinite, where

m

(2.33) or equivalently, L - LR is positive semidefinite, where LR is the Reuss average (Reuss, 1929) (2.34) The observation that the Voigt and Reuss estimates provide bounds for components of E is due to Hill (1952). Its importance lies not in the quality of the bounds themselves, but in the principle established by their existence: given the definition (2.21) of L, with the boundary condition (2.19), for

12

J. R. Willis

example, it follows without approximation that L, - L is positive semidefinite. The result is useful, however, only when some other information is employed in parallel. The definition of Lv involves only volume fractions and makes no allowance for the detailed distribution of phases. Therefore, at the very least, statistical uniformity must be assumed in addition, before a useful L is even defined. Also, a composite could be composed of isotropic phases, distributed anisotropically. In this case, Lv and LR would be isotropic, though L plainly would not be. Bounds for components of L are still obtained, however, provided that its overall symmetry is known from separate considerations. Implications of the Voigt and Reuss bounds for a composite containing parallel fibers have been discussed by Hill (1964). Better bounds, containing more statistical information on the composite, will be discussed in later sections. First, however, a different definition of overall properties will be given. 2. Dejinitions Based on Ensemble Averages Any given boundary value problem for a composite is defined by the equilibrium equation (2.17), together with the constitutive relation (2.14) and the boundary conditions. If the specimen to which the boundary conditions are applied is regarded as one sample from an ensemble, with parameter c( E 9, the stress, strain, and displacement fields depend upon CI and their means (a), (e), (u) could be sought. Then, at any chosen point, overall moduli L could be defined so that (a) = (Le) = €(e>.

(2.35)

For any given boundary conditions, there is no guarantee that the tensor f, does not vary over I/. Furthermore, if solutions at a chosen point x are examined, may be sensitive to the particular boundary conditions selected : an indication that this is so appears in Section III,A, where a formal perturbation solution generates f, as a nonlocal operator. If, however, it is believed that may be adequately approximated by a tensor of overall moduli, it may, as in the preceding section, be estimated by considering particular boundary value problems for which (a) and (e) are uniform over I/. It is, for example, plausible that ( e ) = e when the boundary condition (2.19) is applied, so long as the composite is statistically uniform. This depends, however, on the truth of the ergodic assumption described in Section II,A and will be discussed further when solutions are actually constructed. Finally, for such problems, a further application of the ergodic assumption leads to the ensemble averaged version of (2.26),that the mean

Overall Properties of Composites

13

energy density ( U ) at x is expressible in the form (U)

(2.36)

= +(ae) = +(a)(e),

which will allow the construction of bounds for f, by substituting into the classical energy principles suitable approximations to a and e. Kroner (1977) calls (2.36) the Hill condition. It should be remarked that the calculation of (a), (e), and (u) for any particular boundary value problem could be pursued as an end in itself, without explicit reference to f,. The drawback of the ensemble averaging approach is that mean values such as (a) have no strict meaning in any one sample, though it may be reasonable to regard (a) as some “locally averaged” version of a, provided that it varies slowly relative to the scale of the microstructure. The advantage of the approach, however, is that it can be followed even when (a) and (e) vary macroscopically and strict definitions based upon global volume averages cannot be applied. Usually, in the sequel, the ergodic assumption will be invoked and problems for which (a) and (e) are uniform will be emphasized. For wave propagation, considered in the final section, spatial variation is an intrinsic feature, however, and the ensemble averaging approach is the only one possible. 111. Methods Related to the Classical Variational Principles

A. SMALLVARIATIONS IN MODULI 1. General Perturbation Theory If the tensor of moduli L deviates only slightly from a constant value a formal solution to any given boundary value problem is easily developed in the form of a perturbation series. With the notation

Lo,

6L = L - Lo,

(3.1)

substitution of the constitutive relation (2.16) into the equilibrium equation (2.17)gives (LO)ijklUk,lj

+ (aLijklUk,J,j+ f;. = 0,

x

E

(3.2)

which is to be solved subject to a suitable boundary condition. Inessential complications are avoided by restricting the discussion to the displacement boundary condition u=uo,

x€av.

(3.3)

J. R. Willis

14

The problem defined by (3.2) and (3.3) can be expressed in the form of an integral equation by introducing the Green’s function G(x, x’) for a homogeneous “comparison” medium with moduli Lo. This satisfies (LO)ijklGkp,lj(X,x’) + 6ip 6(x - x’) = 0,

x E V,

(3.4)

Gip(x,x’)= 0, x E 8V. (3.5) If uo is now continued into V as the displacement field that the body force field f and the boundary condition (3.3) would generate in the comparison medium, it follows that the displacement u in the actual medium must satisfy (3.6) or, upon integrating by parts and using (3.5), Ui(X) = UP(X)

-J

aGip

7 (x, x’)8Lpjkl(X’)Uk,l(X’) dx’.

v axj

(3.7)

The symmetry of 6L ensures that only the strain e,, appears under the integral in (3.7). An integral equation for the strain field is produced by differentiation: symbolically, e = e0 - r&e,

(3.8)

where r denotes the linear operator

r:

z(x) +

Jv

r(x,x’)z(x’)dx’

(3.9)

whose kernel is related to G by rijpg(x, x’) = a2Gip/(dXjdXh)I(ij),(pq).

(3.10)

x’I-~

It should be noted that r has a singularity of order Ix at x = x’, which must be interpreted in the sense of generalized functions (Gel’fand and Shilov, 1964). It is convenient for present purposes to regard tensors of elastic moduli as operators with the form (3.9); with this interpretation, the kernel of 6L, regarded as an operator, would be 6L(x, x’) = 6L(x) 6(x - XI).

(3.1 1)

Inversion of (3.8) gives e = [I

+ r6L]-leo

(3.12)

so that u = (Lo

+ &)[I + r6L]-’e0.

(3.13)

15

Overall Properties of Composites

The definition (2.32) off, relates (a) and (e); taking expectations of (3.12) and (3.13) and eliminating e0 shows that f, must be a nonlocal operator, given by

+ @L[I + r6~]-’)([1+

2 = L,

r6~1-l) -I.

(3.14)

The expectations in (3.14) are impossible to evaluate in closed form. However, when 6L is sufficiently small, formal expansion of the right side of (3.14) gives the perturbation series

2 = L~ + ( 6 ~ )+ (hL)r(&)

- (6Lr6L)

+ ’ . ..

(3.15)

For an n-phase material of the type described in Section II,B,l, the last term shown explicitly in (3.15) gives, when applied to (e), n

n

/

\

where 6L, denotes L, - Lo. Equation (3.16) demonstrates the nonlocal nature of the operator 2 and indicates, too, that successive terms in the series (3.15) contain integrals of joint probabilities for increasing numbers of points. Given these joint probabilities up to a certain order, an approximation to 2 follows by truncation of the series (3.15). Its evaluation is still not trivial because the finite-body Green’s function G from which is derived is generally not known. If V is large, it might be suspected that r could be replaced by r“,the corresponding operator derived from the infinite-body Green’s function G “ . This is not so, however, because G” is homogeneous of degree - 1 in (x - x’). Correspondingly, r“ is homogeneous of degree -3 and the integral in (3.16) is no more than conditionally convergent. One approach to the problem is to substitute I?“ into (3.16) and other terms in the series (3.15) and to single out for some special treatment those which involve integrals that do not converge. A “renormalization” is then effected, which amounts to identifying physically what the terms probably ought to contribute and replacing them by convergent terms that make this contribution. A discussion of renormalization is given by McCoy (1979). It will be apparent from the description just given that the author’s view of renormalization is that, while it might be necessary in contexts in which boundary conditions are unknown, it is no satisfactory substitute for straightforward analysis of a problem of which every aspect is fully specified.

J. R. Willis

16

An alternative derivation of is provided by the method of smoothing, given by Beran and McCoy (1970). In this, e is represented in the form e = (e)

+ e’

(3.17)

which, by substituting into (3.8) and averaging, produces (e) = e0 - r(GL)(e) - r(6Le’).

(3.18)

Subtracting (3.18) from (3.8) now gives the equation e’ = -r(6L - (6L))(e)

- T(6Le’

- (6Le’))

(3.19)

for e’, which may be solved by iteration to yield e’ = {-I?(& - (6L)) + r p L r ( 6 L - ( 6 ~ ) ) - (6Lr(6L - (6L)))I - . . .>(e).

(3.20)

It follows from (3.17),(3.20),and (2.35) that f, = L,

+ ( 6 ~ )-

- ( 6 ~ ) ) )+ . . . .

(3.21)

This agrees with (3.151, but terms are automatically grouped so that r operates on a quantity whose mean is zero, wherever it appears. In analogy with (3.16), the last term shown explicitly in (3.21) gives, when applied to (e>, n

(hLr(6L

- ( 6 ~ ) ) )(e) =

c

n

Jdxf 6Lrr(X,x ’ ) ( ~ , ~ xl) (x,

r = l s=l

- P,(X)P,(X’))~L,(e(x’)),

(3.22)

in which the integral stands a reasonable chance of convergence if r is replaced by I‘“. McCoy (1979) regards (3.21) as a correctly renormalized series, but it must be pointed out that if, for a finite body, boundary conditions are applied for which (e(x’)) varies linearly (or worse) with x’, the integral converges only if P,, - PrP, approaches zero sufficiently rapidly at large separation of x and x’. The composite would thus at least be required to possess no long-range order: if p,, - P,P, merely oscillated about zero at large separations, as it would for a periodic composite, the integral would be conditionally convergent or even divergent for many choices of (e(x’)). When the probabilities have the required properties, however, Eq. (3.22) demonstrates that f, can be approximated by a local operator for any field (e(x’)) which displays negligible variation over the range within which P,,- P,P, is effectively different from zero. Furthermore, this local approximation may be obtained by considering a statistically uniform medium (for which P,, P,,,etc., are translation invariant and take values that agree with those given in (3.22)at the chosen point x), subjected to boundary

Overall Properties of Composites

17

conditions that generate a uniform field (e(x')). This restricted problem, for which a variety of other approaches are available, is taken up in sections that follow. The present section is concluded by remarking that the ideas summarized here have been developed and applied in many publications, amongst which those by Fokin and Shermergor (1969), Beran and McCoy (1970), Zeller and Dederichs (1973), Gubernatis and Krumhansl (1975), and Kroner (1977) are representative. Explicit recognition of the convergence difficulty is comparatively recent, and Zeller and Dederichs (1973) and Gubernatis and Krumhansl (1975) ignore it altogether. Fokin and Shermergor (1969) mention renormalization, but apparently mean by this the replacement of operators by formal perturbation series; no reference to convergence is made. Kroner (1977) combines r with a projection operator which has the effect of ensuring that it operates only on quantities with zero mean, as in (3.21). Earlier perturbation studies, for example those summarized in the book by Beran (1968), concentrate on solving the field equations (3.2) directly, though Karal and Keller (1964) performed firstorder perturbation theory, without considering convergence, on an operator equation like (3.8) for wave propagation. 2. Unform Mean Strain The general perturbation theory given above applies without change when the particular boundary condition (2.19) is applied. Then, if the composite is statistically uniform, it is to be expected that (e(x)> = c,

(3.23)

except, perhaps, in a boundary layer close to 8V and the local approximation to which f, reduces should be E,defined by (2.21) from the volume average iT of the stress. Some justification for these remarks is provided in the reasoning to follow, which derives essentially from work of Korringa (1973) and Willis (1977). Properties of the operator r are required and, as already anticipated in (3.9), it is convenient to set z = 6Le = (L - Lo)e.

(3.24)

The tensor z is called the stress polarization tensor. It will find extensive application later but, for now, it is simply a notational convenience. If v is defined so that v = u - uo,

(3.25)

it follows that v satisfies the field equations (LO)ijkluk,lj+ 7ij.j = 0,

E I/

(3.26)

J. R. Willis

18 with

v=o,

XEdV.

(3.27)

Equation (3.6) provides an integral representation for v, but with a view toward the ultimate relation of r to r", an alternative representation is now developed, which employs G" rather than G. It follows, in fact, by the usual argument involving Gauss' theorem, that ug(xr)=

fav -

G;(x - x')[tij

+ ( z i j ( x )- z$]njdS

jvG,POp,j(~- x r ) [ z i j ( x )- zi";.]dx,

(3.28)

where tij

=(WijklUk,l

(3.29)

and z* is any constant. Equation (3.28) represents v as a solution of (3.26) for any specified field z. The boundary condition (3.27) is also satisfied if T i = ( t i j + z i j - z t ) n j solves the boundary integral equation

Lv

G$(x - x ' ) T i ( x ) d S =

sv

GG,j(x - x ' ) [ z i j ( x )- z t ] dx.

(3.30)

Anticipating that the stress and strain n and e will oscillate rapidly about their mean values 8 and 8, z will likewise oscillate rapidly about its mean value 7 and the integrand on the right side of (3.30) oscillates about zero if z* is identified with 7.It follows directly from the field equations (3.26) that

L"

T,(x)dS = 0

(3.31)

and it is plausible that T i ( x ) actually oscillates about zero, when z* = f. Granting this, the boundary integral in (3.28) is significant only in a layer close to aV and so, elsewhere,

(3.32) Correspondingly, if the strain associated with v is denoted by e,

(3.33) the first equality coming directly from (3.8). Thus, when V is large in comparison with microstructure, Eq. (3.33) relates r to the infinite-body operator r"in a way that is consistent with (3.21). It should be noted, however, that the argument given is plausible rather than rigorous and would not apply at all for boundary conditions that would produce stress and strain fields that were not uniform "on average."

Overall Properties of Composites

19

Equation (3.33) approximates r by a translation-invariant operator that depends only on values 2, and so on the configuration of the composite, in a neighborhood of x', because the integral converges. The ergodic hypothesis therefore implies that, as x' ranges over V , the values of e(x') that are found occur with frequencies in proportion to the probabilities with which the associated local configurations occur. It follows that the average iZ of e over V may be identified with the ensemble mean (e) which may, in turn, be verified to be zero by taking the mean of (3.33). Therefore, (e) = V and now, from the mean of (3.12) with e0 = F,

F = ([I

+~ J L I - ~ ) ~ ,

(3.34)

so that this operator reduces to the identity when it is applied to a constant, though not otherwise. Equation (3.14) for f, now gives

fk- = h-,

(3.35)

where

L = L~ + (GL[I

+r6~1-1) = L, + ( 6 ~ )- (6Lr 6 ~ +) . . . .

(3.36)

This result, although superficially simpler than (3.21), is, in fact, identical, since r is evaluated as in (3.33). This section is concluded by noting that perturbation theory may also be performed using elastic compliances rather than moduli. With the notation

M = Mo + 6M,

(3.37)

where

LoMo = (Lo + 6L)(Mo + 6M) = I,

(3.38)

the integral equation (3.8) may be expressed in terms of u and uo by operating on each side with Lo + 6L to produce u = 60

+ (I - Lor)G L ( M+~ ~ M ) G .

(3.39)

AhMa,

(3.40)

It follows, using (3.38), that u =a '-

where

A

= L~ - L o n o .

(3.41)

The subtlety of the process of taking the limit of large V may be emphasized by noting that, as it stands, Eq. (3.40) applies when displacements are prescribed over V so that uo = LoE, which is different from a. It has been shown

J . R. Willls

20

already that (a) = a and so (3.40)gives -

L = ([I

+ A6M]-')Lo,

(3.42)

in which A is related to the Green's function for the displacement boundary value problem by (3.41).If, however, tractions were prescribed over dV, an equation like (3.40) could be derived, with A replaced by some operator A1 and 'a taking the value if. In this case, for consistency,

([I+A16M]-')a=if,

(3.43)

which shows that A, certainly differs from A. In fact, if A1 is used, Eq. (3.43) resembles (3.34) and it is plausible that, when V is large, A, takes the form

A1 : tl

+

LoFLo'l- Lo(tl- ii),

(3.44)

lr being defined by (3.33). If tractions compatible with a are prescribed over dV, reasoning exactly parallel to that leading to (3.26) gives the expression

M=M,+(~M)-(~MA,~M)+-

(3.45)

for the overall tensor of compliances. It has, in fact, been verified that the first few terms of the series (3.42)and (3.45)agree, when A and A, are defined, respectively, by (3.41) and ( 3 4 9 , and r conforms to (3.33). Kroner (1977) introduced operators r and A for boundary value problems that were not precisely specified and related them through (3.41).Subsequent applications showed, however, that his A in fact was A l of the present work. Correct answers were generated because he employed the method of smoothing which has the effect of reducing A to A, in any case, as well as relating (a)and ( e ) independently of the exact choice of ao or eo. B. BOUNDS The formal expansions (3.36)and (3.45)for E and M are of limited practical use because they contain correlation functions of all orders. It was shown in Section 11, A how these correlation functions were related to joint probabilities for finding various phases at specified points, but only very few of these will be known in practice. Therefore, there will be no option but to truncate the series (3.36) and (3.45) after a few terms, to give conflicting estimates of L. If nothing is known about the composite other than the volume fractions c, of the phases, truncation of (3.36)at the term of first order in 6L gives the Voigt estimate -

L

'v

Lv = Lo

+ (6L)

(3.46)

Overall Properties of Composites

21

while a corresponding truncation of (3.45) gives the Reuss estimate -

L

-

LR = [Mo

+ (6M)]-’.

(3.47)

It was shown in Section II,B,l that the estimates Lv and LR at least have the precise status of bounding L from above and below and it will be shown now that other truncations of (3.36) and (3.45) similarly provide bounds. In fact, when e0 = e is prescribed, Eq. (3.8) has formal series solution cn

e=

C (-r 6 ~ ) ~ e .

(3.48)

k=O

Truncating after the mth term gives the approximation e* = B,F

(3.49)

where (3.50) The approximation (3.49) is associated with a displacement u* that satisfies the boundary condition (2.19) exactly. It is, therefore, a candidate for substitution into the minimum energy principle; this gives

ELCI (e*, (Lo + 6L)e*),

(3.51)

in which the “inner product” is defined by (3.52) Inspection of (3.10) shows that the operator r is self-adjoint so that, for any fields z1and t,,

(rz,,t,)= (r,,rz,);

(3.53)

this is also verified directly in Section IV,A. It also follows, from (3.4) and (3. lo), that

n o r= r.

(3.54)

The operators r,6L that act on the left member of the inner product in (3.51) may now be transferred to the right member, to yield

eLe I e(Lo + (6LBz,,,))e,

(3.55)

in which Lo appears by itself because Tz has mean value zero for any z. The ergodic hypothesis was also used to replace the spatial mean by the ensemble mean. The upper bound (3.55) contains correlation functions up to order

J . R. Willis

22

+

+

2m 1 and is precisely (3.36), truncated at 2m 1 terms. An exactly similar development, using (3.40) with goreplaced by 5 and A by A l shows, from the complementary energy principle, that upper bounds for % are obtained l by truncating(3.45)at an odd number of terms. These results are due to Dederichs and Zeller (1973). Improved bounds of odd order may be obtained by adopting, in place of (3.49), the trial field m

e*

=

C (-r6L)kek,

(3.56)

k= 0

where ek are nonrandom “strain tensors,” arbitrary except that e, = e to satisfy the boundary condition. When m = 1, for example, substitution into (3.55) gives

~LizI s(Lo+ ( ~ L ) ) E- 2 ~ ( 6 L r&)el

+ e (6LT 6L + 6Lr 6LT 6L)e, ,

(3.57)

whose right side is minimized when

el

= (6Lr

6~

+ 6Lr 6Lr 6

~- l(6Lr ) b ~ ) ,

to generate the “optimized” third-order bound

€,

+ ( 6 ~ )- (6Lr 61,) x (ax6~ + 6Lr 6Lr 6

(3.58)

e,, where

= L~

~ - y) h L r 61,).

(3.59)

“Optimized” bounds of third order were first developed by Beran and Molyneaux (1966); the present account follows that of Kroner (1977). Bounds of even order have also been constructed by Kroner (1977) who has shown, essentially, that the last term in the series (3.55) is negative definite provided that Lois chosen so that 6L is positive definite: such bounds will be obtained from another standpoint in Section IV,C. Finally, it may be noted that the bounds given in this section improve as rn increases, since the trial form (3.56) includes the optimal form for any smaller value of m. Thus, the best bounds that can be obtained are those which use all of the correlation functions that are known. In practice,however, correlation functions or, equivalently,joint probabilities P,, . . .,may well not be available,though something may be known of their structure. In particular, isotropy of P , s , . can yield a major simplification in reducing the effect of the r operator to that of multiplication by a constant tensor P.Conditions under which this occurs for correlation functions up to a specified value of m define, in effect, Kroner’s (1977) concept of “disorder of grade m.” Kroner ,

Overall Properties of Composites

23

used this in conjunction with trial fields of the form

1

n-1

[

e*= I - r &

C

e

k=l

(3.60)

to construct bounds for “disordered” materials. These will not be reviewed in detail, but bounds produced in Section IV,C for materials that need not be “disordered” reduce to some of those given by Kroner when disorder is also assumed.

IV. Methods Related to the Hashin-Shtrikman Variational Principle EQUATION AND VARIATIONAL PRINCIPLE A. THEINTEGRAL The integral equation (3.8) has so far been regarded as an equation for the strain e, but, as anticipated in the definition (3.24) of the stress polarization t,it can also be regarded as an equation in this variable; explicitly, so long as L - Lo is everywhere nonsingular,

[(L - Lo)-’

+ r]t = eo.

Small variations in moduli are not necessarily envisaged and the tensor of moduli Lo relates to any conveniently chosen homogeneous comparison material. Before considering the solution of (4.1), some of its properties will be noted. First, as indicated in the preceding section, the operator r is selfadjoint. This can be verified directly by employing the “virtual work” equality

ez) = 0 (44 for any stress cl that is divergence-free and any strain e2 derived from a displacement that is zero over i3V. Now let (01,

el = -rt1,

e, =

-rt2

(4.3)

for any chosen tl,t 2 ,and let b l be the stress associated with e l , so that cl = Loel

+ zl.

(4.4)

Then, ( z , , W = (Loel - bl, e2) = (Loel, e2), (4.5) by (4.2). The last expression in (4.5) is symmetric in the suffixes 42 by the symmetry of the elastic moduli Lo ; hence, r is self-adjoint. Next, setting

J . R. Willis

24

r is positive definite, and hence that (L - Lo)- + r is positive definite, so long as (L - Lo)-' is positive definite at each point

z1 = z2 in (4.5) shows that

of I/. If Lo is so restricted, therefore, Eq. (4.1) is equivalent to the minimum principle 2( 0, - 0 )I (z*, (L - Lo)- 'z*)

+ (z*, rz*)- 2(z*,eo)

(4.6)

for any approximation z* to z. The left side of (4.6) represents the extreme value of the functional -(z, eo) = (L,e

- a,eo) = 2(B, - 0),

(4.7)

where 2 0 , = (L,e, eo) = (e, L,eo) = (ao,eo) 2 0 = (a,eo) = (a,e),

(4.8)

(4.9)

by repeated application of (4.2). The functional on the right side of (4.6) is extremized by the solution of (4.1), regardless of whether L - Lo is positive definite. The variational principle is, in fact, a maximum principle when L - Lo is everywhere negative definite. To prove this, it is convenient to introduce a "strain polarization" 11 by z = Lev, 11 = MOT, (4.10) so that 11 = M,a - e.

(4.11)

It now follows, using (4.2) where necessary, that = (11, Loti)

(z,

- (0,Moa).

(4.12)

But also, (z, (L - Lo)-lz) = (Lo%(L - Lo)-lLoll)

(L - Lo)- 'Lo

= Mo(M0 - M)-

- I.

(4.13) (4.14)

Hence, (z, (L - Lo)-lz) =

(119

(M,

- W'11) - (11,LoV)

(4.15)

and, by addition of (4.12) and (4.15), (z, (L

+

- ~ , ) - l t ) (z,rz) = (11, (M, - ~ 1 - 1 1 1 ) - ( a , ~ , a ) . (4.16)

This is negative definite so long as Mo - M or, equivalently, L - Lo, is negative definite; in that case, the minimum principle (4.6)is replaced by the

Overall Properties of Composites

25

maximum principle

2(u0- rli) 2 (t*,(L - Lo)-%*) + (z*,Tt*)- 2(t*,e0).

(4.17)

These derivations were given by Willis (1977),following arguments presented by Hill (1963b). Hill showed, in fact, that the Hashin-Shtrikman principle which is embodied in (4.6) and (4.17)can be derived from the classical energy principles, by discarding a term known to be definite: the even-order bounds of Kroner (1977)referred to in the preceding section are obtained by the same method and will later be derived from (4.6) and (4.17). The principle was originally derived, directly from the field equations, by Hashin and Shtrikman (1962a). B. PERTURBATION THEORY The case of small variations in moduli has, in effect, been adequately covered in Section III,A. It is useful to record, however, that perturbation theory for e, together with the definition t = 6Le, yields the formal series solution m

t = 6L

C (-r&)keo.

(4.18)

k=O

The stress polarization t plays a more distinctive role if a composite is considered which comprises a matrix containing a distribution of inclusions. For simplicity, the inclusions will be taken to be identical in shape, size and orientation and will be assumed to have moduli L1.The moduli of the matrix will be taken as L2 and the inclusions will be distributed at number density n, and associated volume concentration cl. Generalization to the case of several types of inclusion presents no more difficulty than the need for more cumbersome notation. If the comparison material is now identified with the matrix, so that Lo = L2, the definition (3.24) shows that t is nonzero only over the inclusions. Equation (3.8), with t = 6Le, defines e in terms of z anywhere in V , but the integral equation (4.1) now applies only over the regions occupied by the inclusions. It is convenient to call these regions V,, A = 1,2,, . . , as in Section II,A and to define't to be the restriction of z to V,. Equation (4.1) then gives [(L,

-~

~

)

+ r)tA+ C rzB= eo, -

l

xE

v,,

A = 1,2,. . . . (4.19)

B+A

The relation (0)

= Lo(e)

+ (.>

(4.20)

26

J. R . Willis

encourages study of the expectation value of z. This is expressible in the form (4.21)

where zi(x) denotes the expectation value of z,(x), conditional upon an inclusion being centered at x,, and V(x) is the set V(x) = (x,:

x E V,}.

(4.22)

An equation for z<(x) may be obtained by taking the expectation of (4.19), conditional upon V, being fixed. This gives

where '7: denotes the expectation of ,'t conditional upon inclusions being centered at xA and x'. Equation (4.23) thus introduces a new unknown, for which an equation could be generated by taking expectations of (4.19) with V, and V' fixed. This new equation, however, would contain &,, the expectation of zC given V,, V,, and V,, and continuing the process would generate an infinite hierarchy of equations for expectations of this type. The difficulty is resolved if a series solution is sought in powers of the concentration cl, in the limit c1 << 1. The integral in (4.23) may be expected to be of order c1 and its neglect generates the simple zeroth-order equation

[(L, - L ~ ) - I + r]zi

= eO,

x E V,,

(4.24)

which describes the response of a single inclusion V, perturbing a field that, in its absence, would take the value eo. This problem is of fundamental importance and is discussed below. 1. T h e Single Inclusion Problem

If the inclusion A is close to the boundary aV of V , use of the finite-body form of r is essential and little can be deduced from (4.24). If, however, the inclusion is far from aV, since it is isolated, the difficulties mentioned in Section II1,A for the operator disappear, and r may be replaced directly by r",whose kernel is given by (3.10)with G replaced by G". The infinitebody Green's function G" satisfies (3.4) with Lo = Lz and the boundary condition (3.5) is relaxed. G" is a homogeneous function of degree - 1 in x - x' and may determined by placing the point load at the origin, so that x' = 0. A useful representation is obtained by first noting the plane-wave decomposition (4.25)

Overall Properties of Composites

27

of the delta function, given by Gel'fand and Shilov (1964). This motivates study of the equation

+

sl,(g

(LZ)ijkfGfp,fj

*

x) = 0,

(4.26)

in which GCmay be taken to depend only upon (c * x). A particular solution is given by

G6 = - [LAC)] -

*

XI,

(4.27)

where L,(<) is a matrix with components (4.28)

= (LZ)ijkfljcf.

[LZ(C)lik

The superposition indicated by (4.25), applied to Gr, now gives G" in the form

Jla=

G"(x) = (8n2)-'

The integral equation (4.24), with inclusion is considered, becomes (Ll

- LZ)
7:

- (8n2)-1

jvAdx' 8°C

1

dS[Lz({)]-'S(C

x).

(4.29)

now replaced by z since only one

it,=

dSILZ(<)]ik-lcjlf

(X - x')]Tkl(x')l(i,j)= f?:(X),

XE

v,.

(4.30)

Equation (4.30) has a remarkably simple solution for the important special case in which V, is an ellipsoid and e0 is uniform over V,. Suppose first that V, is a sphere with radius a, centered at the origin. If z is taken constant over V,, the integral with respect to x' in (4.30) requires the evaluation of az/apZ

<-

J

VA

dx'qp -

c - xi),

when p = x. The integral over V, gives the area of the disc formed from the intersection of the plane * x = p with V,: this is n(a2 - p z ) if IpI < a and zero otherwise. When x E V', therefore, its second derivative is - 2n and the left side of (4.30) is constant over V,. Thus, when e0 is constant over V,, t is constant and given by [(L, - L 2 ) - l + P]t = eO, (4.31) where the constant tensor P has components 'ijkf

= (4n)-

it,, =

dS[L2(<)1,

'<jt11(ij)(kI)*

(4.32)

A similar result is available when V, is the ellipsoid xTATAx< a. Evaluation of the integral is reduced to the case just considered by the dual transformations y = AX, 5 = ATC (4.33)

J . R. Willis

28 and produces the result f‘ijkl

J

= (4~lAl)-l

151=1

dS[LA<)li ‘tjtl[tT(ATA)-1 < ] - 3 ’ z l ~ i j ~ ~ k ~ ~ .(4.34)

Eshelby (1957) gave an explicit solution for the ellipsoidal inclusion problem when the matrix is isotropic, using methods of potential theory. He also proved that the field within an ellipsoidal inclusion in an anisotropic matrix would be uniform, but gave no detailed prescription for its determination. The earliest published derivation of (4.34) appears to be that of Khatchaturyan (1967), though (4.32) was given by Kneer (1965). Eshelby (1961) showed that, for an inclusion in an isotropic matrix, a polynomial field eo would generate a polynomial field within the inclusion. This was proved for an anisotropic matrix, in an unpublished essay by Willis (1970) and independently by Asaro and Barnett (1975);an application of the result was presented by Willis (1975). In general, evaluation of the tensor P requires a computation. Although progress can be made analytically for a number of special cases, the only one of sufficiently general interest and simplicity is that of a spherical inclusion in an isotropic matrix. If L2 is isotropic, therefore, with Lam6 moduli A, p, (L2)ijkl

= lzsij

6kl

+ p ( 6 i k sjl +

(4.35)

sjk)

and it follows without difficulty that (4.36) For a spherical inclusion, P is given by (4.32);it is isotropic and so may be represented in terms of a “bulk modulus” xP and “shear modulus” pp, so that Pijkl

= K p 6ij

+ pp(bik + 6jl 6jI

6jk

- S b i j 6kl).

(4.37)

The “moduli“ xP,pp may be obtained from the scalars f‘jikk

= 9Kp,

Pijij = 3 K p

+ 1oPp3

(4.38)

for each of which the integrand of (4.32) reduces to a constant. The result is expressible in the form 1

(3KP9

2PP) = (,(A

where K

=1

+ 2p)’ 15y::p))

=

(A. + )

3(K + 2pL) ’ (4.39) 4p)

5p(3k.

+ 2pj3 is the bulk modulus of the matrix. The notation p = ( 3 K P , 2PP)

(4.40)

29

Overall Properties of Composites

was introduced by Hill (1965a).It has the property that, for isotropic tensors A and B, their product AB may be represented as

AB = ((3K.4)(3KB), (2pA)(2pB)).

(4.41)

The identity has the representation (1, l), so that

A - ' = (1/31tA, 1/2pA).

(4.42)

If the inclusion is also isotropic, say with L1 = (3~',2p'),Eq. (4.31) has solution z = TeO,

(4.43)

where 3 K , = ([3(K'

- .,I-'

+ 3Kp}-1,

(4.44) 2.k = P ( P ' - P r l + 2 P P V This applies, of course, only when e0 is constant over the inclusion. Reverting now to the composite problem, overall moduli L may be estimated by choosing e0 = V. Then, ( e ) = C! and E may be deduced directly from (a), as given by Eq. (4.20). To lowest order in cl, (z) = c l z j since, when z i is constant, the integration in (4.21)simply multiplies the integrand by the volume of one inclusion. Thus, to first order in cl,

L = Lz + c,T.

(4.45)

2. Interactions In considering interactions between inclusions, it is helpful to rewrite Eq. (4.23)in the form C(L,- L J - ~+ rl.2 -tJrlx,(~,I, - p,)rz;

+ Jrlx,p,,,r(t.;B

-z ~ ) (4.46)

whose right side is precisely ( e ) , or v if e0 takes this constant value. Estimation of L to order c: requires an estimate of z$ to order c l r which, in turn, requires 0(1) estimates of z i , z;~ for substitution into the integrals in (4.46). z i is known already to lowest order, from the preceding section, while the zeroth-order estimate for zBABis defined by the equation (L, - LZ)-'zjB+ r z j ,

+

= eO,

x E V,

(4.47)

and a similar equation with A and B interchanged; these are obtained by taking expectations of (4.19)with V, and VBfixed and neglecting the integral, ~ ~(4.47) . describes the problem of two inclusions which involves ~ 5Equation

30

J. R. Willis

perturbing a strain field eo. For spheres in an infinite isotropic matrix and e0 = V, it was solved approximately by Willis and Acton (1976). More recently, Chen and Acrivos (1978a,b) have solved the same problem from a different standpoint. Both pairs of authors have produced estimates for the coefficient of c: in the expansion of L. The integrals that appear in (4.46) converge when r is replaced by r";Willis and Acton (1976), and subsequently Willis (1980a), whose formulation is otherwise followed in the present account, employed the explicit representation (3.33) for r from the outset. Chen and Acrivos (1978b) found it necessary to renormalize a conditionally convergent integral, which they attempted by a method successfully developed by Batchelor and Green (1972) for the related problem (formally the incompressible limit of the present one) of the overall viscosity of a dilute suspension. They noted, however, an ambiguity in the renormalizing prescription, which allowed at least two different estimates for L; an explanation of why a particular one (namely, the one consistent with the present development) is correct, formed a major part of their effort. The exact solution of (4.47) will not be considered, but before leaving this problem, a simplification of (4.46) will be discussed because it is relevant later. When the inclusions are spheres of radius a and PBIA is isotropic, and so a function of - xAl only,

,xI

(4.48) for any constant T;, where the tensor P is defined by (4.32). This may be seen by choosing, without loss, xA= 0 and expanding the left side of (4.48) explicitly in the form

(4.49)

XI

(,xI

Now lx" - < 2a and the integral over < 2a is P. The integral over > 2a is zero because PBIA - P, depends upon only and the integral = r is zero for any r > lx" of T"(x, + x" - x) over the surface this follows by differentiating with respect to r the corresponding volume integral over lxBl < r. A corresponding result applies for ellipsoidal inclusions, provided that PElAnow has "ellipsoidal" symmetry so that the composite is, in effect, generated from an affine transformation of an isotropically distributed suspension of spheres; an explicit proof is given by Willis (1978).

Ix,~

Ix,~

Ix,~

XI;

Overall Properties of Composites

31

Finally, Eq. (4.21) shows that ( 7 ) depends only on the mean value 5; of 7; over inclusion A. Integrating Eq. (4.46) with respect to x over V ’ reduces the kernel of the J? operating on 7; to the constant P which, in turn, generates P times the integral of t; over V,. Equation (4.46) therefore implies, to first order in el,

+ PI$

[(L, - LJ1

+ ClP?; - J ~ x B P B ~ S , ( &

=E

- 7;)

(4.50)

where, on the right side of (4.50), 7; and zBAB are required only to zeroth order and represents the mean value of r as x ranges over VA. 3. Closure Assumptions

At high concentrations el, the power series approach outlined above is not helpful. An approximation, whose status will be further clarified later, is obtained by assuming, in (4.23) or (4.46), that 7 BA B

- z BB

(4.51)

exactly. It is related to perturbation theory in the sense that, if the composite has no long-range order, equation (4.51) is asymptotically true at large separations, and few inclusions are close together when c1 << 1. Also, if the composite is perfectly ordered, with the inclusions arranged on a lattice, Eq. (4.51) is true identically at any el. Accordingly, (4.51) is called the quasicrystalline approximation. It was introduced, in the context of scattering theory, by Lax (1952). When it is adopted, Eq. (4.46), with e0 = F, reduces to

[(L, - LZ)-’

+ r]tj + J d x B ( p B 1 A

-pB)rt;

= s,

x E &, (4.52)

which retains some allowance for interactions through the integral. Equation (4.52) has a simple exact solution if the inclusions are spheres, distributed isotropically, or else are ellipsoids, with corresponding “ellipsoidal” symmetry for P B I AIt. is consistent to take 7; constant over VBsince the result (4.48) shows that the integral is then constant over V, . Therefore, (4.52) has solution 7;

= [(L,

-L

p

+ (1 - c,)P]-’e

(4.53)

and, correspondingly, L is estimated as

€ = Lz + C,[(L1

- LJ-1

+ (1 - c,)P]-’E.

(4.54)

It may be noted that (4.54) agrees exactly with (4.45) to first order, but differs from the exact series for L at order c:, the discrepancy corresponding exactly to the integral involving & - 7; in (4.50). Although the limit c1 = 1

J. R . Willis

32

is not allowed, it is interesting that substituting el = 1 into (4.54) yields L = L1, so that there is at least some hope that (4.54) provides a reasonable estimate of L at high concentrations. The other important feature of (4.54) is that it contains no explicit dependence on the conditional probability though its isotropy (or, more generally, “ellipsoidal” symmetry) appears implicitly, through the tensor P which depends upon the matrix A defining the basic ellipsoid. The approximation (4.51) is one of a class whose next simplest member is

-

(4.55)

this provides a good approximation, except when three inclusions are close together and could be used to close the hierarchy of equations at the next ~ however, . is not constage, giving a closed integral equation for ~ 5 This, sidered further. Closure assumptions like (4.51) or (4.55) may also be applied to a general n-phase composite. If the restriction of z to the rth phase is denoted by zr, n

(4.56) (4.57)

where z:(x) now denotes the expectation value of z’, conditional upon phase r being present at x. Substituting (4.53) into (4.1) and taking expectations conditional upon phase r being present at x gives (L, - ~ ~ ) - 1 z+;

f: J d x ‘ q x , x~)z;s(x~)~slr(x~, x) eo, =

s= 1

r = 1,2,. . . ,n

(4.58)

where zss(x’)denotes the expectation value of zs at x’,given the presence of x)is the probability for finding phase phase r at x and phase s at x‘and PSlr(x’, s at x’,given phase r at x. The analog of the quasicrystalline approximation (4.51) is zSs(x’)= z:(x’),

(4.59)

which reduces Eqs. (4.58) to a closed set for the unknowns t:(x). If e0 is uniform, and equal to F, the operator I? takes the form (3.33), except in a boundary layer close to LJV and it is consistent to take z: constant if the medium is statistically uniform. In that case, the equations reduce to the

Overall Properties of Composites

33

algebraic set

in which the probability P, has been identified with the volume concentration c,. The integral is independent of x because PSI,is insensitive to translations. As for the suspension of spheres, further progress is possible if all of the functions PSI,are isotropic. The observation that the integral of r m ( x - x') over any spherical surface Ix = r is zero allows the integral in (4.60)to be replaced by one over an arbitrarily small sphere centered at x, over which PSI,- c, may be replaced by its value at x. Now

x'I

(4.61)

P,lr(X, x) = 6,s

and so, since the integral of Eqs. (4.60)simplify to

[(L, - Lo)-'

rm over the volume of

+ PIT:- P(T) =

e,

the small sphere is P,

r = 1 , 2 , . . . , n.

(4.62)

It follows from (4.62)that T: = T,(C

+ P(T))

(4.63)

where T, = [(L,

- Lo)-'

+ PI-'

(4.64)

and hence that (2)

where (T)

=

= [I - (T)P]-'(T)E,

(4.65)

c:= c,T,. An estimate Z for L is then obtained in the form Z = Lo + [I - (T)P]-'(T).

(4.66)

If the material is in fact composed wholly of phase r, so that c, = 1 and c, = 0 for s # I, it is easy to verify that (4.66) gives f, = L,, as it should. C. VARIATIONAL ESTIMATES

It was shown in Section IV,A that the displacement boundary value problem is solved by the stress polarization z that extremizes the functional

+

F(T)= (T,(L - Lo)- lz) (z,r T ) - 2(t, e').

(4.67)

34

J. R . Willis

Furthermore, its extreme value may be expressed, when e0 = 5, in the form -

2(D0 - D)= Z(Lo - L)Z,

(4.68)

by (2.23). Thus, a variational estimate for L is obtained by substituting any trial field z* into (4.67). The first trial field that will be considered is rn

Z* = 6~

1 (-r6L)ke,

(4.69)

k=O

which is just (4.18), truncated at the mth term. The resulting F(r*) simplifies by use of the self-adjoint property (3.53) of r, and 6L correspondingly, to the form F(T*)=

- (Z,6LBzrn-I?),

(4.70)

where B, is defined by (3.50). The approximation (4.70) to F(T) thus leads, upon replacement of the spatial mean by an ensemble mean, to the estimate

Z = Lo + (6LBz,-l)

(4.71)

for L, which involves correlation functions for up to 2m points and is precisely (3.36), truncated at this order. The results of Section IV,A show, in addition, that - L is positive definite if Lo is chosen so that L - Lo is negative definite at each point of V. In this case, the estimate (4.71) becomes one of Kroner's (1977)bounds of even order. Dually, (4.71)yields a lower bound for L if Lo is chosen is that L - Lo is positive definite; this result was not given by Kroner (1977). He gave, instead, an upper bound of even order for the compliance tensor R. This could be obtained (together with a new lower bound) by working with the strain polarization tf defined by (4.10) and an integral equation like (4.1) for the traction boundary value problem, with I' replaced by A l : the Hashin-Shtrikman principle was given explicitly in terms of t f , as well as z, by Hill (1963b). 1. Hashin-Shtrikman Bounds

The next obvious development is to construct "optimized" bounds by substituting into (4.67) approximations z* that contain parameters. Here, the Hashin-Shtrikman principle displays a clear advantage over the classical principles discussed in Section III,B for, whereas the classical principles require either compatible strain fields or self-equilibrated stress fields, the polarization T* is subject to no such constraint. Considering first an n-phase material, it is natural to choose r* to have the piecewise-constant form (4.72)

35

Overall Properties of Composites

where the constants z, are arbitrary. Equation (3.67) then gives, when e0 = Z,

F(z*) = v-’

zr(Lr- Lo)-’zr Jdxf,(x) r=l

- 2v-’

(4.73)

zr Jdxf,(x)E. r= 1

Now by definition, the mean value of f , ( x ) over V is cr, the concentration of phase r. Also, except when x is in a boundary layer close to aV, the operator r takes the translation-invariant form (3.33) and so, when I/ is large,

v- 1 J d x ~ . ( xJdxrr(x, ) x~)fs(xf)

J d x / ’ r y x y f s ( x+ x y The mean value over x of f,(x)f,(x+ x”) is Prs(x,x + either by definition V - 1 Sd.r,(x)

CJ.

XI’),

of an “empirical” two-point probability or by use of the ergodic hypothesis directly. Hence, when V is large,

qz*) =

C,~,(L,

- L , ) - I ~ ,+

cc n

n

t, J d x ” r y x ~ ~ )

r = l s=l

r=O

+

x (Prs(x,x x”) - crcs)zs- 2

n

1 crz$,

(4.74)

r= 1

which is independent of x because P,, is assumed to be insensitive to translations. This derivation has been given in detail as an explicit example of the ergodic assumption. Earlier, in Section III,A2, reference was made to the sampling by r of local configurations, but precise specification [such as (4.72)] was not possible and the argument necessarily remained qualitative. The best estimate for F(t) is now obtained by extremizing (4.74). This requires that (L, - Lo)-%, +

i:

~ d x ” r ” ( x ~ ~ ) ( P , , r (-x c,~)z,~ , O=)e,

s= 1

r = l , 2 , . . . , n,

(4.75)

having used the relation P,, = Psircrand chosen x = 0. It is interesting to note that Eqs. (4.75) are identical to (4.60), with z: = z,. Furthermore, the extreme value of F(z*) is simply - C:= crz$ and Eqs. (4.60), which were derived on the basis of the ad hoc closure assumption (4.59), in fact have the

36

J. R. Willis

status of providing the “best possible” estimate for L, relative to the HashinShtrikman variational principle. Finally, if this estimate is called L is positive or negative definite, whenever Lo is such that L, - Lo is correspondingly definite for all r. The bounds were given, at this level of generality, by Willis (1977).The original Hashin-Strikman bounds were derived rather differently from the variational principle by Hashin and Shtrikman (1962a,b, 1963) and, by a different method again, by Walpole (1966a,b), for the special case in which PSI,is isotropic and so a function of Ixrr/only. The integral operator in (4.75) then reduces to the constant tensor P, as in (4.62), and is given by (4.66). Willis (1977) noted, in fact, that (4.62) and (4.66) apply to a composite with “ellipsoidal” symmetry, if P is given by (4.34), which allows, for example, the construction of bounds for bodies containing aligned needles of finite length, or parallel cracks, as limiting cases. Kroner (1977) produced a form of (4.66), with the inverse operator expanded as a series, by substituting into his even-order bound the approximating field (3.60), taking the limit m -+ cc and simplifying the result by appeal to the concept of “disorder”; this appears to be realized as a limiting form of the cell model of Miller (1969a,b). The work of Hori and Yonezawa (1974) contains a relevant discussion. In view of the fact that “disorder” implies at least isotropic statistics and produces Hashin-Shtrikan bounds of less general application than (4.66), it receives no detailed treatment here.

z,

2. A Matrix Containing Inclusions Willis (1978,1980a)has considered the construction of bounds of HashinShtrikman type for a matrix containing inclusions. As in Section IV,B, the inclusions occupy regions V,, A = 1, 2, . . . , all of identical size and shape, and have moduli L1 while the matrix has moduli L2. It is convenient, however, to let Lo remain arbitrary and to let z* take the constant values zl and z2 in the inclusions and matrix, respectively. The term (z*, rz*) in F(z*) may be simplified by noting that the mean value of Tz* over V is zero, so that

this gives integrals only over the inclusions. With the infinite-body form (3.33) for r, it follows that

(4.77)

Overall Properties of Composites

37

+ c 2 t 2 .It is useful to define

where Z* = clzl

J

V,’

dx

J

dx’r(x - x’) = P

( A = B),

(4.78) ( A B), (so that P is given by (4.34) when the inclusions are ellipsoids). Then, Eq. (4.77) gives VA

VE

+

= QAB

(r*,rZ*)= (b/v)1( T i - 7 2 ) p ( z 1 - z*) + A

1

(ti

-~ Z ) Q A B (~ ~7 2 )

BfA

+ (71 - t2)J“-“A

dxf,(x’)(z2

(4.79)

- T*).

Replacing the right side of (4.79)by its expectation value and evaluating the other terms in F(t*) now gives

c cr[zr(Lr Lo)-%, 2

F(z*) =

-

r= 1

x

{

czP+

sV

-2

4 + c1(z1 - z2)

~XBCPBIAQAB - clrA(xB)]]

(zl

-

t2)3

(4.80)

since t 1 - t* = c2(t1- t 2 )and t 2- T* = cl(tz - ti).z1 and z2 are now chosen to extremize (4.80). With the notation p’ = p

+

s

v-v,

the associated variational estimate

~XB[PB(AQAB - clfA(xB)],

(4.81)

for L takes the form

2

It can be verified that this is identical to (4.66), with PI = 2 and P replaced by P . For a composite containing more than one type of inclusion, an expression like (4.81)for F(t*)could be derived, but this, generally, would contain more than one tensor P and its extreme value would not yield such a simple expression for Z. Suppose, now, that P B I A is isotropic or, more generally, has “ellipsoidal” symmetry. The reasoning that gave Eq. (4.48) shows that the integral in (4.81) (for which the region VA is excluded) is zero, so that P = P. If, in addition, Lo is chosen as L,, elementary manipulation shows that (4.82) is identical to (4.54),which was obtained by use of the quasicrystalline approximation. Thus, use of this approximation, whose validity can be guaranteed only at low concentrations, in fact yields a variational estimate for L at any concentration, as does (4.59) for the n-phase material.

J . R. Willis

38

It is perhaps worth mentioning that the estimate (4.82),which was given by Willis (1980a),contains allowance for arbitrary inclusion shapes and distributions, through the tensor P'. In the particular case of "ellipsoidal" symmetry, when P = P, (4.82) [or, equivalently, (4.66)] yields bounds for the overall moduli of composites containing aligned ellipsoids. Examples for the related problem of conductivity (reviewed briefly in Section VI,A) were presented by Willis (1977), while for elasticity, estimates for L based upon (4.82)have been obtained by Willis (1980~)in the course of a study of wave propagation. Interactions between inclusions can also be treated from the variational principle. If, as in Section IV,B, the comparison material is identified with the matrix and the restriction of z* to VAis called zA,let (4.83) where the constant zo and the function fare to be determined. Substituting (4.83) into (4.67) gives a long expression with terms involving up to four inclusions, whose joint probability densities are unknown. However, an approximation correct to order cf is obtained by keeping terms involving only one or two inclusions. The resulting estimate for F(z*) has been given by Willis (1978); its extreme value, again correct to order c:, yields the estimate

z = L, + C,[(L,

- Lz)-l

+ PI-' + Cl[(L1 - Lz)-' + PI-'

x P[(L, - LJ-1 + P I - '

- C,[(L,

- LJ'

+PI-'

x Jv-,

~XBCPB~AQAB - CITA(XB)I[(LI- LJ-' +PI-'

+ C'[(L'

- L,)-'

X

Jv-",

+PI-'

~ J ' B IA Q A B [(L - LA-' I

x [(L, - La)-'

+ PI-'.

+ P + QABI-'QAB (4.84)

For a genera1 inclusion shape, (4.84) is not an exact estimate of L, even to order cl, because (4.83)takes z* constant over any inclusion. For ellipsoids, however, (4.84)agrees exactly with (4.45)to order c1 and so provides a variational estimate of the coefficient of cf in the expansion for L. Furthermore, this estimate of c: is a bound whenever (L, - L,) happens to be definite. Willis and Acton (1976) produced an approximation I,for a matrix containing an isotropic dispersion of spheres, by solving (4.47)by iteration. This amounts, in (4.84),to neglecting QAB in the term [(L, - L2)- + P + QAB]in the last integral. The variational estimate (4.84) was plotted out for an

Overall Properties of Composites

39

isotropic distribution of spheres by Willis (1980a). It compared quite well with the exact coefficient of c f computed for particular examples by Chen and Acrivos (1978b) and generally followed the estimate of Willis and Acton (1976), though the latter actually gave an estimate for it just less than that obtained from (4.84)in one case where this was, in fact, a strict lower bound. 3. Examples The various estimates and bounds so far given, although explicit, still usually require a small computation for application to a particular composite. The reader is referred to the original papers cited for such results. Here, a few limiting cases which may be discussed algebraically are presented for illustration. If the comparison material is taken isotropic, so that Lo = (3x0,2p0), and PSIror PBIAis isotropic as well, the Hashin-Shtrikman estimate (4.66) for L involves the tensor P, which is defined by (4.39),with K , p now taking the values K ~po, . For a composite comprising n isotropic phases with moduli L, = (3~,,2p,),(4.66) yields directly ?, = (3i?,2,ii),where (4.85) in which

The estimates R, ,iiprovide lower bounds for R, iiwhenever K~ < K, and p o < p, for all r, and upper bounds whenever K~ > K, and p o > p, for all r. The best lower bounds, correspondingly, are obtained by taking ico = K [ = min(ic,) and po = p l = min(p,); this is a limiting case in which the terms for which K , = K~ and p, = p l in (4.86) are replaced by zero. Likewise, the best upper bounds are those for which K~ = rcg = max(K,) and p o = pLB= max(p,). These bounds were given by Hashin and Strikman (1963) with the implicit [ both to be obtained from the same phase, and K ~ restriction that ~ , , phad pLssimilarly. The restriction was removed by Walpole (1966a), who also gave the bounds in alternative but equivalent forms. Another example for which the isotropic form of P is relevant is provided by an isotropic polycrystal. This is regarded as a limiting case of a multiphase material in which the moduli L, of the rth phase are obtained by a rotation, Q, say, of the moduli L, of a single crystal, taken relative to some convenient reference frame. The parameter r is now a continuous variable

,

J. R . Willis

40

which defines crystal orientation. All orientations have equal probability, which ensures overall isotropy of the polycrystal; polycrystals with texture would either have this restriction relaxed, or the restriction to isotropic PSI, relaxed, or both, with corresponding complications for the analysis. The tensor T, defined by (4.64)is anisotropic but its mean value (T) is isotropic, so that

(T)

= (37%

2pd,

(4.87)

say: with this definition, 2, ji are still given by (4.85).Heavy calculation is avoided by noting that I C ~p ,T can be obtained from the scalars ( T j j k k ) , (Tijij), as in (4.38). The corresponding scalars formed from T, are independent of crystal orientation. They are, therefore, identical to the required mean values and can be calculated for L, = L,. Consider, for illustration, a cubic crystal for which, relative to the cube axes, the constitutive relation takes the form Okk

= 3ucekk9

g11

- 622

= 2pc(ell - e22),

GI2

= 2p:e12,

(4.88)

with equations for the remaining components being obtained by cyclic permutation of the suffixes. It follows immediately that (Lc)iikk

= 9Kc5

(Lc)ijij = 3 K c

+ 4& +

(4.89)

The notation L, = ( ~ I c 2pc,2p3 ,, introduced by Walpole (1966b) allows products and inverses to be worked out directly so that, if T, represents T, referred to the cube axes, Eq. (4.64)gives

+ 3Kp}-l,

T, = ({[3(4 -

{ "4 - P0)I - + 211,)

{[2(pc - p0)l-l

+ 2pP }-' , (4.90)

-

Then, from (4.38)and (4.89), 3% = ([3(ICc -

+ 3Jcp}-1,

2PT = 5{[2(Pc - po11-l

+ 2pp}-l + ${[2(p:

- p(J1-l

+ 2pp}-1

(4.91)

and the estimates (4.85)are completely specified. It is easy to show that the , 2p') is positive definite if and only if all of ic, p, and p' tensor L = ( 3 ~2p, are positive. Bounds for R, p, therefore, follow by substituting into (4.91)the appropriate extreme values for ice, p o . The unique choice for x0 is IC,. This gives K~ = 0 for any p o and correspondingly, from the first of Eqs. (4.85), IC = K , exactly. The two possible extreme choices for p o are p, and pi: the greater provides an upper bound for p and the lesser a lower bound. When p o = pC,the second of Eqs. (4.85)gives, after rearrangement, p = pl, where

Pl = P c + 3{[5/(& - Pc)l + 8PP)

(4.92)

Overall Properties of Composites

41

while, when po = p:, it gives Ji = Ji, , where

p 2 = p: + 2{[51(~, - p:)i + 12~;)- l.

