Asymptotic models of fields in dilute and densely packed composites

A l s y m p t o t i c Models of Fields in Dilute and Densely Packed Composites

A. B. Movchan • N. V. Movchan • C. G. Poulton

Imperial College Press

^ s y m p t o t i c Models ^ of Fields m Dilute and Densely Packed Composites

This page is intentionally left blank

^lsymptotic Models ™ of Fields in Dilute and Densely Packed Composites

A. B. Movchan • N. V. Movchan • C. G. Poulton University of Liverpool, UK

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ASYMPTOTIC MODELS OF FIELDS IN DILUTE AND DENSELY PACKED COMPOSITES Copyright © 2002 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 1-86094-318-7

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Preface

This book is about asymptotic models for problems of elasticity, electrostatics and electromagnetism describing physical phenomena in heterogeneous composite structures. Particular attention is paid to analysis of structures containing inclusions or voids which are either of small relative volume (dilute composites) or are placed close to each other (densely packed composites). The methods described in this text are analytical, and the range of our interests covers two areas: (a) the method of compound asymptotic expansions applied to singularly perturbed boundary value problems and (b) the multipole method which proves to be efficient in analysis of fields for domains containing arrays of inclusions of circular or spherical shapes. The book came as a result of our recent work on mathematical modelling of defects in electromagnetism and elasticity. One simple and efficient method for the study of small defects is via evaluation of their dipole tensors and the corresponding energy change associated with the perturbation field. However, when inclusions are finite in size and interact with each other one needs the high-order multipole approximations of solutions. A particular feature of singularly perturbed problems is the presence of so-called boundary layer fields concentrated in the high-gradient regions. Boundary layers are usually described by solutions of model problems posed in unbounded domains. In some cases one can obtain these solutions explicitly or evaluate their asymptotics at infinity. In this text we study models of solids containing small inclusions or voids, and the boundary layers describe perturbations of elastic fields associated with these inclusions. It is shown that the leading asymptotic representation of a boundary layer at

VI

Preface

infinity is determined by components of a dipole tensor of the inclusion, and by the remote load applied on the exterior boundary of the domain. The analytical technique we use to model small inclusions is known in the literature as the method of compound asymptotic expansions. The theory of compound asymptotic expansions was created during the last two decades, and the key results in this development to date belong to Maz'ya et al. (2001) who have written a two-volume monograph, which is the most comprehensive text on the theory of singularly perturbed elliptic boundary value problems at present. The dipole fields associated with inhomogeneities in electrostatics, electromagnetism, fluid mechanics and elasticity were used in many applications to evaluate the energy of the perturbation fields as well as to determine effective moduli of composites with periodic structures (see for example the classical work of Lord Rayleigh (1982), Polya and Szego (1951) and G.I. Taylor (1928)). A systematic analytical outline of properties of dipole tensors for a class of boundary value problems for the Laplacian was included in a book on isoperimetric inequalities in mathematical physics by Polya and Szego (1951). Further studies of dipole tensors in vector problems of elasticity were published by Movchan and Serkov (1997). The book by Movchan and Movchan (1995) outlines applications of dipole tensors in certain classes of asymptotic models of fracture mechanics. A new "shield effect" was discovered by Valentini, Serkov, Bigoni and Movchan (1999) for coated inclusions in elastic media. It has been shown that one can choose parameters of an elastic coating in such a way that an inclusion becomes "neutral": if placed in a constant stress field the inclusion does not produce any elastic energy change. For scalar problems associated with the Laplace operator, structures of this kind have been described in the books by Cherkaev (2000) and Milton (2002). Dipole tensors were also efficiently used in the asymptotic analysis of a class of eigenvalue problems. In the papers by Movchan (1988, 2001) one can find asymptotic algorithms for models of vibration of domains containing small inclusions. The paper by Movchan and Nazarov (1990) and recent publications by Esparza and Movchan (1998), and Esparza (2002) contain asymptotic studies of singularity exponents at the vertices of conical defects and cones with imperfect bonding over their lateral surface. For inclusions of circular or spherical shapes, the perturbation fields can be constructed explicitly even for the case when a body contains an array of these defects placed close to each other. Inevitably, it involves analysis of

