Heterogeneous Media: Local Fields, Effective Properties, and Wave Propagation 012819880X, 9780128198803

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HETEROGENEOUS MEDIA LOCAL FIELDS, EFFECTIVE PROPERTIES, AND WAVE PROPAGATION SERGEY KANAUN This book outlines new computational methods for solving volume integral equation problems in heterogeneous media. It starts by surveying the various numerical methods of analysis of static and dynamic fields in heterogeneous media, listing their strengths and weaknesses, before moving on to an introduction of static and dynamic Green functions for homogeneous media. Volume and surface integral equations for fields in heterogeneous media are discussed next, followed by an overview of explicit formulas for numerical calculations of volume and surface potentials. The book then covers Gaussian functions for the discretization of volume integral equations for fields in heterogeneous media, static problems for a homogeneous host medium with heterogeneous inclusions, and volume integral equations for scattering problems, and concludes with a chapter outlining solutions to homogenization problems and calculations of effective properties of heterogeneous media. The book also features multiple appendices detailing the code of basic programs for solving volume integral equations, written in Mathematica.

HETEROGENEOUS MEDIA

ELSEVIER SERIES IN MECHANICS OF ADVANCED MATERIALS

ELSEVIER SERIES IN MECHANICS OF ADVANCED MATERIALS

HETEROGENEOUS MEDIA LOCAL FIELDS, EFFECTIVE PROPERTIES, AND WAVE PROPAGATION

About the author

KANAUN

Dr. Sergey Kanaun is a Professor of Mechanical Engineering at the Technological Institute of Higher Education of Monterrey, State Mexico Campus, Mexico. His core areas of research are continuum mechanics, mechanics of composites, micromechanics, elasticity, plasticity, and fracture mechanics. Prior to his current teaching post, he was a Professor at the Technical University of Novosibirsk in Russia and also Chief Researcher at the Institute of Engineering Problems of the Russian Academy of Sciences, Saint Petersburg, also in Russia. He has published over 140 articles in peerreviewed journals and two books.

ISBN 978-0-12-819880-3

9 780128 198803

SERGEY KANAUN

Heterogeneous Media

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Heterogeneous Media Local Fields, Effective Properties, and Wave Propagation

Sergey Kanaun

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-819880-3 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Ana Claudia A. Garcia Production Project Manager: Anitha Sivaraj Designer: Matthew Limbert Typeset by VTeX

TO MY WIFE Araceli

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Contents

Preface Notations 1

Introduction References

2

Homogeneous media with external and internal field sources 2.1 Electro- and magnetostatic fields in homogeneous media 2.2 Green functions of electro- and magnetostatics 2.3 Elastic media with external and internal stress sources 2.4 Temperature fields in a homogeneous medium with heat sources 2.5 Quasistatic fields in poroelastic media 2.6 Acoustic waves in fluids 2.7 The Green function of time-harmonic electromagnetics 2.8 The Green function of time-harmonic elasticity 2.9 The Green function of time-harmonic poroelasticity 2.10 Volume potentials of electrostatics 2.11 Volume potentials of static elasticity 2.12 Surface potentials of electrostatics 2.13 Surface potentials of static elasticity 2.14 Volume and surface potentials of quasistatic poroelasticity 2.15 Time-harmonic potentials 2.16 Notes References

3

Volume and surface integral equations for physical fields in heterogeneous media 3.1 Integral equations for steady electric fields in heterogeneous media 3.2 Thin heterogeneities of small or large electroconductivity in homogeneous host media 3.3 Volume integral equations of static elasticity for heterogeneous media 3.4 Surface integral equations for thin inclusions in homogeneous elastic media 3.5 A crack in a poroelastic medium subjected to surface pressure 3.6 Approximate equations of the crack problem of quasistatic poroelasticity

xi xiii 1 5 7 7 10 11 14 16 19 21 22 24 27 31 33 41 43 46 48 49

51 51 55 62 65 68 73

viii

Contents

3.7 A cavity subjected to pressure of injected fluid in a poroelastic medium 3.8 Integro-differential equations for time-harmonic fields in heterogeneous fluids 3.9 Acoustic wave scattering from a rigid screen 3.10 Volume and surface integral equations of time-harmonic electromagnetics for heterogeneous media 3.11 Integro-differential equations of time-harmonic elasticity for heterogeneous media 3.12 Integro-differential equations of time-harmonic poroelasticity for heterogeneous media 3.13 Notes References 4

5

Numerical calculation of volume and surface potentials 4.1 Gaussian approximating functions 4.2 Fast Fourier transform algorithms for calculation of Gaussian quasiinterpolants and related sums 4.3 Numerical calculation of volume potentials of electrostatics 4.4 Far field asymptotics of static potentials and multipole expansions of the potential densities 4.5 Volume potential of time-harmonic acoustics 4.6 Surface potentials of electrostatics 4.7 Surface potentials of time-harmonic acoustics 4.8 Notes Appendix 4.A Computational programs for fast calculation of sums of identical functions shifted at the nodes of regular grids Appendix 4.B The computational program for numerical calculation of a 3D potential of electrostatics References Numerical solution of volume integral equations for static fields in heterogeneous media 5.1 Discretization of the volume integral equations of electrostatics for heterogeneous media 5.2 Iterative solutions of the systems of linear algebraic equations of large dimensions 5.3 Numerical solution of the volume integral equations of electrostatics 5.4 Numerical solutions of the volume integral equations of static elasticity for heterogeneous media 5.5 Thermo-elastic deformation of heterogeneous media 5.6 Elasto-plastic deformation of heterogeneous media 5.7 Notes

75 82 85 87 89 95 101 101 103 103 108 117 122 124 126 132 135 135 140 143

145 145 151 153 156 169 172 187

Contents

Appendix 5.A The computational program for numerical solution of volume integral equations of electrostatics for heterogeneous media References 6

7

8

Cracks in heterogeneous media 6.1 A planar crack of arbitrary shape in a homogeneous elastic medium 6.2 Cracks with curvilinear surfaces 6.3 Stress intensity factors at the crack edge 6.4 Elastic bodies containing cracks 6.5 A homogeneous elastic medium containing cracks and inclusions 6.6 A planar crack subjected to pressure of injected fluid in a poroelastic medium 6.7 Notes Appendix 6.A The computational program for numerical solution of the crack problem of elasticity References

ix

187 190 193 193 199 207 211 224 231 237 237 242

Time-harmonic fields in heterogeneous media 7.1 Scattering of acoustic waves from heterogeneities in fluids 7.2 Acoustic wave scattering from a rigid screen; direct and inverse problems 7.3 Electromagnetic wave scattering from a heterogeneity of arbitrary shape in dielectric media 7.4 Scattering of elastic waves from heterogeneities of arbitrary shapes 7.5 Scattering of elastic waves from a planar crack 7.6 Scattering from heterogeneous inclusions in poroelastic media 7.7 Notes Appendix 7.A The computational program for solution of the problem of acoustic wave scattering from heterogeneities in fluid Appendix 7.B The computational program for solution of the acoustic wave scattering problem for a rigid screen Appendix 7.C Integrals Fm (ρ, q) associated with the scattering problem of elasticity for a crack References

245 245

Quasistatic crack growth in heterogeneous media 8.1 Crack growth under prescribed pressure applied to the crack surface 8.2 Governing equations of the hydraulic fracture theory 8.3 Hydraulic fracture crack propagation in a homogeneous elastic medium with varying fracture toughness 8.4 The three-parameter model of hydraulic fracture crack growth in heterogeneous elastic media 8.5 Notes

335 335 347

256 275 281 293 310 321 321 327 330 333

356 371 376

x

Contents

Appendix 8.A The computational program for construction of equilibrium crack contours in a homogeneous elastic medium with varying fracture toughness Appendix 8.B Approximating functions for simulation of hydraulic fracture crack propagation in homogeneous elastic media Appendix 8.C Computer simulation of hydraulic fracture crack propagation by the three-parameter model References 9

The homogenization problem 9.1 Effective property tensors of heterogeneous media 9.2 The representative volume elements of heterogeneous media 9.3 The effective field method 9.4 The homogenization problem for elastic heterogeneous media 9.5 Homogenization of elastic media with multiple cracks 9.6 Homogenization of elasto-plastic composites 9.7 The homogenization problem for time-harmonic fields in heterogeneous media 9.8 Notes Appendix 9.A Averaging of rank four tensors over orientations References

Index

377 382 383 384 387 387 393 401 416 432 447 453 471 471 472 475

Preface

Heterogeneous media have been the object of intense theoretical and experimental studies for more than a century. This interest is caused by the importance of heterogeneous materials in engineering applications. Strictly speaking, all materials used in human practice are heterogeneous at some scale, and specific features of their microstructures affect a wide spectrum of the macroscopic properties. Composites and nanomaterials, geological structures, metal alloys, and polymer blends form an inexhaustive list of heterogeneous materials. In the theory of heterogeneous media, two principal trends can be indicated. The first one comprises approximate analytical methods for the evaluation of the effective properties of heterogeneous materials. This trend has been extensively developed in a large part of the 20th century. The emergence and development of the second trend is related to the exponential growth of computer capacity from the second half of the 20th century. Powerful computers and commercial software for the numerical analysis of linear and nonlinear problems of physics and continuum mechanics provide efficient tools for the solution of various problems of heterogeneous media. The background of these programs is mainly the finite element method. This method allows evaluating effective static properties as well as local physical fields in heterogeneous materials, but its application to the analysis of wave propagation problems encounters principal and technical difficulties. Another branch of computational mechanics and physics of heterogeneous media is related to the numerical solution of volume integral equations. It is known that the principal static and dynamic problems of heterogeneous media can be formulated in terms of volume integral equations. In this book, a universal numerical method for the solution of the volume integral equations for static and dynamic fields in heterogeneous media is systematically developed. The method is based on the “approximate approximation” concept introduced by Vladimir Maz’ya. This concept provides robust algorithms for the solution of volume and surface integral equations for fields in heterogeneous media. For static problems, efficiencies of these algorithms and of the finite element method are comparable. But these algorithms can be successfully used for the solution of wave propagation problems. In this book, the numerical method is applied to the solution of various static and dynamic problems of heterogeneous media. Electrostatic and electrodynamic fields, static and dynamic fields in elastic and poroelastic media, quasistatic crack growth in heterogeneous media, and the homogenization problems for static and time-harmonic fields in heterogeneous media are considered. Computational programs for the numerical solution of the basic problems are presented. This book is addressed to students, engineers, and researchers who use numerical methods for the analysis of physical fields in heterogeneous materials.

xii

Preface

The results presented in the book are based on the publications of the author with his students and colleagues. The author thanks Professor Vladimir Maz’ya for discussions, Professor Valery Levin for reading the manuscript and comments, and Dr. Evgeny Pervago for the help in programming. The author thanks the Technological Institute of Higher Education of Monterrey, State Mexico Campus, for the support in the research activities. S. Kanaun Mexico May 2020

Notations

ui , εij , Cij kl , ... σij = Cij kl εkl =

Lower case latin indices are tensorial. 3 

Cij kl εkl

k,l=1

T(ij )kl = 12 (Tij kl + Tj ikl ) x, x x, x a · b ⇒ ai b i a × b ⇒ ij k aj bk A ⊗ B ⇒ Aij...k Blm...n a⊕b⇒ ⇒ (a1 , a2 , .., an , b1 , b2 , ..., bm ) δij ij k E 1 , E 2 , ..., E 6 P 1 , P 2 , ..., P 6 ∇i = ∂x∂ i = ∂i divT ⇒ ∂i Tij k... rotT =  × T ⇒ ⇒ ij k ∂j Tklm... 2 2 2 = ∂ 2 + ∂ 2 + ∂ 2 ∂x1

∂x2

∂x3

f ∗(k) = f (k) = = f (x) exp(ik · x)dx k, k k, k pv F (x)dx  = = limε→0 |x|>ε F (x)dx δ(x) (x) V V (x) = 1 if x ∈ V V (x) = 0 if x ∈ /V f (x) f (x)|x   2 ϕ(x) = (πH1)3/2 exp − h|x| 2H   2 1 ϕ(x) = πH exp − h|x| 2H

Summation with respect to repeated indices is implied. Parentheses in indices mean symmetrization. Point and vector of a point in 3D space. Point and vector of a point in 2D space. Scalar product of vectors and tensors. Vector product of vectors and tensors. Tensor product of vectors and tensors. Direct sum of vectors. The Kronecker symbol. The Levi-Civita symbol. The basic rank four tensors in Eqs. (2.210)–(2.211). The basic rank four tensors in Eqs. (2.212)–(2.213). The Nabla operator. Divergence of a tensor field T(x). Rotor of a tensor field T(x). The Laplace operator. The Fourier transform of a function f (x). Point and vector of a point in the 3D k-space. Point and vector of a point in the 2D k-space. The Cauchy principal value of the integral. Dirac’s delta function. Delta function concentrated on the surface . Region in 3D (2D) space. Characteristic function of the region V . Ensemble average of a random function f (x). Average of a random function f (x) under the condition that x ∈ V . The Gaussian approximating function in 3D space. The Gaussian approximating function in 2D space.

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Introduction

1

In this book, the principal objects of study are physical fields in heterogeneous materials subjected to static or dynamic loading. In many important cases, such fields are described by continuum models and satisfy systems of linear differential equations (elliptic, parabolic, hyperbolic) with space-varying coefficients. Let a field u(x) in the heterogeneous medium with the tensor of material properties C(x) satisfy the equation L(C)u = −f,

(1.1)

where L(C) is a linear differential operator, f(x) is an external field source, and x is a point of the medium. We introduce a homogeneous reference medium with a constant property tensor C0 and present the function C(x) in the form C(x) = C0 + C1 (x),

(1.2)

where C1 (x) is deviation of the heterogeneous medium properties from the properties of the reference medium. In many important cases, the operator L is linear with respect to tensor C, and we can rewrite Eq. (1.1) in the form L(C0 )u = −f − L(C1 )u.

(1.3)

The inverse operator G(C0 ) with respect to L(C0 ) is defined by the equation G(C0 )L(C0 ) = −I,

(1.4)

where I is the identity operator. Applying the operator G(C0 ) to both parts of Eq. (1.3), we obtain the equation for the field u(x) in the form u = u0 + G(C0 )L(C1 )u,

u0 = G(C0 )f.

If the operator G(C0 ) can be presented in the integral form  G(C0 )f(x) = g(C0 , x − x  )f(x  )dx  ,

(1.5)

(1.6)

then Eq. (1.5) is the volume integral equation for the field u(x) in the heterogeneous medium  (1.7) u(x) = u0 (x) + g(C0 , x − x  )L(C1 )u(x  )dx  . The kernel g(C0 , x) of the integral operator in this equation is the Green function of the differential operator L(C0 ). The volume integral equation (1.7) is equivalent to the original differential equation (1.1). In the literature, Eq. (1.7) is called the equation of Lippmann–Schwinger type. Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00008-1 Copyright © 2021 Elsevier Ltd. All rights reserved.

