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\
:xl+xl
(1.1)
= l},
(1.2)
and 9u £
(£1,2:2) = 0
on
dge = {(x1,x2)
:x\ + x22 =£2},
(1.3)
where r is the polar radius, r = [x\ + x2)1/2, and p(xi,x2) is a given function which is assumed to have zero average over the exterior boundary Oil, that is,
J:
p(x\,x2)ds
= 0.
(1.4)
Let UQ be a solution of the unperturbed problem posed in Cl (homogeneous domain, without a void)
V2u0(xi,x2) -7^(xi,X2)=p{x1,x2)
= °>
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<9 {d + eJBi(e) - eA^e)} + sin/9{C2 + sB2(e) = Cicos6 + C2smO. Hence ^• = ^ • = ( 1 ^ 2 ) ^ .
3 = 1,2,
eA2{e)}
4
Long and close range interaction
within elastic
structures
and the solution uE has the form e2 ( 1+ Q_g2)(1+a.2
Ue(x1,X2) = (C1X1+C2X2)
+ a .2)
The energy change associated with the presence of the small "void" ge within the unit disk fi is evaluated as follows: £(ue;Q£) - £(u0;Q.) = -
(ue - u0)p(x)ds. Jda 2
r2n
-2 / 1 — £ ) Jo
(Ci cos 6 + C2 sin Qfd6
= -<^){C*+C>)-
(L8)
This explicit result indicates that the energy E is decreasing when a void occurs within an elastic body subjected to an out-of-plane shear load. It follows from (1.8) that the asymptotic approximation of the energy change is given in the form £(u £ ;Q E ) -S(u0-,n)
27T£2|Vu0(0,0)|2.
It will be shown in the text below that this formula also holds for the case when a small circular void exists in a body of an arbitrary shape. Of course, we were able to produce this simple solution because of a special choice of geometry of the domain. It is quite tempting now to think about the energy change for bodies whose boundaries are not circular. It turns out to be possible to have a general asymptotic algorithm, which would enable one to estimate the change of energy for a body containing a small defect (or a group of defects) of general shape. In further sections, we describe an asymptotic algorithm based on the method of compound asymptotic expansions (see Maz'ya, Nazarov and Plamenevski (2001) and Kozlov, Maz'ya and Movchan (1999)), and furthermore we shall extend this study to vector problems of elasticity. 1.1.2
Asymptotic
algorithm
involving
a boundary
layer
Like in the previous section, we consider an out-of-plane shear of an elastic domain Cle that contains a small cavity ge. However, we no longer assume that n is a disk.
Dilute composite structures.
1.1.2.1
Scalar
problems.
5
Formulation of the problem
Mathematically the problem can be formulated in the following way. We assume that a bounded domain fi C K2 has a smooth boundary and contains the origin. As before, ge is the disk of radius e with the centre at the origin; evidently ge = {x : e~xx G g},
(1.9)
where g is the unit disk. The same notation is adopted ne=Q\gs.
(1.10)
Then the displacement u satisfies the boundary value problem Au(x,e) du dr du dn
= 0, x G O e ,
(1-11)
= 0,
(1.12)
= P(x).
(1.13)
dgs
an
Here d/dn is the outward normal derivative on dCl. The function p(x) is assumed to be smooth, and it is also assumed to satisfy the balance condition / p(x)ds = 0. (1.14) Jan The problem (1.11)-(1.13) has a solution specified up to an arbitrary additive constant. The condition (1.14) means that the total shear force, applied at the exterior boundary, is equal to zero.
1.1.2.2
The leading-order approximation
In the absence of the void we would have the following Neumann boundary value problem in fl: A«(0>(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-
A^dC.j. j = l,
where B[% and B$ are the real and imaginary parts of B-f', respectively. We recall that W& = R e ( / « ) (see (1.52)), and
wW=
-i( m ^G + i) + ^ 2 G-i)) +o (^) as i z i- >00 -
It is noted that
and hence mn
=
-2TTB[JI
mj2
=
-2TTB[JJ\
Finally, we derive the formula for the dipole matrix
V
Im(CiC_i)
-|Ci|2-Re(CiC_i)/
It follows from this formula that the "void" g has the same dipole matrix as an elliptical void characterised by the conformal map z = Ci
a,b are real and positive.
In this case
a = Ci+C-i, 6 = C1-C_1,
Long and close range interaction
16
within elastic
structures
and hence d = ±(a + b),
C_ 1
=
i(a-6).
(1.65)
Note that — |Ci| 2 + Re(CiCLi) = -\(a = -^(a
+ bf + i ( a + b)(a - b) + b),
and
- I C i ^ - R e C d C - ! ) = - | ( a + 6). It follows that the dipole matrix for an ellipse, oriented along coordinate axes, has the form
m = -n(a + b)fbQ0ay
(1.66)
Energy equivalent voids We shall say that two voids are energy equivalent if they possess the same dipole matrix. In this case the energy increment associated with the presence of such voids is the same when they are placed in an infinite plane with a constant load at infinity. It follows from formula (1.64) that only two coefficients C\ and C_ i of the conformal mapping are required to construct the dipole matrix for a scalar problem associated with an out-of-plane shear. Hence any class of energy equivalent voids, for out-of-plane shear, includes a void of elliptical shape.
1.1.5
Dipole matrices
for
inclusions
Dipole matrices can be defined not only for voids but also for inclusions, so that instead of the Neumann boundary condition (this is what we have for a void) we deal with transmission interface conditions on the inclusion boundary.
Dilute composite structures.
Scalar
17
problems.
"Soft" inclusions. Let us assume that the domain w is occupied by an elastic material with the shear modulus fio, and the surrounding matrix is characterised by the shear modulus fi. The following equations hold /iV 2 u(x) = 0,
xeR2\g,
(1.67)
and M
V2/»(x)=0,
xeg.
(1.68)
Conditions of ideal contact (involving continuity of the "displacement" and "tractions") are prescribed on the interface boundary du du^0' H— - fi0——,
(o\ u = u(>,
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
= (— - l\ xit
x G dg.
(1.74)
Then the fields w^ + Xi, to*(*'0) + /xxi/fio satisfy the interface boundary conditions (1.69), and as before u;M satisfies the condition (1.71). In the present case, we have extended the field «;W i n to g in such a way that the shear tractions are continuous on the interface boundary. It can be verified by direct calculations that 1-A Mo
Sg I,
(1.75)
where
Mij = ix I
Vu>« • Vw^dx 2
JM. /K \Q \9
+ MO /" V w ' ^ 0 ' • Vw^'^dx
(1.76)
JJg a
Taking the limit when Mo —> 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 > • Vu<*>ds + 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<3> = 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
(1.89)
with given complex coefficients a, 7. To find the functions <j> and ip, which are analytic in the exterior of the unit circle and satisfy the equation (1.88) and the conditions (1.89), we apply the classical method developed by Muskhelishvili (1953) and reduce the problem to a system of integral equations on the circle L = {£ : |£| — 1}
(1.90)
The complex potentials are represented in the series form
(1.91) (<0
m=^+£r=i $
Dipole fields in vector problems of linear
elasticity
23
Let 5+ = {|£| < 1} and S~ = {|£| > 1}. Using the Cauchy theorem we deduce 2m JLa
2m ™
-£
JL°-£
2m JL
a - £
i
and similarly 2m 1
1
2m JLL ua - £
2mJLJL a-£ a
£
We note here that for a point a on the unit circle, \a\ = 1 and a = 1/er. The following notations are used
fc=l
s
and ~
fl(fc)
fe=i
£*
Also
where N-m-l a m = apN-m
-
2^
PN-m-n-inp
,
n=l
Pfc =
1.^ T7-
Jtfc
Clr
+1
+ E l i c-»^-
(1.92)
^JV
5T - £ „ = 1 " C - „ ^ + 1
«=0
24
Long and close range interaction
within elastic
structures
and
2ir. JL u'(a)(a - 0
u'({)
i i
Then, we obtain a system of linear algebraic equations with respect to unknown coefficients in the expansions (1.91). It is remarked that the coefficients a and 7 are determined by the conditions at infinity. We consider three sets of complex potentials which correspond to the following choice of a and 7: 71 = -na,
72 = MCi, 73 = V2/xcii, (1.93)
a\ = — — 7 , Q2 ""
x—1
a3=0.
x-1
The corresponding complex potentials are , , .
a.jZ
Bj
_ /
1 (1.94)
1
7j* . #.
as 12;I —> 00. The coefficients Bj and .D,, j = 1,2,3, have the form Bj = fijCi — atjC-i = —JJCI — c\bj — otjC-i, Dj — SjCi — 7 / c _ i = —oTjCi — djC\ — 7jC_i,
where JV-2
bj = otjpN-i
+ pN-3(a-ij
+ Ij) + 2 J
kpN-k-2akj,
fc=2
dj = oljPN+i + PN-i(aij
(1.95)
N + Ij) + y^fcpjv-fcOfcj, fc=2
and the quantities akj are defined from the following system of linear algebraic equations:
O-mj — PN-m-2(o-lj
+ Ij) ~
N-m-l 2_^ k=2
^PN-k-m-l^kj
= Ct.PN-
Dipole fields in vector problems of linear
elasticity
25
The dipole fields. In terms of complex potentials the fields W ^ given in the form W?> + iW?' = —
xj(z) - z#Az) -
2/x ~™~'
^ 1
TT
Z
-^
1
2/x xBi-
~ ^
+ 0
T-pr I ,
+ Bi^ - Diz z z The asymptotic representation (1.86) is equivalent to
Mz)
are
(1.96)
\z\ ->• OO.
W[j) + iW?] ~ M,-! ( £ { T n + iT 12 } + j L { T n + zT12}
+ M j 2 ( ^ { - ^ 2 2 + iTi 2 } + ^ { T 2 2 -
iT12}\
+ - ^ M i 3 ( ^ { 3 i i + ^22} + J^{2Ti2 + i(T 22 - T u ) } ) .
(1.97)
The complex representation of Green's tensor T is / . -2xlnlz| +
T =q 2
2zz
2
.z -z 2zz
\
Z2-Z2
(z + ^z)2 2zz -2xln\z\
-
(z-z)2 2zz
I
and the coefficient q is the same as in (1.85). Consequently, the dipole matrix M is denned by /_
M =
4
n +x - l n
(x-l)
(x-1) A
-e + x - l
2
n-
2
-n-
-e
(X-1)2
A
x-l A
x-1
(x-l) A
e+ x - l
2
e+ x - l T (1.98)
Long and close range interaction
26
within elastic
structures
where n = \ci\2 +Re[a{c1\,
S = 2Re[c1c_i] +Re[o"ci] + R e [ a I 1 c i ] , "
E = | C l | 2 + Re[a^ l C l ],
T = - 2 | C l | 2 + 2Im[a[ Cl ],
6 = v^ImKci],
A = V2(Jm[ac-i]
.
(1.99)
+ Im[ a i ci])>
and a], a" and aJ are specified as solutions of the following system of linear algebraic equations (see Movchan and Serkov (1997))
JV-ro-1 a
m ~ PN-m-lO,"
-
2_^
PN-m-k-\ka%
=
PN-m,Cl,
PN-m-k-lkaJ
=
PN-m-k-lka^
= pjV-m-2*Ci,
JV-m-l a
m ~ PN-m-2a{
-
m ~ PN-m-2~a[
~
^ k=2
PN-m-2Cl,
> (1.100)
N-m-1 a
^J fc=2
where the index m may take values — 1 , 1 , . . . , N. The formula (1.98) gives the explicit representation of the dipole matrix for a void whose shape is characterised by the conformal map (1.87). In the following section we give simplified formulae for the dipole matrices for the cases when the conformal mapping functions are represented by sums of three or four terms.
1.2.4
Examples
Here we give examples of the dipole matrices for regions whose shapes are close to polygonal.
Dipole fields in vector problems of linear
27
elasticity
1. Let the conformal mapping w(£) have just three non-zero coefficients c i , c _ i , c _ 2 . In this case Cl
x - 1
M =
A
5
/-
(x-1)2
(x-1) „
Icil2-
Aq
Cl
(x-1)2
-
x - 1
2
x - 1
5 (x-1)2
ci
A
A x - 1 -2|ci|2
x - 1
x-1
\
(1.101) H = 2(|c1|2 + |c_1|2 + 2|c_2|2),
S = 4Re(c_ 1 ci),
A = 2V2Im(c1c_1).
2. For the case of a conformal mapping with non-zero coefficients c i , c _ i , c _ 2 , c - 3 (|ci| ^ jc_ 3 |) the m a t r i x M is given by (1.98), where the coefficients can be represented in the form n =
Re(c3c2)
+
l C l |*
^
4
|ci|2-|c-3|2
Re(cic_i) + |ci|2 |ci|2-|c-3|2
Im(dc_3)
0 = V2
Im(cic_i)
T =2
|ci
2
- |c_3|2
+
+
C!| a -
|C_3|2
4Re(cic 2 _ 1 c_ 3 ) I
10
I
19
\ci\2 - | c _ 3 | 2
2 Cl|,
ci|2-|c_3|2
A = 2^/2
ci|
ci|2 + |c_3|2
S = 2|c!| 2 + 4 | c _ 2 | 2 + 6 | c _ 3 | 2 + 2 | c _ i |
Re(c2c_3c_i)
|2
|ci|2-|c-3|2
2
ci] +
Im(c?c_3c_i) - 19 —I — 19 Cx| 2 -
|C_3|J
Re(c_ 3 cf) - |ci| |ci|2-|c_3|2 (1.102)
1.2.5
The energy
equivalent
voids
Like in t h e scalar case, one can define classes of energy equivalent voids in vector problems of elasticity. Voids t h a t belong to t h e same class possess t h e same dipole m a t r i x M (see (1.98)). In contrast with the scalar case (outof-plane shear) one cannot state t h a t every class of energy equivalent voids
28
Long and close range interaction
within elastic
structures
Fig. 1.3 Energy equivalent voids. The conformal mapping o>(£) is given by u>(£) = £ + ( 7 0 - 1 + ( 3 £ ) - 2 e i a + £ - 3 / 1 3 , where (a) a = 0, (b) a = TT/6, (C) a = TT/2, (d) a = 4TT/3.
contains an ellipse. Figure 1.3 presents examples of energy equivalent voids whose shapes are close to polygonal; they are specified by the conformal mapping w(£) = £ + (70'1 + (3£) - 2 e i a + £ - 3 / 1 3 , for different values of the parameter a.
1.3
Circular elastic inclusions
Here we give examples of dipole matrices for circular elastic inclusions. Two cases are considered: perfectly bonded inclusions (displacement and tractions are continuous at the interface) and inclusions separated from the matrix by an imperfect linear interface.
1.3.1
Inclusions
with perfect
bonding at the 0
0
interface
Let R be the inclusions radius, and A , fj, , A, /x be the Lame elastic moduli of materials of the inclusion and the matrix.
Circular elastic
inclusions
29
1. For the case of out-of-plane shear (Laplace's operator), the dipole matrix can be written as follows m = 27T/U / * - W > # T
M + Mo where I is the identity matrix. It is noted that the matrix m is positive definite when /x0 > /x, and it is negative definite when ^o < M2. For the vector case, when the displacement field satisfies the Navier system, the Kolosov-Muskhelishvili complex potentials are represented by
x)
^
=
1Z
T
+
Rfl(a)fl2/io(x-l)-2/i(x0-l)
—z
(1.103)
= -5- H ;—> R z X/XQ + (x
M*b-1) + 2W,
/ 1
+
°{W
(1.104)
Here the constants a and 7 are chosen in accordance with (1.89). The dipole matrix has the form
/s + e s-e 0 \ M =
H - e s + e
4q V
0
0
(1.105)
0 26/
where 0 =
1 2/i 0 (x - 1) - 2/i(x 0 - 1) (x - l)2 /x(x0 - 1) + 2/x0
The matrix M is positive definite provided fio > fi and /xo(x — 1) > /i(xo — 1), and it is negative definite when /xo < /x and /io(x — 1) < /x(x0 - 1). 1.3.2
Dipole tensors
for imperfectly
bonded
inclusions
We consider a coated elastic inclusion and show that parameters of the interface layer can be chosen in such a way that a coated inclusion becomes neutral. This section is based on the results of the paper by Valentini et al. (1999).
30
Long and close range interaction
within elastic
structures
1.3.2.1 Derivation of transmission conditions at the zero-thickness interface Consider a circular coated inclusion. Let Si stand for the domain occupied by the inclusion, So for the thin interface layer, and S2 for the exterior part of an elastic body. The inclusion and the ambient medium are assumed to be isotropic elastic solids characterised by the Lame constants /ii, Ai, and /X2, A2. The material within the interface layer is assumed to be linear transversely anisotropic, and the components of stress and strain are related by the following linear equation
(
Crr Cr9
Cr$ Cee
0 \ 0 ;
(1.106)
0 0 2G9) in the particular case of isotropic material Cgg — Crr = A + 2/i, Crg — A and GQ = [i. The equilibrium equations are written in the form
Lil>(0),C^Ge)=0, 2
W r„<2) H(o) i a > ^2,A2)=0,
aeSo,
xGS2,
where Lp 0 ' iar (-) is the Lame operator written in the polar coordinate system. In the matrix form it can be represented as
L%r:=D1H^Dl where D i , D2 are the matrix differential operators
Circular elastic
inclusions
31
where H1-0' is given by (1.106) for the interface, whereas for the matrix and inclusion Hij)
/2^')+A« A« \ 0
=
\W 2n& + A « 0
0 0 2|i<J'',
The boundary conditions on the outer and inner boundaries of the interface layer are the conditions of the perfect bonding: ffW(«(i))
= (T(«)
(„(<»),
u
<7< n V 2 ) ) = * { n ) (w ( 0 ) ).
r = R,
«(2)=u(°),
r = fl + £,
(1.107)
where e is the thickness of the interface. Solution of the above formulated problem is sought in the form of the asymptotic series: oo
1
2
oo
2)
«<*> ~ xytii *.
«< ' ~ jryui ,
t=0
i=0
u(3}
oo
~ Xy«i 3 ) t=0
A scaled "fast" variable p is defined within the interface layer
p=(r-R)e~l. Then the equilibrium equation inside Ho can be expanded in powers of £ with the leading term i / r " " v '° 12 1I „W r 9_y,°> "5^ S
0.
(1.108)
C r £ i - 0p ~ -: ^
The solution is sought in the form
Wi)-{c{B)P + D(e))-
(L109)
The leading-order term in the representation of traction at the interface can therefore be rewritten as
.<">(uT>=i(^>).
(I.U0)
32
Long and close range interaction
within elastic
structures
From (1.110) we deduce that the leading-order terms of tractions on the outer and inner boundary of the interface are equal: ^(ll^lr-K = ^ ( ^ )
=
^(uP)\r=R+£.
Let us now analyse the transmission condition for the displacements. Following (1.109), the displacement jump between the outer boundary (p = 1) and inner boundary (p = 0) is specified by the functions A{6) and C{6). The displacement jumps across the interface are given by (2)i
(i)i
Kfi\r=R+e
1
~ Kfi\r=R
(2) I
l
(1)|
/ (o)\
= 7^r<»rK
I
)
(1.111) r=R
(0)x
(1.112) r=R
where we have assumed that the elasticity coefficients of the interface layer are small, i.e. C„ = eC*Tr,
Ge = eG*e.
(1.113)
Relations (1.111) and (1.112) describe a linear, zero-thickness interface, and the stiffness coefficients of the interface can be expressed as Sr =
, £
Sg =
.
(1-114)
e
It is noted that if the elastic coefficients of the interface do remain finite, one would arrive at the conditions of perfect bonding: 0^\vP)\T=R 1.3.2.2
= <7(nV3))|r=fl+e,
U*1' \r=R = U^\r=R+e.
(1.115)
Neutral coated inclusions
Here we consider a particular example for a circular imperfectly-bonded inclusion and give an explicit representation for the corresponding dipole tensor, which characterises the energy change associated with the presence of such an inclusion. The definition of the dipole matrices, and their properties were discussed in details in the earlier text. The paper by Bigoni et al. (1998) also includes detailed study of a dipole tensor for the case when the inclusion is coated by a layer of finite thickness.
33
Circular elastic inclusions
The dipole matrix for a circular inclusion, of radius R, with the linear interface described above has the form a
p
=
49
/ £ + 2r?/(x - l ) 2 -£ + 277/(x-l)2
V
- £ + 2»7/(x - l ) 2 ^ + 277/(x-l)2
0 \ 0 ,
0
2Z)
0
(1.116)
where _ srR(no(x
- 1) - /x(*o ~ 1)) ~ 4/z^o
srR(fi(x0
- 1) + 2/x0) + 4/ijio
. _ srse-R2ro(jU0 - n) + (a r + sg)Rnno(3(n0 - /x) - T 0 ) - 12/J 2 /I§ ^~ srsgTT0R2 + R(sr + sg){3r + To)iJ,fj,Q + 12fi2iJ,l T = x/x0 + /x, T 0 = x 0 /i + /i 0 ,9 = (A + /i)/(87r/i(A + 2/x))). It is remarkable that the appropriate choice of the stiffness parameters may result in the coating corresponding to a "neutral inclusion". Namely, if 4/x/xp
= Sr
~ R(jio{x - 1) - M * - 1))
and «0
=
4/i/io(3(r - Ho + A») - 2f)
fl((/io-Ai)(ro + 3r) + ro(r 0 -r))'
all the entries of the dipole matrix (1.116) become zero. An inclusion which possesses such a dipole matrix is "invisible" - it produces no energy change if placed in an homogeneous stress field. It is also readily verified that in a particular case, when the radial and tangential stiffness coefficients are equal, Sg = Sr ~
S,
the quantity £ can be written as follows: _ ~
(/j. - HQ)RS + 2fj,fj,0 TRs + 2/i/io '
The eigenvalues of the matrix V are Ai = R2£/(2q) and A2 = R2r]/(2q), and it is straightforward to observe that these eigenvalues are negative when |s|«l.
34
1.4
Long and close range interaction
within elastic
structures
Close-range contact between elastic inclusions
The present section is based on the work by McPhedran and Movchan (1994). This research was motivated by an earlier publication of McPhedran, Poladian and Milton (1988). Instead of dilute composites when inclusions/voids in an elastic solid are separated by a large distance, this section presents analysis of problems involving inclusions which are close to touching. The analysis uses the multipole method (for a classical scalar version see Lord Rayleigh, 1892) applied to circular inclusions in the two-dimensional linear elasticity (plane strain).
1.4.1
Governing
equations
We study a plane-strain problem for an isotropic elastic body, with the Lame moduli A, /x, containing a set Q, — l j i f2j of two circular inclusions fli, i = 1,2,... with the Lame moduli A0, / A The displacement fields u, u° in the elastic matrix and elastic inclusions satisfy the system of equilibrium equations /iAu(x) + (A + fj,)VV • u(x) = 0, x e R 2 \ H ,
(1-H7)
/x°Au°(x) + (A0 + /i°)VV • u°(x) = 0 , x e l l ,
(1.118)
and the interface boundary conditions u(x) = u°(x), (u;x) = <7°W(u 0 ;x), x <E dfti,
(1.119)
where a\n' = a^rij ; a^ = ^{dui/dxj + duj/dxi) + AJjjV • u are stress tensor components, and n = (rj.1,712) is the unit normal vector on the boundary of the inclusion. The following conditions are prescribed at infinity an(x)
-> oft, of2
C I 2 ( : E ) —» 0,
as
\x\
—>
00.
(1.120)
Close-range contact between elastic
1.4.2
Complex
inclusions
35
potentials
We use the Kolosov-Muskhelishvili complex potentials to represent the displacement field 2/i(ui + iu2) = X(j>(z) - zcj)'(z) - ip(z),
(1.121)
where z = x\+ ix2, x = (A + 3/i)(A + / i ) _ 1 . One of the model solutions, constructed by Honein and Herrmann (1990), will be used here, and it is related to a perturbation of an elastic field by a single circular inclusion. The perturbation field in the elastic matrix is determined by the complex potentials Mz)
= 0(2) ~ Pf{z),
i M * ) = i>{z) - afa)
(1.122) - /3~f'(z)
+ (a-
27)-*'(0),
(1.123)
where <j>, ip are the complex potentials for the unperturbed configuration, and /(z) = # z ) + - ( 0 ' ( * W ( O ) ) . (1-124) z Here we assume that the centre of the inclusion is located at the origin. The "hat" operation is defined by
kz) = n-).
(i.i25)
z The complex potential (j> is chosen in such a way that <j)'(0) is real, and the constants a, ft and 7 are defined as follows: a
~
_ iix? - fjfix nO + fixO '
P~fi
/i-/i° . + fj,°x'
_ fi(x° - 1) - /i°(x - 1) 7 _
where _ A°+3 M ° * A° + /i° ' 0
2M°+M(^°-1)
'
36
1.4.3
Long and close range interaction
Analysis
for two circular
within elastic
elastic
structures
inclusions
Let S = @ (J H, where ft = fli |J l ^ , © = R 2 \fii U ^2! circular subdomains $7j, occupied by the inclusions with elastic constants A0, /x°, are defined as follows fii = {x e R2 : \x\ < a},
Q2 = {x e R2 : \x - 1| < a},
(1.126)
where a is the radius of a circle, a < 1/2, as shown in Fig. 1.4.
Fig. 1.4
Two circular inclusions at a close distance.
It is noted that the length parameters are normalized in such a way that the distance between the centres of circular inclusions is equal to one. We consider a plane-strain deformation of the domain S by a load applied at infinity in such a way that the corresponding complex potentials are given by 4>aPPi. = E(z - 1/2), i;appl. = A(z - 1/2) - E/2,
(1.127)
where E, A are real constants. We seek the complex potentials 4>(z), tp(z) for the elastic matrix in the following form 00
00
^) = E^-E(i^F+^- 1 /2), 00
,
00
,
-
*(*) = E -f - E 7rf^ - \*w+A^z ~ w - (L128>
37
Close-range contact between elastic inclusions
We would like to have a central symmetry with respect to the point (1/2,0). It is noted that the presence of the third term in (1.128) is due to this symmetry condition. We also note that, if af2, a?? are principal components of the stress tensor at infinity, then h
+^2 );
= -^Pi
-a2)e
where a is the angle of rotation of the principal axes with respect to the system of coordinates Oxy. Here a can take values a = 0, a = n/2. If potentials of the perturbation field are defined as Ppert.
= 4>l+ r, 4>pert. = 1pl + V v ,
where oo C-k
-z2
oo V^
±
zk > ^ -
OO
c
-k
2Z(i_ z) fc'
,
°° oo
1
C-k
,fc+l ! fc=l
k=\
then one can use a representation of the complex potentials in the form (z) = (j>l(z) + 4>r(z)
+
(t>appl.(z),
tp(z)
+
1pappl.{z).
= 1pr{z)
+ i>l{z)
First, consider the left circular inclusion with the centre at the origin. The equations (1.122), (1.123) imply the following relations for the complex potentials in an elastic matrix
M*) = -/?/(*), M*) = -a(<j>r(z)
+ appl.{z))
Vs., ,
-P-lf'(z)
,
„ V
+ (« - 2 7 ) T ( # ( 0 ) + ^ . ( 0 ) ) ,
(1-129)
38
Long
and close range
interaction
within
elastic
structures
where f(z) = ipr(z) + ^appi.(z) + %-(#(*) - # ( 0 ) ) . w e note that the representation (1.129) for the complex potentials guarantees the validity of the interface contact conditions (1.119). Similar to the classical paper by Lord Rayleigh (1892), one can reexpand r and Vv in the neighbourhood of the left cylinder, i.e. the neighbourhood of z = 0. The expansions for <pr and ipr then take the form: oo
+^CkZk,
r(z) = "Co fc=0
oo
+y^jdkz k
tpr(z) = -d0
fc=0
where, for the symmetric case, the coefficients of the series expansions are real and the following relations hold:
c0" =
1
_
( Q / 3 )
-I{-
Q
"
do = 1_^af3)-i^~1(co
1
(2("
4
+
JE)
"
2C2
° 2 ~ d o ) + c ° _ \E}>
- 2E) ~ VA + E) + 2 C 2 ° 2 + do}' oo
co = ~y^c_ fc , fc=l
oo
1
^0 = ^ { - f c c _ f c 2
-
d_fc}.
*-!
Also, c
j
—
/
v
fe=i
fc=l
^
J
/
k + j-1
,c-j
fc=l
J
\
J
/
(1 130)
'
Close-range contact between elastic inclusions
39
Now, we use the contact conditions at the interface between two materials. Collecting coefficients near like powers of z, we obtain from the first relation (1.129) that --(3 + £)-2a2£cTd fc=i ^
2
+£(-fcc^-Q=0, ' fc=i
(1.132)
and c_fe + f3 < aik V
3= 1
'
j=l
X
J
-(H^'l^l^j^U^l. Next, in the second equation in (1.129) the function f'(z)
(1.133)
satisfies
and, therefore V,(z) = -a(Mz)
+(a-2i)
+ ECJ - \)) - ^4>'i(z)+ a2 —
(ci+E).