Preface

vn

interaction between different inclusions within the array and requires highorder multipole representations of solutions. The multipole method for heterogeneous domains was introduced over a hundred years ago by Lord Rayleigh (1892) who studied transport properties of conducting media containing periodic arrays of circular or spherical dielectric inclusions. This study was extensively developed and applied to models of two-phase composite structures in elasticity and electro-magnetism. McPhedran, Milton and Poladian (1988) introduced an asymptotic approximation for multipole coefficients in the representation of an electrostatic potential around circular dielectric inclusions, which are close to touching. This analysis was extended further to vector problems of two-dimensional elasticity by McPhedran and Movchan (1994). A new original development for spectral problems of electromagnetism has been published in a series of papers by McPhedran and his colleagues (1982, 1994, 1995, 1996, 1997) who adopted the multipole method to analysis of dispersion diagrams for photonic band gap composite structures used in the design of photonic crystal fibres. The technological and theoretical motivations are linked to the design of modern fibre-optics communication lines and optical niters. One of the most advanced recent studies of photonic crystal fibres that combines practical implementation of band-gap structures with analytical and numerical modelling was presented by Liu, Russell and Dong (1998), Mogilevtsev, Birks and Russell (1999), Russell and Liu (2000) and Diez et al. (2000). In the paper by Poulton et al. (2000) a generalisation of the original multipole method was developed to analyse propagation of elastic waves through a two-dimensional doubly periodic array of circular inclusions. An homogenised elastic composite material is, in general, anisotropic in the long-wave approximation. Such a material may also exhibit interesting filtering properties for the case when the wavelengths are comparable with the scale size of the periodic structure. In this book we shall show some asymptotic features of multipole solutions for the cases of small inclusions (dilute composites), inclusions close to touching (dense packing) and highcontrast inclusions. The plan of this book is as follows. We begin with a simple introduction where we talk about the compound asymptotic expansions technique applied to boundary value problems posed in domains containing small inclusions. Dipole tensors are defined both for scalar boundary value problems for the Laplacian and for vector problems of elasticity. Constructive methods are presented for evaluation of dipole

Vlll

Preface

tensors, and examples of defects of "equivalent shapes" are discussed in detail. Further, for the case of voids close to touching we introduce an asymptotic algorithm based on a multipole method for circular inclusions. The reader can see a link between Chapter 1 and the final Chapter 3, where the multipole methods are described in detail. A correspondence is also established between arrays of voids close to touching and lattice structures that exhibit filtering properties for waves of certain frequencies. In Chapter 2 we show how dipole tensors can be used in spectral problems involving domains with small defects. The main study is allocated for singularity exponents at the vertices of thin conical inclusions. Examples of "imperfect interfaces" considered in this chapter include the cases of thin and soft elastic coatings. Chapter 3 describes a multipole method, originally due to Lord Rayleigh (1892). The first two sections of this chapter deal with static problems (both electrostatics and elasticity) in composite structures containing doubly periodic arrays of circular inclusions. We also study the asymptotic problems involving dilute densely packed structures and two-phase high-contrast composites. Finally we present a version of the multipole method for eigenvalue problems of electromagnetism and elasticity, and our main aim is to discuss the structure of dispersion diagrams associated with electromagnetic and elastic waves propagating across the composite. Constructive algorithms are also given for evaluation of values of the effective refractive index for doubly periodic composites. We would like to acknowledge many productive and stimulating discussions we had with Prof. D. Bigoni, Prof. L. Botten, Prof. A. Cherkaev, Prof. V.G. Maz'ya, Prof. R.C. McPhedran, Prof. G.W. Milton, Prof. J.R. Willis, and Dr. Y. Antipov, Dr. D. Esparza, Dr. S. Guenneau, Dr. N. Nicorovici, Dr. S. Serkov, Dr. M. Valentini, Dr. V. Zalipaev. Also, we very much appreciate the continuous moral support of all the colleagues at the Division of Applied Mathematics, University of Liverpool.