(4.93)

In (4.92),p p is evaluated with Lo = (3ic,, 2pc),while in (4.93),pb represents pp evaluated with Lo = ( 3 q , 2p3. The bounds (4.92), (4.93) were first given by Hashin and Shtrikman (1962b) and were discussed further by Walpole (1966b). Examples for which P is anisotropic are usually rather complicated, but two simple limiting cases are worthy of mention. Both involve spheroids with axes of symmetry parallel to the 3-axis, distributed with corresponding “spheroidal” symmetry. The tensor P is then transversely isotropic and it is helpful to introduce the notation of Walpole (1969), in which a transversely isotropic tensor A is written

b, b, d, 2f 29)

A=

if the relation a

= Ae

3

(4.94)

between symmetric tensors G, e implies

3(6,, + 6 2 2 ) =

613 =

&ll

+ e22) + be337

2ge13,

With this notation, A-

‘ = (d/2A, - b/2A, - b/2A, a/A, 1/2f, 1/29)

(4.96)

where A = ad - b2. For a composite containing aligned “need1es”of length 21 and radius ~ 2 , with E << 1, in an isotropic matrix with Lame moduli 1, p, Willis (1980b) showed that

which may be substituted into (4.66) to yield estimates for E. The terms of order E’ are insignificant unless the terms of order unity cancel. This occurs when the needle-shaped inclusions are rigid and Lo is taken equal to L2,

J. R. Willis

42

the tensor of moduli for the matrix. This yields a lower bound for E (the upper bound being infinite) which is obtained from (4.54) with L, infinite:

r, = Lz + clP-l/(l

(4.98)

- c,).

If the needles are distributed at number density n,, their volume concentration is c1 = 4nn,Z3~2/3,which tends to zero if n, is taken finite. The tensor P-', however, contains terms of order E - and ~ gives

where 1, ,u here denote the Lame moduli that define L2. This is exactly the low-concentration estimate (4.45), but it provides a bound at any concentration. Similar degeneracy occurs for a matrix containing aligned platelets of a and thickness ~ U E where , E << 1. In this case, Willis (1980a) has shown that P

N

+

(0, 0, 0, (1 2 p , 0,1/2p) x

+ P V + 2P) IIE

($(A + 3 / 4 9 -$(A + PI, 4 1 + PI, $(A - P), &(31 + 7P), Q(31+ 4 d ) . (4.100)

If the platelets are rigid, a lower bound estimate for the tensor of moduli of the composite is

-

2 L2 + (32/3)n,a3p(ll + 2p)([3(1 + 3p)]-l,

O,O, 0,2/(31

+ 7p),0).

(4.101)

For penny-shaped cracks, regarded as vanishingly thin spheroidal cavities, Eq. (4.54) produces a lower bound estimate for L which takes a particularly simple form if it is inverted to give an upper bound M for the tensor of compliances M. This is

M - M 2 + 4(n1a3)(a ") (0,0, 0, (i + p)-', 0,2/(3R + 4p)), (4.102) 3P which is linear in n1u3,although f, is not. The results (4.99), (4.101), and (4.102) were given by Willis (1980~). +

V. Self-Consistent Estimates Consider a composite comprising a matrix with moduli L,,, in which are embedded n different types of inclusions.Phase r is distributed at volume concentration c, and has moduli L,. The inclusions that comprise phase r

Overall Properties of Composites

43

have identical shape, except for r = n + 1 which defines the matrix. Suppose that the composite is subjected to a uniform mean strain Z which generates the mean stress B = LZ.If the average values of the stress and strain over the rth phase are defined as b,, e,, it follows that br

= Lrer

(5.1)

r=l n+ 1

ii =

1 c,L,e,.

(5.3)

r= 1

Eliminating cn+,en+ gives n

s=Ln+,s+

1 cr(Lr-Ln+l)er

(5.4)

r=l

so that, if e, = A,V,

(5.5)

the tensor of overall moduli is given by n

z=Ln+1+

1 cr(Lr-Ln+1)Ar,

(5.6)

r= 1

exactly. Sections IV,B and IV,C give, in effect, estimates for the tensors A,. In particular, z, might be calculated by solving Eqs. (4.75) and then A, deduced from the relation e, = (L, - Lo)-%,. A simpler approximate procedure is to estimate e, as the mean strain in an inclusion of rth type, embedded in an infinite homogeneous matrix with moduli equal to the overall moduli L of the composite. If, in particular, the rth inclusion is an ellipsoid, Eq. (4.31), with L1, L2 replaced by L,, L, respectively, gives z, = [(L, - L)-1

+ Pr]-lZ

(5.7)

and, correspondingly, A, = [I

+ P,(L, - L)]-’,

where P, represents the tensor P, defined by (4.34),for the rth ellipsoid. For inclusion shapes other than ellipsoids, the solution of (4.24) is required. Formally, it is convenient to define z, to be the mean of z over the inclusion and to define P, so that (5.7) is true. With this agreement, substituting (5.8) into (5.6) yields an equation which must be solved for L because it appears also on the right side. This “self-consistent” prescription for estimating L is essentially that given by Hill (1965b) and Budiansky (1965).

J. R. Willis

44

Suppose, however, that the composite has no clearly defined matrix phase to be singled out for special treatment or, equivalently, that c , + ~= 0. If the shapes of the phases can still be distinguished, it remains possible to estimate r, and A, by (5.7) and (5.8), but (5.6) no longer applies. Instead, a variety of desirable restrictions may be identified. First, Eq. (5.2) (with c,+ = 0) requires that n

1 cr[I + Pr(Lr- E)] -

= I.

(5.9)

r= 1

Also, a = LO coupled with (5.3) implies

c c,L,[I + P,(L, - E ) - y n

=E

(5.10)

+ ( r ) , with Lo = E, implies c c,[(Lr - E)-1 + PI]-' = 0.

(5.11)

r= 1

and finally, C = Lo%!

n

rz1

Equations (5.9)-(5.11) provide three alternative methods for estimating L. They are not the same, in general, for Eq. (5.9) may be expressed in the alternative form n

1 crPr[(Lr-

L)-1

+ Pr]-' = 0

(5.12)

r= 1

and Eq. (5.11) is equivalent to

cs[I

+ P,(L, - L)]y

I-

= L.

(5.13)

If, however, Pr = P for all r, Eqs. (5.9)-(5.11) are equivalent. Furthermore, in this case, if cn+ # 0, (5.6) can also be placed in any of the forms (5.9)-(5.11) with n replaced by n + 1, even though the mean strain in the matrix is not estimated from the solution of an inclusion problem. A different perspective on the self-consistent method has been given by Willis (1977). Instead of estimating the tensors P, by solving inclusions problems, the Hashin-Shtrikman variational principle provides the optimal piecewise-constant polarization z* for estimating L, as the solutions r,, r = 1,2,. . . , n (or n + 1 if there is also a matrix phase) of (4.75), for any chosen comparison material with moduli Lo. If these equations give zr = s,e,

(5.14)

Overall Properties of Composites

45

the associated estimate for L is ULO) = Lo + (S),

(5.15)

where (S) = c c , S , . An estimate of self-consistent type for f, now follows from the assumption that f,(Lo) will be closest to E when the comparison material is identical with the “overall” material, that is,

Z(L) = E,

(5.16)

(S) = 0.

(5.17)

or, equivalently,

Equation (5.17)states that the mean polarization is zero. This is also implied by (5.11) although, in that equation, the polarization is estimated differently, so that (5.11) and (5.17) are not equivalent. It may be noted that Eq. (5.17) contains information on the two-point probabilities P,, through the solution of (4.75) and requires no commitment to particular inclusion shapes. Suppose now that all of the probabilities P,, have the same “ellipsoidal” symmetry. The estimate (5.15) then reduces to (4.66) and (5.17) implies that (T) = 0. This is (5.11) precisely, with P, = P for all r. Thus, for such composites, the variational approach leads to any one of the equivalent prescriptions (5.9)-(5.11) without requiring any assumption for the shapes of inclusions. Naturally a problem remains if different inclusion shapes are known to occur. There is then little option but to accept (5.6) with A, estimated by (5.8) although, in principle, a generalization of the variational method given in Section IV,C,2 could be developed, if the relevant conditional probability densities were known. It may be remarked finally that, for the composite treated explicitly in Section IV,C,2, namely, a matrix with a single population of inclusions, the self-consistent equation (5.16), with t(Lo) given by (4.82), leads to any of (5.9)-(5.11) with P, replaced by P , which need not relate to “ellipsoidal” symmetry. Two simple examples of the use of the self-consistent method are now presented, for illustration. First, for a matrix containing spheres, distributed isotropically, L is estimated as the solution of (5.9), (5.10) or (5.11), with P given by (4.32),in general, or by (4.39)if L is isotropic as will now be assumed. If there are n - 1 different inclusion types, all isotropic with L, = (3rc,, 2pJ and the matrix ( r = n) is likewise isotropic, the prescription (5.11) implies that rcT and p T , defined by (4.86),are both zero when rc0, p o are chosen as the “overall” moduli. For a matrix containing a single family of inclusions, so that n = 2, elimination of il from the self-consistent equations leads to a quartic equation for p. This will not be given, but, in the particular case where L1 = 0, so that the inclusions are actually cavities, it can be shown

J . R. Willis

46 that

(5.18) while p satisfies the quadratic equation

16p2 + p[Kc,(3

- c1)

- 4p2(4 - %I)]

- 3pzK2(1 - 2 ~ 1 = ) 0.

(5.19)

In the limit xZ -, 00, corresponding to incompressibility of the matrix, these equations imply 4(1 - ~ 1 ) (-l 2 ~ 1 ) 3(1 - 2 ~ 1 ) (5.20) Pz, F = 3 - c1 PZ > c1(3- c1) as given by Budiansky (1965). Equations (5.20) predict that the overall moduli reduce to zero at c1 = and indicate the need for some caution in applying the self-consistent method at high concentrations of inclusions with such extreme properties. It may be noted, for comparison, that the Hashin-Shtrikman estimates (4.85) give upper bounds for IC, p when Lo is chosen identical with Lz . When the matrix is incompressible, these reduce to (5.21) = 4pz(1 - c1)/3c1, p = 3/lz(1 - C,)/(3 + 2Cl). R=

Lower bounds would require Lo = L1 = 0, giving E = 0. As a second example, a self-consistent estimate is obtained for the overall moduli of the polycrystal for which bounds were given in Section IV,C,3. If the associated single crystal is cubic, with L, = (3rcC,2pc,2p3, the selfconsistent prescription (5.17) requires that and pT are zero when K ~po, take the values R, p, where p T are given by (4.91).The equation for I C gives R = K , and elementary manipulation of the equation p T = 0 then gives the cubic equation

8p3

+ ( 9 ~+, 4p,)ji2 - 3pi.i~~ + 4pC)F- 6~,p,p;= 0

(5.22)

for p. This result was first given by Hershey (1954) who assumed spherical grains and applied the prescription (5.10).This led to a quartic equation for p given, in present notation, by (5.22)multiplied by the factor ( 8 p + 9q). It is perhaps worth emphasizing again that (5.22)follows from the variational approach directly from the assumption of isotropy of the polycrystal, without reference to grain shape. The self-consistent method has been applied to a variety of examples less straightforward than those described above. Kneer (1965) studied textured polycrystals, in which grains were modeled as spheres, but not all crystal orientations were equally likely, so that L was anisotropic and P was computed from (4.32).Walpole (1969) considered composites containing transversely isotropic “needles” and discs whose properties were such that end

~

Overall Properties of Composites

47

or edge effects were not significant and P could be estimated asymptotically from one- or two-dimensional problems. Budiansky and OConnell (1976) studied a body containing flat cracks with elliptical boundaries, oriented randomly so that E is isotropic. Circular cracks oriented to make E transverselyisotropic were consideredby Hoenig(1979),andLaws and McLaughlin (1979) studied composites containing short fibers, modeled as aligned spheroids; here again, P was computed for a transversely isotropic matrix. VI. Generalizations The discussion so far has been expressed in terms of elasticity theory. However, the methods (and, in many cases, even the results) of the preceding sections apply to a variety of mathematically similar, though physically distinct, problems. First, if the phases are taken as incompressible and u is interpreted as velocity rather than displacement, the original problem becomes that of finding the overall viscosities of a fluid suspension in the limit of Stokes flow. Of course, for the fluid problem, the configuration of the suspension evolves with the flow and the question studied here, namely that of finding the viscosity given the configuration, is by far the easiest aspect of the whole problem. Study of the evolution of the configuration is, not surprisingly, still at a primitive state of development; it is discussed in the review by Batchelor (1974), for example. The problem of determining the overall thermal conductivity of a composite is described by Eq. (2.16) and (2.17), if u is interpreted as the scalar temperature field, e is its vector gradient, n is the negative of the heat flux vector, and L is the second-order tensor of conductivities. In (2.17), which is now a scalar equation, f is the scalar heat sink field. The explicit form of the constitutive relation becomes, in place of (2.14), 0 I. =

L IJ. . eJ .= L..u IJ .J ..

(6.1)

The Green’s function G is a scalar and r is a second-order tensor operator whose kernel has components Tij(x,x’) = a2G(x,x’)/(dxiax>),

(6.2)

in place of (3.10). The infinite-body Green’s function G“ remains homogeneous of degree - 1 and every equation in which suffixes are not explicitly given applies directly to the conductivity problem. Willis (1977) has, for example, evaluated P for a spheroid in a transversely isotropic matrix and has calculated the conductivity of a body containing aligned spheroids. With different interpretations of n, e, u, and L, the thermal conductivity problem

48

J . R. Willis

is identical to those for electrical conductivities, dielectric constants and magnetic permeabilities. There are, in addition to these, a number of problems whose structure is similar to that of the elasticity problem but which require more explicit development; two such problems are discussed below.

A. VISCOELASTICITY The constitutive relation for a viscoelastic solid may be expressed in the form (2.16),so long as the tensor L is interpreted as the Stieltjes convolution operator

or, equivalently, in terms of generalized functions, L: e(t)+ J [ d ~ ( t t’)/dr]e(t’)dt’,

(6.4)

the function L(t) taking the value zero for t < 0. Either of (6.3) or (6.4) give, for the particular strain history e(t) = eoH(t), u = Le = L(t)eo,

(6.5)

showing that L(t) is the stress relaxation tensor. For a viscoelastic composite, therefore, the objective is to find the overall modulus operator L or, equivalently, the overall relaxation tensor L(t).Dually, if the operator L is invertible, with inverse M, the corresponding function M(t) is called the creep compliance tensor and the overall compliance operator M or creep compliance tensor M(t) could be sought. Formally, the viscoelastic problem can be reduced to the form of the elastic problem already considered, by application of the correspondence principle: for boundary conditions such as (2.19),with 5 now a function oft, application of either Fourier or Laplace transforms yields an “elastic” problem, in which the moduli are now functions of the transform variable. The elastic problem is solved and finally the transform is inverted. All of the methods given in Section III-V are applicable, therefore, so long as the relaxation tensor L(c) has the symmetry (6.6) which has been implicitly assumed for elastic bodies. The variational estimates of Sections III,B and IV,C are now merely estimates rather than bounds, however, because the transform of L is generally complex and cannot be taken as definite in the manner assumed for elasticity. Lijkl(t)

= Lklij(t)

Overall Properties of Composites

49

It is possible, in fact, to analyze the problem directly in the time domain. For fields 2, q defined over V , the appropriate generalization of (3.52)is the bilinear form

where * denotes the ordinary operation of convolution with respect to time. The definition (6.7) is time-dependent, but involves only values oft, q up to time t if they are both zero for times t < 0. Now define

O(t)= t ( n ,e) = +(Le,e),

(6.8)

in analogy with (2.22).If e is associated with displacements that conform to the boundary condition (2.19), with i? now a function oft, it follows that

o(t)= f ( i 3 , E ) = $(LE,E),

(6.9)

as in (2.23). Also, the symmetry of the bilinear form

(Le,,e,) = 1/V

sv

[(dL/dt) * e,]

* e,dx

(6.10)

yields the variational principle for the displacement boundary value problem, that the functional

8*(t)= +(Le*,e*),

(6.11)

where e* is the strain associated with any displacement taking the prescribed boundary values, is stationary at each t for the actual strain field, e. Variational principles of this type are summarized by Leitman and Fisher (1973). Following Laws and McLaughlin (1978), it is useful to define the Green’s function G(x, x‘, t) for a homogeneous comparison body with modulus operator Lo so that it satisfies

+ 6,6(x

- x’)H(t)= 0,

x E V,

t

> 0,

(6.12)

Gip(x,x’,t) = 0,

x E at‘,

t > 0,

(6.13)

Gip(x,x’,t) = 0,

x E V,

t I 0.

(6.14)

The representations (3.6) and (3.7) then hold for the solution u(x, t ) of the viscoelastic boundary value problem (3.2)and (3.3),so long as G is interpreted as an operator of the same type as L. With this preparation, the whole of Sections 111-V apply also to the viscoelastic case, except that variational

50

J. R . Willis

estimates no longer provide bounds. Of course, the difficult task is to evaluate the operators explicitly.In practice, this is likely to be performed using transforms. The most complete study to date is that of Laws and McLaughlin (1978), who evaluated the operator P for spherical and circular cylindrical elastic inclusions ( E glass) in a viscoelastic matrix (a cold setting epoxy polymer) and obtained corresponding self-consistent estimates for the overall creep compliance tensor.

B. THERMOELASTICITY Static problems of thermoelasticity concern, in addition to stress and strain fields a,e, heat flux and temperature fields q, 8 which are related as follows: u = Le - 18,

(6.15)

q = KV8.

(6.16)

In (6.15) I is the stress-temperature tensor and L is the tensor of isothermal elastic moduli. Temperature 8 is measured relative to a reference temperature TR,say, at which the body is stress-free when its strain is zero. The tensor K in (6.16)is the tensor of thermal conductivities and is positive definite. In the discussion at the beginning of Section VI, it was implicitly assumed that K was symmetric. This is the usual assumption, though Carlson (1972) has pointed out that it is not a requirement of thermodynamics. In the absence of heat sources, divq = 0 and this equation, together with (6.16) and the thermal boundary conditions, define a purely thermal problem which can be solved independently of conditions on stress or displacement. In particular, if 8 takes a prescribed constant value over aV, then it is constant over I/. Whether or not this condition is adopted, 8 may be regarded as known in (6.15) to generate, with (2.17) and the stress or displacement boundary conditions, a purely mechanical problem in which 8 provides one of the sources of stress. Restricting attention now to the case of uniform temperature 8, zero body forces and displacement boundary conditions, the principle of minimum energy states that the actual displacement field is that which minimizes the Helmholtz free energy of the body, whose density +I! is given by

i,b = +eLe - 8k - iff?’.

(6.17)

The scalar f is introduced in (6.17) to allow a complete thermodynamic description. Equation (6.15) is equivalent to a=

(6.18)

51

Overall Properties of Composites

and, if q denotes entropy density, q=

-a$/as

= Ie

+ fe.

(6.19)

The specific heat at constant strain s(e) is then s(e) = T,(aq/ae), = ~ Dually, with the definitions

m = MI,

g =f

~

f

.

+ Im,

(6.20) (6.21)

Eqs. (6.15) and (6.19) may be rewritten

e = Ma

+ em,

(6.22)

q=

+ gf?,

(6.23)

and the specific heat at constant stress s(u) is s(u) = TR(dq/dO),

=

(6.24)

TRg.

The tensor m is the thermal expansion tensor. The only bounds so far published for thermoelastic properties of general composites have been obtained by substituting into the minimum energy principle a uniform strain field and, dually, substituting into the complementary energy principle a uniform stress field (Schapery, 1968; Rosen and Hashin, 1970), generalizing the procedure given in Section II,B,l where the Voigt and Reuss bounds for L were derived. This work will not be reviewed, but rather, an outline of generalizations of the methods of Sections III-V will be given. A uniform comparison material, with properties Lo, I,, fo is introduced, relative to which the constitutive relation (6.15) may be written u = L,e

- 1,s

+ Z,

(6.25)

where the stress polarization z satisfies

z = (L - L,)e - (1 - lo)&

(6.26)

generalizing (3.24). If 6 is uniform and the boundary condition (2.19) is imposed, the strain e may be represented, analogously to (3.8), in the form e = e - Tr,

(6.27)

to which perturbation theory may be applied, as in Section III,A, if L - Lo is small. It is more interesting, however, to eliminate e from (6.26) and (6.27) to give equation [(L - L J - ~+ rlt = E - (L - L,)-'(I

- lop,

(6.28)

J. R . Willis

52

which is (4.1) with a particular e0 on the right side. The Hashin-Shtrikman principle, therefore, shows that the functional ~ ( z * )= (z*, ( L - L,)-'Z*)

-2(r*, E )

+ (z*,rz*)

+ 2(z*, (L - LJl(1

- I,)@

(6.29)

is stationary when z* = z, the actual polarization and, furthermore, that F(z) is a minimum if L - Lo is positive definite or a maximum if L - Lo is negative definite at each point of V . The extreme value F(z) is expressible in terms of the mean $ of the Helmholtz free energy. In fact, ~ ( z= ) -(0,q

+ (2,(L - L,) - ](I - io)e),

(6.30)

which gives, upon substituting for z using (6.25) in the first term and (6.26) in the second,

F(z) = PLOP- 281,C - (a,E)+ (e,le) - ((I

- I,)(L - L , ) - ~ I- io))e2.

(6.31)

Now from (6.17) 2$ = (a,4 - (e,W - ( f > 0 2 ,

(6.32)

where (f ) denotes the mean off over V. The stress a is divergence-free and so (o,e) = (u,E)= (a,E).Therefore, ~ ( 7= )

2+,

+

foe2

- 2~

(6.33) - [(f> + ((1 - lo), (L - Lo1-V - 10))]e2. Estimates for overall properties L, i,7 follow if $ is defined to have the

form (6.17). With a temporary distortion of the notation used in Section IV,C,l, for an n-phase material, let z, denote the mean of z over the phase r and let a,, e, denote the corresponding means of stress and strain. Then, by averaging (6.25) over phase r, a, = LOer- 1,8 + 7,. (6.34) and, averaging (6.26) and rearranging, e, = (L, - L,)-'[z,

+ (1, - l,)O].

(6.35)

Hence, by substituting into (6.32), 2iJ = ZLOE - 281,s -

+ &r,s

ecc,(i, - i,ML, - L,)-

it,

- C c r [ f i + (1, - 1o)tLr - LJ-lOr

- i,)]e2.

(6.36)

Overall Properties of Composites

53

Now z, is a linear function of Z and 8, say zr = S,F - s,e.

(6.37)

If 3 has the form (6.17), it follows then that

-

+ CCrSr, + &[s, + (1, - IO)(L, - Lo)-'S,],

L = Lo

I = lo

9 = C C r C f i + (1,

- Lo)-'(lr - 10 - s r ) ] .

-

(6.38) (6.39) (6.40)

The average of (6.34) gives

a = (L, + ~ C , S , ) F - (I, It is desirable, therefore, if a = a$/%, that Ccrsr

= Ccr(Ir -

+ Cc,s,)e.

(6.41)

- LO)-'sr*

(6.42)

This relation is in fact true: it can be shown to follow from the selfadjointness of (6.28). The relations (6.38)-(6.42) have been given, in terms of stress concentration factors which relate e, to z! and 8, by Laws (1973), who deduced the symmetry (6.42) from the minimum energy principle. The formulas of Laws may be related to those above by expressing e, in the form

e, = A$

- a,&

(6.43)

Equations (6.35) and (6.37) imply

A, = (L, - L0)-'S,,

a, = (L, - Lo)-'(sr + lo - I,),

(6.44)

and (6.42) is equivalent to CcJrAr = C c r ( I r

+ La,).

(6.45)

This follows upon use of the relations Cc,A, = I,

'&a,

= 0,

(6.46)

which are implied by averaging (6.28) over V . If z, is now estimated by substituting z*(x) = C z r f i ( x )

(4.72)

into F(z*) and extremizing, as in Section IV,C,l, a result of the form (6.37) is obtained and the symmetry of the algebraic equations [which are just (4.75) with more complicated right side] implies that the approximate S,, s, so generated satisfy (6.42). Also, the extreme value for F(z*) is given by (6.30), except that z, are now those obtained via (4.72). Completing the

54

J . R . Willis

algebra gives

z(L - L)E - 200 - i)z - ( f - 7 ) I~ o

(6.47)

(20)

z,

whenever (L, - Lo) is positive (negative) definite for all r, where l, and are approximations that correspond exactly to (6.38)-(6.40) with S,, s, now defined as the Hashin-Shtrikman approximations. The inequality (6.47)yields the Hashin-Shtrikman bounds for L that were discussed in Section IV,C,l and also gives directly corresponding bounds for and hence for the specificheat at constant strain. Bounds for 1are obtained indirectly, by considering the quadratic form. To illustrate this, suppose that the composite (though not necessarily its constituent phases) is isotropic, so thatland interact with i?only through its dilatational component. Definiteness of the inequality (6.47) then requires that

f’

7

( B - ZSY I (R - ?s)(7 - TI, where R, iz represent the bulk moduli associated with E, L and B, diagonal entries of ‘i;7. The inequality (6.48)may be written - [(R - E)(f -

Jo]l’z I 7J I B + [(R

- E)(f - f)]l’z

(6.48) are the (6.49)

and bounds not involving E,f (whose values are not known) may be obtained by replacing R - iz, f - f by the differences between the Hashin-Shtrikman bounds for i?, f,respectively. The bounds produced from the classical energy principles by Schapery (1968) and Rosen and Hashin (1970) also required considering a quadratic form but seemed always to involve the exact overall moduli L. It should perhaps be mentioned that the full development of (6.47) is unnecessary for a composite with only two phases. Laws (1973) derived the exact result

-

1 = (L - LJ(L1

- LZ)-’I1

+ (L - L,)(LZ - Ll)-’lz,

(6.50)

by eliminating the tensors A,, a, (or, equivalently, S,, s,) between equations (6.38),(6.39),(6.42), and (6.46), so that a bound for E immediately induces a bound for 1.It may be noted that, since the Hashin-Shtrikman estimates for S,, s, conform exactly to (6.45), (6.46), the estimate corresponding to (6.39) for is expressible in the form of Eq. (6.50), with L replaced by E, so that this estimate provides bounds automatically. Finally, Laws (1973) proposed the estimation of A,, a, (or, equivalently, S,, s,) by the self-consistent method of embedding an inclusion of rth type in a matrix whose properties are those of the composite, as described for the mechanical problem in Section V. The remarks of that section are applicable here : the usual self-consistent method has a variational interpretation for composites with “ellipsoidal” symmetry and, for more general

Overall Properties of Composites

55

composites, a “variational” prescription, which allows explicitly for the statistics through the probability P,, ,is available as an alternative. Explicit self-consistent formulas for a composite containing spherical inclusions were given by Budiansky (1970). VII. Problems Which Lack Convergence Beginning in Section III,A, one of the major points of principle to receive emphasis has been the desirability of formulating a boundary value problem exactly for a finite body and only afterward making any simplification that is allowed when the body is, in fact, large in comparison with microstructural dimensions. This led, for example, to the prescription (3.33) for the limiting form of the operator r which is not just a direct replacement of r by the operator I‘” derived from the infinite-body Green’s function. The kernel P ( x ) decays like 1x1- at large 1x1 and, if applied directly to z, would lead to an integral that is only conditionally convergent. This type of complication is altogether more severe for problems which generate operators whose infinite-body forms decay more slowly at large 1x1. It is the writer’s view that precise formulation of such boundary value problems prior to the development of infinite-body approximations is highly desirable, at the very least. Such initial formulations have, however, been very much in the background if they have been acknowledged at all in most studies carried out to date, both for problems of “overall modulus” type and for the problems that will now be described. The problem with a “badly behaved” Green’s function that has received most attention is that of determining the resistance to flow of viscous fluid through a fixed bed of obstacles. Macroscopically, the resistance takes the form of a body force f related linearly to mean fluid velocity (u). If the fixed bed is statistically isotropic, the relation reduces to

f = -paZ(u),

(7.1)

where p represents the viscosity of the fluid and the Darcy coefficient is to be estimated. A fixed bed of spheres has been considered by several authors, starting with Brinkman (1947) who performed a self-consistent calculation, rather like that described in Section V. Brinkman’s idea has been developed further by Lundgren (1972) and Howells (1974) to make explicit allowance for correlations between sphere positions. Low-concentration approximations for the Darcy coefficient are of considerable theoretical interest (largely because of the convergence difkdty), though probably of limited practical

56

J. R. Willis

value; they have been considered by Childress (1972), Hinch (1977), and others. The convergence difficulty may be explained by discussing the flow of viscous fluid past a single bounded obstacle. Far from the obstacle, the induced perturbation to the mean flow has the “Stokeslet” pattern generated by an isolated body force. At distance r from the obstacle, this is of order r - ’ . Therefore, if a second obstacle is placed in the flow, its interaction with the first is of order r - l if their separation is r, and any attempt to allow for a distribution of obstacles by summing interactions between pairs is doomed to failure through divergence of the integral. The problem is even worse in two dimensions: Green’s function is of order lnr and the problem of an isolated cylinder perturbing a uniform mean flow cannot even be defined. Self-consistent calculations and perturbation expansions are the only type that have been applied so far to the fixed-bed problem. It is likely, however, that a variational approach could be developed: all three methods have been applied by Talbot and Willis (1980) to a problem with similar mathematical structure, which will now be outlined. TO A RANDOM ARRAYOF VOIDS A. A MODELPROBLEM:DIFFUSION

Irradiation of metal by neutrons, electrons, or ions generally produces overall distortion. This is often associated with the atoms rearranging themselves so that the specimen contains voids or gas bubbles, with consequent volume change. The full range of phenomena is complex and cannot be properly described here; the review of Bullough and Hayns (1978) provides background, together with details of a variety of calculations of selfconsistent type. It is apparent, however, that the basic rearrangement takes place by diffusion in the presence of various types of sink distribution, including the distribution of the evolving set of voids or gas bubbles. Talbot and Willis (1980) studied the simple model problem of determining the mean sink strength of a random array of voids in the presence of a single population of diffusing defects (gas atoms, say), when no other sinks were present. The material was taken to occupy the region V and was regarded as a matrix containing identical spherical voids occupying V, with centers xA, A = 1, 2,. . . , as described in Section II,A. Overall, for such a medium, if the defects have concentration c(x, t ) and are introduced throughout V at rate &x, t), the concentration is expected to be described by the diffusion equation aclat = b(v2c- k2c) + I?,

xE

v,

(7.2)

in which b is an overall diffusion coefficient and the sink term -bk2c, which is analogous to the Darcy resistance (7. l), represents the mean effect

Overall Properties of Composites

57

of the voids. If (7.2) is indeed a fair description, then it must apply in particular to the steady-state situation in which & is independent of x and t and no flux is admitted across dV, so that &/an = 0 there. The solution of (7.2) in this case is c = T, a constant, related to I? and k2 by Dk2C = I?.

(7.3)

The simplest way to estimate bk2is by the self-consistent method of considering the flux into a single void of radius a embedded in the “overall” material. If the boundary condition c = 0 is adopted at the void surface, this leads to the solution c = i ~ {1

- (a/r)exp[ - k(r - a)]},

(7.4)

where T is given by (7.3) and r represents distance from the void center. It may be noted at this stage that, if k = 0, (7.4) gives a perturbation of order r - while introduction of the overall sink term gives exponential decay and so would “screen” distant sinks from one another. The flux into the void is now calculated as

F = 4 n ~ (+ l k~)bF,

(7.5)

while (7.2) gives the sink strength per unit volume as bk2T. If the voids are distributed at number density n l , therefore, self-consistency requires that

k 2 = 4nanl(l

+ ka).

(7.6)

The overall diffusion coefficient b is not estimated so easily and it is usually identified with the diffusion coefficient D of the matrix. In contrast with the above, the steady-state problem is described exactly by the equations

DV2c + K = 0,

x E V’

dc/dn = 0,

x E av,

CEO,

X € a v ~ ,

(7.7) A = 1 , 2,...,

where V’ denotes the part of V occupied by the matrix and the flux term K is related to R by

Iz = K(1 - CJ,

(7.8)

where c1 = 4nn,a3/3 represents the volume concentration of voids. The object is to solve the system (7.7) to obtain E, the mean of c over V (with c = 0 over VA),in terms of which (7.3) provides a precise definition of the sink term Bk2, analogous to the definition (2.21) of overall moduli.

J. R . Willis

58

B. AN INTEGRAL EQUATION AND

PERTURBATION

THEORY

The problem defined by Eq. (7.7) may be expressed in the form of an integral equation, analogous to (4.1), if a Green’s function for the region V is defined so that V 2 G ( x , x ‘ ) = 6 ( x - x’) - l/V,

x E V,

aG/an= 0,

x E av.

(7.9) (7.10)

The presence of the distributed source should be noted in (7.9): although it is small when V is large, it is essential for the consistency of (7.9), (7.10). The solution of these equations is made unique by requiring that JV

G ( x , x’) dx = 0,

(7.1 1)

which also ensures that G ( x ,x’) = G(x’, x). Application of Gauss’ theorem in the usual way yields c(x’) = F

+ SaVv G ( x ,x’)q(x)ds - K’

JV,

G ( x , x’) dx,

(7.12)

where V, denotes the union of the regions V,, K‘ = K / D

(7.13)

and q ( x ) = ac/an,

x Ea

v,.

(7.14)

The boundary condition c = 0 at the void surfaces now generates an integral equation for 4 ( x ) in which the additional unknown z is counterbalanced by the consistency condition

J,

q(x)ds = V’K’ = V(l - c,)K‘.

(7.15)

More conveniently, the integral equation may be viewed as an equation for q ( x ) given F, in which K is defined by (7.15). Proceeding as in Section IV,B, the restriction of q(x)to aVAis called qA(x).The integral equation may then be written in the form CGqA+F=O,

X’EaVA,

A = l , 2, . . . ,

(7.16)

A

where operator G is defined by G : qA -+

LVA

+

dsG(x,x’)qA(x) K‘

VA

dxG(x,x’).

(7.17)

Use was made, in deriving (7.16), of (7.11) to replace the integral over V by integrals over V,, A = 1,2, . . . . Taking expectations of (7.16), keeping

Overall Properties of Composites

59

’V fixed, now gives

+

~ 4 2J ~ ~ B P B I , G ~ : B

+ z = 0,

XI

E

av,.

(7.18)

The term K‘ in the definition (7.17) of G now becomes
=[

~ ( l -CAI-’

{L~~+ qj(x)ds

L~~

J~~BPBIA

I

~ B A B w ~ X (7.19)

but, in the limit of large V , this is equivalent to

(K‘)= [v(l - ~ 1 > ] - ’JdxBPB

LvB&(x)ds.

(7.20)

The simplest approach to (7.18) is to apply the quasicrystalline approximation (4.51), which reduces it to an equation for q;. If it is assumed in addition that the operator G defined by (7.17) is translation invariant in the limit of large V , it follows that 4: is insensitive to translations of x A . Finally, if PBIAis isotropic, 42 reduces to a constant, qo say, and (7.20) becomes 3c1q0 ( K ‘ ) = 4na2n1q01 - c1 a(1 - cl)‘

(7.21)

The problem is now to find the appropriate representation for G. When q2 is constant over aV,, Talbot and Willis (1980) simplified the integrals in (7.17) by application of the mean value theorem for harmonic functions. The result is expressible in the form

1 - c1

-a(l

C1 + ic,) + Ix‘ - xA12+ 4na2G,(x,,x‘) 2a

(7.22) where GA(xA,x’)is a function that is essentially an “image” term and is small when V is large and x’ and xA are not close to dV. Substituting into (7.18) now gives, when I/ is large, 1 -c1 +Z=O,

X E ~ V ’ , (7.23)

J. R. Willis

60

in which the term involving V-’ cannot be ignored because it is integrated over I/. The left side of (7.23) is expected to be independent of x’: accepting that this is so, (7.23) may be averaged over dV,, which has the effect of replacing G(x,, x’) by G(x,, xA)- a2/6V. Finally, a convergent integral involving G is obtained by use of (7.11); the result is

{

[qo/(l - cl)] u - 4na2

s

dxB(PBIA

- ~,)G(x,,x,)

I

+ U C , (-~ ~ , / 5 ) = Z.

(7.24) If the voids have no long-range order, only points X~ close to xAcontribute significantly to the remaining integral and so G may be replaced by the infinite-body Green’s function. Thus, finally,

The author must confess to finding the above argument not entirely convincing. While it is true that the quasicrystalline approximation is likely to produce a result correct only to lowest order, the contribution of order c1 from the integral of a term of order V-l hardly lends confidence to the assumption of translation invariance on which the derivation depends. The estimate (7.25) nevertheless does have an honest interpretation, as will be shown below. Before proceeding, however, it may be remarked that interactions can be allowed for, as in Section IV,B,2, by taking expectations of (7.16) with two spheres fixed and making the closure assumption (4.55). A calculation equivalent to this was performed by Brailsford (1976) (who did not consider terms of order V-’, however). The “fixed-bed’’ problem has been approached in a similar way by Hinch (1977). In the present problem, an integral equation for (q:B - q z ) is generated, whose solution shows the same type of exponential decay as (7.4) and yields an estimate for 8 k 2 that agrees with (7.6) to order (c1)’l2.

C. VARIATIONAL FORMULATION

Consider, in place of (7.16), the integral equation

syI

dxf(x)G(x,x’)

+ g(x’)= 0,

X‘ E

Vl,

(7.26)

where G is the Green’s function for V and V, is a subset of V . Equation (7.26) is self-adjoint, from the symmetry of G, and it is easy to prove that (7.27)

Overall Properties of Composites

61

for any function f* defined over V,. Equation (7.26) therefore implies the maximum principle,

(7.28) with equality when f* = f . Now Eq. (7.12) defines c(x‘) as a harmonic function within V,, which is zero because it is zero on the boundary aK. Equation (7.16) therefore applies for x E VA,A = 1,2, . . . and can be considered as a limiting case of (7.26). The corresponding limiting form of (7.28) is

{LVAds’qA(x’)+ K*

VA

dx’

(7.29) B+A

where qA here denotes the restriction to aVAof any approximation q* to q and K* is the corresponding estimate of K’, as in (7.15). Any approximation q* thus bounds K , and hence bk2,from below. If the functions qA are now taken constant over aVA, the functional in (7.29)can be simplified using (7.22).The result is

a’ 15

- - (5 - c ~ ) ( K *+) ~2K*Z 5 K‘Z,

(7.30)

where (7.31) G A denotes G,(xA,xA) and G A B denotes G(xA,xB). The inequality (7.30) applies for any constants qA: they need not be all the same, as was assumed in the preceding section. It is essential, in fact, that they should vary with A for a useful result to be obtained.

J. R. Willis

62

Although the distribution of the voids is assumed statistically uniform (except close to dV), the Green's function is sufficiently unpleasant to discourage the replacement of the spatial mean over A in (7.30) by an ensemble mean over some chosen V,. Still, assuming that, for large V , the left side of (7.30) is independent of the sample CI,it is now replaced by its expectation value. If qAis taken to depend only upon the location x, of V,, the expectation involves integrals of probability densities for the location of up to four inclusions, because K* is defined by a sum. The object is to maximize the expectation of the left side of (7.30)by setting its variation with respect to qR equal to zero. The same result is obtained, in fact, by differentiating the left side of (7.30)with respect to qR and taking the expectation value of the result, conditional upon x R being fixed. With the approximation, valid when V is large, that the conditional expectation of any mean value (that is, V - times a summation) is equal to the unconditional expectation, Talbot and Willis (1980) showed that

''

ha3 -k 3V(1 - cl)

a2(5 - cl)

JdXApA4'- 15(1 -

cl)

-

(K*)

+- 0, 1 - c1 C

(7.32)

where

(7.33) Equation (7.32) shows that 4Ris independent of x R . This allows evaluation of the integral, to simplify (7.32) to

4R- a2(5 - Cl)(K*)/lS

+ I = 0.

(7.34)

Then, with the definition qA = [(l - c,)/3cl]a(K*)

+ 8,

(7.35)

where J d X , P A 8 = 0,

(7.36)

Talbot and Willis (1980) showed that (7.34) requires

-

(a/3cl)(K*)(PB - nl) + eBPB 0

(7.37)

except, perhaps, in a layer close to aV where P, may deviate significantly from n, ; in this layer, the left side of (7.37) will have zero mean. Elsewhere, PB n, and so eB 0 and qA is constant, as assumed in Section IV,B. N

-

Overall Properties of Composites

63

Further analysis is required, however, because these statements are true at any chosen point x A , but terms of order V - may still contribute significantly to the functional. The end result is that

which is precisely consistent with (7.25).Since the expectation of the left side of (7.29) is quadratic, its extreme value is just one half of the linear term, which is ( K * ) l precisely. Hence, from the definition (7.3) of Bk',

bk' 2 (1 - c,)D(K*)R.

(7.39)

Assuming now that PslRis isotropic, it is convenient to express it in the form PBlR

where x = I

x -~ x,1/2a.

[

= nidx),

(7.40)

Then, g(x) = 0 for x 5 1 and (7.38) may be written

a2(K*)/3Z = 1 - SC, - c:/5

+ 12cl

J:

I'

(g(x) - 1)xdx

. (7.41)

The simplest possible choice for g(x)is g(x) = 1:this defines the "well-stirred" approximation, used in the low-concentration limit by Batchelor and Green (1972),Willis and Acton (1976), and others. Unfortunately, the approximation fails at high concentrations, predicting that (K*) becomes negative after passing through a singularity near c1 = 0.2. This is inconceivable, since it can only have resulted from the expectation value of the negative definite quadratic functional (7.27) having become positive. Equation (7.41) thus indicates that g(x) must necessarily rise above 1 when c1 is large. This is plausible at high concentrations, since knowledge that a sphere is located at some point must almost guarantee the presence of spheres centered 2a away. It is also borne out by more realistic models discussed by Talbot and Willis (1980). The most interesting is that derived from the Perkus-Yevick approximation to the pair function g(x)for a statistical-mechanical ensemble of hard spheres. The integral equation for g(x) produced by Perkus and Yevick (1958) was solved by Laplace transforms by Wertheim (1963). The transform requires numerical inversion (Throop and Bearman, 1965), but the integral in (7.41)can be deduced analytically from the small-argument expansion of the transform. With the Perkus-Yevick form for g(x), (7.41) remains finite, but, interestingly, the self-consistent estimate (7.6) lies below the lower bound when c1 > 0.2, if b is identified with D. This contrasts with the results of Sections IV and V which show that the self-consistent estimate of L always lies between the Hashin-Shtrikman bounds.

64

J . R. Willis

VIII. Wave Propagation Considerable effort has been applied to the study of wave propagation in random media: see, for example, the books by Chernov (1960), Uscinski (1977), and Ishimaru (1978). The vast majority of studies have been, however, for electromagnetic or acoustic waves, both of which are described essentially by the scalar wave equation. Methods developed for these fields of application can be carried over to elastodynamics at the expense of algebraic complication, but they need not always be relevant. Whereas, in the electromagnetic case, interest frequently centers on waves whose wavelength is short in comparison with distances over which variations in the properties of the medium are significant, the opposite limit is often the one that is required for elastic wave propagation through a composite. For example, a wave traveling with a speed 3 km s-’ typical of metals or rocks has a wavelength of 3 mm when its frequency is 1 MHz. Also, strong discontinuities in properties across phase boundaries are the rule rather than the exception in solids and limit the utility of methods that assume smooth variations. In any stochastic problem involving wave propagation, the approach invariably adopted is to seek the ensemble mean (u) as discussed in the static context in Section II,B,2 and to hope that this relates in some reasonable, but unspecified way to what would be observed if some ‘local average” were taken. Spatial variation of the mean field is an essential feature of wave propagation and the volume averaging approach given for static problems in Section 11,BJ is simply not available. For applications to composites, for which a transducer is bound to register displacement averaged over a region large relative to the microscale, knowledge of the mean field (u) is likely to be sufficient. For geophysical applications, however, an individual seismogram registers displacement over a scale small relative to the scale of the relevant inhomogeneities and information on fluctuations about the mean is also of interest. A discussion is given by Hudson (1968), for example. Attention will be restricted to the mean wave in this article for the reason that fluctuations have not yet been considered within the very recently developed framework that will be described. Before proceeding with this, brief mention will be made of the longer established methods, which amount to direct adaptations to the elastic case of methods developed for acoustic or electromagnetic waves. First, for a weakly inhomogeneous medium, the displacement field can be described by the appropriate dynamical variant of Eq. (3.6) and this can be solved by perturbation theory. Knopoff and Hudson (1964) and Hudson (1968) performed the most elementary iteration (the Born approximation) to obtain local results that were not valid uniformly over large propagation distances. Karal and Keller (1964) also retained only terms of low order, but obtained

Overall Properties of Composites

65

an integrodifferential equation for (u) by employing the method of smoothing (described in Section 111,AJ) and eliminating uo in favor of (u). Plane-wave solutions were shown to satisfy an eigenvalue problem which gave the wavenumber as a complex-valued function of frequency. Thus, the waves were predicted to be dispersive and also to decay exponentially with distance of propagation. This does not imply physical dissipation since only the mean wave is described in this way: as the wave propagates “coherent” energy in the mean wave is scattered into “incoherent” form by the inhomogeneities. The results of Karal and Keller (1964) are restricted to weak inhomogeneities,but are not limited in frequency. McCoy (1973) considered perturbation series to arbitrary order, obtained by the method of smoothing exactly as in Section III,A,l, to give a formally exact equation governing (u). He then restricted attention to long waves and retained only terms of low order in the ratio microstructural dimension to wavelength. This provided a demonstration that long waves propagate as though the material were uniform, with moduli equal to the overall moduli given by (3.21) and density equal to the mean density of the composite. McCoy also gave the lowest order real and imaginary corrections to the long-wavelength,low-frequency limit of the dispersion relation. An interesting point of principle was settled in this way, but quantitative prediction would still require the summation of series like (3.21). For materials comprising a matrix containing a dispersion of inclusions or fibers, an explicit “multiple scattering” approach is usually adopted, in which the total field is expressed as the sum of an “incident” field uo and fields from each of the scatterers. In calculating the field scattered from inclusion A, the field incident upon A is taken as the sum of uoand the fields scattered from all other inclusions. An infinite set of integral equations (one for each inclusion) is generated by this procedure. It is reduced by taking the expectation value of the equation for inclusion A , conditional upon that inclusion being fixed. This introduces the conditional mean field (u)~, but also, analogously to (4.23), amean field ( u ) ~conditional ~ upon inclusions A and B being fixed. The usual method of closing the heirarchy is to employ Lax’s (1952) quasicrystalline approximation in some form. Datta (1977) took ~ found that long waves did not propagate as essentially ( u ) ~= ( u ) ~and though the material were uniform with overall static moduli and mean density, as predicted by McCoy (1973). However, subsequently (Datta, 1978), he adopted a quasicrystalline approximation on parameters defining the source that generates the field scattered from any chosen inclusion and confirmed McCoy’s finding. Bose and Ma1(1973,1974) simplifiedcalculations by making a “point-scatterer” approximation and again found wave speeds and attenuation coefficients in broad agreement with McCoy (1973). Varadan et al. (1978) and Varadan and Varadan (1979) performed more extensive

J. R. Willis

66

calculations for the propagation of SH waves in a composite containing aligned fibers by expressing both incident and scattered fields as eigenfunction expansions, whose coefficients were related through the T-matrix formalism of Waterman (1969, 1976). The equations were closed by making a quasicrystalline approximation for the coefficients defining the scattered field and solved by truncating the series after several terms. All of this work relies quite heavily upon explicit solutions for point scatterers. It has, consequently, been developed only when the matrix and inclusions are isotropic. The remainder of this section is devoted to an alternative approach, developed by Willis (1980b,c),which comprises a direct generalization of the methods given in Section IV. This does not rely upon isotropy and, although it has so far been applied only to a matrix containing inclusions, it is equally applicable to a composite such as a polycrystal. Thus, it offers a unified approach to wave propagation in any composite. A. POLARIZATION FORMULATION

1. Integral Equations

The equation of motion for any continuum subjected to body force f per unit volume may be given in the form diva + f = p,[,

(8.1)

generalizing (2.17), where p denotes momentum density. Equation (8.1) is independent of material properties. These enter through the constitutive relation (2.16)for the stress and the additional relation

P = PU,t

(8.2)

for the momentum density, where p denotes the mass density of the medium and u is its displacement. Usually, (8.2) is substituted directly into (8.1), but there is some advantage here in treating (8.2) as a constitutive relation, on a par with (2.16).Relative to a comparison material with moduli Lo and density po, the stress polarization z has already been defined by (3.24). The momentum polarization 1 ~ : is now introduced through the corresponding definition (8.3) x = ( P - PO)U,t. Thus, in terms of Lo and p o , a = Loe

+z

and

p = p o ~ ,+t x .

(8.4)

Substitution of these into (8.1) then gives an equation for the displacement in the comparison material, when it is subjected to an extra body force div z -

Overall Properties of Composites

67

z,, . Willis (1980b) showed that the displacement field u so generated could be

expressed in the form, analogous to (3.7),

u = U O - St - Mn,

(8.5) where the field uo is the solution of the boundary value problem for the comparison material in the absence of t and n and S and M are operators related to the Green’s function G for the comparison body: S: zij(x,t ) -+ Jdt’

M: q(x, t )

-+

Jvdx’SPij(x,t, x’, t’)rij(x’,t’),

(8.6)

Jdt’ Jv dx‘Mpi(x,t, x‘,t‘)ni(x’,t‘),

where the kernels of the operators S and M have components SPij(x,t, x’, t’) = dGPi(x,t, x’, t ’ ) / d ~ > I ( ~ ~ ) ,

(8.8)

M J x , t, x’, t’)= - dGpi(x,t, x’, [‘)/at’.

(8.9) It is important to note that the field uo is supposed to satisfy initial conditions of prescribed displacement and prescribed momentum rather than velocity; otherwise, an additional term would appear. Substitution of (8.5) into the definitions (3.24), (8.3) of t, II: now gives the pair of operator equations

+ S,r + M,n = eO, ( p - po)-’n + S,tt + M,+G = ,:u (L - Lo)-’t

(8.10) (8.11)

where e0 is the strain associated with uo and S,, M, have kernels (Sx)pqij= d2Gpi/(dxqdx>)l(ij)(pq), (8.12) (Mx)pqi

= - aZGpi/(dxqW l ( p q ) .

(8.13)

Equations (8.10) and (8.1 1) constitute a generalization of (4.1). Korringa (1972) formulated time-reduced dynamic problems in terms of stress polarization, but did not introduce momentum polarization. Time-reduced versions of (8.10) and (8.11) are obtained by Fourier transformation with respect to time. Now consider a boundary value problem for a random medium which, for definiteness, will be taken as a matrix containing a dispersion of inclusions. As in earlier sections, the matrix will be taken to have moduli L, and density p z . The inclusions, which occupy regions V,, A = 1 , 2 , . . . , have moduli L, and density p1 and are distributed at volume concentration c I . The object will be to determine the mean wave (u) which may be expressed, with the aid of (8.5), in the form (u) = u0 - S(z) - M(a).

(8.14)

68

J . R . Willis

Following the pattern set in Section IV,B, the comparison material is chosen identical with the matrix. The definitions (3.24), (8.3) of the polarizations r, n then show that they differ from zero only over the regions V, occupied by the inclusions and it is useful to define rA,nA as their restrictions to V,. Substitution into (8.10) and (8.11) gives equations similar to (4.19) whose conditional expectations, with V, fixed, may be expressed in forms similar to (4.46):

+

(L, - ~ ~ ) - 1 r : :s,zj

+~

+

, n j JdxB(pB1, - P , ) ( S , ~ ;

+ M,Z;)

+ J’~XBPBIA[S~(~BAB - r;) + M,(~!L- n;)] = eo

+ ~,n;),

- J~X,P,(S,~; (P1

(8.15)

+ S , d + M,PA, + j d x B ( p B I , - P B ) ( s , ~ G + M,~G)

- P2)

-1

A

+ J~x,P,~,[s,~(~:B

-

= upr - J ~ X B P B ( S , ~ ~ ;

+ M,,(~B,,-

+ ~ , ~ n ; ) ,x E v,.

(8.16)

Now it is easy to show that (t) (x, t ) =

j’

d x B p B ~ ( x t), ,

(8.17)

with a similar relation for (n). Thus, from (8.5), the right sides of (8.15) and (8.16) are the mean strain ( e ) and mean velocity (u),,, precisely. Equations (8.15) and (8.16) are not useful as they stand because they involve &, n2B as well as r; , n;. If, however, the quasicrystalline approximation (4.51) is made, for n as well as r, they become approximate but explicit equations for r;, n;. As discussed in Section IV,B,3, the approximation is valid strictly only at low concentrations of inclusions but its alternative derivation in the static case, in Section IV,C, from the Hashin-Shtrikman variational principle, provides some limited encouragement to proceed with it at any concentration. 2. Plane Waves Equations (8.15) and (8.16) apply to any boundary value problem whose solution for the comparison body is uo. The simplest problem in elastodynamics is, however, to seek plane-wave solutions of the differential equations. These waves are usually considered as propagating in an infinite medium. Boundary conditions are necessarily disregarded. The corresponding procedure for Eqs. (8.15) and (8.16) is to set uo = 0 and to relate the operators S, M to the infinite-body Green’s function. Solutions of the homogeneous

69

Overall Properties of Composites

equations are now sought in the form

- x) + wt]},

zi(x,t) = zl(x - x,)exp{-i[k(n ni(x, t) = nl(x - x,)exp{

- i[k(n

x) + ot]},

(8.18) (8.19)

in which the circular frequency w and the unit wave normal n are chosen and the wavenumber k is to be determined. In the same way that elementary plane-wave solutions are defined for a uniform medium, the composite is taken to be statistically uniform. With the forms (8.18) and (8.19), it follows that (z)(x, t ) = clzl exp{ - i[k(n

- x) + ot]}

(8.20)

with a corresponding equation for (n), where TI denotes the mean value of zl(x - xA)over V,. Then, from (8.5) with uo = 0, (u)(x, t ) = -c,[&,

+ MEl] exp{ -i[k(n

*

x) + oil},

(8.21)

where S, M denote the Fourier transforms of S, M, evaluated at (kn,o). Equation (8.21)gives the right sides of the homogeneous equations (8.15) and (8.16), by differentiation with respect to either x or t. The terms involving &, nBABare eliminated by use of the quasicrystalline approximation and the terms that remain are simplified by noting that the time integration in (8.6) or (8.7) produces simply the corresponding Fourier transforms with respect to time; that is, it reduces the operators to time-reduced form. To proceed further, explicit representations of the operators are required. The time-reduced infinite-body Green's function for the comparison material has components Gipthat satisfy the equations (L2)ijklGkp,jl

+ pZwZGip+ 6ip6(x)

=

(8.22)

and it follows immediately by Fourier transforming that

G(kn, w ) = [k2~,(n)- p 2 w 2 ~ ] - ' ,

(8.23)

where L,(n) is the acoustic tensor with components [L2(n)]ik

= (L2)ijklnjnl;

(8.24)

its eigenvalues Ar = p2c? define the speeds c,(n) with which plane waves may propagate in the direction n and the corresponding eigenvectors ur(n) give the wave polarizations. The derivatives with respect to x or t that are required to generate S, S,, etc., from G correspond to multiplication by kn or w for S, S,, etc. A representation for G itself is found by starting from the plane-wave decomposition (4.25)of the delta function. It was shown by Willis(1980b)that

J. R. Willis

70

the eigenvectors u' being taken as orthonormal. Equation (8.25) reduces to (4.29) when w = 0. The low-frequency limit of the dispersion relation for plane waves may now be studied by retaining only terms of order zero in o or k in the homogeneous equations corresponding to (8.15), (8.16). The operators M,, Mtthat appear on the right sides are homogeneous functions of degree zero and so are retained. S , itself, operating on r j or T;, is replaced by the static operator I?", from which it differs by a term of order 0': this follows directly from (8.12) and (8.25). The term of order w 2 in the integral with respect to x, has a bounded coefficient because the integrand contains the factor P,,, - P, which guarantees convergence. The operators M,, S,,, M,, are all at least of order w and so may be disregarded. Thus, to zeroth order, with the quasicrystalline approximation, Eqs. (8.15) and (8.16) reduce to

s,,

s,,

(L,

-~

+ r m r j + Jdx,(~,,, - P,)rmT; = (e),

~)-1rj (p1

- pz)-ln?