Again, substituting the expansions (1.128), (1.131)-(1.133) into the interface contact conditions we obtain - d _ i - aa 2 (cl + E) + a2{a - 2 7 )(ci +E)=0, d2 + i c _ 1 + Q c ^ a 4 = 0, <*_,, + | ( j - l)c_,-+i - a2(j - 2)c_ j + 2 = ~acja2j,j
> 3.
(1.134) (1.135) (1.136)
Further, if c\ and i? are real, oo
(2 7 a 2 )- 1 rf_ 1 + J B - ^ f c c _ f c = 0 ,
(1.137)
40
Long and close range interaction
within elastic
d_2 + - c _ i - a a 4 2 _ , f c c _ W fc=i
J ^
structures
)=0,
(1.138)
'
d
-j + 2^> ~ 1 ) C l -^' ~ a2ti ~ T>C2~i
-aa2^c^(k+jr1)=0,j>3.
(1.139)
We have derived a system of linear equations with respect to the multipole coefficients in the expansions (1.128). The series can be truncated, and a finite system of equations (1.132), (1.133), (1.137)-(1.139) can be solved numerically for different values of the constants A and E, characterizing the loading conditions at infinity, and for different elastic moduli of inclusions and the elastic matrix. With the change of variables D-j = d-j/a2:>, C-j = C-j/api one can represent the truncated system (1.132), (1.133), (1.137)-(1.139) in the form (I + B)X = Y , where I is the identity matrix, X is the vector of unknown multipole coefficients, the right-hand side Y is specified by the loading conditions at infinity, and components Bij decay exponentially as we increase i or j .
1.4.4
Square
array of circular
inclusions
In this section we derive the series expansion, based on the results of the paper by Honein and Herrmann (1990) for the complex potentials associated with a square array of circular inclusions. We assume that the centres of circular inclusions are located at the points z — Zjk = j + ik, j , k £ Z, in the complex plane. The following series expansions are valid for complex potentials in a neighbourhood of the central inclusion: 4>(z) = Q(z) + 5 3 4>jk{z) + Ez,
(1.140)
iP(z) = Mz) + J2^kW
(L141)
j,k
+ Az
'
Close-range contact between elastic inclusions
41
where OO
OO
z*i-x
*-^ 1=1
1=1
,.,
z )
d
-Y-
~(2'-i)
I
Z*
v ^ ( 2 * - l)c-(2t-p
,
,
T h e constants i?, .4 define the load applied at infinity; the t e r m s jk, rpjk characterize t h e contribution from the inclusion with the centre at z = ZjkWe shall apply the relations (1.122), (1.123) to the inclusion at the origin. It follows t h a t <j>0{z) = -Pf(z),
(1.145) 4
a ~ lpo(z)
= -a(rest(z)
+ appl.(z)) - j3—^f'(z)
+
+(a - 2 7 A # e . t ( 0 ) + <j>'appL(0), where
f(z) = West + Ipappl + — ( # „ « ( * ) "
KesM)-
Here rest(z) =^2jk j,k
= ^2 ^ m=l
lc
2m-l,
lprest(z)
= ^2 ^ m=l
ld
2m-l-
T h e coefficients cim-i
= ^tpjk j,k
and cfom-i are defined by
^ Clm-l — — 2_^C-(2l-l) 1=1
/2i + 2 m - 3 \ I „ _1 J
c
i)
2m+2l-2,
(1.146)
42
Long and close range interaction
d2m-i = ~Y1 d-(2i-i) [ 1=1
within elastic
2m-1
structures
i 5 ' 2 m + 2 '- 2
^
+$>-i) C _ ( a l _ 1 ) (
2m _ 1
J4 12l+2m!
The coefficients 52m, 5 2 m a r e lattice sums; the first set of sums was introduced by Lord Rayleigh in 1892 for the conductivity problem and the second set occurs in the plane elasticity formulation. The following definitions are used here: $2m - 2 ^ ~a^ j,k
^
Z
J*
= £^,m>l.
(1-147)
Z h
j,k
i
Following Lord Rayleigh (1892) we calculate the lattice sums S^m, ^ m over an infinite strip along the Ox\ axis, and then we take the limit to infinity with respect to the thickness of the strip. The following system of linear algebraic equations holds for the multipole coefficients c_( 2 m _i), d_(2m-i)> m — 1,2,..., of the expansions (1.142) OO
OO
c_i -pa2J2d-(2i-i)(U
- 1 ) S H +j9a 2 53(2Z - l)c_ ( 2 I _ 1 ) 2i5^ 1 | 2
i=i
i=i
OO
,
2l l
= -f3a2A,
^ )s2l+2
1=1
di - 2 7 a 2 £
(1.148)
^
c_ ( 2 m _ 1 ) (2m - l ) S 2 m •= -2ja2E,
(1.149)
771—1
c_ ( 2 m _ 1 ) - /3a
^
d
-(2i-i)
I
„
,
2m - 1
IS; 2m+2
r
'-2
Close-range contact between elastic
43
inclusions
2 ^->n*-*-&-*{* Z^; )*i +
7 (1) •'2l+2m
-(3a4m(2m
+ 1) | ] c . ^ y
d-(2m-i)
(
%
^
~
l
\ S2m+2l = 0, m > 2, (1.150)
~ a2(2m - 3)c_ ( 2 m _ 3 )
-aa 2 ' 2 ™" 1 ) f > _ ( 2 i _ 1 } ( 2 Z + J ™ -
3
) S 2 m + 2 i _ 2 = 0, m > 2.
(1.151)
For coefficients of the truncated series (1.142)—(1.144) we obtain a finite system of linear algebraic equations. We note that the multipole coefficients decay rapidly as we increase the order of the multipole. 1.4.5
Integral approximation for the multipole Inclusions close to touching
coefficients.
1.4.5.1 Scalar problem Here we describe the idea proposed by McPhedran, Poladian and Milton (1988) for a scalar conductivity problem in a two-dimensional domain containing circular highly conducting inclusions, which are close to touching. First, we consider a pair of circular inclusions of radius a such that 0 < 1/2 — a
c
1 ( l - 2 a ) ( l + 2o)'
1,. 1 °° = 25 ( 1 - 7c) -
The inclusions come close to touching as c —> oo; the points 1 — aoo are shown to be the limits for the "image" positions inside inclusions. For a pair of such inclusions (see Fig. 1.4) the complex potential of an electric field in the matrix is represented by a series oo
V(z) = Y,(Amzm i=l
+ Bm/zm).
(1.152)
44
Long and close range interaction
within elastic
structures
The multipole coefficients are linked via transmission boundary conditions (continuity of potential and normal current flow) at the interface boundary \z\ = a. The idea of McPhedran, Poladian and Milton (1988) was to represent the above coefficients via continuous multipole densities p+, and p _ , that is, p+(x)x~m~1dx,
Am= Jl—aoa
/"Coo
Bm=
p^(x)xm-1dx.
(1.153) Jo In this case, instead of a discrete system of multipoles, distributed on the line connecting the centres of cylinders, with the limit points a^, 1 — a^, a continuous distribution is used. A functional equation holds for the density p_ p-(a2/(l - x)) = (a - l)/{a + l)p_(x), xe (0, a o o ), where a is the conductivity of the inclusion. The above functional equation has an explicit solution P-(x) = const [(doo — x)/(l — ax — x)]s ,
(1.154)
and the exponent s is denned by s = log((a - l ) / ( a + 1))/ l o g ^ / t l - a ^ ) ) . When 0 < s < 1, the above integral representations provide good approximations for high-order multipole coefficients. For the complex potential associated with the case of a square array of circular inclusions, the same representation (1.152) can be used with A2k = B2k=0, k = l,2,.... 1.4.5.2
Vector problem
Next, we obtain the integral approximation for the multipole coefficients in the series approximation of the complex potentials for the plane-strain problem for two circular inclusions.
Close-range contact between elastic
inclusions
The following integral representations hold Muskhelishvili complex potentials when c ^> 1:
4{() =f°"iMdx
m
S— ^
7o
for
+j '
. r- aw - Miw^ + *
45
the
j^dx,
ftw-Kw^. € —x
Jl-ax
Kolosov-
(1.155)
(1.156)
The unknown densities are assumed to possess the following symmetry: Ai(x) = Ar(l - x); Bi{x)=BT(l-x),
(1.157)
and J4J, B\ are assumed to vanish outside the interval (0,aoo). Using the first equation in (1.129) we obtain
r ^ w 1 ^dx
Jo
+
$~x
Ji-aoo
$~x
"/L T^ {B'& + A'&{x -1/2)}'
(L158)
Indeed, if we can find functions which satisfy the relations Mx)
= (3 ( V ( - ) + A'r(-)(x \ X X
(
2
2
- 1/2)) , x G (O.aoo), J \
£ , ( - ) + A[(-)(x - 1/2)) , x G (1 - aoo, 1), (1.159) a; a; / then these densities yield (1.158). Taking into account symmetry (see (1.157)), we re-write equations (1.159) in the form a2 1 a2 Mx) = f3Bt(l ) + f3(- - x)A[(l ), x G (0, aoo), X
Ar(x)
£i
X
= (3Br(l - - ) + /3(x - \)A'T{1 - - ) , x 1 x x e (1 - aoo,l).
(1.160)
46
Long and close range interaction
within elastic
structures
Next, let us consider the second of the field identities (1.129) applied to the left circular inclusion. It can be represented as follows:
Bl{x)-\4l{x)J___ Jo a4 fac"
r~M^idx
Z-x dx
{
Jo
„,,a2.
Z-x
x2 ...a2.
_((«Q -_ 2 27 7 ))^j/f
.,i,a2..
1
^-dx.
When inclusions are close to touching, we neglect the last term and try to find approximations for the density functions, so that
Bt(x) - Ufa) =
aAr(^)+^B'A
~ h)A"€]-
-f**4r&+^h
(L161)
The symmetry relations (1.157) yield
Bt{x) - \^{x) = o^(l - £ ) - ^B\{1 + A;(I
4
- ±)
- £ > + ^ - i K ( i - £>•
(Li62)
With the use of (1.160), (1.162) we obtain the following system of functional equations
Mx) = p | B , ( I - ^ ) - 4 ( 1 - ^ ) ( * - 1 / 2 ) } , l
Bt(x) = aAt(l - - ) - ( - X
X
-)A[{x).
(1.163)
I
It turns out that a solution of the system can be written down explicitly; the density Ai has the form i 4 j ( a o o - y ) = c o n s t ( — ^ — )s(l + ciy), y e ( 0 , a M ) . l + cy
(1.164)
Discrete lattice
Here c = (1 — 2aoo) equation /(s)
= (^_) -L
S / 3
1
47
approximations
, and the exponent s satisfies the transcendental
- i _ (1^2° )'a + ^2L_ _ 1 Z ^
^*oo
"oo
^
^*oo
S
=
o.
(1.165)
^*oo
The coefficient c\ is defined by _ 16c25 (C2-l)2/(S
Cl_
l)'
+
and it provides an asymptotically small contribution as the circular inclusions are close to touching (c —> oo). We note that the representation of the density Ai is similar to the formula (13) in McPhedran, Poladian and Milton (1988) but the exponent s is determined from the more complicated equation (1.165). Note that, as the inclusions come close to touching, the following approximate formula can be used: c(a-l//3) 4(a + l)
+
{
° >•
We remark that the advantage of the multipole method, originally due to Rayleigh, is that it converges exponentially fast and produces good approximation even for a small gap between inclusions.
1.5
Discrete lattice approximations
This section is based on the paper by Movchan, Zalipaev and Movchan (2002). We consider waves propagating within a doubly periodic structure. For certain types of structures there exist intervals of frequencies (so-called photonic/phononic band gaps) that correspond to waves that cannot propagate through. It was noted that the filtering effects associated with wide photonic/phononic band gaps occur when voids within a doubly periodic array are sufficiently close to each other. In this section we show that one can construct a lattice structure whose spectral properties (within certain intervals of frequencies) are similar to those of the continuum structure associated with a doubly periodic array of voids close to touching.
48
1.5.1
Long and close range interaction within elastic
Illustrative
one-dimensional
structures
example
Let us consider propagation of out-of-plane shear elastic waves through a periodic array of homogeneous isotropic layers (see Figure 1.5A). The outof-plane displacements Uj, j = 1,2, inside the layers satisfy the equations mu'i + u>2puj = 0, x e S ] n ) , j = 1, 2,
(1.166)
where S\n' = (—b + nd, nd), S1^™ = (nd, a + nd), n is integer, d — a + b is the period, Uj, j = 1,2, are the shear moduli, and p is the material density, the same for both layers. We assume the ideal contact conditions on the interface between the two layers: u\ = u 2 ,
Mi^'i = A*2U2-
(1.167)
The solution must also satisfy the Floquet's quasi-periodicity condition Uj(x
+ d)= eikodUj{x),
j = 1, 2, |fc0| < -K/d,
(1.168)
where fco is the Bloch parameter (or quasi-momentum). The case of /ii//X2 <S 1 is considered here, and we show that the dispersion curves representing w as a function of fco can be approximated by the corresponding solution for a bi-atomic discrete system. It is evident that the general solution of equation (1.166) has the form Uj
= AjeiklX
+ Bje~ikiX,
x G SJ n) , j = 1,2,
(1.169)
where A,-, Bj, j = 1,2, are constant coefficients, and kj = OJ^J'pj'UJ. Taking into account the boundary conditions (1.167) at the points x = 0 and x = a as well as the Floquet's condition (1.168) referred to x = a, we obtain a homogeneous system of linear algebraic equations with respect to Aj, Bj, j = 1,2. This system has a non-trivial solution, provided the following dispersion equation describing the propagating modes holds 2-\/e(cos(fc16)cos(fc1av/e)-cos(fcocO)
=
(e+i)siD.(kib)sm(kiay/e),
(1.170)
here e = /X1//X2 ^ 1 a n d &2 = k\^/l. Let k\b
(1.171)
Discrete lattice
(A)
approximations
_sL ^f
(B)
Vl
v
s-l us
vs
us+i vs+i
Fig. 1.5 One-dimensional arrays of two layers (A) and two particles with springs (B).
where
Pi =
a2b2e - 4 -
+
64 + a 4 e 2 a3be + ab3,, - ^ 2 - + — — ( 1 + e
p 2 = (&2 + ab(l + e) +
)
.1 r /
2/ 2 f i 1
ea2)p/m.
(1.172)
(1.173)
On the other hand, one can write the equations of motion for a onedimensional periodic array of particles of two different types connected by springs (see Kittel (1996), chapter 4): mius — c(vs + v3-i - 2us), m2va = c(us + u s _i - 2vs),
(1-174)
where c is the stiffness of springs (see Figure 1.5B). Let d be a period of the array. Assume that the functions us and vs allow for the following representations us — uexp(iskod)exp(-itJt),
vs = vexp(isk0d)exp(-iu>t),
(1.175)
with unknown amplitudes u,v. On substitution of (1.175) into (1.174) we have
{
(w 2 m! - 2c)u + c(l + e-ikod)v = 0, ikod )u + (w 2 m 2 - 2c)v = 0. c ( l 4. e
^ ^'
'
50
Long and close range interaction
within elastic
structures
Fig. 1.6 Acoustic and optical bands for a one-dimensional array of particles connected by linear springs a = 1, m i = 1, m,2 — 2 and c = 10.
This system has a non-trivial solution if and only if the following equation holds mim 2 w 4 - 2c(mi + m 2 )w 2 + 2c 2 (l - cos(k0d)) = 0.
(1.177)
This discrete model is defined by the two parameters mi/c and n ^ / c , and the equations (1.171) and (1.177) are identical when Pi = mim2/c2,
p2 = (mi +m2)/c
(1.178)
Of course, in this case the dispersion diagrams also agree. Figure 1.6 shows the graphs for u versus fc0 when a = 1, mi = 1, m® = 2 and c = 10. The diagram clearly displays a band gap between the acoustic and optical modes.
1.5.2
Two-dimensional
array of
obstacles
Discrete model. Consider a discrete model for a system consisting of two types of particles connected by springs. Let us write the equations of motion for a square array of particles with masses mj and m 2 connected
Discrete lattice
approximations
Vm-U+1
Um,n+1
V m , n +i
Um+l,n+l
Um-l/2.n-l/2
V,,,.!^,,-!^
U m+ l/2,n-l/2
Vm+l/2.n-l/2
51 .
Vm+l,n+l
Um+3/2,n-l/2
Fig. 1.7 Two-dimensional array of particles connected by springs, by springs with stiffnesses c\ and c2 (see Figure 1.7) mi«m,n = Cl(u m _l,„ + U m _ l i n _ l - 2 u m , n ) +C2(Vm,n + ^ m _ I i n + i ~ 2 u m , n ) , "^2Vm,n = C i ( u m + i > n + U m + l > n _ i -
2vm,n)
+C2(um,n + M m + I , n + 1 - 2l>m,n)> m
l " m + l , n + ^ = Ci(vm>n+1
m
2 # T O + I > n - 4 = c l ( u m + l , n + l + Um+xn+i
+Vm+i]n+i
-2um+l>n+l)
(1.179)
+C2(Um,n + f m _ l , „ + l - 2 u m + I „ + i ) , +C2(um+l,n + Um+3
n+
-
i - 2um+l
2vm+in+i) n+
i )
Let d be a period of the array. The propagating modes are represented in the form (1.180)
T T ( ° ) exTp[i(k m + k n)d] exp(-iwi), U m ,„ -= U^o 0x 0y
where U m , n = {umtn,vm,n,um+in+±,vrn+±in+±)T. Substituting (1.180) into (1.179) we obtain a homogeneous system of linear algebraic equations with respect to UQ Q, U\ \ , VQ Q, v\ \ . The system '
2 ' 2
'
2 ' 2
has a non-trivial solution if the corresponding determinant vanishes. This yields the dispersion equation q0u)s + qiOJ6 + q2u4 + qzu2 + q± = 0,
(1.181)
Long and close range interaction
52
within elastic
structures
where qo = m\m\,
qi = - 4 {mim 2 (ci + c2)(mi + m2)} ,
12 = 4{-mim 2 c 1 c 2 (cosfco x d + cosfc0j/d) + 8cic 2 (mi + m 2 ) 2 +4(c 2 + cl)(m\ + m 2 + 3m1m2)}
,
93 = 8[cic 2 (cosk 0x d + cosk0yd)(ci + c 2 ) ( m i + m 2 ) - (mi + m2)(c\ + c2) -5cic 2 (ci + c 2 )(mi + m 2 )], 94 = cic 2 [-(cosfc 0x d + cosA;oyrf)(30cic2 + 16(c2 4-c2,)) —8cic2cosfcox^cosfcoyd + 4(8c2 + 8c2 + 17cic 2 )]. This equation can be solved analytically, and the four solutions corresponding to the propagating modes are given by w2 = (m1m2y1({m1
+ m 2 )(ci + c2) ± \S±\),
(1.182)
where S±
=
( " i 2 + m 2 ) ( c i + c 2 ) 2 + 2mim 2 cic 2 (cosfcoa;^+cosfco y d—2)±2mim 2 |5'o|,
"^o = ( c i + c2) + 2(cf c2 + cic 2 ) (cos koxd+cos koyd) + ic^c^ cos koxd cos kovd. The dispersion diagram constructed in accordance with the above formulae exhibits band gaps for the cases of bi-atomic structures of particles of different mass. It can also be shown that a correspondence can be established between this simple lattice structure and a continuum model describing fields in highly porous media. Square array of circular voids. We consider propagation of outof-plane time-harmonic waves within an isotropic elastic medium, which contains an array of infinitely long circular cylindrical cavities of radius r c directed along the z-axis (see Figure 1.8). For the z component of the displacement we have (A + k2b)u = 0,
—
0,
(1.183)
Discrete lattice
approximations
53
0 G © 0 O 0
© O O Fig. 1.8
where kf, =
UJ/VI,
A square array of circular holes.
and v\ = —, the material density is p, and shear modulus
is fi.
It is assumed that u satisfies the following quasi-periodicity condition u(r + R p ) = w(r)
ik e
°' R p,
(1.184)
where R p = p\a.± + P2&2- Here ax,a2 are the fundamental translation vectors of the square lattice (|ai| = |a.21), ko is the Bloch wave-vector, and the multi-index p = (pi,P2) has integer components. According to Poulton et al. (2000), inside every cell of the lattice the solution for the z component of the displacement can be expanded in terms of multipoles «(r) = J2 [AtJiihr)
+ BiYi(kbr)]
eiW.
(1.185)
Using the boundary conditions (1.183), we deduce Ai = -MiBu
Mi =
Y{{hrc) J'l{forc)
Application of the generalised Rayleigh method, after a certain normalization of the coefficients Bi, leads to the following system of algebraic equations (I + Ji(w,ko))Z = 0,
(1.186)
54
Long and close range interaction
within elastic
structures
where I is the identity matrix, and the Rayleigh matrix is given by Rlm = ( - l ) ^ s g n ( M O % i % ^ ,
(1.187)
the components of the vector ZT = (..., z_\, ZQ, ZI, ••) are zm = y/\Mm\Bm. In the expressions for the coefficients of the systems (1.186) only the lattice sums Sf(k, ko) depend on the geometry of the lattice structure. They are given by 5, y (fc,k 0 )=
Yl
Yl(kRp)ei^l+ik°-R",
$ p = arg(R p ).
(1.188)
In the system (1.186) the off-diagonal elements decay exponentially fast away from the main diagonal. Thus, to obtain the photonic band diagrams for plane waves propagating through an array of circular holes, we truncate the above system and evaluate the determinant of the truncated matrix J + i?(a>,ko). Zeros of the determinant of the truncated matrix give the values of u for a given ko: det(/ + .R(w,ko)) = 0.
(1.189)
This yields the dispersion equation (relation between u> and ko) which allows us to obtain the dispersion diagrams and to analyse the band gaps which may exist for a given periodic structure. We evaluate the band gap states corresponding to the range of frequencies for which no elastic waves can propagate through the composite. The corresponding numerical results have been obtained for an out-of-plane shear problem, the dispersion diagram for the square array is shown in Figure 1.10. The elementary cell of the reciprocal lattice is shown in Figure 1.9. The horizontal axis in the dispersion diagram corresponds to the arc length evaluated along the trajectory TMK. The diagram shows the presence of the band gap which occurs when the circular holes are close to touching (rc = 0.49). Comparison of discrete and continuum models. For the sake of simplicity, assume that for the discrete system described above mi = 0. In this case the dispersion equation is bi-quadratic g0w4 + q[w2 + q'2 = 0,
(1.190)
Discrete lattice
b2
approximations
55
i
K
f , \ M
r
Fig. 1.9
Reciprocal square lattice and the Bloch contour TMK.
'
(0
'
'"--...
6 5
—1--.1 mode — ' | 2 mode — i 3 mode"---.-z •
"-
-
-
>,
* 4
-
•
o
3 U n 0) d •t
2
. :
: ^~~"\^
1 4
^ \ r
K
2
/ ^
,
M
0 2 Bloch vector k 0
K
4
6
Fig. 1.10 Dispersion diagram for out-of-plane shear waves in the square array ( | a i | = |a2| = 1) of circular holes of radius rc = 0.49; the normalized shear modulus and density are fj, = 1 and p = 1 respectively. where q'0 = 4 ( c i + c 2 ) 2 ,
q[ = 8 { c i c 2 ( c i + c2)(cos(k0xd)
- 4) - c\ -
c\}
q'2 = 16cic 2 (ci -f- c\){l - cosk0xd) + 0^2(38 - 40cosfc0x
2(ci + ca) m2
(1.191)
56
Long and close range interaction
0
within elastic
structures
2
bloch vector k 0
Fig. 1.11 Acoustic and optical branches of the dispersion relation for two-dimensional array of particles with springs ( | a i | = |aa| = l , m i = 0,7712 = 1 and ci = 14.4, C2 = 0.45).
At the boundary of the Brillouin zone, when kox 8cic2 wr = (ci + c )m ' 2 2
7T, k,0y
7T,
(1.192)
Without loss of generality we can assume that m2 — 1. Then, for the given frequencies wi,a>2 the stiffnesses are determined by (see (1.192) and (1.191)) Cl,2 =
"4{
i±
O2 W2
(1.193)
The corresponding dispersion diagram for the discrete system is shown in Figure 1.11. To adjust the parameters of the discrete system, we set oj\ and W2 to be equal to the values at fcox = TT, koy = n taken from Figure 1.10 (corresponding to the continuum model). In this case, the band gap width is the same for both models. Numerical computations also show a very good agreement of the results for the acoustic modes. At the points A,B,C,D,E,F,G the group velocity is equal to zero, which is consistent with the fact that the particles displacement is described by standing waves.
Chapter 2
Dipole tensors in spectral problems of elasticity
In this chapter* we consider two spectral problems associated with asymptotic fields in the vicinity of a thin conical inclusion: the first problem corresponds to the case of a perfectly bonded inclusion, and the second one to the case when the inclusion is connected to the surrounding medium via a thin and soft "interface" layer. The main steps of the asymptotic algorithm employed in this chapter are described in Section 2.1 for a perfectly bonded inclusion; the generalisation to the case of a "coated" inclusion is discussed in Section 2.2. We remark that the asymptotic expansion of the solution is constructed using the compound asymptotic expansion technique* and the method developed for elliptic boundary value problems in domains with conical points by Kondrat'ev (1967), Maz'ya & Nazarov (1989) and Maz'ya & Plamenevskii (1977).
2.1
2.1.1
Asymptotic behaviour of fields near the vertex of a thin conical inclusion Spectral problem
on a unit
sphere
In this section we consider a model problem for an infinite elastic medium with a thin conical inclusion k£ = { x £ R 3 : xz > 0, e~1x^lx' £ g, x' = (xi, X2)}, where g is a plane domain bounded by a simple smooth contour * We acknowledge the invaluable help of Dr David Esparza in preparation of the text and figures of this chapter. t For a comprehensive study of this technique we refer to Maz'ya, Nazarov & Plamenevskii (2001). 57
58
Dipole tensors in spectral problems of elasticity
dg, and e (0 < e
:= p0Au°(e;x)
L(—)u(e;x):=pAu(e;x)
+ (A0 + /i 0 )VV • u°(s;x)
= 0, x e k£;
+ (A + //)VV • u(e; x) = 0, x G Ks. (2.1)
We assume that the inclusion and the surrounding medium are perfectly bonded, that is, u°(e;x)
=u(e;x),
cro{-n) {u°; e, x) = (T (n) (u;e, as), x e dke,
(2.2)
where (T
3
^ - ^ E ^ + ^ + ^M,^!^^;
(2.3)
n is an outward unit normal vector on dKs, and Sij is the Kronecker delta. We assume that a uniform stress field is applied at infinity. Near the vertex of the inclusion the displacement fields u° and u admit the representations u°(e; x) = pA^v°{e;
6, if), x G ke; u(e; x) = pA^v(e;
0,
where (p, 6,
Asymptotic behaviour of fields near the vertex of a thin conical inclusion
59
The exponent A and the vectors v°, v solve the following spectral problem on the unit sphere S: V°{A{e))v°
= 0 o n g£, V(A(e))v
v° = v, Q°(A(e))v°
= 0 on S/g€,
= Q(A(e))v on dgE.
(2.5) (2.6)
The system above is obtained by transformation of the problem (2.1), (2.2) into the spherical coordinates. In (2.5), (2.6) we denote the matrix differential operators by
and the traction vectors on the surface dke by
and
To simplify the notations we omit the arguments 9, (p as well as d/d
= OonS.
(2.7)
The eigenvalues Ao of this spectral problem are integers. The corresponding eigenvectors are traces on S of homogeneous vector polynomials "V(m>fi (a;) of degree m (m = 0 , 1 , 2 , . . . ; j = 1,2,..., 3(2m + 1)) satisfying the homogeneous Lame system, or traces of the vector fields V ( l j ' ) ( 9 / 9 x ) T ( x ) , where T is the three-dimensional Somigliana tensor. Since we are looking for solutions with a finite elastic energy, we analyse those cases for which Re(A(e)) > —1/2. Because our objective is to study the singularity in the stress field, we shall consider the perturbation of only two eigenvalues Ao = 0 and Ao = 1. The eigenvectors corresponding to Ao = 0 are the rigid-body displacements; they satisfy the problem (2.5), (2.6), and therefore, the eigenvalue
Dipole tensors in spectral problems of elasticity
60
Ao = 0 is not perturbed. The vector polynomials V ^ ^ x ) corresponding to the eigenvalue Ao = 1 have the form VM\x)
= XjeU\
j = 1,2,3; V * 1 - 4 ^ ) = - ^ ( i 2 e ( 1 ) + z i e ( 2 ) ) , v2
VW(x) = -^(x3e^+x2e^),V^(x)
V(UT)(x) = -L(a; 2 C (i) V2
= -^(x3e^
fi)
XleW),VV
(*)
VM\x) = -^(xieW v2
+ Xle^),
= -^fae™ v2 -
(2.8)
~ ^e(3)),
X3eW).
