Self-Consistent Methods for Composites: Vol.2 Wave Propagation in Heterogeneous Materials (Solid Mechanics and Its Applications)

SELF-CONSISTENT METHODS FOR COMPOSITES SOLID MECHANICS AND ITS APPLICATIONS Volume 150 Series Editor: G.M.L. GLADWEL...

Author: S.K. Kanaun | V. Levin

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SELF-CONSISTENT METHODS FOR COMPOSITES

SOLID MECHANICS AND ITS APPLICATIONS Volume 150 Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

Self-Consistent Methods for Composites Vol. 2: Wave Propagation in Heterogeneous Materials

by

S.K. KANAUN Instituto TecnolÓgico y de Estudios Superiores de Monterrey, Campus Estado de México, México and

V.M. LEVIN Instituto Mexicano del PetrÓleo, México

S.K. Kanaun Instituto Tecnolo´ gico y de Estudios Superiores de Monterrey Campus Estado de Mé xico, Mé xico V.M. Levin Instituto Mexicano del Petro´ leo, México

ISBN 978-1-4020-6967-3

e-ISBN 978-1-4020-6968-0

Library of Congress Control Number: 2008921358 c 2008 Springer Science+Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To our teachers: Lazar Markovich Kachanov and Isaak Abramovich Kunin

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2.

Self-consistent methods for scalar waves in composites . . . . 2.1 Integral equations for scalar waves in a medium with isolated inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The eﬀective ﬁeld method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The eﬀective medium method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Version I of the EMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Version II of the EMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Version III and IV of the EMM . . . . . . . . . . . . . . . . . . . . 2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.

Electromagnetic waves in composites and polycrystals . . . . 3.1 Integral equations for electromagnetic waves . . . . . . . . . . . . . . . 3.2 Version I of EMM for matrix composites . . . . . . . . . . . . . . . . . . 3.3 One-particle EMM problems for spherical inclusions . . . . . . . . 3.4 Asymptotic solutions of the EMM dispersion equation . . . . . . . 3.5 Numerical solution of the EMM dispersion equation . . . . . . . . 3.6 Versions II and III of the EMM . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The eﬀective ﬁeld method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 One-particle EFM problems for spherical inclusions . . . . . . . . . 3.9 Asymptotic solutions of the EFM dispersion equation . . . . . . . 3.9.1 Long-wave asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Short-wave asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Comparison of version I of the EMM and the EFM . . . . . . . . . 3.12 Versions I, II, and III of EMM . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Approximate solutions of one-particle problems . . . . . . . . . . . . . 3.13.1 Variational formulation of the diﬀraction problem for an isolated inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.2 Plane wave approximation . . . . . . . . . . . . . . . . . . . . . . . . .

5 7 14 14 16 17 19 21 21 24 27 30 33 36 42 47 48 48 50 51 54 58 59 59 60

viii

Contents

3.14 The EFM for composites with regular lattices of spherical inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Versions I and IV of EMM for polycrystals and granular materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.

5.

Axial elastic shear waves in ﬁber-reinforced composites . . . 4.1 Integral equations of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The eﬀective medium method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The eﬀective ﬁeld method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Integral equations for the local exciting ﬁelds . . . . . . . . 4.3.2 The hypotheses of the EFM . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The dispersion equation of the EFM . . . . . . . . . . . . . . . . 4.4 One-particle problems of EMM and EFM . . . . . . . . . . . . . . . . . . 4.4.1 The one-particle problem of the EMM . . . . . . . . . . . . . . 4.4.2 The one-particle problem of the EFM . . . . . . . . . . . . . . . 4.4.3 The scattering cross-section of a cylindrical ﬁber . . . . . 4.4.4 Approximate solution of the one-particle problem in the long-wave region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Solutions of the dispersion equations in the long-wave region . 4.5.1 Long-wave asymptotic solution for EMM . . . . . . . . . . . . 4.5.2 Long-wave asymptotic solution for EFM . . . . . . . . . . . . . 4.6 Short-wave asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Numerical solutions of the dispersion equations . . . . . . . . . . . . . 4.8 Composites with regular lattices of cylindrical ﬁbers . . . . . . . . 4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 71 75 76 77 78 80 84 84 85 87 89 89 91 92 95 97 98 99 103 105 108 114 115

Diﬀraction of long elastic waves by an isolated inclusion in a homogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1 The dynamic Green tensor for a homogeneous anisotropic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Integral equations for elastic wave diﬀraction by an isolated inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3 Diﬀraction of long elastic waves by an isolated inclusion . . . . . 123 5.4 Diﬀraction of long elastic waves by a thin inclusion . . . . . . . . . 128 5.4.1 Thin soft inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.4.2 Thin hard inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.5 Diﬀraction of long elastic waves by a short axisymmetric ﬁber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.6 Total scattering cross-sections of inclusions . . . . . . . . . . . . . . . . 138 5.6.1 An isolated inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6.2 Long-range scattering cross-sections . . . . . . . . . . . . . . . . 144 5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Contents

6.

7.

Eﬀective wave operator for a medium with random isolated inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Diﬀraction of elastic waves by a random set of ellipsoidal inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Green function of the eﬀective wave operator . . . . . . . . . . 6.3 Velocities and attenuations of long elastic waves in matrix composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Long elastic waves in composites with random thin inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Isotropic elastic medium with random crack-like inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Isotropic elastic medium with a random set of hard disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Long elastic waves in composites with short hard ﬁbers . . . . . . 6.5.1 Random sets of ﬁbers homogeneously distributed over orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Random set of ﬁbers of the same orientation . . . . . . . . . 6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic waves in a medium with spherical inclusions . . . . . . 7.1 Version I of the EMM for elastic waves . . . . . . . . . . . . . . . . . . . . 7.2 The one-particle problems of EMM . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Diﬀraction of a plane monochromatic wave by an isolated spherical inclusion . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 An approximate solution of the one-particle problems in the long-wave region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The dispersion equations of the EMM . . . . . . . . . . . . . . . . . . . . . 7.3.1 The EMM dispersion equation for longitudinal waves . 7.3.2 The EMM dispersion equation for transverse waves . . . 7.3.3 Total scattering cross-sections of a spherical inclusion . 7.3.4 The EMM dispersion equations in the short-wave region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Versions II and III of EMM for long waves . . . . . . . . . . . . . . . . . 7.5 Numerical solution of the EMM dispersion equations . . . . . . . . 7.6 The eﬀective ﬁeld method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 The hypotheses of the EFM . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 The eﬀective ﬁeld for transverse waves . . . . . . . . . . . . . . 7.6.3 The eﬀective ﬁeld equations for longitudinal waves . . . . 7.7 One-particle problems of EFM . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Transverse waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Longitudinal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 EFM dispersion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Long transverse waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Short transverse waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Long longitudinal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.4 Short longitudinal waves . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Numerical solution of the EFM dispersion equations . . . . . . . .

ix

155 155 162 166 168 168 177 182 183 184 187 189 189 194 194 198 200 202 204 205 207 208 213 217 218 220 223 228 228 232 235 235 238 239 240 244

x

Contents

7.9.1 Transverse waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Longitudinal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.3 Longitudinal waves in epoxy-lead composites . . . . . . . . . 7.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

244 247 250 254 256

Elastic waves in polycrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 General consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The eﬀective medium method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The one-particle problem of EMM . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Polycrystals with orthorhombic grains . . . . . . . . . . . . . . . . . . . . . 8.5 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 258 260 262 265 270 272 276 276

A. Special tensor bases of four-rank tensors . . . . . . . . . . . . . . . . . . A.1 E-basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 P -basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Averaging the elements of the E- and P -bases . . . . . . . . . . . . . . A.4 Tensor bases of four-rank tensors in 2D-space . . . . . . . . . . . . . .

279 279 280 282 283

8.

B. The Percus-Yevick correlation function . . . . . . . . . . . . . . . . . . . 285 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Preface

This second volume of this book is devoted to the problem of wave propagation in heterogeneous materials. Self-consistent methods are developed for the analysis of various modes of wave propagation in composite materials and polycrystals. Velocities and attenuation coeﬃcients are calculated of electromagnetic and elastic monochromatic waves propagating in matrix composite materials with random and regular microstructures, and in polycrystals. Predictions of the methods are compared with expreimental data and with exact solutions available in the literature. The volume contains few references to the ﬁrst volume of the book, and may be read independently. We thank Dr. Eugeny Pervago for help in calculations, and deeply appreciate the corrections and suggestions made by the Series Editor, Professor Graham Gladwell, in the revision of the manuscript. The authors thank the Technological Institute of Higher Education of Monterrey (State of Mexico Campus), and Mexican Oil Institute, for supporting this book. Mexico, August 2007

S.K. Kanaun V.M. Levin

Notations

εij , εαβ , Cijkl , Cαβλµ , .... σij = Cijkl εkl T(ij)kl = 12 (Tijkl + Tjikl ) a, b, T or a, b, T, ... x, x ⊗ (a ⊗ b ⇒ ai bj ) a · b (a · b = ai bi ) a × b (a × b ⇒ eijk aj bk ) δij ijk E 1 , E 2 , ..., E 6 P 1 , P 2 , ..., P 6 ∂ ∇i = ∂x i defu ⇒ (i uj) divT ⇒ i Tijk... rotT = × T ⇒ eijk j Tklm... 2 2 2 = ∂2 + ∂2 + ∂2 ∂x ∂x ∂x 2 3 1 f (k) = f (x) exp(ik · x)dx. k, k δ(x) Ω(x) V V (x) = 1 if x ∈ V V (x) = 0 if x ∈ /V f (x) f (x)|x

Lower case Greek or Latin indices are tensorial Summation with respect to repeating indices is implied. Parentheses in indices mean symmetrization. Non-index notations for vectors and tensors. Point and vector of a point in 3D(2D)-space. Tensor product of vectors and tensors. Scalar product of vectors and tensors. Vector product of vectors and tensors. The Kronecker symbol. The Levi-Civita symbol. The basic four-rank tensors (Appendix A.1) The basic four-rank tensors (Appendix A.2) The Nabla operator. Deformation operator. Divergence of a tensor ﬁeld T. Rotor (curl) of a tensor ﬁeld T. The Laplace operator. The Fourier transform of a function f (x). Point and vector of a point in the Fourier transform space. The Dirac delta-function. Delta-function concentrated on a surface Ω. Region in 3D(2D)-space. Characteristic function of a region V. Mean value of a random function f (x). Mean value of f (x) under the condition that x ∈ V.

xiv

Notations

C 0, C C∗ C1 = C − C0 λ0 , µ0 ; λ, µ; λ∗ , µ∗ K0 , K, K∗

Tensors of elastic moduli of the matrix. Inclusions, and the eﬀective medium. Deviation of the elastic moduli in inclusions. The Lame parameters of the matrix. Inclusions, and the eﬀective medium. Bulk moduli of the matrix, inclusions, and the eﬀective medium. The Young moduli of the matrix, inclusions, and the eﬀective medium. The Poisson ratio of the matrix, inclusions and the eﬀective medium. Densities of the matrix, inclusions, and the eﬀective medium.

E0 , E, E∗ ν0 , ν, ν∗ ρ0 , ρ, ρ∗ K 1 = K − K0 µ1 = µ − µ0 λ1 = λ − λ0 ρ1 = ρ − ρ0 M0 = K0 + 43 µ0 M = K + 43 µ, M∗ =K∗ + 43 µ∗ l0 =

M0

ρ0

,l=

Deviation of properties in inclusions.

M ρ ,

∗ l∗ = M ρ∗ t0 = µρ00 , t = µρ , t∗ = µρ∗∗ t2 µ0 η02 = l20 = λ0 +2µ 0 0

Speed of longitudinal waves in the matrix, inclusions, and the eﬀective medium. Speed of transverse waves in the matrix, inclusions, and the eﬀective medium.

2

µ η 2 = tl2 = λ+2µ ω α0 = lω0 , β0 = tω0 α = ωl , β = ωt α∗ = lω∗ , β∗ = tω∗ Jn (z) Hn (z) = Jn (z) − iYn (z) jn (z) hn (z) = jn (x) − iyn (x) Pn (z)

Frequency of oscillations. Wave numbers of longitudinal and transverse waves in the matrix material. Wave numbers of longitudinal and transverse waves in inclusions. Wave numbers of longitudinal and transversel waves in the eﬀective medium. The Bessel function of order n. The Hankel function of order n. The spherical Bessel function of order n. The spherical Hankel function of order n. The Legendre polynomial of order n.

1. Introduction

The problem of wave propagation in composite materials has many important applications. The solution of this problem allows us to predict the response of composite materials to various types of dynamic loadings; this problem forms the theoretical background for non-destructive ultrasonic evaluation of microstructures of composites. The main objectives of the theory in this problem are the dependencies of the phase velocity and attenuation coeﬃcient of the mean (coherent) wave ﬁeld propagating in the composite on the frequency of the incident ﬁeld (dispersion curves) and on the details of the composite microstructure. For composite materials with random microstructures, this problem cannot be solved exactly, and only approximate solutions are available. Self-consistent methods are widely used for the construction of such approximate solutions. In self-consistent methods, complex actual wave ﬁelds propagating in heterogeneous media are approximated by simple ones using physically reasonable hypotheses. All the self-consistent methods are based on two types of such hypotheses. The ﬁrst one reduces the problem of interactions between many inclusions in the composite to a problem for one inclusion (the one particle problem). The second hypothesis is the condition of self-consistency. For application of these methods, the heterogenous medium should have speciﬁc features: a typical element (particle) should exist in the medium. Such a particle may be an inclusion in the matrix-inclusion composites, a grain in random polycrystalline materials, a crack in materials with defects, etc. Application of self-consistent methods to the solution of wave propagation problems for random heterogeneous media has a long history. Self-consistent solutions of the problem of scalar wave propagation through a medium with many particles may be found in the works of Maxwell and Rayleigh. During more than a century, in a number of works, these methods were extensively developed and used for the solution of various wave propagation problems. The hypotheses of the methods were often not formulated explicitly but used as more or less evident assumptions. Although nowadays there exist various modiﬁcations of self-consistent methods, it is possible to point out two main ideas of self-consistency: the eﬀective ﬁeld (EFM) and the eﬀective medium (EMM) methods. These methods are based on diﬀerent hypotheses and, generally speaking, give diﬀerent results when applied to the same composite

2

1. Introduction

material. It is necessary to emphasize that the hypotheses of the methods are introduced in heuristic manners, and one cannot evaluate ad hoc the region of their validity. Only the comparison of predictions of the methods with experimental data or with exact solutions (when the latter exist) allow us to estimate the regions where these predictions are reliable. In this volume, attention is focused on electromagnetic and elastic wave propagation through matrix composites and polycrystals. The diﬃculties in application of self-consistent methods to such materials are connected with the ﬁnite sizes of the particles. The “particles” have their own internal ﬁelds that depend on random shapes and properties of the inclusions. In the framework of self-consistent methods, these ﬁelds should be found from the solutions of so-called one particle problems: diﬀraction of a monochromatic incident wave by an isolated inclusion. Exact solutions of these problems are known only for inclusions of rather simple forms (sphere, cylinder, penny shaped crack). In the long-wave region, the solutions of the one-particle problems may be found for a wider class of the inclusions. In the general case, the one-particle problem may be solved only numerically. In spite of rapid development of computer capacities, numerical simulation of wave propagation in random composites faces essential technical difﬁculties. This problem is more complex than numerical simulations of static ﬁelds in random composites for the following reasons. First, for the numerical simulation of wave ﬁelds, discretization meshes should be much ﬁner than those used for the simulation of static ﬁelds. As a result, the computational cost of the wave problems is essentially higher than the cost of static problems. This cost grows essentially if the propagating waves are comparable or shorter than the linear sizes of inclusions. Secondly, the ergodic hypothesis that is usually used for the solution of the homogenization problems in statics does not hold for wave problems: averaging the detailed wave ﬁeld over the ensemble realization of the random set of inclusions does not coincide with the spatial averaging of a typical realization of this ﬁeld. That is why self-consistent methods remain the only eﬃcient tool for the solution of the homogenization problem in the case of wave propagation. In this volume of the book, we have the following objectives: • To give classiﬁcation of various self-consistent methods applied to the wave propagation problems • To develop general algorithms for the application of the methods to various types of propagating waves • To compare predictions of the methods with experimental data, and indicate the regions of the parameters of the composites where the methods give reliable results The contents of this volume of the book is as follows. In Chapter 2, scalar wave propagation problem is considered. The main hypotheses of the EFM and of various versions of the EMM are formulated

1. Introduction

3

and discussed. Every method leads to a speciﬁc dispersion equation for the wave number of the mean wave ﬁeld propagating in the composite. The general algorithms of the methods developed in this Chapter are used later for the solution of the homogenization problem for electromagnetic and elastic wave propagation. Chapter 3 is devoted to electromagnetic wave propagation in matrix composites and polycrystals. For composites with spherical inclusions, the dispersion equations of the EFM and the EMM derived in this Chapter serve for all frequencies of propagating waves, volume concentrations, and properties of the inclusions. The long and short wave asymptotic solutions of these equations are obtained in explicit analytical forms. The main diﬀerences in predictions of various self-consistent methods for velocities and attenuation coeﬃcients of electromagnetic waves are indicated and analyzed. Chapter 4 is devoted to the problem of axial elastic shear wave propagation in a homogeneous medium reinforced with unidirected cylindrical ﬁbers. For this case, speciﬁc features of elastic wave propagation through composites are indicated. A symbolic form of the integral equations of the problem that is convenient for application of self-consistent methods is introduced. Composites reinforced with random and regular sets of unidirected ﬁbers are considered. For regular composites, the EFM dispersion equation indicates the existence of stop and pass bands in the frequency region for propagating waves. The predictions of the method are compared with exact solutions existing in the literature. In Chapter 5, the one-particle problem for the elastic wave propagation is considered in detail. The problem is the diﬀraction of a plane monochromatic wave by an isolated inclusion in a homogeneous elastic medium. For spherical or cylindrical inclusions, exact solutions of this problem have the form of series of vector harmonics, and serve for all frequencies of the incident wave and properties of the inclusions. For long waves, the solution of the one particle problem may be found for a wider class of inclusions. Such solutions for ellipsoidal inclusions, thin hard disks, cracks, and short axisymmetric ﬁbers are presented. In Chapter 6, the homogenization problem for long-wave propagation is considered. The EFM is used here for the construction of the eﬀective wave operator for a medium with spherical inclusions. This operator turns out to be non-local, and corresponds to the moment theory of elasticity for a medium with constrained rotations. The long-wave asymptotics of the velocities and attenuation coeﬃcients of the mean wave ﬁelds in composites with ellipsoidal inclusions, thin inclusions, cracks, and long ﬁbers are obtained. In Chapter 7, the EFM and EMM are applied to the problem of elastic wave propagation in composites with spherical inclusions. The developed dispersion equations of the methods serve for all frequencies of the incident ﬁeld, properties and volume concentrations of the inclusions. Long- and short-wave asymptotic solutions of these equations are obtained in explicit analytical

4

1. Introduction

forms. Numerical solutions of the dispersion equations are constructed for frequencies that cover long-, middle-, and short-wave regions of propagating waves in comparison with the inclusion sizes. Predictions of the methods are compared with experimental data existing in the literature. For random as well as regular composites, the dispersion equations predict the existence of several modes (branches) of wave propagation that may be interpreted as acoustical and optical branches. Such branches are obtained for electromagnetic as well as for elastic waves. The diﬀerences in the predictions of diﬀerent self-consistent methods are indicated and analyzed. Propagation of the elastic waves in polycrystalline materials is considered in Chapter 8. The version of the EMM developed in this chapter leads to a dispersion equation that describes all the characteristic features of elastic wave propagation in polycrystals. The Rayleigh, stochastic, and diﬀusion frequency regions may be clearly indicated from the analysis of the solution of this dispersion equation. The comparison of the theoretical predictions of the method with experimental data are presented.

2. Self-consistent methods for scalar waves in composites

In this chapter, we consider self-consistent methods in application to a simple model problem: propagation of scalar waves in a medium with isolated inclusions. The main hypotheses of the eﬀective ﬁeld method and various versions of the eﬀective medium method are introduced. For every method, the algorithm of development of dispersion equations for the mean (coherent) wave ﬁeld propagating in the composites is presented.

2.1 Integral equations for scalar waves in a medium with isolated inclusions A function u(x, t) that describes scalar wave ﬁelds in a heterogeneous medium satisﬁes the following diﬀerential equation u −

1 v 2 (x)

∂2u = 0. ∂t2

(2.1)

Here is the Laplace operator, v(x) is the velocity of the wave, x(x1 , x2 x3 ) is a point of 3D-space, t is time. In what follows, we study monochromatic waves of frequency ω; the dependence of u(x, t) on time is deﬁned by the factor eiωt . In this case, u(x, t) = u(x)eiωt and the amplitude u(x) of the ﬁeld satisﬁes the Helmholtz equation u + k 2 (x)u(x) = 0, k(x) =

ω . v(x)

(2.2)

If the medium consists of a homogeneous matrix and a set of isolated inclusions, the parameter k 2 (x) in this equation takes the form k 2 (x) = k02 + k12 V (x), k12 = k 2 − k02 ,

(2.3)

where k0 is the wave number of the matrix, k is the same for the inclusions, V (x) is the characteristic function of the region V occupied by the inclusions (V (x) = 1 if x ∈ V, V (x) = 0 if x ∈ / V ). The original equation (2.2) may be rewritten in the form u + k02 u(x) = −k12 V (x)u(x).

(2.4)

6

2. Self-consistent methods

Let g0 be the inverse of the operator l0 = + k02 on the left-hand side of this equation. The operator g0 is an integral operator the kernel g0 (x) of which (the Green function) satisﬁes the following equation (2.5) + k02 g0 (x) = −δ(x), where δ(x) is the 3D-Dirac delta-function. The solution of this equation has the form g0 (x) =

e−ik0 r , r = |x|. 4πr

(2.6)

After applying the operator g0 to both sides of (2.4) we obtain the integral equation for the wave ﬁeld u(x) in the form (2.7) u(x) = u0 (x) + k12 g0 (x − x )u(x )V (x )dx . Here u0 (x) is the incident ﬁeld that would have existed in the medium without inclusions, and with prescribed conditions at inﬁnity. This ﬁeld satisﬁes the equation l0 u0 = 0, and in what follows the incident ﬁeld u0 (x) is a plane wave with the wave vector k0 propagating in the matrix material: u0 (x) = U0 e−ik0 ·x , k0 ·x =k0 n0 · x.

(2.8)

Here n0 is the wave normal of the propagating wave, dot is the scalar product of two vectors n0 · x =n0i xi ... For a single inclusion (2.7) may be written in the form (2.9) u(x) = u0 (x) + k12 g0 (x − x )u(x )dx , v

where v is the region occupied by the inclusion. Note that the integral term in this equation may be interpreted as the ﬁeld scattered by the inclusion. If the ﬁeld u(x) inside the inclusion is known, the scattered ﬁeld us (x) us (x) = k12 g0 (x − x )u(x )dx (2.10) v

may be obtained by calculating this integral. The far asymptotic of the scattered ﬁeld (approximation of us (x) for large |x|) has the form k12 e−ik0 r s F (n), F (n) = u(x)eik0 n·x dx, (2.11) u (x) = r 4π v x r = |x|, n= . r

2.2 The eﬀective ﬁeld method

7

Here the origin (x = 0) is taken inside the inclusion, and the equation 1/2 1/2 = |x|2 − 2x · x + |x |2 |x − x | = [(x − x ) · (x − x )] x · x ≈ |x|(1 − ) = r − n · x (2.12) |x|2 is taken into account for |x| >> |x |. The function F (n) in (2.11) is called the amplitude of the scattered ﬁeld. If n = n0 , F (n0 ) is the forward scattering amplitude. If there are many inclusions, the function V (x) in (2.7) deﬁnes their spatial positions. In what follows, V (x) is considered as a spatially homogeneous random function. Hence, the wave ﬁeld u(x) is also a random function that may be presented in the form u(x) = u(x) + u1 (x).

(2.13)

Here u(x) is the detailed wave ﬁeld averaged over realizations of the random function V (x), u1 (x) is the ﬂuctuating part of this ﬁeld. The main objective of the theory is the construction of the mean (coherent) part of the ﬁeld propagating in the composite. If the incident ﬁeld is a plane wave, the mean wave ﬁeld u(x) is also a plane wave in the composite with spatially homogeneous random sets of inclusions. In this case, u(x) may be interpreted as a ﬁeld propagating in some homogeneous medium with the eﬀective dynamic properties of the composite. The calculation of these properties is the content of the homogenization problem for wave propagation. Note that for wave problems, averaging u(x) over the ensemble realizations of V (x) cannot be replaced by the spatial averaging of u(x) over the representative volume element of the composite material (the ergodic property does not hold for wave ﬁelds). In the next sections, we consider the application of self-consistent methods for the solution of homogenization problems for wave propagation.

2.2 The eﬀective ﬁeld method Let V (x) correspond to a ﬁxed typical realization of a spatially homogeneous random set of inclusions. Such a function may be presented in the form vi (x), (2.14) V (x) = i

where vi (x) is the characteristic function of the region occupied by the ith inclusion. Equation (2.7) for the wave ﬁeld u(x) in the composite may be rewritten in the form g0 (x − x )u(x )dx , (2.15) u(x) = u∗ (x) + k12 vi g0 (x − x )u(x )dx . (2.16) u∗ (x) = u0 (x) + k12 j=i

vj

8

2. Self-consistent methods

Comparison of (2.9) and (2.15) shows that the inclusion vi may be considered as an isolated one in the matrix material by action of a local exciting ﬁeld u∗ (x). According to (2.16) the ﬁeld u∗ (x) is the sum of the incident ﬁeld u0 (x) and the ﬁelds scattered by the surrounding inclusions except inclusion vi (see Fig. 2.1). Let (2.15) be solved for an arbitrary exciting ﬁeld u∗ (x), and the ﬁeld inside the inclusion vi be presented in the form u(x) = Λi u∗ (x),

(2.17)

where Λi is a linear operator that depends on the physical properties of the matrix and the inclusion, and the shape of the latter. Substituting (2.17) into (2.15) and (2.16) we obtain the following system 2 g0 (x − x )Λi u∗ (x )dx , (2.18) u(x) = u0 (x) + k1 i

u∗ (x) = u0 (x) + k12

vi

j=i

g0 (x − x )Λj u∗ (x )dx .

(2.19)

vi

It is seen from these equations that the ﬁeld u∗ (x) may be considered as the main unknown of the problem. If this ﬁeld is found from (2.19), the detailed wave ﬁeld in the composite may be to constructed from (2.18). The main hypotheses of the eﬀective ﬁeld method (EFM) concern the structure of the local exciting ﬁeld u∗ (x) that acts on every inclusion in the composite. The ﬁrst of these hypotheses (H1 ) is formulated as follows. H1 . Field u∗ (x) is a plane wave in the vicinity of each inclusion in the composite. The actual exciting ﬁeld that acts on the inclusions is random for composites with random microstructures. The assumption that this ﬁeld is a plane wave in the vicinity of each inclusion simpliﬁes the construction of the wave ﬁeld inside this inclusion. According to hypothesis H1 , the local exciting ﬁeld acting on the jth inclusion has the form uj∗ (x) = U∗j e−ik∗ ·(x−xj ) , x ∈ vj .

Fig. 2.1. The one-particle problem for the eﬀective ﬁeld method.

(2.20)

2.2 The eﬀective ﬁeld method

9

Here xj is the center of the inclusion vj , k∗ and U∗j are the unknown wave vector and amplitude of the exciting ﬁeld uj∗ (x). Thus, the ﬁeld inside the jth inclusion is to be found from the solution of (2.15) for the exciting ﬁeld u∗ (x) in form (2.20) (the one-particle problem). Let (2.15) be solved, and u(x) be obtained in the form

j u(x) = Λj U∗j e−ik∗ ·(x−x ) , x ∈ vj . (2.21) Here the operator Λj depends on the properties of the matrix, the inclusion, and the form of the latter. Because of linearity of the operator Λj , the ﬁeld u(x) may be written in the form

j u(x) = Λj U∗j e−ik∗ ·(x−x )

j j j = Λj e−ik∗ ·(x−x ) eik∗ ·(x−x ) U∗j e−ik∗ ·(x−x ) = λj (x − xj )u∗ (x), j

j

−ik∗ ·x

λ (x) = Λ (e

ik∗ ·x

)e

(2.22) .

(2.23)

Note that the function λj (x) may be constructed for the inclusion centered at the origin (x = 0). Let us introduce the function λ(x) that coincides with λj (x − xj ) inside the jth inclusion and is equal to zero in the matrix; and a function u∗ (x) that coincides with the local exciting ﬁeld uj∗ (x) inside every region vj . The function λ(x) composes a stationary random ﬁeld with a constant mean value. Using these functions we can rewrite (2.18) for the total wave ﬁeld u(x) and (2.19) for the local exciting ﬁeld u∗ (x) in the forms 2 )u∗ (x )V (x )dx , (2.24) u(x) = u0 (x) + k1 g0 (x − x )λ(x )u∗ (x )V (x, x )dx . (2.25) u∗ (x) = u0 (x) + k12 g0 (x − x )λ(x Here V (x ; x) is the characteristic function (with argument x ) of the region Vx deﬁned by the equation vi , when x ∈ vk . (2.26) Vx = i=k

The function V (x, x ) is equal to zero when points x and x are inside the same inclusion, and V (x, x ) = V (x ) if x and x belong to diﬀerent inclusions, or x is in the matrix. Averaging (2.24) over the ensemble of realizations of the random function V (x) we ﬁnd the mean wave ﬁeld u(x) in the composite in the form

2 ) u∗ (x )|x dx . (2.27) u(x) = u0 (x) + pk1 g0 (x − x ) λ(x

10

2. Self-consistent methods

Here p = V (x) is the volume concentration of the inclusions, u∗ (x)|x is the mean value of the ﬁeld u∗ (x) by the condition that x ∈ V. To derive (2.27) we used the equation

λ(x)u (2.28) ∗ (x)V (x) = V (x) λ(x)u∗ (x)|x = p λ(x) u∗ (x)|x that holds if all the inclusions are identical and the function λ(x) is the same in every region vj . If there are deviations in the shapes and sizes of the inclusions, we assume that the random functions λ(x) and u∗ (x) are statistically independent. Because λ(x) is a stationary random ﬁeld, averaging over the ensemble realizations may be replaced by spatial integration, and we ﬁnd

1 1 λ(x) = lim (2.29) λ(x)dx = λ(x)dx = λ0 . Ω→∞ Ω Ω v0 v 0 Here Ω is a region that occupies all 3D-space in the limit Ω → ∞, v0 is the volume of a typical inclusion. It follows from the deﬁnition (2.22) of the function λ(x) that the value of the constant λ0 depends on the wave number of the eﬀective ﬁeld: λ0 = λ0 (k∗ ). In order to ﬁnd the mean u∗ (x)|x in (2.27) let us average equation (2.25) for u∗ (x) under the condition that x ∈ V. As a result we obtain u∗ (x)|x = u0 (x) + k12 g0 (x − x )λ0 (k∗ ) u∗ (x )|x , x V (x, x )|x dx . (2.30) Here we use the equation

)u∗ (x )V (x, x )|x = λ0 (k∗ ) u∗ (x )|x , x V (x, x )|x λ(x that follows from the deﬁnition (2.26) of the function V (x, x ). The two-point mean u∗ (x )|x , x is the mean value of the ﬁeld u∗ (x ) under the condition that the points x and x are inside diﬀerent inclusions. The conditional mean V (x, x )|x on the right hand side of this equation depends on the geometrical properties of the spatial distribution of the inclusions. For a spatially homogeneous random function V (x), this mean takes the form < V (x ; x)|x >= pΨ (x − x)

(2.31)

Here Ψ (x −x) is a function of the diﬀerence between vectors x and x . For a spatially isotropic random ﬁeld of inclusions, this function depends only on the distance between x and x (Ψ (x − x) = Ψ (|x − x|)). The properties of this function follow from the deﬁnition (2.26) of the function V (x ; x): Ψ (x) is continuous and

2.2 The eﬀective ﬁeld method

11

Fig. 2.2. Typical behavior of the correlation function Ψ (r) for an isotropic random set of spherical inclusions of radius a.

Ψ (0) = 0,

Ψ (∞) = 1.

(2.32)

Typical behavior of the function Ψ (r) is shown in Fig. 2.2. Sometimes in the physics literature, the function Ψ (r) is called a “correlation hole” for a typical inclusion. As it is seen from (2.30), the conditional mean u∗ (x)|x is expressed via a more complex conditional mean u∗ (x )|x , x (averaging under the condition that points x and x belong to V ). This two-point conditional mean can be expressed via a similar three-point conditional mean using the same equation (2.25). Actually, if we average (2.25) under the condition x, x ∈ V , the mean value of the function u∗ (x) under the condition that x , x , x ∈ V, etc. appears in the right-hand side of this equation. As a result, we come to an inﬁnite chain of equations that connects all the multi-point conditional means of the local exciting ﬁeld u∗ (y). In order to obtain a closed equation for the conditional mean u∗ (x)|x one has to accept an additional hypothesis (H2 ) concerned with the properties of the multi-point conditional means. The simplest hypothesis of this type is called the quasicrystalline approximation, and according to this hypothesis we state that ∗ (x). u∗ (x )|x , x = u∗ (x )|x = u

(2.33)

The ﬁeld u ∗ (x) is called the eﬀective exciting ﬁeld for all the inclusions in the composite. Thus, hypothesis H2 of the EFM may be formulated as follows H2 . The mean value of the local exciting ﬁeld u∗ (x ) under the condition that points x and x are inside diﬀerent inclusions coincides with the same mean under the condition that only point x is inside the region V occupied by the inclusions. For the problem of scalar wave propagation in a medium with point scatterers, the hypotheses of the EFM were formulated explicitly in [26, 70, 71].

12

2. Self-consistent methods

Hypothesis H2 closes the chain of the equations for multi-point conditional means of the eﬀective ﬁeld at the ﬁrst step (2.33) together with (2.30) give us the following integral equation for the eﬀective ﬁeld u ∗ (y) u ∗ (x) = u0 (x) + pk12 g0 (x − x )λ0 (k∗ ) u∗ (x )Ψ (x − x )dx (2.34) From (2.27), we obtain for the mean wave ﬁeld the equation u(x) = u0 (x) + pk12 g0 (x − x )λ0 (k∗ ) u∗ (x )dx .

(2.35)

After excluding the incident ﬁeld u0 (x) from these two equations we go to the closed equation for the eﬀective ﬁeld in the form 2 u∗ (x )Φ(x − x )dx , (2.36) u ∗ (x) = u(x) − pk1 g0 (x − x )λ0 (k∗ ) Φ(x) = 1 − Ψ (x).

(2.37)

Note that the function Φ(x) in (2.37) is equal to zero outside a ﬁnite vicinity of the origin (x = 0). The size of this vicinity has the order of the correlation radius of the random set of inclusions. Equation (2.36) is a convolution equation. Therefore, if the mean ﬁeld ∗ (x) is u(x) is a plane wave with the wave vector k∗ , the eﬀective ﬁeld u also a plane wave with the same wave vector ∗ (y) = U∗ e−ik∗ m·x . u(x) = U e−ik∗ m·x , u

(2.38)

After applying the Fourier transform operator to (2.36) we obtain the following linear algebraic equation connecting the Fourier transforms of the eﬀective and mean wave ﬁelds u∗ (k). u ∗ (k) = u(k) − pk12 GΦ (k)λ0 (k∗ )

(2.39)

Here we denote the Fourier transform of functions by the same letter with argument k: (f (k) = f (x) exp(ik · x)dx), the function GΦ (k) is the following integral (2.40) GΦ (k) = g0 (x)Φ(x)eik·x dx. From (2.39) we ﬁnd the connection between the Fourier transforms of the eﬀective and mean ﬁelds in the form −1 u(k). (2.41) u ∗ (k) = 1 + pk12 GΦ (k)λ0 (k∗ ) Let us return to equation (2.35) for the mean wave ﬁeld. The Fourier transform of (2.35) together with (2.41) give us the equation

2.2 The eﬀective ﬁeld method

u(k) = u0 (k) − pk12 g0 (k)λ0 (k∗ ) u∗ (k) −1 u(k), = u0 (k) − pk12 g0 (k)λ0 (k∗ ) 1 + pk12 GΦ (k)λ0 (k∗ ) 1 g0 (k) = 2 . k − k02

13

(2.42)

Here g0 (k) is the Fourier transform of the Green function in (2.6). Multiplying both parts of (2.42) with the function l0 (k) = k 2 − k0 2 and taking into account the equations l0 (k)g0 (k) = 1,

l0 (k)u0 (k) = 0,

we obtain −1

u(k) = 0. k 2 − k02 − pk12 λ0 (k∗ ) 1 + pk12 GΦ (k)λ0 (k∗ )

(2.43)

(2.44)

Because the Fourier transform of the mean wave ﬁeld in (2.38) has the form u(k) = (2π)3 U δ(k − k∗ m), the multiplier in front of function u(k) in (2.44) should be equal to zero for k = k∗ −1 k∗2 − k02 − pk12 λ0 (k∗ ) 1 + pk12 GΦ (k)λ0 (k∗ ) = 0.

(2.45)

This equation is in fact the dispersion equation for the wave number k∗ of the mean wave ﬁeld in the composite. The functions λ(k∗ ) in this equation should be obtained from the solution of the one-particle problem. The eﬀective wave number k∗ is the main objective of the theory: the phase velocity and attenuation coeﬃcient of the mean wave ﬁeld in the composite are connected with k∗ by the equations v∗ =

ω , Re(k∗ )

γ = − Im(k∗ ).

(2.46)

Note that the hypotheses of the EFM formulated above are equivalent to the following two hypotheses. H1 . Every inclusion in the composite behaves as an isolated one in the original matrix under the action of a plane eﬀective wave ﬁeld u ∗ (x) u ∗ (x) = U∗ exp(−ik∗ ·x).

(2.47)

This ﬁeld is the same for all the inclusions. ∗ (x) is the local exciting ﬁeld u∗ (x) averaged under H2 . The eﬀective ﬁeld u the condition x ∈ V. u ∗ (x) = u∗ (x)|x .

(2.48)

Application of these hypotheses leads to the same one-particle problem and the same dispersion equation (2.45). Thus, these hypotheses are equivalent to the quasicrystalline approximation.

14

2. Self-consistent methods

2.3 The eﬀective medium method In this section, another self-consistent method (the eﬀective medium method) is used for the solution of the homogenization problem for matrix composites and polycrystals. In the literature, the EMM exists in several diﬀerent versions. It is possible to say that the method is in fact a set of methods joined by a common hypothesis that, for the construction of the ﬁeld inside any inclusion in the composite, the inhomogeneous medium outside some vicinity of this inclusion may be replaced by a homogeneous medium with the eﬀective properties of the composite. Various versions of the EMM are considered in this Section. 2.3.1 Version I of the EMM The simplest version of the EMM (version I) is based on the following two hypotheses (See Fig. 2.3). I1 . Each inclusion in the composite behaves as an isolated one in a homogeneous medium with the eﬀective properties of the composite. The exciting ﬁeld for every inclusion is the ﬁeld propagating in the eﬀective medium. I2 . The mean wave ﬁeld in the composite medium coincides with the ﬁeld propagating in the homogeneous eﬀective medium. Hypothesis I1 reduces the problem of interactions between many inclusions in the composite to a one-particle problem. Hypothesis I2 is the condition of self-consistency. According to these hypotheses, the ﬁeld inside each inclusion in the composite is to be found from the solution of the following equation e−ik∗ |x| 2 . (2.49) g∗ (x − x )u(x )dx , g∗ (x) = u(x) = u(x) − k∗1 4π|x| v 2 Here k∗1 = k 2 − k∗2 , k∗ is the wave number of the eﬀective medium, g∗ (x) is the Green function of the operator + k∗2 . Let the mean wave ﬁeld u(x)

Fig. 2.3. The one-particle problem of version I of the EMM.

2.3 The eﬀective medium method

15

be a plane wave with the wave vector k∗ = k∗ m (u(x) = U e−ik∗ ·x ), and the inclusion v be centered at the origin (x = 0). The general solution of (2.49) may be presented in the form u(x) = Λ∗ [u(x)] ,

(2.50)

where Λ∗ is a linear operator. For the inclusion centered at the point xk = 0, the operator Λ∗ should be replaced by a linear operator Λk∗ that corresponds to this inclusion; the ﬁeld u(x) in the region vk takes the form u(x) = Λk∗ [u(x)] = Λk∗ U e−ik∗ ·x

= Λk∗ e−ik∗ ·(x−xk ) eik∗ ·(x−xk ) U e−ik∗ ·x (2.51) = λk∗ (x − xk ) u(x) , λk∗ (x) = Λk∗ (e−ik∗ ·x )eik∗ ·x .

(2.52) (2.53)

Note that the function λk∗ (x) may be constructed from the one-particle problem for the inclusion vk centered at the origin. ∗ (x) that coincides with λk (x − xk ) in the Let us introduce the function λ ∗ region vk occupied by the kth inclusion and is equal to zero in the matrix. Using this function, we can rewrite (2.7) for the wave ﬁeld in the composite in the form ∗ (x ) u(x ) V (x )dx . (2.54) u(x) = u0 (x) + k12 g0 (x − x ) λ After averaging this equation over the ensemble realizations of the random set of inclusions, we go to the closed equation for the mean wave ﬁeld u(x) in the composite 2 (2.55) u(x) = u0 (x) + pk1 g0 (x − x )λ∗ (k∗ ) u(x ) dx .

1 (2.56) Λ∗ (x)dx. λ∗ (k∗ ) = Λ∗ (x ) = v0 v 0 Here we take into account that Λ∗ (x) is a stationary random ﬁeld with a constant mean value; v0 is the region occupied by a typical inclusion. The Fourier transform of (2.55) has the form u(k) = u0 (k) + pk12 g0 (k)λ∗ (k∗ )u(k),

(2.57)

and after multiplying this equation by the function l0 = k 2 − k02 , and taking into account (2.43), we obtain 2 (2.58) k − k02 − pk12 λ∗ (k∗ ) u(k) = 0. Because u(k) = (2π)3 δ(k − k∗ ), the multiplier in front of u(k) in this equation should be equal to zero when k = k∗ k∗2 − k02 − pk12 λ∗ (k∗ ) = 0.

(2.59)

16

2. Self-consistent methods

This is the dispersion equation for the wave number k∗ of the mean wave ﬁeld in the framework of version I of the EMM. This dispersion equation is simpler than the dispersion equation (2.45) of the EFM, but (2.59) does not contain information about spatial distribution of the inclusions. In (2.45), this information is presented in the form of a speciﬁc correlation function Φ(x) of the random set of inclusions.

2.3.2 Version II of the EMM Version I of the EMM was used for the calculation of various physical eﬀective properties of composite materials and polycrystals (see [9, 10, 35, 69, 78, 80]). The comparison of theoretical predictions with experimental data pointed out some shortcomings of this version of the EMM. It turns out that the theoretical values of eﬀective (static) constants of matrix composites with very hard or very soft inclusions do not correspond to experimental data if the volume concentration of the inclusions is more than 0.3. (See [12, 40], where elastic properties of matrix composites were considered.) In order to correct this drawback of the method, another version of the EMM has been used by many authors (see, e.g., [12, 37, 38, 59, 69, 80]). In this version, a layer of the matrix material was introduced in the boundary between the inclusion and the eﬀective medium by the formulation of the one particle problem (see Fig. 2.4, where k0 is the wave number of the the matrix). Thus, the ﬁrst hypothesis of this version of the EMM was formulated as follows (version II): II1 . Every inclusion in the composite behaves as a kernel of a two-layered inclusion embedded in the eﬀective medium. The properties of the kernel coincide with these characteristics of the inclusion, and the properties of the outside layer coincide with the properties of the matrix.

Fig. 2.4. The one-particle problem of version II of the EMM (the Kerner cell).

2.3 The eﬀective medium method

17

The radius r1 of the outside layer depends on the volume concentration p of the inclusions, and for the spherical inclusions of the radii r0 , the radii r0 and r1 are related by the equation:

r0 r1

3 = p.

(2.60)

The condition of self-consistency coincides with that of version I of the EMM II2 = I2 .

(2.61)

(The mean ﬁeld in the composite coincides with the ﬁeld propagating in the eﬀective medium.) This version of the EMM leads to a better agreement with known experimental data than version I when applied to the calculation of static properties of matrix composites. The dispersion equation that corresponds to this version coincides with (2.59) but the function λ(k∗ ) is to be constructed from the solution of the one-particle problem for an inclusion with the layer in the border with the eﬀective medium. The parameter k(x) is equal to k for the kernel part of such an inclusion, and k(x) = k0 for the layer. 2.3.3 Version III and IV of the EMM In a number of works, where the EMM was applied to the calculation of eﬀective dynamic properties of matrix composites, another condition of selfconsistency was used (version III of the EMM) (see, e.g., [95]). III2 . The properties of the eﬀective medium should be chosen so as to reduce the mean forward amplitude of the ﬁeld scattered on a coated inclusion embedded in the eﬀective medium. Here the mean forward amplitude of the scattered ﬁeld is an amplitude averaged over the sizes and properties of the inclusions. (See Fig. 2.5, where F (m) is the forward amplitude of the scattered ﬁeld). Thus, the dispersion equation for the wave number k∗ may be written as follows F (n0 ) =F (k∗ , n0 ) = 0,

(2.62)

where n0 is the wave normal of the incident ﬁeld, F (n0 ) is deﬁned by the integral similar to (2.11) 0 0 2 k∗1 (x)u(x)eik∗ n ·x dx. (2.63) F (n ) = V

Integration here is taken over the region V occupied by the inclusion and the layer,

18

2. Self-consistent methods

Fig. 2.5. The one-particle problem for version III of the EMM. F (n0 ) is the forward amplitude of the scattered ﬁeld us . 2 k∗1 (x) = k2 − k∗2 inside the inclusion, 2 k∗1 (x) = k02 − k∗2 inside the layer.

(2.64)

The function u(x) under the integral in (2.63) is the ﬁeld in the coated inclusion embedded into the eﬀective medium with wave number k∗2 . The radii of the coated inclusion are chosen as in version II (2.60), and thus, the ﬁrst hypothesis of this version of the EMM coincides with hypothesis II1 III1 = II1 .

(2.65)

For polycrystals and granular materials with various types of grains, the matrix phase does not exist, and condition III2 is formulated for an uncoated inclusion embedded in the eﬀective medium. IV2 . The properties of the eﬀective medium should be chosen so as to reduce the mean forward amplitude of the ﬁeld scattered on a grain embedded in the eﬀective medium. This is the fourth version of the EMM existing in the literature, and hypothesis IV1 of this version coincides with I1 IV1 = I1 .

(2.66)

The equation for the wave vector k∗ may be written as follows F (k∗ , n0 ) = 0,

(2.67)

where averaging is performed over the orientation of the crystallographic axes of the crystal inside the grains. Note that the various versions of the EMM are not identical and give diﬀerent results for the wave number of the mean wave ﬁeld propagating in composites.

2.4 Notes

19

There are more complex versions of the EMM, where the ﬁrst hypothesis was formulated for several inclusions embedded in the matrix phase, and the medium outside some vicinity of such a cluster of inclusions was replaced by that of the eﬀective medium [37]. Such generalizations have many degrees of freedom, but introduce technical diﬃculties in the analysis. The self-consistent methods outlined in this chapter will be applied later to problems of electromagnetic and elastic wave propagation in media with isolated inclusions and polycrystals.

2.4 Notes The ﬁrst applications of self-consistent schemes to problems of scalar wave propagation through a medium with many particles may be found in the works by Maxwell and Rayleigh (see [69]). The subsequent development of the eﬀective ﬁeld method to scalar wave propagation through a medium with point scatterers was carried out by Foldy [26], Lax [70, 71], Waterman et al. [25, 112]. Version I of the EMM was used for the calculation of various physical eﬀective properties of composite materials and polycrystals [8,9,35,66,69,78], and many others. Version II of the EMM was proposed in [12, 37, 38, 59, 69, 80]. Version III of the EMM was applied to the calculation of eﬀective dynamic properties of matrix composites; another condition of self-consistency was used in [88, 95].

3. Electromagnetic waves in composites and polycrystals

In this chapter, the self-consistent methods are applied to the solution of the problem of electromagnetic wave propagation in the composites with spherical inclusions and polycrystals with quasispherical grains. For every method, the dispersion equation for the wave number of the mean wave ﬁeld propagating in the composites is derived. The asymptotic solutions of these equations in the long and short-wave regions are obtained in closed analytical forms. Phase velocities and attenuation coeﬃcients of the mean wave ﬁelds are calculated in wide regions of the parameters of the composites and frequencies of the incident waves. Predictions of various self-consistent methods are compared and analyzed.

3.1 Integral equations for electromagnetic waves Let a plane electromagnetic wave of frequency ω propagate through a homogeneous medium with an array of isolated inclusions; let V (x) be the characteristic function of the region V occupied by the inclusions (V (x) = 1, x ∈ V ; V (x) = 0, x ∈ V ). The medium and the inclusions are dielectrics with tensors of dielectric permittivities ε0 and ε, respectively. The magnetic permittivities of the components are equal to 1 (µ0 = µ = 1). For simplicity, we also assume that the dielectric properties of the medium do not depend on the frequency of propagating waves. If dependence on time t is deﬁned by the factor eiωt , the Maxwell equations for the amplitudes of the electric E(x) and magnetic H(x) ﬁelds have the forms [11] rotE(x) = −iωH(x), rotH(x) = iωD(x), divD(x) = 0, divH(x) = 0,

(3.1)

where the electric ﬁeld E(x) and electric displacement D(x) vectors are connected by the relations D(x) = ε(x) · E(x), ε1 (x) = ε1 V (x),

ε(x) = ε0 + ε1 (x), ε1 = ε − ε0 .

(3.2)

Here and farther, dot is the scalar product of vectors and tensors: ε(x) · E(x) ⇒ εij (x)Ej (x); “rot” operator is also called “curl”.

22

3. Electromagnetic waves in composites and polycrystals

For eliminating the magnetic ﬁeld H(x) from (3.1), we apply the rotoperator to both parts of the ﬁrst equation of system (3.1) rot (rotE(x)) = grad divE(x) − E(x) = −iωrotH(x).

(3.3)

Using the second equation of (3.1) and the ﬁrst equation of (3.2), we obtain the equation for the electric ﬁeld E(x) in the form E(x) − grad divE(x) + ω 2 ε0 · E(x) = −ω 2 ε1 (x) · E(x),

(3.4)

where is the Laplace operator. Note that in the regions where ε(x) is constant, the equation divE(x) = 0 holds, and (3.4) takes the form E(x) + ω 2 ε0 · E(x) = −ω 2 ε1 · E(x),

(3.5)

Let G0 (x) be the Green function of the operator on the left-hand side of (3.4), then G0 (x) − grad divG0 (x) + ω 2 ε0 · G0 (x) = −δ(x)1.

(3.6)

Here 1 is the second order unit tensor with the components δij , δ(x) is the 3D-Dirac delta-function. After applying the integral operator with kernel G0 (x) to both parts of (3.5) we ﬁnd the following integral equation for the electric ﬁeld E(x) in the medium with inclusions: (3.7) E(x) − G(x − x ) · ε1 · E(x )V (x )dx = E0 (x), G(x) = ω 2 G0 (x).

(3.8)

Here integration is spread over all 3D-space, E0 (x) is the incident ﬁeld that would have existed in the medium without inclusions and by the given sources of the ﬁeld. For monochromatic plane waves, the ﬁeld E0 (x) takes the form 0

E0 (x) = U0 e−ik

·x

,

k0 = k0 n0 ,

|n0 | = 1,

|U0 | = 1,

(3.9)

where k0 is the wave vector for the matrix material, U0 is the polarization vector. For an isotropic matrix (ε0 = ε0 1, ε0 is a scalar), the function G(x) in (3.7) takes the form G(x) = k02 g(x)1 + ⊗ g(x),

g(x) =

e−ik0 |x| , 4πε0 |x|

k02 = ω 2 ε0

(3.10)

and the vectors U0 and k0 in (3.9) are orthogonal (E0 (x) is the transverse wave).

3.1 Integral equations for electromagnetic waves

23

Equation (3.7) has been considered by many authors, and is in essence the equation for the ﬁeld E(x) inside the inclusions (in the region V ). The ﬁeld outside V may be reconstructed from (3.7) if E(x) inside V is known. This equation serves for a set of inclusions as well as for an isolated inclusion in an inﬁnite homogeneous medium. The kernel G(x) of this equation is a generalized function for which regularization is given, e.g., in [29]. The Fourier transform G(k) of the Green function G(x) has the form 1 k02 1 − 2k⊗k , (3.11) G(k) = ε0 (k 2 − k02 ) k0 where k is the vector parameter of the Fourier transform, k = |k|. Note that for an isolated inclusion occupying a ﬁnite region v, (3.7) may be written in the form 0 s s (3.12) E(x) = E (x) + E (x), E (x) = G(x − x ) · ε1 · E(x )dx , v

where E (x) is the ﬁeld scattered by the inclusion. In the far zone (|x| d, where d is a characteristic size of the inclusion) the following asymptotic representations hold: x (3.13) |x − x | ≈ |x| − n · x , n = , |x| s

g0 (x − x ) ≈

e−ik0 |x| ik0 (n·x ) e , 4πε0 |x|

∇ ⊗ ∇g0 (x − x ) ≈ −k02 n ⊗ n

e−ik0 |x| ik0 (n·x ) e , 4πε0 |x|

(3.14) (3.15)

and the scattered ﬁeld Es (x) in the far zone takes the form e−ik0 |x| F(n), |x| k2 F(n) = 0 (1 − n ⊗ n) · ε1 · E(x )eik0 (n·x ) dx . 4πε0 v

Es (x) ≈

(3.16) (3.17)

Here F(n) is the amplitude of the scattered ﬁeld. If the direction n co0 incides with n0 (the direction of the incident ﬁeld propagation), F(n ) is the forward scattered amplitude. It is known ([2], Chapter 3) that the forward scattered amplitude is connected with the total normalized scattering cross-section Q of the inclusion by the relation (the optical theorem) Q=−

4π Im[U0 · F(n0 )]. k0 S0

(3.18)

Here S0 is the maximal area of the cross-section of the inclusion v by the plane orthogonal to the wave vector k0 ; Im denotes the imaginary part of a complex number.

24

3. Electromagnetic waves in composites and polycrystals

3.2 Version I of EMM for matrix composites In this section we consider the homogenization problem for elelectromagnetic wave propagation and start with the eﬀective medium method. The main hypotheses of the EMM formulated in Section 1.3 may be applied to electromagnetic wave propagation without modiﬁcations. Thus, hypothesis I1 of the method is formulated as follows. I1 . The wave ﬁeld inside every inclusion in the composite coincides with the wave ﬁeld inside an isolated inclusion embedded in the homogeneous medium with the eﬀective properties of all the composite. The exciting ﬁeld E∗ (x) that acts on this inclusion is a plane wave propagating in the eﬀective medium ∗

E∗ (x) = U∗ e−ik

·x

,

k∗ = k∗ n, |U∗ | = 1.

(3.19)

Here k∗ and U∗ are the eﬀective wave vector and the polarization vector of the wave E∗ (x). The second hypothesis is the condition of self-consistency: II1 . The mean wave ﬁeld in the composite medium coincides with E∗ (x) E(x) = E∗ (x).

(3.20)

In order to obtain the ﬁeld inside the inclusions it is necessary to solve the one particle problem: diﬀraction of plane monochromatic wave (3.19) by an isolated inclusion in the homogeneous (eﬀective) medium. Let v be the region occupied by the inclusion. The integral equation of the one-particle problem of this version of the EMM is similar to (3.7) and has the form (3.21) E(x) − G∗ (x − x ) · ε∗1 · E(x )dx = E∗ (x). v

Here ε∗1 = ε − ε∗ , G∗ (x) is the Green function for the eﬀective medium, and G∗ (x) has form (3.10) if ε0 is replaced by the dielectric property of the eﬀective medium ε∗ , and the wave number k0 of the matrix is replaced by the wave number k∗ of the eﬀective medium. Let the general solution of (3.21) be known, and the ﬁeld E(x) inside the inclusion centered at the point x = 0 be presented in the form ∗

E(x) = ΛE∗ (x) = Λ[U∗ e−ik ∗

−ik∗ ·x

λ(x, k ) = Λ[U∗ e

·x

] = λ(x, k∗ ) · E∗ (x),

ik∗ ·x

]e

(3.22)

∗

⊗U .

Here Λ is a linear operator that depends on the dielectric properties of the eﬀective medium and the inclusion, the shape of the latter, and the wave vector k∗ of the exciting ﬁeld.

3.2 Version I of EMM for matrix composites

25

If the inclusion occupies the region vj centred at point xj = 0, the wave ﬁeld inside this inclusion is presented in the form (x ∈ vj ) ∗

E(x) = Λj [U∗ e−ik

·x

] = Λj [U∗ e−ik·

∗

∗

(x−xj ) −ik∗ ·xj

e

∗

= λ (x − x , k ) · E (x), j

j

∗

λj∗ (x, k∗ ) = Λj∗ [U∗ e−ik

·x

∗

]eik

·x

] (3.23)

⊗ U∗ .

(3.24)

k∗ ) that coincides with Let us introduce a stationary random function λ(x, λj (x − xj , k∗ ) inside the inclusion centered at point xj , (j = 0, 1, 2, 3, ...) and is equal to zero in the matrix. Using this function and hypothesis I1 of the EMM, we present the ﬁeld E(x) in the composite in a form that follows from (3.7), (3.23) , k∗ ) · E∗ (x )dx . (3.25) E(x) = E0 (x) + G(x − x ) · ε1 · λ(x In order to ﬁnd the mean value of the ﬁeld E(x) let us average both parts of (3.25) over the ensemble realizations of the random set of inclusions. After taking into account the condition of self-consistency I2 (the mean wave ﬁeld E(x) coincides with the wave ﬁeld E∗ (x) propagating in the eﬀective medium) ∗

E(x) = E∗ (x) = U∗ e−ik

·x

(3.26)

we obtain the following integral equation for the mean electric ﬁeld: (3.27) E(x) = E0 (x) + p G(x − x ) · ε1 · Λ∗ (k∗ ) · E(x ) dx , 1 1 k∗ )dx = 1 lim λ(x, k∗ )dx. (3.28) λ(x, Λ∗ (k∗ ) = p Ω→∞ Ω Ω v0 v 0 k∗ ) is Here Λ∗ (k∗ ) is a constant (with respect to x) tensor because λ(x, a realization of a spatially stationary random function with a constant mean value. Equation (3.27) is a convolution equation. Thus, after applying the Fourier transform to both parts of it we obtain the following algebraic equation for the Fourier transform of the mean wave ﬁeld E(k): E(k) = E0 (k) + pG(k) · ε1 · Λ∗ (k∗ ) · E(k) .

(3.29)

Taking the product of both parts of this equation with the tensor L0 (k) = G(k)−1 L0 (k) =

ε0 2 k − k02 1 − k 2 m ⊗ m , 2 k0

k = km

(3.30)

26

3. Electromagnetic waves in composites and polycrystals

gives us the following equation for the Fourier transform of mean electric ﬁeld E(k): ε1 (k2 − k02 )(1 − m ⊗ m) − k02 m ⊗ m − pk02 · Λ∗ (k∗ ) · E(k) = 0. (3.31) ε0 Here we take into account the equation L0 (k) · G(k)−1 = 1, L0 (k) · E0 (k) = 0,

(3.32)

where 1 is the rank two unit tensor. Because the Fourier transform E(k) of the ﬁeld (3.26) has the form E(k) = U∗ (2π)3 δ(k − k∗ ),

(3.33)

the determinant of the tensor multiplier in front of E(k) in (3.31) should be equal to zero when k = k∗ = k∗ n. det[(k∗2 − k02 )(1 − n ⊗ n) − k02 n ⊗ n − pk02

ε1 · Λ∗ (k∗ )] = 0. ε0

(3.34)

This equation is the condition of existence of non-trivial solutions of (3.31) and is in fact the dispersion equation for the wave vector k∗ = k∗ n of the mean electric ﬁeld (3.26) propagating in the composite. If ε1 is an isotropic tensor (ε1 = ε1 1), the wave vector k0 of the inci∗ dent ﬁeld and the vector k ∗ of∗the mean wave ﬁeld have the same direction 0 n = n , and the tensor Λ (k ) in (3.34) takes the form Λ∗ (k∗ ) = Λ∗t (k∗ )(1 − n ⊗ n) + Λ∗l (k∗ )(n ⊗ n).

(3.35)

Thus, if E(x) is a transverse electric ﬁeld E(x) = U∗ e−ik

∗

n·x

,

U∗ ⊥n,

(3.36)

the equation for the wave number k∗ of the mean wave ﬁeld follows from (3.34), (3.35) in the form k∗2 − k02 − pk02 ε¯1 Λ∗t (k∗ ) = 0,

ε¯1 =

ε − ε0 . ε0

(3.37)

If Λ∗t (k∗ ) = Λ∗l (k∗ ), longitudinal electric waves may propagate in the medium, and the dispersion equation for the corresponding wave numbers takes the form 1 + pε1 Λ∗l (k∗ ) = 0.

(3.38)

The coeﬃcients Λ∗t (k∗ ) and Λ∗l (k∗ ) in these equations should be found from the solution of the one-particle problem (3.21). Thus, we have to go to the consideration of this problem in details.

3.3 One-particle EMM problems for spherical inclusions

27

3.3 One-particle EMM problems for spherical inclusions For spherical homogeneous inclusions, the one-particle problem of version I of the EMM is the solution of integral equation (3.21) when E∗ (x) has the form (3.19) and is a plane transverse wave; v is a spherical region of radius a (a = 1) centered at the point x = 0. The integral equation (3.21) is equivalent to the following system of diﬀerential equations: E + k2 E = 0, |x| ≤ a; E +

k∗2 E

(3.39)

= 0, |x| > a.

√ Here k∗ is the wave number of the eﬀective medium, k = ω ε is the same number for the inclusion. Let us introduce the Cartesian (x1 , x2 , x3 ) and spherical (r, ϕ, θ) coordinate systems with the origin at the center of the inclusion, and let er , eϕ , eθ be unit vectors of the spherical system. The wave vector k∗ of the exciting ﬁeld E∗ (x) = U∗ e−ik∗ ·x is directed along the x3 -axis, and the polarization vector U∗ along the x1 -axis (see Fig. 3.1). The boundary conditions on the surface of the inclusion have the forms (x = rn, E(x) = E(r, n)) E(a+0, n)×n = E(a−0, n)×n,

H(a+0, n)×n = H(a−0, n)×n, (3.40)

where × is the vector product, a± 0 = lim δ→0 (a ± δ), δ > 0. The condition at inﬁnity is: E(x) → E∗ (x) + O e−ik0 r /r when r = |x| → ∞. This problem does not diﬀer from the classical Mie problem that is considered in detail in [2]. The ﬁelds E(x) in the inclusion and the medium are expressed in terms of vector spherical harmonics. Four-vector spherical harmonics are deﬁned by the equations x3

er e3 θ e1

e2

ϕ

eϕ eθ x2

UL

x1 UT

k*

Fig. 3.1. The spherical coordinate system for the solution of the one-particle problem of EMM.

28

3. Electromagnetic waves in composites and polycrystals

M01n = cos(ϕ)πn (θ)zn (r)eθ − sin(ϕ)τn (θ)zn (r)eϕ , Me1n = − sin(ϕ)πn (θ)zn (r)eθ − cos(ϕ)τn (θ)zn (r)eϕ ,

(3.41) (3.42)

zn (r) [rzn (r)] er + sin(ϕ)τn (θ) eθ r r [rzn (r)] eϕ , + cos(ϕ)πn (θ) r zn (r) [rzn (r)] er + cos(ϕ)τn (θ) eθ = n(n + 1) cos(θ)πn (θ) r r [rzn (r)] eϕ . − sin(ϕ)πn (θ) r

N01n = n(n + 1) sin(ϕ)πn (θ)

(3.43)

Ne1n

(3.44)

Here the functions πn (θ) and τn (θ) are expressed via the Legendre polynomials Pn (x) πn (θ) =

dPn (cos θ) dPn (cos θ) d2 Pn (cos θ) , τn (θ) = cos θ − sin2 θ 2 . (3.45) d cos θ d cos θ d (cos θ)

Functions zn (r) are the spherical Bessel function jn (r) or the Hankel (2) function hn (r) = jn (k) − iyn (k) of order n. If zn (r) = jn (r), the vector (2) harmonics have upper index 1; if zn (r) = hn (r), this index is equal to 2. In terms of these harmonics, the electric ﬁelds E(x) inside the inclusion and in the medium take the forms: E(x) =

∞

in

(2n + 1) (1) (1) (1) Xn M01n (x) + Xn(2) Ne1n (x) , |x| ≤ a, n(n + 1)

(3.46)

in

(2n + 1) (3) (2) (2) Xn M01n (x) + Xn(4) Ne1n (x) , |x| > a. n(n + 1)

(3.47)

n=1

E(x) =

∞ n=1

(1)

(2)

(3)

(4)

The coeﬃcients Xn , Xn and Xn , Xn in these series are to be found (1) (2) from the boundary conditions (3.40). As a result, the coeﬃcients Xn , Xn that deﬁne the electric ﬁeld inside the inclusion take the forms Xn(1) = (−ik)[k∗ ξn (k∗ )ψn (k) − kψn (k)ξn (k∗ )]−1 ,

(3.48)

Xn(2) = (−ik)[kξn (k∗ )ψn (k) − k∗ ψn (k)ξn (k∗ )]−1 ,

(3.49)

ψn (k) = kjn (k),

(2)

ξn (k) = kh

(k),

f (k) = df /dk.

(3.50)

Note that here and farther, we imply that k0 , k, k∗ are non dimensional wave numbers (k0 a, ka, k∗ a) because the radius of the inclusions is equal to 1. Equation (3.46) for E(x) together with (3.28), (3.35) give us for the coefﬁcient Λ∗t in (3.37) the equation

3.3 One-particle EMM problems for spherical inclusions

Λ∗t

∞ 3 = (2n + 1) Xn(1) g0n (k∗ , k) 2 n=1 jn (k∗ )jn (k) (2) − g1n (k∗ , k) , + Xn (n + 1) k∗ k

1 [k∗ jn+1 (k∗ )jn (k) − kjn+1 (k)jn (k∗ )], k∗2 − k 2 1 g1n (k∗ , k) = 2 [k∗ jn (k∗ )jn+1 (k) − kjn (k)jn+1 (k∗ )]. k∗ − k 2

g0n (k∗ , k) =

29

(3.51)

(3.52) (3.53)

Here the following formulas of integration of the vector harmonics are used 3in(n + 1) 1 (1) g0n (k∗ , k)n0 , M01n (x)eik∗ ·x dx = (3.54) v v 2 3in(n + 1) jn (k) (k∗ jn (k∗ )) k∗ 1 (1) ik∗ ·x N (x)e dx = − i g0n (k∗ , k) n0. v v e1n 2 k k∗ k (3.55) It follows from (3.16), (3.46), (3.54), and (3.55) that the scalar product 0 of the polarization vector U0 and the forward scattering amplitude F(n ) is ∗ proportional to the coeﬃcient Λt 0

U∗ · F(n ) =

k∗2 ε∗1 Λ∗t , 3

ε∗1 =

ε − ε∗ . ε∗

(3.56)

Let us introduce the function Q(k∗ ) 4 Q(k∗ ) = − k∗ ε∗1 Λ∗t . 3

(3.57)

It follows from (3.18) that the imaginary part of this function coincides with the total normalized scattering cross-section Q of the inclusion in the background medium if k∗ = k0 and ε∗ = ε0 . The graphs in Fig. 3.2 show the ¯ (line 1) and imaginary Im(Q) ¯ (line 2) parts dependence of the real Re(Q) of the function Q on the wave number k0 of the exiting ﬁeld for a spherical inclusion of a unit radius. The inclusion with the dielectric permittivity ε = 5.2 − 0.03i in the matrix with ε0 = 1 is considered. Note that the short-wave limit of the function Q(k0 ) is lim Q = 2i.

k0 →∞

(3.58)

This is a consequence of the extinction paradox ([2]), Chapter 4.4.3). The graphs in Fig. 3.2 show the character of convergence of Q(k0 ) to this limit. (The function Q(k0 ) was calculated by numerical summation of the series

30

3. Electromagnetic waves in composites and polycrystals 6

Q 4 2 0 −2 −4 −6 −0.5

lg(k0a) 0

0.5

1

1.5

2

Fig. 3.2. The dependence of the real (1) and imaginary (2) parts of the function Q in (3.57) on the wave number of the medium k0 a. Im Q coincides with the normalized total scattering (extinction) cross-section of the spherical inclusion.

(3.51) for discrete values of k0 in the region −0.5 < lg(k0 ) < 2 with the step 0.1 in the decimal logarithmic scale. Thus, small-scale oscillations of Q(k0 ) are not shown in Fig. 3.2.) Finally, the dispersion equation (3.37) for the transverse part of the mean wave ﬁeld may be written in the ﬁnal form k∗2 = k02 + pk02 ε1 Λ∗t (k∗ ),

(3.59)

where the function Λ∗t (k∗ ) is deﬁned in (3.51). The phase velocity and attenuation of the mean wave ﬁeld are expressed via the solution of this equation in the form v∗ =

ω , γ = − Im k∗ , Re k∗

(3.60)

and the eﬀective dielectric permittivity of the composite is ε∗ =

k∗2 . ω2

(3.61)

3.4 Asymptotic solutions of the EMM dispersion equation Let us consider the solution of the dispersion equation (3.59) in the long-wave region. To be exact, we ﬁnd the principal terms in the real and imaginary parts of the solution of (3.59) when ω, k0 , k∗ → 0. First, we have to ﬁnd the principal terms of the asymptotic form of the function Λ∗t in (3.51) in the long-wave region. This asymptotic form follows from the integral equation

3.4 Long and short-wave asymptotics of the EMM

31

(3.21) of the one-particle problem if we keep only the principal terms of the kernel G∗ (x) and of the ﬁeld E∗ (x) on the right-hand side of this equation. The Green function of the eﬀective medium has a form similar to (3.10) G∗ (x) = k∗2 g∗ (x)1 + ⊗ g∗ (x),

g∗ (x) =

e−ik∗ |x| , 4πε∗ |x|

k∗2 = ω 2 ε∗ . (3.62)

Let us expand g∗ (x) in a series with respect to the wave number k∗ and keep the ﬁrst four terms of this series 1 1 e−ik∗ r 1 i ≈ − ik∗ − k∗2 r + k∗3 r2 , r = |x|. (3.63) g∗ (x) = 4πε∗ r 4πε∗ r 2 6 Substituting (3.63) into (3.62) and keeping only the terms of order not higher than k∗3 in the equation for G∗ (x), we obtain G∗ (x) ≈ Gs∗ (x) + ik∗3 Gω ∗, 1 1 1 Gs∗ (x) = Gω ∇⊗∇ , 1. ∗ =− 4πε∗ r 6πε∗

(3.64) (3.65)

Here Gs∗ (x) is the “static” part of the Green function (ω = 0). As a result, the integral equation (3.21) of the one-particle problem takes the form s ∗ G∗ (x − x ) + ik∗3 Gω (3.66) E(x) − ∗ · ε∗1 · E(x )dx = U , v

where the right hand side is replaced by the constant (U∗ = E∗ (0)) in the long-wave limit. Looking for the solution of this equation in the form E(x) = E(0) + ik∗3 E(1) ,

(3.67)

we substitute (3.67) into (3.66) and equate the terms of the same order with respect to k∗ . As a result, we obtain the long-wave asymptotics of the ﬁeld E(x) inside the inclusion in the form −1 s G∗ (x − x ) + ik∗3 Gω dx · ε · U∗ = Λ∗ · U∗ , (3.68) E(x) = 1 − ∗1 ∗ v

2k∗3 ε∗1 3 . Λ = 1−i 3 + ε∗1 3(3 + ε∗1 ) ∗

(3.69)

It follows from this equation that the long-wave asymptotics of the coefﬁcients Λ∗t and Λ∗l in (3.37) and (3.38) are 2k∗3 ε∗1 3 ∗ ∗ . (3.70) 1−i Λt = Λ l = 3 + ε∗1 3(3 + ε∗1 )

32

3. Electromagnetic waves in composites and polycrystals

After substituting this equation for Λ∗t into (3.59) and keeping the terms of order k0 in the real part and k04 in the imaginary part, we obtain the equation for the eﬀective wave number k∗ in the form s 2 ε∗ ε − εs∗ εs∗ ε1 εs∗1 4 − ip (k0 a) , εs∗1 = . (3.71) k∗ a = k0 a s 2 ε0 (3 + ε∗1 ) ε0 εs∗ Here εs∗ is the “static” eﬀective dielectric property (ω = 0) of the composite obtained in the framework of the EMM. εs∗ is the solution of the following algebraic equation εs∗ = ε0 + p

3ε1 εs∗ , 2εs∗ + ε

(3.72)

that is a consequence of (3.59). This equation is the well known Bruggeman formula for the static dielectric permittivity of the composite with spherical inclusions [8]. If ε → 0 (optically soft inclusions), the solution of this equation is 3 εs∗ = (1 − p)ε0 . 2

(3.73)

If ε → ∞ (optically hard inclusions), εs∗ takes the form εs∗ =

ε0 . 1 − 3p

(3.74)

Evidently, these equations have physical meanings if p < 2/3 for soft inclusions and p < 1/3 for hard ones. The imaginary part of the eﬀective wave number k∗ in (3.64) is proportional to k04 (ω 4 ) and hence, the attenuation coeﬃcient γ in the long-wave region is deﬁned by the Rayleigh scattering of electromagnetic waves by the inclusions. Let us ﬁnd the short wave asymptotics of the solution of (3.59) when k0 , k∗ → ∞. It follows from (3.57) and (3.58) that the coeﬃcient Λ∗t in this limit takes the form Λ∗t = −

3i . 2k∗ ε∗1

(3.75)

After substituting this equation for Λ∗t into (3.59), we obtain the following short-wave limit value of k∗ 3 k∗ a = k0 a − i p. 4

(3.76)

Thus, in the short-wave limit, the velocity of the mean wave ﬁeld in the composite coincides with the phase velocity of waves in the matrix v∗ =

ω ω = = v0 , Re k∗ k0

(3.77)

3.5 Numerical solution of the EMM dispersion equation

33

and the attenuation coeﬃcient γ of this ﬁeld does not depend on the properties of the matrix and inclusions, and is equal to 34 p. If the inclusions have the radius a, the attenuation coeﬃcient γ is deﬁned by the equation γa = − Im k∗ a =

3 p. 4

(3.78)

This result may be interpreted as follows. In the short-wave limit, the geometrical optics interpretation may be used for the description of the mean wave ﬁeld in the composite. This ﬁeld may be considered as a set of independent rays propagating through the composite medium. Because a continuous component (matrix) exists in the medium, the phase velocity of the mean ﬁeld coincides with the wave velocity in the matrix. The attenuation coeﬃcient γ in the short wave limit does not depend on the frequency and properties of the inclusions and is proportional to the area occupied by the scatterers on a unit length (p). This is a consequence of the paradox of extinction (the short-wave limit of the cross-section of the inclusion depends neither on its properties nor the properties of the matrix).

3.5 Numerical solution of the EMM dispersion equation In this section, the dispersion curves k∗ = k∗ (k0 ) or frequency dependences of the eﬀective wave number k∗ are constructed in a wide region of the wave number k0 of the incident ﬁeld on the basis of the dispersion equation (3.59). Thus, the roots of (3.59) should be found in the complex plane (Re k∗ , Im k∗ ) as functions of the wave number k0 . Note that (3.59) may have many branches of the solutions that correspond to diﬀerent modes of wave propagation. A natural way to solve (3.59) is based on the following iterative procedure (n+1) (n) (n) (n) ∗ − k∗ . (3.79) = k∗ + δ k0 1 + pε1 Λt k∗ k∗ (n)

Here k∗ is the nth iteration for the eﬀective wave number; the parameter δ is to be chosen for convergence of the iterative process. Starting from an (0) initial guess k∗ , we calculate the next approximations using this equation (n+1) (n) − k∗ | becomes suﬃciently small. until the diﬀerence |k∗ Unfortunately, this method has several drawbacks. • Sometimes it diverges even for very small δ, and more sophisticated methods should be used for seeking the roots of (3.59). • It is diﬃcult to ﬁnd all the roots of (3.59) by this method. • The iterative procedure may jump from one branch of the solutions of (3.59) to another branch if these branches are close.

34

3. Electromagnetic waves in composites and polycrystals

In the numerical solution of (3.59), an important problem is to ﬁnd approximate positions of all the roots in the complex plane (Re k∗ , Im k∗ ), and then to calculate the precise values of the roots. Let us consider the function of the complex variable k∗ F (k∗ ) = k∗2 − k02 − pk02 ε1 Λ∗t (k∗ ).

(3.80)

This function is analytic, and the criterion for presence of roots of this function in some region Ω of the complex k∗ -plane is the value of the following integral F (k∗ ) 1 dk∗ (3.81) 2πi F (k∗ ) Γ

taken along the border Γ of Ω. It is shown in the theory of complex variables that this integral is equal to the diﬀerence between the number of roots and the number of poles of F (k∗ ) inside Ω. Thus, dividing the original region into a number of subregions and numerically calculating the integral (3.81) over the boundaries of these subregions, we can ﬁnd the approximate positions of the roots of (3.59) inside this region. After that, the precise values of the roots may be found by the iteration procedure (3.79) or by the Newton method and its modiﬁcations. Note that series (3.46) for Λ∗t converges rather slowly for medium and short waves. It is necessary to keep approximately k0 a + 4 3 k0 a + 2 terms of these series (see [2], Appendix A) to obtain reliable values of the coeﬃcient Λ∗t . Let us consider the phase velocities and attenuation coeﬃcients of the waves propagating in a composite with optically soft inclusions (ε0 = 1, ε = 0.01). In this case, only one branch of the solutions of (3.59) was found. The dependence of the relative phase velocity v∗ /v0 = k0 / Re(k∗ ) and the attenuation coeﬃcient γa = − Im(k∗ a) on the wave number k0 a of the incident ﬁeld are presented in Fig. 3.3 for the volume concentrations of the inclusions p = 0.1, 0.2 and 0.3. Dashed horizontal lines in Fig. 3.3 correspond to the short-wave limit of the attenuation coeﬃcients for each volume concentration of inclusions (3.78). For optically hard inclusions (ε0 = 1, ε = 100), equation (3.59) proves to have several solution branches. For the volume concentration of inclusions p = 0.1, three such branches were found. These branches are indicated by labels 1, 2, and 3 in Fig. 3.4. Branch 1 is the acoustical branch, and for small k0 a, this branch corresponds to the long-wave asymptotics of the solution obtained in Section 3.4 (equation (3.71)). The attenuation coeﬃcient γa along this branch grows rapidly with k0 a, and this branch becomes practically

3.5 Numerical solution of the EMM dispersion equation 0

1.4

lg(γa)

35

v*/v0 0.3

−1

1.3 p=0.1

−2

0.2

0.3 1.2

0.2

1.1

p=0.1

−3 −4 −5 −1

lg(k0a) −0.5

0

0.5

1

1.5

2

1 −1

lg(k0a) −0.5

0

0.5

1

1.5

2

Fig. 3.3. The dependences of the attenuation γa and the relative velocity of the mean electric ﬁeld in the composite with optically soft inclusions (ε = 0.01, ε0 = 1) on their volume concentration p = 0.1, 0.2, 0.3.