2

Heterogeneous Media

The problem of calculation of fields in heterogeneous media with arbitrary varying property tensors C(x) can be solved only numerically. The volume integral equation method is preferable if a finite heterogeneous region V is embedded into an infinite homogeneous host medium. In this case, the problem is reduced to calculation of the field inside the region V only, and if this field is found, the field in the host medium is reconstructed from the same integral equation. It should be noted that the field u(x) in Eq. (1.7) satisfies automatically the conditions at infinity, which is important for solution of the scattering problems. These are advantages of the volume integral equation method over the methods based on the differential equations (e.g., the finite element method, which is better suited for the numerical calculation of fields in finite regions). In this book, the calculation of various physical fields in heterogeneous media is reduced to the solution of volume and surface integral equations. For deriving these equations, the Green functions of the differential operators in the governing equations for physical fields in homogeneous media should be constructed. Chapter 2 is devoted to construction of the Green functions of electro and magnetostatics, thermodynamics, electrodynamics, acoustics, static and dynamic elasticity, and poroelasticity. Volume and surface potentials associated with the Green functions are introduced, and properties of these potentials are indicated. Volume and surface integral equations for physical fields in heterogeneous media are derived in Chapter 3. Surface integral equations are used for the determination of fields in homogeneous host media containing thin heterogeneous with sharp property contrasts from the properties of the host medium. Methods of numerical solution of volume integral equations have been discussed in the literature for decades [1], [2], [3], [4]. In the conventional methods, the solution region V is divided into a finite number of subregions, and in each subregion, the unknown functions are approximated by standard functions (e.g., polynomial splines, wavelets, etc.). After application of the method of moments or the collocation method, the problem is reduced to a finite system of linear algebraic equations for the coefficients of the approximation (the discretized problem) [2]. In the case of integral equations for the fields in heterogeneous media, the matrices of the discretized problems are nonsparse and should have large dimensions for acceptable accuracy of the solution. The elements of these matrices are integrals over the subregions calculated in a number of points in the region V . These integrals can be singular, and complexity of their calculations depends on types of approximating functions and geometry of the subregions. For conventional approximating functions, a great portion of computer time is spent on calculation of these integrals and iterative solution of the discretized problem. In order to reduce considerable computational cost of the numerical solutions, various techniques were proposed (e.g., fast multipole methods [5] and wavelet expansions [6], [7]). The algorithms of these methods are cumbersome and difficult to perform for the solution of static and dynamic problems of elasticity and poroelasticity of heterogeneous media. In Chapters 4 and 5, an efficient method for numerical solution of the volume integral equations for fields in heterogeneous media is presented. In the method, Gaussian radial functions are used for approximation of the fields. The theory of approximation

Introduction

3

by Gaussian and other similar functions was developed in the works of V. Mas’ya, V. Maz’ya, and G. Schmidt and presented in the book [8]. The principal result of this theory can be formulated as follows. Any bounded together with the first derivatives function u(x) can be approximated by the following series:    1 |x|2 um ϕ(x − hm), ϕ(x) = exp − . u(x) ≈ u(h,H ) (x) = (πH )d/2 H h2 d m∈Z

(1.8) Here, m ∈ Z d is a vector with integer components in the space of the dimension d, hm are coordinates of the nodes of the approximation (h is the distance between the neighbor nodes), um = u(hm) is the value of function u(x) at the nodes, and H is a dimensionless parameter of the order of 1. In [8], Eq. (1.8) is called the “approximate approximation” because its error |u(x) − u(h,H ) (x)| does not vanish when h → 0 and can be assessed as follows: |u(x) − u(h,H ) (x)| ≤ C1 ||u||h + O(exp(−π 2 H )).

(1.9)

Here, ||u|| is a norm of the function u(x) and the constant C1 does not depend on h. The second term on the right hand side of this equation is the so-called saturation error, which does not vanish when h → 0. But for H = O(1), this term is small and can be neglected in practical calculations. Gaussian approximating functions are an efficient tool for solution of volume integral equations for the following reasons. • Actions of many integral operators of mathematical physics on Gaussian functions are presented in closed analytical forms and do not require numerical integration. Thus, for these functions, the time of calculation of the elements of the matrices of the discretized problems is substantially reduced in comparison with the methods that incorporate conventional approximating functions. • For discretization of the volume integral equations by the Gaussian functions, the only required information is the coordinates of approximating nodes and material properties at the nodes, but not detailed geometry of the mesh cells (subregions). Thus, the method is mesh-free. • For regular grids of approximating nodes, the matrices of the discretized problems have a Toeplitz structure. Hence, the fast Fourier transform (FFT) algorithms can be used for the calculation of matrix-vector products in the process of iterative solution of the discretized problems. In this book, the numerical method of solution of volume integral equations based on the “approximate approximation” concept of V. Maz’ya is systematically developed. Chapter 4 is devoted to numerical calculations of the volume and surface potentials of the fields in homogeneous media. Basic points of the “approximate approximation” concept are formulated, and the results of action of various integral operators of mathematical physics on the Gaussian functions are obtained. The computational programs adopting the FFT algorithms for fast calculation of the 2D and 3D potentials are presented.

4

Heterogeneous Media

Chapter 5 is devoted to the numerical solution of the volume integral equations for static fields in heterogeneous media. Electrostatic fields, static fields in elastic and elasto-plastic media, and quasistatic fields in poroelastic media are considered. Discretization of the integral equations for these fields by the Gaussian approximating functions is performed. Methods of iterative solution of the discretized problems are discussed, and computational programs incorporating the FFT algorithms for the calculation of matrix-vector products are presented. In Chapter 6, the crack problems for elastic and poroelastic heterogeneous media are considered. The problems are reduced to systems of surface and volume integral equations for the crack opening vectors and the stress tensor in the medium. Then, these equations are discretized by the Gaussian functions, and the numerical solutions are compared with exact solutions and results of other numerical methods presented in the literature. Chapter 7 is devoted to the numerical solution of dynamic (time-harmonic) problems of heterogeneous media. The cases of acoustic, electromagnetic, elastic, and poroelastic wave scattering from heterogeneous inclusions and groups of inclusions are considered. The discretized equations of the problems are presented, and the numerical solutions are compared with exact solutions of the scattering problems for heterogeneities of canonical forms. In Chapter 8, quasistatic crack growth in a heterogeneous elastic medium is considered. First, crack growth by a prescribed increasing pressure applied to the crack surfaces is studied. The principles of linear fracture mechanics are used for the determination of the crack configuration in the process of loading. The method of fast solution of the crack problems developed in Chapter 6 provides efficient numerical algorithms for the simulation of crack growth in heterogeneous elastic media. Then, the crack growth caused by the pressure of a fluid injected inside the crack is studied (hydraulic fracture). The governing equations of the hydraulic fracture theory are presented, and methods of efficient numerical solution of these equations are proposed. Chapter 9 is devoted to solution of the homogenization problem. For a heterogeneous medium, this problem consists in the determination of material parameters of a homogeneous medium with equivalent response to external loading. For solution of this problem, the representative volume element of heterogeneous media is introduced. The effective property tensors of heterogeneous media are expressed in terms of the averages of the fields over the representative volume elements. A combination of the self-consistent effective field method and the numerical methods developed in the previous chapters is used for the calculation of the fields in the representative volume elements. Comparisons of predictions of the method with the results of other numerical and analytical methods are presented. For time-harmonic fields, self-consistent effective medium and effective field methods in application to solution of the homogenization problem are considered. Predictions of both methods are compared for an example of a model heterogeneous medium. Notes for the reader. Chapter 2 presents a survey of Green functions of the differential operators in the governing equations for physical fields in homogeneous media. Volume and surface integral equations for the fields in heterogeneous media are obtained

Introduction

5

in Chapter 3. Chapter 4 is independent and devoted to approximation by the Gaussian radial functions and the FFT algorithms for the calculation of Gaussian quasiinterpolants and related sums. In the first part of Chapter 5, iterative methods for solution of the systems of linear algebraic equations of large dimensions are discussed, and the algorithms of the methods that are appropriate for solution of the volume integral equations for the fields in heterogeneous media are presented. The second part of Chapter 5 and Chapters 6, 7, and 8 are devoted to application of the method to solution of specific problems of heterogeneous media. In Chapter 9, the principal points of the homogenization problem are discussed, and the self-consistent effective field method for solution of this problem is presented. This chapter can be also read independently, and the numerical methods developed in the previous chapters are used as tools for solution of the self-consistent equations for the representative volume elements of heterogeneous media.

References [1] W. Chew, Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, 1990. [2] A. Peterson, S. Ray, R. Mittra, Computational Methods for Electromagnetics, IEEE Press, New York, 1997. [3] A. Samokhin, Integral Equations and Iterative Methods in Electromagnetic Scattering, VSP, Utrecht, Boston, Köln, Tokyo, 2001. [4] L. Tsang, J. Kong, K. Ding, Ch. Ao, Scattering of Electromagnetic Waves, Numerical Simulations, John Wiley & Sons, New York, 2001. [5] H. Chang, L. Greengard, V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, Journal of Computational Physics 155 (1999) 468–498. [6] B. Alpert, G. Belkin, R. Coifman, V. Rokhlin, Wavelet bases for the fast solution of second kind integral equations, SIAM Journal of Scientific and Statistical Computations 14 (1993) 159–184. [7] W. Dahmen, S. Proessdorf, R. Schneider, Wavelet approximation methods for pseudodifferential equations II: matrix compression and fast algorithms, Advances in Computational Mathematics 1 (1993) 259–335. [8] V. Maz’ya, G. Schmidt, Approximate Approximation, Mathematical Surveys and Monographs, vol. 141, American Mathematical Society, Providence, 2007.

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Homogeneous media with external and internal field sources

2

This chapter is devoted to integral presentations of physical fields in homogeneous media caused by external and internal field sources distributed in finite volumes or on surfaces. For point concentrated sources, these fields are the Green functions of the differential operators of the governing equations for the fields. The Green functions of electro- and magnetostatics, electrodynamics, thermo-conductivity, acoustics, static and dynamic elasticity, and poroelasticity are considered. The volume and surface potentials associated with the Green functions are introduced, and regularization formulas for potentials with singular kernels are presented. Discontinuities of the potentials on the boundaries of the regions containing field sources are determined.

2.1 Electro- and magnetostatic fields in homogeneous media We consider a homogeneous dielectric medium with the tensor of dielectric permittivity Cij . The system of differential equations for the electric field Ei (x) and electric displacement Di (x) in the medium with distributed electric charge q(x) has the form [1] ∂i Di (x) = q(x),

Di (x) = Cij Ej (x),

rotij Ej (x) = 0.

(2.1)

Here, rot is the differential operator rotij = ij k ∂k ,

∂i =

∂ , ∂xi

(2.2)

where ij k is the antisymmetric Levi-Civita tensor. Thus, the electric field Ei (x) in dielectric media subjected to electric charge is rotor-free. It is known that any rotor-free vector function can be presented in the form of the gradient of a scalar function, and in particular, the electric field Ei (x) can be presented in the form Ei (x) = −∂i u(x),

(2.3)

where the scalar function u(x) is called the electric potential. For the field Ei (x) in this equation, the third equation of the system (2.1) is automatically satisfied. After substitution of Eq. (2.3) into the system (2.1), we obtain the equation for the potential u(x) in the form Cij ∂i ∂j u(x) = −q(x). Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00009-3 Copyright © 2021 Elsevier Ltd. All rights reserved.

(2.4)

8

Heterogeneous Media

The Green function of the operator Cij ∂i ∂j in this equation is a vanishing at infinity solution of Eq. (2.4) with Dirac’s delta function δ(x) on the right hand side Cij ∂i ∂j g(x) = −δ(x).

(2.5)

If the function g(x) is known, a partial solution of Eq. (2.4) is presented in the form  u(x) = g(x − x  )q(x  )dx  . (2.6) Henceforth, we assume that q(x) is a piece-wise analytical function with a finite support or q(x) vanishes at infinity faster than any negative power of |x|. Such functions will be called finite. The electric field Ei (x) and electric displacement Di (x) in the medium with distributed electric charge are presented in the integral forms that follow from Eqs. (2.1), (2.3), and (2.6)   Ei (x) = ∂i g(x − x  )q(x  )dx  , Di (x) = Cij ∂j g(x − x  )q(x  )dx  . (2.7) The system of equations of magnetostatics of homogeneous media is formulated in terms of the magnetic flux vector Bi and the magnetic field intensity Hi [1] ∂i Bi (x) = 0,

Bi (x) = Mij Hj (x),

rotij Hj (x) = −ηi (x).

(2.8)

Here, Mij is the tensor of magnetic permittivity of the medium and ηi (x) is the so-called free current, which can be considered as a source of magnetic field in the medium. Because rotij Hj is not equal to zero, the field Hi (x) cannot be presented as the gradient of a scalar function. Partial solutions of the system (2.8) can be presented in the integral forms similar to Eq. (2.7)   Hi (x) = sij (x − x  )ηj (x  )dx  , Bi (x) = Mij sj k (x − x  )ηk (x  )dx  . (2.9) In order to determine the kernel sij (x) of the integral operator in these equations, we consider the identity that holds for any vanishing at infinity vector function Ai (x) [2] Mj k ∂j ∂k Ai = Mj k ∂i ∂j Ak − rotik rotkl Al , rotik = ij k Mj m ∂m .

(2.10) (2.11)

If Mij = δij , where δij is Kronecker’s symbol, this equation is the well-known formula of vector analysis A = ∇ div A − rot(rotA).

(2.12)

Here, A = A(x) is a vector function in 3D space,  = ∂i ∂i is the Laplace operator, ∇ = grad is the Nabla operator with the components (∂1 , ∂2 , ∂3 ), and div A = ∂i Ai .

Homogeneous media with external and internal field sources

9

Let g(x) be the Green function of the differential operator on the left hand side of Eq. (2.10) Mij ∂i ∂j g(x) = −δ(x).

(2.13)

Then, for a vanishing at infinity vector field Ai (x) we have the following equation, which follows from Eq. (2.10):     Ai (x) = − g(x − x  ) ∂i ∂j Mj k Ak (x  ) − rotik rotkl Al (x  ) dx  . (2.14) Changing Ai (x) in this equation to the magnetic field Hi (x) and taking into account Eq. (2.8), we obtain       Hi (x) = − g(x − x )rotik ηk (x )dx = − rotik g(x − x  )ηk (x  )dx  . (2.15) Here, we overthrow the operator rotik from a finite function ηk (x) onto the kernel g(x) using Gauss’ theorem and the equation ∂i g(x − x  ) = −∂i g(x − x  ). Overthrowing the derivatives from a finite function on the kernels in the integrals similar to (2.14) and (2.15) will be called integration by parts. Thus, the kernel sij (x) in Eq. (2.9) has the form sij (x) = −rotij g(x).

(2.16)

Let us consider a generalized form of Eqs. (2.1) and (2.8) and introduce vectors ui (x) and σi (x) that satisfy the equations ∂i σi (x) = −q(x),

σi (x) = Mij uj (x),

rotij uj (x) = −ηi (x).

(2.17)

The functions q(x) and ηi (x) can be interpreted as sources of the fields ui (x) and σi (x), q(x) is called the external source, and ηi (x) is the internal source. It follows from Eq. (2.14) that a vanishing at infinity solution of the system (2.17) ui (x) is presented in the form     ui (x) = − g(x − x  ) ∂i ∂j Mj k uk (x  ) − rotik rotkl ul (x  ) dx  =   (2.18) = ∂i g(x − x  )q(x  )dx  − rotik g(x − x  )ηk (x  )dx  , where g(x) is the Green function of the operator Mij ∂i ∂j . Here, integration by parts is used. This equation defines the field ui (x) in terms of known distributions of external and internal field sources in the medium. The system of differential equations for steady electric current Ji (x) and electric field Ei (x) in a conductive medium has the form [1] ∂i Ji (x) = 0,

Ji (x) = Cij Ej (x),

rotij Ej (x) = 0.