(2.9)
The rotations V( 1 , 7 )(x), V( li8 >(x), V"(1,9)(x) are solutions to the problem (2.5), (2.6), with A(e) = 1. The fields (2.8) satisfy both equations (2.5) and the continuity condition (2.6) for the displacements, however they leave a discrepancy in the traction condition. The stresses (here we use the notation 0°® := a J ^ V ^ j x ) , 0 $ := CT^V^X)) produced by the fields V( 1, ')(aj) have the form: a°V
= 2/x0 + Ao, a°}j) = Xo, of
= 2fi + A, «7&'> = A, i ± j , i, j = 1, 2 , 3 ;
4 4 ) = 4 5 ) = 4 6 ) = ^ o , 0® = ^ = ^ = X/2M;
(2.10)
the remaining stresses are equal to zero. For Ao = 1, the leading order approximations $ ° and <& to the vector fields v° and v on the unit sphere S are then expressed as follows: 6
*°(9,V)
= £c,-*<1J>(*,¥>) on &;
6
#(0 )¥ >) = ^ C i $ W ) ( ^ ) o n S / s £ ,
(2.11)
where <&( lj '(0, y>) are the traces of the vectors V ^ ' ^ x ) on the unit sphere S. The coefficients Cj are unknown and will be found later. In order to
Asymptotic
behaviour of fields near the vertex of a thin conical inclusion
61
compensate for the discrepancy left by the fields (2.11) in the traction condition (2.6), we need to construct a boundary layer solution in the vicinity of ge. 2.1.2
Boundary
layer
solution
In the vicinity of the point A/"(0,0,1) on the unit sphere § we introduce new scaled variables: £ = s~lr], where 77 = (771,^2) = 2:3" 1a3/-
(2-12)
We represent the unit normal vector n on dk£ in the form n = (1 + s2(i • v)*)-V2{VleM
+ v2eW _ £ (£ . v)e<%
(2.13)
where v=(yi, v-i) denotes the inward unit normal vector on dg.
Fig. 2.1 The vicinity of the point Af (0,0,1) after the coordinate transformation »:—>£.
In the stretched coordinates £ the domain g£ C S is transformed into a finite domain j c l 2 (see Fig. 2.1), and the Lame operator L and the traction operator B= an have the form
i(£)(p {1+ ° (£2)} *(^)) lllxiHi = £ - 2 £ o ( | ) * ( 0 (2.14) +
£ -l£ l ( £, £)*(£) + 0(1),
Dipole tensors in spectral problems of elasticity
62
S(£,«)(P {1+0(£2)} *(£))|||*||=1 =
e-lB0{&,v)*{t) (2.15)
+ 81& £,!/)*(*) +0(e), where ^ ( C i , C 2 ) = (A + 2/x)d2 + /xCl, 4 2 (Ci,C2) = Cftti,C2)
= (A + M)CIC 2 ,
4 ? K I , C a ) = (A + 2/x)C22 + Ki2, rg3(Ci,C2) = MC? + C22), ^ 3 ( ^ ; C I , C 2 ) = £ 1 1 (C;CI,C2) = - ( A + M)(6CI 2 + 6CIC2),
£ f (€;Ci,C2) = r?2(€;Ci,C2) = -(A + /x)(6CiC2 + 6<22);
(2-16)
^o1(^;Ci,C2) = (A+2/X)I/ 1 CI+M^C 2 , B22(^;Ci,C2) = ^ i C i + (A+2/i)^c 2 , # o V ; f t , C a ) = A^iCa + A ^ f t , ^ o V ; f t , £2) = /W1C2 + A1/2C1, So^;C 1 ,C2)=M(^lCl+^2C2), B f («, " ; ft, C2) = - A ^ f t f t + 6C2 - 1 ) - M(€ • v)<j, Blj(£, u; ft, ft) = - / ^ ( f t f t + 6C2 - 1) - A(£ • i / ) 0 , i = 1,2.
(2.17)
The operators Co, Bo have a block diagonal structure which corresponds to a plane strain and an out-of-plane shear elasticity problems: Co
Co 0 0 /uAf
, Bo
Bo 0 0
(2.18)
H&
where Co and Bo are the 2 x 2 matrix differential operators for a plane strain problem, and
is the Laplace operator in the ^-coordinates. We remark that similar formulae hold for the operators L° and B° associated with the inclusion.
Asymptotic
2.1.2.1
behaviour of fields near the vertex of a thin conical inclusion
63
The leading term
We denote the leading term of the boundary layer within the inclusion by £U>°W(£) and within the surrounding medium by ew^(^). Using (2.14) and (2.15), we derive the following problem for the fields
^ ( ^ ) ™ o ( 1 ) ( £ ) = cUG
(2.19)
S8(^>"K (1) «) - Bo(^)™ ( 1 ) (0 = X > * ( % , „ ) , u> o ( 1 >(0=«> ( 1 ) (0, £ e 3 f l ;
(2.20)
where * ( 1 ) ( £ , v) = {X + 2li-X0* ( 2 ) ( £ , ") = (A - A >
i e
2 M o )^ie ( 1 ) + (A -
X0)u2e^,
« + (A + 2/x - A0 - 2/x 0 )i/ 2 e( 2 \
* ( 3 ) ( ^ ^ ) = (A-A 0 )( I / 1 e( 1 )+ J / 2 e( 2 )),*( 4 )(^,^ = v / 2(/x-Mo)(^e ( 1 ) +^ie( 2 )), *< 5 >(£,i/) = x/2(/i-/XoW* ( 3 ) , * ( 6 ) ( ^ , ^ = x / 2 ( M - / x > 1 e ( 3 ) .
(2.21)
Since the average values of the fields (2.21) on dg are equal to zero, the solution w^1' exists and vanishes at infinity (see, for example, the paper by Kondrat'ev (1967)):
«,«(£) =TW(C) +0(||£||- 2 ) (2.22)
where r ( £ ) = ("Tij (€))? 3 =i i s the tensor with components 7
« « ) = [47r/x(A + 2/x)]- 1 (-%(A + 3 M )ln||C|| + (A + / x ) ^ ^ i r 2 ) , 1zAt) = 7 i 3 ( 0 = 0, t, j , = 1,2; 733(0 = -(27r M )- 1 ln||€il,
and W ^ {d/d£) are the differential operators formed by the vectors W ( 0 ( 0 = 6 e W , i = 1,2; W( 3 >(£) = ^ ( 6 c ( 1 ) + &e<2>),
(2-23)
Dipole tensors in spectral problems of elasticity
64
W^(t)
= £ ie ( 3 >, W(5\Z)
= 6e<3>.
(2.24)
The coefficients af can be written in terms of the elements m ^ of the dipole matrix m = (m^)ij=i obtained by solving a set of model problems (2.19), (2.20) with the following vectors on the right-hand side of (2.20) (A + 2fi - A0 - 2/i 0 )iv 1 e« + (A - A0)zA,e(2\ (A - \o)view V2(V - fi0)(v2e{1)
+ (X + 2fi-X0-
+ vie^),
(JJL - ix0)vxe^,
2fi0)v2e(-2\ {ji - /i0)i^e<3>.
(2.25)
The elements of the dipole matrix m depend on the elastic properties of the inclusion and the surrounding medium, and on the geometry of the domain g. Comparing the right-hand sides (2.21) and (2.25) we deduce the coefficients of': a j)
i
=mji,j
= 1,2, a\3) = (A-A 0 )[2(A + / i - A 0 - ^ 0 ) ] - 1 ( m i i + m 2 i ) ,
c*i = ma, a\> = 0, I = 5,6, i = 1,2,3; af' 4 5 ) = \/2m 5 5 , a[j) = a^
— V2JTI44,
= 0, j = 1, 2,3,4; ajj* +1) = V2m45,
k = 4, 5.
To construct the next terms in the asymptotic approximation to the solution of the problem (2.5), (2.6), we take A(e) and v(e; 9,
ls a
&(6,iP)+eX(9)w(1\e-lr1)+e$il)(0,tp).
n infinitely smooth cut-off function such that
X(9) = 1 for 9 € [0,7r/6], and x (0) = 0 for 0 € [ir/3,ir].
(2.27)
This function is introduced because the boundary layer solution is defined on the upper hemisphere only. Since the vector field w^ decays at infinity as 0 ( | | £ | | _ 1 ) , it gives no contribution into the equation for 3>' '(0,
Asymptotic
behaviour of fields near the vertex of a thin conical inclusion
65
by applying the operator V to the field v(e;6,) = -AiV'(l)*(9,
on S,
(2.28)
where V1 denotes the derivative of V with respect to A. The solvability condition for (2.28) has the form [{-AlV'(l)*(9,lp))-Y<>1>»(e,lp)ds
= 0, j = l , 2 , . . . 6 ,
(2.29)
where Y^tf,
= -A1cj,
j = 1,2,...6.
The latter implies that, for the problem (2.28) to be solvable, the quantity Ai should be equal to zero. Thus, from (2.28) it follows that &1\0,
(2.30)
In the next section we shall construct the second order terms in the asymptotic approximations for A(e) and v(e;0,ip) : A(e) ~ 1 + £ 2 A 2 , v(e; 6,
eX(e)w^(e-1r1)
if) + e2X(9)w^(e-1r1).
(2.31)
Problem for wM
The vector-valued function w^ lem
satisfies the following boundary value prob-
£ > ° ( 2 > ( 0 + c°lW°(l\t)
= o, £ e g,
£0™(2)(£) + A™ (1) (£) = o, £ e R 2 \ s , w°W(t) = ™(2)(£), * W 2 ) ( 0 - B°0w°M($) +
BtwWit)
(2.32)
66
Dipole tensors in spectral problems of elasticity
-B?™°«(£) = £ > * ( 1 , i ) ( 0 . £E Og,
(2.33)
3= 1
where *(M)(£) = *&*>(£) = (A - A0)(£ • i/)e(3), * ( 1 ' 3 ) (£) = (A + 2M - A0 - 2fx0)($ • u)e^\
(2.34)
The asymptotic behaviour of the field w^> as ||£|| —> oo is specified by (see, for example, Kondrat'ev (1967), Movchan and Nazarov (1990), and Movchan and Movchan (1995)):
w^{i)=T^\i)
+ 0{\\i\\-1) (2.35)
= ar(£) + 6 + 3 ( ^ ) + 0 ( | | a - i ) , 11(11 ^ o o ; where
3(
M > = ( "" r , ( i l ( M , e < " + ^ < ra ) e l 2 , + - 3 ( M ) e < 3 , ) 6
2
H^I^E^E-S^.^M;
P-36)
The latter can be shown by using the following argument. By virtue of (2.22) Y^1' is a homogeneous vector function of degree -1. Since £ • Vtd/dii = d/diiWtwdidUw - d/dih
Asymptotic
behaviour of fields near the vertex of a thin conical inclusion
67
then according to (2.16),
A&a/fiflTU = 2(A + M) ( ^ V > + ^ e « 2
~v.(l)
+E-^
\
*
e(3> +0
)
3
2e
( ^ i i " ) = ii«ir (]j|]f)+0(ii«ii"3)-
(2-37)
Looking for a particular solution of the equation
llcll we obtain the equalities (2.36). Note also that Co(aT + b) = 0 for £ ^ 0. The vector field u/ 2 ) is defined up to an arbitrary constant vector 6. It will be convenient to take b=-
^ f - ^ T ( a i e W + a2e<2)) + a s e ^ , 27T/X \ X + 1
/
-1
where x = (A + 3/x)(A + /z) . With such a choice of b, the quantity Y ' 2 ' written in the coordinates 77 = e£ is independent of e. The vector a in (2.35) is unknown. We shall show that the components of a can be written in the form 6
X > / 3 « , fc = 1,2,3,
(2.38)
where the coefficients /9|^ are given by
f%) = JLoW , 2/#> = _^L-a 5B W, j = 5,6; 1 4 A + 3/i
A + 3/i
M) _ M ( A - A 0 ) - ( ^ + /i)(/x-Mo) r Q) (j). ^ 3 - ( A + M - A 0 - / x 0 ) ( A + 2 / ,) ( a i + ° 2 } 2(/x-/Xo)(3A + 2 / x - 3 A 0 - 2 / x 0 ) f « , . , , , . . 77— r —T S&Sg, 3 = 1,2,3,4;
/ 9 »Qx (2.39)
the remaining coefficients p£' are equal to zero. In (2.39) Sg denotes the area of the domain g.
Dipole tensors in spectral problems of elasticity
t2 II „
-
'
'
^
/
-
„
s
s
\
\
1/
\
1 1 1 1 1 1 1
1 „ 1 */ /
1
\
\
\
\ **
1
/ ** ^
6
1 1
\R \
\ \ 1 1
, /
\ \
Fig. 2.2
\\
/
/
/
_ _ - - ' ' ^ > H
The domain of integration DR \ g, where DR is a circle of radius R.
In order to show (2.38), (2.39), we consider the following identity fe ( i » • (Clwo(® +
C\w°w)dt
Ja
e « • (£rV 2) + ClWW)dt = 0, i = 1, 2,3,
f
(2.40)
JDR\g
where DR = {£ G K2 : ||£|| < i?} is a circular domain enclosing g (see Fig. 2.2). Integrating by parts in (2.40) gives f e« • £ > ° < 2 ^ = - /
JDR\5
e « • B>°( 2 )dZ,
Jd(DR\g)
Asymptotic
behaviour of fields near the vertex of a thin conical inclusion
69
= f e®-B0w<2)dI+ f e«.B 0 tt> (2) dZ, JdDR Jdg
I eW • C\w°Mdt = - I cW • B°lW°^dl - 2n0 f w^^dl,
/" e ^ • ^ u ^
1
^ = - f e^ . Blw'Wdl - 2A0 / ( ^
Jg
Jdg
[
e^-B1w^dl+2fx[
w^mdl,
Jd(DR\g)
f
eW-dwWdt=[
JDR\g
+ u2w°2W)dl,
Jdg
e®-CiwWd£= [
JDjt\g
( 1 )
i = 1,2,
i = 1,2,
Jd(DR\g)
e^-BlW^dl+2xf
{n^+^w^^dl,
Jd(DR\g)
Jd(DR\g)
where n = (ni,ri2) is the outward unit normal vector on 8{DR \ g ) , n = (£i/R,£2/R) °n 9DR and n = v on dg. Thus, f e « . (£>°( 2 ) + £?u>o(1))d£ + / Jg
eW • ( / W 2 ) + A«> (1) )d£
JDR\g~
= /"
e « • (BoWM + BlW^)dl
JdDR
+ f e « • ( * W 2 ) + »i«; ( 1 ) Jdg
-B°0w°W
- B°lW°^)dl
+ 2IU
i = 1,2,3,
(2.41)
where Ii = fj, I JODR
w\ 'mdl
+ /
(/XW3 ' — /x0W3
)vidl,
i = l,2;
Jdg
J3 = A / (i«i J ni + «4 ri2)dl JdDR + [ ({Xw^ - A0Wi(1Vi + (^2 1 } - X0w02{l))v2)dl. Jda
70
Dipole tensors in spectral problems of elasticity
Using the interface conditions (2.33) we can re-write (2.41) in the form eW • (£ 0 ™ ( 2 ) + A t » ( 1 ) ) d $
/ e « • ( £ > ° ( 2 ) + C°xw°M)d£ + J Jq
JDRYO
e « • ( * W 2 ) + BlWW)dl
= [
^TcjV^iZ) dl
+ [ e« • (
JdDR
Jdg
\
j = 1
+2Ih t = 1,2,3,
(2.42)
where Ii = — / w^&dl U JdDR
h = ± [ n
(wi'ki
+ (fi - fj,0) / w^Vidl, Jdg
+ wP&W
i = l,2;
+ (A - A0) / (wfVj +
JBDR
w^v2)dl.
Jdg
To evaluate the first integral on the right-hand side of (2.42) we shall use the asymptotic representations (2.35) and (2.22) for «/ 2 ) and w^\ This yields /
e « • (B 0 u> (2) + BlW{1))dl
JdDR
e w • B0(aT)
= [ JdDR
+ f e<*> • (B0S + BiT ( 1 ) )d/ + o(l), i = 1,2,3. (2.43) JdDR Note that, because 6 is a constant vector, Bob = 0. Using Betti's theorem the first integral on the right-hand side of (2.43) can be evaluated as follows: /
cW • B0(aT)dl
JdDR
= - J
= /
eW
• Co(aT)d$
JDR
e ( i ) • a<5(£)d£ = - a i , i = 1,2,3.
(2.44)
Thus, we can re-write (2.42) in the form m=
f
e « • VCj*(1'j)dZ + /
eW • (BoS + B i T ^ d Z + 2J<. (2.45)
Asymptotic behaviour offieldsnear the vertex of a thin conical inclusion Direct calculations of the first integral on the right-hand side of (2.45) together with (2.34) give ,
/
Jdg
6 eW
( V2c 6 (/x- Vo)I, i = 1, c
(1 J)dl
• J2 J* '
= \ V2c5(/i - Mo)/, i = 2, { AI, i = 3,
J=I
(2.46)
where A = (cx + C2)(A — A0) + c3(A + 2/i - A0 — 2/i 0 ), and / = / (£-I/)di. Jdg In order to evaluate J, we apply the Divergence Theorem to obtain / ^ukdl Jdg
= -6jkSg.
(2.47)
The sign minus on the right-hand side of (2.47) is due to the fact that v is an inward normal vector on dg. Thus, 1=1
(&"i + &V2)dl = -2Sg. Jdg Substituting (2.48) into (2.46) gives / Jdg
6 e « • ^cj^^dl i=i
(2.48)
( -2V2c6(n - Ho)Sg, i = 1, = I -2V2c5(fx - n0)Sg, i = 2, [ -2ASg, i = 3.
(2.49)
To calculate the second integral on the right-hand side of (2.45), we first re-write the functions S, in the form S
A
^l
j .A
Zl&
2
£ _
IKII '^
A
£l6
'|l€||
.
2
£|
"IKIF 2
ll£l|2
1£U 2
W
where
A/ =
r^ ( C 5 ^ 5 ) + C 6 Q ^ 6 ) ) ' A u = r^ ( C 5 ^ 5 ) + C 6 a * 6 ) ) ;
5i = 2(A + 2 / x ) [ C l Q ^
+ C2
^2)
+ C3Q
^
+ C4
^4)]' *=
ll2 3
' -
(2 50)
"
71
Dipole tensors in spectral problems of elasticity
72
Applying the operator So to the vector-valued function 3 and integrating along ODR we obtain < ^ /
cW • BoBdl = -
i f
A+i
,
i =
An,
i.
i = 2,
(2.51)
JdDR
0,
i = 3.
The function Y ^ is a homogeneous function of degree -1, and its components can be written as follows
i A
it
£2
—
3£ x
£2 \
TW=i,(6%1]#)+i//(26jI +i/7/(e -(1)
._I_(cs<>f
+
^
tf-3&2
X;
£1 f).
Jj_ _ _ i . ( l w f ,
+c64«)^,
(2.52)
where 2
i ) + c2aY> ( ) +. c„ 3ayJ& ( )> A/ = 2q(Cla\Jl> {> + c4va^>) A// = 2g(citt2 + c2a22' + C3a23' + AM
4
c^a^)
= \f2q(ciot£' + c2af> + c3a$> + c4a$ g = (A + /i)/(87r/x(A + 2/i)).
(2.53)
Asymptotic behaviour of fields near the vertex of a thin conical inclusion
73
Evaluating B\ 'T^1' and integrating over 8DR gives: -(C5CK4 + c6af')A/i_1,
i = 1,
(1) 5) 6) •e W . B ! T ^ = < - ( c 5 4 + c 6 c 4 ) A M - \ i = 2,
/
(2.54)
JdDR k-27T/i(x-l)(i/+i//),
t = 3.
In order to evaluate the 7j integrals, we first note that, although the components of the displacement vector u / 1 ) ^ ) on dg are unknown, the terms (A — A0)(u;^ '1/1+W2 V7), (/J- — Mo)w3 v\ and (/z — \i^)w\ 'v-i can be written as follows: i\ \ \/ (!) , (1) i A — A0 (i).(B0-B0>)(£1e«+6e(2>), (X-X0)(w\ 'Vl+W^ V 2 ) = ;2(A + fi - A0 - /i„) w
(/i - fio^vi
= w ( 1 ) • (Bo - B(|)(&e<3>), i = 1,2.
(2.55)
Thus, w& • B0(&e<3>)
Ii = f J9D H
w*1) • (Bo - B°0)(^e^)dl,
i = 1,2;
J9S
2(A + M) JdDR + 9,
*
^°
r /
i0«-(»o-»S)«ie(1)+6e(9))dI.
Applying Betti's theorem to the region (DR \g)\Jg 0= /
WW(£)
(2.56)
gives
• £ 0 &e< 3 >) - (&e(3>) • A > ™ ( 1 ) m
Jg
/ Jdq
t i / 1 ) ^ ) • (Bo - BD&e^)
- (&e(3)) • ( B 0 ™ « ( 0 -
B°0w°W(£))dl
74
Dipole tensors in spectral problems of elasticity
+ / ™ « ( £ ) • Bo(&e(3>) - (&e<3>) • B0w^{i)dl, JdDR
i = 1,2;
li)' 1 '^) • 4 ( 6 e ( 1 ) + 6 e ( 2 ) ) - ( 6 e ( 1 ) + 6 e ( 2 ) ) • ^
0= /
( 1 )
« ) ^
JDR\g
+ fw°W{t)
• £ S ( 6 e ( 1 ) + 6 e ( 3 ) ) - ( 6 e ( 1 ) + 6 e ( 2 ) ) • C°0w°W(Z)dt
Jg
= f
wW{£) • BofaeW
+ &e<2>) - fae™ + &e<2>) • S 0 « ; ( 1 ) ( € ) *
+ / W W(£) • (Bo - »S)(6e { 1 ) + &e<2>) Ja s - ( 6 e ( 1 ) + 6 e ( 2 ) ) • (»o«> ( 1 ) «) -
B°0w°W(t))dl,
and therefore, to ( 1 ) • (Bo - BS)(6e<3>)di + /
/ Jdg
(&e<3>) • BotaWdl - / (6e ( 3 ) ) • Y > * « d Z , i = 1,2;
= /
/
™(1) • Botee( 3 ))d/
JdDR
u>(1)-(B0-Bo)(6e(1)+&c(2))tfl = - /
v/a g
(2.57)
ti»( 1 >-Bo(6c ( 1 ) +&c ( 2 ) )ca
J9£>R
+ /
(tieW+Z2eW).B0wWdl-
JdDR
[ (fceW+fceWj.^c^dl. Jdg
(2.58)
j=i
Here, to derive (2.57) and (2.58), we used the boundary condition (2.20). Calculating the integrals on the right-hand sides of (2.57) and (2.58) gives /
(tieW)-B0wWdl=
JdDR
[
( & e ( 3 ) ) . B 0 Y « d Z + o(l)
JQDR
= \{^4li
+ <*«$*) + o(l), i = 1,2;
(2-59)
Asymptotic
behaviour of fields near the vertex of a thin conical inclusion
75
6
r {z)
c
/ {£ie )-YJ 3*{3)dl
= -y/2{ii-Vo){csk2
+ c&5il)Sg,i
J
9g
= l,2- (2.60)
j=1
JdDR
JdDR H2e(2))dl
+ o(l) = -4wn(Ai
/ (^ie (1) + 6 e ( 2 ) ) • B0w^dl JdDR
+ An) + o(l);
= f foe'1* JdDR
+ ^e(2)) •
+o(l) = 27r/i(x - l ) ( i j + i / 7 ) + o(l); /• / (ZieW +
(2.61)
B0Twdl
(2.62)
6 6e(2)).
£ C j .*tf)«fl = _ 2 [(A + /i - A0 - /i 0 )(ci + c2)
Jdg
j=1
+(\-\o)cs}Sg.
(2.63)
Substituting (2.59)-(2.63) into (2.56)-(2.58) and taking the limit as R ->• oo we obtain /i = y/2(n - fj.0)c6Sg + -(c5af'
+
c6af'),
h = V2(n - fJ,0)c5Sg + -(c5<4 5) + c 6 o4 6) ),
J 3 = 7T(X - 1)
+
(A + 2/z)(A-A 0 ) (A + // - A0 - /x0)
(A-A0)(Cl+c2) +
(A - A0)2 (\ + fJ.-\0-
(A, +
:c3
fJio)
An)
Sg.
(2.64)
The latter together with (2.45), (2.49), (2.51) and (2.54) gives (2.39).
76
Dipole tensors in spectral problems of elasticity
2.1.3
Stress singularity
exponent
A2
In this section we derive the representations for the second order terms $ ( 2 ) ( 0 , p ) and A2 in the expansion (2.31). We shall use the following approximation: V{1 + £2A2) ~ 7>(1) + e2A2V'(l),
(2.65)
where V(l) admits the representation V{1) = Co(d/dV)
+ dfad/dr,)
+ C2(d/dV)
(2.66) 2
in the vicinity of M. Taking into account the terms of order 0(e ), we derive the following system for the vector field & ' and the quantity A 2 : 7»(1)# (2) = -A 3 7>'(1)* - F on S.
(2.67)
Here F = A x T ( 1 ) + (V(l) - C0)xT{2)
+ [C0 + A , X ] T ( 1 ) +
[£o,x}?(2),
and [A, B] = AB — BA denotes the commutator of operators A and B. As follows from Kondrat'ev (1967), the system (2.67) is solvable if and only if its right-hand side is orthogonal to the traces Y^1'-'' of the fields Y{-1'^{d/dx)T{x) on the unit sphere S, that is, / ( ( - A 2 7 » ' ( l ) * - F) • Y^ds
= 0, j = 1,2,..., 6.
(2.68)
j , k = 1,2,..., 6,
(2.69)
j = 1 , 2 , . . . , 6,
(2.70)
We shall show that / yd.*) . 7>'(l)#(W>
6
• Fds = V Mjkck, fe=i
where M lfc = q[-(2 - x ) a f ) - 4fc> + /?<*> + 2 ( x + l)-\aW M2k = q[-a™ - (2 - *)a 2 fc) + 4 f c ) + 2(x + l ) "
+ a™)}, 1
^ + af)],
M3fc = - g ( x + l ) ^ f c ) , M 4 f c = g ( x - l ) 4 f c ) , fc = 1,2,3,4;
Asymptotic behaviour of fields near the vertex of a thin conical inclusion
M5p = 2-1/2g[(3 + x)4 p ) - (x - l ) / ^ ] , M 6 p = 2- 1 /2 (? [(3 + x ) 4 p ) - (x-l)(3[p)},p
= 5,6;
(2.71)
the remaining elements Mjk of the matrix M are equal to zero. We shall first verify (2.69). We denote Vd = {xeM.3 : \\x\\ < d}, a ball of radius d centred at the origin, and introduce an infinitely smooth function C(||a;||) S Co°(^i) which vanishes on dT>\ together with its derivatives and is equal to 1 near the origin. Using properties of the Somigliana tensor we derive the following identity v(1,fc)
jv
( ^ ) T ( a ! ) -£(^)(C(P)V< 1J >(*))<*=
= J C(P)V(l'j)(x) •
V(1 fc)(
'
L(^)V^(^)T(x)d:x
£ ) V ( U ) ( 0 ) = = ^ f c ' i.* = 1.2,---,6.
(2.72)
Since V(Pyp)(p*^)
= PV(1 + P§-)*llli\
V(A + 1) = P(l) + AP'(l) + 1/2A2V"{1), we also have
lim5^o|
jp-2Y^k\0,^(<:<j>)P*M\0,
(2-73)
(2.74)
77
78
Dipole tensors in spectral problems of elasticity
+\^py{1'k)(o^)v"(m(hjHe,ip))dPds = - l i m ^ o C(*)
JY^k\e,V)V'{l)^j\9,V)ds
Js
_ f Y{1'k\e,>p) •V\l)^{1'j\e,ip))ds. (2.75) Js, The latter together with (2.72) gives (2.69). The equalities (2.70) follow from the following relations based on (2.66) and (2.68): =
/ Y ^ ) • 7>(1)( X T W + xT<2>)ds = 0, k = 1,2,..., 6, Js
d
w
3
0 ) cW -x{9) jI= l > ( iE= l ^ ( £ ) +^ E ( t& + ^ ) )%). =l
(2-76)
where
4 J ) = — - T a g i , i = l,2,j = 5,6;
4')=2
(Q ( 0 + a ( 0 ) j z = lj2)3 .
x+ r
the remaining quantities af' are equal to zero. As follows from (2.68)-(2.70), the solvability condition of the problem (2.67) for the vector field &2' can be written as an eigenvalue problem for the matrix M: Mc
= A2c,
(2.77)
where c = (ci, C2,..., c&) is the vector of the coefficients in the representation (2.11), and A2 is the spectral parameter.