3

0.5

Re(k*a)

2.5

p=0.1

0.4

2

0.3

1.5

3 0.2

1 2

0.5 0

γa

p=0.1

1 0

0.5

3

0.1 1

1.5

2

k0a 2.5 3

1 0

0

2 0.5

k0a 1

1.5

2

2.5

3

Fig. 3.4. The dependence of the real Re(k∗ a) and imaginary γa = − Im(k∗ a) parts of the wave number of the mean electric ﬁeld propagating in the composite with optically hard inclusions (ε = 100, ε0 = 1) for the volume concentration p = 0.1.

invisible when k0 a > 0.3. Note that if the parameter γa is about 1, the amplitude of the wave decreases by a factor of 10 in a distance about a diameter of inclusions. Branches 2 and 3 are the optical branches. Branch 2 is visible in the interval 0.3 < k0 a < 0.6. If k0 a > 0.6, the attenuation along this branch becomes large, and this branch practically disappears. Branch 3 starts from k0 a about 0.6, and the attenuation along this branch is moderate until high frequencies. At high frequencies, the attenuation along this branch corresponds to the short-wave asymptotic solution of (3.59) obtained in Section 3.4 (3.75). The dashed parts of the dispersion curves in Fig. 3.4 correspond to large attenuation coeﬃcients of the corresponding waves. The number of diﬀerent branches of the solutions of (3.56) increases with the volume concentration p of the inclusions. For p = 0.3, the branches of the solutions of (3.56) in the region 0 < k0 a, k∗ a < 3 are presented in Fig. 3.5. Every branch is essential in some frequency interval, and the attenuation

36

3. Electromagnetic waves in composites and polycrystals 3

2.5

Re(k*a)

γa

p=0.3

2.5

p=0.3

2

2

1.5

1.5 1 1 0.5

0.5 0

k0a 0

0.5

1

1.5

2

2.5

3

0

k0a 0

0.5

1

1.5

2

2.5

3

Fig. 3.5. The variation of the real Re(k∗ a) and imaginary γa = − Im(k∗ a) parts of the wave number of the mean electric ﬁeld propagating in the composite with optically hard inclusions (ε = 100, ε0 = 1) for the volume concentration p = 0.3.

coeﬃcient is large outside this interval. Note that a common envelop of these branches reﬂects general behavior of the mean wave ﬁeld propagating in the composite.

3.6 Versions II and III of the EMM Let us turn to versions II and III of the EMM discussed in Section 2.3. The ﬁrst hypothesis of these versions is formulated as follows. II1 , III1 . Every inclusion in the composite behaves as a kernel of a twolayered inclusion embedded in the homogeneous medium with the eﬀective properties of the composite. The size and the properties of the kernel coincide with these characteristics of the inclusion, and the properties of the outside layer coincide with the properties of the matrix. Thus, the one-particle problems of versions II and III of the EMM are the diﬀraction problem for a coated spherical inclusion embedded in the eﬀective medium. The dielectric property of the external layer of the inclusion coincides with the property of the matrix, and its external radius r1 (see Fig. 3.4) is deﬁned by (2.60). For r0 = 1, the radius r1 takes the form 1

r1 = p− 3 .

(3.82)

Here p is the volume concentration of inclusions. Let v be the volume occupied by the inclusion, and vl be the volume of the matrix layer around the inclusion. The one-particle problem of these versions of the EMM is the solution of the following integral equation (3.83) E(x) − G∗ (x − x ) · ε∗1 (x) · E(x )dx = E∗ (x). V

3.6 Versions II and III of the EMM

37

Here V = v ∪ vl , ε − ε∗ if x ∈ v, , ε∗1 (x) = ε0 − ε∗ if x ∈ vl G∗ (x) is the Green function of the eﬀective medium that has the form (3.62). The solution of this problem is presented in ([2], Chapter 4.5). The electric ﬁeld in the medium with a coated inclusion has the form E(r, ϕ, θ) =

∞

in

n=1

E(r, ϕ, θ) =

(2n + 1) (1) (1) Y M01n (r, ϕ, θ), n(n + 1) n

(1) + iYn(2) Ne1n (r, ϕ, θ) , 0 r r0 ;

(3.84)

(2n + 1) (3) (1) (1) Yn M01n (r, ϕ, θ) + iYn(4) Ne1n (r, ϕ, θ) n(n + 1) n=1 (3) (3) + Yn(5) M01n (r, ϕ, θ) + iYn(6) Ne1n (r, ϕ, θ) , r0 < r < r1 ; ∞

in

(3.85) (2n + 1) (3) (3) −iYn(7) Ne1n (r, ϕ, θ) − Yn(8) Me1n (r, ϕ, θ) in E(r, ϕ, θ) = n(n + 1) n=1 (1) (1) (3.86) + M01n (r, ϕ, θ) + iNe1n (r, ϕ, θ) , r r1 . ∞

Here the spherical Bessel functions in the vector spherical harmonics (1) (1) (3) (3) M01n , Ne1n , M01n and Ne1n have the arguments kr if 0 ≤ r ≤ r0 , the arguments k0 r if r0 < r < r1 , and k∗ r if r ≥ r1 . (i) Eight constants Yn , (i = 1, 2, ..., 8) for every n(n = 1, 2, ...) in (3.84)(3.86) are to be found from the boundary conditions on the two boundaries: the kernel - layer (r = r0 ) and layer - eﬀective medium (r = r1 ). These conditions have the forms [E(r, n)]j × n = 0,

[H(r, n)]j × n = 0,

j = 0, 1;

n=

x , r

(3.87)

where [f (r)]j = f (rj +0)−f (rj −0) is a jump of the function at the boundaries, rj ± 0 = limδ→0 (rj ± δ), δ > 0. (q) The system of linear algebraic equations for the constants Yn follows from these conditions in the form Mn Yn = fn ,

(3.88)

38

3. Electromagnetic waves in composites and polycrystals

⎡

m11 0 m13 0 m15 ⎢ m21 0 m23 0 m25 ⎢ ⎢ 0 m32 0 m34 0 ⎢ ⎢ 0 m42 0 m44 0 Mn = ⎢ ⎢ 0 0 m53 0 m55 ⎢ ⎢ 0 0 m63 0 m65 ⎢ ⎣ 0 0 0 m74 0 0 0 0 m84 0

Yn = Yn(i) , fn = fn(i) , m11 = jn (kr0 ),

0 0 m36 m46 0 0 m76 m86

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥, m58 ⎥ ⎥ m68 ⎥ ⎥ 0 ⎦ 0

0 0 0 0 0 0 m77 m87

i = 1, 2, 3, ..., 8,

m13 = −jn (k0 r0 ),

m25 = k0 Dhn (k0 r0 ),

m34 = −Djn (k0 r0 ), m36 = Dhn (k0 r0 ),

m42 = kjn (kr0 ),

m44 = −k0 jn (k0 r0 ),

m53 = jn (k0 r1 ),

m55 = −hn (k0 r1 ),

m46 = k0 hn (k0 r0 ),

=

fn(2)

=

fn(3)

m68 = k∗ Dhn (k∗ r0 ),

m76 = −Dhn (k0 r1 ), m77 = k0 jn (k0 r1 ),

m74 = Djn (k0 r1 ), fn(1)

(3.91)

m58 = hn (k∗ r1 ),

m63 = k0 Djn (k0 r1 ), m65 = −k0 Dhn (k0 r1 ), m84 = k0 jn (kr1 ),

(3.90)

m15 = hn (k0 r0 ),

m21 = kDjn (kr0 ), m23 = −k0 Djn (k0 r0 ), m32 = Djn (kr0 ),

(3.89)

m86 = −k0 hn (k0 r1 ), =

fn(4)

m87 = k∗ hn (k∗ r0 ),

= 0,

fn(5) = jn (ke r1 ), fn(6) = k∗ Djn (k∗ r1 ), fn(7) = Djn (k∗ r1 ), fn(8) = k∗ jn (k∗ r1 ). Djn (z) =

jn (z) + jn (z), z

Dhn (z) =

hn (z) + hn (z). z

(3.92) (3.93)

The function Λ∗t (k∗ ) in the dispersion equation (3.59) for version II of (1) (2) the EMM takes the form (3.51), where the coeﬃcients Xn , Xn should be (1) (2) replaced by Yn , Yn that are found from system (3.88) Λ∗t (k∗ ) =

∞ 3 (2n + 1) Yn(1) g0n (k∗ , k) 2 n=1 jn (k∗ )jn (k) − g1n (k∗ , k) . + Yn(2) (n + 1) k∗ k (q)

(3.94)

The equations for the coeﬃcients Yn are rather huge, and they are not written here, but they can be calculated in analytical forms or numerically from (3.88). Let us ﬁnd the long-wave asymptotics of the solution of the dispersion equation (3.59) of version II of the EMM. Keeping only the principal terms

3.6 Versions II and III of the EMM

39

in the real and imaginary parts of the kernel G∗ (x) in (3.83) and changing its right hand side E∗ (x) to a constant (E∗ (x) ≈ E∗ (0) = U∗ ), we ﬁnd the solution of the resulting equation (the electric ﬁeld inside the coated inclusion) in the form that is the consequence of the asymptotic representation (3.64) E(x) = Es (x) + ik∗3 Eω (x).

(3.95)

Here Es (x) is a static electric ﬁeld (ω = 0) inside the coated inclusion. After substituting this equation into (3.83) and gathering the terms that have the same order in respect to k∗ we go to the following equations for the functions Es (x) and Eω (x) in (3.95) Gs∗ (x − x ) · ε∗1 (x ) · Es (x )dx = U∗ , (3.96) Es (x) − V Eω (x) − Gs∗ (x − x ) · ε∗1 (x ) · Eω (x )dx = Eω (3.97) ∗, Eω ∗ =−

V

G∗ ω · ε∗1 (x ) · Es (x )dx ,

x ∈ V,

V = v + vl .

(3.98)

V

Using the results of Section 6.3 for the solution of static problems for a layered inclusion in a homogeneous medium we present the solutions of (3.96) in the forms Es (r, n) = [1 + A(r, n)] · U∗ ,

Eω (r, n) = [1 + A(r, n)] · Eω ∗.

d α(r), A(r, n) = 1 + (n ⊗ n)r dr ⎧ (1) ⎪ if 0 ≤ r ≤ r0 , ⎨ Ys α(r) = Ys(2) + Ys(3) r−3 if r0 < r < r1 , ⎪ ⎩ (4) −3 if r1 ≤ r. Ys r

(3.99) (3.100)

(3.101)

(q)

The equations for the constants Ys (q = 1, 2, 3, 4) are the consequence of the general conditions on the borders of the inclusion, the layer, and the medium (r = r0 and r = r1 ) dα [α(r)]j = 0, ε(r)r + [ε(r)α(r)]j + [ε(r)]j = 0, j = 0, 1. (3.102) dr j As the consequence of these conditions, the equation for the constants takes the form ⎡ ⎤ ⎡ (1) ⎤ ⎡ ⎤ Ys 1 −1 −r0−3 0 0 (2) ⎥ ⎢ 0 1 r−3 −r−3 ⎥ ⎢ ⎥ ⎥ ⎢ 1 1 ⎥ ⎢ Ys ⎢ ⎢ 0 ⎥, = (3.103) ⎥ ⎢ (3) ⎣ 1 0 m33 0 ⎦ ⎣ Ys ⎦ ⎣ −1 ⎦ (4) −1 0 1 m43 m44 Ys

(q) Ys

40

3. Electromagnetic waves in composites and polycrystals

m33 = ε¯∗ =

3r0−3 , ε¯1

ε∗ , ε0

m43 =

ε¯1 =

ε¯∗ + 2 −3 r , ε¯∗ − 1 1

m44 =

3¯ ε∗ −3 r , 1 − ε¯∗ 1

(3.104)

ε − ε0 . ε0

The solution of this system for r0 = 1, r1 = p−1/3 is (3 + 2p¯ ε1 )(¯ ε∗ − 1) − ε¯1 (1 + 2¯ ε∗ ) , (3 + ε¯1 )(1 + 2¯ ε∗ ) + 2p¯ ε1 (1 − ε¯∗ ) [3 + ε¯1 (1 + 2p)](¯ ε∗ − 1) , = (3 + ε¯1 )(1 + 2¯ ε∗ ) + 2p¯ ε1 (1 − ε¯∗ ) −3¯ ε∗ ε¯1 , = (3 + ε¯1 )(1 + 2¯ ε∗ ) + 2p¯ ε1 (1 − ε¯∗ ) (3 + ε¯1 )(¯ ε∗ − 1) − p¯ ε1 (2 + ε¯∗ ) . = p(3 + ε¯1 )(1 + 2¯ ε∗ ) + 2p2 ε¯1 (1 − ε¯∗ )

Ys(1) =

(3.105)

Ys(2)

(3.106)

Ys(3) Ys(4)

(3.107) (3.108)

(q)

After substituting these equations for Ys into (3.99), (3.100) and calculating the integral Eω we obtain the principal terms of the long-wave asymptotics of the ﬁeld E(x) inside the kernel of the coated inclusion. The principal terms of such asymptotics of the coeﬃcient Λ∗t in (3.94) follow from (3.95), (3.99), (3.105) and (3.106) in the form Λ∗t ≈ Λs (ε∗ ) − ik∗3 Λω (ε∗ ),

(3.109)

Ys(1) , (1) + Ys )

(3.110)

s

Λ (ε∗ ) = 1 + Λω (ε∗ ) =

2(1

[(ε − ε∗ )(1 + Ys(1) ) 9ε∗ + (ε0 − ε∗ )(1 + Ys(2) )(p−1 − 1)].

(3.111)

After substituting this equation for Λ∗t into the dispersion equation (3.59) and solving the latter with the accuracy of k0 in the real part, and k04 in the imaginary part, we obtain s i εs∗ 4 ε ε1 − p (k0 a) ∗ 2 Λω (εs∗ ). (3.112) k∗ a = k0 a ε0 2 ε0 Here εs∗ is the static dielectric property of the composite that is the solution of the following equation εs∗ = ε0 + pε1 Λs (εs∗ ),

(3.113)

that is a consequence of (3.59). After substituting Λs into this equation from (3.110) we obtain a square matrix equation for the eﬀective dielectric permittivity εs∗ of the composite, and the proper solution of this equation has the form

3.6 Versions II and III of the EMM

εs∗ = ε0 +

3pε1 . 3 + (1 − p)¯ ε1

41

(3.114)

This equation coincides with the result of the Maxwell-Garnet theory (see [69,80]). This equation has physical meaning for all values of the volume concentration of inclusions and their properties. Therefore, (3.114) is free from the shortcomings of (3.72) for εs∗ obtained in the framework of version I of the EMM. The lines with dots in Fig. 3.6 present εs∗ (p) (3.114) for optically hard (ε = 100) and optically soft (ε = 0.01) inclusions in the homogeneous matrix (ε0 = 1). The solid lines in this ﬁgure correspond to the solutions of (3.66) of version I of the EMM for this cases. Substituting (3.114) for εs∗ into the right hand side of (3.105)–(3.108) for (q) Ys (ε∗ = εs∗ ) we obtain ε1 (1 − p) , 3ε0 + (1 − p)ε1 ε1 =− , 3ε0 + (1 − p)ε1

pε1 , 3ε0 + (1 − p)ε1

Ys(1) = −

Ys(2) =

Ys(3)

Ys(4) = 0.

(3.115) (3.116) (4)

As it is seen from (3.99), (3.100), the constant Ys deﬁnes the static electric ﬁeld in the eﬀective medium caused by the presence of the coated inclusion. The equivalence of this ﬁeld to zero means that this inclusion does not disturb the external ﬁeld applied to the medium. Thus, the condition of self-consistency II2 gives the same result as the condition of self-consistency III2 (see Section 1.3.3). III2 . The properties of the eﬀective medium should be chosen so as to decrease the disturbed ﬁeld caused by the presence of the coated inclusion. 4 εs

ε=100

*

3

2

1 ε=0.01 0

0

0.2

0.4

0.6

p

0.8

εs∗

Fig. 3.6. The dependence of the static dielectric permittivity of a matrix composite with spherical inclusions on the volume concentrations of inclusions. The properties of the inclusions are ε = 100 (the upper part of the ﬁgure) and ε = 0.01 (the low part of it), the property of the matrix ε0 = 1. Solid lines correspond to version I of the EMM and lines with circles to versions II and III of the EMM and to the EFM.

42

3. Electromagnetic waves in composites and polycrystals (4)

(Note that the equivalence to zero of the numerator in (3.108) for Ys immediately gives equation (3.114) for εs∗ .) Therefore, for spherical inclusions, versions II and III of the EMM give the same result, at least in the long-wave limit. Let us calculate the factor Λω in the imaginary part of the eﬀective wave (1) (2) number in (3.112). After substituting Ys , Ys from (3.105), (3.106) into (3.111) for Λω we obtain Λω = 0.

(3.117)

It means that the imaginary part of the eﬀective wave number k∗ has an order higher than ω 4 when ω → 0. Thus, version II of the EMM does not describe the attenuation caused by the Rayleigh scattering of waves on inclusions. This conclusion does not depend on the volume concentrations of inclusions, and therefore, version II does not give the correct long wave asymptotics of the attenuation coeﬃcients even for small concentrations of inclusions. Let us consider version III of the EMM. As is shown in [2], (Chapter 8), the forward amplitude F(k∗ , n0 ) of the wave ﬁeld scattered by a coated inclusion is presented in the form F(k∗ , n0 ) = −

∞ i (2n + 1) Yn(7) (k∗ ) + Yn(8) (k∗ ) U∗ . 2k∗ n=1

(3.118)

According to the condition of self-consistency III2 the eﬀective wave number k∗ should be chosen in order to decrease the vector F(k∗ , m). As was said above, the static limit of this version of the method coincides with the results of version II. It is possible also to demonstrate that the attenuation coeﬃcients obtained in the framework of version III of the EMM also does not describe the Rayleigh scattering of waves on inclusions.

3.7 The eﬀective ﬁeld method As in the previous sections, we consider propagation of a plane monochromatic wave through the medium with a random set of isolated inclusions. The main hypotheses of the EFM were formulated in Section 2.2 and may be applied to electromagnetic waves without corrections. We introduce the hypotheses of the quasicrystalline approximation as they were formulated at the end of Section 2.2. H1 . Each inclusion in the composite behaves as an isolated one in the ∗ (x). This ﬁeld background medium (matrix) aﬀected by the eﬀective ﬁeld E is a plane wave that is the same for all the inclusions.

3.7 The eﬀective ﬁeld method

43

∗ (x) is the average of local exciting ﬁeld E∗ (x) by the condiH2 . Field E tion x ∈ V. ∗ (x) = E∗ (x)|x . E

(3.119)

∗ (x) that acts on each inclusion in the composite Thus, the eﬀective ﬁeld E is taken in the form ∗ (x) = U∗ e−ik∗ ·x , E

(3.120)

where U∗ and k∗ are unknown amplitude and wave vector of this ﬁeld. The wave ﬁeld inside the inclusion that occupies a region v satisﬁes an integral equation (the equation of the one particle problem of the EFM) that is a consequence of (3.7): ∗ (x) + G(x − x ) · ε1 · E(x )dx , x ∈ v. (3.121) E(x) = E v

Let the general solution of this equation be known, and the electric ﬁeld inside the inclusion v0 with the center at the point x = 0 be presented in the form ∗ (x), ∗ (x) = λ(x, k∗ ) · E E(x) = ΛE ∗ ∗ ik ·x ∗ (x) ⊗ U e λ(x, k∗ ) = ΛE |U∗ |−2 .

x ∈ v0 ,

(3.122) (3.123)

In this equation, Λ is a linear operator; λ(x, k∗ ) is a second rank tensor function that depends on the properties of the matrix and the inclusion. The electric ﬁeld inside the inclusion vj centered at the point xj takes a form similar to (3.122): ∗ (x) = Λj U∗ e−ik∗ ·(x−xj ) e−ik∗ ·xj E(x) = Λj E ∗ (x), x ∈ vj ; = λj (x − xj , k∗ ) · E ∗ (x) ⊗ U∗ eik∗ ·y |U∗ |−2 . λj (x, k∗ ) = Λj E

(3.124) (3.125)

Here the linearity of the operator Λj is taken into account. Note that the function λj (x, k∗ ) depends on the shape and properties of the jth inclusion and does not depend on the position of the inclusion in space. This function is to be constructed from the solution of the one-particle problem for the jth inclusion centered at the point x = 0. k∗ ) that coincides with Let us introduce a stationary random function λ(x, the function λj (x − xj , k∗ ) inside the region vj (j = 0, 1, 2, ...) and equal to zero in the matrix. Using this function, we may write the ﬁeld E(x) in the composite medium in the form 0 , k∗ ) · E ∗ (x )V (x )dx . (3.126) E(x) = E (x) + G(x − x ) · ε1 · λ(x

44

3. Electromagnetic waves in composites and polycrystals

This is a consequence of the integral equation (3.7) of the one-particle problem, and representation (3.124) of the electric ﬁelds inside the inclusions in the framework of the EFM. In order to ﬁnd the mean electric ﬁeld E(x) in the medium we average (3.126) over the ensemble realizations of the random set of inhomogeneities. ∗ (x) is the same for all the inclusions, we obtain Because the ﬁeld E ∗ (x )dx , E(x) = E0 (x) + p G(x − x ) · ε1 · Λ0 (k∗ ) · E (3.127) 1 1 k∗ )dx = 1 lim λ(x, k∗ )dx. (3.128) λ(x, Λ0 (k∗ ) = Ω→∞ p Ω Ω v0 v 0 k∗ ) is a Here Λ0 (k∗ ) is a constant (with respect to x) tensor because λ(x, realization of a stationary random function, Ω is a region that occupies all 3D-space in the limit Ω → ∞, p is the volume concentration of inclusions. It is taken into account that for a stationary random function, the ensemble and spatial averages coincide. Let us ﬁnd a local external ﬁeld E∗ (x) acting on the inclusion that covers an arbitrary point x. In the framework of the EFM, this ﬁeld takes the form , k∗ ) · E ∗ (x )V (x ; x)dx . (3.129) E∗ (x) = E0 (x) + G(x − x ) · ε1 · λ(x Here V (x ; x) is the characteristic function (with argument x ) of the region Vx deﬁned by the relation (2.26) Vx =

∪ vi , i = k

when x ∈ vk .

(3.130)

∗ (x) we use (3.119) (condition of selfIn order to ﬁnd the eﬀective ﬁeld E consistency) ∗ (x) =< E∗ (x)|x > . E

(3.131)

and average (3.129) under the condition x ∈ V . As a result, we obtain a ∗ (x) closed equation for the eﬀective ﬁeld E 0 ∗ (x )Ψ (x − x)dx , E∗ (x) = E (x) + p G(x − x ) · ε1 · Λ0 (k∗ ) · E (3.132) Ψ (x − x) =

1 V (x ; x)|x . p

(3.133)

Here the function Ψ (x) depends on the geometric characteristics of the random set of inclusions, as discussed in Section 2.2.

3.7 The eﬀective ﬁeld method

45

The diﬀerence between (3.127) and (3.132) gives ∗ (x) = E(x) − p G(x − x )Φ(x − x ) · ε1 · Λ0 (k∗ ) · E ∗ (x )dx , E (3.134) Φ(x) = 1 − Ψ (x),

(3.135)

where the function Φ(x) is equal to zero outside a certain vicinity of the origin (x = 0). Equation (3.134) is a convolution equation. This fact has two conseque ∗ (x) have the same wave vector k∗ . nces. First, the plane waves E(x) and E E(x) = Ue−ik∗ ·x ,

∗ (x) = U∗ e−ik∗ ·x . E

(3.136)

∗ (k) of these ﬁelds are Secondly, the Fourier transforms E(k) and E connected by the algebraic equation ∗ (k) = Π(k) · E(k) , E

(3.137) 0

−1

Π(k) = [1 + pG (k) · ε1 · Λ (k∗ )] Φ G (k) = G(x)Φ(x)eik·x dx. Φ

,

(3.138) (3.139)

After applying the Fourier transform operator to (3.127) and using (3.134) we obtain the following equation for the Fourier transform of the mean wave ﬁeld E(k): E(k) = E0 (k) + pG(k) · ε1 · Λ0 (k∗ ) · Π(k) · E(k) .

(3.140)

Let us multiply both parts of this equation with the tensor L0 (k) = G−1 0 (k) deﬁned in (3.30): L0 (k) =

ε0 2 k − k02 1 − k 2 m ⊗ m . k02

Because L0 (k) · E0 (k) = 0, the Fourier transform of the mean electric ﬁeld in the composite should satisfy the equation: [(k2 − k02 )1 − k 2 m ⊗ m − pk02 ε¯1 · Λ0 (k∗ ) · Π(k)] · E(k) = 0,

1 ε1 . ε0 (3.142)

ε¯1 =

The Fourier transform of the mean wave ﬁeld (3.136) has the form E(k) = (2π)3 Uδ(k − k∗ ). Thus, the determinant of the tensor in front of the vector E(k) in (3.142) should be equal to zero for k = k∗ = k∗ n0 ∗

det[(k∗2 − k02 )1 − k∗2 n0 ⊗n0 − pk02 ε¯1 · Λ0 (k∗ ) · Π(k )] = 0.

(3.143)

46

3. Electromagnetic waves in composites and polycrystals

This is the condition of existence of a non-trivial vector E(k), and in essence, the dispersion equation for the wave vector k∗ = k∗ m of the mean electric ﬁeld in the framework of the EFM. If ε1 is an isotropic tensor (ε1 = ε1 1), the the spatial distribution of inclusion is homogeneous and isotropic (Φ(x) = Φ(|x|)), the wave vectors k0 and k∗ should have the same direction (m), and the tensors Λ0 (k∗ ) and ∗ Π(k ) in (3.128), (3.138) take the forms Λ0 (k∗ ) = Λ0t (k∗ )(1 − n0 ⊗n0 ) + Λ0l (k∗ )(n0 ⊗n0 ), 0

0

0

0

Π(k∗ ) = Πt (k∗ )(1 − n ⊗n ) + Πl (k∗ )(n ⊗n ).

(3.144) (3.145)

For a transverse electric ﬁeld propagating in the composite E(x) = Ue−ik∗ n

0

·x

,

U⊥n0 ,

(3.146)

we obtain from (3.143) the dispersion equation for the wave number k∗ in the form k∗2 − k02 − pk02 ε1 Λ0t (k∗ )Πt (k∗ ) = 0.

(3.147)

If Λ0t (k∗ )Πt (k∗ ) = Λ0l (k∗ )Πl (k∗ ), a longitudinal wave may propagate through the inhomogeneous material. The dispersion equation for the wave numbers of longitudinal waves is 1 + pε1 Λ0l (k∗ )Πl (k∗ ) = 0.

(3.148)

In this study, we consider mainly transverse electromagnetic waves, and focus attention on (3.147). The solution of the dispersion equation (3.147) is the ﬁnal aim of the theory: the eﬀective dielectric permittivity of the composite ε∗ , the phase velocity v∗ and the attenuation γ of the mean wave ﬁeld are deﬁned via the wave number k∗ by the equations ε∗ =

k∗2 , ω2

v∗ =

ω , Re(k∗ )

γ = − Im(k∗ ).

(3.149)

The mean wave ﬁeld from a point source (the mean Green function < G(x) >) may be also constructed from (3.142). If one accepts that E0 (x) = G(x), where G(x) has the form (3.10) the mean wave ﬁeld coincides with the mean Green function, and for the Fourier transform G(k) of this function we obtain the equation −1

G(k) = [L0 (k) − pε1 · Λ∗ (k∗ ) · Π(k)]

.

(3.150)

3.8 One-particle EFM problems for spherical inclusions

47

3.8 One-particle EFM problems for spherical inclusions The one-particle problem of EFM is the solution of the integral equation ∗ (x) has form (3.120), and is a plane transverse wave; v is a (3.121) when E spherical region of radius a (a = 1) centered at point x = 0. The integral equation (3.121) is equivalent to the following system of diﬀerential equations: E + k2 E = (k02 − k∗2 )U∗ e−ik∗ ·x , E +

k02 E

=

(k02

−

k∗2 )U∗ e−ik∗ ·x ,

|x| ≤ a, |x| > a.

(3.151)

The boundary conditions on the surface of the inclusion the forms −ikhave 0r e (3.40), and the condition at inﬁnity is: E(x) → E∗ (x)+O if |x| → ∞. r This problem diﬀers from the Mie problem by the terms on the right-hand side of (3.151). The same technique as for the Mie problem may be used for solution of (3.151), (3.40). In particular, the ﬁeld inside the inclusion takes the form (x ∈ v): E(x) = |U∗ |

∞ n=1

in

(2n + 1) (1) (1) (1) Xn M01n (x) + Xn(2) Ne1n (x) n(n + 1)

k 2 − k∗2 ∗ −ik∗ ·x + 02 U e . k − k∗2

(3.152) (1)

Here the notation is the same as in (3.46), and the coeﬃcients Xn and of this series are obtained from the boundary conditions in the forms

(2) Xn

Xn(1) =

k(k02 − k 2 ) [k0 ξn (k0 )ψn (k∗ ) − k∗ ψn (k∗ )ξn (k0 )] , k∗ (k∗2 − k 2 ) [k0 ξn (k0 )ψn (k) − kψn (k)ξn (k0 )]

(3.153)

Xn(2) =

k(k02 − k 2 ) [k∗ ξn (k0 )ψn (k∗ ) − k0 ψn (k∗ )ξn (k0 )] , k∗ (k∗2 − k 2 ) [kξn (k0 )ψn (k) − k0 ψn (k)ξn (k0 )]

(3.154)

where the functions ψn and ξn are deﬁned in (3.50). Equation (3.152) for E(x) together with (3.128), (3.144) give us the following equation for the coeﬃcient Λ0t in (3.147) ∞ 3 (2n + 1) Xn(1) g0n (k∗ , k) 2 n=1 jn (k∗ )jn (k) k 2 − k∗2 − g1n (k∗ , k) + Xn(2) (n + 1) + 02 , k∗ k k − k∗2 1 g0n (k∗ , k) = 2 [k∗ jn+1 (k∗ )jn (k) − kjn+1 (k)jn (k∗ )], k∗ − k 2 1 g1n (k∗ , k) = 2 [k∗ jn (k∗ )jn+1 (k) − kjn (k)jn+1 (k∗ )]. k∗ − k 2

Λ0t =

(3.155) (3.156) (3.157)

48

3. Electromagnetic waves in composites and polycrystals

Let us consider the integral GΦ in (3.139) for a homogeneous and isotropic spatial distributions of inclusions. In this case Φ(x) = Φ(|x|), and after integration over the unit sphere we present GΦ in the form of one-dimensional integrals: GΦ = Gt (k0 , k∗ )(1 − n ⊗ n) + Gl (k0 , k∗ )(n ⊗ n),

(3.158)

Gt (k0 , k∗ ) = k02 qt (k0 , k∗ ) + Jt (k0 , k∗ ), ∞ qt (k0 , k∗ ) = e−ik0 r Φ(r)j0 (k∗ r)rdr,

(3.159) (3.160)

0

∞ −ik0 r 1 e Jt (k0 , k∗ ) = − + 3 r 0 j1 (k∗ r) (k0 r)2 Φ(r)dr, × j2 (k∗ ζ)(1 + ik0 r) − k∗ r Gl (k0 , k∗ ) = −

1 − 3

0

∞

e−ik0 r j2 (k∗ ζ) 2(1 + ik0 r) − (k0 r)2 r j1 (k∗ r) (k0 r)2 Φ(r)dr. + k∗ r

(3.161)

(3.162)

The functions Πt and Πl in (3.145) are expressed via the integrals Gt and Gl in the forms Πt = [1 + pε1 Λ0t (k0 , k, k∗ )Gt (k0 , k∗ )]−1 ,

(3.163)

pε1 Λ0l (k0 , k, k∗ )Gl (k0 , k∗ )]−1 .

(3.164)

Πl = [1 +

Thus, dispersion equation (3.147) of the EFM for the transverse part of the mean wave ﬁeld may be written in the ﬁnal form k∗2 = k02 + pk02 ε1 [Λ0t (k0 , k, k∗ )−1 + pε1 Gt (k0 , k∗ )]−1 .

(3.165)

3.9 Asymptotic solutions of the EFM dispersion equation 3.9.1 Long-wave asymptotics Consider the EFM dispersion equation (3.165) for very long waves when k0 a, k∗ a << 1. In this case, we can use the asymptotic equation for G(x) similar to (3.64) in the integral equation (3.121) of the one-particle problem as well as for the calculation of the integral GΦ in (3.139) and (3.158). As the result, the long wave asymptotics for the integral GΦ and the coeﬃcients Λ0t and Λ0l in (3.163), (3.164) take the forms

3.9 The long and short-wave asymptotics of the EFM

∞ 2 1 3 + i (k0 a) JΦ 1, JΦ = Φ(ζ)ζ 2 dζ, 3 3 0 r 3 2ε1 0 0 3 , ζ= . 1 − ik0 Λ t = Λl = 3 + ε1 3(3 + ε1 ) a

GΦ = −

49

(3.166) (3.167)

After substituting these equations into (3.165) and keeping the terms of order k0 in the real part and of order k04 in the imaginary part, we obtain the solution in the form 4 (k0 a) (εs − ε0 )2 εs f √ s , −i (3.168) k∗ a = k0 a ε0 9p ε0 ε∗ ε0 f = 1 − 3pJΦ .

(3.169)

Here εs∗ is the static eﬀective dielectric property of the composite obtained in the framework of the EFM 3pε1 . (3.170) εs = ε0 + 3 + (1 − p)ε1 This equation for εs coincides with the Maxwell-Garnet formula [28] and with equation (3.114) of version II of the EMM that was discussed in Section 3.6. The imaginary part of the eﬀective wave number k∗ in (3.168) is proportional to k04 (ω 4 ) and describes the attenuation caused by the Rayleigh scattering of electromagnetic waves by inclusions. The dependence of the attenuation coeﬃcient on the space distribution of the inclusions in the longwave region is deﬁned by the integral JΦ in (3.166). In order to have physically correct values of γ (γ > 0) the structural factor f in (3.169) should be positive f = 1 − 3pJΦ > 0.

(3.171)

Let V (x) be the characteristic function of the region occupied by a spatially homogeneous random set of spherical inclusions of unit radii (a = 1). The covariance S2 (x) = V (y)V (y + x) of the random function V (x) is connected with the correlation function Φ(|x|) in (3.133), (3.135) by the equation 3p S0 (|x|) + p2 (1 − Φ(|x|)), (3.172) 4π where S0 (|x|) is the volume of the intersection of two spheres of unit radii if |x| is the distance between their centers 4 |x|3 4π 1 − |x| + if |x| ≤ 2; S0 (|x|) = 3 3 16 S2 (x) =

S0 (|x|) = 0 if |x| > 2.

(3.173)

It is shown in [99] that for any realizable random function V (x) the following inequality holds S2 (x) − p2 dx ≥ 0. (3.174)

50

3. Electromagnetic waves in composites and polycrystals

After substituting in this equation function S2 (x) from (3.172) and taking into account that

2

S0 (|x|)dx = (4π/3) we obtain

4 S2 (x) − p2 dx = πp − p2 Φ(|x|)dx 3 ⎛ ⎞ ∞ 4 ⎝ 4 = πp 1 − 3p Φ(r)r2 dr⎠ = πpf. 3 3

(3.175)

0

This equation together with (3.174) show that the structural factor ∞ f = 1 − 3p Φ(ζ)ζ 2 dζ

(3.176)

0

is non-negative if the function Φ(x) corresponds to a realizable spatial distribution of the inclusions. 3.9.2 Short-wave asymptotics Let us ﬁnd the short-wave limit (k0 , k∗ → ∞) of the solution of (3.165) in a form similar to (3.76) k∗ = k0 − iγ.

(3.177)

Here γ does not depend on k0 . The short wave asymptotics of the functions Λ0t in (3.155) have the form 3 i − 2γ . (3.178) Λ0t = − k0 ε1 2 We take into account that the principal term of the short wave asymptotic of the series in (3.155) is equal to −3i/ (2k0 ε1 ) , and the principal term of such an asymptotic of the last term in this equation is i2γ/ (k0 ε1 ) . For the integral Gt in (3.165) we have the following short wave asymptotic equation: Gt = k02 q(k0 , k∗ ) + Jt (k0 , k∗ ) ∞ = k02 e−ik0 ζ Φ(ζ)j0 (k∗ ζ)ζdζ + O(1) 0 ∞ k2 = 0 e−i(k0 −k∗ )ζ Φ(ζ)dζ + O(1) 2ik∗ 0 k0 sin(k∗ ζ) , = −i IΦ (γ) + O(1), j0 (k∗ ζ) = 2 (k∗ ζ) ∞ IΦ (γ) = eγζ Φ(ζ)dζ. 0

Here the limit form (3.177) of k∗ is used.

(3.179) (3.180)

3.10 Numerical solution

51

After substituting (3.178), (3.179) into the dispersion equation (3.165) and solving the resulting equation with respect to k∗ we obtain k∗ a = k0 a − ip

−1 3 3 − γa 1 − p − γa IΦ (γa) . 4 4

(3.181)

Comparing (3.177) and (3.181) we see that the limit value γ of the attenuation coeﬃcient γ is the solution the following equation γa = p

−1 3 3 − γa 1 − p − γa IΦ (γa) . 4 4

(3.182)

Thus, in the short-wave limit, the attenuation coeﬃcient γ does not depend on the properties of the inclusions and is a function only of their volume concentration p and the pair correlation function Φ(r). This function is given by (3.182) via the integral IΦ deﬁned in (3.180). If p is small, the solution of (3.182) is γa =

3 p. 4

(3.183)

This asymptotic value of γ corresponds to the case of independent scatterers, and coincides with the short-wave asymptotic solution of the dispersion equation (3.78) of the EMM. If p is not small, the attenuation coeﬃcient γa becomes smaller than 34 p; this is a result of the spatial correlation in the positions of the inclusions.

3.10 Numerical solution For the numerical solution of the EFM dispersion equation (3.165), the correlation function Φ(x) of the random set of inclusions should be deﬁned.The most reliable correlation function of a homogeneous set of non-overlapping spheres is constructed from the so-called Perkus-Yevick pair correlation function of the centers of spheres. Let ψ(|x|) be the normalized probability to ﬁnd the center of some inclusion at a point x if the point x = 0 is occupied by the center of another inclusion. The function Φ(x) in (3.133), (3.135) is connected with the function ψ(|x|) by the relation Φ(x) = 1 −

3 4π

2

v0 (x )dx

ψ(|x − x |)v0 (x − x)dx ,

(3.184)

where v0 (x) is the characteristic function of the spherical region with unit radius, and center at the point x = 0. The double integral over 3D-space in this equation is reduced to a single (one-dimensional) integral.

52

3. Electromagnetic waves in composites and polycrystals

3 Φ(r) = 1 − 2

r+2 f (r, s)ψ(s)s2 ds,

(3.185)

2

if r > 2 and 0 < s < r − 2, 1 f (r, s) = 2 − [20(|r + s|3 − |r − s|3 ) 80rs + |r + s|5 − |r − s|5 ] if 0 < r < 2 , 1 f (r, s) = [40(4 − |r − s|2 ) − 20(8 − |r − s|3 ) 80rs + 32 − |r − s|5 ] if r > 2 and r − 2 < s < r + 2 . f (r, s) = 0

The function ψ(r) satisﬁes the so-called Ornstein-Zernike integral equation. An analytical solution of this equation was found in [110] and is presented in Appendix B. In order to construct the function Φ(r) for various p, the integral in (3.185) was calculated numerically using the deﬁnition of the function ψ(r) in Appendix B. The graphs of the functions ψ(r) and Φ(r) are presented in Fig. 3.7 for the volume concentrations p = 0.1; 0.3; 0.5, ζ = r/a. After substituting Φ(r) from (3.184) into (3.169) for the structural coeﬃcient f in the long-wave asymptotic of the eﬀective wave number (3.168), we obtain the following equation ⎤ ⎡ ∞ r (3.186) f = 1 − p ⎣8 + 3 (1 − ψ(ζ)) ζ 2 dζ ⎦ , ζ = . a 2

For the Percus-Yevick correlation function ψ(r), the integral in this equation is calculated, and the structural factor f takes the form (see [106, 111])

5 Ψ (ζ)

Φ (ζ)

1 0.5

0.8

4

0.5

0.6

3

0.3

2

p=0.1

0.3

0.4 p=0.1 0.2

1

0 0

0 0

2

4

6

8

1

2

3

4

ζ

ζ −0.2

Fig. 3.7. The Percus-Yevick pair correlation function ψ(ζ) and the correlation function Φ(ζ) in (3.185) for a non-overlapping random set of spheres of unit radii for the volume concentrations of spheres p = 0.1; 0.2; 0.3, ζ = r/a.

3.10 Numerical solution

f=

(1 − p)4 . (1 + 2p)2

53

(3.187)

The dependence of the short wave limit γ of the attenuation coeﬃcient (the solution of (3.182) for the function Φ(r) deﬁned in (3.185)) on the volume concentrations of inclusions is presented in Fig. 3.8 by the line with dots. The solid line is the short wave limit of the attenuation coeﬃcient in the framework of the EMM: γa = 3p/4. Let us consider optically soft inclusions (ε0 = 1, ε = 0.01). In this case, there is only one branch of the solutions of (3.165), and the dependences of the relative velocity and attenuation of the mean wave ﬁeld on the parameter k0 a are presented in Fig. 3.9. 0.3

γa 0.2

0.1

0 0

0.1

0.2

0.3

p

0.4

Fig. 3.8. The dependence of the short-wave limit γ of the attenuation coeﬃcient on the volume concentration p of inclusions. The line with dots corresponds to the EFM: the solution of (3.182) for the correlation function Φ(r) in (3.185); the solid line to the EMM (3.78). 0

lg (γa) 1.3

−1 p=0.1 0.2

v*/v0 0.3

0.3

−2

1.2

−3

0.2 1.1

−4

p=0.1

lg(k0a) −5 −1

−0.5

0

0.5

1

1.5

2

1 −1

−0.5

lg(k0a) 0

0.5

1

1.5

2

Fig. 3.9. Dependences of the attenuation coeﬃcients γa and the relative velocities v∗ /v0 of the mean wave ﬁeld in the medium with spherical inclusions, on the wave number of the matrix k0 a, and on the volume concentration p, for optically soft inclusions (ε0 = 1, ε = 0.01). (The EFM with the Percus-Yevick correlation function.)

54

3. Electromagnetic waves in composites and polycrystals 3

0.4 Re(k*a)

γa

p=0.1

2.5

p=0.1

0.3

2 2

0.2 1.5 2

1 0.5 0

3

k0a 0.5

1

1.5

2

2.5

2 k0a

1

0

1 0

3

0.1

0

0.5

1

1.5

2

2.5

3

3 −0.1

Fig. 3.10. The dependences the real Re k∗ a and imaginary γa = −Imk∗ a parts of the wave number of the mean wave ﬁeld in the composite with optically hard (ε0 = 1, ε = 100) spherical inclusions, on the wave number k0 a of the matrix for the volume concentration of inclusions p = 0.1. 1,2,3 are diﬀerent branches of the solution of the dispersion equation (3.165). Dashed parts of the branches correspond to waves with large attenuations. (The EFM with the Percus-Yevick correlation function.)

For optically hard inclusions (ε0 = 1, ε = 100), there are several branches of the solutions of (3.165). For p = 0.1, three branches of the solutions are shown in Fig. 3.10. Branch 1 is the acoustical branch. This branch exists for k0 a < 0.3. After that, attenuation γ along this branch becomes negative, which is physically incorrect. In the region 0.3 < k0 a < 0.6, the mean wave ﬁeld is described by branch 2. Outside this region, branch 2 has large attenuation and is practically invisible. Branch 3 is an optical branch, and the attenuation along this branch is moderate till high frequencies. At high frequency, the attenuation γ along this branch tends to the short wave limit γ deﬁned by (3.182). The phase velocity along this branch tends to its value in the matrix material v0 . The number of diﬀerent branches of the solutions of (3.165) dramatically increases with the volume concentration of inclusions. The branches inside the interval 0 < k0 a < 3 for p = 0.3 are presented in Fig. 3.11. Thus, diﬀerent branches of the solution describe the mean wave ﬁeld in diﬀerent frequency intervals. The common envelope of these branches (the line with dots in Fig. 3.11) may be considered as one dispersion curve that describes the mean wave ﬁeld propagating in the composite.

3.11 Comparison of version I of the EMM and the EFM For comparison of predictions of the eﬀective ﬁeld and eﬀective medium methods, we consider a composite material experimentally studied in [76]. In these experiments, the dielectric permittivities of the matrix and the inclusions

3.11 Comparison of predictions of the EMM and EFM 4

0.6

Re(k*a)

3.5

p=0.3

γa

55

p=0.3

0.5

3

0.4

2.5 0.3

2 1.5

0.2

1

0.1

0.5 0

0

0.5

1

1.5

2

2.5

k0a

0

k0a

0 3 −0.1

0.5

1

1.5

2

2.5

3

Fig. 3.11. The dependences the real Re k∗ a and imaginary γa = −Im k∗ a parts of the wave number of the mean wave ﬁeld in the composite with optically hard (ε0 = 1, ε = 100) spherical inclusions, on the wave number of the matrix k0 a for the volume concentration of inclusions p = 0.3. The line with dots is the common envelope of diﬀerent branches of the solution of the dispersion equation (3.165). Dashed parts of the branches correspond to the waves with large attenuations. (The EFM with the Percus-Yevick correlation function.) 1.1 v*/v0

4 10xγa 0.0118

1

3

p=0.109

0.056 2 0.9

0.056 1

p=0.109

0.0118 0.8

0

0.5

1

1.5

k0a 2

0

0

0.5

1

1.5

k0a 2

Fig. 3.12. Dependences of the velocities v∗ and attenuation coeﬃcients γa of the mean wave ﬁeld in the medium (ε0 = 1) with spherical inclusions (ε = 5.2−0.03i) of various volume concentrations p = 0.0118; 0.056; 0.109, on the wave number k0 a of the matrix. Short lines with crosses (p = 0.0118), squares (p = 0.056), and triangles (p = 0.109) at the ends are experimental data of [76]. The solid lines corresponds to version I of the EMM, the lines with dots to the EFM with the Percus-Yevick correlation function.

were: ε0 = 1, ε = 5.2 − 0.03i. The experiments were carried out for frequencies that correspond to the value of the non-dimensional wave numbers k0 a in the region 1.08 < k0 a < 1.23. The volume concentrations of spherical inclusions were p = 0.018; 0.056; 0.109. The results are presented in Fig. 3.12. The lines with dots are predictions of the EFM, solid lines are version I of the EMM, the short lines with crosses, triangles, and squares at the ends are experimental data of [76] (∗−p = 0.0118; − p = 0.056; − p = 0.109).

56 3

3. Electromagnetic waves in composites and polycrystals Re(k*a)

p=0.109

10xγa

p=0.109

4 2

2 1

0

k0a 0

0.5

1

1.5

2

2.5

k0a

0 3

0

0.5

1

1.5

2

2.5

3

Fig. 3.13. The dependences the real Re k∗ a and imaginary γa = −Imk∗ a parts of the wave number of the mean wave ﬁeld in the composite with spherical inclusions(ε0 = 1, ε = 5.2 − 0.03i) on the wave number of the matrix k0 a for the volume concentration of inclusions p = 0.109. The solid line correspond to version I of the EMM, the lines with dots to the EFM. White dots indicate the acoustical branch and black dots the optical branch. Dashed parts of the branches correspond to the waves with large attenuations.

For p = 0.0118 and 0.056, the dispersion equation of the EFM, as well as the EMM, has only one branch. For p = 0.109, the EFM indicates two branches: the acoustical branch that is essential in the interval 0 < k0 a < 1.7 and then disappears, and the optical branch that describes the mean wave ﬁeld when k0 a > 1.7. This branch is shown by lines with black dots in Fig. 3.13. Meanwhile the dispersion equation of the EMM has only one branch for p = 0.109. These branches are presented in Fig. 3.13, in the interval 0 < k0 a < 3. Dashed parts of the lines in Figs. 3.12 and 3.13 correspond to the regions of the dispersion curves with large attenuations. For large volume concentrations of inclusions, the EMM dispersion equation also indicates the existence of two branches of the solutions. The dispersion curves for the composite with the considered properties of the components and p = 0.3 are presented in Fig. 3.14. Lines 1 in this ﬁgure correspond to the acoustical branches obtained from (3.59) of the EMM (solid and dashed line) and (3.165) of the EFM (line with dots). Lines 2 are optical branches obtained from the same equations (line with triangles corresponds to the EFM). Dashed parts of the branches are regions with large values of the attenuation coeﬃcients. Thus, when k0 a < 1.5, the mean wave ﬁeld is deﬁned by the acoustical branch because the attenuation along the optical branch is large in this region. For k0 a > 1.5, attenuation along the acoustical branch becomes large, and the optical branch deﬁnes the mean wave ﬁeld because attenuation along this branch is much smaller. In Fig. 3.14, transition from branch 1 to branch 2 of the EMM is shown by the line with arrow connected the dispersion curves correspond to these branches near the point k0 a = 1.5.

3.11 Comparison of predictions of the EMM and EFM 8

57

p=0.3

Re(k*a)

1 6

4 2 2

0

k0a 0

1

2

3

4

5

Fig. 3.14. The dependences of the real Re k∗ a part of the wave number of the mean wave ﬁeld in the composite with spherical inclusions (ε0 = 1, ε = 5.2 − 0.03i) on the wave number of the matrix k0 a for the volume concentration of inclusions p = 0.3. The solid line correspond to version I of the EMM, the lines with dots and triangles to the EFM. 1 is the acoustical branch and 2 is the optical branch. Dashed parts of the branches correspond to the waves with large attenuation. The arrow indicates the jump from the acoustical branch to the optical branch.

15

2 v*/ v0

p=0.3

2

10xγa

2

p=0.3 1

1.5

1

10 1

2 1

k 0a

0 0

1

2

3

4

2

5

1

0.5

1

k0a

0 5

0

1

2

3

4

5

Fig. 3.15. The dependences of the velocities and attenuation of the mean wave ﬁeld in the composite with spherical inclusions (ε0 = 1, ε = 5.2 − 0.03i), on the wave number of the matrix k0 a for the volume concentration of inclusions p = 0.3. The solid line correspond to version I of EMM, the lines with dots and triangles to EFM. 1 indicates the acoustical branch, and 2 the optical branch. Dashed parts of the branches correspond to waves with large attenuation. The arrow indicates the jump from the acoustical branch to the optical branch.

The dependences of the velocities and attenuation coeﬃcients of the mean wave ﬁeld on parameter k0 a for these branches are presented in Fig. 3.15. As is seen from these ﬁgures, both methods predict a transition from the acoustical branch to the optical one in the interval 1.5 < k0 a < 2. The acoustical branches of the EMM and EFM are close in the region k0 a < 1.5, and their optical branches are close for k0 a > 1.5. The phase velocities of

58

3. Electromagnetic waves in composites and polycrystals

the mean wave ﬁeld predicted by EMM and EFM are very close, but the attenuations deviate essentially in the region 0.5 < k0 a < 2. EFM predicts smaller attenuation values than EMM.

3.12 Versions I, II, and III of EMM Let us consider a composite material with a set of optically soft inclusions: ε0 = 1, ε = 0.1. In this case, the dispersion equations of the EMM and its versions has only one branch. Figure 3.16 shows the dependence of the attenuation coeﬃcient γ and velocity v∗ of the mean wave ﬁeld on the wave −1 lg(γ)

1.08 v *

p=0.1

p=0.1

1.06 −2

1.04 1.02

−3

1 −4 0.5 0

lg(k0) 0

0.5

0.98 −0.5

1

p=0.3

lg(γ)

1.25

lg(k0) 0

v*

0.5

1

p=0.3

−1 1.15

−2

1.05

−3 −4 −0.5 0

lg(k0) 0

lg(γ)

0.5

1

p=0.5

−0.95 −0.5 1.6

−1

1.4

−2

1.2

−3

1

−4 −0.5

lg(k0) 0

0.5

1

lg(k0) 0

v*

0.8 −0.5

0.5

1

p=0.5

lg(k0) 0

0.5

1

Fig. 3.16. The dependences of the attenuation coeﬃcients γ and the eﬀective phase velocities v∗ on the wave number k0 of the matrix for the composites with optically soft inclusions (ε = 0.1, ε0 = 1). Lines with crosses correspond to version I, lines with dots to version II, and lines with triangles to version III.

3.13 Approximate solutions of one-particle problems

59

number k0 of the matrix material, for various values of the volume concentrations of inclusions p. Solid lines correspond to version I, the lines with circles to version II. In order to obtain the eﬀective wave number k∗ of a composite medium in the framework of version III it is necessary to ﬁnd the roots of the function F(k∗ ) in the complex plane (Re k∗ , Im k∗ ), where F(k∗ ) is the forward scattering amplitude. This function is deﬁned by the right-hand side of (3.56), where Λ∗t has the form (3.94). The corresponding dependences of γ and v∗ on the wave number k0 of the matrix material are the lines with triangles in Fig. 3.16. Horizontal dashed lines in the graphs for γ in Fig. 3.16 correspond to the short-wave asymptotics of γ for the considered volume concentrations of inclusions (γa = 34 p). These graphs show that the principal discrepancies between diﬀerent versions of the method take place in the region of long and medium waves (0 < k0 r0 < 5). In the short-wave region (k0 r0 > 5) these discrepancies are small. In the long wave region, the slopes of the dependences lg γ(lg k0 ) in the logarithmic scales are diﬀerent for diﬀerent versions of the EMM. The correct slope (4) that corresponds to the Rayleigh wave scattering is observed only for version I. For version II and III, this slope is larger, and the attenuation is much less than version I predicts. This fact was discussed in Section 3.10.

3.13 Approximate solutions of one-particle problems In this section, approximate solutions of the one-particle problems are constructed on the basis of the variational formulation of the diﬀraction problem. 3.13.1 Variational formulation of the diﬀraction problem for an isolated inclusion Let us consider integral equation (3.121) of the one-particle EFM problem. ∗ (x) + G0 (x − x ) · ε1 · E(x )dx , (3.188) E(x) = E v

ε1 = ε − ε 0 ,

∗ (x) = U∗ e−ik∗ ·x . E

∗ (x) is the eﬀective ﬁeld Here v is the region occupied by the inclusion, E that acts on this inclusion. We show in this Section that the solution of this equation is a stationary point of the following functional JQ(E) = (E, ε1 · E) − (G0 ε1 E, ε1 · E) − (E∗ , ε1 · E) − (E∗ , ε1 · E),

(3.189)

60

3. Electromagnetic waves in composites and polycrystals

G0 (x − x ) · ε1 · E(x )dx , 1 (f, φ) = f (x) · φ(x)dx. v v

(G0 ε1 E)(x) =

v

Here the bar over the functions means complex conjugation. The variational derivative of the functional JQ has the form δ(JQ) ¯ − ε1 ·E ¯ − ε1 ·G0 ε1 E ¯ ∗ , δE , δE = ε1 ·E δE ¯ . + ε1 ·E − ε1 ·G0 ε1 E − ε1 ·E∗ , δ E

(3.190)

(3.191)

Here we take into account that ε1 is a rank two symmetric tensor, and the kernel G0 (x) of the operator G0 has the property G0 (−x) = G0 (x). The real Re (δE) and imaginary Im (δE) parts of the variation δE of the electric ﬁeld inside the inclusion may be considered as independent functions. Thus, substituting into (3.191) E = Re E+i Im E, E∗ = Re E∗ +i Im E∗ , and δE = Re δE+i Im δE we ﬁnd the equation δ(JQ) , δE = (ε1 · Re E − ε1 ·G0 ε1 Re E − ε1 · Re E∗ , Re (δE)) δE ¯ . + ε1 · Im E − ε1 ·G0 ε1 Im E − ε1 · Im E∗ , Im δ E (3.192) At a stationary point of the functional JQ, this variational derivative ¯ should be equal to zero. Thus, the multipliers in front of Re (δE) and Im δ E in this equation should vanish ε1 · Re E − ε1 ·G0 ε1 Re E − ε1 · Re E∗ = 0, ε1 · Im E − ε1 ·G0 ε1 Im E − ε1 · Im E∗ = 0.

(3.193)

Combining these two equation into one, and eliminating the common multiplier ε1 we go to (3.188). It was shown in [43] that if E∗ (x) = U0 e−ik0 ·x , the value of the functional JQ for the exact solution Ee (x) of (3.188) is proportional to the forward amplitude F(m) of the scattered ﬁeld (3.16). The imaginary part of JQ is proportional to the total normalized scattering cross-section Q of the inclusion 0

JQ(Ee ) = −U0 · F(n ),

Im[JQ(Ee )] =

k0 S0 Q. 4π

(3.194)

3.13.2 Plane wave approximation The variational formulation of the diﬀraction problem allows us to construct approximate solutions of the one-particle problem of self-consistent methods.

3.13 Approximate solutions of one-particle problems

61

Let us consider the so-called plane wave approximation when the electric ﬁeld E(x) inside the inclusion is taken in the form of a plane wave E(x) = Ee−il·x ,

l = lτ,

x∈v

(3.195)

with unknown amplitude E and wave vector l, |τ | = 1. After substituting (3.195) into the functional JQ (3.189) and using the Ritz scheme we obtain the following equation for the constant vector E E − IG · ε1 · E = F (l − k∗ )U∗ , 1 F (k) = eik·x dx, v v IG = IG (k0 , l) = G0 (x)eil·x f (x)dx, 1 f (x) = v(x + x )v(x )dx , v v

(3.196) (3.197) (3.198) (3.199)

where v(x) is the characteristic function of the region occupied by the inclusion with the center at the origin x = 0. The same equation for E may be obtained by the Galerkin scheme if we substitute (3.195) into (3.188), multiply both parts with eil·x , and then average the resulting equation over the volume v of the inclusion. For a spherically isotropic inclusion of radius a = 1 in an isotropic medium, it is natural to assume that the vectors l and k∗ have the same direction n0 . For this case, we have 1 4 f (x) = 1 − r + r3 , r < 2; f (x) = 0, r ≥ 2, r = |x|; 3 16 j1 (k) , F (k) = F (|k|) = 3 k IG = [k02 q(k0 , l) + K1 (k0 , l)]1 + K2 (k0 , l)n0 ⊗ n0 , 2 q(k0 , l) = e−ik0 r j0 (lr)f (r)dr, 0

Ki (k0 , l) =

0

2

e−ik0 r i (r)dr,

(i = 1, 2),

(3.200) (3.201) (3.202) (3.203) (3.204)

j1 (lr) , (3.205) lr 2 (r) = [f (r) − rf (r)]j2 (lr) − 2f (r)lrj1 (lr) − f (r)l2 rj0 (lr). (3.206) 1 (r) = f (r)j0 (lr) + [rf (r) − f (r)]

Thus, the solution of (3.196) for E takes the form −1 E = 1 − εˆ1 k02 q(k0 , l) + K1 (k0 , l) F (|l − k∗ |)U∗ .

(3.207)

If l = k∗ , this equation is simpliﬁed, and for the ﬁeld inside the inclusion, we have

62

3. Electromagnetic waves in composites and polycrystals

E(x) = Ee−ik∗ ·x ,

(3.208)

−1 E = 1 − εˆ1 k02 q(k0 , k∗ ) + K1 (k0 , k∗ ) U∗ .

(3.209)

This approximate solution of the one-particle problem may be used in the framework of the EFM as well as of the EMM. For the EFM, the coeﬃcient Λ0t (k∗ ) in the dispersion equation (3.165) coincides with multiplier in front of U∗ in (3.209) −1 Λ0t (k∗ ) = 1 − εˆ1 k02 q(k0 , k∗ ) + K1 (k0 , k∗ ) .

(3.210)

For the EMM, the similar coeﬃcient Λ∗t in (3.59) coincides with (3.210) if k0 = k∗ . As a result, the dispersion equation (3.59) for the eﬀective wave number k∗ that corresponds to version I of the EMM takes the form k∗2 = k02 + pk02 ε1 Λ∗t (k∗ ), −1 , Λ∗t (k∗ ) = 1 − ε¯∗1 k∗2 q(k∗ ) + K (1) (k∗ )

(3.212)

K (j) (k∗ ) = Kj (k∗ , k∗ ).

(3.213)

qˆ(k∗ ) = q(k∗ , k∗ ),

(3.211)

where q(k, l), Kj (k, l), (j = 1, 2) are deﬁned in (3.203), (3.204). The plane wave approximation essentially simpliﬁes solutions of the dispersion equations. The predictions of EFM and EMM based on the exact solution of the one-particle problem prove to be close to the solution of the dispersion equation with the approximate solution of the one-particle problem in the long- and medium-wave regions. In the short-wave region, these results may deviate essentially.

3.14 The EFM for composites with regular lattices of spherical inclusions One of the advantages of EFM in comparison to EMM is the possibility to take into account the inﬂuence of the peculiarities in spatial distributions of inclusions on the mean wave ﬁeld propagating in the composite. In this section, composites with regular lattices of spherical inclusions are considered. It is known that for such composites, there are stop bands in the frequency region where the waves exponentially attenuate, and pass-bands where they propagate without attenuation. In recent years, such composites have been of interest because of the possibility of creating new types of wave ﬁlters on the basis of such artiﬁcial materials (see, e.g., [39, 114], where some other areas of applications of these materials are discussed). In EFM, the spatial distribution of inclusions inﬂuence the form of the speciﬁc two point correlation function Φ(x) of random ﬁeld of inhomogeneities. This function is deﬁned in (3.133), (3.135) and may be constructed for random

3.14 The EFM for composites with regular lattices of spherical inclusions

63

sets as well as for regular lattices of inclusions. For the latter, the corresponding correlation function is the result of averaging the original regular lattice over spatial translations. The resulting mean wave ﬁeld may be interpreted as averaging of the detailed ﬁeld over such translations. Let V (x) be characteristic function of the region occupied by identical spherical inclusions of unit radii, the centers of which compose an inﬁnite regular lattice in 3D-space. If q is the vector of this lattice, the function V (x) takes the form v(x + q), q = ja1 + sa2 + ta3, (3.214) V (x) = q

where a1 , a2 , a3 are the vectors of an elementary cell of the lattice j, s, t = 0, ±1, ±2, ...; v(x) is the characteristic function of the region occupied by a sphere of the unit radius with center at the point x = 0. If r is a random vector homogeneously distributed in space, the realizations of the random function V (x + r) are various translations of the original regular lattice. In this case, the second statistical moment V (x)V (x + r) of the function V (x) is a periodic function. After averaging the product V (x)V (x + r) over the random vector r we obtain 1 v(x + x )v(x )dx , (3.215) f (x + q), f (x) = V (r)V (x + r) = v0 v 0 q where v0 is the volume of a unit sphere; the function f (x) has an explicit form (3.200). From (3.215) and using deﬁnition (3.133), (3.135) of the function Φ(x) we obtain

Φ(x) = 1 −

1 f (x + q). p q

(3.216)

Here the prime over the summation sign means omitting of the term with q = 0. For this function Φ(x), the integral GΦ (k) in (3.139) takes the form Φ Φ (3.217) G (k) = G(x)Φ(x) exp(ik·x)dx = GΦ 0 (k) + G1 (k), 1 G(x)f (x) exp(ik·x)dx, (3.218) GΦ 0 (k) = p

GΦ 1 (k)

1 ˜ =− f (µ)G(k − µ). v0 µ

(3.219)

The term µ = 0 is omitted in this sum, f˜(k) and G(k) are the Fourier transforms of functions f (x) and G(x) deﬁned in (3.215) and (3.11), µ is the vector of the inverse lattice with respect to the original one,

64

3. Electromagnetic waves in composites and polycrystals 2

j (µ) f˜(µ) = 12π 1 2 , µ

j1 (µ) =

sin(µ) cos(µ) , − µ2 µ

µ = |µ|.

(3.220)

Let us consider propagation of electromagnetic waves in a homogeneous dielectric medium with the tensor of dielectric permittivity ε∗ . The Fourier transform E(k) of the electric ﬁeld in such a medium satisﬁes the equation 2 (3.221) k (1 − m ⊗ m) − ω 2 ε∗ · E(k) = 0. Equation (3.142) for the mean electric ﬁeld in the composite medium may be rewritten in the form 2 k (1 − m ⊗ m) − k02 1 − p k02 ε1 · Λ0 (k∗ ) · Π(k) · E(k) = 0. (3.222) Comparing (3.221), (3.222) we note that the latter describes propagation of waves in a medium with the eﬀective tensor of dielectric properties ε∗ that has the following form ε∗ (k) = ε0 + pε1 · Λ0 (k∗ ) · Π(k).

(3.223)

In the static limit (ω, k → 0), for the medium with spherical inclusions, this tensor takes the form εs = ε0 + pε1 · Λ0s · Π s ,

(3.224)

3ε0 −1 1, Π s = Π(0) = [1+pε1 · GsΦ · Λs0 ] , Λ0s = Λ0 (0) = 3ε0 + ε1 1 Φ s s . G (x) = ⊗ Gs = G (x)Φ(x)dx, 4πε0 |x|

(3.225) (3.226)

For an isotropic and homogeneous random set of inclusions, the function Φ(x) depends only on |x| = r, and the integral GsΦ takes the form GΦ s =−

1 1. 3ε0

(3.227)

Thus, the value of this integral does not depend on the detailed behavior of the function Φ(r), and the eﬀective dielectric permittivity of the composite takes the form of the Maxwell-Garnet formula (3.170) 3pε1 . (3.228) ε s = ε0 1 + 3ε0 + (1 − p)ε1 For regular lattices of spherical inclusions, the integral GΦ has the form (3.217), and for the static tensor εs , we obtain the equation εs = ε0 + pε1 · [1 + pε1 · Γ ]

−1

,

(3.229)

Γ =−

1 ˜ ˜s ˜ s0 (µ) = − µ ⊗ µ . f (µ)G (µ), G v0 µ ε0 µ2

(3.230)

3.14 The EFM for composites with regular lattices of spherical inclusions

65

The tensor Γ has the symmetry of the lattice, and for an orthorhombic lattice this tensor is Γ = α1 e1 ⊗ e1 + α2 e2 ⊗ e2 + α3 e3 ⊗ e3 , 3λ1 λ2 λ3 j12 ( (λ1 i1 )2 + (λ2 i2 )2 + (λ3 i3 )2 ) 2 αj = 2 (λj ij ) , (2π)2 ε0 i ,i ,i [(λ1 i1 )2 + (λ2 i2 )2 + (λ3 i3 )2 ] 1

λj =

2

(3.231) (3.232)

3

2π . Lj

(3.233)

Here (i1 L1 e1 + i2 L2 e2 + i3 L3 e3 ) are the vectors of the elementary cell of the lattice (|ej | = 1), i1 , i2 , i3 = 0, ±1, ±2, ...; the prime over the summation sign means omitting the term with i1 = i2 = i3 = 0. For a cubic lattice (L1 = L2 = L3 = L0 ), this tensor is isotropic and takes the form Γ =

1 α0 (λ0 )1, ε0

(3.234)

3λ0 j1 (λ0 i2 + j 2 + m2 ) α0 (λ0 ) = , (2π)2 i,j,m i2 + j 2 + m2

λ0 =

2π , L0

(3.235)

where L0 is the distance between the centers of neighboring inclusions. For inclusions of unit radius, the parameter λ0 is connected with the volume concentration of inclusions p by the relation 1 λ0 = 6π 2 p 3 .

(3.236)

The corresponding dependence of the eﬀective dielectric permittivity of the composite for a cubic lattice of inclusions is presented in Fig. 3.17

3

s

ε *

2.5

2

1.5 p 1

0

0.1

0.2

0.3

0.4

0.5

Fig. 3.17. The dependence of the eﬀective static dielectric permittivity εs∗ of the composite for a cubic lattice of spherical inclusions, on their volume concentration p (ε = 10, ε0 = 1). The dashed line is the same dependence for an isotropic distribution of inclusions.

66

3. Electromagnetic waves in composites and polycrystals

(ε = 10, ε0 = 1) by the solid line. Note that the dependence εs∗ (p) for the cubic lattice almost coincides with that for the isotropic distribution of inclusions in space given by (3.228) (the dashed line in Fig. 3.17). As is seen from (3.168), the long wave limit of the attenuation coeﬃcient of the mean wave ﬁeld in the composites has the form 4 (k0 a) (εs∗ − ε0 )2 (3.237) 1 − n0 Φ(x)dx . γa = √ s 9pε0 ε∗ ε0 This equation describes the attenuation caused by the Rayleigh wave scattering by the inclusions. Here n0 is the numerical concentration of inclusions, integration is spread over all 3D-space, and the volume of the elementary cell of the lattice is equal to n−1 0 . For regular structures, the function Φ(x) has the form (3.216), and integration of the function f (x) in (3.215) gives 4π f (x + q)dx = . (3.238) 3 Thus, the integrals vq Φ(x)dx over the elementary cell vq take the form vq

Φ(x)dx = n−1 0 −

4π . 3p

(3.239)

For the medium with spheres of unit radius we have p = 4π 3 n0 , and all these integrals disappear except the one calculated over the cell with q = 0 because the corresponding term is not present in the sum (3.216). The integral over this cell is equal to n−1 0 , and for the integral in (3.237) we have ﬁnally −1 (3.240) Φ(x)dx = n0 , 1 − n0 Φ(x)dx = 0. Thus, for regular composites, the long wave limit (3.237) of the attenuation coeﬃcient is equal to zero, and γ has an order at least higher than k04 . This means that the Rayleigh scattering of waves is absent in this case. This fact is well-known for periodic structures. Let us consider wave propagation through a medium with a cubic lattice of spherical inclusions of unit radius. In this case, the tensor GΦ (k) in (3.217) takes the form Φ k = km, m = mi ei . GΦ (k) = GΦ 0 (k) + G1 (k), 1 GΦ G0t (k0 , k)(1 − m ⊗ m) + G0l (k0 , k)m ⊗ m , 0 (k) = p 1 2 0 k0 qt (k0 , k) + Jt (k0 , k) , Gt (k0 , k) = ε0 1 G0l (k0 , k) = G0t (k0 , k) + Gl (k0 , k), ε0

(3.241) (3.242) (3.243) (3.244)

3.14 The EFM for composites with regular lattices of spherical inclusions

67

where the integrals qt (k0 , k), Jt (k0 , k), Gl (k0 , k) are deﬁned in (3.158)– (3.162), 1 1 GΦ (3.245) 1 (k) = Gt (k0 , k)(1 − m ⊗ m) + Gl (k0 , k)m ⊗ m, 9 j12 (µ(r, s, t)) 2k02 − µ2 (r, s, t) + µm (r, s, t) , G1t (k0 , k) = − 2ε0 r,s,t µ2 (r, s, t) [k 2 + µ2 (r, s, t) − k02 − 2kµm (r, s, t)] 9 j12 (µ(r, s, t)) k02 − k 2 + 2kµm (r, s, t) − µ2m (r, s, t) 1 , Gl (k0 , k) = − ε0 r,s,t µ2 (r, s, t) [k 2 + µ2 (r, s, t) − k02 − 2kµm (r, s, t)] µ(r, s, t) = λ0 r2 + s2 + t2 , µm (r, s, t) = λ0 (rm1 + sm2 + tm3 ).

The functions Π(k) in (3.138) for a cubic lattice take the form Π(k) = Πt (k)(1 − m ⊗ m) + Πl (k)m ⊗ m, −1

Πt (k) = [1 + pε1 Λ0t (k0 , k∗ )Gt (k0 , k)]

−1

Πl (k) = [1 + pε1 Λ0l (k0 , k∗ )Gl (k0 , k)]

(3.246)

,

(3.247)

.

(3.248)

The dispersion equation (3.222) is divided into two equations: for the transverse part of the mean wave ﬁeld k∗2 − k02 − pk02 ε1 [Λ0t (k0 , k∗ )−1 + pε1 G0t (k0 , k∗ ) + pε1 G1t (k0 , k∗ )]−1 = 0 (3.249) and for its longitudinal part 1 + pε1 [Λ0l (k0 , k∗ )−1 + pε1 G0l (k0 , k∗ ) + pε1 G1l (k0 , k∗ )]−1 = 0.

(3.250)

Here Λ0t (k0 , k∗ ) and Λ0l (k0, k∗ ) are the transverse and longitudinal parts of the tensor Λ0 (k∗ ) in (3.144). Note that the solutions of the dispersion equations (3.249), (3.250) depend on the direction m of the mean wave ﬁeld propagation. For a cubic structure, the directions of the vectors k0 and k∗ coincide, but the form of the dispersion equations depends on this direction via the coeﬃcients mi that are the projections of the vector m on the unit vectors ei of the elementary cell. Let us consider the transverse part of the mean wave ﬁeld that propagates along a side of the cube (m = e1 , and m1 = 1, m2 = m3 = 0). For the coeﬃcient Λ0t (k0, k∗ ) one can use the exact solution (3.155) or the approximate solution (3.210). If the approximate solution is used (the wave ﬁeld inside every inclusion is a plane wave with the wave vector of the eﬀective ﬁeld), the dispersion equation is simpliﬁed dramatically and takes the forms k∗2 = k02 + pk02 ε1 Γt (k0, k∗ ), Γt (k0, k∗ ) = [1 + pε1 G1t (k0 , k∗ )]−1 .

(3.251) (3.252)

68

3. Electromagnetic waves in composites and polycrystals

Here we take into account that for the plane wave approximation 0 −1 Λ (k∗ ) = 1 − IG (k0 , k∗ ) · ε1 = 1−ε1 pGΦ (3.253) 0 (k∗ ). The numerical analysis of (3.251) shows the existence of an inﬁnite set of diﬀerent branches of its solutions. Two branches inside the ﬁrst Brillouin zone for the composite with the dielectric permittivities ε0 = 1 of the matrix, and ε = 5 of the inclusions, are presented in Fig. 3.19 for the volume concentrations of the inclusions p = 0.3. The elementary cell and the corresponding Brillouin zone are shown in Fig. 3.18. The end of the wave vector k∗ of the mean wave ﬁeld moves ﬁrst from the origin O to the point M, then to the point R, and after that, to the point Γ. The corresponded dispersion curves are presented on the right hand side of Fig. 3.19. If the end of the vector k∗ moves from the origin O directly to the point Γ , the corresponding dispersion curves are on the left-hand side of this ﬁgure.

Γ

o M

R

λ 0=2π / L0

L

Fig. 3.18. Elementary cubic cell and the Brillouin zone of the cubic lattice. 2.5

k0a

2

1.5

1

0.5

Γ

0

O

M

R

Γ

k*a

Fig. 3.19. The acoustical and optical branches of the wave propagation inside the Brillouin zone. On the right-hand side, the wave vector k∗ moves from the point O to point M , then to point R, and after that to point Γ . These points of the Brillouin zone are indicated in Fig. 3.8. On the left-hand side, the wave vector moves directly from point O to point Γ.

3.14 The EFM for composites with regular lattices of spherical inclusions 7

Re(k*a) 10x γ a

6

69

p=0.3

5 4 3 2 1 0

k0a 0

1

2

3

4

5

Fig. 3.20. The acoustical branch of wave propagation in the medium with a cubic lattice of inclusions. The solid line is the real Rek∗ a part of the eﬀective wave number, and the dashed part is the attenuation coeﬃcient along this branch. There are three stop bands in the frequency region where the waves exponentially attenuate. The high-frequency waves go through the composite without attenuation.

Let us consider the main acoustical branch of (3.251). This branch is presented in Fig. 3.20. The eﬀective wave numbers k∗ that correspond to this branch are real, except for several regions where the attenuation coeﬃcient γ turns out to be not equal to zero. The dependence of attenuation γa on k0 a for this branch is shown by the dashed line in Fig. 3.20. The regions of frequencies where the attenuation coeﬃcient is not equal to zero may be considered as the stop bands for this wave. These bands narrow, and the attenuation inside these bands decreases, when k0 a increases. When k0 a is more than 4, the stop bands disappear, and the corresponding waves propagate through the medium without attenuation. Note that the ﬁrst stop band is situated in the vicinity of Bragg’s frequency (wave numbers) that is deﬁned by the equation k0 =

1 λ0 , 2

(3.254)

and λ0 = 2.609 for p = 0.3. In order to understand the input of the acoustical and optical branches into the mean wave ﬁeld, let us consider the Fourier transform (3.150) of the mean Green function or the mean wave ﬁeld from a concentrated harmonic force in the medium. For the composites with a regular lattice of inclusions, the detailed form of (3.150) is G(k) = gt (k)(1 − m ⊗ m) − gl (k)m ⊗ m, k02 , gt (k) = 2 2 ε0 [k − k0 − p¯ ε1 k02 Γt (k, k0 )] 1 gl (k) = , ε0 [1 + p¯ ε1 Γl (k, k0 )]

(3.255) (3.256) (3.257)

70

3. Electromagnetic waves in composites and polycrystals

where gt (k), gl (k) are the transverse and longitudinal parts of the Fourier transform of the mean Green function. After application of the inverse Fourier transform and integration over the unit sphere in the k-space we obtain for G(x) G(x) = Gt (x) + Gl (x), ∞ 1 Gt (x) = gt (k)eikr kdk1 4π 2 ri −∞ ∞ 1 ikr dk , gt (k)e +∇⊗∇ 4π 2 ri −∞ k ∞ 1 dk . Gl (x) = ∇ ⊗ ∇ gl (k)eikr 2 4π ri −∞ k

(3.258)

(3.259) (3.260)

The theory of residues may be applied to calculate these integrals. For small k0 , the poles of the function gt (k) are situated at the point k = k∗ = √ k0 εs∗ and close to the points k = ks± = λ0 s ± k0 , s = 1, 2, 3, .... Other poles of the functions gt (k) and gl (k) are situated close to the points (3.261) k = ksl = λ0 (s − i j 2 + m2 ), √ (3.262) s, j = 1, 2, 3, ...; m = 0, 1, 2, 3, ...; i = −1. As a result, the equation for G(x) takes the form 1 k∗2 −ik∗ r e +∇⊗∇ e−ik∗ r G(x) = 4πrεs 4πrεs ∞ 2 + − k0 + (R+ e−iks r + Rs− e−iks r ) 4πrε0 s=1 s , ∞ k02 1 + −iks+ r − −iks− r + ∇⊗∇ (R e + Rs e ) + .... 4πε0 r s=1 s

(3.263)

Here Rs± are the residues of the functions gt (k) at points ks± . For large s, the numbers Rs± may be estimated as follows . ± . 9 p2 ε21 k02 .Rs . | cos(λ0 s)|. 2 ε20 (λ0 s)5

(3.264)

Thus, the picture of the mean wave ﬁeld that propagates from a point source in the medium with a regular lattice of inclusions has the following structure. The ﬁrst two terms in (3.263) describe propagation of waves in the homogeneous medium with the eﬀective static properties εs of the composite (compare with the equation (3.64) for G∗ (x)). This is the main wave that corresponds to the acoustical branch shown in Fig. 3.20. The other waves generated in the medium have wave numbers ks± , their amplitudes are proportional to Rs± and are much less than the amplitude of the main branch. For

3.15 Versions I and IV of EMM for polycrystals and granular materials

71

large s, the amplitudes of these waves rapidly tend to zero. The terms that are not written in (3.263) attenuate exponentially, and the corresponding waves disappear in a length of the order of the distance L0 between inclusions. Note that this picture of the mean wave ﬁeld in the composite with a cubic lattice of inclusions was obtained by using the plane wave approximation for the solution of the one-particle problem. If the exact solution (3.155) of this problem is used, the results change. In this case, the dispersion equation has no real solutions, and all the propagating waves attenuate. There is attenuation for all frequencies, but in the vicinities of Bragg’s frequencies the attenuation coeﬃcients are much higher than outside these regions. It is possible to say that the plane wave approximation is compatible with the eﬀective ﬁeld method for regular structures. Only in the framework of such an approximation does the EFM predict the existence of pass and stop bands in the frequency region.

3.15 Versions I and IV of EMM for polycrystals and granular materials Let us consider propagation of electromagnetic waves through polycrystals and granular materials. The microstructure of such materials may be treated as a set of grains ideally conjugated along the borders. For polycrystals, all the grains are the same crystals with diﬀerent orientations of crystallographic axes inside diﬀerent grains. For granular materials, the properties of grains may vary arbitrarily from grain to grain but they are constant inside each grain. The ﬁrst hypothesis of the EMM for such materials is formulated like hypothesis I1 of version I. I1 . Each grain in a polycrystalline or granular material behaves as isolated one in the homogeneous medium with the eﬀective properties of the composite. The exciting ﬁeld that acts on each grain coincides with the mean wave ﬁeld in the inhomogeneous medium (Fig. 3.21). This assumption reduces the problem of interaction between many particles to the one-particle problem. But for anisotropic grains, the solution of the one-particle problem cannot be found in the form of a series similar to (3.51). At this step, one has to use some approximate solutions. Let us assume that the wave ﬁeld inside each grain is a plane wave with the eﬀective wave vector of the composite (l = k∗ ) ∗

E(x) = Ee−k

·x

,

k∗ = k∗ m.