(2.19)

10

Heterogeneous Media

Here, Cij is the tensor of electroconductivity. In this case, the sources of the fields are on the region boundary or at infinity (for an infinite medium). Similar to the case of electrostatics, the field Ej (x) is expressed in terms of the electric potential u(x), Ei (x) = −∂i u(x),

(2.20)

and the equation for u(x) follows from the system (2.19) and has the form Cij ∂i ∂j u(x) = 0.

2.2

(2.21)

Green functions of electro- and magnetostatics

For isotropic media, the tensor of electric permittivity Cij has the form Cij = cδij ,

(2.22)

where c is a scalar. As a result, Eq. (2.5) for the Green function of electrostatics takes the form c∂i ∂i g(x) = cg(x) = −δ(x).

(2.23)

The 3D Fourier transform g ∗ (k) of the Green function g(x) is defined by the equation  ∗ g (k) = g(x)eik·x dx, (2.24) where k · x = ki xi is the scalar product of √ the vector xi of the point x and the vector parameter ki of the Fourier transform, i = −1. The inverse 3D Fourier transform is the following integral:  1 g(x) = g ∗ (k)e−ik·x dx. (2.25) (2π)3 Applying the Fourier transform operator to both parts of Eq. (2.23) and taking into account that in the Fourier transform space, the partial derivative ∂i is converted in the multiplier (−iki ) and δ ∗ (k) = 1, we obtain ck 2 g ∗ (k) = 1,

g ∗ (k) =

1 , ck 2

k 2 = |k|2 = ki ki .

(2.26)

Application of the inverse Fourier transform to g ∗ (k) yields the explicit equation for the Green function  −ik·x 1 e 1 g(x) = dk = . (2.27) 4πc|x| c(2π)3 k2

Homogeneous media with external and internal field sources

11

In the case of an anisotropic medium, linear transformation of the Cartesian coordinates xi yi = Aij xj ,

xi = A−1 ij yj

(2.28)

converts Eq. (2.5) into the following: Cij Aik Aj l ∂k ∂l g(A−1 y) = −δ(A−1 y).

(2.29)

If the tensor Aij satisfies the equation Cij Aik Aj l = δkl ,

(2.30)

we obtain for  g (y) = g(A−1 y) the equation ∂i ∂i  g (y) = − det Aδ(y),

(2.31)

where det A is the determinant of the tensor Aij . Here, the following property of the delta function is used [3]: δ(A−1 y) = det Aδ(y).

(2.32)

Eq. (2.31) is similar to Eq. (2.23), and therefore,  g (y) has the form  g (y) =

det A . 4π|y|

(2.33)

For a positive symmetric tensor Cij , the solution of Eq. (2.30) is Aij =



Cij−1 ,

det A =

1 √ , det C

(2.34)

and as a result, we obtain the Green function g(x) of electrostatics for anisotropic media in the form  1 (2.35) , r¯ (x) = det C(Cij−1 xi xj ). g(x) = 4π r¯ (x)

2.3 Elastic media with external and internal stress sources We consider a homogeneous elastic medium with stiffness tensor Cij kl subjected to body forces of density qi (x). The stress σij (x) and strain εij (x) tensors in the medium satisfy the system of differential equations [2] ∂j σij (x) = −qi (x),

σij (x) = Cij kl εkl (x),

Rotij kl εkl (x) = 0,

(2.36)

12

Heterogeneous Media

where Rotij kl is the incompatibility operator Rotij kl εkl (x) = ikl j mn ∂k ∂m εln (x).

(2.37)

The tensor εij (x) can be presented as the symmetrized gradient of a vector potential ui (x) εij (x) = ∂(i uj ) (x) =

 1 ∂i uj (x) + ∂j ui (x) . 2

(2.38)

For εij (x) in this equation, the third equation in the system (2.36) is automatically satisfied [2]. The vector ui (x) is the displacement vector of a point x. The equation for the field ui (x) follows from the system (2.36) in the form Cij kl ∂j ∂k ul (x) = −qi (x).

(2.39)

The Green function gij (x) of the operator Cij kl ∂j ∂k is a vanishing at infinity solution of the equation Cij kl ∂j ∂k glm (x) = −δim δ(x).

(2.40)

If the tensor gij (x) is known, the displacement vector and the strain and stress tensors in the medium are presented in the integral forms  (2.41) ui (x) = gij (x − x  )qj (x  )dx  ,  εij (x) = ∂(i gj )m (x − x  )qm (x  )dx  ,  (2.42) σij (x) = Cij kl ∂k glm (x − x  )qm (x  )dx  . Application of the Fourier transform operator to Eq. (2.40) results in the equation for the Fourier transform gij∗ (k) of the Green function of elasticity gij∗ (k) = (Ciklj kk kl )−1 .

(2.43)

It is known [4] that for an arbitrary anisotropic homogeneous medium, gij (x) is an even homogeneous function of the order of −1 gij (x) = gij (|x|n) =

1 gij (n) , |x|

ni =

xi . |x|

(2.44)

Explicit forms of gij (x) can be obtained for isotropic, transverse isotropic media, and for media with hexagonal symmetry [4]. For of an isotropic medium with Lame constants λ and μ, the tensors gij∗ (k) and gij (x) have the forms gij∗ (k) =



ki k j 1 − κ δ , ij μk 2 k2

κ=

λ+μ , λ + 2μ

(2.45)

Homogeneous media with external and internal field sources

gij (x) =

 xi x j  1 (2 − κ)δij + κ 2 . 8πμ|x| x

13

(2.46)

Stresses in elastic media can exist without external sources qi (x) (body forces). Such stresses are called internal, and their origins can be inhomogeneous temperature fields, plastic deformations, phase transitions accompanied by altering of crystalline lattices, etc. Let a finite region V in a homogeneous elastic medium be plastically deformed, and let mij (x) be the tensor of plastic deformations. Because the region V is constrained by the surrounding material, there appear stresses σij in the medium. e is defined by Hooke’s law: ε e = C −1 σ . The corresponding elastic strain tensor εij ij ij kl kl e and inelastic m strains composes the total strain tensor ε , The sum of elastic εij ij ij which should satisfy the compatibility equation e + mkl ) = 0 . Rotij kl εkl = Rotij kl (εkl

(2.47)

In the absence of body forces, the stress tensor satisfies the homogeneous equilibrium  equation ∂j σij = 0 . As a result, the system of equations for internal stresses takes the form ∂j σij = 0,

e σij = Cij kl εkl ,

e Rotij kl εkl = −ηij ,

ηij = Rotij kl mkl .

(2.48) (2.49)

In this equation, ηij (x) is called the tensor of the dislocation density and mij (x) is the tensor of the dislocation moments [2], [5]. If ηij (x) is a finite function, a partial solution of the system (2.48) is presented in the integral form [2]  (2.50) σij (x) = Zij kl (x − x  )ηkl (x  )dx  , where Zij kl (x) is the Green tensor for internal stresses. The explicit form of this tensor for isotropic media is presented in [2], [6]. After taking Eq. (2.49) into account and integrating by parts, the stress tensor in Eq. (2.50) can be presented in the form  (2.51) σij (x) = S ij kl (x − x  )mkl (x  )dx  , Sij kl (x) = Rotij mn Zmnkl (x) .

(2.52)

The kernel Sij kl (x) in this equation can be expressed in terms of the Green function of static elasticity gij (x) in Eq. (2.40). Let a homogeneous medium be subjected to external qi (x) and internal ηij (x) stress sources. It follows from Eqs. (2.36) and (2.48) that the system of equations for the stress field σij can be written in the form −1 σmn = −ηij . ∂j σij = −qi , Rotij kl Cklmn

(2.53)

Using the Green functions gij (x) and Zij kl (x), we can present the solution of this system in the integral form      σij (x) = Cij kl ∂k glm (x − x )qm (x )dx + Zij kl (x − x  )ηkl (x  )dx  . (2.54)

14

Heterogeneous Media

Substituting in this equation the left hand sides of Eq. (2.53) for qi (x) and ηij (x) and integrating by parts, we obtain  σij (x) = −Cij kl ∂k glm (x − x  )∂n σnm (x  )dx  −  −1 σrs (x  )dx  = − Zij kl (x − x  )Rotklmn Cmnrs     −1   σmn x  dx  . =− Cij kl ∂k ∂m gln x − x  +S ij kl x − x  Cklmn (2.55) Comparing the left and right parts of this equation, we obtain the identity −1 , Iij mn δ(x) = −Cij kl ∂k ∂m gln (x) − Sij kl (x)Cklmn

(2.56)

where Iij kl = δi(j δk)l is the unit four rank tensor. Thus, the equation for the function Sij kl (x) takes the form Sij kl (x) = Cij mn Kmnrs (x)Crskl − Cij kl δ(x) , where the function Kij kl (x) is defined by the equation   Kij kl (x) = − ∂i ∂k gj l (x) (ij )(kl) .

2.4

(2.57)

(2.58)

Temperature fields in a homogeneous medium with heat sources

The temperature field T (x, t) in a homogeneous medium with heat sources of the density q(x, t) satisfies the system of differential equations [7] ∂i Ji (x, t) + cρ

∂T (x, t) = q(x, t), ∂t

Ji (x, t) = −Cij ∂j T (x, t).

(2.59)

Here, Ji (x, t) is the heat flux, Cij is the tensor of thermo-conductivity, c and ρ are heat capacity and density of the medium, respectively, and t is time. The equation for the temperature field follows from the system (2.59) in the form Cij ∂i ∂j T (x, t) − cρ

∂T (x, t) = −q(x, t). ∂t

(2.60)

The Green function g(x, t) of the operator Cij ∂i ∂j − cρ ∂t∂ is a vanishing at infinity solution of the equation Cij ∂i ∂j g(x, t) − cρ

∂g(x, t) = −δ(x)δ(t). ∂t

(2.61)

Homogeneous media with external and internal field sources

15

If the function g(x, t) is known, a partial solution of Eq. (2.60) is presented in the integral form  (2.62) T (x, t) = g(x − x  , t) ∗ q(x  , t)dx  , where the symbol (∗) means the convolution operator with respect to time  t f (t) ∗ u(t) = f (t − t  )u(t  )dt  .

(2.63)

0

If the medium is isotropic, the tensor of thermo-conductivity has the form Cij = κδij ,

(2.64)

where κ is a scalar. In this case, Eq. (2.61) takes the form a 2 ∂i ∂i g(x, t) −

∂g(x, t) 1 = − δ(x)δ(t), ∂t cρ

a2 =

κ . cρ

(2.65)

Application of the Fourier transform with respect to spatial variables xi yields the equation a 2 k 2 g ∗ (k, t) +

∂g ∗ (k, t) 1 = δ(t), ∂t cρ

the solution of which has the form 

1 g ∗ (k, t) = H (t) exp −a 2 k 2 t . cρ

(2.66)

(2.67)

Here, H (t) is the Heaviside function: H (t) = 0, t < 0, H (t) = 1, t > 0. After application of the inverse Fourier transform with respect to the ki -variables, we obtain the equation for g(x, t)   √ cρH (t) cρ|x|2 exp − . (2.68) g(x, t) = √ 4κt (2 πκt)3 In the case of an anisotropic medium, we introduce linear transformation of spatial variables xi yi = Aij xj ,

xi = A−1 ij yj

(2.69)

with the tensor Aij taken from the condition Cij Aik Aj l = δkl .

(2.70)

In the y-variables, Eq. (2.61) is converted into (g(y, ˜ t) = g(A−1 y, t)) 1 ∂ g(y, ˜ t) 1 ˜ t) − ∂i ∂i g(y, = − (det A)δ(y)δ(t). cρ ∂t cρ

(2.71)

16

Heterogeneous Media

The solution of this equation is similar to (2.68), √ g(y, ˜ t) =

  cρH (t) det A cρ|y|2 exp − . √ 3 4t (2 πt)

(2.72)

For a symmetric positive tensor Cij , the tensor Aij in Eq. (2.70) is −1/2

Aij = Cij

,

1 det A = √ . det C

(2.73)

Thus, for an anisotropic medium, the Green function g(x, t) of thermo-conductivity in the (x, t)-presentation takes the form √

cρ  cρH (t) g(x, t) = √  √ exp − Cij−1 xi xj . (2.74) 4t det C (2 πt)3

2.5 Quasistatic fields in poroelastic media The theory of fluid-saturated porous media of M. Biot [8], [9] is an adequate model of mechanical behavior of many geologic structures. It is assumed in the model that the medium consists of a solid skeleton and a porous space filled with fluid. The theory provides a coupled system of differential equations for the vector of displacements ui (x, t) of the solid skeleton and fluid pressure p(x, t) in the porous space. For an isotropic homogeneous medium, the system of equations of quasistatic poroelasticity has the form (λ + μ)∂i ∂j uj + μ∂j ∂j ui − α∂i p = −Fi , ∂ ∂ −α ∂j uj +κ∂j ∂j p − β p = −f, ∂t ∂t

(2.75) (2.76)

where λ and μ are the effective Lame constants of the solid skeleton with dry pores and α and β are Biot’s parameters α=1−

K , Ks

β=

α−φ φ + . Ks Kf

(2.77)

Here, Ks and Kf are the bulk moduli of the solid and fluid phases, K is the effective bulk modulus of the skeleton with dry pores, and φ is the porosity of the medium. The coefficient κ reflects mobility of the fluid in the porous space, and κ=

κ , η

(2.78)

where κ is the permeability of the medium and η is the fluid viscosity. The right hand sides Fi and f of Eqs. (2.75) and (2.76) are the field sources. The stress tensor σij (x, t)

Homogeneous media with external and internal field sources

17

in the solid skeleton is defined by the equation σij = λ∂k uk δij + 2μ∂(i uj ) − αpδij .