Asymptotic
behaviour of fields near the vertex of a thin conical inclusion
79
Due to (2.4), the stress field is singular at the cone vertex if A(e) — 1 < 0. Thus, as follows from (2.31), a necessary condition for the stress field to become singular at the vertex of the notch is A2 < 0.
(2.78)
The eigenvector c determines the stress field generated by the external load in the vicinity of the origin in the elastic medium without a cavity. By solving the eigenvalue problem (2.77) we find the set of 6 eigenvalues A 2 and the corresponding eigenvectors or the stress modes that generate these singularities. The elements of the matrix M depend on the elastic constants A, fi and the geometry of the domain g. Let k£ = {xeM3 : x$ > 0, y/xf + x\ < ex^}, so that g is a unit circle. The non-zero elements of the matrix M for the unit circle are M n = M 2 2 = ~{1 - 2v), Mu = M 2 1 = i ( l - 2i/), M13 = M 2 3 = - „ " > , M 3 i = M 3 2 = - - , M 3 3 = 1, 4(1 -v) 2 M55 = M66 =
-4(l-,)(3-4,)-
The eigenvalues and the corresponding stress modes are given below: eigenvalues
stress modes
A 1}
= 5(1 + yf&) *11 = ^22 < 0,(733 > 0
A 2)
= 5(1 - }/S)
2
2
A23) = - ( 1 - 2v)
(Tii < 0, (722 > 0
A24) = - ( 1 - 2v)
axi > 0
A (5) — 5-2i/-8i/ 2 2 - ~ 4(3-4v)(l-v)
A
A (6) _ A
* " =CT22> 0,(733 > 0
2
-
5-2I/-8I/2
-4(3-4„)(l-«/)
-.
a23
(2.79)
^ n > ° ^
n
0-13 > U
From (2.79) it follows that A22), A23) = A24) and A25) = A26) are negative for all v € (0,1/2), while A2 ' is positive. The eigenvalue A2 ' is associated
80
Dipole tensors in spectral problems of elasticity
I
u
I
I
I
I
I
I
I
I
'
/ -0.1 s
-0.2 -0.3 -0.4
-
/
"-
^^^^
^-0.5 -0.6 -0.7 A(3) 2=
A
Nw
'
AW A
2
V
, '
.
A< 2
•
-0.8 -0.9
5
) =A < 6 > 2
/
1 f
0
//
^
v.
' N. I
I
I
I
I
I
I
I
I
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
'-*
0.5
V
Fig. 2.3 Negative eigenvalues for a circular domain g: A2 A^4) (dashed line), A^5) = A26) (dot-dashed line).
(solid line), A!2(3)
with shearCT23in the yz-plane, and the eigenvalue A2 with shear a\z in the xz-plane. The eigenvalues A2 ' and A2 are generated by different kinds of load: A2 is associated with tension along the y-axis and compression along the a;-axis, and A2 with shear <Ti2 in the xy-plane. The remaining (2) eigenvalues A,(D 2 ' (positive) and A2 ' (negative) correspond to axisymmetric k
modes: the eigenvalue A2 ' is associated with biaxial compression in the xyplane and tension along the z-axis, while the eigenvalue A2 ; with biaxial tension in the a;y-plane and tension in the z direction. The negative eigenvalues A2 , A2 = A2 , and A2 = A2 f° r the unit circle are shown in Fig. 2.3. One can see that, for the Poisson ratio v G (0,1/2), the strongest singularity is always associated with a nonaxisymmetric load.
Imperfect interface.
2.2
"Coated" conical
inclusion.
81
Imperfect interface. "Coated" conical inclusion.
In this section we consider a model problem for a circular conical inclusion connected to an elastic medium via a thin and soft layer. We assume that the elastic properties of this "coating" layer differ from those of the inclusion and the medium, and there is a perfect bonding between the layer and the inclusion as well as between the layer and the surrounding medium. Our objective is to study the effect of the "coating" on the behaviour of the elastic stress field near the vertex of the inclusion. The material presented in this section is divided into four parts. The formulation of the problem is given in Section 2.2.1, the boundary layer solution and the constitutive equations of the imperfect interface are discussed in Section 2.2.2. Section 2.2.3 deals with the spectral problem for the matrix M and the stress singularity exponent A2. Section 2.2.4 considers some examples which highlight the effect of the "coating" on the stress distribution near the vertex of the inclusion.
2.2.1
Formulation
of the
problem
Consider a "coated" conical inclusion embedded in an elastic matrix. Let us define the following domains: k£ = {x S R 3 : £3 > 0, e^a^Ha:'!! < Rltx' = (xi, x?)}, occupied by a thin circular conical inclusion with elastic constants A 0 ,/i 0 , the "coating" k* = {x £ R 3 : x3 > 0, e - 1 ^ 1 ^ ' ! ! e (Ri,R\ +eRo),x' = (xi,X2)} with elastic constants A*,/x*, and the elastic matrix Ke = R 3 \ (k£ U k*) with elastic constants A,/x (see Fig. 2.4). The elastic materials are all homogeneous and isotropic. We assume that the "width" £2Ro of the layer k* is sufficiently small in comparison with the radius of the inclusion ER\. The displacement fields u° within the inclusion, it* within the "coating" layer and u within the surrounding matrix satisfy the homogeneous Lame equations: L°(^-)u°(e;x)
= 0, x e ke, L*Au*(e;x)
L(—)u(e;x)
= 0,xeKE,
= 0,
xek*£,
(2.80)
82
Dipole tensors in spectral problems of elasticity
and the interface conditions u°(e;x)
= u*(e;x),
u*(e;x)=u(e;x),
x^Wx'W = sRi,
x^\\x'\\
= eRl+e2Ro,
(2.81)
and o- o(n) (u°;£,x) =
x^\\x!\\=eRx,
o-*( n )(u*;£,x)=o-W(u;£,x), xf\\x!\\=£R1+^Ra.
(2.82)
We also assume that a uniform stress field is applied at infinity. In the vicinity of the origin the displacements u°, u* and u admit the representations (compare with (2.4)): u°(e;x) = pA^v°(e;9,ip),
x G jfee, u*{e; x) = pA^v*(e;
u(e; x) = pA^v{s;
6, ip), x G k*,
9,
(2.83)
where (p, 9, tp) are spherical coordinates centred at the vertex of the inclusion ke, p= ||x||,v G \0,2TV),9 G [0,7T]. The exponent A(e) and the vector fields v°,v* and v solve the following problem on the unit sphere § (compare with (2.5), (2.6)) which is obtained by transforming (2.80)-(2.82) into spherical coordinates: V°(A(e))v°
= 0 on ge, V*(A(e))v* = 0 on
g\,
V{K{e))v = 0 on S/g^Ug!,
(2.84)
w° = «*, Q°(A(£))w° = Q*(A{s))v* on 3g £ .
(2.85)
t;* = v, Q*(A(e))v* = Q(A(e))v on 3(S/&Ug*),
(2.86)
where g£,g*,g°, are the regions of intersection of the unit sphere § and the cones K£, k* and kE, respectively. To construct an asymptotic expansion of the solution to (2.84)-(2.86) we shall use a procedure similar to that of Section 2.1.
Imperfect interface.
"Coated" conical X
3
inclusion.
83
S2Ro
«^e [Ao J /X,
k:[\;f
Fig. 2.4
Conical inclusion fce with "coating" fc* embedded in an elastic matrix
Ks.
We take the leading order approximation (e — 0) to the fields v, v*, v° in the form similar to (2.11): v ~ #(0, p) on S \ ()onge
where
(2.87)
Dipole tensors in spectral problems of elasticity
84
3= 1
(2.88) 3= 1
The leading terms satisfy equations (2.84) and the displacement conditions in (2.85), (2.86). However, they leave a discrepancy in the stress conditions. To compensate for this discrepancy we construct a boundary layer solution in the vicinity of the point Af on the unit sphere S.
2.2.2
Boundary
layer
solution
In the vicinity of the point j\f we shall use the scaled variables £ given by (2.12). In the ^-coordinates the vicinity of the point Af is transformed into an infinite plane with a circular inclusion of radius R\ surrounded by a thin layer g* of width eRo (see Fig. 2.5).
Fig. 2.5
The region after the change of coordinates.
The Lame operator L and the traction operator B — an are given by (2.14)-(2.15), similar representations hold for L°, L* and B°, B*.
Imperfect interface.
2.2.2.1
"Coated" conical
inclusion.
85
Change of coordinates for the "coating" layer
In the previous section we described the coordinate transformation from ^ ( 0 , 0 , 1 ) into R2 with a "coated" circular inclusion. Now we need another transformation to deal with the coating. We introduce the new variables £ and if' and consider the following change of coordinates:
C
l€ll--Ri
-, '€[0,27r),
(2.89)
and so the derivatives are 8 dtp1
d d d
1 d e <9£'
r = ||£|| =eC + i*i-
\{g\Jg')
(2.90)
[X,fA
zRo
Fig. 2.6
The inclusion with the thin coating in the (£1,62) coordinate system.
We shall write the representation of the differential operators L and B in the new system of coordinates, and thus formulate the relevant boundary value problems. First, we use the relation R\ > ££(C G [0, Ro\), in the approximation of the derivatives: 1 . 1 £(, + i?i Rl
eC tl\
+ ...
Dipole tensors in spectral problems of elasticity
86
The Laplacian operator written in the ((,
+
r 2 8
+
efli dC +
W
'
and therefore, •C*(33)=^^.
(2-91)
The operator CQ depends on the elastic constants A*, /x*. The transformation of the operator with such constants, into the new operator with the new coordinates (£, (f) gives
where the operators IQ,1\ are written in the coordinates (C,y)- Consequently, in the Cartesian system of coordinates the operators have the following form: f (A* + n*) cos2 ip-§^ + \x*-^ — i/* 0 —
0
(X*+fi*) sin2 ip^+fi*^
(A*+/i*)sin<£cos<^
V
(A* + n*) sin
0
0
^
0 M *JW
(2.92)
and 11 is a symmetric matrix operator (in the ^-coordinates) with components ,*(11) h
,*(12)
h
1 , . » . *w . 2 + /x)(sm
-W^
,*(21)
d
^dc'
1 /N* ,
*w
0
92 )
° d^
^
, /
2
fJL* 8 +
R'id'C • 2 \
®
\
#~(A +/^ )(-smv?cos^—+ (cosV-sin y ) — - ) ,
=h
,*(13) ,*(31) ,,» , * M J ^ l = V = - ( A + M )-RlCOS(^^-2,
l
Imperfect interface.
i;03> =Z jC»»>
=
"Coated" conical
d2
_(A.+/Ofll8in
inclusion.
*(33) jZj
°
87
—.
(2.93)
fli"#C Now we shall express this operators in the polar system of coordinates*, given that the elements depend on £ and
(2.94)
and therefore, after substituting the values of the derivatives given by (2.90) and the canonical vectors given by (2.94), we express the operator IQ in the cylindrical system of coordinates: (A*+2/z*)^
0
0
l7*0 — —
\
0 0
(2.95)
d2
0
n* a? /
Following a similar procedure, we also express the symmetric matrix operator 11 in the cylindrical system of coordinates: j * ( l l ) j*(12) j * ( 1 3 ) \
;* _
z *(12)
£*(22)
*(13)
0
0 I-
•(22)
where ^(11) = ^ ( A * + 2 ^ ) ^ , ^ ( 1 2 ) = ^ ( A *
,.(13) _
n
—
-(A*
52
.VD
L
ac
2
a2 V ) 3Cfy>
..(22) _
M* a
x
i?iac
(2.96)
For the boundary conditions operators we can write e~lBl + B\ -4 £ - 2 6 S + £ _ 1 6 i + 0 ( l ) . *Here, we remind the reader that the operators have a block diagonal structure, therefore, one can treat the problem in the polar system of coordinates for the first two components of the displacement and the third component separately.
Dipole tensors in spectral problems of elasticity
88
By using the expression for the derivatives in B£ (see (2.90)), we obtain the operator 6Q in the Cartesian coordinate system:
b*0 =
/-(A*+/i*)cos2<^-M*^
-(A*+/i*)sin^cos<^
0
-(\*+H*)sm
-/x*^-(A*+Ai*)sin>^
0
0
\
0 (2.97)
and the operator 6j has the the following form: 'b.(ll)
6 *(12)
fc.(13)^
6 .(12)
_6,(11)
6.(13)
6.(23)
6 *(23)
0
/
where (11) &i
1 />*
,
*N •
^
1 — (/z* sin 2
L*(12)
= -5-(A +/x )sin<£cos
6 *(i3) =
( A * + f)Rl
c o s y ^ , 6^(23) = (A* + / / ) i ? i s i n ^ ^ -
(2.98)
Similarly, using the transformation of coordinates (2.94) we obtain the operators &o a n d b\ in the cylindrical coordinate system:
(A*+2/z')£
0
0
/**&
K
0
(2.99)
0
/;* —
KTa£ ( A * + M * ) - R l ^ \
'Hi
61 =
(2.100)
fll
\(A*+/x*)i?1^
0
0
/
Thus, we have written the stress operators in the new sets of coordinates for the thin layer. We shall assume here that the elastic constants A*,/x* are of order e, that is, A* = e\*0, M* = e\*0,
(2.101)
Imperfect interface.
"Coated" conical
inclusion.
89
where A*,/i* are of the same order as A,/i, A0,/z0. As a consequence of (2.101), the stress conditions operator for the thin layer will be written in the form e~xBl +B\^
e~% + €'%
-»• e^&S + b{,
(2.102)
where the matrix differential operators by, b\ have the same form as bg, b\, respectively, but with the elastic constants A*,/i*:
((K + Wo)Tk ° 0 /i*£
K
° 0
\ (2.103)
and
6J =
o
' H i d
V(A:+ M :)i?i^
o
(2.104)
o
The same can be said about the Lame differential operator, which can be written as e~'C*0 + e-'-C'i -> e~T0 + e-%
-3i* -»• e~% + e- 2"i *^ .
(2.105)
The operators l*0 and ij for the thin layer are given by (2.95), (2.96) with the elastic constants A*,/i* (due to the assumption (2.101) and the change of coordinates (2.89)). 2.2.2.2
Problem for to*1)
In the previous section we obtained the expressions for the Lame and stress differential operators for the thin layer. We now write the problem for the leading terms of the boundary layer, eto*1)(^), ew°^(^), and etw**1)^,
Dipole tensors in spectral problems of elasticity
90
layer. The displacement fields are thus given by v° ~ # > , * ) + eX(0)w°W(t),
|£| G g,
v* ~ **(^e)+ex(^K(1)(C,^).Ce[o,i?o],¥'e[o,27r)I v ~ *(
(2.106)
|£| e K2 \ (ffUj*),
where the functions 3>°, ** and * are given by (2.87), and x(#) is the cutoff function5 defined in Section 2.1. The fields w^\ w°^ and u>*(1) satisfy the following equations: £°0{%)w°M(t)
= 0,
M\\
* o ( ^ ) « ' * ( 1 ) ( C , ¥ ' ) = 0 ) C e ( 0 , i 2 o ) , ¥>€[0,27r)
/: 0 (^)ti;( 1 )(0=0 ) ||{||>i2i+eieo. Here, the operators £., CQ have the form (2.16), while IQ has the form (2.95), with elastic constants A*,/x*. We can see that the latter operator depends only on the derivatives with respect to £. This means that l^w^ can be written in the following way: ^wf\Cp)=0,
^ < ( 1 ) ( C , V ) = 0 , ^W*M(£,
(2-108)
From (2.108) we can see that the dependence upon £ is linear for all components of w*^(C,ip), therefore, w*^ can be written as w*{1)tt,
+ U<1\cp),
(2.109)
where W ^ ) , U*M(
Imperfect interface.
"Coated" conical
inclusion.
91
In Fig. 2.7 we present both interfaces. We can see that for the first interface w°W is evaluated at ||£|| = i?j, while w*^ is evaluated at £ = 0. Similarly, for the second interface ti/ 1 ) is evaluated at ||£|| = Ri + sRo, while w*W is evaluated at ( = Ro. From now on, we keep in mind that the evaluation of a function at a point will be given either with respect to the coordinates (^1,^2), for w°^\ w^\ or with respect to ( for w*^. We thus re-write the displacement conditions, and consider their relation to w*^ as given by (2.109): Interface 1: w°W{Rlttp)
•(i)/ ,„*(!) =w*W(0,
-
ipe
[0,2TT)
Interface 2: w(1)(R1+eR0,
= R0W*{1X
«1,«2)
«1.«2)
w*^> ( R 0 , ¥>) = ™ ( 1 ) ( H i + £ ^ 0 . V>) ^ ' ( ^ ( O , v)
=
w01-1)^,^)
< = Ro
Fig. 2.7 The coating of width eRo between the inclusion and the surrounding matrix. The numbers 1 and 2 denote the interfaces. The relevant coordinates and elastic constants are shown for each layer, as well as the displacement conditions for wW.
The function w^ is assumed to be smooth for all values of ||£||, and therefore, it admits a Taylor expansion. We consider the Taylor expansion for wW at Rx+eRo: wW{R1+eR0,
dwW + £R0—— (Rlf
(2.111)
92
Dipole tensors in spectral problems of elasticity
so to the first-order approximation, w^(R1+eR0)=w^\R1,V>).
(2.112)
Jump in the displacement We now evaluate the jump in the displacement across the interface (denoted here by [w](Ri,(p)) by subtracting left-hand sides of (2.110), and using (2.112): [w](R!,
= RoWWfr),
(2.113)
so that the function W*' '(
W*W(
= ^-(w^(R1,ip)~w°^(R1,tp)).
(2.114)
We can also write from (2.110) the function U*^l\
B0&,v)u>W(e)(R1+£Ro,
- Kw^HRoM
= -$>*&)(„), (2.115)
Imperfect interface.
"Coated" conical
inclusion.
93
where * # ( " ) = ((A + 2 / i K A * , 2 ) 0 ) r , *£>(!/) = (A^, (A + 2 ^ 2 , Of, *£>(i/) = A(«/i,^,0) T , *£>(!/) *
=
V2M^2^I,0)T,
= ^ ( 0 , 0 , K,) r , *£>(,/) = ^ / x ( 0 , 0 , i ^ ) r .
(2.116) The functions S&J^ (i/) are written here in the Cartesian system of coordinates. It is important to note that because we assumed (2.101), the contribution of BQ<&* within the coating layer will appear at the next step of the asymptotic algorithm. Using (2.103) and (2.109) we obtain the following representation for the second term in (2.115)
{(\*0+2n*0)W^\v)\ bZw'W(Ro,
M:^ ( 1 V)
(2.117)
the latter yields that the stress field is independent of £ throughout the thin layer. We now write the stress condition at the interface 1. We evaluate the corresponding stresses to obtain 8_ bZw*(1\0,v)-B°o(-,u)w°W(R1,v)
= ^cj*°^(v),
(2.118)
where
*°W(v)
= ((A O + 2 M < > I , A O ^ , 0 ) T , * O ( 2 V ) = {\ou1,{Xo +
* o ( i V ) = A o (i/i,i*,0) r ,* o < 4 >(i') =
2^o)v2,0)T,
y/2iio{v2,vi,0)T
¥°<6>(»/) = V2fi o (0,0 ) i* 2 ) T ,* o < 6 >(i/) = V2/x o (0,0,^ 2 ) T . (2.119) We now combine the stress conditions (2.115) and (2.118) by using the fact that the stress field is independent of £, and in particular, b*0w*W(0,
=bm0w«»(Ro,
(2.120)
Dipole tensors in spectral problems of elasticity
94
Thus, we obtain
(B0£,u)wM ^ '
- B ^ ^ w ^ i R u v )
= -^>*^V),
"d?
(2-121)
3=1
where \0)v2,0)T,
* < V ) = ((A + 2fi - A0 - 2/x0)i/!, (A -
* ^ ( i / ) = ((A - A > 1 ; (A + 2^ - A0 - 2M<>2, 0 ) T , ¥<3>(I/)
= (A - A o ) ( ^ 2 , 0 ) T , *W(i/) = V2(M - / O K ^ i , 0 ) T ,
* < 5 V ) = y/2{» - /io)(0,0, ^ 2 ) T , *<6>(i/) = ^ ( / i - / O ( 0 , 0 , ^ ) r . (2.122) Taking into account (2.114) we re-write (2.115) in the form:
/^K(w^{Rl>lf)^woW{Ru(fi))\
^wPiR^-w^iR!,?)) \
(2.123)
^(wi1\R1,
J
Here, the right-hand side of (2.123) is written in the cylindrical system of coordinates, therefore, we shall write the operators BQ and BQ as: ' arr(wr i
',w]p') ' (i)
W\
(2.124)
B0(^,u)wW(t)
\
/
Imperfect interface.
"Coated" conical
inclusion.
95
and
/<,K (1) X (1) )\ B°0{l,v)w°W(£)
(2.125)
= -
\
»o-it-
)
where a^,.,a°v are the stress operators in the polar system of coordinates. In (2.118), (2.121) we have written the contribution of the leading order to the stress field in Cartesian coordinates. We now write the \I> functions in cylindrical coordinates, noting that SE^', y!/0"' are of a similar form because ^ 0 ) _ q,U) _ q,°U)
(2.126)
therefore *(1)(V)= |
-(\ + H - A0 - fJ-o) - ( M ~ Mo) cos 2 ^ ' {fi-li0)sm2ip 0
' - ( A + /i - A0 - fj,0) + (/i - /z 0 )cos2<^ * (y>) = I -(n-ii0)Bm2ip 0 (2)
*(3)(v) = (
(A —A 0 )\ /sin2
*&(/2(/z - Mo) ( 0 \ sin , sin2<£ = 1/2 — 1/2 cos 2y?, sin 2
Dipole tensors in spectral problems of elasticity
96
Hence, we have derived all t h e terms in (2.121), (2.123) in t h e cylindrical system of coordinates. Therefore, we substitute t h e relevant terms in these coordinates. We shall analyse t h e problem for t h e first two components, which is t h e plane strain problem and obtain t h e functions wr ' , w\p ' , w° , Wp . We also analyse t h e out-of-plane shear problem t o obtain urz , w° . Plane strain problem We will now analyse t h e problem for t h e first two components of t h e displacement fields in cylindrical coordinates. In t h e stress conditions given by (2.123), we substitute (2.124), (2.125) a n d ^ in t h e polar coordinates (written similarly as in (2.127)), to obtain:
-arr{w^\w^;Ri,ip)
- (ci + C 2 ) ( A + /J) + (c 2 - ci)/xcos2<£> - Ac3
-X/2/XC4sin2y = -
K
+ O 2 M ° (wPiRuip)
- ^(fr,y>)),
(2.128)
-(Tr
= -£(w£\R1,v)-w°vW(R1,
(2.129)
We re-write b o t h conditions (keeping in mind t h e notation for t h e j u m p in the displacement) as follows
M^ 1 ),«#>;£!,¥>) = ^ I M ^ C D ] ^ ^ ) _ (Cl +c2)(X + fx) - Ac 3 ito
+(c2 — ci)ficos2ip
arv(w(1\wW]R1,¥>)
— v2(iC4sm2
= !^[wW}(R1,¥>)
+
-\/2/xc 4 cos2(/?,
(2.130)
(c1-c2)tism2tp
(2.131)
We also re-write t h e stress condition (2.121) a n d use (2.122), (2.124), (2.125) a n d (2.127) t o obtain: arr(wi1\wW-,Ru
= -(\ +
Li-\0-ti0)(c1+c2)
Imperfect interface.
"Coated" conical
- ( A - A 0 )c 3 - ( / i - n0)(ci - c2)cos2(parv(w?\wW;Ri,
-
G0T^W0^\W0^;RX^)
inclusion.
y/2(n-
/x0)c4sin2,
97
(2.132)
= -(/x - fi0)(c2 - c 1 )sin2^
-V2(/x - /x0)c4 cos 2p, v? € [0, 2TT).
(2.133)
The system of equations (2.130)-(2.133) describes the boundary value problem for the functions {wr , w\p , u>r > ifJ }• We have incorporated the information about the thin layer, such as the elastic constants A*,/j*, and the width RQ into the interface condition. We need to construct the solution in order to find the asymptotics of wr ', wlp at infinity. We shall solve the problem by using the complex potentials. Complex potentials We shall solve the system of equations by representing the displacement fields in the form of the Kolosov-Muskhelishvili complex potentials (we refer to the book by Muskhelishvili, 1953), <j> and tp: wP + iw{^ = ^e-^ixcpiz)
- z(j)'(z) - ip'{z)) (2.134)
o(1)
«V where wr w° , w^ ip are the potentials
+ iw°v
{1)
= 5 ^ e - ^ ( x o 0 o ( z ) - zWM
-
WM),
,w\p' are the components of displacement in the matrix, while ' are the components of displacement within the inclusion, , complex potentials for the matrix and (f>0, xp0 are the complex for the inclusion, and _ A 4- 3/x A + /i
_ A0 + 3fi0 A0 + Ho
The representations for the stresses in terms of the complex potentials ip, <j>, are given below (see, for example, Muskhelishvili (1953), eq.(39.4)): arr + iarv = 4f(z) + '0(z) + Vjz) - e^[z
98
Dipole tensors in spectral problems of elasticity
field u>°W i s non-singular, while the displacement field w^ decays at infinity. Therefore the corresponding complex potentials must satisfy these constraints: oo
oo
~
oo
00
* = £ ^> ^ = £ §r' *° = £««*"> ^ = £ b ^ n > n=l
n=l
n=0
(2-136)
n=0
where all the coefficients are complex. We substitute these expressions into equations (2.134) to obtain the displacement fields and the stresses as a sum of complex power series. We now substitute these expressions into the system (2.130)-(2.133) and solve a recurrent system of equations to find the coefficients. The main relations between the coefficients are ttn+l
= /?n+3 = a n + 3 = 6 „ + i = 0 , 71 = 1, 2 . . . ,
0-2 =
0,
/3 2 =
0,
b0 -
XQCLQ =
0.
Thus, by solving the system (2.130)-(2.133) we obtain a 4 x 4 linear algebraic system for the coefficients a.\, fa, 03, b\, and a 2 x 2 system for the coefficients Pi,a\. As a consequence, the complex potentials have then the following form: h —, <j>0(z) = a0+ aiz + a3z3, i^0{z) = b0 + b\z. z° (2.137) Here we write the constants: _ /Jo(2/ifl0 + (A; + 2/Qfli)[(A - An)(ci + c2 + c3) + (M ~ A»Q)(CI + <%)] 4fifi0Ro + (A* + 2fi*)R1(x0fi + fj,0- (fi- (j,0)) (j>(z) = — , tp(z) = z z
2/x/xOJR0[A(ci + c2 + c3) + /i(ci + c2)] 4/i/i0i?o + (AJ + 2fi*0)Ri(x0fj, + ix0 - (fi - Ho))' 01 = i ? j ( 2 a i - ( A - A0)(ci + c 2 + c3) - ( / i - / i 0 ) ( c i + c 2 )), "l =
:—Rl(fJ. - Ho)(ci - c 2 - v ^ i c 4 ) , a 3 =/3 3 i2f 6 - a i - R f 4 , X/X0 + /X
#3 = « ! a i - x 0 (3 H I
— Ro
Rx) 4/x/z0
Rx] R0
Afifi0
Imperfect interface.
"Coated" conical
inclusion.
99
: (ai + fJ.Rl(c2 - ci - i\Z2c4)) 4ai 3/3, yh = -52- _ 3 T + (^ - Mo)(c2 - ci + iV2c 4 ).
(2.138)
Therefore, the problem for the plane strain for the first two components of the functions w^\ w0^ has been solved. The displacements wr ,w\p' can be evaluated using (2.134). Since we are interested in the behaviour of these functions at infinity, we consider the asymptotics shown in Section 2.1 (see (2.23)), and write them in the cylindrical system of coordinates. From here, we equate the coefficients of order 0(l/r) of wi- ,w\p and the asymptotics at infinity. We first write the vectors present in (2.23) in the cylindrical coordinates. Thus, we have /_£cos2^+2!Eii£\ r
(1
W >(^)r«) = A ( rW(£))=2 9
(
w ( 2 ) ( | ) r « ) = ^(r( 2 )(0) = 2g
r
\-x
-i
r
sin tp cos
sin ip +
l-x
>
i*\ (2.139)
sin
\
/
w(3, r
# <«=^<4 r<1,K)+ 4 r<2,( «> /
= V2q
V
l+x
sin 2(p \ (2.140)
-^COs2(£
/
We consider the linear combination given by (2.23) for the vectors W{i)(£) and equate the coefficients in the expansion (2.23) and the displacement field (2.134).
100
Dipole tensors in spectral problems of elasticity
Thus, t h e coefficients af'
are given by
nW
- „(2) _
Cl-i
—
(Jen
^ i ( M ~ Mo)
4q(xfi0 + n)
Rl[(X*0 + 2/x 0 )#i(x 0 - 1)(A - AQ + xx - xxQ) + 4xxofl0(A + xx)] 4g(x - l)[4/xiXo#o + (AS + 2n*0)R1(x0n + 2/x0 - /x)] ,(2) 1
=
a (i)
2
=
Ri(ti-l*o)
4g(x/x0 + /x)
iff [(A; + 2/x0)iZi(*o - 1)(A - Aa + n - /i0) + 4/i0.Ro(A + /*)] 4g(x - l)[4/Jiz0.Ro + (A* + 2/x*)fli(xo/x + 2/xD - /x)] v<3) - J?)