Contents

Preface

v

Chapter 1 1.1

1.2

Long and close range interaction within elastic structures Dilute composite structures. Scalar problems 1.1.1 An elementary example. Motivation 1.1.2 Asymptotic algorithm involving a boundary layer . . . . 1.1.2.1 Formulation of the problem 1.1.2.2 The leading-order approximation 1.1.2.3 Asymptotic formula for the energy 1.1.3 The dipole matrix 1.1.3.1 Definition of the dipole matrix 1.1.3.2 Symmetry of the dipole matrix 1.1.3.3 The energy asymptotics for a body with a small void 1.1.4 Dipole matrix for a 2D void in an infinite plane 1.1.5 Dipole matrices for inclusions 1.1.6 A note on homogenization of dilute periodic structures Dipole fields in vector problems of linear elasticity 1.2.1 Definitions and governing equations 1.2.2 Physical interpretation 1.2.3 Evaluation of the elements of the dipole matrix 1.2.4 Examples 1.2.5 The energy equivalent voids ix

1 1 1 4 5 5 6 8 8 9 10 12 16 18 19 19 21 22 26 27

x

1.3

1.4

1.5

Contents

Circular elastic inclusions 1.3.1 Inclusions with perfect bonding at the interface 1.3.2 Dipole tensors for imperfectly bonded inclusions . . . . 1.3.2.1 Derivation of transmission conditions at the zero-thickness interface 1.3.2.2 Neutral coated inclusions Close-range contact between elastic inclusions 1.4.1 Governing equations 1.4.2 Complex potentials 1.4.3 Analysis for two circular elastic inclusions 1.4.4 Square array of circular inclusions 1.4.5 Integral approximation for the multipole coefficients. Inclusions close to touching 1.4.5.1 Scalar problem 1.4.5.2 Vector problem Discrete lattice approximations 1.5.1 Illustrative one-dimensional example 1.5.2 Two-dimensional array of obstacles

30 32 34 34 35 36 40 43 43 44 47 48 50

Chapter 2 2.1

2.2

Dipole tensors in spectral problems of elasticity Asymptotic behaviour of fields near the vertex of a thin conical inclusion 2.1.1 Spectral problem on a unit sphere 2.1.2 Boundary layer solution 2.1.2.1 The leading term 2.1.2.2 Problem for w& 2.1.3 Stress singularity exponent A2 Imperfect interface. "Coated" conical inclusion 2.2.1 Formulation of the problem 2.2.2 Boundary layer solution 2.2.2.1 Change of coordinates for the "coating" layer 2.2.2.2 Problem for w^ 2.2.2.3 Problem for u/ 2 ) 2.2.2.4 Asymptotic behaviour of w^ at infinity . . . 2.2.3 Stress singularity exponent A2 2.2.4 Some examples. Discussion and conclusions

28 28 29

57 57 57 61 63 65 76 81 81 84 85 89 102 109 115 117

Contents

Chapter 3 3.1

3.2

3.3

3.4 3.5

3.6

Multipole methods and homogenisation in two-dimensions The method of Rayleigh for static problems 3.1.1 The multipole expansion and effective properties . . . . 3.1.2 Solution to the static problem The spectral problem 3.2.1 Formulation and Bloch waves 3.2.2 The dynamic multipole method 3.2.3 The dynamic lattice sums 3.2.4 The integral equation and the Rayleigh identity 3.2.5 The dipole approximation The singularly perturbed problem and non-commuting limits . 3.3.1 The Neumann problem and non-commuting limits . . . 3.3.2 The Dirichlet problem and source neutrality Non-commuting limits for the effective properties Elastic waves in doubly-periodic media 3.5.1 Governing equations 3.5.2 Convergence of the Rayleigh matrix 3.5.3 Numerical results and comments Concluding remarks

xi

125 125 126 130 138 139 141 143 147 152 158 160 162 165 168 169 174 176 182

Bibliography

185

Index

189

Chapter 1

Long and close range interaction within elastic structures

In this chapter we discuss an asymptotic scheme developed for perturbation problems modelling small defects in solids, as well as "thin bridge" problems associated with inclusions close to touching. 1.1

Dilute composite structures. Scalar problems.

We begin* with elementary examples and illustrations related to boundary value problems for the Laplacian. Further, we introduce the definition of dipole matrices and show their applications in asymptotic models. This study is extended to problems of elasticity. 1.1.1

An elementary

example.

Motivation.

Consider an out-of-plane shear^ of a body Cl containing a small void gs (see Fig. 1.1), where 0 < e -C 1 is a small parameter characterising the relative size of the void. For the sake of simplicity, we assume that 0 is a disk of radius 1, and ge is also a disk of small radius e: ft = {(xi, x 2 ) : x\ + x\ < 1}, gs = {(xltx2)

:x\-\-x22

<e).