(3.265)

The amplitude of this wave E is a stationary point of a variational functional that is similar to functional (3.189) of the one-particle problem. If the distribution of grains over orientations is homogeneous, and the form of grains is quasispherical, the eﬀective medium is isotropic. In this case, the amplitude E(x) of the wave ﬁeld inside a grain has the form (3.209)

72

3. Electromagnetic waves in composites and polycrystals

E0

<E>

ε∗

Fig. 3.21. The one-particle problem of version I of the EMM for polycrystalline materials.

E(x) = H−1 (x) · U∗ e−ik∗ ·x , H(x) = H0 (x) − K (2) (k∗ )m ⊗ (m · ε∗1 (x)), H0 (x) = 1 − k∗2 q(k∗ ) + K (1) (k∗ ) ε∗1 (x), ε∗1 (x) = ε(x) − ε∗ .

(3.266) (3.267) (3.268)

Here q(k∗ ) and K (j) (k∗ ) are deﬁned in (3.213). In order to obtain an approximation to the detailed ﬁeld E(x) in the composite, one has to choose the so-called reference medium. For matrix composites, it is natural to take the matrix phase as the reference medium, and the corresponding equation for the electric ﬁeld takes the form (3.25). For polycrystals, there is no matrix phase. The most reasonable choice is to take the eﬀective medium as a reference medium. As a result, the wave ﬁeld in the polycrystal is presented in the form (3.269) E(x) = E∗ (x) + G∗ (x − x ) · ε∗1 (x ) · H−1 (x ) · E∗ (x )dx , ε∗1 (x) = ε(x) − ε∗ . Here the integrand function ε∗1 (x) · H−1 (x) is constant in each grain but changes from grain to grain, G∗ (x) is the Green function of the eﬀective medium. The second hypothesis of the method is the condition of self-consistency. I2 . The ﬁeld E∗ (x) in the eﬀective medium coincides with the mean wave ﬁeld propagating in the composite. E(x) = E∗ (x).

(3.270)

Averaging (3.269) over the ensemble realizations of the polycrystal microstructures, we obtain E(x) = E∗ (x) + G∗ (x − x ) · ε∗1 (x ) · H−1 (x ) · E∗ (x )dx , (3.271)

3.15 Versions I and IV of EMM for polycrystals and granular materials

and comparing this equation with (3.270) we ﬁnd the equation G∗ (x − x ) · ε∗1 (x ) · H−1 (x ) · E∗ (x )dx = 0.

73

(3.272)

Therefore, the mean under the integral in this equation should be equal to zero: ε∗1 (x ) · H−1 (x ) = 0. (3.273) Substituting ε∗1 (x) = ε(x) − ε∗ in this equation we ﬁnd the equation for the eﬀective dielectric permittivity of the polycrystal in the form −1 ε∗ = H−1 (x) · ε(x) · H−1 (x) . (3.274) Note that the eﬀective dielectric permittivity ε∗ appears on the right hand side of this equation, and the latter should be solved with respect to ε∗ . Let us consider a typical grain embedded in a homogeneous medium with the eﬀective properties of the polycrystal, when the ﬁeld inside the grain take in the form (3.266). The electric ﬁeld in the medium is presented in the following integral form: (3.275) E(x) = E∗ (x) + G∗ (x − x ) · ε∗1 (x ) · H−1 (x ) · E∗ (x )dx , v

where v is the volume of the grain. The integral term in this equation is the ﬁeld Es (x) scattered by the grain. The mean value of this ﬁeld has the form s E (x) = G∗ (x − x ) · ε∗1 (x ) · H−1 (x ) · E∗ (x )dx . (3.276) v

Here the averaging is carried out over the orientations of the crystallographic axis inside the grains. If the condition of self-consistency is taken in the form IV2 (see Section 2.3: the mean scattered ﬁeld on the grain embedded into the eﬀective medium is equal to zero), the mean under the integral in this equation should disappear, and we return to equation (3.273) for the eﬀective parameters of the composite. Thus, for polycrystals and granular materials, the conditions of self-consistency I2 and IV2 give the same ﬁnal equations (3.273). Let us consider (3.274) for polycrystalline materials with a homogeneous distribution of crystallographic axes of grains over orientations. All the grains have approximately the same size (a = 1). In this case (3.274) takes the form ε∗ =

α2 β2 (1 − m ⊗ m) + m ⊗ m, γ0 + α1 γ0 + β1

(3.277)

where m is the wave normal of the incident ﬁeld, the coeﬃcients γ0 , α1 , α2 , β1 , β2 are functions of the eﬀective wave number k∗ and have the forms

74

3. Electromagnetic waves in composites and polycrystals

h0i γ1 γ2 γ3 γ4 α1 β1 α0

1 1 1 , + + h01 h02 h03

(ε − ε ) i ∗ = 1 − k∗2 q(k∗ ) + K (1) (k∗ ) , (i = 1, 2, 3); ε∗ 1 ε 1 − ε∗ ε 2 − ε∗ ε 3 − ε∗ , = + + 3 h01 ε∗ h02 ε∗ h03 ε∗ 1 ε 1 − ε∗ ε 2 − ε∗ ε 3 − ε∗ , = + 2 + 2 3 h201 ε∗ h02 ε∗ h03 ε∗ 1 ε1 ε2 ε3 , = + + 3 h01 h02 h03 1 ε1 (ε1 − ε∗ ) ε2 (ε2 − ε∗ ) ε3 (ε3 − ε∗ ) , = + + 3 h201 ε∗ h202 ε∗ h203 ε∗ 1 1 = (3α0 γ2 − β1 ) , α2 = (3γ3 + 3α0 γ4 − β2 ) , 2 2 3 3 = α0 (3γ0 γ1 + 2γ2 ) , β2 = γ3 + α0 (3γ3 γ1 + 2γ4 ) , 5 5 π 2π K (2) (k∗ ) 1 + D2 + (D3 − D2 ) cos2 (θ) = sin(θ)dθ 12π 0 0 −1 dϕ, + (D1 − D2 ) sin2 (θ) cos2 (ϕ)

γ0 =

1 3

Di = −K (2) (k∗ )

(εi − ε∗ ) , h0i ε∗

(i = 1, 2, 3).

(3.278) (3.279) (3.280) (3.281) (3.282) (3.283) (3.284) (3.285)

(3.286) (3.287)

Note that the integral α0 is a combination of elliptical functions. The equation for the eﬀective wave number k∗t of the transverse electromagnetic wave propagating through the polycrystalline medium takes the form α2 (k∗t ) ε∗t (k∗t ) 2 , (3.288) = k02 , ε∗t (k∗t ) = k∗t ε∗v γ0 (k∗t ) + α1 (k∗t ) 1 k02 = ω 2 ε∗v , ε∗v = (ε1 + ε2 + ε3 ) . (3.289) 3 Here ε1 , ε2 , ε3 are the eigenvalues of the dielectric property tensor of an anisotropic monocrystal, ε∗v is the averaging of this tensor over orientations (Voigt’s tensor). Figure 3.22 shows the dependence of the normalized velocities and attenuation coeﬃcients of transverse electromagnetic waves in an isotropic polycrystal with the eigenvalues of the monocrystal ε1 = 1, ε2 = 2, ε3 = 3. In these graphs, γt = − Im(k∗t )a,

λt =

v∗t k0 . = v0 Re(k∗t )

(3.290)

3.16 Conclusion 1.1

lg(γ t) 0

75

λt

1 −1 0.9

−2

0.8

−3 −4 −0.5

0

0.5

1

1.5

lg(kc)

0.7 −0.5

0

0.5

1

1.5

lg(k0)

Fig. 3.22. The dependence of the attenuation coeﬃcient γ and the normalized phase velocity λt = vt /v0 of the transverse part of electromagnetic wave, on the non-dimensional wave number k0 , for an isotropic polycrystal (eigenvalues of the permittivity tensor of the grain are: ε1 = 1, ε2 = 2, ε3 = 3).

where v0 is Voigt’s velocity of the transverse electromagnetic waves in polycrystal. Thus, equation (3.288) allows us to describe all the important features of the phenomenon of electromagnetic wave propagation through polycrystals. This equation correctly describes the attenuation in the Rayleigh (long-wave) region where γ ∼ k04 , gives of quantitatively correct result in the stochastic (medium-wave) region γ ∼ k02 , and in the diﬀusive (short-wave) region γ ∼ const(k0 ). The dashed lines in Fig. 3.22 correspond to these three asymptotes.

3.16 Conclusion For the comparison of predictions of the self-consistent methods applied to the problem of electromagnetic wave propagation, let us consider the following parameters of the problem: volume concentration p of the inclusions, dielectric permittivities of the medium and the inclusions, and frequency ω of the incident ﬁeld or a nondimensional wave number k0 a. Version I of the EMM allows us to describe qualitatively correctly the velocities and attenuation coeﬃcients of the mean wave ﬁeld in the composites with weakly contrasting properties of the components in the long and short-wave regions. This method gives physically correct dependence of the eﬀective velocity v∗ and the attenuation coeﬃcient γ on the frequency ω of the incident ﬁeld if the appropriate one-particle problem is solved with suﬃcient accuracy. For strongly contrasting components, this version of the EMM leads to essential errors for static properties if the volume concentration of inclusions exceeds 0.25–0.3.

76

3. Electromagnetic waves in composites and polycrystals

Version II and III of the EMM improve the predictions for static dielectric properties but do not correctly describe the dependence of the attenuation coeﬃcients on frequency in the long-wave region. Version I and IV of the EMM applied to granular and polycrystalline materials give close results. The predictions of these versions of the EMM are physically correct for all frequencies of the propagating waves. The version of the EFM developed in this chapter allows us to describe many important features of the wave propagation through media with isolated inclusions. It gives a good correspondence with experimental data in the region of long waves, and is physically correct in the short-wave region. The diﬀerences between predictions of various versions of the EMM and the EFM are most essential in the region of the middle-wave lengths. In this region, the lengths of the waves are comparable with the sizes of the inclusions as well as with the distances between them. As a result, the wave ﬁeld in the medium has a complex structure, and the principal hypotheses of both methods may become invalid. The absence of experimental data for velocities and attenuation coeﬃcients of electromagnetic waves in this region does not allow to decide between all the various methods. For optically hard inclusions, the EMM and EFM predict the existence of diﬀerent branches of wave propagation. The number of these branches depends on the contrast of the phases and the volume concentration of the inclusions. Usually, each branch describes the mean wave ﬁeld in a certain frequency region. The common envelop of the branches may be considered as the dispersion curve that describes the mean wave ﬁeld in the whole frequency region. Note that the regions of applications of the EMM and EFM depend also on the shape of inclusions. Application of the methods to composites with inclusions of non-canonical shapes faces the diﬃculties associated with the solution of the corresponding one-particle problems. The approximate solution of the one-particle problem proposed in Section 3.13 is called the plane wave approximation. Of course, it is naive to expect that with the help of one plane wave one could describe a complex wave ﬁeld inside an inclusion in the region of middle and short waves. But use of this approximation in the framework of the EFM and the EMM allows to describe satisfactorily the behavior of the wave velocities and attenuation coeﬃcients in the long-wave region.

3.17 Notes The material of Chapter 3 is based on the works [45, 46, 48, 49]. Electromagnetic wave propagation through the dielectric media with spherical inclusion were considered in [38, 65, 76, 101–104].

4. Axial elastic shear waves in ﬁber-reinforced composites

The self-consistent methods developed in Chapters 2 and 3 may be applied to the analysis of elastic wave propagation in composites without essential modiﬁcations. Nevertheless, elastic waves introduce speciﬁc diﬃculties. First, two types of elastic waves (longitudinal and transverse waves of various polarizations) may propagate in the composites, and the dispersion equations for each wave should be derived by the methods. Secondly, elastic waves oblige us to consider a system of two integral equations for the displacement and strain ﬁelds, and this makes the analysis more cumbersome than that for scalar or electromagnetic waves. In this Chapter we consider a relatively simple case: propagation of axial elastic shear waves through composites reinforced with long unidirectional ﬁbers. The wave vector of these waves is orthogonal to the ﬁber axes, and the polarization vector coincides with the ﬁber directions. In this case, there is only one nonzero component of the displacement ﬁeld in the composite, and only one type of wave propagates in the composite. This makes the algorithm of the self-consistent methods more transparent than this for other composites in which a wave of one type generates waves of other types. The structure of this chapter is as follows. In Section 4.1, the integral equations of the axial shear wave propagation problem are considered. In Section 4.2, the general scheme of the EMM is developed for construction of the dispersion equation for the mean wave ﬁeld in the composite. In Section 4.3, the EFM is applied to the solution of the same problem. Section 4.4 is devoted to the solutions of the one-particle problems of both methods. In Sections 4.5 and 4.6, the solutions of the dispersion equations in the long and short-wave regions are constructed. In Section 4.7, the results of numerical solutions of the dispersion equations of both methods are compared in a wide region of frequencies of the incident ﬁeld. Section 4.8 is devoted to wave propagation in composites with periodic arrangements of cylindrical ﬁbers. We show that the EFM predicts the existence of pass and stop bands in the frequency region for the propagating waves.

78

4. Axial elastic shear waves in ﬁber-reinforced composites

4.1 Integral equations of the problem Let an inﬁnite homogeneous medium (matrix) with elastic moduli tensor C 0 and density ρ0 contain a random set of unidirectional cylindrical ﬁbers with moduli tensor C and density ρ. All the ﬁbers have the same radius a and occupy a region V with characteristic function V (x). If a monochromatic elastic wave of frequency ω propagates in such a medium, the displacement ﬁeld ui (x, t) takes the form ui (x, t) = ui (x)eiωt , and the amplitude ui (x) of this ﬁeld satisﬁes the equation ∂ , ∂xi

∇i Cijkl (x)∇l uk (x) + ρ(x)ω 2 uj (x) = 0,

∇i =

C(x) = C 0 + C 1 (x),

C 1 = C − C 0,

ρ(x) = ρ0 + ρ1 (x),

C 1 (x) = C 1 V (x), ρ1 (x) = ρ1 V (x),

ρ1 = ρ − ρ0 .

(4.1)

(4.2)

If the ﬁbers are directed along the x3 -axis, V (x1 , x2 , x3 ) = S(x1 , x2 ) is a function of only two coordinates x1 , x2 . In this Chapter, propagation of the axial shear waves in the composite with transversely isotropic components is considered. In this case, the wave normal n of the waves is orthogonal to the x3 -axis, and the polarization vector e is directed along this axis (see Fig. 4.1). For these waves, only the component u3 of the displacement vector is not zero u1 = u2 = 0,

u3 (x) = u(y),

y = y(x1 , x2 ),

(4.3)

Fig. 4.1. The axial shear wave with the wave normal n and the polarization vector e in the ﬁber reinforced composite.

4.1 Integral equations of the problem

79

and (4.1) may be rewritten in the form (i, j = 1, 2). 0 1 ∇j u(y) + ρ0 ω 2 u(y) = − ∇i Ci33j (y)∇j u(y) + ρ1 (y)ω 2 u(y) . ∇i Ci33j

(4.4)

For a transversely isotropic matrix and inclusions with the axis of isotropy 0 1 and Ci33j (y) of the elastic moduli tensor are x3 , the components Ci33j 0 = µ0 δij , Ci33j

1 Ci33j (y) = µ1 S(y)δij ,

µ1 = µ − µ0 ,

(4.5)

where µ0 and µ are the shear moduli of the medium and the inclusions in the direction of the x3 -axis. In this case (4.4) takes the form µ0 u(y) + ρ0 ω 2 u(y) = −µ1 ∇i [S(y)∇i u(y)] − ρ1 S(y)ω 2 u(y),

(4.6)

where is the Laplace operator. After applying the operator g = −1 µ0 +ρ0 ω 2 to this equation we obtain the integral equation for the scalar function u(y) in the form ρ1 ω 2 g(y − y ) + ∇i g(y − y )µ1 εi (y ) S(y )dy , (4.7) u(y) = u0 (y) + εi (y) = ∇i u(y). Here u0 (y) is the amplitude of the incident ﬁeld that would have existed in the medium without inclusions (µ1 = 0, ρ1 = 0) by the given sources of the waves, and g(y) is the Green function that satisﬁes the equation (4.8) µ0 +ρ0 ω 2 g(y) = −δ(y), where δ(y) is the 2D-Dirac delta-function. The explicit form of g(y) is i ρ0 H0 (k0 r), r = |y|, k0 = ω , (4.9) g(y) = − 4µ0 µ0 where H0 (z) is the second kind Hankel function of order 0. The Fourier transform of the Green function g(y) takes the form 1 , k 2 = k12 + k22 , (4.10) g(k) = g(y)eik·y dy = 2 µ0 k − ρ0 ω 2 where k(k1 , k2 ) is the vector parameter of the Fourier transform. The incident ﬁeld u0 (y) in (4.7) is an axial plane shear wave with the wave vector k0 = k0 n0 orthogonal to the x3 -axis and the polarization vector directed along the x3 -axis. For such a wave, we have 0

u0 (y) = U0 e−ik

·y

,

0

ε0i (y) = ∇i u0 (y) = −ik0 n0i U0 e−ik

·y

.

(4.11)

80

4. Axial elastic shear waves in ﬁber-reinforced composites

The equation for the strain ﬁeld εi (y) = ∇i u(y) in the composite follows from (4.7) in the form 0 ρ1 ω 2 ∇i g(y − y )u(y ) + Kij (y − y )µ1 εj (y ) S(y )dy . εi (y) = εi (y) + (4.12) Kij (y) = ∇i ∇j g(y). It is convenient to rewrite the two equations (4.7) and (4.12) as one symbolic equation (4.13) F(y) = F0 (y) + K(y − y )L1 F(y )S(y )dy , where the symbolic matrices K(y) and L1 , and the vectors F(y), F0 (y) are introduced 2 ω g(y), ∇g(y) ρ1 , 0 1 , (4.14) K(y) = , L = ω 2 ∇g(y), K(y) 0, µ1 0 u (y) u(y) . (4.15) F(y) = , F0 (y) = ε0 (y) ε(y)

4.2 The eﬀective medium method If the incident ﬁeld is a plane wave, the ﬁeld propagating in the composite with a random set of cylindrical ﬁbers is a non-plane random wave ﬁeld. Our objective is to evaluate the mean value of this ﬁeld. Strictly speaking, we have to solve the multi-particle problem (the problem of diﬀraction of the incident ﬁeld by many ﬁbers) for every realization of a random set of ﬁbers, and then average the solutions over the ensemble realizations of this set. The diﬃculties of this problem oblige us to ﬁnd approximate solution. Let us start with the eﬀective medium method. The hypotheses I1 and I2 of version I of the EMM formulated in Section 2.3 are applied to the elastic wave without any change. I1 . Each inclusion in the composite behaves as an isolated one embedded in the homogeneous medium with the overall (eﬀective) properties of the composite. The ﬁeld F∗ (y) that acts on this inclusion is a plane wave propagating in the eﬀective medium. I2 . The mean wave ﬁeld propagating in the composite coincides with the ﬁeld propagating in the homogeneous eﬀective medium. F∗ (y) = F(y)

(4.16)

The hypothesis I1 reduces the problem of interactions between many inclusions in the composite to a one-particle problem; hypothesis I2 is the condition of self-consistency.

4.2 The eﬀective medium method

81

The one-particle problem of the EMM is diﬀraction of a plane monochromatic wave by an isolated ﬁber embedded in an eﬀective homogeneous medium. In the framework of the EMM, the solution of this problem gives the wave ﬁeld inside every inclusion in the composite. The integral equation of this problem is similar to (4.7), (4.12), and in the symbolic notations (4.14), (4.15) has the forms (4.17) F(y) = F∗ (y) + K∗ (y − y )L∗1 F(y )dy , L∗1 =

s

ρ∗1 , 0 0, µ∗1

.

(4.18)

Here s is the area of the ﬁber cross-section, the matrix K∗ (y) has the form (4.14), where g(y) should be replaced by the Green function g∗ (y) of the homogeneous medium with the eﬀective dynamic properties µ∗ and ρ∗ of the composite, µ∗1 = µ − µ∗ , ρ∗1 = ρ − ρ∗ , and u∗ (y) and ε∗i (y) are plane waves with the eﬀective wave vector k∗ u∗ (y) U∗ ∗ −ik∗ ·y ∗ , (4.19) = f F∗ (y) = e , f = −ikj∗ U∗ ε∗j (y) ρ∗ k∗ = k∗ n, k∗ = ω . µ∗ Here U∗ is the amplitude of the displacement ﬁeld u∗ (y). If the distribution of ﬁbers in the matrix is homogeneous and isotropic, the eﬀective medium is transversely isotropic and the wave vectors k∗and k0 of the mean and the incident ﬁelds have the same direction n = n0 . Note that µ∗ , ρ∗ and k∗ are unknown parameters in (4.19). Let the general solution of (4.17) be known, and the ﬁeld u(y) inside the inclusion with the center at the point y j have the form ∗

u(y) = Λj [u∗ (y)] = Λj [U∗ e−ik j

−ik ·(y−y )

j

= Λ [e u

∗

−ik∗ ·y

λ (y) = Λ[e

ik∗ ·y

]e

·y

]

ik ·(y−yj )

]e

∗

∗

U∗ e−ik

·y

= λu (y − y j )u∗ (y),

,

(4.20) (4.21)

Here Λ is a linear operator that depends on the properties of the eﬀective medium and the inclusion. Note that the function λu (y) does not depend on the coordinates of the center of the inclusion. These functions may be constructed from the solution of the one-particle problem for an inclusion centered at the point y = 0 . Similarly, for the strains inside the inclusion centered at point y j we obtain εi (y) = ∇i Λ[u∗ (y)] = Λεij (y − y j )ε∗j (y),

ik∗ ∗ ∗ ∗ ∗ j Λεij (y) = ∇i Λ[U∗ e−ik ·y ]eik ·y + Λ[e−ik ·y ]eik ·y δij . 2 k∗

(4.22) (4.23)

82

4. Axial elastic shear waves in ﬁber-reinforced composites

In symbolic form, the ﬁeld F(y) inside the inclusion centered at y j is F(y) = ΛF∗ (y) = Λ(y − y j )F∗ (y), u λ (y), 0 . Λ(y) = 0 Λεij (y)

(4.24) (4.25)

Let us introduce stationary random functions Λ(y) that coincide with i Λ(y − y ) if y is inside the inclusion centered at the point y i (i = 1, 2, 3, ...), and is equal to zero in the matrix. Using this function and hypothesis I1 of the EMM we present the wave ﬁeld F(y) in the composite in the form that follows from (4.13) and (4.24): )F∗ (y )dy . (4.26) F(y) = F0 (y) + K(y − y )L1 Λ(y In order to ﬁnd the mean wave ﬁeld F(y), let us average both parts of (4.26) over the ensemble realizations of the random set of inclusions and take into account the condition of self-consistency (hypothesis I2 ) F∗ (y) = F(y) .

(4.27)

As a result, we obtain the closed integral equation for the mean wave ﬁeld in the composite 0 (4.28) F(y) = F (y) + p K(y − y )L1 Λ∗ F(y ) dy , 1 1 lim Λ(y)dy, (4.29) Λ(y)S(y)dy = Λ∗ = pΩ Ω→∞ s s Ω 1 1 λ , 0 ∗ Λ∗ = , λ∗ = λu (y)dy, Λ∗ij = Λεij (y)dy (4.30) ∗ 0, Λij s s s

s

∗

Here Λ is a constant (with respect to spatial coordinates) matrix, Ω is a region that occupies the entire 2D-plane (x1 , x2 ) in the limit Ω → ∞, p = S(y) is the volume concentration of the inclusions, s is the area of the inclusion centered at the origin. Note that Λ∗ depends on the wave vector of the mean wave ﬁeld k∗ : Λ∗ = Λ∗ (k∗ ). Apply the Fourier transform to (4.26) F(k) = F0 (k) + pK(k)L1 Λ∗ (k∗ ) F(k) , and multiply the resulting equation with the matrix L0 (k) l (k), 0 L0 (k) = 0 , l0 (k) = µ0 k 2 − ρ0 ω 2 . 0, l0 (k)

(4.31)

(4.32)

4.2 The eﬀective medium method

83

Taking into account the equations that follow from (4.14), (4.11) L0 (k)F0 (k) = 0, M(k) =

L0 (k)K(k) = M(k), −iki ω2 , . −iki ω 2 , −ki kj

(4.33) (4.34)

We obtain the equation for the Fourier transform F(k) of the mean wave ﬁeld in the composite: L∗ (k) = L0 (k) + pM(k)L1 Λ∗ (k ∗ ).

L∗ (k) F(k) = 0,

(4.35)

This symbolic equation is equivalent to two algebraic equations for the Fourier transforms of the mean displacement u(k) and deformation εj (k) = −ikj∗ u(k) ﬁelds. The ﬁrst of these equations is l∗ (k) u(k) = 0,

l∗ (k) = l0 (k) − pρ1 ω 2 λ∗ (k ∗ ) + pµ1 iki kj∗ Λ∗ij (k ∗ ), (4.36)

and the second equation is (4.36) multiplied with −ikj . Because the tensor Λ∗ij in (4.30) is a function of the vector k∗ only, the product Λ∗ij (k∗ )kj∗ is presented in the form Λ∗ij (k ∗ )kj∗ = −iki∗ H(k∗ ),

(4.37)

where H(k∗ ) is a scalar function. ∗ If the mean wave ﬁeld u(y) is a plane wave (u(y) = U∗ e−ik ·y ), its 2 Fourier transform is u(k) = (2π) U∗ δ(k − k∗ ). Thus, equation (4.36) takes the form l∗ (k)δ(k − k∗ ) = 0

(4.38)

and the multiplier l∗ (k) in front of δ(k − k∗ ) in this equation should be equal to zero when k = k∗ l∗ (k∗ ) = l0 (k∗ ) − pρ1 ω 2 λ∗ (k∗ ) + pµ1 k∗2 H(k∗ ) = 0.

(4.39)

This equation may be rewritten in the canonical form µ∗ (k∗ )k∗2 − ρ∗ (k∗ )ω 2 = 0, µ∗ (k∗ ) = µ0 + pµ1 H(k∗ ),

(4.40) ρ∗ (k∗ ) = ρ0 + pρ1 λ∗ (k∗ ).

(4.41)

Equation(4.40) is in fact the dispersion equation for the eﬀective wave number k∗ of the mean wave ﬁeld in the composite. Note that the functions H and λ∗ are to be found from the solution of the one-particle problem (4.17). Thus (4.40), (4.41) compose a system for the eﬀective parameters µ∗ , ρ∗ and k∗ of the composite in the framework of the EMM. The phase velocity v∗ and the attenuation coeﬃcient γ of the mean wave ﬁeld are connected with the wave number k∗ by the equations ω , γ = − Im(k∗ ). v∗ = (4.42) Re(k∗ )

84

4. Axial elastic shear waves in ﬁber-reinforced composites

4.3 The eﬀective ﬁeld method Let us reconsider a realization of a random set of cylindrical ﬁbers in a homogeneous matrix. The characteristic function S(y) of the region occupied by the inclusions is the following sum: sk (y), (4.43) S(y) = k

where sk (x) is the characteristic function of the region occupied by the k-th inclusion. Equation (4.13) for the wave ﬁeld F(x) in the medium with inclusions may be rewritten in the form ∗ K(y − y )L1 F(y )dy , (y ∈ sk ), (4.44) F(y) = F (y) + sk

F∗ (y) = F0 (y) +

n=k

K(y − y )L1 F(y )dy .

(4.45)

sn

Equation (4.44) shows that every inclusion in the composite may be considered as an isolated one in the original matrix under the action of a local exciting ﬁeld F∗ (y). The ﬁeld F∗ (y) does not coincide with the incident ﬁeld F0 (y), and consists of F0 (y) and the ﬁelds scattered by surrounding inclusions. 4.3.1 Integral equations for the local exciting ﬁelds Let (4.44) be solved for arbitrary exciting ﬁelds F∗ (y), and similar to (4.24), the ﬁelds F(y) inside the inclusion centered at the point y k be presented in the form F(y) = Λk F∗ (y).

(4.46)

Here Λk are the linear operators of the solution of the diﬀraction problem for one inclusion (the one-particle problem). It follows from (4.13), (4.46) that the ﬁeld F(y) in the medium is expressed via the ﬁeld F∗ (y) in the form 0 F(y) = F (y) + P(y − y )L1 ΛF∗ (y )S(y )dy (4.47) Here function ΛF∗ (y) coincides with Λk F∗ (y) inside the kth inclusion (k = 1, 2, 3, ...). Note that the linear operator Λk may be presented in the form of an integral operator Λk (y, y )F∗ (y )dy , (4.48) (Λk F∗ )(y) = sk

4.3 The eﬀective ﬁeld method

85

where Λk (y, y ) is a generalized function known from the solution of the oneparticle problem. Points y, y belong to the same domain sk because the ﬁeld inside the k-th inclusion depends only on the values of the local exciting ﬁelds in region sk . The equation for the exciting ﬁeld F∗ (y) that acts on every inclusion in the composite follows from its deﬁnition (4.45) in the form ∗ 0 (4.49) F (y) = F (y) + K(y − y )L1 ΛF∗ (y )S(y, y )dy , where S(y, y ) is characteristic function (with argument y ) of region Sy deﬁned by the equation Sy = si if y ∈ sk . (4.50) i=k

The function S(y, y ) is equal to zero if points y and y are within the same inclusion. Thus, the integral term in (4.49) is the ﬁeld scattered on all the inclusions except the one that occupies region sk if y ∈ sk . The equation for the mean wave ﬁeld follows from (4.47) after averaging the latter over the ensemble realizations of the random set of inclusions (4.51) F(y) = F0 (y) + K(y − y )L1 ΛF∗ (y )S(y ) dy . 4.3.2 The hypotheses of the EFM The principal hypotheses of the EFM concern the local exciting ﬁeld F∗ (y) acting on each inclusion in the composite. These hypotheses are formulated in Sections 2.2 and 3.7 and may be applied to the case of axial elastic shear wave propagation without modiﬁcations. H1 . The ﬁeld acting on each inclusion in the composite is a plane axial ∗ (y) (the eﬀective ﬁeld) that is the same for all the inclusions shear wave F ∗ (y) is the local exciting ﬁeld F∗ (y) averaged over H2 . The eﬀective ﬁeld F the ensemble realizations of the random ﬁeld of inclusions under the condition y ∈ S. ∗ (y) = F∗ (y)|y F

(4.52)

The ﬁrst hypothesis reduces the problem of interaction between many inclusions in the composite to a one-particle problem. The latter is the solution of the following integral equation ∗ (y) + K(y − y )L1 F(y )dy , (y ∈ sk ), (4.53) F(y) = F sk

∗ (y) is the plane wave where F

86

4. Axial elastic shear waves in ﬁber-reinforced composites

∗ (y) = F

u ∗ (y) ε∗j (y)

∗

= f ∗ e−ik

·y

, f∗ =

U∗u −ikj∗ U∗ε

,

(4.54)

Here k∗ is the wave vector of the eﬀective ﬁeld. Because the eﬀective ﬁeld is the conditional mean (4.52), the amplitudes U∗u and U∗ε in this equation do not coincide. Generally speaking, the conditional mean of the derivative does not coincide with the derivative of a conditional mean (∇i u∗ (y)|y = ∇i u∗ (y)|y). As in (4.24), the solution of this problem for an inclusion centered at the point y k may be presented in the form ∗ (y), F(y) = Λ(y − y k )F u λ (y), 0 , Λ(y) = 0 Λεij (y)

(4.55) (4.56)

where functions λu (y), Λεij (y) are deﬁned in (4.21), (4.23). Using (4.55), we can express the ﬁeld F(y) in the composite and the local ∗ exciting ﬁeld F∗ (y) in (4.47), (4.49) via the eﬀective ﬁeld F 0 ∗ (y )S(y )dy , )F F(y) = F (y) + p K(y − y )L1 Λ(y (4.57) ∗

0

F (y) = F (y) +

∗ (y )S(y, y )dy , )F K(y − y )L1 Λ(y

(4.58)

) coincides with Λ(y − y k ) inside the k-th inclusion. where the function Λ(y After averaging (4.57) over the ensemble realizations of the random ﬁeld of inclusions we obtain for the mean wave ﬁeld F(y) the equation ∗ (y )dy , (4.59) F(y) = F0 (y) + p P(y − y )L1 Λ0 F

Λ0 = Λ(y)|y , p = S(y) . (4.60) Averaging equation (4.58) for the local exciting ﬁeldunder the condition ∗ (y) = F∗ (y)|y y ∈ S and taking into account of the hypothesis H2 F ∗ (y) : we ﬁnd a closed equation for the eﬀective ﬁeld F ∗ 0 ∗ (y )Ψ (y − y )dy . (4.61) F (y) = F (y) + p P(y − y )L1 Λ0 F Here Ψ (y − y ) is the following two-point correlation function Ψ (y − y ) =

1 S(y, y )|y p

(4.62)

that depends only on the geometrical properties of the random set of inclusions.

4.3 The eﬀective ﬁeld method

87

After excluding the incident ﬁeld F0 from (4.59) and (4.61) we obtain ∗ (y) = F(y) − p K(y − y )L1 Λ0 F ∗ (y )Φ(y − y )dy , F (4.63) Φ(y) = 1 − Ψ (y).

(4.64)

Equation(4.63) is a convolution equation. Therefore, if the mean ﬁeld ∗ (y) is also a F(y) is a plane wave with wave vector k∗ , the eﬀective ﬁeld F ∗ plane wave with the same wave vector k ∗ u (y) ∗ , (4.65) F (y) = ε∗i (y) u ∗ (y) = U∗u e−ik∗ n·y ,

ε∗i (y) = −iki∗ U∗ε e−ik∗ n·y .

(4.66) 0

Let us return to deﬁnition (4.60) of the matrix Λ u 1 λ (y) 0 . Λ0 = Λ(y)dy, Λ(y) = 0 Λεij (y) s s

(4.67)

Here the functions λu (y) and Λεij (y) are connected with the displacement u(y) and strain εi (y) ﬁelds inside the inclusion centered at the origin by the equations u∗ (y), εi (y) = Λεij (y) ε∗j (y). u(y) = λu (y)

(4.68)

Thus, the matrix Λ0 takes the form λ0 (k∗ ) 0 Λ0 = , 0 Λ0ij (k∗ ) 1 λ0 (k∗ ) = s

s

1 u(y) dy = u ∗ (y) sU∗u

Λ0ij (k∗ ) = H(k∗ )

(4.69)

u(y)eik∗ ·y dy,

s

ki∗ kj∗ i , H(k∗ ) = k∗ k∗2 sU∗ε k∗2 i

εi (y)eik∗ ·y dy.

(4.70) (4.71)

s

4.3.3 The dispersion equation of the EFM Because (4.59) and (4.63) are convolution equations, the Fourier transform to these equations gives us a system of linear algebraic equations for the Fourier transform of the unknown functions ∗ (k), F(k) = F0 (k) + pK(k)L1 Λ0 F

(4.72)

∗ (k) = F(k) − pKΦ (k)L1 Λ0 F ∗ (k). F

(4.73)

88

4. Axial elastic shear waves in ﬁber-reinforced composites

Here KΦ (k) is the following integral KΦ (k) = K(x)Φ(x)eik·x dx.

(4.74)

Equation (4.73) may be written in the form 1 0 Φ 1 0 ∗ I + pK (k)L Λ F (k) = F(k), I = . 0 δik

(4.75)

∗ (x) are plane Because the mean wave ﬁeld F(x) and the eﬀective ﬁeld F axial shear waves, their Fourier transforms are proportional to δ(k − k∗ n) U 2 F(k) = (2π) δ(k − k∗ n), (4.76) −iki∗ u ∗ (k) = −(2π)2 U∗ (k)ε δ(k − k∗ n), (4.77) F −iki U∗ where n is the wave normal of these waves. ∗ (k) in (4.69), (4.77) is the The product of the matrix Λ0 and vector F following symbolic vector λ0 (k∗ )U∗u ∗ (k) = (2π)2 δ(k − k∗ n), Λ0 F (4.78) −ik∗ mi H(k∗ )U∗ε where scalar functions λ0 (k∗ ) and H(k∗ ) are deﬁned in (4.70), (4.71) via the solution of the one-particle problem. Resolving (4.75) and substituting the result into (4.72) we obtain the equation for the Fourier transform of the mean wave ﬁeld −1 F(k), F(k) = F0 (k) + pK(k)L1 Λ0 I + pKΦ (k)L1 Λ0

(4.79)

Multiplying this equation with the matrix L0 (k) in (4.32) and taking into account (4.33) we obtain the equation for F(k) in the form L∗ (k)F(k) = 0,

L∗ (k) = L0 (k) − pM(k)L1 Λ0 I + pKΦ (k)L1 Λ0

−1

(4.80) .

(4.81)

Because F(k) is proportional to δ(k − k∗ n), it follows from this equation that L∗ (k∗ ) = 0. This symbolic equation is equivalent to the equation µ1 ρ1 t−π Π −T 2 2 − k0 1 + p λ0 (k∗ ) = 0, k∗ 1 + p H(k∗ ) µ0 ρ0

(4.82)

(4.83)

4.4 One-particle problems of EMM and EFM

89

and is in fact the dispersion equation for the eﬀective wave number k∗ of the mean wave ﬁeld in the composite. Here the coeﬃcients π, t, Π and T are functions of the eﬀective wave number k∗ 1 pρ1 ω 2 Γ Φ (k∗ ), t = 1 + pρ1 ω 2 λ0 (k∗ )GΦ (k∗ ), ik∗ Π(k∗ ) = 1 + pµ1 H(k∗ )K Φ (k∗ ), T (k∗ ) = −pik∗ µ1 H(k∗ )Γ Φ (k∗ ), π(k∗ ) = −

= Πt − πT.

(4.84) (4.85) (4.86)

GΦ (k∗ ), Γ Φ (k∗ ), K Φ (k∗ ) are the following integrals that depend on the speciﬁc correlation function Φ(r) deﬁned in (4.62), (4.64) ∞ iπ H0 (k0 r)J0 (k∗ r)Φ(r)rdr, (4.87) GΦ (k∗ ) = − 2µ0 0 ∞ πk0 Γ Φ (k∗ ) = − H1 (k0 r)J1 (k∗ r)Φ(r)rdr, (4.88) 2µ0 0 1 iπk02 ∞ K Φ (k∗ ) = − + [H0 (k0 r)J0 (k∗ r) + H2 (k0 r)J2 (k∗ r)] Φ(r)rdr. 2µ0 2µ0 0 (4.89) In these equations, Jn (z) is the Bessel function and Hn (z) is the Hankel function of the second kind of the order n. Finally, the dispersion equation for the eﬀective wave number k∗ of the mean wave ﬁeld may be written in the canonical form −1 µ1 t − π ρ1 Π − T 2 2 1+p H = 0. (4.90) k∗ − k0 1 + p λ0 ρ0 µ0 Functions λ0 (k∗ ) and H(k∗ ) in this equation are to be found from the solution of the one-particle problem that is considered in Section 4.4.

4.4 One-particle problems of EMM and EFM 4.4.1 The one-particle problem of the EMM The one-particle problem of the EMM is the solution of the integral equation (4.17) if s is a disk of radius a centered at the origin y = 0. The symbolic integral equation (4.17) is in fact two integral equations for the displacement and strain ﬁelds in the medium with an inclusion ρ1 ω 2 g∗ (y − y ) + ∇i g∗ (y − y )µ1 εi (y ) dy , (4.91) u(y) = u∗ (y) + s ρ1 ω 2 i g∗ (y − y ) + i j g∗ (y − y )µ1 εj (y ) dy . εi (y) = ε∗i (y) + s

(4.92)

90

4. Axial elastic shear waves in ﬁber-reinforced composites

Here g∗ (y) is the Green function of the eﬀective medium, u∗ (y) = U∗ exp(−ik∗ · y) is a plane wave in the eﬀective medium that coincides with the mean wave ﬁeld in the composite u∗ (y) = u(y). Integral equations (4.91), (4.92) are equivalent to the following system of diﬀerential equations ∆u + k2 u = 0,

|y| ≤ a,

∆u +

k∗2 u

|y| > a,

(4.93)

k2 =

ρω 2 , µ

ρ∗ , µ∗

(4.94)

= 0,

k∗2 = ω 2

with the following conditions on the boundary of the inclusion (r = a) u(a − 0, ϕ) = u(a + 0, ϕ),

. . ∂u(r, ϕ) .. ∂um (r, ϕ) .. µ = µ∗ . . ∂r .r=a−0 ∂r r=a+0 (4.95)

Here r and ϕ are the polar coordinates in the y-plane. The ﬁeld u(y) should also satisfy the radiation condition at inﬁnity u(y) − u∗ (y) ∼

exp(−ik∗ r) √ if r = |y| → ∞. r

(4.96)

The solution of this problem has the form (see [21]) u(y) = U∗ u(y) = U∗

∞ m=0 ∞

am Jm (kr) cos(mϕ),

|y| ≤ a,

(4.97)

[m (−i)m Jm (k∗ r) + bm Hm (k∗ r)] cos(mϕ), |y| > a, (4.98)

m=0

where m = 1 if m = 0 , m = 2 if m > 0, Jm (z) are Bessel functions, and Hm (z) are Hankel functions of the second kind. The coeﬃcients am in (4.97) are to be found from the conditions (4.95) and take the forms am = (−i)m m

µ∗ k∗ , ∆m

iπ k∗ a [µ∗ k∗ Hm (k∗ a)Jm (ka) − µkHm (k∗ a)Jm (ka)] , 2 df . f (z) = dz ∆m =

(4.99) (4.100) (4.101)

The coeﬃcients λ∗ (k∗ ) and H(k∗ ) in the dispersion equation (4.40) are calculated from (4.70), (4.71)

4.4 One-particle problems of EMM and EFM

λ∗ (k∗ ) = H(k∗ ) =

1 πa2

ini πa2 k∗

u(y)eik∗ ·y dy =

s

s

∞

am gm (k, k∗ ), m=0 ∞

εi (y)eik∗ ·y dy =

am gm1 (k, k∗ ),

91

(4.102) (4.103)

m=0

gm (k, k∗ ) = 2im [ak∗ Jm−1 (ak∗ )Jm (ak) − akJm−1 (ak)Jm (ak∗ )] , a2 k 2 − k∗2 2im Jm (ka)Jm (k∗ a). gm1 (k, k∗ ) = gm (k, k∗ ) + k∗ a

=

(4.104) (4.105)

Here, for the calculation of the integral in (4.103), the equation εi (y) = i u and the Gauss theorem were used. 4.4.2 The one-particle problem of the EFM The one-particle problem of EFM is the solution of the integral equations (4.53) when u ∗ (y) and ε∗i (y) are plane waves with wave vector k∗ that does not coincide with wave vector k0 in the matrix. For an inclusion of radius a centered at the origin y = 0, the integral equation (4.53) is equivalent to the following system of diﬀerential equations µ0 2 (k − k∗2 )e−ik∗ ·y , |y| ≤ a, µ 0 u + k02 u = (k02 − k∗2 )e−ik∗ ·y , |y| > a u + k 2 u =

(4.106)

with conditions (4.95) on the boundary of the matrix and the inclusion, where µ∗ should be changed for µ0 . The solution of (4.106) may be found in the same form as the solution of the classic diﬀraction problem (see [21]) and takes the form ∞ −ik∗ ·y , |y| ≤ a, (4.107) am Jm (kr) cos(mϕ) + ζe u(y) = m=0

u(y) =

∞

[m (−i)m Jm (k∗ r) + bm Hm (k0 r)] cos(mϕ), |y| > a,

(4.108)

m=0

ζ=

µ0 k02 − k∗2 . µ k 2 − k∗2

(4.109)

The constants am and bm are to be found from the boundary conditions (4.95), m = 1 if m = 0, m = 2 if m > 0. In what follows we need only the constants am that deﬁne the ﬁeld u inside the inclusion

92

4. Axial elastic shear waves in ﬁber-reinforced composites

am = (−i)m m

A(k∗ )Jm (k∗ a)Hm (k0 a) − B(k∗ )Jm (k∗ a)Hm (k0 a) , µ0 k0 Jm (ka)Hm (k0 a) − µkJm (ka)Hm (k0 a)

A(k∗ ) = (1 − ζ)k0 µ0 ,

(4.110)

µ B(k∗ ) = 1 − ζ k∗ µ0 . µ0

(4.111)

It follows from (4.70), (4.71) that functions λ0 (k∗ ) and H(k∗ ) in the dispersion equation (4.90) of the EFM have the following form ∞ 1 ik∗ ·y u(y)e dy = am gm (k, k∗ ) + ζ, (4.112) λ0 (k∗ ) = πa2 s0 m=0 ∞ ini ik∗ ·y H(k∗ ) = εi (y)e dy = am gm1 (k, k∗ ) + ζ, (4.113) πa2 k∗ S0 m=0 gm (k, k∗ ) = 2im

ak∗ Jm−1 (ak∗ )Jm (ak) − akJm−1 (ak)Jm (ak∗ )

gm1 (k, k∗ ) = gm (k, k∗ ) +

2

2

(ka) − (k∗ a) 2im Jm (ka)Jm (k∗ a). k∗ a

,

(4.114) (4.115)

4.4.3 The scattering cross-section of a cylindrical ﬁber Let us consider the diﬀraction of a plane axial shear wave propagating in the original matrix (u0 (y) = U0 exp(−ik0 · y)) by an isolated cylindrical ﬁber. The ﬁeld us (y) = u(y) − u0 (y) scattered by such an inclusion is the integral terms in (4.91): ρ1 ω 2 g(y − y )u(y ) + ∇i g(y − y )µ1 εi (y ) dy . (4.116) us (y) = s

Because the integration is over the region s, (4.116) deﬁnes the scattered ﬁeld through the ﬁelds u and εi inside the inclusion. Using the standard technique of evaluation of the integral in (4.116) [2] we ﬁnd that the following equations hold for large |y|: e−ik0 r y r = |y|, (4.117) n= , us (y) ≈ A(n) √ , r r ⎡ ⎤ 3 iπ ρ1 k0 iµ1 A(n) = i e4 ⎣ nk εk (y)eik0 (n·y) dy − u(y)eik0 (n·y) dy ⎦ , 8π k0 µ0 ρ0 s

s

(4.118) where A(n) is the scattering amplitude. If the vector n coincides with the wave normal n0 of the incident ﬁeld, A(n0 ) is the forward scattering amplitude. The long-distance asymptotics of the stress and strain scattered ﬁelds follow from (4.117) in the forms

4.4 One-particle problems of EMM and EFM

e−ik0 r εsi = −ik0 ni A(n) √ , r

e−ik0 r σis = −ik0 ni µ0 A(n) √ . r

93

(4.119)

The scattering cross-section Q(k0 ) of the ﬁber is deﬁned by the equation [2] 1 Im JQ(k0 ), JQ(k0 ) = (σi0 u ˜s + σis u ˜0 )ni dΓ, (4.120) Q(k0 ) = k0 µ0 Γ

where u0 (y) and σk0 (y) are plane incident waves with the wave vector k0 = k0 n0 u0 = e−ik0 n

0

·y

,

σi0 = −ik0 n0i µ0 e−ik0 n

0

·y

.

(4.121)

The tilde in (4.120) means complex conjugation, Γ is a circle of a large radius R. From (4.120) and (4.121) we obtain 0 ik0 µ0 A(n)e−ik0 r eik0 (n ·y) ˜∗ + σis u ˜0 )ni = − √ (σi0 u r

ik0 r −ik0 (n0 ·y) ˜ . +(n0 · n)A(n)e e

(4.122)

For large R, the integral JQ(k0 ) in (4.120) is evaluated by the saddle-point method [23]

˜ 0 )eiπ/4 JQ(k0 ) ≈ iµ0 2πk0 A(n0 )e−iπ/4 + A(n

= −iµ0 8πk0 Re A(n0 )e−iπ/4 , and for Q(k0 ) in (4.120) we obtain

8π Re A(n0 )e−iπ/4 . Q(k0 ) = − k0

(4.123)

Thus, the scattering cross-section Q(k0 ) is deﬁned via the forward scattering amplitude A(n0 ). This fact may be called the optical theorem for an inﬁnite cylinder. (For electromagnetic waves, the optical theorem is proved in [2].) From (4.102), (4.103) and (4.118) it follows that the function A(n0 ) has the form ρ1 k03 µ1 0 2 iπ/4 (4.124) H − λ0 . A(n ) = iπa e 8π µ0 ρ0 0 i 1 0 ik0 n0 ·y H= n εi (y)e dy, λ0 = u(y)eik0 n ·y dy. (4.125) πa2 k0 i πa2 s

s

94

4. Axial elastic shear waves in ﬁber-reinforced composites

From this equation and (4.123), we obtain the equation for the scattering cross-section Q(k0 ) of the ﬁber: µ1 ρ1 Q(k0 ) = πa2 k0 Im (4.126) H − λ0 , µ0 ρ0 ∞ ∞ λ0 = am gm , H= am g1m , (4.127) m=0

gm =

m=0

kJm+1 (ka)Jm (k0 a) 2im 2

g1m = gm +

− k0 Jm (ka)Jm+1 (k0 a)

(ka) − (k0 a)

2im Jm (ka)Jm (k0 a), k0 a

,

(4.128)

dJm (z) . dz

(4.129)

2

Jm (z) =

Here the coeﬃcients am have the forms (4.99) when k∗ = k0 . Using the technique presented in [2] we can show that the short-wave limit of Q(k0 ) (k0 → ∞) is lim

k0 →∞

4 Q(k0 ) = . πa π

(4.130)

This limit depends on neither the properties of the matrix nor those of the inclusion (the paradox of extinction for a cylindrical inclusion). The character of convergence of the function Q(k0 ) to this limit may be seen from Fig. 3.2. In this ﬁgure, ∞ µ1 µ1 ρ1 ρ1 Q(k0 ) = k0 a H − λ0 = k0 a am gm1 − gm . (4.131) µ0 ρ0 µ0 ρ0 m=0 The left-hand side of Fig. 4.2 corresponds to the case µ/µ0 = 100, ρ/ρ0 = 10 (a hard and heavy inclusion), and the right-hand side to the case 3 2

1.5

Q Im(Q)

1

1

Q Im(Q)

0.5

Re(Q) 0

0

−1

−0.5

−2 −2 −1.5 −1 −0.5 0

lg(k0a) 0.5

1

1.5

2

−1 −2 −1.5 −1 −0.5 0

Re(Q) lg(k0a) 0.5

1

1.5

2

Fig. 4.2. The dependence of the normalized forward scattered amplitude Q of shear waves scattered by an isolated cylindrical ﬁber of unit radius a = 1, on the dimensionless frequency of the incident ﬁeld k0 a; the left part corresponds to a hard and heavy cylindrical inclusion (µ/µ0 = 100, ρ/ρ0 = 10); the right part to a soft and light inclusion (µ/µ0 = 0.01, ρ/ρ0 = 0.1).

4.4 One-particle problems of EMM and EFM

95

µ/µ0 = 0.01, ρ/ρ0 = 0.1 (a soft and light inclusion). It follows from (4.130) that the short-wave limit of the series on the right-hand side of (4.131) is 4i/π, and this limit depends neither on the shear modulus and the density of the matrix nor on these parameters of the inclusion. 4.4.4 Approximate solution of the one-particle problem in the long-wave region Let us consider the detailed form of integral equation (4.53) of the one-particle problem of the EFM ρ1 ω 2 g(y − y )u(y ) + ∇i g(y − y )µ1 εi (y ) dy , u(y) = u ∗ (y) + s

εi (y) = ε∗i (y) +

(4.132)

ρ1 ω 2 ∇i g(y − y )u(y ) + Kij (y − y )µ1 εj (y ) dy .

s

(4.133) If the length of the eﬀective waves u ∗ (x) and ε∗i (y) is much larger than the diameter of the inclusion, the displacement u(y) and deformation εi (y) inside the region s occupied by the inclusion do not vary essentially and may be considered as constants. To obtain these constants, let us substitute them into (4.132), (4.133) and integrate these equations over the area s (the Galerkin scheme). Because for a circular ﬁber the following equation holds dy ∇i g(y − y )dy = 0, (4.134) s

s

the cross-terms in the system (4.132) (4.133) disappear. Thus, in the longwave region, we can neglect the cross-terms in (4.132), (4.133) and consider the system in two independent equations for the displacement u(y) and strain ε(y) ﬁelds (4.135) u(y) = u∗ (y) + ρ1 ω 2 g(y − y )u(y )dy , s εi (y) = ε∗i (y) + µ1 Kij (y − y )εj (y )dy . (4.136) s

Consider this system, but assume that ﬁelds u(y) and εi (y) inside the inclusion are plane waves with the wave vector k∗ of the eﬀective ﬁeld: u(y) = U e−ik∗ ·y ,

εi (y) = εi e−ik∗ ·y .

(4.137)

The constant amplitudes U and εi may be found from (4.135) and (4.136) by the Galerkin procedure. Substitute u(y) and εi (y) from (4.137) into (4.135) and (4.136), multiply both parts of the resulting equations with eik∗ ·y , and integrate them over the region s. Finally, for U and εi we obtain

96

4. Axial elastic shear waves in ﬁber-reinforced composites

(1 − ρ1 ω 2 G)U = U∗u ,

(δij − µ1 Pij )εj = −ik∗ ni U∗ε .

Here constants G and Pij are the following integrals 1 ik∗ ·y e G= S(y)dy g(y − y )e−ik∗ ·y s(y )dy , 2 πa 1 ik∗ ·y e S(y)dy Kij (y − y )e−ik∗ ·y s(y )dy . Pij = πa2

(4.138)

(4.139) (4.140)

After the introduction of a new variable y − y = R, the integral G takes the form i −ik∗ ·R H (k R)e dR s(R + y)s(y)dy . (4.141) G=− 0 0 4πa2 µ0 Here the last integral is calculated explicitly: s(R + y)s(y)dy = πa2 f (R), (4.142) ⎞ ⎤ ⎡ ⎛ / / 2 2 R R R 2a ⎠− ⎦, R ≤ 2a; 1− 1− f (R) = 2/π ⎣arctan ⎝ R 2a 2a 2a f (R) = 0, R > 2a. After integration over the unit circle in the outer integral in (4.141), G is presented as the one-dimensional integral 2a iπ H0 (k0 r)J0 (k∗ r)f (r)rdr. (4.143) G=− 2µ0 0 The same procedure gives us the equation for the tensor Pij in (4.140) (4.144) Pij = P1 δij + P2 ni nj , 2 2a 1 iπk0 P1 = − + [H0 (k0 r)J0 (k∗ r) − H2 (k0 r)J2 (k∗ r)]f (r)rdr, 2µ0 4µ0 0 (4.145) 2 2a iπk0 P2 = H2 (k0 r)J2 (k∗ r)f (r)rdr. (4.146) 2µ0 0 Finally, we obtain the amplitudes of the displacement and strain ﬁelds inside the inclusion in the forms (4.147) U = (1 − ρ1 ω 2 G)−1 U∗u , εi = −ik∗ ni (1 − µ1 P )−1 U∗ε , 2a 1 iπk02 P =− + [H0 (k0 r)J0 (k∗ r) + H2 (k0 r)J2 (k∗ r)]f (r)rdr. 2µ0 4µ0 0 (4.148)

4.5 Solutions of the dispersion equations in the long-wave region

97

The functions λ0 (k∗ ) and H(k∗ ) in the dispersion equation (4.90) that correspond to approximate solution (4.137), (4.147) are −1 ρ1 iπk02 2a λ0 (k∗ ) = 1 + H0 (k0 r)J0 (k∗ r)f (r)rdr , ρ0 2 0 µ1 µ1 iπk02 2a H(k∗ ) = 1 + − H0 (k0 r)J0 (k∗ r) 2µ0 2µ0 2 0 −1 + H2 (k0 r)J2 (k∗ r)f (r)rdr .

(4.149)

(4.150)

As in system (4.135), (4.136), let us neglect the cross-terms in the equation (4.75) for the eﬀective ﬁeld and go to the approximate system t(k∗ )U∗u = U,

Π(k∗ )U∗ε = U,

(4.151)

where the coeﬃcients t(k∗ ) and Π(k∗ ) are deﬁned in (4.84), (4.85). As a result, the dispersion equation of the EFM (4.90) is essentially simpliﬁed, and transformed into the following one −1 µ1 ρ1 1 1 1+p = 0, k∗2 − k02 1 + p ρ0 ρ (k∗ ) µ0 µ (k∗ ) ρ (k∗ ) = 1 +

ρ1 iπk02 ρ0 2

0

(4.152)

2a

H0 (k0 r)J0 (k∗ r)ξ(r)rdr,

µ1 iπk02 − 2µ0 4 2a µ1 × H0 (k0 r)J0 (k∗ r) + H2 (k0 r)J2 (k∗ r)]ξ(r)rdr, µ0 0

(4.153)

µ (k∗ ) = 1 +

ξ(r) = f (r) − pΦ(r).

(4.154) (4.155)

Here the function f (r) is deﬁned in (4.142).

4.5 Solutions of the dispersion equations in the long-wave region In this section, we study the solutions of the dispersion equation of the EMM and EFM in the long-wave region, where the wave numbers k0 a and k∗ a are small (k0 a, k∗ a 1).

98

4. Axial elastic shear waves in ﬁber-reinforced composites

4.5.1 Long-wave asymptotic solution for EMM The parameters λ∗ and H in the dispersion equation (4.40), (4.41) of the EMM have the forms (4.102), (4.103). In the long-wave region, only the principal terms in the real and imaginary parts of the coeﬃcients am in (4.99) and the coeﬃcients gm , g1m in (4.104), (4.105) should be taken into account. Because the principal terms of the Bessel and Hankel functions for small values of arguments are Jn (z) ≈

1 z n , n! 2

z iπ 2n z 2n iπ n z Hn (z) ≈ −2n−1 (n − 1)! + + ln 2 n! 2 2 2

(4.156)

,

(4.157)

we obtain the principal terms of the coeﬃcients am , gm in the forms 2 −1 k∗ iπ µ 2 a0 = 1 − (ka) + , 4 k02 µ∗ −1 µ∗1 iπ 2ik∗ a1 = − 1+ 1 − (k∗ a)2 , k 2µ∗ 4 ik . g0 = 1, g10 = 0, g11 = 2k∗

(4.158) (4.159) (4.160)

The other coeﬃcients am , gm , g1m may be neglected in the long-wave region. As a result, the principal terms of the long-wave asymptotics of the coeﬃcients λ∗ and H take the forms iπ ρ∗1 (k∗ a)2 , 4 ρ∗ −1 −1 µ iπ µ∗1 µ ∗1 ∗1 1+ 1 + (k∗ a)2 . H = 1+ 2µ∗ 8 µ∗ 2µ∗

λ∗ = 1 −

(4.161) (4.162)

After substituting these equations into the dispersion equation of the EMM (4.40), (4.41) and taking into account only the principal terms in the real and imaginary parts of its solution, we obtain k∗ = ks − iγ, pπ 2µ1 (µ − µs ) ρs 3 ρ1 (ρ − ρs ) (ks a) . , γ= + ks = ω µs 8a ρ2s (µ + µs )2

(4.163) (4.164)

Here µs and ρs are the static values of the eﬀective shear modulus and density when ω, k0 = 0 . These parameters have the following forms ρs = ρ0 + p(ρ − ρ0 ),

µs = µ0 + 2p

(µ − µ0 )µs . µ + µs

(4.165)

4.5 Solutions of the dispersion equations in the long-wave region

99

The last equation is the algebraic equation for the eﬀective static elastic shear modulus µs of the composite. The dependence of µs /µ0 on the volume concentrations p of the inclusions obtained from the solution of (4.165) are presented in Fig. 4.5 by solid lines. The upper part of the ﬁgure corresponds to hard inclusions (µ/µ0 = 100), and the lower part to soft inclusions (µ/µ0 = 0.01). 4.5.2 Long-wave asymptotic solution for EFM Consider the solution of the dispersion equation (4.90) of the EFM in the long-wave region. Using the solution of the one-particle problem of the EFM (4.112), (4.113), we can present the parameters λ0 , H in the dispersion equation (4.90) in the forms λ0 = H=

∞ m=0 ∞

am gm +

µ0 k02 − k∗2 , µ k 2 − k∗2

am g1m +

m=0

µ0 k02 − k∗2 , µ k 2 − k∗2

(4.166) (4.167)

where the coeﬃcients am are deﬁned in (4.110) and gm , g1m have forms (4.114), (4.115). The long-wave asymptotics of the coeﬃcients λ0 and H, and integrals GΦ , Γ Φ and K Φ in (4.87)–(4.89) are iπ ρ1 (k0 a)2 , 4 ρ0 −2 iπ µ1 µ1 1 + 1 + (k0 a)2 , H = 1+ 2µ0 4 2µ0 ∞ iπa2 1 Φ JΦ , JΦ = 2 Φ(r)rdr, Γ Φ = 0, G =− 2µ0 a 0 iπ 1 KΦ = 1 − (k0 a)2 JΦ , 2µ0 2 λ0 = 1 −

(4.168) (4.169) (4.170)

(4.171)

Using these equations in the solution of dispersion equation (4.90) with respect to k∗ and keeping only the principal terms in the real and imaginary parts of k∗ we obtain ρs ks = ω , (4.172) k∗ = ks − iγ, µs µs = µ0 + pµR ,

ρs = ρ0 + pρ1 ,

(4.173)

100

4. Axial elastic shear waves in ﬁber-reinforced composites

−1 µ1 µR = µA , µA = µ1 1 + . 2µ0 2 2 2 µ k0 ρ πp ρ 0 R (1 − 2pJΦ )(ks a)3 . +2 1 γ= 16a ρs µ0 ρ0 ρs ks µA 1−p 2µ0

−1

(4.174) (4.175)

Here µs , ρs are the static elastic modulus and density of the composite, γ is the principal term of the attenuation coeﬃcient in the long-wave region. As follows from (4.175), the principal term of γ is proportional to the factor 1 − 2pJΦ that depends on the correlation function Φ(r) of the random ﬁeld of inclusions via the integral JΦ in (4.170). For calculation of this integral 1 JΦ = 2 a

∞

∞ Φ(r)rdr =

0

Φ(ζ)ζdζ, ζ = 0

r a

(4.176)

we have to choose a concrete form of the correlation function Φ(r). Here this function is constructed using the Percus-Yevick distribution function of the centers of non-overlapping disks. The Percus-Yevick function ψ(r) is the probability density to ﬁnd a center of an inclusion at a point y if the center of another inclusion is situated at the origin (y = 0). The integral equation for this function is presented in Appendix B. The connection between the function Φ(r) and ψ(r) is given by the equation 1 s0 (y )dx ψ(|y − y |)s0 (y − y)dy , (4.177) Φ(y) = 1 − 2 π where s0 (y) is the characteristic function of the disk of the unit radius centered at the point y = 0. Note that the double integral over 2D-space in this equation is reduced to the following integral r+2 2π 2 Φ(r) = 1 − 2 dϕ f r2 + s2 − 2rs cos(ϕ) ψ(s)sds, π 0 2 r r r 2 − f (r) = 2 arccos if r ≤ 2 1− 2 2 2

f (r) = 0 if r > 2 .

(4.178)

(4.179)

This integral may be easily evaluated numerically for the given ψ(r). Note that detailed tables of the function ψ(r) are presented in [36]. For the volume concentrations of the inclusions p = 0.1; 0.2; 0.3, the graphs of the function ψ(r) and Φ(r) are presented in Fig. 4.3. The dependence of the factor 1 − 2pJΦ on the volume concentration of the inclusions p for the Percus-Yevick correlation function is presented in Fig. 4.4.

4.5 Solutions of the dispersion equations in the long-wave region 3

ψ(ζ)

1

Φ(ζ)

0.6

p=0.1

101

p=0.5 2 p=0.3

p=0.5

1

p=0.3

0.2 p=0.1 −0.2

0 0

1

2

3

0

1

2

3

4

ζ

ζ

4

Fig. 4.3. Two-point correlation function ψ(ζ) of the centers of non-overlapping disks for the Percus-Yevick model, and the corresponding correlation functions Φ(ζ); p is the volume concentration of ﬁbers, ζ = r/a. 1 1-2pJΦ(p)

0.8 0.6 0.4 0.2 0

0

0.2

0.4

p

Fig. 4.4. The dependence of the factor 1 − 2pJΦ (p) on volume concentrations of inclusions p for the Percus-Yevick correlation functions.

The static elastic shear modulus µs of the composite takes a form that follows from (4.173), (4.174) 2p(µ − µ0 ) . (4.180) µs = µ0 1 + 2µ0 + (1 − p)(µ − µ0 ) The dependence of this modulus on the volume concentrations of ﬁbers p is presented in Fig. 4.5 by lines with dots. The upper part of Fig. 4.5 corresponds to hard inclusions (µ/µ0 = 100) and the lower part to soft inclusions (µ/µ0 = 0.01). It is seen from this ﬁgure that the diﬀerence in the predictions of the two self-consistent methods for this type of inclusions is appreciable only in the region of high volume concentrations (p > 0.3). For low contrasting properties of the phases this diﬀerence is negligible. In Fig. 4.6, the experimental and theoretical dependence of the shear modulus µs of the composite reinforced with cylindrical ﬁbers on volume

102

4. Axial elastic shear waves in ﬁber-reinforced composites 5

m*/m 0

4 3 2 1 0

0

0.2

p

0.4

Fig. 4.5. The dependence of the static elastic shear modulus µ∗ = µs of the ﬁber composites on volume concentration of inclusions p. The upper part of the ﬁgure corresponds to µ/µ0 = 100, the lower part to µ/µ0 = 0.01. Solid lines are the predictions of the EMM, lines with dots are the predictions of the EFM. m*, GPa 6

4

2

0

0.2

0.4

0.6

p

Fig. 4.6. The dependence of the static elastic shear modulus µ∗ = µs of the ﬁber composites on the volume concentrations of inclusions p; the solid line corresponds to the EMM, the line with dots to the EFM, squares are experimental data from [18].

concentrations of the ﬁbers p is presented. For this composite µ0 = 2.03GP a, µ = 12.5GP a. The solid line in Fig. 4.6 is the prediction of the EMM, the line with dots is the prediction of the EFM, squares are experimental data presented in [18]. It is seen that the predictions of the EFM are closer to the experimental data than those from EMM. The values of the attenuation coeﬃcients obtained by the EMM and EFM are compared in Fig. 4.7 (the left-hand side corresponds to hard and heavy ﬁbers, and the left-hand side to soft and light inclusions). In the long-wave region the dimensionless parameter γa/(k0 a)3 does not depend on the frequency of the incident ﬁeld, and is a function of volume concentration p and properties of inclusions. Solid lines in Fig. 4.7 show the dependence of this parameter on p obtained by the EMM (4.165), and lines with dots are predictions of the EFM (4.180). In both cases, the EMM gives values of γ that are

4.6 Short-wave asymptotics 2

1.5

γa/(k0a)3

103

γa /(k0a)3

1.5 1 1 0.5 0.5

0

0

0.2

0.4

p

0

0

0.2

0.4

p

Fig. 4.7. The dependence of normalized attenuation coeﬃcients of the mean wave ﬁeld on volume concentrations of inclusions p in the long wave region; solid lines correspond to EMM, lines with dots to EFM. The left-hand side corresponds to hard and heavy inclusions (µ/µ0 = 100, ρ/ρ0 = 10), the right-hand side to soft and light inclusions (µ/µ0 = 0.01, ρ/ρ0 = 0.1).

higher than the EFM predictions. Only for small values of p do the methods give close results in the long-wave region. Deviation in predictions of the methods are appreciable only for inclusions that are softer and lighter than the matrix.

4.6 Short-wave asymptotics Let us consider the solution of the dispersion equation (4.40) of the EMM in the short-wave limit when ω, k0 → ∞ and λ∗ , H → 0. From (4.40), (4.41) we obtain the following asymptotic equation for k∗ : / p µ1 ρ1 ρ0 + pρ1 λ∗ ≈ k0 1 − (4.181) k∗ = ω H − λ∗ . µ0 + pµ1 H 2 µ0 ρ0 Hence, in the short-wave limit, the attenuation coeﬃcient takes the form pQ(k0 ) p µ1 ρ1 H − λ∗ = , (4.182) γ = − Im k∗ = k0 Im 2 µ0 ρ0 2πa2 where Q(k0 ) is the total scattering cross-section of the inclusion deﬁned in (4.126). Taking into account (4.130) we obtain the short-wave limits of the attenuation coeﬃcient γ and the phase velocity v∗ of the mean wave ﬁeld in the forms 2p , πa ω µ0 lim v∗ = lim = = v0 . k0 →∞ k0 →∞ Re k∗ ρ0 lim γ = γ¯ =

k0 →∞

(4.183) (4.184)

104

4. Axial elastic shear waves in ﬁber-reinforced composites

Here we account that k0 Re µµ10 H − ρρ10 λ∗ → 0 if k0 → ∞. Let us turn to the solution of the dispersion equation of the EFM in the short-wave region and ﬁnd the limiting value of the eﬀective wave number in the form k∗ = Re k∗ − iγ,

(4.185)

where γ does not depend on k0 , and Re k∗ = O(k0 ). If ω, k0 → ∞, the following equations hold: K1Φ (k∗ )

+

K2Φ (k∗ )

iπk02 → 2µ0

∞

H0 (k0 r)J0 (k∗ r)Φ(r)rdr = −k02 GΦ (k∗ ),

0

(4.186) 1 ∆ → 1 + ipk0 J(k0 , k∗ )Λ, 2

Λ=

µ1 ρ1 H − λ0 , µ0 ρ0

(4.187)

∞ H0 (k0 r)J0 (k∗ r)Φ(r)rdr → IΦ (γ),

J = πk0 ∞

eγr Φ(r)dr.

IΦ (γ) =

(4.188)

0

(4.189)

0

Let us consider the short-wave limit of the function Λ in (4.187). From (4.102), (4.103) we obtain the equation for k0 Λ in the form k0 Λ = k0

∞

am

m=0

µ1 µ1 ρ1 ρ1 µ0 k02 − k∗2 · gm1 − gm +k0 − . (4.190) µ0 ρ0 µ0 ρ0 µ k 2 − k∗2

As is follows from (4.126), (4.130), the short-wave limit of the inﬁnite sum in this equation is 4i/ (πa), and this limit of the last term in the right-hand side of (4.190) is (−2iγ) . As a result, the short-wave limits of the functions k0 Λ and ∆ in (4.187) take the forms lim k0 Λ = 2i(

k0 →∞

2 − γ), πa

lim ∆ = 1 − p(

k0 →∞

2 − γ)IΦ (γ). πa

(4.191)

Thus, in the short-wave region, the EFM dispersion (4.90) is transformed into the following one: pΛ Λ 2 2 2 Λ or k∗ ≈ k0 1 − p ≈ k0 1 − . (4.192) k∗ − k0 ≈ −k∗ p ∆ ∆ 2∆

4.7 Numerical solutions of the dispersion equations 0.6

105

γa

0.4

0.2

0 0

0.2

0.4

0.6

p

Fig. 4.8. The dependence of the short-wave limits of the attenuation coeﬃcients γ¯ on the volume concentrations p of inclusions; solid lines correspond to EMM, lines with dots to EFM.

Hence, the short-wave limit γ of the attenuation coeﬃcient γ is p Λ k0 Im . (4.193) γ¯ = − lim (Im k∗ ) = lim k0 →∞ k0 →∞ 2 ∆ After substituting in this equation the limits from (4.191) we obtain γ¯ = p

2 − γ¯ πa

−1 2 1−p − γ¯ IΦ (¯ γ) . πa

(4.194)

Equation (4.194) is in fact an equation for the short-wave limit value γ¯ of the attenuation coeﬃcient. For small volume concentrations of inclusions p, the value of γ¯ coincides with γ¯ in (4.183) obtained in the framework of the EMM: γ¯ = 2p/ (πa). For large p, the solution of (4.194) depends on the correlation function Φ(r) via the integral IΦ (γ) in (4.189). The dependence of γ¯ on p for both methods are presented in Fig. 4.8. The solid line corresponds to the EMM and the line with dots to the EFM for the Perkus-Yevick correlation function of random set of ﬁbers. The short-wave limit of the velocity of the mean wave ﬁeld in the framework of the EFM coincides with its velocity v0 in the matrix, as well as for the EMM.

4.7 Numerical solutions of the dispersion equations The dispersion equation (4.40) of the EMM is in fact the system of the following three equation µ∗ = µ0 [1 + pµ1 H (k∗ , µ∗ )] ,

(4.195)

ρ∗ = ρ0 [1 + pρ1 λ∗ (k∗ , µ∗ )] ,

(4.196)

106

4. Axial elastic shear waves in ﬁber-reinforced composites

k∗ = ω

ρ∗ , µ∗

µ1 =

µ1 , µ0

ρ1 =

ρ1 , ρ0

(4.197)

where the functions H (k∗ , µ∗ ) and λ∗ (k∗ , µ∗ ) are deﬁned in (4.102)–(4.105). The numerical solution of this system may be found by the iterative procedure based on the following equations

(n) (n−1) (n−1) (n−1) (n−1) − µ∗ , (4.198) + δ µ0 1 + pµ1 H k∗ , µ∗ µ∗ = µ∗

(n) (n) (n−1) (n−1) (n−1) − ρ∗ , (4.199) , µ∗ ρ∗ = ρ∗ + δ ρ0 1 + pρ1 λ∗ k∗ -1/2 , (n) ρ∗ (n) k∗ = ω . (4.200) (n) µ∗ (n)

(n)

(n)

Here k∗ , µ∗ , ρ∗ are the nth iterations of the eﬀective parameters; δ is a numerical parameter chosen for the convergence of the iterative process. The dispersion equation (4.90) of the EFM may be also solved by the iterative procedure (n+1)

k∗

(n)

= k∗ ⎡ /

+ δ ⎣k0

ρ1 Π − T 1 + p λ0 ρ0

⎤ −1 µ1 t − π (n) 1+p H − k∗ ⎦ , µ0

(4.201)

where H, λ, Π, T, π, t, and are the functions of k∗ deﬁned in (4.84)–(4.86) and (4.112)–(4.115). Because (4.90) is the equation for one the complex wave number k∗ , any method of seeking roots of the function of complex variable −1 µ1 t − π ρ1 Π − T F (k∗ ) = k∗2 − k02 1 + p λ 1+p H (4.202) ρ0 µ0 may be used in this case. The drawbacks of the iterative procedures were mentioned in Section 2.5. When there are several branches of the solutions of the dispersion equation, the iterative procedure may jump from one branch to another. If the dispersion equation of the EFM is considered, the detailed analysis of the positions of the roots of the function F (k∗ ) in the complex k∗ -plane may be carried out. Such an analysis is more diﬃcult to perform for the system (4.195)–(4.197) that involves in fact two complex variables ρ∗ and µ∗ . In Figs. 4.9–4.11, the numerical solutions of the dispersion equations of the EMM and EFM for composites with an epoxy matrix (µ0 = 1.25GP a and ρ0 = 1,202 kg/m3 ) and glass ﬁbers (µ0 = 26.25 GPa and ρ0 = 2,502 kg/m3 ) are presented for the volume concentration of the ﬁbers p = 0.1; 0.3, and 0.5. For the solution of the dispersion equation of the EFM, the two point correlation functions Φ(r) described in Section 4.4 was used. The solid lines correspond to the EMM, line with dots to the EFM and the dashed lines to

4.7 Numerical solutions of the dispersion equations 5

0.15

p=0.1

Re (k*a)

γa

107

p=0.1

4 0.1

3 2

0.05 1 k0a

0 0

1

2

3

4

k0a

0 5

0

1

2

3

4

5

Fig. 4.9. The acoustical branch of the solutions of dispersion equation of the EMM (solid lines) and the EFM (lines with dots) for the epoxy-glass composite with the volume concentration of ﬁbers p = 0.1. Dashed lines is the EFM with the approximate solution of the one-particle problem. 5

0.6

p=0.3

Re(k*a)

γa

p=0.3

0.5

4

0.4

3

0.3 2 0.2 1

0.1 k0a

0 0

1

2

3

4

5

k0a

0 0

1

2

3

4

5

Fig. 4.10. The same as in Fig. 4.9 for p = 0.3. 5

Re(k*a)

1.2

p=0.5

4

γa

p=0.5

1

2

2

0.8

3

0.6

1

2

1

0.4

1

2

1

0.2 k 0a

2

0 0

1

2

3

4

1

k0a

0 5

0

1

2

3

4

5

Fig. 4.11. The same as in Fig. 4.9 for p = 0.5. Two branches of the solution of the dispersion equation of the EMM are indicated. 1 is the acoustical branch, and 2 is the optical branch.

108

4. Axial elastic shear waves in ﬁber-reinforced composites

the EFM with the approximate solution of the one-particle problem described in Section 4.5. For p = 0.1 and 0.3 the EMM and EFM give close predictions for the real and imaginary parts of the eﬀective wave number. For p = 0.5, the EMM indicate the existence of two branches of the solutions. The acoustical branch (1 in Fig. 4.11) describes the mean wave ﬁeld in the long-wave region, and attenuation along this branch becomes large in the region of medium and short waves. The optical branch (2) has large attenuation for long waves, and its attenuation is moderate for medium and short waves, where this branch deﬁnes the parameters of the mean wave ﬁeld. The EFM predicts only one branch, the lines with dots in Fig. 4.11.

4.8 Composites with regular lattices of cylindrical ﬁbers In the framework of the EFM, the peculiarities in spatial distribution of inclusions are taken into account via the two-point correlation function Φ(r) of the set of inclusions. This function can be constructed for random as well as for regular (periodic) sets of ﬁbers. In the latter case, Φ(r) is obtained by averaging the original lattice of inclusions over its spacial translations. The resulting mean wave ﬁeld may be interpreted as the detailed ﬁeld averaged over such translations. Let S(y) be the characteristic function a regular 2D-lattice of cylindrical ﬁbers of radii a. If q is the vector of this lattice, the function S(y) takes the form s(y + q), q = iL1 e1 + jL2 e2 , (4.203) S(y) = q

where e1 , e2 are the unit vectors of the elementary cell of the lattice, L1 , L2 are the sizes of the cell along the basic vectors, i, j = 0, ±1, ±2, ..., s(y) is the characteristic function of the unit circle centered at the origin (y = 0). If r is a random vector homogeneously distributed in space, realizations of the random function S(y + r) are translations of the original regular lattice in the vector r. The second moment S(r)S(y + r) of the random function S(y + r) is a periodic function. Here averaging is carried out over all possible positions of vector r. The explicit equation for this function takes the form S(r)S(y + r) 1 S(r)S(y + r)dr = s0 f (y + q), = lim Ω→∞ Ω Ω q 1 s(y + y )s(y )dy , f (y) = s0

(4.204) (4.205) (4.206)

where s0 is the area of a unit circle, Ω is a region that occupies all the y-plane in the limit Ω → ∞. The explicit equation for function f (y) in (4.206) coincides with (4.142).