= -iw(u),

x

E

v,

(8.26) (8.27)

where (u) is given by (8.21) and (e) has components (eij) = -ikni(uj)/(ijj.

(8.28)

Furthermore, to lowest order, the right-hand sides of (8.26) and (8.27) may be taken constant over V,. Equation (8.26) is identical to (4.52), with z replaced by the local mean strain (e), since the operator r has the same effect as rm on a function such as z; which has bounded support. Given, therefore, that (4.52) generates the estimate f, of the overall moduli, the solution of (8.26) by definition implies (T) =

(L - L,)(e).

(8.29)

Also, if ij is defined as the mean density of the composite, so that p" = ClPl

+ (1 - C J P 2 ,

(8.30)

Eq. (8.27) can be placed in the similar form

(n) = - i o ( p - p&u).

(8.31)

Equations (8.29) and (8.31) show that the mean polarizations (T), (n) are those that would be generated if a wave (u) propagated in homogeneous material with moduli and density p. This agrees with the finding of McCoy (1973).The result can be deduced algebraically by eliminating (2) and (n) between Eqs. (8.21), (8.29),and (8.31).

Overall Properties of Composites

71

Corrections to the low-frequency limit of the dispersion relation are found by retaining some allowance for the exponential in the integrand of (8.25) which defines G. The lowest order imaginary correction is of particular interest, since this determines the rate of decay of themean wave. It is obtained essentially by approximating the exponential that appears in S,, M,t by 1. These constant terms in S,, M,t, when applied to r i , ni, have the effect of producing the meansT,, E l , over V’, premultiplied by the constants and the volume V, of the inclusion. The result of retaining just these terms in addition to those of lowest order in (8.15) and (8.16) is to generate the equations

= (e) (PI

- io3AV, AS,T,,

- pz)-’n;

= -~o(u)

(8.32)

+ ~ c o ~ AAM?rI, V~

(8.33)

where ASx, AM are constant tensors defined in the work of Willis (1980b) and the statistics of the inclusion distribution enter the perturbation just through the factor A =1

+ Jdx,(P,la

-p ~ ) .

(8.34)

Equations (8.32) and (8.33) were derived by Willis (1980c), who deduced from them the correction (8.35) (k2/k?)- 1 = (iAM,)Q , , where k, is the zeroth-order approximation to the wave number, n, represents the number density of inclusions and Q, has the form of a scattering cross-section for an inclusion perturbing a wave of rth type. The detailed formulas are given by Willis (1980~) and are not quoted. It may be remarked, however, that the “cross-section” Q, reduces exactly to the scattering crosssection of an isolated inclusion embedded in the matrix, in the limit of low concentrations. The result (8.35) then reduces to the widely accepted form given by Waterman and True11 (1961),so long as A = 1. A similar conclusion has been reached, for acoustic and electromagnetic waves propagating in a medium with isotropic phases, by Twersky (1977, 1978). Equation (8.34) 1 the well-stirred approximation used, for a corshows that A = 1 - 8 ~ for responding two-dimensional problem, by Varadan, et al. (1978).This approximation gives, however, A < 0 for c1 > i, which would correspond to growth rather than decay of the wave. This could imply failure of the quasicrystalline approximation, but the discussion of Section VI1,C has already shown that the well-stirred approximation is untenable at high concentrations. Twersky (1975) has evaluated A for a Perms-Yevick distribution of spheres : this remains positive at all concentrations.

J. R . Willis

72

The main conclusion of this section is that plane waves propagate as predicted by McCoy (1973) at low frequencies, but, in contrast to the work of McCoy, the approach leads directly to useful estimates of wave speeds and overall moduli that are actually those predicted from the HashinShtrikman formalism. Such estimates are given, together with scattering cross-sections, for composites containing aligned spheroids, by Willis (1980~). It will be apparent, too, that polycrystals could be dealt with in a similar way, following the static reasoning of Section IV,B,3, and self-consistent estimates could be developed by extending the arguments of Section V. Such calculations are in progress at the moment. The problem that remains is to complete the analogy with elastostatics by relating the results to a variational principle. At the time of writing, this has not been accomplished. However, a variational principle, whose implications have not yet been explored, has just been derived; this is outlined in the section that follows. B. VARIATIONAL PRINCIPLE This article is concluded with a derivation of a variational principle, akin to Hamilton’s principle, that is associated with Eqs. (8.15) and (8.16). It is obtained by the same type of reasoning that was applied in Section IV,A to generate the Hashin-Shtrikman principle. First, let b l , e l , p1 be the stress, strain, and momentum fields associated with the displacement u1 generated by polarizations zl, zl,so that ~1

= - S Z ~- M z 1 ,

nl = Loel P1

+ zl,

= POUlJ

+ Zl?

(8.36) (8.37) (8.38)

and define stress, strain, momentum, and displacement fields a2,e 2 ,p2, u2 similarly. It will be assumed, for simplicity,that (8.15)and (8.16)are associated with a boundary value problem with prescribed displacements. Then, u1 = u2 = 0 over aV and the analog of the virtual work equality (4.2) is (8.39) b1,ez) + ( P l , t , U 2 ) = 0, the inner products being defined as spatial means, as in (3.52).It follows now that

J; dt{(z,,ez) -

(7bU2,Jl

Overall Properties of Composites

73

by expressing zl,n1 in terms of cl,p1 as in (8.4) and using (8.39). This is the dynamic analog of (4.5). For any z, n, a Lagrangian functional 9 is defined by the equation

' + S&) +

2 9 = (z,[(L - Lo)--

(7, M,n)

- Po)-' + M J l 4 - (.,S,,t) - 2(2,eo) 2(n,u:). - (n,[(P

+

(8.41)

This generalizes the right side of (4.6). A form of Hamilton's principle for the functional (8.41) may be obtained by considering the variation of the "action" J9ddt. It follows, with the aid of (8.40),that

where Y =

-SZ

Q = POVJ

- Mn

(8.43)

+ a.

(8.44)

The integral in (8.42) vanishes when z, n satisfy (8.15) and (8.16);thus,

6

jd'9 d t = l[ (Q, w 2

- (v,

Wl'd.

(8.45)

Formally, therefore, the requirement that the variation (8.42) should vanish generates the operator Eqs. (8.15) and (8.16).The usual form of Hamilton's principle is expressed in terms of displacement and variations are only allowed which vanish at times 0 and tl . This is reflected by the right side of (8.45) which, however, is less easily reduced to zero by choice of 67, Sn. It is remarked, finally, that the stationary value of the action functional is expressible in the form

ji'9 d t = Ji' (9- go)& - $[(p + po,u - uo)]b',

(8.46)

where 9, are the classical Lagrangian functionals, given by

2 2 = (P,u,,)- (c,e), 2P0= (PO, - (ao,eo),

(8.47) (8.48)

the fields with superscript 0 relating to the solution of the boundary value problem for the comparison material.

74

J . R. Willis

IX. Recent Developments There have been a number of developments, particularly in the area of wave propagation, since this article was completed. For present purposes, the most noteworthy is that a variational structure, related to that indicated in Section VIII,B, is now fully developed (Willis, 198Od). Also, wave propagation has been studied, using the quasi-crystallineapproximation coupled with quantum-mechanical formalism, by Devaney (1980). His work can be related to that described in Section VIII,A, by noting that his “transition operator,” when applied to the field uo in (8.5), generates a source term (div t - z,*).The novelty of Devaney’s scheme is that he proposes to estimate the dynamic Green’s function self-consistently which amounts, in present notation, to defining operators Lo, po, relative to which (t) = (z) = 0. This generalizes the static prescription given by Eq. (5.16). Devaney did not relate his formulation to a variational principle, but this could now be done using the variational principles developed by Willis (1980d). REFERENCES Asaro, R. J., and Barnett, D. M. (1975). The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion. J. Mech. Phys. Solids 23, 77-83. Batchelor, G. K. (1974). Transport properties of two-phase materials with random structure. Annu. Rev. Fluid Mech. 6, 227-255. Batchelor, G. K., and Green, J. T. (1972). The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401-427. Bensoussan, A., Lions, J. L., and Papanicolaou, G. (1978). “Asymptotic Analysis for Periodic Structures,” North-Holland Publ., Amsterdam. Beran, M. J. (1968). “Statistical Continuum Theories,” Wiley (Interscience), New York. Beran, M. J., and McCoy, J. J. (1970). Mean field variations in a statistical sample of heterogeneous linearly elastic solids. Int. J. Solids Struct. 6,1035-1054. Beran, M. J., and Molyneux, J. (1966). Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Q. Appl. Math. 24,107-1 18 Bose, S . K., and Mal, A. K. (1973). Longitudinal shear waves in a fiber-reinforced composite. Int. J. Solids Struct. 9, 1075-1085. Bose, S . K., and Mal, A. K. (1974).Elastic waves in a fiber-reinforced composite. J. Mech. Phys. Solids 22, 217-229. Brailsford, A. D. (1976).Diffusion to a random array of identical spherical sinks. J . Nucl. Mafer. 60,257-278. Brinkman, H. C. (1947).A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles Appl. Sci. Res., Sect. A 1, 27-34. Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. J . Mech. Phys. Solids 13, 223-227. Budiansky, B. (1970). Thermal and thermoelastic properties of isotropic composites. J . Compos. Mater. 4, 286-295. Budiansky, B., and OConnell, R. J. (1976). Elastic moduli of a cracked solid. Int. J . Solids Struct. 12, 81-97.

Overall Properties of Composites

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Bullough, R., and Hayns, M. R. (1978). Continuum representation of the evolving microstructure prevailing during the irradiation of crystalline materials. In “Continuum Models of Discrete Systems” (J. W. Provan, ed.), pp. 469-502. Univ. of Waterloo Press. Carlson, D. E. (1972). Linear thermoelasticity. In “Encyclopedia of Physics” (C. Truesdell, ed.), Vol. VI a/2, pp. 297-345. Springer-Verlag, Berlin and New York. Chen, H. S., and Acrivos, A. (1978a). The solution of the equations of linear elasticity for an infinite region containing two spherical inclusions. Int. J . Solids Struct. 14, 331-348. Chen, H. S., and Acrivos, A. (1978b). The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Int. J. Solids struct. 14,349-364. Chernov, L. A. (1960). “Wave Propagation in a Random Medium” (Engl. transl. by R. A. Silverman). McGraw-Hill, New York. Childress, S. (1972). Viscous flow past a random array of spheres. J. Chem. Phys. 56,2527-2539. Cole, J. D. (1968). “Perturbation Methods in Applied Mathematics.” Ginn (Blaisdell), Boston, Massachusetts. Datta, S. K. (1977). A self-consistent approach to multiple scattering by elastic ellipsoidal inclusions. J . Appl. Mech. 44,657-662. Datta, S. K. (1978). Scattering by a random distribution of inclusions and effective elastic properties. In “Continuum Models of Discrete Systems” (J. W. Provan, ed.), pp. 11 1-127. Univ. of Waterloo Press. Dederichs, P. H., and Zeller, R. (1973). Variational treatment of the elastic constants of disordered materials. Z . Phys. 259, 103-116. Devaney, A. J. (1980). Multiple-scattering theory for discrete, elastic, random media. J. Math. Phys. 21,2603-2611. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. SOC.London, Ser. A 241, 376-396. Eshelby, J. D. (1961). Elastic inclusions and inhomogeneities. In “Progress in Solid Mechanics” (1. N. Sneddon and R. Hill, eds.), Vol. 11, pp. 87-140. North-Holland Publ., Amsterdam. Fokin, A. G. (1979). Statistical properties of inhomogeneous solid media: Central moment functions of material characteristics. J. Appl. Math. Mech. (Engl. Trunsl.) 42, 570-578. Fokin, A. G., and Shermergor, T. D. (1969).Calculation ofeffective elastic moduli ofcomposite materials with multiphase interactions taken into account. J . Appl. Mech. Tech. Phys. (Engl.Trans(.) 10, 48-54. Gel’fand, I. M., and Shilov, G. E. (1964). “Generalized Functions,” Vol. 1. Academic Press, New York. Gubernatis, J. E., and Krumhansl, J. A. (1975). Macroscopic engineering properties of polycrystalline materials: Elastic properties. J. Appl. Phys. 46, 1875-1883. Hashin, Z., and Shtrikman, S. (1962a). On some variational principles in anisotropic and nonhomogeneous elasticity. J . Mech. Phys. Solids 10, 335-342. Hashin, Z . , and Shtrikman, S. (1962b). A variational approach to the theory of the elastic behaviour of polycrystals. J . Mech. Phys. Solids 10, 343-352. Hashin Z., and Shtrikman, S. (1963). A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11,127-140. Hershey, A. V. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. J . Appl. Mech. 21, 236-240. London, Sect. Hill, R. (1952). The elastic behaviour of a crystalline aggregate. Proc. Phys. SOC., A 65, 349-354. Hill, R. (1963a). Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11, 357-372. Hill, R. (1963b). New Derivations of some elastic extremum principles. In “Progress in Applied Mechanics,” The Prager Anniversary Volume, pp. 99-106. Macmillan, New York.

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Hill, R. (1964). Theory of mechanical properties of fibre-strengthened materials. I. Elastic behaviour. J . Mech. Phys. Solids 12, 199-212. Hill, R. (1965a). Continuum micromechanics of elastoplastic polycrystals. J . Mech. Phys. Solids 13, 89-101. Hill, R. (1965b). A self-consistent mechanics of composite materials. J . Mech. Phys. Solids 13, 21 3-222. Hinch, E. J. (1977).An averaged-equation approach to particle interactions in a fluid suspension. J . Fluid Mech. 83, 695-720. Hoenig, A. (1979). Elastic moduli of a non-randomly cracked body. Int. J . Solids Struct. 15, 137-154. Hori, M., and Yonezawa, F. (1974). Statistical theory of effective electrical, thermal and magnetic properties of random heterogeneous materials. 111. Perturbation treatment of the effective permittivity in completely random heterogeneous materials. J . Math. Phys. 15, 2177-21 85. Howells, I. D. (1974). Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. 1. Fluid Mech. 64,449-475. Hudson, J. A. (1968).The scattering of elastic waves by granular media. Q. J. Mech. Appl. Math. 21,487-502. Ishimaru, A. (1978). “Wave Propagation and Scattering in Random Media,” Vol. 2. Academic Press, New York. Karal, F. C., and Keller, J. B. (1964). Elastic, electromagnetic and other waves in a random medium. J . Math. Phys. 5 , 537-547. Khatchaturyan, A. G. (1967). Some questions concerning the theory of phase transformations in solids. Sou. Phys.-Solid State (Engl. Transl.) 8, 2163-2168. Kneer, G. (1965). Calculation of elastic moduli of polycrystalline aggregates with texture. Phys. Status Solidi 9, 825-838. Knopoff, L., and Hudson, J. A. (1964).Scattering of elastic waves by small inhomogeneities. J . Acoust. Soc. Am. 36, 338-343. Korringa, J. (1972). Propagating modes of a heterogeneous, macro-homogeneous continuum. J . Phys. (Orsay, Fr.) 33, C6, 117-122. Korringa, I. (1973). Theory of elastic constants of heterogeneous media. J . Math. Ph,ys. 14, 509-513. Kroner, E. (1977). Bounds for effective elastic moduli of disordered materials. J . Mech. Phys. Solids 25, 137 - 155. Laws, N. (1973). On the thermostatics of composite materials. J . Mech. Phys. Solids 21, 9-17. Laws, N., and McLaughlin, R. (1978). Self-consistent estimates for the viscoelastic creep compliances of composite materials. Proc. R. Soc. London, Ser. A 359, 251-273. Laws, N., and McLaughlin, R. (1979). The effect of fibre length on the overall moduli of composite materials. J . Mech. Phys. Solids 27, 1-13. Lax, M. (1952). Multiple scattering of waves. 11. The effective field in dense systems. Phys. Rev. 85, 621-629. Leitman, M. J., and Fisher, G. M. C. (1973).The linear theory of viscoelasticity. In “Encyclopedia of Physics” (C. Truesdell, ed.), Vol. VI a/3, pp. 1-123. Springer-Verlag, Berlin and New York. Lundgren, T. S. (1972). Slow flow through stationary random beds and suspensions of spheres. J . Fluid Mech. 51, 273-299. McCoy, J. J. (1973). On the dynamic response of disordered composites. J . Appl. Mech. 40, 511-517. McCoy, .J. J. (1979). On the calculation of bulk properties of heterogeneous materials. Q.Appl. Math. 36, 137-149.

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Miller, M. N. (1969a). Bounds for effective electrical, thermal and magnetic properties of heterogeneous materials. J . Math. Phys. 10, 1988-2004. Miller, M. N. (1969b). Bounds for effective bulk modulus of heterogeneous materials. J. Math. Phys. 10, 2005-2013. Perkus, J. K., and Yevick, G. J. (1958). Analysis of classical statistical mechanics by means of collective co-ordinates. Phys. Rev. 110, 1-13. Reuss, A. (1929). Calculation of the flow limits of mixed crystals on the basis of the plasticity of mono-crystals. Z . Angew. Math. Mech. 9, 49-58. Rosen, B. W., and Hashin, Z. (1970). Effective thermal expansion coefficients and specific heats of composite materials. Int. J. Eng. Sci. 8, 157-173. Sanchez-Palencia, E. (1974). Comportements local et macroscopique d’un type de milieux physiques hkterogenes. Znt. J. Eng. Sci. 12, 331-351. Schapery, R. A. (1968). Thermal expansion coefficients of composite materials based on energy principles. J . Compos. Mater. 2, 380-404. Talbot, D. R. S., and Willis, J. R. (1980). The effective sink strength of a random array of voids in irradiated material. Proc. R. SOC.London, Ser. A 370, 351-374. Throop, G. J., and Bearman, R. J. (1965). Numerical solutions of the Perkus-Yevick equation for the hard-sphere potential. J. Chem. Phys. 42,2408-241 1. Uscinski, B. J. (1977). “The Elements of Wave Propagation in Random Media.” McGraw-Hill, New York. Twersky, V. (1975). Transparency of pair-correlated, random distributions of small scatterers, with applications to the cornea. J. Opt. Soc. Am. 65, 524-530. Twersky, V. (1977). Coherent scalar field in pair-correlated random distributions of aligned scatterers. J . Math. Phys. 18, 2468-2486. Twersky, V. (1978). Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers. J. Math. Phys. 19, 215-230. Varadan, V. K., and Varadan, V. V. (1979). Frequency dependence of elastic (SH)-wave velocity and attenuation in anisotropic two-phase media. Wave Motion 1, 53-63. Varadan, V. K. Varadan, V. V., and Pao, Y. H. (1978). Multiple scattering of elastic waves by cylinders of arbitrary cross-section. I. SH waves. J. Acoust. SOC.Am. 63, 1310-1319. Voigt, W. (1889). Ueber die Beziehung zwischen den beiden Elasticitatsconstanten isotroper Korper. Ann. Phys. (Le@zig)[3] 38, 573-587. Walpole, L. J. (1966a). On bounds for the overall elastic moduli of inhomogeneous systems. 1. J . Mech. Phys. Solids 14, 151-162. Walpole, L. J. (1966b). On bounds for the overall elastic moduli of inhomogeneous systems. I f . J. Mech. Phys. Solids 14,289-301. Walpole, L. J. (1969). On the overall elastic moduli of composite materials. J . Mech. Phys. Solids 17, 235-251. Waterman, P. C. (1969). New formulation for acoustic scattering. J. Acoust. SOC.Am. 45, 1417 - 1429. Waterman, P. C. (1976). Matrix theory of elastic wave scattering. J. Acoust. Soc. Am. 60, 56 7 -5so.

Waterman, P. C . , and Truell, R. (1961). Multiple scattering of waves. J . Math. Phys. 2, 512537. Wertheim, M. S. (1963). Exact solution of the Perkus-Yevick integral equation for hard spheres. Phys. Rev. Lett. 10, 321-323. Willis, J. R. (1970). “Asymmetric Problems of Elasticity,” Adams Prize Essay. University of Cambridge, Cambridge, England. Willis, J. R. (1975). The interaction of gas bubbles in an anisotropic elastic solid. J . Mech. Phys. Solids 23. 129-138.

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Willis, J. R. (1977). Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25, 185-202. Willis, J. R. (1978). Variational principles and bounds for the overall properties of composites. h “Continuum Models of Discrete Systems” (J. W. Provan, ed.), pp. 185-215. University of Waterloo Press. Willis, J. R. (1980a). Relationships between derivations of the overall properties of composites by perturbation expansions and variational principles. In “Variational Methods in Mechanics of Solids” ( S . Nemat-Nasser, ed.), pp. 59-66. Pergamon, Oxford. Willis, J. R. (1980b). A polarization approach to the scattering of elastic waves. I. Scattering by a single inclusion. J . Mech. Phys. Solids 28, 287-305. Willis, J. R. (1980~).A polarization approach to the scattering of elastic waves. 11. Multiple scattering from inclusions. J . Mech. Phys. Solids 28, 307-327. Willis, J. R. (198Od). Variational principles for dynamic problems for inhomogeneous elastic media. Wave Morion 3, 1- 11. Willis, J. R. and Acton, J. R. (1976). The overall elastic moduli of a dilute suspension of spheres. Quart. J . Mech. Appl. Math. 29, 163-177. Zeller, R. and Dederichs, P. H. (1973).Elastic constants of polycrystals. Phys. Stat. So/idi(b)55, 831-842.

ADVANCES IN APPLIED MECHANICS. VOLUME

21

Elastic Wave Propagation in Stratified Media B. L. N . KENNETT Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge. England

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Previous Work on Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. ScopeofArticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Elastic Waves in Stratified Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Coupled Equations for Displacements and Tractions . . . . . . . . . . . . . . . B. Boundary Conditions on the Elastic Wavefield . . . . . . . . . . . . . . . . . . . C . Propagator and Fundamental Matrices . . . . . . . . . . . . . . . . . . . . . . . D. The Introduction of a Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. The Formal Response of a Half Space . . . . . . . . . . . . . . . . . . . . . . . . 111. Reflection and Transmission of Elastic Waves. . . . . . . . . . . . . . . . . . . . . . A . Decomposition of the Seismic Wavefield in a Uniform Medium . . . . . . . . . B. Reflection and Transmission at an Interface . . . . . . . . . . . . . . . . . . . . . C . A Vertically Inhomogeneous Region . . . . . . . . . . . . . . . . . . . . . . . . . D. Free Surface Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. The Factorization of Reflection and Transmission Matrices . . . . . . . . . . . F. Recursive Approach for a Stack of Uniform Layers . . . . . . . . . . . . . . . . G . Comparison between Recursive and Propagator Methods . . . . . . . . . . . . H . Turning-Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Mixed Solid and Fluid Stratification . . . . . . . . . . . . . . . . . . . . . . . . . J. Wave Vector Representation of a Source . . . . . . . . . . . . . . . . . . . . . . IV . Half-Space Response in Terms of Reflection Matrices . . . . . . . . . . . . . . . . . A. The Response via a Surface Source Vector . . . . . . . . . . . . . . . . . . . . . B. Buried Sources and Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Surface Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Factorization of the Full Medium Response. . . . . . . . . . . . . . . . . . . . . E . Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Inversion of the Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Slowness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80 80 82 83

83 88 89 93 98 100 100 107 111 115 117 121 122 123 124 125 127

128 131 135 137 143 152 152 159

79 Copyright Q 1981 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-002021-1

B. L. N . Kennett

80

VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164

I. Introduction

A. PREVIOUS WORKON STRATIFICATION The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. The first study of the interaction of elastic waves with a planar interface between two dissimilar elastic media was, however, made by Green (1838) in an attempt to model light propagation. Seismic problems led Knott (1899) to derive the reflection and transmission coefficients for incident plane waves on such an interface via energy arguments, and Zoeppritz (1919) to find the corresponding amplitude coefficients. Rayleigh (1885) showed that it was possible for a free surface wave to propagate along the surface of an isotropic elastic half space without dispersion, and pointed out that such a wave channeled along the surface would at large distances dominate any bodily elastic waves traveling through the bulk of the solid. The forced motion of an isotropic elastic half space was investigated in detail by Lamb (1904). This work showed that the surface Rayleigh wave could be significantly excited, but the computed displacements did not show the oscillatory character nor the large transverse component of motion seen on contemporary seismograms. Lamb suggested that the oscillatory nature arose from the behavior of the recording instruments and this was in part true of the lightly damped systems then in use, although dispersion probably dominated this effect. To explain the transverse surface wave component, Love (1911) introduced stratification into the model. He showed that with a superposed layer on a uniform half space there could be a free transverse surface wave (a Love wave) with amplitude which decayed exponentially into the half space, which showed frequency dispersion with horizontal wavenumber. For purely analytic developments of elastic wave propagation the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space; such problems are considered in some detail in the book by Ewing et al. (1957), together with a comprehensive bibliography. If uniform layers are not used then exact solutions are only available for specialized choices of velocity distribution. Some progress was, however, made with general stratification problems, notably by Jeans

Elastic Wave Propagation in Stratijied Media

81

(1923) who considered elastic wave propagation in a sphere, and Jeffreys (1928) who looked at Love wave propagation in a model with a uniform layer overlying a nonuniform half space. During the years up to 1940, the analysis of the arrival times of seismic pulses using ray theoretical methods lead to the construction of models of the velocity distribution for compressional (P) waves within the Earth (Jeffreys and Bullen, 1940).The analysis technique, and the smoothing applied to the observational data, led to a continuously varying velocity profile interrupted only by boundaries between the mantle of the Earth and the fluid core and between the core and inner core. The construction of such models stimulated the development of methods to handle wave propagation in realistic media. For a model consisting of a number of uniform layers, Thomson (1950) introduced a transfer matrix technique to relate the displacements and tractions at the top and bottom of a layer and thus, using the continuity conditions at the interfaces, through the entire stack. This work was corrected and extended by Haskell (1953) and has been extensively applied to the calculation of Love and Rayleigh wave dispersion on multilayered half spaces. With even a hand calculator dispersion calculations for three or four layers over a half space were practicable. The development of electronic computers meant that soon considerably larger numbers of layers could be employed and the models used began to approximate continuous stratification. The excitation of surface waves by realistic sources was considered by Haskell (1964) and Harkrider (1964), and this work led the way to the construction of theoretical seismograms for the surface wave train (e.g., Knopoff et al., 1973). These matrix methods may also be used to solve for the reflection and transmission effects of a stack of uniform layers, and this procedure together with a Weyl integral representation of the source forms the basis of the “reflectivity” technique for the calculation of body wave seismograms (Fuchs and Muller, 1971). When the elastic parameter distribution depends only on one coordinate, Alterman et a/. (1959)showed how the stress-strain relations and the elastic equations of motion can be reduced by transform techniques to a set of first-order differentialequations, to be solved subject to free surface boundary conditions and excitation by the source. This approach has been the basis of most purely numerical work for smoothly varying velocity profiles for both surface waves (Takeuchi and Saito, 1972) and body waves (Chapman and Phinney, 1972). Since the numerical methods require considerable computational effort, in recent years approximate methods have been developed which are designed to isolate individual seismic phases for continuously stratified velocity

82

B. L. N . Kennett

models. Thus Richards (1976) has advocated the use of uniform asymptotic approximations (Langer, 1937) to model the depth variation of the seismic displacements and Chapman (1978) has produced a simple method of computing theoretical seismograms based on assumptions about the reflection properties of the medium.

B. SCOPEOF ARTICLE In this article we consider the excitation of elastic waves in a horizontally stratified medium by a point source within the stratification. The aim is to present a unified treatment, in terms of the reflection properties of the stratification, following an approach introduced by Kennett (1974) and extended by Kennett and Kerry (1979). We will usually consider a stratified half space bounded above by a free surface, but will indicate how the results may be extended to a stratified plate. The propagational characteristics of the medium are most easily extracted in the transform domain and we therefore take a Fourier transform with respect to time and a Hankel transform with respect to radial distance from the source. Rather than working directly with horizontal wavenumber k, we find it more convenient to introduce the horizontal slowness p = k / o , where o is the angular frequency. The slowness has the merit of being a conserved quantity for all wave types in coupled wave interaction (e.g.,with an interface), and indeed this property just represents an extension of Snell’s law. ) we are able to construct the entire response of a In the ( p , ~ domain stratified elastic medium in terms of reflection and transmission matrices for the constituent parts of the medium. This matrix approach allows a systematic treatment of the coupling between longitudinal and transverse waves, and leads to convenient representations of the elastic wavefield for arbitrary location of source and receiver within the stratification. The expressions for the excitation have a ready physical interpretation and enable systematic development of useful approximations to the response. This reflection matrix method leads to an effective computational scheme for finding the excitation due to general sources embedded in models composed of a stack of uniform layers, and also for piecewise smooth models in a high frequency asymptotic development. We may, therefore, simulate the effect of, for example, an explosion or earthquake in a realistic Earth model. A variety of computational procedures exist for inverting the transforms numerically which allow the construction of theoretical “seismograms” for the time response at the receiver. Such seismograms may be calculated for the full response or for approximations emphasising certain features of the records.

Elastic Wave Propagation in Stratijied Media

83

For scalar wave propagation (e.g., horizontally polarized shear waves in an isotropic medium),the reflection matrix technique is equivalent to results presented by Brekhovskikh (1960). The present approach does, however, allow an effective treatment of coupled wave equations. Although we shall normally consider coupled compressional and vertically polarized shear waves in an isotropic medium, the results extend directly to the fully anisotropic case. 11. Elastic Waves in Stratified Regions

A. COUPLED EQUATIONS FOR DISPLACEMENTS AND TRACTIONS We will consider a horizontally stratified medium in which the elastic properties depend only on the depth coordinate z, and for simplicity will confine our attention to the excitation produced by a point source. More complex sources may then be simulated by superposition of point source solutions. We take a cylindrical set of coordinates (r,4, z ) with the vertical axis passing through the location of the source. Then for a small dynamic displacement w(r,4, z, t ) and a density distribution p(z), the corresponding stress tensor z i j satisfies the equations of motion azrrz

+ a r z r r + r - ' a+zr+ + r-'(zrr - 74), dzz,z + &zrg + r - l a,zss + 2r-'z,, a,r,,

+ &T,, + r-la,r,z + r-%,,

- ~5 = pa,,w, - pf, = p attwr

(2.1)

= pattw, - pfi

in the presence of a body force per unit mass f = (L,f,,fz).If the elastic properties are isotropic, we have the linear constitutive relation zij

= 1dijekk

+ 2p?jj,

(2.2)

in terms of the Lame moduli 1,p, where the components of the strain tensor eij are err = a,w,,

e,4

= r-'(a,w,

+ w,),

err = azwz, e,z = i(r-'8,wz

+ azw,),

(2.3)

erz = t(azwr + arwz), e,, = $(arw, - r - l w ,

+ r-'

a,w,).

B. L . N.Kennett

84

As the elastic parameters depend only on z, we would like to find an arrangement of the elastic equations C(2.1)-(2.3)] which emphasizes the z dependence of the displacements and stresses. To this end we introduce the quantities (cf. Hudson, 1969.) Wv

= r-'[ar(rWr)

WH =

r-

~,W,]Y

l[ar(rwb)

- a,wr],

zvz = r - ' [ d r ( r r r 2 )

a,T,,],

zHz = r - l[ar(rT&z)-

a+~rz],

with similar definitions for fv, fHy and define the operator

Vl$

= r - l ar(r a,$)

+ r-'

(2.4)

Vi such that

a,,+.

(2.5) We may now combine and rearrange the equations of motion to give a form in which derivatives with respect to z appear only on the left-hand side of the equations : a,w, =

-n(n

+ 2p)-1wv + ( I + 2CL)-

iTz2,

and

a,%

= P - l%z

Y

a,%,

= (P

a,, - CL V

h l - PfH,

(2.7)

where we have introduced the composite modulus v =I

+ 2p -- P / ( A + 2p) = 4p(A + p ) / ( I + 2p).

(2.8)

We note that the elastic equations have separated into two groups of coupled equations. The first set (2.6) couple compressional (P) waves, with local wave speed a = [(I

+ 2P)/PI1/',

(2.9)

to shear disturbances (SV) involving vertical displacement, with wave speed

P = CP/Pl"'.

(2.10)

The second set comprises shear disturbances entirely confined to a horizontal plane (SH) with the same wave speed. This decomposition into coupled P-SV and SH wave systems also occurs for a transversely isotropic medium with a vertical symmetry axis (Takeuchi and Saito, 1972). For a generally anisotropic medium it is still possible to arrange the elastic equations in a

Elastic Wave Propagation in Stratified Media

85

form where derivatives with respect to z appear only on the left-hand side of the equations (Woodhouse, 1974),but now there is no decoupling. Since we have assumed that the elastic properties vary only with the depth z , we may generate a suite of coupled ordinary differential equations by applying a Fourier-Bessel transform over time and horizontal coordinates to (2.6) and (2.7).We define

%,[$I

=

$(k,m,4

and then

Instead of the horizontal wavenumber k, it often proves more convenient to use the horizontal slowness p = k / o , with units of reciprocal velocity. For each azimuthal order m we introduce a set of variables related to the transforms of the displacements and stresses appearing in (2.6) and (2.7) by

u = i?jz, V = -k-'i?j ",

P = fzz, S = -k-'f

W = -k-'i?j

T = -k-'fHZ.

H,

VZ?

(2.12)

The scaling factors are designed to give a set of variables in each group with equal dimensionality; the scaling via the horizontal wavenumber arises from the horizontal differentiation in (2.4). In terms of the new displacement variables U(k,m,z, w),etc., we obtain the coupled set of ordinary differential equations: for P-SV waves

(2.13)

86

B. L. N . Kennett

and for SH waves

We note that for an isotropic medium the coefficients appearing in (2.13)and (2.14)areindependent of the azimuthal order rn and the azimuthal dependence of U(k,rn,z,w), etc., will arise only from the nature of the force system f. The elements of the coupling matrices involve only the values of the elastic parameters and not their vertical derivatives. This desirable property was first pointed out by Alterman et aJ. (1959) in an analogous development for a sphere and makes (2.13)and (2.14)well suited to numerical solution. Both of the sets of coupled equations C(2.13)and (2.14)] can be written in the form a,B(k,rn,z, w ) = wA(p,z)B(k,rn,z,w) + pw-'f(k, rn, z,w)

(2.15)

in terms of the stress-displacement vector B; where for P-SV waves B(k,rn,z,w)= [ U , V,w-'P,o-'S]*

(2.16)

and for SH waves

'

BH(k, rn, z , w ) = [W ,w - TI '.

(2.17) Subsequently, when we wish to look at the general structure of the results we will write

B(k,rn,z,w)= [W,w-'T]'

(2.18)

and we will specialize when appropriate to the P-SV and SH systems. We suppose the elastic properties to be piecewise continuous functions of depth z and at any discontinuity we assume that there is welded contact. Thus we require continuity of displacement and the normal and tangential components of traction across any horizontal plane and so w,, w4, w, and z~,, T+,, t,, must be continuous. Since the operations in (2.4) only involve horizontal derivatives w y ,w H and z y Z , rH2must also be continuous. The transform operator Fmpreserves the continuity property and thus the stressdisplacement vectors €3 will be continuous across any plane of discontinuity as well as all other planes z = const. Between discontinuities in elastic properties we may, therefore, use (2.15) to construct the stress-displacement vectors B and then we are able to use the continuity of B to carry the solution across the level of any jump in the elastic parameters. The displacements in the medium may be recovered from the transformed quantities by inverting the Fourier-Bessel transform, making use of the

Elastic Wave Propagation in Stratijied Media definition (2.4) of w y ,w, to give

wz(r,(6, z, t) = (271)w,(r, 4, z, t) = (271)-

J:

m

s"

-m

doe-'"'

doe-'"'

som

87

m

dkk

U(k,m,z, o)J,(kr)eim@ m=-m

Somdkk (2.19)

w,(r, (6, z, t ) = (274-

' Jm

doe-'"'

-00

som

dkk

With the assumption of a point source we can restrict the summation over angular order to Imf < 2. These expressions for the components of displacement may alternatively be written as a surface harmonic expansion w(r, (6, z, t ) = w,i'

+ w46 + wz4

lom dkk (URF + V S ; + WT?), m

= (24-l

Jm -

doe-'"' m

m=-a)

(2.20)

in terms of the vector surface harmonics (Takeuchi and Saito 1972)

Rr = 2YF(r,4), Sr = k-' V, YF(r,4),

(2.21)

Tr = -k - '2 A V, Yr(r,(6), where Y?(r, 4) = Jm(kr)eim,,and V 1 is the horizontal gradient operator

V, = ' i d ,

+ 4 r - l a,.

In the course of this article we will concentrate on point sources and employ the cylindrical representation in the stratified half space. It is, however, interesting to note that if we consider a two-dimensional situation where all stresses and displacements are independent of the coordinate Y and take a Fourier transform (2.22)

B. L. N . Kennett

88

then the sets of equations C(2.13) and (2.14)] are recovered if we work in Cartesian coordinates and set U = iEz,

P = iTz,

I/= Ex,

s =Txz

W = W,,

T = Zyz.

(2.23)

We have here made a plane wave decomposition rather than the cylindrical wave decomposition of (2.19) and (2.20). B. BOUNDARY CONDITIONS ON THE ELASTIC WAVEFIELD We consider an elastic half space ( z 2 0) with a point source embedded at a depth zs (Fig. 1). At the free surface of the half space ( z = 0) no tractions can be supported so that z, z+,, ,z, must all vanish. After Fourier-Bessel transformation we therefore require the stress variables P , S , T to vanish at z = 0. The stress-displacementvector thus has the form

B(k m, 0 , ~=) [Wo,o]',

(2.24)

or explicitly for P-SV waves

BP(k m,0

9

4

=

[U,,

9

0, O],'

(2.25)

'.

(2.26)

and for SH waves

BH(km, 0 , 4

= [Wo,O]

We assume that the elastic parameters are constant below some plane z = zL lying beneath the source (this can be arranged by suitable choice of zL), then we must apply a radiation condition at z = zL. We require that the Free Surface

+r RECEIVER

*

'R

-1

FIG.I Configuration of elastic half space with a source at depth zs and receiver at depth zR.Beneath zL the medium has uniform properties so that a radiation condition has to be

applied at this level. The conventions for up- and down-going waves are also indicated.

Elastic Wave Propagation in StratiJied Media

89

wavefield in the underlying half space ( z > zL) be either purely down-going waves or evanescent with depth, the character depending on the slowness. The convention we have adopted for up-going waves (U) and down-going waves (D)is shown in Fig. 1, and these directions will be indicated by suffices on appropriate quantities. We shall see in Sections II,E and IV how this boundary condition can be included in the calculation of the response of a half space. For a stratified plate we would just have another condition of the form of (2.24) at the lower surface of the plate z = zL (say). C. PROPAGATOR AND FUNDAMENTAL MATRICES 1. Source-Free Regions

If we consider a portion of the stratification without any sources, then the stress-displacement vector B, for given angular order m, horizontal wavenumber k, and frequency o,satisfies the coupled differential equations a,B(z) = oA(z)B(z).

(2.27)

We define a fundamental matrix solution B(z) of the corresponding matrix equation d,B(z) = oA(z)B(z),

(2.28)

as a matrix whose columns are independent solutions of (2.27). Usually these solutions are taken to have some common characteristics; they may for example represent up- and down-going waves at the level z. From such a fundamental matrix we may construct a propagator matrix P(z, zo) (Gilbert and Backus, 1966) for a portion of the medium as P(z, z0) = B(z)B- ' ( z 0 )

(2.29)

so that the propagator is a fundamental matrix satisfying the constraint P(Z,,ZO)

=I

(2.30)

where I is the unit matrix of appropriate dimensionality. In terms of this propagator the solution of (2.27) with the stress-displacement vector specified at some level zo is B(z) = P(Z, zo)B(zo).

(2.31)

The continuity of the stress-displacement vector across discontinuities in the elastic parameters means that the idea of a propagator in the sense of (2.31) is not confined to a region of continuous elastic properties and may be extended to the whole stratification.

B. L. N . Kennett

90

If we take, for a moment, the simple scalar equation d,Y =may,

where a is constant, then the solution of the initial value problem is Y(Z) = exp{oa(z - ZO))Y(ZO). In a similar way if A is constant (i.e., we are considering a portion of the medium with uniform properties), the solution of (2.27) has the form (2.32)

B(z) = exp{wA(z - zo)}B(zo)

where the matrix exponential may be defined by its series expansion. We see that the propagator in (2.31) constitutes a generalization of (2.32) to the case where A is no longer necessarily constant. In the context of general matrix theory, the propagator is usually referred to as the “matricant” (see, e.g., Frazer et al., 1938). Consider a finite interval a Iz, zo I b, and construct the recursive scheme pk

+

zO)

=I

+0

dcA(c)pk([,zO),

with Po(z,zo) = I, derived from the equivalent Volterra integral equation to (2.28). This procedure will converge uniformly as k ..+ co,provided that all the elements in A are bounded. For the elastic case this will always be true for solids, and so the propagator is given by P(z, zo) = 1 + w

s’ dz’A(z’)+ m2 LO

JZ 20

dz’A(z’)

s” dz”A(z”)+ . LO

*

. (2.33)

For a uniform medium we see that t h s series is just the expansion of an exponential, as expected from (2.32). If we differentiate det P row by row, making use of the fact that P satisfies (2.28), we find that d,{det P> = tr A det P,

(2.34)

and thus (2.35) since det P(z, zo) = I. For our coupled set of elastic equations tr A = 0, and so det P(z,zo) = I throughout the medium. Thus the propagator can never be singular. The two propagators P(z, zo) and P(z,zl) are both fundamental matrix solutions of (2.28), with continuity at any jumps in elastic parameters. This means that each column in P(z, zo)can be expressed as a linear combination

Elastic Wave Propagation in Stratijied Media

91

of the columns of P(z, zl), i.e., (2.36) for some constant matrix C. When we set z = z1 in (2.36) P(z1,zo) = P(z,,z,)C

=

c

since P(z,, zl) = I, and so P(Z, zo) = P(Z, Zl)P(Z, zo). 9

(2.37)

This result leads to a simple relation between the propagator and its inverse P- yz, 20) = P(z0, z).

(2.38)

Let us take a portion of the stratification between the levels zn and zo, and divide this into n parts by introducing intermediate levels z I z2 I ...< zn- 1. Successive use of (2.37) then enables us to represent the overall propagator as a continued product of the propagators for the subdivisions (2.39) If the subdivisions are sufficiently fine, we may take the matrix to be essentially constant in an interval, and so by the mean value theorem A(z) N A(5jh

zj-1 < 5j, z < zj,

(2.40)

so that from (2.32) n

exp{NCj)(zj - Zj-l)}.

P(zn,zo) N

(2.41)

j= 1

The approximation of taking A constant over subintervals in z is equivalent to the approximation of assuming that the elastic properties are uniform in (zj, zj- 1), i.e., that we have a homogeneous layer in this region. The representation (2.41) for the overall propagator is thus equivalent to the matrix method due to Thomson (1950) and Haskell (1953) and exp(oA(Cj)(zjzj- ,)} is just thejth layer matrix. 2. Propagators and Sources

The presence of some form of source in an elastic medium introduces a forcing term into the coupled elastic differential equations and so we need to be able to solve equations of the type

a,W) = oA(z)B(z) +

(2.42)

B. L. N . Kennett

92

for given initial conditions on the stress-displacement vector B. We premultiply (2.42) by P-l(z,z0), the inverse of the propagator for the homogeneous equations (2.27), so that P-'(z, z0) d,B(z) - wP-'(z, z0)A(z)B(z) = P-'(z, zo)g(z).

(2.43)

Now P- '(2, zo) satisfies the equation d,P-

'(2,

zO) = -UP-

'(2, zO)A(Z),

(2.44)

so that we can recognize the left-hand side of (2.43) as a differential u p - '(2, zo)B(z)l = Pbo, z)g(z).

On integrating with respect to z we obtain

w.4 = P(Z, zo)B(zo) +

P(Z, Mi)di,

(2.45)

which displays rather attractively the cumulative effect of the forcing term as we move away from the reference plane z = zo. We now specialize to a source confined to a plane g ( 4 = g16(z - zs) + gz 6'b - zs)

(2.46)

which, as we shall see, corresponds to a wide class of seismic sources. Then the integral in (2.45) takes the form P(Z, i)g(O 4 = H(z - ZS)P(Z, Z S Y ,

(2.47)

with the vector Y specified by = P(Zs,~s)g1- di-P(zs,i)l~=,sg,.

(2.48)

We recall that P(zs,i) = P-'([,zs), and so from (2.44) = g1 + @A(ZSkZ.

(2.49)

When we combine the results of (2.47) and (2.49) we see that the effect of the forcing term (2.46) only becomes effective for depths z below the level of the source. Further the stress-displacement vector suffers a discontinuity at the source level with a jump B ( z ~ + )- B ( z ~ - ) = Y = g1

+ wA(zs)g,.

(2.50)

Above the source the characteristics of the elastic wave field are governed by the initial conditions on the stress-displacement field which normally involve the source indirectly via the boundary conditions on the field.

Elastic Wave Propagation in Strat$ed Media

93

D. THEINTROIXJCTION OF A SOURCE In order to allow a general form of point source, we adopt the force system referred to Cartesian axes, where we have combined a simple force F with a set of couples specified by the source moment tensor (Gilbert, 1971). We are particularly interested in the effect of this force system embedded in the stratification, but in order that the character of the source may be appreciated we will first consider the source radiation expected in an unbounded medium. 1. Radiation Patterns

The displacement produced by a point force acting at some point in an unbounded elastic medium was first given by Stokes (1849) and the results were extended to couple sources by Love (1903), and we shall use these results as the basis of our discussion. Suppose that we have a point force with component F j ( t ) acting in the j direction at the origin and we observe the ith component of displacement at R in a direction specified by direction cosines y i . The response may be found by convolving the force with the dynamic Green’s tensor g i jto give 4npwi(R,t ) = 4npYij(R,x)* F j 6 ( x )

JR:y sFj(t

= (3yiyj - 6 i j ) R - 3

- s)ds

+ yiyj(~?R)-’Fj(t- R/u) - (Yiyj - 8ij)(PZR)-’Fj(t- R/P).

(2.52)

The “far-field” contribution decaying as R - follows the same time dependence as the force F. The P-wave arrival travels with the velocity c1 and the S wave with velocity B, and between them is a disturbance which decays rapidly as we move away from the source. The moment tensor M , may be thought of as specifying the relative weighting to be applied to the nine elements of an array of couples and dipoles illustrated in Fig. 2. The diagonal elements of M correspond to dipolar effects and the offdiagonal elements to pure couples. The response to this type of point source may once again be obtained by convolution with the Green’s tensor wi(R,

l ) = gij(R,

* dk[Mjk 6(x)] * k f j k 6(x)

= ak??ij(R,x)

94

B. L. N . Kennett Moment Tensor Elements

FIG.2 The first row of the moment tensor, showing dipole diagonal element and couples off the diagonal.

by the properties of a convolution and the distribution akh(x).Explicitly the displacement is

SR”,;

4npwi(R, t ) = 3(5yiyjyk - 1ij~,)R-~

SMjk(t - S ) dS

- (6yiyjYjYk - lijk)(aZR2)-‘Mjk(t - R / a ) + ( @ i Y j Y k - l i j k - h i j Y k ) ( P 2 R 2 ) - l M j k ( l - R/P) ‘)‘il/jyk(a3R)-‘jifjk(t- R/M) - ( Y i Y j - hij)yk(fi3R)-’fijk(t-

‘/P),

(2.53)

with l i j k = yihjk + yjhik -k Ykhjk. The “far-field’’ terms now behave like the derivative of the moment tensor components. The radiation characteristics of (2.52) and (2.53) are rather different for P and S waves. If we assume a single force component or a moment tensor with a single nonzero entry, we may extract from (2.52) and (2.53)the radiation patterns for a 3 force, a 33 dipole, and a 31 couple. These patterns are illustrated in Fig. 3. For applications in seismology we are particularly interested in “indigeneous” sources (i.e., those for which no change in total linear or angular momentum is produced by the source). In terms of our choice (2.51) this requires F to be identically zero and M to be symmetric. One such source may be obtained by taking an isotropic moment tensor Mjk = MoJjk for which (2.53)yields 471pwi(R,t)= yi{a2R2)-’M0(t- R/a) + ( a 3 R ) - ’ f i o ( t- R/or)} (2.54) an isotropic, purely radial, P disturbance with no S wave part, modeling an explosion.

P

S

FIG.3 Radiation patterns for simple point sources: (a) single 3 force, (b) 33 dipole, (c) 31 couple, and (d) 31 double couple.

B. L. N . Kennett

96

A simple model of an earthquake source is to consider a point dislocation in the direction e along a fault plane specified by a normal n, so that e n = 0. Then from the work of Burridge and Knopoff (1964) we know that an equivalent force system is a double couple without moment, specified by the directions e, n with strength M,(t) = p A d ( t ) where A is the area of slip and d(t) the averaged displacement. The corresponding moment tensor is

-

Mjk(t)

= Mo(t)(ejnk

+ eknj)

(2.55)

The displacement at R due to this double couple may be found from (2.53)as

4npwi(R,t ) = (9Qi - 6 T i ) R P 4

Ria

sM,(t

- s) ds

+ (4Qi - 2Ti)(dR2)-'MO(t - R/R) - (3Qi + 3Ti)(P2R2)-1Mg(t- R/P) + Q i ( a 3 R ) - ' h 0 ( t- R/u) + Ti(P3R)-'h0(t- R/P) where

(2.56) Ti = ni(yjej) -t ei(yfcnk)- Q i

-

We note that Q is purely radial, lying in the direction of R , and that Q T = 0, so that T is a purely transverse field. For both the explosion and the double couple the radiation for the whole field may be expressed in terms of the far-field parts. When sources such as these occur in an inhomogeneous medium the radiation patterns may be modified slightly by inhomogeneity in the vicinity of the source, but will qualitatively, at least, resemble those in an unbounded medium. It is interesting to compare the radiation pattern for a double couple with that for a single couple (Fig. 3). If we take n in the 3 direction and e in the 1 direction

IQI3' = 2y3y1 = sin 28coscp, IT131= (yf

+ y$ - 4yiy$)"2 = (c0s228cos2cp + cos2@sin2cp)'/',

(2.57)

in local spherical polar coordinates. The far-field P radiation pattern 1QI3' for the 31 double couple is just twice that for either of the constituent couples, but the S-wave radiation is four lobed rather than two lobed.

Elastic Wave Propagation in Stratified Media

97

The use of seismic moment tensors is discussed at some length with a number of examples by Backus and Mulcahy (1976a,b). 2. The Source as a Stress-Displacement Vector Discontinuity We wish to relate the general point source system(2.51)(fj = F j 6(x - x,) a k [ M j k6(x - x,)]} to the stress-displacement vectors introduced in Section II,A and so we need to generate the appropriate cylindrical force components appearing in (2.6) and (2.7). Thus we must construct (2.58) as well as f,. The original force system (2.51) is related to a fixed set of Cartesian axes, but the radial and azimuthal components can be found from

J

i& = , e+'+(f, k if,).

(2.59)

are all singular at xs = (O,O,z,) where zs is the depth The elements f,,f,,f, of the source, and to evaluate the Fourier-Bessel transformation we have to make use of the expansion of J,(kr)e'"~near the origin. The integration over the horizontal plane during transformation will leave the z dependence of the source terms unaffected. Thus we anticipate that F,, F,,, F , and M,,, M x y , My,, M y , will appear with a 6(z - zs) dependenceand the remaining moment tensor components (M,,, M,,, M y , ,M z y,M,,) with a 6'(z - zs) term. The forcing terms appearing in (2.13)and (2.14), for each angular order m, will, therefore, all be of the form

T"@,m, z, 4 = f?" 6(z - zs) + f?" 6'(z - zs).

(2.60)

Now we have already discussed the solution of the coupled differential equations for the stress-displacementvector (2.42) with a term of this type, (2.46), in Section II,C,2. The solution of (2.13) and (2.14) with our general point source excitation will thus lead to a discontinuity in the stress-displacement vector across the source plane

B(k,m,z,+,o) - B(k,m,z,-,w) = 9 ( k , m , 0 ) = -w-'[f'; + wA(p,z,)f?].

(2.61)

The presence of the S'(z - zs) terms means that although the only nonzero entries in the forcing terms in (2.13) and (2.14) are in the stress terms, the discontinuity in the stress-displacement vector extends also to the displacement terms because of the coupling via the A matrix. The jump in the components of the stress-displacement vector B across zs depends strongly

B. L. N . Kennett

98 on azimuthal order:

m

= 0,

m = +1, m = f1, m = 0, m = +1, m = 0, m = +1, m = _+2,

m = 0, m = +1, m = _+2. (2.62) As mentioned above, the assumption of a point source limits the azimuthal order to Iml < 2. These results for a general moment tensor generalize Hudson's (1969a) analysis of an arbitrarily oriented dislocation. We recall that for an "indigeneous" source the moment tensor is symmetric and thus only has six independent components, and also F will vanish, so that this leads to a significant simplification of these results. In particular, for a point source the stress variable P will always be continuous, excitation for m = 1 is confined to the horizontal displacement terms and Twill only have a jump for m = +2.

+

E. THEFORMAL RESPONSE OF A HALFSPACE As previously, we consider a stratified half space, bounded above by the free surface z = 0, with a point source at the level z = zs. Below the plane z = zL we take the elastic properties to be uniform. The boundary conditions discussed in Section II,B for z > zL may be accommodated by taking the stress-displacement vector, for any angular order m and fixed slowness p , frequency w, in the form W L ) = BL(zL

+NO,C,IT,

(2.63)

where two of the four independent solutions are excluded from some suitable choice of fundamental matrix BL. For a uniform half space in z > zL we would take the up-going and down-going P and S waves as independent solutions and the choice (2.63)will leave only downward propagating waves or decaying evanescent waves.

Elastic Wave Propagation in Stratijed Media

99

At the free surface the stress-displacement vector satisfies (2.24) B(O) = [WO,~]', since no traction may be present. The stress-displacement vector just below the source B(zs+) may be related to the wavefield in z > zL by the propagator matrix P(zs, zL), B(z,+) = P(~,,ZL)B(ZL) = P(ZS,ZL)BL(ZL+)[O, CD]'.

(2.64)

At the level of the source we have a discontinuity in the B vector and so the stress-displacement vector just above the source is from (2.61) B ( z ~ - ) = P(z~,zL)B(zL)- 9.

(2.65)

The surface displacement may thus be found from B(O) = P(0, zs)B(zsP(0, zL)B(zL) - P(0, z S ) ~ . )7

(2.66)

We introduce the vector

s = P(0,zs)Y = [S,,S,]T,

(2.67)

which represents the displacement and traction effect of the entire discontinuity due to the source propagated up to the surface. Thus S includes not only the direct radiation from the source to the surface, but also part of the source field interaction with the entire stratified half space. The surface displacement may now be written in terms of the boundary conditions at zL B(0) = P(0, ZL)BL(ZL+)[O, CD]'

-

s.

(2.68)

We set

4 0 , ZL + 1 = P(0, z,)BL(zL + 1

(2.69)

and the free surface condition requires

F,,

);( T+x;-(ji =

I

F,,

(;I );( -

(2.70)

where we have introduced the subpartitions of the matrix F (ie., Fij are 2 x 2 matrices for the P-SV wave case and scalars for SH waves. From (2.70)we see that the wavefield beneath zLis controlled by

C,

= F,-,'S,

(2.71)

100

B. L . N . Kennett

Thus in arranging to neutralize the traction ST induced at the surface by the source discontinuity we obtain formally (2.72) WO= F12F;;s~ - s, provided that the secular function for the half space det F,, does not vanish. As we can see the condition det F,, = 0 is just that for the existence of free waves on the stratified structure, without forcing. The nature of our boundary conditions means that these waves will be confined to a finite depth interval and so have the character of “surface” waves. The corresponding poles in W, will give rise to the surface wave train in the response (see Section IV,E). Once we have found the surface displacement, the stress-displacement vector at any other level may be found from

B(z) = P(z, 0)[wo 01T , = P(z,O)[w,,O]= + P(z,z,)Y,

z < zs, z > zs.