R\ 4g(x-l)
-
a (i)
a3
(A* + 2 M *)i?i(x 0 - 1)(A - A0) + 4/x0A/i!o AfifioR0 + (A* + 2/x*)/?i(x0/x + 2/x0 - /x)
_ a (2) _ a (3) _ o a ffl -a3 -a3 - U, a3 -
^
~ PQ)
2g(x/Xo+/i)
-
foul) U-141J
Thus, we have obtained the coefficients characterising the behaviour at infinity of the first two components of the function w^1'. We shall now analyse the problem for the third component of the displacement w^\ Out-of-plane shear problem We consider the problem for the functions w3 '(£),w3
{£):
A^3o(1)(£) = 0, 11(11 < Rlt A ^ K ) = 0, 11(11 > Ru (J.—^—(Ri,
= -V2(fi-
fi0)(c5 simp+ c6 cos if),
dw(1) pi 3 {Ri,
+ ^{41)(Ri,
^e[0,27r).
(2.142)
Imperfect interface.
"Coated" conical inclusion.
101
We take the solution in the form «£> = c5V<5>(0 + <%V<>6\£), w°3{1) = c5V°'(5>(() + c6V°^(t),
(2.143)
where V°'( 5 )((), V°'( 6 )((), and V^(£),V( 6 H€) are harmonic functions in {||(|| < Rx} and R 2 \ {||(|| < i?i}, respectively, and V&{$), V<6>(£) have the following asymptotics as ||(|| —» oo:
v<5> ~ a f ^ 4 ) ( | ) r ( 4 ) + 4 5 ) ^ ( 5 ) (|)r(0 + --., V ( 6 ) ^ 4 6 ) W ( 4 ) ( ^ ) r ( ^ ) + a i 6 ) W ( 5 ) ( | ) r ( | ) + . . . , as | | ( | | ^ o o . (2.144) Using (2.143), (2.144), we reformulate (2.142) and write it as a set of two problems:
A?V°(5)(0 = 0, 11(11 < RU A^ 5 )(() = 0, 11(11 > RU dV^ H—^-(Ri,
dy o ( 5 ) -Ho
dr
r
(Ri,
/ i ^ - ^ i . y . ) = -^/isin^+g^5)^!,V)-V°(5)(iJi,¥»)),
V
G
[0,2TT),
(2.145) and A?V°<6>(() = 0, 11(11 < Ru A^6\0
av (6 > M-
fi^(RuV) or
or
av o ( 6 )
(i?l ,tp)-fJ,o —5
or
=
= 0, 11(11 > Rlf
(-Rl» V) = - V 2 (M - Mo) COS V?,
-V2»cos
Now, we know that
w (4) (i)r(() = -^rafran ll(ll) = - a f e ^ = - ^ £ , (2.147) (5)
w (£)r(() = - ^ 4 ( i n 11(11) = - a f e ^ = -g*.
Dipole tensors in spectral problems of elasticity
102
Thus, we write v°(5)=di6+^2, (2.148) V(5)
^3$!
d.4£2
Substituting these functions into the system of equations we obtain the coefficients d\, ^2,^3,^4, and therefore, the coefficients a 4 , a 4 ,a$ and
5
4
P
VMo-Ro+M^i(M + Mo)
4
5
(2.149) 2.2.2.3
Problem for
w^
Once we have solved the problem for the functions w°^l\w*^-\ it/ 1 ), we now formulate the problem for the next order terms in the asymptotic expansion of the displacement fields for each layer. We consider the secondorder terms of the boundary layer, e2w°(2\£), s2wW(£), and e2w*(2\(,(p) for the displacement fields. We take the solution in the form v0~&°(
11(11
+ w*W((,ip)+e2w*W((,
v ~ * ( ^ , 6) + ex(6)wW(£)
+ e2x(9)w(2\£),
Ce[0,J2o], ||(|| >R1+
(2.150) eR0.
where x(#) is the cut-off function mentioned in Section 2.1 (see (2.27)). We now apply the corresponding Lame operator (L°{\0,(j,0], L*[X*,n*], L[X, fi]) in each region, and take into account the second-order terms. We thus obtain £ > o ( 2 ) ( 0 + £i«> o(1) (£) = 0, 11(11 < Ru l*0w*W((;,
Ce[0,i2o]^6[0,27r),
£ 0 ™ ( 2 ) (£) + A t » ( 1 ) ( 0 = 0, 11(11 >R1+
(2.151)
eR0.
The operators £ 0 ,£i,£i have the form (2.16), and IgJl are given by (2.95), (2.96), respectively.
Imperfect interface.
"Coated" conical
inclusion.
103
Since we know the function w*^(£,
Z*tu*«
MK + ^W^M
+
^W^M
f^*(1)(v) (2.152) Now we can write the expression for
IQW*^:
'(A: + 2 M ;)|Uf)\ l*Qw*M
=
* d2
*(2)
* a2
*(2)
(2.153) /
therefore, solving the second equation in (2.151) is equivalent to solving the following system of equations:
(A; + 2f0){$*wf) + £ < % ) ) + i(AJ + M + W^b)
/AodC2
=0
.(2)
V" + %Wt(
(2.154) From (2.154) it follows that w*W is a quadratic function in C, and it can be written as
w*W = _F»C 2 + W*(2)MC + [T(2)(^)>
(2.155)
where W*(2) (ip), U*(2\
Dipole tensors in spectral problems of elasticity
104
Displacement boundary conditions We shall now analyse the displacement conditions at each boundary, retaining the notion of interface 1 and interface 2 given in the previous section. We consider the Taylor approximation for w^1' at R± + eRo given by (2.111). Because there is a term of order 0(e), we shall take into account that term for the second order in the asymptotic expansion for the displacement. We will also incorporate it in the conditions at each boundary. From the conditions (2.81), (2.110) and (2.154) we obtain: Interface 1:
= w^2\0)
w^iRup)
= U*{2\if), ip e
[0,2TT).
Interface 2:
w^(Ri
+eRo,v) +Ro-^r(Ri,
= RlF*&)
+ RoW<2\ip)
+ C/*(2)(V),
At the second interface we also consider the Taylor expansion for at i?i + eRo ft
(2.156) w(2\£)
(2)
™(2)CRi + £#o, f) = u>(2)CRi, v>) + eRo—^iRuip)
+ ...
(2.157)
Thus, we re-write the displacement conditions at the second interface as follows wW(R1,lp)
+ R0-^r(R1,
= R20F*(
=
w<2\R0,ip)
+ W
(2.158)
We want to find the function W*^2\tp). Subtracting the condition (2.158) at the second interface from the condition (2.156) at the first interface we obtain W& (Rl,
V)
- wo<2) {Rl ,
(Rl, y,)
= R20F*{
(2.159)
Imperfect interface.
"Coated" conical
inclusion.
105
Taking into account (2.113), we can then write W<2\v)
= J-[«,< 2 >](i2 1>v 0 + ^-{RiM
- RoF*(
[0,2TT).
(2.160) On the other hand, F*(
F'(
--1±ir[wWm,
-5^35^5 (l^-'T
<2 I61)
'
Finally, from (2.156) it follows that the function U*^2\ip) is equal to the function io°(2)(£) evaluated at ||£|| = R\. Therefore, we have written the functions that are present in the representation of the function w*(2\(,
o{1 J
B°0w°M(R1,cp)+B°1w°W(R1,
6
= J2cjy*(-j){v)
+ b*1w*(1\0,
where *0(1,3>(J2i,¥>) = -(A„ + 2no)£ • ve& = (A0 + 2/i0)fl1e<3>,
(2.162)
Dipole tensors in spectral problems of elasticity
106
*°( 1 ' 4 ) (i?i, ¥ )) = 0, *o(1>5)(#!,) = -V2LI0£
• ve^
= V2/xOJRie(2),
* o ( 1 ' 6 ) ( i ? i , ,
(2.163)
are the contributions from the function 3>° into the stress field for the inclusion, while the functions ^*^\u) are the contributions of 3>* for the stress field for the coating layer. They have the same form as those shown in (2.119), (2.122), but with elastic constants A*,/x*. Interface 2:
Bow^^)(R1,^)+B1wW(R1,
^^(Ruf) j=i
6
= ^ C j * * « ( i / ) + 6 ^ * ( 1 ) ( ^ o , v ) +b*0w*(2\Ro,
(2.164)
where
^•3\RUip)
= -(X + 2A*)€ • " e ( 2 ) = (A + 2/i)i? ie ( 2 ), (2.165)
4
6
2)
¥& >(/2i,¥>) = 0, ¥& >(i2i,¥>) = - V 2 / x | • i/eW = V2/xflie( ,
We now combine the two stress conditions (2.162), (2.163). Within the coating layer, the term Ylj=i cj^* (u) does not depend on £ and is present in both conditions. Thus, (2.162) and (2.164) give
BoWV\Ru
-blw*W(Ro,
- b*0w*W(Ro,ip) = BZw°V>{R1,ip) + B01w°W(R1,
+ J2cj^°{1'j)(Ri,'p)-b*1w<1\0,lp)-b*ow*^(0,ip), 3= 1
tpe
[0,2TT).
(2.166)
Imperfect interface.
"Coated" conical
inclusion.
107
We can write the last equation as follows *W 2 >(fli,¥>) - B°0w°W(Ru
+ Y,cJ*{1J)(Ri,
d_ - Ro^Bow^iRuv)) ' dr'
-
+
Z W
1
^ , ^
blw<1\Ro,Lp)
- 6 X ( 1 ) ( 0 , ^ ) + 6>*(2)(i?o^)-b>*(2)(0,
(2.167)
where
J2*{1'j)(Ri,) - *{n\'j)(Ri,
0= 1
Evaluating b*0w*W(R0,ip) - b;$w>*(2)(0,
bZw<2\Ro,
&[41)](*i,v)
(2.168)
V f [wPiR!,?) ) On the other hand, 6Iw*<1>(flo,¥>)-&!«>*(1)(0,¥0
/ ~K\Wf\{R,M
f -A^{[4 1 >](i2 1 ) V )} \
\
+ Ri
-Ro
V
o
j
-^{[^(Ruf)}
.
0
\
Thus, we can re-write (2.167) as follows Bow^(R1,)
=
d -RofriBowMKRutp)
-(B1«;«(i?l¥p)-B>0«(i?1,V)) + ^Cj*(1^(i?1,¥))
(2.169)
Dipole tensors in spectral problems of elasticity
108
n^htn. ,„\\ /2[w^](R lt
2[w$)]{R1,
+ Ri \
„(i)
/ . , * d L„(l)l
+ R~i
]{Ri,
(2.170)
K&v^KRi,*)
V
o
J
We now re-write the condition for the jump in the displacement. From the condition in (2.164), we shall calculate blw*(2\Ro,(p). The latter can be written as:
( (XI + 2fi)(2RoFZ(
ti(2RoF;{
(2.171)
We re-write the condition (2.164) at the second interface using (2.171), (2.160) and the expression for b{w*(1\R0,(p). This yields
B0w^(R1,lp)
=
B1w^{R1,
((Xt + 2f0){£[w?)}(R1,
+ RoFZ^)} \
»:{±jwr2)}(Ru
6
6
(2.172) 3=1
3=1
Therefore, we have now combined the stress and displacement conditions at the boundaries and obtained a system of two equations for the two unknown functions w^ and w°^: £o«> (2) + A » ( 1 ) = 0, ||£|| > Ri, C°0w°W + c°w°W with the interface conditions (2.170) and (2.172).
= 0, 11(11 < Ru (2.173)
Imperfect interface.
2.2.2.4
"Coated" conical
inclusion.
109
Asymptotic behaviour of w(2> at infinity
The function w^ oo:
admits the following asymptotic representation as ||£|| —•
« ; ( 2 ) ( C ) = T P ) ( € ) + 0(||«||- 1 ) (2.174) 1
= a r « ) + b + Efa) + OflKir ), as ||£|| -» oo, where d, 5 are constant vectors. To evaluate a we use the same procedure as that of Section 2.1. We consider the dot product of (2.173) and the unit vector eW and integrate it over a circle of radius R. We evaluate the integrals by parts and use the conditions (2.170), (2.172). Then we consider the limit as R —> oo. This yields 0= /
eW • (C0wM (£) + £lWW
(£))d£
JDR\g
= f
eW
+ [e^-{Bo0w°M($)
• (B 0 u> (2) (£) + BlU>W(£))dZ
+ Bo1w°W(Z))dl + 2Ii, 1 = 1,2,3,
(2.175)
where Ij = — /
£JU>3
K
'dl + fi /
i^«4
dl, j = 1,2,
Jdg
JDR
-Xo f {viwl{l)
+v2w°2{1))dl,
Jdg
where v\, V2 are the components of the normal v.
(2.176)
110
Dipole tensors in spectral problems of elasticity
We re-write (2.175) as follows: e « • (B 0 ™ (2) (£) + B i i u « ( £ ) - J3°0w°W(Z) - B?u; 0 «(£))dZ
0= / Jdg
+ [ e^.(B0wM(Z) JdDR
+ B1wW{£))dl
+ 2Ii, i = 1,2,3.
(2.177)
The first integral on the right-hand side of (2.177) can be evaluated using the interface condition (2.170). The latter yields d
Ra-{B0w^){RlM^Y,c^i\Rx^)
,(*).
0: Jda
i=l
,(Di
dl
o + [
e®.(B0w<-2\t)
+ B1w(1\t))dl
+ 2Ii, i = l , 2 , 3 .
(2.178)
In order to evaluate the second integral in (2.178), we first consider the asymptotic at infinity of w(2\£) given by (2.174) and of u / 1 ) ^ ) , given in Section 2.1 (see (2.22)). The integrals relating the stress operators BQ,B\ acting on *&' ' , E , respectively, have been evaluated before in Section 2.1 (see (2.51) and (2.54)). Combining these two expressions gives e(i).(g1T(i)+B03)^
x-2 x
/ .dDR
L
e(2) . ( B l T (D + BoS)dl =
^_?rf,
dDR
*
f e^ . (Bj Y(1> + B0S)dl = JdDR V ((„.(JX) c ^ +• QJX) ^ )+ e r f A+M
(6)
-ce,a\
2TT/X(1
- x)(i/ +
Au)
+ 4 2 ) ) + c 3 (af ) + a f ) ) .
(2.179)
Imperfect interface.
"Coated" conical
inclusion.
Ill
We shall next calculate the Ij integrals. First, we consider the expressions for u>3 ', u>3 ': (5) .
in
(6)
Qr sinw
a\
Z7r/i
27T/X
r
cos(p r
(6)
+ce
^lr+v^^cos^
(2 180)
-
Substituting these expressions into the representations for the integrals h,I2 yields ^ c e ^ a f + ^ O u - Z v K l ? ? ] , I2 = c^a^
+ V2(fx-^nRl}.
(2.181)
To evaluate the integral Is, we consider the following identity: (!) ,
•
(1)
cos
(1)
=w).',
which implies v^wp + v2w{2] = ~wrl)
on
%
(2.182)
The minus sign on the right-hand side is due to the fact that v is the inward normal vector on dg. Thus, I3 = X f
w^dl
JdDR
-X
f Jdg
w^dl
w°{1)dl.
+ \0 f
(2.183)
Jdg
From (2.137), it follows that the radial component of the displacement field in" has the form: < ( 1 ) (r ) ¥ >) = - ^ - [ ( x 0 - l ) a 1 r + (x 0 -3)r- 3 (i?e{a 3 }cos2^ -Im{a3}sm2
(2.184)
and therefore, / Jdg
w0r^dl = TLR?(X 0 - 1 ) ^ . Vo
(2.185)
Dipole tensors in spectral problems of elasticity
112
Here, the constant a\ is given by (2.138). Now, we consider wT '(Ri,
±[^(Re{a1}coS2
+
Im{a1}sm2
-^j(Re{f33}
cos2ip + Im{f33} sm2
Thus, (2.187) Jda Idg
M
Therefore, from (2.183), (2.185) and (2.187) we obtain: (2.188)
•^3 = — (x0 - l)7ri?Jai. Mo
We have calculated the Ij integrals. Then we shall evaluate Jg S&'• 'J'dl, keeping in mind that £ • v = — R\:
f I>* ( 1 J 3 {Ri,
^0)2TTRIC5, -V2{H
+ c2)(A - A0) + c3(A + 2/x - A0 - 2/x0)])T.
-
/X 0 )2TT^C 6 ,
(2.189)
We now analyse the remaining integrals in (2.178). We will show that
j
&''
e^-{-Ro-^(B -\-Ro 0wW)(Ruip)
Idg
, • 2[wW]{Ru
2
+
* l KK M * ](/2i,v) ;
{ ti£[*>PKRi,d
J_ r,„(i)i
V
0
/
(2.190) First, we consider the components of w^\ w°^\ given by (2.180), (2.184), (2.186). For the first two components, taking into account the equalities (2.124), (2.125), we deduce that
Imperfect interface.
"Coated" conical
inclusion.
113
If we write this vector in the Cartesian coordinates, differentiate it with respect to r and integrate along 8DR, we shall obtain for the first two components integrals of the type
J0
* \ sin2
J0
* \ sm2ip J
J0
\ simp J
but all these integrals are zero. Therefore, f2* d i \ I eW • -^ \B0wW) (Ri,
(2.191)
For the third component, using (2.180), we have a „
,
„
W _i? oe (3) . (—BowW{R1 ltV ))
(5)
r
„
„
=
d2
(i)
^ - ^ - ( f l ^ )
.
(6)
a^sinyj
a^cos^
Thus,
iao
; ( 3 ) • —(B 0 to ( 1 ) )(i2i, v)«fl = 0.
(2.193)
The remaining terms in (2.190) involve the jump in the displacement across the interface, evaluated at r = R\. All the functions that are present in those two vectors are of the type {sin2y>, cos2<^}. By the same argument as explained in (2.191) (once they have been expressed in Cartesian coordinates), the remaining two terms in (2.190) are equal to zero. We now determine the components of the vector a. From (2.170), taking into account (2.190), (2.193), we have:
*i = I -'dff
eU)
•J2ci*™J)(Ri>^dl j=l
+ I
eU)
• (B°H + 8 i T ( 1 ) ) ^
JdDR
+2lj, j = 1,2,3.
(2.194)
114
Dipole tensors in spectral problems of elasticity
Thus, 4//
(6)
.
4/x
(6)
+ a<2>) + c3(a^
as = - y ^ - I c i C a W + <Jj») + c2(a?
+ a<8>)]
A + /i
4A ° -Tr/ijai - 2nR21[(\ - A 0 )( Cl + c2 + c3) + 2(/i - /i 0 )c 3 ]. A + /x0
(2.195)
,(J) We obtain the relevant constants p£' if we write the components of d in the form
6
ak = Y,CjPk)> * = 1,2,3; 3= 1
then the non-zero coefficients are ^
A + 3/z
4
2
A + 3/x
5
3
A + 2/T *
2
;
^ (A + n - A0 - HO)(H0(2HRQ + (A; + 2/x;)i?i) - 2(A + riwoRp , *• 4/x/i0fio + (A* + 2fi*0)Ri (x0fi + fa - (/i - n0))
-2nRl(X - A0) + - i ^ - T r i i ? , j - 1,2,
^3) = - A T ^ 3 ) + a 2 3 ) ) _ 2nRi{x+" - A ° - Mo)+ ^rk 7 ^ „ r (A - Ao)(/xo(2/xfi0 + (A; + 2/x;)fl!) - 2Xfiti0Ro X 4WoRo + (K + 2fJ,*0)R1(xofi + Ho - {ft - Ho)) >' Therefore, we have evaluated the components of the vector d.
Imperfect interface.
2.2.3
Stress
singularity
"Coated" conical
exponent
inclusion.
115
A2
The derivation of the eigenvalue problem for the stress singularity exponent A2 was done in Section 2.1 for the case of a conical cavity. In Section 2.1.3 we derived the corresponding boundary value problem for the function &2>((p) on the unit sphere S. The solvability condition for this problem is equivalent to the matrix eigenvalue problem (2.77), where the matrix M depends upon the coefficients of', pf . By solving the eigenvalue problem for M we obtain the stress singularity exponent A2 and the corresponding stress mode. We evaluated the coefficients or?' ,p£' in the previous section, and therefore, we can now write the corresponding eigenvalues of the matrix M. In this section we shall obtain the eigenvalues A2 and the associated eigenvectors. The matrix M has a block-diagonal structure, and is given by (2.71). Due to this structure there are three non-zero eigenvalues which are clearly identifiable, A2 , A2 , and A2 ', which are equal to M44, M55, M&Q, respectively. Also,
Therefore, taking into account the representation (2.71) of the matrix M, we have M12 = M 2 i, M i 3 = M 2 3 , M31 = M 3 2 , M11 = M 2 2 ,
(2.197)
and therefore, in order to obtain the remaining three eigenvalues, we need to solve the eigenvalue problem for the following 3 x 3 matrix: / M u Mia M13 \ M12 M n M13 • \ M 3 i M31 M 3 3 /
(2.198)
The eigenvalues of this matrix are 1,., ,, ., , 1 A2 ' = 2 ( M H + M 12 + M 33) + 5 V W l l + M12 - M 3 3 ) 2 + 8M13M31, A22) = \(Mn
+ M12 + M33) - \y/{Mu
+ Mi 2 - M 3 3 ) 2 + 8M13M31,
A^3) = M n - Mi 2 . Note that A23) = A 2 4) .
(2.199)
116
Dipole tensors in spectral problems of elasticity
We have seen part of the structure of the matrix M, and now we shall analyse the behaviour of its eigenvalues. We assume that the matrix, the inclusion and the coating layer materials have the same Poisson's ratio, that is, v = v0 = v*.
(2.200)
This relation allows us to write A =
2^_ 1 — 2i/
2 ^ . K 1 — 2f
=
W 1 — 2v
(2.201)
Therefore, all the eigenvalues will depend on the shear moduli /i, fi0, and AC We now consider (2.71) which defines the elements of M, the expressions for the eigenvalues (2.199), and relations (2.201) to calculate the eigenvalues. The eigenvalues are given by
(22o2>
w-WsW^^i *=m-m
+li±
-
^^f}
23
<->
where
/ i = (1 - v)^Ri(l
- /V/x)[—(1 + ") + ( ! " 2^)]
-^L(i-2v)R0[^(l
+
v)-l},
h = - ( 1 - 2u)R0 + (1 - v)^.Rx{^ + 1 - 2v), M M M
h = (1 - i / A ( l /x
MO/M)2#I
+ —(1 /x
MO/M)#O.
(2.204)
Imperfect interface.
"Coated" conical
inclusion.
117
From (2.204) we can see that (2.202), (2.203) depend on the ratios ^f and —, and (after re-arranging terms) ^ . The next eigenvalue is of multiplicity 2, that is,
w-^-Ww-
(2205)
Note that A2 > A2 ^° n o * depend on RQ or ^ . We can verify that when \i0 = 0 we recover the expression for a cavity. The following non-zero eigenvalue also has multiplicity 2: L(5) = A (6) = 2 2
5-2^-Si/2 fR0 + ^Ri(l-Ho/ri 4(1 - t/)(3 - 41/) ^ + ^ ( 1 + ^ ) '
which depends on ^ and ^-. We can easily verify that when Ro = 0 (no coating), this eigenvalue is the one corresponding to a perfectly bonded conical inclusion. 2.2.4
Some examples.
Discussion
and
conclusions.
In this section we analyse the eigenvalues for the imperfect interface which were evaluated in the previous section. The eigenvalues were evaluated under the assumption that all three materials have the same Poisson ratio. We study the effect of the coating on the eigenvalues, and therefore, we will analyse how A2 depend upon the ratio /x*//x keeping fJ,0/fi fixed. Due to (2.200), ti/n = Kl\
Mo/M = V A ,
(2.207)
which means we can write the eigenvalues as functions of /x,/i0,/x* and v, only. We shall assume that 0 < ^ < 10, 0 < — < 10. fj,
(2.208)
a
We consider the eigenvalues given by (2.202)-(2.206). The eigenvalues Aj , A2 , Aj , and Aj depend on the ratio /io/V, while the eigenvalues A 2 3) , A24) do not. It is important to note that for the "limit cases", we shall recover the eigenvalues we already know (for those cases). For instance, in the case when there is no coating (RQ = 0), we should obtain the same
Dipole tensors in spectral problems of elasticity
118
Fig. 2.8 Plots of the eigenvalue Aj , for the case when Ho/fi — 0.2,0.4,0.6,0.8, and Po/f1 = °- 5 i li l-5> 2> 2.5. The curve for the ideal contact (Bo = 0) is shown by a dotted line.
eigenvalue as those for a perfectly bonded circular inclusion. If we take Mo/M = 0>.Ro/-Ri = 0, then we will recover the eigenvalues for a circular cavity. We shall now consider the following cases: (1) (2) (3) (4)
Both coating and inclusion "softer" than matrix. Coating "softer" than matrix, and matrix "softer" than inclusion. Matrix "softer" than coating, and inclusion "softer" than matrix. Matrix "softer" than both coating and inclusion.
We proceed as follows: we will consider the set of values of \i0j{i = 0.2,0.4,0.6,0.8, when the inclusion is softer than the matrix, and the
Imperfect interface.
"Coated" conical
inclusion.
119
H>=2,R/R=1
0.1 ideal contact
0.05 M'D/(i=0.5
0 1»',/H=TT-
-0.05 t»' 0 'f"2
-0.1
/
M*a/(«2.5
0.1
0.2
0.3
> /
0.4
0.5
v
^=e,
^/^6,R 0 /R,=1
0.2
ideal contact
0.15 0.1 -
VRr1
0.2 ideal contact
/ --..../
(1*^=0.5
0.15 0.1 ~
0.05
<
0.05 0
M'n/n=1.5
-0.05
"
"
t»->=2
0.1
0.2
0.3
0.4
0.5
•
•
•
•
•
.
/
.
"
•
•
.
.
/
S^ ^~~
/ -0.1
•
MVM-0.5
-0.05
-0.1
"
~ 0.1
iFli^2
— 0.2
/
-P^-as 0.3
0.4
0.5
Fig. 2.9 Plots of the eigenvalue Aj , for the case when /io/A4 = 2 , 4 , 6 , 8 , and MS/M : 0.5,1,1.5, 2,2.5. The curve for the ideal contact (.Ro = 0) is shown by a dotted line.
other set of values of //<>/// = 2,4,6,8, when the matrix is softer than the inclusion. For every value of /x0//i we will plot the eigenvalue, taking /^//u = 0.5,1.0,1.5,2.0,2.5. In Figures 2.8 and 2.9 we plot Aj as a function of v for different values of n*0/H- We first consider the case when the inclusion is softer than the matrix and also plot the curve for a perfectly bonded inclusion for the same value of /X0/M- The eigenvalue A2 is positive for all considered cases. The curve for fi^/y. = 0.5 is always the closest to the dotted curve for the case of an ideal contact. In Fig. 2.9 we consider the case when the matrix is "softer" than the inclusion. The eigenvalue Ag is either positive or negative, depending on the ratio /i*//i 0 . The ideal contact curve for each case is positive and greater than Aj for the imperfect interface.
Dipole tensors in spectral problems of elasticity
120
u /U=0.4,R /R =1
M.o/|i=0.2,Ro/R,=1 0|»V„=0.5
-0.05 .
*
U'Jfl
ideal contact
M'o/H=1.5
ideal contact
-0.1
\
M'u/H=2.5
-0.15 > -0.2 -0.25 •
.if
0.1
*Xj 0.2
\
• / / '
0.3
0.4
-0.15 n'o/M=2.5
0.5
-0.2
*Ssy/I
/
"**"*>--^. 1,s
0.1
0.2
0.3
0.4
0.5
0.4
0.5
v H /u=0.6,Ro /R,=1 ^ o *^ 1
0.1
0.2
0.3
u Ai=0.8,R /R =1 r •(J
0.4
0.5
0.1
0
0.2
1
0.3
Fig. 2.10 Plots of the eigenvalue Aj , for the case when MO/M = 0.2,0.4,0.6,0.8, and MO/M = 0.5,1,1.5, 2, 2.5. The curve for the ideal contact (Ro = 0) is shown by a dotted line.
In Figures 2.10 and 2.11 we plot A^ \ First, we consider the case when /x 0 //i < 1, that is, the inclusion softer than the matrix. From Fig. 2.10 we see that the eigenvalue Aj is negative for all cases considered, and the strongest singularity is found for /i 0 //x = 0.2. The ideal contact curve in every case lies above the curves corresponding to the coated inclusion. In Fig. 2.11 we plot A^2) for the case of the matrix being "softer" than the inclusion. The eigenvalue A)> ' is negative for all the cases considered. The curve for the ideal contact is the lowest of all curves in each case and the strongest singularity is obtained for /U 0 /M = 8. The eigenvalue Aj does not depend on Ro or on \i*0jp. The plot for the eigenvalue A2 is shown in Fig. 2.12. We also show the graph for \i0 — 0, which corresponds to the case of a circular cavity. In Figures 2.13 and 2.14
Imperfect interface.