We adopt the notation 0 £ = fl \ gE. *We would like to thank Mr S. Malik for his help with the typesetting of this section, t i n this case the displacement vector has the form (0, 0,u(o;i,X2)). 1

Long and close range interaction

2

Fig. 1.1

within elastic

structures

A domain with a small void.

A function ue(x\, x2) is assumed to satisfy the following boundary value problem; V2us(xi,x2) -jr1(x1,x2)

= 0

=p(x!,x2)

£2<x\

when

on

dil = {{xi,x2)

+

x\

K^JEfl, on

(xi,x2)edfl.

(1.5)

(1.6)

Dilute composite structures.

Scalar

3

problems.

The energy functional is defined as follows: £(u;fl) = - / u(x) —dn (x)dsx, Jan where n is the unit outward normal on dQ,. We would like to know the energy change E(u£;9,£)

-£(u0;ft)

as we "replace" the field UQ in 0 by the field ue in the domain Ct£ containing a small void g£. For the illustrative purpose, we assume that p(x\,x2)

— C\ cos 6 + C2 sin 8,

where 6 is the polar angle, and C\, C2 are given constant coefficients. In this case UQ{X\,X2)

— C1X1

+C2X2,

and the solution ue of (1.1)-(1.3) has the form ue(xi,x2)

= u0(xi,x2)+ewE

(xi,x2),

(1.7)

where the function w£ is given by the formula w£(xi,x2)

. . . cos 0 , . . sin 9 „ . . _ .. = Ai{e) +A2(e) h B i ^ i + B2{e)x2. r r

The coefficients Aj (s) and Bj (e) are chosen in such a way that the harmonic field (1.7) satisfies the boundary conditions (1.2) and (1.3), that is, cos0 j d + eBi(e) - ^

H

+ sin0 j c 2 + eB2{e) - ^

H

=0

and cos(a;) = 0 , i £ f i ,

(1.15)

dvW —— (as) = p(x),

(1.16)

x e an.

6

Long and close range interaction

within elastic

structures

Outside a neighbourhood of dge, the function u can be approximated by the field v^\ However, v^ does not necessarily satisfy the boundary condition (1.12). This error in the Neumann boundary condition is given in the form

! > - " ( o ) ) = - i y o ) °n^£.

(i.i7)

To compensate for the leading part of the error in the boundary condition, we construct a boundary layer ew^(x/e) and seek the asymptotic approximation for the function u in the form:

u{x,e)~vW(x)+ew(0)(X), where X = x/e and the function u/°)(X) is defined as solution of a model problem posed in the exterior of the unit circle: A«,(°>(X) - 0, ||X|| > 1, —

(X) = - c o s * — ( O ) - s i n * — ( 0 ) , 11X11 = 1,

u>(°>(X) -> 0, as ||X|| ->• oo,

(1.18) (1.19) (1.20)

where p = ||X||, and 6 is the polar angle. The solution of (1.18)—(1.20) can be represented in the explicit form: w (°)(X)

= ^cos0—(0)+Sme—(0)J.

(1.21)

Direct substitution of u(x,e) = u(x,e) — u(0)(sc) — ew^(x/e) into the governing equations shows that it satisfies Laplace's equation and that the right-hand sides in the boundary conditions are small. It can also be verified that the energy norm of the function u is small, as e —> 0.

1.1.2.3

Asymptotic formula for the energy

Similar to Section 1.1.1, we show how the presence of the small void ge affects the energy £(u; Q,e) in the domain fl£. In contrast with Section 1.1.1, we do not have an explicit formula for the solution. Instead, we shall use its asymptotic approximation.

Dilute composite structures.

Scalar

problems.

7

First, we note that an additional term e2V(x) will be involved in the asymptotic approximation: u(x,s) ~ vW(x)

+ ew(°\x/e)

+ e2V(x),

(1.22)

where the role of the function V is to "remove" the error created by the boundary layer term ew^0' (x/e) in the Neumann boundary condition (1.13). The function V is defined as a solution of the following model problem: AV(x) = 0 in

^

=

fi,

-Tn\W{Xl^+X2^])

(1.23)

°na

(L24)

The energy increment is given by the formula S(u;Vle)-S{vW;Vl)

= /

p{x){v^{x)

-

u(x,e))ds

JdQ

~ - /

p{x)\ew(-°\x/e)+E2v(2)(x)\ds.