(2.73)

Thus we can get a complete formal specification of the seismic wavefield by using the propagator solution, but it is difficult to get any physical interpretation of the results. There are also some computational disadvantages in working directly with the propagator. In the following sections we will establish equivalent results to (2.72) in which the physical character is more clearly seen and many of the numerical problems are removed. The procedure we have just described is easily adapted to a stratified plate, for which the lower boundary condition at z = zL would be B(zL) = [W’,O]’.

(2.74)

The displacements on the two faces of the plate may then be found in terms of the subpartitions of the propagator P(0, zL)for the plate, WL = P;,’ST,

WO = P , l P ; / s T - s w

(2.75)

and det P , , is just the secular function for the plate. 111. Reflection and Transmission of Elastic Waves

A. DECOMPOSITION OF THE SEISMIC WAVEFIELD IN A UNIFORM MEDIUM We have seen that the stress-displacementvector introduced in Section I1 provides a convenient framework for looking at elastodynamic problems, but it is also desirable to be able to study the nature of the seismic wavefield

Elastic Wave Propagation in Stratified Media

101

more directly. The approach we choose enables us to recover the displacement and stress components conveniently when they are needed, as for example to satisfy boundary conditions. For a particular combination of wave number k, frequency o and azimuthal order rn we consider a transformation from the stress-displacement vector to a new vector

B = DV. Then in a source-free region, V will satisfy

(3.1)

d,(DV) = wADV,

(3.2)

d,V = {wD-'AD - D - ' a,D}V.

(3.3)

i.e., we have

Following Dunkin (1965) we now choose D to be the local eigenvector matrix for A(p, z), and then D-'AD

= iA,

(3.4)

where iA is a diagonal matrix whose entries are the eigenvalues of A. For P-SV waves AP = diag{-qu,

-4p,4a,qp},

(3.5)

qp = ( 6 - 2 - p y ,

(3.7)

and for SH waves

with qa = (a-2 - p 2 ) ' / 2 ,

in terms of slowness p = k/w, and we will normally choose branch cuts for qa,qs such that Im wqu,Im oqp2 0. In a uniform medium A is constant and so D will be independent of z ; in this case the second term on the right hand side of (3.3) vanishes and so d,V

= iwAV,

(3.8)

with a solution V(z)

= exp{iwA(z - z,)}V(z,)

= Q(z,z,)V(z,).

(3.9)

The stress-displacementvector field may be recovered from (3.9) in the form B(z) = Dexp{ioA(z - Z,)}D-'B(Z~),

(3.10)

B. L . N . Kennett

102

and so we can identify the propagator for the uniform medium as (cf. 2.32) = exp{wA(z - zo)} = Dexp{ioA(z - zo)}D-’.

P(z,z,)

(3.11)

From the explicit representations for A C(3.6) and (3.7)], we may construct “wave propagators” Q(h, 0) = exp{ioAh} from (3.9) to give for P-SV waves QP

( h3 0) = diag{e-iW=h, e-iwqah, eiwqah eiw#), 7

(3.12)

and for SH waves QH(h,

0) = diag{e-iwqflh, eiwqflh).

(3.13)

With our convention that z increases with increasing depth, these exponentials correspond to the phase increments we would expect for the propagation of upward and downward traveling P and S waves traveling a vertical distance h. For example, if we have a plane P wave traveling at an angle 9 to the z axis, then p = sin $/a,

qa = cos $/a

(3.14)

and the phase difference we expect to be introduced in traversing a depth interval h is exp(iocos %/a)

= exp(iwq,h)

(3.15)

and the inverse of this for up-going waves. With these identifications for the phase shifts in (3.12) and (3.13) we see that from (3.9) we must identify the elements of the “wave vector” V with upor down-going P or S waves. For P-SV waves P‘

= [h,$U,+D,$DIT,

(3.16)

where 4, II/ are the “amplitudes” of P and S waves, respectively, and the suffices U, D represent up-going and down-going waves (Fig. 1). For SH waves vH

= [XU,

XDIT

(3.17)

writing x for the SH wave “amplitude.” We may summarize the behavior of the wave vector V by introducing partitions V,, VD corresponding to upand down-going waves

v = [V,,

VJT.

(3.18)

The choice of the radicals qa, qp made above means that in the evanescent regime VDcorresponds to the amplitude of exponentially decaying waves and V, to those of exponential growing terms.

Elastic Wave Propagation in Stratijied Media

103

The columns of the matrix D are the eigenvectors of the coefficient matrix A and from the form of V may be identified as “elementary” stress-displacement vectors corresponding to the different wave types. For P-SV waves D, = rb:, b ~ , ,

b;, b;i,

where

(3.19)

bL,D = Ea[ Tiq,,P,P(2p2PZ - I ) , T2iPP2PqalT, bt,D = &p[P, Tiqp, T2iPp2Pqp,P(2p2P2 For SH waves with b:,D

= EHIP-l,

T iP@qp]T

to achieve comparable dimensionality with the choices for bP, bS; the SH wave slowness /?-’ appears here in a similar role to the horizontal slowness p in (3.19). We have a free choice of the scaling parameters E ~ , E ~ ,and E ~ would like the quantities $”, $”, etc., to have comparable meanings. It is, therefore, convenient to normalize the b vectors so that in the propagating regime each of them carries the same energy flux in the z direction. The energy flux in this direction is given by the scalar product of the traction across planes of constant z and the velocity of deformation in the medium

9=

c wjzjz. j

In terms of our cylindrical representation of the elastic field, this energy flux averaged over a cycle, for frequency o,takes the form

F(B) = +(io)[UP* + VS* + WT* - U*P - V*S - W*T],

(3.21)

which may be conveniently split into contributions from the P-SV wave and SH wave systems. Let us consider, for example, the down-going P wave elementary stressdisplacement vector b; in the case of a propagating wave (i.e., qa real), then

4~-’F(b;)

=~

22%~.

(3.22)

We may, therefore, choose a convenient, frequency-independent, normalization by taking &a

= (2Pqa)-

’”,

(3.23)

104

B. L. N . Kennett

and then the energy flux carried in the z direction by bL is 02/4. We make a corresponding choice for the normalizations for both SV and SH waves by choosing Eg = EH = (2pqp)-"?

(3.24)

Then for propagating P and S waves

40>-2@(bzS9H)= 1,

40-2g(b5S9H)= - 1,

(3.25)

so that the sense of the energy transport does indeed correspond to our convention that z increases with depth. If, however, we consider evanescent P or S waves (i.e., q. or qa imaginary), then, e.g.,

F(bL)= 0,

(3.26)

confirming that evanescent waves carry no energy in the z direction. The nature of the expression (3.21) for the vertical energy flux suggests the introduction of a composition of two stress-displacement fields

wX(B1,BJ = UlPf+ ViSt + WIT: - UTP1 - V,*Si - WfT1, (3.27) which for B, and B, identical reduces to a multiple of the energy flux. We have already seen that for propagating P and S waves X applied to b;, for example, will give unity so that from (3.25)

X ( b L , g ) =-1,

X(bL,bL)= 1,

X ( b t , b c ) = -1,

XG,b;)=1,

%(b!, bz) =

X@!,b;) = 1.

- 1,

(3.28)

Of greater interest is the result of taking cross compositions between the various elementary stress displacement vectors for which we find that we have a set of orthogonality relations X(bc, bL) = 0,

X(bS,, b;) = 0,

X(bc, bS,) = 0,

X(bL, b;) = 0,

X(bP,, bS,) = 0,

X(bpD, g)= 0,

X(bE, b;) = 0.

(3.29) (3.30)

This set of orthogonality relations immediately indicates that, with our choice of normalization, the inverse of the eigenvector matrix D-' must contain the same elements as D. We are thus able to find the rows of the inverse matrix D-' as the reciprocal vectors to the elementary stress-displacement vectors b:, etc. Thus

Elastic Wave Propagation in Strat$ed Media

105

we may write, for P-SV waves, D i l = [sLR,sLs;]'

in terms of the reciprocal vectors 2pP2pqa, - q a , rt ipIT s L , D = ES[2pp2pq,, T i p ( 2 p 2 p 2 - l), kip, - q p ] * . jL,D

= &a[ T i p ( 2 P 2 ~ ' -

(3.31)

For SH waves

DH1 =

[s;,s;l'

with

I,=" E b [, P f l, q a , kip-']'.

(3.32)

Now that we have been able to construct explicit forms for the eigenvector matrix D and its inverse we are in a position to construct the propagator matrix for the uniform medium from (3.11). For P-SV waves we have pP (h? 0) = DP diag{e-iW=h,e-kah,

eiwq.h,

eiWBh}Dp 1

(3.33)

B. L. N . Kennett

106 For SH waves,

P,(h,O) = DHdiag(e-i"4~h,eiwq~h}D;1,

P,,

= P22 =

CB,

P,,

= (pb2)-'SB,

(3.35)

P,, = pb2qiSB. (3.36)

These uniform layer propagators are identical to the layer matrices of Haskell (1953) although they were derived by a different route. We have followed Dunkin (1965) in choosing a diagonalising transformation via the eigenvector matrix D, but other choices are possible and lead to the same result. For example, Hudson (1969a) describes a transformation to work in terms of variables +u T +D (in our notation) and t h s is closely related to the original treatment of Haskell. Hudson is able to calculate the exponential of h s transformed matrix by summing the series (2.33) since direct exponentiation is only convenient for a diagonal matrix. In our construction of the propagator matrix for stress and displacements associated with P-SV waves via (3.33) we have split the wavefield in the uniform medium into its component parts via D - I . We have then propagated the separate up and downgoing P and SV wave contributions through the depth interval h and then finally reconstituted displacements via the matrix D. The propagator includes the entire wave propagation effects for its interval, but as we shall see later particular features such as reverberations within a layer depend on the nature of the stress-displacement fields at the boundaries. The fundamental matrix for the up- and down-going P and S waves in the uniform medium is easily extracted from (3.10), and (3.34), (3.36) as Bp(z)= D, exp { iwA,z}, ~ H ( Z= ) DH exp { iwAHZ},

(3.37)

so that the elementary stress displacement vectors are just enhanced by appropriate phase factors. The matrix D was introduced in (3.1) as a transformation connecting the wave vector V to the stress-displacement vector D. In consequence, its subpartitions play the role of transforming up- and down-going wave components into displacement and stress. We may display this relation by writing (3.38)

so that mu,mD are the displacement transformations and nu, n, the stress transformations. For the P-SV wave system mu,etc., will be 2 x 2 matrices and for SH waves simply scalars.

Elastic Wave Propagation in Stratijied Media B. REFLECTION AND TRANSMISSION AT AN

107

INTERFACE

Consider two different elastic media separated by the planar interface z = zl. As a consequence of the conditions of welded contact at the interface we have continuity of the stress-displacement vectors B across zl. In the two media we will, however, have a different representation of the stressdisplacement field into its component up and downgoing wave parts. In medium 0, z < zl, ml-) =

Do(z1 - )VO(Z, - 1,

(3.39a)

and in the medium 1, z > zl, (3.39b) B(z1 +I = DI(Z1 +)Vl(Zl+). Once we equate the two representations of the stress-displacement field at z1 (3.39a,b) we may connect the wavefields in the two half spaces by

Vo(zi -1 = Di’(zi -)Di(zi +)Vi(zi +) = QVi(zi+),

(3.40)

and may now extract the reflection and transmission coefficients for the inter face. We will first illustrate the procedure for SH waves, for which we get rather simple results, and then extend the method to allow for the coupling between P and SV waves. 1. SH Waves

Consider SH waves with horizontal slowness p incident upon the interface. By prescribing p to be the same in the two media we will automatically satisfy Snell’s law and the inclinations of the waves to the vertical in the two media will depend on the vertical slownesses

-p y . When we make use of the explicit expressions for VH (3.17), D H (3.20) and qpo = ( p i 2 - p2)1/2,

qp1 = ( p ; 2

DH1 (3.32) we find that (3.40) may be written as

(;I:)

popl

= EBOEBl (P0Ypo Po4po

-

:)(-

icllqpl 1

iP19p1

)(;;),

(3.41)

~= (2p0qs0)~ / p in terms of the shear moduli po = pop;, p1= p$:. Since ~ we see that the equations (3.41)can be recast entirely in terms of the products pqp,which play the role of impedances for this oblique incidence of SH waves,

(3.42)

~

B. L. N . Kennett

108

Suppose we have only an incident down-going wave in medium 0, then we must require the up-going wave in medium 1 to vanish (i.e., xUl = 0). We then define reflection and transmission coefficients for downward propagation by xuo = RDXDO ,

XDI

= TDXDO,

(3.43)

so that from (3.44) we find RD = q 1 2 q i 2 ,

TD

= qi2-

(3.45)

In terms of the SH impedances introduced above, (3.42)yields RD = P04po - P14Sl > P O r l S O + P14Sl

T,= 2(PoPlq~oq@1)1’2 POclj30 + PlqS1

(3.46)

As the contrast in properties across the interface becomes very small RD

--*

‘(vqfl)/Pqp,

TD

--*

1*

(3.47)

We may proceed in a similar way to derive reflection and transmission coefficients for upward propagation from medium 1, but in this SH case they are most easily obtained by exchanging the suffices 0 and 1 so that

2. P-SV Waves We now need to account for the coupling between P and SV waves introduced by the interface. Once again we satisfy Snell’s law by taking all waves to have horizontal slowness p . We split the wave vectors into up- and down-going parts on each side of the interface (3.18) and partition the coupling matrix Q = D, ‘(zl -)D,(zl +) into 2 x 2 submatrices Qij so that (3.40)becomes

6;)

I (z;-j-g;j Qii

=

Q12

(-2;)

(3.49)

Consider an incident wave system comprising only down-going waves in medium 0, so that V,, = 0 (Fig. 4). Matrices of transmission and reflection

Elastic Wave Propagation in StratiJied Media

109

p = 0.1

a : 6.0 P:3.4

a : 8.0 P : L . 6

24 FIG.4 The configuration of reflected and transmitted waves at an elastic interface for slowness p = 0.1.

coefficients for downward propagation may then be defined by

Note that with this convention the reflection and transmission coefficients follow the standard indexing of matrix elements, which is very useful in manipulation. With these definitions for R,, TDwe see that by analogy with (3.44),(3.45) for the SH wave case TD = QT2,

RD = QizQ,;'.

(3.52)

In terms of the elements qij of Q the individual coefficientsmay be recovered from

110

B. L. N . Kennett TABLE I ELEMENTS OF THE INTERFACE MATRIXQ APPEARING IN THE DOWNWARD REFLECTION AND TRANSMISSION MATRICES R,, TI,

The reflection coefficients are, therefore, ratios of second-order minors of the matrix Q. The matrix elements in Q appearing in (3.53) and (3.54)are presented in Table I and we notice the significance of the shear modulus and density contrasts across the interface, Ap = po - p1 and Ap = p o - p l . The denominator which appears in all the coefficients (det Qz2)is given by

and, as noted by Stoneley (1924),if this denominator vanishes, then we have the possibility of free interface waves with evanescent behavior in each of the half spaces. These Stoneley waves have a rather restricted range of existence, for most reasonable contrasts in density the shear velocities Po and P1 must be nearly equal for (3.55). The slowness is always greater than bin(Po,P1))-'* If we now consider upward incident waves from medium 1, with VDo = 0, then (3.49)may be written as (3.56)

Elastic Wave Propagation in Stratijed Media

111 I

a-

FIG.5 The magnitudes of the interfacial reflection coefficients for the model of Fig. 4 as a

i RSS

Ii

D

i

The upward reflection and transmission matrices are given by

and so from (3.56) we find

Tu = Q i i - QizQ2;fQzi,

Ru = -Q,;'Qzi.

(3.57)

For this single interface we would, of course, obtain the same results by interchanging the suffices 0 and 1 in the expressions for RD, TD ; but as we shall see, the present method may easily be extended to more complex cases. In terms of the reflection and transmission matrices we may represent the interface matrix Q as

Since Q was frequency independent at fixed slowness p , all the interface coefficients share this property. In Fig. 5 we illustrate the behavior of the reflection coefficients at an interface as a function of slowness. C. A VERTICALLY INHOMOGENEOUS REGION 1. The Wave Propagator

We consider an arbitrary vertically inhomogeneous region in z1 < z < z3 sandwiched between two uniform half spaces in z < zl, z > z3. The stressdisplacement vectors at the top and bottom of this region are related by B(z1)

= P(Z,,Z3)B(Z3).

(3.59)

B. L. N . Kennett

112

In the two bounding half spaces we once again represent the stress-displacement fiekls in terms of the appropriate wave vectors so that (3.59) becomes D(z1 -)V(Z1-) = P(Z1, z3)D(z3 + )

W 3

+1.

(3.60)

Thus the wave vectors are connected by w 1 -

1 = D - %1= Q(Z1-

)P(Zl, z3)D(z3 2

z3

+)V(Z, +),

+ ) W 3 +),

(3.61)

and by analogy with (3.59) we may term Q the wave propagator. This wave propagator has similar properties to P(zl, z3) since from (2.37) Q(z1 - , z 3 + ) = D-'(zi

= Q(Z1-

-)P(zl,zz)D(zz)D-'(zz)P(zz,z3+)D(z,+), 9

zmzz

Y

z3

+ 1,

(3.62)

where, e.g., Q(z, -, z2) corresponds to the introduction of a half space in z > zz with the same elastic properties as at z2. A consequence of (3.62) is that we have a simple expression for the inverse of the wave propagator Q ( z ~ - , z ~ +=) Q - l ( ~ 3 + , ~ 1 - ) .

(3.63)

Although P(z,, [) is continuous across the plane z = [, Q([,z3 +) will not be unless there is no discontinuity in the elastic parameters across this plane; hence the + , - indicators appearing in (3.60) are strictly necessary. The relation between the wave vectors in the bounding half spaces (3.61) has the same form as (3.40) for the interface problem, and indeed we may identify Q in the previous section as the wave propagator Q(z, -, z1 +) for the interface. We have, therefore, already established the formal basis for the determination of the reflection and transmission coefficients for the inhomogeneous region. Thus we introduce the subpartitions of Q(z, - , ~ +)3 and write

In the case of the single interface, Q(zl -, z1+) is independent of frequency at fixed slowness, but for the inhomogeneous region Q(z, -, z3 +) contains a frequencydependent propagator term and so TD(Zl-, 23 +),RD(zl-, z3 +) will be frequency dependent.

Elastic Wave Propagation in Stratijied Media

113

For the region z1 < z < z3 we may specify the entire reflection and transmission effect via a matrix, for either P-SV waves or SH waves

:(--_-I--; );

R(z - z3+) = 1

,

9

(3.66)

I

in terms of the partitions of the wave propagator. As a consequence of the nature of the coefficient matrices A(p,z) for elastic waves (2.13) and (2.14), Kennett et al. (1978) have shown that R is symmetric, even for dissipative media, i.e.,

RD = R:,

R , =RE,

TD= TE.

(3.67)

If further the medium is perfectly elastic R is a unitary matrix and this entails further interrelations between the coefficients. The wave propagator may be expressed in terms of the reflection and transmission properties of the inhomogeneous region as

The block matrix form of (3.68) may be explicitly inverted to give

where we have used (3.63). If we exchange subscripts U, D and reflect the matrix (3.69) blockwise about its center we recover the matrix (3.68). This is equivalent to saying that the upward reflection and transmission matrices are just the downward matrices from the inverted structure. Our reflection and transmission matrices are only well defined for z3 2 z1 and so when we in terms of reflection and wish to represent the wave propagator Q(z,,z,) transmission coefficients we will use (3.68) for Z, 2 Z, and (3.69) for zB < zA. The stress-displacement propagator P may be recovered from the wave propagator Q by using our definition (3.61) thus P(z~,z~ =) D(z1-)Q(zi - , z ~ + ) D - ' ( z ~ + )

(3.70)

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B. L. N . Kennett

as noted by Kennett (1974).As in Section II1,A we may use this relation to look at the physical nature of the propagator. The operator D-’(z, +) takes the stress-displacement field into its up- and down-going wave parts at z = 2 3 +. These then interact with the structure via the operation of Q which we note requires knowledge of propagation in both directions through the layering. The up- and down-going waves at z1 - are then used to reconstruct the stress-displacementfields at zl. The continuity of P(z,, 23) at zl, z3 also gives us the representation

If there are strongly reflecting interfaces at zl, z3 (3.70) and (3.70a) will provide strikingly different pictures of the wave propagation pattern. Q(z, -, z3 +) will include all the reverberations between the bounding interfaces, but these will be excluded from Q(z, + ,z3 - ). The stress displacementpropagator will be the same since it is insensitive to the elastic properties outside its range of definition.

2. Invariant Imbedding If the elastic parameter distribution is piecewise continuous, we may use the approach discussed in Section III,C,l to characterize the reflection and transmission properties of a portion of the stratification by making use of the technique of “invariant imbedding.” Consider now the region zA I z I zB, we isolate this from the rest of the stratification by introducing hypothetical elastic half spaces in z < zA,z > zB with elastic properties equal to those at z = zA + ,z = zB -, respectively. We may now apply the theory in the previous section and find the wave propagator and reflection and transmission matrices. The continuity of the elastic properties at zA, zB means that no interface terms are introduced by these planes, and the reflection properties are entirely controlled by the structure within (zA, zB).We may, therefore, write, e.g., (3.71) for this class of coefficients. Consider the reflection properties of an ensemble of models as zA is reduced with zB fixed; as in radiation transport theory (see, e.g., Wing, 1962) we may think of the real medium as “imbedded” in this ensemble (hence the term “invariant imbedding”).This procedure provides a unique and unambiguous definition of the reflection and transmission matrices RkB, TkB,etc., and so avoids many problems which arise with less restrictive definitions. We have

Elastic Wave Propagation in Stratijied Media

115

already, in fact, used this approach in deriving the chain rule for the wave propagator (3.62). 3. A Unform Region If we consider the special case of uniform elastic properties in Z, 5 z I zB, then we have already established the form of the wave propagator in (3.12), (3.13) Q(Z,,Z,) = exp{iwA(zA- zB)} (3.72) where EABis the phase income matrix for downward propagation from Z, to zB. For P-SV waves

E7B = diag[ei"4dzB-zA)

eiW#(ZB-z~)

I,

(3.73a) (3.73b)

If we now compare the expressions (3.69) and (3.71) for the wave propagator, we see that as expected both RtB,RCB vanish, and so there is no reflection from the region. We may also identify

TtB = EAB,

TCB = EAB,

(3.74)

for this uniform zone. D. FREESURFACE REFLECTIONS We have so far confined our attention to reflection effects from the stratification, but we also have the possibility of reflection from the boundaries of the stratification. Consider the region 0 Iz 2 z2, bounded above by the free surface. In order to isolate the surface generated effects we adopt the imbedding technique of Section II,C,2 above and take a "fictitious" half space to lie in z > z2 with continuity of elastic properties across z = z2. We take an incident upward wavefield from z > z2 and this will be reflected from the stratification and the surface to give a down-going field characterized by the reflection matrix R3z2)defined as (3.75) w 2 +) = R%2)C.;(z2 + 1.

116

B. L. N . Kennett

At the free surface, the displacement may be found in terms of the wavefield at z1 as (cf. 2.69)

(3.76) and so in terms of the subpartitions of F

(!;’I-;)(??). F’Z

=

____ FZZ

(3.77)

vD(zZ)

The vanishing stress condition thus requires

R:(ZZ) = - Fi;FzI.

(3.78)

If, in particular, we move zz right up to the surface, i.e., zz = 0+, we have F(O,O+) = D(0-t) and

-

R[(O+) = R = -n-’n D

U

(3.79)

in terms of the partitions of D(O+) (Kennett, 1974). The procedure we have just described may be easily modified to deal with alternative linear boundary conditions expressed in terms of the stress and displacement behavior (e.g., a rigid boundary for which the displacement vanishes). As a simple example we consider the construction of the free surface reflection coefficients appearing in = -nilnu. For SH waves, from (3.20) nu = - iEppPqp,

nD= iEppPqp, and thus RH = 1. For P-SV waves, from (3.19)

and so the free surface reflection matrix is

Elastic Wave Propagation in Stratified Media

117

These reflection coefficients become singular at a slowness such that the denominator vanishes; i.e.,

(2p2 - p

-y

+ 4p2q,qs = 0,

and this is just the condition for the existence of free Rayleigh surface waves on a uniform half space with the surface properties (cf. Section IV,E).

E. THEFACTORIZATION OF REKECTION A N D TRANSMISSION MATRICES 1. Superimposed Stratification

We consider the stratified region z1 5 z I z3 bounded by half spaces with continuity of elastic parameters across z = z l , z = z 3 . The region is then subdivided by the introduction of a horizontal plane z = z2 such that z1 2 z2 5 2 3 . From the chain rule for the wave propagator (3.62) we have Q(z1 +

9 ~ 3 - )

(3.80)

= Q(z1+,~2)Q(z2,~3-).

We may now use the invariant imbedding approach to represent Q(zl +, z 2 ) in terms of the reflection and transmission matrices for z1 5 z 2 z 2 ,RA2, TA2 from (3.68) and, similarly, for Q ( z 2 , z 3 - ) in terms of Ri3, Ti3, etc. The overall wave propagator is once again given by (3.68) in terms of R63, Ti3, etc., so that (3.80) can be written 13 ~ 1 3 -RD( D )

I I

U

_____________ I

1 3 ~ 1 - 13 RD( D )

I

/T62

- RA2(Tk2)-1R12 U II

1 2 T12 - 1 RD( D )

iI

i

Tt3 - R63(T63)-1~23

x

U

\

( T A y I

I

--________--_-___I-_----____ I - (T63)- 1R23 U

I

2 3 ~ 2 -31 RD( D )

(T;3)-1

i

. (3.81)

If we identify the partitions of the matrix products as their counterparts in RA3, etc., we find, for example,

B. L. N . Kennett

118

The overall reflection and transmission matrices can, therefore, be found in the form RA3 = RA2 + Th2Ri3[I - RhZRi3]-1 T12 D9 Tb3 = Ti3[Z

- RhZRi3]-'TiZ,

2 3 ~ 1 21 -Ri3R12]-1T23 U U Rh3 = R63 + TD u [ Th3 = ThZ[l - Ri3RA2]-'Th3,

(3.83) ?

so that we have been able to factor the response. Equations (3.83) were given by Kennett and Kerry (1979) and generalize relations deduced for a layer sandwiched between two half spaces by Kennett (1974). If the level z = z2 should be one at which a discontinuity in elastic parameters occurs, then Eqs. (3.83) are still valid, provided that the split is made consistently on one side of the discontinuity.

2. Factorization of Free Surface Rejections As in Section III,C we consider the portion of the stratification 0 5 z I z2 adjacent to the free surface, but now split it up by introducing the plane z = z1 (0 < z1 5 z2). In (3.76) we have defined the matrix F(O,z,) as a product of the propagator P(O,z,) and the eigenvector matrix D(z2). We factor the propagator at the level z1 and so may represent F(O,z,) as a product of F(0, zl) and the wave propagator Q(zl, z2) from F(0, z2) = P(0, z,)D(zz), = P(0, zl)D(zlP- '(Zl)P(Zl, Z z ) W 2 ) , = F(0, zi)Q(zi ~ 2 ) . 9

(3.84)

Now we have defined the free surface reflection matrix RE2(= RF(z,)) via the subpartitions of F(0, z2) as RE2 = - (F$J-'F;, . The relevant partitions F?,, Fgl are given by the products Fi2 = F i 2 [ l

F;l

1

= -F22{

- RE1RA2](TA2)-', ~ F 1 ~ 1 2 U U f

2 - RUF 1 ~D1I(2 ~ D1) - 21 ~ U1 1,

(3.85)

since RE2 = -(Fi2)-'Fi1, and so we may write RE2 = RA2 + TA2[Z - RE1RA2]-1RE1Th2, = RA2 + TA2R:'[I - RA2RE1]-1T'z U .

(3.86)

This factored form has exactly the same structure for REZ in terms of RA2, RE1 as the expression (3.83) for Rh3 in terms of R i 3 ,RhZ.It is thus immaterial which linear boundary condition is applied at the edge of the stratification, provided that the appropriate coefficient matrices are employed.

Elastic Wave Propagation in Stratijied Media

119

3. Interpretation of Factorization We have now been able to obtain convenient expressions for the overall reflection response of a stratified region in terms of the properties of the upper and lower part of that region, but have yet to associate any physical interpretation to these results. To simplify our discussion we will confine our attention to the down-going matrices

Rb3 = Rb2 + Th2Ri3[Z- Rh2Ri3]-'Th2,

(3.87a)

TA3 = Ti3[Z - Rh2R63]-1Th2.

(3.87b)

In each of these expressions there appears the matrix inverse [ I - Rh2Ri3]-' , and we would like to be able to specify the action of this operator. At first sight we might expect that we should perform a direct evaluation of the inverse in terms of det(Z - Rh2Ri3)and the matrix adjugate. However, as pointed out by Cisternas et al. (1973),the determinant includes combinations of 12 and 23 reflection coefficients which could not be produced by any physically realizable system of wave interactions. The difficulty may be resolved if we instead make a series expansion of the matrix of the form

[ I - A]-'

=I

+ A + A2 +

* * * .

With this representation of the inverse the expressions for the downward matrices become 1 3 - R12 + T12R23T12 + ... 12R23RlZR23TA2 RD - D U D D + T U D U D (3.88a) 13 TD

-~ 2 - D

. .,,

3 ~ 1 2 23 1 2 ~ 2 3 ~ + h 2 D +TDRU D

(3.88b)

and higher terms have further powers of the compound matrix Rh2Ri3 introduced. The sequences (3.87) are illustrated schematically in Fig. 6. The total response to some incident downgoingwavefield vD(z1)can be considered as the sum of contributions from each term in the series. The action of any individual term may be seen by reading it from right to left. If we consider first the reflected part of the field (3.88a), then RA2 corresponds to the entire reflectivity of the upper zone in the region. The second term Th2Ri3Tb2arises from transmission down through the upper zone, reflection by the lower zone and transmission back up through the upper region. In Th2Ri3Rb2Ri3TA2 we have the same set of interactions as previously, but in addition have included waves reflected back from the upper zone which once again are reflected by the lower region before passage to zl. Further terms include higher order internal interactions between the upper and lower portion of the region. The total response (3.87a)includes all these

120

B. L. N . Kennett

FIG.6 Graphic representation of the first few terms of the expansions of the reflection and transmission matrices for superposed,media (3.88) showing schematically the interactions undergone by the waves with the regions z1 < z < z2 and z2 < z < z 3 .

internal reverberations so that we can term [ I - Rh2Rh3]-' the "reverberation" operator for the region. A similar pattern can be seen in the transmitted part of the field (3.88b). The first term T i 3 T b 2corresponds to direct transmission through the region, whilst the second and higher terms include the successive internal reverberations in the region. With this interpretation of the downward matrices (3.88), we see that if we truncate the reflection series after ( M + 2) terms we will have included M internal reverberations. This device is very convenient when one is studying the effect of multiple reflections and has been used by Kennett (1975, 1979a) and Stephen (1977). From a computational viewpoint it is interesting to compare the reflection response at the three levels of truncation: (i) M = 0,

R , = RA2 + Th2Ri3TA2,

(3.89a)

R , = RA2 + Th2Rh3[l+ Rh2Rh3]-1TA2,

(3.89b)

(ii) M = 1, (iii) M = 00, (3.87a),

R,

= RA2

+ TA2Rk3[Z- RL2Rk3]-' TA2.

(3.89~)

We see that the same combinations of terms appear in (3.89b) as (3.89~)and even for the P-SV case we are only dealing with 2 x 2 matrices so that

Elastic Wave Propagation in Stratified Media

121

matrix inversion is straightforward. For more than a single internal multiple it is, therefore, more efficient to compute the entire response.

F.

RJ3CURSIVE APPROACH FOR A STACK OF UNIFORM

LAYERS

From the factorization equations for a region we are able to deduce a convenient recursive approach to the calculation of reflection and transmission matrices in a medium composed of a stack of uniform layers. Consider a uniform layer in z1 < z < z2 overlying an inhomogeneous region in z2 < z < z3. We suppose the reflection and transmission matrices RD(z2-) = RD(zz-, z3+), etc., are known, then we may add in the propagation through the uniform layer using (3.83) : RD(z,+)

= E12RD(z2-)E12,

TD(z1 +)

= TD(zZ-)E",

TU(Z1+)

= E12T"(z2-),

(3.90)

where we have introduced the phase income E l 2 for downward propagation through the layer [(3.73) and (3.74)]. A second application of (3.83) now allows us to include the effect of the interface at z = zl, to calculate the set of reflection matrices at z = z1 - . We will write, e.g., YD(z1) = rD(zl - , z l +) for the interfacial matrix and then RD(zl

-) = rD(Z1)

TD(zl

-1

+ tU(zl)RD(zl

= TD(zl

- rU(zl)RD(zl

- rU(zl)R(zl +)]-ltD(zl),

Rdzl-)

= RU(Z2-)

+ TD(zl

TU(z1-1

= TU(zl)[l

- RD(zl +)rU(zl)]-'TU(zl

+)rU(zl)[z

+)]-ltD(zl),

(3.91)

- RD(zl +)rU(Z1)]-lTU(Z1

+),

+).

The two sets of equations C(3.90)and (3.91)] constitute our recursive scheme. We start at the base of the layering, z3 say, with the interfacial matrices at z3, RD(z3), etc. These we now recognize as R D ( z 3 -), etc. We may then use (3.90)to step the stack reflection and transmission matrices to the top of the lowest layer in the stack. Then using (3.91) we can bring our stack matrices to the upper side of this interface. The cycle (3.90) followed by (3.91) then allows us to work up the stack, a layer at a time, through an arbitrary number of layers. For the downward matrices R D , T,, (3.90) and (3.91) require only downward stack matrices to be held during the calculation. However, for upward quantities, where in effect we are adding a layer at the most complex level

122

B. L. N. Kennett

of the wave propagation system both downward and upward quantities are required. This form of recursive solution may easily be extended to the free surface reflection matrix (3.78), although here the recursive scheme would start at the surface with the free surface reflection coefficients rather than from the base of the layering. This recursive scheme has two important computational characteristics. First, at fixed slowness p , the interfacial matrices are frequency independent, so that at each layer step frequency dependence only enters via the phase income E l 2 . If, therefore, the interfacial coefficients are stored, calculations may be rapidly performed at many frequencies for one slowness p . Second, if we have evanescent waves in any layer we get no exponentially growing solutions. When, for example, the P wave is evanescent and so qz is negative our choice of radicals (3.7) ensures that exp{iwq,(z,

- Z1)> = eXP{-w/q,/(z,

- ZA>,

(3.92)

and in (3.90) we will always have z2 > zl. The scalar versions of the recursive forms for downward reflection coefficientshave been known for a long time and widely used in acoustics and physical optics. The extension to coupled waves seems first to have been used in plasma studies (e.g., Altman and Cory, 1969) and was idependently derived for the seismic case by Kennett (1974).The recursive scheme [(3.90) and (3.91)] and truncated forms of the recursion including only a limited number of reverberations per layer, followingthe treatment of Section 111,E,3, have been used for calculations by Kennett (1975, 1979a, 1980), Stephen (1977), and Kerry (1981). G. COMPARISON BETWEEN RECURSIVE AND PROPAGATOR METHODS

In the preceding section we have established a recursive calculation scheme [(3.90) and (3.91)] for the reflection and transmission coefficients for elastic waves incident upon a stack of uniform layers. However, this same case allows us to use the analytic form of the stress-displacement propagator in each layer (3.34)and then to find the reflection and transmission coefficients from the corresponding wave propagator [(3.61) and (3.66)]. For purely propagating waves throughout the stack both approaches perform well and the propagator method presents the advantage of working only with real quantities, for a perfectly elastic medium. Once waves become evanescent in any part of the stack the differences become much more marked. As we have noted, the recursive scheme C(3.90) and (3.91)] completely avoids the introduction of any exponentially growing

Elastic Wave Propagation in Stratijied Media

123

terms during the calculation of the reflection coefficients. This result has the corollary that no exponentially growing terms should be present in any representation of the reflection coefficients. In the propagator method, however, from (3.34) we see that we have terms involving CA, CB, SA, SB in each layer and for evanescent P, for example, C A = cosh w4,h,

S A = 4; sinh wq,h.

(3.93)

The growing exponentials in these terms must cancel in the final calculation of the reflection coefficients.In finite accuracy computations the cancellation is not complete since the growing exponentials swamp the significant part of the calculation. Compared with this the complex arithmetic needed for the recursive scheme seems a small handicap. Now we have already seen from (3.54) that the reflection coefficients are given by ratios of minors of the wave propagator. This suggests that calculations should be made directly with minors of the propagator matrices, a procedure which was independently suggested by Molotkov (1961) and Dunkin (1965). The minor matrix method works well for reflection coefficients, but gives rise to some difficulties in transmission since here individual matrix elements are needed as well as a minor (3.53) (Fuchs, 1968; cerveny, 1974). An efficient computational procedure for the minor matrix method has been given by Kind (1976). H. TURNING-POINT PROBLEMS In Section II1,E we have presented a convenient approach to the calculation of reflection and transmission matrices in media composed of a stack of uniform layers. We would also like to be able to consider models with piecewise smoothly varying properties interrupted by only a few major discontinuities. Such a model can be approximated by a fine cascade of uniform layers, but then the process of continuous refraction by parameter gradients is represented by very high-order multiple reflections within the uniform layers. It is, therefore, desirable to find a more direct representation. The task is simplified somewhat by using a combination of invariant imbedding and factorization. Consider the region z1 Iz Iz 3 with an interface at z 2 (Fig. 7). We may factor the reflection response into three parts The arising from the zones I: (z1,z2-), 11: ( z 2 - , z 2 + ) , and 111: (z,+,z,). zones I and I11 correspond to reflection from a smoothly varying profile and zone I1 is just the usual interface term for media with the properties immediately contiguous to the interface. We are, therefore, left with the problem of finding the reflection and transmission properties of a region of parameter

124

B. L. N. Kennett

FIG.7 The decomposition of a piecewise, smooth medium into convenient zones for the calculation of the overall reflection and transmission matrices.

gradients sandwiched between two uniform half spaces with continuity at the limits of the region [e.g., (zl, z2 -)I. A variety of techniques may be used to extract the reflection properties of this zone [e.g., numerical integration of the governing equations (2.13) to construct a propagator and thus recover the reflection matrix via (3.66)]. A very promising approach is to use fundamental stress-displacement matrices based on a uniform asymptotic approximation to the solution of (2.13) via the Airy functions Ai(x), Bi(x). Richards (1976) introduced this technique, but neglected any interaction between P and S waves. Woodhouse (1978) has systematized the approximation and shown how to derive the higher order coefficients in an asymptotic expansion in inverse powers of frequency. An alternative approach with the same leading order matrix, but corrections determined by a convergent iterative scheme of Picard type has been proposed by Kennett and Illingworth (1981). The successive correction terms in this scheme can be identified with higher order wave interactions with the medium. Rather than isolate the effect of the interface, Richards (1976) has chosen to consider the inhomogeneous media and the interface as a single unit. He is, therefore, led to introduce frequency-dependent "generalized" interface terms at fixed slowness p which include gradient effects at the interface. This procedure has the disadvantage that a wavefield decomposition is made within inhomogeneous regions. I. MIXEDSOLIDAND FLUIDSTRATIFICATION We have hitherto confined our attention to wave propagation in stratified elastic solids, but there are some circumstances for which we need to be able to allow for the possibility of fluid zones within, or bounding, the stratifica-

Elastic Wave Propagation in Strat$ed Media

125

tion. Such a situation arises, for example, in the case of oceanic regions, and in models of the Earth's deep interior where the region of the core is taken to behave as a fluid. Within the fluid regions we have no shear strength and thus only P waves propagate, but we have the possibility of conversion to SV shear waves at boundaries with solids. No SH wave propagation is possible within the fluid and so solid-liquid boundaries in this case behave like a free surface, since the shear stress must vanish. For the P-SV system we may use our previous development of reflection and transmission matrices throughout the stratification, with care as to the treatment where shear waves are absent. The simplest consistent formalism is to maintain a 2 x 2 matrix system throughout the stratification, and in fluid regions just have a single nonzero entry, e.g., (3.94) If the inverse of such a reflection or transmission matrix is required it should be interpreted as the matrix with the single inverse entry, e.g., (3.95) Within fluid stratification this choice used in the 2 x 2 matrix schemes for the factorization of the reflection and transmission response C(3.83) and (3.86)] will yield the correct behavior. When a fluid-solid boundary is encountered (e.g., in the recursive scheme of Section III,F), then the interfacial reflection and transmission coefficients used must be those appropriate to such a boundary. These forms may be obtained by a careful limiting process from the solid-solid coefficients discussed in Section III,C,2, by forcing the shear wave speed in one of the media to zero. With these coefficients the upward and downward transmission matrices for the interface will only have a single nonzero row or column so that we link directly the reflection matrices in the fluid [e.g., (3.94) with the full 2 x 2 form in the solid]. J. WAVEVECTOR REPRESENTATION OF A SOURCE We have so far represented the action of sources within the stratification in terms of a discontinuity Y in the stress-displacement vector at the level of the source. An alternative approach is to regard the source as giving rise to a discontinuity in the wave vector V. Such an approach has been used by Haskell(l964) and Harkrider (1964) to specify their sources.

B. L. N . Kennett

126

Consider a source in a locally homogeneous region about the source plane then for a stress-displacement discontinuity Y the discontinuity in the wave vector V is given by z,,

[V(Z,)]!

= Z(zS)= D- '(Z,)~(Z,).

(3.96)

This discontinuity may, as in (3.18), be partitioned into up- and down-going wave parts (3.97) The significance of these terms is most readily seen if we consider a source embedded in a uniform medium of infinite extent. Above such a source we would expect only up-going waves and below only down-going waves so that the wavefield solution in this case is

P(z)= [qJ(Z),o]=, =

[O,

WIT,

z < zs,

z > z,,

(3.98)

and so the discontinuity Z has the representation (3.99)

A comparison of the two forms for E, (3.97) and (3.99) shows that a source will radiate a wavefield -Xu upward and CD downward. For a source in a vertically inhomogeneous region we may still apply (4.18) by breaking the medium at the source level and considering the two halves of the stratification to be extended by uniform half spaces with the properties at zs. We thus use the technique we have found so successful in the treatment of reflection and transmission effects, and as we shall see the wavefield discontinuity Z is most useful when used with reflection matrices. For the general point source (2.51) we may form the jump vector E corresponding the stress-displacement discontinuity Y (2.62). The expressions for the components of Y show that the source radiation is determined by the following combinations of moment tensor components and forces

and

Elastic Wave Propagation in Stratified Media

127

For P-SV waves we write E,

= (4%

4;, $;IT,

4%

(3.101)

and for SH waves S

S T

EH= (XU, XD)

>

(3.102)

for up-going components, for down-going components.

(3.103)

and we also introduce %JD =

1,

= - 1,

Then the components of the jump vector E are for m

= 0,

+ suDi[$~'Mo + Mzz/a2]},

&,D

= &a{ - q a W - ' F z

$L,D

= Ep{SUDipw-lFz

&,D

= Ep{SUDtiPP-

- $PqpMO)?

lN0}

(3.104)

;

form = 1 1 , &,D

+ PqaLM1 f - iNl*I>%

= Ea{SUDiiPW-lFT

$L,D = ~ p { - $ q p ~ -F, ~

+ ~ u D + i ( p - -~ 2p2)M1i + s ~ ~ ~ P ~ N (3.105) ,,},

T ~ u ~ P - ' $ w - ~TF$iqpP-'M,+}; ,

X ~ , D =~ p {

for m = +2, 1' 2M @J,D = Ea{SUD$P 2

* 1,

$L,D = Ep{-ipqaM2f}? ,&,D

(3.106)

= EP{kSUDPP-1M2f.}.

For an indigenous source with no net resultant force or couple the terms No, N, and the force contributions F, ,F, all vanish.

,

IV. Half-Space Response in Terms of Reflection Matrices In Section II,E we established an expression for the displacement and stress field induced by a source within an elastic half space in terms of propagator matrices. However, as discussed in Section III,G, this approach suffers computational disadvantages as well as offering little in the way of direct physical insight.

I28

B. L. N . Kennett

We shall here show how the displacement field can be recast into a form which depends on the reflection properties of the half space. By this means we are able to find convenient forms of the response for buried sources or receivers, for which the numerical problems may be avoided, and also the characteristics of the wave propagation process are clearly displayed. A. THERESPONSE VIA

A

SURFACE SOURCE VECTOR

We introduce a source into the half space by means of a discontinuity in the stress-displacement vector 9, and then this may be projected to the surface by the action of a propagator to give the surface vector S (2.67):

s = P(0, z,)Y

=

[s,,

ST]

.'

(4.1)

For simplicity we will restrict our attention to the case when the stratification is underlain by a uniform half space in z > zL. With a suitable choice of reference level the fundamental matrix at zL+ may be taken as BL(ZL+ 1 = DL(ZL+ h

(4.2)

and we seek to include only down-going or exponentially decaying waves in z > zL.The surface displacement may now be found from (2.72)in the form

wo = F~~F;,'ST- sw,

(4.3)

where F12,F22are the partitions of F(O,zL+) (2.69) which with the choice (4.2) is given by F(O,zL+) = P(O,zL)D(zL+).

(4.4)

We now recall the definition of the wave propagator Q in terms of the propagator Q ( O + , zL+) = D-'(O+)P(O,zL)D(zL+),

(4.5)

and see that we have an alternative representation of the matrix F,

+

+

F(0, ZL ) = D(O )Q(O

+ ,ZL +).

(4.6)

The representation (3.68) for the wave propagator connects F to the reflection and transmission matrices for the whole half space (z > 0)RgL, TELL, etc. We write the transformation matrix D(O+) just at the surface in the form (3.38)and then F is given by the partitioned product

. (4.7)

Elastic Wave Propagation in Stratijied Media

129

The partitions of F appearing in (4.3) are just

+

FIZ

=(mD

FZZ

= (nD f

OL TOL - 1 )( D ) > OL TOL -1 )( D >

(4.8)

and depend exclusively on the reflection and transmission properties for incident downward waves at the top of the stratification. F12involves only the displacement transformations for up- and down-going waves (mu,mD) while F,, depends only on the stress terms. This is to be expected since from the original definition of F(0,zL)we see that F(z,ZL) = P(z, zL)BL(zL)= BL(z),

is just a fundamental stress-displacement matrix whose column vectors correspond to up and downgoing waves in the underlying half space. F I 2 , FZ2are just the displacement and stress components, respectively, of the “down-going” vectors. From (4.8) we have the alternative form for the free surface displacement (Kennett and Kerry, 1979; Kennett, 1979a)

WO= (mD

muRgLL)(nD n&L)-lST

-

sw.

(4.9)

Also from (3.79), the reflection matrix for the free surface itself has the form = - n; In,, so that we have the equivalent form

WO= (mD+ m,R~L)[Z - RRgL]-’nD1ST - Sw

(4.10)

as in the treatment of Kennett (1974). We have seen in Section I1 the role of det Fzz as the secular function for the half space and now from (4.9) det F22= det(nD+ n,RgL)/det TgL.

(4.11)

The reflection and transmission matrices RD, TD can be calculated without the need of introducing the growing exponentials present in the propagator representation of F1,, Fzz. Thus (4.9) is quite a convenient form for the entire half-space response. There is, however, a possibility of loss of precision via growing terms in the propagator P(O,z,), appearing in the definition of the surface source vector S. This causes very little difficulty for sources close to the surface and (4.9) has been used successfully by Kennett (1979a) in the calculation of complete synthetic seismograms, including surface generated multiples, for small offsets between source and receiver. We will see in the next section how the growing solutions with the propagator approach can be avoided for a buried source or buried receiver at an arbitrary depth in the half space.

B. L. N . Kennett

130

We introduce the quantity 9

+

+

9 = (mD muRgL)(nD nuRgL)-',

(4.12)

which is bilinearly related to the downward reflection matrix REL, and then we have the reciprocal relation (Kennett, 1979b)

REL = -(mu - Fnu)-l(mD - 9 n D ) .

(4.13)

Thus, if 9 can be determined, we can recover the reflection matrix for the half space RgL from surface observations (i,e., we have effectively removed the free surface). The expression (4.9) for the surface displacement is

wo = 9

(4.14)

s T -SW,

and so the matrix 2F may be recovered in principle if we know the displacements for given surface excitations (Sw,ST).For SH waves only one such source is needed since % is a scalar, but for P-SV waves we need two independent sources to find the 2 x 2 matrix %. For near vertical incidence an approximate technique may be found to recover RLp from just vertical component seismograms (Kennett, 1979b). From our previous construction we see that 9may be obtained directly from the fundamental matrix F(0, zL) which connects to upward and downward waves in the lower half space. In terms of the "downward" displacements F 1 2 and stresses F Z 2 ,

9 = F12F2;' This gives a general connection between the reflection matrix and the type of solution most conveniently employed in straight numerical integration of the governing equations [(2.13) and (2.14)]. A further representation of the surface displacement may be found by using the identity (I - A ) - l = z

+ A(Z - A ) - 1

(4.15)

in (4.10).We thus have

+

W, = (mD muRE)[I

+ @R;~(L(I- @RE~)-']II,'s~

- s,,

(4.16)

and after regrouping of terms we find (Kennett and Kerry, 1979)

Wo = (mu

+ m,ii)REL(Z- fiREL)-ln;lS, + {m&'S,

- S,}. (4.17)

For a purely uniform half space REL = 0 and so the surface displacement solution to the Lambs problem with surface source vector (Sw,S,)T is just the expression in braces {m,& 'ST - S,}. The first term in (4.17) displays the part of the response which arises from the departure of the half space from uniformity. The operator (mu + m$) simply converts an up-going

Elastic Wave Propagation in Strat@ed Media

131

wave vector into the displacement expected at a free surface. Once we have extracted this conversion operator, we see that R;L(Z - #REL)-' (=[ I RELk]-lRgL)will, from the discussion in Section III,D, include all the reverberations between the half-space layering and the free surface. B. BURIEDSOURCES AND RECEIVERS At a level z within the stratification we represent the wavefield in terms of its up- and down-going wave parts

v(z)

=

[vU(z), vD(Z)]T.

(4.18)

The vanishing of the stress at the free surface requires that for positions above the source (0 I z < z,) the up- and down-going parts of the field are connected by (3.75)

vD(z) = RE(z)vu(z),

0 5 z < ZS.

(4.19)

In a similar way the lower boundary condition, that we should have no upcoming wave field in z > zL, gives a field relation for positions below the source VU(4 = R d z , ZL + ) W ) ,

zs < z I

ZL 3

(4.20)

directly from the definition of the reflection matrix R,(z, zL+) for the structure between z and zL. We now follow Kennett and Kerry (1979) and show how we may represent the wavefields at source and receiver in terms of the reflection properties of the stratification. 1. Source Wauejields The wavefields just above and below the source are related by the jump vector I; (3.96)which represents the radiation from the source

V(ZS+) - V(Z,-)

(4.21)

= Z(z,).

We now aim to represent the wavefields above and below the source in terms of the reflection properties of the regions above and below the source. For notational simplicity we follow (3.71)and write

RLL = R,(z,+,z,+),

RES = R ~ Z , - ) .

(4.22)

The wavefield just below the source at z,+ and that in the lower half space can be related by the wave propagator Q(z, ,zL+), and then, introducing the jump across the source level (4.21),we find

+

V(Z,-)

= Q ( z ~ + ZL , +)V(zL+) - Z(zs).

(4.23)

B. L. N. Kennett

132

In the lower half space there is only a down-going wave, so V(zL+) = [0, VD(zL+)]T, and the up- and down-going components just above the source are connected by R, from (4.19). Using the partitioned form of the wave propagator, Eq. (4.23)thus may be written as

(4.24) and so we find, eliminating V,(Z, +), that V,(Z,-)

= R;L{R;SV,(z,-)

+ EL} - x;.

Thus just above the source the wavefield generated by the source in the stratification is given by (Kennett and Kerry, 1979)

V,(zs -) = [Z

- R;LR;s]

- ‘(RS,LEL - EL)

(4.25)

V,(Zs-) = R~SV,(Zs-).

An attractive feature of the representation (4.25) is that we may attach a physical significance to each of the contributions. As noted in Section 111,E,3, a physically meaningful sequence is obtained by reading (4.25) from right to left (cf. Fig. 8). We consider first the expression (RLLCS,- EL)which corresponds to the entire upward effect of the source at the plane z = zs. This arises in part by direct upward radiation (-Xi)(3.99)and in part by reflection of energy which initially departed downwards from the source (RtLE;). This excitation term, for an isotropic elastic medium, will depend on azimuthal order whereas the reflection matrices are independent of azimuth.

I

L

FIG.8 Schematic representation of the wave interactions in the passage from a buried source to a buried receiver.

Elastic Wave Propagation in Stratijied Media

133

’

The term [ I - RS,LRL’]- corresponds to a reverberation operator through the whole half space coupling the upper and lower regions at the source level z = zs. This is most readily seen by expanding the matrix inverse in an “interaction” series [cf. (3.88)]

[ I - RLLR5SI-1

=

I

+ RSLRFS D U + R D~LRFSRSLRFS U D U +

. ...

(4.26)

The leading term has no effect, but subsequent terms in the expansion correspond to reflection by the layering, and the free surface, above the source (RE’) followed by reflection from the lower part of the half space, beneath the source (RkL).Each successive term introduces a further internal reverberation. As we shall see, free surface waves correspond to the singularities of the whole stack reverberation operator, and thus to the vanishing of the secular function det(f - RSd-RL’) = 0.

(4.27)

The composite action of [ I - RS,LREs]-l(RLLLCS,- EL) is thus to produce, just above the source level zs, a sequence of wave groups corresponding to radiation from the source subjected to successively higher order multiple reverberations within the layering. If we alternatively wish to look at the wavefield just below the source, we may make an analogous development to (4.23)-(4.25), but now satisfy the free surface boundary condition and so obtain

VD(zS+)= [ I

- REsRS,L]-’(ZS, - RF,SZS,),

v,(z,+ ) = R E VD(zS+ ).

(4.28)

We may interpret (4.28) in a similar manner to our discussion for (4.25), with the difference that we now concentrate on the down-going effects of the source.

2. Receioer Wavejields With the two expressions [(4.25) and (4.28)] for the wavefield in the neighborhood of the source, we may use the wave propagator to construct the wavefield at any receiver level and thus determine the stress-displacement field. a. Receiver Above the Source (0 I zR < zs) The wavefield at the receiver zR is related to the wavefield just above the source by

B. L. N . Kennett

134

and from (4.20) the up- and down-going components at zR are linked by

VD(ZR) = RLRVU(ZR).

(4.30)

Since the receiver lies above the source, we use the form (3.68) for the wave propagator to yield

VU(zR) = VD(zR)

=

fTtS-

RS RD

RS - 1

)

RS - 1

)

RS (RU

- REs)}VU(zS-),

(Rts - RLS)VU(zS-),

and so we find

VU(ZR)= [ I - R;SRER]

- T;sVu(zs -).