4.8 Composites with regular lattices of cylindrical ﬁbers

109

For regular lattices, function Φ(y) deﬁned in (4.62), (4.64) takes the form

Φ(y) = 1 −

1 f (y + q). p q

(4.207)

Here the prime over the summation sign means omitting the term q = 0. For a square lattice of ﬁbers with distance L0 between the neighbor ﬁber centers, the correlation function Φ(y) takes the form

Φ(y) = 1 −

1 f (y+L0 (ie1 + je2 )), p i,j

(4.208)

where summation is carried out over integers r and s, from -∞ to ∞, prime over the sum sign means omitting the term r = s = 0. In what follows, the system (4.75) for the amplitudes of the eﬀective ﬁelds is used in the simpliﬁed form (4.151) that was used for approximate solution of the dispersion equation (4.152): u∗ (k) = u(k), [1 + pρ1 ω 2 GΦ (k)λ0 (k∗ )] Φ [δik + pµ1 Kik (k)H(k∗ )] ε∗k = εi (k),

(4.209) (4.210)

For the functions λ0 (k∗ ) and H(k∗ ), we take approximate equations Φ (k) in this equation are (4.149), (4.150), and the integrals GΦ (k) and Kij Φ GΦ (k) = g(y)Φ(y)eik·y dy = GΦ (4.211) 0 (k) + G1 (k), Φ 0 1 (k) = ∇i ∇j g(y)Φ(y)eik·y dy = Kij (k) + Kij (k), (4.212) Kij Here Φ(y) has form (4.208), and 1 g(y)f (y)eik·y dy, GΦ (k) = 0 p 1 GΦ f (α) g (k − α), 1 (k) = − s0 µ 1 0 ∇i ∇j g(y)f (y)eik·y dy, Kij (k) = p 1 1 Kij (k) = g (k − α). f (α)(ki − αi )(kj − αj ) s0 µ

(4.213) (4.214) (4.215) (4.216)

Here the prime over the sum means omitting the term α = 0, f(k), g(k) are the Fourier transforms of functions f (y) and g(y), α is the vector of the inverse lattice with respect to the original one. Direct calculation of the Fourier transform of function f (y) in (4.206) gives 2

J (α) f(α) = 4π 1 2 . α

(4.217)

110

4. Axial elastic shear waves in ﬁber-reinforced composites

where J1 (z) is Bessel function of the ﬁrst kind. As a result, the system (4.209), (4.210) for the amplitudes of the eﬀective ﬁelds takes the form u Φ 1 + pρ1 ω 2 λ0 (k∗ ) GΦ (4.218) 0 (k∗ ) + G1 (k∗ ) U∗ = U, Φ 1 + pµ1 H K0 (k∗ ) + K1Φ (k∗ ) U∗ε = U, (4.219) Φ where the scalar coeﬃcients GΦ 0 (k∗ ) and K0 (k∗ ) are the following integrals ∞ iπ (k ) = − H0 (k0 r)J0 (k∗ r)f (r)rdr, (4.220) GΦ ∗ 0 2pµ0 0 1 K0Φ (k∗ ) = − 2pµ0 iπk02 ∞ + [H0 (k0 r)J0 (k∗ r) + H2 (k0 r)J2 (k∗ r)] f (r)rdr. 4pµ0 0 (4.221) Φ Functions GΦ 1 (k∗ ) and K1 (k∗ ) depend on the symmetry of the inclusion lattice. For a quadratic lattice with the step L0 , these functions are

GΦ 1 (k∗ ) = − K1Φ (k∗ ) =

λ20 J12 [α(r, s)] , π r,s D(k0 , k∗ , r, s)

λ20 J12 [α(r, s)(k∗ − αm (r, s))2 , π r,s D(k0 , k∗ , r, s)

D(k0 , k∗ , r, s) = α2 (r, s)[k∗2 + α2 (r, s) − k02 − 2kαm (r, s)], α(r, s) = λ0 r2 + s2 , αm (r, s) = λ0 (rm1 + sm1 ), 2π λ0 = . L0

(4.222)

(4.223)

(4.224) (4.225)

Here m is the wave vector of the propagating ﬁeld, mi = m · ei , (i = 1, 2). The ﬁnal dispersion equation for the wave number k∗ of the mean wave ﬁeld propagating in the composite takes the form −1 µ1 ρ1 1 1 1+p , (4.226) k∗2 = k02 1 + p ρ0 ρ (k∗ ) µ0 µ (k∗ ) where coeﬃcients ρ (k∗ ) and µ (k∗ ) are ρ (k∗ ) = 1 + p

ρ1 2 Φ µ1 k0 G1 (k∗ ), µ (k∗ ) = 1 + p K1Φ (k0 ). ρ0 µ0

(4.227)

In Figs. 4.12–4.14, the numerical solutions of dispersion equation (4.226) when the wave vector k∗ belongs to the ﬁrst Brillouin zone (elementary cell of the inverse lattice with the size λ0 /2) are presented for the medium with

4.8 Composites with regular lattices of cylindrical ﬁbers

111

k0a p=0.123

1.2 1 λ0 /2

0.8 0.6

O

Γ

M

λ0 /2

0.4 0.2 Γ

M

0

Γ

O

k*a

Fig. 4.12. The acoustical and optical branches of wave propagation for the composite with a square lattice of cylindrical cavities by the volume concentration p = 0.123. The wave vector of the propagating wave belongs to the ﬁrst Brillouin zone shown in the inset. Bold lines are the acoustical and the ﬁrst optical branches presented in [85], thin lines and the dashed line are the solutions of (4.226). k0a p=0.38 2

1.5

λ0 /2

1 O

Γ λ0 /2 M

0.5

Γ

M

0 O

Γ

k*a

Fig. 4.13. The same as in Fig. 4.12 for p = 0.38. k0a 2.5

p=0.503

2 λ 0/ 2

Γ

1.5 1 O

λ 0/ 2 M

0.5

Γ

M

0 O

Fig. 4.14. The same as in Fig. 4.12 for p = 0.503.

Γ

k*a

112

4. Axial elastic shear waves in ﬁber-reinforced composites

a square lattice of cylindrical cavities (µ0 = 1, ρ0 = 1, p = 0.123 (Fig. 3.12); p = 0.38 (Fig. 3.13), p = 0.503 (Fig. 3.14)). The interval OM in these ﬁgures corresponds to the direction along the side OM of the inverse lattice, and 0 ≤ k∗ ≤ λ0 /2 inside this interval, the interval OΓ corresponds to the direction √ along the diagonal of the inverse lattice, and therein 0 ≤ k∗ ≤ λ0 / 2. For the interval M Γ , the end of the √ vector k∗ moves along the M Γ -side of the Brillouin zone, λ0 /2 ≤ k∗ ≤ λ0 / 2 for the M Γ -interval. The bold lines in these ﬁgures correspond to the exact acoustical and the ﬁrst optical branches presented in [85]. Thin lines are the acoustical and optical branches of the solutions of dispersion equation (4.226). This equation indicates the existence of an acoustical and two close optical branches, one of the latter is shown by the dashed lines in these ﬁgures. The acoustical and optical branches of the solutions of (4.226) in the ﬁrst Brillouin zone for epoxy-glass composites with square lattice of ﬁbers are shown in Fig. 4.15 for p = 0.5. It is seen from this ﬁgure that, strictly speaking, (4.226) does not indicate existence of a stop band in the frequency region for all directions of wave propagations. (The acoustical or optical branches exist for all values of frequencies (k0 a)). Nevertheless, for every ﬁxed direction, the stop bands exist. Let us consider waves propagating in the direction of a side of the lattice square (k0 = k0 e1 ). For such waves, the acoustical branches of the solutions of (4.226) for the epoxy-glass composites with the volume concentrations of inclusions p = 0.1; 0.3; 0.5 and 0 < k0 a < 3 are shown in Figs. 4.16–4.18. It turns out that for every acoustical branch, there exist series of the intervals in the frequency region, where the attenuation coeﬃcient (imaginary part of the solutions of (4.226) is not equal to zero. (Attenuations along the acoustical branches are shown by dashed lines in Figs. 4.16–4.18.) Thus, the waves of k0a 2.5 p=0.5 2

1.5

1

λ0 / 2

0.5 O Γ

M

0 O

Γ λ0 / 2 M Γ

k*a

Fig. 4.15. The acoustical and optical branches of the solution of the dispersion equation (4.226) inside the ﬁrst Brillouin zone for an epoxy-glass composite with a square lattice of cylindrical ﬁbers of the volume concentration p = 0.5.

4.8 Composites with regular lattices of cylindrical ﬁbers 3

Re(k*a)

0.06

γa

p=0.1

2.5

0.05

2

0.04

1.5

0.03

1

0.02

0.5

0.01

0

0

0.5

1

1.5

2

113

0 k0a

2.5

Fig. 4.16. The acoustical branch (bold line) of the solutions of the dispersion equation (4.226) for the epoxy-glass composites with square lattice of ﬁbers by the volume concentration of the latter p = 0.1, dashed line is attenuation γa along this branch. 3

Re(k*a)

0.2

γa

p=0.3

2.5

0.16

2 0.12 1.5 0.08 1 0.04

0.5 0

0

0.5

1

1.5

2

2.5 k0a

0

Fig. 4.17. The same as in Fig. 4.16 for p = 0.3. 3

Re(k*a)

γa

p=0.5

2.5

0.25

2

0.2

1.5

0.15

1

0.1

0.5 0

0.3

0.05 0

0.5

1

1.5

2

Fig. 4.18. The same as in Fig. 4.16 for p = 0.5.

2.5

0 k 0a

114

4. Axial elastic shear waves in ﬁber-reinforced composites

the corresponding frequencies attenuate exponentially, and these zones may be considered as stop bands or phononic gaps. When frequency (parameter k0 a) increases, the stop bands narrow and ﬁnally disappear for suﬃciently short wave lengths (k∗ a > 3λ0 a). This fact has clear physical interpretation: suﬃciently short waves, similar to straight rays, might go through the square lattice of inclusions without attenuation. The same result may be obtained by analysis of equation (4.194) for the short-wave limit of the attenuation coeﬃcient. For regular lattices of ﬁbers, the integral I(Φ) in (4.194) diverges for any positive value of γ and for the correlation function Φ(r) in the form (4.207). As a result, the only solution of (4.194) is γ = 0. Qualitatively, similar behavior of stop bands were observed experimentally in [61] in composites with a regular grid of spherical inclusions.

4.9 Conclusion The main characteristics of self-consistent methods in application to the axial elastic shear wave propagation in composites with cylindrical ﬁbers are similar to those of the electromagnetic case. Compared with EMM, EFM has several advantages. Its predictions correspond better to experimental data in the long-wave region, and it allows us to take into account the peculiarities in spatial distributions of inclusions in composites on their eﬀective properties. But unlike the EMM, the EFM requires more precise statistical information about the random ﬁeld of inclusions in the composite. This information is contained in the speciﬁc two-point correlation function of such ﬁelds. The construction of such a function is an independent and laborious problem. Note that the value of the attenuation coeﬃcient γ is very sensitive to the shape of the correlation function Φ(r). For regular composites, the dispersion equation (4.90) combined with the exact solution of the one-particle problem does not predict existence of pass bands of wave propagation. This result is natural because the ﬁrst hypothesis (H1 ) of the method (see Section 3.2.2) does not serve for regular composites. In the latter case, the ﬁeld that acts on each inclusion is not a plane wave, but a wave of more complex structure. Nevertheless, the approximate dispersion equation (4.226) predicts the existence of pass bands for acoustical branches, where the waves can propagate without attenuation, as well as a series of stop bands, where the waves attenuate exponentially. The position of the ﬁrst stop band in the frequency region is close to the Bragg frequency k∗ a ≈ λ0 /2; when the frequency of the incident ﬁeld increases, the stop bands narrow and ﬁnally disappear. The width of the stop bands grows with the volume concentration p of inclusions. All these facts hold also for the solutions of the exact dispersion equations for regular lattices of inclusions, but the development of such equations as well as their solution is a much more diﬃcult task than the solution of the approximate dispersion equation (4.226).

4.10 Notes

115

The versions of the eﬀective ﬁeld and eﬀective medium methods developed in this chapter may be considered as the simplest ones. But the dispersion equations developed in Section 3.4 are valid for more complex versions of the methods. For instance, for EMM, versions II and III of the method give diﬀerent equations for the coeﬃcients H and λ∗ in the dispersion equation (4.90), but the general form of this equation remains unchangeable.

4.10 Notes The material of this chapter is based on the work [54, 55]. Self-consistent solutions of the problem in the long-wave region where obtained in [4,6,13,98].

5. Diﬀraction of long elastic waves by an isolated inclusion in a homogeneous medium

We consider the one-particle problem for self-consistent methods. The solution of this problem is constructed for incident waves longer than characteristic sizes of the inclusion (long-wave approximation). Explicit equations for the elastic ﬁeld inside an ellipsoidal inclusion and its limiting forms (oblate and prolate spheroids) are obtained in Sections 5.1–5.3. Thin soft and hard inclusions, and hard axisymmetric ﬁbers are considered in Section 5.4. The ﬁnal Section 5.5 is devoted to the proof of the optical theorem for diﬀraction of elastic waves, and the calculation of the total scattering cross-sections of inclusions of various forms.

5.1 The dynamic Green tensor for a homogeneous anisotropic medium We consider an inﬁnite elastic medium with the tensor of elastic moduli C 0 and density ρ0 subjected to body forces q(x, t). For harmonic vibrations of frequency ω, qα (x, t) is a time-periodic function qα (x, t) = qα (x)eiωt , and the vector of displacements in the medium has a similar form uα (x, t) = uα (x)eiωt . The amplitude uα (x) of the displacement ﬁeld satisﬁes the equation 0 ∇µ uβ (x) + ρ0 ω 2 uα (x) = −qα (x). ∇λ Cαλβµ

(5.1)

If q(x) is a function with a ﬁnite support, the partial solution of this equation is presented in the form (5.2) uα (x) = gαβ (x − x )qβ (x )dx , where gλβ (x) is the Green tensor of the operator L0αβ 0 L0αβ = ∇λ Cαλβµ ∇µ + ρ0 ω 2 δαβ .

(5.3)

The tensor gλβ (x) is the solution of the following equation: L0αλ gλβ (x) = −δαβ δ (x) .

(5.4)

118

5. Diﬀraction of long elastic waves

The aim of this section is to obtain a series expansion of the tensor gαβ (x) with respect to frequency ω s (x) + gαβ (x) = gαβ

∞

(k)

ω k gαβ (x) .

(5.5)

k=1

In the next sections, this expansion is used for the construction of longwave approximate solutions of the diﬀraction problems for an isolated inclusion. A similar expansion of the scalar Green function (see (3.63)) was used in Chapter 3 for the construction of the long-wave approximate solution of electromagnetic wave diﬀraction problems. For an elastic anisotropic medium, expansion (5.5) is more diﬃcult than for the scalar case. The Fourier transform of (5.4) gives us an algebraic equation for the Fourier transform gλβ (k) of the Green tensor gλβ (x): 0 kµ − ρ0 ω 2 δαβ . L0αλ (k, ω) gλβ (k) = δαβ , L0αλ (k, ω) = kβ Cαβλµ

(5.6)

The solution of this equation is −1 gαβ (k) = k 2 Λαβ (ξ) − ρ0 ω 2 δαβ ,

(5.7)

0 ξλ ξµ , ξα = Λαβ (ξ) = Cαλβµ

kα k

, k = |k| .

(5.8)

Application of the inverse Fourier transform to (5.7) leads to the representation of the tensor gαβ (x) in the form gαβ (x) =

1 (2π)

∞ dΩξ

3

2 −1 k Λαβ (ξ) − ω 2 ρ0 δαβ exp (−ikrn · ξ)k2 dk . (5.9)

0

Ω1

Here Ω1 is the surface of the unit sphere in the k-space of the Fourier transforms, r = |x|, n = x/r. Introduce new variables t = kr and s = n · ξ, and rewrite the internal integral in (5.9) in the form ∞ Jαβ (ξ) = 0

= 2

1 r

2 −1 k Λαβ (ξ) − ω 2 ρ0 δαβ exp (−ikrs) k 2 dk

∞

−1 t2 t2 Λαβ (ξ) − λ2 δαβ exp (−its) dt ,

(5.10)

0 2

λ = ρ0 (rω) . Using the equation −1 −1 −1 2 2 2 = Λ−1 Λλβ , t2 t2 Λαβ (ξ) − λ2 δαβ αβ + λ t Λαλ (ξ) − λ δαλ

(5.11)

(5.12)

5.1 The dynamic Green tensor for a homogeneous anisotropic medium

we transform the integral (5.10) into the sum of two integrals: 1 (0) (1) J δαλ + Jαλ Λ−1 Jαβ = λβ (ξ) , r ∞ ∞ 2 −1 −its (1) (0) −its 2 t Λαλ (ξ) − λ2 δαλ dt , Jαλ = λ e dt . J = e 0

(5.13) (5.14)

0

The integral J J (0) =

119

(0)

is the following generalized function [7]

i + πδ (s) . s

(5.15) (1)

For the calculation of the integral Jαλ , let us present the two rank symmetric tensor Λαβ (ξ) in the orthogonal basis of its eigenvectors e(i) (i = 1, 2, 3) Λαβ (ξ) =

3

(i)

vi2 (ξ) e(i) α (ξ) eβ (ξ) .

(5.16)

i=1

Here vi2 (ξ) (i = 1, 2, 3) are the eigenvalues of the tensor Λαβ (ξ). −1 in (5.12) is presented In the basis e(i) , the tensor t2 Λαβ (ξ) − λ2 δαβ in the form 3 −1 2 2 −1 (i) 2 (i) 2 t vi (ξ) − λ2 t Λαβ (ξ) − λ δαβ = eα (ξ) eβ (ξ) .

(5.17)

i=1 (1)

As a result, the integral Jαβ in (5.13) is reduced to the sum of three similar integrals (i) (i) 3 eα (ξ) eβ (ξ)

∞

(1) Jαβ

=λ

2

i=1

vi2 (ξ)

0

λ e−its . dt , µi (ξ) = 2 2 t − µi vi (ξ)

(5.18)

Because the µi are real numbers, the integrands in this equation are singular, and regularizations of these integrals should be deﬁned. This regularization (the rule of bypassing the singular points) should be chosen from the condition that the Green function gαβ (x) describes the outgoing waves from the point source (the causality condition). This condition leads to the following regularization of the integrals in (5.18) ∞ 0

∞ e−its cos ts cos ts dt =p.v. dt + lim dt + f (s, µi ). ρ→∞ t2 − µ2i t2 − µ2i t2 − µ2i 0

(5.19)

γρ

∞ Here p.v. ...dt is the Cauchy principal value of the integral, γρ is the 0

semicircle of radius ρ in the lower half-plane of the t-complex plane. This

120

5. Diﬀraction of long elastic waves

semicircle connects the points (µi − ρ) and (µi + ρ) that are on the line of integration, f (s, µi ) is an odd function of the argument s. The ﬁrst integral in the right-hand side of (5.19) is [7] ∞ π cos ts dt = − sin µi |s| , p.v. 2 t − µ2i 2µi

(5.20)

0

and the integral over the semicircle γρ has the value iπ cos ts lim dt = cos µi s . ρ→0 t2 − µ2i 2µi

(5.21)

γρ

By integration over the unit sphere in (5.9), the term proportional to f (s, µi ) vanishes because this function is odd with respect to the argument s. (1) Thus, omitting inessential terms, we have for the integral Jαβ the following equation (1)

Jαβ = λ2

(j) (j) 3 eα (ξ) eβ (ξ) j=1

vj2

(ξ)

·

iπ exp (−iµj |s|) , 2µj

(5.22)

where the coeﬃcients µj are deﬁned in (5.18). Because e(i) is an orthogonal basis, this equation may be rewritten in the form ⎡ ⎤ 3 3 (j) (j) iπλ ⎣ eα (ξ) eλ (ξ) ⎦ iλ|s| (1) (k) (k) e (ξ) eβ (ξ) . exp − Jαβ = 2 j=1 vj (ξ) vk (ξ) λ k=1

(5.23) Note that if a two rank symmetric tensor T has eigenvectors e(i) and eigenvalues ti (i = 1, 2, 3), the function exp(T) in the basis of eigenvectors is the two rank tensor deﬁned by the equation exp(T)αβ =

3

(k)

exp (tk ) e(k) α eβ .

(5.24)

k=1

Application of this deﬁnition to the sum in (5.23) gives (k) (k) 3 eλ (ξ) eβ (ξ) iλ|s| (k) (k) e (ξ) eβ (ξ) = exp −iλ|s| exp − vk (ξ) λ vk (ξ) k=1 k=1

−1 = exp −iλ|s|Λλβ2 (ξ) , (5.25) 3

5.1 The dynamic Green tensor for a homogeneous anisotropic medium

121

(1)

where the tensor Λλβ (ξ) has the form (5.16). Thus, the integral Jαβ in (5.23) may be written in the form

iπλ − 12 −1 (1) Λαλ (ξ) exp −iλ|s|Λλβ2 (ξ) . (5.26) Jαβ = 2 Substituting (5.15) and (5.26) into (5.13) we ﬁnd the value of the internal integral in (5.9): 1 Jαβ (ξ) = r

1 iπλ − 32 − 12 −1 + πδ (s) Λαβ (ξ) + Λ (ξ) exp −iλ|s|Λλβ (ξ) , s 2 αλ (5.27)

and the equation for the Green function may now be written as follows: 1 Jαβ (ξ)dΩξ . (5.28) gαβ (x) = 3 (2π) Ω1

Consider the integral over the unit sphere Ω1 that corresponds to the ﬁrst term in the right-hand side of (5.27). Because s = n · ξ, we have

−1

[(n · ξ) Λαβ (ξ)] Ω1

Ψαβ (s) =

1 dsξ = −1

1 Ψαβ (s) ds, s

Λ−1 αβ (ξ) δ[(n · ξ) − s]dΩξ .

(5.29) (5.30)

Ω1

Since Ψαβ (s) is an even function, we have 1 1 Ψαβ (s) ds = 0 , −1 s

(5.31)

and the equation for the Green function gαβ (x) takes the form s ω (x) + gαβ (x, ω) . gαβ (x) = gαβ

(5.32)

s (x) is the “static” Green tensor, i.e., the Green tensor of the Here gαβ 0 operator L in (5.4) when ω = 0: 1 s (x) = Λ−1 (5.33) gαβ αβ (ξ) δ (n · ξ) dΩξ . 3π 2 r Ω1 ω (x, ω) of the Green tensor in (5.32) is The frequency-dependent part gαβ the following integral:

iω √ − 32 − 12 ω Λ (ξ) exp −iωr|ξ · n| ρ Λ (ξ) dΩξ . (5.34) gαβ (x, ω) = 0 αλ λβ 16π 2 r Ω1

122

5. Diﬀraction of long elastic waves

Expanding the exp-function under the integral in a Taylor series, we ﬁnd the equation: ∞

ω (x, ω) = gαβ k/2

(k)

gαβ (n) =

ρ0 16π 2

1 (−iωr)k (k) g (n) , r (k − 1)! αβ

(5.35)

k=1

− 1 (k+2)

Λαβ2

k−1

(ξ) |n · ξ|

dΩξ .

(5.36)

Ω1

For an isotropic medium with Lam´e constants λ0 and µ0 , the following equation holds: k

ρ02 Λ−k αβ (ξ) = t0 =

µ0 , ρ0

1 1 (δαβ − ξα ξβ ) + 2k ξα ξβ , t2k l 0 0 / λ0 + 2µ0 l0 = . ρ0

(5.37)

(5.38)

where t0 and l0 are the velocities of transverse (shear) and longitudinal waves propagating in the medium. In this case, we have from (5.33) s (x) = gαβ

1 t0 1 + η02 δαβ + 1 − η02 nα nβ , η0 = , 8πµ0 r l0

(5.39)

(k)

and the functions gαβ (n) in the series (5.35) takes the forms (k)

gαβ (n) =

−(k+2)

t0 1 + k + η0k+2 δαβ 4πρ0 k (k + 2) +(k − 1) η0k+2 − 1 nα nβ .

(5.40)

5.2 Integral equations for elastic wave diﬀraction by an isolated inclusion Let an inﬁnite homogeneous elastic medium with the tensor of elastic moduli C 0 and density ρ0 , contain a region V (inclusion) with elastic moduli C and density ρ. For monochromatic excitation of frequency ω, the amplitude uα (x) of the displacement ﬁeld in the medium satisﬁes the equation ∇β Cαβλµ (x)∇µ uλ (x) + ρ(x)ω 2 uα (x) = 0

(5.41)

0 1 + Cαβλµ V (x), Cαβλµ (x) = Cαβλµ

(5.42)

1 0 = Cαβλµ − Cαβλµ , Cαβλµ

ρ(x) = ρ0 + ρ1 V (x),

ρ1 = ρ − ρ0 .

Here V (x) is the characteristic function of the region V .

(5.43)

5.3 Diﬀraction of long elastic waves by an isolated inclusion

123

The diﬀerential equation (5.41) is equivalent to the following integral equation uα (x) = u0α (x) + ρ1 ω 2 gαµ (x − x ) uµ (x ) dx +

V 1 ∇λ gαµ (x − x ) Cλµρτ ερτ (x ) dx ,

(5.44)

V

ερτ (x) = ∇(ρ uτ ) (x) .

(5.45)

Here u0α (x) is the incident ﬁeld that would have existed in the matrix material without the inclusion under the prescribed conditions at inﬁnity, gαµ (x) is the dynamic Green tensor that was considered in Section 5.1. To derive this equation we use the same procedure as in the static case (see Chapter 2). The equation for the strain tensor εαβ (x) follows from (5.44) in the form εαβ (x) = ε0αβ (x) + ρ1 ω 2 ∇(α gβ)λ (x − x ) uλ (x ) dx V

1 − K αβµλ (x−x ) Cµλρτ ερτ (x ) dx ,

(5.46)

V

Kαβλµ (x) = −∇λ) ∇(β gα)(µ (x) , ε0αβ (x) = ∇(β u0α) (x) .

(5.47)

We should add to these equations the integral equation for the stresses 0 2 0 σαβ (x) = σαβ (x) + ρ1 ω Cαβλµ ∇µ gλρ (x − x ) uρ (x ) dx +

V 1 Sαβλµ (x − x ) Bλµρτ σρτ (x ) dx ,

(5.48)

V 0 0 0 Sαβλµ (x) = Cαβρτ Kρτ δν (x) Cδνλµ − Cαβλµ δ (x) . 0 σαβ

(x) =

0 Cαβλµ ε0λµ

(x) , B = C

−1

1

(5.49) 0

, B =B−B .

(5.50)

Equations (5.44), (5.46), and (5.48) are in fact the equations for the ﬁelds uα (x), εαβ (x), and σαβ (x) inside the inclusion. The ﬁelds outside V may be reconstructed from (5.44) and (5.46) if uα (x) and εαβ (x) inside the inclusion are known.

5.3 Diﬀraction of long elastic waves by an isolated inclusion In what follows in this chapter, we suppose that the length of the incident wave is longer than the characteristic linear sizes of the inclusion. In this case, the so-called long-wave approximation solution of (5.44), (5.46), and (5.48)

124

5. Diﬀraction of long elastic waves

may be constructed. In order to obtain this approximation we keep only the ﬁrst three terms in the series (5.32), (5.35) for the Green function, and take the latter in the form (1)

(3)

s (x) − iωgαβ + iω 3 r2 gαβ (n) . gαβ (x) = gαβ

(5.51)

By the construction of the long-wave approximation, ω should be considered as a small parameter, and the solutions of (5.44), (5.46), and (5.48) are (1) to be found in the form of series similar to (5.51). Note that gαβ in (5.51) is a constant tensor. Therefore, by diﬀerentiation of the tensor gαβ (x) and construction of the kernels of the integral operators in (5.44), (5.46), and (5.48), the term linear with respect to ω disappears, and the main imaginary parts of the tensors ∇g, K, and S are deﬁned by the last term in expansion (5.51). Equation (5.51) gives the long-wave approximation of the kernel K (x) in (5.47) in the form K ω = iω 3 H ,

K (x) = K s (x) + K ω , K sαβλµ (x)

=

s −∇λ) ∇(β gα)(µ 3 2

Hαβλµ =

ρ0 16π 2

(x) ,

−5

2 ξ(β Λα)(λ (ξ) ξµ) dΩξ ,

(5.52) (5.53) (5.54)

Ω1

where H is the constant tensor. The kernel S(x) in the long-wave region is deﬁned by the equation S(x) = S s (x) + iω 3 S ω ,

S ω = C 0 HC 0 .

(5.55)

Note that the static components K s (x) and S s (x) of the functions K(x) and S(x) are homogeneous generalized functions of the degree (−3). Regularizations of these functions were deﬁned in Section 2.2. In the long-wave region, the solutions of (5.44), (5.46), and (5.48) are to be found in the forms 3 ω uα (x) = uR α (x) + iω uα (x) ,

(5.56)

3 ω εαβ (x) = εR αβ (x) + iω εαβ (x) ,

(5.57)

R ω (x) + iω 3 σαβ (x) . σαβ (x) = σαβ

(5.58)

Substituting these equations into (5.44) and (5.46) and equating the terms of the same order with respect to ω leads to the following system R 0 1 εR (5.59) uα (x) = uα (x) + ∇λ g s αµ (x − x ) Cλµρτ ρτ (x ) dx , (1)

uω α (x) = −ρ1 gαβ

V

uR β (x) dx, V

(5.60)

5.3 Diﬀraction of long elastic waves by an isolated inclusion 0 εR αβ (x) = εαβ (x) −

1 K sαβλµ (x−x ) Cλµρτ εR ρτ (x ) dx ,

V

εω αβ

(x) =

1 Hαβλµ Cλµρτ

εR ρτ

(x) dx −

V

125

(5.61)

1 K sαβλµ (x − x ) Cλµρτ εω ρτ (x) dx .

V

(5.62) For the stress tensor we have R 0 1 R (x) = σαβ (x) + S sαβλµ (x−x ) Bλµρτ σρτ (x ) dx , σαβ ω σαβ

V R Sω αβρτ σρτ

(x) = V

(x) dx +

(5.63)

1 S sαβλµ (x − x ) Bλµρτ εω ρτ (x) dx .

V

(5.64) Because the incident ﬁelds u0α and ε0αβ are plane harmonic waves, the altering of these ﬁelds inside the inclusion may be neglected in the long-wave region, and u0α and ε0αβ may be considered as constant vector and tensors. Let the region V be an ellipsoid with surface described by the equation xα dαβ xβ = 1 , dαβ =

1 δαβ (no summation with α!), a2α

(5.65)

where aα (α = 1, 2, 3) are the ellipsoid semi-axes. In this case, the system (5.59)–(5.64) has an analytical solution. If ε0λµ (x) = const inside V , tensor εR αβ in (5.61) is also constant and has the form (see Section 3.3 of Volume 1) 0 0 0 0 1 −1 εR , αβ = Λαβλµ (a) ελµ , Λ (a) = I + A (a) C

(5.66)

where tensor A0 (a) is deﬁned in (3.82) of Volume I. Substituting (5.66) into the right-hand side of (5.62) we obtain ω 0 ω 0 1 0 εω αβ = Λαβλµ (a) ελµ , Λ (a) = vΛ (a) HC Λ (a) ,

(5.67)

where v = 43 a1 a2 a3 is the volume of the ellipsoid. Note that only average characteristics of the ﬁelds over the volume of the inclusions present in the dispersion equations of the self-consistent methods considered in this book. By averaging equation (5.59) for the displacement vector uR α (x) over the volume of the ellipsoid, the integral term in this equation disappears s dx ∇λ gαµ (x − x ) dx = 0. (5.68) V

V

126

5. Diﬀraction of long elastic waves

This is a consequence of the facts that the internal integral in this equation is a linear function of x inside V, and the ellipsoidal region V is symmetric with respect to the origin. That is why, in what follows we neglect the term 1 R R ∇ λ g αµ (x − x ) Cλµρτ ερτ dx in (5.59) for u (x) and accept the equation V 0 uR α (x) = uα (x).

(5.69)

From (5.59) and (5.60), we ﬁnd for the imaginary part uω α of the displacement vector in (5.56) (1)

0 uω α = −vρ1 gαβ uβ .

(5.70)

Finally, the long-wave approximations for the displacement and strain ﬁelds inside the ellipsoidal inclusion take the forms uα (x) = λαβ (ω, a) u0β (x) , εαβ (x) = Λαβλµ (ω, a) ε0λµ , (1)

λαβ = δαβ − iω 3 vρ1 gαβ ,

Λ = Λ0 + iω 3 Λω .

(5.71) (5.72)

Substituting these equation for ε(x) and u(x) into the right-hand side of (5.44) and (5.46) leads to the integral representation of the wave ﬁelds outside the inclusion. Thus, (5.71), (5.72) are the principal terms of the solution of the problem of diﬀraction of long elastic waves on an isolated ellipsoidal inclusion in a homogeneous medium. The tensor A0 in (5.66) depends on the form and orientation of the ellipsoidal inclusion. If the medium is isotropic, the components of A0 in the basis of the main axes of the ellipsoid are deﬁned in (3.85), (3.86) of Volume I. Meanwhile the tensors g (1) and H in (5.51), (5.54) are independent on the ellipsoid parameters, and for the isotropic matrix have the forms (1)

2 + η03 , 12πρ0 t30 1 2 2 1 = H1 Eαβλµ + H2 Eαβλµ − Eαβλµ , 3

gαβ = g1 δαβ , g1 =

(5.73)

Hαβλµ

(5.74)

H1 =

η05 3 + 2η05 , H2 = . 5 36πρ0 t0 60πρ0 t50

If the inclusion is a sphere, the tensor A0 is isotropic: 1 2 2 1 , + A02 Eαβλµ − Eαβλµ A0αβλµ = A01 Eαβλµ 3 (3 − 4η02 ) (3 + 2η02 ) A01 = , A02 = , 27K0 15µ0

(5.75)

(5.76) (5.77)

Here K0 is the bulk elastic modulus of the matrix. In this case, the tensors λαβ and Λαβλµ in (5.71) are also isotropic and take the forms

5.3 Diﬀraction of long elastic waves by an isolated inclusion

λ = 1 − iω 3 vρ1 g1 = 1 − i(β0 a)3 1 2 2 1 , = Λ1 Eαβλµ + Λ2 Eαβλµ − Eαβλµ 3

λαβ = λδαβ , Λαβλµ

ρ1 (2 + η03 ) , 9ρ0

127

(5.78) (5.79)

Λ1 = Λ01 + i(α0 a)3 Λω Λ2 = Λ02 + i(β0 a)3 Λω (5.80) 1, 2, −1 K1 1 K1 0 2 4 Λ1 , M0 = K0 + µ0 , (5.81) 1+ , Λω Λ01 = 1 = 3 M0 M0 3 −1 2µ1 2 µ1 0 2 3 + 2η02 Λ 3 + 2η05 , Λω . Λ02 = 1 + 2 = 15µ0 45 µ0 2 (5.82) Let an isotropic medium contain a layered spherical inclusion. Thus, the elastic moduli and densities of the inclusion are piecewise constant functions of the distance r from the inclusion center. In this case, the long-wave solution of (5.46) has the form (1) R 0 ω (5.83) uα (x) = uα (x) , uα (x) = −gαβ ρ1 (x) u0β (x) dx , 0 εR αβ (x) = εαβ (x) −

V εω αβ

(x) = −Hαβλµ −

V S 1 Kαβλµ (x − x ) Cλµρτ (x ) εR ρτ (x ) dx ,

(5.84)

1 Cλµρτ (x) εR ρτ (x) dx

V S 1 Kαβλµ (x − x ) Cλµρτ (x ) εω ρτ (x ) dx ,

(5.85)

V (1)

¯1 gαβ u0β (x) , ρ¯1 = uω α (x) = v ρ

N i=1

ρ1i

vi , v

(5.86)

where v is the volume of the inclusion, vi is the volume of the ith layer, ρ1i = ρi − ρ0 , ρi is the density of the ith layer. Equation (5.84) for a constant tensor ε0αβ (x) was solved in Section 3.9 of Volume 1. Because the tensor H is constant, (5.85) has the following solution: 1 R (x) = A (r, n) H Cρτ (5.87) εω αβλµ λµρτ αβ δν (x) εδν (x) dx , V

where r = |x|, n = x/r and the origin of the coordinate system is at the inclusion center. The tensor A(r, n) is deﬁned in Section 3.8 of Volume 1. The equation for this tensor contains a set of constants, and the algorithm for computing these constants is presented in Section 3.9 of Volume 1. Substituting εR (x) = A(r, n)ε0 into integral (5.87), we obtain the tensor ω εαβ (x) in the form

128

5. Diﬀraction of long elastic waves

εω αβ (x) = A(r, n)HPv , Pv =

C 1 (x)A(x)dx .

(5.88)

v

Thus, in the long-wave region, the strain ﬁeld inside the multi-layered spherical inclusion is connected with the incident ﬁeld by (5.71), where Λ = Λ(ω, x) = Λ0 (x) + iω 3 Λω (x) , Λ0 (x) = A (r, n) , (1)

Λω (x) = A (r, n) HPυ , λαβ (ω) = δαβ − iω 3 v ρ¯1 gαβ .

(5.89) (5.90)

This long-wave approximate solution of the diﬀraction problem for an ellipsoidal inclusion includes the solutions for its limiting forms (oblate and prolate spheroids). The cases of thin inclusions and long ﬁbers that are not limiting forms of an ellipsoidal inclusion are considered in the next sections.

5.4 Diﬀraction of long elastic waves by a thin inclusion In this section, we consider an inclusion V that occupies a thin region with one of the characteristic sizes much smaller than two other ones. Let Ω be the middle surface of the inclusion, n(x) be the normal to Ω, and h(x) = δ1 l(x) be the transverse size of V along the normal. Here δ1 is a small dimensionless parameter (δ1 << 1), and l(x) has the order of the maximal size of the inclusion. We suppose that h(x) is a suﬃciently smooth function everywhere, except maybe in a small vicinity of the border contour Γ of Ω (see Fig. 5.1). The asymptotic method that was developed in Chapter 4 of Volume 1 for the solution of static problems for thin inclusions is used in this Section for the construction of the long-wave approximate solutions of the corresponding diﬀraction problems.

n(x)

h(x) Ω

Fig. 5.1. A thin inclusion in a homogeneous medium.

5.4 Diﬀraction of long elastic waves by a thin inclusion

129

5.4.1 Thin soft inclusion We start with the inclusion that is much softer than the matrix. In this case CC0−1 = O(δ2 ),

(5.91)

and δ2 is another small parameter of the problem. Suppose that the density of the material of the inclusion does not appreciable exceed the density of the matrix. The asymptotic analysis of the strain ﬁeld inside a thin soft inclusion is similar to analysis the statics. Using the results of Chapter 4 (Volume 1), it is possible to show that the strain ﬁeld inside the inclusion is evaluated as follows εαβ (x) =

1 n(α (x)bβ) (x) + O(δ1 , δ2 ), x ∈ Ω, h(x)

(5.92)

where b(x) is the jump of the displacement vector by intersection of the region of the inclusion along the normal n(x). As in (5.56)–(5.58), the vector b(x) is to be found in the form b(x) = bs (x) + iω 3 bω (x).

(5.93)

The equation for the real part bR (x) of this vector follows from the integral equation (5.63) for stresses in the form (see Section 4.5 of Volume I) s 0 (x, x )bsβ (x ) dΩ = nβ (x) σαβ (x), (5.94) λαβ (x) bsβ (x) + Tαβ Ω

T s (x, x ) = −n (x) S s (x − x ) n (x ) , 1 λαβ (x) = nα (x)Cλαβµ nµ (x). h(x)

(5.95) (5.96)

The imaginary part of b is the solution of an equation that follows from (5.64): s ω ω (x) + T (x, x ) b (x ) dΩ = Tαβ (x, x ) bsβ (x ) dΩ , λαβ (x) bω β αβ β Ω

Ω

(5.97) ω

0

0

T (x, x ) = n (x) C HC n (x ) .

(5.98)

For a thin ellipsoidal inclusion, the middle surface Ω of the inclusion is an ellipse with semi-axes a1 , a2 , and the function h(x) takes the form h (x) = 2hz (x) , 2 2 12 x1 x2 h h z (x) = 1 − − , , << 1 . a1 a2 a1 a2

(5.99) (5.100)

130

5. Diﬀraction of long elastic waves

If the incident stress ﬁeld σ 0 (x) is constant on Ω, (5.94) has the exact solution (see Section 4.6 of Volume I) 2a2 a2 b (x) = b z (x) , b = 1 Bnσ 0 , B = 2 a2 2a1 T 0 = T 0 (x) [z (x) − 1]dx , x ¯ = (x1 , x2 ) . R

0

0

nCn +T 0 h

−1 ,

(5.101) (5.102)

Here the function z(x) is equal to zero outside Ω, and the integral is calculated over the whole (x1 x2 ) plane. For a plane surface Ω, the normal vector n and the tensor T ω do not depend on x, and we ﬁnd from (5.97) that bω =

2a21 4 vBT ω Bnσ 0 , v = πa31 (a1 ≥ a2 ) . a2 3

(5.103)

Equations (5.101)–(5.103) present the long-wave approximation of the ﬁelds inside a thin soft ellipsoidal inclusion. According to (5.44) and (5.96), the displacement u(x) outside a thin crack-like inclusion coincides with the incident ﬁeld, and the strain ﬁeld is presented in the form 0 εαβ (x) = εαβ (x) + Kαβλµ (x − x ) Λλµρτ (a1 , a2 ) Z (x ) ε0ρτ (x ) dx , Ω

2a2 Z (x) = 1 z (x) , a2 Λ (a1 , a2 ) = Λ0 (a1 , a2 ) + iω 3 Λω (a1 , a2 ) , 0

0

0

Λ (a1 , a2 ) = C nBnC , ω

0

(5.104) (5.105) (5.106) (5.107)

0

Λ (a1 , a2 ) = vΛ (a1 , a2 ) HΛ (a1 , a2 ) .

(5.108)

If the matrix is isotropic, the tensor H is deﬁned in (5.74). The constant tensor T 0 was calculated in Section 4.6 of Volume I and has the form 0 Tαβ = T10 e1α e1β + T20 e2α e2β + T30 nα nβ ,

(5.109)

where e1α and e2α are the unit vectors of the principal axes of the ellipse Ω. The coeﬃcients Ti0 (i = 1, 2, 3) are presented in (3.233)–(3.237) of Volume I. If the material of the inclusion is also isotropic, we have Bαβ = B1 e1α e1β + B2 e2α e2β + B3 nα nβ , −1 −1 a2 µ a2 µ + T10 + T20 , B2 = 2 , 2 2a1 h 2a1 h −1 a2 λ + 2µ 0 + T3 . B3 = 2 2a1 h B1 =

(5.110) (5.111) (5.112)

5.4 Diﬀraction of long elastic waves by a thin inclusion

131

For a thin soft spheroidal inclusion (a1 = a2 = a), (5.109), (5.110) are simpliﬁed: 0 = T10 θαβ + T30 nα nβ , θαβ = δαβ − nα nβ , Tαβ

πµ0 πµ0 3 − 2η02 , T30 = 1 − η02 , 8a 2a Bαβ = B1 θαβ + B3 nα nβ , −1 1 2µ a π 3 − 2η02 · + , B1 = B2 = 2µ0 µ0 h 4 T10 =

−1 1 2 (λ + 2µ) a 2 · + π 1 − η0 . B3 = µ0 µ0 h

(5.113) (5.114) (5.115) (5.116)

(5.117)

In the P -basis (see Appendix A.2), the tensors Λ0 and Λω in (5.106) are presented in the forms

2 µ0 Λ0 = µ0 B1 P 5 + 4 B3 1 − 2η02 P 2 + 1 − 2η02 P 3 + P 4 + P 6 , η0 (5.118) 2 µ0 h1 (η0 ) 2 5 h0 (η0 ) 2 B1 P + Λω = B3 (1 − 2η02 )2 P 2 πρ0 t50 8 η04 0 (5.119) + 1 − 2η02 P 3 + P 4 + P 6 , 2 4 1 2 3 8 1 − η05 , h0 (η0 ) = + η0 − 2η03 + η05 . (5.120) h1 (η0 ) = 5 3 3 5 4 5 5.4.2 Thin hard inclusion Let us consider a thin inclusion with material that is much harder than the material of the matrix, i.e., C −1 C 0 = O(δ2 ), δ2 << 1, and δ2 /δ 1 = O(1). Here δ1 is a small parameter introduced in the previous section that reﬂects geometrical properties of the inclusion. Asymptotic analysis similar that in statics (see Chapter 4, Volume 1) shows that the stress tensor inside the inclusion is deﬁned mainly by the components tangential to Ω σαβ (x) =

1 qαβ (x) + O(δ1 , δ2 ) , Θαβλµ (x)qλµ (x) = qαβ (x) , h(x)

Θ(x) = E 1 − 2E 5 (n) + E 6 (n) = P 1 (n).

(5.121) (5.122)

Here Θ(x) is the projector on the surface Ω. The ﬁeld q(x) is to be found in the form q(x) = q s (x) + iω 3 q ω ,

(5.123)

132

5. Diﬀraction of long elastic waves

and the real part q s (x) of this tensor is the solution of an integral equation that follows from (5.62): s s s (x) + Uαβλµ (x, x ) qλµ (x ) dΩ = Θε0 αβ (x) , (5.124) µαβλµ (x) qλµ Ω

ω µαβλµ (x) qλµ

(x) +

s ω ω Uαβλµ (x, x ) qλµ (x ) dΩ = Uαβλµ

Ω

s qλµ dΩ, Ω

(5.125) −1

µαβλµ (x) = h (x) Θαβρτ (x) Bρτ δν Θδνλµ (x) , U s (x, x ) = Θ(x)K s (x − x ) Θ(x ) , U ω = ΘHΘ.

(5.126) (5.127)

Here the integral with the kernel Uαβλµ (x, x ) in (5.124), (5.125) is understood in the sense of regularization (4.27) of Volume I. If C → ∞ (B → 0), (5.124), (5.125) are transformed into the equations for a inextensible membrane embedded in the matrix medium. Let us ﬁnd the solution of (5.124) for a thin hard ellipsoidal inclusion. If ε0 (x) is constant on Ω, (5.124) has an exact solution (see Section 4.6 of Volume 1): R 0 (x) = qαβ z (x) , qαβ

0 qαβ =

2a21 Gαβλµ Θε0 λµ , a2

−1 −1 a2 1 0 ΘC Θ + U , 2a21 h 0 U = pv U s (x) [z (x) − 1]dx .

G=

(5.128)

(5.129) (5.130)

Ω

z(x) = 1 − (x1 /a1 )2 − (x1 /a1 )2 if x ∈ Ω, z(x) = 0 if x ∈ / Ω.

(5.131)

From (5.125) we ﬁnd q ω (x) = vGU ω Gz (x) Θε0 (x) .

(5.132)

The strain ﬁeld outside a thin hard inclusion takes the forms 0 εαβ (x) = εαβ (x) + Kαβλµ (x − x ) Λλµρτ (a1 , a2 ) Z (x ) ε0ρτ (x ) dx . Ω

(5.133)

5.5 Diﬀraction of long elastic waves by a short axisymmetric ﬁber

133

Here the tensor Λ is Λ (a1 , a2 ) = − Λ0 (a1 , a2 ) − iω 3 Λω (a1 , a2 ) ,

(5.134)

Λ0 = Θ (n) GΘ (n) , Λω = vΛ0 HΛ0 .

(5.135) 0

For an isotropic medium, the tensor U is deﬁned in Section 4.6 (equations (4.125)–(4.128)) of Volume I. For an oblate spheroid (a1 = a2 = a), the tensor U 0 in the P -basis has the form π 1 2 3 + η02 P 1 (n) + η0 − 1 P 2 (n) . (5.136) U0 = 32µ0 a 2 If the material of the inclusion is also isotropic with the shear modulus µ0 and Poisson ratio v0 , the tensor G in the P -basis is 1 2 2 1 (5.137) G = G1 P + G2 P − P , 2 −1 π µ0 aµ0 1 − ν · + 1 + η02 , 2 2hµ 1 + ν 8 −1 π aµ0 + (3 + η02 ) . G2 = µ0 2hµ 16 G1 =

(5.138) (5.139)

In this case, the tensor Λ0 coincides with the tensor G, and the tensor Λω is deﬁned by the equation 1 2 2 ω 1 P P , (5.140) P + Λ − Λω = Λ ω 1 2 2 Λω 1 =

vµ20 1 + 4η05 G21 , 5 30πρ0 t

(5.141)

Λω 2 =

vµ20 3 + 2η05 G22 . 5 60πρ0 t

(5.142)

Here v is the volume of the inclusion.

5.5 Diﬀraction of long elastic waves by a short axisymmetric ﬁber In this section, we consider an isotropic medium containing an inclusion in the form of a prolate body of rotation with the axis Γ , radius a(z) (z ∈ Γ ), and the length 2l. Here z is the coordinate of the Cartesian coordinate system (x1 , x2 , z) with the origin at the center of the inclusion, and the zaxis directed along the axis Γ (see Fig. 5.2). If the parameter δ1 = a/l is

134

5. Diﬀraction of long elastic waves z

m x2 u0

x1

Fig. 5.2. Diﬀraction on a ﬁbre.

small, the inclusion is a ﬁber, and the material of this ﬁber is supposed to be much harder than that of the matrix, i.e., C −1 C 0 = O(δ2 ), where δ2 << 1 is a small parameter. It is assumed that the densities of the inclusion and the matrix have the same order. In this case, the solutions of integral equations (5.44), (5.46) are to be found in the form (5.56), (5.57) uα (x) = usα (x) + iω 3 uω α (x) ,

(5.143)

εαβ (x) = εsαβ (x) + iω 3 εω αβ (x) .

(5.144)

From (5.59) and (5.60) we obtain that the principal terms of the displacement ﬁeld uα (x) have the forms (1) usα (x) = u0α (x) , uω (x) = −ρ g u0β (x) dx . (5.145) 1 α αβ V ω For the construction of the strains εR αβ (x) and εαβ (x), we neglect the variation of these ﬁelds over the cross-section of the ﬁber (εR (x) = εR (z), εω (x) = εω (z)). After integration over the cross-sections s(z) of the ﬁber in (5.61) and (5.62), we obtain one-dimensional integral equations for εR (z) and εω (z):

1 εsαβ

(ξ) +

1 K0αβλµ (ξ, ξ ) Cλµρτ εsρτ (ξ ) dξ = ε0αβ (ξ) ,

−1

1 εω αβ

(ξ) + −1

1 K0αβλµ (ξ, ξ ) Cλµρτ εω ρτ (ξ ) dξ

(5.146)

5.5 Diﬀraction of long elastic waves by a short axisymmetric ﬁber

= −πl

3

1 Hαβλµ Cλµρτ

1

δ12 (ξ) εsρτ (ξ) dξ,

135

(5.147)

−1

z a(ξ) , ξ= . δ1 (ξ) = l l

(5.148)

Here the kernel K0αβλµ (ξ, ξ ) is K0αβλµ (ξ, ξ ) =

Ksαβλµ [(ξ − ξ ) m − ξ ]dSξ ,

(5.149)

S(ξ )

and mα is the unit vector along the z-axis. As in Section 5.2 of Volume 1, the kernel K0αβλµ (ξ, ξ ) is presented in the form K0αβλµ

1 (ξ, ξ ) = 2π

∞

e−ik3 (ξ−ξ ) K0αβλµ (k3 , ξ ) dk3 .

(5.150)

−∞

Expanding the symbol K0 (k3 , ξ) of the operator K0 over the small parameter δ1 and keeping only the principal terms of this expansion, we obtain the following equation for the tensors εR (ξ) and εω (ξ): d2 P0−1 εR (ξ) − δ12 ln δ1 A1 C 1 2 α2 (ξ)εR (ξ) = ε0 (ξ) , dξ d2 P0−1 εω (ξ) − δ12 ln δ1 A1 C 1 2 α2 (ξ)εω (ξ) dξ 1 = −πl3 δ12 HC 1 α2 (ξ)εR (ξ) dξ.

(5.151)

(5.152)

−1

Here the tensors P0 and A1 have the forms −1 P 0 = I + A0 C 1 , 1 1 1 − η02 P 2 + 2P 5 , 1 + η02 P 1 − A0 = 4µ0 2 A1 =

(5.153) (5.154)

1 2(1 + 3η02 )P 1 − 3(1 − η02 )P 2 + 8 1 − η02 P 3 + P 4 32µ0 (5.155) + 8 3 − 2η02 P 5 − 16P 6 .

If we introduce the new functions τR (ξ) and τI (ξ) τR (ξ) = C 1 εR (ξ) , τI (ξ) = C 1 εω (ξ) .

(5.156)

136

5. Diﬀraction of long elastic waves

equations (5.152) and (5.153) take the forms 1 0 d , τR (ξ) = C 1 P 0 ε0 (ξ) + δ12 ln δ1 A1 D2 α2 (ξ)τR (ξ) , D = dξ

τI (ξ) = C 1 P 0

(5.157)

δ12 ln δ1 A1 D2 α2 (ξ)τI (ξ)

− πl3 δ12 H

1

α2 (ξ)τR (ξ) dξ .

(5.158)

−1

In the long-wave approximation, the incident ﬁeld ε0αβ (ξ) can be considered as a constant, and we can write τR (ξ) = Λ0 (ξ) ε0 , τI (ξ) = Λω (ξ) ε0 , 2µ0 ν 3 4 6 Λ0 (ξ) = Λ0θ + λR (ξ) P + P , + P η02 E µ0 1 − η02 2 4µ0 0 1 P + 8µ0 P 6 , Λθ = P + 2 1 + η02 η0 (1 + η02 )

(5.159) (5.160) (5.161)

Λω = −λI (ξ) P 6 .

(5.162)

Here the functions λR (ξ) and λI (ξ) satisfy the following system of diﬀerential equations that follows from (5.157), (5.158): d2 2 α (ξ)λR (ξ) − q 2 λR (ξ) = −q 2 E , dξ 2

(5.163)

l3 2 + 3η05 d2 2 2 α (ξ)λ (ξ) − q λ (ξ) = · I I dξ 2 ρ0 t50 30 ln δ1

1

α2 (ξ)λR (ξ) dξ,

(5.164)

−1

q 2 = −2

E0 (1 + ν) . Eδ12 ln δ1

(5.165)

Here E and ν are the longitudinal Young modulus and Poisson’s ratio of the ﬁber, E0 is the Young’s modulus of the matrix. (ω) For long incident waves, the imaginary part uα (x) of the displacement ﬁeld in (5.145) takes the form 1 3 2 (1) 0 (x) = ρ l δ g u (x) α2 (ξ)dξ. (5.166) u(ω) 1 α 1 αβ β −1

(ω)

Thus uα (x) has the order of δ12 , and may be neglected. Hence, the strain tensor (or functions τR (ξ) and τI (ξ)) is the only unknown of the problem.

5.5 Diﬀraction of long elastic waves by a short axisymmetric ﬁber

137

Equation (5.163) was studied in Section 5.2 of Volume 1, where the solution of this equation was obtained for ﬁbers in the form of a cylinder, a prolate spheroid, and a double-cone. Since the right-hand side of equation (5.164) is constant, this equation is identical to equation (5.163). Therefore, from the results of Section 5.2 of Volume 1 the forms of the functions λR (ξ) and λI (ξ) for the ﬁbers are as follows: Cylindrical ﬁber (α = 1). cosh qξ E, λR (ξ) = 1− cosh q λI (ξ) =

tanh q l3 δ12 E 5 λR (ξ) . 2 + 3η 1− · 0 t30 30µ0 q

(5.167)

(5.168)

Note that these equations hold if |(δ12 ln δ1 )E/E 0 | << 1 (q → ∞). Except for δ1 -vicinities of the ﬁber ends (5.167) leads to the results λR (ξ) = E , λI (ξ) = o (1) . If the ﬁber is absolutely rigid (q → 0), (5.167), (5.168) give µ0 1 − ξ 2 l3 2 + 3η05 λR (ξ) = , λI = 3 · λR (ξ) . 2 δ1 ln δ1 t0 45 ln δ1−1

(5.169)

(5.170)

Ellipsoidal ﬁber.In this case δ1 = a/l, where a and l are the ellipsoid semi-axes, f (ξ) = 1 − ξ 2 . A bounded solution of (5.164) for the ellipsoidal ﬁber turns out to be constant and is determined by the equations 2 2 + 3η05 q2 E l3 , λI = 3 · . (5.171) λR = 2 + q2 t0 45 (2 + q 2 ) ln δ1−1 A ﬁber in the form of a double-cone. If the inclusion has the form of a double-cone, then δ1 = a/l, where a is the radius of the middle cross-section, f (ξ) = 1 − |ξ|. For a ﬁber of such a shape, we obtain the solution of (5.164): 2 1 q2 E β 1− 1 − |ξ| −3 + 1 + 4q 2 . , β= λR (ξ) = 2 q −2 2+β 2 (5.172) λI (ξ) =

l3 2 + 3η05 β (5 + β) λR (ξ) , · · t30 45 (q 2 − 2) ln δ1 (2 + β) (3 + β)

(5.173)

Thus the principal term of the function τ (ξ) is presented in the form τ (ξ) = Λ (ξ) ε0 (ξ) , Λ (ξ) = Λ0 (ξ) + iω 3 Λω (ξ) ,

(5.174)

where the scalar functions λR (ξ) and λI (ξ) for the ﬁbers of various forms of are deﬁned in (5.167), (5.170) and (5.172).

138

5. Diﬀraction of long elastic waves

Let us calculate the principal terms of the asymptotes of the displacement and strain ﬁelds outside the inclusion. Because ∇g(x) and K(x) in the integral equations (5.44), (5.46) are smooth bounded functions, the wave ﬁelds far from the ﬁber axis are presented in the forms (5.175) uα (x) = u0α (x) + ∇µ gαλ (x − mz ) s (z ) τλµ (z ) dz , Γ

εαβ (x) =

ε0αβ

(x) +

Kαβλµ (x − mz ) s (z ) τλµ (z ) dz ,

(5.176)

Γ

K (x) = Ks (x) + iω 3 H , s (z) = πa2 (z) ,

(5.177)

where τ (x) is the slowly varying part of the principal asymptotic term of the ﬁeld C 1 ε(z) inside the inclusion that has the form (5.174). It was shown in Section 5.2 of Volume 1 that for a cylindrical ﬁber and a ﬁber in the form of a double-cone these equations describe only the slowly changing parts of the ﬁelds inside the ﬁbers. In the vicinities of the ﬁber ends or the ribs on the ﬁber surfaces, boundary layer functions should be added to these solutions. These additions are important for the detailed analysis of the elastic ﬁeld distributions inside the inclusions. For the solution of the homogenization problems, only integral characteristics of the elastic ﬁelds inside the inclusions are essential. The boundary layer functions give a small contribution to these characteristics and can be neglected.

5.6 Total scattering cross-sections of inclusions In the process of diﬀraction of the incident plane waves by an isolated inclusion, the elastic energy is scattered in all directions (Fig. 5.3). An important integral characteristic of the scattered energy is the total scattering

Fig. 5.3. Scattering of energy by an isolated inclusion.

5.6 Total scattering cross-sections of inclusions

139

cross-section of the inclusion. This parameter deﬁnes the attenuation of the waves propagating in the medium with many inclusions. For instance, for a small volume concentration of inclusions, the attenuation coeﬃcient γ of the mean wave ﬁeld in the composite is connected with the total scattering cross-section Q by the following equation γ=

1 n0 Q, 2

(5.178)

where n0 is the numerical concentration of inclusions (the number of the inclusion in the unit volume). In this section, we calculate the long-wave asymptotics of the total scattering cross-sections of the inclusions of various forms by diﬀraction of longitudinal and transverse waves. 5.6.1 An isolated inclusion In this section, we develop a general equation for the total scattering crosssection of an arbitrary inclusion in an isotropic medium. To this end we use the basic integral equation (5.41) of the diﬀraction problem 2 ω gαλ ( x − x ) ρ1 ( x ) uλ ( x ) uα (x) = u0α (x) + V

1 +∇µ gαλ (x − x ) Cλµρτ (x ) ερτ (x ) dx .

(5.179)

Here the density and elastic moduli of the inclusion may be functions of coordinates (inhomogeneous inclusion). The dynamic Green function gαβ (x) for the isotropic medium may be presented in the form −iα0 |x| −iβ0 |x| ∂2 e−iβ0 |x| e 1 2e − − , (5.180) δαβ β0 gαβ (x) = 4πρ0 ω 2 |x| ∂xα ∂xβ |x| |x| ω ω α0 = , β0 = . (5.181) l0 t0 The integral terms in (5.179) is the ﬁeld usα (x) scattered on the inclusion 2 ω gαλ (x − x ) ρ1 (x ) uλ (x ) usα (x) = V

1 (x ) ερτ (x ) dx . +∇µ gαλ (x − x ) Cλµρτ

(5.182)

For large |x| and x ∈ V , the following equations hold |x − x |

−1

∇γ1 ∇γ2 ...∇γm

,

−1

, |x − x | ∼ |x| − (n · x ) , n = x/ |x| , (5.183) −iq|x| e−iq|x−x | m e ∼ (−iq) nγ1 nγ2 ...nγm eiqn·x . (5.184) |x − x | |x|

∼ |x|

140

5. Diﬀraction of long elastic waves

As the result, the scattered ﬁeld usα (x) far from the inclusion may be represented in the form usα ≈ Aα (n)

e−iα0 |x| e−iβ0 |x| + Bα (n) . |x| |x|

(5.185)

Here Aα (n) is the vector amplitude of the longitudinal wave and Bα (n) is the corresponding amplitude of the shear wave scattered in the direction n. These amplitudes are expressed via the displacement and strain ﬁelds inside the inclusion: Aα (n) = nα nβ fβ (α0 n) , Bα (n) = (δαβ − nα nβ ) fβ (β0 n) , q2 2 fα (qn) = [ω uα (x ) ρ1 (x ) exp (iqn · x ) dx 4πρ0 ω 2

(5.186)

V

− iqnβ

1 ελµ (x ) Cαβλµ (x ) exp (iqn · x ) dx ] , (q = α0 , β0 )

V

(5.187) 0 (x) and The stress ﬁeld is also represented as the sum of the incident σαβ s s 0 s scattered σαβ (x) stress ﬁelds. For σαβ (x) = Cαβλµ ∇λ uµ (x), we ﬁnd from (5.185) that

s σαβ ∼ −iλ0 α0

e−iα0 |x| Aτ nτ δαβ |x|

e−iα|x| e−iβ0 |x| A(α0 nβ) + β0 B(α nβ) . − 2iµ0 α0 |x| |x|

(5.188)

Let S be an arbitrary closed surface containing the inclusion (see Fig. 5.3). The elastic energy radiation Q through this surface is deﬁned by the equation 1 ∗ σαβ + σαβ u˙ β + u˙ ∗β mα dS . (5.189) Q=− 4 S

Here the superscript ∗ denotes complex conjugation, mα are the components of the external normal to the surface S. Since the displacements and the stresses are time-periodic, i.e. uα (x, t) = uα (x)eiωt , σαβ (x, t) = σαβ (x)eiωt , we can write 1 ∗ ∗ σαβ uβ e−2iωt + σαβ u∗β e2iωt + σαβ u∗β − σαβ uβ mα dS . Q = − iω 4 S (5.190)

5.6 Total scattering cross-sections of inclusions

141

Averaging this equation over the time period T of oscillations, we obtain 1 Qt = T

T 0

1 Q (t) dt = ωIm σαβ u∗β mα dS . 2

(5.191)

S

Here ·t denotes the time average. Taking into account that uα = u0α + 0 s , σαβ = σαβ + σαβ we present the mean Qt as the sum of three terms: the energy of the incident ﬁeld radiation Q0 t , the energy of the scattered s ﬁeld radiation Q t ,and the energy of the interference of the incident and int : scattered ﬁelds Q t 0 Qt = Q t + Qs t + Qint t , (5.192) 0 1 0 Q t = ω Im σαβ u0∗ (5.193) β mα dS , 2

usα

S

Qs t =

1 ω Im 2

s σαβ us∗ β mα dS , S

int 1 Q = ω Im t 2

0 s 0∗ σαβ us∗ β + σαβ uβ mα dS .

S

Because of the energy conservation law Qt = (5.192) Qs t = − Qint t .

(5.194)

(5.195) 0 Q t , we have from (5.196)

The total scattering cross-section Q(ω) is the ratio of the average radiation energy of the scattered ﬁeld Qs t and the average energy I 0 t radiating from the unit surface orthogonal to the direction of the incident wave propagation, i.e., Q (ω) =

Qs t . I 0 t

(5.197)

0 If we introduce 0 the unit vector n of the direction of the incident wave propagation, I t takes the form

0 0 0∗ 0 1 I t = ωIm σαβ uβ nα . 2

(5.198)

In what follows, the incident ﬁeld is a plane wave with the wave number q (5.199) u0α = eα exp −iqn0 · x , 0 = −iq λ0 eρ n0ρ δαβ + 2µ0 e(α n0β) exp −iqn0 · x . (5.200) σαβ

142

5. Diﬀraction of long elastic waves

Here eα is the polarization vector. For a longitudinal wave we have q = α0 , eα = n0α , and for a transverse wave q = β0 and e·n0 = 0. Substituting (5.199) into (5.198), we ﬁnd

2 0 1 I t = − ωq (λ0 + µ0 ) e · n0 + µ0 . 2

(5.201)

Equation (5.196) shows that there are two options for the calculation of Q (ω): through the mean scattered ﬁeld Qs t and through the mean inter int . Substituting (5.185) and (5.188) into (5.194) for Qs t ference ﬁeld Q t we obtain 3 1 12 2 2 (λ0 + 2µ0 ) α0 |Aα | + µ0 β0 |Bα | dS . (5.202) Qs t = ω 2 2 x S

Another deﬁnition of Qs t is based on the “optical theorem” for the elastic wave diﬀraction problems. Let us take the surface S in (5.195) as a sphere of a large radius R. Using (5.185), (5.188), (5.199), and (5.200) we obtain that on S the following equations hold: i 0 0 uS∗ nα σαβ β = − [F1 (n) exp (iα0 r) + F2 (n) exp (iβ0 r)] exp −iqrn · n , r (5.203) i S 0 u0∗ nα σαβ β = − [Φ1 (n) exp (−iα0 r) + Φ2 (n) exp (−iβ0 r)] exp iqrn · n , r (5.204) ∗ 0 ∗ 0 (5.205) F1 (n) = q λ0 (A · n) e · n + 2µ0 (A · e) n · n , F2 (n) = qµ0 B ∗ · n0 (e · n) + (B ∗ · e) n · n0 , (5.206) Φ1 (n) = α0 (λ0 + 2µ0 ) (A · e) , Φ2 (n) = β0 µ0 (B · e) .

(5.207)

Let us substitute these equations into the following integral (mα = nα ) 0 S∗ S σαβ uβ + σαβ J (ω) = u0∗ (5.208) β nα dS, S

presented in equation (5.192) for Qint t , and use the saddle point method for evaluation of this integral for large R (see, e.g., [23]). For an arbitrary function f (n), the integral over the sphere S of a large radius R may be evaluated as follows (see [3]) 2πi 0 1 f −n exp (−iqr) − f n0 exp (iqr) . f (n) exp iqrn0 ·n dS ∼ r q S

(5.209)

5.6 Total scattering cross-sections of inclusions

143

Using this equation for the calculation of the integral J (ω) in (5.208) we obtain J (ω) =

2π 1 q (λ0 + 2µ0 ) exp (iα0 r) e · A∗ n0 exp (−iqr) q + e · A∗ −n0 exp(iqr) + qµ0 exp (iβ0 r) e · B ∗ n0 exp (−iqr) + e · B ∗ −n0 exp (iqr) − α0 (λ0 + 2µ0 ) exp (−iα0 r) × e · A n0 exp (iqr) − e · A −n0 exp (−iqr) − β0 µ0 exp (−iβ0 r) e · B n0 exp (iqr) − e · B −n0 exp (−iqr) , (q = α0 , β0 ) . (5.210)

Let the incident ﬁeld be a longitudinal wave, i.e., q = α0 , e = n0 . Since n · B(n0 ) = 0, (5.210) is transformed into the following equation: 1 J (ω) = 4π (λ0 + 2µ0 ) −i Im n0 · A n0 (5.211) + Re n0 · A −n0 exp (−2iα0 r) , 0

and the total scattering cross-section QL (ω) takes the form 4π Im n0 · A n0 . QL (ω) = −Im [J (ω)] / I 0 t = − α0 a

(5.212)

If the incident wave is a transverse wave (T -wave), q = β0 , e · n0 = 0. In this case, Aα (±n0 )eα = eα n0α nββ fβ (±αn0 ) = 0 and (5.210) gives 1 0 J (ω) = 4πµ0 −i Im e·B n0 + Re e·B −n0 exp (2iβ0 r) .

(5.213)

As the result, we obtain the total scattering cross-section of an inclusion by diﬀraction of the T -wave QT (ω) = −

4π Im e·B n0 . β0 a

(5.214)

In order to calculate QL,T (ω) it is necessary to know the displacement and strain ﬁelds inside the inclusion (see equations (5.186), (5.187) for A and B). If these ﬁelds are found approximately, the obtained equations allow us to ﬁnd approximate values for the total scattering cross-sections. For the construction of the long-wave approximations considered in the previous sections, we have to accept in the equations for the QL,T (ω) exp(iqn0 ·x) ≈ 1 , exp(−iqn · x) ≈ 1 (x ∈ V ) .

(5.215)

Let us use these equations to calculate the total scattering cross-section for inclusions of various shapes.

144

5. Diﬀraction of long elastic waves

5.6.2 Long-range scattering cross-sections Spherical inclusion. Let (5.202) be used for the calculation of the total scattering cross-section Q(ω) for long waves; the static values of the ﬁelds inside the inclusion are taken for the calculation of the integrals Aα and Bα in (5.186) (the “quasi-static” approximation) uα (x) ≈ u0α (x) , εαβ (x) ≈ Aαβλµ (x)ε0λµ (x) ,

(5.216)

The tensor A(x) in this equation is found for a spherical inclusion in Section 3.8. For the longitudinal wave, we have u0α = n0α , ε0αβ = −iαn0α n0β ,

(5.217)

and the corresponding function fα (qn) in (5.187) is: α02 v 2 ω nα ρ¯1 − α02 Pαβλµ nβ nλ nµ , 4πρ0 ω 2 1 1 = Cαβρτ (x) Aρτ λµ (x) dx v V 1 2 2 1 = Kp Eαβλµ + 2µp Eαβλµ − Eαβλµ , 3

fα (αn) = Pαβλµ

Kp = K0 (π1 − q1 ) , µp = µ0 (π2 − q2 ) .

(5.218)

(5.219) (5.220)

In these equations πi , qi (i = 1, 2) and ρ¯1 are deﬁned in (7.43)–(7.46) of Volume I and (5.87). From (5.185) we ﬁnd the amplitudes Aα and Bα in the form , 3Kp + 2µp 3 cos2 θ − 1 α2 v ρ¯1 nα , cos θ − (5.221) Aα = 4π ρ0 3 (λ0 + 2µ0 ) Bα =

β2v 4π

ρ¯1 µp − 2η0 cos θ n0α − nα cos θ , ρ0 µ0

(5.222)

where θ is the angle between the wave normal n0α of the incident wave and an arbitrary direction nα of the scattered wave propagation. For the total scattering cross-section in (5.202), we obtain 4 4πa2 8 2 1 2 1 4 2 (α0 a) µ K QL (ω) = + + p 2 9 15 p η05 3 (ρ0 l02 ) 2 5 2 1 ρ¯1 +1 . (5.223) + 3 ρ0 η03

5.6 Total scattering cross-sections of inclusions

145

For a homogeneous inclusion, the coeﬃcients Kp and µp are deﬁned by the equations −1 1 3 + , (5.224) Kp = K1 3K0 + 4µ0 −1 1 6 (K0 + 2µ0 ) + . (5.225) µp = µ1 5µ0 (3K0 + 4µ0 ) For an absolutely rigid sphere, we have Kp =

1 5µ0 (3K + 4µ0 ) (3K0 + 4µ0 ) , µp = , 3 6 (K0 + 2µ0 )

(5.226)

and for a spherical cavity Kp = −

K0 (3k0 + 4µ0 ) 5µ0 (3K0 + 4µ0 ) , µp = − . 4µ0 9K0 + 8µ0

(5.227)

In order to consider diﬀraction of shear waves, we introduce the Cartesian coordinate system (x1 , x2 , x3 ) and the spherical coordinate system (r, θ, ϕ) with a common origin at the inclusion center. The x3 -axis of the Cartesian system is the polar axis of the spherical system (see Fig. 5.4). Let the wave vector of the incident ﬁeld be directed along the x3 -axis, and the polarization vector along the x1 -axis. In this system, the amplitudes of the incident displacement and strain ﬁelds are u0α = e1a , ε0αβ = −iβ0 e1(α e3β) ,

(5.228)

where e1α and e3a are the unit vectors of axes x1 and x3 , respectively. In this notation we have

q2 v 1 ρ¯1 ω 2 eα −qβµp e1α cos θ + e3α sin θ cos θ . (5.229) fα qn0 = 2 4πρ0 ω x3 eγ θ

eϕ

eθ

ϕ

x1 n e1

Fig. 5.4. Diﬀraction by a spherical inclusion.

x2

146

5. Diﬀraction of long elastic waves

The amplitudes of the scattered waves are calculated from (5.186) in the form α02 v ρ¯1 sin θ − η0 µp sin 2θ nα cos ϕ , (5.230) Aα = 4π ρ0 ρ¯1 β02 v ρ¯1 ϕ θ − + µp cos θ eα sin ϕ + Bα = cos θ − µp cos 2θ eα cos ϕ . 4π ρ0 ρ0 (5.231) Here eϕ and eθ are unit vectors of the spherical coordinate system, er = n. For shear waves, equation (5.199) for the total scattering cross-section takes the form 4 1 ρ¯1 2 2µ2p 4π (β0 a) 2 5 3 a 2 + η0 . QT (ω) = 2 3 + 2η0 + 3 9 ρ0 15 (ρ0 t20 ) (5.232) Let us ﬁnd the cross-sections QL (ω) and QT (ω) from the “optical theorem”. In the long-wave approximation we have

q 2 ωv 2 2 2 (1) 0 ω ρ¯1 g eα + k 0 qnβ (P HP )αβλµ e0λ e0µ , (5.233) Im fα qn0 = − 4πρ0 P = C 1 Λ0 .

(5.234)

In this equation, k0 is the wave number of the incident wave, n0 is the wave normal, e0 is the unit polarization vector, g (1) and H are deﬁned in (5.73) and (5.74). The imaginary part Im Aα (αn0 ) of the vector amplitude of the longitudinal wave (e0α = n0α , k0 = α0 ) is α2 ωv 2 2 2 (1) 8 Im Aα αn0 = − 0 ω ρ¯1 g + α02 9kp2 H1 + µ2p H2 n0α . (5.235) 4πρ0 3 Substitution of this equation into (5.212) leads to equation (5.223) for the total scattering cross-section of a spherical inclusion for longitudinal waves. Similarly, for the vector amplitude of the transverse scattered wave we obtain β 2 ωv 2 2 2 (1) ω ρ¯1 g + 2β 2 µ2p H2 e0α , (5.236) Im Bα β0 n0 = − 0 4πρ0 and using (5.214) we ﬁnd the same equation (5.232) for the total scattering cross-section of a spherical inclusion for T -waves. Note that the comparison of these two approaches for calculation of the total scattering cross-sections shows that the optical theorem requires careful calculation of the ﬁelds inside the inclusion. For instance, the quasi-static approximation (5.216) leads to correct results if equation (5.202) is used, but (5.212) and (5.214) give the values of QL and QT equal to zero in this case (the imaginary part of the “static” ﬁelds is zero).

5.6 Total scattering cross-sections of inclusions

147

Ellipsoidal inclusion. For ellipsoidal inhomogeneity, the long-wave approximation for Im fα (qn) has the same form (5.233), where the tensor P is −1 P = vC 1 E 1 +A (a) C 1 ,

(5.237)

and the tensor A(a) is deﬁned in (3.85), (3.86) of Volume 1. Hence, in accordance with (5.212) and (5.214), we have to know the explicit form of the tensor P to ﬁnd the the total scattering cross-section of the inclusion. In what follows, we consider only an ellipsoid of rotation (spheroid) with semi-axes a1 = a2 = a, a3 (the x3 -axis coincides with the axis of rotation). Let the inclusion be an oblate spheroid (a a3 ). In this case, all the integrals in equation (5.74) for the tensor A(a) may be calculated, and this tensor in the P -basis takes the form: 1 A = A1 P 2 + A2 P 1 − P 2 + A3 P 3 + P 4 + A5 P 5 + A6 P 6 , (5.238) 2 1 2 η0 f0 (γ) + f1 (γ) , 2µ0 1 1 + η02 f0 (γ) + f1 (γ) , A2 = 2µ0

A1 =

f1 (γ) 1 , A5 = [1 − f0 (γ) − 4f1 (γ)] , µ0 µ0 1 2 a η0 (1 − 2f0 (γ)) + 2f1 (γ) , γ = , A6 = µ0 a3

A3 = −

(5.239) (5.240) (5.241) (5.242)

where functions f0 (γ) and f1 (γ) are presented in (3.96) of Volume I. Calculating the tensor in the right-hand side of (5.237), we ﬁnd 1 P = v kp P 2 + 2mp P1 − P2 +lp P 3 + P 4 + 4µp P 5 + np P 6 , 2 (5.243) λ1 + µ1 1 + A6 , (5.244) kp = 2∆ µ1 (3λ1 + 2µ1 ) λ1 1 − A3 , (5.245) lp = ∆ 2µ1 (3λ1 + 2µ1 ) −1 1 λ1 + 2µ1 1 1 + 2A1 , (5.246) + A2 , np = mp = 2 2µ1 ∆ 2µ1 (3λ1 + 2µ1 ) −1 1 1 [1 + (λ1 + 2µ1 ) A6 + A5 , ∆= µp = µ1 2µ1 (3λ1 + 2µ1 ) + 4 (λ1 + µ1 ) A1 + 4λ1 A3 + 2A1 A6 ] − 2A23 .

(5.247)

148

5. Diﬀraction of long elastic waves

These equations can be used for the calculation of the total scattering cross-sections of a very thin spheroidal cavity (crack) or a hard circular disk. When γ → ∞, f0 ≈ π/(4γ), f1 ≈ (1 − η02 )π/(8γ), and the principal terms of the coeﬃcients Ai in (5.238) are π(1 + η02 ) π(3 + η02 ) π(1 − η02 ) , A2 = , A3 = − , 16µ0 γ 16µ0 γ 8µ0 γ π 1 1 2 π 2 2 3 − 2η0 , A6 = 1 − η0 . 1− η + A5 = µ0 4γ µ0 0 4γ A1 =

(5.248)

(5.249)

Let the inhomogeneity be a penny-shaped crack (C 1 = −C 0 ). From (5.243) we ﬁnd the principal terms of the coeﬃcients in (5.243) 2 kp = 1 − 2η02 np , mp = 0 , lp = 1 − 2η02 np ,

(5.250)

4µ0 a3 16a3 µ0 , µ , = − p 3η04 (1 − η02 ) 3 (3 − 2η02 )

(5.251)

np = −

and the product P HP in (5.233) takes the form

2 P HP = R1 1 − 2η02 P 2 + 1 − 2η02 P 3 + P 4 + P 6 + 2R2 P 5 , (5.252) n2p η04 h0 (η0 ) , R1 = πρ0 t50

µ2p 3 + 2η05 R2 = . 15πρ0 t50

(5.253)

Here the function h0 (η0 ) is deﬁned in (5.119). Let us consider a longitudinal wave (e0α = n0α , k = α), and let θ be the angle between the wave normal of the incident wave and the axis of the spheroid (Fig. 5.5). From (5.233), (5.186) and (5.213), in the long-wave limit (a3 → 0) we obtain x3

θ

n

ej

x2

j e e0

x1

Fig. 5.5. The system of coordinates for an oblate spheroid.

5.6 Total scattering cross-sections of inclusions

QL =

16 4 (α0 a) a2 9π

1 − 2η02 sin2 θ 2

η05 (1 − η02 )

h0 (η0 ) +

2 sin2 2θ η0 (3 − 2η02 )

149

2 h1

(η0 ) , (5.254)

where the functions h0 (η0 ) and h1 (η0 ) are deﬁned in (5.119). The total scattering cross-section of the inclusion by diﬀraction of transverse waves depends on the orientation of the polarization vector. This vector is located in the plane that is orthogonal to the wave normal, and can be deﬁned by the polar angle ϕ in this plane (Fig. 5.5). It allows us to represent the polarization vector e0α of the incident transverse wave in the form θ e0α = eϕ α sin ϕ + eα cos ϕ,

(5.255)

θ where eϕ α and eα are the unit vectors of the spherical coordinate system with the origin at the center of the inclusion and the polar axis directed along the normal to the middle plane of the inclusion. In this case, the product P HP takes the form 2 (P HP )αβλµ n0λ n0µ = R2 n0(α mβ) sin θ + e1(α mβ) cos θ − mα mβ sin 2θ

3 + R1 η02 mα mβ + 1 − 2η02 θαβ sin 2θ sin ϕ + R2 e2(α mβ) cos θ cos ϕ.