\l /H=2,R /R,=1
"Coated" conical inclusion.
121
u. Al=4,R /R =1
H /n=6,R /R,=1
n /u=8,R /R =1
n
^0 ^
0 ^
O
1
0
1
Fig. 2.11 Plots of the eigenvalue A^ , for the case when /lo/A4 = 2,4, 6,8, and li*0lp. = 0.5,1,1.5, 2, 2.5. The curve for the ideal contact (Ro = 0) is shown by a dotted line.
we plot the eigenvalue A2 . In the case when the inclusion is "softer" than the matrix (Fig. 2.13), the eigenvalue A2 is negative. The ideal contact curve lies above the curves for imperfectly bonded inclusions. In Fig. 2.14 we plot A25) for the case when the matrix is softer than the inclusion. In this section we formulated the problem for a "coated" conical inclusion, with a circular cross-section, embedded in an elastic matrix. We assumed that the elastic materials of the coating, inclusion, and surrounding matrix are homogeneous and isotropic, and have different elastic properties. The coating was taken to be thin and soft. We assumed perfect bonding on both interfaces, dk£ and dKe. In the analysis of the eigenvalues, we assumed that all three materials (i.e., coating, inclusion and surrounding matrix) had the same Poisson ratio,
Dipole tensors in spectral problems of elasticity
122
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Fig. 2.12 Plot of the eigenvalue Aj ' = Aj , for the case when fJ,0/^ = 0.5,1,1.5,2, 2.5,3,3.5,4,4.5. The case for the circular cavity (notch) corresponds to the case when tx0/lj, = 0.
v, and we derived the representations for the eigenvalues in terms of v and the elastic constants //, /x0, and /i*. We plotted the eigenvalues for different values of the ratios (J.0/n and \i*0jii. For comparison, we also plotted the eigenvalues for a perfectly bonded inclusion. We analysed the effect of the coating on the stress singularity for different values of the ratio /i 0 //i. We have found that if the inclusion is "softer" than the matrix (/X0/M < 1); the soft coating "increases" the singularity (see Figures 2.10, 2.13 for A^2), A^5)). If the matrix is "softer" than the inclusion, then the coating "decreases" the singularity associated with A2 . In the case of Aj , which does not depend on the coating parameters, we found that for /i Q //i < 1 there will be a singularity, still less than the one for a cavity, while for Ho/fJ. > 1 there will be no singularity in the stress field.
Imperfect interface. "Coated" conical inclusion. H0/M>.2, FyR,=1
|i o /n=0.4
123 R
c/R,=1
•s^/ • „ / H U \ />:>*•<
/•.*•'5
. A*-! /fr>.2.5
v ( 5 ;) i (6) Fig. 2.13 Plots of the eigenvalue A^ = A£', for the case when /*0//i = 0.2,0.4, 0.6, 0.8, and MO/M — 0.5,1,1.5,2,2.5. The curve for the ideal contact (Ro = 0) is shown by a dotted line.
Finally, in t h e following table we summarise t h e results. We indicate t h e effect of t h e soft t h i n coating (compared t o t h e case of a perfectly bonded inclusion) on t h e stress singularity exponent for each eigenvalue considered in this section. Mo/M < 1
Mo/M > 1
A«
no singularity
increased
A<2>
increased
decreased
Af = A?
no effect
no effect
A ? > = A<6>
increased
increased
Dipole tensors in spectral problems of elasticity
124
0.6 ideal contact
0.2
•
1
W
^ = 4 ,
t\/H=2, R
ideaf contact
0.4
\
(i' o /(i=0.5
0
;...V---
/lixO.S
JI.*J»I
v7 ^ r -0.2
t>7n="
%«
~———-________^^
[iVtel.5
° M-0/M=2
-0.2
-0.4
0.1
0.2
0.3
0.4
0.5
Tji^S——..
-0.4
0.1
0.2
0.3
0.4
0.5
v
v Ho/M5, FVH1=1 1 0.8
ideal contact
.
.
.
.
•
\
0.6 •
.
.
.
>
•
•
-
"
"
'
ji" /ji=0.5
P?^ M/M.5
0 I»VM=2
-0.2 0.2
0.3
0.4
0.5
,'jr*r-
—.
-0.4
0.1
0.2
0.3
0.4
0.5
v I (6)-1, for the case when ^ O / M = 2,4,6,8, and Fig. 2.14 Plots of the eigenvalue Aj = A^" p.%/H = 0.5,1,1.5,2,2.5. The curve for the ideal contact (Ro = 0) is shown by dotted line.
Chapter 3
Multipole methods and homogenisation in two-dimensions
3.1
The method of Rayleigh for static problems
We consider an array of infinitely long circular cylinders, periodically spaced in two-dimensions, as shown in Figure 3.1. Each cylinder has radius a and possesses a transport property (such as, say, dielectric constant) which is different from the surrounding matrix. The field, whether it is electrostatic or elastic, can be represented by a scalar potential V. If the potential between the cylinders is labeled V, while the potential within the cylinders themselves is labeled V1, then in both regions the potential satisfies Laplace's equation: AV = 0
(3.1)
AV* = 0
(3.2)
in fi, and
in C. Here A is the two-dimensional Laplacian operator. On the edge of each cylinder we prescribe the boundary conditions V = Vi
(3.3) i
dV
dV
In this case a can denote the conductivity of the cylinders, in the case of electrostatics. The surrounding matrix material has been normalised so that its conductivity is equal to 1. 125
126
Multipole methods and homogenisation
Fig. 3.1
in
two-dimensions
The array of cylinders, in direct space.
Finally, we impose a potential gradient on the array of cylinders, in the direction of the x-axis. This can be expressed by boundary conditions which apply on the edge of the unit cell: V^,y)=V(~,y)
+ SVx,
(3.5)
V(x,^) = V(x,~) + SVy.
(3.6)
The derivatives of V are periodic in both the x and y directions.
3.1.1
The multipole
expansion
and effective
properties
If we take the centre of one of the cylinders as the origin of polar coordinates, then the potential external to the cylinders can be expanded in the series oo
oo
A
e
V = Yl eR (*') + ]C BeRe ( z ~0
(3-7)
The method of Rayleigh for static problems
127
where z = x+iy = re%e. We now split the applied field into two components, one which is odd in the a;-direction and even in y, and another which is odd in y and even in x. Bearing in mind that the inhomogeneous problem cannot exhibit symmetries which are not possessed by the driving potential, we can expand the potential function V as cosn# n odd oo
+ nEodd K(D" + B »(7) n ] s i n n *-
<3-8)
Here and henceforth we write the vector r in polar coordinates as the ordered pair r = (r, 9). Within the central cylinder the potential must be non-singular, so we can expand it as
oo
=
C
Q " ICn cosnfl + C°n sinnfl] .
o + E
(3.9)
n odd
where once again we have split the field into its even and odd components. Substituting (3.8) and (3.9) into the boundary conditions (3.3) and (3.4), an algebraic manipulation allows us to discard the coefficients C|'°, hence
where, once again, a is the radius of the cylinders. For brevity, it is convenient to frame this condition in terms of the 'conductivity contrast' v = ( l + o - ) / ( l -&), thus AT = vB?
.
(3.11)
In a paper written in 1892, Lord Rayleigh notes that the values of the coefficients Ai'°, Bl'°, A^'°, B^'", • • • are necessarily the same for each cylinder. The coefficient AQ however changes as one moves 'upstream' with the potential gradient. Without loss of generality, we set A% = 0 within the central cell.
Multipole methods and homogenisation
128
in
two-dimensions
Fig. 3.2 The central unit cell, showing the contour within which Green's theorem is to be applied.
We now apply Green's theorem to the potential V and a function U, which we define to be U = x = rcos#.
(3.12)
Choosing the contour to encompass the region between the central cylinder and the outer boundary of the unit cell (see Figure 3.2) we obtain / {VAU - UAV) dA=
[
(v
9U__dV_x U\ds dn dn
(3.13)
Both U and V satisfy Laplace's equation, hence the left-hand side of (3.13) vanishes and we are left with the line integrals around the outer boundary and around the central cylinder. Evaluating the integral around the circular path dC first, we obtain by direct substitution idC \\ JdC
dn
dn fl-K
C S6
Jo
+
(-)"
°
S
[(-)n (AnCOSne + An Sin n0)
n odd
(B„ cosn6 + B°nsinn6)
zos6 J2 n\(-Y
(A^ cos nO +
A^smne)
n odd
(~Y
{Ben cosn6 + B°n sinnO) ) add = 2iraBt
(3.14)
The method of Rayleigh for static problems
129
The integral around the outer boundary is meanwhile
a
on
on
rd/2
dx
J-d/2
~ y\x=d/2
c=-d/2
~~ V\x=-d/2j
dy (3.15)
The remaining integral on the right-hand side of this expression is equal to the total current, or flux, across the side of the unit cell in response to the applied potential. If we define d/2
gV
•d/2
dx
(sx
dy,
(3.16)
c
(3.17)
=-d/2
t h e n we find
a
v
r
ir \ on
u
ir)ds = on J
**d~6V*d
where SVX is the drop in potential from left to right. When we add in the contribution from around the cylindrical boundary, we obtain the expression Cxxd - 8Vxd + 2a-nB\ = 0 .
(3.18)
The equation (3.18) can now be re-arranged to give an expression for the effective conductivity of the central rectangle in the x direction: a
xx
-
Wx
=
l-
2TTOBI
d6Vx
(3.19)
Using exactly the same reasoning, the effective conductivity in the y direction is VV
d6Vy
(3.20)
The problem is now reduced to calculating the relationship between the dipole coefficients B\'° and the external gradient (6Vx,6Vy).
130
3.1.2
Multipole methods and homogenisation
Solution
to the static
in
two-dimensions
problem
For convenience in notation we introduce the direct lattice {RP}, which is the set of vectors in two dimensions, each of which points to the centre of the pth cylinder. For brevity we will say that p = 0 labels the central cylinder. Formally, we define Rp = (nd, md) , n, m e Z and p = (n, m) .
(3.21)
Much of the analysis which follows uses sums over functions in Fourier space; the vectors from the origin to the hth node of the reciprocal array are defined to be
(
27T
n—,m—
27r\
J , n, m € Z and h = (n,m) .
(3.22)
The subscript h in the vector {K-h} is a label similar to p in direct space; the vector K/j points to the centre of the hth cell in the reciprocal lattice, and, as with the direct lattice, h = 0 corresponds to the central lattice point in reciprocal space. The unit cell in reciprocal space is also a square, with side length 2n/d. We introduce the periodic Green's function for this problem as the solution of the equation AG(r;r')=£*(r-r'-.Rp)-±
(3.23)
p
where r and r' are vectors within the region fi and where A is the area of the unit cell. The constant term 1/A on the right-hand side of (3.23) removes the net 'charge' from the unit cell. This is necessary to avoid conditionally convergent terms in the expansion of the Green's function. We can express G as a sum over the reciprocal lattice
G{£) = Y,9{Kh)eiK*Z h
=
J29(Kh)eiK^+g(0)
(3.24)
h^iO
where £ = r — r'. The Poisson Summation Formula (see, for example, Jones 1966) relates sums over the direct lattice to sums over the reciprocal lattice via the equation
£*«--Rp) = ^ E e J i r f c ' * ' p
h
(3-25)
The method of Rayleigh for static
problems
131
and so (3.23) becomes
A
h^o
A
h
7 $>**-*.
(3.26)
Hence we have 9{Kh)
=
~l^
ioih
1 *—r
e^h'^
( 3 " 27 )
*°
and the Green's function is
MO
h
with g(0) being some constant which has no effect in subsequent field computations, and so can be ignored. Glasser (1974) obtains the exact closed form for G
xS-sr--(5i) ll+c - (,r),+ s ta2 1 ^ e
i K h
^
f Z\2„
_
_„_ / 0 ..M
, 1
h^O
^ > »
6\ (K£ sin 7 + in cos 7,3) [(9i(0,g)]V3
(3.29)
where q = e _7r , 7 = arg£ and 6i(z,q) is the elliptic Theta function of the first kind. The practical use of this representation is that the Theta functions are well tabulated and this ameliorates numerical construction of the Green's function. In addition, one can use (3.29) to obtain analytic expressions for a wide class of sums over the reciprocal lattice, a technique which we will employ later on. For problems involving arrays of circular cylinders, it is convenient to express the Green's function in terms of Bessel functions: 1
1
h^O
h
°°
t=-oo
=-E^7£ £=-00
h^O
{KhdY
132
Multipole methods and homogenisation
£
in
two-dimensions
iee-^SeA2(0
(3-30)
t=-oo
where we have defined
Fortunately these 'lattice sums' (3.31) have been studied by many authors (most notably by Ewald 1921). At the worst they can be summed directly, and in fact for many instances of the index I the sums can be expressed as a terminating polynomial in £. For a square array, we have by symmetry that Se,e,2{0 — 0 unless I is an integer multiple of 4, and also that 5_^_^2(0 = S(,e,2(0- The non-trivial sums can be evaluated by expanding (3.29) in powers of the argument of £. The ones which directly concern us are:
So,o^) = - ^ 4 1
|,+7o
+
1
p4inep
P¥=O
2TT
(3.32)
F
2n
where 70 « —0.318895593319827 is a constant and the quantities Se are the static lattice sums inBp
^n=EV P^0
K
(3 34)
"
P
originally introduced by Rayleigh. Substituting these results into (3.30) and neglecting such constants as UJQ we obtain the final expression for the static Green's function:
<™-**(h)-(h)'-•*£%("<««>)•
^
which is unique to within the addition of a constant. We show in Figure 3.3 a surface plot of the function G obtained using equation (3.29); the series in (3.35) has a radius of convergence equal to d.
The method of Rayleigh for static
Fig. 3.3 origin.
problems
133
Graph showing the static Green's function with the singularity placed at the
Applying Green's theorem within the unit cell, in the region fl shown in Figure 3.2, we have / / (VA'G - GA'V) dA' = f J Jo. Jrr u s e
™V-G^ dn'
dn'
ds'
(3.36)
where T and dC are the contours shown in Figure 3.5. We already know from our definition of the Green's function (3.23) that If
[VA'G - GA'V] dA' = V(r) - \ \ \
VdA' .
(3.37)
The second term on the right hand side can be set to zero without any loss of generality and so we obtain
V(r) = f TUdC
™V-G™ ds' . dn' dn'
(3.38)
Periodicity of the functions G and dV/dn' guarantees that
L
8V
(3.39)
and it is a straightforward matter to show that
L
dG
*rA
i
nr
r c o s
#
, rT/
rsinfl
(3.40)
Multipole methods and homogenisation
134
in
two-dimensions
where 6VX and 5Vy are the differences in the potential across the unit cell in the x and y directions respectively. Thus we have the functional form of the potential V(r)=5Vx.
—
+5Vy.
—
jdc
+
dr'
dr'
ds'.
(3.41)
We expand both the potential and its radial derivative as Fourier series on the boundary of the circular inclusion: .imO
(3.42)
= Yl P™eim$
(3.43)
V(r)\
dv dr
m = — oo oo
(r) r—a
m = — oo
The Fourier components am and f3m are related to the unknown coefficients in the full expansion (3.7) by the equations " 2 m - 1 = l^A\m-\ ft
-(2m-l)
ftm-1
P-(2m-\)
=
^ [ - A | m - 1 + B2m-l]
~ ^[A2m-1
2 2m-1 B 2m-li = [A-2m-1 2a 2m - 1 + 2ia K*2m-1
—
^
2a
B
2m-l]
(3.44)
+B2m-l]
(3.45)
+ #2m-l] + 7^[^2m-l +
[^2m-l
-°2m-l]
(3.46)
^2m-l] >
(3.47)
B
~
2m-l]
2m - 1 2ia [•^2m-l
—
with all the even coefficients being zero due to the symmetry of the array. We now expand the Green's function (3.35), choosing that r' < r: G(r;r')
= -lnr--Y,-
e
(j)
coe[/(« - V)]
- — (r2 + r'2 - 2rr' cos{9 - tp))
U=0m=0
'
\
(3.48)
/
)
The method of Rayleigh for static problems
135
where we have defined lattice sums ae to be (Se \0
=
Ol
if I = 4m , m € Z o1 otherwise
(3.49)
for ease of notation. The radial derivative of the Green's function can be written Bd
r'
1 .21 /r'\e
r
1
Re e e+m w_i m i x me i (-iy
e+m
(3.50)
l i=0 m=0 '
We are now in a position to evaluate the integrals around the boundary of the inclusion. The first of these appearing in (3.41) is
Jdc
Br
'
-l^ +
Jo 1
/r>\e
°°
v
e=i
^cosie-tp)
i
'
(-l)V<+nrnr"-1ein*e"v U=0n=0
v
E a«
0imifi
ad(p
z=0 (
OO
OO
= ^ EEn +^ e V( Tl^+ e \ a
n+er
The second integral in (3.41) is r2ir
( r 2 + r' 2 - 2 r r ' cos(6> - <£>))
_n„tAnO,
a e
ct-e
(3.51)
136
Multipole
methods
IL^=0 c—t\ „n— = n0
and homogenisation
'
\
E A-
in
two-dimensions
/
.tmip
ad
m = —oo
oo
2
+ii:e
n+ n +1 infl ra^e"'Q_\ i E E n^^ ^ ^ +e n
ma
.
(3.52)
Ln=01=0
Interchanging the coefficients am and 3m with the more familiar Am° and B^° by means of the equations (3.47), we obtain the functional form of the potential T
T
7T&
V(r) = SVX - cos 0 + SVy - sin 0 - — [B\r cos 0 + B°r sin 0] 271—1
OO
+ E
[ B a»-i cos(2n - 1)0 + B°n_x sin(2n - 1)0]
(;)
71=1 OO
f ^
OO
21-1
/2n + 2 l - 2 \ x
n=l^=1
, „ + 2 € - 2 f/ \n 2n—1
'
x [ 5 ^ _ x sin(2n - 1)0 - B\t_x cos(2n - 1)0] .
(3.53)
On comparison with the expansion for the potential (3.7) we obtain two systems of infinite linear equations, for n — 1,2,... lln-X + *n,l^a 2 Bi e +
^ ^
(2n + 2 l - 3 ) ! , a»,™ n2n+«-2Re Bs (2n-2)!(2^-l)!(J2n+2'-2a ""1 = ^ ^ (3.54) 7T
^2n-l + *n,l^a25f y " ^
(2n + 2 l - 3 ) ! fljB+M.2™ (2n-2)!(2£-l)!CT2"+2^2a
, i?2
'~1 "
a% „ ^ d ^ (3.55)
The method of Rayleigh for static
problems
137
These identities determine the solution of the problem uniquely, and can be given a physical interpretation: the non-singular part of the potential at the origin must be equal to the net effect of singular sources located elsewhere in the array. If one absorbs the dipole term into the sum by setting <72 = IT/A in the first instance and 02 = —n/A in the second, then one recovers precisely the identities due to Lord Rayleigh (1892), the first being associated with an applied field along the £-axis and the second with an applied field oriented along the y-axis. If we concentrate firstly on the field applied along the a;-axis, then the relations (3.54) constitute an infinite linear algebraic system which links the multipole coefficients A\ and B^. If we ignore all coefficients higher than dipoles, then we obtain a first-order approximation to the system: 2
A\ + ^-B{
« -d5Vx
(3.56)
If we substitute the equation for the boundary condition (3.11), then we obtain {v + f)B\
= -5Vx + 0{f) (3.57) a where / = 7ca2/A is the filling fraction of the cylinders and v is the dielectric contrast. This is enough to determine a first order approximation for the effective conductivity: in conjunction with equation (3.19) we have efx = l-~r}+0{f) =
(3.58) (3 59)
VT~f+°^
-
This is the same result deduced by Lorentz (1952) and by Lorenz (1869). We note that the same formula has been discovered independently by researchers such as Clausius, Mossotti and J.C.Maxwell-Garnett. The treatment given by Rayleigh also allows the possibility of deriving correction terms for inclusions which have a higher filling fraction. This would involve including higher order terms in the system (3.54). If we include the quadrupole terms then we obtain the two equations Al + fBt + a^Bt A% + 3a4a4Bt
« °j5Vx « 0.
(3.60)
138
Multipole methods and homogenisation
in
two-dimensions
Using the boundary condition (3.11) to eliminate A\ and A%, leads to the approximation Bi
*/ + / - ^ ( < r 4 a 4 ) 2
-6VX + O (f) d
.
(3.61)
The next approximation to the effective conductivity is then
The lattice sum can be evaluated either directly or by expansion of the theta function in (3.29). The result, given to 13 decimal places, is 04 = 3.1512120021539. This gives a correction term which is directly dependent on the filling fraction / of the inclusions. It is easy to see that this process can be continued as long as necessary in order to achieve accuracy to an arbitrary order.
3.2
The spectral problem
We now present a generalization of the method given in the previous section to dynamic problems. The physical principle behind the method is the same, relying on an identity between the singular and non-singular parts of an expansion of a potential function: the non-singular part of the potential around the central cylinder must be equal to the superposed effect of the singular sources arising from all the other cylinders in the array. As in the static case, we formulate the problem in terms of an integral equation involving a Green's function which has sources at all the lattice points in the array. We then expand both the potential and the Green's function in terms of basis functions which are appropriate to the geometry of the problem. Since the cylinders which we consider are circular, we expand the potential in terms of Bessel functions rather than in a Laurent series. We then project these expansions back onto the surface of the cylinder, applying the boundary conditions in order to solve the integral equation. The derivation given here follows that given by McPhedran et al. (1997), having been developed in a series of papers (McPhedran & Dawes 1992, Nicorovici & McPhedran 1994, Nicorovici et al. 1994, 1995a, 1995b, 1995c, McPhedran & Movchan 1994, Movchan et al. 1997). Most of the essential points have been compiled from these references; a few of the intermediary
The spectral problem
139
steps and comments which are necessary to make the procedure clearer to the reader who may not be familiar with the subject have also been added. 3.2.1
Formulation
and Bloch
waves
We consider the case of waves moving in an infinite square array of cylinders in two dimensions, the periodicity of which is of length d. Each cylinder has radius a and possesses a transport property (such as, say, refractive index) which is different from the surrounding matrix. If we assume that the wave is time-harmonic, then it may be described at each point by a scalar potential u, which satisfies the Helmholtz equation
(A + n 2 fc 2 )u i (r) = 0,
(3.63)
within the central cylinder, and (A + A:2) u(r) = 0,
(3.64)
in the region between the cylinders. Here n is the relevant transport property for the wave, such as the refractive index in the context of electromagnetism. For the case of anti-plane shear elastic waves, n = \fn/p where /i is the shear Lame coefficient and p is the density of the material. In all cases we assume that we have normalised the transport properties relative to the background material. We now assume that on the boundary of each cylinder the following hold: u = u\ du 1 dul dr n 2 dr
(3.65) . .
This is consistent with the boundary conditions which arise from continuity of field components across the cylinder interface in the case of electromagnetism, and with the continuity of traction in the case of elasticity. Note that when n —> oo the problem becomes singularly perturbed; this is an issue which will be discussed in Section 3.3. For the moment we can assume that n is finite.
140
Multipole methods and homogenisation
in
two-dimensions
As in the static case, we will find it convenient to work with both direct and Fourier lattices. The array of cylinders is again described by a set of lattice vectors { R p } , with p = 0 labeling the central cylinder. The unit cell is once again a square of side length d centred on the origin. In order to solve the problem completely we must specify boundary conditions to be satisfied on the edge of the unit cell. In order to find these, we note that because the potential u is a field component in a periodic medium then it must satisfy a periodicity condition, known as the Bloch or Floquet condition. This is written u(r + R P ) = u(r) e ik °- R * ,
(3.67)
where ko is the Bloch wavevector, which lies in the x-y plane and characterises the wave which is propagating through the material. Its origin in the field of Solid State Physics means that ko is often referred to as the crystal momentum. Functions which satisfy this kind of periodicity condition are often known in the literature as 'quasi-periodic' functions. Because the Bloch condition (3.67) applies to the value of the potential function over the entirety of the lattice, it also applies to its Cartesian derivatives. For this reason, it supplies both of the boundary conditions required on the edge of the unit cell. The relationship between the bloch vector fco and the frequency A; of a propagating wave is known as a dispersion relation. The slope at any point on the dispersion curve defines the group velocity of the wave. Hence in the frequency domain where the dispersion relation is 'flat', the group velocity is zero and no energy is transmitted by the wave as it moves through the lattice. One can also observe the phenomenon of band-gaps in the material. These are domains of frequency for which no propagating modes are possible. Any wave of this frequency would be heavily damped immediately upon entering the material. In the limit when the wavelength in the material is long and the frequency is low (i.e. when both fc and fco are small parameters) then the dispersion relation can be used to define an effective phase refractive index for the material. By homogenising the material in this way any effects arising from the boundary of the material can be ignored. If the change in phase of a propagating mode across the unit cell is denoted A $ = fco^, then the effective phase refractive index neff is defined by the equation A $ = hod — neffkd
(3.68)
The spectral problem
141
and so neff
—
.. dk~ hm -77fc,fc0-+o dk
= (3.69) a where a is the slope of the lowest frequency mode in the dispersion diagram as the frequency of the wave tends towards zero. 3.2.2
The dynamic
multipole
method
We begin by expanding the potential function u in terms of a set of basis functions (or multipoles) which fit the geometry of the problem. In cylindrical polar coordinates the Helmholtz equation (3.63) separates in terms of Bessel functions, and so we can expand u within the unit cell in terms of its singular and non-singular parts: oo
u(r,6)=
^
[aeJe{kr) + beYe(kr)}exp(i£9),
(3.70)
f=-oo
between the cylinders, and oo
^(^0)=
^2
ceJe(nkr)exp(ie6),
(3.71)
^=-oo
within the central cylinder, where only non-singular terms can occur. By applying the boundary condition (3.65) we can eliminate the coefficients ce from the expression for the potential:
=
aiJt(ka) + b(Jt(ka) J((nka)
From the remaining boundary condition (3.66) we obtain the relationship
ae = -Mebe
,
where M
=
nJe(nka)Y;(ka) nJe(nka)J'e(ka)
J^nka)Ye(ka) — J'e(nka)Je(ka)
(3.73)
142
Multipole methods and homogenisation
in
two-dimensions
The quantities Mi will, in general, be referred to as the 'boundary condition terms' or the 'boundary coefficients', for obvious reasons. From (3.74) it can be seen directly that Me. = M-e. We would now like to incorporate the effect of the other inclusions in the array. To do this, it is best to make use of a Green's function which obeys the same type of periodicity condition as the function u itself, i.e. the Green's function must be quasiperiodic. This Green's function will satisfy the differential equation (A +fc2)g(r- r') = £ ) 6{r - r' -
flp)eifc°-
(3.75)
p
where r and r' are vectors in the region £1 shown in Figure 3.5 and the operator A applies to the unprimed variables. The Green's function has been defined so as to automatically satisfy the quasiperiodicity condition g(r + Rp- r1) = g(r; r')eik°-R'
,
(3.76)
as well as a conjugate condition on its second variable, Rp) = g(r;r')e-ik°R*
g(r;r'+
.
(3.77)
It is a useful observation that we can immediately write down a solution for the Green's function: it is the sum of an infinite set of outgoing waves, spreading out from each point of the lattice. Thus, g(r;r') = -l-J2^l)(k\r~r'-Rp\)eik°R'>
.
(3.78)
p
The task is first to obtain an expansion of the expression (3.78) in terms of the appropriate functions; this will enable the Green's function to mesh cleanly with the expansion of the potential (3.70). Defining £ = r - r', we note that |i? p | > |£| for all p ^ {0,0}, and so in this region we can use Graf's addition theorem (see Abramowitz & Stegun 1972) to expand the Hankel function in (3.78): oo
Ho^im-Rpl)^
E l=-oo
HtMikRJJtikQeW'-i)
(3.79)
The spectral problem
143
where 7 = arg£ and <j>p = argJ2 p . Hence, omitting the central term,
£#o ( 1 ) (fc|£ -
Rp\)eik°-R>
p/0
,-H-r 12 [HtW(kRp)eit+'eik'>-R'] Ji{K)e~
= S
e=-ocp^£0
= St(k,ko)Je{kt)e-uT
,
(3.80)
where = '%2HiW(kRp)eit+>eiko-R'.
St(k,k0)
(3.81)
p^O
Thus we can express the Green's function in terms of Bessel functions: 00
l
W
g(r;r') = - -H0 (kO
~\Y.
^(fc,fc0)JK^e^7 .