(4.31)

Thus to construct the receiver wavefield each of the wave groups in the source field is projected up to the receiver level by Tts, and then there is the possibility of further reverberations in the zone between the source and the surface which is allowed for by the reverberation operator [ I - R;sREs]-l (Fig. 8). From the wavefield components (4.25),(4.30),(4.31)we may generate the receiver displacement from B(zJ = D(zR)V(ZR) and so

W(Z,)

+ rn\RER)[I - R;sRLR]-lTts

= (m;

x [I

- P;R3 - '(RS,LES, - EL),

(4.32)

using the representation (3.88) for D(zR).

b. Receiver Below the Source We now start with the wavefield just below the source and construct the receiver wavefield from v(zR)

=

o(zR zS +)v(zS + 9

(4.33)

13

and to satisfy our radiation condition in the lower half space we require VU(zR)

=

RRd. VD(zR).

(4.34)

As now the receiver lies below the source, we use the representation (3.69) for the wave propagator and, by an analogous treatment to (4.31),we find VD(ZR)= [ I - RR,SR;L]

-

T;sVD(zs

+ ).

(4.35)

We here include the downward transmission to the receiver allowing for the presence of reverberations between the source level and the base of the stratification. From the wavefield components (4.28),(4.34),and (4.35)the displacement at the receiver can be found in the form

+ m$RkL)[I

- RbsR$L]-lT;s x [ I - R:sRS,L]-1(z;SD - R:%:).

W(Z,)= (m;

(4.36)

Elastic Wave Propagation in Stratijied Media

135

We have thus been able to construct the displacements and wavefields for arbitrary depths of source and receiver within the stratification entirely in terms of the reflection and transmission matrices for the various portions of the stratification. Since we have seen that these reflection and transmission matrices may be constructed without introducing any contribution from growing exponential terms in the evanescent regime (4.32), Eq. (4.36) gives rise to no numerical problems when calculating the response (Kennett, 1980). The results we have obtained for the source and receiver wavefields and the receiver displacements are quite general and not restricted to our choice of boundary conditions. Thus, if, throughout, RE is taken as the reflection matrix for the linear boundary condition imposed at the top of the stratification, and R b as the reflection matrix for the boundary condition at the base, our wavefields will still have the same structure. We may, therefore, for example, use (4.32) and (4.36) for a stratified plate if Rb is taken as the reflection matrix for a free surface at z = zL.

C. SURFACE DISPLACEMENTS

If we consider initially a surface source with a receiver positioned at a level just below the surface, then from (4.36) the displacement field is given by (4.37) W(O+ ) = (m, + rn,RgL)[I - h g L ]- ‘(c;- RE:). We recall that

k = -nD1nU, so that (4.37) may alternatively be written as

W(O+) = (mD+ rn,REL)(n,

+ n,RgL)-l(n&E + n&).

(4.38)

The definition of the wavefield jump vector is Zo = -DP19’(O) where Y ( 0 ) is the traction-displacement discontinuity at the surface, which we may write in the form Y ( 0 ) = (S,, ST).The combination (nDCg+ n&) just recovers STand the displacement on the surface is related to W(0+) by the displacement discontinuity Sw : W(0)= W(O+)- s*.

(4.39)

The surface displacement obtained from (4.38) and (4.39) is just in the form (4.9), and so we see the equivalence of the techniques discussed in Sections IV,A and IV,B. For a source at depth we see that the surface source vector S is constructed by propagating the original stress-displacementdiscontinuity to the surface. We thereby create a source giving rise to an equivalent stress-displacement field in the half space, but we pay the price of propagating evanescent fields in their direction of growth.

B. L. N . Kennett

136

For a buried source this problem may be avoided if we recover the surface displacement from (4.32) as W(O+) = MFs[Z - RS,LRF,S]-l(RkLZk - ZE),

(4.40)

where MFS = (mu

+ mDR)[I - RE'R] - TE',

(4.41)

is the transfer operator between the up-going wave vector just above the source and the displacement at the free surface. There is a further form for the surface displacement which is of interest since it provides a different picture for the free surface reflections. To derive (4.3) we wrote the equation for the forced motion in the half space in the form B(O) = P(0, zL)DL(zL+ )VL(zL + 1 - P(0, zs)Y(zs), = FLV,(zL

+ 1 - P(0, zs)Y(zs),

(4.42)

with VL(zL+) = [O,C,IT; but we may also write

B(O) = F,(VL(zL+) = F,{VL(Z,

- DL'P(zL, ~s)DsDi'.liP(zJ>,

+ 1 - W L , ZS)WS)}?

(4.43)

in terms of the wave propagator Q(zL, zs). There is no up-coming field in z > zL and so the upward part of (V, Q(zL, zs)Z} is entirely determined by the source contribution. The surface displacement may thus be found from (4.44) w,= ( P I 1 - F12F;21Fz1)(TSTL)-l(RSDLZ=SD - Z",, in terms of the partitions of F. From (4.7)we find F~~ - F,zP;2iF21 = {mu - (m,

+ m,RF)

+

OL - 1

(4.45) where RgL, TELare the matrices for the whole half space. The upward transmission matrix may be factored as in (3.83): (nD

TEL = TEs[f - R;LR$s]

)

-

nU}T:L,

TLL.

(4.46)

From (4.44H4.46)we find the surface displacement

+

+

Wo= (mu - (mD muRgL)(nD nuR~L)-'nu}~(zs),

(4.47)

with G(z,) = TES[f - R;LR:s] - 1 ( RDS L PD - x : ) ,

(4.48)

a form which is very similar to (4.9), but which avoids numerical problems. The field G includes all interactions of the source with the stratification above

Elastic Wave Propagation in StratiJed Media

137

and below the source, but unlike (4.25), does not allow for reflections generated at the free surface which are entirely represented by the term in braces. This may be clearly seen if we recognize that and so (4.47) may also be written as

W, = (mu + m,i?)[Z

- RELi?]-lG(zs).

(4.50)

The reverberation operator between the free surface and the stratification is now clearly displayed. We note that the surface displacement in the form (4.47) and (4.50) can be used conveniently for both buried and surface sources since only G(zs)needs to be recalculated when the source depth is varied. As the source approaches the surface G(zs) takes a particularly simple form, since T t s tends to the unit matrix and REs vanishes, so that from (4.48) Q(O+)

= RELEE - Et.

(4.51)

These various representations of the surface displacementallow us a choice of the most convenient form for different calculations. The form (4.40) has been used by Kennett (1980) in the calculation of complete theoretical seismograms at moderate ranges, and Kerry (1980) has used (4.47) as a basis for modal synthesis of surface waves. As we shall see Eqs. (4.40) and (4.50) enable us to construct convenient factorizations of the response and isolate physically significant contributions.

D. FACTORIZATION OF THE FULLMEDIUM RESPONSE In our discussions of the synthesis of the response of a stratified medium excited by a source, we have seen the important role played by reverberation operators of the general form ( I - RiJRA,J)-l.We have previously been able to give a physical interpretation to our results by identifying the action of the successive terms in the power series expansion of the inverse. Even when this expansion cannot be guaranteed to converge, we may make a partial expansion of the reverberation operator by means of the identity M

(I

+ 1 (RiJJRA,J)k k= + (R;JRgJ)M+yZ - R;JJRA,J)-,.

- REJRgJ)-’ = Z

1

(4.52)

The terms in the expansion may be identified with k-fold internal reflections between the two zones separated by the plane z J . The remainder represents

138

B. L. N . Kennett

those parts of the wavefield which have suffered more than M internal reflections. With the aid of this partial expansion, we may isolate particular contributions from the full medium response and still retain the singular part within the remainder term.

1. Surface Rejections We consider initially the representation (4.50)for the surface displacement, and use the partial expansion (4.52) for the half-space reverberation operator (I - REL@-’ out to the first surface reflection to give

W, = (mu + m&{f

+ R;L:LITR;LR(f

+ RkL@ - R;Lk)-1)Q(Zs),

(4.53)

and now examine the various contributions. If no surface reflection has occurred we have

wL’)

= (mu

+ mJl)TEs[I

- RS,~R:’]

-

‘(RS,~XS, - E”,,

(4.54a)

using the explicit form for Q(z,) (4.48). This contribution allows for all internal reflections within the stratification. We may alternatively express (4.54)as Who)= (mu + m$?)TCs{f + RS,LR;s(f - RS,LREs)-l)(RS,LLI;S,- Ei) (4.54b) and the term TEs(RtLES,-EL) contains the direct propagation from a buried source to the surface. Kennett and Simons (1976) have used just this part of the response to calculate theoretical seismograms for the earliest arrivals from a source model of the July 30,1970 Columbian earthquake at a depth of 650 km. The remainder in (4.54b) corresponds to internal multiple reflections before the wavefield is projected to the surface by the transmission term T&. For a surface source the part of the response which has been reflected once by the half space and undergone no surface reflections is just (4.55) who)= (mu + m,R)(R;LZk -

EE)

using (4.51)for C(O+).This approximation to the surface response is similar to the form used by Kennett (1979a), in calculations of theoretical seismograms at small offsets from the source (4.56) wL’) = (mu + m,,d)R;%, ‘sT, derived from the surface source vector representation (4.17). For a near surface source (4.56) allows surface “ghost” reflections just above the source to be included via ST, but excludes surface multiples of reflected events generated beneath the source level.

Elastic Wave Propagation in StratiJied Media

139

A representation for the surface displacement including up to a single free-surface reflection is from (4.53) W f )= (mu + mDk)(Z+ RgLk)O(zs).

(4.57)

The source-stratification vector Q(zs) may be expanded as in (4.54b)and the reflection matrix R E factorized at the source in the form OL - R O S OSRSLTOS + TOSRSLROSRSL(I - RSLROS D u)-’TgS (4.58) R D -

D

D + T U

U

D

D

U

D

and then we may extract the simplest contributions which involve no internal multiples in the stratification. We find that

w&’) = (mu + mJ?)[TEsRS,L(ZS,- T;’RT~Z:SU)- TE’zL + TOSRSLTOSRTOSR~L~S

U

D

D

U

D

D

+ Rgs&’Es(RS,LZS, - Z:) + Rem]

(4.59)

where the remainder includes all those contributions with further internal multiples of the type RkLRES.The nature of the three types of entry in (4.59) is indicated schematically in Fig. 9.

A.

FIG.9 Contributions to the surface response from a buried source. (a) P + pP + sP; S + sS pS, (b) PP, SS, and (c)reverberation near the source.

+

‘

B. L. N . Kennett

140

Direct upward propagation gives ( - TgsX$), whilst all energy which has been returned once from beneath the level of the source occurs in TESRLL(XL- TE'6TfX;). These two terms together constitute the main arrivals characterized in seismology as P and S; through TEs(RS,LZS,- X;), including energy which left both upward and downward from the source and returned by the nature of the half space. The contribution T$'SRS,L6T$Z$ represents energy which started off upward, but which is reflected by the free surface before reflection beneath the source. This term thus corresponds to the surface reflected terms pP, sP, sS, and pS generated near the source (Fig. 9a). The next contribution with just a single surface reflection T;SRS,L

T;S@TOSRSLXS U D

D

has been reflected back twice below the level of the source (Fig. 9b). The first part is similar to that in straight P or S propagation and then at the free surface conversion may occur to give PP, PS, SP, and SS contributions. The final class of contribution Rgs~T~s(RS,LXS, - Xi) arises from the beginning of a reverberation sequence, near the receiver, between the free surface and the layering above the source (Fig. 9c). The higher terms are contained within the residue of (4.53) which includes all contributions with two or more free-surface reflections

Wb"' = (mu + m D 6 ) { R ~ L 6 R ~ L-6RELLw]-lG(~s). [z

(4.60)

They are, however, more easily envisaged if we use the representation [(4.40) and (4.41)] for the surface displacement. Here the above-source operator ( I - RES6)-' occurs in the displacement factor MFs modulating all reverberations involving the stratification and the free surface arising from (Z - RbLRFs)-'. Comparable results to those above may be derived from (4.40) and (4.41) by taking a partial expansion

(I

- RSLRFS) D U -1 = I

+ RSLRFS D

u

+ RD

SLRFSRSLRFS u D u

( I - RS,LRES)-l, (4.61)

and then splitting REs just below the free surface so that

RES = REs + TgsR(Z - REsR)-'TEs,

(4.62)

which we see gives a further set of reverberative effects above the source level but this time associated with the source. 2. Partitioned Stratijcation In some cases the response from a particular part of the half space may be of interest. The influence of the properties of this zone may be examined by partitioning the half space and then recasting the response in terms of the reflection and transmission properties of the partitioned stratification.

Elastic Wave Propagation in Stratijied Media

141

For a surface source the surface displacement may be found from (4.50) and (4.51)as W, = (mu

+ mDR)[z - R ; ~ R ] I - ' ( R ~ ~-z ~c:).

We take the half space to be split at the level z (3.83) OL RD

- ROJ -

D + T U

= RgJ

OJRJL D(

- ROJRJL U

= zJ (zJ < zL), and

D)-

1

(4.63)

then by

01 TD,

+ YgJL.

(4.64)

We may also factor the half-space reverberation operator to display the reverberative sequence in the upper part of the layering, above zJ, [ I - RELR]I-l = [ I - RgJR]-'(Z

+ YgJLR[Z - Rp,Li?]-l).(4.65)

With the substitutions (4.64) and (4.65) we may reorganize the surface displacement into the form

W, = (mu + m,R)[Z - R ; ~ W ] - ~ { ( R ~ ~ C ~ c:) + YgJLR[I - R;LR]-yR;LX; - CE)}.

+ ygJL~g (4.66)

Ifjust the first term in.braces were present, the resulting displacement would be just that given by the analog of (4.63) for the stratification in the half space truncated at the level z,. The remaining terms couple waves into the lower part of the half space with surface reverberations still included. Equation (4.66) indicates the nature of the errors we commit when we set up a stratified half space with a uniform lower region by truncating a velocity profile. We then ignore energy reflected from the removed velocity structure and internal reflections between the two parts of the structure, and the free surface. The portion of the surface displacement which does not contain any free-surface reflections from the deeper part of the layering ( z > zJ) is

w0= (mu + mDR)[z - R~~R]-'{R;~Z~- X: + T?RLL(Z

- R;JRJ,L)- TOJZO}. D D

(4.67)

Further restrictions on the portion of the surface displacement to be calculated require specific decisions about the effects to be included, and these may be well illustrated by (4.67). a. Full Response for Part of the Medium A good example of this approach is provided by the "reflectivity method of Fuchs and Miiller (1971). In this approximate technique designed originally for use in seismic refraction studies, where the first part of the waveform was of interest, a number of simplifications were made to (4.67).

142

B. L . N . Kennett

First, the upper and lower parts of the half space were assumed to have no mutual interaction [i.e., ( I - R;’R’,)-’ is replaced by the unit matrix which we recognize as the leading term in an expansion of the type (4.52)]. This enables the lower part of the layering to be considered in isolation. Second, all free-surfacereflections were neglected.This led to the “reflectivity” approximation for the response of the deeper part of the layering: (4.68)

In the original treatment further approximations were made to the transmission terms to allow only for direct interfacial transmission losses and only a single (P) wave component was included. Later Fuchs (1975) allowed for P to S conversions. Nevertheless, through the term R’, all internal multiples and interconversions were included in z, < z < zL. The practical application of (4.68) requires an adroit choice of the level zl at which the half space is separated, so that the mutual interaction between OJ and JL is kept to a minimum. It is also necessary to avoid interference with free surface reflections. In some cases a natural break point in the structure occurs [e.g., in the calculation of energy return from rocks beneath the seabed, z, is taken at the seabed (Orcutt, et al., 1976; Fowler, 1976)l. This leaves the water column in OJ and the entire sub-seafloor structure in JL. The “reflectivity” approximation is then good for the first seven or eight seconds of arrivals in water of 4-km depth which includes most of the useful information. Miiller and Kind (1976)have restored in part the surface reflection terms, as in (4.67),by taking a model with a very dense fluid with the acoustic wave speed of air overlying an elastic half space. This gives a reasonable approximation to the free surface reflection coefficients, but does not lead to any singularities on the real slowness axis. b. Approximations Throughout

An alternative to retaining a full response in any part of the medium is to approximate the reflection and transmission matrices in (4.67)and to consider only limited levels of interaction in any reverberation operator, including that for the free surface. For a multilayered medium we may follow Kennett (1974) to see how “ray expansions” for the reflection and transmission matrices may be generated. In Section II1,F we have shown how the reflection and transmission matrices may be incremented through a layer, and if we now make a partial expansion of the layer operator [ I - rU(zI)RD(zl+)I-’ as in (4.52)we find, e.g.,

Elastic Wme Propagation in Stratified Media

143

The phase delays for each layer are contained in R,(z, +) so that (4.69) gives rise to a suite of contributions with phase delays appropriate to multiple passage of the layer at the particular slowness. If an overall accuracy level E is desired, a convenient working criterion for the truncation level is that for an N-layered stack

[RURLIMI &IN,

i.e., M 2 ln(c/N)/ln(R,R,),

where R,, R, are the moduli of the largest reflection coefficients at the roof and floor of the layer (Kennett, 1974). Thus in a velocity inversion, for example, M should be quite high. Commonly, a fixed expansion level (e.g., M = 1) is maintained in “ray” expansions (see, e.g., Helmberger, 1968), but Muller (1970) has forcibly demonstrated the necessity for including highorder reflections for accurate results in even simple models. For continuous stratification approximations are commonly made in representing the reflection response. Away from turning points a WKBJ type approximation is appropriate (Chapman, 1974),and a uniform asymptotic approximation may be found across a turning point using Langer’s method (Richards, 1976). Higher order reflection terms can be calculated by an iterative development [e.g., with the WKBJ solutions (Chapman, 1976)], but these are usually neglected. 3. Other Approximations

In our treatment above we have shown how a number of approximations to the full elastic response can be systematically generated with a clear presentation of the neglected terms. The methods we have used may easily be extended to deal with many other situations. Many other authors have, however, proceeded on an ad hoc basis for the problem in hand without making any assessment of the accuracy of their approximation. Such approximations usually arise from an interest in particular classes of wave arrivals [e.g., Faber and Muller (1980)have examined Sp converted phases from upper mantle discontinuities by a modification of the reflectivity technique].

E. SURFACE WAVES The various expressions which we have derived for the receiver response for general point source excitation have singularities associated with the properties of the reflection and transmission matrices for portions of the stratification and the corresponding reverberation operators. In particular, we have a set of poles associated with the vanishing of the secular function

B. L. N . Kennett

144

for the half space detF,, [(2.72) and (4.11)]. This secular function is independent of the depth of the source and depends on the elastic properties in the half space. For the combinations of frequency and slowness for which detF,, vanishes we have nontrivial solutions of the equations of motion satisfying both the vanishing of traction at the surface and decaying displacement at depth (Jw1-i 0 as z -i co). For our choice of structure this latter property arises from the presence of only exponentially decaying waves in the lower half space z > zL (2.63). For perfectly elastic media the poles reside in the real slowness axis, and the work of Sezawa (1935) and Lapwood (1948) show that at large distances the polar residue response corresponds to the surface wave train. The stressdisplacement fields associated with these residues show no discontinuity across the source level (Harkrider, 1964), so that the excitation of the surface waves should not be thought of as occurring directly, but by the interaction of the entire wavefield with the stratification and the surface. 1. Surface Wave Dispersion The location of the surface wave poles in frequency wave number space is given by the vanishing of the secular function, ie., det F,,(o, p ) = det(n,

+ n,RgL)/det TkL= 0.

(4.70)

If we have a uniform half space we would require detn, = 0, no root is possible for SH waves, but for P-SV waves we require -

Po”)2 + 4P24a04/?o= 0,

(4.71)

and this is the usual equation for the Rayleigh wave slowness pRO on a half space with properties u0,Po, p o . For any increase in velocity within the half space we have the possibility of dispersive wave propagation with the slowness depending on frequency. We may rewrite (4.70)in the form

A = det n, det(Z - kRgL),

(4.72)

although the impression that there is always a root when detn, = 0 is illusory since it is matched by a singularity in k at the same slowness. However, with increasing frequency for slownesses such that all wave types are evanescent at the surface, RgL -i 0 and so A-idetn,,

as o-+co, p>P;’.

(4.73)

Thus the limiting slowness for the dispersive Rayleigh waves, for P-SV propagation, is the Rayleigh slowness pRO for a half space with the surface properties throughout. For smaller slownesses than pROwe may use the

Elastic Wave Propagation in Stratijied Media

145

simpler dispersion relation (Kennett, 1974) det(Z - k g L )= 0.

(4.74)

This relation constitutes a constructive interference requirement between waves reflected down from the surface and then back from the stratification. a. Love Waves For SH waves the free surface reflection coefficient is unity so that (4.74) becomes

(4.75) i.e., the Love wave secular relation requires us to seek the combination of frequency and slowness for which a wavefield is reflected from the stratification without change of amplitude or phase. It is interesting to see how more familiar Love wave dispersion relations may be obtained from this result. Consider the simple example of a layer with density po shear wave speed Po and thickness ho overlying a uniform half space with density p1 and shear wave speed pl. The reflection coefficient at ho is given by (3.46) and propagating this to the surface we require exp(2ioqpoho)(poqpo- p1qpJ/{poqpo + plqpl)

= 0,

(4.76)

and with a slight rearrangement we have tan wqpoho = - ~rulqpl/ruo4po,

(4.77)

the conventional dispersion relation (Ewing et al., 1957). We can see directly from (4.76) that roots will only be possible when the interface coefficient has modulus unity and waves propagate in the upper layer, i.e., the Love wave slowness is restricted to

p;' < p < p,?

(4.78)

Also since the reflection coefficient is independent of frequency, the roots of the dispersion equation in frequency at fixed slowness are just on = (nn - i$(P))/4poho

(4.79)

where $(y) is the phase of the interface reflection coefficient. Thus, since $ ( p ) decreases steadily with increasing p approaching - n when p .+ and is zero when p = fly1, we have a sequence of modes with a lower cut off in frequency at (4.80) on0= nn/(Py2- ~;')'''h0, and with slownesses which approach p;' at very high frequencies. For frequencies less than the cut off for any mode, roots cannot be found on our

B. L. N . Kennett

146

chosen Riemann sheet Im(oqpl)2 0, but move offto complex p values on the lower sheet (Gilbert, 1964).In this case the reflectionform (4.76)considerably simplifies the analysis of the dispersion. b. Rayleigh Waves In the case of P-SV waves the dispersion relation has a rather more complicated form. From the original form (4.70)we have the equation R2P2 - P02)(1+ RPP)&, - 2iPqfloRs+,1 x "2P2 - P02)(1 + RSS)&fl- 2iPq,oRPs&a1 - [2iP4ao(l - RPP)Ea + (2P2 - PO2)Rsp&pl x [2ipqpo(l - RSS)&p+ C2P2 - Po2)RsP&al= 0,

(4.81)

where RPP,etc., are the components of RgL. This form shows the departure of the dispersion relation from the Rayleigh function (4.71)due to the presence of inhomogeneity in the half space. For slownesses rather less than pRO we may use the form (4.74),which may be written as 1 - r?,(Rpp

+ R,)

- 2r?,R,

+ (RppRss -

= 0,

(4.82)

where we have used the symmetry of RgL and the free surface matrix

(itpp= RSS,r?,

=

ESP).

'

If we have both evanescent P and S waves at the surface (ie., PO < p < pRO), then at reasonably high frequency lRPPl, /RBI will be very small. Thus we have an approximate form for the dispersion of the fundamental Rayleigh mode (2P2 - Bo2)2(1 + Rss) - 4P24dfl(l- Rss) = 0,

(4.83)

with Rss real and less than unity. For higher mode Rayleigh waves we have a limit point at a slowness PO (i.e., we now have propagating S waves at the surface). When P waves are evanescent at the surface (i.e., a;' < p < PO'), for a perfectly elastic medium we may use the unitarity relations of Kennett et al. (1978)to express the determinant of RgL as

'

R,,Rss

- R& = R,R$p

with lR,l

=

1.

(4.84)

Thus from (4.82)we may write the dispersion relation in the form RSS = [ 1 - (2r?PSRPS + ~

+

s S ~ P P ) l / [ ~ S R$Pl S

(4.85)

which begins to resemble the Love wave dispersion relation (4.75). The similarity is enhanced for slownesses well into the evanescent range for P when IRppl,lRpsl are small and diminish with frequency so that asymptotically

Elastic Wave Propagation in StratiJied Media

147

we have Rss = (&)-'

+ O(l/w)

(4.86)

and for p > @;I, & has modulus unity. We have, therefore, just a phase shift from the Love wave relation and so the dispersion curves for the first few Rayleigh wave higher modes are very similar to those for Love wave curves. When both P and S waves can propagate at the surface, the simple character of a surface wave begins to be lost and we must use the full form (4.82).

2. Surface Wavesand Channel Waves We have so far used surface forms for the dispersion relation for the half space, alternative forms may be derived by introducing a notional receiver at some level Z, within the half space. From (4.4) (4.87)

F(O,zL+) = F(O,zJQ(z,,zL+),

and thus, writing FL for the partitions of F(0, zL), we have det F :2 = det { F ; RkL(TRd,)-

+F

:2(

TkL)- '},

(4.88)

in terms of the partitions of the wave propagator Q(z,, zL+). The free surface reflection matrix RLRis defined as (3.78),

RER = -(F!2)-1F;1, and thus (4.88)may be written as det Fk2 = det F52 det(Z - RLRRZL)/detTZL.

(4.89)

Thus we may represent the secular function for the whole half space in terms of the secular function above Z, and a shifted half-space operator. Thus if we choose 2, so that det F:2 and (det TZL)- are both nonzero, we may use the secular equation (Kennett and Kerry, 1979)

A = det [ I - RERRkL]= 0,

(4.90)

for the dispersion and thus generalize (4.74).The restriction on 2, is to ensure that for a particular frequency-slowness pair, surface waves are not possible on the structure above Z, nor channel waves on the region below zR. Channel waves are localized nontrivial solutions of the elastic equations of motion in a stratified full space in which the displacement IwI + 0 as z + & co. Consider the stratified region Z, < z < zL bounded above and below by uniform half spaces, as in our invariant imbedding approach. For a channel wave to exist V(zR-1

=

[VU(zR-),O]T,

V(zLf)

=

[O, VD(zLf)]T,

(4.91)

148

B. L. N . Kennett

and so since V(Z, -) = Q(zR -, zL+)V(z,+), we must require the channel wave secular function Y=detQ,,=(detTRd,)-l = O . (4.92) When this condition is satisfied, then the reflection and transmission coefficients across the region have no meaning. We may once again visualize a receiver at a level zK within the channel, and then factoring the wave propagator Y = det(Z - RERRbL)/det(TER)det(TEL). (4.93) Thus, assuming that the partial transmission terms exist, we may redefine the channel wave secular function as (Kennett and Kerry, 1979)

7 = det(Z - RtRREL)= 0.

(4.94)

This represents a constructive interference condition for waves successively reflected above and below the level zK,and will only be attainable if there is an inversion in the velocity profile and the wavefield is evanescent in both z < ZR and z > ZL. 3. Decomposition of the Surface Wave Secular Function

Dunkin (1965) has demonstrated that at high frequencies in a model composed of a stack of uniform layers the secular function tends to factor into a secular term for the near surface layering and Stoneley functions for the deeper interfaces. In our present treatment we see that the Stoneley functions arise from the denominators of the interface reflection. We may generalize Dunkin’s result to a stratified region by considering the factorization of the secular function. We have already shown that the dispersion is controlled by (4.95) A = det(n: + nERgR)det(Z - RfPRLL) = 0, from (4.89) when the half space is divided at zR. For a velocity profile for P and S waves which is monotonically increasing with depth we may choose zRto lie deep in the evanescent region. In this case IRtL\ will be very small and so (4.95) approximates

A N det(n:

+ n\RgR),

(4.96)

i.e., the secular equation for the truncated structure down to the level zR terminated by a uniform half space in z > z,, with continuity of properties at z = zR. The equivalence of (4.95) and (4.96) enables the technique of “structure reduction” in the calculation of dispersion curves (Schwab and Knopoff, 1972; Kerry, 1980). In this technique the reflectivity of the deep evanescent region is neglected.

Elastic Wave Propagation in Stratijed Media

\

149

I

I

FIG. 10 Decomposition of the surface wave secular function in the presence of a velocity inversion. For a slowness corresponding to the short dashed line S waves are evanescent in the shaded regions, and so the crustal waveguide and the deeper velocity inversion are partially decoupled.

With a velocity inversion in the structure the situation is rather more complex. For slownesses such that the turning point for S waves lies well below the low-velocity channel, a choice of zRin the evanescent regime at the base of the layering will give (4.96) once again. When, however, the slowness is such that there are propagating S waves within the inversion, but evanescent waves in a region outside (Fig. lo), we must be more careful. Consider a level zc lying in the evanescent region for S waves above the low velocity zone, then RgR = Rgc FR RU

- RCR

-

U

+ TCURRgR(Z- RECRgR)-'TgR CRRFC +TD U(

- RCRRFC D

U)-

1 CR

(4.97)

TU.

In each of these expressions we have both upward and downward transmission through the evanescent region contained in the matrices TER, TER. At moderate frequencies these transmission terms will be quite small and diminish rapidly as the frequency increases. We thus have the high frequency approximations RgR N Rgc,

RER N RSR,

(4.98)

and so in this limit we have

A

31

det(n,

+ nuRgc) det(Z - RERRbL),

(4.99)

which is the product of the dispersion equation for surface waves on the structure above z = zc, det(n,,

+ n,Rgc)

= 0,

(4.100)

and the dispersion equation for channel waves on the structure below z = zc, det(2 - RERRLL)= 0.

(4.101)

150

B. L. N . Kennett

At intermediate frequencies there will coupling between the channel and the near surface through the transmission matrices TgR, TER.A given surface wave mode will in certain frequency ranges be mainly confined to the near surface region, and then (4.100)will be approximately satisfied, and in others it will be mostly a channel wave when (4.101) is approximately satisfied (Frantsuzova et al., 1972; Panza et al., 1972). 4. Computation of Surface Wave Dispersion The reflection matrix representation of the secular equation for surface waves presented in the previous sections based on the work of Kennett (1974),and Kennett and Kerry (1979) can be used as the basis of an efficient scheme for calculating surface wave dispersion, as shown by Kerry (1981). For a model composed of uniform layers there are considerable advantages in fixing slowness p and varying frequency co when seeking for roots of the secular equation. We may first make use of the frequency independence of the interface reflection and transmission coefficients to reduce the computational effort required to calculate the reflection matrices required in the secular function (4.70).Second, the zeroes of the secular function are approximately evenly spaced in frequency o,whereas the spacing in slowness p is rather irregular. This approach is well suited to finding the dispersion curves for a large number of higher mode surface waves. Conventional calculations with fixed frequency and varying slowness have typically been used for a limited number of modes (e.g., Schwab and Knopoff, 1972). With a stored set of interface coefficients we use the recursive relations for the reflection matrices [(3.90) and (3.91)] and so introduce the frequency dependent phase terms in each layer, and thus calculate the downward reflectionmatrix at the surface RgL and det 7’:. We then evaluate the secular function and use quadratic interpolation to iteratively refine an estimate of the frequency of the root. A suitable termination procedure for the iteration (Kerry, 1980)is when two estimates of the secular function with opposite sign are less than some threshold and the smallest eigenvector of (n, + n,RgL) evaluated at the estimated root is also less than a preassigned threshold. This procedure works very well most of the time, but when we are close to becomes nearly singular a root of the channel wave function ( I - eRRF:L) and it is very difficult to get accurate numerical evaluation of REL with finite accuracy arithmetic. To test for this possibility a “receiver” level in the lowvelocity zone is chosen and in a single pass through the structure REL, REL, det(TgL),det(TiL),and det(Tp) are calculated. We test the quantity q = det(Z -

RERRP) = det(TgR)det(TEL)/det(TgL).

If q is large, then we may use the secular function (4.70). If, however, q is small, we have a channel mode and then we calculate downward from the

151

Elastic Wave Propagation in Stratijied Media

0

1

0

,

,

I

~

I

I

200

400

,

I

600 Depth (km)

l

,

2

~

~

~

1

~

3

~

4

~

1

1

800

1000

~

~

'

5

~

~

~

6

w (radls)

FIG.11 The velocity model T7 (Burdick and Helmberger, 1978) after "earth flattening" and a portion of the corresponding dispersion curves for frequencies up to 1 Hz. For p > 0.22 s/km the dispersion arises in the crustal waveguide, but for smaller slownesses, cear horizonatal portions of the dispersion curves occur corresponding to propagation in the velocity inversion.

surface to find RER.We then terminate the iteration by requiring the smallest eigenvalue of ( I - RERRRtL)to be less than our threshold. Once all the roots in a given frequency band are found, the slowness is incremented by a small amount and the new interface coefficients calculated. A fine illustration of the success of this technique is provided by Fig. 11, which illustrates all the higher mode branches with frequencies between 0.01 and 1 Hz and phase velocities between 4.2 and 5.2 km/s for a recently proposed continental upper mantle structure (T7-Burdick and Helmberger,

~

~

.

~

B. L. N . Kennett

152

1978). The portions of the modes with strong dispersion correspond to near surface propagation and those with hardly any dispersion to channel modes. V. Inversion of the Transforms

In Section IV we have shown how the displacement scalars at the surface of the half space, as a function of frequency, slowness and azimuthal order can be represented in terms of the nature of the source and the propagational characteristics of the half space, as represented by reflection and transmission matrices. In this section we will discuss the inversion of the transforms to give time series for the displacement at positions on the surface of the half space. With our assumption of a point source the azimuthal summation is restricted to angular orders Iml 5 2, and this presents no complication once the integrals over frequency and slowness have been performed for each m. We will, therefore, take as an example the case of azimuthal symmetry and consider the construction of the vertical component of the surface displacement,which by (2.19) is given by

woz(r,t) = (274-lm: J

dwe-’”* JOm dkkU,(k,co)J,(kr).

(5.1)

In terms of the horizontal slowness p , (5.1) takes the form

J:

woz(i-,t ) = ( 2 7 ~ ) ~ ’ dww2e-iW‘So”dppU,(p,co)J,(opr), 00

(5.2)

for real w. We now have a choice of the order in which the slowness and frequency integrals are undertaken. If the slowness integral is calculated first, then the intermediate result is the complex frequency spectrum iToz(r,o)at a particular location, and so this approach may be designated the spectral method. When alternatively the frequency integral is evaluated first, we have a final integral over slowness and we follow Chapman (1978) by calling this approach the slowness method. Within each of these broad classes a variety of techniques are used to evaluate the slowness integrals.

A. THESPECTRAL METHOD We here consider the construction of the spectrum of a seismogram in terms of the slowness integral

Elastic Wave Propagation in StratiJied Media

153

and the character of the half space contribution will determine the nature of evaluation scheme. The symmetry properties of U,(p, o)enable us to recast (5.3) in terms of the outgoing Hankel function Hb’)(opr) (Hudson, 1969b), rather than the standing wave representation in terms of J , ( o p r ) . Thus, Roz(r,4 =

)ol~lJ:

W

~PPUO(P, o)Hbl’(wpr),

(5.4)

taking the contour above the branch point at the origin, for large values of the argument Hb’fopr) may be replaced by its asymptotic form Hb”(wpr)

-

(2/7cwpr)”’ exp{i(opr - 7c/4)}.

(5.5)

If U,(p, o)represents the full response of an elastic half space, we have rather different problems from the case where U,(p, w ) is an approximation designed to give a good representation of an individual seismic phase. 1. The Full Response of a Haw Space

Consider initially the full half space response for a surface source which by (4.9) may be written as W,(p, o)= (m,

+ muRKL)(n, + nuREL)-‘S,

- Sw .

(5.6)

The boundary condition of only outgoing or decaying evanescent waves in the lower half space imposed the conditions Im(oq,,) 2 0, Im(oqsL)2 0 and so we have branch points in W, and hence U , at Ip( = a;’, IpI = /I;’. In terms of the propagator solution, these branch points may be seen to arise from the presence of D ( z L + ) in (4.4) F(0, ZL) = P(0, Z

d W L

+)

(5.7)

and the minors of F enter into the expression for (5.6). No other branch points are present when the free surface is taken fully into account since for a uniform layer P(zj, z j - is symmetric in qaj, qaj and this property will transfer to the continuous case. However, we do have additional singularities as we have seen in Section IV,E for slownesses in the range /I; < Ip( < ,’;I/ where Fminis the smallest shear wave speed anywhere in the half space (this is normally attained at the surface). In this slowness interval we have higher mode Rayleigh wave poles on the real p axis and for the fundamental Rayleigh mode the limit point is p = pRO.The distribution of poles will depend on the frequency being considered. We thus have the set of singularities sketched in Fig. 12, and the contour of integration for o > 0 runs just below the singularities for p > 0 and just above for p < 0. This may be justified by including some slight attenuation in the medium (as is physically reasonable) in which case the poles move into the first and third quadrants of the complex p plane. The line of the

B. L. N . Kennetl

154

Re P

p

plane

FIG. 12 Representation of singularities, branch cuts, and integration contours for the full response of a half space.

branch cuts from M ; * , pi' is not critical provided that the conditions Im(oq,,) 2 0, Im(wqsL)2 0 are maintained along the real p axis. We, therefore, follow Lamb (1904) by taking cuts parallel to the imaginary p axis. The most direct attack on the evaluation of (5.3) would be to perform numerical integration along the real axis, but the presence of the poles is a major obstacle to such an approach. One possibility would be to deform the contour in (5.4) to C (Fig. 12)and so pick up polar residue contributions from all the poles which lie to the right of and add to the real axis integral up to p; a line segment off into the first quadrant, where Ht)(wpv)is a decaying function of complex p . At even moderate frequencies the number of modal residue contributions to be evaluated becomes very large indeed (cf. Fig. ll), and locating all the poles is a major computational problem. The first few modal residues ( p ?: ') give the major contribution to what would generally be regarded as the surface wave train with relatively low group velocities. The summation of the modes with smaller slownesses just synthesizes S body wave phases by modal interference (see, e.g., Nolet and Kennett, 1978). The addition of modal residue contributions from a portion of the real p axis, neglecting the remainder of the contour can give good results for the S wave coda (Kerry, 1980)(Fig. 13). An alternative to evaluating the contour integral plus residue contribution described above is to arrange to move the poles off the contour of integration. As we have noted this may be achieved by introducing physical attenuation. The modification to our calculation is that now the velocity distribution must be taken to be complex (Schwab and Knopoff, 1972; Kennett, 1975). Thus allowing for the velocity dispersion required to achieve a causal attenuation behavior we have, e.g.,

'

'

Elastic Wave Propagation in Stratijied Media

I

200

LOO

600

BOO

1000

1200

l

155

l

lLOO

1600

X (km)

Source Depth 60km FIG.13 Vertical component S wave seismograms calculated by superposition of higher mode Rayleigh wave pole contribution. The seismograms show distinct surface generated S multiples and a clear late fundamental Rayleigh mode (Ro).

(O’Neill and Hill, 1979) where a,(z), Q,, are the P wave velocity and quality factor at the reference frequency 0,. Q, is found to be virtually independent of frequency so that for low loss media, i.e., Q, large, and a moderate band of frequencies around the reference level, the frequency dependence of the complex velocities are weak. In this case a satisfactory approximation is provided by -

a(z) 2: a(z)(l

+ i sgn w/2Q,).

(5.9) If Q, is small, i.e., Q, 5 100, the frequency correction to the velocities is necessary (O’Neilland Hill, 1979).Kennett (1980)discusseshow the frequency dependence of the velocities may be incorporated into the computation scheme without excessive cost. With the complex velocity distribution we may calculate reflection and transmission matrices as before and so calculate the surface response. Now, however, if we try to integrate along the real p axis the poles have moved off the contour although their influence is strongly felt on the axis. Kennett (1980) has used the original form (5.3)for the slowness integral, in a weakly attenuative medium, with a Chebyshev approximation for the Bessel function J,(opr) and an arbitrary moment tensor source. The integration is performed over panels in slowness using the trapezium rule with spacing matched to the behavior of the integrand. The integration is truncated at a slowness of (0.85/?,)- well beyond the surface Rayleigh slowness. The oscillatory nature of the Bessel function J , ( o p r ) provides a limitation on the

’,

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B. L. N . Kennett

1

1

r

S

I

I

100

F

1 200

X (km)

FIG. 14 Three component sets of theoretical seismograms from a double couple source at a depth of 4 km in a simple crustal model. The P and S wave trains are accompanied at large distances by phases P,, S. traveling in the higher velocity material beneath the crust. Clear Rayleigh waves (R) are seen on vertical and radial components (2,R ) and Love waves (L) on the transverse component (T).

range r of useful calculation for a fixed upper frequency limit. Nevertheless, good results may be obtained and Fig. 14 illustrates well-developed P,S and dispersive Rayleigh wave trains from a shallow source in a simple crustal model. Kind (1978) has considered the full response of an attenuative medium to a vertical force, but has taken the asymptotic form of the Hankel function in (5.4)so excluding near field effects, and has also neglected the contribution from negative slownesses. An advantage of these procedures is that once U,(p, w ) has been calculated each seismogram requires only numerical integration for the appropriate range.

2. Rejlectivity Approximations For all approximations to the half-space response which retain a finite number of free surface reflections, the singularities in the complex p plane are markedly altered from the preceding case. There are now no poles and instead we have, at least, branch points at the surface P and S wave slow-

157

Elastic Wave Propagation in Stratified Media

’,

In the original reflectivity method (Fuchs and nesses, i.e., lpl = a; ( p ( = Muller, 1971)only primary transmission effects are included down to the top of the reflection zone ( z > z J ) and so branch points arise at the P and S wave slownesses for each of the overlying layers. The absence of poles means that numerical integration along the real p axis in (5.3) or (5.4) presents no difficulty. Fuchs and Muller (1971) mapped (5.4) from slowness p to the angle of incidence 9 for the wave type of interest at the top of the reflection zone. They then used a trapezium rule and restricted the integration to a limited range of real angles. Injudicious choice of the integration interval can give quite large numerical arrivals at the limiting slownesses, these may be muted by tapering the behavior of V,(p,w) near the limits. ‘For many problems where there is a relatively low-velocity overburden of reasonable thickness the exclusion of evanescent waves in the overburden is satisfactory since all the arrivals of interest occur in a limited range of slowness. Such is, for example, the case for the compressional arrivals returned from the structure beneath the sea bottom in the deep ocean (Orcutt et al., 1976).When, however, S waves are of particular interest it is probably most convenient to integrate directly in terms of slowness and so include the contribution from evanescent waves above the reflection zone. The oscillatory nature of the Bessel functions poses numerical problems. These can sometimes be alleviated by using a modification of Filon’s (1928) method to handle the asymptotic phase behavior of the Bessel function when U,(p, w ) is slowly varying. As in the full response physical attenuation may be included by taking a complex velocity model (Kennett, 1975). In Fig. 15 we show an example of the successful use of the reflectivity method in matching the character of theoretical seismograms to experimental records from a marine refraction seismic survey on the Reykjanes Ridge, southwest of Iceland (Bunch and Kennett, 1980).

P (km/r/

LO

60

T

2

FIG. 15 Comparison of experimental and theoretical seismograms for a marine seismic refraction line on the Reykjanes Ridge (Bunch and Kennett, 1980).

B. L. N . Kennett

158

3. Complex Slowness Contours

When attenuation is concentrated on the displacement due to a particular phase on the recorded seismograms, then the properties of U,(p, o)become simpler and there are fewer interference effects. In these circumstances it is advantageous to deform the contour in the p plane so that it roughly corresponds to the steepest descent path for the phase under consideration. For a single geometric ray between source and receiver the path crosses a saddle on the real p axis at the geometric slowness at an angle of k n/4.The sense of travel is determined by the nature of the ray. If the range increases with slowness then we take the positive sign. When the range decreases with slowness, as in a reflection, then the negative sign is appropriate. When a number of raypaths arrive at a receiver location, as, for example, in a triplication in the travel time curve caused by a strong velocity gradient then we have a number of closely spaced saddles. For numerical calculation if the contour roughly follows the steepest descent path the segments off the real axis move into regions of rapid decay of the integrand so that the convergence of the integral is improved. For example, within a triplication with three saddles, the path shown in Fig. 16 would be useable for a range of distances. The characteristics of the saddle points vary with range so that the contour must be chosen to match the behavior and singularities of the integrand, since discontinuities in elastic parameters and parameter gradients introduce strings of poles into the complex p plane (see, e.g., Chapman and Phinney, 1972; Cormier and Richards, 1977). The contour calculations have been successfully used, with full numerical solutions of the governing equations (2.13)to find U,(p, o), by Chapman and Phinney (1972). More recently, this approach has been used with considerable success in conjunction with the Langer uniform asymptotic approximation for calculating reflection properties in an inhomogeneous medium (Cormier and Richards, 1977; Choy, 1977). A disadvantage of this method is that the integrand is required for complex values of p and the arguments

C

11

u

II

FIG. 16 Contour in complex p plane

Elastic Wave Propagation in Stratified Media

159

of the functions in the Langer treatment (Richards, 1976) require analytic continuation of the velocity distribution which must be rather carefully performed. However, attenuative effects may be directly included. Although the ideal choice for the contour of integration varies with range, good results can be obtained by using the same contour for a span of receiver positions so that U,(p, o)along the contour has only to be evaluated once. 4. Inversion of the Frequency Transform

The commonest method used to perform the final frequency integral is to use a fast Fourier transform (Cooley and Tukey, 1965) over a set of discrete frequencies. The finite bandwith of practical recording equipment sets an upper limit on frequency and this must be taken below the Nyquist frequency for the transform. The time series obtained after transformation is of fixed length and is cyclic in nature. There is, therefore, the possibility of time “aliasing,” energy which would arrive after the end of the allotted time interval is wrapped back over the early part of the series. When significant reverberation is possible in the seismograms, as, for example, when all free surface reflections are included, a very long time interval must be used with the consequence of fine frequency sampling. The inclusion of physical attenuation helps by damping reverberation in the structure. In order to follow arrivals at varying ranges with a fixed time interval, it is convenient to calculate Woz(r,t - PRr) by multiplying the spectrum at r by eWiopRr (Fuchs and Muller, 1971). The reduction slowness pR is chosen to confine the arrivals most conveniently, and so if at each frequency the slowness integral is split into a number of parts, a different reduction slowness may be used to compute a time series for each part. The final seismogram may then be obtained by superposition with appropriate time delays (Kennett, 1980). When only a limited duration pulse is to be calculated,it is computationally efficient to interpolate the amplitude and phase behaviour of the spectrum. This ?equires sorting out the 2n radian ambiguities in the phase spectrum and is discussed in detail by Choy (1977). B. THESLOWNESS METHOD 1. Convolution Representation of the Surface Response We consider now the construction of an intermediate representation for the surface displacement in time-slowness space by evaluating the frequency integral, at a specified range. ~ , ~ (r,pt ), = (2z)-1m: J

d o e - i ” W J o( oP r)Uo(P,o).

(5.10)

B. L . N , Kennett

160

This transform of a product may be expressed in terms of a convolution as (Chapman, 1978) @-oz(P, r, t ) = -n-’ 8t,{i70(p,t) * [ ~ ( t pr) - H ( t - pr)](p’r’ - t ’ ) - ~ / ~ ) ,

+

=

-nu’ a,,

J:‘,

dmb,(p, t - m)(p2r2 - m’)-’/’

(5.11)

where Do@, t ) is the inverse Fourier transform with respect to frequency of U,( p , w). The original frequency integration is thus replaced by calculating the inverse transform Do@, t) and a convolution, albeit over a finite interval. Once @,,(p, r, t) has been calculated, then the displacement may be found from the integral woz(r, t ) = Jom dPP@O,(P? r, 0,

(5.12)

and if we use the form (5.11) for the intermediate result, wOz(r,t)= -n-‘8,,

JOm

dpp

-pr

dmo,(p, t

- m)(pzr2 - m2)-112. (5.13)

When the portion of the seismogram which is of interest is composed of relatively low-frequency arrivals or when a long time duration is required at considerable range, as in the synthesis of surface wave trains, it is probably most convenient to evaluate fioz(p,r, t ) from (5.10). For relatively high-frequency arrivals we may approximate the convolution kernel in (5.11) by the contribution from the two integrable singularities (Chapman, 1978) [H(t

+ pr) - H(t - pr)J(p2r2- tz)’/’ N_

(2pr)-’/’(H(pr - t)(pr - t)-’/’

+ Hft + pr)(pr + t)”’}.

(5.14)

With this approximation the vertical displacement (5.13) becomes

(5.15)

When there is a common time dependence s(t), for all the components of the moment tensor defining the source, so that fio(p,t ) = s ( t ) * O0(p,t), the double integral for the displacement may be written as

Elastic Wave Propagation in StratiJied Media

161

where o o ( p , t) is the Hilbert transform of o o ( p , t) and o(t) =

Ji dld,,s(l)H(t - l)(t- l)-’”

(5.17)

may be regarded as the “effective” source function (Chapman, 1978). In general, the main contribution in (5.16) comes from o o ( p , t - pr) and the incoming part o o ( p , t + pr) can be neglected. The approximation (5.14) is equivalent to taking asymptotic forms (5.5) for the Hankel functions-a “plane wave” expansion. The representation (5.16) is currently of greatest utility when U,(p, w ) has a simple form so that Oo(p,t) may be easily calculated. However, if the portion of the response represented by Uo(p,w ) does not include strong reverberation, a direct Fourier inversion (e.g., by an FFT) could be used to construct o o ( p , t) and then we find (5.18) by numerical integration. A convolution with the “effective” source may then be performed at leisure :

Lmdma(t

woz(r,t ) 2 - 7 ~ - ’ ( 2 r ) - ’ ’ ~

- m)V(r,m).

(5.19)

This procedure is under active development. 2. Generalized Ray Theory Simplifications of the expressions for the surface displacement only emerge when U,(p, w ) has particular forms. Thus if U,(p, w ) = s(w)G(p)eiWr(”),

(5.20)

we may construct useful displacement representations using either a complex p contour (Cagniard, 1939) or an integral along the real p axis (Chapman, 1978). Each of these approaches may be used for a piecewise continuous medium by representing the full response as a sum of “generalized rays” so that Uo(p,w ) = s(o)

1 GI(p)eiorr(P),

(5.21)

I

where G,(p)represents the product of reflection and transmission coefficients on a particular passage path through the medium, characterized by phase delay zI(p).The displacement wz(r,t) is then approximated by a finite subset of the infinite expansion (5.21).Efficient combinatorial techniques for choosing such a set have been given by Hron (1972). The application of Cagniard’s method to a medium composed of a stack of uniform layers was first made by Helmberger (1968),and Chapman (1976) has shown how this approach may be extended to smoothly varying media.

B. L. N . Kennett

162

a. The Cagniard Method We consider the inversion of an individual "generalized ray" contribution, having abstracted the source time function. Thus we construct W(r,t ) = (271)-' = (4n)-'

Somd pmp: J JOm

eiwr(p)

dpp

doe-i"'G")eiw''P'J,(wpr),

:j

m

doe-'"'G(p)

{Hb"(opr) - Hbl'( - o p r ) } .

(5.22)

We now consider separately the two parts depending on the Hankel functions Hb''(wpr) and Hb"( - o p r ) . For the first part we distort the p contour to the Cagniard path r such that Im{z(p) + Pr}

(5.23)

= 0,

+

which lies in the fourth quadrant and quits the real axis when z(p) pr is equal to the geometric ray travel time. Then we may invert the Fourier transform to give

x [(t - z(p))' - p2rZ]-1'2.

The second part is to be taken over the mirror image of and so using the Schwarz reflection principle

(5.24)

r in the real axis

W-(r,t ) = -(W+(r,t))*.

(5.25)

The generalized ray contribution is thus given by W(r,t ) = 71-l Im

Jr dppG(p)H(t - 9)[(t - 9)(t - 9 + 2pr)]-';',

(5.26)

where we have introduced the time variable

9 = T ( P ) + Pr,

(5.27)

so that the contour r is specified by Im 9 = 0. We may cast (5.26) into a convolution integral in time by changing the variable of integration to 9 W(r,t ) = z - ' Im

J:o,dSp(dp/dS)G(p)[(t - 9)(t - 3 + 2pr)]-'",

(5.28)

where p is an implicit function of 9. If we wish to reinstate the source time function, we must convolve W(r,t)with &s(t). The detailed behavior of (5.28) depends on the nature of the reflection and transmission product G(p) and the phase delay z(p). A survey of the application of (5.28) and its extensions has been given by Pao and Gajewski (1977),and they also discuss numerical implementation. Alternative numerical schemes are presented by Wiggins and Helmberger (1974), and Ben-Menahem and Vered (1973).

Elastic Wave Propagation in Stratlfed Media

163

For all but the closest range the early part of the generalized ray arrival can be adequately modeled by replacing (t - 9 + 2pr)’I2 in (5.28)by ( 2 ~ r ) ” ~ . If we include the source time function, the resulting approximation for the generalized ray is W(r,t)= 71-l

Jam d9a(t - 9) Im {(dp/dL9G(p)(p/2r)”2}

(5.29)

with the “effective” source function a(t) defined as in (5.17). Equation (5.29) provides a convenient means of calculating a generalized ray response, but the whole of this development depends on the medium being perfectly elastic. An empirical allowance has, therefore, been made by some authors to try to account for the effect of attenuation in the medium (see, e.g., Burdick and Helmberger, 1978).

6. The Chapman Method If we take the outgoing part of the expression (5.16), we find that the displacement contribution for an individual generalized ray (5.20) is given by W(r,t)= - a ( t )

* n - 2 Jom

d p ~ ”Re[G(p)/(t ~ - 9(p,x))].

(5.30)

Chapman (1978) has shown that, if attention is restricted to values of p on the real axis such that z ( p ) is real, an effective approximation to (5.30) is

Wk, t ) =

-4,m * Im[L(t) *

1 G(P)(dP/dS)I

(5.31)

1=9

where L(t)is the analytic time function L(t)= H(t)t-’i2 + iH(-t)(-t)-1’2.

(5.32)

The sum in (5.31) is over all values of 9 for which t

= 9 = z(p)

+ pr.

(5.33)

To avoid numerical complications care is needed in calculating the convolutions and smoothing over a short interval At removes artificial singularities. The approximation (5.32) may be used for a turning ray for which the Cagniard method fails and gives good results in many other cases. Once again (5.32) is only valid for a perfectly elastic medium, but here (5.16) could be used for a numerical solution in an attenuative medium. VI. Conclusion We have shown how the excitation of elastic waves within a horizontally stratified structure can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the

164

B. L. N . Kennett

full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although our development has been for isotropic media nearly all the results apply directly to the case of full anisotropy if 3 x 3 reflection and transmission matrices allowing coupling between all wave types are employed. Indeed our development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation (e.g., radio waves in an anisotropic ionosphere). The reflection techniques are also not restricted to horizontal stratification since they may also be used for spherical stratification. Here Legendre transformation replaces the Hankel transform and the slowness p = (1 + $)/am in terms of the angular order I for a sphere of radius a. The basis functions for a uniform layer are now the spherical Hankel functions rather than the exponentials, but an analog of the recursive scheme for the reflection matrices may be constructed for models composed of a sequence of uniform spherical shells. The Poisson summation formulation enables the summation over Legendre functions to be mapped into an integral and the inversion of the transforms proceeds in similar ways to those discussed above.