(2.79)

The system (2.75)–(2.76) follows from the complete system of equations of dynamic poroelasicity (Section 2.9) by neglecting the inertial terms proportional to the densities of the solid and fluid phases. After application of the Laplace transform operator to Eqs. (2.75) and (2.76), we obtain the system of equations of poroelasticity in the (x, ω)-presentation (λ + μ)∂i ∂j uj (x, ω) + μ∂j ∂j ui (x, ω) − α∂i p(x, ω) = −Fi (x, ω), κ 1 −α∂j uj (x, ω) + ∂j ∂j (x, ω) − βp(x, ω) = − f (x, ω). ω ω In these equations,  ui (x, ω) =

∞

ui (x, t)e

−ωt

 dt, p(x, ω) =

0

∞

p(x, t)e−ωt dt,

(2.80) (2.81)

(2.82)

0

and Fi (x, ω), f (x, ω) are the Laplace transforms of the source functions. It is assumed that at the initial moment t = 0, u(x, 0) = 0, and p(x, 0) = 0. For finite functions Fi (x, ω) and f (x, ω), a partial solution of the system (2.80)–(2.81) is presented in the integral form as follows:   1    i (x − x  , ω)f (x  , ω)dx  , ui (x, ω) = Gij (x − x , ω)Fj (x , ω)dx + ω (2.83)   1 p(x, ω) = i (x − x  , ω)Fi (x  , ω)dx  + g(x − x  , ω)f (x  , ω)dx  . ω (2.84) Here, Gij (x, ω), i (x, ω), and g(x, ω) are the Green functions of poroelasticity in the (x, ω)-presentation. Applying the Fourier transform operator with respect to spatial variables xi to Eqs. (2.83) and (2.84) and using the convolution property, we obtain algebraic equations for the Fourier transforms u∗i (k, ω) and p ∗ (k, ω) of the functions ui (x, ω) and p(x, ω): 1 ∗  (k, ω)f ∗ (k, ω), ω i 1 p ∗ (k, ω) = i∗ (k, ω)Fi∗ (k, ω) + g ∗ (k, ω)f ∗ (k, ω). ω u∗i (k, ω) = G∗ij (k, ω)Fj∗ (k, ω) +

(2.85) (2.86)

Here, G∗ij , i∗ , g ∗ , and Fi∗ , f ∗ are the Fourier transforms of the Green and source functions with respect to spatial variables. Applying the Fourier transform operator to Eqs. (2.80) and (2.81) and substituting u∗i (k, ω) and p ∗ (k, ω) from Eqs. (2.85) and (2.86) into the transformed equations, we obtain the following system for the Fourier transforms of the Green functions:

18

Heterogeneous Media



 1 1 (λ + μ)ki kj G∗j k Fk∗ + j∗ f ∗ + μk 2 G∗ik Fk∗ + i∗ f ∗ − ω ω

1 ∗ ∗ ∗ ∗ ∗ (2.87) − α(iki ) j Fj + g f = Fi , ω

  κ  1 1 1 k 2 + β j∗ Fj∗ + g ∗ f ∗ = f ∗ . − α(iki ) G∗ik Fk∗ + i∗ f ∗ + ω ω ω ω (2.88) In this system, the functions G∗j k , j∗ , and g ∗ are unknowns. Equating expressions in front of Fi∗ and f ∗ in the left and right hand sides of Eqs. (2.87) and (2.88), we obtain the following system of equations for the Fourier transforms of the Green functions:   (2.89) (λ + μ)ki kk + μk 2 δik G∗kj − α(iki )j∗ = δij ,   (2.90) (λ + μ)ki kj + μk 2 δij j∗ − α(iki )g ∗ = 0,

κ 

κ  k 2 + β j∗ = 0, α(iki )i∗ − k 2 + β g ∗ = −1. (2.91) α(iki )G∗ij − ω ω Looking for G∗ij (k, ω) and i∗ (k, ω) in the forms G∗j k

  k i kj ki k j = A 2 + B δij − 2 , k k

i∗ = (−iki )C,

(2.92)

where A, B, C are scalar functions of k and ω, we find explicit expressions for the Green functions in the (k, ω)-presentation: q 2 ki kj 1 (λ + μ) ki kj  , δ − − ij μ(λ + 2μ) k 4 (λ + 2μ) k 4 (k 2 + q 2 ) μk 2 1 ∗ iki  1 i∗ (k, ω) = 2 2 , , g (k, ω) = 2 2 ω αk (k + q ) κ(k + q 2 ) α 2 + β(λ + 2μ) α2 q 2 = Q2 ω, Q2 = . , = 2 κ(λ + 2μ) α + β(λ + 2μ) G∗ij (k, ω) =

(2.93) (2.94) (2.95)

Application of the inverse Fourier transform to G∗ij (k, ω), i∗ (k, ω), and ω1 g ∗ (k, ω) yields the (x, ω)-presentations of the Green functions of quasistatic poroelasticity

r  1 (λ + μ) δij − ∂i ∂j − 4πμr μ(λ + 2μ) 8π    1 − e−qr r − ∂i ∂j + , (λ + 2μ) 8π 4πq 2 r  (1 − e−qr ) 1 e−qr , g(x, ω) = , i (x, ω) = − ∂i α 4πr ω 4πrκ

Gij (x, ω) =

(2.96) r = |x|.

(2.97)

After application of the inverse Laplace transforms to Eqs. (2.83) and (2.84), we obtain the (x, t)-presentation of displacements and pressure in the medium with source

Homogeneous media with external and internal field sources

19

functions Fi (x, t) and f (x, t)   ui (x, t) = Gij (x − x  , t) ∗ Fj (x  , t)dx  +  i (x − x  , t) ∗ f (x  , t)dx  ,  p(x, t) =

i (x − x  , t) ∗ Fi (x  , t)dx  +



(2.98)  g (x − x  , t) ∗ f (x  , t)dx  . (2.99)

In these equations, the kernels Gij (x, t), i (x, t) are the originals (the inverse Laplace transforms) of the Green functions Gij (x, ω), i (x, ω) in Eqs. (2.96) and (2.97) and  g (x, t) are the originals of ω1 i (x, ω) and ω1 g(x, ω). Explicit expressions of i (x, t),  the originals have the forms

r 

1 1 Gij (x, t) = δij − δ(t)− (λ + μ(1 + ))∂i ∂j 4πμr μ(λ + 2μ) 8π     1 Qr  ∂i ∂j H (t) − erf c , (2.100) − √ (λ + 2μ) 4πQ2 r 2 t  

 Q2 r 2 Q 1 exp − δ(t) − , (2.101) i (x, t) = − ∂i α 4πr 4t 8(πt)3/2      Qr 1  , (2.102) H (t) − erf c √ i (x, t) = − ∂i α 4πr 2 t   Q Q2 r 2  g (x, t) = exp − . (2.103) 4t 8κ(πt)3/2 Here, erf c(z) = 1 − erf(z), erf(z) is the error function  z 2 2 e−t dt, erf(z) = √ π 0

(2.104)

H (t) is Heaviside’s function, and δ(t) is Dirac’s delta function.

2.6

Acoustic waves in fluids

For compressible fluids, the linearized equations of motion are formulated in terms of fluid pressure p(x, t) and velocity υi (x, t) of fluid particles [10] ∂p + K∂i υi = 0, ∂t

ρ

∂υi + ∂i p = qi . ∂t

(2.105)

Here, ρ is the fluid density, K is the fluid bulk modulus, and qi is the body force acting on fluid particles. Applying the time derivative to the first equation and using

20

Heterogeneous Media

the second one, we obtain the equation for acoustic pressure p(x, t) in the form ∂i ∂i p −

1 ∂ 2p = −∂i qi , c2 ∂t 2

c2 =

K . ρ

(2.106)

Here, c is the sound velocity in the fluid. Applying the time derivative to the second equation of the system (2.105) and using the first one, we obtain the equation for the velocity υi of fluid particles in the form ∂i ∂j υj −

1 ∂ 2 υi 1 ∂qi =− . K ∂t c2 ∂t 2

(2.107)

If dependence on time is defined by the multiplier eiωt , where ω is frequency (time-harmonic acoustics), then qi (x, t) = qi (x)eiωt , p(x, t) = p(x)eiωt , υi (x, t) = υi (x)eiωt , and the equation for the pressure amplitude p(x) takes the form ∂i ∂i p(x) + κ 2 p(x) = −∂i qi (x),

κ2 =

ω2 . c2

(2.108)

The Green function g(x) of the operator ∂i ∂i + κ 2 (the Helmholtz operator) is the solution of the equation ∂i ∂i g(x) + κ 2 g(x) = −δ(x),

(2.109)

and the explicit form of g(x) is well known, i.e., g(x) =

e−iκ|x| . 4π|x|

The partial solution of Eq. (2.108) can be presented in the integral form  p(x) = ∂i g(x − x  )qi (x  )dx  .

(2.110)

(2.111)

For time-harmonic acoustics, the Green tensor Gij (x) of the operator ∂i ∂j + κ 2 δij in Eq. (2.107) for the velocity of fluid particles is the solution of the equation ∂i ∂k Gkj + κ 2 Gij = −δij δ(x).

(2.112)

The Fourier transform G∗ij (k) of the Green function satisfies the equation ki kk G∗kj (k) − κ 2 G∗ij (k) = δij ,

(2.113)

and its solution is G∗ij (k) =

ki k j 1 − δij . κ 2 (k 2 − κ 2 ) κ 2

(2.114)

Homogeneous media with external and internal field sources

21

After application of the inverse Fourier transform, we obtain the x-presentation of this function

e−iκ|x| 1 . (2.115) Gij (x) = − 2 δij δ(x) + ∂i ∂j 4π |x| κ

2.7 The Green function of time-harmonic electromagnetics We consider equations of time-harmonic electromagnetics for an isotropic homogeneous medium with dielectric permittivity c and magnetic permittivity μ. In this case, the electric field Ei , the current Ji (x), and the magnetic field Hi are defined by the equations Ei (x, t) = Ei (x)eiωt ,

Ji (x, t) = Ji (x)eiωt ,

Hi (x, t) = Hi (x)eiωt , (2.116)

and Maxwell equations for the amplitudes of these fields take the forms [1] rotij Ej (x) + iωμHi (x) = 0,

(2.117)

rotij Hj (x) − iωcEi (x) = Ji (x),

(2.118)

∂i (cEi (x)) = ρ(x),

(2.119)

∂i (μHi (x)) = 0.

Applying the operator rot to the first equation and using the second one, we obtain −rotij rotj k Ek (x) + κ 2 Ei (x) = iωμJi (x),

κ 2 = ω2 cμ.

(2.120)

The Green function Gij (x) of the operator −rotij rotj k + κ 2 δik is the solution of the equation −rotij rotj k Gkl (x) + κ 2 Gil (x) = −δil δ(x).

(2.121)

Using the identity (2.12) we can rewrite this equation in the equivalent form −∂i ∂j Gj k (x) + ∂j ∂j Gik (x) + κ 2 Gik (x) = −δik δ(x).

(2.122)

Application of the Fourier transform operator yields the equation for the Fourier transform G∗ij (k) of the Green function Gij (x) ki kk G∗kj (k) − (k 2 − κ 2 )G∗ij (k) = −δij .

(2.123)

The solution of this equation has the form G∗ij (k) =

k2

ki k j 1 . δij − 2  2 2 −κ κ k − κ2

(2.124)

22

Heterogeneous Media

After application of the inverse Fourier transform, we obtain the explicit form of the Green function Gij (x) Gij (x) = g(x)δij +

1 ∂i ∂j g(x), κ2

g(x) =

e−iκ|x| . 4π |x|

(2.125)

For the current amplitude Ji (x), the electric Ei (x) and magnetic Hi (x) fields in the medium are presented in the integral forms  (2.126) Ei (x) = −iωμ Gij (x − x  )Jj (x  )dx  ,  (2.127) Hi (x) = − rotij Gj k (x − x  )Jk (x  )dx  .

2.8 The Green function of time-harmonic elasticity We consider an infinite homogeneous elastic medium with stiffness tensor Cij kl and density ρ subjected to body forces of density qi (x, t). For time-harmonic elasticity, qi (x, t) = qi (x)eiωt , the vector of displacements is ui (x, t) = ui (x)eiωt , and the amplitude ui (x) satisfies the equation ∂j Cij kl ∂k ul (x) + ρω2 ui (x) = −qi (x).

(2.128)

For a finite function qi (x), the partial solution of this equation is presented in the integral form  ui (x) = gij (x − x  )qj (x  )dx  , (2.129) where gij (x) is the Green function of the operator ∂k Cikj l ∂l + ρω2 δij . Thus, gij (x) is a vanishing at infinity solution of the equation Lik (∂)gkj (x) + ρω2 gij (x) = −δij δ (x) ,

Lij (∂) = ∂k Cikj l ∂l .

(2.130)

For an arbitrary anisotropic medium, an elegant method of construction of gij (x) was proposed in [11]. The method is based on the plane-wave decomposition of the 3D Dirac delta function δ(x) indicated in [3]  1 d 2 δ(z) δ(x) = − 2 δ  (ξ · x)dSξ , δ  (z) = . (2.131) 8π |ξ |=1 dz2 Here, Sξ is the surface of a unit sphere. It follows from Eqs. (2.130) and (2.131) that the tensor gij (x) can be found in the form of the plane-wave decomposition similar to (2.131)  d 2 Fij (z) 1   F (ξ · x)dS , F (z) = , (2.132) gij (x) = ξ ij 8π 2 |ξ |=1 ij dz2

Homogeneous media with external and internal field sources

23

where the function F (z) satisfies the equation Lik (ξ )Fkj (z) + ρω2 Fij (z) = δij δ(z),

z = ξ · x.

(2.133)

For any fixed vector ξi , the tensor Lij (ξ ) = ξk Cikj l ξl is positive and symmetric. Therefore, there exists a basis of orthogonal normalized eigenvectors eiα (ξ ) (α = 1, 2, 3) such that this tensor is presented in the form Lij (ξ ) =

3 

lα (ξ )eiα (ξ )ejα (ξ ),

lα (ξ ) > 0,

α = 1, 2, 3.

(2.134)

α=1

Here, lα (ξ ) are the eigenvalues of Lij (ξ ). Presenting the tensor Fij (z) in the basis eiα , Fij (z) =

3 

fα (z)eiα (ξ )ejα (ξ ),

(2.135)

α=1

and using Eq. (2.133), we obtain the equations for the functions fα (z) in the form 

lα (ξ )fα (z) + ρω2 fa (z) = δ(z),

α = 1, 2, 3.

(2.136)

The solutions of these equations are   1 iω|z| fα (z) = − exp − , 2iρωυα (ξ ) υα (ξ )

 υα (ξ ) =

lα (ξ ) . ρ

(2.137)

Then, using Eq. (2.132), we obtain the equation for the Green function gij (x)

gij (x) =

 

3  eiα (ξ )ejα (ξ ) 1  ω|ξ · x| iω exp −i dSξ . δ(ξ · x) − 2υα (ξ ) υα (ξ ) 8π 2 ρ υα (ξ )2 |ξ |=1 α=1

(2.138) This presentation provides direct decomposition of the Green function of timeharmonic elasticity as the sum of the static part gijs (x) (for ω = 0) and the dynamic part gijω (x) gij (x) = gijs (x) + gijω (x), 3  eiα (ξ )ejα (ξ ) 1  δ(ξ · x)dSξ = gijs (x) = 8π 2 ρ υα (ξ )2 α=1 |ξ |=1  1 = L−1 (ξ )δ(ξ · x)dSξ , 8π 2 ρ |ξ |=1 ij

(2.139)

(2.140)

24

Heterogeneous Media

  3  eiα (ξ )ejα (ξ ) iω  ω|ξ · x| exp −i dSξ = υα (ξ ) 16π 2 ρ υα (ξ )3 α=1 |ξ |=1 √ 

iω ρ √ −1/2  −3/2 L (ξ ) exp −iω|ξ · x| ρLkj (ξ ) dSξ . =− 16π 2 |ξ |=1 ik

gijω (x) = −

(2.141)

Here, the function exp(Tij ) is defined for any symmetric tensor Tij with eigenvalues tα and eigenvectors eiα by the equation exp(Tij ) =

3 

exp(tα )eiα ejα .