(5.256)

Substituting this equations into (5.233) and using (5.186) and (5.214), we obtain QT (θ, ϕ) = QT 1 (θ) sin2 ϕ + QT 2 (θ) cos2 ϕ,

(5.257)

where the coeﬃcients QT 1 (θ) and QT 2 (θ) correspond to the total scattering θ cross-sections of the transverse waves with the polarization vectors eϕ α and eα . 16 2 cos2 2θ sin2 2θ 4 2 (β0 a) a QT 1 = (5.258) 2 h0 (η0 ) + 2 h1 (η0 ) , 9π (1 − η02 ) (3 − 2η02 ) QT 2 =

16 2 cos2 θ 4 (β0 a) a2 2 h1 (η0 ) . 9π (3 − 2η02 )

(5.259)

Let the inclusion be an absolutely rigid (C 1 )−1 = 0) spheroidal disk. In this case, we have 1 32a3 µ0 2 1 2 2 1 P P P , (5.260) + − P = vA−1 = 3 2 (1 + η02 ) 3 + η02 2 4 3 + 2η05 1 2 256a6 µ20 1 + 4η05 2 1 P − P . (5.261) P + P HP = 2 135πρ0 t50 2 (1 + η02 )2 2 (3 + η02 )

150

5. Diﬀraction of long elastic waves

If the incident wave is longitudinal, then 256a6 µ20 1 + 4η05 0 0 (P HP )αβλµ nλ nµ = θαβ sin2 θ 135πρ0 t50 2 (1 + η02 )2 4 3 + 2η05 1 2 0 0 0 2 + nα nβ −2m(α nβ) cos θ + mα mβ cos θ − θαβ sin θ , 2 2 (3 + η02 ) (5.262) θαβ = δαβ − mα mβ .

(5.263)

The total scattering cross-section of the inclusion in the case of the L-wave takes the form 5 128 1 + 4η05 4 2 4 3 + 2η0 4 (α0 a) a (5.264) QL = 2 + 2 sin θ. 2 2 135πη0 (3 + η0 ) (1 + η0 ) For a transverse wave, the tensor (P HP )n0 e0 for the rigid disk takes the form 1 + 4η05 256a6 µ20 0 0 − (P HP )αβλµ nλ eµ = 2 θαβ sin 2θ cos ϕ 135πρ0 t50 4 (1 + η02 ) 4 3 + 2η05 0 1 (n(α eβ) − m(α e1β) cos θ − n0(α mβ) sin θ + 2 (3 + η02 )

1 + (2mα mβ − θαβ ) sin 2θ) sin ϕ + n0(α e2β) − m(α e2β) cos θ cos ϕ 4

. (5.265)

This equation leads to the following equations for the coeﬃcients QT 1 and QT 2 in (5.257) 5 128 1 + 4η05 4 2 4 3 + 2η0 2 2 QT 1 = (β0 a) a (5.266) 2 + 2 sin θ cos θ , 135π (3 + η02 ) (1 + η02 ) QT 2

5 128 4 2 4 3 + 2η0 2 (β0 a) a = 2 sin θ . 135π (3 + η02 )

(5.267)

Let consider the inclusion in the form of a prolate spheroid. Suppose that x3 -axis is directed along the rotation axis, and a1 = a2 = a, a3 > a. In this case, tensor A in (5.237) is represented in the form

5.6 Total scattering cross-sections of inclusions

151

1 1 4 (π − ν0 ϕ2 ) − γ 2 ϕ1 P 2 16πµ0 (1 − ν0 ) 2 γ2 1 2 2 1 1 + γ ϕ1 + + (1 − 2ν0 ) ϕ2 P − P + 2π − 4 2 3 4 2 + 2 (ϕ1 − ϕ2 ) P + P + 2 1 − γ ϕ1 + (1 − 2ν0 ) (4π − ϕ2 )]P 5 0 (5.268) +8 (1 − ν0 ) (2π − ϕ2 ) − γ 2 ϕ1 P 6 , 3ϕ2 − 4π 2π 1 , (5.269) , ϕ2 = 1 − γ 2 − γ 2 arch ϕ1 = 1 − γ2 1 − γ2 γ a γ= < 1, (5.270) a3 A=

where ν0 is Poisson coeﬃcient of the matrix. If γ → 0, we go to an inﬁnitely long cylindrical ﬁber. After taking into account only the main terms of the coeﬃcients ϕ1 , ϕ2 : ϕ1 ∼ 2π(1 + 3γ 2 ln γ), ϕ2 ∼ 2π(1 + γ 2 ln γ), (5.268) is transformed into the following equation 1 η02 2 1 + η02 1 P + P1 − P2 A= µ0 4 4 2 2 1 1 − η0 2 + γ ln γ P 3 + P 4 + P 5 − γ 2 ln γ P 6 . (5.271) 2 2 If the ﬁber is absolutely rigid (P = A−1 ), we have 2 4πa63 2 + 3η0 55 µ0 4πa33 µ0 6 P , P HP = P 2, P = 3 ln γ 135ρ0 t50 ln γ 2 4πa63 2 + 3η05 µ0 0 0 mα mβ cos2 θ , (P HP )αβλµ nλ nµ = 135ρ0 t50 ln γ 2 4πa63 2 + 3η05 µ0 0 0 (P HP )αβλµ nλ eµ = mα mβ sin ϕ . 135ρ0 t50 ln γ

(5.272)

(5.273) (5.274)

The total scattering cross-section of the ﬁber by diﬀraction of longitudinal and two transverse waves take the forms QL =

5 4πa23 4 2 + 3η0 4 (α0 a3 ) 2 cos θ , 135η0 (ln γ)

QT 1 =

5 4πa23 4 2 + 3η0 2 2 (β0 a3 ) 2 sin θ cos θ , QT 2 = 0 . 135 (ln γ)

(5.275)

(5.276)

Thin crack-like inclusion. If the material of the oblate spheroid is very soft, we have a crack-like inclusion. However, for a thin cavity ﬁlled with soft material, it is more convenient to use the approach developed in Section 4.3.

152

5. Diﬀraction of long elastic waves

Let us ﬁnd the total scattering cross-section of the crack-like spheroidal defect. The imaginary part of the vector fα (qn) in (5.233) is presented in the form Im fα (qn) = −

q 3 ωv 2 0 0 knβ Λω αβλµ nλ eµ , 4πρ0

(5.277)

4 3 πa . (5.278) 3 Here the tensor B is deﬁned in (5.110), and for a spheroidal cavity has form (5.113). Substitution of equation (5.118) for Λω into (5.277) leads to the following equation for the total scattering cross-sections of the inclusion for the longitudinal and transverse waves: 4 2 1 2 16πa2 (α0 a) h0 (η0 ) 2 2 2 1 − 2η , B1 h1 (η0 ) sin2 2θ + B sin θ QL = 3 0 9 8η0 η05 (5.279) 4 2 16πa (β0 a) 1 2 B h1 (η0 ) cos2 2θ + h0 (η0 ) B22 sin2 2θ , QT 1 = (5.280) 9 8 1 Λω = Λ0 HΛ0 , Λ0 = C 0 nBnC 0 , v =

4

QT 2 =

2πa2 (β0 a) 2 B1 h1 (η0 ) cos2 θ. 9

(5.281)

When the Lam´e parameters λ and µ tend to zero, these equations are transformed into equations (5.254), (5.258) and (5.259) that correspond to the elliptical crack. Thin hard disk. Let us consider a thin hard spheroidal inclusion and use the results of Section 5.4. For such an inclusion, the imaginary part of the vector fα (qn) is: Im fα (qn) = −

q3 ω 0 0 kvΛω αβλµ nβ nλ eµ , 4πρ0

(5.282)

where tensor Λω is deﬁned in (5.140). As the result, we ﬁnd the following equations for the total scattering cross-sections of the inclusion: QL =

2π 1 4 1 + 4η05 G21 + 3 + 2η05 G22 sin4 θ , (α0 a) a2 27 5η0

(5.283)

2π 1 4 (β0 a) a2 3 + 2η05 G22 + 1 + 4η05 G21 sin2 θ cos2 θ , (5.284) 27 5 2π 4 21 (β0 a) a 3 + 2η05 G22 sin2 θ. QT 2 = (5.285) 27 5 If the inclusion is absolutely rigid (µ = ∞) but its density has the same order as the density of the surrounding medium, these equations are transformed into (5.264) and (5.266). QT 1 =

5.7 Notes

153

Short axisymmetric ﬁber. For a hard ﬁber, (5.175) gives the long-wave approximation for the vector fα (qn) (q = α0 , β0 ) in the form 3

i (ql) fα (qn) = − nβ δ12 4ρ0 ω 2

1

α2 (ξ) ταβ (ξ) dξ .

(5.286)

−1

Here δ1 = a/l, and tensor ταβ (ξ) is deﬁned in (5.174) ταβ (ξ) = ikΛαβλµ (ξ) n0λ e0µ .

(5.287)

Let a longitudinal wave propagate in the medium with the ﬁber (e0α = n0α , k = α). Substituting this equation into (5.286) and using the optical theorem (5.212) we ﬁnd QL =

α04 vf2 2 + 3η05 2 · ϕ (q) cos4 θ . 60π η0 µ20

(5.288)

Here vf is the ﬁber volume, ϕ(q) = E(1 − tan(q)/q), ϕ(q) = Eq 2 (2 + q 2 )−1 , Eq 2 β(5+β) , ϕ(q) = (q2 −2)(2+β)(3+β)

for cylinder; for long spheroid; for double cone;

(5.289)

and the parameters q and β are deﬁned in Section 3.4, E is the Young’s modulus along the ﬁber axis. The total scattering cross-section of the ﬁber for transverse waves is QT 1 =

β04 vf2 2 + 3η05 2 · ϕ (q) sin2 θ cos2 θ , QT 2 = 0 . 60π µ20

(5.290)

If the ﬁber is absolutely rigid, (E → ∞, q → 0), (5.288) and (5.290) are transformed into (5.275) that are the same for the ﬁbers of all three forms.

5.7 Notes The solution of the problem of diﬀraction of elastic waves by inclusions of canonical shapes (sphere, ellipsoid) in an inﬁnite isotropic homogeneous medium may be solved by the method of separation of variables in the wave equations and expansion of the solution in series with respect to the eigenfunctions of the problem. This method was used for the solution of the problem of diﬀraction of monochromatic elastic waves by a spherical inclusion in [20, 77, 115] and by an ellipsoidal inclusion in [16, 27]. Another approach to the solution of the one-particle problem for an isolated inclusion is based on the integral equations of the diﬀraction problem.

154

5. Diﬀraction of long elastic waves

The kernels of the operators in these equations are derivatives of the Green function of the background medium. Representation of the Green function of the wave operator for an anisotropic medium in the form of an integral over unit sphere was obtained in [111]. In this Chapter, this representation is obtained by the method proposed in [50]. The analogue of the polynomial conservation theorem for an ellipsoidal inclusion in dynamics was proved in [24]. Systematic study of the solutions of the integral equation of the diﬀraction problem in the long-wave region is contained in [30–32]. In these works, the wave ﬁeld inside the inclusion is replaced by its static limit, and the ﬁeld outside the inclusion is reconstructed from the integral equation of the problem. In the work [50] these results are improved by taking into account the principal terms in the imaginary part of the wave ﬁeld inside the inclusion. The general solution of the problem of diﬀraction of long waves by ellipsoidal inclusion was used for construction of the wave ﬁelds in the vicinity of prolate and oblate spheroids in [19, 33, 111]. In [33] and [84], the solution of the problem of diﬀraction of long elastic waves by a penny-shape crack in an isotropic medium was obtained. The problem of diﬀraction of long elastic wave on a thin inclusions that are not limit forms of an ellipsoid was considered in [51, 52], and the problem of diﬀraction of long elastic waves by a short axisymmetric ﬁber was solved in [53]. Calculation of the total scattering cross-sections of inclusions of various shapes has been considered in many works (see, e.g., [30, 31, 64, 100]). The proof of the optical theorem for elastic waves is presented in [30]. In this chapter, another proof of this theorem proposed in [72] is presented. The equation for the cross-section of the inclusion by diﬀraction of longitudinal waves obtained in [30] and [72] are identical. But the equation for the cross-section by diﬀraction of transverse waves proposed in [30] contains “cross-term” connected with interference of the incident transverse wave and scattered longitudinal wave. As is shown in [72] and in this chapter, this cross-term is absent in the equation for the transverse cross-section QT (ω), and the optical theorem is similar to its classical formulation also in the case of transverse waves. The total scattering cross-section of a spherical inclusion in the long-wave region was found in [20, 100, 115] on the basis of the exact solution of the diﬀraction problem, and in [30] with the help of quasi-static approximation. It was pointed out in [30] that the equations for QT (ω) presented in [20,100,115] contain a mistake. The equations for QL (ω) and QT (ω) presented in this Chapter coincide with the equations in [30], and in [111]. The latter work presents the scattering cross-sections of a circular crack, a hard circular disk, and a hard ellipsoidal ﬁber by diﬀraction of three diﬀerent types of waves.

6. Eﬀective wave operator for a medium with random isolated inclusions

In this chapter, we consider a homogeneous medium containing a random set of isolated inclusions. The eﬀective ﬁeld method is applied to the solution of the homogenization problem for wave propagation. The dispersion equation for the mean wave ﬁeld in the composite is derived using the long-wave solutions of the one-particle problem . We show that this dispersion equation corresponds to a homogeneous medium (eﬀective medium) with attenuation and dispersion. The Green function of the wave operator for the eﬀective medium is constructed and analyzed. The velocities and attenuation coefﬁcients of the long waves propagating in the composites with inclusions of various forms are calculated in the framework of the EFM.

6.1 Diﬀraction of elastic waves by a random set of ellipsoidal inclusions We consider an inﬁnite homogeneous medium with the tensor of elastic property C 0 and density ρ0 , containing a random set of isolated ellipsoidal inclusions with the properties C and ρ. As before, V (x) is the characteristic function of the region V occupied by the inclusions. For harmonic vibrations, the amplitudes of the displacement uα (x) and strain εαβ (x) ﬁelds satisfy equations similar to (5.44) and (5.46): uα (x) = u0α (x) + [ρ1 ω 2 gαβ (x − x ) uβ (x ) 1 ελµ (x )]V (x ) dx , +∇ρ gαβ (x − x ) Cρβλµ

εαβ (x) = ε0αβ (x) +

[ρ1 ω 2 ∇(β gα)ρ (x − x ) uρ (x )

1 ερτ (x )]V (x ) dx . −K αβλµ (x − x ) Cλµρτ 1

0

(6.1)

(6.2)

Here C = C − C , ρ1 = ρ − ρ0 . If vk is the region occupied by the kth inclusion, and vk (x) is characteristic function of this region, the function V is the following sum vk (x). (6.3) V (x) = k

156

6. Eﬀective wave operator

It follows from these equations that the wave ﬁelds inside the inclusion that occupies an arbitrary region v satisfy the equations uα (x) = u∗α (x) + [ρ1 ω 2 gαβ (x − x ) uβ (x ) v 1 +∇ρ gαβ (x − x ) Cρβλµ ελµ (x )]dx , εαβ (x) = ε∗αβ (x) + [ρ1 ω 2 ∇(β gα)ρ (x − x ) uρ (x )

(6.4)

v 1 −K αβλµ (x − x ) Cλµρτ ερτ (x )]dx .

(6.5)

The ﬁelds u∗α (x) and ε∗αβ (x) in these equations have the forms ∗ 0 uα (x) = uα (x) + [ρ1 ω 2 gαβ (x − x ) uβ (x ) 1 ελµ (x )]V (x; x ) dx , x ∈ V , +∇ρ gαβ (x − x ) Cρβλµ ρ1 ω 2 ∇(β gα)ρ (x − x ) uρ (x ) ε∗αβ (x) = ε0αβ (x) + 1 ερτ (x ) V (x; x ) dx , x ∈ V . −K αβλµ (x − x ) Cλµρτ

(6.6)

(6.7)

and may be interpreted as the local exciting ﬁelds acting on the inclusion v. The function V (x; x ) in these equations is the characteristic function (with argument x ) of the region Vx Vx = vi if x ∈ vj . (6.8) i=j

Thus, the function V (x; x ) is equal to zero if the points x and x are inside the same inclusion. The local exciting ﬁelds in (6.6), (6.7) are the sums of the incident ﬁelds u0 , ε0 and the ﬁelds scattered by all the inclusions except the one that occupies region v if x ∈ v. In this chapter, we consider propagation of waves that are longer than the characteristic sizes of the inclusion. That is why the ﬁelds u∗ (x) and ε∗ (x) are assumed to be constant in the region of each inclusion. If the inclusions are ellipsoids, the displacement and strain ﬁelds inside the inclusions are expressed via the local external ﬁelds u∗ (x) and ε∗ (x) by equations (5.71). These equations give the solution of the diﬀraction problem for an isolated inclusion (the one particle problem) in the long-wave region uα (x) = λαβ (x) u∗β (x) , εαβ (x) = Λαβλµ (x) ε∗λµ (x) .

(6.9)

Here λαβ (x) and Λαβλµ (x) coincide with the tensors λαβ (ak ) and Λαβλµ (ak ) when x ∈ vk in (5.72). Substituting these equations into the right-hand side of (6.1)–(6.7), we ﬁnd the wave ﬁelds in the medium with inclusions in the forms

6.1 Scattering of elastic waves on a set of ellipsoidal inclusions

uα (x) = u0α (x) +

157

ρ1 ω 2 gαβ (x − x ) λβµ (x ) u∗µ (x )

1 (x ) Λλµρτ (x ) ε∗ρτ (x ) V (x ) dx , (6.10) +∇ν gαβ (x − x ) Cνβλµ 0 ρ1 ω 2 ∇(β gα)λ (x − x ) λλµ (x ) u∗µ (x ) εαβ (x) = εαβ (x) − 1 (x ) Λρτ δν (x ) ε∗δν (x ) V (x ) dx , (6.11) −K αβλµ (x − x ) Cλµρτ The local exciting ﬁelds u∗α (x) and ε∗αβ (x) inside the region occupied by the inclusions satisfy the following system of integral equations: ρ1 ω 2 gαβ (x − x ) λβµ (x ) u∗µ (x ) u∗α (x) = u0α (x)+ 1 (x ) Λλµρτ (x ) ε∗ρτ (x )V (x; x ) dx , (6.12) +∇ν gαβ (x−x ) Cνβλµ

ρ1 ω 2 ∇(β gα)λ (x − x ) λλµ (x ) u∗µ (x ) 1 (x ) Λρτ δν (x ) ε∗δν (x ) V (x; x ) dx . −K αβλµ (x − x ) Cλµρτ (6.13)

ε∗αβ (x) = ε0αβ (x)+

If the solutions of these equations are found, and the ﬁelds u∗ (x) and ε (x) are determined as functions of the incident ﬁelds u0 (x) and ε0 (x), we can substitute them into (6.10) and (6.11) and construct the detailed wave ﬁelds in the composite. Let us average (6.10) and (6.11) over the ensemble realizations of the random set of inclusions. As the result, the equations for the mean displacement u (x) and strain ε (x) ﬁelds take the forms 0 ∗µ (x ) dx uα (x) = uα (x) + p ω 2 gαβ (x − x )ρλβµ u Λ +p ∇ν gαβ (x − x )Cνβλµ ε∗λµ (x ) dx , (6.14) ∗µ (x ) dx εαβ (x) = ε0αβ (x) + p ω 2 ∇(β gα)λ (x − x )ρλλµ u Λ −p K αβλµ (x − x )Cλµρτ ε∗ρτ (x ) dx , (6.15) ∗

u ∗ (x ) = u∗ (x) |x , ε∗ (x ) = ε∗ (x) |x .

(6.16)

Here the symbol ·|x means averaging under the condition x ∈ V . We suppose statistical independence of the random functions u∗ (x) and ε∗ (x) on the property and size of the inclusion subjected with these ﬁelds. In these equations, C Λ and ρλ are constant tensors CΛ =

n0 1 1 1 C Λ (x) V (x) = vC Λ (x) , p p

(6.17)

158

6. Eﬀective wave operator

ρλ =

1 n0 ρ1 λ (x) V (x) = vρ1 λ (x) , p p

(6.18)

n0 is the numerical concentration of the inclusions, and p = V (x) is their volume concentration. Averaging in the right-hand side of (6.17) (6.18) is carried out over the ensemble realizations of the random sizes, orientations, and properties of the ellipsoidal inclusions. The equations (6.14) and (6.15) imply that the conditional means u ∗ (x) ∗ and ε (x) deﬁne the mean displacement and strain ﬁelds in the composite. To ﬁnd the ﬁelds u ∗ (x) and ε∗ (x), let us average both sides of equations (6.12) and (6.13) under the condition x ∈ V . As the result, we ﬁnd the following equations: 2 ω gαβ (x − x ) ρλβλ u∗λ (x ) |x , x u ∗α (x) = u0α (x) + p ∗ Λ ελµ (x ) |x , x Ψ (x − x ) dx , (6.19) +∇ν gαβ (x − x ) Cνβλµ 2 ω ∇(β gα)λ (x − x ) ρλλµ u∗µ (x ) |x , x ε∗αβ (x) = ε0αβ + p ∗ Λ ερτ (x ) |x , x Ψ (x − x ) dx , (6.20) −K αβλµ (x − x ) Cλµρτ Ψ (x − x ) =

V (x; x ) |x . V (x)

(6.21)

In order to obtain a closed system for the conditional means u ∗ (x) and ε (x), we use the quasicrystalline approximation (see Section 2.2). According to this approximation we assume that

∗ (x ) , (6.22) u∗ (x ) |x , x = u∗ (x ) |x = u ∗

ε∗αβ (x ) |x , x = ε∗αβ (x ) x = ε∗αβ (x ) .

(6.23)

Excluding the incident ﬁelds u0 (x) and ε0 (x) from (6.14), (6.19), (6.15), and (6.20) we obtain the equations for the eﬀective ﬁelds in the forms 2 ω gαβ (x − x ) ρλβλ u ∗λ (x ) u ∗α (x) = uα (x) − p Λ ε∗λµ (x ) Φ (x − x ) dx , (6.24) +∇ν gαβ (x − x ) Cνβλµ

2 ω ∇(α gβ)λ (x − x ) ρλλµ u ∗µ (x ) Λ ε∗ρτ (x ) Φ (x − x ) dx , −K αβλµ (x − x ) Cλµρτ

ε∗αβ (x) = εαβ (x) − p

Φ (x) = 1 − Ψ (x) .

(6.25) (6.26)

Equations (6.24) and (6.25) are convolution equations. Thus, applying the Fourier transform to (6.24), (6.25) we ﬁnd the system of linear algebraic

6.1 Scattering of elastic waves on a set of ellipsoidal inclusions

159

equations for the Fourier transforms of the eﬀective ﬁelds (for the Fourier transforms of functions we keep the same notation with argument k). u ∗α (k) = uα (k) − ptαβ (k) u ∗β (k) − pTαλµ (k) ε∗λµ (k) ,

(6.27)

ε∗αβ (k) = εαβ (k) − pπαβλ (k) u ∗λ (k) − pΠαβλµ (k) ε∗λµ (k) .

(6.28)

Here the tensors tαβ (k) , Tαλµ (k) , παβλ (k) , Παβλµ (k) are the following integrals Λ ∇ρ gαβ (x) Φ (x) eik·x dx , (6.29) Tαλµ (k) = Cρβλµ tαβ (k) = ω 2 ρλλβ

gαλ (x) Φ (x) eik·x dx ,

(6.30)

Παβλµ (k) =

Λ Cρτ λµ

παβλ (k) = ω 2 ρλµλ

K αβρτ (x) Φ (x) eik·x dx ,

(6.31)

∇(α gβ)µ (x) Φ (x) eik·x dx .

(6.32)

For a statistically isotropic set of inclusions, Φ(x) in these equations is a continuous function that disappears outside a region, the linear size l of which has the order of the correlation radius of a random set of inclusions. Suppose that in the long-wave region the ﬁelds u(x), ε(x) , u ∗ (x), and ε∗ (x) change slowly, and their Fourier transforms disappear outside the region of the k-space deﬁned by the condition |k|l << 1. Therefore, for calculation of the integrals (6.29)–(6.32) in this region, we can approximate the function eik·x by the ﬁrst terms of its Taylor series 1 eik·x ≈ 1 + ikα xα − kα kβ xα xβ . 2

(6.33)

Let us calculate the integrals in (6.29)–(6.32) taking into account (6.33), spherical symmetry of the function Φ(x) = Φ(|x|), and the principal terms of the expansion of the function g(x) (5.51) with respect to ω. After that we can solve the system (6.27), (6.28) and obtain the functions u ∗α (k) and ε∗ (k). Keeping only the principal terms with respect to ω in the real and imaginary parts of u ∗α (k) and ε∗ (k) we obtain the equations u ∗α (k) = dαβ (ω) uβ (k) ,

(6.34)

ε∗αβ (k) = Dαβλµ (k, ω) (−ikλ ) uµ (k) .

(6.35)

Here the tensors dαβ (ω) and Dαβλµ (k, ω) have the forms (1) dαβ (ω) = δαβ + iω 3 pρ1 gαβ J , J = Φ (x) dx ,

(6.36)

160

6. Eﬀective wave operator

1 D (k, ω) = D0 I − iω 3 n0 A0 v 2 C 1 Λ0 HC R − JHC R 0 −l2 A1 · (ik ⊗ ik) C R , −1 D0 = I − n0 A0 vC 1 Λ0 , R 1 0 0 R C (a) = C Λ (a) D , C = n0 vC 1 Λ0 (a) D0 , A1αβλµρτ =

1 2

(6.37) (6.38) (6.39)

K sαβλµ (n) nρ nτ dΩn ,

(6.40)

Ω1

l2 =

∞

r Φ (r) dr , n = 0

k . |k|

(6.41)

The tensors g (1) , H, A0 , Λ0 are deﬁned in (5.73)–(5.81). Applying the Fourier transform to (6.14) Λ ∗ρ (k)+p (−ikρ ) gαβ (k) Cβρλµ ε∗λµ (k) . uα (k) = u0α (k) + pω 2 gαβ (k) ρλβρ u (6.42)

and using equations (6.34), (6.35) to give u ∗α (k) and ε∗αβ (k), we obtain the equation for the Fourier transform of the mean wave ﬁeld in the composite: uα (k) = u0α (k) + pω 2 gαβ (k) ρλβµ dµρ uρ (k) Λ Dλµντ (−ikν ) uτ (k) . +p (−ikρ ) gαβ (k) Cβρλµ

(6.43)

Multiplying both sides of this equality with the tensor L0αβ (k, ω) 0 kµ − ρ0 ω 2 δαβ , L0αβ (k, ω) = kλ Cαλβµ

(6.44)

and taking into account the equations L0αβ (k, ω) u0β (k) = 0 , L0αλ (k, ω) gλβ (k, ω) = δαβ ,

(6.45)

we transform (6.43) into the following equation for the Fourier transform of the mean wave ﬁeld uα (k) L∗αβ (k, ω) uβ (k) = 0 , L∗αβ

(k, ω) =

∗ kλ Cαλβµ

(k, ω) kµ −

ω 2 ρ∗αβ

(k, ω) .

Here the tensors C ∗ (k, ω) and ρ∗ (k, ω) have the forms C ∗ (k, ω) = C s − l2 C R A1 · (ik) ⊗ (ik) C R + iω 3 n0 v 2 C R HC R − JC R HC R , s

0

R

C = C + pC , ρ∗αβ = ρs δαβ +

(1) iω 3 pvρ1 f gαβ

(6.46) (6.47)

(6.48) (6.49)

, ρs = ρ0 + pρ1 ,

(6.50)

6.1 Scattering of elastic waves on a set of ellipsoidal inclusions

f=

161

1−

p J . v

(6.51)

It follows from (6.46) that the x-representation of the mean wave ﬁeld u(x) satisﬁes the equation L∗ u (x) = 0.

(6.52)

Here the action of the operator L∗ on the function u(x) is deﬁned by the equation 1 L∗ (k) u (k) e−ik·x dk . (6.53) L∗ u (x) = 3 (2π) The function L∗ (k) in this equation has the form (6.47), and is the symbol of the operator L∗ . The operator L∗ is in fact the wave operator of the homogeneous eﬀective medium that is equivalent to the given composite material. It follows from (6.53) and (6.47) that in the x-space, the operator L∗ is represented in the form ∗ ∇µ − ρ∗αβ ω 2 . L∗αβ = −∇λ Cαλβµ

(6.54)

This is a canonical form of a wave operator for some homogeneous elastic medium. Note that in this equation, C ∗ is not a constant tensor but an operator that depends on the frequency ω of the propagating wave. The density tensor ρ∗αβ in (6.54) is also an operator of the same type. As it is seen from (6.48), C ∗ is the sum of the the operator of multiplication with the constant tensor C s − iω 3 C ω and the diﬀerential operator of the second order (6.55) C ∗ = C s + iω 3 C ω + l2 C R A1 · (∇ ⊗ ∇) C R , C ω = n0 v 2 C R HC R − JC R HC R .

(6.56)

In the static case (ω = 0), the symbol of the operator C ∗ coincides with that obtained in Section 8.4 (see equation (8.108)) of Volume I. As was mentioned in this Section, such an operator corresponds to the moment theory of elasticity for the medium with constrained rotation [68]. The correlation radius l of the random set of inclusions plays the role of a small parameter with dimensions of length that appears in the equations of any moment theory of elasticity. Thus, the eﬀective medium that is described by the operator L∗ in (6.54), (6.56) is a medium with week spatial dispersion. The velocity of the wave propagation in such a medium is deﬁned by the real parts of the tensors C ∗ and ρ∗ . Existence of the imaginary components of these operators leads to attenuation of propagating waves due to scattering by the inhomogeneities. In the next section, these eﬀects are considered in more details for composite materials with isotropic components.

162

6. Eﬀective wave operator

6.2 The Green function of the eﬀective wave operator Let us consider a composite material with isotropic components, and suppose for simplicity that all the inclusions are spheres of the same radius and properties. Therefore, only the spatial distribution of inclusions is random. In this (1) case, the tensors gαβ and Hαβλµ have the forms (5.73) and (5.74), and the tensor A0αβλµ is deﬁned in (5.76). The tensor [A1 · (ik ⊗ ik)] takes the form 1 1 k 2 5 3 + 4η02 Eαβλµ − Eαβλµ (n) A · (ik ⊗ ik) αβλµ = 105µ0 3 4 2 , n = k/ |k| . (6.57) (n) − 3Eαβλµ (n) − 2Eαβλµ + 1 − η02 3Eαβλµ Substitution of these equations in (6.54) and (6.56) transforms L∗ into the wave operator for an isotropic medium. The function L∗ (k, ω) in (6.47) takes the form 2 ∗ 2 K∗ − µ∗ nα nβ + 2µ∗ δαβ − ρ∗ ω 2 δαβ . (6.58) Lαβ (k, ω) = k 3 K∗ = Ks + p2 k 2 l2 Kl + iω 3 pvf Kω , Ks = K0 + pKR , 2 2 2

3

µ∗ = µs + p k l µl + iω pvf µω , µs = µ0 + pµR , −1 −1 1 1 0 0 + 3 (1 − p) A1 , µR = + 2 (1 − p) A2 , KR = K1 µ1 4µR 1 µ2R 2 3 + 4η02 , 14KR + µR 3 + 4η0 , µl = Kl = 105µ0 3 105µ0 3 − 4η02 3 + 2η02 2 , A02 = , Kω = 9kR H1 , µω = 2µ2R H2 , 9k0 15µ0 ρ2 2 + η03 . ρ∗ = ρs − iω 3 pvf ρω , ρω = 1 12πρ0 t30

A01 =

(6.59) (6.60) (6.61) (6.62) (6.63) (6.64)

The projection operators associated with the unit vector n παβ (k) = nα nβ , θαβ (k) = δαβ − nα nβ , n = k/|k| ,

(6.65)

T split the vector uα (k) into longitudinal uL α and transverse uα components. ∗ The operator Lαβ in (6.58) may be also presented in the form of the sum of two orthogonal components proportional to the projectors παβ (k) and θαβ (k)

L∗αβ (k, ω) = L∗L (k, ω) παβ (k) + L∗T (k, ω) θαβ (k) ,

(6.66)

4 L∗L = k 2 K∗ + µ∗ − ρ∗ ω 2 , L∗T = k 2 µ∗ − ω 2 ρ∗ . 3

(6.67)

6.2 The Green function of the eﬀective wave operator

163

In these equations, the scalar coeﬃcients L∗L (k, ω) and L∗T (k, ω) determine propagation of longitudinal and transverse waves through the eﬀective medium. Let us go to the construction of the Green tensor of the operator (6.58). The decomposition (6.66) corresponds to the representation of the Fourier ∗ (k, ω) of the operator L∗ in the form of transform of the Green function gαβ two orthogonal components ∗ ∗L ∗T (k, ω) = gαβ (k, ω) + gαβ (k, ω) , gαβ −1

∗L gαβ (k, ω) = [L∗L (k, ω)]

−1

∗T gαβ (k, ω) = [L∗T (k, ω)]

(6.68)

παβ (k) ,

(6.69)

θαβ (k) .

(6.70)

The longitudinal component of this tensor ∗L gαβ (k, ω) =

L∗L

kα kβ 1 · 2 . (k, ω) k

(6.71)

in the x-representation has the form 1 exp (−ik · x) dk ∗L . gαβ (x, ω) = 3 ∇α ∇β 2 (−k 2 M∗ + ω 2 ρ∗ ) k (2π) 2

M∗ = Ms + (kpl) Ml − iω 3 pf Mω ,

(6.72) (6.73)

4 4 4 Ms = Ks + µs , Ml = Kl + µl , Mω = Kω + µω , (6.74) 3 3 3 where the coeﬃcients Ks , Kl , Kω , µs , µl and µω are deﬁned in (6.59)–(6.64). For calculation of the integral in (6.72), we introduce the spherical coordinate system (r, θ, ϕ) in the k-space with the polar axis directed along the vector x. In this coordinate system, the integral in (6.72) takes the form ∗L (x, ω) = gαβ

1

3 ∇α ∇β (2π)

∞ 0

dk −k 2 M ∗ +ω 2 ρ∗

exp [−ikr (n · e)]dΩn ,

Ω1

(6.75) where e = x/|x|, and the internal integral is calculated over the unit sphere Ω1 in the k-space. After calculation of the internal integral, (6.75) is reduced to the one-dimensional integral ⎤ ⎡ ∞ 1 exp (−ikr) 1 ∗L dk ⎦ . (6.76) (x, ω) = ∇α ∇β ⎣ gαβ 2π 2 ir k (−k 2 M∗ +ω 2 ρ∗ ) −∞

164

6. Eﬀective wave operator

The integral in this equation is presented in the form ∞ exp (−ikr) dk k (−k 2 M∗ +ω 2 ρ∗ ) −∞ ⎡ ∞ 1 1 1 exp (−ikr) dk ⎣ = 2 − 2 l Ml (k32 − k12 ) k32 k1 k −∞

1 + 2 k1

∞

−∞

1 k exp (−ikr) dk − 2 k 2 − k12 k3

∞

−∞

⎤ k exp (−ikr) ⎦ dk , k 2 − k32

(6.77)

where k1 and k3 are the roots of the equation k4 +

1 p2 l2 Ml

Ms + iω 3 pf Mω k 2 −

ω2 p2 l2 Ml

ρs − iω 3 p v f ρω = 0 (6.78)

located in the low half-plane of the complex k-plane. If ω is considered as a small parameter, the principal terms of these roots are 1 3 ρω Mω ρs 1 − iω p v f , (6.79) k1 = ω + Ms 2 ρs Ms Mω f ω3 i Ms ·√ − . (6.80) k3 = v 2l lp Ml Ms Ml The ﬁrst integral in the right-hand part of (6.77) is the Fourier transform of the generalized function k −1 , and is equal to πi. Two other integrals are calculated by residue theory. The result is 1 ir

∞

−∞

π 1 − e−ik3 r 1 − e−ik1 r exp (−ikr) dk = 2 − . kL∗L (k, ω) l Ml (k12 − k32 ) k32 r k12 r

(6.81)

∗L Finally, the function gαβ (x, ω) takes the form

∗L (x, ω) = gαβ

p2 l2 Ml 1 1 −ik1 r −ik3 r ∇α ∇β 1 − e 1 − e + . 4π ρs ω 2 r M2s r (6.82)

∗T The equation for the transverse component gαβ (x, ω) of the Green tensor may be obtained by the same technique, and takes the form 1 −ik2 r 1 ∗T e − e−ik1 r δαβ gαβ (x, ω) = 4πµs r l2 µl µs −ik2 r −ik4 r 1−e 1−e + . (6.83) − ∇α ∇β ρs ω 2 r µs r

6.2 The Green function of the eﬀective wave operator

165

Here k2 and k4 are the roots of the equation k4 +

1 p2 l2 µl

ω2 µs + iω 3 p v f µω k 2 − 2 2 ρs − iω 3 p v f ρω = 0 (6.84) p l µl

located in the low half-plane of the complex k-plane. The principal terms of these roots are 1 ρω µω ρs 1 − iω 3 p v f , (6.85) + k2 = ω µs 2 ρs µs µω f ω3 i µs ·√ − . (6.86) k4 = v 2l µs µl pl µl ∗ ∗L ∗T (x, ω) = gαβ + gαβ of the wave operator L∗ Finally, the Green tensor gαβ in (6.58) is presented in the form −ik2 r −ik1 r e−ik2 r e e 1 µs ∗ δαβ − − gαβ (x, ω) = ∇α ∇β 4πµs r ρs ω 2 r r 4 µl 1 e−ik4 r 2 δαβ − (pl) ∇α ∇β 1 − e−ik4 r − 4πµs r µs r 5 Ml µs . (6.87) − 2 1 − e−ik3 r Ms r s If ω tends to zero, we obtain the static Green function gαβ (x) of the homogeneous medium that is equivalent to the composite material r µs 1 s 1 − exp − δαβ gαβ (x) = 4πµs r pl µl 4 r µs 2l2 µs 1 1 − exp − r+ 1− − ∇α ∇β 8πµs Ms µs r pl µl , -5 2 r Ms 2 (pl) Ml µs 1 − exp − . (6.88) − 2 Ms r pl Ml

In contrast with the classical Green function of a homogeneous elastic medium, the right-hand side of this equation has no singularities when r → 0 (Fig. 6.1). The bounded value at the origin is the characteristic property of the Green function of a quasicontinuum or a non-local model of a medium with microstructure (see [67, 68]). For large r, the Green function (6.88), in addition to having the classical asymptotic ∼ r−1 , has terms of order r−3 , and terms that decrease exponentially with r. For r l, (6.88) is transformed into the classical static Green function for the homogeneous medium with the bulk and shear elastic moduli Ks and µs 2 1 µs s δαβ − 1− ∇α ∇β |x| . gαβ (x) = (6.89) 8πµs |x| Ms This function is the principal term of the Green function (6.88) for large r.

166

6. Eﬀective wave operator

−3

−2

−1

1 −1 −2

2

3

pl=0.6 pl=0.4

−3 −4

pl=0.2 pl=0

−5

Fig. 6.1. Schematic distribution of the vertical component of the mean displacement ﬁeld by application of a concentrated force to the medium with the Green function in (6.88); p is the volume concentration of inclusion, l is the correlation radius of the random ﬁeld of inclusions. If pl = 0, (6.88) is the Green function for a classical model of the elastic medium. If pl > 0, the Green function (6.88) corresponds to a non-local model of an elastic medium with microstructure.

6.3 Velocities and attenuations of long elastic waves in matrix composites The longitudinal (6.82) and transverse (6.83) parts of the Green function describe propagation of two types of waves from a point force source in a homogeneous medium equivalent to the considered composite material. The ﬁrst type of wave is characterized by the wave numbers k1 and k2 deﬁned in (6.79) and (6.85). The attenuation coeﬃcients of these waves (the imaginary parts of the wave numbers) are proportional to ω 4 . This means that the attenuation of these waves is caused by Rayleigh scattering by the inhomogeneities. The waves of the second type with the wave numbers k3 and k4 attenuate much more rapidly than the ﬁrst ones. Because the imaginary parts of these wave numbers are proportional to 1/l, these waves practically disappear in a distance of several correlation radii of the random set of inclusions. Existence of such waves is typical for a medium with spatial dispersion. At a large distance from the source, the contribution of these waves in the total wave ﬁeld can be neglected. If l tends to zero in (6.87), the principal term of the Green tensor takes the form −ik2 |x| −ik2 |x| e−ik1 |x| e e 1 µs ∗ δαβ + − , ∇α ∇β gαβ (x, ω) = 4πµs |x| ρs ω 2 |x| |x| (6.90) This function is similar to the Green function of a homogeneous isotropic medium with elastic moduli Ks , µs and the density ρs , but the wave numbers

6.3 Velocities and attenuations of long elastic waves

167

k1 and k2 in this equation are complex numbers. Their real parts deﬁne the velocities of longitudinal l∗ and transverse t∗ waves in the medium with inclusions / ω ω Ms µS = = , t∗ = . (6.91) l∗ = Re(k1 ) ρs Re(k2 ) ρs As is seen from these equations, the eﬀective velocities l∗ and t∗ do not depend on the frequency in the long-wave limit. The principal terms in the imaginary parts of the wave numbers are proportional to ω 3 . Using (6.64) we ﬁnd that the equations for the attenuation of the longitudinal (γL = − Im k1 ) and transverse (γT = − Im k2 ) waves are represented in the forms 4 4 l0 1 pf 8 2 2 3l∗ 2 4 kR + µR (α0 a) + γL a = 18ρ0 ρs l∗ l05 15 η05 3 5 3 2 l∗ , (6.92) ρ21 1 + 3 + l0 η0 4 t0 pf 2t∗ 2 4 γT a = (β0 a) µ 3 + 2η05 18ρ0 ρs t∗ 5t50 R 3 t∗ 2 3 ρ1 2 + η0 . + t0

(6.93)

Here a is the radius of the inclusions Because the eﬀective medium cannot be active, the attenuation coeﬃcients should be positive. Therefore, the factor f deﬁned in (6.51) and presented in (6.92), (6.93) should satisfy the condition ∞ p f = 1− J = 1 − 3p Φ (ζ) ζ 2 dζ ≥ 0 , v ◦

ζ=

|x| . a

(6.94)

It is shown in Section 3.9 that this condition is fulﬁlled for any physically realizable random set of inclusions. If the volume concentration of inclusions is small, interaction between them may be neglected, and the attenuation coeﬃcients γL and γT are deﬁned by the equations γL =

1 n0 QL , 2

γT =

1 n0 QT , 2

(6.95)

where QL and QT are the total scattering cross-sections of a single inclusion by diﬀraction of longitudinal and transverse incident waves. These crosssections are deﬁned in (5.223) and (5.232).

168

6. Eﬀective wave operator

6.4 Long elastic waves in composites with random thin inclusions In this section, we consider propagation of long elastic waves in a medium containing a random set of thin ellipsoidal inclusions. The ratios of semi-axes a1 , a2 , a3 of such inclusions are as follows: a1 a3 a3 , 1; = O(1). a1 a2 a2

(6.96)

As for the spherical inclusions considered in the previous Sections, the eﬀective wave operator of a medium with thin inclusions may be derived by the eﬀective ﬁeld method. For thin ellipsoidal inclusions with density comparable with the density of the matrix, the eﬀective density of the composite is close to the density of the matrix ρ∗αβ ≈ ρ0 δαβ , and the equation for the tensor of eﬀective elastic moduli takes the form 2 (6.97) C ∗ = C s + iω 3 n0 v C ω , C s = C 0 + n0 C R , −1 C R = vΛ0 (a1 , a2 )D0 , D0 = I + n0 K Φ Λ0 (a1 , a2 ) , C ω = C R HC R − n0 J C R H C R , 4 3 πa , (a1 ≥ a2 ), 3 1 K Φ = K s (x)Φ(x)dx, v=

(6.98) (6.99) (6.100)

J=

Φ(x)dx.

(6.101)

For crack-like inclusions, the tensor Λ0 in these equations is determined in (5.118), and for hard thin inclusions, in (5.134) and (5.135). Let us consider some special cases of spatial distributions of thin inclusions in an isotropic homogeneous medium. 6.4.1 Isotropic elastic medium with random crack-like inclusions Homogeneous distribution of inclusions over orientations. Suppose that the correlation function Φ(x) is spherically symmetric: Φ(x) = Φ(|x|). In this case, the tensor K Φ coincides with the tensor A0 deﬁned in (5.76). If thin are homogeneously distributed over orientations, the tensor 0 inclusions vΛ = vC 0 nBnC 0 is also isotropic: 0 1 2 2 1 vΛαβλµ = KP Eαβλµ + 2µP Eαβλµ − Eαβλµ . (6.102) 3 Here the coeﬃcients KP and µP are 2 1 4 − µ20 vB3 , KP = η02 3

(6.103)

6.4 Long elastic waves in composites with thin inclusions

1 2 B2 µ0 v[4B3 + 3B1 (1 + ξ)] , ξ = , 15 B1

µP =

169

(6.104)

where B1 , B2 and B3 are deﬁned in (5.110)–(5.112). The tensor of the static elastic moduli of the medium with a crack-like inclusions follows from (6.97) in the form 1 2 s 2 1 (6.105) Cαβλµ = Ks Eαβλµ + 2µs Eαβλµ − Eαβλµ , 3 Ks = K0 − n0 v KR , KR = KP D1 , µs = µ0 − n0 v µR , µR = µP D2 , −1 −1 D1 = 1 + 9n0 v A01 KP , D2 = 1 + 4n0 v A02 µP .

The tensor C R HC

C R HC R

R

(6.107) (6.108)

in (6.99) is

αβλµ

=

(6.106)

1 2 KH Eαβλµ + 2µH πρ0 t50

1 2 1 Eαβλµ − Eαβλµ 3

,

2 1 4 − µ40 B32 h0 , 2 η0 3 4 4 2 1 µ B h + µ4 B 2 1 + ξ 2 h1 , µH = 15 0 3 0 40 0 1 2 1 1 2 2 2 2 5 3 − 4η0 η0 D1 + 3 + 2η0 D2 , h1 = h1 D22 , h0 = 9 4 5

(6.109)

KH =

(6.110) (6.111) (6.112)

where coeﬃcient h1 is deﬁned in (5.120). Hence, the eﬀective medium is isotropic with the elastic moduli tensor 1 2 ∗ 2 1 (6.113) Cαβλµ = K∗ Eαβλµ + 2µ∗ Eαβλµ − Eαβλµ , 3 2

2

K∗ = Ks − iω 3 n0 v Kω , µ∗ = µs − iω 3 n0 v µω , 4 2 5 1 1 η05 4 2 µ0 B3 D1 Kω = KH − n0 J , − πρ0 t50 4 η02 3

(6.114) (6.115)

4 1 µω = µH πρ0 t50

2 5 1 2 n0 5 J 3 + 2η0 µ v (4B3 + 3B1 (1 + ξ))D2 . − 30 15 0

(6.116)

170

6. Eﬀective wave operator

In this case, the wave equation in k-representation has the form ∗ Cαβλµ kβ kµ − ω 2 ρ0 δαλ Uλ (k) = 0, (6.117) and splits into two independent equations for the longitudinal UαL (k) and transverse UαT (k) components of the wave ﬁeld: 2 k M∗ (ω) − ρ0 ω 2 UαL (k) = 0, (6.118) 2 (6.119) k µ∗ (ω) − ρ0 ω 2 UαT (k) = 0 , 4 4 M∗ = Ms + iω 3 n0 Mω , Ms = Ks + µs , Mω = Kω + µω . (6.120) 3 3 As the result, the dispersion equations for the wave numbers of longitudinal α∗ and transverse β∗ waves take the forms α∗2 M∗ (ω) − ρ0 ω 2 = 0,

β∗2 µ∗ (ω) − ρ0 ω 2 = 0.

(6.121)

In the long-wave region, the solutions of these equations are α∗ = ω

ρ0 Ms

1 1 Mω ρ0 µω 1 − iω 3 n0 , β∗ = ω 1 − iω 3 n0 . 2 Ms µs 2 µs (6.122)

The real parts of the wave numbers deﬁne the velocities of longitudinal (l∗ ) and transverse (t∗ ) waves propagating in the medium with crack-like inclusions / Ms µs , t∗ = . (6.123) l∗ = ρ0 ρ0 The imaginary parts of α∗ and β∗ are the attenuation coeﬃcients of these waves 3 n0 4 l0 Mω α , (6.124) γL = − Im(α∗ ) = 2π 0 l∗ η0 µ20 3 n0 4 t0 µω γT = − Im(β∗ ) = β0 . (6.125) 2π t∗ µ20 The attenuation coeﬃcients γL and γT are proportional to ω 4 , i.e., the attenuation is caused by the Rayleigh scattering of the propagating wave by the inclusions. If the concentration of the inclusions is small (n0 v << 1), (6.123)– (6.125) are transformed into the following equations: l∗ = l 0

n0 v µ0 1− 15

6

B3 15 − 40η02 + 32η04 + 4η02 B1 (1 + ξ) η02

7 12 , (6.126)

6.4 Long elastic waves in composites with thin inclusions

12 n0 v µ0 4B3 + 3B1 (1 + ξ) , t∗ = t∗ 1 − 15 6 2 n0 v 4 2 B32 α0 µ0 15 + 40η02 + 32η04 h0 (η0 ) γL = 5 30π η0 7 16 2 2 + B1 1 + ξ h1 (η0 ) , η0 2 n0 v 4 2 β0 µ0 32B32 h0 (η0 ) + 3B12 1 + ξ 2 h1 (η0 ) , 240π where the functions h0 (η0 ) and h1 (η0 ) are deﬁned in (5.120). For thin spheroidal voids, (6.126)–(6.129) take the forms

171

(6.127)

(6.128)

γT =

(6.129)

12 8η02 n0 v 15 − 8η02 + 32η04 + l∗ = l 0 1 − , 15π η02 (1 − η02 ) (3 − 2η02 )

(6.130)

5 12 9 − 8η02 4n0 v 1− , 15π (1 − η02 ) (3 − 2η02 )

4 t ∗ = t0

(6.131)

2 16h1 (η0 ) n0 v 4 h0 (η0 ) 2 4 15 − 40η0 + 32η0 + γL = α 2 , 30π 3 0 η02 (1 − η02 )2 η0 (3 − 2η02 ) (6.132) 2 n0 v 4 h1 (η0 ) h0 (η0 ) β + γT = 2 . 10π 3 0 3 (1 − η02 )2 (3 − 2η02 )

(6.133)

Thin soft inclusions with the same orientation. For a medium with thin inclusions of the same orientation, the tensor C ∗ is deﬁned in (6.97), where averaging is carried out over random sizes of the inclusions. In what follows, we consider spheroidal soft inclusions (a1 = a2 = a), and suppose that the correlation function Φ(x) has the symmetry of an ellipsoid with semiaxes b1 = b2 = b, and b3 directed along the normal m to the inclusion middle surface (see Fig. 6.2). In this case, the tensor K Φ has the form (6.101), and its components are deﬁned by the parameter γ = b/b3 . If the positions of the inclusion centers are independent, γ has the order of the mean aspect ratio of the inclusions a/a3 . For spheroidal inclusions, tensor Λ0 is deﬁned in (5.118). A medium with such inclusions is transversely isotropic, with the symmetry axis directed along the vector m. In the P -basis, the tensor C ∗ is represented in the form 1 C ∗ = k∗ P 2 + 2µ0 P 1 − P 2 + λ∗ P 3 + P 4 + 4µ∗ P 5 + n∗ P 6 , (6.134) 2

172

6. Eﬀective wave operator x3 b3

a3

b2 a2

x2

a1 b1 x1

Fig. 6.2. Thin ellipsoidal inclusion with semi-axes (a1 , a2 , a3 ) inside the correlation hole of ellipsoidal symmetry with semi-axes (b1 , b2 , b3 ) .

k∗ = ks + iω 3 n0 kω , λ∗ = λs + iω 3 n0 λω ,

(6.135)

µ∗ = µs + iω 3 n0 µω , n∗ = ns + iω 3 n0 nω .

(6.136)

i

i

Here P = P (m), the subscript “s” means static values of the elastic moduli. The explicit equations for the static elastic moduli are presented in Section 7.10 and may be written as follows ks = λ0 + µ0 − n0 vkR ,

(6.137)

λs = λ0 − n0 vlR , µs = µ0 − n0 vµR ,

(6.138)

ns = λ0 + 2µ0 − n0 vnR , 2 kR = 1 − 2η02 nR , λR = 1 − 2η02 nR , 2 −1 µ0 vA3 2 nR = 4 A3 1 + n0 2 1 + 8η0 η0 − 1 (f0 − f1 ) , η0 η0 1 0−1 µR = µ0 A1 1 + n0 vA1 1 − f0 + 4 η02 − 1 f1 ,

(6.139) (6.140) (6.141) (6.142)

where functions f0 and f1 are deﬁned in (3.126) of Volume I. The coeﬃcients in the imaginary parts of the elastic moduli in (6.135), (6.136) have the forms 2 h (η0 ) 2 2 2 v nR − n0 vnR J , kω = 1 − 2η02 nω , nω = 0 5 πρ0 t0

(6.143)

h (η0 ) 2 2 2 v µR − n0 vµR J , λω = 1 − 2η02 nω , µω = 1 5 16πρ0 t0

(6.144)

where functions h0 (η0 ) and h1 (η0 ) are the same as in (6.112).

6.4 Long elastic waves in composites with thin inclusions

173

If a plane wave Uα (x) = Uα e−iqn·x

(6.145)

with the wave vector q propagates in a medium with inclusions, the polarization vector Uα of this wave satisﬁes the equation 2 ∗ q Λαλ − ρ0 ω 2 δαλ Uλ = 0. (6.146) ∗ nβ nµ is the acoustic tensor. For C ∗ in (6.134), Λ∗αλ has Here Λ∗αλ = Cαβλµ the form

Λ∗αλ = Λ∗0 δαλ + Λ∗1 nα nλ + Λ∗2 mα mλ + Λ∗3 m(α nλ) ,

(6.147)

Λ∗0 = µ0 + δ1 cos2 θ , Λ∗1 = k∗ ,

(6.148)

Λ∗2 = δ1 + δ3 cos2 θ , Λ∗3 = 2δ2 cos θ ,

(6.149)

δ1 = µ∗ − µ0 , δ2 = µ∗ − λ∗ − k∗ ,

(6.150)

δ3 = µ0 + n∗ + k∗ − 4µ∗ − 2λ∗ ,

(6.151)

where θ is the angle between the vectors m and n. Introducing the vector e = m × n, we rewrite the tensor Λ∗αλ in the form [22] that is more convenient for analysis (see Fig. 6.3) Λ∗αλ = λ∗0 δαλ + λ∗1 nα nλ + λ∗2 mα mλ + λ∗3 eα eλ ,

(6.152)

λ∗0 = µ0 + δ1 cos2 θ − δ2 sin2 θ , λ∗1 = δ2 + k ∗ ,

(6.153)

λ∗2

2

= δ1 + δ2 + δ3 cos θ ,

λ∗3

2

= δ2 sin θ .

m

(6.154)

n

e

Fig. 6.3. The basis vectors for presentation of the parameters of the mean wave ﬁeld in a medium with a set of thin inclusions of the same orientation; m is the normal to the middle planes of inclusions, n is the wave normal, e = m × n.

174

6. Eﬀective wave operator

For the tensor Λ∗αλ in (6.152), equation (6.146) can be solved explicitly for an arbitrary direction of the wave normal n. If the polarization vector of the propagating wave is proportional to the vector e (Uα = U eα ) and m · e = n · e = 0, equation (6.146) takes the form (6.155) q 2 µ0 + δ1 cos2 θ − ρ0 ω 2 = 0. As the result, the wave number q of the wave with the wave normal n is obtained from this equation in the form q2 = ω

ρ0 µs (θ)

1 3 µω 2 1 − iω n0 cos θ , 2 µs (θ)

µs (θ) = µ0 − n0 vµR cos2 θ.

(6.156) (6.157)

Since the polarization vector Uα is orthogonal to the wave normal, the wave is transverse, and its phase velocity t∗2 and attenuation coeﬃcient γT 2 are deﬁned by the equations 1/2 , (6.158) t∗2 = t0 1 − n0 vµR /µ0 cos2 θ γT 2 =

n0 β4 32πµ20 0

t0 t∗2

3

2 v 2 µ2R − n0 vµR J h1 (η0 ) cos2 θ .

(6.159)

If n × m = 0, i.e., θ = 0, these equations take the forms 1/2

t∗2 = t0 (1 − n0 vµR /µ0 ) γT 2 =

n0 β4 32πµ20 0

t0 t∗2

3

,

2 v 2 µ2R − n0 vµR J h1 (η0 ) .

(6.160) (6.161)

When n · m = 0, and θ = π/2, the shear wave propagates in the composite without attenuation and with the velocity t0 = µ0 /ρ0 of the transverse wave in the matrix material. Because the polarization vectors of three isonormal waves are orthogonal, the amplitudes of the two other waves are located in a plane that is orthogonal to the vector e. Thus, for each of these vectors we have Uα = η1 mα + η2 nα ,

(6.162)

where η1 and η2 are unknown coeﬃcients. Substituting this equation into (6.146), we obtain q 2 (λ∗0 δαβ + λ∗1 nα nβ + λ∗2 mα mβ ) (η1 mβ + η2 nβ ) −ρ0 ω 2 (η1 mα + η2 nα ) = 0 .

(6.163)

This vector equation is equivalent to two scalar equations for the coeﬃcients η1 , η2 :

6.4 Long elastic waves in composites with thin inclusions

175

(λ∗0 + λ∗2 ) η1 + λ∗2 η2 cos θ − λ∗ η1 = 0,

(6.164)

λ∗1 η1 cos θ + (λ∗0 + λ∗1 ) η2 − λ∗ η2 = 0,

(6.165)

∗

2

2

where λ = ρ0 ω /q . The characteristic equation of this system is . . ∗ . λ + λ∗ − λ∗ , λ∗2 cos θ . . = 0, det .. 0 ∗ 2 λ1 cos θ, λ∗0 + λ∗1 − λ∗ . and its solution has the form 1 ∗ λ1 + λ∗2 ± (λ∗1 − λ∗2 )2 + λ∗1 λ∗2 cos2 θ . λ∗± = λ∗0 + 2

(6.166)

(6.167)

Hence, the quasi-longitudinal and quasi-transverse waves can propagate in the medium with the ﬂattened crack-like inclusions. The wave numbers of these waves are ρ0 ρ0 , q = ω . (6.168) q1 = ω 2 λ∗+ λ∗− For an arbitrary angle θ, the explicit equations for the wave numbers q1,2 are rather huge. Let us consider only two speciﬁc cases: θ = 0 and θ = π/2. If θ = 0, i.e., n × m = 0, we obtain from (6.167) and (6.168) that the longitudinal wave that can propagate in this direction has the wave number || qL1 1 3 nω ρ0 || 1 − iω n0 . (6.169) qL1 = ω ns 2 ns Another wave that can propagate in this direction is the transverse wave with parameters deﬁned in (6.160) and (6.161). The wave with the wave number (6.169) propagates along the axis of material symmetry of the composite, and its phase velocity and the attenuation coeﬃcient are / 1/2 vnR (λ0 + 2µ0 ) || , l0 = , (6.170) l∗1 = l0 1 − n0 λ0 + 2µ0 ρ0 || γL1

n0 4 α = 2πη 0

,

l0 ||

l∗1

-3

2 h0 (η0 ) v 2 n2R − n0 J vnR . µ0

(6.171)

If θ = π/2, the longitudinal wave can propagate in this direction, and its wave number is 1 3 ρ0 kω ⊥ 1 − iω n0 . (6.172) qL1 = ω µ0 + ks 2 µ0 + ks

176

6. Eﬀective wave operator

The velocity and attenuation coeﬃcient of this wave follow from this equation in the form vnR || ⊥ ⊥ , γL1 = l0 1 − n0 (1 − 2η02 )2 = (1 − 2η02 )γL1 . (6.173) l∗1 λ0 + 2µ0 The parameters of the transverse wave that can propagate in this direction coincide with the parameters of such a wave in the matrix material. Consider now a medium containing random parallel spheroidal cracks. Suppose that the crack centers are statistically independent and homogeneously distributed in space (the Poisson set). For such a stochastic model of the spatial distribution of the inclusions, the tensor K Φ takes the form KΦ =

1 5 P + η02 P 6 . µ0

(6.174)

This equation follows from equation (6.101) for K Φ when b3 → 0. As the result, f0 , f1 → 0 in (5.238), and the integral J in (6.101) disappears. The equations for the static eﬀective elastic moduli of such a medium take the form Ks = λ0 + µ0 − n0 v µ0 d2 (1 − 2η02 )2 , λs = λ0 − n0 v d2 (1 − 2η02 ), d1 =

π

−1 3 − 2η02 + n0 v , 4

µs = µ0 (1 − n0 v d1 ) , (6.175)

ns = λ0 + 2µ0 − n0 v µ0 d2 , d2 =

(6.176)

−1 1 2 πη0 (1 − 2η02 ) + n0 v . η02 (6.177)

The imaginary parts of the elastic moduli in (6.135) and (6.136) are µω =

1 2 h1 (η0 ) v (µ0 d1 )2 , 16 πρ0 t50

kω = (1 − 2η02 )2 nω ,

h0 (η0 ) nω = v 2 (µ0 d2 η02 )2 , πρ0 t50

lω = (1 − 2η02 )nω .

(6.178) (6.179)

Here the functions h0 (η0 ) and h1 (η0 ) are deﬁned in (5.119). These equations give the velocities and attenuation coeﬃcients of the longitudinal and transverse waves propagating along the material symmetry axis in the forms 1/2 1/2 || || , t∗ = t0 1 − n0 v η02 d1 , (6.180) l∗ = l0 1 − n0 v η02 d2 || γL

n0 v 2 4 l 0 3 3 2 α0 || = η0 d2 h0 (η0 ), 2π l∗

(6.181)

6.4 Long elastic waves in composites with thin inclusions

n0 v 2 4 t 0 4 2 || β γT = d1 h1 (η0 ). 32π 0 t||∗

177

(6.182)

For longitudinal waves propagating in the direction orthogonal to the material symmetry axis, the velocity and attenuation coeﬃcients are 1/2 , l∗⊥ = l0 1 − n0 v d2 η02 (1 − 2η02 )2 γL⊥

n0 v 2 4 l 0 4 2 α0 ⊥ = d2 (1 − 2η02 )2 h0 (η0 ). 2π l∗

(6.183) (6.184)

In this direction, two transverse waves can also propagate. One of them has the same parameters as the transverse wave propagating along the material symmetry axis, and the other wave does not attenuate and has the velocity t0 = µ0 /ρ0 of the transverse waves in the matrix material. For small crack concentrations, the attenuation coeﬃcients of three isonormal waves (one longitudinal and two transverse) can be found from the general equations where only the terms of the order n0 v should be kept. As the result, we obtain the attenuation coeﬃcients of these waves in the forms n0 v 2 4 (1 − 2η02 sin2 θ)2 8 sin2 θ cos2 θ α0 h0 (η0 ) + h1 (η0 ) , (6.185) γ1 = 2π 3 η05 (1 − η02 )2 η0 (3 − 2η02 )2 n0 v 2 4 sin2 2θ 2 cos2 2θ β h (η ) + h (η ) , γ2 = 0 0 1 0 0 2π 3 (1 − η02 )2 (3 − 2η02 )2 γ3 =

n0 v 2 4 2 cos2 θ β0 h1 (η0 ). 2π 3 (3 − 2η02 )2

(6.186)

(6.187)

For cracks of the same radius a, these equation take the forms γi =

1 n0 Qi 2

(i = 1, 2, 3),

(6.188)

where Qi (i = 1, 2, 3) are the total scattering cross-sections of an isolated crack by diﬀraction of every such wave. These cross-sections are deﬁned in 5.279). 6.4.2 Isotropic elastic medium with a random set of hard disks Consider now the isotropic elastic medium containing a random set of spheroidal hard disks. The eﬀective dynamic elastic moduli C ∗ of such a medium may be found from equations (6.97). The tensors Λ0 and Λω in these equations are presented in (5.134) and (5.135).

178

6. Eﬀective wave operator

Homogeneous distribution of rigid disks over the orientation. For a homogeneous distribution of disks over the orientations, the medium is macro isotropic with the bulk K∗ and shear µ∗ elastic moduli in the forms K∗ = Ks + iω 3 n0 Kω ,

µ∗ = µs + iω 3 n0 µω ,

(6.189)

4 1 Ks = K0 + n0 vG1 D10 , µs = µ0 + n0 (vG1 + 3 vG2 ) D20 , (6.190) 3 15 2 0 2 Kω = 4 vΛω (6.191) 1 − 4n0 J vG1 H1 (D1 ) , 1 2n0 2 ω v (Λω J v (G1 + 3G2 ) (D20 )2 , µω = (6.192) 1 + 3Λ2 ) − 15 15 −1 1 1 − 4n0 A01 vG1 , 3 −1 2 , D20 = 1 − A02 (vG1 + 3 vG2 ) 15

D10 =

(6.193) (6.194)

ω 0 0 where the coeﬃcients G1 , G2 , Λω 1 , Λ2 , A1 , A2 are deﬁned in (5.137), (5.140) and (5.77). The wave numbers of the longitudinal and transverse waves that can propagate in the medium are iω 3 ρs Mω 1− , (6.195) α∗ = ω Ms 120πρ0 t50 Ms iω 3 ρ0 µω β∗ = ω 1− , (6.196) µs 120πρ0 t50 µs

where Ms = Ks + 4µs /3 , Mω = Kω + 4µω /3. The phase velocities and attenuation coeﬃcients of these waves take the forms / Ms µs , t∗ = , (6.197) l∗ = ρ0 ρ0 3 l0 1 Mω , l∗ η0 ρ0 ρs t40 3 t0 1 1 4 β γT = µω . 120π 0 t∗ ρ0 ρs t40

γL =

1 α4 120π 0

(6.198) (6.199)

For absolutely rigid inclusions, the coeﬃcients G1 and G2 in (6.190)– (6.194) do not depend on the geometrical parameters of the inclusions and take the forms G1 =

8µ0 , π (1 + η02 )

G2 =

32µ0 . π (3 + η02 )

(6.200)

6.4 Long elastic waves in composites with thin inclusions

179

If the concentration of the inclusions is small, the velocities and attenuation coeﬃcients of the propagating waves are 16η02 2 1 n0 v , (6.201) + l∗ = l 0 1 + 15π 3 + η02 1 + η02 2 12 1 n0 v , (6.202) + t∗ = t 0 1 + 15π 3 + η02 1 + η02 32n0 v 2 4 4 3 + 2η05 1 + 4η05 (6.203) α + γL = 2 , 225π 3 η0 0 (3 + η02 )2 (1 + η02 ) 4n0 v 2 4 24 3 + 2η05 1 + 4η05 (6.204) β + γT = 2 . 225π 3 0 (3 + η02 )2 (1 + η02 ) Hard disks of the same orientation. For hard disks of the same orientation, the averaging over orientation in (6.97) is absent, and the tensor of the eﬀective elastic moduli C ∗ in the P -basis takes the form 1 2 ∗ 2 1 C = k∗ P + 2m∗ P − P + λ0 P 3 + P 4 2 + 4µ0 P 5 + (λ0 + 2µ0 ) P 6 , k∗ = ks +

(6.205)

iω 3 n0 iω 3 n0 5 kω , m∗ = ms + 30πρ t5 mω , 30πρ0 t0 0 0

(6.206)

2 2 2 ks = λ0 + µ0 + n0 vKR , kω = 1 + 4η05 v kR − n0 vkR J , (6.207) ms = µ0 + n0 vmR , mω = 3 + 2η05

2 v 2 m2R − n0 vmR J

, (6.208)

1 −1 G2 (1 − n0 vG2 A2 ) . (6.209) 2 where A1 , A2 are deﬁned in (5.239) and (5.240) and we suppose that the aspect of the correlation hole independent on the inclusion size. If the plane wave Uα (x) = Uα exp(−iqn · x) propagates in the medium with such inclusions, the polarization vector Uα satisﬁes the equation 2 ∗ q Λαβ − ω 2 ρ0 δαβ Uβ = 0, (6.210) KR = G1 (1 − 4n0 vG1 A1 )

−1

, mR =

where the acoustic tensor Λ∗αβ has the form (6.152) with the coeﬃcients λ∗0 = µ0 + n0 v (kR + mR ) +

iω 3 (k + m ) sin2 θ , ω ω 30πρ0 t50

(6.211)

180

6. Eﬀective wave operator

λ∗1 = λ0 + µ0 , iω 3 n0 ∗ ∗ λ2 = λ3 = − n0 v (mR + kR ) + (mω + kω ) sin2 θ . 30πρ0 t50

(6.212) (6.213)

Let a transverse wave polarized orthogonal to the vector of the material symmetry m propagate in an arbitrary direction. The wave number of this wave has the form ω − iγT 2 , (6.214) q2 = t∗2 where the phase velocity t∗2 and the attenuation coeﬃcient γT 2 are

t∗2

γT 2

µ0 = ρ0

vmR 1 + n0 sin2 θ µ0

n0 β04 = 60πρ20 t40

t0 t∗2

3

1/2 ,

mω sin2 θ.

(6.215)

(6.216)

The wave numbers of two other isonormal waves q1 and q3 coincide with (6.167) and (6.168) where λ∗i (i = 0, 1, 2, 3) are deﬁned in (6.211). Let the wave normal be parallel to the material symmetry axis (θ = 0). In this case, the longitudinal and transverse waves do not attenuate, and have the velocities / λ0 + 2µ0 µ0 || || , t∗ = . (6.217) l∗ = ρ0 ρ0 Let the wave normal be orthogonal to the material symmetry axis (θ = π/2). In this case, the transverse wave polarized along the symmetry axis || || has the wave velocity and attenuation coeﬃcient coinciding with t∗ and γT . The polarization vector of the longitudinal and other transverse waves are situated in the plane orthogonal to the symmetry axis. Their velocities l∗⊥ , ⊥ ⊥ t⊥ ∗ and attenuation coeﬃcients γL , γT are deﬁned by the equations / ks + ms ms ⊥ l∗ = , t⊥ = , (6.218) ∗ ρ0 ρ0 γL⊥ γT⊥

3 l0 n0 1 4 = α0 ⊥ (kω + mω ), 2 12πρ0 η0 l∗ l04 3 t0 n0 1 4 = β mω . 0 2 ⊥ 60πρ0 t∗ t40

(6.219) (6.220)

If the disks are absolutely rigid, and their centers compose the Poisson random set, the tensor K Φ takes the form (6.174), and the integral J is zero. The static elastic moduli ks and ms of such a composite take the forms

6.4 Long elastic waves in composites with thin inclusions

ks = λ0 + µ0 1 +

181

4n0 v 8n0 v , ms = µ0 1 + . π (1 + η02 ) π (3 + η02 )

(6.221)

Note that these coeﬃcients are linear with respect to n0 v, and therefore, have the same forms as for a small concentration of inclusions. The coeﬃcients kω and µω are 2 µ20 1 + 4η02 2 µ20 3 + 2η05 (6.222) kω = 16 v 2 , µω = 64 v 2. π 2 (1 + η02 ) π 2 (3 + η02 ) The phase velocity and attenuation coeﬃcient of a wave with a polarization vector orthogonal to the vectors n and m are 1/2 8 sin2 θ , π (3 + η02 ) 16n0 v 2 4 t0 3 3 + 2η02 2 = β 0 2 sin θ . 2 t∗2 15π 3 (3 + η0 )

t∗2 =

γT 2

µ0 ρ0

1 + n0 v

(6.223)

(6.224)

If the waves propagate along the material symmetry axis, their velocities are deﬁned in (6.217). For waves propagating in the direction orthogonal to the material symmetry axis (6.218) and (6.220) are transformed into the following equations 1/2 2 4η 2 1 || + , l∗⊥ = l∗ 1 + n0 v 0 π 3 + η02 1 + η02 1/2 8 1 + n0 v = , π (3 + η02 ) 16n0 v 2 4 l0 3 4 3 + 2η05 1 + 4η05 ⊥ α γL = 2 + 2 , 15π 3 η0 0 l∗⊥ (3 + η02 ) (1 + η02 ) 16n0 v 2 4 t0 3 3 + 2η05 ⊥ γT = β0 ⊥ 2. t∗3 12π 3 (3 + η02 )

t⊥ ∗3

|| t∗

(6.225)

(6.226) (6.227)

(6.228)

For small concentrations of inclusions, the velocities of three isonormal waves corresponding to the wave normal n are 2 2η 2 1 4 sin + θ , (6.229) l∗1 = l0 1 + n0 v 0 π 3 + η02 1 + η02 4 sin2 θ , (6.230) l∗2 = t0 1 + n0 v π (3 + η02 ) 2 n0 v 1 2 sin 2θ , l∗3 = t0 1 + + (6.231) 2π 3 + η02 1 + η02

182

6. Eﬀective wave operator

The attenuation coeﬃcients of these waves are 4n0 v 2 4 4 3 + 2η05 1 + 4η05 4 γ1 = α + 2 sin θ, 2 15π 3 η0 0 (3 + η02 )2 (1 + η0 ) 16n0 v 2 4 3 + 2η05 2 β0 γ2 = 2 sin θ, 15π (3 + η02 ) 2n0 v 2 4 4 3 + 2η05 1 + 4η05 2 γ3 = β + 2 sin 2θ. 15π 3 0 (3 + η02 )2 (1 + η02 )

(6.232)

(6.233)

(6.234)

If the rigid disks have the same radius, the attenuation coeﬃcients γ1 , γ2 , γ3 take the forms (6.188), where Qi are the total scattering crosssections of a rigid disk by diﬀraction of three isonormal waves. These crosssections are deﬁned in (5.283)–(5.285).

6.5 Long elastic waves in composites with short hard ﬁbers In this section, we consider propagation of long elastic waves in a medium containing a random set of short hard axisymmetrix ﬁbers. The eﬀective ﬁeld method applied to the solution of the homogenization problem for such a composite gives the following expression for the tensor of the eﬀective elastic moduli of the medium with short ﬁbers C ∗ = C s + iω 3 C ω ,

C s = C 0 + C R,

C R = D 0 ΛR

(6.235)

C ω = D0 Λω D0 − JC R HC R

(6.236)

Here ΛR , Λω and D0 are the following tensors ΛR = L(x)Λ0 (x) , Λω = L(x)Λω (x) , −1 D 0 = I − K Φ ΛR , K Φ = K s (x)Φ(x)dx,

(6.237) (6.238)

functions Λ0 (x) and Λω (x) are deﬁned in (5.160)–(5.162), L(x) is the deltafunction concentrated on the ﬁber axes Li (x), Li (x)φ(x)dx = s(z)φ(z)dz, (6.239) L(x) = i

Γi

φ(x) is an arbitrary smooth function, Γi is the middle line of i-th ﬁber, s(z) is the function of the ﬁber form (deﬁned in (5.177)). The scalar J are determined as follows

6.5 Long elastic waves in composites with short hard ﬁbers

183

J=

Φ(x)dx.

(6.240)

Here Φ(x) = 1 − Ψ (x), and Ψ (x) is the two-point correlation function of the random set of ﬁbers deﬁned in (7.374) of Volume I. Note that the following equations hold L(x)Λ(x) = n0 Λm ,

Λm = πa2 L

1

f 2 (ξ)Λm (ξ)dξ,

(6.241)

−1

where n0 is the numerical concentration of the ﬁbers, f (x) is the function of the ﬁber form and the tensor Λm depends on the ﬁber sizes and their orientations in space. The orientations of the ﬁbers are deﬁned by the random vector m that appears in the basic tensors P i (m). The other means in (6.237) are calculated similarly. 6.5.1 Random sets of ﬁbers homogeneously distributed over orientations Let the centers of the ﬁbers form a statistically homogeneous and isotropic set, and their orientations be distributed homogeneously. Suppose that the correlation function Φ(x) is spherically symmetric: Φ(x) = Φ(|x|). In this case, the tensor K Φ takes the form η02 3 + 2η02 1 Φ Iαβλµ − δαβ δλµ (6.242) Kαβλµ = δαβ δλµ + 9µ0 15µ0 3 Then, the means in (6.235) become isotropic tensors, and the eﬀective elastic moduli tensor C ∗ can be written as 1 ∗ Cαβλµ (6.243) = K ∗ δαβ δλµ + 2µ∗ Iαβλµ − δαβ δλµ 3 K ∗ = Ks + iω 3 Kω , µs = µ0 + n0 v µR ,

µ∗ = µs + iω 3 µω , KR = Λ1 D1 ,

Ks = K0 + n0 v KR

µR =

1 Λ2 D2 2

−1 −1 η02 3 + 2η02 , D2 = 1 − n0 v Λ2 D1 = 1 − n0 v Λ1 µ0 15µ0

2 2 Kω = n0 v 2 H − 3n0 v JH1 (λ6 D1 ) 2 2n0 2 2 2 v H − n0 v JH2 (λ6 D2 ) µω = 15 15

(6.244) (6.245) (6.246) (6.247) (6.248)

184

6. Eﬀective wave operator

In these equations v is the ﬁber volume, Λ1 =

1 [4(λ1 + λ3 ) + λ6 ] , 9

2 H = H1 + H2 3

1 (2λ1 + 6λ2 − 4λ3 + 3λ5 + 2λ6 ) 15 H1 and H2 are determined in (5.75) and Λ2 =

λ1 =

µ0 , η02

λ5 = 8µ0 ,

λ2 =

4µ0 , 1 + η02

λ3 =

µ0 ν ϕ(q), η02 E

(6.249) (6.250)

(6.251)

λ6 = ϕ(q)

where the function ϕ(q) was deﬁned in (5.289). For a macroscopically isotropic medium, wave equation (6.270) splits into two independent equations for the longitudinal and shear components of the displacement vector. The corresponding wave numbers kL and kT are determined by the equations kL = α∗ − iγL , kT = β ∗ − iγT , α∗ = ω/l∗ , β ∗ = ω/t∗ , / 4 ∗ Ks + µs /ρ0 , t∗ = µs /ρ0 , l = 3

(6.252) (6.253)

where l∗ and t∗ are the eﬀective velocities of the longitudinal and transverse waves, γL and γT are their attenuation coeﬃcients 1 4 1 4 4 (α∗ ) l∗ Kω + µω , γT = (β ∗ ) t∗ µω . (6.254) γL = 2ρ0 3 2ρ0 6.5.2 Random set of ﬁbers of the same orientation Let all the ﬁbers have the same orientation. In what follows, we suppose that the function Ψ (x) has the symmetry of a prolate spheroid that is coaxial with the ﬁber and has semi-axes b1 = b2 = b, b3 > b, the b3 semi-axis is directed along the vector m. As a result, the tensor K Φ has the form (5.268), and its components are determined by the ratio γ = b/b3 < 1. If we suppose in addition that the ﬁber locations in the space are statistically independent, the ratio γ has the order (a/L)|m that is the mean aspect ratio of the ﬁbers with orientation m, L is the ﬁber length. In this case, tensor K Φ is deﬁned in (5.271). The composite is transversely isotropic, and its tensor of eﬀective elastic moduli takes a form that follows from (6.235)–(6.240): 1 C ∗ = ks P 2 + 2ms P 1 − P 2 2 3 4 + ls P + P + 4µs P 5 + ns + iω 3 nω P 6 . (6.255)

6.5 Long elastic waves in composites with short hard ﬁbers

185

Here the elastic coeﬃcients are µs = µ0 +

2n0 vµ0 2n0 vµ0 , ms = µ0 + , 1 − n0 v (1 + η 2 ) (1 − n0 v)

ns = λ0 + 2µ0 + n0 vnR ,

ks = λ0 + µ0 +

η2

n0 vµ0 , (1 − n0 v)

n0 v µ0 n0 v 1 − η02 Ed (q, γ) + 2νϕ(q) ls = λ0 + , (1 − n0 v) η02 [1 − n0 vd (q, γ)] E ϕ(q)γ 2 ln γ d(q, γ) = − , µ0 ϕ (q) 2 , nω = n0 v 2 − n0 v J Hn2R . nR = 1 − n0 vd (q, γ)

(6.256)

(6.257)

(6.258) (6.259)

The isotropy axis of the tensor C ∗ coincides with the vector m. The dispersion equation for the wave numbers of the mean wave ﬁeld propagating in the composite has the form 2 ∗ (6.260) k Λαβ − ω 2 ρ0 δαβ Uβ = 0, where the acoustic tensor Λ∗αβ is deﬁned in (6.152), λ∗0 = ms + (µs − ms ) cos2 θ − (µs − ks + ls ) sin2 θ , λ∗1 = µs + ls , (6.261) λ∗2 = 2µs − ms − ks + ls + ms − 4µs + ks − 2ls + ns − iω 3 nω cos2 θ , (6.262) λ∗3 = (µs − ks + ls ) sin2 θ .