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Chapman, C. H. (1974). Generalized ray theory for an inhomogeneous medium. Geophys. J . R. Astron. Soc. 36, 673-704. Chapman, C. H. (1976). Exact and approximate generalized ray theory in vertically inhomo46, 201-234. geneous media. Geophys. J . R. Astron. SOC. Chapman, C. H. (1978). A new method for computing synthetic seismograms. Geophys. J . R . Astron. SOC.54, 481-518. Chapman, C.H., and Phinney, R. A. (1972). Diffracted seismic signals. Methods Comput. Phys. 12, 166-230. Choy, G. L. (1977). Theoretical seismograms of core phases calculated by frequency-dependent full wave theory, and their interpretation. Geophys. J . R. Astron. Soc. 51, 275-312. Cisternas, A,, Betancourt, O., and Leiva A. (1973). Body waves in a ”real Earth.” BUN.Seismol. Soc. Am. 63, 145-156. Cooley, J. W., and Tukey, S. W. (1965). An algorithm for the machine calculation of complex Fourier series. Math. Compur. 19, 297--301, Cormier, V. F., and Richards, P. G. (1977). Full wave theory applied to a discontinuous velocity increase: The inner core boundary. J . Geophys. 43, 3-31. Dunkin, J. W. (1965). Computations of modal solutions in layered elastic media at high frequencies. Bull. Seismol. Soc. Am. 55, 335 -358. Ewing, W. M., Jardetsky, W. S . , and Press F. (1957). “Elastic Waves in Layered Media.” McGraw-Hill, New York. Faber, S., and Muller, G. (1980). Sp phases from the transition zone between the upper and lower Am. 487-508. mantle. Bull. Seismol. SOC. Filon, L. N. G. (1928). On a quadrature lbrmula for trigonometric integrals. Proc. R. Soc. Edinburgh 49, 38-47. Fowler, C. M. R. (1976). Crustal structure of the Mid-Atlantic ridge crest at 37”N. Geophys. J. R. Astron. Soc. 41, 459-492. Frantsuzova, V. I., Levshin, A. L., and Shkadinskaya, G. V. (1972). Higher modes of Rayleigh waves and upper mantle structure. In “Computational Seismology” (V. I. Keilis-Borok, ed.), pp. 93-100. Consultants Bureau, New York. Frazer, R. A,, Duncan, W. J., and Collar, A. R. (1938). “Elementary Matrices.” Cambridge Univ. Press, London and New York. Fuchs, K. (1968). Das Reflexions-und Transmissionsvermogen eines geschichteten Mediums mit belieber Tiefen-Verteilung der elastischen Moduln und der Dichte fur schragen Einfall ebener Wellen. Z . Geophys. 34,389-411. Fuchs, K. (1975).Synthetic seismograms of PS-reflections from transition zones computed with the reflectivity method. J . Geophys. 41, 445-462. Fuchs, K., and Muller, G. (1971). Computation of synthetic seismograms with the reflectivity method and comparison with observations. Geophys. J . R . Astron. Soc. 23, 417-433. Gilbert, F. (1964). Propagation of transient leaking modes in a stratified elastic waveguide. Rev. Geophys. 2, 123-153. Gilbert, F. (1971). The excitation of the normal modes of the Earth by earthquake sources. Geophys. J . R. Astron. Soc. 22, 223-226. Gilbert, F., and Backus, G. E. (1966). Propagator matrices in elastic wave and vibration problems. Geophysics 31, 326-332. Green, G . (1838). On the laws of reflexion and refraction of light at the common surface of two non-crystallized media. Trans. Cambridge Philos. Soc. I, 245-285. Harkrider, D. G. (1964). Surface waves in multi-layered elastic media. 1. Rayleigh and Love waves from buried sources in a multilayered half space. Bull. Seismol. Soc. Am. 54,627-679. Haskell, N. A. (1953). The dispersion of surface waves on multilayered media. Bull. Seismol. Soc. Am. 43. 17-34.

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Haskell, N. A. (1964). Radiation pattern of surface waves from point sources in a multi-layered medium. Bull. Seismol. Soc. Am. 54, 377-393. Helniberger, D. V. (1968).Thecrust-mantle transition in the Bering Sea. Bull. Seismol. Soc. Am. 58. 179-214. Hron, F. (1972).Numerical methods of ray generation in multilayered media. Methoak Comput. Phys. 12, 1-34. Hudson, J. A. (1969a). A quantitative evaluation of seismic signals at teleseismic distances. I. Radiation from a point source. Geophys. J. R. Astron. Soc. 18, 233-249. Hudson, J. A. (1969b). A quantitative evaluation of seismic signals at teleseismic distances. 11. Body waves and surface waves from an extended source. Geophys. J . R. Astron. SOC.18. 353-370. Jeans, J . H. (1923). The propagation of earthquake waves. Proc. R. Soc. London, Ser. A 102, 554-574. Jeffreys, H . (1928). The effect on Love waves of heterogeneity in the lower layer. Mon. N o / . R . Astron. Soc., Geophys. Suppl. 2, 101-111. Jeffreys, H., and Bullen, K. E. (1940). “Seismological Tables.” Br. Assoc., Gray-Milne Trust. Kennett, B. L. N. (1974). Reflections, rays and reverberations. Bull. Seismol. Soc. Am. 64, 1685 1696. Kennett, B. L. N. (1975). The effects of attenuation on seismograms. Bull. Seismol. Soc. Am. 65, 1643-1651. Kennett, B. L. N. (1979a). Theoretical reflection seismograms for an elastic medium. Geophys. Prosp. 27, 301-321. Kennett, B. L. N. (1979b). The suppression of surface multiples on seismic records. Geophys. Prospect. 27, 584-600. Kennett, B. L. N. (1980). Seismic waves in a stratified half space. 11. Theoretical seismograms. Geophys. J . R. Astron. SOC. 61, 1-10, Kennett, B. L. N., and Illingworth, M. R. (1981). Seismic waves in a stratified half space. 111. Piecewise smooth models. Geophys. J . R. Astron. Soc. (in press). Kennett, B. L. N., and Kerry, N. J. (1979). Seismic waves in a stratified half space. Geophys. J . R. Astron. Soc. 57, 557-583. Kennett, B. L. N., and Simons, R.S. (1 976).An implosive precursor to the Columbia earthquake 1970 July 31. Geophys. J . R. Asrron. Soc. 44,471-482. Kennett, B. L. N., Kerry, N. J., and Woodhouse, 1. H. (1978). Symmetries in the reflection and transmission of elastic waves. Geophys. J . R. Astron. Soc. 52, 215-229. Kerry, N. J. (1981). The synthesis of seismic surface waves. Geopkys. J . R. Astron. Soc. 64, 425-446. Kind, R. (1976). Computation of reflection coefficients for layered media. J . Geophys. 42, 191-200. Kind, R. (1978). The reflectivity method for a buried source. J. Geophys. 44, 603-612. Knopoff, L., Schwab, F., and Kausel, E. (1973). Interpretation of Lg. Geophys. J . R. Astron. SOC.33, 389-404. Knott, C. G . (1 899). Reflection and refraction of elastic waves, with seismological applications. Philos. May. [ 5 ] 48,64-97,567-569. Lamb, H. (1904). On the propagation of tremors over the surface of an elastic solid. Philos. Trans, R. Soc. London, Ser. A 203, 1-42. Langer, R. E. (1937). On the connection formulas and the solution of the wave equation. Phys. Rev. 51,669-676. Lapwood, E. R. (1948). The disturbance due to a line source in a semi-infinite elastic medium. Philos. Trans. R. Soc. London, Ser. A 242,63-100. -

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Love, A. E. H. (1903). The propagation of wave motion in an isotropic solid medium, Proc. London Math. Soc. 1, 291-316. Love, A. E. H. (1911). “Some Problems of Geodynamics.” Cambridge Univ. Press, London and New York. Molotkov, L. A. (1961). On the propagation of elastic waves in media consisting of thin plane parallel layers. “Problems in the Theory of Seismic Wave Propagation,” pp. 240 - 281, Nauka, Leningrad (in Russian). Miiller, G. (1970). Exact ray theory and its application to the reflection of elastic waves from vertically inhomogeneous media. Geophys. J . R. Astron. SOC.21, 261-283. Miiller, G., and Kind, R. (1976). Observed and computed seismogram sections for the whole earth. Geophys. 1. R. Astron. Soc. 44,699-716. Nolet, G., and Kennett, B. (1978). Normal-mode representations of multiple-ray reflections in a spherical earth. Geophys. J . R. Astron. Soc. 53,219-226. O’Neill, M. E., and Hill, D. P. (1979). Causal absorption: Its effect on synthetic seismograms computed by the reflectivity method. Bull. Seismol. Soc. Am. 69, 17-26. Orcutt, J. A., Kennett, B. L. N., and Dorman L. M. (1976). Structure of the East Pacific Rise from an Ocean bottom seismometer survey. Geophys. J. R. Astron. SOC.45, 305-320. Panza, G. F., Schwab, F. A., and Knopoff, L. (1972). Channel and crustal Rayleigh waves. Geophys. J . R. Astron. SOC.30, 273-280. Pao, Y.-H., and Gajewski, R. R. (1977). The generalised ray theory and transient responses of layered elastic solids. Phys. Acoust. 13, 183-265. Rayleigh, Lord (1885). On waves propagated along the plane surface of an elastic solid. Proc. London Math. SOC.17, 4-11. Richards, P. G. (1976). On the adequacy of plane wave reflection/transmission coefficients in the analysis of seismic body waves. Bull. Seismol. SOC.Am. 66,701-717. Schwab, F., and Knopoff, L. (1972). Fast surface wave and free mode computations. Methods Comput. Phys. 11, 87-180. Sezawa, K. (1935). Love waves generated from a source of a certain depth. BUN. Eurthquake Res. Inst., Univ. Tokyo 13, 1-17. Stephen, R. A. (1977). Synthetic seismograms for the case of the receiver within the reflectivity zone. Geophys. J . R. Astron. SOC.51, 169-182. Stokes, G . G. (1849). Dynamical theory of diiraction. Trans. Cambridge Phiios. Soc. 9. Stoneley, R. (1924). Elastic waves at the surface of separation of two solids. Proc. R. SOC. London, Ser. A 106, 416-420. Takeuchi, H., and Saito, M. (1972). Seismic surface waves. Methods Compur. Phys. 11, 21 7-295. Thomson, W. T. (1950). Transmission of elastic waves through a stratified solid medium. J . Appl. Phys. 21, 89-93. Wiggins, R. A., and Helmberger, D. V. (1974). Synthetic seismogram computation by expansion in generalised rays. Geophys. J . R. Astron. SOC.37, 73--90. Wing, G. M. (1962). “An Introduction to Transport Theory.” Wiley, New York. Woodhouse, J. H. (1974). Surface waves in a laterally varying layered structure. Geophys. J. R. Astron. Soc. 37, 461-490. Woodhouse, J. H. (1978). Asymptotic results for elastodynamic propagator matrices in plane stratified and spherically stratified earth models. Geophys. J . R. Astron. SOC.54, 263-280. Zoeppritz, K. (1919). Erdbebenwellen VIIIB : Uber Reflexion und Durchgang seismicher Wellen durch Unstetigkeitsflachen. Goetringer Nachr. 1, 66-84.

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ADVANCES IN APPLIED MECHANICS. VOLUME

21

Elastic Behavior of Composite Materials : Theoretical Foundations L . J . WALPOLE School of Mathematics and Physics University of East Anglia Norwich. England

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Preliminary Analysis of Tensors and Elastic Behavior . . . . . . . . . . . . . . . . . A. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Decomposition: Geometric Symmetry . . . . . . . . . . . . . . . . . . . . . . . . C . Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Surface Decomposition of the Constitutive Law . . . . . . . . . . . . . . . . . . E . Interfacial Stress and Strain Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 111. The Elastic Field of an Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Transformed Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Homogeneous Inclusion: Ellipsoidal Shape . . . . . . . . . . . . . . . . . . . . . C . Inhomogeneous Ellipsoidal Inclusion . . . . . . . . . . . . . . . . . . . . . . . . IV . The Elastic Field of a Composite Body . . . . . . . . . . . . . . . . . . . . . . . . . A . Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. The Overall Elastic Behavior of a Composite Body . . . . . . . . . . . . . . . . . . A . Overall Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Self-Consistent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Isotropic Phases with Common Shear Modulus . . . . . . . . . . . . . . . . . . E . Weakly Inhomogeneous Composite . . . . . . . . . . . . . . . . . . . . . . . . . F. Hashin-Shtrikman Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Self-consistent Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 171 171 173 181 182 186 188 188 193 200 202 202 203 208 208 215 218 220 223 226 230 236

I . Introduction Especially in recent years. when practical applications have been increasingly called for. an extensive literature has been assembled to assess theoretically the various aspects of the elastic. and wider mechanical. 169 Copyright 0 1981 by Academic Press. Inc . All rights of reproduction in any form reserved .

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behavior of composite materials. The subject has proved amply fertile and intricate enough to foster a diversity of interpretations and procedures. This article attempts to blend and consolidate the more fundamental, mathematically rigorous, principles into a general theoretical framework from which more specialized developments can proceed. A composite material is a heterogeneous solid continuum that bonds together a number of discrete homogeneous continua, each of which has a well-defined sharp boundary. It is understood that the bonding at the interfaces (and the continuity of each region) remains intact in the present circumstances where the entire mixture is to be placed in an equilibrated state of infinitesimal elastic strain by external loads and constraints. Each separate homogeneous region has its characteristic tensor of elastic moduli (in the stress-strain relation), which when anisotropic reflects a particular alignment of the crystallographic axes relative to the fixed Cartesian ones. A single phase consists of all those regions that share the same tensor of elastic moduli and perhaps also the same alignment of a geometrical shape and of crystallographic axes; in particular, it might stand as a connected matrix embedding the other phases or as the empty interior of a porous or cracked medium. In practical, natural or man-made, examples, the heterogeneous structure is generally a “microscopic” one appreciated on a scale far larger than the molecular dimensions yet very small compared with the “macroscopic” (or “overall”) one of engineering purposes. For example, a metallic specimen may appear in the large to be a single homogeneous and possibly isotropic medium, from both a geometrical and physical point of view, while under microscopic examination it is revealed as a firm mixture of innumerable, variously oriented, anisotropic crystals each large enough to be regarded as a continuum. Several types of manufactured materials are instilled deliberately with a reinforcing microstructure, for example, as a high concentration of dispersed particles or fibers whose individual diameters or cross sections, and spacing, are large compared to the molecular dimensions, but small compared to the overall ones. In seeking to prescribe some macroproperty accurately enough for practical purposes, perhaps in some optimal or economical manner, it is necessary to know how far it is influenced by the the more local, “microscopic” physical and geometrical properties, more especially by those that can be readily measured, or controlled beforehand in a manufacturing process. We are directed here toward circumstances where the composite material can be looked on as a homogeneous continuum whose tensor of elastic moduli is to be evaluated as precisely as possible, between upper and lower limits, when much of the finer geometrical detail remains unspecified, or in any case too intricate or too special to allow for. A sufficient level of generality must be maintained in the analysis, although it

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can specialize to any particular results that illuminate the wider area. After bringing together some alternative mathematical procedures, we prefer to employ that which is named “self-consistent” and which is foreshadowed elsewhere in the literature in what we may recognize and support as its approximate versions. Earlier on we find it appropriate to examine a number of features of the detailed, ‘‘microscopic’’ elastic field, especially to identify those that become transmitted in some way to the macroscopic level, including, firstly, those whose origin can be traced back aptly enough to the prime characteristic of a composite heterogeneity, the abrupt interfaces between phases. It is well known that in the manner illustrated by Batchelor (1974) and Hashin (1964,1970)the linearly elastic analysis can be viewed in the abstract as that shared analogously by other linear, “transport” phenomena, to which we shall refer briefly on appropriate occasions. Each different physical context offers its own kinds of insight and composite structures. The elastic one can lay claim to special analyses that find no counterpart anywhere else, as they stem in essence from the additional flexibilitiesconferred by its fourthorder tensor of elastic moduli (or “transport coefficient”).

11. Preliminary Analysis of Tensors and Elastic Behavior

A. NOTATION Cartesian tensors will be employed and their components will be denoted by the customary suffix notation, with reference to the coordinates x l , x2, x j . Repetition of the same suffix in a single tensor (or product) implies by the usual convention that there is a summation of that suffix over the values 1, 2,3. Differentiationswith respect to the coordinates are often denoted simply by attaching the appropriate suffixes to a tensor by means of a preceding comma. We shall have to deal extensively with tensors of the second order, which may denote a stress or a strain for example, and with fourth-order tensors of the kind which make up a linear relation between two second-order tensors that are each symmetric with respect to an interchange of the two suffixes. Every fourth-order tensor is made symmetric with respect to interchange of its leading pair of suffixes and of its terminal pair, so that it retains no more than 36 independent components. Fourth-order tensors are given Latin capitals as kernel letters: lower case, Latin or Greek, light face letters are kept for second-order tensors and vectors. The tensors of second and fourth orders form certain inner products frequently enough to call for a

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L. J. Walpole

simplified symbolic notation that denotes them just by the juxtaposition of their kernel letters. For example, AB will stand for the fourth-order inner product (generally noncommutative) whose components are AijmnBmnkl, while Au and U A denote the second-order inner products AijklUkl and Ok[Ak[ij, respectively. Repeated inner products with other tensors of the second or fourth orders can be symbolized in the same associative manner without the need for intervening brackets, unless a particular juxtaposition is to be emphasized. The notation uv denotes the scalar inner product uijuijof two symmetric second-order tensors. A scalar prefix indicates that each component of a tensor is to be multiplied by that factor. Two tensors of the same order can be equated to one another in the symbolic notation (if they have the same components), or they may be added together to stand for the tensor formed by the sum of their corresponding components. Two fourth-order tensors A and B are said to be inverse to one another if u = Aw,

w

= Bu,

for any choice of u or of w as a symmetric second-order tensor. In other words, AB=BA=I where 1 is the fourth-order "unit" tensor made suitably symmetric by having the components

in terms of the symmetricsecond-order unit tensor 6, known as the Kronecker delta. The inverse of a tensor is denoted sometimes by the same kernel letter with superscript - 1 attached. The formula for the inverse of a product is (AB)-' = B-'A-' where A and B are any two invertible tensors. However, especially where this superscript would frequently appear rather ungainly, we prefer sometimes to adopt a fresh symbol for an inverse. It may be shown that AZ = A

= ZA

for any fourth-order tensor A (with the assumed symmetries).When a fourthorder tensor possesses the further symmetry that allows its leading and terminal pairs of suffixes to be interchanged, so that, for example,

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173

then it is described as “diagonally symmetric” (corresponding to a matrix array of the components) or simply as “symmetric,”and it can retain no more than 21 independent components. If such a symmetric tensor A makes the scalar quantity uAu strictly positive for every nonvanishing symmetric second-order tensor u, then A is said to be positive definite, but only positive semidefiniteif the scalar can take the value zero for some such u. A symmetric positive definite tensor possesses a unique inverse which is also symmetric and positive definite. If the difference A - B, between two symmetricpositive definite tensors, is itself positive definite {or semidefinite) then it may be confirmed (by the simultaneous reduction of A and B to “diagonal form”) that so also is the difference B-’ - A - The statement that A - B is positive definite (or semidefinite) will be written more concisely as the “inequality” A > B (or A 2 B). The preceding theorem may be displayed thereby for reference as

’.

A>0,

B>0,

A2B+B-’2A-’.

(2.2)

B. DECOMPOSITION : GEOMETRIC SYMMETRY

A fourth-order tensor will generally have a structure that reflects some underlying geometric symmetry like that of a crystal. Its number of independent components can be reduced, therefore, in the well-known manner explained for the various crystal symmetries by, for instance, Nye (1960), or more recently by Srinivasan and Nigam (1969)in terms of components that are invariant with respect to the coordinates. An appropriate decomposition of the structure of a tensor can reflect valuable physical insight while offering to simplify greatly the calculation of the various inverses and inner products. Isotropic tensors are the principal ones to be prepared for in our detailed calculations and there is a smaller role for anisotropic tensors that reflect the symmetries of cubic and hexagonal crystals. By the decomposition, Z=J+K where

and hence where JJ=J,

KK=K,

JK=KJ=Q,

L. J . Walpole

174

the fourth-order unit tensor is expressed as the sum of two other tensors which are likewise isotropic and idempotent: each is equal to the inner product with itself. This decomposition reflects the one which splits a symmetric second-order tensor (v, say) into its isotropic and deviatoric parts. For, we observe that u = JV

where

-=

+ KV = $S + ‘v, Iv..

Ukk,

1J

= 0.. --V ... 1J i*’lJ

On the other hand, when the decomposition is recast as I = JJ

+ KK,

it delivers the convenient expression

vw = VIW = +ijE + ’vlw for the scalar inner product of two symmetric second-order tensors, in which there are no cross-products left between the isotropic and deviatoric parts. The general isotropic tensor of fourth order (L, say) has the (“spectral”) decomposition L = aJ

+ bK

(2.5)

where the coefficients a and b are arbitrary scalars (cf. Hill, 1965a).The inner product of two isotropic tensors can be calculated readily by means of the “multiplication table” (2.4)for the products between the elementary idempotent tensors. It is found to be isotropic and commutative, with coefficients that are just the respective products of those of the two original tensors. The inverse of the tensor L is obtained simply when each coefficient (a,b # 0) is replaced by its reciprocal. The decomposition for L splits an inner product Lv into its isotropic and deviatoric parts, while a quadratic form vLv is given the convenient expression

vLv

= u(aJJ

+ bKK)v = aij2 + b‘v’u,

from which it may be inferred that L is positive definite if and only if its coefficients a and b are both positive. The coefficients can be expressed, independently of each other and of the particular coordinate system, as linear combinations of the components of L. First, J (and, second, K ) is allowed to make an inner product with L from the left or the right or both sides, so leaving an equation for a (and, second, forb) which can be contracted with respect to the suffixes by noting that J ..._= 3 , llJJ

J _ _ .= . I, 1JlJ

K 1lJJ ..._= O ,

K I.... =5. JlJ

(2.6)

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Thus, as expressions which exhibit a and b as positive scalars when L is positive definite, we have

a = "J 3 mmijLijklJklnn = i L i i j j ? b = ~ K m n i j L i j k l K= k l$(L.-. m ni j i j - "L 3 i i j j ).

To proceed to a corresponding analysis for anisotropic tensors, it is necessary to construct the relevant, linearly independent, elementary tensors and then to combine them together in the appropriate way. For cubic anisotropy, we need the further idempotent tensor S defined so that with respect to a particular system of Cartesian coordinates (whose axes coincide with those of a cubic crystal) Sijkt = aPid P j,a,

=o

a,

=

if i = j = k otherwise.

1

= 1,

We may verify that in this coordinate system (and hence in any other)

JS

SS = S,

= SJ = J .

A linear combination of the three elementary tensors I , J , and S would express the general tensor with cubic anisotropy in a form which was recommended by Thomas (1966) and by Srinivasan and Nigam (1969), and which was employed also by Lifshitz and Rosenzweig (1947) and by Dederichs and Leibfried (1969). However, we shall find it more convenient to combine three tensors which are not only each idempotent, but which also have vanishing inner products with each other (and which hence make for a closer parallel with the preceding isotropicanalysis).We establish the decomposition K

= K'

-+ K '

where

K'

=S -J,

K" = I - S,

in order that

The general fourth-order tensor for cubic anisotropy is constructed, therefore, as L = aJ + b'K' + b"K" where the coefficients a, b', and b" are arbitrary scalars. The inner product of two such tensors, calculated by means of the multiplication table (2.7), is commutative and of the same anisotropic form, and its coefficients are just

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L. J . Walpole

the respective products of those of the two original tensors. The inverse of the tensor L is obtained by replacing each of its coefficients (a,b', b" # 0) by its reciprocal, since the unit tensor I , decomposed in the same manner, has coefficients of unity. The quadratic form vLv receives the natural expression vLv = v(aJJ

+ b'K'K' + b"K"K")v

and, with the brackets removed, it may be inferred that L is positive definite if and only if all its coefficients a, b', b" are positive. We may write down expressions for these coefficients, which exhibit them as positive scalars when when L is positive definite, as a ='J3

..L.. J lJkl klnn,

mmij

b'

= 3KmnijLijklKklmn,

b"

= $KknijLijklKLlmn.

After taking the inner products with L , the suffixes have been contracted here by noting that

Last, we consider the class of anisotropic tensors which reflect a transverse isotropy (for example, of a hexagonal crystal) and for which, in other words, all directions perpendicular to some unit vector n are equivalent. First, we observe that there are two elementary second-order tensors of this type, which may be identified as the idempotent parts that decompose the Kronecker delta as

6.. 1J = a ,1J. + b.. 1J

(2.8)

where aCJ, . = y1.n. 1 J'

b.. = 6.. - n.n.. 1J 1J I J

It may be verified that

b.tk bkJ. = b.. IJ' aikakj= aij, at.k bkJ. = O = a J.kb kr. . The most general second-order tensor reflecting the transverse isotropy is defined by the construction c.. ZJ = aa.. 1J + Bb.. 1J

(2.9)

where CI and fl are arbitrary coefficients. If a prime is attached to each of the coefficients to define another tensor c', then we find that c.ik c'. = c. c'. kJ J k kt

= aa'a.. 1J

+ pp'b... IJ

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In particular, we deduce that c' is the inverse of c when a' and /3' are reciprocals of tx and fi, respectively. The set of elementary tensors of fourth order may be constructed as the various independent "outer" products of a and b. Part of the set is presented as idempotent and diagonally symmetric tensors when the unit tensor is decomposed as I

= E'

+ E2 + E 3 + E4

(2.10)

where

Ehkl = *bijbkl,

E2. lJkl = a..a 1J k l ,

E;k1 = $ ( b i k b j l

-I-bjkbil

- bijbkl),

E!jki = $(bikajl

+

-k

bi1Ujk

bjlaik

-k

bjkail).

It may be confirmed that EPE4 = E4 if p = q ,

EPE4 = O

if p f q ,

(2.1 1)

where p and q can take any of the values 1, 2, 3, or 4.There remain two elementary tensors which may be defined, without diagonal symmetry of their suffixes, as ES. lJkl = a I,J &kl2

E$d

= bijakl

9

and although each has a vanishing inner product with E3 and with E4, their inner products with each other and with E' and E 2 are nonvanishing and moreover noncommutative. This less congenial multiplication table is displayed most conveniently as the square array

E' E2 E' E4 E5 E6

E' 0 O O E5 0

0 0 0 E2 0 0 O E 3 0 O O E 4 0 0 0 E6 0 0

0 E5 0 0 0 2E'

E6 0 0 0 2E2 0

(2.12)

where the outside column and row indicate the first and second factor, respectively, of the inner product. The general fourth-order tensor for a transverse isotropy may be constructed as an arbitrary linear combination of the six elementary tensors as L

= cE'

+ dE2 + eE3 + f E 4 + gE5 + hE6,

(2.13)

L. J . Walpole

178

say. By letting the elementary tensors make inner products with L from the left and the right, followed by an appropriate contraction of the suffixes, we can solve for each scalar coefficient in turn (by eliminating all of the others each time) in terms of the components of L. The first four coefficients can be given the forms = *EhnijLijklEhn,

= E&nijLijklE~hn~

which show them to be all positive when L is positive definite, while the remaining two may be shown as = iEkijLijklE/!lnn,

= iE#!mijLijklE&n.

By picking out each of the arbitrary coefficients in order, we may resort to the briefer symbolic expression L = (c, d7 e,f,9, h).

(2.14)

Diagonal symmetry prevails when the coefficients g and h are the same, and our expression can be recognized then as a rearranged version (with a different set of elementary tensors) of that given by Synge (1956) and by Srinivasan and Nigam (1969). However, we cannot confine attention to tensors with the full diagonal symmetry, for this property is generally lost immediately in an inner product, even though it might be regained in a repeated product. Any inner product can be calculated readily now by means of the multiplication table for the elementary tensors. Thus if L' is the tensor defined by attaching a prime to each of the coefficients of L , we find that the inner product L L has the (generally noncommutative) symbolic form

LL'

= (cc'

+ 2hg', dd' + 2gh, ee', fs',gc' + dg', hd' + ch').

By reducing this expression to that for the unit tensor, namely (2.10), we may calculate the inverse of L as (2.15)

where 1 = cd - 2gh. When L and L' both have the diagonal symmetry (so that we replace h and h' by g and g', respectively), then this property is restored in the repeated inner product LL'L, which we may calculate for future reference as LL'L = [c2ct + 4cgg' + 2g2d', d2d' + 4dgg' + 2g2c', e2e1,f 'f dgd' + cgcl + (2g2 + cd)g', dgd' + cgc' + (2g2 + cd)g']. (2.16) I,

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179

Alternative forms for L and its inverse when there is the diagonal symmetry (h = g ) are given by Lijkl = $cSijs,,

where sij = b,

+ (d - 2g2/c)aijaki+ eE&l + fE$kl,

+ (2g/c)aij,

tij = aij - (g/c)bij.

E 3 and E4 may be replaced here by E3E3 and E4E4,respectively, especially to help construct the quadratic forms vLu and uMv. It may be inferred that this L (and likewise M ) is positive definite if and only if e, f , c, d - 2g2/c, and hence d, are all positive quantities. We focus now on the particular, arbitrarily anisotropic, fourth-order tensor L of elastic moduli (or of “stiffness”) and on its inverse tensor M (of “compliance”)which make up a linear constitutive relation E = Mo,

o = LE,

(2.18)

between the symmetric second-order tensors o and E of stress and strain respectively. They both always possess the diagonal symmetry and the other symmetries of (2.1). L is assumed to be positive definite for a real elastic medium and so, therefore, is M . If the medium happens, for example, to be isotropic, we can write

+

M = (1/31~)3 + (1/2p)K,

L = ~ I C 2pK, J

(2.19)

to introduce as positive scalars the bulk modulus IC and the shear (or rigidity) modulus p, which may be related to alternative pairs of independent elastic moduli; the connections with Young’s modulus E and Poisson’s ratio v are

_3 --_ 1 + -,1 E

p

31c

v

=

($ - ;)I(; + ;)*

The constitutive relation breaks down as Jo = ~ I C J E , K a = 2 p K ~ ,

that is, as two direct, invertible relations between the hydrostatic part of the stress and the dilatational part of the strain and between the respective deviatoric parts. If, however, the medium has a cubic anisotropy we can define the tensors K‘ and K” with respect to it and write L

= 31cJ

+ 2p‘K’ + 2p”K”,

M

1 3K

= -J

1 1 +K‘ + __ K”, 2p‘ 2p”

(2.20)

L. J . Walpole

180

in terms of the bulk modulus K and the two shear moduli p’ and p”, defined as three independent positive scalars. The constitutive relation decomposes to the readily invertible form J o = 3tiJE,

K’o = 2p’K‘~,

K“o = 2p”K”~.

If, lastly, the medium is transversely isotropic about some axis (in the direction of the unit vector n) we may express L and M in the forms (2.14) and (2.15),with h = g, and then set down the connections d - 2g2/c = E,

c = 2k,

g/c = v,

e = 2m,

f = 2p

with five familiar independent elastic moduli, namely with the plane-strain bulk modulus k, the transverse shear modulus m, the axial shear modulus p , and the axial Young’s modulus and Poisson’s ratio E and v, respectively (where k, m, p , and E are all positive). Alternative expressions in terms of this latter set of moduli derive from (2.17) as Lijki Mijki

= kSijSki

+ EaijUki + 2mE;ki + 2pE$..,

= (l/4k)bijbkI

+ (l/E)lijtki + (1/2m)E& + ( 1 / 2 p ) E : k I ?

(2.2I)

where t . . = a,. - vb ...

s.. 1J = b.. V + 2va.. 1J ’

lJ

CJ

1J

With any anisotropic elastic medium whose tensor of elastic moduli is L with inverse M , there are associated immediately two particular isotropic media defined by Voigt (1928) and Reuss (1929) so as to have tensors of elastic moduli L, and L R , respectively (which are “averages” of L in a sense and for a purpose explained in Section V,A). Writing L,

+ 2pvK,

=~K-,J

LR = ~ K , J +2pRK,

the respective bulk and shear moduli are specified by 9K,

= Liijj,

(1/~,)= M i i j j ,

l o pv - L lJlJ . . . .- +Liijj,

(5/2pR)= M.... I I Y - $Miijj.

That ti, and p, (and, similarly, tiR and pR)are thereby made positive scalars is displayed more explicitly by writing Liijj

= JmmijLijkIJkinn,

Lijij

- i L i i j j = KmnijLijklKkImn.

When L and M have the isotropic forms (2.19),their bulk and shear moduli coincide (as intended) with those of Voigt and Reuss. On the other hand, when L and M have the expressions (2.20)for cubic anisotropy, we calculate

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181

that Ic" = K = KR,

5PV = 211'

+ 3P",

+

5/PR = 2/p1 3/P",

and for the transversely isotropic forms (2.21)that

+ 4k(l + v)', 159" = E + 6(m+ p ) + k(1 - 2v)', 1 / 1 c ~= ( l / k ) + (1 - ~ v ) ~ / E ,

9 ~ "= E

+ 6[(1/m) + (l/p)] + ( 4 / E ) ( 1+ v)2.

5/pR =

C. RESOLUTION

The decomposition (2.8)of the Kronecker delta brings about the familiar resolution (2.22)

of a vector v as the sum of two component vectors, one parallel and one perpendicular to the unit vector n. The decomposition (2.10) of the fourthorder unit tensor offers likewise a quadripartite resolution of any symmetric second-order tensor w as w = E'w

+ E2w + E3w + E4w

where the four parts are mutually orthogonal in the sense implied by the multiplication table (2.11). The significance of this resolution can be illustrated by a particular choice of w and of n. At a point of a surface the resolution may be made with respect to the local unit normal identified as n and with (for example) the local stress tensor u identified as w. The traction vector oijnjcan be given the resolution (2.22)into the sum of its normal and tangential components and we may establish the connections (2.23)

together with, on the other hand, E$klokl

= ninjnknlckl

= ni(ajk'klnl),

E?jkl'kl

= ni(bjkoklnl)

+

(2.24)

nj(bikaklnl).

It is, therefore, appropriate to refer to E2a and E40 as the normal and tangential parts, respectively, of the stress tensor, for each both derives from, (2.24),and gives rise to, (2.23),just the normal and tangential components,

L. J. Walpole

182

respectively, of the traction vector. By the combination El

+ E 3 = A,

E 2 + E4 = B

(2.25)

say, by which A

+B =I,

BB = B,

AA = A,

AB = BA = 0,

(2.26)

we may recognize A a and Ba as those orthogonal “interior” and “exterior” parts of the stress tensor (and, similarly, of any symmetric second-order tensor) that were introduced by Hill (1972) in order to accomplish the bipartite tensorial resolution c.. = A lJkl , . (T kl ZJ = (dik

+ Bijkl‘kl

- nink)(6jl

- njnl)akl

+

(nidjk

+

njdik

- ninjnk)nlokl

(2.27)

as a natural, physically significant counterpart of the vectorial resolution (2.22). The interior and exterior parts of the infinitesimal strain tensor can be defined at the surface point and, by means of the relation &.. ij = L( 2 ui,j

+ Uj,A

(2.28)

they can be calculated from the gradient of the displacement vector u or from its normal and tangential components specified by the resolution U. i,j . = aj.k U i,k .

+ bJ.kU.i , k .

(2.29)

The connection AijklEkl

= t(bikbjl

+ bjkbil)uk,l

= 3bik(bjlUk,l)

+ 3bjk(bilUk,l)

(2.30)

reveals that if the tangential component of the displacement gradient is known, or, therefore, if the displacement components are known about the surface point, then the interior part of the strain is determined fully (while the exterior part remains indeterminate). When some combination of the stress and strain components is determined, then the complementary combination may be specified, in turn, if the appropriate decomposition and rearrangement of the constitutive law is available. Our next purpose is to effect a decomposition which is found to be particularly appropriate at a bonded interface in a composite medium and at a boundary where all the individual traction and displacement components are determined.

D. SURFACE DECOMPOSITION OF THE CONSTITUTIVE LAW When the resolution (2.29) of the gradient of the displacement vector is inserted in the relation (2.28),the infinitesimal strain tensor is decomposed at

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183

a point of the surface as

+ Jjni)+ E$

(2.31)

g i j = +(Ainj

where and where the unit normal n to the surface can be assigned either sense of direction. If the displacement components are supposed known along one side of the surface, we may evaluate there the tangential components of their gradient and, hence, also that part of the strain denoted by E * . The vector , I (which denotes the normal component of the displacement gradient) still remains undetermined and leaves an undetermined part in the strain components and hence in the stress components defined by the constitutive law. If, however, the components of the traction vector are also supposed known at the surface point, then A can be determined completely so as to leave explicit expressions for all the strain and stress components [as in the related context of Hill (1961, Section 6)]. The arbitrarily anisotropic constitutive law (2.18) expresses the traction vector as

+ L.. E*n. lJkl kl J

g . . n . = ]i..l. IJ

J

EJ J

where hij = Likj,nk?ll.

The symmetry and positive definiteness of the tensor L ensures that the second-order tensor h is symmetric ( k i j = k j i ) and positive definite [since oikijuj

= $(Dink

+ vkni)&kji(ujnl +

snj)

>0

for any nonvanishing vector u]. Therefore, there exists a second-order tensor k inverse to h which satisfies the relation

k im . hmi. = 6.. LJ

(2.32)

and which is also symmetric and positive definite. [This inverse tensor plays a fundamental role in the theory of anisotropic elastic media, for instance (with a suitable interpretation of the unit vector n) in the calculation of the Green’s tensor that we meet in Section III,B and as the “Kelvin-Christoffel stiffnesses” in the propagation of plane waves.] The relation (2.32) allows the determination of l and hence of the components of strain and stress in terms of the known components of traction and displacement.We find in turn that

‘6 == ( k j n k

f kiknj)(ojk

+ MQ*E*, a = LP*a + Q*E*, E

= P*a

- LjklmEL), (2.33) (2.34)

184

L. J . Walpole

where P&, = 3kiknjnl -k kjfljnk -k kjkninl -k kjlnink), Q*

=L

- LP*L.

(2.35)

Next we examine the properties of the various fundamental tensors that have been introduced at the surface, to help make for further interpretation of these final expressions. The asterisk attached to the fourth-order tensors may serve to mark their dependence on the direction n (as well as on the elastic moduli): it is to be removed in due course to define related tensors without this directional dependence. The symmetries of the tensor L are transferred in the same way to the fourth-order tensors P* and Q*. Moreover, P* is made positive semidefinite by its definition in terms of the positive definite k. A monotonic dependence on the elastic moduli is shown by imagining that L is changed slightly to L + A L , say. There are, consequently, the small changes given by Ahij = (ALikjl)nknl, Akij = - kipnq(ALpqrs)nskrj, AP* = - P*(AL)P*,

(2.36)

AQ* = -Q*(AM)Q* Therefore, if L is increased by any infinitesimal (and hence any finite) positive definite amount, so that A L is a positive definite tensor, then h and Q* increase while k and P* decrease by amounts which are positive definite or at least positive semidefinite. Next we take the tensor A from (2.25) and note the implications of applying it as an operator to (2.33) and (2.31) in turn (so eliminating 1 from the latter), while at the same time we let its companion B operate on (2.34). We are thereby led to confirm algebraically that AP* = P*A = 0,

BQ* = Q*B = 0,

(2.37)

BP*

AQ* = Q*A = Q*.

(2.38)

and hence = P*B = P*,

Furthermore, we deduce that (BLB)P* = B = P*(BLB),

(AMA)Q* = A = Q*(AMA).

(2.39)

The first member of this pair may be verified by expressing the tensor BLB in terms of h and n, after which the second member follows much more easily from the relations (2.26), (2.35), and (2.37). We may derive at once the identities P* = P*LP*, Q* = Q*MQ* (2.40)

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185

to thereby confirm that P* is positive semidefinite and that this property is shared by Q* as well, and hence, according to the relation (2.35),also by the differences L - Q* and M - P*. If the first and second terms of (2.40) are multiplied by L and M , respectively, from the left and then from the right, the tensor products P*L, LP*, MQ*, and Q*M are all revealed to be idempotent tensors that are seen to participate in the decomposition of I when the relation (2.35)is given the alternative forms P*L + MQ* = I = LP* + Q*M. (2.41) Furthermore, by letting P* make inner products with (2.35), it is seen to have a vanishing inner product with Q* : p*Q* = 0 = Q*p*. It may be hoped that these various properties make for some structural simplicity in the tensors P* and Q*, which otherwise must seem rather uncongenial individuals especially for a generally anisotropic medium. In the constitutive relation (2.18),the stress and strain tensors may be each decomposed into the sum of interior and exterior parts. Then after the operators P*B and Q*A are allowed to form inner products with the first and second terms of (2.18), respectively,we may appeal to the identities (2.37), (2.38),and (2.39) to show, in turn, that BE = P*(Ba) - P*L(A&),

AC = Q*(AE)- Q*M(Bc),

and hence E (T

+ B)E= P*(Ba) + MQ*(A&), = ( A + B)c = Q*(AE)+ LP*(Bo).

=(A

(2.42)

These two pairs of expressions bring out the theorem stated by Hill (1972, in full detail for an isotropic medium) and by Laws (1975, for an anisotropic medium); namely, that if both the exterior part of the stress tensor and the interior part of the strain tensor (or if all the traction and displacement components) are supposed known at a point of a surface, then the interior part of the stress and the exterior part of the strain and hence, also, the entire stress and strain tensors are determined uniquely at that point. The earlier pair of expressions [(2.33) and (2.34)] and the latter pair (2.42) may be regarded as equivalent methods of decomposing the constitutive law, if allowance is made for the identities (2.38)and for the derivation of the strain tensor from the displacement vector (without which the latter pair would remain available, however, merely by the algebraic manipulation from the constitutive relation). For, we have noted already that the vector 1 is eliminated when operator A is allowed to make an inner product with the decomposition (2.31),and so E and E* share the same interior part.

L. J. Walpole

186

The individual components of the various tensor operators can be specified explicitly when the form of the tensor of elastic moduli L is known. For instance, if we have the isotropic (2.19),then, with respect to the unit normal n, the second-order tensors h and k and the fourth-order tensors P* and Q* will take the transversely isotropic forms of (2.9) and (2.13),respectively [and the fully isotropic L may be recast in this form for an appeal to the product formula (2.16) when Q* is evaluated]. All their various anticipated properties will be thereby brought out. We find in turn that

h = (ti + 4p)u + pb, k P*

= a/(. =

+ 4p) + b/p,

ti + $11) + ~

4 1 2 ~ ~

(2.43)

Q* = 9E1/2[(1/lc) + (l/$p)] + 2pE3 = 2p[(1

+ V ) ~ l /-( iV) + ~ 3 1 ,

where ti, p, and v are the bulk modulus, shear modulus, and Poisson’s ratio, respectively. It is of interest to calculate further that

P*L - vES/(1 - V) = E2 MQ*

+ V ~ 5 / ( 1- V) = ~1

+ E4 = LP* - vE6/(1 - v), + 133 = Q*M + V . E / (-~ V),

(2.44)

in order to find the tensor products of (2.41) and to verify their idempotent property by means of the multiplication table (2.12)forthe elementary tensors.

E. INTERFACIAL STRESS AND STRAIN JUMPS When the surface in question is in fact an interface between two different elastic media, then the preceding analysis may be carried out similarly on the other side where the tensor of elastic moduli is L,, with inverse MI say. All tensors defined in terms of the elastic moduli of this other side will be given a suffix 1 to distinguish them from the corresponding ones already defined on the original side. Next we allow for the overriding physical relations at the interface. Suppose that there is perfect bonding in that the displacement and traction vectors at a point on one side are precisely the same as those at the adjacent point on the other side. It follows then, after reference to (2.30) and (2.27), that the interior part of the strain and also the exterior part of the stress are the same on each side, as was shown by Hill (1972), while the remaining part of each suffers a discontinuity or “jump.”

Elastic Behavior of Composite Materials

187

In other words, at the interface A&= A E ~ BC , = Bo,.

The entire strain (or stress) tensor on one side is now directly and uniquely related to that on the other side, since it can be expressed in terms of the common interior strain and exterior stress parts (or in terms of the common displacement and traction components). From the pair of relations (2.42), we find at once that E

= P*(Bol)

+ MQ*(A&,),

o = Q*(Ae1) + LP*(Bo,).

After due appeal to the identities (2.38) and (2.41), and to the constitutive , two equivalent relations can be recast to relation between o1 and E ~ these express the strain and stress jumps concisely by

P*(L, - L)E1= &

- E1 =

-PT(L - L1)&,

Q"(M1 - M ) c ~= c - 01 = -QT(M - M,)c.

(2.45)

where each equality on the right is set down simply as the dual of the one on the left, in accord also with the identities P*(L1 - L)Pf = P" - P*1,

Q*(A41 - M)QT

=

Q* - QT.

(2.46)

The algebraic derivation of the first (and similarly of the second) of these equivalent identities is reached most easily by deploying the operators A and B to verify that

P*(L, - L)PT = P*(A + B)(L, - L)(A + B)PT = P*BLIBPF - P*BLBPT = P*B - BP*1 -- P* - PT. The earlier pair of identities (2.36) can be recovered now when the two tensors of elastic moduli differ only slightly. The relations (2.45) help to determine the tensors of strain and stress at one side of an interface, especially when they can be calculated comparatively easily on the other side [as in examples of Hill (1972) and Walpole (1978)l. It is appropriate next to turn to a particular interface in order to construct the full elastic field at points on and away from it. An early example was given by Michell (1899, 1964) where two half-spaces of dissimilar isotropic materials were bonded (or were slipping) over the plane interface. In the present context it is of interest to proceed to a single closed interface eventually of ellipsoidal shape, in preparation for the much more complex, multiphase circumstances.

L. J. Walpole

I88

111. The Elastic Field of an Inclusion

A. TRANSFORMED INCLUSION Suppose a closed interface separates the whole outer region (the “matrix”) made of a homogeneous anisotropic material from the inner region (the “inclusion”) made of a different homogeneous anisotropic material. This composite body is in a self-equilibrated state of infinitesimal elastic strain for which the constitutive law remains as O =

LE,

E=

Mo,

(3.1)

in the matrix, while in the inclusion it is transformed (for the present) as

+ q1)

O

= L1(E

E

= M,(o

= LIE

+ z1,

- 71) = M1O - q1,

in terms of prescribed constant second-order symmetric tensors connected by the relation = Llrll,

q1 = MlTl.

tl

and q l (3.3)

At each point of the inclusion and matrix there are the equations 0.. r3.J ’ = 0’

Eij

+

= +(ui,j

UjJ,

which specify, respectively, that the symmetric stress tensor satisfies the equations of equilibrium in the absence of body forces and that the strain field is derived in this way from a displacement vector u which is continuous across the interface and continuously differentiable everywhere away from it, and which vanishes at all remote points. The stress field o is to have its traction components at a point on one side of the interface the same as those at the adjacent point on the other side. The tensor - q l , may be identified accordingly as the uniform “stress-free” infinitesimal strain (of physical origin) which the “transformed” inclusion would have undergone in the absence of the surrounding matrix, whereas E is the measure of the strain actually attained at any point of the inclusion or of the matrix. From an alternative viewpoint, it can be anticipated that if L were to be made vanishingly small to remove the matrix completely in subsequent formulae, then the constrained strain E would be brought to coincidence with the stress-free strain -ql. The tensors z1 and ql may be called the “polarization stress” and “polarization strain,” respectively, to point to an analogy with electrostatic potential theory. We shall in due course assign an ellipsoidal shape to the interface in order to construct the detailed elastic field using the method explained by Eshelby

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189

(1957,1959,1961),to whom reference can be made for a wealth of discussion, application, and intuitive appeal; less complete approaches to the problem were made by Myklestad (1942) and Robinson (1951). In the meantime though, the shape is left unspecified,perhaps even to be an unknown in some applications. While full calculations are then precluded, it is worth seeking any general features that persist not only for the ellipsoidal shapes. It is of primary importance to calculate the total strain energy and, consequently,the mean strain and stress within the inclusion. By the connections E

= -p171,

(3.4)

5 = Q1~1,

where the overbar signifies a mean (volume average) over the inclusion, fourth-order tensors P , and Q1 are introduced, as “mean polarization factors.” They have a dependence on the elastic moduli of the inclusion and the matrix, and on the shape but not on the size of the inclusion (and nor do they retain any dependence on 71 and ql). Along with (3.4),we can form the expressions E

+ ~1

= (MI

-PI)~I,

5 - 71 =

--WI

-QI~VI,

for the mean of the bracketed quantities in (3.2); E + ‘1, is the purely elastic part of the strain field within the inclusion. Either of P1 or Q1 can be determined from the other by means of their interrelation which can be conveniently stated in two ways as L1 - Q1 = L1P1L1,

Mi - P i = M1QIM1.

(3.5)

In order to calculate the strain energy, the integral of i a M , a over the inclusion (whose volume is V , say) is added to the integral of goMa over the matrix (whose volume is VM,say). After substitution of the relations (3.2), we observe (omitting the factor i)that

and also, dually, that

= VT,P,Z,.

(3.7)

The integral of 08 over W , the whole of space, has disappeared finally from these expressions since, after using the equations of equilibrium (and the symmetry of a) to write Q..&.. 1J

IJ

= a..u. . = (a..u.) 1J 1.J IJ I P J Y

190

L. J. Walpole

an application of the divergence theorem leaves no surface integrals at the interface nor in the remote region where the elastic field is assumed to vanish sufficiently rapidly. The displacement vector is related linearly to the polarization stress tensor z1 (or to q , ) by means of a third-order tensor field, in terms of which P , and Q, can be hence expressed after substitution for the displacement components in the left-hand sides of (3.7)and (3.6),and after elimination of the arbitrary symmetric tensor z,. The expressions show immediately that both Pi and Q , are diagonally symmetric and positive definite, since these properties are possessed by both L , and L (and by M , and M ) . This procedure is akin to the familiar one of potential theory, whereby analogous second-order tensors of “polarization” and “virtual mass” are shown to be symmetric and positive definite, for instance, by Schiffer and Szego (1949), Payne (1967), and BatcheloG (1967). An appeal to Betti’s reciprocal theorem furnishes another observation of the diagonal symmetry. For if the. polarization stress is adjusted to zo, say, in order to generate another elastic field, it follows that

since each side of this equation can be expressed as the same volume integral over the whole of space. The symmetry of P , is now a consequence of the arbitrariness of zo and z,, and it is passed on to Q1 by the relation (3.3, while the positive definiteness remains displayed by (3.6) and (3.7). There remain further properties to be extracted from (3.5).Both the tensors L , Q1 and M , - P , are seen to be positive definite and, consequently, by theorem (2.2), so are the tensors LT and MT which are defined by

so as to be made diagonally symmetric and inverse to one another, and so as to appear in the relations

between the means of the stress and strain in the inclusion. The tensors LT and MT have a special significance for which they deserve their own symbols, written either in this compact way or in the more elaborate bracketed notation, L*(L1,L ) and M*(M,, M ) , respectively, whenever it is necessary to indicate the particular elastic moduli of the inclusion and the matrix. The preceding statements of positive definiteness may be brought together in terms of the notation of Section II,A as in order to add that while (at least for finite L , and M , ) they place limits on the components of P,, Q1 and related tensors, there is room for much

Elastic Behavior of Composire Materials

191

more restrictive limitations (if L and M remain finite) to disclose eventually (as “isoperimetric inequalities”) how optimal values of important overall parameters can be attained by particular shapes of the inclusion, as exemplified with respect to the strain energy by Khachaturyan (1967) when the inclusion and the matrix are made of the same material. We shall be content with what is gained, by Walpole (1970a), when a monotonic dependence on the tensor of elastic moduli is extracted. If one or other (or both) of L , and L is imagined to increase by a positive definite amount to become a “greater” positive definite tensor, then an immediate application of the familiar extremum principle of minimum energy shows that the positive scalars calculated in (3.6) and (3.7) are increased and decreased, respectively. In other words, P1 decreases and Q1 increases by positive definite amounts. A deeper and stronger consequence (of the same extremum principle, but with a more flexible approximation that adjusts also the polarization tensors) is that LT increases and, consequently, MT decreases by positive definite amounts. Bounds on these tensors (that is, on their components) are made available by calculating them (in the next Section III,B) when L , and L are brought to (the best possible) coincidence. In particular, when L , - L is positive definite we have that L*(Ll,L,)> L*(L,,L) > L*(L,L)

(3.10)

and the inequality signs are reversed when L - L , is positive definite (but an indefinite L - L1 needs special consideration). These “inequalities” may appear still rather coarse and deprived of any geometrical ingredient, but in fact each one has an “isoperimetric” status that (in Section II1,C) makes it an equality for a particular ellipsoidal shape of the inclusion. At points in the matrix remote from the inclusion there is not yet any elastic field of stress or strain. However, we can introduce there immediately a uniform strain and a uniform stress oA, subject to the constitutive relation (3.11)

cA= L E ~ , E~ = MoA,

merely by a superposition and by an adjustment of z1 and q l to new constant values, in terms of a new polarization stress zA and polarization strain qA which are prescribed to be constant and subject to the relation zA = ~

~

qA q = ~M , T, ~ .

The definitions T I = (L1 - L

) E+~rA,

~1

=(M -

M,)oA

+ qA,

eliminate z1 and q l from the elastic field and ensure that the relation (3.3) is met and, moreover, that the strain field E , and the stress field o1 attained

L . J. Walpole

192 at all points is given by E, = E

+

&A,

o, = d

+ oA.

For, these definitions provide for the required behavior of the far field, and for the appropriate constitutive relations, namely o1 = L E , ,

E,

= Mu,,

in the matrix, and o, = L&, E, =

+ ?f) = LIE1 +

TA,

M,(o, - z A ) = M , o , - q A ,

(3.12)

in the inclusion.The tensor - qAis the uniform "stress-free'' strain prescribed for the inclusion (to which the actual strain E , would tend there when L becomes vanishing). The mean strain and stress in the inclusion can be evaluated as 01

+ =5 + =E

&A

=

gA =

- P I T ~+ &A = Q1ql

- P1rA,

+ oA = BloA+ QlqA,

(3.13)

respectively, in terms of the prescribed stresses and strains, where the fourthorder tensors A , and B , are defined by A,

=I

+ P,(L - Li),

B, = I

+ Q,(M - M I ) ,

(3.14)

and hence also by A,

= (LT

+ LJ'(LT + L),

B,

+ M,)-'(MT + M ) ,

= (MT

and are subject to an interrelation which can be written in two ways as

L,A,

= B,L,

M,B,

=

A,M.

If the inclusion and the matrix have almost the same elastic properties, we may replace (3.14) by A,

=I

+ P(L - Li),

+

(3.15) B , = I Q(M - M I ) , approximately, where P , and Q, are evaluated as P and Q, respectively, when L , is identified with L (and M , with M ) . In particular, z A and qA can be made to vanish so as to leave the inclusion as a perfectly fitting one for which (3.12) becomes the customary constitutive relation and for which oA and remain as the remote, uniform stress and strain fields, perturbed at the nearer points by the inclusion. The mean strain and mean stress in the inclusion are left specified by -

E , = A,EA,

respectively.