(2.142)

α=1

Taking into account the equations δ(ξ · x) = δ(ξ · n|x|) =

1 δ(ξ · n), |x|

ni =

xi , |x|

(2.143)

the static part of the Green function is presented as the integral over the unit sphere  1 s L−1 (ξ )δ(ξ · n)dSξ . (2.144) gij (x) = 8π 2 ρ|x| |ξ |=1 ij Thus, the static part gijs (x) is a homogeneous function of the order of |x|−1 , while the dynamic part gijω (x) has no singularity at x = 0. For an isotropic medium with Lame constants λ, μ, the Green function of timeharmonic elasticity takes the form 

 −iα|x| −iβ|x| e−iβ|x| e 1 2e − ∂ ∂ δ − , β ij i j |x| |x| |x| 4πμβ 2   ρ ρ α=ω , β =ω . λ + 2μ μ

gij (x) =

(2.145) (2.146)

2.9 The Green function of time-harmonic poroelasticity For an isotropic homogeneous medium, the complete system of equations of dynamic poroelasticity in terms of the displacement vector ui of the solid skeleton and fluid pressure p in the porous space has the form [8], [9] (λ + μ)∂i ∂j uj (x) + ∂j ∂j ui (x) − ρ u¨ i + ρf2 − α∂i u˙ i + ρf

χ ... χ u i − α∇p + ρf ∂i p˙ = −Fi , η η (2.147)

χ χ ∂i u¨ i + ∂i ∂i p − β p˙ = −f. η η

(2.148)

Homogeneous media with external and internal field sources

25

In these equations, λ and μ are effective Lame constants of the solid skeleton with dry pores, α and β are Biot’s parameters defined in Eq. (2.77), η is the fluid viscosity, χ is the medium permeability, ρ is the effective density ρ = φρf + (1 − φ)ρs ,

(2.149)

where ρf and ρs are the densities of the fluid and solid phases, φ is the medium poros... ity, u˙ i , u¨ i , and u i are the first, second, and third time derivatives of the displacement vector, p˙ is the first time derivative of the pressure, and Fi and f are the field sources. For time-harmonic poroelasticity, Fi (x, t) = Fi (x)eiωt , f (x, t) = f (x)eiωt , and the displacement vector ui (x, t) and pressure p(x, t) are presented in the forms ui (x, t) = ui (x)eiωt , p(x, t) = p(x)eiωt . From Eqs. (2.147) and (2.148), we obtain the system of equations for the amplitudes ui (x) and p(x) α ∂i p(x) = −Fi (x), (λ + μ)∂i ∂j uj (x) + μ∂j ∂j ui (x) + ρt ω2 ui (x) − 

(2.150)

− α ∂i ui (x) − (κ∂i ∂i + β) p(x) = −fˆ(x),

(2.151)

 α=α−

ρf2 ρf η 1 1 ,ρ = , ρt = ρ − , κ= , fˆ(x) = f (x). ρ  ρ  iωχ iω ρ ω2

(2.152)

For finite functions Fi (x) and f (x), a partial solution of this system is presented in the integral form   (2.153) ui (x) = Gik (x − x  )Fk (x  )dx  + i (x − x  )fˆ(x  )dx  ,   (2.154) p(x) = k (x − x  )Fk (x  )dx  + g(x − x  )fˆ(x  )dx  , where Gik (x), i (x), g(x) are the Green functions of the system (2.150)–(2.151). Substituting Eqs. (2.153) and (2.154) into the system (2.150)–(2.151) and equating the expressions in front of Fi and f in the left and right hand sides of the resulting equations, we obtain the systems of partial differential equations for the Green functions   (λ + μ)∂i ∂k + (μ∂j ∂j + ρt ω2 )δik Gkj (x) −  α ∂i j (x) = −δij δ(x), (2.155)  (2.156) − α ∂j Gj i (x) − κ∂j ∂j + β i (x) = 0,   α ∂i g(x) = 0, (2.157) (λ + μ)∂i ∂j + (μ∂k ∂k + ρt ω2 )δij j (x) −   (2.158) − α ∂j j (x) − κ∂j ∂j + β g(x) = −δ(x). Application of the Fourier transform operator to these equations yields the system   α ki j∗ = −δij , (2.159) −(λ + μ)ki kk + (−μk 2 + ρt ω2 )δik G∗kj + i

 (2.160) i α kj G∗j i − −κk 2 + β i∗ (x) = 0,

26

Heterogeneous Media



 α ki g ∗ = 0, −(λ + μ)ki kj + (−μk 2 + ρt ω2 )δij j∗ (x) + i

 i α kj j∗ − −κk 2 + β g ∗ = −1.

(2.161) (2.162)

Here, G∗ij , i∗ , and g ∗ are the Fourier transforms of the Green functions Gij (x), i (x), and g(x). The functions G∗ij (k) and i∗ (k) can be found in the forms   k i kj ki k j ∗ (2.163) Gij = A 2 + B δij − 2 , i∗ = (−iki )C, k k where A, B, C are scalar functions of the variables ki . Substitution of these expressions into the system (2.159), (2.162) yields the following equations for these functions: κk 2 − β  α (λ + 2μ)k 2 − ρt ω2 1 ∗ , C = − = , B= , g ,    μk 2 − ρt ω2 (2.164)     = (λ + 2μ)k 2 − ρt ω2 −κk 2 + β +  α2k2. (2.165) A=−

After application of the inverse Fourier transforms, the explicit equations for the Green functions take the forms  −iκ r  −iκ r e t − e−iκf r e t − e−iκs r e−iκt r Gij (x) = δij + g1 ∂i ∂j + g2 ∂i ∂j , 4πμr 4πμr 4πμr (2.166)     2 κf μ μ 1 1 κ2 g1 = 2 − s2 , g2 = − 2 − 2 , (2.167) (κf − κs2 ) M κt (κf − κs2 ) M κt   −iκf r κf2 κs2 − e−iκs r  α e i (x) = γ ∂i , (2.168) , γ= 4πr βμ κt2 (κf2 − κs2 )   2 κf2 κs2 e−iκf r e−iκs r 1 M κf

 1− − b2 , b1 = , g(x) = b1 4πμr 4πμr βμ κ 2 − κ 2 μ κt2 f

b2 =

κf2 κs2



2

1 M κs

 1− , βμ κ 2 − κ 2 μ κt2 s f

Here, κt is the wave number of shear waves,  ρt κt = ω , μ

s

(2.169) r = |x| ,

M = λ + 2μ.

(2.170)

(2.171)

and κf , κs are the wave numbers of the so-called fast and slow longitudinal waves that can propagate in the poroelastic medium. The numbers κf and κs are the solutions of

Homogeneous media with external and internal field sources

27

the dispersion equation [16] ε2 κ 4 + i

 μ 2 2 iμκt2 α2 κ 2 − iκt ε + Mβ +  (μβ) = 0, M M

ε2 =

μχ ωη

(2.172)

with negative imaginary parts.

2.10 Volume potentials of electrostatics The Green functions provide natural integral presentations of fields in homogeneous media with external and internal field sources. For instance, the electric potential u(x) in the medium with the electric charge q(x) distributed in the region V is the integral  1 g(x − x  )q(x  )dx  , g(x) = u(x) = , (2.173) 4πc|x| V which is called the volume potential of electrostatics. Let the charge with intensity m h be concentrated at the point x = 0, and let the charge with the opposite intensity − m h be concentrated at the point x = he, where h is a distance between the charges and e is a unit vector. If h tends to zero, the resulting charge q(x) takes the form m [δ(x) − δ(x − he)] = mei ∂i δ(x). h→0 h

q(x) = lim

(2.174)

The electric charge (2.174) is called the dipole of vector intensity Mi = mei concentrated at the point x = 0. The electric potential that corresponds to this charge is defined by the equation  (2.175) u(x) = g(x − x  )Mi ∂i δ(x  )dx  = Mi ∂i g(x). If dipoles with density Mi (x) are distributed in a region V , the electric potential u(x) takes the form  ∂i g(x − x  )Mi (x  )dx  , (2.176) u(x) = V

and the corresponding electric field Ei (x) is  Ei (x) = Kij (x − x  )Mj (x  )dx  , Kij (x) = −∂i ∂j g(x).

(2.177)

V

The kernels of the integral operators in Eqs. (2.173) and (2.176) have weak singularities when x → x  (g(x) = O(|x|−1 ) and ∂i g(x) = O(|x|−2 )), and these integrals exist in the ordinary sense. The integrand function in Eq. (2.177) has a strong singularity when x → x  (Kij (x) = O(|x|−3 )), and this integral needs regularization. Let

28

Heterogeneous Media

υε (x) be a sphere of a small radius ε with the center at point x. The integral (2.177) can be presented in the form      Kij (x − x )Mj (x )dx ≈ Kij (x − x  )Mj (x  )dx  + V V \υε (x)  Kij (x − x  )dx  Mj (x). (2.178) + υε (x)

Here, V \υε (x) is the region V without the excluded sphere υε (x). In the variable y = x − x  , the last integral in this equation is     Aij = Kij (x − x )dx = Kij (y)Vε (y)dy, (2.179) υε (x)

where Vε (y) is the characteristic function of a sphere of radius ε with the center at y = 0: Vε (y) = 1 if y ∈ υε (0), Vε (y) = 0 if y ∈ / υε (0). The integral on the right hand side in Eq. (2.179) is calculated over the entire 3D space. Using the Parseval formula, we change the integrand functions with their Fourier transforms and write   1 Kij∗ (k)Vε∗ (k)dk. Aij = Kij (y)Vε (y)dy = (2.180) (2π)3 In this equation, Kij∗ (k) and Vε∗ (k) are the Fourier transforms of the functions Kij (y) and Vε (y) mi mj , mk Ckl ml j1 (ε|k|) Vε∗ (k) = 4πε 2 , |k|

Kij∗ (k) = Kij∗ (m) =

mi =

ki , |k|

(2.181) (2.182)

where j1 (z) is the spherical Bessel function of the first kind. Then, after introducing a spherical coordinate system in the k-space, we obtain the equation for the tensor Aij Aij =

1 (2π)3

 S1

Kij∗ (m)dSm

 0

∞

Vε∗ (|k|)|k|2 d|k|,

(2.183)

where S1 is the surface of a unit sphere in the k-space. Taking into account the equation 1 Vε (0) = (2π)3



 

Vε∗ (|k|)e−ik·y dk  y=0

=

4π (2π)3

 0

∞

Vε∗ (|k|)|k|2 d|k| = 1, (2.184)

we obtain the following expression for the tensor Aij :  1 Aij = K ∗ (m)dSm . 4π S1 ij

(2.185)

Homogeneous media with external and internal field sources

29

Finally, regularization of the formally divergent integral in Eq. (2.177) follows from Eq. (2.178) by ε → 0  (2.186) Ei (x) = pv Kij (x − x  )Mj (x  )dx  + Aij Mj (x). V

Here, pv

 V



...dx is the Cauchy principal value of the integral defined by the equation Kij (x − x  )Mj (x  )dx = lim

pv



ε→0 V \υε (x)

V

Kij (x − x  )Mj (x  )dx  .

(2.187)

2.10.1 Discontinuity of the volume potential in Eq. (2.177) The potentials (2.173) and (2.176) are continuous functions in the entire space, but the potential (2.177) is discontinuous on the boundary  of the region V . In order to determine the jumps of this potential on , we define the function Mi (x) outside V by an arbitrary smooth continuation and present the integral in Eq. (2.177) in the form   Kij (x − x  )[Mj (x  ) − M(x)]dx  + Kij (x − x  )dx  Mj (x) . Ei (x) = V

V

(2.188) The first integral on the right hand side is continuous on  because the integrand function has a weak singularity. Let us consider the limit of the second integral when x tends to a point x 0 ∈  from outside or inside of the region V . We introduce Cartesian coordinates (y1 , y2 , y3 ) with the origin at the point x 0 and the y3 -axis directed along the external normal n0i = ni (x 0 ) to  and consider the limits of the integral  Jij (y) =

Kij (y − y  )dy 

(2.189)

V

when y tends to zero from the side of the normal n0i or from the opposite side. Let a point y =y 0 , y 0 = 0, be fixed. Then, we introduce the dimensionless coordinates ξi = yi / y 0 , (i = 1, 2, 3). Because Kij (y) is a homogeneous function of the degree of −3, the integral (2.189) in the new coordinates takes the form      0   Kij (ξ − ξ )dξ = Kij (ξ − ξ  )V (ξ  )dξ  , (2.190) Jij (y) = Jij (y ξ ) = V

where V (ξ ) is the characteristic function of the region V in the coordinates ξi . If y 0  0 0 0 tends to zero, ξi = yi / y  is the unit vector of the direction along which the point y 0 tends to the origin. In the limit y 0 → 0, the region V (ξ ) is transformed (in the coordinates ξi ) into the half-space ξ3 < 0, i.e., V (ξ1 , ξ2 , ξ3 ) → H1 (ξ1 , ξ2 , ξ3 ) = 1 − H (ξ3 ), where H (ξ3 ) is the Heaviside function (Fig. 2.1).

30

Heterogeneous Media

Figure 2.1 The local coordinate system at the boundary of the region containing field sources.

Thus, the following equation holds:  0 lim Jij (y ) = Kij (ξ 0 −ξ  )H 1 (ξ  )dξ  = y 0 →0  1 Kij∗ (k)H1∗ (k) exp(−ik · ξ 0 )dk . = (2π)3

(2.191)

In this equation, Kij∗ (k) and H1∗ (k) are the Fourier transforms of the functions Kij (ξ ) and H1 (ξ ) that have the forms [14] Kij∗ (k) =

ki kj , ck 2

  H1∗ (k) = (2π)2 δ(k1 )δ(k2 ) πδ(k3 ) − ik3−1 ,

and the last integral in Eq. (2.191) is calculated explicitly, i.e.,  1 Kij∗ (k)H1∗ (k) exp(−ik · ξ 0 )dk = (2π)3  1 ∗ Kij (0, 0, 0) − Kij∗ (0, 0, 1)sign(ξ30 ) . = 2

(2.192)

(2.193)

Taking into account these equations, we obtain for the limit Jij+ (0) of the integral Jij (y 0 ) at the point y 0 = 0 from outside of V (y 0 → +0) 1 Jij+ (0) = lim Jij (y 0 ) = [Kij∗ (0) − Kij∗ (n0 )], 0 2 y →+0

y0 ∈ /V.

(2.194)

The same procedure gives us the limit of Jij (y 0 ) when y 0 → −0 from inside V , i.e., 1 Jij− (0) = lim Jij (y 0 ) = [Kij∗ (0) + Kij∗ (n0 )], 0 2 y →−0

y0 ∈ V .

(2.195)

Hence, the jump of the integral Jij (y) in Eq. (2.189) on the boundary  of the region V is [Jij (0)] = Jij+ (0) − Jij− (0) = −Kij∗ (n0 ) .

(2.196)

It follows from Eq. (2.188) that the jump of the potential Ei (x) on the boundary  of the volume V is defined by the equation

Homogeneous media with external and internal field sources

[Ei (x)] = Ei+ (x) − Ei− (x) = −Kij∗ (n)Mj (x).

31

(2.197)

Here, ni = ni (x) is the external normal to  at point x ∈ . Note that the function Kij∗ (k) is not defined at k = 0. Below, the constant Kij∗ (0) will be determined, but its value does not affect the jump of the potential Ei (x) in Eq. (2.177) on the boundary of the region V .