Jdo.

y

(1.25)

j

Using the formula for w^ we obtain £{u; fie) - 5(«(°); ft) ~ -e2 / - ^ — JdQ ®n

(x)T(x)ds,

where T(x) is a singular field defined as follows

This function is harmonic in fle, and it satisfies the homogeneous Neumann boundary condition dT —~(x)=0 on

ondfl.

Green's formula applied to the functions v^ and T in flE yields dv{0 \ w JJ^-{x)T(x)ds=J -—(x)T(x)ds fan on

dv{0} w f (^(x)T(x)-v(°H*)^(*)jds < — * «», , a r ,

= Jdg.

=

2K\Vv^H0)\2.

8

Long and close range interaction

within elastic

structures

The approximation for the change of energy is S(u; n e ) - £(v(0); fi) ~

-2TT£2|V1;(0)(0)|2.

(1.26)

It is consistent, of course, with Section 1.1.1; one would need to know the solution y(0' of the model problem in the unperturbed domain fi. We note that this model problem is independent of the small parameter e and its solution (analytical or numerical) can be considered as a standard exercise. Naturally, the result (1.26) yields that the energy is reduced when a small void is introduced into the solid.

1.1.3

The dipole matrix

The problem discussed above looks more complicated if the small void ge is not circular. In this case the boundary layer u/°) (X) would not allow for such a simple explicit representation as in (1.21). However, we are going to show here that there is a good way forward. We shall introduce a matrix characteristic of a defect known as a dipole matrix, this matrix can be used when one needs to evaluate the energy change associated with the presence of a small inclusion or void in an elastic solid. 1.1.3.1

Definition of the dipole matrix

One can represent the boundary layer field w(£) as a linear combination

™(0 = £ - 5 l - ( 0 H « ) '

(1-27)

where the functions Wj, j = 1,2, are solutions of the following model problem

Aw i (O=0, C e R 2 \ s ,

(1-28)

5 % ) = -n 3 -(0, £ e dg,

(1.29)

wjit) -)• 0, as IKK -> oo.

(1.30)

Dilute composite structures.

Scalar

9

problems.

The right-hand side in the Neumann boundary condition (1.29) has zero average over the boundary dg I nj{$)ds = 0, j = l,2. Jdg

In this case there exists a solution of (1.28)-(1.30), which has a finite energy and decays at infinity like 0 ( | | £ | | _ 1 ) . It is possible to represent the leading-order part as a linear combination of first-order derivatives of the fundamental solution for Laplace's equation:

^•(€) = - ^ E m ^ ] r | 2 + c , ( i i € i r 2 ) .

a-31)

IKII^ 0 0 -

where my are constant coefficients. The matrix m = {rnij)\j=i of coefficients from the expression (1.31) is called the dipole matrix. 1.1.3.2

Symmetry of the dipole matrix

The fields Wj in (1.31) are chosen in such a way that the functions Wj(£) = tj + Wj(t),

£ G R 2 \ g, j = 1,2,

(1.32)

satisfy the homogeneous equations (1.28) and (1.29) (the right-hand side should be replaced by zero) and behave like linear functions at infinity. It will be shown that the matrix m is symmetric. Green's formula, applied to the functions Wj and wk, in a large domain Dr \g = {£ : ||£|| < r}\g gives 0= /

(wkAWj

-

WjAwk)dS

JDr\g

= L {-^-Wk ' -0r-Wj)dS + L9 \-^Wk '

^WT

= I$+I&), where

i$ = -^j[2'j^E^ilpi^ii^+ow^-^.