(6.263)

One of the wave number corresponding to the wave normal n we obtain from (6.260) by putting Uα = eα (e · n = 0) k22 ms + (µs − ms ) cos2 θ − ω 2 ρ0 = 0 . (6.264) The root of this equation is k2 = ω

1 ms + (µs − ms ) cos2 θ ρ0

1/2 .

This wave is purely longitudinal. Its velocity is 2n0 v 1 + η 2 cos2 θ ∗ , t2 = t 1 + (1 + η 2 ) (1 − n0 v) and its attenuation factor is zero.

(6.265)

(6.266)

186

6. Eﬀective wave operator

Two other wave numbers are to be found from (6.168), where the parameters λ∗i has the form (6.263). If θ = 0, i.e., n = m, then along this symmetry axis the pure longitudinal and transverse waves can propagate. The expression for the wave number k1 of the longitudinal wave is 1 3 nω ρ0 1 − iω , (6.267) k1 = ω ns 2 ns from here we ﬁnd the phase velocity of this wave, l1∗ = l0 1 +

n0 vη 2 ϕ (q) µ0 (1 − n0 vd (q, γ))

1/2 (6.268)

and its attenuation factor γL1

∗ n0 ∗ 4 l1 = (α ) 120πρ20 m l05

2 2 ω 2 ∗ − n n2R , αm + 3 v Jv = ∗. 0 5 η l1 (6.269)

Transverse wave propagating along the symmetry axis (Uα = eα ) does not attenuate, and its phase velocity t∗1 coincides with t∗2 when θ = 0, i.e., 1/2 2n0 v t∗1 = t0 1 + . 1 − n0 v

(6.270)

Let now θ = π/2 (the wave normal is perpendicular to the axis of isotropy). The longitudinal wave propagating in this direction does not attenuate, and its velocity is l3∗

= vL

1/2 n0 v 1 + 3η 2 1+ , (1 − n0 v) (1 + η 2 )

(6.271)

One of the transverse waves that can propagate in this direction has the polarization vector Uα = eα , and its velocity is to be obtained from (6.266) when θ = π/2: t∗3 = t0 1 +

2n0 v (1 − n0 v) (1 + η 2 )

1/2 .

(6.272)

The velocity of the second transverse wave, polarized along the axis of isotropy coincides with t∗1 . Let us consider the case of the small ﬁber concentration (n0 v 1). The attenuation coeﬃcients of three isonormal waves (quasi-longitudinal (L), quasi-transverse (T ) and pure transverse (T 1)) propagating in the direction n that has the angle θ with the isotropy axis can be obtained from the general

6.6 Notes

187

dispersion equation in which the terms of the order higher than n0 v are neglected. As the result, we obtain for the attenuation coeﬃcients α4 v 2 2 + 3η 5 2 · ϕ (q) cos4 θ , (6.273) γL = n0 120π µ20 η β 4 v 2 2 + 3η 5 2 γT = n0 · ϕ (q) sin2 θ cos2 θ , γT 2 = 0 . (6.274) 120π µ20 If in addition the ﬁbers have the same sizes, these equations are transformed into equations similar to (6.188) where Qi are the total scattering cross-sections of a ﬁber deﬁned in (5.290) and (5.288). Note that for the ﬁbers of the same sizes, the imaginary part C ω of the tensor C ∗ takes the form ω 6 = n0 v 2 (1 − n0 J) H2 n2R Pαβλµ (m) , Cαβλµ

(6.275)

where P 6 (m) is the element of the P i -basis. If the centers of the ﬁbers form a periodic lattice, the integral J is equal to ω the volume (n−1 0 ) of the periodic cell, and C = 0. Thus, as was mentioned before, long waves propagate through periodic structures without attenuation. Hence, the parameter 1 − n0 J may be considered as the measure of deviation of the spatial distribution of random set of inclusions from a periodic distribution that aﬀects the attenuation coeﬃcients.

6.6 Notes The problem of calculation of the eﬀective dynamic properties of matrix composites for the propagation of elastic waves was considered in a number of works. Mainly, the attention was focused on the long-wave region of wave propagation. The eﬀective medium method was used for the solution of this problem in [14–16], and in the following series of the works [86, 92, 93]. In implicit forms, the eﬀective ﬁeld method was used for the solution of this problem in [75, 108, 109]. Remarkable results were obtained in the works of [106,107,111], where the technique of integral equations combined with the assumptions of the eﬀective ﬁeld method was developed for the solution of the wave propagation problem in particulate composites in the long wave region. This technique turned out to be more eﬃcient for carrying out the averaging procedure. Variational bounds for dynamic eﬀective dynamic properties of matrix composites with ellipsoidal inclusions in the long-wave region were obtained in [96–98]. The results of these works coincide with the results of application of the eﬀective ﬁeld method developed in [50–52,73]. This chapter is based on the latter works.

7. Elastic waves in a medium with spherical inclusions

In this chapter, the eﬀective medium and eﬀective ﬁeld methods are applied to elastic wave propagation in composites with a set of spherical inclusions. For spherical isotropic inclusions in an isotropic matrix, the series solutions of the one-particle problems of the self-consistent methods may be obtained for any frequency of the incident ﬁeld. As a result, the predictions of the methods may be analyzed and compared in a wide region of frequencies of the incident ﬁelds that covers long, medium and short waves. The contents of the chapter is as follows. In Section 7.1, version I of the EMM is developed in a form that may be applied for propagating waves of any length. The solutions of the one-particle problems are considered in Section 7.2, and the ﬁnal forms of the dispersion equations of the EMM are presented in Section 7.3. The long- and short-wave asymptotic solutions of the EMM dispersion equations are obtained in this section. In Section 7.4, versions II and III of the EMM are considered in the long-wave region. Numerical solutions of the EMM dispersion equations, and comparison of predictions of the three versions of the EMM in a wide region of frequencies are presented in Section 7.5. In Section 7.6, the eﬀective ﬁeld method is developed for the problem. Speciﬁc features of the EFM for longitudinal wave propagation are indicated. It is shown that in this case, the local exciting ﬁeld acting on each particle is not a plane wave but a sum of plane and radial waves. Solutions of the one-particle problems of the EFM are presented in Section 7.7. The dispersion equations of the EFM in the long- and short-wave region are considered in Section 7.8. Numerical solutions of the dispersion equations of the EFM, and comparison of predictions of the EMM and EFM with experimental data for epoxy-lead composites are considered in Section 7.9.

7.1 Version I of the EMM for elastic waves The starting point of our consideration is system (6.1), (6.2) of the integral equations for the amplitudes of the displacement and strain ﬁelds in a composite with isolated inclusions

190

7. Elastic waves in the medium with spherical inclusions

ui (x) = u0i (x) + ω 2 gik (x − x )ρ1 uk (x )V (x )dx 1 + j gik (x − x )Ckjmn εmn (x )V (x )dx , εij (x) = ε0ij + ω 2 (j gi)k (x − x )ρ1 uk (x )V (x )dx 1 + Kijkl (x − x )Cklmn εmn (x )V (x )dx , Kijkl (x) = i k gjl |(ij)(kl) .

(7.1)

(7.2) (7.3)

The matrix material has the tensor of the elastic moduli C 0 and the mass density ρ0 , and these parameters for the material of the inclusions are C and ρ. The inclusions occupy region V, and V (x) is the characteristic function of this region, ρ1 = ρ − ρ0 , C 1 = C − C 0 .

(7.4)

In (7.1), (7.2), u0 (x) and ε0 (x) are the incident displacement and strain ﬁelds, gij (x) is the Green function of the matrix material. For an isotropic medium with the Lam´e parameters λ0 , µ0 , gik (x) has the explicit form −iα0 r −iβ0 r e−iβ0 r 1 e 2e − − , (7.5) gik (x) = δ β ik 0 i k ρ0 ω 2 r r r α02 =

ρ0 ω 2 , λ0 + 2µ0

β02 =

ρ0 ω 2 . µ0

(7.6)

In what follows, the incident ﬁeld is a plane monochromatic wave, and the vector u0i (x) and tensor ε0ij (x) = (i u0j) (x) are 0

u0i (x) = Ui0 e−iq q0 = q0 n0 ,

·x

,

0

0 0 −iq ε0ij (x) = −iq(i Uj) e

·x

,

q0 ·x = q0 n0i xi ,

(7.7) (7.8)

where q0 is the wave number of the wave in the matrix, n0 is the wave normal, and Ui0 is the polarization vector (q0 = α0 for longitudinal waves, and q0 = β0 for transverse waves). Symbolically, the two equations (7.1) and (7.2) may be written as one equation (7.9) F(x) = F0 (x) + K(x − x )L1 F(x )V (x )dx . Here the following symbolic vectors and matrices are introduced 0 u(x) u (x) 0 F(x) = , F (x) = , ε(x) ε0 (x)

(7.10)

7.1 Version I of the EMM for elastic waves

K(x) =

ω 2 g(x), g(x) ω g(x), K(x) 2

, L1 =

ρ1 , 0 0, C 1

191

.

(7.11)

In this chapter, self-consistent methods are used for the calculation of the velocity and attenuation coeﬃcient of the mean wave ﬁeld F(x) propagating in the composite (the homogenization problem). Let us start with the eﬀective medium method. Hypotheses I1 and I2 of version I of the EMM formulated in Section 2.3 may be applied to elastic waves without modiﬁcations. Thus, to construct an approximate solution of the homogenization problem we accept the following assumptions. I1 . Each inclusion in the composite behaves as an isolated one embedded in the homogeneous medium with the overall (eﬀective) properties of the composite. The ﬁeld that acts on this inclusion is a plane wave propagating in the eﬀective medium. I2 . The mean wave ﬁeld in the composite coincides with the ﬁeld propagating in the homogeneous eﬀective medium. The ﬁrst hypothesis reduces the problem of interaction between many inclusions in the composite to a one-particle problem. The second hypothesis is the condition of self-consistency. The one-particle problem of this version of the EMM is the problem of diﬀraction of a plane monochromatic wave by an isolated inclusion embedded in the homogeneous eﬀective medium. The integral equation of this problem is similar to (7.9) and has the form (7.12) F(x) = F∗ (x) + K∗ (x − x )L∗1 F(x )V (x )dx , v

u∗ (x) ρ − ρ∗ , 0 ∗1 , , L = ε∗ (x) 0, C − C ∗ 2 ∗ ω g (x), g ∗ (x) ∗ . K (x) = ω 2 g ∗ (x), K ∗ (x) F∗ (x) =

(7.13) (7.14)

∗ Here v is the region occupied by the inclusion, gik (x) is the Green’s function of the homogeneous medium with the eﬀective dynamic properties C ∗ and ρ∗ of the composite, and F∗ (x) is a plane wave with the eﬀective wave vector q∗ = q∗ n ∗ Ui∗ . (7.15) F∗ (x) = U∗ e−iq ·x , U∗ = ∗ −iq∗ n(i Uj)

This ﬁeld plays the role of the exciting ﬁeld in the one-particle problem. If the spatial distribution of the inclusions in the matrix is homogeneous and isotropic, the eﬀective medium is also isotropic, and the eﬀective wave vector q∗ and wave vector q0 of the incident ﬁeld have the same direction (n = n0 ). Note that C ∗ , ρ∗ and q ∗ are unknown parameters in (7.12). Let the general solutions of (7.12) be known, and the ﬁeld F(x) inside the inclusion centered at the point x = xj be presented in the form

192

7. Elastic waves in the medium with spherical inclusions ∗

F(x) = Λ(x − xj )U∗ e−iq

·xj

∗

= Λ(x − xj )eiq

·(x−xj )

F∗ (x).

(7.16)

Here the function Λ(x) = Λ(x, C ∗ , ρ∗ , C, ρ) depends on the dynamic properties of the eﬀective medium and the inclusion. Note that the functions Λ(x) may be constructed from the solution of the one-particle problem for an inclusion centered at the point x = 0. Λ(x) = Λ [F∗ (x)] .

(7.17)

In this equation, Λ is the linear operator of the solution of the one-particle problem for the inclusion centered at the origin. Let us introduce the stationary random function Λ(x) in 3D-space. This j iq∗ ·(x−xj ) inside the inclusion centered at function coincides with Λ(x − x )e the point xi (i = 1, 2, 3, ..), and is equal to zero in the matrix. Using these functions and hypothesis I1 of the EMM, we present the wave ﬁeld F(x) in the composite in a form that follows from (7.9), (7.16): 0 ∗ (x )V (x )dx , (7.18) F(x) = F (x) + K∗ (x − x )L1∗ Λ(x)F v

where F0 (x) are the plane wave (7.10), (7.7). In order to ﬁnd the mean value of the wave ﬁeld F(x), let us average both sides of (7.18) over the ensemble realizations of the random set of inclusions, and take into account the condition of self-consistency (hypothesis I2 ) F∗ (x) = F(x).

(7.19)

As a result, we ﬁnd the integral equation for the mean wave ﬁeld F(x) in the form (7.20) F(x) = F0 (x) + p K(x − x )L1 Λ∗ F(x )dx , where p = V (x) is the volume concentration of inclusions, Λ∗ is a symbolic matrix with constant components 1 Λ(x)dx. (7.21) Λ∗ = v v Here v is the volume of a typical inclusion. The detailed equation for the matrix Λ∗ has the form ∗ λij , 0 ∗ , (7.22) Λ = 0, Λ∗ijkl where the constant tensors λ∗ij and Λ∗ijkl are the following integrals over the volume of the inclusion

7.1 Version I of the EMM for elastic waves

λ∗ik =

1 v

λuik (x)eiq∗·x dx, Λ∗ijkl =

v

1 v

193

Λεijkl (x)eiq∗·x dx.

(7.23)

v

In these equations, the tensors λuik (x) and Λεik (x) deﬁne the displacement and strain ﬁelds inside the inclusion centered at the origin by the action of the exciting ﬁeld F∗ (x) ui (x) = λuik (x)u∗k (x),

εij (x) = Λεijkl (x)ε∗kl (x), x ∈ v.

(7.24)

Application of the Fourier transform to (7.20) leads to an algebraic equation for the Fourier transform of the mean wave ﬁeld in the form F(k) = F0 (k) + pK(k)L1 Λ∗ F(k). Here k is a vector parameter of the Fourier transform. Multiplying (7.25) with the matrix L0 0 Lik (k), 0 0 , L (k) = 0, L0ik (k) 0 L0ik (k) = Cijkl kj kl − ρ0 ω 2 δik ,

(7.25)

(7.26) (7.27)

and using the properties of the tensor L0ik (k) ω2 , −iki , L0 (k)F0 (k) = 0, L0 (k)K(k) = −iki ω 2 , −ki kj

(7.28)

we obtain the equation for the Fourier transform of the mean wave ﬁeld in the composite in the form 0 (7.29) L (k) − pM(k)L1 Λ∗ F(k) = 0, ω2 , −iki . (7.30) M(k) = L0 (k)K(k) = −iki ω 2 , −ki kj The symbolic equation (7.29) is a system of two equations, and the detailed form of the ﬁrst is 0 1 Lik (k) − pρ1 ω 2 λ∗ik + pkj Cijmn Λ∗mnkl kl∗ uk (k) = 0. (7.31) The second equation of the system (7.29) is equation (7.31) multiplied by ikj . If the mean wave ﬁeld is a plane wave uk (x) = Ui∗ e−iq∗ ·x , its Fourier transformation is uk (k) = (2π)3 Uk∗ δ(k − q∗ ), and for k = q∗ , the determinant of the tensor in front of uk (k) in (7.31) should be equal to zero. This condition may be written in the form ∗ det qj∗ Cijkl (q ∗ )ql∗ − ω 2 ρ∗ik (q ∗ ) = 0, (7.32) ∗ 0 1 = Cijkl + pCijmn Λ∗mnkl , Cijkl

ρ∗ik = ρ0 δik + pρ1 λ∗ik .

(7.33)

194

7. Elastic waves in the medium with spherical inclusions

Equation (7.32) is the condition of existence of non-trivial solutions of homogeneous equation (7.31). On the other hand (7.31) is the dispersion equation for the wave number q∗ of the mean ﬁeld propagating in the composite medium. The tensors Λ∗ and λ∗ are functions of the dynamic parameters of the eﬀective medium, and should be be found from the solution of the oneparticle problem (7.12). Thus, (7.32), (7.33) are in fact a system of equations ∗ , ρ∗ik and of the composite in the for the unknown eﬀective parameters Cijkl framework of version I the EMM. The phase velocity and attenuation coefﬁcient of the mean wave ﬁeld are connected with the eﬀective wave number q∗ by the equations v∗ =

ω , Re(q∗ )

γ = −Im(q∗ ).

(7.34)

7.2 The one-particle problems of EMM The one-particle problems of the EMM are in fact the problems of diﬀraction of longitudinal and transverse plane monochromatic waves by a spherical inclusion embedded in a homogeneous isotropic eﬀective medium. These problems are considered in detail in [21], and in this section, we present their solutions in forms convenient for the use in self-consistent methods. 7.2.1 Diﬀraction of a plane monochromatic wave by an isolated spherical inclusion For presentation of the solutions of the diﬀraction problems, we introduce the Cartesian (x1 , x2 , x3 ) and spherical (r, ϕ, θ) coordinate systems with the origin at the center of a spherical inclusion of radius a (see Fig. 7.1). The wave x3

er e3

θ

e2

e1 ϕ

eθ

eϕ x2

UL

x1

k* UT

Fig. 7.1. Coordinate systems for the solution of the one-particle problem.

7.2 The one-particle problems of EMM

195

normal n of the incident wave is directed along the x3 -axis that is also the polar axis of the spherical system. The integral equations of the diﬀraction problem have a form similar to (7.1): 0 2 [gik (x − x )ρ1 uk (x ) ui (x) = ui (x) + ω v 1 + j gik (x − x )Ckjmn εmn (x ) dx , (7.35) (i gj)k (x − x )ρ1 uk (x )dx εij (x) = ε0ij (x) + ω 2 v 1 +Kijkl (x − x )Ckjmn εmn (x ) dx , (7.36) where v is the volume occupied by the inclusion. These equations are completely equivalent to the following system of diﬀerential equations µui + (λ + µ) i k uk + ρω 2 ui = 0, µ0 um i

+ (λ0 +

µ0 ) i k um k

+

ρ0 ω 2 um i

= 0,

r ≤ a,

(7.37)

r > a.

(7.38)

Here ui (x) is the displacement ﬁeld inside the inclusion, um i (x) is this ﬁeld in the matrix. The vector of displacements as well as the normal component of the stress tensor should be continuous on the border r = a between the matrix and inclusion, and therefore, we can write ui (a − 0, ϕ, θ) = um i (a + 0, ϕ, θ),

(7.39)

m (a + 0, ϕ, θ). ni σij (a − 0, ϕ, θ) = ni σij

(7.40)

Here ni is the external normal to the surface of the inclusions, f (a ± 0) = limδ→0 f (a ± δ), δ > 0. Let us consider the diﬀraction problem for longitudinal and transverse incident waves separately. Longitudinal wave (L-wave). If the incident wave u0 is longitudinal, it is presented as the series in the spherical vector harmonics L10n [21] u0 = e3 exp(−iα0 x3 ) =

L10n = er

∞ 1 n+1 (−i) (2n + 1)L10n , α0 n=0

djn (α0 r) jn (α0 r) dPn (cos θ) Pn (cos θ) + eθ . dr r dθ

(7.41)

(7.42)

Here e3 is the unit vector along the x3 -axis; er , eθ are the basic vectors of the spherical coordinate system (r, ϕ, θ); jn (z) is the spherical Bessel function, and Pn (cos θ) is the Legendre polynomial of order n. The scattered ﬁeld us = um − u0 in the medium, and the ﬁeld u inside the inclusion (transmitted ﬁeld) may be also presented as series in spherical harmonics

196

7. Elastic waves in the medium with spherical inclusions

us = u=

∞

an L30n + bn N30n ,

(7.43)

n=0 ∞

1 1 an L0n . + bn N0n

(7.44)

n=0

Here the vector harmonics L30n and N30n are L30n = er

dhn (α0 r) hn (α0 r) dPn (cos θ) Pn (cos θ) + eθ , dr r dθ

N30n = er

(7.45)

dPn (cos θ) n(n + 1) 1 d hn (β0 r)Pn (cos θ) + eθ [rhn (β0 r)] , r r dr dθ (7.46)

hn (z) = jn (z) − iyn (z) is the spherical Hankel function of the second kind, 1 1 3 3 and N0n in (7.44) are the harmonics L0n and N0n where the functions L0n hn (z) are replaced by the spherical Bessel function jn (z), and prime denotes that α0 and β0 are replaced by the wave numbers α and β of the material of the inclusion. In (7.43) and (7.44), the constants an , bn , an , bn are to be found from the boundary conditions (7.39), (7.40). For longitudinal waves, these conditions take the forms u0r (a) + usr (a) = ur (a),

u0θ (a) + usθ (a) = uθ (a),

0 s (a) + σrr (a) = σrr (a), σrr

(7.47)

0 s σrθ (a) + σrθ (a) = σrθ (a).

(7.48)

Calculating the components of the displacement and stress ﬁelds from (7.43), (7.44) and substituting them into conditions (7.47), (7.48) we obtain four equations for the constants an , bn , an , bn . This system may be written in the matrix form 1 an an f1 (α0 a) n+1 2n + 1 − Ln = − (−i) Ln , (7.49) bn bn f31 (α0 a) α0 Mn

an bn

−

µ M µ∗ n

an bn

= − (−i)

n+1

2n + 1 α0

f51 (α0 a) f71 (α0 a)

where the matrices Ln and Mn are 2 2 f1 (α0 a), f22 (β0 a) f5 (α0 a), f62 (β0 a) Ln = , M . = n f32 (α0 a), f42 (β0 a) f72 (α0 a), f82 (β0 a)

,

(7.50)

(7.51)

The matrices Ln and Mn in (7.49), (7.50) are deﬁned by the same equa2 1 (α0 a), fl2 (β0 a) are replaced by fm (αa), tions (7.51), but the functions fm 1 i fl (βa). Here fm (qr) (m = 1, 2, ..., 8; i = 1, 2) are the following radial functions

7.2 The one-particle problems of EMM i f1i (αr) = nyni (αr) − αryn+1 (αr),

f2i (βr) = n(n + 1)yni (βr),

i f3i (αr) = yni (αr), f4i (βr) = (n + 1)yni (βr) − βryn+1 (βr), 1 i (αr) f5i (αr) = n2 − n − (βr)2 yni (αr) + 2αryn+1 2 i (βr) , f6i (βr) = n(n + 1) (n − 1)yni (βr) − βryn+1

= (n − − (βr)2 i 2 i yni (βr) + βryn+1 f8 (βr) = n − 1 − (βr), 2

f7i (αr)

1)yni (αr)

i αryn+1 (αr),

197

(7.52) (7.53) (7.54) (7.55) (7.56)

(i = 1, 2).

(7.57)

Here the wave numbers α and β remain without indices for the waves in the inclusion, and take index “0” for the matrix. If i = 1, the functions yn1 (z) are the spherical Bessel functions jn (z), for i = 2, these functions are the spherical Hankel functions hn (z). Transverse waves (T-waves). For a transverse incident wave that propagates along the x3 -axis and is polarized along the x1 -axis, the incident monochromatic wave u0 is presented as the series u0 = e1 e−iβ0 x3 =

∞ n i (−i) (2n + 1) M101n + N101n , n(n + 1) β0 n=1

(7.58)

where the spherical vector harmonics M101n and N1e1n are Pn1 (cos θ) dP 1 (cos θ) cos ϕ − eϕ jn (β0 r) n sin ϕ, (7.59) sin θ dθ n(n + 1) jn (β0 )Pn1 (cos θ) cos ϕ = er r 1 1 1 d θ dPn (cos θ) ϕ Pn (cos θ) + e cos ϕ − e sin ϕ [rjn (β0 r)] . dθ sin θ r dr (7.60)

M101n = eθ jn (β0 r) N1e1n

In this case, the scattered and transmitted ﬁelds have the forms us =

∞

cn L3e1n + dn M3o1n + en N3e1n ,

(7.61)

cn L1e1n + dn M1o1n + en N1e1n .

(7.62)

n=1

u=

∞ n=1

Here M3o1n and N3e1n are the right-hand sides of (7.59), (7.60) where functions jn (z) are replaced by hn (z),

198

7. Elastic waves in the medium with spherical inclusions

d L3e1n = er [hn (α0 r)]Pn1 (cos θ) cos ϕ dr 1 1 hn (α0 r) θ dPn (cos θ) ϕ Pn (cos θ) + e cos ϕ − e sin ϕ . dθ sin θ r

(7.63)

The equations for the vector harmonics L1e1n , M1o1n and N1e1n are determined by the right-hand sides of (7.63), (7.59), and (7.60) where the arguments of the spherical Bessel functions are replaced by αr and βr. The boundary conditions on the surface of the inclusion for the transverse wave follow from (7.39), (7.40) in the forms u0r (a) + usr (a) = ur (a),

u0θ (a) + usθ (a) = uθ (a),

u0ϕ (a) + usϕ (a) = uϕ (a), 0 s (a) + σrr (a) = σrr (a), σrr

(7.64) 0 s σrθ (a) + σrθ (a) = σrθ (a),

0 s σrϕ (a) + σrϕ (a) = σrϕ (a).

(7.65)

These conditions give six equations for the constants in (7.61), (7.62). Four of these equations may be written in a form similar to (7.49), (7.50) 1 cn cn f2 (β0 a) n+1 2n + 1 1 − Ln = (−i) , (7.66) Ln en en n(n + 1) β∗ f41 (β0 a) Mn

cn en

µ − M µ∗ n

cn en

= (−i)

n+1

2n + 1 1 n(n + 1) β∗

f61 (β0 a) f81 (β0 a)

.

(7.67)

Two equations for dn and dn are hn (β0 a)dn − jn (βa)dn = − (−i)

n

f62 (β0 a)dn −

2n + 1 jn (β0 a), n(n + 1)

1 µ 1 n 2n + 1 2f6 (β0 a) . f6 (βa)dn = − (−i) µ∗ n(n + 1) a

(7.68)

(7.69)

The constant dn follows from these equations in the form dn

=

(−i)

−1 µ (2n + 1) 2 1 jn (βa)f6 (β0 a) − h(β∗ a)f6 (βa) . β∗ a µ∗

n+1

(7.70)

7.2.2 An approximate solution of the one-particle problems in the long-wave region In the long-wave region, the elastic ﬁelds ui (x) and εij (x) inside the inclusion may be considered as constant. The approximate solution of the one-particle problem (7.35), (7.36) is based on this assumption. After substituting the

7.2 The one-particle problems of EMM

199

constant ui and εij into the right-hand sides of (7.35), (7.36) and averaging the results over the volume of the inclusion (Galerkin’s scheme) we ﬁnd the following system of linear algebraic equations: 1 εmn , ui = u0i + ρ1 ω 2 Gik uk , εij = ε0ij + Pijkl Cklmn 1 dx gik (x − x )dx , Gik = v v v 1 Pijkl = dx Kijkl (x − x )dx . v v v

(7.71) (7.72) (7.73)

Here the overbar means the mean value over the spherical region v 3 3 0 −iq·x 0 u0i = U e dx, ε = −i U 0 qj) e−iq·x dx. (7.74) ij 4πa3 v i 4πa3 v (i By such averaging of (7.35), (7.36), the double integral over a spherical region 1 dx j gik (x − x )dx = 0 (7.75) v v v disappears. The integrals in (7.72) and (7.73) are reduced to the calculation of the following integrals

e−iq|x−x | dx , I = dx v v |x − x | −iq|x−x | e dx dx. i j Iij = v v |x − x |

(7.76) (7.77)

These integrals are I=

16π 2 a3 [(qa) − 1] , 3

Iij = −

16π 2 a3 (qa)δij , 9

(qa) = −3iqaj1 (qa)h1 (qa).

(7.78) (7.79)

The ﬁnal equations for the tensors Gik and Pijkl take the forms 1 [3 − (α0 a) − 2(β0 a)] , 3ρ0 1 2 2 1 Pijkl = P1 Eijkl + P2 Eijkl − Eijkl , 3 (α0 a) 1 (β0 a) 2(α0 a) P1 = − , P2 = − . + 3(3K0 + 4µ0 ) 5 µ0 3K0 + 4µ0

ω 2 Gik = gδik ,

g=−

(7.80)

(7.81) (7.82)

200

7. Elastic waves in the medium with spherical inclusions

The mean value of the incident ﬁelds over the inclusion are 3j1 (qa) 0 0 3 , u i = Ui e−iq·x dx = Ui0 h(qa), h(qa) = 4πa3 v qa ε0ij = −iq(i Uj) h(qa),

(q = α0 or β0 ).

(7.83) (7.84)

As the result, the solution of (7.71) takes the form −1

Ui0 h(qa),

(7.85)

−1 1 1 εij = −i Eijkl − Pijmn Cmnkl qk Ul0 h(qa).

(7.86)

ui = (1 − ρ1 g)

7.3 The dispersion equations of the EMM For version I of the EMM, the one-particle problem is diﬀraction of a plane monochromatic wave by an isolated spherical inclusion embedded in the homogeneous medium with the eﬀective properties of the composite. The incident ﬁeld acting on the inclusion is the mean wave ﬁeld uk (x) propagating in the composite medium. Thus, in the equations for the displacement and strain ﬁelds inside the inclusion obtained in the previous section, index “0” should be replaced by the index “∗” that corresponds to the eﬀective medium. The explicit forms of the tensors λ∗ik and Λ∗ijkl in the dispersion equation (7.32), (7.33) of the EMM are to be found from the equations 1 ui (x) exp(iq∗ · x)dx, (7.87) λ∗ik uk (x) = λi = v v 1 ∗ Λijkl εkl (x) = Λij = εij (x) exp(iq∗ · x)dx. (7.88) v v Here ui (x), εij (x) are the displacement and strain ﬁelds inside the inclusion, where q∗ =α∗ for longitudinal and q∗ = β ∗ for transverse waves. Let us consider the vector λi and tensor Λij separately for the longitudinal and transverse waves. Longitudinal waves. In accordance with (7.44), the vector λi in (7.87) is the sum of the following integral λ = λL =

∞ 3 1 iα∗ ·x 1 iα∗ ·x a L (x)e dx + b N (x)e dx . n 0n 4πa3 n=0 n v 0n v (7.89)

After calculating these integrals we obtain (n is the wave normal of the incident ﬁeld)

7.3 The dispersion equations of the EMM

λL = λL n, λL = −3α∗ a

∞

(αa)2

f31 (αa)f11 (α∗ a) + gn (α, α∗ ) (α∗ a)2 f 1 (βa)f31 (α∗ a) +bn 2 , (α∗ a)2

(−i)n+1 an

n=0

gn (α, α∗ ) =

1 [αajn+1 (αa)jn (α∗ a) − (α∗ a)2 −α∗ ajn+1 (α∗ a)jn (αa)] .

201

(7.90)

(7.91)

(7.92)

Using equation (7.44) for the wave ﬁeld inside the inclusion, we ﬁnd that the tensor Λij in (7.88) takes the form ∗ 1 L Λij = Λij = (i uj) (x)eiα ·x dx v v 1 L L 3 3 . (7.93) = −iα∗ a Λ1 δij + Λ2 ei ej − δij 3 L Here the coeﬃcients ΛL 1 and Λ2 are the following series

ΛL 1 =−

2 ∞ (αa) (−i)n+1 an gn (α, α∗ ), α∗ a n=0

ΛL 2 =−

∞ 3 9 (−i)n+1 an Fna (α, α∗ ) + bn Fnb (α, α∗ ) − ΛL , (7.95) 3 2(α∗ a) n=0 2 1

(7.94)

Fna (α, α∗ ) = f11 (αa) f21 (α∗ a) − 2f11 (α∗ a) − (α∗ a)2 f31 (α∗ a) + f31 (αa)f61 (α∗ a) + (α∗ a)2 f31 (αa)f11 (α∗ a) + (α∗ a)2 gn (α, α∗ ) , (7.96) Fnb (α, α∗ ) = f21 (βa) f21 (α∗ a) − 2f11 (α∗ a) − (α∗ a)2 f31 (α∗ a) + f41 (βa)f61 (α∗ a) + (α∗ a)2 f21 (βa)f31 (α∗ a).

(7.97)

Transverse waves. In this case, the vector λi and tensor Λij in (7.87) and (7.88) are λ = λT = λT e1 , ∞ 3 n+1 f 1 (αa)jn (β∗ a) i n(n + 1) cn 3 λT = 2 n=1 β∗ a jn (βa)f41 (β∗ a) + (β∗ a)gn (β, β∗ ) , −iadn gn (β, β∗ ) + en β∗ a

(7.98)

(7.99)

202

7. Elastic waves in the medium with spherical inclusions

Λij = ΛTij = −iβ∗ aΛT e1(i e3j) , ΛT = −

3 2(β∗ a)3

∞

(7.100)

in+1 n(n + 1) (cn Hcn + idn Hdn + en Hen ) ,

(7.101)

n=1

Hcn = 2 f11 (αa)f71 (β∗ a) + f31 (αa)f81 (β∗ a) + (β∗ a)2 f31 (αa)f31 (β∗ a), (7.102) (7.103) Hdn = −(β∗ a) jn (βa)f71 (β∗ a) + (β∗ a)2 gn (β, β∗ ) , Hen = 2 f21 (βa)f71 (β∗ a) + f41 (βa)f81 (β∗ a) + (β∗ a)2 jn (βa)f41 (β∗ a) + (β∗ a)2 gn (β, β∗ ) .

(7.104)

In the long-wave region (ω → 0), the principal terms of the asymptotics of the tensors λ∗ij and Λ∗ijkl are the same for the longitudinal and transverse waves. The explicit equations for these tensors were obtained in Chapter 5 (see (5.78)–(5.82)). λ∗ij = λ∗ δij , λ∗ = 1 − i (βs a)

3

ρ − ρs 2 + ηs3 , 9ρs

(7.105)

1 2 2 1 Λ∗ijkl = Λ∗1 Eijkl + Λ∗2 (Eijkl − Eijkl ), (7.106) 3 Λ∗1 = Λs1 + i(α0 a)3 Λω Λ∗2 = Λs2 + i(β0 a)3 Λω (7.107) 1, 2 −1 K − Ks 1 K − Ks 2 Λs1 = 1+3 , Λω (Λs ) , (7.108) 1 =3 3 3Ks + 4µs 3Ks + 4µs 1 −1 2(µ − µs ) s 2 3 + 2η0 , (7.109) Λ2 = 1 + 15µs µ − µs s 2 2 3 + 2η05 Λω (Λ2 ) . (7.110) 2 = 45 µs In these equations, the index “s” (statics) means the static values (ω → 0) of the bulk Ks and shear µs moduli of the eﬀective medium. 7.3.1 The EMM dispersion equation for longitudinal waves For longitudinal waves, the dispersion equation (7.32) of the EMM takes the form 4 − ω 2 ρL α∗2 K∗ + µL (7.111) ∗ = 0, 3 ∗ L K∗ = K0 + 3pK1 ΛL 1 , µ∗ = µ0 + pµ1 Λ2 , ρ∗ = ρ0 + pρ1 λL , L and λL , ΛL 1 and Λ2 are determined in (7.91), (7.94) and (7.95).

(7.112) (7.113)

7.3 The dispersion equations of the EMM

203

L In the long-wave region, the coeﬃcients λL , ΛL 1 and Λ2 should be replaced ∗ ∗ by their asymptotics λ∗ , Λ1 and Λ2 in (7.105), (7.107), and the principal terms of the eﬀective elastic moduli and density of the composite take the forms 3

K∗ = Ks + i (αs a) Kω ,

3

µ∗ = µs + i (αs a) µω , ω 3 ρ∗ = ρs − i (αs a) ρω , αs = , ls / 4 Ms , Ms = Ks + µs . ls = ρs 3

(7.114) (7.115) (7.116)

Substituting these equations into (7.111)–(7.113) we obtain a system of algebraic equations for the eﬀective static moduli Ks and µs of the composite Ks = K0 + pK1 µs = µ0 + pµ1

3Ks + 4µs , 3K + 4µs

(7.117)

5µs (3Ks + 4µs ) , µs (9Ks + 8µs ) + 6µ(Ks + 2µs )

(7.118)

ρs = ρ0 + pρ1 .

(7.119)

Equations (7.117) and (7.118) are a system of algebraic equations for the eﬀective static bulk Ks and shear µs moduli. The equations for the coeﬃcients Kω and µω in (7.114) follow from the dispersion equation (7.111) in the forms µωL Ms K − K , (7.120) KωL = 3pK1 + 3K s (3K − 4µs )2 µs 10p (7.121) µ1 (µ − µs ) 3 + 2ηs5 , ω ηs3 2 2(3 + 2ηs2 ) µ (µ − µs ) − 30pµ1 (3 + 2ηs2 ), (7.122) ω = µs 15 + µs µs µs ρ − ρs 2 + ηs3 , ηs2 = ρωL = pρ1 . (7.123) 3 9ρs ηs Ms Finally, in the long-wave region, the solution of the dispersion equation (7.111) may be written as follows µωL =

α∗ = αs − iγL , ω , ls

(7.124) /

Ms , ρs 1 ρωL 3KωL + 4µωL 4 . + γL a = p (αs a) 2 3Ks + 4µs ρs αs =

ls =

(7.125) (7.126) 4

The attenuation coeﬃcient γL a is proportional to (αs a) , and thus, the attenuation in the long-wave region is caused by the Rayleigh wave scattering on inclusions.

204

7. Elastic waves in the medium with spherical inclusions

7.3.2 The EMM dispersion equation for transverse waves Now consider propagation of transverse waves in the composite. In this case, the dispersion equation (7.32) for the eﬀective wave number β∗ of the transverse waves propagating in the composite takes the form β∗2 µT∗ − ω 2 ρT∗ = 0,

(7.127)

µT∗ = µ0 + pµ1 ΛT2 ,

ρT∗ = ρ0 + pρ1 λT ,

(7.128)

λT , ΛT2

and the coeﬃcients are determined in (7.99), (7.101). In the long-wave region, the principal term of the asymptotic solution of this equation has the form β∗ = βs − iγT , βs2 =

(7.129)

ω 2 ρs 1 4 , γT a = (βs a) µs 2

µωT ρωT + µs ρs

,

(7.130)

and the eﬀective shear modulus and density of the composite are 3

µT∗ = µs + i (βs a) µωT ,

3

ρT∗ = ρs − i (βs a) ρωT .

(7.131)

In these equations, µs and ρs are the eﬀective static shear modulus and static density that coincide with these parameters (7.118), (7.119) for longitudinal waves, µωT =

10p µ1 (µ − µs )(3 + 2ηs5 ), ω

ρωT = pρ1

(7.132)

ρ − ρs (2 + ηs3 ), 9ρs

(7.133)

where ω is determined in (7.122). If the approximate solution of the one-particle problem is used, the coefﬁcients in the dispersion equations (7.111), (7.127) take the forms λL = (1 − ρ∗1 g ∗ ) ΛL 1 =

−1

h2 (α∗ a),

(7.134)

1 −1 (1 − 9P1∗ K∗1 ) h2 (α∗ a), 3

∗ ΛL 2 = (1 − 2P2 µ∗1 )

−1

h2 (α∗ a), (7.135)

λT = (1 − ρ∗1 g ∗ )

−1

h2 (β∗ a),

−1

ΛT = (1 − 2P2∗ µ∗1 )

h2 (β∗ a).

(7.136)

The constants g ∗ , P1∗ , P2∗ are deﬁned in (7.80), (7.82), where the parameters of the matrix should be replaced by the parameters of the eﬀective medium, ρ∗1 = ρ − ρ∗ ,

K∗1 = K − K∗ ,

µ∗1 = µ − µ∗ .

(7.137)

7.3 The dispersion equations of the EMM

205

7.3.3 Total scattering cross-sections of a spherical inclusion Let us consider the long-distant asymptotics of the scattered ﬁeld usi from an isolated inclusion. This ﬁeld is the integral terms in (7.35), and for large |x|, it takes the form (see (5.185)) usi (x) ≈ Ai (n)

e−iα0 r e−iβ0 r + Bi (n) , r r

n=

x , |x|

r = |x|

(7.138)

Here Ai (n) and Bi (n) are the vector amplitudes of the longitudinal and shear waves, scattered in the n-direction. These amplitudes are expressed via the ﬁelds inside the inclusion Ai (n) = ni nk fk (α0 n),

Bi (n) = (δik − ni nk )fk (β0 n)

(7.139)

q2 2 ρ1 ω fk (qn) = uk (x ) exp(iqn·x )dx 4πρ0 ω 2 V 1 −iqnl Clkmn εmn (x ) exp(iqn·x )dx , (q = α0 or β0 ) V

(7.140) The equation for the normalized (divided on πa2 ) total scattering crosssection QL (ω) of a spherical inclusion by diﬀraction of longitudinal waves follows from (5.212) in the form 4 QL (ω) = − Im n0 · A(n0 ) . (7.141) 2 α0 a The normalized total scattering cross-section QT (ω) by the diﬀraction of transverse waves is a consequence of (5.214): 4 Im e0 · B(n0 ) , (e0 · n0 = 0) (7.142) QT (ω) = − 2 β0 a Here n0 is the wave normal of the incident ﬁeld, e0 is a unit vector of the direction of polarization of the transverse incident wave. Thus, the scattering cross-sections of the inclusion for the waves of both types are proportional to the forward scattering amplitudes A(n0 ) and B(n0 ) (the “optical theorem”, Section 5.5). For the L-waves, the integrations in (7.140) and (7.141) give L 4 ρ1 9K1 ΛL 10 + 4µ1 Λ20 , (7.143) λL0 − QL = − (α0 a) Im 3 ρ0 3K0 + 4µ0 L where λL0 , ΛL 10 , Λ20 are deﬁned by (7.90)–(7.95) when α∗ = α0 , β∗ = β0 . For transverse waves, we have 4 ρ1 µ1 (7.144) λT 0 − ΛT 0 , QT = − (β0 a) Im 3 ρ0 µ0

206

7. Elastic waves in the medium with spherical inclusions

where λT 0 and ΛT 0 are determined in (7.99) and (7.101) when α∗ = α0 , β∗ = β0 . The short-wave limit (ω → ∞) of the normalized cross-sections QL (ω) and QT (ω) has the same value 2 (the paradox of extinction) lim QL (ω) = lim QT (ω) = 2.

ω→∞

(7.145)

ω→∞

The character of convergence of the functions L 4 ρ1 9K1 ΛL 10 + 4µ1 Λ20 , QL = − α0 a λL0 − 3 ρ0 3K0 + 4µ0 4 ρ1 µ1 QT = − β0 a λT 0 − ΛT 0 . 3 ρ0 µ0

(7.146) (7.147)

to this limit may be seen from the graphs in Figs. 7.2 (hard and heavy inclusion) and 7.3 (soft and light inclusion). 5

10

QL

4

8

3

6

2

4

1

2

0 −2

QT

Im(QL)

−1.5

−1

−1

−0.5

0

0.5

1

1.5

0 −2

−1.5

−2

Re(QL) lg(α0a)

−2

Im(QT)

−1

−0.5

0 0.5 Re(Q)

1

1.5 lg(β0α)

−4

Fig. 7.2. The real and imaginary parts of the functions QL and QT in (7.146) for a hard and heavy inclusion (E/E0 = 50, ρ/ρ0 = 10). 5

QL

3

Im(QL)

4

2.5

3

2

2

1.5

1

1

0 −2 −1.5 −1 −0.5 −1 −2

0

0.5

1

1.5

Re(QL) lg(α0α)

Im(QT)

QT

Re(QT)

0.5 0 −2 −1.5 −1 −0.5 0 −0.5

0.5

1

1.5 lg(β0a)

Fig. 7.3. The real and imaginary parts of the functions QL and QT in (7.146) for a soft and light inclusion (E/E0 = 0.02, ρ/ρ0 = 0.1)

7.3 The dispersion equations of the EMM

207

7.3.4 The EMM dispersion equations in the short-wave region Consider the solution of the dispersion equation of the EMM in the shortwave limit. In this case ω, α0 , β0 → ∞, and (7.90)–(7.101) imply that L λ L , ΛL 1 , Λ2 , λT , ΛT → 0. As a result, the solution of the dispersion equation (7.111) for longitudinal waves takes the following form

ρ0 + pρ1 λL L K0 + (4/3)µ0 + 3pK1 ΛL 1 + (4/3)pµ1 Λ2 L p ρ1 9K1 ΛL 1 + 4µ1 Λ2 ≈ α0 1 − λL − 2 ρ0 3K0 + 4µ0 3p Q , = α0 1 − 8α0 a L

1/2

α∗ = ω

(7.148)

where QL has the form (7.146). Hence, when α0 → ∞ the phase velocity l∗ and attenuation coeﬃcient γ of the mean wave ﬁeld are l∗ =

ω ω = = l0 , Reα∗ α0

3 p, (7.149) 4a where l0 is the velocity of longitudinal waves in the matrix. Here we take into account the equation (7.150) lim Re QL = 0, lim Im QL = 2. γL = −Imα∗ =

ω→∞

ω→∞

For transverse waves, we obtain in the short-wave region p ρ1 3p µ1 = β0 1 − β∗ ≈ β0 1 − λT − ΛT QT . 2 ρ0 µ0 8β0 a

(7.151)

Because of (7.144), we can write the short wave limits of the velocity and attenuation coeﬃcient in this case ω 3p . (7.152) = t0 , γT = − Im β∗ = t∗ = Re β∗ 8a Thus, the attenuation coeﬃcient γT of transverse waves has the same value as γL γT a = γL a =

3 p, 4

(7.153)

and as for longitudinal waves, the wave velocity of the transverse waves in the short-wave limit coincides with their velocity in the matrix t∗ = t0 .

208

7. Elastic waves in the medium with spherical inclusions

7.4 Versions II and III of EMM for long waves Let us consider versions II and III of the EMM for medium with spherical inclusions. The one-particle problem in these versions is formulated as follows (see Section 2.3). II1 , III1 . Each inclusion in the composite behaves as a kernel of a two layered inclusion embedded in the eﬀective medium. The size and the properties of the kernel coincide with these characteristics of the inclusion, and the properties of the outside layer coincide with the properties of the matrix. The wave that acts on this inclusion is a plane wave propagating in the eﬀective medium. Application of versions II and III of the EMM to elasticity faces the difﬁculty of the solution of the diﬀraction problem for a layered inclusion. Such a solution is presented in [1] and may be used inside the general algorithm of the EMM developed in this chapter. For version II, the integrals λ∗ and Λ∗ in (7.87), (7.88) should be calculated over the kernel of the layered inclusion. The radius a of the kernel coincides with that of the inclusions; the outside √ radius of the layer is a2 = a/ 3 p. The medium outside the layer is the eﬀective medium with the overall properties of the composite. Let us consider version II of the EMM in the long-wave region. The integral equation of the one-particle problem is similar to (7.12), and in the detailed form, it is the system of two integral equations ∗ ω 2 gik (x − x )ρ∗1 (x )uk (x ) ui (x) = u∗i + VK

∗ 1∗ (x − x )Ckjmn (x )εmn (x ) dx , (7.154) + j gik εij (x) =

ε∗ij

∗ ω 2 (j gi)k (x − x )ρ∗1 (x )uk (x )

∗ 1∗ (x − x )Cklmn (x )εmn (x ) dx . + Kijkl

+

VK

(7.155)

Here V K is the region occupied by the inclusion with a layer (Kerner’s cell). In the long-wave region, the incident ﬁelds u∗i and ε∗ij are constant inside ∗ (x) of the eﬀective medium in these equations V K . The Green function gik can be expanded in a series with respect to the frequency ω of the incident ﬁeld. The principal terms of this expansion are (see Section 5.1) (1)

(3)

∗ s (x) = gik (x) − iωgik + iω 3 r2 gik (n), gik

(7.156)

s where gik (x) is the “static” Green function for the eﬀective medium, and (1)

gik = g1 δik , (3)

gik =

g1 =

2 + ηs3 , 12πρs t3s

1 (4 + ηs5 )δik + 2(ηs5 − 1)ni nk . 5 60πρs ts

(7.157) (7.158)

7.4 Versions II and III of EMM for long waves

209

The principal terms of the solution of (7.154), (7.155) are to be found in the forms 3 ω ui (x) = uR i (x) + iω ui (x),

3 ω εij (x) = εR ij (x) + iω εij (x).

(7.159)

Substituting these equations into (7.154), (7.155), and equating the terms of the same power with respect to ω in the left and right hand sides of these equations, we obtain R ∗ ω ∗ ρ1 (x)dx, (7.160) ui = ui , ui = −g1 ui VK

∗ εR ik (x) = εik −

VK

s 1 Kikml (x − x )C∗mlrs (x )εR rs (x )dx ,

1 εω (x) = H C∗rspq (x)εR ikrs ik pq (x)dx VK s 1 − Kikmn (x − x )C∗mnrs (x )εrs (x )dx ,

(7.161)

(7.162)

VK

where the isotropic tensor Hikrs is 1 2 2 1 , + H2 Eijkl − Eijkl Hijkl = H1 Eijkl 3 H1 =

ηs5 , 36πρs t5s

H2 =

3 + 2ηs5 . 60πρs t5s

(7.163)

(7.164)

It follows from (7.160) that K ∗ uω i = g1 v [pρ + (1 − p)ρ0 − ρs ] ui ,

(7.165)

where v K is the volume of the Kerner cell. Equation (7.161) is the equation for the static elastic strain ﬁeld εR ij (x) inside the layered inclusion subjected to a constant incident ﬁeld ε∗ik . The complete solution of this problem is presented in Chapter 3 of Volume 1. As is shown there, the solution of (7.161) has the following form ∗ εR ij (x) = Λijkl (x)εkl ,

1 Λijkl (x) = Eijkl + Aijkl (x),

(7.166)

where the tensor A(x) is A(x) = (E 1 + E 5 (n)D)(5 + D)α1 (r) + (E 2 + E 4 (n)D)α2 (r) + [E 2 + 2E 1 + E 3 (n) + E 4 (n) + 4E 5 (n) D + E 6 (n)D(D − 2)] [α3 (r) − α1 (r)] .

(7.167)

210

7. Elastic waves in the medium with spherical inclusions

d Here r = |x|, n = x/r, D = r dr , E i is the elements of the E-basis deﬁned in Appendix A.1. The three scalar function αk (r) in (7.167) have the following forms inside the i-th layer (ai−1 < r < ai ) :

α1 (r) = Y1i + Y2i r2 + Y3i r−3 + Y4i r−5 ,

(7.168)

α3 (r) = Y5i + Y6i r2 + Y7i r−3 + Y8i r−5 (7.169) 1 i −3 r , α2 (r) = [β(r) − (5 + D)α3 (r)] . (7.170) β(r) = Y9i + Y10 3 Thus, ten constants Yki (k = 1, 2, .., 10) deﬁne the solution inside the ith layer. The algorithm for the construction of all the constants Yki is presented in Chapter 3 of Volume 1. Once the solution of (7.161) is found, the equation for the tensor εω ij (x) follows from (7.162) in the form (x) = Λ (x)H [Cmnrs (x) − C∗mnrs ] Λrspq (x)dx·ε∗pq . (7.171) εω ijkl klmn ij VK

The integral on the right-hand side of this equation can be presented as follows 1 Rijmn = K [Cijkl (x) − C∗ijkl ] Λklmn (x)dx v VK 1 2 2 1 + S2 (Eijmn − Eijmn ), (7.172) = S1 Eijmn 3 S1 = p(K − K∗ )(1 + Y91 ) + (1 − p)(K0 − K∗ )(1 + Y92 ), 7 1 1 1 1 S2 = 2 (µ − µ∗ ) 1 + 3Y1 + 2Y5 + (3Y2 + 2Y6 ) p 5 + 2 (µ0 − µ∗ ) 1 + 3Y12 + 2Y52 (1 − p) 7 2 2 −2/3 − p) . + (3Y2 + 2Y6 )(p 5

(7.173)

(7.174)

The integral in the deﬁnition (7.88) of the tensor Λ∗ij in the dispersion equation (7.32) takes the form 1 0 C 1 Λklmn (x)dx = (Cijkl − Cijkl )ΛK (7.175) klmn , v V ijkl ΛK = p(K − K∗ )(1 + Y91 )E2 7 1 + 2 (µ − µ∗ ) 1 + 3Y11 + 2Y51 + (3Y21 + 2Y61 ) (E 1 − E 2 ), 5 3 (7.176) where V is the region of the kernel of the Kerner cell, and v is its volume.

7.4 Versions II and III of EMM for long waves

211

The ﬁnal equations for the tensors λ∗ij and Λ∗ijkl in the dispersion equation (7.32) are λ∗ij = λρ δij ,

λρ = 1 + iω 3 g1 v [pρ + (1 − p)ρ0 − ρs ] , 1 3 = ΛK ijrs Erskl + iω Hrsmn Rmnkl .

(7.177)

Λ∗ijkl

(7.178)

From these equations and (7.32), we obtain ρs = ρ0 + p(ρ − ρ0 ),

(7.179)

∗ 0 0 = Cijkl + p(Cijmn − Cijmn )ΛK Cijkl mnkl .

(7.180)

The ﬁrst of these equations is the eﬀective density ρs of the composite in the long-wave region, the second is the system of equations for the static elastic moduli. Note that tensor ΛK in (7.176) depends on the elastic constants of the eﬀective medium, and therefore, (7.180) is an equation for these constants. The attenuation of the mean wave ﬁeld is proportional to the imaginary parts of the tensors λ∗ and Λ∗ , and the latter are linear functions of the tensor ΛK . Numerical solutions of (7.180) were used to construct the dependence of the eﬀective Young modulus of the composites with spherical inclusions on their volume concentration p. Such solutions are lines with triangles in Fig. 7.4. The solid line in this ﬁgure shows the results of the numerical solution of the system (7.117), (7.118) for the static constants of the composite in the framework of version I of the EMM. Dots are experimental data presented in [91]. The upper part of the ﬁgure corresponds to hard spherical inclusions 5 E*/E0 4 3 2 1 0

0

0.1

0.2

0.3

0.4

0.5

p

Fig. 7.4. The dependence of the eﬀective Young’s modulus of the composites with spherical inclusions on their volume concentration p. Lines with triangles are predictions of version II of the EMM (7.180), solid lines are predictions of version I of the EMM (7.117), (7.118). Dots are experimental data presented in [91]. The upper part of the picture corresponds to the case of hard spherical inclusions (E/E0 = 28.7, ν0 = 0.394, ν = 0.23), and the lower part to spherical pores.

212

7. Elastic waves in the medium with spherical inclusions

(E/E0 = 28.7, ν0 = 0.394, ν = 0.23), and the lower part to spherical pores. It is seen from this ﬁgure that version II of the EMM allows us to describe static experimental data in the region of high volume concentration of inclusions (p > 0.3) better than version I does. After solution of (7.180) and calculation of the eﬀective elastic moduli tensor C ∗ and density ρs , we can ﬁnd the imaginary parts of the tensors λ∗ and Λ∗ in (7.177), (7.178). The imaginary part of the tensor λ∗ in (7.177) is zero because of (7.179). Direct calculation of the components of the tensor R in (7.172), where C ∗ is found from (7.180), shows that R is also zero for arbitrary elastic moduli of the matrix and inclusions. In order to understand this result, let us rewrite equation R = 0 (see (7.172)) in the following form 1 1 Cijkl (x)Λklmn (x)dx = C∗ijkl K Λklmn (x)dx. (7.181) vK V K v VK After multiplication of both sides of this equation by an arbitrary external ﬁeld ε∗ we ﬁnd that its left hand side is the mean stress ﬁeld averaged over the Kerner cell (σV K ). The right hand side is the product of the eﬀective moduli tensor C∗ with the mean strain ﬁeld averaged over the Kerner cell (εV K ). Thus the mean stress and strain ﬁelds over the Kerner cell are connected by the equation σV K = C∗ εV K

(7.182)

that is valid for the eﬀective medium. Note that (7.182) may be considered as an independent condition of self-consistency, and it was proposed in this form in [12]. Direct calculation shows that condition (7.182) gives the same results for the eﬀective elastic moduli of composites as version II of the EMM. Equivalence of the tensor R to zero means that the imaginary part of the tensor Λ∗ in (7.178) and the tensor εω ij (x) in (7.171) disappear. As a result, the terms proportional to ω 3 are absent from the long-wave asymptotics of the mean wave ﬁeld in the composite medium. Thus, the attenuation of the mean wave ﬁeld does not correspond to the Rayleigh scattering of waves on inclusions in the composite (γ is equal to zero in the long wave region). The same result was obtained in Chapter 3 when version II of the EMM was applied to electromagnetic wave propagation through composites. Another important consequence of the equation R = 0 may be obtained if we consider the far ﬁeld scattered by the Kerner cell. Equations (7.154) and (7.179) imply that the far scattering ﬁeld us (x) has the form ∗ us (x) ≈ ∂j gik (x)Rkjmn ε∗mn (0).

(7.183)

The ﬁrst integral term in (7.154) disappears because of (7.179) (ρ(x) − ρs )dx = v l ρ0 + v k ρ − (v l + v k )(ρ0 + p(ρ − ρ0 )) = 0.

(7.184)

V

k

7.5 Numerical solution of the EMM dispersion equations

213

Here v l is the volume of the layer, v is the volume of the kernel, p = v/(v l +v). Thus, if R = 0 (or if tensor C∗ satisﬁes (7.180)), the scattered ﬁeld in the far zone disappears. It is clear that if the condition of self-consistency is formulated as in version III of the EMM (parameters of the eﬀective medium should be chosen in order to eliminate the ﬁeld scattered by the layered inclusion), tensor R should be zero, and this means that the tensor C∗ should satisfy (7.180). Thus, in the long-wave region, versions II and III of EMM give the same results. They improve the predictions for the static elastic constants for high volume concentrations of inclusions. But these versions do not correctly describe the attenuation of the waves even for small volume concentrations of the inclusions.

7.5 Numerical solution of the EMM dispersion equations The iterative procedure that was used for the solution of the dispersion equation of the electromagnetic wave propagation may be applied to the solution of the longitudinal wave dispersion equation (7.111). For the application of this procedure, these equations should be rewritten in the form 8 8 9 L(n) 9 L(n) 9 ρ∗ 9 ρ∗ L(n) L(n) , β∗ = ω: = ω : L(n) , (7.185) α∗ L(n) L(n) µ∗ K∗ + 43 µ∗ L(n)

K∗

L(n) µ∗

L(n)

ρ∗

L(n−1)

= K∗

=

L(n−1) µ∗

L(n−1)

= ρ∗

+δ

L(n−1)

K0 + 3pK1 ΛL 1 (α∗

+δ

µ0 +

L(n−1) L(n−1) pµ1 ΛP , β∗ ) 2 (α∗

+δ

L(n−1)

, β∗

L(n−1)

ρ0 + pρ1 λL (α∗

L(n−1)

, β∗

L(n−1) , ) − K∗ −

L(n−1) µ∗

L(n−1)

) − ρ∗

(7.186)

, (7.187) . (7.188)

Here the index n corresponds to the number of the iteration, the parameter δ (|δ| < 1) is to be chosen for convergence of the iterative process, the L functions λL and ΛL 1 , Λ2 are deﬁned in (7.91)–(7.95). L The eﬀective elastic moduli K∗L , µL ∗ and density ρ∗ of the composite medium in the nth step of the iterations take the forms L(n)

K∗L = K0 + 3pK1 ΛL 1 (α∗ µL ∗

=

ρL ∗ =

L(n)

, β∗

L(n) L(n) µ0 + pµ1 ΛP , β∗ ), 2 (α∗ L(n) L(n) ρ0 + pρ1 λL (α∗ , β∗ ).

),

(7.189) (7.190) (7.191)

214

7. Elastic waves in the medium with spherical inclusions

For the transverse incident ﬁeld, the iterative scheme is carried out on the basis of (7.127) and (7.128) 8 8 9 T (n) 9 T (n) 9 ρ∗ 9 ρ∗ T (n) T (n) , β∗ = ω: = ω : T (n) . (7.192) α∗ T (n) T (n) µ∗ K∗ + 43 µ∗ T (n+1)

K∗

T (n)

= K∗

+δ

T (n) T (n) T (n) − K∗ , K0 + 3pK1 ΛT1 α∗ , β∗

(7.193) T (n+1)

µ∗

T (n+1)

ρ∗

T (n)

T (n)

T (n)

T (n)

− µ∗ , µ0 + pµ1 ΛT2 α∗ , β∗

T (n) T (n) T (n) T (n) − ρ∗ , = ρ∗ + δ ρ0 + pρ1 λT α∗ , β∗ = µ∗

+δ

(7.194) (7.195)

and the eﬀective parameters of the composite are calculated from the equations T (n) T (n) , (7.196) K∗T = K0 + 3pK1 ΛT1 α∗ , β∗ T (n) T (n) , µT∗ = µ0 + pµ1 ΛT2 α∗ , β∗ T (n) T (n) . ρT∗ = ρ0 + pρ1 λT α∗ , β∗

(7.197) (7.198)

The results of calculations of the phase velocities and attenuation coeﬃcients of longitudinal and transverse waves in composites with inclusions of various properties are presented in Figs. 7.5–7.8. Composites with two types of inclusions were considered: inclusions that are heavier and harder than the l*

p=0.3

lg(γ La) −1

1.5

p=0.1 −3

1.3 p=0.1

−5

1.1 p=0.3 0.9 −2

−1

lg(α0a) 0

1

−7 −2

lg(α0a) −1

0

1

Fig. 7.5. The dependence of vL and attenuation coeﬃcient γ of the longitudinal waves on the frequency of the incident ﬁeld (wave number of the matrix material α0 ) for a medium with heavy and hard inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4).

7.5 Numerical solution of the EMM dispersion equations 0.65

t*

215

p=0.3

0 lg(γTa)

0.6 p=0.1

−2

0.55 p=0.3 0.5

−4

p=0.1 0.45 0.4 −2

lg(β0a) −1

0

1

lg(β0a)

−6 2 −1.5

−0.75

0

1.5

0.75

Fig. 7.6. The dependence of the velocity vT attenuation coeﬃcient γ of the transverse mean wave ﬁeld on the frequency of the incident ﬁeld (wave number of the matrix material β0 ) for the medium with heavy and hard inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4). 1.6

0

l*

p=0.1

1.3

lg(γ La)

−2 p=0.3

1

0.7 −2

−1

p=0.1

−4

p=0.3

0

lg(α0α)

1

−6 −2

−1

0

lg(α0α)

1

Fig. 7.7. The dependence of the velocity vL and attenuation coeﬃcient γ of the longitudinal waves on the frequency of the incident ﬁeld (wave number of the matrix material α0 ) for the medium with light and soft inclusions (ρ/ρ0 = 0.1, E/E0 = 0.02, ν = 0.3, ν0 = 0.4). lg(γ T a)

t* −1

0.6 p=0.1

−3

0.55

p=0.3 0.5

0.45 −2

−1

p=0.1

−5

p=0.3

0

1

lg(β0a)

−7 −2

−1

0

1

lg(β0a)

Fig. 7.8. The dependence of velocity vT attenuation coeﬃcient γ of the transverse mean wave ﬁeld on the frequency of the incident ﬁeld (wave number of the matrix material β0 ) for the medium with heavy and hard inclusions (ρ/ρ0 = 0.1, E/E0 = 0.02, ν = 0.3, ν0 = 0.4).

216

7. Elastic waves in the medium with spherical inclusions

matrix (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4); inclusions that are lighter and softer than the matrix (ρ/ρ0 = 0.1, E/E0 = 0.02, ν = 0.3, ν0 = 0.4). Here E, E0 are the Young’s moduli of the inclusions and the matrix, ν, ν0 are the Poisson’s ratios of the latter. The calculation for two volume concentrations of the inclusions p = 0.1 and p = 0.3 are presented in the ﬁgures. The lines with dots and triangles in these ﬁgures are the result of the use of the approximate solution of the one-particle problem obtained in Section 7.2.2 in the framework of the EMM. The exact and approximate solutions give close predictions for the velocities of the mean wave ﬁelds, but their predictions for the attenuations are close only in the long-wave region. The dashed horizontal lines in these pictures are the short-wave asymptotics for the velocities and attenuation coeﬃcients in the framework of the EMM (Section 7.3.4). Note that “longitudinal” (7.190) and “transverse” (7.197) eﬀective parameters of composites coincide only in the long-wave region. The dependence T of the real and imaginary parts of the eﬀective shear moduli µL ∗ and µ∗ of the composite with heavy and hard inclusions on the longitudinal wave number for the matrix are presented in Fig. 7.9. The dependence of the bulk moduli T (K∗L and K∗T ) and densities (ρL ∗ and ρ∗ ) behave similarly: they coincide in the long-wave region, deviate in the medium-wave region, and become close in the short-wave region. Comparisons with experimental data are shown in Fig. 7.10. The experimental data for the velocities and attenuation coeﬃcients of longitudinal incident waves were presented in [62] for composites with a matrix of PMM (polymetilmetacrelate) and steel spherical inclusions of radii a1 = 0.55 mm and volume concentration p = 0.115. The densities of these materials are ρP M M = 1,160kg/m3 , ρST = 7,800kg/m3 . Velocities of longitudinal waves lP M M = 2,630m/s, lST = 2,630m/s, and velocities of transverse waves tP M M = 1,320m/s, tST = 3,220m/s. p=0.1

Re(µL), Re(µT) 0.45

0.1

p=0.1

Im(µL), Im(µT)

0 −0.1

0.35 −0.2

0.25 −2 −1.5 −1 −0.5

lg(α0a) 0

0.5

1

1.5

−0.3 −2 −1.5 −1 −0.5

lg(α0a) 0

0.5

1

1.5

Fig. 7.9. The dependence of the real and imaginary parts of the eﬀective “longiT tudinal” µL ∗ and “transverse” µ∗ shear moduli of the composite with heavy and hard inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4) on the longitudinal wave number for the matrix.

7.6 The eﬀective ﬁeld method 1.1

217

γ La

l*/l0 0.2 1

0.1

0.9

0.8 0

0.5

1

1.5

2

2.5 α0a

0

0

0.5

1

1.5

2

2.5

α0a

Fig. 7.10. The dependence of the velocity and attenuation coeﬃcient of the longitudinal wave on the wave number of the matrix material for a composite with spherical inclusions. Dots are experimental data in [62].

7.6 The eﬀective ﬁeld method Using the symbolic integral equation (7.9) we may present the wave ﬁeld inside an arbitrary inclusion vj in the composite in the form F(x) = F∗ (x) + K(x − x )L1 F(x )dx , (7.199) vj

F∗ (x) = F0 (x) +

K(x − x )L1 F(x )V (x , x)dx .

(7.200)

Here V (x; x ) is the characteristic function (with argument x ) of the region Vx deﬁned by the equation Vx = vi if x ∈ vj . (7.201) i=j

As follows from (7.201), the function V (x; x ) is equal to zero if the points x and x are inside the same inclusion. Thus, the integral term in (7.200) is the sum of ﬁelds scattered by all the inclusions except the one that occupies region vj if x ∈ vj . The ﬁeld F∗ (x) may be considered as the local exciting ﬁeld for the inclusion that occupies region vj . Let (7.199) may be solved for an arbitrary exciting ﬁeld F∗ (x), and the ﬁeld F(x) inside the inclusion centered at the point xj can be presented in the form F(x) = Λj F∗ (x),

(7.202)

where Λj is the linear operator of the solution of the diﬀraction problem for the inclusion vj . equations (7.199), (7.202) imply that the ﬁeld F(x) in the medium is expressed via the ﬁeld F∗ (x) in the form

218

7. Elastic waves in the medium with spherical inclusions

F(x) = F0 (x) +

K(x − x )L1 ΛF∗ (x )V (x )dx .

(7.203)

Here the function ΛF∗ (x) coincides with Λj F∗ (x) in the jth inclusion (j = 1, 2, 3, ...). The linear operator Λj may be presented in the form of an integral operator Λj (x, x )F∗ (x )dx , (7.204) Λj F∗ (x) = vj

where Λj (x, x ) is a generalized function known from the solution of the oneparticle problem. Points x and x belong to the same region vj because the ﬁeld inside jth inclusion depends only on the values of the local external ﬁelds in the region vj . The equation for the local exciting ﬁeld F∗ (x) that acts on an arbitrary inclusion follows from its deﬁnitions as a sum of the incident ﬁelds and the ﬁelds scattered by the surrounding inclusions (7.205) F∗ (x) = F0 (x) + P(x − x )L1 ΛF∗ (x )V (x , x)dx . Here the function V (x , x) is deﬁned in (7.201). 7.6.1 The hypotheses of the EFM Equations (7.203) and (7.205) show that the exciting ﬁeld F∗ (x) may be considered as the principal unknowns of the problem. The hypotheses of the EFM concern the structure of the ﬁeld F∗ (x). Let us introduce the ﬁrst of these hypotheses (H1 ). H1 . Each inclusion in the composite behaves as an isolated one in the matrix under the action of the local exciting ﬁeld Fe∗ (x). In the vicinity of the inclusion centered at the point x = xj , this ﬁeld has the form j

Fe∗ (x) = f ∗ (x − xj )e−iq∗ ·x .

(7.206)

Here the function f ∗ (x) and vector q∗ are the same for all the inclusions. This hypothesis reduces the problem of interaction between many inclusions to the solution of the equation K(x − x )L1 F(x )V (x )dx , x ∈ vj (7.207) F(x) = Fe∗ (x) + vj

that deﬁnes the wave ﬁeld inside any inclusion v j . After solution of this equation, the ﬁeld inside the inclusion centered at the point x = xj may be presented in the form similar to (7.202): F(x) = Λj Fe∗ (x),

(7.208)

7.6 The eﬀective ﬁeld method

219

where the operator Λj depends on the properties of the matrix and the inclusion. e∗ (x) that coincides with Λj Fe∗ (x) when If we introduce the function ΛF x ∈ vj , (7.203) for the wave ﬁeld F(x) in the composite is presented in the form e∗ (x )V (x )dx . (7.209) F(x) = F0 (x) + p K(x − x )L1 ΛF Averaging this equation over the ensemble realization of the random set of inclusions we obtain the mean wave ﬁeld F(x) in the form

e (x )|x dx . (7.210) F(x) = F0 (x) + p K(x − x )L1 ΛF ∗ Equation (7.200) for the local exciting ﬁeld that acts on the inclusion v takes the form ∗ )Fe (x )V (x ; x)dx . (7.211) F (x) = F0 (x) + K(x − x )L1 Λ(x ∗ The second hypothesis of the EFM is formulated as follows. H2 . The conditional mean Fe∗ (x), and the conditional mean of the local exciting ﬁelds F∗ (x) averaged over the ensemble realization of the random set of inclusions, coincide ∗ (x) = F∗ (x)|x = Fe (x)|x . F ∗

(7.212)

Note that the hypotheses H1 , H2 are equivalent to the quasicrystalline approximation discussed in Section 2.2. It follows from (7.206) that the mean Fe∗ (x)|x is presented in the form ; < e ∗ j iq∗ ·(x−xj ) f (x − x )e |x e−iq∗ ·x , (7.213) F∗ (x) = F (x)|x = ∗

j

; where

<

j f ∗ (x − xj )eiq∗ ·(x−x ) |x

is a constant vector because the aver-

j

aged ﬁeld is stationary in space. Averaging (7.211) under the condition x ∈ V and using (7.212), we ﬁnd the following equation

∗ (x) = F0 (x) + p K(x − x )L1 Λ(x )Fe∗ (x )|x Ψ (x − x )dx , (7.214) F Ψ (x − x ) =

1 V (x; x )|x. p

(7.215)

After excluding the incident ﬁelds F0 (x) from (7.210) and (7.214) we ∗ (x) in the form obtain the equation for the ﬁeld F

220

7. Elastic waves in the medium with spherical inclusions

∗ (x) = F(x) − p F

)Fe∗ (x )|x Φ(x − x )dx , (7.216) K(x − x )L Λ(x

Φ(x) = 1 − Ψ (x).

(7.217)

Here the function Φ(x) is equal to zero outside a ﬁnite vicinity of the origin (x = 0). The size of this vicinity has the order of the correlation radius of the random set of inclusions. 7.6.2 The eﬀective ﬁeld for transverse waves For a shear (transverse) incident wave, the displacement vector has the form u0 (x) = U0 m exp(−iβ0 n0 ·x),

(7.218)

where n0 is wave normal, β0 is the wave number of the waves in the matrix, U0 is the amplitude of the incident ﬁeld, and the vectors m and n0 are orthogonal. For a homogeneous and isotropic random set of inclusions, it is reasonable to consider the local exciting ﬁeld Fe∗ (x) for any inclusion as a plane wave with an unknown wave number β∗ , the wave normal n = n0 , and the direction of the polarization vector m coinciding with those of the incident ﬁeld. Such a wave may be presented in the form j

Fe∗ (x) = f ∗ (x − xj )e−β∗ n·x , mi U∗u exp(−iβ∗ n · x). f ∗ (x) = −iβ∗ n(i mj) U∗ε

(7.219) (7.220)

Note that amplitudes U∗u and U∗ε in these equations do not coincide because, generally speaking, the conditional mean of a derivative does not coincide with the derivative of the conditional mean (∂j u∗i (x)|x = ∂j u∗i (x)|x). ∗ in (7.213) For the local exciting ﬁeld (7.219), (7.220), the eﬀective ﬁeld F takes the form mi U∗u ∗ exp(−iβ∗ n · x). (7.221) F∗ (x) = f (x) = − iβ∗ n(i mj) U∗ε Because (7.216) is a convolution equation, the mean ﬁelds F(x) is also a plane shear wave with the wave number β∗ of the eﬀective ﬁeld mi U ui (x) = exp(−iβ∗ n · x). (7.222) F(x) = εij (x) −iβ∗ n(i mj) U

e For the mean Λ(x)F ∗ (x)|x in (7.216) we have

∗ (x). e (x)|x = Λ0 F ΛF ∗

(7.223)

7.6 The eﬀective ﬁeld method

221

The constant matrix Λ0 in this equation is to be found from the solution of the one particle problem (7.199). The detailed form of this matrix is 0 λij , 0 , (7.224) Λ0 = 0, Λ0ijkl where the tensors λ0ik and Λ0ijkl are the integrals over the volume v of the inclusion centered in the origin 1 λu (x) exp(iβ∗ ·x))dx, (7.225) λ0ik = v v ik 1 Λ0ijkl = Λε (x) exp(iβ∗ ·x)dx. (7.226) v v ijkl The functions λuik (x) and Λεijkl (x) in this equations deﬁne the ﬁeld inside the inclusion by diﬀraction of the eﬀective ﬁelds u ∗k (x) and ε∗kl (x) u∗k (x), ui (x) = λuik (x)

εij (x) = Λεijkl (x) ε∗kl (x),

x ∈ v.

(7.227)

Because (7.210) and (7.216) are convolution equations, the Fourier transforms of these equations give us a system of linear algebraic equations with respect to the Fourier transforms of the unknown functions ∗ (k), F(k) = F0 (k) + pK(k)L1 Λ0 F

(7.228)

∗ (k) = F(k) − pKΦ (k)L1 Λ0 F ∗ (k), F KΦ (k) = K(x)Φ(x) exp(ik · x)dx.

(7.229) (7.230)

Equations (7.228) and (7.229) imply that the Fourier transform of the mean wave ﬁeld in the composite satisﬁes the following equation F(k) = F0 (x) + pK(k)L1 Λ0 D−1 F(k) , Φ

1

0

D = I + pK (k)L Λ .

(7.231) (7.232)

Multiplying (7.231) by the matrix L0 (k) deﬁned in (7.26), and using properties (7.28) of this matrix, we ﬁnd the equation (7.233) L0 (k) − pM(k)L1 Λ0 D−1 F(k) = 0. Here the matrix M(k) is deﬁned in (7.30), the symbolic matrix D has the following detailed form tij (k), Tijk (k) , (7.234) D= πijk (k), Πijkl (k)

222

7. Elastic waves in the medium with spherical inclusions 0 tik (k) = δik + pρ1 ω 2 GΦ ir (k)λrk ,

(7.235)

Φ 1 Tikl (k) = pΓijr (k)Cjrmn Λ0mnkl ,

(7.236)

Φ (k)λ0rk , πijk (k) = pρ1 ω 2 Γ(ij)r

(7.237)

1 Φ 1 Πijkl (k) = Eijkl + pKijrs (k)Crspq Λ0pqkl .

(7.238)

Φ Φ In these equations, GΦ ik (k), Γijk (k), Kijkl (k) are the following integrals

GΦ ik (k) = Φ (k) = Γijk Φ (k) = Kijkl

gik (x)Φ(x)eik·x dx,

(7.239)

∂j gik (x)Φ(x)eik·x dx,

(7.240)

Kijkl (x)Φ(x)eik·x dx.

(7.241)

For transverse wave propagation, mk 3 U δ (k − β∗ n) , F(k) = (2π) −iβ∗ n(i mj)

(7.242)

and the symbolic equation (7.233) is in fact two equations. The ﬁrst has the form 2 (7.243) k µ∗ (k) − ω 2 ρ∗ (k) U δ(k − β∗ n) = 0, µ1 0 ρ1 Λ (t − π), ρ∗ (k) = ρ0 + p λ0 (Π − T ), ∆ ∆ and the second is this equation multiplied with ikj . Here µ∗ (k) = µ0 + p

λ0 = mi λ0ik mk ,

Λ0 = mi nj Λ0ijkl mk nl ,

t(β∗ ) = mi tik (β∗ )mk , T (β∗ ) = −iβ∗ mi Tikl (β∗ )mk nl , i ni mj πijk (β∗ )mk , Π(β∗ ) = ni mj Πijkl (β∗ )mk nl , π(β∗ ) = β∗ ∆ = Πt − T π.

(7.244)

(7.245) (7.246) (7.247) (7.248)

The multiplier in front of the delta-function in (7.243) should be zero for k = β∗ . This condition gives the following dispersion equation for the eﬀective wave number β∗ β∗2 µ∗ (β∗ ) − ω 2 ρ∗ (β∗ ) = 0, pµ1 0 Λ (t − π), µ∗ (β∗ ) = µ0 + ∆ pρ1 0 λ (Π − T ), ρ∗ (β∗ ) = ρ0 + ∆

(7.249) (7.250) (7.251)

7.6 The eﬀective ﬁeld method

223

where scalar coeﬃcients t, T, π and Π and functions tik (β∗ ), Tikl (β∗ ), πijk (β∗ ), and Πijkl (β∗ ) are deﬁned in (7.246), (7.248) and in (7.235)–(7.238). Equation (7.249) is the equation for the unknown eﬀective wave number β∗ of the mean shear wave propagating through the composite medium. Note that Λ0 and λ0 are functions of the wave number β∗ , and these functions have to be found from the solution of the one-particle problem (7.199). 7.6.3 The eﬀective ﬁeld equations for longitudinal waves The longitudinal incident wave has the form u0 (x) = U0 n0 exp(−iα0 n0 ·x),

(7.252)

where n0 is the wave normal, α0 is the wave number of the longitudinal waves in the matrix material, U0 is the amplitude of this ﬁeld. For a homogeneous and isotropic random set of inclusions, the mean wave ﬁeld F(x) should be also a plane longitudinal wave with an unknown wave number α∗ , wave normal n = n0 , and polarization vector U = U n, ui (x) , (7.253) F(x) = εij (x) ui (x) = ni U exp(−iα∗ n · x), εij (x) = −iα∗ ni nj U exp(−iα∗ n · x).