2.11 Volume potentials of static elasticity The displacement vector and strain and stress tensors in a homogeneous elastic medium subjected to body forces with density qi (x) distributed in a region V are presented in the form of the volume potentials   gij (x − x  )qj (x  ), εij (x) = ∂(i gj )k (x − x  )qk (x  ), (2.198) ui (x) = V V    σij (x) = Cij kl ∂k glm (x − x )qm (x )dx , (2.199) V

where gij (x) is the Green function of static elasticity. Since the function gij (x) and its first derivatives ∂i gj k (x) have weak singularities when x → x  (|x − x  |−1 and |x − x  |−2 , respectively), these integrals exist in the ordinary sense and are continuous functions on the boundary  of the region V . The stress field in the medium with distribution of dislocation moments mij (x) (sources of internal stresses) is presented in the form (2.51)  S ij kl (x − x  )mkl (x  )dx  , (2.200) σij (x) = V

  Sij kl (x) = Cij mn Kmnrs (x)Crskl − Cij kl δ(x) , Kij kl (x) = − ∂i ∂k gj l (x) (ij )(kl) . (2.201) The integrand functions in the potential (2.200) and in the potential for the total strain εij (x) = Cij−1kl σkl (x) + mij (x) =  = K ij kl (x − x  )qkl (x  )dx  , qij (x) = Cij kl mkl (x)

(2.202)

V

have strong singularity |x − x  |−3 when x → x  , and these integrals need regularization. The same technique that was used for derivation of Eq. (2.186) yields the regularization formulas for the potentials (2.200) and (2.202) [13], i.e.,  (2.203) εij (x) = pv K ij kl (x − x  )qkl (x  )dx  + Aij kl qkl (x), V σij (x) = pv S ij kl (x − x  )mkl (x  )dx  + Dij kl mkl (x). (2.204) V

32

Heterogeneous Media

The tensors Aij kl and Dij kl in these equations have forms of the integrals over a unit sphere S1 in the k-space   1 1 Kij∗ kl (k)dSk , Dij kl = S ∗ (k)dS1 , (2.205) Aij kl = 4π |k|=1 4π |k|=1 ij kl where Kij∗ kl (k), Sij∗ kl (k) are the Fourier transforms of the functions Kij kl (x) and Sij kl (x). In the case of an isotropic medium, Kij∗ kl (k) and Sij∗ kl (k) have the forms  1 5 λ+μ Eij kl (m) − κEij6 kl (m) , κ = , μ λ + 2μ   ki , Sij∗ kl (k) = −2μ Pij1 kl (m) + (2κ−1) Pij2 kl (m) , mi = |k|

Kij∗ kl (k) =

(2.206) (2.207)

and the tensors Aij kl and Dij kl in Eq. (2.205) are defined by the equations   1−κ 2 5 − 2κ 1 2 1 Aij kl = E + Eij kl − Eij kl , 9μ ij kl 15μ 3   4μ(1 − 4κ) 2 2μ(5 + 4κ) 1 2 1 Eij kl − Eij kl − Eij kl . Dij kl = 9 15 3

(2.208) (2.209)

Here, the tensors Eijα kl (m) (α = 1, 2, ..., 6) compose a tensor basis in the space of rank four transverse isotropic tensors (E-basis) Eij1 kl = δi(k δj )l , Eij2 kl = δij δkl , Eij3 kl (m) = δij mk ml , (2.210)  Eij4 kl (m) = mi mj δkl , Eij5 kl (m) = mi mk δj l (i,j )(k,l) , Eij6 kl (m) = mi mj mk ml , (2.211) and the tensors Pij1 kl and Pij2 kl are the elements of another basis of the same tensor space (P -basis) P 1 = E 1 − 2E 5 + E 6 , P =E −E , 3

3

6

P 2 = E2 − E3 − E4 + E6,

P =E −E , 4

4

6

P =E −E , 5

5

6

(2.212) P =E . 6

6

(2.213)

The tensors Eijα kl (m) as well as the tensors Pijα kl (m) compose a closed algebra, and the multiplication tables of the tensors of E- and P -bases are presented in [13]. The potentials (2.200) and (2.202) are discontinuous on the boundary  of the region V , and the values of the jumps of these potentials are defined by the equations [13] + − [εij (x)] = εij (x) − εij (x) = −Kij∗ kl (n)qkl (x) ,

(2.214)

[σij (x)] = σij+ (x) − σij− (x) = −Sij∗ kl (n)mkl (x).

(2.215)

+ Here, ni = ni (x) is external normal to  at point x, εij (x) and σij+ (x) are limit values − of the potentials from the side of the normal n, and εij (x) and σij− (x) are the limits from the opposite side.

Homogeneous media with external and internal field sources

2.12

33

Surface potentials of electrostatics

Let the electric charge of density q(x) be concentrated on a smooth surface  in 3D space. In this case, the electric potential u(x) in the medium is presented in the form of the surface integral  1 u(x) = g(x − x  )q(x  )d , g(x) = . (2.216) 4πc|x|  This integral is called the potential of the simple layer of electrostatics. Because g(x) has a weak singularity when x → x  , this integral exists in the ordinary sense, and it is a continuous function on the surface . The electric field that corresponds to the potential (2.216) has the form  Ei (x) = − ∂i g(x − x  )q(x  )d . (2.217) 

Here, the integrand function has strong singularity |x − x  |−2 when x → x  , and this integral needs regularization for x ∈ . The potential Ei (x) turns out to be discontinuous on , and in order to determine the jumps of Ei (x), we present this potential in the form     Ei (x) = − ∂i g(x − x  ) q(x  ) − q(x) d − ∂i g(x − x  )d q(x) 



(2.218) and consider the limits of the integrals in this equation when x →  from the side of the normal ni (x) to  and from the opposite side. The function q(x) outside  is defined by arbitrary smooth continuation. The first integral in Eq. (2.218) is continuous on  as the integral with a weak singularity. Thus, the jump of the potential Ei (x) is defined by the second integral. In the limit x → x 0 ∈ , the integral  Ji (x) = ∂i g(x − x  )d (2.219) 

can be presented in the form  lim Ji (x) = lim x→x 0 ∈

x→x 0 \ε (x 0 )

∂i g(x − x  )d + lim



x→x 0 ε (x 0 )

∂i g(x − x  )d . (2.220)

Here, ε (x 0 ) is a part of the surface  inside a sphere of a small radius ε centered at the point x 0 and \ε (x 0 ) is the complement of ε (x 0 ) to the entire surface . The integral over \ε (x 0 ) is a continuous function at the point x 0 , and the jump of Ji (x) at x 0 is defined by the integral over ε (x 0 ). We introduce the local Cartesian coordinate system (y1 , y2 , y3 ) with the origin at the point x 0 and the y3 -axis directed

34

Heterogeneous Media

along the normal n0i = ni (x0 ) to . In these coordinates, the integral over ε (x 0 ) in Eq. (2.220) takes the form   lim ∂i g(x − x  )dx  = lim ∂i g(y − y  )d . (2.221) x→x0  (x 0 ) ε

y→0 ε (0)

In what follows, the parameter ε tends to zero. Therefore, ε (0) can be considered as a planar disk defined by the equation ε (0) = (y12 + y22 < ε 2 , y3 = 0). In the integral  Ji0 (y) = ∂i g(y − y  )d , (2.222) ε (0)

the integrand function ∂i g(y − y  ) is a homogeneous function of the order of −2. Thus, the limit value of this integral at y = 0 does not depend on the radius ε of the disc ε (0). To study the limits of Ji0 (y) when y → 0, we can consider the equivalent integral over the entire 3D space  Ji0 (y) = ∂i g(y − y  )δ(y3 )dy  . (2.223) Using the convolution property and changing the integrand functions with their Fourier transforms, this integral is calculated explicitly, i.e.,  ∞ 1 0 Ji (y) = (−iki )g ∗ (k 1 , k2 , k3 )δ(k1 )δ(k2 ) exp(−iy j kj )dk 1 dk2 dk3 = 2π −∞  (−i) ∞ 1 1 = exp(−iy 3 k3 )dk 3 = − δ3i sign (y3 ) . (2.224) 2πc −∞ k3 2c It is taken into account that the Fourier transforms of the functions ∂i g(y) and δ(y3 ) i) are (−ik and (2π)2 δ(k2 )δ(k3 ), respectively [14]. In the invariant form, Eq. (2.224) ck 2 can be written as follows: Ji0 (y) = −

1 0 n sign (n0 · y) . 2c i

(2.225)

Let ε tend to zero in Eq. (2.220). Taking into account Eqs. (2.218) and (2.225), we obtain the limit values of the integral Ji (x) in Eq. (2.219) at a point x 0 ∈  from the side of the normal n0i (Ji+ (x 0 )) and from the opposite side (Ji− (x 0 )) in the forms  1 (2.226) Ji± (x 0 ) = pv ∂i g(x 0 −x  )d ∓ n0i , 2c  where the principal value (pv) of the integral exists because ∂i g(x) is an odd function of x. In Eq. (2.218) for Ei (x), the first integral exists in the ordinary sense when x ∈ , and therefore, it can be understood in the sense of the principal value. Thus, for the limits of the potential Ei (x) on , we can write the equation    lim Ei (x) = −pv ∂i g(x − x  ) q(x  ) − q(x) d − x→



Homogeneous media with external and internal field sources

 1 − pv ∂i g(x − x  )d q(x) ± ni (x)q(x) = 2c  1 = −pv ∂i g(x − x  )q(x  )d ± ni (x)q(x). 2c 

35

(2.227)

The limits Ei+ (x) and Ei− (x) of the potential (2.217) on  follow from this equation in the forms  1 ± (2.228) Ei (x)| = −pv ∂i g(x − x  )q(x  )d ± ni (x)q(x). 2c  Thus, the jump of this potential on  is determined by the equation 1 [Ei (x)] = Ei+ (x) − Ei− (x) = ni (x)q(x) . c

(2.229)

Let mi (x) be the density of dipoles concentrated on a smooth surface  in 3D space. The corresponding electric potential has the form of the surface integral  u(x) = − ∂i g(x − x  )mi (x  )d , (2.230) 

and for x ∈ , the regularization formula of this integral follows from Eq. (2.228) in the form  1 ± u (x)| = −pv ∂i g(x − x  )mi (x  )d ± ni (x)mi (x). (2.231) 2c 

2.12.1 The potential of the double layer of electrostatics If the vector of dipole density mi (x) in Eq. (2.230) is mi (x) = cμ(x)ni (x),

(2.232)

where ni (x) is the normal vector to  and μ(x) is a scalar function, the potential (2.230) takes the form  u(x) = − ∂i g(x − x  )cni (x  )μ(x  )d . (2.233) 

This potential is called the potential of the double layer of electrostatics. The limit values of this potential on the surface  follow from Eq. (2.231), i.e.,  1 (2.234) u± (x) = −pv ∂i g(x − x  )cni (x  )μ(x  )d ± μ(x), x ∈ , 2  and the jump of u(x) on  is equal to the scalar function μ(x).

36

Heterogeneous Media

The electric field that corresponds to the potential (2.233) has the form     Ei (x) = ∂i ∂j g x − x  cnj x  μ x  d .

(2.235)



The potential Ei (x) as well as u(x) is discontinuous on . Let us consider the potential (2.235) when μ = const,    0 Ii (x) = ∂i ∂j g x − x  cnj x  d μ. (2.236) 

Using the identity (2.12) and Eq. (2.23) for the Green function g(x), we obtain 1 ∂i ∂j g(x) = rotik rotkj g(x) + ∂k ∂k g(x)δij = rotik rotkj g(x) − δ(x)δij . c (2.237) Then, application of Stokes’ theorem to the integral in Eq. (2.236) yields    1 0   rotik rotkj g(x − x ) − δ(x − x )δij cnj (x  )d μ = Ii (x) = c     = − rotik g x − x  τ k x  d  cμ−μni (x)  (x) .

(2.238)



Here,  is the boundary contour of the surface , τi (x) is a unit vector tangent to , and (x) is the delta function concentrated on . Let μ(x) in Eq. (2.235) be a linear function on ,   μ x  = di x  −x i . (2.239) Here di is a constant vector. For such a density, the potential in Eq. (2.235) takes the form     1 Ii (x) = ∂i ∂j g x − x  cnj x  dk x  −x k d . (2.240) 

We consider the limits of this potential when x → x 0 ∈ . Let ε (x 0 ) be a part of the surface  inside a sphere of a small radius ε with the center at the point x0 , and let \ε (x 0 ) be the complement of ε (x 0 ) to the entire surface . Introducing the local Cartesian coordinates (y1 , y2 , y3 ) with the origin at the point x 0 and the y3 -axis directed along the normal n0i = ni (x 0 ) to , we present the integral (2.240) in the form    ∂i ∂j g y − y  cnj y  dk yk d = Ii1 (y) =    = ∂i ∂j g y − y  cnj y  dk yk d + \ε (0)    + ∂i ∂j g y − y  cnj y  dk yk d . (2.241) ε (0)

Homogeneous media with external and internal field sources

37

The integral over \ε (0) is a continuous function at the point y = 0 (x = x 0 ). Thus, the jump of the function Ii1 (y) is defined by the limits of the integral over ε (0) at y = 0. Because in what follows ε → 0,  (0) can be considered a planar circular disk of radius ε with the normal n0i . Introducing the variables ξi = yi /|y|, we obtain  Iiε (y) =

ε (0)



= ∂i ∂j

 ∂i ∂j g y − y  cn0j dk yk d =

   g ξ − ξ  cdk ξk n0j δ n0 ·ξ  dξ ,

(2.242)

where the last integral is calculated over the entire 3D space. It is taken into account that ∂i ∂j g (x) is an even homogeneous function of degree −3, and therefore, the limits of this integral at y = 0 do not depend on the disk radius ε. Presenting the function n0i δ(n0 · ξ  ) in Eq. (2.242) in the form



 n0i δ n0 ·ξ = −∂i H n0 ·ξ ,

(2.243)

where H (z) is the Heaviside function, substituting this equation into Eq. (2.242), and integrating by parts, we find  Iiε (ξ ) = −∂i ∂j + ∂i ∂j



   ∂  j g ξ − ξ  cdk ξk H n0 ·ξ  dξ  +

  [∂  j g ξ − ξ  cdk ξk ]H n0 ·ξ  dξ  .

(2.244)

Using Gauss’ theorem the first integral in this equation is transformed into the integral over a sphere of a large radius R that vanishes when R → ∞. Thus, the integral Iiε (ξ ) takes the form 

  Iiε (ξ ) = −∂i ∂j ∂j g ξ − ξ  cdk ξk H n0 ·ξ  dξ  + 

  + ∂i ∂j g ξ − ξ  cdj H n0 ·ξ  dξ  . (2.245) Because 1 ∂j ∂j g(ξ − ξ  )= − δ(ξ − ξ  ), c

(2.246)

we obtain for Iiε (ξ ) the equation 



  Iiε (ξ ) = H n0 ·ξ di − Kij ξ − ξ  cd j H n0 ·ξ  dξ  , Kij (ξ ) = −∂i ∂j g(ξ ).