(1-33)

10

Long and close range interaction

within elastic

structures

as r —> oo,

(1-34)

and

Here we use the notation (wk,Wj)=

/

-^—Wjds,

Jda dn

and note the symmetry (Wk,Wj)

=

(Wj,Wk)

and the fact that the matrix {Mjk)2j k=\ with components M.jk = (wj,u>k) is positive definite. Taking the limit as r —»• co in (1.33), we derive Mjk = -rrijk - Sjk Sg, j , k = 1,2,

(1.36)

where 6jk is the Kronecker delta, and Sg denotes the area of g. It follows from (1.36) that the dipole matrix m, defined for a void in a solid, is symmetric negative definite. 1.1.3.3

The energy asymptotics for a body with a small void

We note that the matrix m provides a canonical representation for the energy increment when a defect is introduced in the elastic medium. Assume that one would like to evaluate the energy for the problem similar to the one discussed in Section 1.1.2, for a body containing a non-circular void. The asymptotic algorithm would have to be modified, and the boundary layer problem would not allow for a simple explicit solution. We shall show here how the dipole matrix can be used to provide the required asymptotic formula for the energy. The potential energy associated with the state of out-of-plane shear of an elastic body f2e, with a small void ge, is defined by Se(u; n e ) := - / p(x)u3(x,e)ds, JdV

(1.37)

Dilute composite structures.

Scalar

problems.

11

where T is the exterior part of the boundary, and p is the applied shear traction. It is remarked that the three-term asymptotic relation should be used: u3(x,e) ~vw{x)+sw(x/e)+e2V{x),

xen\g£.

(1.38)

The function V is defined in Cl and "removes" an error produced by the boundary layer w in the Neumann boundary condition at the exterior surface. It satisfies the following boundary value problem AV(x) = 0, x £ 0 ,

(1.39)

J,AC—1

The function V can be written in the form

v W =£„4r«(*) + i ^ ) ^ < o , . U

J

j,k=l

(1.4D

where T^ >, k = 1,2, are singular at the origin, and the second term of the expression inside the brackets compensates the singularity. The functions T(fc) are defined as solutions of the Neumann boundary value problems in 0. nAT^ix) + s—(x) = 0, x G n, dxk

(1.42)

——(x) = o, x e an.

(i.43)

on Here S(x) is the Dirac delta function, and solutions of (1.42), (1.43) should be understood in the sense of distributions. We recall that the function v^ satisfies the Neumann boundary value problem in the region fi Au (0) (a;) = 0 , i € f i ;

/x-|—(a:) = p(x), x € dSl.

(1.44)

On the boundary dfl we have e2 J2 ^JkT(h)(x)—-(0), i,fe=i

Xj

xedtl.

(1.45)

Long and close range interaction

12

within elastic

structures

Consequently, the leading part of the potential energy increment is specified by £E(u;£l)-£0(vW;n)~-e2

V m j f c ~ ( 0 ) / p(x)T^(x)ds. dXj

j,k=i

(1.46)

Jdn

Using Green's formula, one can simplify the last integral (

p{x)T{k){x)ds=

(T^k\x)Av^(x)-v^(x)ATw(x))dx

f

Jdg

Jdfl

The formulae (1.46), (1.47) imply ££(u;n)-£0(vW;n)

~ e2 ] T ^ - ( 0 ) m j f c — ( 0 ) J,fc —1 2

= £ Vt; (0) (0) • m V / » ( x ) ( 0 ) ,

(1.48)

i.e. the leading part of the potential energy increment is represented as a quadratic form with the matrix m. Thus, in order to evaluate the energy change one needs a solution of the boundary value problem (1.44), corresponding to the unperturbed region fi. The quantity (1.48) is invariant with respect to rotation of the coordinate system, and it is readily verified that (rrijk)2:fc=1is the rank-2 Cartesian tensor. 1.1.4

Dipole matrix for a 2D void in an infinite

plane

In this section we give a constructive algorithm, which can be used for evaluation of dipole matrices for two-dimensional voids (see Fig. 1.2). Consider the following boundary value problem: V2W0) = 0

dn

0

in R

on

W^=tj+w^(0, where w^((,)

-> 0 as |£| ->• oo.

2y-

T = dg, J = 1,2,

(L49)

(1.50) (1.51)

Dilute composite structures.

Fig. 1.2

Scalar

problems.

13

A void g in an infinite plane.

Introduce analytic functions

z = £\+ *6, where the functions W^ equations

and V^

(1.52)

satisfy the Cauchy-Riemann system of

(1.53) 9£2

9€i '

Note that dWW dn

n\——

56

d6~ dvw

ni

hn2n2-

36

96~ (1.54)

where r = (—n2, rii) is the tangent vector to the boundary dg. The boundary condition (1.50) can be rewritten in the form f(j){z)

~ f(j)(z)

= constant.