(7.254) (7.255)

The matrix equation (7.216) for the eﬀective ﬁeld has the following detailed form u ∗i (x) = ui (x) − pω 2 gij (x − x )ρ1 λjk ue∗k (x )|x Φ(x − x )dx − p k gij (x − x )ρ1 Λjklm εe∗lm (x )|x Φ(x − x )dx , (7.256) ∗ 2 (i gj)k (x − x )ρ1 λjk ue∗k (x )|x Φ(x − x )dx εij (x) = εij (x) − pω − p Kijkl (x − x )ρ1 Λjklm εe∗lm (x )|x Φ(x − x )dx . (7.257) Let us take the mean local eﬀective displacement ﬁeld ue∗i (x) in the form of a plane longitudinal wave with unknown wave number α∗ and amplitude U∗u propagating in the direction n of the incident ﬁeld j

ue∗i (x) = fi (x − xj )e−iα∗ ·x = U∗u ni e−iα∗ ·x ,

α∗ = α∗ n.

(7.258)

The local eﬀective strain ﬁeld in the vicinity of the inclusion with the center at point xj will be taken in the form

224

7. Elastic waves in the medium with spherical inclusions

Uv

Up

Fig. 7.11. The local exciting ﬁeld acting on an inclusion for longitudinal wave propagation in the composite.

εe∗ij (x)

−iα∗ n·xj

= −iα∗ U∗p ni nj e−iα∗ n·(x−x j j s v (x − x )i (x − x )j d (j1 (α∗ r)) e−iα∗ n·x , +U∗ j 2 |x − x | dr

= Fij (x − x )e j

j

)

(7.259)

Here the ﬁrst term on the right hand side corresponds to a plane longitudinal wave that is similar to (7.258), the second term is a radial wave that acts on the inclusion with the center at point xj , j1 (z) is the spherical Bessel function of the ﬁrst kind, U∗p , U∗v are unknown amplitudes of these ﬁelds (see Fig. 7.11). The function f ∗ (x) that corresponds to (7.256) and (7.257) has the form U∗u ni e−iα∗ ·x p . (7.260) f ∗ (x) = x x d −iα∗ U∗ ni nj e−iα∗ n·x + U∗v ri 2 j dr (j1 (α∗ r)) It should be emphasized that the local exciting strain ﬁeld εe∗ij (x) cannot be taken in the form of the ﬁrst term in (7.259) only, because the system (7.256), (7.257) has no solutions for such a εe∗ij (x). If εe∗ij (x) = −iα∗ U∗p ni nj e−iα∗ n·x , the eﬀective strain tensor ε∗rs (x) coincides with εe∗ij (x), 1 Λjklm εe∗lm (x )|x Φ(x − x )dx and the integral term Kijkl (x − x )Cklmn in (7.257) consists of two terms. The ﬁrst is proportional to U∗p δij and the second to U∗p ni nj . All other terms in this equation turn out to be proportional to the tensor ni nj only. Because δij and ni nj are linearly independent tensors, the coeﬃcient in front of δij should be equal to zero. Thus U∗p = 0, and the tensor ε∗rs (x) disappears. As a result, the system (7.256), (7.257) will have no solution. The choice of εe∗ij (x) in the form (7.259) allows us to obtain non-trivial values for the coeﬃcients U∗u , U∗p and U∗v , and to ﬁnd unique solutions for the eﬀective displacement and strain ﬁelds u ∗i (x), ε∗ij (x). ∗ (x) takes It follows from (7.213) and (7.260) that the eﬀective ﬁeld F the form

7.6 The eﬀective ﬁeld method

∗ (x) = 1 F v

∗

f ∗ (x )eiα

·x

∗

dx e−iα

·x

=

v

u ˆ∗i (x) ε∗ij (x)

225

,

∗

u ˆ∗i (x) = U∗u ni e−iα ·x ,

ε∗ij (x) = −iα∗ ni nj U∗p + Fij (α∗ )U∗v e−iα∗ n·x ,

(7.261) (7.262) (7.263)

Fij (α∗ ) = F1 δij + F2 ni nj , (7.264) 2 j1 (α∗ a) j1 (α∗ a) 3j1 (α∗ a) j0 (α∗ a) − , F2 = F1 = 3 α∗ a α∗ a α∗ a 3 + 2((α∗ a)2 − 1) + 2 cos(2α∗ a) + α∗ a sin(α∗ a) . 4 8(α∗ a) (7.265)

e for the local exciting ﬁeld (7.260) takes the The mean Λ(x)F ∗ (x)|x following form

λ ue (x)|x ij ∗i e Λ(x)F∗ (x)|x = . (7.266) Λijkl εe∗kl (x)|x Here the operator λij and Λijkl deﬁne the displacement ui (x) and strain εij (x) ﬁelds inside the inclusion by the action of the exciting ﬁelds ue∗i (x) and εe∗kl (x) ui (x) = λij ue∗i (x), εij (x) = Λijkl εe∗kl (x).

(7.267)

Equation (7.216) is a convolution equation, and application of the Fourier transform to this equation gives us the following equation ∗ (k) = F(k) − pKΦ (k)ΛFe (k). F ∗ KΦ (k) = K(x)Φ(x)eik·x dx, e )Fe∗ (x )|x eik·x dx. Λ(x ΛF∗ (k) =

(7.268) (7.269) (7.270)

The Fourier transforms of the components of the eﬀective ﬁeld in (7.261) have the forms u ∗k (k) = (2π)3 nk U∗u δ(k−α∗ n), ε∗kl (k) = −(2π)3 iα∗ nk nl U∗p + Fkl U∗v δ(k−α∗ n). The Fourier transform ΛFe∗ (k) in (7.270) is nk λU∗u e 3 δ(k−α∗ n), ΛF∗ (k) = (2π) −iα∗ Λpij U∗p + Λvij U∗v

(7.271) (7.272)

(7.273)

226

7. Elastic waves in the medium with spherical inclusions

Λpij = Λp1 δij + Λp2 ni nj − Λvij = Λv1 δij + Λv2 ni nj −

1 δij , 3 1 δij . 3

(7.274) (7.275)

Explicit equations for the scalar coeﬃcients λ, Λp1 , Λp2 and Λv1 , Λv2 are obtained in the next section after solution of the one-particle problems. Taking into account that the Fourier transforms of the mean displacement and strain ﬁelds are uk (k) = (2π)3 nk U δ(k − α∗ n), 3

εij (k) = −(2π) iα∗ ni nj U δ(k − α∗ n).

(7.276) (7.277)

from (7.268), we obtain a system of linear algebraic equations that connects scalar amplitudes U∗u , U∗p , U∗v of the eﬀective ﬁelds and the amplitude U of the mean displacement ﬁeld Tp (α∗ )U∗p + Tv (α∗ )U∗v + t(α∗ )U∗u = U, Π1p (α∗ )U∗p + Π1v (α∗ )U∗v + π1 (α∗ )U∗u = U, Π2p (α∗ )U∗p + Π2v (α∗ )U∗v + π2 (α∗ )U∗u = U.

(7.278) (7.279) (7.280)

The coeﬃcients of these system are t = 1 + pρ1 ω 2 λGΦ , 2 Tp = iα∗ p 3K1 Λp1 − µ1 Λp2 Γ1Φ + 2µ1 Λp2 Γ2Φ , 3 2 v v Φ v Φ Tv = iα∗ p 3K1 Λ1 − µ1 Λ2 Γ1 + 2µ1 Λ2 Γ2 , 3 ω2 π1 = −ipρ1 λΓ1Φ , α∗ 2 p p p Φ Φ Π1p = 1 + p 3K1 Λ1 − µ1 Λ2 P4 + 2µ1 Λ2 P3 , 3 2 v v Φ v Φ Π1v = 3F1 + F2 + p 3K1 Λ1 − µ1 Λ2 P4 + 2µ1 Λ2 P3 , 3 ω2 π2 = −ipρ1 λΓ2Φ , α∗ 2 p p p Φ Φ Π2p = 1 + p 3K1 Λ1 − µ1 Λ2 P1 + 2µ1 Λ2 P2 , 3 2 Π2v = F1 + F2 + p 3K1 Λv1 − µ1 Λv2 P1Φ + 2µ1 Λv2 P2Φ . 3

(7.281) (7.282) (7.283) (7.284) (7.285) (7.286) (7.287) (7.288) (7.289)

7.6 The eﬀective ﬁeld method

In these equations, GΦ , Γ1Φ , Γ2Φ , PiΦ (i = 1, 2, 3, 4) are integrals pend on the correlation function Φ(x) ∞ 1 GΦ = [G1 (r)j0 (α∗ r) + G2 (r)j1 (α∗ r)] Φ(r)rdr, µ0 0 ∞ i Φ Γ1 = − [2G2 (r) + G3 (r) + G4 (r)] j1 (α∗ r)Φ(r)dr, µ0 0 ∞ i [2G2 (r) + G3 (r)] j1 (α∗ r) Γ2Φ = − µ0 0 2j2 (α∗ r) − [2G2 (r) − G4 (r)] j1 (α∗ r) − Φ(r)dr, α∗ r ∞ 1 j1 (α∗ r) Φ P1 = − 2G2 (r) µ0 0 α∗ r

P2Φ

+ [G3 (r) + G4 (r)] j1 (α∗ r)} Φ (r)dr + iα∗ Γ2Φ , ∞ 1 2j1 (α∗ r) [2G2 (r) − G3 (r) − 2G4 (r)] =− µ0 0 α∗ r + [G3 (r) + G4 (r)] j0 (α∗ r) − [2G2 (r) − G4 (r)]

P3Φ

1 =− µ0

P4Φ = −

1 µ0

∞

0

0

8j2 (α∗ r) (α∗ r)2

Φ (r)dr + iα∗ Γ2Φ ,

227

that de-

(7.290) (7.291)

(7.292)

(7.293)

(7.294)

[2G2 (r) + G3 (r) + G4 (r)] j1 (α∗ r)Φ (r)dr + iα∗ Γ1Φ , (7.295)

∞

[2G2 (r) + G3 (r) + G4 (r)] j0 (α∗ r)Φ (r)dr + iα∗ Γ1Φ .

Here functions Gi (r)(i = 1, ..., 4) are 0 1 1 (β0 r)2 − iβ0 r − 1 e−iβ0 r + (iα0 r + 1)e−iα0 r , G1 (r) = 2 (β0 r) 1 1 (α0 r)2 − 3(iα0 r + 1) e−iα0 r G2 (r) = 2 (β0 r) 0 + 3(iβ0 r + 1) − (β0 r)2 e−iβ0 r , 1 1 G3 (r) = 3(1 + iβ0 r) − 2(β0 r)2 − i(β0 r)3 e−iβ0 r (β0 r)2 0 − 3(1 + iα0 r) − (α0 r)2 e−iα0 r , 1 1 G4 (r) = 9(1 + iα0 r) − 4(α0 r)2 − i(α0 r)3 e−iα0 r 2 (β0 r) 0 − 9(1 + iβ0 r) − 4(β0 r)2 − i(β0 r)3 e−iβ0 r .

(7.296)

(7.297)

(7.298)

(7.299)

(7.300)

228

7. Elastic waves in the medium with spherical inclusions

The solution of the system (7.278)–(7.280) is presented in the form U∗p = Dp U,

U∗v = Dv U,

U∗u = Du U,

(7.301)

where coeﬃcients Dp , Dv and Du are expressed via the coeﬃcients on the left-hand side of system (7.278)–(7.280) according to the Cramer’s rule. It follows from (7.210) and (7.273) that the Fourier transform of the mean displacement ﬁeld in the composite takes the form F(k) = F0 (k) + K(k)L1 ΛFe∗ (k).

(7.302)

Multiplying both parts of this equation by the matrix L0 (k) deﬁned in (7.26), and using properties (7.28) of this matrix, we ﬁnd the equation L0 (k)F(k) − M(k)L1 ΛFe∗ (k) = 0.

(7.303)

Here the matrix M(k) is deﬁned in (7.30). The ﬁrst equation of this system is 2 k M∗ (k) − ω 2 ρ∗ (k) U δ(k − iα∗ ) = 0, (7.304) and the dispersion equation for the wave number α∗ of the longitudinal mean wave ﬁeld takes the form α∗2 M∗ (α∗ ) − ω 2 ρ∗ (α∗ ) = 0.

(7.305)

The coeﬃcients in this equation are M∗ (α∗ ) = M0 + pM∗1 (α∗ ), 4 4 M∗1 (α∗ ) = 3K1 Λp1 + µ1 Λp2 Dp + p 3K1 Λv1 + µ1 Λv2 Dv , 3 3 ρ∗ (α∗ ) = ρ0 + pρ1 λDu . Λp1 ,

(7.306) (7.307) (7.308)

Λp2 ,

Note that λ, Λv1 , Λv2 are functions of the wave number α∗ , and the detailed form of these functions have to be found from the solution of the one-particle problems (7.207) for the local exciting ﬁelds ue∗i and εe∗ij in (7.258), (7.259).

7.7 One-particle problems of EFM 7.7.1 Transverse waves The one-particle problem of EFM is diﬀraction of a plane transverse wave (7.218) by a spherical inclusion with elastic moduli λ, µ and density ρ embedded in the matrix material with the dynamic characteristics λ0 , µ0 , ρ0 . If

7.7 One-particle problems of EFM

229

the inclusion of the radius a is centered at point x = 0, the integral equation (7.207) is equivalent to the following system of partial diﬀerential equations µui + (λ + µ) i k uk + ρω 2 ui = qiT (x), m 2 m µ0 um i + (λ0 + µ0 ) i k uk + ρ0 ω ui qiT (x) = µ0 (β02 − β∗2 )e1i exp(−iβ∗ x3 ).

=

qiT (x),

r ≤ a,

(7.309)

r > a.

(7.310) (7.311)

Here ui is the displacement vector inside the inclusion, um i is this vector in the matrix, is the Laplace operator. These equations diﬀer from the equations of the classical problem of diﬀraction of a plane monochromatic wave by a spherical inclusion because their right-hand sides are not zero: this is because the wave number β∗ of the eﬀective ﬁeld in the one particle problem of EFM does not coincide with the wave number of the matrix β0 . The solution of (7.309)–(7.311) may be found as was the solution of the classical diﬀraction problem (see [21]). The choice of the solution in the form of the following series u= um =

∞

cn L1e1n + dn M1o1n + en N1e1n + ζ ∗ e1 exp(−iβ∗ x3 ),

(7.312)

n=1 ∞

cn L3e1n + dn M3o1n + en N3e1n

n=1 n

(−i) (2n + 1) n(n + 1) 2 2 µ0 β0 − β∗ ζ∗ = µ β 2 − β∗2 +

i M1o1n + N1e1n , β∗ a

(7.313) (7.314)

satisﬁes the diﬀerential equations and the conditions at inﬁnity for the scattered ﬁeld. Here d [hn (α0 r)]Pn1 (cos θ) cos ϕ dr 1 1 hn (α0 r) θ dPn (cos θ) ϕ Pn (cos θ) cos ϕ − e sin ϕ . + e dθ sin θ r

L3e1n = er

(7.315)

The constants cn , dn , en and cn , dn , en in (7.312) and (7.313) are to be found from the conditions (7.39), (7.40) on the boundary between the inclusion and the matrix (r = a). These conditions give a system of linear algebraic equations for the constants that may be presented in the matrix form [17] cn cn − Ln = FL (7.316) Ln n, en en µ cn cn − = FM Mn (7.317) Mn n , en en µ∗

230

7. Elastic waves in the medium with spherical inclusions

1 2n + 1 1 f2 (β∗ a) (1 − ζ ∗ ) , f41 (β∗ a) n(n + 1) β∗ a 1 µ ∗ 1 f6 (β∗ a) n+1 2n + 1 1− , = (−i) ζ f81 (β∗ a) n(n + 1) β∗ a µ0

FL n = (−i) FM n

n+1

(7.318) (7.319)

where the matrices Ln and Mn are 2 2 f1 (α∗ a) f22 (β∗ a) f5 (α∗ a) f62 (β∗ a) Ln = , M . = n f32 (α∗ a) f42 (β∗ a) f72 (α∗ a) f82 (β∗ a) L n and M n in (7.316), (7.317) have forms (7.320) 2 2 1 1 (β0 a) are replaced with fm (αa), fm (βa). fm (α0 a), fm The system for the constants dn and dn is

(7.320)

if radial function

2n + 1 (1 − ζ ∗ )jn (β∗ a), (7.321) n(n + 1) µ ∗ µ 1 n 2n + 1 2 1− f (βa)dn = − (−i) ζ f61 (β∗ a). f6 (β0 a)dn − µ0 6 n(n + 1) µ0 (7.322) hn (β0 a)dn − jn (βa)dn = − (−i)

n

i (qr), (m = 1, 2, ..., 9, i = 1, 2) are deﬁned in (7.52) The radial functions fm In these equations, the wave numbers α and β are without indices for the ﬁelds inside the inclusion, they have index “0” for the ﬁelds in the matrix and “∗” for the medium with the eﬀective properties. If i = 1 functions yn1 (z) are the spherical Bessel functions jn (z), for i = 2 these functions are the Hankel functions hn (z). Tensors λ0ik and Λ0ijkl in (7.225), (7.226) are expressed via integrals from the solution of the one-particle problem. Let us begin with the equation for λ0ik 1 0 ∗ λik u k = ui (x) exp(iβ∗ n · x)dx. (7.323) v v

Here ui (x) is given by (7.312), where vector e1 has to be replaced by mU∗u . After integration of the spherical vector harmonics in (7.323) we obtain ∗k = λT u ∗i (x) or λ0ik = λT δik , (7.324) λ0ik u ∞ 1 3 f (αa)jn (β∗ a) λT = − in+1 n(n + 1) cn 3 2 n=1 β∗ a jn (βa)f41 (β∗ a) +(β∗ a)gn (β, β∗ ) +ζ ∗ , +iadn gn (β, β∗ ) + en β∗ a (7.325) gn (β, β∗ ) =

(βa) jn+1 (βa)jn (β∗ a) − (β∗ a) jn+1 (β∗ a)jn (βa) . (βa)2 − (β∗ a)2

(7.326)

7.7 One-particle problems of EFM

The deﬁnition (7.226) of the tensor Λ0ijkl implies 1 (j uεj) (x) exp(iβ∗ n · x)dx, Λ0ijkl ε∗kl (x) = v v

231

(7.327)

where uεj (x) has the form in (7.312) where vector e1 is replaced with mU∗ε . After calculating the integrals in this equation we obtain 1 Λ0ijkl ε∗kl (x) = ΛT ε∗ij (x) or Λ0ijkl = ΛT Eijkl ,

ΛT = −

∞ 3 in+1 n(n + 1)ΛTn + ζ ∗ , 2(β∗ a)3 n=1

ΛTn = cn Hcn + idn Hdn + en Hen , Hcn = 2 f11 (αa)f71 (β∗ a) + f31 (αa)f81 (β∗ a)

Hdn Hen

(7.328)

+ (β∗ a)2 f31 (αa)f31 (β∗ a), = −a(β∗ a) jn (βa)f71 (β∗ a) + (β∗ a)2 gn (β, β∗ ) , = 2 f21 (βa)f71 (β∗ a) + f41 (βa)f81 (β∗ a) + (β∗ a)2 jn (βa)f41 (β∗ a) + (β∗ a)2 gn (β, β∗ ) .

(7.329)

(7.330) (7.331)

(7.332)

The coeﬃcients in the dispersion equation (7.249) for the transverse waves take the forms ρ1 µ1 2 t = 1 + p (β0 a) GΦ λT , T = −pβ∗ a Γ Φ ΛT , (7.333) ρ0 µ0 µ1 p ρ1 2 (β0 a) Γ Φ λT , Π = 1 + p K Φ ΛT , (7.334) π= β∗ ρ0 µ0 where GΦ , Γ Φ and K Φ are the following integrals ∞ j1 (β∗ r) Φ(r)rdr, GΦ = G1 (r)j0 (β∗ r) + G2 (r) β∗ r 0 ∞ 4j2 (β∗ r) + G3 (r)j1 (β∗ r) G2 (r) j1 (β∗ r) − ΓΦ = − β∗ r 0 2j2 (β∗ r) Φ(r)dr, +G4 (r) β∗ r ∞ 9j1 (β∗ r) 32j2 (β∗ r) Φ + G2 (r) j0 (β∗ r) − K =− β∗ r (β∗ r)2 0 j1 (β∗ r) + G3 (r) j0 (β∗ r) − β∗ r j1 (β∗ r) 4j2 (β∗ r) − Φ (r)dr + β∗ Γ Φ +4G4 (r) β∗ r (β∗ r)2

(7.335)

(7.336)

(7.337)

Here the functions Gi (r) are deﬁned in (7.297)–(7.300). These equations deﬁne all the coeﬃcients in the dispersion equation (7.249–7.251), and we now construct their solution.

232

7. Elastic waves in the medium with spherical inclusions

7.7.2 Longitudinal waves For longitudinal wave propagation, the choice of the local exciting ﬁeld in the forms (7.261) and (7.263) leads to two diﬀerent one particle problems. The ﬁrst problem is diﬀraction of a plane longitudinal wave with wave number α∗ by an isolated spherical inclusion embedded in the background matrix, and the second is diﬀraction of a radial wave with the same wave number by this inclusion. In this section, the solutions of both these problems are constructed. Diﬀraction of a plane longitudinal wave by a spherical inclusion. Let us consider an incident longitudinal wave e3 exp(−iα∗ x3 ) with unit amplitude propagating in the x3 -direction (see Fig. 7.1). Such a wave may be presented in the form of a series in the spherical vector harmonics [21] ∞ 1 (−i)n+1 (2n + 1)L∗1 (7.338) u0 = e3 exp(−iα∗ x3 ) = − on (r), α∗ n=0 djn (α∗ r) jn (α∗ r) dPn (cos θ) Pn (cos θ) + eθ . (7.339) dr r dθ The ﬁrst one-particle problem of EFM is the problem of diﬀraction of the plane wave (7.338) by a spherical inclusion with elastic modules K, µ and density ρ embedded in a matrix material with properties K0 , µ0 , ρ0 . If the inclusion has radius a and is centered at point x = 0, the one-particle problem is equivalent to the following system of partial diﬀerential equations: 1 (7.340) µui + K + µ i k uk + ρω 2 ui = qiL (x), r ≤ a, 3 1 m 2 m L (7.341) µ0 um i + K0 + µ0 i k uk + ρ0 ω ui = qi (x), r > a, 3 r L∗1 on (r) = e

qiL (x) = M0 (α02 − α∗2 )e3i exp(−iα∗ x3 ).

(7.342)

Here ui is the displacement vector inside the inclusion, um i is this vector in the matrix, is the Laplace operator, e3i = δi3 are the components of the basis vector e3 . These equations diﬀer from the equations of the classical problem of diﬀraction of a plane longitudinal wave by a spherical inclusion because their right-hand sides are not zero; this is because the wave number α∗ of the eﬀective ﬁeld does not coincide with wave number α0 of the matrix. The solution in the form of the following series ∞ 1 ∗ 3 an Lon + bn N1 (7.343) u= on + ζ e exp(−iα∗ x3 ), um =

n=0 ∞

−

n=0

ζ∗ =

1 n+1 3 3 (−i) (2n + 1)L∗1 + a L +b N n on n on , on α∗

M0 α02 − α∗2 , M α2 − α∗2

4 M = K + µ. 3

satisﬁes the diﬀerential equation and the conditions at inﬁnity.

(7.344) (7.345)

7.7 One-particle problems of EFM

233

The spherical harmonics L3on and N3on are deﬁned in (7.45), (7.46), L1 on 3 3 and N1 on are obtained from Lon and Non by replacing the spherical Hankel functions of the ﬁrst kind hn (z) by the spherical Bessel function jn (z); prime denotes that α0 and β0 are to be replaced by the wave numbers α and β of the material of the inclusion. Constants an , bn and an , bn in (7.343), (7.344) have to be found from the conditions (7.39), (7.40) on the boundary between the inclusion and the matrix (r = a). These conditions give us a system of linear algebraic equations for constants an , bn and an , bn ⎛

f12 (α0 a), ⎜ f 2 (α a), ⎜ 3 0 ⎜ 2 ⎝ f5 (α0 a), f72 (α0 a),

f22 (β0 a), f42 (β0 a), f62 (β0 a), f82 (β0 a),

−f11 (αa), −f31 (αa), − µµ0 f51 (αa), − µµ0 f71 (αa),

⎞⎛ ⎞ ⎛ ∗⎞ −f21 (βa) an F1 ⎟ ⎜ ⎟ ⎜ 1 −f4 (βa) ⎟ ⎜ bn ⎟ ⎜ F2∗ ⎟ ⎟ ⎟⎜ ⎟ = ⎜ ⎟. − µµ0 f61 (βa) ⎠ ⎝ an ⎠ ⎝ F3∗ ⎠ − µµ0 f81 (βa) bn F4∗ (7.346)

Here the components of the vector on the right-hand side are n+1

F1∗ =

(−i) α∗ a

F2∗ =

(−i) α∗ a

(2n + 1)(1 − ζ ∗ )f11 (α∗ a),

(7.347)

n+1

(2n + 1)(1 − ζ ∗ )f31 (α∗ a), n+1 (−i) µ ∗ (2n + 1) gn1 (α∗ , η0 ) − F3∗ = ζ gn1 (α∗ , η0 ) , α∗ a µ0 n+1 µ ∗ (−i) (2n + 1) 1 − F4∗ = ζ f71 (α∗ a). α∗ a µ0

(7.348) (7.349) (7.350)

From (7.266), (7.267) and (7.343) we obtain the mean λik ue∗k (x)|x in the form u(x)|x = λik ue∗k (x)|x 1 = ui (y)eiα∗ n·y dye−iα∗ n·x = λni U∗u e−iα∗ n·x , v v

(7.351)

where the scalar coeﬃcient λ is ∞ 3 n+1 1 1 an f3 (αa)f11 (α∗ a) i (α∗ a) n=0 0 +(α∗ a)2 gn (α, α∗ ) + b f21 (βa)f31 (α∗ a) + ζ ∗ ,

λ=−

gn (α, α∗ ) =

(αa) ajn+1 (αa)jn (α∗ a) − (α∗ a) ajn+1 (α∗ a)jn (αa) . (αa)2 − (α∗ a)2

Here the functions fk1 (r) are deﬁned in (7.52)–(7.57).

(7.352) (7.353)

234

7. Elastic waves in the medium with spherical inclusions

Similarly, for Λijkl εe∗kl (x)|x we derive 1 Λijkl εe∗kl (x)|x = (j ui) (y)eiα∗ n·y dye−iα∗ n·x v v = iα∗ Λpij U∗p e−iα∗ n·x , 1 Λpij = Λp1 δij + Λp2 ni nj − δij , 3 ∞ 2 (αa) 1 Λp1 = − in+1 an gn (α, α∗ ) + ζ ∗ , (α∗ a) n=0 3 3 (D − Λp1 ) , 2 ∞ 3 n+1 D=− i [an Fan (α, α∗ ) + bn Fbn (β, α∗ )] + ζ ∗ , (α∗ a)3 n=0

Λp2 =

(7.354) (7.355) (7.356) (7.357) (7.358)

Fan (α, α∗ ) = f11 (αa) f21 (α∗ a) − 2f11 (α∗ a) − (α∗ a)2 f31 (α∗ a) +f31 (αa)f61 (α∗ a)+(α∗ a)2 f31 (αa)f11 (α∗ a)+(α∗ a)2 gn (α, α∗ ) , (7.359) 1 1 1 2 1 Fbn (β, α∗ ) = f2 (βa) f2 (α∗ a) − 2f1 (α∗ a) − (α∗ a) f3 (α∗ a) + f41 (βa)f61 (α∗ a) + (α∗ a)2 f21 (βa)f31 (α∗ a).

(7.360)

Diﬀraction of a radial wave by a spherical inclusion. The local exciting displacement ﬁeld that corresponds to the second term in (7.259) for the exciting strain ﬁeld εe∗ij has the form ue∗ (x) = j1 (α∗ r)U∗v er ,

(7.361)

where r = |x − xs |, xs is the center of the inclusion, and er is the basis vector of the spherical coordinate system connected with the center of the inclusion. The local exciting ﬁeld (7.361) produces only radial displacements in the inclusion u(r) and in the matrix um (r). These displacements satisfy the following diﬀerential equations (U∗v = 1) M0 2 ∂ 1 ∂ 2 r α0 − α∗2 j1 (α∗ r), r ≤ a, u + α2 u = (7.362) 2 ∂r r ∂r M ∂ 1 ∂ 2 m r + α02 um = α02 − α∗2 j1 (α∗ r), r > a. u (7.363) 2 ∂r r ∂r The solution of these equations has the following form u(r) = ζ ∗ j1 (α∗ r) + c j1 (αr), m

u (r) = j1 (α∗ r) + ch1 (α0 r),

r ≤ a, r > a.

(7.364) (7.365)

7.8 EFM dispersion equations

235

The constants c and c in these equations have to be found from the boundary conditions (7.39), (7.40); this leads to the system (7.366) c j1 (αa) − ch1 (α0 a) = (1 − ζ ∗ ) j1 (α∗ a), 2 2 µ (βa) (β0 a) j0 (αa) − 4j1 (αa) − c h0 (α0 a) − 4h1 (α0 a) c µ0 αa α0 a

1 = (α∗ a) 2 η0

M ∗ µ ∗ j1 (α∗ a) . (7.367) 1− ζ j0 (α∗ a) − 4 1 − ζ M0 µ0 α∗ a

The product Λ0ijkl ε∗kl for the radial exciting wave (7.361) takes the form 1 e ∂(j ui) (y)eiα∗ n·x dye−iα∗ n·x Λijkl ε∗kl (x)|x = v v 1 v v −iα∗ n·x v v v = −iα∗ Λij U∗ e , Λij = Λ1 δij +Λ2 ni nj − δij , 3 (7.368) where scalar coeﬃcients Λv1 and Λv2 are j1 (α∗ a) j1 (αa) + c Λv1 = j0 (α∗ a) ζ ∗ α∗ a α∗ a 1 ∗ + [ζ f (α∗ a) + 3c g1 (α, α∗ )] , 3 3j1 (α∗ a) j1 (α∗ a) j1 (αa) v ζ∗ + c Λ2 = 3 j0 (α∗ a) − α∗ a α∗ a α∗ a + ζ ∗ f (α∗ a) + 3c g1 (α, α∗ ), 3 f (α∗ a) = 2 (α∗ a)2 − 1 + 2 cos(2α∗ a) 4 4(α∗ a) +(α∗ a) sin(2α∗ a)] .

(7.369)

(7.370) (7.371) (7.372)

These equations determine all the coeﬃcients in the dispersion equation (7.306) for the longitudinal waves, and we now construct its solution.

7.8 EFM dispersion equations 7.8.1 Long transverse waves The long-wave asymptotic solution of the one-particle problem of EFM was obtained in Chapter 4, and is the same for the transverse and longitudinal waves. As a result, the tensors λ0ij and Λ0ijkl in (7.225), (7.226) in the longwave limit have the forms (5.78)–(5.81).

236

7. Elastic waves in the medium with spherical inclusions

Let us consider the coeﬃcients T, t, Π and π in (7.246), (7.247) in the long-wave limit. With the accuracy (β0 a)3 we obtain T = π = 0,

(7.373)

and the coeﬃcients Π and t take the forms

µ1 3 Π ≈1−p (Ps − i (β0 a) JΦ Pω )Hs + i(β0 a)3 Ps Hω , µ0 2(3 + 2η02 ) 2(3 + 2η05 ) , Pω = , 15 15 −1 2µ1 2 µ1 2 (3 + 2η02 ) (Hs ) , Hs = 1 + (3 + 2η02 ) , Hω = 15µ0 45 µ0 ρ1 3 t ≈ 1 − ip (2 + η03 ) (β0 a) JΦ , 3ρ0 ∞ 1 Φ(r)r2 dr. JΦ = 3 a 0 Ps =

The eﬀective shear modulus µ∗ and eﬀective density ρ∗ in the dispersion equation (7.249), (7.251) in the long-wave are µ∗ = µs + ipβ03 vf µω , ρ∗ = ρs − iβ03 pvf ρω , 4 v = πa3 , f = 1 − 3pJΦ , 3

(7.374) (7.375)

where µs is the static eﬀective shear modulus of the composite (ω → 0), and ρs is a “static” density µs = µ0 + pµR ,

ρs = ρ0 + pρ1 , −1 µ1 Hs 1 2(3 + 2η02 ) = + (1 − p) . µR = 1 − pµ1 Hs Ps µ1 15µ0

(7.376) (7.377)

The factors µω and ρω in the imaginary parts of µ∗ and ρ∗ in (7.374) are µω = µ2R

3 + 2η05 , 30πµ0

ρω = ρ21

2 + η03 . 12πρ0

(7.378)

Finally, we obtain the long-wave asymptotic solution of the dispersion equation (7.249) in the form ρs β∗ = βs − iγ, βs = ω , (7.379) µs Here the attenuation coeﬃcient γ is 4 βs 2 µ2R pf (β0 a) ρ21 5 3 γa = (3 + 2η0 ) + (2 + η0 ) . 18 β0 5 µ0 µs ρ0 ρs

(7.380)

7.8 EFM dispersion equations 4

1

m*/m0

237

m*/m0

3.5 0.8 3 2.5

0.6

2 0.4 1.5 1

0

0.1

0.2

0.3

p

0.4

0.2

0

0.1

0.2

0.3

0.4

p

Fig. 7.12. The dependence of the eﬀective static shear modulus of a composite with spherical inclusions on their volume concentration p. Solid lines correspond to (7.376), line with black dots are numerical data from [89]. The left-hand ﬁgure corresponds to absolutely rigid inclusions, the right-hand ﬁgure to spherical pores.

In Fig. 7.12, equation (7.376) for the eﬀective shear modulus µs of the composites is compared with the numerical computation of these moduli presented in [89]. The solid line in the left part of Fig. 7.12 shows the dependence (7.376) of µs on the volume concentration p of the inclusions for absolutely rigid inclusions (µ = ∞); the line with black dots presents the numerical results of [89]. The corresponding dependences for the material with spherical pores are shown in the right-hand part of Fig. 7.12. The numerical results of [89] were obtained by the ﬁnite element technique applied to the solution of the elasticity problem for a representative volume of the composite material. The number of inclusion inside the representative volume, and the sizes of the ﬁnite elements used by the calculations in this work allow us to consider this result as virtually an exact solution of the homogenization problem. Thus, the graphs in Fig. 7.12 may be interpreted as a comparison of the predictions of the EFM and the exact values. The attenuation coeﬃcient γ of the mean wave ﬁeld in the long-wave 4 region is proportional to the factor (β0 a) (the Rayleigh scattering) and to the structural factor f ∞ f = 1 − 3p Φ(ζ)ζ 2 dζ, 0

ζ=

r . a

(7.381)

The factor f and the function Φ(r) depend only on geometrical properties of the random ﬁeld of inclusions. It is shown in Section 3.9 that the factor f is non-negative (f ≥ 0) for any realizable correlation function of a random set of inclusions. 4 The dependence of the attenuation coeﬃcient γa/ (β0 a) calculated from (7.380) on the volume concentration of inclusions p is presented in Fig. 7.13 for the correlation function Φ(ζ) that corresponds to the Percus-Yevick correlation function of the centers of nonoverlap spheres. The solid line in

238

7. Elastic waves in the medium with spherical inclusions 0.6

g a/(b0a)4

0.4

0.2

0

0

0.1

0.2

0.3

0.4

p

Fig. 7.13. The dependence of the attenuation in the long-wave region on the volume concentration p of spherical inclusions. The solid line corresponds to absolutely rigid inclusion, the dashed line to porous media.

this ﬁgure is for the medium with absolutely rigid inclusions, the dashed line is for a porous medium. 7.8.2 Short transverse waves Consider the solution of the dispersion equation of the EFM in the short-wave limit. In this case ω, α0 , β0 → ∞ and, as follows from (7.325) and (7.329) λT , ΛT → 0. In the short-wave limit, the eﬀective wave number of the mean wave ﬁeld has the form β∗ = Re β∗ − iγ,

(7.382)

where γ does not depend on β0 , and Re β∗ = O(β0 ). The integrals GΦ , Γ Φ and K Φ in (7.335)–(7.337) in the short wave limit take the forms ∞ 1 lim β02 GΦ (β∗ ) = β02 e−iβ0 r j0 (β∗ r)Φ(r)dr = − iβ0 aI(γa), (7.383) ω→∞ 2 0 ∞ lim K Φ (β∗ ) = lim β0 Γ Φ (β∗ ) = −iβ02 e−iβ0 r j1 (β∗ r)Φ(r)dr ω→∞ ω→∞ 0 ∞ 1 r = − iβ0 aI(γa), I(γa) = eγaζ Φ(ζ)dζ, ζ = . 2 a 0 (7.384) Here the limiting form of β∗ (7.382) together with asymptotic formulas j0 (β0 r) ∼ sin(β∗ r)/β∗ r, j1 (β∗ r) ∼ − cos(β∗ r)/β∗ r for large β∗ were used. Taking these relations into account we obtain from (7.306)–(7.308) that pµ1 pρ1 ΛT , ρ∗ = ρ0 + λT , (7.385) µ∗ = µ0 + ∆ ∆ ρ1 µ1 1 (7.386) λT − ΛT . ∆ = 1 − ipβ0 aI(γ)Λ, Λ = 2 ρ0 µ0 Note that Λ/∆ → 0 when ω → ∞.

7.8 EFM dispersion equations

239

Equations (7.325) and (7.385) imply that in the short-wave limit the equation for the eﬀective wave number β∗ may be written in the form Λ . (7.387) β∗ = β0 1 + p 2∆ It follows from (7.325)–(7.329) that, for large ω (β0 ), the equation for β0 Λ in (7.386) takes the form ρ1 ρ1 µ1 µ1 µ0 β02 − β∗2 λT 0 − Λ T 0 + β0 − . (7.388) β0 Λ = β0 ρ0 µ0 ρ0 µ0 µ β 2 − β∗2 Here λT 0 and ΛT 0 are the same as in (7.325) and (7.329). Equation (7.145) implies that the limiting value of the ﬁrst term on the right-hand side of (7.388) is −3i/2a, and the limit of the last term is 2iγ. As a result, we obtain the short-wave limits of the functions β0 Λ and ∆ in (7.386) in the form 2i 3 − γa , (7.389) lim β0 Λ = − ω→∞ a 4 3 − γa I(γa). (7.390) lim ∆ = 1 − p ω→∞ 4 Thus, in the short-wave limit, (7.387) takes the form 3 − γa 4 . β∗ a = β0 a − ip 1 − p 34 − γa I(γa)

(7.391)

This equation implies that the phase velocity of the mean wave ﬁeld coincides with the velocity of the transverse waves in the matrix material v∗ =

ω ω = = v0 , Re(β∗ ) β0

(7.392)

and from (7.382) and (7.391) we ﬁnd that the short-wave limit γ of the attenuation coeﬃcient γ is −1 3 3 − γa 1 − p − γa I(γa) γa = − Im (β∗ a) = p . (7.393) 4 4 Equation (7.393) is in fact the equation for the short-wave limit γ of the attenuation coeﬃcient. 7.8.3 Long longitudinal waves Omitting long but evident calculations similar to those presented in Section 7.8.1 for transverse waves, we ﬁnd the long-wave asymptotics of the coeﬃcient M∗ in dispersion equation (7.306) in the form

240

7. Elastic waves in the medium with spherical inclusions

1 2 8 f 2 + 5 KR , + µ2R 3M0 15 3 η0 ∞ r JΦ = Φ(ζ)ζ 2 dζ, ζ = . a 0 3

M∗ ≈ Ms + ip (α0 a) f = 1 − 3pJΦ ,

(7.394)

Here the static elastic constant Ms of the composite is deﬁned by the equation 4 Ms = M0 + pMR , MR = KR + µR , 3 −1 −1 1 1−p 1 2(3 + 2η02 ) + , µR = + (1 − p) . KR = K1 M0 µ1 15µ0

(7.395)

The long-wave asymptotics of the eﬀective density ρ∗ coincides with (7.374) 2 ρ21 3 1 + 3 , ρs = ρ0 + pρ1, (7.396) ρ∗ ≈ ρs − i (α0 a) pf 9ρ0 η0 and the solution of the dispersion equation (7.306) with accuracy of the principal terms with respect to ω is / ω Ms αs = , ls = , (7.397) α∗ = αs − iγ, ls ρs where the attenuation coeﬃcient γ is 4 5 4 3 ls 3 1 ρ21 (αs a) 2 8 2 2 2 . γa = pf 1+ 3 + 2 KR + 15 µR η 5 + 3 18ρ0 ρs l0 η0 (v0 vs ρ1 ) 0 (7.398) 7.8.4 Short longitudinal waves Consider the solution of the dispersion equation (7.306) in the short-wave region (ω → ∞). In this case α0 a, β0 a, αa, βa, → ∞, and coeﬃcients Λpi , Λvi (i = 1, 2), λ in the dispersion equation (7.306) tend to zero. In the short-wave limit, the eﬀective wave number of the mean wave ﬁeld has the form α∗ = Re α∗ − iγ,

(7.399)

where γ does not depend on the frequency ω of the incident ﬁeld, and Re α∗ = O(ω). Consider the short-wave asymptotics of the coeﬃcients of the system (7.278) for the amplitudes of the eﬀective ﬁelds. The principal terms of the integrals GΦ and ΓiΦ (i = 1, 2) that appear in (7.281)–(7.289) for the coeﬃcients of system (7.278) in the short-wave limit take the forms

7.8 EFM dispersion equations

241

ρ1 ∞ 1 1 iα∗ r (α0 r)2 e−iα0 r e − e−iα∗ r Φ(r)dr ρ0 0 r 2iα0 r ∞ ρ1 α0 ρ1 ≈ eγr Φ(r)dr = − i (α0 a) I(γ), (7.400) ρ0 2i 0 2ρ0

ρ1 ω 2 GΦ (α∗ ) ≈

Γ1Φ

≈

I(γ) =

Γ2Φ 1 a

η2 ≈ 0 µ0 ∞

0

∞

cos(α∗ r)Φ(r)e−iα0 r dr =

η02 aI(γ), 2µ0

eγr Φ(r)dr.

(7.401) (7.402)

0

Coeﬃcients PiΦ (i = 1, ..., 4) in (7.290)–(7.296) in the short-wave limit take the forms PiΦ ≈ iα∗ Γ1Φ ≈ i (α∗ a)

η02 I(γ), i = 1, 2, 3, 4. 2µ0

(7.403)

Here the limiting form (7.399) of α∗ together with the asymptotic formulas for the spherical Bessel functions (j0 (α∗ r) = sin(α∗ r)/α∗ r, j1 (α∗ r) ≈ − cos(α∗ r)/α∗ r) for large α∗ were used. Taking into account that in the shortwave limit coeﬃcients λ, Λp1 , Λp2 in (7.352)–(7.357) have the order ω −1 , and Λv1 , Λv2 in (7.369) and (7.370) have the order ω −2 , we obtain the principal terms of the asymptotics of the coeﬃcients in system (7.278)–(7.280) for the amplitudes of the eﬀective displacement and strain ﬁelds in the forms d b1 a2 a1 , Tv = + 2 , t = t0 + , (7.404) ω ω ω ω Π1p = Π2p = 1 + Tp , π1 = π2 = t − 1, (7.405) di a2 + 2 , i = 1, 2. Πiv = (7.406) ω ω where a1 , a2 , b1 , d, d1 , d2 are some diﬀerent constants independent of ω, and the coeﬃcients t0 , Tp0 have the forms Tp = Tp0 +

ρ1 i (α0 a) I(γ)λu , t0 = 1 − p 2ρ0 i (α0 a) η02 4 I(γ). Tp0 = p 3K1 Λp1 + µ1 Λp2 3 2µ0 Thus, keeping only the terms of order ω −2 in (7.278)–(7.280) rewrite this system in the form d b1 a2 a1 u U∗p + + 2 U∗v + t0 + U∗ = U, Tp0 + ω ω ω ω d1 b1 a2 a1 u 1 + Tp0 + U∗p + + 2 U∗v + t0 − 1 + U∗ = U, ω ω ω ω d2 b1 a2 a1 u 1 + Tp0 + U∗p + + 2 U∗v + t0 − 1 + U∗ = U. ω ω ω ω

(7.407) (7.408) we can

(7.409) (7.410) (7.411)

242

7. Elastic waves in the medium with spherical inclusions

The solution of this system is U∗v

= 0,

U∗p

=

U∗u

−1 b1 + a1 0 0 = t + Tp + . ω

(7.412)

Thus, in the limit ω → ∞, the system of three equations (7.278)–(7.280) is transformed into a system of two equations (equations (7.410) and (7.411) coincide): Tp0 U∗p + t0 U∗u = U, 0 Tp + 1 U∗p + t0 − 1 U∗u = U,

(7.413)

and the amplitude U∗v of the “radial” wave disappears from the original system (7.278)–(7.280). Thus, in the short-wave limit, the radial local exciting waves take no part in the dispersion equation (7.307), and the amplitudes U∗p and U∗u of the eﬀective ﬁelds coincide U i , ∆L = Tp0 + t0 = 1 − p (α0 a) I(γ)ΛL (α∗ ), ∆L 2 ρ1 η02 4 p p 3K1 Λ1 + µ1 Λ2 . ΛL (α∗ ) = λ − ρ0 µ0 3

U∗p = U∗u =

(7.414) (7.415)

¯ L of an isolated spherical The total normalized scattering cross-section Q inclusion by diﬀraction of longitudinal plane waves has the form ¯ L = Im (QL ) , QL = − 4 (α0 a) ΛL (α0 ) Q 3 4 ρ1 η2 4 = − (α0 a) λ0 − 0 3K1 Λp10 + µ1 Λp20 . 3 ρ0 µ0 3

(7.416)

Here the coeﬃcients λ0 , Λp10 , Λp20 coincide with λ, Λp1 , Λp2 in (7.352)–(7.357) if the eﬀective wave number α∗ is replaced by the wave number α0 of the ¯ L tends matrix in these equations. According to the extinction paradox, Q to 2 when ω → ∞, and this limit does not depend on the properties of the medium and the inclusion. Thus, the limit of the function (α0 a) ΛL (α0 ) in (7.416) when ω → ∞ is 3 lim (α0 a) ΛL (α0 ) = − i. 2

ω→∞

(7.417)

Note that Re (QL ) tends to zero when ω → ∞. Taking into account (7.416) and (7.417) we ﬁnd that, in the short-wave limit, the coeﬃcients in dispersion equation (7.306) take the forms Dv = 0, Dp = Du = −1 L ,

(7.418)

7.8 EFM dispersion equations

and (7.306) is transformed into the following equation: ΛL (α∗ ) ΛL (α∗ ) or α∗ a = (α0 a) 1 + p . α∗2 = α02 1 + p ∆L 2∆L

243

(7.419)

This last equation holds because the ratio ΛL (α∗ )/∆L tends to zero as when ω → ∞. For large ω, the product (α0 a) ΛL (α∗ ) takes the form ρ1 u 1 4 3K1 Λp10 + µ1 Λp20 λ0 − (α0 a) ΛL (α∗ ) = (α0 a) ρ0 M0 3 2 ρ1 M1 M0 α0 − α∗2 + (α0 a) − ρ0 M0 M α2 − α∗2 2 α − α02 α02 − α∗2 . (7.420) = (α0 a) ΛL (α0 ) + (α0 a) (α2 − α∗2 )

−1

(ω)

Equation (7.417) implies that the ﬁrst term in the right-hand side of this equation tends to −3i/2 when ω → ∞, and the limit of the second term is equal to 2iγa since (7.399) yields α∗ = α0 − iγ. Hence, the limiting values of (α0 a) ΛL (α∗ ) and L in (7.415), (7.420) are 3 − γa , (7.421) lim (α0 a) ΛL (α∗ ) = −2i ω→∞ 4 3 − γa I(γ). (7.422) lim L = 1 − p ω→∞ 4 Thus, in the short-wave limit, (7.419) for the eﬀective wave number α∗ a takes the form α∗ a = α0 a − iγa = α0 a − ip

−1 3 3 − γa 1 − p − γa I(γa) . 4 4 (7.423)

From this equation we obtain the ﬁnal equations for the real and imaginary parts of the wave number of the mean wave ﬁeld: Re α∗ = α0 ,

p 34 − γa . Im(α∗ ) = γ = a 1 − p 34 − γa I(γa)

(7.424) (7.425)

Thus, if ω → ∞, the velocity of the mean wave ﬁeld coincides with the velocity of longitudinal waves in the matrix material v∗ =

ω ω = = v0 , Re(α∗ ) α0

(7.426)

244

7. Elastic waves in the medium with spherical inclusions

and the short-wave limit γ of the attenuation γ is the solution of the following equation p 34 − γa . (7.427) γa = 1 − p 34 − γa I(γ) this coincides with equation (7.393) for the short-wave limit of the attenuation coeﬃcient for transverse waves.

7.9 Numerical solution of the EFM dispersion equations 7.9.1 Transverse waves The following iterative procedure was used for the numerical solution of the dispersion equation (7.249) ⎡ 8 ⎤ 9 (n) 9 ρ∗ β∗ ⎢ 9 (n+1) (n) − β∗(n) ⎥ = β∗ + δ ⎣ω : β∗ (7.428) ⎦. (n) µ∗ β∗ Here the index n corresponds to the number of the iteration, parameter ε (|ε| < 1) is to be chosen for convergence of the iterative process. Functions ρ∗ (β∗ ) and µ∗ (β∗ ) are deﬁned in (7.249)–(7.251). The long-wave asymptotic solution (7.379) was taken as the “zero ” iteration. Numerical analysis of the dispersion equation (7.249) revealed several branches of its solutions. Three such branches in the region 0 < β∗ a, β0 a < 3 for the medium with hard and heavy inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4, p = 0.3) are presented in Fig. 7.14. The dependences of the 2.4

Re(b*a) 3

Im(b*a) 3

2 1.6

2

2

3 1.2 1 0.8

1 2

0.4

0 0

0.5

1

1.5

2

b0a

0

1 0

0.5

1

1.5

2

b0a

Fig. 7.14. The dependence of the real Re(β∗ a) and imaginary Im(β∗ a) parts of the wave number of the mean wave ﬁeld on the wave number in the matrix material β0 for a medium with hard and heavy inclusions. 1 is the quasi-acoustical branch, 2 is quasi-optical branch, and 3 is the branch typical for a non-local medium.

7.9 Numerical solution of the EFM dispersion equations

245

real parts of the eﬀective wave number on the wave number β0 of the matrix are shown on the left-hand ﬁgure, and the dependence of the imaginary parts of β∗ (−Im β∗ ) are shown on the right hand. In the long- and short-wave regions, the behavior of branch 1 coincides with the asymptotic solutions obtained in Sections 7.8.1 and 7.8.2. This branch may be called acoustical (quasiacoustical). The second branch (2) goes lower than the acoustical branch and starts with ﬁnite values of frequency (β0 a ≈ 0.7); this branch may be called optical (quasioptical). The third branch (3) goes higher than the acoustical branch and starts with the point that correspond to the root of (7.249) for ω = 0. The existence of non-trivial roots of the dispersion equations for ω = 0 is typical for a medium with microstructure (see [67] pp. 51–58, [68] pp. 33–37). Non-trivial roots of (7.249) for ω = 0 are complex numbers with non-zero real parts. The attenuation coeﬃcients γa of the waves that correspond to branches 2 and 3 are several orders of magnitude larger than the attenuation coeﬃcients of the acoustic waves, and are approximately 2 along these branches. Thus, these waves practically disappear in a distance equal to the diameter of the inclusions. The solutions corresponding to the second and third branches were found by seeking the roots of the function F (β∗ ) = β∗2 − β02

µ0 ρ∗ (β∗ ) ρ0 µ∗ (β∗ )

(7.429)

in the complex plane (Re β∗ , Im β∗ ). Note that in the medium-wave region and for high volume concentrations of inclusions, the eﬀective wave numbers of these three branches are close, and the iterative procedure (7.428) may jump from branch 1 to 2 or 3. In this region, these three branches should be carefully separated. The results of calculation of the phase velocities and attenuation coeﬃcients of shear waves that correspond to the acoustical branch of the solutions of the dispersion equation are presented in Figs. 7.15 and 7.16. Figure 7.15 shows results for composites with hard and heavy inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4, and E, ν and E0 , ν0 are Young’s moduli and Poisson’s ratios of the inclusions and the matrix) and volume concentrations of inclusions p = 0.1 and p = 0.3. The region of the wave number β0 covers the long, medium and shortwave regions of the propagating wave (0 < β0 a < 100, logarithmic scale is used in Fig. 7.15). The detailed behavior of this dependence in the region (0 < β0 a < 3) is shown in Fig. 7.16, where a non-logarithmic scale is used. The dashed horizontal lines in these ﬁgures are the short-wave asymptotics of the velocities and attenuation coeﬃcients of waves obtained in Section 7.8. Figure 7.17 shows the results of calculation of the phase velocities and attenuation coeﬃcients of shear waves for the composite with soft and light inclusions (ρ/ρ0 = 0.1, E/E0 = 0.02, ν = 0.3, ν0 = 0.4). The graphs in this ﬁgure are constructed with the step 0.25 in the logarithmic scale and do not reﬂect small-scale oscillations of the velocity and attenyation coeﬃcients that actually take place.

246

7. Elastic waves in the medium with spherical inclusions

1.1

0

t*/t0

1

lg(γ a)

−2

0.9 p=0.1

−4

p=0.3

−6

0.8 0.7 0.6 −2

−1

0

1

lg(β0a)

−8 −2

−1

0

1

lg(β0a)

Fig. 7.15. The dependence of the relative velocity v∗ /v0 (v0 is the velocity of shear waves in the matrix) and attenuation coeﬃcient γ of the mean wave ﬁeld on the frequency of the incident ﬁeld (wave number of the matrix material β0 ) for the medium with heavy and hard inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4). 1.1

0.6

t∗/t0

γa

1

p=0.1 0.4

p=0.1

0.9

p=0.3

0.8

0.2 p=0.3

0.7 0.6 0

0.5

1

1.5

2

2.5

β0a

0 0

0.5

1

1.5

2

2.5

β0a

Fig. 7.16. The same dependence as in Fig. 7.15 in the non-logarithmic scale. 1

1.05 t*/t0

lg(γ a) p=0.3

−1

1

0.85 −1.5

p=0.1

−3

0.95

0.9

p=0.3

−0.5

p=0.1

p=0.3

p=0.1

−5

0.5

lg(β0a)

−7 −1.5

−0.5

0.5

lg(β0a)

Fig. 7.17. The dependence of the relative velocity t∗ /t0 and the attenuation coeﬃcient γ of the mean transverse wave ﬁeld on the wave number of the matrix material β0 for a medium with light and soft inclusions (ρ/ρ0 = 0.1, E/E0 = 0.02, ν = 0.3, ν0 = 0.4).

7.9 Numerical solution of the EFM dispersion equations 1.05 t*/t0

247

γa

0.3

1

p=0.3 p=0.1

0.15

0.95

p=0.1 0

p=0.3

0.9

0

1

2

3

β0a

p=0.3 0.85

0

1

2

3

β0a

−0.15

Fig. 7.18. The same dependence as in Fig. 7.17 in a non-logarithmic scale.

The detailed dependence of the velocities and attenuation coeﬃcients on the mean wave ﬁeld for frequency (β0 a) in the region 0 < β0 a < 4 are presented in Fig. 7.18. Note that in some region of medium wave lengths (0.6 < β0 a < 1.8 for p = 0.3) the imaginary part of the solutions of the dispersion equation (7.428) (attenuation coeﬃcient) becomes negative. In this region, the method overestimates interactions between inclusions, and its predictions for the attenuation coeﬃcients become physically incorrect. For small volume concentrations of inclusions (p < 0.1), the attenuation coeﬃcient is positive. For hard and heavy inclusions, the method predicts positive values of the attenuation coeﬃcients for all frequencies of the incident ﬁeld and volume concentrations of inclusions. 7.9.2 Longitudinal waves The following iterative procedure was used for the numerical solution of dispersion equation (7.306) ⎡ 8 ⎤ 9 (n) 9 ρ∗ α∗ ⎢ 9 (n+1) (n) −α∗(n) ⎥ = α∗ + δ ⎣ω : (7.430) α∗ ⎦. (n) M∗ α∗ Here the index n corresponds to the number of the iteration, parameter ε (|ε| ≤ 1) is to be chosen for convergence of the iterative process. Functions ρ∗ (α∗ ) and M∗ (α∗ ) are deﬁned in (7.306)–(7.308). The static limits Ms and ρs of M∗ , ρ∗ may be taken as the “zero” iteration. Figure 7.19 shows the results of calculation of the phase velocities and attenuation coeﬃcients of longitudinal waves in composites with volume concentration of inclusions p = 0.1. Lines with black dots in this ﬁgure correspond to hard and heavy inclusions (E/E0 = 50, ρ/ρ0 = 10, ν = 0.3, ν0 = 0.4), lines with clear dots to soft and light inclusions (E/E0 = 0.25, ρ/ρ0 = 0.1, ν = 0.3, ν0 = 0.4).

248

7. Elastic waves in the medium with spherical inclusions 1.1

0

p=0.1

l*/l0

lg(g a)

p=0.1

−1 −2

1

−3 −4

0.9

−5 −6

0.8

0.7 −2 −1.5 −1 −0.5

lg(a 0a) 0

0.5

1

1.5

−7 −8 −2 −1.5 −1 −0.5 2

lg(a 0a) 0

0.5

1

1.5

2

Fig. 7.19. The dependence of the attenuation coeﬃcient γa and relative velocity l∗ /l0 of the mean wave ﬁeld on the non-dimensional wave number α0 a, for the matrix for volume concentration of inclusions p = 0.1 (v0 is the velocity of longitudinal waves in the matrix). The line with black dots corresponds to a medium with heavy and hard inclusions (ρ/ρ0 = 10, E/E0 = 50, ν = 0.3, ν0 = 0.4), and the line with clear dots to a medium with soft and light inclusions (E/E0 = 0.25, ρ/ρ0 = 0.1, ν = 0.3, ν0 = 0.4). 1.1

0.2

l*/l0

ga

p=0.1

p=0.1

1

0.1

0.9

0.8

0.7

a 0a 0

0.4

0.8

1.2

1.6

2

0

a 0a 0

0.4

0.8

1.2

1.6

2

Fig. 7.20. The same dependenceas in Fig. 7.19 in a non-logarithmic scale.

The considered region of wave number α0 a covers long, medium and shortwave regions of propagating waves (0 < α0 a < 100, logarithmic scale along α0 a axis is used in both parts of Fig. 7.19). The graphs in this ﬁgure are constructed with the step 0.25 in the logarithmic scale and do not reﬂect small-scale oscillations of the velocities and attenuation coeﬃcients. Detailed behavior of this dependence in the region 0 < α0 a < 2 is shown in Fig. 7.20, where non-logarithmic scales along the α0 a-axes are used. The dashed horizontal lines on both sides of Fig. 7.20 are the short wave asymptotics of the velocities and attenuation coeﬃcients of the mean wave ﬁelds given in Section 7.5.4. For high volume concentrations of inclusions, the dispersion equation (7.306) has several diﬀerent branches in the region Im(α∗ a) < 1. To

7.9 Numerical solution of the EFM dispersion equations

249

ﬁnd all these solutions, we construct, ﬁrst, a 2D-map of the function F(Re α∗ , Im α∗ ) = |α∗2 M∗ (α∗ ) − ω 2 ρ∗ (α∗ )| in a wide region of the complex plane (Re α∗ , Im α∗ ), and indicate approximate positions of the roots of this function in this region. After that, we use the iterative scheme (7.430) to ﬁnd the precise values of the roots, choosing the initial guess in the vicinity of each root. For p = 0.3, the graphs of the phase velocities and attenuation coeﬃcients of the mean wave ﬁeld for the composite with hard and heavy inclusions are presented in Fig. 7.21 by the lines with black dots. The detailed behavior of the dependence in the region 0 < α0 a < 2 is presented in Fig. 7.22. In this case, we ﬁnd two main branches of the solutions of dispersion equation (7.306). In the long wave region, only one of these branches coincides with the long-wave asymptotic solution obtained in Section 7.8. This branch can be called acoustical, and it is indicated as the A-branch in Fig. 7.22. The attenuation of the waves that correspond to this branch rapidly grows with frequency, and these waves become practically invisible when α0 a > 0.8. The second branch appears at α0 a ≈ 0.85, and the 2

l*/l0

0

p=0.3

p=0.3

lg(g a)

−1

1.75

−2

1.5

−3 1.25

−4

1

−5

0.75 0.5 −2 −1.5 −1 −0.5 0

−6

lg(a 0a) 0.5

1

0.5

2

−7 −2 −1.5 −1 −0.5 0

lg(a 0a) 0.5

1

0.5

2

Fig. 7.21. The same dependence as in Fig. 7.19 for p = 0.3. 2.2

0.6

p=0.3

l*/l0

p=0.3

γa

0.5

O

1.6

0.4

A

0.3 0.2

1

O

0.1

A

α0a

0.4 0

0.4

0.8

1.2

1.6

α0a

0 2

0

0.4

0.8

1.2

1.6

2

Fig. 7.22. The same dependences as in Fig. 7.21 in a non-logarithmic scale.

250

7. Elastic waves in the medium with spherical inclusions

1.1

γ La

l*/l0

1 0.1

0.9

0.8

0

0.5

1

1.5

2

2.5 α0a

0 0

0.5

1

1.5

2

2.5

α0a

Fig. 7.23. The comparison of the theoretical predictions (solid lines) for the velocities and attenuation coeﬃcients of longitudinal waves in the composites with PMM matrix and steel spherical inclusions with experimental data in [62] (circles).

attenuation coeﬃcient γa along this branch is about 0.2. In the short-wave region, the attenuation along the second branch corresponds to the shortwave asymptotics of γa obtained in Section 7.8. This branch may be called optical, and it is indicated as the O-branch in Fig. 7.22. Thus, in the region α0 a > 0.9, the waves of the O-branch attenuate much less than the waves of the A-branch, and the mean wave ﬁeld is mainly deﬁned by the parameters of the O-branch. The transition from the A-branch to the O-branch takes place in the region 0.8 < α0 a < 0.95 (dashed solid lines in Fig. 7.22). For soft and light inclusions and p = 0.3, there are several branches of the solution of the dispersion equation (7.306). But in this case a branch that coincides with the long wave asymptotic in the long-wave region and with the short-wave asymptotic in the short-wave region may be indicated. The velocities and attenuation coeﬃcients of the waves that correspond to this branch are shown in Figs. 7.21 and 7.22 (lines with clear circles). The comparison of the prediction of the theory with experimental data may be seen in Fig. 7.23. The experimental data for the velocities and attenuation coeﬃcients of longitudinal incident waves were presented in [62] for composites with PMM (polymetilmetacrelate) matrix and steel spherical inclusions of radii a1 = 0.55 mm, and volume concentration p = 0.115. The kg kg densities of these materials are ρpmm = 1160 m 3 , ρst = 7800 m3 ; the velocities m m , of longitudinal waves in these materials are lpmm = 2630 sec , lst = 5940 sec m m . and the velocities of transverse waves are tpmm = 1320 sec , tst = 3220 sec Solid lines in these ﬁgures are obtained by numerical solution of the dispersion equation (7.306), circles are experimental data presented in [62]. 7.9.3 Longitudinal waves in epoxy-lead composites Let us go to the results of the numerical solution of the dispersion equation (7.306) for a composite that consists of an epoxy matrix and lead

7.9 Numerical solution of the EFM dispersion equations 3

Re(α*a)

0.3

p=0.159

γa

251

p=0.159

2.5 2

0.2

1.5 1

0.1

0.5 α0a

0 0

0.5

1

1.5

2

2.5

3

α0a

0 0

0.5

1

1.5

2

2.5

3

Fig. 7.24. The dispersion curves for composites with an epoxy matrix and lead inclusions of volume concentration p = 0.159. The solid line is the acoustical branch of the solution of the dispersion equation (7.305) of the EFM; black points are experimental data of [63]. The dashed line is the optical branch of wave propagation which attenuation γa ≈ 1.

spherical inclusions. Such composites were experimentally studied in [62, 63]. The elastic moduli of the epoxy resin are K0 = 6.069 GPa,µ0 = 1.739 GPa and its density is ρ0 = 1,202 kg/m3 . The corresponding parameters of lead are K = 44.047 GPa,µ = 8.357 GPa, and ρ = 11,300 kg/m3 . The dependence of the real part of the dimensionless eﬀective wave number Re(α∗ a) of the propagating wave on the wave number α0 a of the longitudinal waves in the matrix (dispersion curve) is presented in the left-hand side of Fig. 7.24 for the volume concentrations of inclusions p = 0.159. The dependence of the attenuation coeﬃcient γa = Im(α∗ a) on α0 a is shown in the right-hand side of this ﬁgure. Black dots in Fig. 7.24 are experimental data from [62, 63]. The dispersion equation (7.306) has only one essential branch. The solid lines in Fig. 7.24 correspond to this (acoustical) branch. Other branches (see, e.g., the dashed line in the left hand side of Fig. 7.24) have attenuation coeﬃcients close to 1(γa ≈ 1), and the corresponding waves eﬀectively disappear in a distance of several diameters of the inclusions. This waves cannot be observed in the experiments with a representative volume element of the composite with linear sizes much more than the diameter of a typical inclusion. Note that rapid change of the phase velocity in the region 0.4 < α0 a < 0.7 is described by this acoustical branch, and the attenuation coeﬃcient γa is maximal in this region. For a volume concentration of inclusions p = 0.26, the situation changes dramatically (see Fig. 7.25). For such a composite, the dispersion equation (7.306) has four diﬀerent branches in the region (0 < α0 a < 3). The attenuation coeﬃcient γa alters along each branch, and the dashed parts of the branches indicate regions where γa is greater than 0.5. Solid parts of the branches correspond to small values of γa (γa < 0.5). Branch 1 in Fig. 7.25 is the acoustical branch;

252

7. Elastic waves in the medium with spherical inclusions 3

Re(α∗a)

1.5

p=0.26

γa

p=0.26 1

2.5 2

2

1 3

1.5 2 1

4

4

2

0.5 3

0.5 0

1

4

1 0

2 0.5

3 1

4 1.5

1

α0a 2

2.5

3

0

0

2 0.5

1

α0a

3 1.5

2

2.5

3

Fig. 7.25. The dispersion curve for epoxy-lead composites by the volume concentration of inclusions p = 0.26. The dashed parts of the dispersion curves correspond to values of attenuation coeﬃcient γa more than 0.5.

it corresponds to the asymptotic solution of (7.306) in the long-wave region (Section 7.8). The attenuation along this branch is small in the region (0 < α0 a < 0.8) but after that frequency it grows dramatically, and the wave corresponding to this branch practically disappears. The second branch (2) is the optical branch, and it appears at α0 a = 0.7. At ﬁrst, the attenuation coeﬃcient along this branch is large, and the corresponding waves cannot be observed experimentally. In the interval 0.9 < α0 a < 1.3, the attenuation coeﬃcient of this branch becomes suﬃciently small, and the corresponding waves become visible in experiments. For α0 a > 1.3, the attenuation coeﬃcient of branch 2 becomes large again, and the corresponding wave disappears. The third branch (3) appears at α0 a = 1.2, and at ﬁrst, the attenuation along this branch is large. For α0 a > 1.25, the attenuation becomes less than 0.3 (γa < 0.3) and this branch becomes visible. For larger values of α0 a (α0 a > 1.5) this branch is situated close to the diagonal Re(k∗ a) = α0 a of the square in the left-hand side of Fig. 7.25. Thus, the phase velocity of the corresponding waves become close to its velocity in the matrix. The attenuation coeﬃcient along branch 3 is small and tends to the short wave limit indicated in Section 7.8 as α0 a grows. Thus, in the region (0.8 < α0 a < 1.2) , transition from the acoustical branch 1 to the optical branch 3 (through branch 2) takes place. Experimental points (black dots) in this region are between the two branches if both of them exist, and close to one of them if another branch is not visible. The fourth branch (4) of the solutions of (7.306) was also found in the considered region, and attenuation along this branch turns out to be small in the short region 2.3 < α0 a < 2.6. But experimental data are closer to branch 3 in this region. Note that every branch corresponds to diﬀerent forms of oscillations of the inclusions in the process of wave propagation, and the higher the frequency of the branch appearance the more complex is the form of these oscillations. In order to generate such a complex form of inclu-

7.9 Numerical solution of the EFM dispersion equations

253

sion oscillations, a speciﬁc excitation should be applied to the medium; for plane monochromatic waves, a particular form of oscillation may not reveal. The case of volume concentration p = 0.34 is presented in Fig. 7.26, and it does not diﬀer qualitatively from p = 0.26. Four diﬀerent branches of the solutions of the dispersion equation (7.306) were found in this case but their relative position are not the same as for p = 0.26. Application of the dispersion equation (7.111) of the EMM to the analysis of the epoxy-lead system is presented in Figs. 7.27–7.29. The principal diﬀerence from the EFM results is that the EMM predicts the existence of only two branches. For small volume concentration, only acoustical branch is visible, and it deﬁnes the mean wave ﬁeld for all frequencies of the incident ﬁeld similar to the predictions of the EFM dispersion equation. 3

Re(α∗a)

1.2

p=0.34

2.5

2

2

γa

p=0.34

4

1

2

1

1

0.8

1

4 0.6

1.5 1

2 3

1

0.4 3

4 1

0.2

0.5 0

0.5

1

1.5

2

2.5

4

3

α0a 0

4

2

2

3

0

0

0.5

1

α0a 1.5

2

2.5

3

Fig. 7.26. The same dependence as in Fig. 7.25 for p = 0.34. 3

Re(a*a)

2.5

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2

0

a 0a 0

0.5

1

1.5

ga

1.2

p=0.159

2

2.5

p=0.159

a 0a

0 3

0

0.5

1

1.5

2

2.5

3

Fig. 7.27. The dispersion curves for composites with an epoxy matrix and lead inclusions of volume concentration p = 0.159. The solid line is the acoustical branch of the EMM dispersion equation (7.111), black points are experimental data from [63]. The dashed line is the optical branch of wave propagation with attenuation γa > 1.

254

7. Elastic waves in the medium with spherical inclusions

3

Re(a*a)

1

0.6

p=0.26 2

p=0.26

2.5 0.4 2 1.5

0.2 1

1 0 0

0.5 0

1 0

2 0.5

a 0a 1

1.5

2

2.5

3

2 0.5

a 0a 1

1.5

2

2.5

3

−0.2

Fig. 7.28. The same as in Fig. 7.27 for p = 0.26. The EMM predicts the existence of two branches of wave propagation. 3

0.8

p=0.34

Re(a*a)

2.5

ga

p=0.34

0.6

2 0.4

1 1.5

0.2

1

1

2 0

0.5 0

a 0a

0

0.5

1

1.5

2

2.5

3

0

2 0.5

a 0a

1

1.5

2

2.5

3

−0.2

Fig. 7.29. The same as in Fig. 7.27 for p = 0.34.

For volume concentrations p = 0.26 and 0.34, the acoustical branch deﬁnes the mean wave ﬁeld in the region 0 < α0 a < 1. For higher frequencies, the attenuation along the acoustical branch becomes large, and the mean wave ﬁeld is deﬁned by the optical branch. Note that in the region 0 < α0 a < 1 the attenuation coeﬃcient along the optical branch is negative; this part of the branch has no physical meaning.

7.10 Conclusion The comparison of predictions of various versions of the EMM and the EFM with experimental data and numerical solutions of the homogenization problem in statics allows us to evaluate the results of the application of selfconsistent methods to the problem of elastic wave propagation in a medium with spherical inclusions.

7.10 Conclusion

255

Version I of the EMM is in a good agreement with experimental data in the long-wave region if the volume concentration of inclusions is small (less than 0.3). This version gives correct values of the velocities and attenuation coeﬃcients of the mean wave ﬁelds in the limit of small volume concentrations of inclusions (non-interacting scatterers). In the short-wave region, this version gives physically reasonable results for the velocities and attenuation coeﬃcients of propagating waves. The errors of the predictions of this version may be appreciable for higly contrasting properties of the matrix and inclusions, and for high volume concentrations of the latter. Versions II and III of the EMM improve the predictions of version I for the elastic moduli and velocities of the mean wave ﬁeld in the long-wave region, but they do not describe correctly the attenuation of the mean wave ﬁeld in this region. The dependence of the attenuation on frequency turns out to be of the order higher then ω 4 , and does not describe Rayleigh wave scattering by inclusions. The analysis of predictions of versions II and III for elastic wave propagation in the medium and short-wave regions faces diﬃculties of complex solutions of the one-particle problem. In Chapter 3, versions II and III were carried out for electromagnetic waves, where the one-particle problem is much simpler. It is shown in this chapter that in the medium- and short-wave regions, the predictions of versions II and III of the EMM are close to the predictions of version I. It is reasonable to expect similar correspondence between various versions of the EMM for elastic waves. The general drawback of all the versions of the EMM is that they are unable to describe the inﬂuence of peculiarities in spacial distributions of inclusions on the eﬀective properties of composite. The EFM developed in this chapter opens the possibility of such a description. In the long-wave region, the EFM gives physically correct values of the velocities and attenuation coeﬃcients of the mean wave ﬁeld in the composites. Its predictions for the static moduli of the composites correspond to the numerical calculations and experimental data. The error of the EFM in this region is appreciable if the inclusions are much harder than the matrix, and its volume concentration is more than 0.4. In the medium-wave region, the method gives physically reasonable values of the phase velocities of the mean wave ﬁelds, but it predicts negative values of attenuation coeﬃcients for composites with high volume concentrations of inclusions that are much softer and lighter than the matrix. Apparently, the picture of the detailed wave ﬁeld in the medium-wave region is very complex, and cannot be described by the relatively simple hypotheses of the EFM. In the short-wave region, the method gives physically correct results for all types of inclusions. Comparison of the EFM with other self-consistent methods (various versions of the eﬀective medium method (EMM)) shows that these methods give close results in the long-wave region and small volume concentrations of

256

7. Elastic waves in the medium with spherical inclusions

the inclusions, but in the medium- and short-wave regions the predictions of the EFM and EMM may deviate essentially. These predictions are also different for composites with high volume concentrations of highly contrasting inclusions (see the comparison of the EFM and EMM predictions for shear wave propagation in ﬁber composites (Chapter 4). Generally speaking, the predictions of the EFM are more reliable than those of EMM, but the EMM is technically easier.

7.11 Notes This chapter is based on the works [56–58].

8. Elastic waves in polycrystals

The problem of wave propagation through polycrystalline materials has attracted much attention due to its important theoretical and practical aspects. For instance, the solution of this problem gives a theoretical foundation for a nondestructive analysis of the microstructure of real metals. It is known that the propagation of elastic waves through polycrystals is accompanied by the dispersion and attenuation of the waves. The cause behind these phenomena is the inhomogeneity of such a medium, which consists of many monocrystalline grains with distinct orientations. It is also known (see, e.g., [82, 83]) that elastic scattering by grains contributes a large part of the ultrasonic attenuation in polycrystalline metals, and that the dependence of attenuation coeﬃcients on frequency ω has three characteristic regions. These are as follows: the Rayleigh or long-wave region, where the length of elastic waves λ is larger than the diameters 2a of grains (λ/2a > 1), the stochastic region where the ratio λ/2a is less than 1 but not very small, and the diﬀusive region where λ/2a 1. In all these regions, the attenuation coeﬃcient γ is proportional to some power α of ω (γ ∼ ω α ) where α = 4 for the Rayleigh region, α = 2 for the stochastic region and α = 0 for the diﬀusive region. Internal friction and viscosity can contribute to attenuation. Appropriate terms in the attenuation coeﬃcients depend linearly on frequency, and their contribution is particularly important for very low frequencies. In this chapter, the problem of elastic wave propagation in polycrystals is solved by the eﬀective medium method. The general scheme of the method is developed in Sections 8.1–8.3. A special basis of four-rank tensors is proposed in Section 8.4 in order to perform the method for polycrystals with orthorhombic symmetry of the monocrystals. Another way to construct the solution of the homogenization problem for polycrystals is based on perturbation theory. The ﬁrst approximation of this theory, the so-called “Born approximation”, is considered in Section 8.5. The Born approximation satisfactorily describes the Rayleigh and stochastic regions of frequencies of the propagating waves if the elastic properties of monocrystals have small deviations from isotropy (small anisotropies). The diﬀusive region cannot be described within the framework of the Born approximation.

258

8. Elastic waves in polycrystals

In Section 8.6, the EMM is used to calculate the phase velocities and attenuation of longitudinal and transverse waves in some polycrystalline metals. It is shown that the method allows us to describe physically correctly the behavior of the mean wave ﬁeld in the long-wave as well as in stochastic and diﬀusive regions. Predictions of the method are compared with experimental data and with the results of the Born and long-wave approximations.

8.1 General consideration Let us describe the model of the polycrystal under consideration. Such materials consist of a set of grains ideally conjugated along their interfaces. Every grain is a monocrystal with a constant elastic moduli tensor C. The orientation of the crystallographic axes of these monocrystals randomly changes from one grain to another. Therefore, inside each grain, the components of the elastic moduli tensor of the medium are constant but they are diﬀerent in diﬀerent grains. The distribution of orientations of the crystallographic axes of the monocrystals inside grains is assumed to be homogeneous. The shapes of the grains are quasispherical and their random radii are described by some distribution function which does not depend on the position of the grains in space. Thus, the eﬀective or macroscopic elastic properties of the polycrystals are isotropic. Note that the densities ρ of the grains are constant and the same for all grains. Let a polycrystal occupy 3D-space, and let x(x1 , x2 , x3 ) be a point of this space. If the medium oscillates harmonically with frequency ω and the dependence on time t in deﬁned by the factor eiωt , the amplitude u(x) of the displacement vector in the medium satisﬁes the equation ∇i Cijkl (x)∇k ul (x) + ρω 2 uj (x) = 0,

(8.1)

where C(x) is the tensor of elastic moduli of the medium. C(x) is a random function with the above mentioned properties that may be presented in the form C(x) = C 0 + C 1 (x).

(8.2)

Here C 0 is a constant tensor, and C 1 (x) describes deviation of the moduli inside the grains. After rewriting (8.1) in the form 0 1 (x)∇k ul (x) + ρω 2 uj (x) = −∇i Cijkl (x)∇k ul (x), ∇i Cijkl

(8.3)

−1 0 and applying the operator ∇i Cijkl (x)∇k + ρω 2 δil to this equation, we ﬁnd the integral equation for the wave ﬁeld u(x) in the polycrystal 0 0 1 (x − x )∇k Cjklm (x )∇ l um (x )dx . (8.4) ui (x) = ui (x) + gij

8.1 General consideration

259

Here g 0 (x) is Green’s function for a homogeneous medium with elastic moduli tensor C 0 . This function satisﬁes the equation 0 0 ∇i Cijkl (x)∇k + ρω 2 δjl glm (x) = −δim δ(x) (8.5) and “radiation” conditions at inﬁnity. Here δim is Kronecker’s symbol, and δ(x) is Dirac’s delta-function. Note that u0 (x) in (8.4) is the “incident” ﬁeld which would have existed in the homogeneous medium with the properties C 0 and ρ and the same sources of waves. The integral equation (8.4) is totally equivalent to the original diﬀerential equation (8.1). For a homogeneous isotropic elastic medium with bulk modulus K0 and shear modulus µ0 , the tensor g 0 (x) is represented in the form 1 xi [h(r)δij + g(r)ni nj ] , r = |x| , ni = r |x| 1 ωr ωr h(r) = 1+i exp −i 4πρω 2 r2 l0 l0 ωr ωr ω 2 r2 exp −i , − 1− 2 +i t0 t0 t0 1 ω 2 r2 ωr ωr g(r) = 3 − 2 + 3i exp −i 4πρω 2 r2 t0 t0 t0 ω 2 r2 ωr ωr − 3 − 2 + 3i exp −i . l0 t0 l0

0 gij (x) =

(8.6)

(8.7)

(8.8)

where l0 and t0 are the velocities of longitudinal and transverse waves in the medium l02 =

3K0 + 4µ0 µ0 , t20 = , 3ρ ρ

(8.9)

Equation (8.4) implies that amplitude ε(x) of the strain tensor in the polycrystalline medium εij (x) = ∇(i uj) (x) satisﬁes the equation 0 εij (x) = ε0ij (x) − Kijkl (x − x´)C1klmn (x´)εmn (x´)dx´; (8.10) 0 K0ijkl (x) = − ∇i ∇j gjl (x) (ij)(kl) ,

ε0ij (x) = ∇(i u0j) (x).