(2.247)

38

Heterogeneous Media

Using the convolution property, the integral in this equation is calculated as follows:    0    −3 Kij ∗ (k) H ∗ (k) e−ik·ξ dk = Kij ξ − ξ H n ·ξ dξ = (2π ) =

 

1 ∗ Kij (0) + sign n0 ·ξ Kij∗ (n0 ) , 2

Kij∗ (k) =

ki k j . ck 2

(2.248)

Here, it is taken into account that the Fourier transform of the function H (n0 · ξ ) has the form [14]   (2.249) H ∗ (k) = H ∗ (k1 , k2 , k3 ) = (2π )2 δ (k1 ) δ (k2 ) πδ (k3 ) + ik3−1 . Let Iiε+ (0) be the limit of the integral (2.242) when ξ tends to zero from the side of the normal n0i , and let Iiε− (0) be the same limit from the opposite side of . It follows from Eq. (2.248) that these limits have the forms  1 Iiε+ (0) = di − Kij ∗ (0) +Kij ∗ (n0 ) cdj , 2  1 ∗ ε− Ii (0) = − Kij (0) −Kij ∗ (n0 ) cdj . 2

(2.250) (2.251)

While changing n0i to (−n0i ), the integral (2.242) changes the sign, but its limiting values are defined by the same equations, (2.250) and (2.251). As a result, the equation Iiε+ (0) = −Iiε− (0) holds, and it leads to the following identity: Kij∗ (0) = c−1 δij . Finally, we obtain for the limit values Iiε± (0) the equations

 1 Iiε± (0) = ± ik n0 dk , 2 ij (n) = δij − cKij∗ (n) = δij − ni nj .

(2.252) (2.253)

Let us consider the limits of the potential (2.235) with the density μ(x) in Eq. (2.239). If ε tends to zero in Eq. (2.241), we obtain

from  Eq. (2.252) that the limits of the potential Ii1 (x) from the side of the normal Ii1+ and from the opposite 

side Ii1− have the forms  Ii1± (x) | = −pv

 1 Kij (x − x  )cnj x  dk (x − x  )k d ± ik (n) dk , 2  (2.254)

where ik (n) is defined in Eq. (2.253), ni = ni (x). Let the density μ(x) in Eq. (2.235) be a smooth function on the surface . The potential Ei (x) in Eq. (2.235) can be presented in the form        Ei (x) = − K ij x − x  cnj x  μ x  − μ (x) − ∂¯k μ (x) x  − x d − 

k

Homogeneous media with external and internal field sources

 −



− 

39

  K ij x − x  cnj x  d μ (x) −    K ij x − x  cnj x  ∂¯k μ (x) x  − x k d .

(2.255)

Here ∂¯i μ (x) is the derivative of the density μ (x) along the surface  ∂¯i μ (x) = ∂ i μ (x) − ni (x)nj (x)∂j μ (x).

(2.256)

The first integral in Eq. (2.255) converges absolutely for x ∈ , and it is a continuous function on . When x ∈ , it can be understood in the sense of the principal value and presented in the form of the sum of two integrals with the densities μ(x  ) − μ(x) and ∂¯i μ(x)(x  − x), respectively. Each of these integrals exists in the mentioned sense. The second integral in Eq. (2.255) is the generalized function (2.238), and the last integral can be considered as the potential (2.241) with the density in Eq. (2.239). The limit values of this potential on  are defined in Eq. (2.254). Thus, it follows from Eq. (2.255) that the limits of the potential Ei (x) on  are defined by the equation Ei± (x) | 

−



     K ij x − x  cnj x  μ x  − μ (x) d −

= −pv 



 1 rotij g x − x τ j x  d  cμ(x) ± ij (n(x)) ∂¯j μ (x) , 2  

(2.257)

where the tensor ij (n) is defined in Eq. (2.253). The jump of the field Ei (x) on  has the form [Ei (x)] = ij (n) ∂¯j μ (x) .

(2.258)

Because ni ij (n) = 0, the normal component ni (x)Ei (x) of the electric field is continuous on , so [ni (x)Ei (x)] = 0.

(2.259)

Note that if  is a closed surface, the contour integral in Eq. (2.257) vanishes. In particular, if  is a part of a plane P , the normal component ni (x)Ei (x) of the electric field on  can be defined by the following regularized equation:  ni Ei (x)| = −pv

    ni K ij x − x  nj c μ x  − μ (x) dP  .

(2.260)

P

Here, integration is spread over the entire plane P that the surface  belongs to, and the function μ(x) is continued by zero outside .

40

Heterogeneous Media

2.12.2 The surface potential of electrostatics with the density belonging to the surface Let the vector density mi (x) of the potential in Eq. (2.230) be tangent to the surface , ni (x)mi (x)| = 0,

θij (x)mj (x)| = mi (x),

θij (x) = δij − ni (x)nj (x).

(2.261) (2.262)

It follows from Eq. (2.231) that this potential is continuous on , [u(x)] = ni (x)mi (x) = 0.

(2.263)

The electric field Ei (x) that corresponds to this potential has the form  Ei (x) =

∂i ∂j g(x − x  )mj (x  )d .

(2.264)



Let x → x 0 ∈  and let n0i be the normal to  at point x 0 . The potential Ei (x) is presented in the form 

Ei (x) =

∂i ∂j g(x − x  )×   × mj (x  ) − θj k (x  )mk (x) − ∂¯k mj (x)θkl (x  )(x  − x)l d +  + ∂i ∂j g(x − x  )θj k (x  )d mk (x)+   ∂i ∂j g(x − x  )θkl (x  )(x  − x)l d ∂¯k mj (x). (2.265) + 



Here, ∂¯i = θij (x)∂j is the derivative along the surface . Using Gauss’ theorem for surfaces, the second integral on the right hand side can be presented in the form 

 ∂i ∂j g(x − x  )θj k (x  )d = −∂i ∂¯k g(x − x  )d =    = − ∂i g(x − x  )ek (x  )d  .

(2.266)



Here,  is the boundary contour of the surface  and ei (x) is the normal vector to  in the tangent plane to . The technique used in Section 2.12.1 yields the equations for the limit values of the third integral in Eq. (2.265),  Iij1 k (x) =



∂i ∂j g(x − x  )θkl (x  )(x  − x)l d,

(2.267)

Homogeneous media with external and internal field sources

41

 Iij1±k (x)| = pv

1 ∂i ∂j g(x − x  )θkl (x  )(x  − x)l d ± ni (x)g ∗ (n)θj k (x). 2  (2.268)

Here, g ∗ (k) is the Fourier transform of the function g(x). The limit values of the potential (2.265) are defined by the equation    ± Ei (x)| = pv ∂i ∂j g(x − x  ) mj (x  ) − mj (x) d x¯  −   1  (2.269) − ∂i g(x − x )ej (x  )d  mj (x) ± ni (x)g ∗ (n)∂¯k mk (x), 2  because θij (x)nj (x) = 0, the tangent component θij (x)Ej (x) of the potential Ei (x) is continuous on , so   (2.270) θij (x)Ej (x)  = 0. If  is a closed surface, the contour integral in Eq. (2.269) vanishes. Let  be a part of the plane P and let the function mi (x) be continued by zero outside . In this case, the limit values of the tangent component of the electric field on  can be presented by the following regularized integral:    θij Ej (x)| = −pv θij Kj k (x − x  )θkl ml (x  ) − ml (x) dP  . (2.271) P

Here, integration is spread over the entire plane P , Kj k (x) = −∂i ∂j g(x), θij = δij − ni nj , and ni is the normal to P . The function mi (x) is continued by zero outside .

2.13 Surface potentials of static elasticity In this section, we consider a homogeneous elastic medium with external or internal field sources concentrated on a smooth surface  with the normal ni (x). The potential of the simple layer of static elasticity is the displacement vector in the medium subjected to body forces of density qi (x) concentrated on ,  ui (x) = g ij (x − x  )qj (x  )d . (2.272) 

Here, gij (x) is the Green function of a homogeneous medium with the elastic stiffness tensor Cij kl . If qi (x) is a smooth function, the field ui (x) is continuous everywhere since gij (x − x  ) has weak singularity |x − x  |−1 when x → x  . The strain field εij (x) that corresponds to this potential has the form  εij (x) = ∂(i g j )k (x − x  )qk (x  )d , (2.273) 

42

Heterogeneous Media

and this field is discontinuous on . The limit values of εij (x) at a point x ∈  from + − the side of the normal ni (x) to  (εij (x)) and from the opposite side (εij (x)) are similar to the limits of the derivatives of the potential of the simple layer of electrostatics given in Eq. (2.254) and have the forms [13]  1 ± εij (x)| = pv ∂(i g j )k (x − x  )qk (x  )d ± ij k (n)qk (x) . (2.274) 2  In this equation, ij k (n) = n(i gj∗)k (n) ,

ni = ni (x) ,

(2.275)

and gij∗ (k) is the Fourier transform of the Green function gij (x). Thus, the jump of the potential in Eq. (2.273) is defined by the equation [εij (x)] = ij k (n)qk (x) .

(2.276)

The potential of the double layer of static elasticity is defined by the equation  (2.277) ui (x) = − ∂j gik (x − x  )Cj klm nl (x  )bm (x  )d , 

where bi (x) is the vector density of the potential. The properties of this potential and the potential of the double layer of electrostatics are similar. The limit values of the potential (2.277) at a point x 0 ∈  are determined by the equation [13]  1 ± 0 ui (x ) = −pv ∂j Gik (x 0 − x  )Ckj mn nm (x  )bn (x  )d ± bi (x 0 ) . (2.278) 2  As a result, the jump of the potential on  is equal to the density bi (x), − [ui (x)] = u+ i (x) − ui (x) = bi (x).

(2.279)

The strain and stress tensors that correspond to the potential of the double layer have the forms     (2.280) εij (x) = K ij kl x − x  Cklmn nm x  bn x  d ,     Cij mn K mnpq x − x  Cpqrs nr x  bs x  d , (2.281) σij (x) = 

  Kij kl (x) = − ∂i ∂k gj l (x) (ij )(kl) .

(2.282)

The field εij (x) is discontinuous on , and the limit values of the potential εij (x) are defined by the equations [13]         ± εij (x) = pv Kij kl x − x  Cklmn nm x  bn x  − bn (x) d∓ 



1 ∓ ij kl (n) ∂¯k bl (x) , 2

(2.283)

Homogeneous media with external and internal field sources

ij kl (n) = Kij∗ mn (n) Cmnkl − Eij1 kl . Consider the surface potential σij (x),     σij (x) = S ij kl x − x  nk x  bl x  d ,

43

(2.284)

(2.285)



Sij kl (x) = Cij mn Kmnpq (x) Cpqkl − Cij kl δ (x) ,

(2.286)

where  is a closed surface. The potentials (2.280) and (2.285) relate by the equation σij (x) = Cij kl εkl (x) − Cij kl nk (x) bl (x)  (x) ,

(2.287)

where (x) is the delta function concentrated on . The potential (2.285) is the stress field in the medium with the dislocation moments of the density ni (x)bj (x) distributed on the surface . The potential (2.285) coincides with the potential in Eq. (2.281) everywhere except the points of the surface . The potential (2.281) is singular – it contains the delta function (x) concentrated on , while the potential (2.285) does not have singular terms. It follows from Eq. (2.283) that the limit values of σij (x) on  are determined by the equation       ± σij (x) = pv Sij kl x − x  nk x  bl x  − bl (x) d ∓ 

1 ∓ Sij∗ kl (n) ∂¯k bl (x) , 2

x ∈ .

(2.288)

Taking into account Eq. (2.207) for Sij∗ kl (n), we obtain that the vector ni σij (x) is continuous on  (ni Sij∗ kl (n) = 0). When x ∈ , the equation for the stress vector ni (x)σij (x) takes the form      nj (x) σij (x) | = pv Tij x, x  bj x  − bj (x) d , (2.289)     Tij x, x  = −nk (x) Skij l x − x  nl x  . (2.290) For an unclosed surface, the integral presentation of the vector nj (x) σij (x) contains a contour integral over the boundary contour  of  [13].

2.14 Volume and surface potentials of quasistatic poroelasticity The Green functions of quasistatic poroelasticity in the (x, ω)-presentations are obtained in Section 2.5 in the forms (2.96) and (2.97),

r  1 (λ + μ) δij − ∂i ∂j − Gij (x, ω) = 4πμr μ(λ + 2μ) 8π    1 − e−qr r − ∂i ∂j + , (2.291) (λ + 2μ) 8π 4πq 2 r

44

Heterogeneous Media

i (x, ω) = −

 (1 − e−qr ) ∂i , α 4πr

1 e−qr g(x, ω) = , ω 4πκr

r = |x|.

(2.292)

We consider the volume potentials associated with the Green functions     Gij (x − x , ω)f (x)d , i (x) = i (x − x  , ω)f (x)d , ij (x) = V

ϑ(x) =

1 ω

V



(2.293) g(x − x  , ω)f (x)d .

(2.294)

V

These potentials and their first derivatives have integrable kernels, and therefore, they are continuous functions in the entire 3D space. The second derivatives of the potentials in Eqs. (2.293) and (2.294) are the potentials with the kernels ∂i ∂j Gkl (x, ω), ∂i ∂j k (x, ω), and ∂i ∂j g(x, ω). When x → 0, the second derivatives of the principal terms of the Green functions Gij (x, ω), i (x, ω), and g(x, ω) are assessed as follows: 1 r = O(r −3 ), Kij(4)kl (x) = ∂i ∂j ∂k ∂l = O(r −3 ), (2.295) Kij(2) (x) = ∂i ∂j 4πr 8π   1 − e−qr 1 − e−qr r −1 ∂i ∂j ∂k ∂l ), ∂ ∂ ∂ + = O(r −1 ). (2.296) = O(r i j k 8π 4πr 4πq 2 r Thus, only the potentials with the kernels Kij(2) (x) and Kij(4)kl (x) need regularizations. (2)

The regularization formula of the integral operators with the kernel Kij (x) has the form   (2) (2) (2)    Kij (x − x )f (x )dx = pv Kij (x − x  )f (x  )dx  + Aij f (x), πij (x) = V

V

(2.297) (2)

where pv is the principal value of the integral and the tensor Aij is the integral over a unit sphere S1 ,  k i kj 1 1 (2) (2)∗ (2)∗ K (k)dSk = − δij , Kı¨j (k) = − 2 . (2.298) Aij = 4π |k|=1 ij 3 k If x ∈ V , the regularization formula for the potential with the kernel Kij(4)kl (x) has the form  ij kl (x) = V

(4) Kij kl (x − x  )f (x  )dx 



Kij kl (x − x  )f (x  )dx  + Aij kl f (x), (4)

= pv V

(4)

(2.299) (4)

where the tensor Aij kl is

Homogeneous media with external and internal field sources

(4)

Aij kl =



1 4π

(4)∗

|k|=1

(4)∗

Kij kl (k) = −

Kij kl (k)dSk = −

45

 1 2 Eij kl + 2Eij1 kl , 15

k i kj kk kl . k4

(2.300)

 r is k −4 Here, it is taken into account that the Fourier transform of the function − 8π (2)∗ 1 2 [14], Eij kl and Eij kl are the elements of the E-basis in Eq. (2.210), and Kij (k) (4)∗

(2)

(4)

and Kij kl (k) are the Fourier transforms of the functions Kij (x) and Kij kl (x) in Eq. (2.295). The jumps of the potentials πij and ij kl in Eqs. (2.297) and (2.299) on the boundary  of the region V are defined by the equations [πij (x)] = ni (x)nj (x)f (x),

[ij kl (x)] = ni (x)nj (x)nk (x)nl (x)f (x), (2.301)

where ni (x) is the external normal to  at the point x. The surface potentials associated with the Green functions of quasistatic poroelasticity are     i (x) = Gij (x − x , ω)bj (x)d , i (x) = i (x − x  , ω)ν(x)d , 

ϑ(x) =

1 ω





(2.302) g(x − x  , ω)ν(x)d .