The constant on the right-hand side can be taken as zero.

Long and close range interaction

14

within elastic

Thus we are looking for such functions f^(z),j analytic in R 2 \ g, satisfy the boundary condition /«>(*)-/tt)(z) = 0

on

structures

= 1,2, that fW> are dg,

(1.55)

and have the form °o

B(J)

z

fe=i

where ^(1)

=

A{2)

1;

=

_L

(L57)

Let us introduce a conformal map F oo

z = F{(7)=Cla

_

+ Y,^t, fc=i

(1-58) °

which establishes a correspondence between the complex variable z in the exterior of the void u) and the complex variable a in the exterior of the unit circle. Note that on the unit circle L = {a : \a\ = 1} the following relation holds a = a~x.

(1.59)

We shall use the notation JP0')(CT) = /«(F((7)). Then, as a -> oo,

J&X&) = A^da +

IA^C-!

+ ^-

) i + O (j^\

(1.60)

Let (T° be a complex number in the exterior of the unit circle, | 1. Then 2m JL a — (Tu

2iri JL a — cru

(1.61)

Using (1.60), (1.61) and applying the Cauchy theorem we deduce A ( j ' ) C_ 1 +

RC0 7

f--A«C1=0,

(1.62)

Dilute composite structures.

Scalar

problems.

15

so that

B[j) = B^R + iBtf = AU)\d\2 -

"{

|Ci|3-,

x€dg.

(1.69)

Solutions are sought in the class of functions which allow for the representation 2

xeR2\g

u{x) = Y^Ci{xi+w^{x)}, i=l 2

u^{x)

= YJCi{xi

+ w^°\x)},

xGg,

i=i

where the fields w(l\x) and w^'°\x) are harmonic in R 2 \ 5 and g, respectively, and satisfy the following transmission conditions on dg: dw® V-Q^-

dw^ ~ Mo

dn

( i ) n

~ (Mo ~ MK,

( i 0

)

M

WW(X)=W^">(X),

xedg.

(1.70) 0

Then it follows that the harmonic fields w^ + Xi, w^' ** + Xi satisfy the interface conditions (1.69), and u;W(x)-*-0, as\x\-+oo.

(1.71)

The following equality is verified by direct calculation M = - m + (/uo - fJ.) Sg I,

(1.72)

where Mij = fx f Vw ( i ) • Vw^dx 2 JR \g

+ Mo / Vw ( i ' 0 ) • Vu>(j'-0)dx. h

(1.73)

18

Long and close range interaction

within elastic

structures

Note that, taking the limit as Mo —>• 0 we arrive at the problem for a solid containing a void g (with the Neumann boundary condition on dg). "Stiff" inclusions. However, the above formula cannot be used when the material of the inclusion is "much stiffer" compared to the material of the matrix (when the shear modulus Mo is large compared to /x). Consider an alternative way of extending the polynomial fields x\, x-i inside the inclusion g. A harmonic function u>*(*'°) can be defined on g in such a way that M - g j r ( * ) - « . - ^ - ( x ) = Ol w®(x) - w oo we obtain m = Af + n SgI. It is noted that the dipole matrix m is negative definite for a void (with the boundary conditions of the Neumann type; see formula (1.72)), and it is positive definite for a stiff inclusion (with the Dirichlet boundary condition which allows for a "rigid-body translation" of the inclusion). 1.1.6

A note on homogenization structures

of dilute

periodic

The notion of dipole matrices can be very useful in the homogenization theory for dilute composite structures.

Dipole fields in vector problems of linear

elasticity

19

It is known in the literature (see, for example, the book by Bensoussan, Lions and Papanicolaou, 1978) that the effective moduli for a periodic dilute composite under the conditions of out-of-plane shear can be evaluated in the form j}nk = H f

W n > • Vuds + no f W n ' 0 ) • Vuda;

JQ\g,

(1.77)

Jge

where Q, is an elementary cell [—1/2,1/2] x [—1/2,1/2] containing a small inclusion ge (as in the text above, e is a small non-dimensional parameter characterising the relative size of the inclusion), and u^n\ u( n '°) satisfy the equations (1.67)- (1.69) in the matrix and the inclusion, periodic boundary conditions, and u^ admits the representation of the type (1.38). Integrating by parts we can write (1.77) in the form:

-Ho / n i U ^ ° ) ^ ds. (1.78) Jdgc 9xt Due to the interface contact conditions, the integrals over dge cancel, and the above formula leads to the following asymptotic approximation Hnk ~ fJ-Snk + £2rnnk.