(3.82)
The radius of convergence of this expression is determined by the limit of applicability of Graf's addition theorem. In this case, the formula (3.82) will converge whenever |£| < Rp for all p, that is, whenever £ < d. 3.2.3
The dynamic
lattice
sums
The task now becomes one of finding a more effective expression of the lattice sums appearing in formula (3.81). As expressed over the direct lattice, the sum converges very slowly, in fact the terms in the sum are -I
try
of the order O {Rp) "'. Ewald (1921) noted that sums which converge slowly over the direct lattice may converge much quicker if expressed as a sum in reciprocal space. To this end, we employ the reciprocal lattice defined earlier in (3.22): Kh = — (m, n)
m,neZ
(3.83)
The reciprocal lattice vectors, with the Bloch wavevector ko, are shown in Figure 3.4. One way of computing quickly converging expressions for the lattice sums is to examine the form of the Green's function when expressed as a
144
Multipole methods and homogenisation
•
\
w
\
_y
two-dimensions
•
•
•
•
•
•
\ .
•
in
Kh\
•
Fig. 3.4
The reciprocal lattice.
sum in reciprocal space. This is easily calculated if we start with the generic expansion g(^) = Y,9(QhyQA
(3-84)
h
where Qh — Kh + fco- We can now employ the incredibly useful Poisson Summation Formula to obtain a connection between sums in reciprocal space to their counterparts in direct space:
V] 6(£ - Rp)eikoR" P
= ^ V e^S A
( 3 . 85 )
h
By substituting this last expression into the Helmholtz equation, (3.84) becomes
£ ( - Q » 2 + k*)g{QhyQ^ h
= jY,jQ^ h
.
(3.86)
The spectral problem
145
Hence we obtain the spectral domain form of the Green's function 1 v-^
e'^ft'^
We now expand the relevant quantities in the spectral domain representation in terms of Bessel functions. Thus we find oo
eiQA =
£
i'JtiQhOeW*-^ ,
(3.88)
£=-oo
and so
e=-oo
h
^h
K
One can then formally equate the two representations of the Green's function (3.82) and (3.89): OO
^*=-oo
h
Vh
-K
By equating the powers of 7 in (3.90) and rearranging, we obtain absolutely convergent expressions for the lattice sums Se(k, ko)Jt(kO
= -H^\ki)6tfi
- A.*" £
^ % e
i i 9 h
• (3.91)
We can further simplify this expression by splitting the lattice sums into two parts, corresponding to sums over Ji and Ye Bessel functions respectively. Thus, Se = Sj + iSj, where
5/(fc,fc0) = YJJ^kRp)eU4'P^ko'R''
'
(3'92)
SY(k,k0) = Y,Y*(kRP)eit+pJko-R'
.
(3.93)
pjto
Multipole methods and homogenisation
146
in
two-dimensions
In order to evaluate the lattice sums S / , we again use a form of Poisson's summation formula £
g-ii^.^-fco)
=
( 2 J ! J- 6(ks -fco- Kh)
p
,
(3.94)
h
and set the magnitude of the vectorfcsequal to k and its argument equal to some variable 6S. Noting that within the first Brillouin zone ks—ko^= Kh for any h, we have ^e-i(ks-k0)Rp
(395)
=Q
p
We then use the expansion oo
e-ik3Rp
^
=
(-ifj^klQe^^-8^ ,
(3.96)
i=-oo
and substitute into (3.95) to find oo
]T ]T (-i)tJt(kRp)eit^*'-e^eiko-R'
= -1 .
(3.97)
p#0«=-oo
Hence we discover the identity oo
5 3 {-i)eS}!{k,ko)e-ue> = -1 .
(3.98)
f=-oo
By integrating over the dummy variable 6S we then obtain Sj(k,fco)= Se,o •
(3.99)
If we substitute this expression into (3.82) then we find that the lattice sums 5/make no contribution at all*; the Green's function can now be written 1
g{r. r') = _-y0(fc£)
1 ° °
~-J2 # ( * . k0)Jt{ki)e-^
,
(3.100)
«=-oo
*One viewpoint on this is that we need only have included the singular Bessel functions Y in (3.82).
The spectral problem
147
where, from (3.90), the remaining lattice sums are
Sj(k, k0)Je(H) = -Y0(H)Se,0 - ±i< £ •gp^e™" 2 Qh
k
. (3.101)
By using this derivation, the magnitude £ has become unstuck from its original designation as the difference between the two vectors in the Green's function. This arbitrariness can be exploited in order to accelerate the convergence of the lattice sums. By making use of the recurrence formulae (see Abramowitz & Stegun 1972) (3.102) (3.103) then we find that after p successive integrations of equation (3.101), we obtain
(p-n)\
SY(k,k0)Je+p(k£) n=l
"*?(*
P
v
(2_
p-2n+2'
5e,o
'
Je+P(QhZ) 2
jeeh
(3.104)
Ql-k
Usually, for analytic purposes, p is set to zero, while for numerical applications p is set in the range 5-7. The lattice sums are absolutely convergent, with order 0(Qflp~ ' ) for large Qh- One can also see from (3.101) that S}^e = (Sj)*, where the asterisk denotes complex conjugation. Thus, only positive order lattice sums need be considered. 3.2.4
The integral equation
and the Rayleigh
identity
Applying Green's theorem within the unit cell, in the region ft shown in Figure 3.5, we have [uA'g(r; r') - g(r; r')A'u(r')] dA! l
^(r;r'Hr')-9(r;r')^(r')
r u dc
ds'
(3.105)
Multipole methods and homogenisation
148
in
two-dimensions
r-«
a r
\ C
))
fc-J I
I
i\
J/&C
»~
Fig. 3.5 The central unit cell, showing the contour within which Green's theorem is to be applied.
where T and dC are the contours shown in Figure 3.5. Here the primes indicate that the operators act on the primed coordinates only. We already know from the definition of the Green's function (3.75) that ff
[u(r')A'g(r; r') - g(r; r')A'u(r')] dA' = u{r) .
(3.106)
In addition, the quasiperiodicity of both u (3.67) and g (3.77) guarantees that the integral around the outer boundary T is zero. Thus we have the integral equation u(
r)= [ Jec
dg *\ du /_/<> {r') 7 (r; r')u{r') - g(r; r') — dn dn'
(3.107)
In order to solve this, we expand both the potential and its radial derivative as Fourier series on the boundary of the circular inclusion:
,(r)|_ = £ «**» ,
(3.108)
£=-oo
du (r) dr
r=a
E ^
^=_
(3.109)
The spectral
problem
149
Using Graf's addition formula and noting that r > r' everywhere outside the inclusions, we expand the Green's function (3.100) in coordinates centred on the origin: oo
9(r;r')
= ~
Yn(kr)Jn(kr')ein^-^
£ TI — — OO
-
OO
~7
OO
£
( - l ) m + " ^ _ n J „ ( f c r - ) J m ( f c r ' ) e i n 0 - ^ , (3.110)
£
m= — oo n=—oo
where 6 = a r g r and (f> = argr'. (The arguments of the lattice sums have been dropped for the sake of simplicity.) The radial derivative of the Green's function can be written ^ r ' ) dr
= - \
n= — oo oo
oo
"I
Yn(kr)JUkr>)e^e-<»
£
£
(-ir+nSZ-nUkr)J'm(kr>)eine-im*
£
.
(3.111)
m = — oo n = — oo
We can now evaluate the integrals around the boundary of the inclusion. The first of these, from (3.107), is
jNMgcr^-jf
£
aeel
.l=-oo
Yn(kr)J^kr')ein^^
£
(3.112)
n = —oo ,
oo
4
oo
£
{-l)m+nSl-nUkr)J'm{kr'yne-im* ad<j>
£
m = — o o n = —oo oo
=
£
y*(fcr)e*
* ) .I »^ ( f c a ) -«< (I —^-
£=-oo
oo + £
oo £
n=—oom=—oo
r J„(fcr)e^(-ir+"^_„
- a L
m
/ / \ " ( ^ W a ) . \
/
(3.113)
150
Multipole
methods
and homogenisation
in
two-dimensions
The second integral in (3.107) is £4>
E A''
Jt=-oo 1
°°
- Y, Yn(kr)Jn(kr')ein(9-^
(3.114)
n=—oo oo oo
~ I
E
(-Vm+nS}n-Mkr)Jm(kr')eine-i
E
m=—oo
= oo
ad(j>
n=—oo
£
y,(fcr)e^[-/3,(^)j,(fca)
oo
+ E
^ ( * r ) e < n * ( - i r + n S £ _ n [-/3m ( y ) Jro(fca)' .
E
n=—oo m=—oo
(3.115) We can now write out the potential explicitly: oo
oo
n = —oo oo
+ E
(-l)m+nSm-nJn(kr)eind
E
n= — oo m=—oo
x ( Y ) [-o<mkJm(ka) + (3mJm(ka)}
(3.116)
On comparison with the original expansion of the potential (3.70), we now make the identification Bn = ( ^ ) [-OnkJ'^ka)
+ (3nJn{ka)}
(3.117)
OO
An =
£
( - l ) m + n S £ _ „ ( y ) [-a ro fcJ^(fca) +/3 m J m (fca)]
m= — oo
(3.118) Hence it immediately follows that oo
An = £
(-ir+"S£_ n £ m .
(3.119)
The spectral problem
151
This equation has become known as the Dynamic Rayleigh Identity. By using the definition (3.73) which incorporates the boundary conditions, we can write the Rayleigh Identity in the form oo
MeBe+ ] T (-l)i+mSl_e(k,k0)Bm
= 0.
(3.120)
m=—oo
This is an infinite linear system, which can be truncated and solved for the coefficients Be. Zeros in the determinant of the above system then correspond to propagating modes, and becausefcoand k are the only free parameters in the system, setting the determinant to zero will generate the dispersion relations, or photonic band structure, of the material. The structure of (3.120) is important. The quantities Me in the first term depend on the cylinder radius and refractive index, but not on the parameters of the array itself. Conversely, the lattice sums Sj depend on the geometry of the array but not on the characteristics of the cylinders. This separation is useful in analytic studies of the system. Due to the symmetry both of the cylinders and of the lattice, we also have the relations M_ £ = Me
(3.121)
SZt = Sj*
(3.122)
where the star (*) denotes complex conjugation. One can immediately see that the system (3.120) is Hermitian. The system (3.120) is readily normalised; by setting zm = y/\Mm\Bm we can re-write it in the form (7 + R(w,fco))Z = 0 ,
(3.123)
where I is the identity matrix, Z is the vector of unknown coefficients Zm and R(u>,fco)is a matrix whose components are Rem = (-lY+msgn(Me)
!^±^ . yJ\MlMm\
(3.124)
When written in this way it can be shown that the coefficients of the matrix R(u>,fco)decay exponentially away from the main diagonal. From this observation it is apparent that the coefficients Ze decay very rapidly with increasing £, and so it is not always necessary to consider a large number of terms in the calculations.
152
3.2.5
Multipole methods and homogenisation
The dipole
in
two-dimensions
approximation
For arrays of cylinders which are not too concentrated, we can neglect all terms in the system (3.120) which are of higher order than the dipole (B±i) terms. If we truncate the system (3.120) to include only terms from m = —1,0,1 and equations from (. = —1,0,1 we obtain the 3 x 3 system
Mi + Si
-S\
-SY*
Mo + SX
SY*
-sY*
sY
'
-sY
•B_r (3.125)
B0
MX + SY _
.Bi
.
Specifying a direction for ko, i.e., the value of 6Q, we can evaluate \V\ as a function of k and ko. Then, the condition \T>\ = 0 defines k as a function of ko. We are particularly interested in this function in the neighbourhood of ko = 0, since this is intimately connected with the behaviour of the system of cylinders for wavelengths A > ( i and thus with the homogenisation of the material. It has already been mentioned that the lowest frequency mode, known as the 'acoustic mode', determines the homogenisation of the material. For the acoustic mode, we will assume that k oc ko for ko —>• 0, for the reason that if we do otherwise the material cannot be homogenised at all. Hence, we will define: k = ako,
(3.126)
and assume that a has the following expansion: a = a0 + a2(k0d)2 + O{k0d)i.
(3.127)
Here, ao and a? are quantities we seek to determine: ao is related to an equivalent (phase) refractive index for the array, through the relationship (see equation (3.69)) 1
(3.128) a If one wanted to measure neg experimentally, then one could insert a sample of the composite in one arm of an interferometer, and then measure the resulting phase shift. a2 is the first estimate for the coefficient specifying the nen =
The spectral problem
153
wavelength dependence of a, and so will determine the first finite frequency correction term to the effective refractive index. We now employ the substitution (3.126), and seek to determine the small fco expansion of a from the equation \T>\ — 0, i.e.
(M0 + SX)[(M1 + SZ)2-\sZf -2 \SY\2 (M1 + Sf) + (SY)2Sr + (SV)2SZ = 0. (3.129) We have from (3.74) the following series expansions for small fc:
Mo Mi =
32 ir{n - l)(fca)4 2
l-rt-QW+Ojka)* 1 , M
7r(fca)
2
In I y J + 7 •
(3.130) n2r + 3 4(n2 - 1)
0(ka)4 (3.131)
where 7 represents the Euler constant. It should be noted that this asymptotic limit depends both on the material properties of the inclusions and on the type of wave propagating through the array. In this case we consider, as before, a Transverse Electric wave moving through an array of cylinders with refractive index nr. We also obtain from (3.101) for small k and fco: ^0
—
J
IT
1 4iexp(i#o) &o 2 k V k — k2
s\ = qy
_ 1M
*
~ d
2
— + 47o^ - - In (fed) + O(feod), (3.132)
2
k
2
IT
-£.\+0{kod), 4-7T j
(3.133)
d2\ k2-k2
AIT2
8TTJ exp (2i00)
a\ ' exp (—2i#o) + O(k0d),
(3.134)
where 70 = -0.318895593319827 and
154
Multipole methods and homogenisation
in
two-dimensions
portional to cos (40o). This is a consequence of the four-fold symmetry of the square array, i.e., the invariance of the array with respect to rotations of an angle which is an integer multiple of 7r/2. If we express the cylinder radius a in terms of the area fraction of the cylinders (/): 2\l/2
(3.135)
a =
•K
and replace n^ by: T+ l
(3.136)
with 1/r representing the dielectric contrast across the interface boundary, then in the expansion of (3.129) the leading term (of order l/A;®) is: _256TT(T-1)(T + / ) 2 8
~
ag(a§ -
l)fd*k*
«8
/ r +f
(3.137)
Hence, we obtain for ao: T-f an = T + f
(3.138)
Equation (3.128) gives us the leading order estimate for the effective phase refractive index (or dielectric constant eeff):
KV) 2 =
£
e« —
T+f T-f
(3.139)
This is, of course, the Clausius-Mossotti or Lorentz-Lorenz equation, derived earlier in section 3.1.2. Having found a 0 , we now seek to determine »2 by setting equal to zero the coefficient of 1/fcg in \V\. Thus, we find: Mi + 5 0 y =
2(r + / ) 2 _ _ (T - f)fd?k2Q "+ 7T ' " \A-K
UJl
2r-l
+
2-K
^-2-p^^2+O(k0df,
(3.140)
The spectral problem
155
and:
™ (Mi + S*)
|0y.2
# r=
l l 4(r + / ) 2 2 2 IT (r - /)/d fc
1 T+ f
-470 + r
1 T+ /
7~ -j
/ / \
n
, 2r-l
\ 4-K J (4)
7 ^2
cos
2lT
(4(?0
16a 2 (r + / ) 9 / 2 + O(fcorf)0. 3 / (T - /) 3 / 2 d 2 fc 2
(3.141)
The coefficient of 1/fcg in (3.141) vanishes by virtue of (3.138). Since we want the coefficient of l/A;® in (3.129), we need to take into account only the leading order term in MQ + S%: y 167r(T + / ) 2 ( T - l ) Mo + 50y = + c?(Mr (*
(3.142)
7)W^
The calculation of the leading-order contribution from the second and the third terms in (3.129) is simpler than the treatment of the first term. We need only take the leading term in each quantity, and replace a by ao throughout. We find:
-2 \s\|2 (Ml + s*) = (sl)2 sr + (sV) si72 y }
16(T + / )
5
(3.143)
= -(T_/)2/3rf6fc6+<W) By substituting (3.142) and (3.143) in (3.129), we have:
(3.144) Then, by equating (3.141) and (3.144) we obtain: / «2
4
a
(T-/)
(T
+
1 / a
/)5/2 ^
n
Jl
l n
m,
L
^47f
+
1 r + / 27TT-/
_ _ _ _ 4 2
T2-/2
1
2T-l 7 0
-2TT ^ T - 1
11
T (4)J 1 - - cr^ cos (40 o )
7T
(3.145)
156
Multipole methods and homogenisation
in
two-dimensions
Consequently, we obtain from the dipole Rayleigh equations, the effective refractive index of the array: (n e ff) 2 =
-470 -
7S 1
1 r1
TT 27T
f
T - 1
/;2i M
2/2(T-/)1/2 2(T + /)5/2
, 1 T+f
+ Z7T T — /,
2 T - 1
ir
\47ry
+
(^ 2 (4 ) .l--<#'cos(40o) \
2?r
(3.146)
7T
for |T| > / and r ^ 1. For complex r there is a branch-cut on the real axis, for — / < r < / and a simple pole at r = 1. The branch-cut corresponds to real and negative values of the relative dielectric constant of the cylinders sr = r?r (lying between - ( 1 — / ) / ( l + / ) and its reciprocal). Such values of er are of course not physically realisable. By means of (3.145) we may study the validity of the Lorentz-Lorenz formula for square arrays of cylinders, having different area fractions, refractive indices of cylinders or orientation angles of the Bloch momentum. In the graphs displayed in Figures 3.6-3.8, the value of area fraction has been limited to / < 0.5 as all the calculations have been made in the dipole approximation. For concentrated systems one would have to extend
o.oi
-0.01 Fig. 3.6 The first dynamic correction 0:2 as a function of area fraction ( / ) , for 9Q = ir/3. The dashed lines correspond to a square array of cylinders, having eT = 5 (short-dashed curve), eT = 10 (medium-dashed curve) and eT = 20 (long-dashed curve), while the solid curve corresponds to an array of perfectly conducting cylinders.
The spectral problem
it/4
"0
Fig. 3.7 The first dynamic correction a ] a s a function of orientation angle (60), for / = 0.4. The dashed lines correspond to a square array of cylinders, having ET = 5 (short-dashed curve), er = 10 (medium-dashed curve) and e r = 20 (long-dashed curve), while the solid curve corresponds to an array of perfectly conducting cylinders.
the matrix in (3.125) for \P\ > 1, in order to include the contribution of high-order multipoles. Thus, in Figure 3.6 one can see the dependence of ai on the area fraction (/), for a fixed orientation angle (#o = TT/3) and various relative dielectric constants (er) of the cylinders. One interesting feature of these curves is that they cross the /-axis (c*2 — 0) even for relatively high contrasts. This could explain why the Lorentz-Lorenz formula works well in such cases. Actually, this result depends on the orientation angle of ko, as can be seen in Figure 3.7. In some cases, like / = 0.4 and e r = 20, a2 is different from zero for all the values of orientation angle in the range 0 < 60 < 7r/4. We may say that 0*2 is small for / < 0.4 and er < 10, independently of the orientation angle #0 (see also Figure 3.8). A number of comments can be made about the form of (3.146). Firstly, m if we replace r by —T in the leading term, e°ff is replaced by l/e\, accordance with Keller's Theorem (Keller 1964). The dynamic correction term gives a form not obeying Keller's Theorem: this we would expect since Keller's Theorem is based on the Laplace's equation, and dynamic fields come from potentials obeying the Helmholtz equation. Secondly, we may enquire as to the values of / and r for which the static result (3.139)
158
Multipole methods and homogenisation
in
two-dimensions
Fig. 3.8 The first dynamic correction 02 as a function of relative dielectric constant of the cylinders (er), for 0o = ir/3. The different curves correspond to a square array of cylinders, having / = 0.1 (short-dashed curve), / = 0.2 (medium-dashed curve), / = 0.3 (long-dashed curve) and / = 0.4 (solid curve).
is a good approximation. The requirement is then: (k0d)2
<
4TT(T +
/)5/2
-2-/2 r-1
2 In
, r +f /
+
w
+ 2r - 1 - 8TT7O
1 - - 4 4 ) cos (40o)
(3.147)
TV
This requirement becomes less stringent as / decreases. However, as nr increases, r decreases toward one, and the term (r 2 — / 2 ) / ( T — 1) in the denominator makes the requirement stronger. Thus, if nr is sufficiently large, it may require very low frequencies before the quasistatic region indicated by the inequality (3.147) is attained. Finally, the first dynamic correction to the effective dielectric constant involves a quadrupolar dependence on direction within the array.
3.3
The singularly perturbed problem and non-commuting limits
An interesting mathematical quandry arises when we attempt to take the limit when the transport property n becomes infinite. We might proceed
The singularly perturbed problem and non-commuting
limits
159
to take the limit as follows: One reason that the potential u is unaffected by this limit in the region between the cylinders, and continues to satisfy the Helmholtz equation: (A + A;2)w = 0.
(3.148)
Meanwhile, it is apparent that the boundary conditions (3.65) and (3.66) are changed into the single Neumann condition du dr
= 0,
(3.149)
dtvo
This corresponds to the case when the cylinders are 'perfectly insulating' or 'vacuous'. On the other hand, if we write the expression for the potential inside the inclusions as (A + n2fc2) u = 0,
(3.150)
one can see that when n —> oo the term on the right hand side dominates, and hence u —> 0. In this case the boundary conditions (3.65) and (3.66) are converted into the single Dirichlet condition u\d„0
= 0.
(3.151)
This corresponds to the case when the cylinders are 'perfectly conducting' or 'rigid'. In fact the problem when n is large is of the singularly perturbed type. If we write the expression for the potential inside the inclusions as fc2n2
A + l ) u = 0,
(3.152)
then one can see that when the quantity kn is large there exists a small parameter near the higher order derivatives in the equation. Solutions to this type of problem typically involve large derivatives and possibly highly oscillating fields inside the inclusion and in its immediate neighbourhood in the matrix. As we shall see, the occurrence of such large derivatives does not preclude a satisfactory mathematical treatment of the problem. It is instructive meanwhile to continue with both our naive models (3.148), (3.149) and (3.150),(3.151), because this highlights an interesting 'paradox' which occurs when the situation is treated non-rigorously. We proceed as follows:
Multipole methods and homogenisation in two-dimensions
160
3.3.1
The Neumann
problem
and non-commuting
limits
The potential is still quasiperiodic (satisfies (3.67)), and in the material outside the cylinder has the general form (3.70). It follows that the coefficients A( and Be from (3.70) satisfy equations of the form (3.73), with: M!°
=
y/(fcq) J't{ka) •
(3.153)
Consequently, we have to change only the coefficients Me in the Rayleigh identities (3.120), while the lattice sums SY remain unchanged. This gives the dipole approximation of the Rayleigh system in the form (3.125), with the determinant:
(MST + SY)[(M? + SY)2-\SY\2] -2 |SrI* {M? + SY) + (Si)2 Sf + (SV)2SZ = 0. (3.154) Now, from (3.153) we have the following series expansions for small k: 4 7r(A;a)2 4 M1°°
=
^
+
5 + 47
JbW 2,
+
^N
(ka\
_,
2
. ,2
(3-155) (3.156)
The lattice sums Sj, of order £ = 0,1,2, have the expressions (3.132-3.134). The determinant (3.154) also depends on #o through terms proportional to cos (4#o )• Again, we employ the substitution (3.126) and assume that a has the expansion (3.127). Then, in the expansion of (3.154) the leading term is of order 1/fco, and is:
T_6
64(1-/2)(1 + /) a g ( l - a g ) / 2 d 8 * g an
1+ /
(3.157)
Hence, we obtain: o& =
1 + /'
(3.158)
The singularly perturbed problem and non-commuting
limits
161
with / — Tra2/d?. We can now see that (3.158) cannot be obtained as the limit of (3.138) for r —> 1. That is, the limits k -4 0 and n —> oo do not commute. The 'paradox' can be stated in the following way: If the refractive index n of the inclusions is fixed, then the problem formulated is a regularly perturbed boundary value problem and can be homogenised in the same way as in the previous section. The limit of large n can then be taken to obtain the 'perfectly conducting' result. On the other hand, if we allow the product nk to be large when we first start to treat the problem, and then homogenise the material, then we obtain a different answer. That is, the final result for the homogenised material depends on the order in which the two limits are taken. If we plot the limit taken on the coordinate axes 1/n and k then we might anticipate that the result for a depends on the trajectory as we approach the origin in the plane (1/n, k). In particular, when 1/n = kp, with p > 1 (the curve 3 shown in Figure 3.9) then the parameter kn in (3.152) is indeed small and we deal with the case of a singularly perturbed boundary value problem. If n is complex with a sufficiently large imaginary part, then we expect the field to attenuate as one moves away from the interface k 0.8
^ ^
0.6
0.4
0.2/
-*— n 0
S'
2
/
s^
s'
-1 , , —T"-^!^
0 1 0 2 0 3
X/N
0 4 0 5
Fig. 3.9 Trajectories in the (1/n, k) plane corresponding to a regular perturbation (1), transition region (2) and a singular perturbation (3). The three trajectories correspond to the power laws with respective exponents less than one, equal to one and greater than one.
162
Multipole methods and homogenisation
in
two-dimensions
boundary, and a thin layer (the boundary layer) would exist in the vicinity of the interface where the field is characterised by relatively large values of its gradient. If n is real and large the field oscillates rapidly inside the dielectric, and it is more difficult to characterise the boundary layer. 3.3.2
The Dirichlet
problem
and source
neutrality
Starting with the Dirichlet problem (3.150),(3.151), we note once more that the coefficients Ae and Be from (3.70) satisfy equations of the form (3.73), with:
Consequently, we have to change only the coefficients Me in the Rayleigh identity (3.120), while the lattice sums Sj remain unchanged. The central term in the Rayleigh system (3.120) is M0 + S% ,
(3.160)
and when we take the limit k —¥ 0, this becomes
I (ta+->) ~ a^W) - ( T + 4 7 ° ) - 1 l n { k d ) + ° { k d )
(3 161)
-
It is readily verified that no other term in the system can cancel out the constant term 2/n\n(a/d) appearing in (3.161), with the effect that k will not necessarily vanish as ko —¥ 0. We can see therefore that the presence of the lowest band is strongly dependent on the coefficient Mo. This coefficient is related to the flux across the cylinder boundary, through the equation
-^dl = 2nkaB0 [-M0J0(ka)
+ Y0'(ka)]
(3.162)
In the context of electromagnetism, the flux integral on the left hand side is proportional to the charge on the cylinder. This suggests that source neutrality is a pre-requisite for homogenisation of the material. In fact if we set the problem up so that it is source neutral, i.e. so that
/
du —dl = 0 ,
(3.163)
The singularly perturbed problem and non-commuting
limits
163
then the boundary coefficients become
(3-164)
Mo = ^ g 4 = °^ • J'0[ka) and M„ = ^ M = 0 ( f c 2 " ) ,
(3.165)
as k -> 0, so that the terms of order fc2 (and smaller ones) cancel the singularities in S^, S j ' , 5 ^ (all of order (fc2 — fcg)-1). This gives rise to a linear acoustic band k = ako, and it can be verified that the expression for the slope a is the same as for the Neumann problem, i.e.
a=
(3 166)
\lT^J
-
where £eff is this time the effective dielectric constant for an array of perfectly conducting cylinders. Source neutrality, amples
non-commuting
limits and some model ex-
By way of illustration, and in order to demonstrate that these effects occur even in simple materials which are non-periodic, we now present three elementary examples. Example 1 - Dirichlet problem with a constant boundary value. Consider an infinite plane, and let v(x, y) be a linear function that describes a potential of an applied electrostatic field. Introduce a perfectly conducting infinite cylinder of circular cross section (of radius a), and assume that the potential of the electrostatic field is constant on the boundary. Thus the problem can be characterised as Au(x, y) — 0 u
x2-\-yz=az
C,
(3.167) 2
2
u(x, y) — v(x, y) + w(x, y) as x + y —>• oo ,
164
Multipole methods and homogenisation
in
two-dimensions
where \Vw\ —>• 0 at infinity. The solution of (3.167) has the explicit form u(r) = v ( r ) - ( l o g a ) - 1 ( i > ( 0 , 0 ) - C ) l o g r
(1
01)
I
01) \
( cos6»—+sin6>—- J , (3.168)
where r = (x,y) = (rcos6,rsin9). Note that this representation includes a logarithmic term unless C = v(0,0). For the latter case the integral of du/dn over any closed contour (which does not intersect the inclusion) is equal to zero. Thus, an appropriate choice of the constant C provides source neutrality. Example 2 - The Neumann problem for a perfectly insulating inclusion. Now assume that the circular inclusion is perfectly insulating, and, therefore, replace the boundary condition in (3.167) by du dn
= 0
(3.169)
x2+y2=a2
The field u then admits the following explicit representation: a2 / dv u(r) = «(r) + - ^CoS9~+Sm9-j
v
n®
\
,
(3.170)
and it is straightforward to verify that this automatically satisfies the source neutrality condition. Example 3 - Oscillation in a body with a small perfectly conducting or perfectly insulating inclusion. Consider the eigenvalue problem Au + fc2w = 0 in n \ D e fill
/3u +
^ - = 0 on a n on -^ = 0 on dV£, on
(3.171)
where 0 represents a bounded region and V£ is a small disk of radius s. Let uo, ko be the eigenfunction and the eigenvalue respectively that solve the following spectral problem in £): Au 0 + kl = 0 in fi 5 ^ = 0 on dQ. on
(3.172)
Non- commuting
limits for the effective
properties
165
When (3 — 0, we deal with the homogeneous Neumann condition prescribed on dVs (perfectly insulating inclusion), whereas for (3 —• oo we have prescribed a homogeneous Dirichlet condition (perfectly conducting grounded inclusion). As fco —• 0, the function UQ is constant, say 1, to leading order, and the derivatives duo/dx(xo,yo), duo/dy(xo,yo) are of order O(fco); here (xo, yo) are the coordinates that correspond to the center of the small inclusion. The asymptotic problem (3.171) is classified as a singularly perturbed problem, and it requires a boundary layer which describes the field in a neighbourhood of the inclusion (and depends on scaled coordinates). This problem was studied by Ward and Keller (1993), who derived the following asymptotic formulae for the eigenvalues: fc2 «fcg-(loge) _1 27r[uo(a:o,J/o)] 2 , when j3 -)• oo,
(3.173)
and
fc2«fc02jl+£2
•K\uo{xo,yo)\
-2ir-
k0
when (3 = 0. (3.174)
We note that for the case of the perfectly conducting grounded inclusion the source neutrality condition is not satisfied, and even if fc2 —• 0 the eigenvalue fc2 does not necessarily vanish for any fixed e. When, however, the inclusion is perfectly insulating, the quantities fcg and A;2 have the same order of magnitude (for a small fco we can guarantee that ke will also be small).