(8.11)

Let us consider the mean displacement u(x) and strain ε(x) wave ﬁelds in the polycrystal. Here the angle brackets means the average over the ensemble realizations of random function C(x). The general theory of elastic media with microstructure [68] states that the ﬁeld u(x) should satisfy the following equation of motion:

260

8. Elastic waves in polycrystals

∇C ∗ ∇ u(x) + ρω 2 u(x) = 0

(8.12)

which is similar to (8.1), but C∗ here is some integral operator. The kernel C∗ (x) of this operator is a fourth rank tensor function with complex components ∗ (8.13) (C ε) (x) = C ∗ (x − x´)ε(x´)dx´. If the mean ﬁeld u(x) is a plane wave u(x) = U0 e−iq∗ .x , the wave vector q∗ = q∗ m of this wave satisﬁes the following dispersion equation: ∗ ∗ (q ∗ )qi∗ ql∗ − ρω 2 δjk = 0; C ∗ (q ∗ ) = C ∗ (x)eiq ·x dx. (8.14) Cijkl The real part of the wave number q∗ determines the phase velocity of the mean wave ﬁeld u(x) (v∗ = ω/ Re(q∗ )) and the imaginary part (− Im(q∗ )) is the attenuation coeﬃcient of this wave. Thus, the problem of calculating velocities and attenuation coeﬃcients of waves in polycrystals is reduced to constructing the function C ∗ (q ∗ ) which is a Fourier transform of the kernel C ∗ (x) of the operator C∗ in (8.13). For a ﬁxed wave vector q ∗ , this function may be interpreted as an eﬀective dynamic tensor of elastic moduli of the polycrystal. The eﬀective medium method will be used to construct this tensor.

8.2 The eﬀective medium method Let the incident ﬁeld be a plane wave. According to version IV of the EMM (Section 1.3) we accept hypothesis IV1 . IV1 . The wave ﬁeld inside each grain of the polycrystal coincides with the wave ﬁeld inside an isolated grain embedded in the homogeneous medium with the eﬀective dynamic properties of the polycrystal (see Fig. 8.1).

ε0

<ε∗>

Fig. 8.1. The one-particle problem of the EMM for polycrystals.

C∗

8.2 The eﬀective medium method

261

This hypothesis reduces the problem of calculation of the elastic ﬁeld inside each inclusion to a one-particle problem.The integral equation of this problem is similar to (8.10), and has the form C 0 = C ∗ . ∗ ∗1 εij (x) = ε∗ij (x) − Kijkl (x − x´)Cklmn (x´)εmn (x´)dx´, (8.15) v

∗ ∗ (x) = − ∇i ∇k gjl (x) (ij)(kl), Kijkl

(8.16)

where g ∗ (x) is the Green function for a homogeneous medium with the eﬀective elastic properties C ∗ of the polycrystal, C ∗1 (x) = C(x) − C ∗ , ε∗ (x) is a plane wave with the wave vector q ∗ , and v is the volume of the grain. Let the general solution of this equation be known and presented in the form ε(x) = Λε∗ (x);

Λ = Λ(C ∗ , C, a, q ∗ ).

(8.17)

Here Λ(C ∗ , C, a, k∗ ) is a linear operator which depends on the elastic moduli tensor C ∗ of the eﬀective medium, moduli of the grain C, radius a of the grain and wave vector q ∗ of the exciting ﬁeld. Thus, according to the ﬁrst hypothesis of the eﬀective medium method, the ﬁeld ε(x) inside each grain has the form (8.17). The strain ﬁeld in the polycrystal may now be presented in the form ∗ (8.18) ε(x) = ε (x) − K ∗ (x − x´)C ∗1 (x´)Λε∗ (x´)dx´ Here the eﬀective medium is taken as the reference medium, the kernel K ∗ (x − x´) and the incident ﬁeld ε∗ (x) are the same as in (8.15). The second hypothesis of this version of the EMM, we take in the form I2 . I2 .The mean wave ﬁeld in the polycrystal coincides with the ﬁeld propagating in the eﬀective medium. ε(x) = ε∗ (x).

(8.19)

Averaging (8.18) over the ensemble realization of the polycrystalline microstructures we ﬁnd the equation (8.20) ε(x) = ε∗ (x) − K ∗ (x − x´) C ∗1 (x´)Λ ε∗ (x´)dx´. Equation (8.19) implies that the integral term in this equation should be zero ∗1 (8.21) C (x)Λ = (C(x) − C ∗ ) Λ (C ∗ , C(x), a(x), k ∗ ) = 0. The average here is carried out over the random orientations of the crystallographic axes of the crystals in the grains and random grain sizes a. This

262

8. Elastic waves in polycrystals

equation yields the equation for the tensor of the eﬀective elastic moduli of the composite in the form −1

C ∗ = Λ (C ∗ , C(x), a(x), k ∗ )

C(x)Λ (C ∗ , C(x), a(x), k ∗ ) .

(8.22)

Let us consider a typical grain v embedded in the homogeneous medium with the eﬀective properties of the polycrystal, and the ﬁeld inside the grain take in the form (8.17). The strain ﬁeld in the medium is presented in the following integral form (8.23) ε(x) = ε∗ (x) − K ∗ (x − x )C ∗1 (x ) · Λε∗ (x´)dx , v

where v is the volume of the grain. The integral term in this equation is the ﬁeld εs (x) scattered on the grain. The mean value of this ﬁeld has the form (8.24) εs (x) = − K ∗ (x − x ) C ∗1 (x )Λ ε∗ (x´)dx . v

Here the averaging is carried out over the orientations of the crystallographic axis inside the grains. If the condition of self-consistency is taken in the form IV2 (see Section 2.3.3: The tensor of the eﬀective properties of the polycrystal should be chosen in order to diminish the mean ﬁeld scattered on a typical grain embedded in the eﬀective medium), the mean under the integral in this equation should disappear, and we go again to equation (8.21) for the eﬀective parameters of the composite. Thus, for polycrystals, the conditions of self-consistency I2 and IV2 give the same ﬁnal equations (8.21) and (8.22).

8.3 The one-particle problem of EMM The one-particle problem of the EMM is the problem of diﬀraction of a plane elastic waves by an anisotropic grain embedded in the homogeneous medium with the eﬀective elastic properties of the polycrystal. The integral equation of this problem has the form (8.15), where the ﬁeld ε∗ (x) in the right-hand side is a plane wave ∗

ε∗ (x) = D∗ e−iq

·x

.

(8.25)

Let us ﬁnd the approximate solution of (8.15) in the same form: ∗

ε(x) = De−iq

·.x

.

(8.26)

After substituting (8.25) and (8.26) in equation (8.15) we ﬁnd the following equation ∗ (8.27) D + K ∗ (x − x´)C ∗1 Deiq ·(x−x´) dx´= D∗ , v

8.3 The one-particle problem of EMM

263

where the tensor D∗ in the right-hand side is constant. Of course, the lefthand side of this equation is not constant. In order to obtain an approximate value of D in (8.26) let us average both parts of (8.27) over the volume v of the inclusion, and then over all possible directions of the wave vector k∗ . As a result, we obtain the tensor D in the form −1 Λ = E 1 + K(k ∗ )C ∗1 , 1 K(k ∗ ) = K ∗ (x − x´)eiq∗ m·(x−x´) dx´dxdm, 4πv Ω v v D = ΛD∗ ,

(8.28) (8.29)

qi∗ = q∗ mi . Here Ω is the surface of a unit sphere in x-space, m is a vector normal to Ω, and qi∗ is the wave number of the eﬀective medium. After introducing a new variable R = x − x´, using (8.16) for K ∗ (x), and the Gauss theorem, we can the rewrite the integral K(q ∗ ) in the form 1 ∗ gik (R)∇j ∇l f (R)eik∗ m.R dRdm . (8.30) Kijkl (q ∗ ) = − 4π Ω v (ij)(kl) Here f (R) is the scalar function which depends only on |R| and has the form 4 3 3 |R| 1 |R| 1 v(x)(x + R)dx = 1 − 4 a + 16 a3 , |R| ≤ 2a , (8.31) f (R) = v 0, |R| > 2a where v(x) is the characteristic function of spherical area v of radius a with the center at the point x = 0. Let us substitute in (8.30) the function g ∗ (x) from (8.6) and introduce a spherical coordinate system in R-space. After calculating the integrals over Ω and over the unit sphere in R-space, we can represent the tensor K(q ∗ ) in the form 1 (8.32) K(q ∗ ) = P1 (q∗ a)E 2 + 2P2 (q∗ a) E 1 − E 2 , 3 where q∗ a is the dimensionless wave number, E 1 and E 2 are the fourth-rank isotropic tensors deﬁned in Appendix 1, and P1 (q∗ a) and P2 (q∗ a) are the scalar integrals 4π 2 1 Φ0 (ζ) h(aζ) + g(aζ) P1 (q∗ a) = − 3 0 3 1 (8.33) + Φ1 (ζ) (h(aζ) + g(aζ)) ζdζ, 3

264

8. Elastic waves in polycrystals

4π P2 (q∗ a) = − 2

2 0

1 Φ0 (ζ) h(aζ) + g(aζ) 3 2 1 + Φ1 (ζ) h(aζ) + g(aζ) ζdζ. 3 5

(8.34)

Here the functions h(r) and g(r) are deﬁned in (8.7), (8.8) but K0 and µ0 should be replaced by the eﬀective bulk modulus K∗ and the shear modulus µ∗ of the polycrystal. Functions Φ0 and Φ1 in (8.33), (8.34) take the forms Φ0 (ζ) = Q00 (ζ)j0 (q∗ ζ) + Q01 (ζ)q∗ j1 (q∗ ζ),

(8.35)

Φ1 (ζ) = Q10 (ζ)j0 (q∗ ζ) + Q11 (ζ)q∗ j1 (q∗ ζ) + Q12 q∗2 j2 (q∗ ζ),

(8.36)

where jn (z) are the spherical Bessel functions, and the functions Qrs (ζ) are 3 1 3 1 3 , Q01 (ζ) = − + − ζ 2 , Q00 (ζ) = − + 4 16ζ ζ 4 16 Q10 (ζ) = −

3 3 3 3 + ζ, Q11 (ζ) = − ζ 2 , 4ζ 16 2 8

(8.37) (8.38)

1 3 Q12 (ζ) = 1 − ζ + ζ 2 . 4 16

(8.39)

Let us introduce longitudinal (α∗ ) and transverse (β∗ ) wave numbers of the eﬀective medium: ρ ρ , β∗ = ω , (8.40) α∗ = ω M∗ µ∗ 4 (8.41) M∗ = K∗ + µ∗ . 3 The integrals P1 (k∗ a) and P2 (k∗ a) should be calculated for q∗ a = α∗ α or q∗ a = β∗ a. The asymptotics of P1 and P2 for small and large values of frequencies ω (small and large values of α∗ a and β∗ a) are obtained on the basis of the general expressions for P1 and P2 . 1. The long-wave asymptotics of P1 and P2 (small ω). These asymptotics have the same forms for longitudinal and transverse waves. ρω 2 a2 P1 (αa) = ρω 2 a2 P1 (βa) =

2

4

2 (αa) i (αa) 5 + − (αa) + O(ω)6 , 9 45 27 (8.42)

ρω 2 a2 P2 (αa) = ρω 2 a2 P2 (βa) =

(βa)4 i (βa)2 2 i 2 4 5 + − (αa) + (αa) − (αa) + O(ω)6 . 10 25 15 75 45

(8.43)

8.4 Polycrystals with orthorhombic grains

265

2. The short-wave asymptotics of P1 and P2 (large ω). Longitudinal waves: ρω 2 a2 P1 (αa) =

i 5(αa)2 − (αa)3 + O(ω), 36 24

(8.44)

ρω 2 a2 P2 (αa) =

i 5(αa)4 + (αa)2 (βa)2 − (αa)3 + O(ω). 60((αa)2 − (βa)2 ) 40

(8.45)

Transverse waves: ρω 2 a2 P1 (βa) =

i(αa)5 (αa)2 (βa)2 − (αa)3 + O(ω), 9((αa)2 − t2 ) 6((αa)2 − (βa)2 )2

(8.46)

ρω 2 a2 P2 (βa) =

3i 7(αa)2 (βa)2 − 15(βa)4 − (βa)3 + O(ω). 120((αa)2 − (βa)2 ) 80

(8.47)

In these equations α = α∗ and β = β∗ . Let us return to the solution of the one-particle problem. It follows from (8.17), (8.25) and (8.28) that in the framework of the considered approximation, the wave ﬁeld ε(x) inside each grain is expressed via the incident ﬁeld ε∗ (x) in the form −1 ε(x) = Λε∗ (x), Λ = I + K(q ∗ )C ∗1 .

(8.48)

This is the ﬁnal form of the solution of the one-particle problem. Therefore, operator Λ in (8.17) coincides with the operator of multiplication on the tensor Λ in (8.48), and the equation for eﬀective moduli tensor C ∗ (q ∗ ) follows from (8.22) in the form

−1 −1 −1 I + K(q ∗ )C 1 (x) . (8.49) C ∗ (q ∗ ) = C(x) I + K(q ∗ )C 1 (x) Here q∗ = α∗ for longitudinal waves, and q∗ = β∗ for transverse waves. Before going to the numerical solution of (8.49) we have to develop a technique for operating with the anisotropic tensors on the right-hand side of this equation.

8.4 Polycrystals with orthorhombic grains In this section we consider polycrystals with orthorhombic symmetry in the elastic moduli of the grains; n1 , n2 and n3 are three mutually orthogonal vectors of the main crystallographic axes of the orthorhombic crystal. Let us introduce twelve linear independent fourth-rank tensors T i that provide a basis in the space of tensors of the given symmetry:

266

8. Elastic waves in polycrystals 1 2 Tijkl = n1i n1j n1k n1l , Tijkl = n2i n2j n2k n2l , 3 4 Tijkl = n3i n3j n3k n3l , Tijkl = n1i n1j n2k n2l , 5 6 Tijkl = n2i n2j n1k n1l , Tijkl = n1i n1j n3k n3l , 7 8 Tijkl = n3i n3j n1k n1l , Tijkl = n2i n2j n3k n3l , 9 10 Tijkl = n3i n3j n2k n2l , Tijkl = 4(n1i n2j n1k n2l )(ij)(kl) , 11 12 Tijkl = 4(n1i n3j n1k n3l )(ij)(kl) , Tijkl = 4(n2i n3j n2k n3l )(ij)(kl) .

(8.50)

Tensor C of the elastic moduli of an orthorhombic crystal is presented in this basis T i in the following form C = c1 T 1 + c2 T 2 + c3 T 3 + c4 (T 4 + T 5 ) + c5 (T 6 + T 7 ) + c6 (T 8 + T 9 ) + c7 T 10 + c8 T 11 + c9 T 12 .

(8.51)

The connection between the scalar coeﬃcients ci in this equation and the elastic constants cij in the matrix representation of Hooke’s law for an anisotropic body [81] is given by the relations c1 = c11 ,

c2 = c22 ,

c3 = c33 ,

c6 = c23 ,

c7 = c55 ,

c8 = c66 ,

c4 = c12 ,

c5 = c13 ,

c9 = c44 .

(8.52)

For crystals of a higher symmetry, some of the nine constants in the righthand side of (8.51) are the same: for trigonal crystals only six constants are independent: c1 = c2 , c5 = c6 , c8 = c9 ;

(8.53)

hexagonal crystals have ﬁve constants: c1 = c2 , c5 = c6 ,

c7 =

1 (c1 − c4 ), 2

c8 = c9 ;

(8.54)

cubic crystals have three constants: c1 = c2 = c3 ,

c4 = c5 = c6 ,

c7 = c8 = c9 ;

(8.55)

an isotropic body has two constants: c1 = c2 = c3 ,

c4 = c5 = c6 ,

c7 = c8 = c9 =

1 (c1 − c4 ). 2

(8.56)

The twelve tensors T i constitute a closed algebra with respect to product of two tensors r s Tnmkl , (T r T s )ijkl = Tijmn

r, s = 1, 2, ..., 12.

(8.57)

8.4 Polycrystals with orthorhombic grains

267

The multiplication table of the tensors T i has the form 1

T T2 T3 T4 T5 T6 T7 T8 T9 T 10 T 11 T 12

T1 T1 0 0 0 T5 0 T7 0 0 0 0 0

T2 0 T2 0 T4 0 0 0 0 T9 0 0 0

T3 0 0 T3 0 0 T6 0 T8 0 0 0 0

T4 T4 0 0 0 T2 0 T9 0 0 0 0 0

T5 0 T5 0 T1 0 0 0 0 T7 0 0 0

T6 T6 0 0 0 T8 0 T3 0 0 0 0 0

T7 0 0 T7 0 0 T1 0 T5 0 0 0 0

T8 0 T8 0 T6 0 0 0 0 T3 0 0 0

T9 0 0 T9 0 0 T4 0 T2 0 0 0 0

T 10 0 0 0 0 0 0 0 0 0 2T 10 0 0

T 11 0 0 0 0 0 0 0 0 0 0 2T 11 0

T 12 0 0 0 0 0 0 0 0 0 0 0 2T 12

For our purposes we have to average these tensors over the orientations of the mutually orthogonal vectors n1 , n2 and n3 in 3D-space. Let us assume that the distribution of these axes over theorientations is homogeneous and denote such an average as ·Ω . The means T i Ω of the basic tensors T i take the forms i 1 (E2 + 2E1 ), i = 1, 2, 3; (8.58) T Ω= 15 i 1 (2E2 − E1 ), i = 4, 5, 6, 7, 8, 9; (8.59) T Ω= 15 i 2 1 T Ω = (E1 − E2 ), i = 10, 11, 12. (8.60) 5 3 Here E1 and E2 are the isotropic four-rank tensors deﬁned in Appendix 1. Representations of these tensors in the T -basis take the forms E1 = T 1 + T 2 + T 3 +

1 10 T + T 11 + T 12 , 2

E2 = T 1 + T 2 + T 3 + T 4 + T 5 + T 6 + T 7 + T 8 + T 9 .

(8.61) (8.62)

Note that tensor E1 plays the role of the unit tensor I in the set of tensors. Thus, if tensor A has the form A=

12

ai T i ,

(8.63)

i=1

where ai are some scalar coeﬃcients, its average over the orientations is AΩ = C =

12 i=1

ci T i ,

(8.64)

268

8. Elastic waves in polycrystals

c1 = c2 = c3 ,

c4 = c5 = c6 = c7 = c8 = c9 ,

c10 = c11 = c12 ,

1 [3(a1 + a2 + a3 ) + a4 + a5 + a6 + a7 + a8 + a9 + 4 (a10 + a11 + a12 )] , 15 1 [a1 + a2 + a3 + 2 (a4 + a5 + a6 + a7 + a8 + a9 ) − 2 (a10 + a11 + a12 )] , c4 = 15 1 [2(a1 + a2 + a3 ) − a4 − a5 − a6 − a7 − a8 − a9 + 6 (a10 + a11 + a12 )] . c10 = 30

c1 =

If a tensor A has the form (8.63), the inverse tensor B = A−1 deﬁned by relations AA−1 = A−1 A = I is represented in the similar form B=

12

ci T i .

(8.65)

i=1

The ﬁrst nine scalar coeﬃcients, ci are the solutions to the following system of linear algebraic equations a1 c1 + a4 c5 + a6 c7 = 1, a3 c3 + a7 c6 + a9 c8 = 1, a5 c1 + a2 c5 + a8 c7 = 0, a7 c1 + a9 c5 + a3 c7 = 0, a9 c2 + a7 c4 + a3 c9 = 0.

a2 c2 + a5 c4 + a8 c9 = 1, a4 c2 + a1 c4 + a6 c9 = 0, a6 c3 + a1 c6 + a4 c8 = 0, a8 c3 + a5 c6 + a2 c8 = 0, (8.66)

The other three coeﬃcients have the forms: c10 =

1 , 4a10

c11 =

1 , 4a11

c12 =

1 . 4a12

(8.67)

For crystals of cubic symmetry, it is possible to introduce only three basis tensors H i to represent the elastic moduli tensor C: C = c1 H 1 + c2 H 2 + c3 H 3 .

(8.68)

The connection between coeﬃcients ci in the right-hand side and the components cij in the matrix form of Hooke’s law for the cubic crystals has the form: c1 = c11 , c2 = c12 , and c3 = c44 . The tensors H i are expressed via the basic tensors T i : H 1 = T 1 + T 2 + T 3, H 3 = T 10 + T 11 + T 12 ,

H 2 = T 4 + T 5 + T 6 + T 7 + T 8 + T 9, (8.69)

and again constitute a closed algebra with respect to the product operation (8.57). This algebra is essentially simpler than the algebra of the T i -tensors. Let tensors A and B be linear combinations of tensors H i (ai and bi are scalars)

8.4 Polycrystals with orthorhombic grains

A=

3

ai H i ,

B=

i=1

3

bi H i .

269

(8.70)

i=1

The product of these two tensors has the form AB = (a1 b1 + 2a2 b2 )H 1 + (a1 b2 + a2 b1 + a2 b2 )H 2 + 2a3 b3 H 3 .

(8.71)

The formula for averaging the tensor A is 1 2 1 2 2 1 AΩ = (a1 + 2a2 )E + (a1 − a2 + 3a3 ) E − E , 3 5 3

(8.72)

1 E1 = H 1 + H 2, E2 = H 1 + H 2. 2

(8.73)

The inverse tensor A−1 is deﬁned by the equation A−1 =

a1 + a2 1 a2 2 1 H − H + H 3, ∆ ∆ 4a3

∆ = (a1 − a2 )(a1 + 2a2 ). (8.74)

This analysis allows us to construct a numerical algorithm for calculating the right-hand side of (8.49). It is necessary to take into account that the averaging in this equation is carried out over orientations of crystallographic axes of grains, and over radii of the latter. Orientations and radii are considered as independent random variables. Hence, these two averages can be made separately. For polycrystals with cubic symmetric of the monocrystals, equation (8.49) is simpliﬁed. For the eﬀective bulk modulus K∗ of such polycrystals, the solution of (8.49) has the explicit form K∗ =

1 (c1 + 2c2 ), 3

(8.75)

and the eﬀective shear module µ∗ satisﬁes the equation (c1 − c2 )S1 (µ∗ ) + 3c3 S2 (µ∗ ) , 2S1 (µ∗ ) + 3S2 (µ∗ ) 7 6 1 S1 (µ∗ ) = , 1 + 2P2 (q∗ )(c1 − c2 − 2µ∗ ) 7 6 1 S2 (µ∗ ) = . 1 + 4P2 (q∗ )(c3 − 2µ∗ ) µ∗ =

(8.76)

(8.77) (8.78)

Here q∗ = α∗ for longitudinal waves, and q∗ = β∗ for transverse waves. The integral P2 (k∗ ) in (8.77), (8.78) is deﬁned in (8.34). The averaging in (8.77), (8.78) is carried out only over the radii of the grains.

270

8. Elastic waves in polycrystals

8.5 The Born approximation Another approximate solution of the homogenization problem for polycrystals may be obtained for small deviations of the elastic moduli inside the grains from the mean value of the tensor of the elastic moduli. Let C 0 be the elastic moduli of the grain averaged over the ensemble realizations of the grains in the polycrystal C 0 = C(x) .

(8.79)

If we introduce the wave operator L0 L0 = C 0 +ω 2 ρ,

(8.80)

propagation of monochromatic waves of frequency ω in the polycrystal is described by equation (8.3) that may be written in a symbolic form as follows L0 u = −L1 u.

(8.81)

Here the operator L1 has the form L1 = C 1 (x), C 1 (x) = C(x) − C 0 .

(8.82)

Applying the operator L−1 0 to (8.81) we ﬁnd equation (8.4) for the detailed wave ﬁeld in the polycrystal; the symbolic form is u = u0 + g 0 L1 u.

(8.83)

Here g 0 is the integral operator with the kernel g 0 (x), u0 is the wave ﬁeld propagating in the medium with the wave operator L0 that satisﬁes the equation L0 u0 = 0.

(8.84)

−1 1 If C 0 C (x) = O(δ), and δ 1, the solution of (8.83) in the form of the Neumann series converges and has the form u = u0 + g 0 L1 u0 + g 0 L1 g 0 L1 u0 + .... =

∞

g 0 L1

n

u0 .

(8.85)

n=0

This equation may be written also in the form u = u0 + g 0 L1 u0 + g 0 L1 g 0 L1 u.

(8.86)

Averaging this equation over the ensemble realization of the random polycrystalline structure we obtain for the mean wave ﬁeld u(x) (8.87) u = u0 + g 0 L1 g 0 L1 u .

8.5 The Born approximation

271

Here we take into account that 1 L = C 1 (x) = 0. (8.88) 1 0 1 L g L u under the operator in the right-hand side of (8.87) Changing 1 0 1 to L g L u we go to the equation u = u0 + g 0 L1 g 0 L1 u .

(8.89)

The solution of this equation is called the Born approximation for the mean ﬁeld u(x). Applying the operator L0 to (8.89) and taking into account the equations L0 u0 = 0,

L0 g 0 = I

we go to the equation for the mean wave ﬁeld in the form L0 − L1 g 0 L1 u = 0.

(8.90)

(8.91)

The Fourier transform of (8.91) leads to the following algebraic equation ∗ ki Cijkl (k)kl − ω 2 ρδjk uk (k) = 0, (8.92) where C ∗ (k) is the k-representation of the operator of the eﬀective elastic dynamic moduli of the composite (8.93) C ∗ (k) = C 0 + g 0 (x) [R(x)eik·x ]dx. Here k is the vector parameter of the Fourier transform, R(x) is the correlation function of the random ﬁeld C 1 (x) (8.94) R(x − x ) = C 1 (x) ⊗ C 1 (x ) . The wave vector q∗ of the mean wave ﬁeld in the polycrystal satisﬁes the equation ∗ (8.95) (q∗ )ql∗ − ω 2 ρδjk = 0. det qi∗ Cijkl For homogeneous distribution of grains over orientations, the polycrystal is isotropic, and the vector q∗ = qn has the direction n of the incident wave. ∗ (q∗ )ql∗ is a two rank symmetric tensor, and for an isotropic The tensor qi∗ Cijkl eﬀective medium, it depends only on the eﬀective wave vector q∗ . Thus, the structure of this tensor is deﬁned by the equation 1 ∗ ∗ ∗ ∗ (8.96) qi Cijkl (q )ql = K∗ (q∗ ) + µ∗ (q∗ ) qj∗ qk∗ + q∗2 µ∗ (q∗ )δjk . 3

272

8. Elastic waves in polycrystals

For a longitudinal wave, uk (k) = Uk δ(k − q∗ ), Uk = U nk , and (8.92) and (8.96) imply that the wave number q∗ = α∗ of this wave is the solution of the equation 4 (8.97) α∗2 K∗ (α∗ ) + µ∗ (α∗ ) − ω 2 ρ = 0. 3 As the result, the velocity l∗ and the attenuation coeﬃcient γl of the longitudinal wave are l∗ =

ω , γl = − Im α∗ . Re α∗

(8.98)

For transverse waves uk (k) = Uk δ(k − q∗ ), Uk qk∗ = 0, and the equation for the wave number q∗ = β∗ of the transverse wave has the form β∗2 µ∗ (β∗ ) − ω 2 ρ = 0.

(8.99)

The velocity and attenuation coeﬃcient of this wave are t∗ =

ω , γt = − Im β∗ . Re β∗

(8.100)

Thus, the problem is reduced to the calculation of the integral on the right-hand side of (8.93), and the determination of the functions K∗ (q∗ ) and µ∗ (q∗ ) in (8.96). If the correlation function R(x) in (8.94) may be presented in the form R(x) = R(0)ϕ(x),

(8.101)

where a scalar function ϕ(x) describes the coordinate dependence of R(x), the integral in (8.93) can be rewritten as follows: g 0 (x) [R(x)eiq·x ]dx|ijmnpqkl = Rijmnpqkl (0)

0 gnq (x) p l [ϕ(x)eiq·x ]dx.

(8.102)

The integral on the right-hand side of this equation is similar to that in (8.30), and may be calculated by the same technique. The details of the calculation of this integral are presented in [90].

8.6 Numerical results The results of calculations of phase velocities and attenuation coeﬃcients of elastic waves in some polycrystalline metals are presented in this section.

8.6 Numerical results

273

Let us consider polycrystalline aluminium, where the tensor of elastic moduli of the grains has cubic symmetry. Three elastic moduli for Al are: c1 = 108.2 GPa, c2 = 61.3 GPa, c3 = 28.5 GPa, ρ = 2.7 · 10−6 GPa · s2 /m2 . The radii of grains a are assumed to be approximately equal. Figure 8.2 shows the dependence of the eﬀective phase velocities l∗ and t∗ , and dimensionless attenuation coeﬃcients γl a and γt a γl a = − Im(α∗ a),

γt a = − Im(β∗ a)

(8.103)

of longitudinal and transverse waves on the real part of corresponding wave numbers (ka = Re(α∗ a) for longitudinal, and ka = Re(β∗ a) for transverse waves lie along the horizontal axes in Fig. 8.2). Because ka = ωa/l∗ or ka = ωa/t∗ , the same graphs describe the frequency dependence of the velocities and attenuation coeﬃcients. The curves in Fig. 8.2 do not depend on an exact value of the radius a, but the region of frequencies that corresponds to the region of dimensionless wave numbers ka in Fig. 8.2 depends on a. If a = 10−4 m the region of frequencies in Fig. 8.2 changes from several megahertz (MHz) to several thousand MHz. Parameters λl and λt in Fig. 8.2 are deﬁned by the following equations: λl =

l ∗ − l0 t ∗ − t0 , λt = , l0 t0

(8.104)

102 λ

γa

3

γt

10−1

λt

γl

2 10−3 1

λl 10−5

0 −1 10−1

1

10

102

ka

10−7

10−1

1

10

102

ka

Fig. 8.2. Dependence of normalized variations (λl , λt ) of phase velocities (left part) and attenuation coeﬃcients (γl a, γt a) (right part) on dimensionless wave numbers for aluminium with ﬁxed grain radii; ka = l∗ a for longitudinal waves and ka = t∗ a for transverse waves. Lines with circles and triangles correspond to Born approximations of the attenuation coeﬃcients with the correlation function of elastic properties in the form (8.105). Dashed lines correspond to the long-wave approximate solution of the one-particle problem presented in [87].

274

8. Elastic waves in polycrystals

where l0 and t0 are Voigt’s velocities of longitudinal and transverse waves. The latter are the velocities of elastic waves in a homogeneous medium with a tensor of elastic constant CΩ that is equal to the tensor of elastic moduli of the monocrystal averaged over the orientations of the crystallographic axes. For Al, l0 = 6,447.51 m/s and t0 = 3,131.68 m/s. To obtain the graphs in Fig. 8.2, equation (8.76) was solved by iterations. The iteration was started by taking C ∗ = CΩ − Voigt’s approximation of elastic moduli. In the Rayleigh (qa < 1) and stochastic (1 < qa < 102 ) regions, it is suﬃcient to carry out ﬁve iterations to get results with an accuracy of 1%, but in the diﬀusive region (qa > 102 ), such an accuracy is reached only after 20–30 iterations. The lines with circles and triangles in Fig. 8.2 correspond to the values of attenuation coeﬃcients obtained from the Born approximation presented in [90]. The correlation function of the elastic properties of the polycrystal R(x) in (8.94) was taken in the form [90] 1 1 1 |x − y| 1 . (8.105) R(x − y) = C (x)C (y) = C C Ω exp − a The dashed lines in Fig. 8.2 correspond to the solution of [87]. In this work, the ﬁeld inside every gain was assumed to be constant (long-wave approximation), and the value of this constant was found by the Galerkin method from the original integral equation of the one-particle problem. The long-wave approximation coincides with the present calculations in the Rayleigh regions but does not describe the wave ﬁeld in stochastic and diﬀusive regions. Calculations of the velocities and attenuation coeﬃcients in polycrystalline nickel with some distribution of the sizes of grains are presented in Fig. 8.3. A monocrystal of nickel has cubic symmetry, and its elastic constants and density are: c1 = 250 GPa, c2 = 160 GPa, c3 = 118 GPa, γ(a)

102λ 16

1 12

λt

γt

γl

10−1

6 10−2 4 10−3 0 10−4 −4 10−1

1

10

102

k(a)

10−5 10−1

1

10

102

k(a)

Fig. 8.3. Dependence of normalized variations of phase velocities (left part) and attenuation coeﬃcients (γl a , γt a) (right part) on dimensionless wave numbers for nickel with a distribution in the sizes of grains. Circles and triangles represent experimental points.

8.6 Numerical results

275

ρ = 8.9 · 10−6 GPa · s2 /m2 . Voigt’s velocities for this material are: l0 = 5,886.57 m/s, and t0 = 3,158.72 m/s. Experimental data of the distribution of radii of grains in polycrystalline nickel are given in [83]. For calculations, the following approximation to the distribution function f (a) of radii of the grains was used: f (a) =

β α αα−1 e−βα . Γ (α) − Γ (α, βA)

(8.106)

Here A is the maximum value of the grain radii: f (a) = 0, when a > A. Γ (α) and Γ (α, βA) are the Euler gamma function and incomplete gamma function, respectively. For the nickel grains, the parameters in (8.106) are: α = 2.65, β = 4.78 · 106 m−1 , A = 1.86 · 10−4 m and a = 5.54 · 10−5 m. The circles and triangles in Fig. 8.3 are experimental data for the attenuation coeﬃcients of longitudinal () and transverse (∆) waves given in [83]. The values of k a = Re(α∗ a) for longitudinal and k a = Re(β∗ a) for transverse waves lie along the horizontal axes in Fig. 8.3: / 3K∗ + 4µ∗ µ∗ , β∗ = ω . (8.107) α∗ = ω 3ρ ρ The values of γ a = − Im(α∗ ) a for longitudinal and for γ a = −Im(β∗ ) a transverse waves lie as ordinates on the left part of Fig. 8.3. The results of the calculations of the velocities and attenuation coeﬃcients for stainless steel considered in [83] are presented in Fig. 8.4. The elastic moduli and density of this steel are: c1 = 216.7 GPa, c2 = 126.8 GPa, c3 = 110 GPa, ρ = 7.8 · 10−6 GPa · s2 /m2 . Voigt’s velocities for this material are: l0 = 5,869.74 m/s, and t0 = 3,281.26 m/s. 102λ 16

γ(a) λt

1

12

γt

γl

10−1 8

λl

10−2

4 10−3 0 10−4 −4 10−1

1

10

102

k(a)

10−5 −1 10

1

10

102

k(a)

Fig. 8.4. Dependence of normalized variations of phase velocities and attenuation coeﬃcients on dimensionless wave numbers for a stainless steel with a distribution of the sizes of grains. Circles and triangles are experimental points.

276

8. Elastic waves in polycrystals

The parameters of the distribution of grain’s radii in the law (8.106) are: α = 2.33, β = 77, 688 m−1 , A = 1.33 · 10−4 m and a = 3.02 · 10−5 m. The circles and triangles in Fig. 8.4 are the experimental data from [83] for the attenuations coeﬃcients of longitudinal and transverse waves, respectively.

8.7 Conclusions Version IV of the EMM allows us to describe the phenomenon of elastic wave propagation in polycrystals in a wide region of frequencies in the exciting ﬁeld. Possible errors in the algorithm proposed here may arise from two sources. The ﬁrst is the approximate solution of the one-particle problem (Section 8.4), and the second source is the hypotheses of the EMM (Section 8.3). It is diﬃcult to estimate precisely the region of application of all these hypotheses. But the theory gives satisfactory agreement with experimental data in the Rayleigh region of frequencies, and it is physically correct in the stochastic and diﬀusive regions. Further development of this approach can be connected with a more precise solution of the one-particle problem. However, increasing the accuracy of the solution of the one-particle problem may have little inﬂuence on the ﬁnal result. In fact, mean wave ﬁelds in polycrystals are extremely rough statistical characteristic of the exact solution, and a large portion of the information about detailed behavior of these ﬁelds inside grains is lost here. Therefore, a detailed consideration of the state of each grain is not reasonable. Of course, this argument can be conﬁrmed only by comparing the derived results with experiments in a wide region of frequencies. Note that the only information about the geometry of polycrystals that is used in the framework of the EMM is the distribution function of the grains over the radii. If the Born approximation is applied for the solution of the problem, we have to know the pair correlation function of the random elastic moduli of the polycrystal (this function is similar to (8.105)). But the construction of this function requires much more statistical information about the geometry of the polycrystal than the size distribution of the grains. Note that the results of the Born approximation strongly depend on the chosen correlation function (see [90]).

8.8 Notes This chapter is based on the work [42]. The Born approximation was ﬁrst used for the solution of the homogenization problem for polycrystals in [74]. Further developments of this approach may be found in [83,90]. An attempt to consider the higher terms of the perturbation series of the solution of the wave

8.8 Notes

277

problem for polycrystals was made in [94]. Overcoming essential technical diﬃculties, the authors showed that the second term of the perturbation series allows us to describe the transition from the stochastic to the diﬀusive region in polycrystals. The eﬀective medium method was applied to the calculation of static overall elastic moduli of polycrystals in [34, 66]. For the solution of the homogenization problem in the long-wave region, the EMM was used in [87] where the eﬀective velocities and attenuation coeﬃcients of elastic waves in polycrystals with cubic symmetry were calculated. In [10], the EMM was applied to the problem of propagation of electromagnetic waves of any length through polycrystals with small anisotropy in dielectric properties of the grains. It was shown that the method allows us to describe all the main features of the phenomenon of electromagnetic wave propagation through polycrystalline media.

A. Special tensor bases of four-rank tensors

Four-rank tensors of special symmetry are used for the description of elastic properties of solids. For the presentation of these tensors and operations with them, it is convenient to introduce special bases. In this appendix, we present some such bases and give explicit equations for the products and inversions of the tensors that belong to the linear shells of these bases. The basis for the four-rank tensors of orthorhombic symmetry is considered in Chapter 8, Section 8.4.

A.1 E-basis Let us introduce six four-rank tensors constructed from a unit vector n, and the two-rank unit tensor δαβ . 1 2 3 = δα(λ δµ)β , Eαβλµ = δαβ δλµ , Eαβλµ = δαβ nλ nµ , Eαβλµ

(A.1)

4 5 6 = nα nβ δλµ , Eαβλµ = nα) n(µ δλ)(β , Eαβλµ = nα nβ nλ nµ . Eαβλµ

(A.2)

(the brackets mean the symmerization over corresponding indices). These tensors are symmetric with respect to the ﬁrst and second pairs of indices, and form a closed algebra with respect to the product operation deﬁned by the equation (convolution over two indices) i j j i Eρτ (A.3) E E αβλµ = Eαβρτ λµ . The multiplication table of tensors E i has the form

E1 E2 E3 E4 E5 E6

E1 E1 E2 E3 E4 E5 E6

E2 E2 3E 2 E2 3E 4 E4 E4

E3 E3 3E 3 E3 3E 6 E6 E6

E4 E4 E2 E2 E4 E4 E4

E5 E5 E3 E3 E6 5 (E + E 6 )/2 E6

E6 E6 E3 E3 E6 E6 E6

(A.4)

280

A. Special tensor bases of four-rank tensors

Elastic properties of isotropic materials are described by isotropic tensors symmetric with respect to the ﬁrst and second pair of indices, as well as over transposition of a pair of indices. The tensors E 1 and E 2 may be used as the basis of such set of tensors. To construct an inverse tensor with respect to a tensor in this tensor space, it is convenient to introduce a basis that consists of two orthogonal tensors E 2 and E 1 − 13 E 2 : 1 1 E2 E1 − E2 = E1 − E2 E2 = 0 . (A.5) 3 3 1 1 1 E1 − E2 = E1 − E2 . (A.6) E1 − E2 E 2 E 2 = 3E 2 , 3 3 3 For an arbitrary tensor A of this subspace, we have the presentation 1 2 2 1 (A.7) A = a1 E + a2 E − E , 3 where a1 , a2 are scalar coeﬃcients. From (A.5), (A.6) it is easy to obtain the equation for the inverse tensor A−1 (AA−1 = A−1 A = E 1 ) 1 2 1 1 E1 − E2 . (A.8) E + A−1 = 9a1 a2 3

A.2 P -basis This basis is constructed similarly to (A.1) and (A.2), but the role of the tworank unit tensor δαβ is played by the projector θαβ on the plane orthogonal to a unit vector m (|m| = 1) θαβ = δαβ − mα mβ .

(A.9)

The P -basis consists of the following six four-rank tensors: 1 2 = θα(λ θµ)β , Pαβλµ = θαβ θλµ , Pαβλµ 3 4 = θαβ mλ mµ , Pαβλµ = mα mβ θλµ , Pαβλµ 5 6 = mα) m(λ θµ)(β , Pαβλµ = m α mβ mλ mµ . Pαβλµ

(A.10) (A.11) (A.12)

The elements of the P - and E-bases (A.1) and (A.2) are connected by the equations P j = αji E i ,

ji E j = α−1 P i ,

where the matrices α and α−1 have the following forms

(A.13)

⎡

10 ⎢0 1 ⎢ ⎢0 0 αij = ⎢ ⎢0 0 ⎢ ⎣0 0 00

0 0 −2 −1 −1 0 1 0 0 0 1 0 0 0 1 0 0 0

⎡

⎤

1 1 ⎥ ⎥ −1 ⎥ ⎥, −1 ⎥ ⎥ −1 ⎦ 1

1 ⎢0 ⎢ −1 ij ⎢ 0 =⎢ α ⎢0 ⎢ ⎣0 0

A.2 P -basis

0 1 0 0 0 0

0 1 1 0 0 0

0 1 0 1 0 0

2 0 0 0 1 0

281

⎤

1 1⎥ ⎥ 1⎥ ⎥. 1⎥ ⎥ 1⎦ 1

(A.14)

The six tensors P i also form a closed algebra with respect to product operation (A.3). The multiplication table of these tensors has the form 1

P P2 P3 P4 P5 P6

P1 P1 P2 0 P4 0 0

P2 P2 2P 2 0 2P 4 0 0

P3 P3 2P 3 0 2P 6 0 0

P4 0 0 P2 0 0 P4

P5 0 0 0 0 5 P /2 0

P6 0 0 P3 0 0 P6

.

(A.15)

Note that the tensor P 1 − 12 P 2 is orthogonal to all the tensors of the P -basis except P 1 , i.e., 1 2 1 2 1 2 1 1 1 1 = P − P P = P − P , P P − P 2 2 2 1 1 P i P 1 − P 2 = P 1 − P 2 P i = 0 , i = 1 , 2 2 1

1 P1 − P2 2

1 P1 − P2 2

=

1 P1 − P2 . 2

(A.16)

(A.17)

(A.18)

The linear shells of the E- and P -bases coincide. Let the tensor A in the P -basis be presented in the form 1 A = a1 P 2 + a2 P 1 − P 2 + a3 P 3 + a4 P 4 + a5 P 5 + a6 P 6 , 2

(A.19)

where ai (i = 1, 2, ..., 6) are scalar coeﬃcients. Then, the inverse tensor A−1 (AA−1 = A−1 A = E 1 ) takes the following form a3 a6 2 1 1 2 a4 4 2a1 6 −1 1 P + P , P − P − P3 − P4 + P5 + A = 2∆ a2 2 ∆ ∆ a5 ∆ (A.20) ∆ = 2 (a1 a6 − a3 a4 ) .

(A.21)

282

A. Special tensor bases of four-rank tensors

If tensors A and B are presented in the P -basis in a form similar to (A.19) with the coeﬃcients ai and bi , the product AB of these tensors is deﬁned by an equation that follows from (A.15) and (A.18): 1 AB = (2a1 b1 + a3 b4 ) P 2 + a2 b2 P 1 − P 2 + (2a1 b3 + a3 b6 ) P 3 2 1 + (2a4 b1 + a6 b4 ) P 4 + a5 b5 P 5 + (a6 b6 + 2a4 b3 ) P 6 . (A.22) 2

A.3 Averaging the elements of the E- and P -bases For the solution of the homogenization problem, one has to average the elements of the bases over a unit sphere in 3D-space, or over a circle in 2D-space. We present the values of the corresponding means. E-basis.

1 E (n) = 4π i

E i (n) dΩn , i = 1, 2, ..., 6 ,

(A.23)

Ω1

Here Ω1 is the surface of the unit sphere in 3D-space; integration is performed over the vector n on this sphere. 1 E 1 = E 1 , E 2 = E 2 , E 3 (n) = E 4 (n) = E 2 , 3

1 1 1 E 5 (n) = E 1 , E 6 (n) = 2E + E 2 , 3 15 P-basis. i 1 P (m) = P i (m) dΩm , i = 1, 2, ..., 6 , 4π

(A.24) (A.25)

(A.26)

Ω1

Here m is the vector on the unit sphere Ω1 .

1 2 2 1 E + 7E 1 , P 2 (m) = E + 3E 2 , P 1 (m) = 15 15

2 2 P 3 (m) = P 4 (m) = 2E − E 1 , 15 5 1 1 1 1 1 2 P (m) = E − E 2E + E 2 . , P 6 (m) = 5 3 15

(A.27) (A.28) (A.29)

A.4 Tensor bases of four-rank tensors in 2D-space

283

A.4 Tensor bases of four-rank tensors in 2D-space Let us consider the E- and P -bases in 2D-space. The E-basis has form (A.1) and (A.2), but the Greek indices take the values 1, 2. Note that for 2D, not all the tensors E i (n) are linear independent, because the following equation holds E 1 − E 2 + E 3 + E 4 − 2E 5 = 0.

(A.30)

Therefore, only ﬁve of the six tensors E i (n) (A.1) and (A.2) are linearly independent (any four tensors from the ﬁrst ﬁve, together with E 6 (n)). For the P -basis, linear dependence of the tensors (A.10) and (A.12) is obvious because the tensors P 1 and P 2 coincide for 2D. The multiplication table for the ﬁve linearly independent elements of the P -basis has the form 1

P P3 P4 P5 P6

P1 P1 0 P4 0 0

P3 P3 0 P6 0 0

P4 0 P1 0 0 P4

P5 0 0 0 P 5 /2 0

P6 0 P3 0 0 P6

(A.31)

Product of two elements (A and B) of the linear shell of the P -basis A = a1 P 1 + a3 P 3 + a4 P 4 + a5 P 5 + a6 P 6 ,

(A.32)

B = b1 P 1 + b3 P 3 + b4 P 4 + b5 P 5 + b6 P 6 ,

(A.33)

where ai , bi are scalar coeﬃcients, takes the following form AB = (a1 b1 + a3 b4 ) P 1 + (a1 b3 + a3 b6 ) P 3 1 + (a4 b1 + a6 b4 ) P 4 + a5 b5 P 5 + (a4 b3 + a6 b6 ) P 6 . 2 −1 The inverse tensor A is deﬁned by the equation a6 1 a3 3 a4 4 4 a1 P − P − P + P 5 + P 6, ∆ ∆ ∆ a5 ∆ ∆ = a1 a6 − a3 a4 .

A−1 =

(A.34)

(A.35) (A.36)

The connection between the elements of the E- and P -basis in 2D is determined by the relations: ⎧ 1 ⎨ E = P 1 + 2P 5 + P 6 , E 2 = P 1 + P 3 + P 4 + P 6 , E3 = P 3 + P 6, E4 = P 4 + P 6, (A.37) ⎩ 5 E = P 5 + P 6, E6 = P 6,

284

A. Special tensor bases of four-rank tensors

P 1 = E 1 − 2E 5 + E 6 , P 3 = E 3 − E 6 , P 4 = E4 − E6, P 5 = E5 − E6, P 6 = E6.

(A.38)

The averaging formulas for the E- and P -basis in 2D-case have the forms i 1 E (n) = E i (n) dln , i = 1, 2, ..., 6 , (A.39) 2π l1

E1 = E1,

1 E 2 = E 2 , E 3 (n) = E 4 (n) = E 2 , 2 5 1 1 6 1 1 E (n) = E , E (n) = 2E + E 2 , 2 8 i 1 P (n) = P i (n) dln , i = 1, 3, 4, 5, 6 , 2π

l1

(A.40) (A.41) (A.42)

1 2 1 2 E + 2E 1 , P 3 (n) = P 4 (n) = 3E − E 1 , P (n) = 8 8 (A.43) 5 1 1 1 2 P (n) = 2E − E 2 , P 6 (n) = E + 2E 1 . (A.44) 8 8

1

B. The Percus-Yevick correlation function

The 3D-case. The Percus-Yevick function ψ(x) is the normalized probability of distribution of the centers of non-overlapping spheres of radii a (under the condition that the origin (x = 0) is occupied by the center of another sphere. This function is deﬁned by the equations [110] 1 ψ(ζ) = 1 + 12πpζ ψ(ζ) = 0, if

∞ 0

w2 (s) sin 1 − w(s)

ζ < 2,

ζ=

sζ 2

sds

if

ζ ≥ 2,

|x| . a

(B.1)

The function w(s) in this equation is the following integral 24p w(s) = s a1 =

1 q(t) sin(ts)tdt,

q(t) = −a1 (1 + p

0

(1 + 2p)2 4

(1 − p)

, a2 =

−3p(2 + p)2 , 2(1 − p)4

t3 ) − a2 t, 2 (B.2)

The correlation function Φ(|x|) in (3.133), (3.135) is connected to the function ψ(x) by the equation 2 3 v0 (x )dx ψ(|x − x |)v0 (x − x)dx . (B.3) Φ(x) = 1 − 4π The double integral over 3D-space is reduced to a one-dimensional integral (Section 3.9). The 2D-case. For 2D, the Percus-Yevick correlation function is presented in the form 1 ψ(ζ) = 1 + 8p ψ(ζ) = 0,

∞ ζx xdx, |ζ| ≥ 2; h(x)J0 2 0

|ζ| < 2.

(B.4)

286

B. The Percus-Yevick correlation function

Here J0 (z) is the Bessel function of order 0, and the function h(x) is h(x) =

s(x)2 . 1 − s(x)

(B.5)

The function s(x) is the solution of the following non-linear integral equation s(x) = −8p

J1 (x) 1 − x 2

∞ 0

s(t)2 2 J0 (t) + J12 (t) dt. 1 − s(t)

(B.6)

This equation may be solved by iteration, and its numerical solution is used for the construction of the function ψ(ζ) in Chapter 4.

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Index

Acoustical branch – of elastic wave propagation – – in ﬁber composites, 107, 112 – of electromagnetic wave propagation, 34, 54, 56, 57, 68–70 – of longitudinal elastic wave propagation – – in composites with spherical inclusions, 251 – of shear elastic wave propagation – – in composites with spherical inclusions, 245 Approximate solution – Born, 257, 270, 271 – Galerkin, 61, 95, 199 – long-wave, 117, 123, 124, 126, 130, 136, 143, 153, 274 – quasi-static, 144, 146 – Ritz, 61 – Voigt, 274 Approximation – plane wave, 60, 61, 68, 76 – quasicrystalline, 11, 42, 158 Attenuation coeﬃcient – elastic waves, 83, 100, 102, 103, 105, 114, 139, 166, 167, 170, 174–181, 184, 186, 187, 194, 203, 207, 214, 236, 237, 239, 240, 257, 260, 272, 273 – electromagnetic waves, 13, 32–34, 42, 49, 51, 53, 66, 74 Basic tensors, 279 – E-basis, 279 – P-basis, 280 – T-basis, 265 Bragg frequency, 69 Brillouin zone, 68 Characteristic function, xiii Correlation function – Percus-Yevick, 51

Decomposition of elastic ﬁeld, 162 Dispersion equation, 3, 4 – elastic longitudinal waves, 202, 207, 213, 228, 239, 240, 247 – elastic transverse waves, 83, 87, 89, 97–99, 103, 105, 204, 235, 238, 244 – – periodic structure, 109, 110, 114 – elastic waves, 194, 200, 260 – electromagnetic waves, 26, 30, 33, 46, 48, 51 – – periodic structure, 67 – long elastic waves, 170, 185 – scalar waves, 13, 16, 17 Eﬀective wave operator, 155 Ergodic property, 7 Fourier transform – dynamic Green function of elasticity, 79, 118 – electrodynamic Green function, 23 Green function of – dynamic elasticity, 79, 121, 139 – eﬀective elastic wave operator, 163 – electromagnetic wave equation, 22 – scalar wave equation, 6 Integral equations for – wave ﬁelds – – elastic, 79, 123, 132, 134, 189, 258 – – electromagnetic, 22 – – scalar, 6 One-particle elastic problem – diﬀraction of long waves – – on ellipsoidal inclusion, 123 – – on short hard ﬁber, 133 – – on thin hard inclusion, 131 – – on thin soft inclusion, 129 – diﬀraction of longitudinal wave – – on spherical inclusion, 194, 232

294

Index

– diﬀraction of radial waves – – on spherical inclusion, 234 – diﬀraction of transversal wave – – on continuous ﬁber, 89 – – on spherical inclusion, 197, 229 One-particle electromagnetic problem, 26, 36, 47, 61 Optical branch – of elastic wave propagation – – in ﬁber composites, 112 – of electromagnetic wave propagation, 35, 55–57, 68, 69 – of longitudinal elastic wave propagation – – in composites with spherical inclusions, 251–254

Scattering cross-section – elastic waves, 143 – – spherical inclusion, 205, 242 – electromagnetic waves, 23, 29 – long elastic waves – – short axisymmetric ﬁber, 153 – – spherical inclusion, 144, 146 – – spheroidal inclusion, 148 – – thin hard inclusion, 152 – – thin soft inclusion, 152 – transverse elastic waves – – cylindrical inclusion, 93 Statistical moment, 63

Periodic structure – cubic, 68 – square, 110 Poisson – set of cracks, 176 – set of rigid disks, 180 Polycrystals, 71, 258

Velocity of waves – elastic, 194 – elastic transverse, 83, 103 – electromagnetic, 30, 46, 58, 75 – long elastic longitudinal, 175–177, 186 – long elastic transverse, 174, 180, 181, 185, 186 – scalar, 5

Regular lattice of inclusions – spherical, 62 – unidirected ﬁbers, 108 Representative volume element, 7

Wave equation – elastic, 78, 90, 117, 195, 204, 229, 258 – electromagnetic, 22 – scalar, 5

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

R.T. Haftka, Z. G¨urdal and M.P. Kamat: Elements of Structural Optimization. 2nd rev.ed., 1990 ISBN 0-7923-0608-2 J.J. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. 1990 ISBN 0-7923-0712-7 P. Karasudhi: Foundations of Solid Mechanics. 1991 ISBN 0-7923-0772-0 Not published Not published. J.F. Doyle: Static and Dynamic Analysis of Structures. With an Emphasis on Mechanics and Computer Matrix Methods. 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-2 O.O. Ochoa and J.N. Reddy: Finite Element Analysis of Composite Laminates. ISBN 0-7923-1125-6 M.H. Aliabadi and D.P. Rooke: Numerical Fracture Mechanics. ISBN 0-7923-1175-2 J. Angeles and C.S. L´opez-Caj´un: Optimization of Cam Mechanisms. 1991 ISBN 0-7923-1355-0 D.E. Grierson, A. Franchi and P. Riva (eds.): Progress in Structural Engineering. 1991 ISBN 0-7923-1396-8 R.T. Haftka and Z. G¨urdal: Elements of Structural Optimization. 3rd rev. and exp. ed. 1992 ISBN 0-7923-1504-9; Pb 0-7923-1505-7 J.R. Barber: Elasticity. 1992 ISBN 0-7923-1609-6; Pb 0-7923-1610-X H.S. Tzou and G.L. Anderson (eds.): Intelligent Structural Systems. 1992 ISBN 0-7923-1920-6 E.E. Gdoutos: Fracture Mechanics. An Introduction. 1993 ISBN 0-7923-1932-X J.P. Ward: Solid Mechanics. An Introduction. 1992 ISBN 0-7923-1949-4 M. Farshad: Design and Analysis of Shell Structures. 1992 ISBN 0-7923-1950-8 H.S. Tzou and T. Fukuda (eds.): Precision Sensors, Actuators and Systems. 1992 ISBN 0-7923-2015-8 J.R. Vinson: The Behavior of Shells Composed of Isotropic and Composite Materials. 1993 ISBN 0-7923-2113-8 H.S. Tzou: Piezoelectric Shells. Distributed Sensing and Control of Continua. 1993 ISBN 0-7923-2186-3 W. Schiehlen (ed.): Advanced Multibody System Dynamics. Simulation and Software Tools. 1993 ISBN 0-7923-2192-8 C.-W. Lee: Vibration Analysis of Rotors. 1993 ISBN 0-7923-2300-9 D.R. Smith: An Introduction to Continuum Mechanics. 1993 ISBN 0-7923-2454-4 G.M.L. Gladwell: Inverse Problems in Scattering. An Introduction. 1993 ISBN 0-7923-2478-1

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G. Prathap: The Finite Element Method in Structural Mechanics. 1993 ISBN 0-7923-2492-7 J. Herskovits (ed.): Advances in Structural Optimization. 1995 ISBN 0-7923-2510-9 M.A. Gonz´alez-Palacios and J. Angeles: Cam Synthesis. 1993 ISBN 0-7923-2536-2 W.S. Hall: The Boundary Element Method. 1993 ISBN 0-7923-2580-X J. Angeles, G. Hommel and P. Kov´acs (eds.): Computational Kinematics. 1993 ISBN 0-7923-2585-0 A. Curnier: Computational Methods in Solid Mechanics. 1994 ISBN 0-7923-2761-6 D.A. Hills and D. Nowell: Mechanics of Fretting Fatigue. 1994 ISBN 0-7923-2866-3 B. Tabarrok and F.P.J. Rimrott: Variational Methods and Complementary Formulations in Dynamics. 1994 ISBN 0-7923-2923-6 E.H. Dowell (ed.), E.F. Crawley, H.C. Curtiss Jr., D.A. Peters, R. H. Scanlan and F. Sisto: A Modern Course in Aeroelasticity. Third Revised and Enlarged Edition. 1995 ISBN 0-7923-2788-8; Pb: 0-7923-2789-6 A. Preumont: Random Vibration and Spectral Analysis. 1994 ISBN 0-7923-3036-6 J.N. Reddy (ed.): Mechanics of Composite Materials. Selected works of Nicholas J. Pagano. 1994 ISBN 0-7923-3041-2 A.P.S. Selvadurai (ed.): Mechanics of Poroelastic Media. 1996 ISBN 0-7923-3329-2 Z. Mr´oz, D. Weichert, S. Dorosz (eds.): Inelastic Behaviour of Structures under Variable Loads. 1995 ISBN 0-7923-3397-7 R. Pyrz (ed.): IUTAM Symposium on Microstructure-Property Interactions in Composite Materials. Proceedings of the IUTAM Symposium held in Aalborg, Denmark. 1995 ISBN 0-7923-3427-2 M.I. Friswell and J.E. Mottershead: Finite Element Model Updating in Structural Dynamics. 1995 ISBN 0-7923-3431-0 D.F. Parker and A.H. England (eds.): IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics. Proceedings of the IUTAM Symposium held in Nottingham, U.K. 1995 ISBN 0-7923-3594-5 J.-P. Merlet and B. Ravani (eds.): Computational Kinematics ’95. 1995 ISBN 0-7923-3673-9 L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Mechanics and Inverse Problems. 1996 ISBN 0-7923-3849-9 J. Menˇcik: Mechanics of Components with Treated or Coated Surfaces. 1996 ISBN 0-7923-3700-X D. Bestle and W. Schiehlen (eds.): IUTAM Symposium on Optimization of Mechanical Systems. Proceedings of the IUTAM Symposium held in Stuttgart, Germany. 1996 ISBN 0-7923-3830-8 D.A. Hills, P.A. Kelly, D.N. Dai and A.M. Korsunsky: Solution of Crack Problems. The Distributed Dislocation Technique. 1996 ISBN 0-7923-3848-0 V.A. Squire, R.J. Hosking, A.D. Kerr and P.J. Langhorne: Moving Loads on Ice Plates. 1996 ISBN 0-7923-3953-3 A. Pineau and A. Zaoui (eds.): IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials. Proceedings of the IUTAM Symposium held in S`evres, Paris, France. 1996 ISBN 0-7923-4188-0 A. Naess and S. Krenk (eds.): IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics. Proceedings of the IUTAM Symposium held in Trondheim, Norway. 1996 ISBN 0-7923-4193-7 D. Ie¸san and A. Scalia: Thermoelastic Deformations. 1996 ISBN 0-7923-4230-5

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J.R. Willis (ed.): IUTAM Symposium on Nonlinear Analysis of Fracture. Proceedings of the IUTAM Symposium held in Cambridge, U.K. 1997 ISBN 0-7923-4378-6 A. Preumont: Vibration Control of Active Structures. An Introduction. 1997 ISBN 0-7923-4392-1 G.P. Cherepanov: Methods of Fracture Mechanics: Solid Matter Physics. 1997 ISBN 0-7923-4408-1 D.H. van Campen (ed.): IUTAM Symposium on Interaction between Dynamics and Control in Advanced Mechanical Systems. Proceedings of the IUTAM Symposium held in Eindhoven, The Netherlands. 1997 ISBN 0-7923-4429-4 N.A. Fleck and A.C.F. Cocks (eds.): IUTAM Symposium on Mechanics of Granular and Porous Materials. Proceedings of the IUTAM Symposium held in Cambridge, U.K. 1997 ISBN 0-7923-4553-3 J. Roorda and N.K. Srivastava (eds.): Trends in Structural Mechanics. Theory, Practice, Education. 1997 ISBN 0-7923-4603-3 Yu.A. Mitropolskii and N. Van Dao: Applied Asymptotic Methods in Nonlinear Oscillations. 1997 ISBN 0-7923-4605-X C. Guedes Soares (ed.): Probabilistic Methods for Structural Design. 1997 ISBN 0-7923-4670-X D. Fran¸cois, A. Pineau and A. Zaoui: Mechanical Behaviour of Materials. Volume I: Elasticity and Plasticity. 1998 ISBN 0-7923-4894-X D. Fran¸cois, A. Pineau and A. Zaoui: Mechanical Behaviour of Materials. Volume II: Viscoplasticity, Damage, Fracture and Contact Mechanics. 1998 ISBN 0-7923-4895-8 L.T. Tenek and J. Argyris: Finite Element Analysis for Composite Structures. 1998 ISBN 0-7923-4899-0 Y.A. Bahei-El-Din and G.J. Dvorak (eds.): IUTAM Symposium on Transformation Problems in Composite and Active Materials. Proceedings of the IUTAM Symposium held in Cairo, Egypt. 1998 ISBN 0-7923-5122-3 I.G. Goryacheva: Contact Mechanics in Tribology. 1998 ISBN 0-7923-5257-2 O.T. Bruhns and E. Stein (eds.): IUTAM Symposium on Micro- and Macrostructural Aspects of Thermoplasticity. Proceedings of the IUTAM Symposium held in Bochum, Germany. 1999 ISBN 0-7923-5265-3 F.C. Moon: IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics. Proceedings of the IUTAM Symposium held in Ithaca, NY, USA. 1998 ISBN 0-7923-5276-9 R. Wang: IUTAM Symposium on Rheology of Bodies with Defects. Proceedings of the IUTAM Symposium held in Beijing, China. 1999 ISBN 0-7923-5297-1 Yu.I. Dimitrienko: Thermomechanics of Composites under High Temperatures. 1999 ISBN 0-7923-4899-0 P. Argoul, M. Fr´emond and Q.S. Nguyen (eds.): IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics. Proceedings of the IUTAM Symposium held in Paris, France. 1999 ISBN 0-7923-5450-8 F.J. Fahy and W.G. Price (eds.): IUTAM Symposium on Statistical Energy Analysis. Proceedings of the IUTAM Symposium held in Southampton, U.K. 1999 ISBN 0-7923-5457-5 H.A. Mang and F.G. Rammerstorfer (eds.): IUTAM Symposium on Discretization Methods in Structural Mechanics. Proceedings of the IUTAM Symposium held in Vienna, Austria. 1999 ISBN 0-7923-5591-1

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P. Pedersen and M.P. Bendsøe (eds.): IUTAM Symposium on Synthesis in Bio Solid Mechanics. Proceedings of the IUTAM Symposium held in Copenhagen, Denmark. 1999 ISBN 0-7923-5615-2 S.K. Agrawal and B.C. Fabien: Optimization of Dynamic Systems. 1999 ISBN 0-7923-5681-0 A. Carpinteri: Nonlinear Crack Models for Nonmetallic Materials. 1999 ISBN 0-7923-5750-7 F. Pfeifer (ed.): IUTAM Symposium on Unilateral Multibody Contacts. Proceedings of the IUTAM Symposium held in Munich, Germany. 1999 ISBN 0-7923-6030-3 E. Lavendelis and M. Zakrzhevsky (eds.): IUTAM/IFToMM Symposium on Synthesis of Nonlinear Dynamical Systems. Proceedings of the IUTAM/IFToMM Symposium held in Riga, Latvia. 2000 ISBN 0-7923-6106-7 J.-P. Merlet: Parallel Robots. 2000 ISBN 0-7923-6308-6 J.T. Pindera: Techniques of Tomographic Isodyne Stress Analysis. 2000 ISBN 0-7923-6388-4 G.A. Maugin, R. Drouot and F. Sidoroff (eds.): Continuum Thermomechanics. The Art and Science of Modelling Material Behaviour. 2000 ISBN 0-7923-6407-4 N. Van Dao and E.J. Kreuzer (eds.): IUTAM Symposium on Recent Developments in Non-linear Oscillations of Mechanical Systems. 2000 ISBN 0-7923-6470-8 S.D. Akbarov and A.N. Guz: Mechanics of Curved Composites. 2000 ISBN 0-7923-6477-5 M.B. Rubin: Cosserat Theories: Shells, Rods and Points. 2000 ISBN 0-7923-6489-9 S. Pellegrino and S.D. Guest (eds.): IUTAM-IASS Symposium on Deployable Structures: Theory and Applications. Proceedings of the IUTAM-IASS Symposium held in Cambridge, U.K., 6–9 September 1998. 2000 ISBN 0-7923-6516-X A.D. Rosato and D.L. Blackmore (eds.): IUTAM Symposium on Segregation in Granular Flows. Proceedings of the IUTAM Symposium held in Cape May, NJ, U.S.A., June 5–10, 1999. 2000 ISBN 0-7923-6547-X A. Lagarde (ed.): IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics. Proceedings of the IUTAM Symposium held in Futuroscope, Poitiers, France, August 31–September 4, 1998. 2000 ISBN 0-7923-6604-2 D. Weichert and G. Maier (eds.): Inelastic Analysis of Structures under Variable Loads. Theory and Engineering Applications. 2000 ISBN 0-7923-6645-X T.-J. Chuang and J.W. Rudnicki (eds.): Multiscale Deformation and Fracture in Materials and Structures. The James R. Rice 60th Anniversary Volume. 2001 ISBN 0-7923-6718-9 S. Narayanan and R.N. Iyengar (eds.): IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Proceedings of the IUTAM Symposium held in Madras, Chennai, India, 4–8 January 1999 ISBN 0-7923-6733-2 S. Murakami and N. Ohno (eds.): IUTAM Symposium on Creep in Structures. Proceedings of the IUTAM Symposium held in Nagoya, Japan, 3-7 April 2000. 2001 ISBN 0-7923-6737-5 W. Ehlers (ed.): IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials. Proceedings of the IUTAM Symposium held at the University of Stuttgart, Germany, September 5-10, 1999. 2001 ISBN 0-7923-6766-9 D. Durban, D. Givoli and J.G. Simmonds (eds.): Advances in the Mechanis of Plates and Shells The Avinoam Libai Anniversary Volume. 2001 ISBN 0-7923-6785-5 U. Gabbert and H.-S. Tzou (eds.): IUTAM Symposium on Smart Structures and Structonic Systems. Proceedings of the IUTAM Symposium held in Magdeburg, Germany, 26–29 September 2000. 2001 ISBN 0-7923-6968-8

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Y. Ivanov, V. Cheshkov and M. Natova: Polymer Composite Materials – Interface Phenomena & Processes. 2001 ISBN 0-7923-7008-2 R.C. McPhedran, L.C. Botten and N.A. Nicorovici (eds.): IUTAM Symposium on Mechanical and Electromagnetic Waves in Structured Media. Proceedings of the IUTAM Symposium held in Sydney, NSW, Australia, 18-22 Januari 1999. 2001 ISBN 0-7923-7038-4 D.A. Sotiropoulos (ed.): IUTAM Symposium on Mechanical Waves for Composite Structures Characterization. Proceedings of the IUTAM Symposium held in Chania, Crete, Greece, June 14-17, 2000. 2001 ISBN 0-7923-7164-X V.M. Alexandrov and D.A. Pozharskii: Three-Dimensional Contact Problems. 2001 ISBN 0-7923-7165-8 J.P. Dempsey and H.H. Shen (eds.): IUTAM Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics. Proceedings of the IUTAM Symposium held in Fairbanks, Alaska, U.S.A., 13-16 June 2000. 2001 ISBN 1-4020-0171-1 U. Kirsch: Design-Oriented Analysis of Structures. A Uniﬁed Approach. 2002 ISBN 1-4020-0443-5 A. Preumont: Vibration Control of Active Structures. An Introduction (2nd Edition). 2002 ISBN 1-4020-0496-6 B.L. Karihaloo (ed.): IUTAM Symposium on Analytical and Computational Fracture Mechanics of Non-Homogeneous Materials. Proceedings of the IUTAM Symposium held in Cardiff, U.K., 18-22 June 2001. 2002 ISBN 1-4020-0510-5 S.M. Han and H. Benaroya: Nonlinear and Stochastic Dynamics of Compliant Offshore Structures. 2002 ISBN 1-4020-0573-3 A.M. Linkov: Boundary Integral Equations in Elasticity Theory. 2002 ISBN 1-4020-0574-1 L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Mechanics and Inverse Problems (2nd Edition). 2002 ISBN 1-4020-0667-5; Pb: 1-4020-0756-6 Q.P. Sun (ed.): IUTAM Symposium on Mechanics of Martensitic Phase Transformation in Solids. Proceedings of the IUTAM Symposium held in Hong Kong, China, 11-15 June 2001. 2002 ISBN 1-4020-0741-8 M.L. Munjal (ed.): IUTAM Symposium on Designing for Quietness. Proceedings of the IUTAM Symposium held in Bangkok, India, 12-14 December 2000. 2002 ISBN 1-4020-0765-5 J.A.C. Martins and M.D.P. Monteiro Marques (eds.): Contact Mechanics. Proceedings of the 3rd Contact Mechanics International Symposium, Praia da Consola¸ca˜ o, Peniche, Portugal, 17-21 June 2001. 2002 ISBN 1-4020-0811-2 H.R. Drew and S. Pellegrino (eds.): New Approaches to Structural Mechanics, Shells and Biological Structures. 2002 ISBN 1-4020-0862-7 J.R. Vinson and R.L. Sierakowski: The Behavior of Structures Composed of Composite Materials. Second Edition. 2002 ISBN 1-4020-0904-6 Not yet published. J.R. Barber: Elasticity. Second Edition. 2002 ISBN Hb 1-4020-0964-X; Pb 1-4020-0966-6 C. Miehe (ed.): IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Proceedings of the IUTAM Symposium held in Stuttgart, Germany, 20-24 August 2001. 2003 ISBN 1-4020-1170-9

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 109. P. St˚ahle and K.G. Sundin (eds.): IUTAM Symposium on Field Analyses for Determination of Material Parameters – Experimental and Numerical Aspects. Proceedings of the IUTAM Symposium held in Abisko National Park, Kiruna, Sweden, July 31 – August 4, 2000. 2003 ISBN 1-4020-1283-7 110. N. Sri Namachchivaya and Y.K. Lin (eds.): IUTAM Symposium on Nonlinear Stochastic Dynamics. Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26 – 30 August, 2000. 2003 ISBN 1-4020-1471-6 111. H. Sobieckzky (ed.): IUTAM Symposium Transsonicum IV. Proceedings of the IUTAM Symposium held in G¨ottingen, Germany, 2–6 September 2002, 2003 ISBN 1-4020-1608-5 112. J.-C. Samin and P. Fisette: Symbolic Modeling of Multibody Systems. 2003 ISBN 1-4020-1629-8 113. A.B. Movchan (ed.): IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics. Proceedings of the IUTAM Symposium held in Liverpool, United Kingdom, 8-11 July 2002. 2003 ISBN 1-4020-1780-4 114. S. Ahzi, M. Cherkaoui, M.A. Khaleel, H.M. Zbib, M.A. Zikry and B. LaMatina (eds.): IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials. Proceedings of the IUTAM Symposium held in Marrakech, Morocco, 20-25 October 2002. 2004 ISBN 1-4020-1861-4 115. H. Kitagawa and Y. Shibutani (eds.): IUTAM Symposium on Mesoscopic Dynamics of Fracture Process and Materials Strength. Proceedings of the IUTAM Symposium held in Osaka, Japan, 6-11 July 2003. Volume in celebration of Professor Kitagawa’s retirement. 2004 ISBN 1-4020-2037-6 116. E.H. Dowell, R.L. Clark, D. Cox, H.C. Curtiss, Jr., K.C. Hall, D.A. Peters, R.H. Scanlan, E. Simiu, F. Sisto and D. Tang: A Modern Course in Aeroelasticity. 4th Edition, 2004 ISBN 1-4020-2039-2 117. T. Burczy´nski and A. Osyczka (eds.): IUTAM Symposium on Evolutionary Methods in Mechanics. Proceedings of the IUTAM Symposium held in Cracow, Poland, 24-27 September 2002. 2004 ISBN 1-4020-2266-2 118. D. Ie¸san: Thermoelastic Models of Continua. 2004 ISBN 1-4020-2309-X 119. G.M.L. Gladwell: Inverse Problems in Vibration. Second Edition. 2004 ISBN 1-4020-2670-6 120. J.R. Vinson: Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials, Including Sandwich Construction. 2005 ISBN 1-4020-3110-6 121. Forthcoming 122. G. Rega and F. Vestroni (eds.): IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics. Proceedings of the IUTAM Symposium held in Rome, Italy, 8–13 June 2003. 2005 ISBN 1-4020-3267-6 123. E.E. Gdoutos: Fracture Mechanics. An Introduction. 2nd edition. 2005 ISBN 1-4020-3267-6 124. M.D. Gilchrist (ed.): IUTAM Symposium on Impact Biomechanics from Fundamental Insights to Applications. 2005 ISBN 1-4020-3795-3 125. J.M. Huyghe, P.A.C. Raats and S. C. Cowin (eds.): IUTAM Symposium on Physicochemical and Electromechanical Interactions in Porous Media. 2005 ISBN 1-4020-3864-X 126. H. Ding, W. Chen and L. Zhang: Elasticity of Transversely Isotropic Materials. 2005 ISBN 1-4020-4033-4 127. W. Yang (ed): IUTAM Symposium on Mechanics and Reliability of Actuating Materials. Proceedings of the IUTAM Symposium held in Beijing, China, 1–3 September 2004. 2005 ISBN 1-4020-4131-6

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 128. J.-P. Merlet: Parallel Robots. 2006 ISBN 1-4020-4132-2 129. G.E.A. Meier and K.R. Sreenivasan (eds.): IUTAM Symposium on One Hundred Years of Boundary Layer Research. Proceedings of the IUTAM Symposium held at DLR-G¨ottingen, Germany, August 12–14, 2004. 2006 ISBN 1-4020-4149-7 130. H. Ulbrich and W. G¨unthner (eds.): IUTAM Symposium on Vibration Control of Nonlinear Mechanisms and Structures. 2006 ISBN 1-4020-4160-8 131. L. Librescu and O. Song: Thin-Walled Composite Beams. Theory and Application. 2006 ISBN 1-4020-3457-1 132. G. Ben-Dor, A. Dubinsky and T. Elperin: Applied High-Speed Plate Penetration Dynamics. 2006 ISBN 1-4020-3452-0 133. X. Markenscoff and A. Gupta (eds.): Collected Works of J. D. Eshelby. Mechanics of Defects and Inhomogeneities. 2006 ISBN 1-4020-4416-X 134. R.W. Snidle and H.P. Evans (eds.): IUTAM Symposium on Elastohydrodynamics and Microelastohydrodynamics. Proceedings of the IUTAM Symposium held in Cardiff, UK, 1–3 September, 2004. 2006 ISBN 1-4020-4532-8 135. T. Sadowski (ed.): IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials. Proceedings of the IUTAM Symposium held in Kazimierz Dolny, Poland, 23–27 May 2005. 2006 ISBN 1-4020-4565-4 136. A. Preumont: Mechatronics. Dynamics of Electromechanical and Piezoelectric Systems. 2006 ISBN 1-4020-4695-2 137. M.P. Bendsøe, N. Olhoff and O. Sigmund (eds.): IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Status and Perspectives. 2006 ISBN 1-4020-4729-0 138. A. Klarbring: Models of Mechanics. 2006 ISBN 1-4020-4834-3 139. H.D. Bui: Fracture Mechanics. Inverse Problems and Solutions. 2006 ISBN 1-4020-4836-X 140. M. Pandey, W.-C. Xie and L. Xu (eds.): Advances in Engineering Structures, Mechanics and Construction. Proceedings of an International Conference on Advances in Engineering Structures, Mechanics & Construction, held in Waterloo, Ontario, Canada, May 14–17, 2006. 2006 ISBN 1-4020-4890-4 141. G.Q. Zhang, W.D. van Driel and X. J. Fan: Mechanics of Microelectronics. 2006 ISBN 1-4020-4934-X 142. Q.P. Sun and P. Tong (eds.): IUTAM Symposium on Size Effects on Material and Structural Behavior at Micron- and Nano-Scales. Proceedings of the IUTAM Symposium held in Hong Kong, China, 31 May–4 June, 2004. 2006 ISBN 1-4020-4945-5 143. A.P. Mouritz and A.G. Gibson: Fire Properties of Polymer Composite Materials. 2006 ISBN 1-4020-5355-X 144. Y.L. Bai, Q.S. Zheng and Y.G. Wei (eds.): IUTAM Symposium on Mechanical Behavior and Micro-Mechanics of Nanostructured Materials. Proceedings of the IUTAM Symposium held in Beijing, China, 27–30 June 2005. 2007 ISBN 1-4020-5623-0 145. L.P. Pook: Metal Fatigue. What It Is, Why It Matters. 2007 ISBN 1-4020-5596-6 146. H.I. Ling, L. Callisto, D. Leshchinsky and J. Koseki (eds.): Soil Stress-Strain Behavior: Measurement, Modeling and Analysis. A Collection of Papers of the Geotechnical Symposium in Rome, March 16–17, 2006. 2007 ISBN 978-1-4020-6145-5 147. A.S. Kravchuk and P.J. Neittaanm¨aki: Variational and Quasi-Variational Inequalities in Mechanics. 2007 ISBN 978-1-4020-6376-3

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 148. S.K. Kanaun and V.M. Levin: Self-Consistent Methods for Composites. Vol. 1:Static Problems. 2008 ISBN 978-1-4020-6663-4 149. G. Gogu: Structural Synthesis of Parallel Robots. Part 1: Methodology. 2008 ISBN 978-1-4020-5102-9 150. S.K. Kanaun and V.M. Levin: Self-Consistent Methods forComposites.Vol. 2:Wave Propagation in Heterogeneous Materials. 2008 ISBN 978-1-4020-6967-3

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