(1.79)

This suggests that the dipole matrix can be interpreted as a normalized correction term in the representation for a matrix of effective moduli for a dilute composite containing small inclusions (or voids). Analysis of properties of dilute composites, based on the dipole matrices is presented in papers by Movchan and Serkov (1997), and Cherkaev et al. (1998). 1.2 1.2.1

Dipole fields in vector problems of linear elasticity Definitions

and governing

equations

Consider an infinite elastic plane containing a finite void g C R2. Let u(x) = (ui(x),U2(x)) be the displacement field which satisfies the homogeneous Navier system /xV2u + (A + / J ) V V • u = 0,

xeR2\g,

(1.80)

Long and close range interaction

20

within elastic

structures

and the homogeneous traction boundary condition (n)

(u;a5) = 0, xedg.

CT

Here A and /i are the Lame elastic moduli, and a^ with the components (n)

(1.81) is the vector of tractions

V^ fc=i

where rik are components of the unit outward normal on the boundary dg. We also assume that the field u(x) has the following asymptotic representation at infinity: 3

u(x)~X)dpVW(x),

(1.82)

P=i

where di are constant coefficients, and the vectors V^' are defined by the formulae V « = (xu0)T,

V™ = ( 0 , z 2 ) r , V = l/y/2(x2,Xl)T.

(1.83)

The solution of the problem (1.80)-(1.83) can be represented in the form 3

u(as) = ^ d p ( V W ( i ) + W W ( i ) ) ,

(1.84)

P=i

where the dipole fields W^ p '(a:) satisfy homogeneous Navier equations and decay at infinity. Let T represent Green's tensor for the system of Navier equations in two dimensions 1+

T(aj) = q

*2

1+

2

„

2

here the constants q and x are defined by

«

=

5 ^ A W * =3-4''

where i> is the Poisson ratio.

(L85>

Dipole fields in vector problems of linear

elasticity

21

The dipole fields W(-P\x) from (1.84) decay with the same rate as the first-order partial derivatives of T, and moreover we can write 3

3

WW ~ £Mpfc £ fc=i

Vf>(A)Ttf)(x),

(1.86)

j=i

where T^\ j = 1,2, represent the columns of Green's tensor T; VJ (gj) are the first-order differential operators. Definition. The matrix {Mpkjt fc=j of coefficients in the asymptotic representation (1.86) is said to be the dipole matrix of the void g. 1.2.2

Physical

interpretation

Let S denote the vector of strain defined by the formula 5 ( u ) = (en(u),e 2 2 (u), V2e 1 2 (u)) T , where the strain components tij are defined in a standard way

Let u° denote the unperturbed diplacement field, before the inclusion/void is introduced in the elastic plane. We assume that it is linear in x\ and x 2 (like in (1.82)). When a void, characterised by the dipole matrix M , is introduced into an elastic plane, the change of elastic energy is given by SS = 5 T ( u ° ) M 5 ( u ° ) . For details of this technical derivation we refer to the paper by Movchan and Serkov (1997). One can also introduce a Cartesian tensor of rank 4, denoted by (M-ijki), in such a way that 5r(u°)M5(u°) = £

e i j (u

0

)M i j f c ( e f c i (u°).

iyj,k,l

Such a tensor is said to be the dipole tensor.

22

1.2.3

Long and close range interaction

Evaluation

of the elements

within elastic

structures

of the dipole

matrix

This section is based on the results of the paper by Movchan and Serkov (1997). Complex potentials. We introduce the Kolosov-Muskhelishvili complex potentials <j>, ip (see the book by Muskhelishvili, 1953) in such a way that u± + iu2 = (2fj.) 1{x(j>(z) - z(j>'{z) - ip(z)}. Here z = xi + 1x2. Let z = w(£) be the conformal mapping function N

-(o = ^ + E | ? n=l

(1.87)

S

which establishes a correspondence between points of the R 2 \ g and points in the exterior of the unit disk |£| > 1. The boundary is assumed to be traction free, which means that (1.88) As £ —>• 00, the complex potentials