3.4
Non-commuting limits for the effective properties
Consider a periodic structure such that each cell contains a highly conducting inclusion. The region occupied by the inclusion is denoted by fi^; the remaining part of the unit cell is designated as Jl( m ). We also consider two problems for the fields V and V, each satisfying the continuity condition (3.65) and the interface condition for the normal derivatives dV(i) dV(m) C ^ T (*>*)= *T 0^)'
(*,3/)€3fi(i),
(3-175)
166
Multipole methods and homogenisation
in
two-dimensions
where £ = 1 for the case of TM polarization and ( = 1/N2 for the case of TE polarization. V satisfies the Helmholtz equations (3.63) and (3.64), while V satisfies corresponding equations, with k replaced by k. The quasiperiodicity conditions are ,
(3.176)
exp(-ik0d)V(x,y)
(3.177)
V(x + d,y) = exp(ik0d)V(x,y) V(x + d,y) =
In what follows, we assume that an acoustic band exists, i.e. (3.178)
lim fc/fco < M , feo->0
where M is constant. The following identity then holds:
JJu(m)
+c =
y( m )AV( m ) - y( m )AV( m ) dA
11
y-WAyW - y W A V ^ dA
(k2-k2) N2(
IJ
V^V^dA
m
+ J Jn (m) V(
)v(m)dA (3.179)
On the other hand, it follows from Green's theorem that y(m) ^
y(m) dn
I +C
y(m)
y-(m) dr
L
dl
dn
y(i) — dr
dl
dr
vw
dr
(3.180)
dl
where 7 denotes the exterior boundary of the unit cell, and Y — dil^y Due to the continuity conditions (3.65) and (3.175), the last two integrals in (3.180) cancel and the quantity I in (3.180) involves only the line integral over the boundary 7 of the unit cell. Only the lines x = ±d/2 of 7
Non-commuting limits for the effective properties
167
contribute to the line integral I and it follows from the quasiperiodicity of V and V ((3.176) and (3.177)) that y W _
j l - exp [-i(k0 - k0)d] \ I
y("») dn
v
dn
dl. (3.181)
where 7^ = {(x,y) : x = d/2, -d/2 < y < d/2}. When k0 ->• 0 the fields V and V in both the matrix and the inclusion may be represented in the form: W i,m > = exp [iJfcou,i>m>] + 0{kl) , {i m
V ' ^ = exp [-ikou^] where Us satisfies:
+ O(fcg) ,
(3.182) (3.183)
is a static field. The field us is a harmonic function which u
du dn
=u®{x,y),
6V(*) = C" dn
on the interface a;2 + y2 = a 2 , and may be written in the form us = x + Us , where Us is a doubly periodic function. The TM case (C = 1) corresponds to an ideal contact (a homogeneous plane without any inclusions at all), while the TE case (£ = 1/N2) describes an array of circular dielectric cylinders of 1/iV2 dielectric constant embedded in a matrix of unit dielectric constant. Using (3.182) and (3.183), equation (3.181) reduces to: ,{m)
/7r .~2
(ko
i{k0 + * o ) - ^ - exp [t(*o - fco)«.m) + 0(*g)J
rd/2 rd/2
to 2 )/
Q 0Us
dx
J-d/2 ,~2
(ko -k2)d'een
(m)
+
dl + 0{kl) x=d/2
0{kl),
(3.184)
where eeff denotes the effective dielectric constant for the static problem, defined using (3.16), with the driving field perpendicular to the cylinder
Multipole methods and homogenisation
168
in
two-dimensions
axes. Note that in the TM case Us — x and therefore £^ff = 1, independent of the value of N, while in the TE case eeff depends on N, and lies in the range (0,1). We now introduce the following notation: a=
k lim — .
(3.185)
fco-s-0 fco
Then -l
k2 - k2 —2
=
2
{i)
N ([[
2
d eefi
ko — k0
V^V dA+
ff
«(<)
m
m)
v( W dA
•'•'"(m)
(3.186) Consequently, as ko, ko —> 0, a2 = eeff [N2Cf + (1 - / ) ]
1
+ O(fc02+fc02),
(3.187)
where / , one may recall, is the area fraction of the inclusions. One can see that, as ko, ko —*• 0 and N —» oo, for a fixed area fraction 2
_ (eeff for the TE polarization, ~ \ 0 for the TM polarization.
, ^ '
. '
The interpretation of this result is that the energy integrated over the unit cell in (3.186) may affect the calculation of the effective properties of the material even if the field within the central cylinder is asymptotically small.
3.5
Elastic waves in doubly-periodic media
In this section we will use the Rayleigh multipole method, developed in the previous sections, to derive the dispersion curves for elastic waves moving through an inhomogeneous material. We restrict our attention to the doubly periodic array of cylindrical cavities which are circular in cross-section and are not permitted to touch, although they can be arbitrarily close. Such materials are common in existing mechanical devices and are easily fabricated, and can also occur naturally (Martinez-Sala et al. 1995).
Elastic waves in doubly-periodic
Fig. 3.10
3.5.1
media
169
Cross-section of a doubly periodic array of cavities in an elastic material
Governing
equations
We consider the problem of in-plane propagation of harmonic waves within an isotropic elastic block of density p and Lame coefficients A and fi. This block contains a set of infinitely long circular cylindrical cavities of radius rc which are parallel to the z-axis and are evenly spaced in two-dimensions. As in the previous section we introduce the direct lattice vector Rp, which points to the pth cavity in the array, as well as the reciprocal lattice vector Kh, given in equation (3.22). The propagation of waves in the elastic material is described by the Navier vector equations (A + 2/x)VV • u ( r ) - MV x V x u ( r ) + poJ2u{r) = 0, r e 1Z3
(3.189)
where u(r) is the displacement vector at position r and o> is the frequency of vibration. The periodicity of the problem implies that u must also satisfy the condition u ( r + Rp) = u ( r ) exp(ifc0 • Rp) ,
(3.190)
170
Multipole methods and homogenisation
in
two-dimensions
where the Bloch wavevectorfcolies entirely in the x-y plane. For a given feo we can find a set of eigenvalues u> of the system (3.189) which correspond to propagating modes in the lattice. The boundary conditions for the problem come from the zero-traction condition on the surface of each cavity. Expressed in cylindrical coordinates, the components of the stress tensor arr, arg and arz must vanish. As the cylinders are considered to be infinitely long the problem is inherently twodimensional, and u = u(x, y). In this case, on the boundary of the circular cavity we require that
(3.191)
. (dur ur ldue\ „ dur {-drL + V + rW)+2^=0^
= X
^
(due ue ldur\ + =»{-dr--T r-de)=0-
(Tre
.„ „ „ . -
(3 192)
(3 193)
"
Also, the Navier equations (3.189) can be split into two sets of equations which can be solved separately. The first (and simplest) involves only the z-component of the displacement vector uz{x,y). The equation for this scalar function can be written Auz + -w2uz P
=0,
(3.194)
with the boundary condition duz or
0
(3.195)
arising from the zero-traction condition (3.191). The boundary conditions on the edge of the unit cell are given by the Bloch condition (3.190) and thus the problem is specified completely. We note that the scalar problem has been covered in the previous section, and is formally identical to the electromagnetic problem of TE waves moving through an array of perfectly conducting cylinders. For the full phononic structure of the material to be calculated, the more intricate problem of in-plane elastic waves must also be solved. However unlike uz, the components ux and uy cannot be easily separated; the problem is inherently
Elastic waves in doubly-periodic media
171
a vector problem. In two dimensions we introduce the Lame potentials <j> and St, such that u = V0 + V x *
(3.196)
For a piece-wise homogeneous material in two dimensions we can write * ( z , y ) = (0,0,V(z,y))
(3.197)
and it is easily verified that d<j>
dip
(3.198)
ux = — + dx dy d
(3.199)
It then follows from (3.189) that the potentials satisfy Hemholtz equations (A + ka2)<j> = 0
(3.200)
2
(A + kb )ip = 0 ,
(3.201)
where ka — u)/va andfc&= ui/vb- Here the quantities va and Vb are the velocities of the dilational and shear waves respectively, when moving through the homogeneous medium. In terms of the Lame coefficients, A + 2/x
2 P (3.202) vb = P The zero traction conditions (3.192), (3.193) lead to the boundary conditions for the potentials P
2 dcj) rc2 dG 1 d2i>
2 3V rc dddr
, 2,
2 d2ip rc d62
2 dip rc dr
0
(3.203)
2.
= 0 (3.204)
In addition both potentials inherit the Bloch quasiperiodicity condition from (3.190): <j>(r + vRp) = (r) exp(ik0 • Rp)
(3.205)
ip(r + vRp) = ip(r) exp(ifc0 • Rp)
(3.206)
Multipole methods and homogenisation in two-dimensions
172
The mixed nature of the boundary conditions forces the vectors fco in (3.205), (3.206) to be identical. The Lame potentials can be expanded in terms of multipoles: oo
[Aila)Je(kar)+BeMYe(kar)]eU9
(r)= J2
,
fc-oo oo
^(r) =
\Ae(b)Je(kbr)+Be{b)Ye(kbr) „ue
J2
(3.207)
£=-oo
The boundary conditions (3.203), (3.204) can be satisfied automatically by setting 'A^a)\
_ (Mt(aa)
Aew J " 1 M^
M/o6)\
(Bt{a)
M^ ) \ B^
(3.208)
where the formulae for the coefficients Mt follow from equations (3.203), (3.204); these coefficients were obtained in Movchan et al. (1997) and can be evaluated explicitly. The result is: M(aa)
=
_(£6£3
+
E2E7)/(E1E6
Af£*6> = i(E6Ei - E2ES)/(E1E6 MW
= -i{E5E3
Mf*) = -{E5E4
+ E2E5), + E2E5),
(3.209) (3.210)
- E1E7)/{E1E6
+ E2E5),
(3.211)
+ E1E8)/(ElE6
+ E2E5).
(3.212)
Here the n—dependence of the coefficients Ei has been suppressed. The coefficients Ei are Ei
E2
E3 =
EA
2nfi
»{ 2n\i
^a'Jn\liarc)
(3.213)
Jn\kaTc)
h2 - 2 ^ Jn{kbrc) + 2 y/nihrc)} ka.In\ka.rc)
h2-2-
~ '
(3.215)
In\kafc)
Yn(kbrc) +
(3.214)
2^Yl(kbrc)
}•
(3.216)
Elastic waves in doubly-periodic
E5
2fi
~kaJn\karc) -
rc
H
Jn{karc
media
173
- ka2(2(i +
X)Jn(karc), (3.217)
n
EQ =
2(i—
kbJ'n(kbrc)
Jn{kbrc)
E7
2(1
- kaln\karc)
rc
(3.218)
T c.
rc T
Yn{kar,
-ka2(2(i
+
X)Yn(karc (3.219)
Es =
n
r,
2ur
kbY^(kbrc)
Yn(kbrc)
(3.220)
The problem has now been reduced to two sets of unknowns: the coefficients Be' and Bt • The objective is thus to construct a system of equations which can be used to find non-trivial values for these coefficients. We now note that the problem for each of the Lame potentials <j> and ip is almost of the same type as the spectral scalar problem (3.63) - (3.67) discussed in the previous sections. The difference is that in the elastic case the two scalar fields are coupled via the conditions imposed on the boundary of the central cylinder. However the arguments which lead up to the dynamic Rayleigh identity (3.119) are independent of these boundary conditions and so a Rayleigh identity may be derived independently for each of the scalar fields in the multipole expansion (3.207). Another way of putting this is: given any scalar field V which obeys both the Helmholtz equation (A + fc2)^ = 0
(3.221)
and a quasi-periodicity condition of the type
V(r + Rp) = V(r)eikoR"
,
(3.222)
then this field can be represented as a multipole expansion oo
V(r)=
Y,
WJt{kr)+YtYt{kr)}eue
(3.223)
Multipole methods and homogenisation
174
in
two-dimensions
where the coefficients At and Bt are related via the equation oo
At=
Y,
(-l) £ + m S£-,(fca,fco)B m
•
(3.224)
m = — oo
The quantities Sj are the lattice sums defined in (3.101). A complete derivation of this identity can be found in section 3.2.2. Because the Lame potentials <j> and ip are both quasiperiodic and obey the Helmholtz equation, we obtain the two distinct identities oo
A^a)
=
Y.
{-l)t+mSl-t{KM)Bja)
,
(3.225)
(--L)e+mSl-e(h,k0)BmV
.
( 3 . 2 26)
m=—oo oo
Ae(b) =
Y m=—oo
These two sets of equations are linked via the boundary conditions (3.208), and upon substitution become the infinite linear system oo
Me^Be^
iab)
+ Me
W
Be -
Y
{-lf+mSl^{kaM)Bm(a)
= 0 ,
m = — oo oo
M^B^+M^BeM-
Y
(-l)'+m^_,(fc6,fc0)5mW
=0 .
m=—oo
(3.227)
This system has the aesthetically pleasing quality of separating the properties of the cavity surface (contained in the coefficients Mta ') from the effect of the geometry of the lattice (contained in the lattice sums Sj). It is remarked that the coefficients Mn and are real, whereas M„ ' and are pure imaginary, making the matrix M Hermitian. The system can be truncated; some justification for this is given in the next section. Zeros in the determinant of the truncated system correspond to non-trivial solutions of the Navier equations, and hence correspond to propagating modes. 3.5.2
Convergence
of the Rayleigh
matrix
Using the large order approximations for Bessel functions together with the definitions of the boundary terms given in equations (3.209-3.220), one can
Elastic waves in doubly-periodic
media
175
show that, as n —> oo, Mn(aa)
= o^r 2 (n)n(^) _2n )
Mjab)
= O (r2(n)n
(3.228) -In
USElA
M„ ( 6 a ) = O (r2{n)n (^Ei^
(3.229)
-In
(3.230)
M„W = C ? ( r 2 ( n ) n ( ^ ) " 2 n ) .
(3.231)
Similarly one can show from (3.93) that, for the lattice sums,
sY(ka,k0) = o(r(e)(^ye\
as^-s-oo.
(3.232)
This causes numerical difficulties when attempting to solve the system (3.227) directly, particularly when (kad/2) « 1, since the off-diagonal terms increase extremely rapidly with index I. One solution to this problem is to rescale the terms of the system so as to put it in a more amenable form. Specifically, we can introduce new unknown coefficients xn and yn such that _ I M (aa) \ B (a) , Mn{ab) „ - y Mn
(b)
1 >,
' Mn(ba) ( o ) B„ +B„(6) MjbV
(3.233)
(3.234)
Upon substituting this into (3.227), one can re-write the Rayleigh identity as oo
Xt + Y, (DiJaa)^m + DtJ^ym) = 0 m = —oo oo
yt+ Y, (DeJba)Xm + DimWym) = 0
(3.235)
Multipole methods and homogenisation
176
where the new coefficients D^01^'
in
two-dimensions
are defined as
(3.236) (ab)
n
s g n M/ a a > S gnM m ( O Q >M m W , oa
ue+mqY
,,
, ,
aa
{|M/ >||M m ( )|} A m (3.237) (4o)
^ m
_ s g nM/
6fc
b
>sgnM m WM m ( ">
——
-J/2
{|M/6b)||MmW|}
, V
- 1
;
^m-fV^i^o;
,
Am (3.238)
(66)
sgnM/ 6 6 ) / | M j 0 ° ) | \ 1 / 2
e+mY
(3.239) with
Aro = J\Mm™\\Mm™\ (l - ^ V ^ ) ) •
(3-240)
It is a straightforward matter to show that, if m is fixed as I —¥ oo, JW°*> = O (
^
ft)')
.
(3.241)
Similarly, if I is fixed as m —¥ oo, DeJal3)=0(m-\^{r-i)m)
•
(3-242)
Thus the new system has matrix elements which decay exponentially away from the main diagonal, giving rise to higher order multipole coefficients which decay similarly quickly. 3.5.3
Numerical
results and
comments
We begin with the analysis of the phononic band structures corresponding to the scalar problem (3.194), (3.195) of anti-plane shear oscillations. For all calculations we used normalised Lame coefficients A — 2.3, \i = 1.0.
Elastic waves in doubly-periodic
media
177
Fig. 3.11 Phononic band structure for anti-plane shear waves in a square array of circular cavities of radius r c = 0.2. The inset shows the irreducible segment of the first Brillouin zone; the horizontal axis is the arc-length traced around the boundary T - M
-K-T.
In Figures 3.11, 3.12(A) and 3.12(B) we show three sets of dispersion curves for different values of radius of a circular cavity in the unit cell (r c = 0.2,0.35,0.4). Each diagram exhibits a single acoustic mode, represented by a curve in the neighbourhood of the point T. The abscissa in these diagrams is the Bloch wavevector fco, according to the scheme depicted in the inset. This curve is symmetric under rotation about Y, which reflects the fact that the homogenised material corresponding to a square array of circular cavities is isotropic with respect to anti-plane shear. The diagrams in Figure 3.12 also show the presence of phononic band gaps, corresponding to frequencies of shear-waves which cannot propagate through the composite material. The phononic band gap increases in width when the area fraction of the cavities is increased. Next, we turn to the analysis of the elastic vector problem in twodimensions. As shown in the previous section, the computation of the phononic band structures requires evaluation of the determinant of the Rayleigh matrix. Direct calculation of the determinant from the system (3.227) is problematic due to the rapid growth both of the coefficients M n ' " ^ and of the dynamic lattice sums. Transformation of the system to the form (3.235) is essential for the accurate evaluation of the determinant.
178
Multipole methods and homogenisation
(A)
in
two-dimensions
(B)
Fig. 3.12 Phononic band structure for anti-plane shear waves in a square array of circular cavities of radius (A) rc = 0.35, and (B) rc = 0.40.
Fig. 3.13 Phononic band structure for shear waves (solid lines) and dilation waves (dashed lines) in a homogeneous elastic matrix. The letter S denotes a non-degenerate mode, while D denotes a mode of degeneracy two.
We commence with the dispersion curves for the unperturbed matrix (Figure 3.13). Here, to each dispersion curve is attached a label showing its degeneracy. Each matrix curve is split by the inclusion of cavities into the appropriate number of non-degenerate modes.
Elastic waves in doubly-periodic
media
179
Fig. 3.14 Phononic band structure for a square array of circular cavities of radius rc = 0.1.The letters a and b near lines denote modes which are mainly of dilational type, and shear type respectively.
Dispersion curves associated with different values of filling fraction are shown in Figures 3.14, 3.15, 3.16, and 3.17. Each of the modes appearing in these figures is of a hybrid type, being neither purely a shear wave nor a pure dilational wave but rather a mixture of the two, depending on the relative magnitude of the potentials cf> and tp. We expect that, in the limit of small filling fraction, these modes decouple completely into pure shear and dilational waves, and this hypothesis is borne out in the analysis of Figure 3.14, where the modes all appear in general to be slight perturbations on plane waves, which have also been marked on the diagram. Note the interesting non-crossing behaviour in the neighbourhood of the point P. Here a shear wave changes into a dilational wave, and a dilational wave evolves into a shear wave, while a second shear wave (originally degenerate with the first) passes through P virtually unperturbed. Compare this with the much simpler behaviour near the point Q. We have labeled curves in Figure 3.14 by their type (a for dilational, b for shear) where they lie close to the free-space lines of Figure 3.13: they are then close to being unalloyed. For small radii, the extent of the mixed nature of a mode is indicated by its distance from the matrix line. As the filling fraction increases (Figures 3.15 and 3.16) then the overall displacement of dispersion lines from the free-space lines can be seen to increase; the modes contain strong mixtures of shear and dilational components and so are more problematic to classify.
180
Multipole methods and homogenisation
K
Fig. 3.15 r c = 0.2.
r
M
in
two-dimensions
K
Phononic band structure for a square array of circular cavities of radius
(A)
(B)
Fig. 3.16 Phononic band structure for a square array of circular cavities of radius (A) rc = 0.30, and (B) rc = 0.35.
Compared to the scalar case (see Figures 3.11, 3.12), we note the presence of two acoustic modes in the vicinity of the point T. This is not surprising because the composite elastic structure can support both shear and pressure waves. Even though these waves do not decouple in the static limit it is to be expected that the dynamic solution will inherit some of the characteristics of two different static problems with the same geometry.
Elastic waves in doubly-periodic
K
Fig. 3.17 r c = 0.4.
r
M
media
181
K
Phononic band structure for a square array of circular cavities of radius
It can also be observed that the modes are no longer symmetric around r , in fact when approached from the M direction the slope of the lower curve is measured to be «Mr = 0.871, whereas approaching from the K direction the slope is a x r = 0.468. This disparity reflects the fact that the homogenised doubly-periodic elastic structure is not isotropic. We also observe (see Figure 3.17) a phononic band gap, specifying the range of frequencies for which no in-plane elastic waves can propagate within the composite. As in the scalar case, the gap is non-existent for
r
M
K
Fig. 3.18 Full phononic band structure for a hexagonal array of cavities, with a filling fraction of / = 0.502. The right-hand diagram shows the first irreducible segment of the Brillouin zone.
182
Multipole methods and homogenisation
in
two-dimensions
the lower values of filling fraction, and increases in width as the radii of the cavities grow larger. We now turn our attention to the calculation of the phononic band structure of a hexagonal structure. The formulation for this problem is identical to that for the square array, except that the basis vectors for the direct and reciprocal lattices must be changed. We choose a structure with the same material properties as discussed previously for the square array, and with a filling fraction of / = 0.502, which is the same as that for the square array shown in Figure 3.17. We note that the slopes in the vicinity of the T point for the hexagonal array are the same. For the lower curve, when approaching from the M direction we measure CCMG = 1.180 whereas when approached from the K direction we measure acK = 1-175. This is a difference of about 1%, which is within the computational accuracy of the estimate. We can contrast this with the case for the square array shown previously, where we obtained estimates for the slopes differing by about a factor of two for the same filling fraction. This is further evidence that the isotropy of the hexagonal structure under a shear load carries over into dynamical situations. It is also interesting to note the absence of a full phononic band gap for the hexagonal structure, for the same filling fraction. It is possible that this is because the presence of a gap depends on the relative distance between adjacent inclusions, which is smaller for a square lattice of a given filling fraction because the inclusions are stacked less densely. A full and comprehensive study of the evolution of the band structure for the hexagonal case would be necessary in order to confirm this.
3.6
Concluding remarks
In this chapter we have been primarily concerned with multipole methods to periodic problems. For each problem we start with a field which obeys either Laplace's equation or the Helmholtz equation, within a periodically replicated cell which contains some circular region of differing material properties. The periodicity of the arrangement allows us to specify the boundary conditions on the edge of the cell, and the boundary conditions on the circular region we can specify as we like without affecting the convergence of the solution. In each situation this approach leads to a 'semi-analytic' solution to the problem, in terms of a series of basis
Concluding remarks
183
functions which 'fit' the circular geometry. This is particularly useful when trying to manipulate the formulation so as to deduce asymptotic properties (such as the effective dielectric constant) of the material in question, and we have seen that it can also lead to finite-frequency correction terms to these normally static properties. In addition, the solution can lead to the understanding of the mathematical phenomenon of non-commuting limits, which can catch the unwary offguard when simultaneous limits are applied haphazardly. An interesting aspect of this method is that it 'splits' up the solution into different fields which are all microscopically uncoupled. All the interaction between the fields, either between T E / T M waves or shear/dilatational waves, occurs at the circular boundary. This means that any problem of interaction on this type could be studied, via a Helmholtz decomposition of the type (3.196). The case of conical incidence in electromagnetism, where an electromagnetic wave propagates obliquely through a periodic lattice is readily tackled in this way (in fact it is formally equivalent to the case of in-plane elastic waves) and the problem of obliquely propagating elastic waves is also soluble. Our choice of basis functions places a restriction on the type of problem that we are able to solve. While it is possible to use other sets of basis functions for differently shaped inclusions (e.g. one could use Mathieu functions for elliptical inclusions) this could be judged an excessive use of one's mathematical resources, and perhaps a better approach is to modify the approach to the integral equation ( (3.38) or (3.107)) so as to employ a more traditional boundary-element method. Alternatively one can obtain an asymptotic solution for a regular perturbation of the circular boundary. Both these approaches are, unfortunately, slightly out of the scope of this book.
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Index
band gaps, 140 Bloch-Floquet condition, 140 Bloch waves, 139, 140 Bloch wavevector, 140 boundary layer, 4, 6, 8, 61, 84
energy, asymptotics, 4, 6, 10 functional, 3 energy equivalent voids, 16 Green's function, dynamic, 142 Green's function, quasiperiodic, 142 static, 130
circular elastic inclusions, 28, 40 Clausius-Mossotti equation, 137 conformal map, 15, 14, 27 corrections to effective properties, 138
Helmholtz equation, 139 imperfectly bonded inclusions, 29, 81 inclusions with perfect bonding, 28 integral approximation for the multipole coefficients, 43
dispersion diagram, 56 dipole approximation, 152 dipole matrix, 8, 9 for inclusions, 16 for voids, 12, 16 symmetry of, 9 dispersion relation, 140 dynamic correction terms to static properties, 154-158
Keller's theorem, 157 Kolosov-Muskhelishvili's complex potentials, 22, 97 dipole fields, in terms of, 25 Lame potentials, 171 lattice approximations, 47 lattice, direct, 130 reciprocal, 130 lattice sums, dynamic, 143, 145 static, 132, 42 Lorentz-Lorenz equation, 137
effective properties, 18, 129 array of cylinders, 129 effective refractive index, 140 item eigenvalues, 59, 78, 115, 117 elasticity, boundary conditions for, 171-172 elasticity, 19, 168 electromagnetism, 138 electrostatics, 125
Maxwell-Garnett equation, 137 multipole coefficients, 44, 44 189
190 multipole expansion, static, 40, 126 dynamic, 141 Navier equations, 19, 168 neutral coated inclusions, 32 non-commuting limits, 158 phononic band structure, 176 Poisson summation formula, 130, 146 Rayleigh identity, static, 136 dynamic, 147-151 Rayleigh identity, for problems of elasticity, 174 source neutrality, 162, 163 spectral problem for an array of cylinders, 138 spectral problem on a unit sphere, 57 square array of circular inclusions, 40 square array of circular voids, 52 stress singularity exponent, 76, 115 thin conical inclusion, 57 traction boundary condition, 20 transmission conditions, 30
Index
Asymptotic Models of Fields in Dilute and Densely Packed Composites
his monograph provides a systematic study of asymptotic models of continuum mechanics for composite structures, which are either dilute (for example, two-phase composite structures with small inclusions) or densely packed (in this case inclusions may be close to touching). It is based on the results of recent research and includes a comprehensive analysis of dipole and multipole fields associated with defects in solids. The text covers static problems of elasticity in dilute composites as well as spectral problems. Applications of the mathematical models included in the book are in damage mechanics and in problems of design of composite structures that can be used as filters or polarisers of elastic waves.
P2Mhc ISBN I IM.IXM I1M /
Imperial College Press www.icpress.co.uk
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