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


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)


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


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


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


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


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


2.3 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


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).


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


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


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


3.2 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


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


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


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


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


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


⎡

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


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


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


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.


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


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.


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


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


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 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


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 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


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


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


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


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


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)


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


(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)


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


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


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


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


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


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


4.1 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


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.


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


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)


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


4.3 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


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


∗ (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.


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


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


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)


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


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


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


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).


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


(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


4.5.1 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


−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


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


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


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


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


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


2.5

0 k 0a

114


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


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


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 Lamé 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


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


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


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 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


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 → ∞.


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


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


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


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)


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


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


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


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


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-


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


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


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


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


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


∇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

Fig. 8.1. The one-particle problem of the EMM for polycrystals.

C∗


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


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


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)


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


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)


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


8.5 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


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


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


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


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


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


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. Gürdal 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. López-Cajún: 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. Gürdal: 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

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 24. 25. 26. 27. 28. 29. 30. 31. 32.

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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. González-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. Kovács (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. Mróz, 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 Sèvres, 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

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 49. 50. 51. 52.

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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. Frémond 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 Göttingen, 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. Burczyński 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-Göttingen, Germany, August 12–14, 2004. 2006 ISBN 1-4020-4149-7 130. H. Ulbrich and W. Günthner (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. Neittaanmäki: 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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