-

0, =

Bid,

(3.16)

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193

B. HOMOGENEOUS INCLUSION : ELLIPSOIDAL SHAPE To make way for detailed calculations, the preceding analysis is specialized first by giving the inclusion the same elastic properties as the matrix, so as to leave the whole of space filled with a homogeneous, anisotropic medium which has the tensor L of elastic moduli, and which contains the “transformed’ inclusion whose polarization stress and strain are z and q, respectively. The constitutive relation (3.2) for the inclusion is left expressed, therefore, as CT

+ q ) = L&+ z,

= L(&

E=

(3.17)

M((T-T)=McT-~,

where Z=

Ly,

(3.18)

q = Mz,

by removing the suffix 1 entirely (and hence from all succeeding places). We recall that the constitutive law (3.1) on the matrix side of the interface can be decomposed as in (2.42) in terms of A, B, P*, and Q*, which are tensors derived from the local unit normal. On the inclusion side there is a corresponding decomposition in which E + q replaces E or (T - z replaces 0.There remain interfacial jumps in the strain and stress tensors, which can be annihilated, respectively, by the operators A and B. It follows that these jumps are determined explicitly now as

[&I= -P*z,

[.I

(3.19)

= Q*q

where the square brackets denote the increase in their contents from a point just inside the matrix to the adjacent point just inside the inclusion. The entire elastic field may be derived from the fundamental one due to a lone point force in an infinite mass of the homogeneous medium. Let u’, E‘ and CT’be the displacement, strain and stress respectively, at the field point with coordinates xi (and the position vector x) due to a single point force f acting at the point with coordinates xi (and position vector x‘).Their definition is determined by the equations elj = ;(u{,j

+ uJ,~),

(T’

= LE‘,

+

~ ~ i j , j fid(x

- x’) = 0,

(3.20)

in terms of Dirac’s delta function, and by the requirement that the displacement components are to be continuously differentiable and proportional to the reciprocal of the intervening distance (x- x’( when the point x is sufficiently remote from x’.Now, by appealing to the divergence theorem to convert an integral over both sides of the interface S into a volume integral

L. J . Walpole

194

over the whole of space, we may derive the reciprocal relation

=

-s&ijLijkl&kldl/ =

-

s

s

d . & ” d l / = d. V J.u.dl/, I V V

(3.21)

where dSj stands for njdS, the product of the element dS of the interfacial surface and the component nj of its unit normal directed into the matrix. The symmetries of L and of the other tensors are allowed for in the summations over repeated suffixes. The expression Uj(X)

= S i j ( X ’ - x)&,

(3.22)

introduces a symmetric Green’s tensor g which depends on the difference x’ - x , and which by further analysis (of, for instance, Eshelby, 1951, and Synge, 1957) can be given the form g = d/(x’- XI

(3.23)

where the second-order tensor d is a function of the components ( x ; xk)/Ix’- and of the elastic moduli. However, it may be noted that continued efforts, particularly of Fredholm (19OO), Lifshitz and Rosenzweig (1947), Eshelby et al. (1953), Kroner (1953), Leibfried (1953), Stroh (1958), Mann et al. (1961), Willis (1965), Lie and Koehler (1968), Dederichs and Leibfried (1969), Mura and Kinoshita (1971), Barnett (1972), Bacon et al. (1978), and Head (1979a,b),have made it clear that explicit expressions of finite form for the tensor d are forthcoming for transverse isotropy (hexogonal anisotropy) and for full isotropy but not for other real anisotropies of the elastic medium. If we return to the reciprocal relation (3.21),to eliminate the arbitrary components of the point force, and to introduce the properties of the delta function and to remove the constant z outside the integration and give new labels to the suffixes, the displacement components at any point x’in the inclusion or the matrix acquire the explicit form

XI

n

or, after an appeal to the divergence theorem to convert this integral over the interface into one over the volume V of the inclusion, the alternative form (3.24) in which the differentiation in the integrand is with respect to the unprimed coordinates of integration. At points x’ remote from the inclusion, it can

Elastic Behavior of Composite Materials

195

be confirmed that the displacement components are proportional to the inverse square of the intervening distance. The ensuing calculations have the greatest interest and simplicity when the interface has the shape of an ellipsoid. It was shown by Eshelby (1957) how the form (3.23) of the Green's tensor makes for displacement components that are linear functions of the coordinates throughout the interior of an ellipsoidal inclusion and hence for strain and stress fields that are simply uniform there (while at exterior points the calculations remain comparatively difficult to complete). There is no dependence in these interior fields on the size of the ellipsoid, but only on the axial ratios (for a fixed orientation). It was shown by Walpole (1977) that, in the present (and likewise in much wider) circumstances, the ellipsoidal shape possesses the special property that (3.25)

where the strain and stress components in these integrals over the interface S are evaluated on its matrix side, and where p is the length of the perpendicular to the center of the ellipsoid from the local tangent plane (or in other words just the constant radius for a sphere). The equations x:/a2

+ xf/b2 + x;/c2 = 1,

p ( x ) = 1/(x:/a4+ xi/b4 + x23 1c4) ' I 2

may be taken to specify the ellipsoidal interface and hence the quantity p ; coincidence of the ellipsoidal and Cartesian axes does not lose any generality, since the crystallographic axes of the anisotropy remain at arbitrary orientation. The property (3.25) opens a simplified access to the uniform strain and stress inside the inclusion, in that it enables the matrix contributions to disappear from the jump formulas (3.19) (to be returned afterwards for their own evaluation), after multiplication by p and integration over the interface. We are left with the conclusion that at any point inside an ellipsoidal inclusion E =

-Pz,

(T=

QV,

(3.26)

where

and where Q = L - LPL.

(3.28)

L. J. Walpole

196

The integral of p over the interface is evaluated here as three times the volume of the ellipsoid, which is a well-known result that can be confirmed by replacing p by xini and applying the divergence theorem. The expressions (3.26), together with (3.28), may be recognized as the special form taken now by (3.4), with (3.5), since the present uniformity of stress and strain obviates the need for an overbar. The tensors P and Q (to which P , and Q , have specialized) are identified fully in (3.27) as interfacial means of the tensors P* and Q*, from which the established properties of symmetry, positive definiteness, and monotonic dependence on elastic moduli, are seen to derive. For a sphere, the weighting factor p of the interfacial means cancels out as a constant, so that Pijkl

= (1/4.a2)

ss

P& ds,

Qijkl

= (1/4.a2)

ss

QZkl d s ,

(3.29)

where a is the radius of the sphere, which in turn will become cancelled out. From (3.26) and (3.28) we may extract the relations g

= -L*E,

E =

-M*

0

(3.30)

where L* = p-'

- L,

M*

= Q-' - M ,

(3.31)

between the uniform stress and strain, which are compatible with the traction and displacement, respectively, on the matrix side of the ellipsoidal interface. These tensors L* and its inverse M* were introduced in this manner along with P and Q by Hill (1965a,c). We recognize (3.30)and (3.31)as a specialized form of (3.9) and (3.8),respectively; the present L* would be distinguished as L*(L,L ) in the previous notation which we shall contract now as L*(L).In place of theorem (3.10) we have that L*(L)> L*(Lo)

or L*(Lo)> L*(L)

(3.32)

according as the tensor Lo is chosen such that L - Lo or Lo - L is positive definite. This property of L* (together with its symmetry and positive definiteness) is found quickly on its own, as was shown by Walpole (1966b), once it is established that the strain energy in the matrix is (apart from a multiplying factor) EL*&,where E is the uniform strain in the inclusion. The tensor L* for a sphere is a fundamental one that depends only on the tensor of elastic moduli of the medium, and which reflects the same elastic symmetries. From (3.18) and (3.26)we may set down the relation of Eshelby (1957, 1961) between the constrained and stress-free strains, E and -9, respectively, as E

= -Sq,

where S = PL.

(3.33)

The expressions (3.27) and (3.29) for P may be converted accordingly into ones for S, as interfacial means of the idempotent tensor P*L, which (2.44)

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197

evaluates for an isotropic medium. Diagonal symmetry of the suffixes is not generally available for S. In S (or in I - S) there is an indication of the extent to which the matrix is able to accommodate the transformed inclusion, in that complete accommodation with no elastic straining would be realized when I - S is a vanishing tensor. The explicit solution reached here, as (3.27) or (3.29) for P and Q, amounts to that found otherwise (after a circuit of Fourier transformations) by Kneer (1965), Morris (1970), Willis (1970),Faivre (1971),Kinoshita and Mura (1971), Lin and Mura (1973), Kunin and Sosnina (1972,1973), and Laws (1977).The double integral calls generally for a numerical computation and to this end Ghahremani (1977) makes available a computer program for arbitrary anisotropy of the medium and arbitrary ellipticity of the inclusion. When the medium is transversely isotropic, Kneer (1965) has shown how the calculations proceed analytically, but his results for a spherical inclusion need the corrections stated by Hutchinson (1970, 1976), Lin and Mura (1973), and by Gubernatis and Krumhansl(l975);an alternative approach is developed by Chen (1968). When the ellipsoid is a “thin plate” with one of its axes very much shorter than both of the other two axes, then P* (and similarly Q*) becomes a constant that can simply be removed outside the integral of (3.27) to show for arbitrary anisotropy of the medium that

Pijkl= +(kiknjnl

+ kilnjnk+ kjkninl4- kjlnink)

(3.34)

where k is the inverse of the tensor h whose components are Likjlnknl, and ti is a unit vector in the direction of the short axis. This result was derived by Kinoshita and Mura (1971) in essentially the same way, and by Walpole (1967)in a more direct way which would allow also a nonellipsoidal approach to the limit of a thin plate. It should be noted that the neglected terms of the order of the thickness-diameter ratio may need to be restored for some purposes. When the elastic medium is transversely isotropic about the axis of the plate, P simplifies to the transversely isotropic form which was given by Walpole (1969), along with a corresponding form for a “needle-shaped’’ inclusion; both these results are reproduced by Laws and McLaughlin (1979). When the medium is an isotropic one whose tensor of elastic moduli is defined by (2.19) in terms of the bulk modulus K and shear modulus p, the preceding expressions simplify and, moreover, we can construct the entire elastic field for any shape of the inclusion in the manner explained by Eshelby (1957, 1959, 1961). The Green’s tensor now receives the familiar form 1 g i j = 4npIx’

-

XI

6.. IJ

K + i p a2

+

8 n p ( ~ - $p) axi a x j

/XI - XI

(3.35)

L. J. Walpole

198

as may be verified at once by the properties V21X’ - XI

2

= IXI

- XI

V2

’

1 Ix‘ - XI

=

- 4 n 6 ( ~ ‘- X )

(3.36)

of the Laplacian operator V2 (= d2/dxiaxi).By noting that d/dxi and - d/ax; operate equivalently on l/lx’ - to allow removal of all the differentiations as primed ones outside the integrals, to be denoted there by a comma, the expression (3.24)simplifies to

XI

(3.37)

in which there remain the integrals &x’)

= --

1

dV(x),

* ( X I ) = --

8n:

s”

Ix’ - xldV(x), (3.38)

that define in turn, in suitable units, the “harmonic” (or “inverse”) potential and the “biharmonic” (or “direct”) potential of attracting matter of unit density which is imagined for this purpose to fill the volume V of the inclusion (of arbitrary shape). We place the negative factors in front of these integrals in preference to letting them obtrude more frequently elsewhere (and now it can be shown that the second derivatives of 4 and the fourth derivatives of $, from which the stress and strain components are composed, each form positive definite tensors uniformly within an ellipsoidal inclusion and on volume average in any other inclusion). It is worth recording the various properties of the potentials, to aid a particular calculation of them, and to examine the properties passed on to the elastic field, and to be ready as well for calculation elsewhere in terms of gravitational harmonic and biharmonic potentials. In view of (3.36), the Laplacian V 2 ( = d2//ax;ax:) operates with the effect that v2*

=4,

v24 =

1 inside I/ 0 outside V ,

(3.39)

and so interfacial discontinuities in the second derivatives of 4 and in the fourth derivatives of $ are anticipated, although there is nowhere any discontinuity in 4 or its first derivatives or in $ and its first, second, and third derivatives, and nor, therefore, in the displacement field (3.37).In terms of the square-bracketed notation of (3.19), and in conformity with (3.39), it may be shown that [4,ij]

= ninj,

[$,ijkl]

= ninjnknL,

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199

and hence that [aij]nj = 0. At points at a large distance r from the inclusion, it follows that (b = - v/4 nr,

$ = - Vr/8n.

All these properties stated for a harmonic potential are well known, given, for instance, by MacMillan (1958).Those of the biharmonic can be deduced from harmonic ones by means of (3.39)and the relations

which stem simply from the scalar product (x’ - xI2 = (xi - xi)(x: - xi). Inside an ellipsoidal inclusion, (b takes the well-known form of a polynomial of second degree in the coordinates, as quoted by MacMillan (1958), and Eshelby (1957, 1961), while cc/ is the fourth-degree polynomial derived by Routh (1895) and by Eshelby (1959), and quoted by Walpole (1967). The components of the consequent uniform strain in the ellipsoid are evaluated by Eshelby (1957, 1961) and they are specialized to prolate and oblate spheroids by Kroner (1958) and to “slender” prolate spheroids by Russel (1973), and to other limiting cases by Brown and Clarke (1975, 1977).These authors have all tabulated Eshelby’s tensor S of (3.33),which has a transverse isotropy for a spheroid, but not diagonal symmetry. Related tensors, such as P and L*, can be readily calculated in terms of the notation of (2.14),but the full table of components need not be recorded here. When, in particular, the inclusion is a sphere of radius a, whose center is at the origin of the coordinates, it is comparatively straightforward to calculate with Eshelby (1961) that at a distance r from the origin 4=+a2(J;;i-l), 1 r2

+=hu4(----+!) 1 r4 1oa4

r2 a2

for r g a 2

u3 ($= - -1_ 3 r’ and hence that at any point inside the sphere

4 . . = :dij, - . ,I,

*

,rjkI ’.

--(1’5 -

dijdkl

+ dikdjl

-k

dil

djk),

in conformity with (3.39) and with the necessary symmetries of the suffixes. Substitution in (3.37) confirms the uniformity of stress and strain within the inclusion. By this means, or by a return to the alternative calculations of (2.43) and (3.29),the fourth-order tensors of (3.26)and (3.31)are calculated

L. J. WaIpole

200 for a sphere as

L* = ~ K *+J 2p*K,

M*

+ (1/2p*)K,

= (1/3~*)J

(3.41)

where 4

K*

= - 3p ,

p* =

3

1

10

+ --) 9K + 8p

’

(3.42)

where J and K are the fundamental isotropic tensors defined by (2.3), and where inverses of isotropic tensors have been calculated in the manner explained for the general tensor (2.5). We may note that each of these coefficients of J and K has the predicted positive value and monotonic increasing or decreasing dependence on K and p. The related tensor S of (3.33) may be given the expression

which is recognized as the interfacial mean of the tensor P*L of (2.44). C. INHOMOGENEOUS ELLIPSOIDAL INCLUSION As was shown by Eshelby (1961), the uniformity of the strain and stress fields within the ellipsoidal inclusion allows the polarization stress and strain z and q to be defined in terms of new prescribed constant values, the z1 and ql, respectively, of ( 3 4 , in such a way that the constitutive law (3.17) reverts to (3.2), namely to that appropriate for an “inhomogenous” ellipsoidal inclusion whose tensor of elastic moduli differs from that of the matrix. In keeping with (3.18), the appropriate substitutions for z and q are given by z = ( L , - L)E z,, q = ( M - M,)o + q1,

+

where E and 0 remain as the strain and stress fields attained in the inclusion. From (3.26)and (3.31)it follows in terms of the new polarization tensors that E

=

-PITI,

0

= Qiv~i,

(3.43)

where PI

= (L*

+ LJ-’,

Q1 = (M*

+ All)-’

(3.44)

Elastic Behavior of Composite Materials

20 1

and where Q1 = L1 - L,P,L,.

We have, therefore, evaluated for an ellipsoid the tensors PI and Q, which were introduced by (3.4), since the averaging overbar is not needed now. We discover next that the related tensors LT and MT of (3.8) are coincident with the present L* and M* and hence are independent of the tensor L , of the inclusion. This outcome is consistent with both “inequalities” (3.10): indeed it serves to show how an ellipsoid enables the right-hand one to be just attained as an equality. If the connections (3.44) are recast in terms of P and Q as P ( L , - LIP, = P

- Pi,

Q(Mi - M ) Q = Q - Q1,

their structure may be compared with (2.46). It follows that for the thin plate PI is obtained from P of (3.34) simply by substituting the tensor of elastic moduli of the inclusion in place of that of the matrix, to leave P , independent of the latter, but it should be noted that there are neglected terms of the order of the thickness-diameter ratio which depend on the matrix, and which may need to be restored for extreme values of the moduli. The left-hand “inequality” of (3.10) is seen now to be attainable as an equality by making the inclusion take the shape of a thin plate. By the procedure of Section III,A, the elastic field can be modified next so that at points remote from the inclusion there is the uniform strain and the uniform stress crA of (3.11). The resulting field of strain E~ and stress cr, is uniform within the ellipsoidal inclusion and so the overbars of (3.13) and (3.16) may be deleted. By bringing together (3.14)and (3.44),the tensors A , and B , may be evaluated explicitly for an ellipsoid. The elimination of P , and Q, leaves them expressed either as A , = (L*

+ L,)-’(L* + L),

B , = (M* + M,)-’(M*

+ M),

(3.45)

in terms of L* and M*, or as A,

= [Z

+ P(LI - L)]-’,

B,

= [I

+ Q(M1 - M)]-’,

(3.46)

in terms of P and Q. These final expressions cannot remain valid for a nonellipsoidal shape, except when the inclusion and the matrix have almost the same elastic properties, in which case they revert approximately to (3.15). Calculation of the elastic field in the matrix remains comparatively arduous to complete at all points, but just outside the inclusion there is the immediate connection (of Section II,E) with the interior field, which has been exploited in the examples of Eshelby (1957, 1961), Hill (1972), Kunin and Sosnina (1973), Kunin et al. (1973), Laws (1975, 1977), Levin (1977), and Walpole (1977, 1978).

L. J. Walpole

202

IV. The Elastic Field of a Composite Body A. FIELD EQUATIONS We are concerned now with the elastic behavior of a composite body made: up of an arbitrary number (n say) of different solid phases bonded firmly together to fill a total volume V. Each phase is assumed to be homogeneous, while unrestricted in its geometrical structure and in its relative concentration by volume; it may consist of a number of separate parts or of a continuous whole. Its constitutive relation between the stress cr and the strain E may allow for an arbitrary anisotropy. In all parts of the rth phase say, this relation is expressed as = L,E,

=M p ,

(4.1) so as to introduce the constant tensor of elastic moduli L, and its inverse M,, both of which are fully symmetric in the manner of (2.1) and are positive definite, for a real material. Any one phase is allowed to be made of the same elastic material as another (or all of the others as in a polycrystal for which n is indefinitely large) but, in having its own orientation of the crystallographic axes, it is distinguished by its appropriate tensor of elastic moduli (relative to the fixed Cartesian axes). In every phase there are the customary equilibrium equations, without body forces, and also the equations which derive the strain components from the components of the displacement vector u, namely 5

’

0.. V.I‘ = 0

Eij

E

=3(qj

+ uj,i),

(4.2)

respectively. It is assumed that the bonding at the interfaces between phases always remains intact whenever the composite body is placed in an equilibrium state of infinitesimal elastic strain: all the traction and displacement components remain continuous across every interface. The displacement components are to be continuously differentiable within each phase (for there are no “stress-free” strains or “dislocations”).At the outer boundary of the body, or in any part of it that is imagined to recede to an infinite distance, the appropriate requirements must be met so as to leave the elastic field uniquely determined, in the strict mathematical sense or perhaps in a more practical, “Saint-Venant” sense. Needless to say, however, a complete description of the internal elastic state is quite beyond the reach of any analytical or numerical approach, while the geometry remains prescribed arbitrarily to the utmost intricacy or is perhaps even unknown to some detailed extent. We must turn eventually to special cases or to approximate approaches or, for practical purposes, to a description of the grosser rather than the finer features of the elastic behavior. More as an aid toward modeling this last

Elastic Behavior of Composite Materials

203

alternative, than toward particular solutions, we first convert the mathematical problem into an “integral equation.” The method of integral equations, or what might otherwise be called the “method of singularities,” following the classification of Love (1954: Historical Introduction), is of wide, albeit usually arduous, application. Although it has not been very fully exploited yet in the present context, we may select the examples of Kupradze (1963), Knops (1964), Rizzo and Shippy (1968),and Willis and Acton (1976) to provide some inspiration. When the geometry is greatly simplified, with a strictly limited number of usually no more than two phases, the alternative “method of series” has proved more popular in giving rise to an extensive literature which is still pursued, for example, recently by Chen and Acrivos (1978a), and which was surveyed at an earlier period by Sternberg (1958). Approximate approaches can be assisted often by an appeal to “extremum principles,” especially to those developed for the present context by Hashin and Shtrikman (1962a), and reappraised by Hill (1963b), and by Walpole (1974).

B. INTEGRALEQUATIONS Considerable insight into the elastic behavior of the composite body can be gleaned eventually to a surprising extent indeed if the constitutive relations are first simply recast as

+

a = Lo& z

(4.3)

in all phases, where in the rth phase say (at a point x) the connection z = (L, - Lo)&

(4.4)

identifies implicitly the symmetric, second-order, “polarization-stress”tensor z. The tensor of elastic moduli Lo, assumed to be symmetric and positive

definite, is introduced as that of some “comparison” body, chosen for convenience to be entirely homogeneous (and perhaps in due course isotropic). If it is imagined that the tensor z is known explicitly in (4.3), so that (4.4) can be discarded, then all reference to the elastic moduli of the composite body is removed in favor of the more amenable comparison body. Since o is defined in the composite body as a stress field satisfying the equilibrium equations, with a continuous traction across the interfaces and perhaps with a prescribed traction at parts of the exterior boundary, we can envisage that E is preserved as the strain field in the comparison body, along with the stress field Lo&,provided that appropriate distributions of body force are derived from z and are introduced throughout the volume, over the interfaces and where necessary over parts of the boundary. From the displacement

L. J. Walpole

204

field of a single point force in the comparison body, there will be derived an integral expression for the displacement field u in the composite body, in terms of z and of the displacements and tractions at the boundary. When the implicit definition(4.4)is restored for z, an “integral equation” will remain. First, we appeal again to the definitions (3.20),with a suffix o attached now to the tensor of elastic moduli, in order to establish the “primed” field of a point force acting at a point X I , in an unbounded mass of the comparison body. Then, by resorting to the divergence theorem to convert the integrals over the exterior boundary S into integrals over the volume V occupied by the composite body, we may derive, by means of (4.2) and (4.3) and the symmetry of Lo,the reciprocal relation

=

sv

&‘Lo& dV

=

sv sv

n

dadV -

dzdV

n

where d S j stands for njdS, the product of the component nj of the outward unit normal and the boundary element dS, and where the middle equations take the opportunity to suppress a clash of suffixes. In order to eliminate the arbitrary components fi of the point force, we can make the substitutions u:(x)

a i j ( x ) = hEij(X’ - x ) f k

= gyj(x’ - x ) f j ,

say, where a superscript o has been attached to the Green’s tensor of (3.22) and where hiij replaces for concisenessthe unwieldy product Loijmlg&, whose comma denotes a differentiation with respect to the unprimed coordinates, in terms of which the integrations are performed. It only remains to let the delta function of (3.20) make its presence felt, in order to transform the reciprocal relation to Ui(X’) = -

+

sv

&j,k(X’

- X)‘Cjk(X) d V ( x )

ss

&j(x’ - X)ajk(X) d S k ( x )

n

An alternative but less compact statement is reached after separating the first integral over the volume I/ into the difference of two such integrals by writing its integrand as (g?.‘C. ) V ~k ,

- g’.z.J k , k , 1.J

Elastic Behavior of Composite Materials

205

and after converting the first of these by the divergence theorem into a sum of integrals over the exterior boundary S and over all the individual internal interfaces (to allow for the discontinuity there of the “traction” components of z): the anticipated way in which the polarization stress z induces distributions of body force in the comparison body is thereby exhibited explicitly. The constant elastic moduli of the comparison body are best left so far as unspecified parameters, for, although they cannot persist as such in the exact calculation of the elastic field, an appropriate choice of them may assist a particular method of solution, or improve a means of approximation. For a similar purpose, we may employ the divergence theorem (and again the properties of the Dirac delta function and the symmetry of Lo) to recast (4.5) as

in order to incorporate uA as an arbitrary (nonsingular and continuously differentiable) displacement field and aA as any stress field satisfying the equations a+.= gA g. .IJ J1’ $,J - 0, of symmetry and equilibrium, while the relations OR

= Lo&A

+ TA,

;&

= $(uC

+ u$)

(4.7)

define the symmetric tensor T~ as an accompanying polarization stress. We proceed to the special case for which everywhere outside some internal interface the composite body consists of a single homogeneous phase, the matrix, whose tensor of elastic moduli is L with inverse M say, while there remains within the interface an “inhomogeneity” of n phases. The displacement field uA(x)is identified now as that toward which u(x) is prescribed to tend at all infinitely remote points, with the accompanying strain cA. Correspondingly, we choose the definition oA= L E A ,

T* = ( L - L,)&A,

which equates cA to the (equilibrated) stress at remote points. We return next to the integral equation (4.6)to let the surface S recede to an indefinitely remote position in the matrix so that both integrals over it are made to vanish, provided (as is soon verified) that the remote values of the elastic

206

L. J. Walpole

field are attained rapidly enough. There remains the statement that

where the integration spreads right out to infinity (and remains convergent), except when Lois chosen coincident with L to make the constant T~ disappear entirely and make zvanish everywhere in the matrix. It is convenient to make this choice of Lo by which the integration remains extended only over the “inhomogeneity” whose volume is denoted anew by V. When the suffix o is removed everywhere to identify the Green’s tensor as that of (3.22), we have that

where (4.3) and (4.4) now specify that

z = 0 - L&= (L, - L)&

(4.10)

in the typical rth phase of V.We are left accordingly with an integral equation, since z cannot be specified explicitly in advance, except perhaps in an approximate fashion to generate an approximate solution. For example, when all of the phases (including the matrix) have almost the same elastic properties then can replace E approximately on the right-hand side of (4.10). We may observe when x’ is a point sufficiently remote from V , that (with the neglect of terms of order l/lxf - xi3) ui(x’)= Uf(X’) - vzjkgij,k(x’- x)

(4.11)

where x stands now for any chosen point within or near the volume V (for instance, the coordinate origin) and where

z = (1/V) j z ( x )dV(x) = a - LF. The remote elastic field is expressed in this way in terms of the average polarization stress or of the average stress and strain within V , where such averages are denoted by the overbar. A helpful relation between these averages (and hence between the averages over the individual phases) may be found at once when the outer boundary of V that separates it from the matrix is of a spherical or ellipsoidal shape, for which the tensors P and Q of (3.26) are defined for a homogeneous material coinciding with that of the present matrix. First, (4.9) is differentiated through with respect to xi to obtain what may be regarded as an integral equation for the displacement gradient ui,j, which converts into one for uj,i and (by addition) into one for the strain field E. This last integral equation for E is integrated over the volume V and then the order of integration in the repeated integral is interchanged

Elastic Behavior of Composite Materials

207

in order to take advantage of the special simplicity of the integration of (3.24) at points inside an ellipsoidal volume. There follows, consequently, the relation & = B A - p y = p - p( 0 - LZ) (4.12) and hence the equivalent relation -

G = iTA - Q ( B - M S ) .

(4.13)

If the volume V consists of just a single homogeneous phase of ellipsoidal shape, as an inclusion surrounded by a matrix, then the relation (4.12) is that obtained from the integral equations by Knops (1964, for isotropic materials).It can be solved to relate the average strain or stress in the inclusion to the averages there of the fields that remain at infinity, in the same way as (3.16) and (3.46)do for the case of the uniform fields at infinity. It is worth emphasizing the one special, yet significant, example, brought out in turn by Eshelby (1957,1961), Hill (1963a),and Knops (1964),for which an exact solution of the integral equation (4.9)can be constructed remarkably enough for any shape of the volume I/ and for any distribution of the phases within V. It requires that the matrix and all the individual phases be made of isotropic materials which all have the same shear modulus p. With K denoting the bulk modulus of the matrix and K, that of the rth phase of I/, we may substitute the form (3.35) for the Green’s tensor and specify in place of (4.10) that z = (K, - K)Eh

in the rth phase, where E ( = E ~ is~ the ) dilatational part of the strain. Differentiation of 1/lx’ - with respect to the unprimed and primed coordinates can be interchanged, by insertion of a minus sign, to allow the removal of the primed differentiation outside the integrand of (4.9),which simplifies now to

XI

(4.14)

where the suffix V, confines an integration to the rth phase, while the summation extends over all the phases of Y. The integral over the points x of V , may be recognized as the harmonic potential at points x’ of matter of density E ( X ) imagined to fill the volume V,. It is well known, as a consequence of (3.36), that the result of letting the Laplacian a2/ax:ax: operate on this integral is -4nEW) or zero according to whether the point x’ lies inside or outside V,. Differentiation of (4.14) with respect to xi implies that K, - K

E(x’) = EA(x’) - -E(xt) K

+ +p

208

L. J. WalpoIe

at points x’in the rth phase, and so the dilatational part of the strain can be solved for explicitly in each phase, in terms of the dilatational part EA ( = E;) of the prescribed strain field E ~ ( x ’ )Substitution . in the integrands of (4.14) leaves, finally, an expression for the displacement field at all points x’ inside or outside V, namely,

V. The Overall Elastic Behavior of a Composite Body A. OVERALL ELASTICMODULI First we need to decide for which circumstances an “overall” (or “macroscopic”) description of the elastic behavior of a composite body is appropriate, as a preferable practical alternative to the detailed pointwise (or “microscopic”)one that was left in Section IV. At the same time the ground must be prepared for a mathematical model which, perhaps couched in several alternative forms, relates the two points of view. The microscopic length scale (I say) is an order-of-magnitude measure of the heterogeneous spacial structure of the composite body, for instance, the distance between inclusions or the diameter of polycrystalline grains. We seek an overall description that refers to a whole range of practical examples of composite solids undergoing infinitesimal elastic straining, rather than one that would attempt to apply to every conceivable case, or just to special untypical ones. The composite material is understood to be “macroscopicallyhomogeneous” (or “statistically homogeneous”) in its elastic behavior. It is implied thereby that, in response to appropriate (“macroscopically uniform”) external loads or constraints, an ample sample can sustain stress and strain fields which when viewed on a scale sufficiently large (in comparison with the microscopic level at which rapid fluctuations occur) appear to have spacially uniform values coinciding with their respective averages over any (“representative”) volume V , where V1I3 >> 1. Either of the average (“macroscopic”) strain or stress can be prescribed arbitrarily, but they are to remain related linearly by means of a fixed fourth-order tensor of “overall elastic moduli,” which retains no dependence on the dimensions of V, and which characterizes uniquely a particular “macroscopically uniform” composite material. If the sample of composite material is imagined to be replaced by a purely homogeneous material whose tensor of elastic moduli coincides with the overall one then the uniformity of the stress and strain fields is sustained precisely, not just

Elastic Behavior of Composite Materials

209

in the statistical sense. This replacement may also remain appropriate (to determine the macroscopic field) in circumstances where the apparent statistical uniformity is relaxed moderately to a nonuniformity perceptible only over distances very large compared to V1/3. Our postulates could naturally be set in terms of more explicit statistics, especially to bring out alternative viewpoints (of ensemble averaging)together with any limitations and refinements; surveys of progress in this direction are offered by Beran (1968, 1971) and Kroner (1972).We follow the foundations laid down in the present elastic context by Hill (1963a, 1967) and Hashin (1964, 1965a), and in an analogous one by Batchelor (1970,1974).Naturally not all the numerous developments can be pursued here; we do not venture into nonlinearly elastic or inelastic domains. The reviews of Hashin (1964) and W&t et al. (1976) can supplement our list of references. For our primary purpose of determining the tensor of overall elastic moduli, we suppose that over the bounding surface S of a (macroscopically homogeneous)representative volume V ,the displacement components ui and the traction components crijnj fluctuate “microscopically”in a wavelike way about the “macroscopic” values Bijxj and C i j n j ,where 5 and 0 are constant symmetric tensors and n, is the component of the outward unit normal. (There is no loss in not letting the displacement components allow for an arbitrary rigid-body translation and rotation of the whole volume.) The microscopic fluctuations have a wavelength which is to be very small compared with the overall dimensions (such as the “diameter”) of V, while their amplitude is independent of these dimensions. At the boundary S it is presumed that both the displacement and traction vectors will take simultaneously the “macroscopically uniform” form just described whenever by the external loading or constraints one or other of them is chosen to do so, possibly without any fluctuation at all. At a11 interior points, except in a narrow boundary layer, the fields of stress cr and strain E are effectively determined uniquely, in terms of whichever of F and F is prescribed (and eventually in terms of the other after the tensor of overall elastic moduli is calculated). In these circumstances, we may select the four integrals

J [ ( &- F i k X k ) n j f J(crij - Fij)njdS, s(cij

J[(bik

(uj

- zjkxk)ni]

- 8ik)xj

+

(ajk

ds, - 8jjk)xi]n,dS,

(5.1)

- @ij)nj(ui - F i k X k ) dS

as ones that can be each assigned a negligibly small, possibly vanishing, value when the integration is confined on S to any small surface element whose diameter is of the order of the wavelength of the microscopic fluctuations.

L. J . Walpole

210

Furthermore, when the integrations are all extended over the whole of S the resulting values remain negligible in comparison to the terms retained, the volume V having been made sufficiently large. We apply the divergence theorem to convert each of these four integrals over the closed surface S into ones over V , while we refer back to (4.2),to derive the strain from the displacement field and to make the stress field satisfy the equations of equilibrium in the absence of body forces. The second integral then vanishes automatically while the vanishing of the others implies in turn that

$ J C r i j E i j dV

= $ V 8EJ. 8lJ...

(5.2)

In other words, B and 8 are to be identified as the overall average strain and stress, respectively, in V (as the overbar has anticipated). They are expressed here entirely in terms of surface values of the displacement and traction, respectively (pointing to an experimental determination). The other result (5.2) is to be returned to presently to evaluate the total strain energy in terms of the overall quantities. The overall averages can be derived of course from the averages over individual phases. A suffix r will distinguish an average over just the volume V, occupied by the rth phase. Then E

= Ccp,,

(5.3)

where c, (= V,/V)is the fractional concentration by volume of the rth phase and the summation is over all n phases (so that Cc, = 1). The average strain and stress in the rth phase have a unique linear dependence on the overall values which we can write as -

E, =

A&

8, = B,B,

(5.4)

in terms of fourth-order, “concentration-factor” tensors A, and B,, either of which need only be determined for any n - 1 of the n phases, in view of the immediate connections C

Cr B = I .

(5.5)

By recalling the stress-strain relation (4.1) for each phase, (5.3) and (5.4) together allow the relation between the overall average stress and strain to be expressed as 8=

LZ,

8 = M8,

(5.6)

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21 1

say, where L , designated as the tensor of overall elastic moduli, and its inverse M are specified in terms of the phase tensors by L = Cc,L,A,,

M = Cc,M,B,.

(5.7)

The interrelation between A , and B, that was implicit in (5.4) can be put simply now in two ways as L,A, = B,L, M,B, = A,M. (5.8) A, and B, do not generally have (for any Y) the diagonal symmetry, though they must comply with the overriding requirement that I and also (we shall see) L and M take their place in (5.5) and (5.7) as diagonally symmetric and positive definite tensors. Furthermore, they must reflect in L and M any appropriate symmetry thrown up by the elastic properties of the phases and the geometrical structure of the composite (from the details of which they are to be calculated).It can be helpful to combine the relations (5.5) and (5.7) as

where the tensors Lo and M , are chosen in any convenient way, say as some “average” of the various L,.‘s and M,’s, or simply as coincident with a particular L, and M , , respectively, so that the corresponding A, and B, disappears to leave only the remaining n - 1 A,% and B,’s to be determined. The overall tensor L and its inverse M are defined by (5.7) as “weighted averages” of the individual L,’s and M,’s, respectively. The necessary connections (5.5) are satisfied on the left-hand or the right-hand side when every A, or every B,, respectively, is equated to the identity I , as simple approximations. The first of these assumptions entails a completely uniform strain fieldB throughout the composite (but leaves an artificial,unequilibrated stress field). It was proposed originally by Voigt (1928) for a polycrystalline aggregate. In that context, Reuss (1929) put forward the dual assumption that the stress field is uniform (while the necessary bonding between the different phases is not maintained). When all the A,‘s and B,’s are replaced by I , there remain in (5.7) two different (diagonally symmetric and positive definite), so-called Voigt and Reuss estimates of the overall tensor L ; namely, the Lv and LRdefined as Lv

= CcrLr,

L , = (CcrMr)-

>

(5.10)

so as to retain a weighting simply with respect to volume concentration only. It may be shown that they have a little more than mere simplicity to support

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L. J. Walpole

their early following in the literature. We may note for the present that the Voigt and Reuss estimates reach together their greatest accuracy when the elastic properties of all the phases (and hence of the overall behavior as well) are almost the same. Moreover, substitution for A, and B, in (5.9) indicates then that only second-order infinitesimals are given up, whereas first-order ones might seem to have been lost by the substitution in (5.7).However, when the differences between the phases become more pronounced, accuracy is inevitably sacrificed. It is significant that the evaluation of the overall elastic moduli and, in1 (5.2), that of the total strain energy E are brought together now by the connection 2E/V = (l/V)JoedV

= OE

=EL% = @ M S .

First, there is the opportunity to confirm quite generally that L and M are diagonally symmetric and positive definite, just like all the individual phase tensors of elastic moduli. It is only necessary to follow the procedures that revealed these very properties in the fourth-order polarization tensors Pl and Q1of Section II1,A. Second, an alternative route to earlier relations such as (5.9) can be followed. Third, familiar elastic extremum principles can be called upon to restrict the magnitude of the energy and to reveal, by gradually deeper and more specialized analyses, further fundamental properties of the overall elastic tensors, beyond those which intuition might envisage. The overall elastic moduli are found to have a monotonic dependence on the individual phase moduli, as shown by Hill (1963a)in a theorem that extends beyond the present heterogeneously elastic circumstances,with implications drawn for a few examples by Walpole (1970b). Thus if the tensors of elastic moduli of any number of the phases are all imagined to increase (or to decrease) by positive definite amounts, to “strengthen” (or to “weaken”) the composite body, then, whatever be the particular geometrical configuration, the overall tensor is made to increase (or to decrease) likewise by a positive definite amount. By imagining that all the phases are first “strengthened and, secondly,“weakened”just far enough to leave a purely homogeneous medium (whose elastic moduli coincide with its overall ones), it is shown at once that L is at least restricted to lie intermediate to the various L,’s. For example, if all the phases are isotropic and if L is likewise isotropic then its overall bulk and shear moduli are increasing functions of all the individual bulk and shear moduli and are placed between the greatest and least bulk and shear modulus, respectively, to make (in particular) for a coincidence with a modulus that is common to all phases. A rather stronger restriction on the position of L can be brought out when the volume concentrations are known, by introducing the Voigt and Reuss approximating fields into extremum principles

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in the way shown by Hill (1952, 1963a). It is expressed by the “inequalities” (5.11) LR < L < Lv which place the estimates (5.10) at positive definite amounts above and below L. The magnitudes of the individual components of L are confined accordingly to the extent allowed by the difference between Lv and LR, which is of second order when the phases differ only slightly in their elastic properties, but which can be otherwise unhelpfully large. It may be possible to reduce the number of independent components of L that must be dealt with if enough is known or presumed known about the general structure of the composite body. Such limited information has not entered the derivation of (5.11)and so it remains to be shown how it can improve both the Voigt and Reuss estimates and restrict further the position of L. At the same time (Section V,F) the theorem (5.11) can be recovered and confirmed to remain always deficient in that neither the Voigt nor the Reuss estimate can ever be attained precisely by a particular example (except trivially by a homogeneous body). Suppose for illustration that L takes the isotropic form written as L = ~ J C-I-J2pK,

(5.12)

where J and K are the elementary isotropic tensors of (2.3) and JC and p are called the overall bulk and shear moduli respectively. Lv and L, will be calculated then presumably to have the likewise isotropic forms Lv = 3 ~ v Jf 2pVK,

L, = ~ K R fJ 2 p ~ K ,

where the suffixesV and R distinguish the Voigt and Reuss moduli, respectively, which estimate and bound according to (5.11) the overall moduli:

< IC < ICV,

(5.13) pR < p < p V . For example, if all the phases are isotropic, with bulk modulus K , and shear modulus pr for the rth phase, we calculate from (5.10)that ICR

(5.14) Another example is that of a polycrystalline aggregate of single crystals that are oriented quite randomly with no general preference for any particular orientation. The tensor of elastic moduli of a crystal, denoted by Lc with inverse M C ,will then have components that vary with orientation. However, the two invariants L:jj and LSij remain unvarying and can therefore be evaluated at any particular orientation. Moreover, it follows that

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L. J. Walpole

and by means of (2.6) we arrive at two equations, 9KV = LCiij,, ..

3 ~+ v lop, = Lijij, C

to solve immediately for the moduli defined by Voigt (1928). Similarly, we obtain two equations 1/KR = MEjj,

113k-R

+ 5/2pR = M&j,

to solve for the moduli defined by Reuss (1929),for any type of anisotropic crystal. The Voigt and Reuss moduli have been calculated already in Section II,B for the two familiar examples of a cubic and a hexagonal crystal. The case of only two distinct phases ( n = 2) is a commonly occurring one worth distinguishing for its individuality. It is easy to adapt the analysis of Section III,A to show that A , and B, (and similarly A , and B,) can be constructed as A1

=I

+ Pi(L2 - Li),

B1 = I

+ Ql(M2 - M i ) ,

in terms of “polarization” tensors P , and Q1 akin to those of (3.14)and possessing similar general properties, such as the diagonal symmetry and the positive definiteness. Then after substitution in the two-phase counterpart of (5.9) with L, and M , identified as L , and M,, that is in L = L2

+ C,(Li - L,)AI,

M = M2

+

CI(M1

-MJB,,

we find that L and M can be given a somewhat more specific structure as

+ c,L, - c,(L, - L,)P,(L, - L,), c , M , + c ~ M Z- cI(M1 - Mz)Q,(M, - M2).

L = c,L, M

-Z

They are displayed thereby as diagonally symmetric and positive definite tensors, since these properties are possessed already by P,, Q L,, and L,. Indeed the positive definite property of P , and Q1goes farther on its own to establish the theorem (5.1 1) for a two-phase body, while the algebraic calculation

,,

Lv - LR

= (Ll - L,)[(l/C,)L

+ (l/Cz)L23- Wl - L2)

indicates explicitly the extent of the restrictions that it implies. For a composite body of two isotropic phases whose overall tensor is likewise isotropic (as are therefore the tensors A , , B , , P,, and Q,) the consequent upper and lower bounds on the overall bulk, shear, and related moduli are set out in full detail by Hill (1963a),with a reference to Paul (1960).Hill (1964)considers in detail as well an important example where the overall tensor of elastic moduli is only transversely isotropic about some given axis and where the phases

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215

need only be likewise transversely isotropic, but where there remain, remarkably, only three rather than five independent overall moduli. This example is that of a fiber-strengthened material whose phases extend continuously in the axial direction, with one phase embedded as aligned cylindrical fibers in the other. B. INTEGRAL EQUATIONS It is rewarding to return to the integral equation (4.6) to underpin the proceedings and to be guided toward various alternative models that evaluate the overall behavior. This integral equation refers to a quite general problem for a composite body and it may be transferred (in the same notation) to the present problem where a “macroscopically homogeneous” body is assumed to have “macroscopically uniform” surface displacements and surface tractions. We shall take it for granted that the approximations within this assumption give rise to no inconsistency and that, in principle, they are capable of rigorous mathematical support by means of the integral equation. It remains to examine the consequences and to be satisfied as far as possible of their consistency. First, by a suitable selection of the A field, it is found possible to greatly simplify the integral equation by the removal of both its surface integrals. For we may let this field have a completely uniform strain and stress at all points x‘ by the definition EA

= s,

(T*

= 3,

UP 1 = E..x’. U J

(5.15)

by which, with the help of (4.3) and (4.7), the accompanying polarization stress becomes defined as T* = 5 -

=T ;

that is, as the overall average of the polarization stress for the composite. Each of the surface integrals in (4.6)can be written as the sum of individual integrals, each one of which is over a small surface element whose size is of the microstructural dimensions. During the integration over such an element, the factors g$(x‘ - x) and hijk(x’ - x) can be considered to have a constant value at all interior points x’, except those very near the element; that is, within the microstructural distance of it. But then each constant factor is left multiplying one of the integrals of (5.1) to which we assigned a negligibly small value. In conclusion, as a reflection of the assumed “macroscopic homogeneity,” the two surface integrals in (4.6)may be deleted, provided that the total interior volume V is taken large enough (with a sufficiently smooth and not excessively re-entrant boundary S ) . Their only significant contribution could be at points x’very near the boundary S, but (in the spirit of a

216

L. J. Walpole

St. Venant principle) these contributions would be lost in the volume averages in terms of which the overall behavior is assessed. The integral equation (4.6)is superseded by u.(xl)= 2.r j.XI.j - Jv gfj,k(X’- x)Czjk(x) - yjkl d v ( x )

(5.16)

where z = 0 - Lo&= (L, - Lo)&

in the rth phase.

(5.17)

There is an alternative definition of the A field which brings about this same outcome. We imagine that all the composite material within the boundary S is replaced by any completely homogeneous material with the retention of either the same traction vector or the same displacement vector, at all points of the boundary. We identify ~’(x’)with the stress field in the homogeneous material in the first case and uA(x’)with the displacement field of the second case. Both the surface integrals of (4.6) can be deleted, since each has a completely vanishing integrand. Moreover, the definitions (5.15) will still hold, except at points x’ very near the boundary. Hence it is found that the final integral equation may be set down once again, with the neglect of terms that are significant at points x’ near S, but which would make a negligible contribution to the phase averages. Next we note that by a sequence of immediate operations (beginning with a differentiation with respect to xi), (5.16) can be converted into other forms which can be regarded in turn as integral equations for the strain field E or for the polarization field z, for example, and which can each be made to yield in principle the required averages over the phases. Naturally, the arbitrary tensor Lo must disappear when exact calculations are finally completed, though it might well remain in approximate ones, in which case (sooner or later) some convenient or perhaps some optimal definition can be specified. The integral equation (5.16),converted to one or other of its equivalent forms (in the present or in analogous physical contexts), has been arrived at in various ways (with or without the arbitrariness of Lo) by a number of authors, for instance, Korringa (1973), Willis and Acton (1976), Willis (1977) and O’Brien (1979); we shall also examine an unrecognized connection with the earlier approach of Brown (1955, 1965). Zeller and Dederichs (1973) and Kroner (1977) point to a resemblance with the Lippmann-Schwinger equation of quantum mechanics. Willis and Acton (1976)and OBrien (1979)consider the detailed application to particular examples (for which Lo is fixed as the tensor of elastic moduli of a matrix phase). O’Brien (1979) is able to place the application of the integral equations on a sound footing, which enables him to correct the shortcomings of earlier authors. The volume of the integration in (5.16)may

Elastic Behavior of Composite Materials

217

be extended indefinitely; it can be verified that the infinite integral converges at all finite points x’in a way that keeps the overall average of the strain field E coincident with8: (cf. Korringa, 1973;Willis and Acton, 1976; OBrien, 1979). We may note that if the boundary S of the volume V is of (finite) spherical or ellipsoidal shape, then the integral equation (5.16) will in any event (even without the “macroscopic homogeneity”) give to the calculated strain field an overall average equal to 8. This result follows if we apply to (5.16)the same steps which took the integral equation (4.9)to the result (4.12), since in the present instance the average of z - Z is a vanishing quantity. The integration over an indefinitely large volume V in (5.16) may, in fact, be replaced conveniently by an integration over a “large” sphere or ellipsoid inscribed inside V (though generally we shall retain the arbitrary shape). For in excluding only integrals over “narrow” boundary layers, such a replacement may be expected to be consistent with the approximations already made. When an arbitrarily ellipsoidal shape is assigned to the boundary of the volume V, the volume integral in (5.16) can be written as the difference of two integrals (after removing the square brackets in the integrand), the second of which is known (from Section II1,B) to be a linear function of x’.By combining the two linear terms, the integral equation (5.16) may be taken to the form Ui(X’)

= E0.X‘. 1’ J -

J”

g;j,k(X’ - X ) T j k ( X ) d V ( X ) .

(5.18)

The constant tensor E’ is given in terms of 8 and 7 or, alternatively, it may be regarded as an unknown constant tensor in terms of which 8 and i3 are to be evaluated. Elimination of E’ will then leave the desired relation (5.6)which determines the tensor of overall elastic moduli. The present integral equation (5.18) can be compared with the earlier one (4.9) in order to infer that it could be noted at once if (instead of prescribing the kind of displacement and traction to be found at the boundary S of V )we imagine that the volume Y is bonded at its ellipsoidal boundary S to some homogeneous matrix whose tensor of elastic moduli is Loand whose strain field at remote points tends to the uniform one E*. This latter model, with its accompanying integral equation, was postulated intuitively in its own right (with an early special choice of Lo and without the present mathematical basis) in the approach of Brown (1955, 1965) in an analogous physical context, which Davies (1971a,b) transferred to the present elastic context. The foregoing procedures are designed to make explicit calculation of the elastic field, or of its required averages, and hence eventually of the tensor of overall elastic moduli. However, there is some advantage in diverting to an implicit approach which defines Lo in terms of L and which incorporates

L. J. Walpole

218

the relation (5.6) earlier on, rather than leaving it to be called upon at the ultimate stage. First, we note that the average of z, from (5.17),may be placed in an implicit form as -

z = B - LoS = ( L - L,)B

Next we identify Lo with the overall tensor L, as a natural definition which, conveniently allows Z to be assigned a vanishing value. The integral equation (5.16) is therefore recast now with the suffix o omitted as n

(5.19) where

z = 0 - LE= (L, - L)E

in the rth phase,

and where B is to remain identified as the average strain in the volume V., to ensure that L is indeed the tensor of overall elastic moduli. Significantly. this integral equation has reverted in its final guise to the earlier (4.9), with the strain field there, ~ ~ ( x made ’ ) , uniform and identical to B for the present purposes. Not only does it offer on its own a means of solution, but by referring back to its earlier derivation it makes way next for a new versatile model which assesses the overall behavior and calculates the detailed elastic field at all points, either exactly or by its own approximate approaches, and which, when postulated in its own right, recovers the integral equation. C. SELF-CONSISTENT MODEL The volume V of composite material (of a “macroscopically homogeneous” structure) is imagined now to be bonded at its boundary S to an unbounded homogeneous matrix whose tensor of elastic moduli is identical to L , the overall one to be determined. The elastic field at all remote points is to be one of a uniform strain accompanied by the uniform stress nA,where uA= L E ~ ,

=M G ~ .

(5.20)

The full strain and stress fields, E and 0,conform to this same constitutive relation in the matrix and to the earlier (4.1)in the rth phase of V ;E is derived by (4.2)from the continuous displacementfield u, while 0 satisfies the equilibrium equations in the absence of body forces. At the interface S, just as at all interior interfaces, the components of traction (and of displacement) are required to be continuous. With the overbar denoting an average over the volume V , the relations E

=

O = o*,

5 = LZ,

T = Ma,

(5.21)

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219

are introduced, compatibly with (5.201, so as to each provide a tensor equation to be solved uniquely (it is presumed) for the symmetric, positive definite tensor L. That these relations are indeed compatible with each other can be observed more explicitly (without even requiring the macroscopic homogeneity) when the interface S is assigned a spherical or ellipsoidal shape, since then there are the consequences(4.12) and (4.13)of the integral equation (4.9). The average strain and stress in the rth phase of the volume V can be placed as E , = AleA, 5, = BpA, (5.22) in terms of concentration-factor tensors which are related immediately by (5.8) and which are identical to those introduced previously by (5.4),although they are now expressed implicitly in terms of the overall elastic moduli. This implicit dependence gives a different status to the four relations (5.5)and (5.7) which are recovered, in turn, from (5.21), since a unique solution for L is offered by each one as it stands or in the combination (5.9). There are yet other noteworthy statements of the condition that determines L. First, we observe that the displacement components (4.11), at all points x’ which are remote from every point x of the volume V , have lost their leading term of order l/lx’ - xI2, because the coefficient Z has been made to vanish. The significance of the disappearance of this term, as a means of determining the overall elastic moduli, was postulated in greatly simplified circumstances, where spherical inclusions of one phase are dispersed dilutely in a matrix of a second phase, by Frohlich and Sack (1946) and (we may perhaps infer) by Maxwell (1873, in an analogous physical context). By the vanishing of the term it is implied that in the entire volume enclosed by an interface C lying remotely distant from I/ we can evaluate the average strain, average stress and total strain energy simply in a manner not dependent on the shape of C. Let W denote the volume intervening between the interfaces S and C. We calculate that

3 S(uinj t-ujni)dC= (V + W)E& JCTijdV = 4S(OikllkXj + ajknkxi)dC= (V + w,CT;, SCTijEijdV = JoijnjuidZ = (V + W)CT!.&!. JEijdV =

IJ

rJ’

(5.23)

where each volume integral over the entire region inside C is converted to an integral over C by the divergence theorem, after which the remote form of the elastic field is substituted and negligible terms are omitted. We are brought finally to the conclusion that the volume V , the volume W , and the entire volume of V and W together, each have the same average strain cA, the same . these average stress oA,and the same average strain energy $ 0 ~ 6 ~From

220

L. J . Walpole

various equivalent statements, we choose the evaluation (5.23) of twice the strain energy to be returned to with extremum principles to seek bounds om the components of the tensor L (which relates uA and rA). Our sequel departs from the developments elsewhere in the literature by proceeding in terms of the problem just described for an infinite elastic: body, whose interfacial requirements of continuity can be met readily for our purposes, in preference to the earlier alternative of a finite composite body subject to boundary requirements that generally cannot be met precisely and which call for an extended argument in order to be met in the correct macroscopic sense. We are rewarded with a briefer and clearer presentation even though it must carry an implicit dependence on the undetermined overall elastic moduli. The procedure that embeds the composite volume V in a homogeneous matrix with tensor L of elastic moduli (in place of an extension of the composite material) has an intuitive appeal that was perceived to some extent, first, by Bruggeman (1935, 1937) and by Frohlich and Sack (1946). However, it seems to have lacked hitherto a secure mathematical foundation, and to have been exploited only in the approximate manner of Section V,G. It is described then as “self-consistent,” and that term may be applied just as aptly, or even more so, at the present juncture. We shall be content in the sequel to seek the predictions available, beyond those associated with Voigt and Reuss, when the symmetry of the tensor of overall elastic moduli can be assigned (as isotropic in our calculations), while the geometrical structure remains otherwise unrestricted (until specialibied somewhat in Section V,G). The elastic properties and volume concentrations of the phases are prescribed arbitrarily, but to make allowance for more detailed information (if it be available) is considered impracticable. Except in special cases, we anticipate that the overall moduli can only be estimated approximately or restricted to a range of values, but we may hope thereby to provide a benchmark for practical or further theoretical considerations.

D. ISOTROPICPHASESWITH COMMONSHEARMODULUS The particular case where all II phases of the composite body are made of isotropic materials each having the same shear modulus is well worth investigating to the full extent that it allows, before the more general, more intractable, cases are faced. For, as was revealed by Hill (1963a), the necessary analysis can be carried out in an exceedingly elegant and exemplary fashion to reach an explicit expression for the overall bulk modulus, when the geometry has an overall isotropic (or merely cubic) structure reflected in a corresponding symmetry of the tensor of overall elastic moduli, but is

Elastic Behavior of Composite Materials

22 1

otherwise quite arbitrary. It leaves also wider implications. In the first place, stringent upper and lower bounds on the overall bulk modulus are forthcoming (in Section V,F) when the individual shear moduli of the phases differ from one another. Furthermore, by its special simplicity and certainty, the analysis stands alone, to be complied with by future refinements (and to be envied perhaps in the analogous contexts of potential theory and slow viscous flow where it has no counterpart). The tensor of overall elastic moduli has the isotropic form (5.12) in the present circumstances where we may assert naturally enough that the overall shear modulus ,u coincides with that which is prescribed in common to all the phases of I/ [as was remarked earlier in Section V,A and as implied by the inequalities (5.13)] and which is assigned, therefore, to the homogeneous matrix beyond. We allow for this coincidence at the outset so as not to complicate the initial analysis. The bulk modulus of the rth phase is K,, while the matrix shares the overall one, K. Then, after reference to (4.15), the appropriate displacement, strain, and stress fields can be constructed in turn at a point x’ (in any phase or in the outer matrix) as (5.24)

(5.25)

where the summations are over all r from 1 to n,where EA = E; and C A = 05, and where is identified as the harmonic potential at x‘ of matter of unit density imagined to fill the volume of the rth phase, and is expressed as in (3.38) as an integral over that volume. From the properties of such a harmonic potential which were described in Section III,B, the elastic field receives its intended behavior. The displacement field is continuously differentiable in each phase and in the matrix beyond, and is continuous across every interface, just like the first derivatives of the potentials. Since 4r,ijjumps in value by ninjwhen passing through an interface (with unit normal ni) into the rth phase from another phase, or perhaps from the matrix, the traction components are kept continuous, even though there are stress and strain jumps across every interface. Since V2r$r is unity at points in the rth phase and is zero elsewhere,the stress field certainly satisfies the equations of equilibrium. At the same time, the dilatational part of the strain field and the hydrostatic part of the stress field are made constant within each phase. In the rth phase

L. J . Walpole

222

say, we have from (5.25)and (5.26)in turn that

where B = eii and d = c i i . Consequently, the four relations (5.21) yield (from the equality of their isotropic rather than their deviatoric parts) the equivalent expressions

cl C r + 4 p - I C C x 3 crKr

'

-

c,

(5.28) cr/Kr 1 Cr (l/Kr) + ( ~ / Q P ) K ( 1 / K r ) + (1/$P)' each of which can be solved immediately for a unique overall bulk modulus in terms only of the elastic moduli and the volume concentrations of tht: phases, irrespective of the finer geometrical details. The result is in agreement with that derived from an alternative standpoint by Hill (1963a)for a composite of two phases. It can be expressed in the two further ways as

=-I

which, in common with the preceding pairs of equations, enjoy the striking property that each is converted to the other when all the elastic moduli [including 4p] are replaced by their reciprocals. The first pair (5.27) can be recast as

to confirm next that they are equivalent to the second pair (5.28) and to each other and that the elastic field (5.24)thereby loses the leading member of the terms that perturb the remote field of uniform stress and strain. The average of +s,ij over the volume V , of the rth phase, which will determine the full average strain Er and average stress iTr in that phase, is calculated from the geometry alone as necessarily an isotropic tensor which is some multiple of aij. An anisotropic contribution would upset the compatibility of the relations (5.21) with the assumed isotropy of the tensor of overall

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moduli. The particular factor which multiples aij can be fixed at once as 6,,/3, without reference to the details of the geometry, since & i i is known to be unity at all points of the sth phase and zero everywhere else. Accordingly,

The averages of the deviatoric parts of the strain and stress coincide in every phase with the deviatoric parts of and d,respectively, and so the relations (5.21) remain fully satisfied in their deviatoric as well as in their isotropic parts, as a consequence of the particular evaluations of the overall bulk and shear moduli.

E. WEAKLY INHOMOGENEOUSCOMPOSITE Another special case within our horizons is the weakly inhomogeneous one where all the n phases of V have almost the same, arbitrarily anisotropic elastic properties, so that their tensors of elastic moduli differ only infinitesimally from one another and, therefore, from the overall tensor. Such an approximation may look very inappropriate where phases are greatly dissimilar and even where identical but strongly anisotropic crystals make up a polycrystal, but the analysis does prepare for the more universal one of Section V,F. An improved order of accuracy seems to call for geometrical details outside those we assume. We may extract from the integral equation (5.19)the approximation

to the displacement field, where the summation takes r from 1 to n, and where zr is an appropriate constant approximation to the polarization stress z in the rth phase. We set zr = (L, - L)E

in order that only second-order infinitesimals are lost, whereas the Voigt approximation (of zr vanishing) banishes first-order ones. Accompanying strain and stress fields can be constructed accordingly as & =8

+

El,

cr = 3

+ LE’ + z

(5.30)

say, where z is to coincide with zr in the rth phase while it vanishes everywhere in the outer matrix. This stress field is equilibrated (like the Reuss approximation which allows E’ and z to vanish) and is related to the strain field in the appropriate way in the matrix and (to the order of approximation) in the individual phases of I/. These expressions for the elastic field

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can be made explicit if L is replaced in them by its Voigt or Reuss estimate or by some other suitable mean from which it differs by only second-order quantities (but we defer this choice). In order to calculate fully the required average strain Fr and average stress 3r in the rth phase, let it be presumed that the isotropic structure (5.12) is taken by the tensor of elastic moduli. With the explicit Green’s tensor (3.35)once again, we have that

where there remain the integrals

that calculate the harmonic and biharmonic potentials of matter of w i l t density imagined to fill the volume V, occupied (perhaps in separate portions) by the rth phase. We note the particular property

lC/r,icjj .... = @

..=

r,rr

1 0

inside V,, outside V,,

(5.32:)

where the repeated suffixes form the Laplacian. The second derivatives of each harmonic potential and the fourth derivatives of each biharmonic one are taken to form isotropic tensors (which are symmetrical with respect to interchanges of all their suffixes) when averaged over any phase, in which case, in conforming with (5.32), they are determined fully without reference to the finer details of the geometry by

( l i ~ Jvs , ) @r,ijdV

(l/E)

svs

$r,ijkl

dv

=+6rs6ij, = h6rs(dij6kl

+ dik d j l + 6il ‘ j k ) .

No new assumption is implicit here at least if there are only a finite number of distinct (possibly isotropic) phases and if the isotropy of the overall tensor can be assumed quite reasonably to persist when all the individual elastic moduli of the phases are varied independently to some extent. Subsequent formulas could not then endure anisotropic factors specified entirely by the geometry, alongside those depending on the variable elastic properties. However, in a polycrystalline aggregate where an effectively infinite number of anisotropic phases differ only in the orientation of their crystallographic axes, it may be necessary to bring in some further assump-

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tion to modify the formulas, in the manner indicated by the statistical analyses of Molyneux (1969, 1970) and of Kroner (1977, 1978).We calculate now on return to (5.30) that -

Er

=

z- PZ'

= [I

+ P(L - L,)]P = ArE

and, similarly, to the present order of approximation that 5, = [ I

+ Q ( M - Mr)]F = Bra,

where P and Q coincide precisely with those tensors of (3.40) that belong to a spherical inclusion in the isotropic medium with tensor L of elastic moduli (and it can be shown incidentally that this coincidence prevails with the same expressions for the average strain and stress even when L is anisotropic). The factors A, and B, are identified here so as to be substituted in (5.9) to reach the relations

L = Lv - Ccr(Lr- Lo)P(Lr- L), M

= MR

- Ccr(Mr - Mo)Q(Mr - MI,

which enable the Voigt and Reuss estimates (5.10) for L to be adjusted in the direction allowed by the inequalities (5.11); MR denotes the inverse of L R . In these right-hand sides, it is appropriate to substitute Lv and MR in place of L and M , respectively (so attaching a suffix to P and Q ) and also in place of Lo and M , (although this last pair will anyhow disappear from the summations). There remain the two explicit evaluations

L = L, - Ccr(Lr - LV)Pv(Lr - LVL M = M R - xcr(Mr - MR)QR(Mr - M R ) ,

of the tensor of overall moduli, each containing only third-order errors. From say the first one, when for example all the phases are isotropic, we may express the overall bulk and shear moduli in terms of their Voigt estimates of (5.14) as

+

~c= KV

- x c r [ ( K r - ~ c v ) ~ / I ~~vt ) ] 3

P = PV

- C c r [ ( p r - pVI2/(pv + ~

(5.33)

$ 1 1 5

where the suffix v has been attached to all the moduli of (3.42). These results were first arrived at in the literature as an implicit consequence of the bounds on K and p that are stated in Section V,F. Subsequently, Molyneux and Beran (1965) and Lomakin (1965, 1966) examined explicitly the weakly inhomogeneous elastic composite. The corresponding outcome in the analogous context of potential theory is arrived at (independently of bounding

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226

procedures) by Brown (1955), Landau and Lifshitz (1960), Prager (1960), Beran and Molyneux (1963), and Batchelor (1974).

F. HASHIN-SHTRIKMAN BOUNDS We return now to the general case where no restriction is placed on the prescribed elastic moduli and volume concentrations of the phases, nor on their unknown geometrical disposition, so long as the tensor of overall moduli keeps to an assigned type of anisotropy (or isotropy). With the help of extremum principles and of elastic fields that approximate the exact one (of Section V,C), we wish to determine to what extent the magnitudes of the overall moduli can be confined to a range narrower than that left in (5.1 1) by the Voigt and Reuss calculations. The pair of inequalities a*&*+ ( T E >

20&* {2.*&

can be established simply by dropping (a* - a)(&*- 8) from the right-hand sides as a positive scalar, as long as the approximate strain and stress fields (distinguished by the asterisk) meet everywhere (after one or other of them is defined first) the same constitutive relations as the exact fields. We also require them to reach the same uniform state at infinity, though we must allow them not to do so quite as rapidly as the exact fields. We set E* =

+ E’,

a* = aA+ a’

in the first and second inequality, respectively, where E’ is any strain derived from a continuous displacement field that vanishes at a great distance from V to the same order as the inverse square of the distance, and where a’ is any stress field that satisfies the equations of equilibrium, while not suffering traction jumps at any interface and while vanishing at a remote distance like the inverse cube of that distance. Both inequalities are then integrated over the whole volume V + W inside the interface C that lies entirely in some remote part of the matrix. Every integration sign will refer to this entire volume unless an attached suffix confines it to a smaller part. The result (5.23) is recalled and next in the same manner, by converting the volume integral into one over the interface C at which negligible terms can be removed, we establish that

+ E’) dV = (V + W)OA&A+ aAf&’ dV, f(0” + (T’)&dV= (V + W)OA&A+ Ja’dV. fO(&A

&A

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The remaining integration of G*E* is merely broken down over the phases and the matrix in the appropriate way for each inequality. After substitution of the relation (5.20),the inequalities can be simplified as

(5.34) respectively, where the overbar and the suffix r average E’ and o’ over the volume V , of the rth phase in V. We attempt to assign (still implicitly in terms of L) precise positive values to these right-hand sides, or at least to quantities definitely smaller than them, for otherwise (with vanishing primed fields) only the earlier inequalities (5.11) will remain. We complete the definition of the primed field by forming the connections

in the typical rth phase and

d

= Lad,

E’ = Mod

in the matrix, where Lo is to be chosen as the tensor of elastic moduli of a homogeneous “comparison” body, with inverse M,, and where one or other of z“ or qr can be chosen arbitrarily as a constant, symmetric, second-order, “polarization” tensor. Then E‘ is defined in the same way as its namesake of the preceding Section V,E in terms of the 7,’s and of the arbitrary Lo in place of L for increased flexibility. When Lo is chosen isotropic (with bulk modulus ic, and shear modulus p,) E‘ has the same form as (5.31), with a suffix o attached to the elastic moduli and with the zr’s left arbitrary. Its averages over the individual phases can be evaluated in the same manner from those geometrical ones of the derivatives of the potentials, again when the overall tensor L is kept isotropic and at least when there are a finite number of distinctly independent (rather than “polycrystalline”) phases. The averages of E‘ and o’over the rth phase are then calculated as -

E: =

-Po?,

8; = Q o ~ ‘ ,

where the suffix o is attached throughout (3.40) to define Po and Q, and hence throughout (3.31), (3.41),and (3.42)to define the accompanying tensors L,* and M,$ which are inverse to each other. Next, it is necessary to accept some weakening of the inequalities (5.34) since the calculation of the terms

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228

that are quadratic in the primed field would seem to call for further knowledge of the geometry. First, we can arrange that Jvr &'(Lo - Lr)E'dV - @;(Lo - L,)B; = Jvr

(E'

- B;)(L, - L,)(E' - EL) dV

> 0,

Jw

&'(Lo- L)EfdV > 0,

by selecting Lo so that the difference Lo - L, is positive definite for all r, in which case Lo - &.L, and hence Lo - L are positive definite. We have already ensured that

c

Jvr E'LoEf dV

+ Jw

EfL,&'dV=

-cv,zrs;,

since the missing integral of O'E' over the whole volume inside C is made negligible, after converting it to a surface integral over 2. The first inequality can now have all the quadratic terms eliminated in favor of the calculated linear ones and can be recast, after some manipulation, as

+

&A(CC,L,- L ) E A - E A C C , ( L , - L)(L, L,*)-'(L,- L)&A

'

> - C c r Y r ( L r + L,*)- y r = 0, by finally making the best choice of zr as that for which

y, 3 (L, + L,*)z*+ (L, - L ) & A = 0, since any other choice would leave an inferior inequality with a negative right-hand side. The left-hand side can be simplified by noting that

+

(L, - L)(L, L , y ( L , - L) = L, - 2 L + (L,*+ L)[(L,+ L

y

- (L

+ L,*)-'](L,*+ L).

Since is arbitrary, it remains only to resort to the theorem (2.2) and to its notation in order to extract finally an explicit "inequality" on L , namely [Cc,(Lr

+ L,*)-']-'

- L,* > L,

if L, > L, for all r.

(5.35)

After corresponding manipulation which eventually makes the best choice of each qr, the second inequality (5.34) implies that

[zc,(M,

+ M,*)-']-'

- M,* > M ,

if M , > M , for all r. (5.36)

By recourse again to the theorem (2.2) and to some algebra, we can confirm the symmetry of this outcome, in that either one of these final statements (5.35) and (5.36) is equivalent to the other with both its inequality signs reversed, and so together they confine L or M (and their components) symmetrically from both sides. It is proposed by Walpole (1966a,b) and by

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229

Willis (1977) that the two statements also have validity when the overall tensor L is anisotropic and when Lo is chosen likewise anisotropic so as to replace L,* and M,* by their anisotropic counterparts defined by attaching the suffix o throughout (3.29) and (3.31). The right-hand inequality of (5.35) will only just hold when the consequent left-hand inequality has its best choice of L o , and there is a corresponding remark to be made for (5.36). For if instead Lo is replaced by a “larger” Lo(>Lo > L, for all r), then the theorem (3.32) gives L,* a larger replacement after which it can be verified by some algebra, of Walpole (1966b), that the left-hand inequality deteriorates, to revert eventually to the right-hand, “Voigt” inequality (5.11)when Lo is made infinite. When, for example, L is isotropic and also when all the phases are isotropic, with bulk modulus IC,and shear modulus p, for the rth phase, then for the statement (5.39, first as it stands and secondly with both its inequality signs reversed, we choose Lo isotropic with its bulk and shear moduli identified first with the largest of the xr’s and the p,’s, say rcL and p L , respectively, and, second, with the smallest of them, I C and ~ ps. The upper and lower bounds on the overall bulk modulus K: and on the overall shear modulus p are delivered then as the inequalities

[-c,/(K, + Its*)] -

-

.:

< IC < [ C C r / ( K r

+

-I(- +

lo

Kt)] -

-Kt,

where

.s*

= gps,

P: =

3

1

).

2 Ps 9% + 8Ps They were first brought to light by Hashin and Shtrikman (1963, though apparently then with the unnecessary restriction that I C ~and p L should belong to the same phase, and likewise J C ~and ps). The upper and lower bounds on K can be identified as just the overall bulk moduli that can be calculated by (5.29) when the shear moduli of the phases are all imagined to increase or decrease, respectively, just far enough to reach a common value. Indeed those calculations, taken with the “strengthening” theorem of Section V,A, provide in the method originally suggested by Hill (1963a) an attractively direct derivation of the bounds on IC.When the phases all have almost the same elastic properties, each bound approaches the overall modulus to within a third-order infinitesimal, as the approximations (5.33) did in their own manner. However, when the phases are more dissimilar, the margin between the bounds can become undesirably extended until eventually an upper bound is taken trivially to positive infinity by a perfectly

230

L. J. Walpole

rigid phase or a lower bound is taken to zero by an empty phase. Such an extreme overall modulus could be attained if the phase in question fills a matrix surrounding the others, but not if, instead, it were surrounded by less extreme phases. No such distinction is allowed for by these bounds (nor yet by any others in the literature) because they accept every phase on the same footing. Nonetheless, a notable improvement on the earlier bounds of (5.13) has been achieved. It may seem likely, moreover, that there is room for further improvement. The bounds still have a remarkably simple and explicit form, while their derivation does not indicate the existence of any ways by which they might be attained, nor does it exact exceptionally esoteric or heavy numerical costs. However, for a composite of two phases, Hashin and Shtrikman (1963) have found an example (an assemblage of spherical composite elements) whose overall bulk modulus coincides with the upper bound or (reversing the numbering of the phases) with the lower bound, over the whole range of phase moduli and concentrations, and so these bounds are left unequivocally as the best possible in the circumstances. No such coincidences with the bounds on the overall shear modulus have yet been found except by Hashin and Shtrikman (1963)and more widely by Walpole (1 972) when one phase has a comparatively low concentration, but especially as their distance apart is of the same order and form as that for the bulk modulus, if would not seem likely that they can be improved dramatically or even marginally until, of course, further details of a composite are allowed for. Beran and Molyneux (1966) and Miller (1969) are able to improve the bounds on the bulk modulus, and McCoy (1970) those on the shear modulus, when the composite body has a more special statistical structure. Experimental results have been compared with the bounds of Hashin and Shtrikman (1963)for a two-phase composite, recently by Watt and O’Connell(l980) and by the earlier authors to whom they refer. The theorems (5.35) and (5.36)lend themselves to other types of composite materials. Hashin and Shtrikman (1962b) propose the bounds for a polycrystalline solid, although their status is qualified by the statistical considerations of Hashin (1965a), Molyneux (1969, 1970), and Kroner (1977, 1978). Application to a transversely isotropic, fiber-strengthened material is offered by Hill (1964), Hashin (1965b, 1979), and Walpole (1969).

G. SELF-CONSISTENT APPROXIMATIONS We wish to take account of particular shapes that may be assigned to the individual phase boundaries, to the grains of a polycrystal, or to the inclusions in a matrix for example, while the geometry remains otherwise unrestricted. The elastic moduli and volume concentrations of the phases remain pre-

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231

scribed arbitrarily. Naturally the overall elastic moduli can be confined still only to individual ranges rather than to precise values. There is something to be said, however, for arriving at particular estimates of them, in the hope that they will fall squarely within ranges that are not too far reaching, and of course already within whatever bounds are available for wider circumstances. An approximate method of determining the average strain and stress in every phase, and hence all the factors A , and B, of (5.22),may well leave each of the relations (5.21), or their replacements [(5.5) and (5.7)], satisfied only approximately by the same diagonally symmetric and positive definite tensor L of overall elastic moduli. If a common solution is left by some “selfconsistent” approach, it would seem to deserve special attention. Even when the appropriate constitutive relation is met in every phase, by satisfying the interrelation (5.8), the self-consistency may not be forthcoming. For after summing both forms of that relation as (5.37) we observe that when one of the four relations [(5.5) and (5.7)] is satisfied precisely, then a second one will be also, but perhaps not the remaining two, unless it can be so arranged.

1. Polycrystal Suppose first that as in a polycrystal, for example, every phase consists of a large number of individual particles, each of comparatively small dimensions. We assume (as an approximation) that all the particles are either spheres of various sizes or similar ellipsoids all with their corresponding axes aligned. Relative to fixed axes, the tensor of elastic moduli varies in any piecewise constant way from one phase to another, while remaining constant within each. It is assumed that the same average strain and stress can be assigned to all the particles of each phase, and hence identically to the whole phase. The averages are estimated for a single particle in a way which meets the relation (5.8) and hence (5.37), which also contains no undetermined parameters, and which may be regarded as a first approximation to the solution of the integral equation (5.19).The outer homogeneous matrix (with tensor L of elastic moduli) is allowed simply to extend right up to the particle in place of the intervening composite material. By reference to (3.46), the consequent A, and B, can be expressed explicitly in terms of the related tensors P and Q that are the same for all phases. These expressions are recast for the moment as

232

L. J . Walpole

in order to sum them to the forms CcrAr -

= P[L(CcrAr)- CcrLrAr],

CcrBr -

= Q[M(CcrBr) - CcrMrBr],

which together with (5.37) allow the conclusion that all four relations (5.5) and (5.7) will be satisfied simultaneously.This “self-consistent” property was first fully ascertained and named by Hill (1965a).We may also construct A , and B, in the form A, = I

+ Pr(L - Lr),

B, = Z

+ Qr(M - Mr),

which corresponds to (3.14),and which is available even when the particles are nonellipsoidal, although the self-consistency may not be produced so precisely. Pr and Qr are positive definite tensors that vary from phase to phase with the tensor of elastic moduli. By substitution in the relations (5.5) and (5.7), we can replace them by CCrPr(L - 4)= 0,

CcrQr(M - Mr) = 0,

L = CcrLr - Ccr(Lr - L)Pr(Lr - L), M

= CcrMr - CcAMr -

M)Qr(Mr -

respectively, where the second pair have taken notice of the first pair so as to show that a solution for L (if it exists) is always related to its Voigt and Reuss estimates in the way demanded by (5.1l), and so is certainly diagonally symmetric and positive definite. Finally, we recall a third method which expresses A , and B, as in (3.45),with a suffix r replacing the 1. Substitution in (5.5) brings about the equalities L = [Ccr(Lr+ I,*)-’]-’

- L*,

M

=

[Cc r ( M , + M * ) - ’ ] - l - M *

(5.39)

whose structure can be compared with the left-hand inequalities of the theorems (5.35) and (5.36), respectively, especially when the particles are spherical. For it is possible to infer then, as argued by Walpole (1966b), that a solution for L will comply with the more universal consequences of those theorems (when they are available). For application to a polycrystal that has the isotropic tensor (5.12) of overall moduli, L* and M* are evaluated by (3.41)and the summations may be carried out just as were those of (5.10).Any of the relations, say either of (5.39), are reduced then to a pair of simultaneous algebraic equations to solve for the overall moduli k: and p. For example, if each individual crystal has the same cubic anisotropy, the equations correctly identify k: with the bulk modulus of the crystal and leave p as the only positive root of the cubic 8p3

+ ( 9 +~ 4p’)p2- 3 p ” ( ~+ 4p‘)p - 6k:p’p’’ = 0,

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233

where p‘ and p” are the shear moduli of the crystal that have their place (2.20) in the tensor of elastic moduli. This cubic was given equivalently by Kroner (1958),Eshelby (1961),and Hill (1965a),while Hershey (1954)found a quartic which has the same positive root. 2. Dispersion Composite

Especially when one phase (the nth say) has a status different from the others, as a matrix in which inclusions of the other phases are dispersed, for example, its average stress or strain can be calculated from that allotted to them, in place of the more direct calculation that may be inconvenient or precluded. Of course, these averages for the first n - 1 phases may be estimated independently of one another, without the greater demand for an elastic field that maintains its accuracy in all phases at the same time. The relations (5.5) specify then the factors A , and B , (since c, # 0) and allow them to be eliminated, therefore, from (5.7) to leave the two equations L

= L,

+ Cc,.(L,. - &,)A,,

M

=M,

+ CC,(M, - M,)B,,

(5.40)

either of which may be solved for L in terms of the remaining known quantities. If the average stress and strain assigned to each of the first n - 1 phases meets each constitutive relation, by means of the relation (5.8), then that relation and (5.5) allow both of (5.40) to be recast at once in turn as

L,A, = B,L,

M,B, = A,M,

so as to confirm that they are equivalent to each other and that the average stress and strain for the nth phase keep to the constitutive relation. Suppose that the typical rth phase is embedded in the matrix of the nth phase and is made up of inclusions which all have the same shape and alignment (otherwise they would belong to another phase perhaps with the same elastic moduli) and which can be all given the same average stress and strain, common to the whole phase and obedient to the constitutive relation. These averages can be estimated by simply once again allowing the outer homogeneous matrix to extend right up to a single inclusion, in place of all the intervening composite material. When, for example, all the particles of every phase are spheres or aligned ellipsoids, the pair (5.38) still specify A , and B, for all r from 1 to n - 1, and consequently, after multiplying them by c,, summing and appealing to the relations (5.5) and (5.7), for r equal to n also. That is, as would be expected for consistency, the formulae then revert at the general level to all of those of the preceding Section V,G,l, in which the nth phase enters on the same footing as the others. Hill (1965~)and Budiansky (1965) noted such a consequence, and for the case of an isotropic matrix containing spherical inclusions of another isotropic phase they bring out a

234

L. J . Walpole

pair of simultaneous equations which have unique positive solutions for the overall bulk and shear moduli. Hill (196%) confirms that these solutions comply with the bounds of Hashin and Shtrikman (1963), and that they take the correct form at a dilute concentration of the inclusions. Such an assurance is forthcoming in other cases as well, to confirm what the method of approximation itself would suggest, namely that the results are to be relied on when the phases are not too dissimilar in their elastic properties or when the and Budiansky inclusions are at a low concentration. However, as Hill (1965~) (1965) illustrate, exaggerated effects are found at the opposite extremes. For instance, rigid or vacuous inclusions induce correspondingly extreme overall elastic moduli well before the maximum possible concentration is reached. When the inclusions have a nonellipsoidal shape, or a variously oriented ellipsoidal one, the nth (matrix) phase remains distinguished from the others in the formulas. Such possibilities are examined, to compare one with another, by Wu (1966), Walpole (1969),Walsh (1969), Boucher (1974, 1975a,b), and Budiansky and O’Connell (1976). Hill (1965b) gives an application to fiber-strengthened materials. We may also refer to the analogous physical contexts of potential theory where the present approach has a longer history which dates back to Bruggeman (1935), Bottcher (1945), Polder and Van Santen (1946), and Niesel (1952), and which can be placed, likewise, in a modern perspective. The calculations can be diverted through much greater complications by leaving behind, realistically, an arbitrary portion of the composite material between the single inclusion and the outer matrix. Undetermined parameters are inevitably introduced into the formulas, but that may be all to the good if they have restricted ranges that confine the overall elastic moduli more narrowly than the more universal bounds of Hashin and Shtrikman (1963) or if they can assuage the excessive estimates at the higher concentrations of inclusions. Hashin (1968,1970) proposes to leave just a spherical shell of the nth phase (where now n = 2) immediately outside the inclusion of spherical shape, where the outer radius of the shell remains undefined as a disposable parameter, for although it can be restricted by a finite upper limit, as otherwise the Hashin-Shtrikman bounds would be transgressed, in the circumstances there can be no uniquely appropriate choice of it. However, the lengthy calculations of Van der Poel(1958), corrected by Smith (1974,1975) and equivalently by Christensen and Lo (1979), do fix this outer radius so as to give the inclusion and its shell the same relative volumes as the phase of inclusions and the matrix phase respectively in the composite material. Kerner (1956) makes this same choice of the outer radius of the shell, but leaves beyond it an arbitrary shell of the composite material, while claiming nonetheless to calculate precise overall bulk and shear moduli (not however

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235

for any precisely defined composite material). We can accept that this calculation of the overall bulk modulus is appropriate to the special assemblage of spherical composite elements for which Hashin (1962) derived otherwise the same precise result, to be coincident with the universal bound on the bulk modulus (as referred to in Section V,F). Accurate results are well known to be available when the inclusions have only a dilute concentration in the matrix of the nth phase, so that the elastic field in any one inclusion takes only a negligible notice of the remote presence of the others. The numerical range allowed then to the total concentration depends to an extent on the shapes of the inclusions and on the relative magnitudes of their elastic moduli and those of the nth phase, but in the most adverse circumstances it must remain below one or two percent. The present calculations would simply make the tensor of elastic moduli of the outer matrix coincident with L, (in place of L ) to remove their implicit dependence on L, while however keeping only a limited “self-consistency” and a narrower numerical accuracy. It suffices now to construct the factors A, and B, (for r < n) using the method indicated by (3.14);namely as A, = I

+ Pr(L, - Lr),

B, = I

+ Q,(M, - M,),

where the relation L,A, = B,L,,

M,Br = A,M,

is satisfied, as only an approximation to (5.8), and where P, and Q, are related positive definite tensors that vary from one phase to another, as they depend on (and only on) the shape and orientation of the inclusions of a phase, and on their elastic moduli and those of the matrix. The formulas (5.40), which have eliminated A,, and B, from (5.5) and (5.7), then evaluate L and its inverse M by the two expressions

L

= CcrLr

- Ccr(Lr - Ln)Pr(Lr - L A

M = CcrMr - Ccr(Mr - M n ) Q r ( M r - MA

which are obedient to the “inequalities” set by the Voigt and Reuss estimates even when the c,’s are not necessarily small. However, the L defined by the first equation is only inverse to the M of the second one (for “self-consistency”) when all the c, (r < n) are much smaller than unity (or when all the n phases have almost the same elastic properties). Thus, accuracy cannot be expected in wider circumstances.When all the inclusionshave a spherical or ellipsoidal shape, the calculations may be completed in the manner indicated (for the individual P, and Qr) by Section III,C and by Eshelby (1957,1961).Russel and Acrivos (1972)and Russel (1973)give full details for “slender” ellipsoidal and

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L. J . Walpole

nonellipsoidal shapes. The overall elastic moduli for spherical inclusions were found first by Bruggeman (1937), the bulk modulus, and by Dewey (1947),the shear modulus. Accuracy for larger total concentrations is available to the order of the second rather than the first power. Calculations have been carried out analytically for “well-separated spherical or aligned ellipsoidal inclusions by Walpole (1972)and by Willis and Acton (1976),respectively. For other, more general and realistic, forms of separation, numerical calculations are called for and are presented by Willis and Acton (1976) and by Chen and Acrivos (1978b).O’Brien (1979) appraises the rigor of their mathematical techniques, whose development was initiated in analogous contexts by Batchelor and Green (1972) and by Jeffrey (1973,1974).

ACKNOWLEDGMENTS Those references to Rodney Hill have particularly lent inspiration. I owe to him my introduction to the subject as a student at the time of his 1963-1965 papers. Jock Eshelby is another to whom I am grateful for early encouragement. This article has some origins in earlier unpublished material (Walpole, 1970a).

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Author Index Numbers in italic refer to the pages on which the complete references are listed. A

C

Acrivos, A,, 30, 39, 75, 203, 235, 236, 237, 241 Acton, J. R., 203, 216, 217, 236,242 Alterman, Z., 81, 86, 164 Altman, C., 122, 164 Asaro, R. J., 28, 74

Cagniard, L., 161, 164 Carlson, D. E., 50, 75 eervenq, V., 123,164 Chapman, C. H., 81,82, 143, 152, 158, 160, 161, 163, 165 Chen, H. S., 30,39, 75, 203, 236,237 Chen, W. T., 197,237 Chernov, L. A., 64,75 Childress, S., 56, 75 Choy, G. L., 158, 159, 165 Christensen, R. M., 234,237 Cisternas, A., 119, 165 Clarke, D. R., 199,237 Cole, J. D., 8, 75 Collar, A. R., 90,165 Cooley, J. W., 159,165 Cormier, V. F., 158,165 Cory, H., 122,164

B Backus, G. E., 89, 97, 164, 165 Bacon, D. J., 194,236 Bamett, D. M., 28, 74, 194,236 Batchelor, G. K. 30, 47, 63, 74, 171, 190, 209, 226, 236 Bearman, R. J., 63, 77 Ben-Menahem, A., 162,164 Bensoussan, A., 8, 74 Beran, M. J., 16, 17, 22, 74, 209, 225, 226, 230,236, 240 Betancourt, 0. 119,165 Bose, S. K., 65, 74 Bottcher, C. J. F., 233,236 Boucher, S., 234,236, 237 Brailsford, A. D., 60,74 Brekhowskikh, L. M., 83,164 Brinkman, H. C., 55, 74 Brown, L. M., 199,237 Brown, W. F., 216, 217, 226,237 Bruggeman, D. A. G., 220, 233, 236, 237 Budiansky, B., 43, 46, 47, 55, 74, 233, 234, 237 Bullen, K. E., 81, 166 Bullough, R., 56, 75 Bunch, A. W. H., 157, 164 Burdick, L. J., 151, 152, 163, 164 Burridge, R., %, 164

D Datta, S. K., 65, 75 Davies, G. F., 209,242 Davies, W. E. A., 217,237 Dederichs, P. H., 17, 22, 75, 78, 216,242 Dederichs, P. M. 175, 194,237 Devaney, A. J., 74, 75 Dewey, J. M., 236,237 Dorman, L. M., 142, 157, 167 Duncan, W. J., 90,165 Dunkin, J. W., 101, 106, 123, 148, 165

E Eshelby, J. D., 28, 75, 188, 189, 194, 195, 197, 199, 200, 201, 207, 233, 235,237 Ewing, W. M., 80, 164

243

Author Index

244 F

Faber, S., 143, 165 Faivre, G., 197,237 Filon, L. N. G., 157, 165 Fisher, G. M., 49, 76 Fokin, A. G., 5 , 17, 75 Fowler, C. M. R., 142, 165 Frantsuzova, V. I., 150, 165 Frazer, R. A., 90,165 Fredholm, I., 194,237 Fuchs, K., 81, 123, 141, 157, 159, 165 G

Gajewski, R. R., 162, 167 Gel’fand, I. M., 14, 27, 75 Ghahremani, F., 197,238 Gilbert, F., 89, 93, 146, 164 Green, G., 80, 165 Green, J. T . , 20, 63, 74, 236 Gubernatis, J. E., 17, 75, 197,235

H Harkrider, D. G., 81, 125, 144, 165 Hashin, Z., 25, 36, 39, 41, 51, 54, 75, 77, 171, 209, 230, 234, 235, 238 Haskell, N. A., 81, 91, 106, 125, 165, 166 Hayns, M. R . , 56, 75 Head, A. K., 194,238 Helmberger, D. V., 143, 151, 152, 161, 163, 164, 166, 167 Hershey, A. V., 46, 75, 233,238 Hill, D. P., 155, 167 Hill. R., 10, 11, 12, 25, 29, 34, 43, 75, 76, 174, 182, 183,185, 186,187,1%, 201,203, 207,209,212,213,214,220,222,229,230, 232, 233, 234,238 Hinch, E. J. 56, 60,76 Hoenig, A., 47, 76 Hori, M., 36, 76 Howells, I. D., 55, 76 Hron, F., 161, 165 Hudson, J. A., 64, 76, 84, 98, 106, 153, 166 Hutchinson, J. W., 197,239

I

Illingworth, M. R., 124, 166 Ishimaru, A., 64, 76

J Jan, R. V., 194,240 Jardetsky, W. S., 80, 165 Jarosch, H., 81, 86, 164 Jeans, J. H., 81, 166 Jeffrey, D. J., 236,239 Jeffreys, H., 81 Jobert, G., 166 K

Karal, F. C., 17, 64, 65, 76 Kausel, E., 81, 166 Keller, J. B., 17, 64, 65, 76 Kennett, B. L. N., 82, 113, 114, 116, 118, 120, 122, 124, 129, 130, 131, 135, 137, 138, 142, 143, 145, 146, 147, 148, 150, 154, 155, 157, 159,164, 166, 167 Kerner, E. H., 234,239 Kerry, N. J., 82, 113, 118, 122, 129, 131, 137, 146, 147, 148, 150, 154, 166 Khachaturyan, A. G., 76, 191,239 Kind, R., 123, 142, 156, 166, 167 Kinoshita, N., 194, 197,239, 240 Kneer, G., 28,46, 76, 197,239 Knopoff, L., 64, 76, 81, 96, 148, 150, 154, 164, 166, 167 Knops, R. J., 203, 207,239 Knott, C. G., 80, 166 Koehler, J. S . , 194, 239 Korringa, J., 17, 67, 76, 216, 217,239 Kroner, E., 13, 17, 20, 22, 25, 34, 36, 76, 194, 199, 209, 216, 225, 230, 233,239 Krumhansl, J. A., 17, 75, 197,238 Kunin, I. A., 197, 201,239 Kupradze, V. D., 203, 239

L Lamb, H., 80, 154, 166 Landau, L. D., 226,239

Author Index

245

Langer, R. E., 166 0 Lapwood, E. R., 144, 166 Laws, N., 47, 49, 50, 53, 54, 76, 185, 197, 201,239 O’Brien, R. W.,216, 217,236,240 Lax, M., 31, 65, 76 O’Connell, R. J., 47, 74, 209, 230, 234,237, Leibfried, G., 175, 194,237, 239 242 Leitman, M. J., 49, 76 O’Neill, M. E., 155, 167 Leiva, A., 119, 165 Orcutt, J. A., 142, 157, 167 Levin, V. M., 201,239 Levshin, A. L., 150, 165 P Lie, K., 194,239 Lifshitz, E. M., 226,239 Lifshitz, I. M., 175, 194,240 Panza, G. F., 150,167 Lin, S. C., 197,240 Pao, Y. H., 65, 71, 77, 162, I67 Lions, J. L., 8, 74 Papanicolaou, G., 8, 74 Lo, K. H., 234,237 Paul, B., 214,240 Lomakin, V. A., 225,240 Payne, L. E., 190,240 Love, A. E. H., 80, 93, 167, 240 Pekeris, C. L., 81,86,164 Lundgren, T. S., 55, 76 Perkus, J. K., 63, 77 Phinney, R. A., 81, 158, 165 Polder, D., 234,240 M Prager, S., 226,240 Press, F., 80, 165 McCoy, J. J., 15, 16, 17, 65, 70, 72, 74, 76, 230,240 McLaughlin, R., 47,49,50,53,54, 76, 197, R 239 MacMillan, W. D., 199,240 Rayleigh, Lord, 80,167 Mal, A. K., 65, 74 Read, W. T., 194,237 Mann, E., 194,240 Reuss, A., 11, 77, 180, 211, 214,241 Maxwell, J. C., 219, 240 Richards, P. G., 82, 124, 143, 158, 159,165, Michell, J. H., 187,240 167 Miller, M. N., 5 , 36, 77, 230,240 Rizzo, F. J., 203, 241 Mirenkova, G. N., 201,239 Robinson, K., 189,241 Molotkov, L. A., 123, 167 Rosen, B. W., 51, 54, 77 Molyneux, J. E., 22, 74, 225, 226, 230,236, Rosenzweig, L. N., 175, 194,240 240 Routh, E. J., 199,241 Moms, P. R., 197, 240 Russel, W. B., 199, 235,241 Muller, G., 81, 141, 142, 143, 157, 159,165, 167 Mulcahy, M., 97,164 s Mura, T., 194, 197,239, 240 Myklestad, N. O., 189,240 Sack, R., 219, 220,237 Saito, M., 81, 84, 87, 167 Sanchez-Palencia, E., 8, 10, 77 N Scattergood, R. O., 194,236 Schapery, R. A., 51, 54,77 Niesel, W., 234,240 Schiffer, M., 190,241 Nigam, S. D., 173, 175, 178,241 Schwab, F. A., 81, 148, 150, 154, 166, 167 Nolet, G., 154, 167 Seeger, A., 194,240 Nye, J. F., 173,240 Sezawa, K., 144,167

Author Index

246

Shermergor, T. D., 17, 75 Shilov, G. E., 14, 27, 75 Shippy, D. J., 203, 241 Shkadinskaya, G. V., 150,165 Shockley, W., 194, 237 Shtrikman, S., 25, 36, 39, 41, 75, 203, 229, 230, 234,238 Simons, R. S., 138, 166 Smith, J. C., 234,241 Sosnina, E. G., 197, 201,239 Srinivasan, T. P., 173, 175, 178,241 Stephen, R. A., 120, 122,167 Sternberg, E., 203,241 Stokes, G. G., 93, 167 Stoneley, R., 110, 167 Stroh, A. N., 194, 241 Synge, J. L., 178, 194. 241 Szego, G., 190,241

T Takeuchi, H., 81, 84, 87, 167 Talbot, D. R. S., 56, 57, 62, 63, 77 Thomas, T. Y., 175,241 Thomson, W. T., 81, 91, 167 Throop, G. J., 63, 77 Truell. R., 11, 77 Tukey, S. W., 159, 165 Twersky. V., 71, 77

Van Santen, J. H., 234,240 Varadan, V. K., 65, 71, 77 Varadan, V. V., 65, 71, 77 Vered, M.,162, 164 Voigt, W., 180, 211, 214, 241 Voight, W., 11, 77

W Walpole, L. J., 36, 39, 40, 41, 46, 77, 187, 191, 195, I%, 197, 199,201,203,212,228, 229, 230, 234,236,241 Walsh, J. B., 234,242 Waterman, P. C., 11, 66, 77 Watt, J. P., 209, 230, 242 Wertheim, M. S., 63, 77 Wiggins, R. A., 152, 167 Willis, J. R., 17,25,28,30,36,38,39,41,42, 44,47,56,57,62,63,66,67,69,71,72,74, 77, 78, 194, 197, 203, 216, 217, 229, 236, 242 Wing, G. M.,167 Woodhouse, J. H. 85, 113, 146, 166, 167 Wu, T. T., 234,242

Y. Z

u.v Uscinski, B. J., 64,77 Van der Poel, C., 234.241

Yevick, G. J., 63, 77 Yonezawa, F., 36, 76 Zeller, R., 17, 22, 75, 78, 216,242 Zoeppritz, K., 80,167

Subject Index dispersion, self-consistent approximation, 233-236 distribution of inclusions, 5-6 energy density, 9- 10 equilibrium equation, 8 indicator function, 4-5 isotropic, common shear modulus, 220223 joint probability density, 6-7 microstructure, idealized, 4 moduli tensor, see Elastic moduli tensor modulus, 84 properties, 1-78 classical variation principles, 13-23 definitions, 7- 13 diffusion to random array of voids, 56- 57 elastic behavior, see Elastic behavior ensemble average, 12- 13 generalization, 47- 55 Hashin-Shtrikman variational principles, 23-42 lack of convergence and, 55-63 mean value, 4 piecewise-constant polarization, 4445 plane-wave decomposition, 26 self-consistent estimates, 42-47 thennoelasticity, 50- 55 virtual work equality, 23 viscoelasticity, 48-50 volume averages, 9- 12 wave propagation, see Wave propagation tensor, see specific tensor types, 3-7 volume integral, 219

A

Acoustic tensor, 69

B

Bihannonic potential, 198- 199 weakly inhomogeneous composite, 224 Bounds, 20-23, see also HashinShtrikman bounds optimized, 22 Bulk modulus, 28, 39-40, 180 matrix containing spheres, 45 overall, 213, 223 weakly inhomogeneous composite, 225

C

Cagniard method, 162-163 Channel wave secular function, 148 Chapman method, 163 Closure assumptions perturbation theory and, 31-33 probability and, 33 quasicrystalline approximation, 3 1- 32 Compliance tensor, 11, 19-20, 179 anisotropic constant tensor and, 42 Composite bounds, 20-23, see also HashinShtrikman bounds conditional density, 6 correlation function, 5 defined. 3-4 247

248

Subject Index

Composite (Conr.): weakly inhomogeneous, 223-226 Compressional waves, 84-86 Concentration-factor tensors, 210, 231232, 235 Constant tensor, 27-28 anisotropic, 41-42 isotropic, 39-40 matrix containing inclusions, 37 Constitutive law surface decomposition, 182- 186 transformed inclusion and, 188 Constitutive relation decomposition, 179- 180 field equations, 202 homogeneous ellipsoidal inclusion, 193 integral equations, 203 linear, 83 Correlation function, 5

D Darcy coeficient, 55 Defect, concentration, 57 Density conditional, 6 conditional probability, 63 joint probability, 6-7 mean, 70 spherical inclusion, 30 Diffusion equation, 56 to random array of voids, 56-57 Displacement coupled equation, 83-88 field polarization formulation, 67 variational principle, 72 weakly inhomogeneous composite, 223 isotropic composite, common shear modulus, 221 receiver, 134 surface buried source, 136, 139-140 convolution representation, 159- 161 elastic half space, 99

free, 116 full half space response, 138. 140 full medium response, 153 half space response and, 128, 35- 137 partitioned stratification, 140- 143 reverberation near source, 139- 140 slowness method, 159- 163 spectral method, 152- 159 transform inversion, 152- 163 vector gradient, 182 homogeneous ellipsoidal inclusion, 193- 194 integral equations, 204-206 wave propagation and, 64-65

E

Earthquake, source model, 96 Elastic behavior, 169-242 field equations, 202-203 integral equations, 203-208 overall, 208-215 Hashin- Shtrikman bounds, 226- 230 integral equations, 209-210, 215-218 isotropic phases, common shear modulus, 220-223 self-consistent approximations, 230236 self-consistent model, 218-220 remote, 206 weakly inhomogeneous composite, 223- 226 Elastic equation, 83 Elastic moduli, tensor of, 8, 179, 184 anisotropic constant tensor and, 42 approximation, 3 1-32 average, 9 crystal, 213-214 dispersion composite, 233, 235 ensemble average, 12 Hashin-Shtrikman bounds and, 228 matrix containing inclusions, 37-38 overall, 208-215 polarization tensors and, 214 Reuss estimate, 21 1-214 Voigt estimate, 211-214 self-consistent approximation, 232

Subject Index self-consistent estimates, 43-44 small variations in, 13-20 general perturbation theory, 13- 17 local approximation, 16 uniform mean strain, 17-20 transformed inclusion, 190- 192 weakly inhomogeneous composite, 225 Elastic wave, see Wave propagation Ellipsoidal inclusion problem, 27-28 Energy density average, 9- 10 ensemble averaged, 13 Hashin- Shtrikman variational principle and, 24 flux, seismic wavefield, 103- 104 functional, 10- 11 return, 140, 142 strain, total, 212 Entropy, density, 51 Equilibrium equation, 8 Eshelby’s tensor, 196. 200

F Field equations, 202-203 Flux, 57-59 variational formulation, 61 Force, body, 55 Frequency transform method, 159 Functional Hashin-Shtrikman bounds, 35 matrix containing inclusions, 37 stress polarization tensor and, 33-34 trial field stress polarization, 34 variational formulation, 61 Fundamental matrix seismic wavefield, 106 solution, 89 propagator matrix as, 90-91

G

Green’s function, 47 infinite body, wave propagation, 69 lack of convergence and, 55, 58-60

249

moduli tensor and, 14-15 quasicrystalline approximation and, 5960 viscoelasticity, 49 Green’s tensor homogeneous ellipsoidal inclusion, 197 symmetric, 194

H Hankel function, 153 Harmonic potential, 198- 199, 224 Hashin- Shtrikman bounds, 34- 36, 226230 dispersion composite, 234 matrix containing inclusions, 36- 39 thermoelasticity and, 54 Hashin- Shtrikman variational principle, 23- 42 closure assumptions, 31-33 estimates, 33-42 functional, thermoelasticity, 52 integral equations and, 23-25 perturbation theory and, 25-33 self-consistent method matrix containing spheres, 45-46 polycrystal, 46 Heat flux, 50 Helmholtz free energy, density, 50

I Inclusion, 188-201 homogeneous ellipsoidal shape, 193200 inhomogeneous ellipsoidal, 200-201 problem interactions, 29-33 single, 26-29 transformed, 188- 192 Indicator function, 4-5 Integral equations, 203- 208 Hashin- Shtrikman variational principle, 23-25 lack of convergence, 58-60 overall elastic behavior, 215-218

Subject Index

250

Interface, stress and strain discontinuity, 186- I87 Invariant imbedding, 114- 115 superimposed stratification, 117

Jump vector, source wavefields, 131- 132 Kronecker delta, decomposition, 181

L Lagrangian functional, variational principle, 73 Laws, formula of, 53 Love wave, 145- 146

M Matricant, see Propagator matrix Moment tensor earthquake source, model, 96 elements, 94 source radiation and, 126 Momentum density, 66 polarization, 66 plane wave, 69-70 variational principle, 72 Motion, equation of, 66

Polarization formulation, 66-72 integral equations, 66-68 plane waves, 68-72 Polarization strain tensor field equation, 202 inhomogeneous ellipsoidal inclusion, 200 integral equations, 203, 205-206 transformed inclusion, 188- 192 Polarization stress tensor inhomogeneous ellipsoidal inclusion, 200 overall, 215-216, 218 transformed inclusion, 188- 192 weakly inhomogeneous composite, 223 Polarization tensors Hashin-Shtrikman bounds, 227 overall elastic moduli tensor and, 214 Polycrystal self-consistent approximation, 23 1-233 self-consistent method, 46 Propagator matrix, 89-92 as fundamental matrix solution, 90-91 relation with inverse, 91 seismic wavefield, 102 source and, 91-92 source free regions, 89-91 uniform layer, 105- 106 Propagator method, recursive method comparison, 122- 123

Q Quasicrystalline approximation, 3 1 analog, 32 Green’s function and, 59-60

P

Perkus- Yevick approximation, 63 Perturbation theory closure assumptions, 31-33 Hashin-Shtrikman variational principle and, 25-33 lack of convergence, 58-60 moduli variations, 13- 17 single inclusion problem, 26-31 Poisson’s ratio, 179- 180

R Radiation patterns, 93-95 Ray expansions, 142- 143 Ray theory, generalized, 161- 163 Cagniard method, 162- 163 Chapman method, 163 Rayleigh wave, 146- 147 slowness, 144, 146

Subject Index Reciprocal relation, 204 Recursive method, 90, 121- 122 propagator method comparison, 122123 Reflection, 100-127 coefficient, 108- 109 magnitude, 11 1 configuration, 209 free surface, 115- 117 factorization, 118 matrix. 147 at interface, 107- 11 1 invariant imbedding, 114- 115 matrix, 112-113 decomposition of medium, 124 elements, 109- 110 factorization, 117- 121 free surface, 147 graphic representation, 120 half space response, 127-152 invariant imbedding, 114 turning point problem, 123- 124 mixed solid and fluid stratification, 124I25 recursive approach, 121- 122 surface, full medium response, 138- 140 SH waves, 107- 108 vertically inhomogeneous region, 111I I5 Reflectivity method, 141- 142 spectral method, 156- 157 Renormalization, moduli variations, I5 Reuss average, 11 Reuss estimate, 21 bulk modulus, 213 overall elastic moduli tensor, 21 1-214 shear modulus, 213 Reuss inequality, 213 Reverberation operator, 133, 137, 141

S

Schwarz reflection principle, 162 Self-consistent approximation, 230-236 dispersion composite, 233-236 polycrystal, 231-233 Self-consistent method, 45-46

25 1

Self-consistent model, 218-220 Seismic wavefield, seewavefield, seismic Seismogram double couple source, 156 experimental vs. theoretical, 157 vertical component S wave, 155 Shear disturbances, coupled equations, 84-86, 88 Shear modulus, 39-41, 180 common, isotropic composite, 221222 matrix containing spheres, 46 overall, 213 Reuss estimate, 213 Voigt estimate, 213 weakly inhomogeneous composite, 225 Single inclusion problem, 26-31 Sink term, 5 6 5 7 Slowness, 144- 146, 159- 163 contours, 158- I59 horizontal, 82 velocity inversion and, 149 Slowness method generalized ray theory, 161- 163 surface response, convolution representation, 159- 161 Source-stratification vector, 136 Spectral method, 152- 159 complex slowness contours, 158- 159 contours, 154 frequency transform inversion, 159 full half space response, 153- 156 reflectivity approximation, 156- 157 Spherical inclusion problem, 28-29 Stieltjes convolution operator, 48 Stoneley wave, 110 Stratification history, 80-82 mixed solid and fluid, 124- 125 partitioned, 140- 143 superimposed, 117- 118 Strain energy, total, 212 Strain field, 14 approximation, 226 isotropic composite, common shear modulus, 221-222 Strain polarization, 24 Strain tensor, 7-8 components, 83, 183 constitutive relation, 185

252

Subject Index

Strain tensor (Cont.): discontinuity homogeneous ellipsoidal inclusion,

193 interfacial, 186-187 field equation, 202 Hashin-Shtrikman bounds, 227 Hashin- Shtrikman variational principle, 23 homogeneous ellipsoidal inclusion, 193,

195-196 infinitesimal, 182-183 inhomogeneous ellipsoidal inclusion,

200-201 integral equation, 203,205 overall average, 210,218-219 specific heat and, 5 1 transformed inclusion, 188-192 uniform mean, 17-20,43 volume average, 9 weakly inhomogeneous composite,

223-225 Stress-displacement vector, 86,88 discontinuity, 97-98,126 elastic half space, 99-100 elementary, 103 fundamental matrix solution, 89 at interface, 107 propagator matrix and, 89-92 seismic wavefield, 101 surface source vector and, 128 vertically inhomogeneous region, 111-

112 Stress field approximation, 226 isotropic composite, common shear modulus, 221-222 thermoelasticity, 50 Stress tensor, 7-8 components, 183 constitutive relation, 185 discontinuity homogeneous ellipsoidal inclusion, 193 interfacial, 186-187 Hashin- Shtrikman bounds and,-227 Hashin-Shtrikman variational principle, 23 homogeneous ellipsoidal inclusion,

193,195-196 inhomogeneous ellipsoidal inclusion,

200-201

overall average, 210,218-219 perturbation theory, 25 resolution, 181-182 specific heat and, 51 thermoelasticity, 5 1 transformed inclusion, 188-192 uniform mean, 43 variational principle, 72 weakly inhomogeneous composite,

233-225 Stress polarization tensor, 17 functional extremes, 33-34 perturbation theory, 25 plane wave, 69-70 quasicrystalline approximation, 31- 32 self-consistent estimate, 43 thennoelasticity, 51-52 trial field, 34-36 Stress relaxation tensor, 48 Stress-temperature tensor, 54 Surface source vector, half space response via, 128-131 Surface wave, dispersion, 150- 152 Surface wave secular function, 147 decomposition, 148-150

T Tensors, see also specific tensor components, 186 decomposition, 173-181 anisotropic, 175- 181 isotropic, 173-175 geometric symmetry, 173-181 cubic crystal, 175-176 hexagonal crystal, 176-179 notation, 171-173 resolution, 181-182 Thermal expansion tensor, 5 1 Thermoelasticity, 50- 55 Traction vector, 181,183 coupled equations, 83-88 Transmission, 100- 127 coefficient, 108-109 configuration, 109 at interface, 107- 11 1 invariant imbedding, 114-115 matrix, 112-113 decomposition of medium, 124

Subject Index elements, 109- 110 factorization, 117-121 graphic representation, 120 invariant imbedding, 114 turning point problem, 123- 124 uniform region, 115 mixed solid and fluid stratification, 124125 recursive approach, 121- 122 SH waves, 107- 108 vertically inhomogeneous region, 111115

V Variational formulation, lack of convergence, 60-63 Variational principle, 72-73, see also Hashin-Shtrikman variational principle Velocity fluid, 55 model T7, 151 Virtual work equality, 23 Viscoelasticity, 48-50 Voigt average, 11 Voigt estimate, 20 bulk modulus, 213, 225 overall elastic moduli tensor, 21 1-214 shear modulus, 2 13 Voigt inequality, 213

W

Wave propagation, 64-73 channel, surface and, 147- 148 displacements, 83-88 elastic half space, 98- 100 response, 127- 152 force system, 93 full medium response factorization, 137- 143 partitioned stratification, 140- 143 surface reflection, 138- 140 fundamental matrix, 89-92 interactions, buried source to buried receiver, 132

253

interface matrix, 111 matrix containing dispersion, 65-66 momentum density, 66 plane waves, 68-72 decomposition, 26 polarization formulation, 68-72 stress polarization, 69-70 polarization formulation, 66- 72 reflection, see Reflection source, 93-98 far-field terms, 93-94 radiation patterns, 93- 97 as stress-displacement vector discontinuity, 97-98 speed, 84 stratification, 80-82 stratified regions, 83- 100 surface channel and, 147- 148 dispersion, 144- 147 half space response, 143-152 tractions, 83-88 transmission, see Transmission variational principle, 72-73 wavefield boundary conditions, 88-89 Wave propagator, 102 chain rule, 112 matrix, 89-92 uniform region, 1 15 vertically inhomogeneous region, 1 1 1114 Wave vector seismic wavefield, 102 source representation, 125- 127 vertically inhomogeneous region, 112 Wavefield boundary conditions, 88-89 buried receiver, 133- 135 above source, 133- 134 below source, 134- 135 buried source, 131- 133 elastic half space, 88 seismic, decomposition, 100- 106 Well-stirred approximation, 63

Y Young’s modulus, 179- 180

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