(2.303)



The densities bj (x) and ν(x) of these potentials are assumed to be integrable functions on . These potentials are similar to the potentials of the simple layers of electrostatics and elasticity. Since the kernels Gij (x, ω), i (x, ω), and g(x, ω) in Eqs. (2.291) and (2.292) have weak singularities, i (x), i (x), and ϑ(x) are continuous functions in the entire space. First derivatives of i (x) and ϑ(x) have jumps on . The jumps of ∂i j (x) are defined by the two first terms in Eq. (2.291) for Gij (x, ω). These terms correspond to the Green function of an elastic medium with Lame constants λ, μ. The first derivatives of the third term in Eq. (2.291) have no singularities and therefore, they do not make any contribution in the jumps of ∂i j (x). The jumps of the derivatives ∂i j (x) are defined by the equation   + − ∂i j (x)  = ∂i j (x) − ∂i j (x) =

(λ + μ) 1 ni (x)δj k − =− ni (x)nj (x)nk (x) bk (x). μ (λ + 2μ) +

(2.304)

−

Here, ∂i j (x) and ∂i j (x) are the limit values of ∂i j (x) from the side of the external normal to  and from the opposite side. The jumps of the derivatives of the potential ϑ(x) on  are defined by the equation   1 + − ∂i ϑ(x)  = ∂i ϑ (x) − ∂i ϑ (x) = ν(x)ni (x), κ

(2.305)

46

Heterogeneous Media

whereas the first derivatives of the potential i (x) in Eq. (2.302) are continuous on , +

−

∂i j (x) − ∂i j (x) = 0,

x ∈ .

(2.306)

Let us consider the potential  1 i (x) = − ∂k Gij (x − x  , ω)Cj kmn nm (x  )bn (x  )d ,

(2.307)



where Cij kl is the stiffness tensor of the solid skeleton with dry pores. The properties of this potential are similar to the properties of the double layer potential of elasticity: It has the jump on  that is equal to the density bi (x): 1+

1−

i (x) − i (x) = bi (x),

x ∈ .

(2.308)

1

The combination of the derivatives of i (x)

 1 1 T  (x) = nj (x)Cij kl ∂k l (x)

(2.309)

is continuous on ,

  1 + 1 − T (x) − T  (x) = 0,

(2.310)

i

i

i

x ∈ .

2.15 Time-harmonic potentials The Green function of time-harmonic acoustics is presented in Eq. (2.110), g(x) =

e−iq|x| , 4π|x|

q=

ω , c

(2.311)

where ω is the frequency and c is the wave speed in the fluid medium. Expansion of this function in the series with respect to the parameter q has the form g(x) =

∞ 1 (e−iq|x| − 1) 1 1  (−iq)n n−1 + = + |x| . 4π|x| 4π|x| 4π |x| 4π n!

(2.312)

n=1

The first term on the right hand side is the static part g s (x) of the Green function, and the series is the dynamic part g ω (x) that vanishes if ω → 0. The dynamic part does not contain singular terms, and as a result, the properties of the volume potentials the kernels of which are the derivatives of the Green function g(x) are determined by the static part of this function g s (x). In particular, the volume potential  Ei (x) = Kij (x − x  )qj (x  )dx  , Kij (x) = −∂i ∂j g(x) (2.313) V

Homogeneous media with external and internal field sources

when x ∈ V should be understood in the following sense:  Ei (x) = pv Kij (x − x  )qj (x  )dx  + Aij qj (x),

47

(2.314)

V

where the tensor Aij is defined by the Fourier transform of the static part of the second derivatives of the Green function g(x),  k i kj 1 1 Aij = Kijs∗ (k)dSk = δij , K s∗ (k) = 2 . (2.315) 4π S1 3 k Here S1 is the surface of a unit sphere in the k-space of the Fourier transforms. The jumps of the potential Ei (x) on the boundary  of the region V are defined by an equation similar to Eq. (2.197), [Ei (x)] = −Kijs∗ (n)qj (x) = −ni (x)nj (x)qj (x),

(2.316)

where ni (x) is the external normal to  at the point x ∈ . It is shown in Section 2.8 that in the case of time-harmonic elasticity, the Green function gij (x) can be divided into a static part that has singularity of the order of |x|−1 and a dynamic part that does not have singularity at x = 0. Thus, the formulas of regularization of the potentials with the kernels Kij kl (x) = −∂i ∂k gj l (x)|(i,j )(k,l) , Sij kl (x) = Cij mn Kmnpq (x)Cpqkl − Cij kl δ(x),

(2.317)

where gij (x) is the dynamic Green function, have the same forms (2.203) and (2.204) as in the static case. The tensors Aij kl and Dij kl are calculated according to Eq. (2.205) with the functions K s∗ (k) and S s∗ (k) under the integrals that are the Fourier transforms of the static parts of Kij kl (x) and Sij kl (x). The jump of the potential  εij (x) = Kij kl (x − x  )qkl (x  )dx  (2.318) V

on the surface  of the region V is also determined by the static part Kijs kl (x) of the function Kij kl (x). In the time-harmonic case, the jumps of the derivatives of the potentials of the simple and double layers are also defined by the static parts of the dynamic Green functions, and the equations for the jumps of these potentials coincide with the equations obtained in the previous sections. Time-harmonic fields far from the field sources are of interest in hydro-acoustics, exploration seismology, and radiolocation. Let us consider the volume potential  u(x) = g(x − x  )q(x  )dx  , (2.319) V

48

Heterogeneous Media

where g(x) is the Green function of time-harmonic acoustics in Eq. (2.311). For large |x| and x  ∈ V (|x| |x  |), the following asymptotic equations hold:      x − x  −1 ≈ |x|−1 , x − x   ≈ |x| − n · x  , ni = xi , |x|    −iq x−x   −iq|x| e e  ∂i1 ∂i2 ...∂im ≈ (−iq)m ni1 ni2 ...nim eiqn·x .  |x − x | |x|

(2.320) (2.321)

As a result, the principal term of the field u(x) far from the region V is presented in the form  e−iq|x| 1 q(x) exp (iqn·x) dx. (2.322) u(x) ≈ A (n) , A(n) = |x| 4π V Here, A(n) is the amplitude of the far field in the direction ni . For the volume potential of time-harmonic elasticity  gij (x − x  )qj (x  )dx  , ui (x) = V 

 −iα|x| −iβ|x| e−iβ|x| e 1 2e δij − ∂i ∂j − , β gij (x) = |x| |x| |x| 4πρω2   ρ ρ α=ω , β =ω , λ + 2μ μ

(2.323) (2.324) (2.325)

Eqs. (2.320) and (2.321) result in the following principal terms for the field ui (x) far from the region V : ui (x) ≈ Ai (n)

e−iα|x| e−iβ|x| + Bα (n) . |x| |x|

(2.326)

Here, Ai (n) is the vector amplitude of the longitudinal wave and Bi (n) is the amplitude of the shear wave in the direction ni far from the region V . These amplitudes are expressed in terms of the integrals over the region V ,  Ai (n) = ni nj fj (αn) , Bi (n) = δij − ni nj fj (βn) , (2.327)  2 η q i (x) exp (iη(n·x)) dx, η = α, β. fi (ηn) = (2.328) 4πρ V

2.16 Notes The Fourier transform method is a standard way of construction of fundamental solutions (Green functions) of differential operators with constant coefficients (see, e.g., [3], [15]). Regularization of the integral operators the kernels of which are derivatives

Homogeneous media with external and internal field sources

49

of the Green functions is defined in [3]. The Green functions of the system of equations of quasistatic and time-harmonic poroelasticity in terms of displacements of the solid skeleton and fluid pressure in the porous space are obtained in [12], [16], [17]. Determination of the jumps of volume and surface potentials on the boundary of the regions containing field sources is performed by the method proposed in [13].

References [1] D. Cheng, Field and Wave Electromagnetics, Addison-Wesley Publishing Co., Massachusetts, 1992. [2] I. Kunin, Methods of Tensor Analysis in the Theory of Dislocations, US Department of Commerce, Clearing House for Fed. Sci. Techn. Inform., Springfield, VA 22151, 1965. [3] I. Gelfand, G. Shilov, Generalized Functions, Academic, New York, 1964. [4] I. Lifshitz, L. Rosenzweig, Green’s tensor for an anisotropic infinite elastic medium, Journal of Experimental and Theoretical Physics 17 (1947) 83–791. [5] R. De Wit, Linear theory of static disclinations, in: J. Simmons, R. De Wit, R. Bullough (Eds.), Fundamental Aspects of Dislocations, in: Nat. Bur. Stand., vol. 317/I, 1970, pp. 651–673. [6] E. Kröner, Kontinuumsteorie der Versetzungen und Eigenshpannungen, Springer, Berlin, 1958. [7] A. Tikhonov, A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. [8] M. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lowfrequency range, The Journal of the Acoustical Society of America 28 (1956) 168–178. [9] M. Biot, Mechanics of deformation and acoustic propagation in porous media, Journal of Applied Physics 33 (4) (1962) 1482–1498. [10] A.D. Pierce, Acoustics, an Introduction to Its Physical Principles and Applications, McGrow-Hill, New York, 1981. [11] J.R. Willis, A polarization approach to the scattering of elastic waves – I. Scattering by a single inclusion, Journal of the Mechanics and Physics of Solids 28 (1980) 287–306. [12] A. Norris, Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and porelastic solids, Proceedings of the Royal Society of London. Series A 447 (1994) 175–188. [13] S. Kanaun, V. Levin, Self-Consistent Methods for Composites, vol. 1, Static Problems, Springer, Dordrecht, 2008. [14] Y. Brichkov, A. Prudnikov, Integral Transforms of Generalized Functions, CRC Press, New York, 1989. [15] V. Vladimirov, Generalized Functions in Mathematical Physics, Mir Publishers, Moscow, 1979. [16] S. Kanaun, V. Levin, M. Markov, Volume integral equations of the scattering problem of poroelasticity and their properties, Mathematical Methods in the Applied Sciences (2018) 1–17. [17] S. Kanaun, Cavities and cracks subjected to pressure of injected fluid in poroelastic media, International Journal of Engineering and Sciences 137 (2019) 73–91.

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Volume and surface integral equations for physical fields in heterogeneous media

3

This chapter is devoted to integral equations for physical fields in heterogeneous media. Steady electric fields in conductive media, electrostatic and time-harmonic fields in heterogeneous dielectrics, acoustic fields in heterogeneous fluids, and static and dynamic fields in elastic and poroelastic heterogeneous media are considered. For a homogeneous medium containing thin heterogeneities with sharp property contrasts from the host medium, the volume integral equations are reduced to surface integral equations. Examples of analytical solutions of the integral equations for heterogeneities of canonical forms are presented.

3.1 Integral equations for steady electric fields in heterogeneous media Let an infinite homogeneous electroconductive medium with the conductivity tensor Cij0 (the host medium) contain a region V with the conductivity tensor Cij0 + Cij1 (x). The region V can be singly connected or consist of a finite number of singly connected regions. The conductivity tensor Cij (x) of the medium can be presented in the form Cij (x) = Cij0 + Cij1 (x)V (x),

(3.1)

where V (x) is the characteristic function of the region V V (x) = 1

if x ∈ V ,

V (x) = 0 if x ∈ / V.

(3.2)

The system of differential equations for the electric field Ei (x) and current Ji (x) in the heterogeneous medium has the form [1] ∂i Ji (x) = 0,

Ji (x) = Cij (x)Ej (x),

rotij Ej (x) = 0.

(3.3)

The fields Ei (x) and Ji (x) are expressed in terms of the electric potential u(x), Ei (x) = −∂i u(x),

Ji (x) = −Cij (x)∂i u(x),

(3.4)

and the equation for u(x) follows from the system (3.3) in the form ∂i Cij (x)∂j u(x) = 0. Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00010-X Copyright © 2021 Elsevier Ltd. All rights reserved.

(3.5)

52

Heterogeneous Media

Substituting in this equation Cij (x) from Eq. (3.1) and moving the term ∂i Cij1 (x)V (x)∂j u(x) to the right hand side, we obtain   Cij0 ∂i ∂j u(x) = −∂i Cij1 (x)V (x)∂j u(x) .

(3.6)

Let g(x) be the Green function of the operator Cij0 ∂i ∂j . Application of the integral operator with the kernel g(x) to Eq. (3.6) yields the integro-differential equation for the potential u(x)    (3.7) u(x) = u0 (x) + g(x − x  )∂i Cij1 (x  )V (x  )∂j u(x  ) dx  . Here, the potential u0 (x) corresponds to the field that would have existed in the homogeneous host medium for the same conditions at infinity; this potential satisfies the equation Cij0 ∂i ∂j u0 (x) = 0. Integration by parts transforms Eq. (3.7) in the equation  u(x) = u0 (x) + V

∂i g(x − x  )Cij1 (x  )∂j u(x  )dx  .

(3.8)

Applying the derivative operator ∂i and using Eq. (3.4), we obtain the integral equation for the electric field Ei (x) in the heterogeneous medium  Kij (x − x  )Cj1k (x  )Ek (x  )dx  = Ei0 (x), (3.9) Ei (x) + V

Kij (x) = −∂i ∂j g(x − x  ),

Ei0 (x) = −∂i u0 (x).

(3.10)

The tensor Bij (x) of the electric resistivity of the medium is defined by the equations  −1 Bij (x) = Cij−1 (x) = Bij0 + Bij1 (x), Bij0 = Cij0 . (3.11) Taking into account Eq. (3.4) and the equations   Ei (x) = Bij0 + Bij1 (x) Jj (x), Cj1k (x  )Ek (x  ) = −Cij0 Bj1k (x)Jk ,

(3.12)

Eq. (3.9) can be transformed in the equation for the electric current Ji (x) in the heterogeneous medium  Si j (x − x  )Bj1k (x  )Jk (x  )dx  = Ji0 (x), (3.13) Ji (x) − V

0 Sij (x) = Cik Kkl (x)Clj0 − Cij0 δ(x),

Ji0 = Cij0 Ej0 .

(3.14)

The right hand sides Ei0 (x) and Ji0 (x) of Eqs. (3.9) and (3.13) can be interpreted as the external fields that act on the heterogeneous region V .

Volume and surface integral equations for physical fields in heterogeneous media

53

Consider some properties of the integral equations (3.9) and (3.13). 1. When x → x  , the kernels of the integral operators in these equations have strong singularities |x − x  |−3 , and the corresponding integrals should be understood in the sense of the regularizations indicated in Section 2.10. Thus, for an arbitrary finite function Fi (x), the following equations hold:   Kij (x − x  )Fj (x  )dx  = pv Kij (x − x  )Fj (x  )dx  + Aij Fj (x), (3.15) V V    Sij (x − x )Fj (x )dx = pv Sij (x − x  )Fj (x  )dx  + Dij Fj (x), (3.16) V V   1 1 Aij = K ∗ (m)dSm , Dij = S ∗ (m)dSm . (3.17) 4π |m|=1 ij 4π |m|=1 ij Here, pv is the principal value of the integral and Sm is the surface of a unit sphere in the k-space of the Fourier transforms, and Kij∗ (k) and Sij∗ (k) are the Fourier transforms of the kernels Kij (x) and Sij (x) Kij∗ (k) =

0 k k C0 Cik ki k j k l li ∗ , S (k) = − Cij0 . ij 0 k 0 km Cmn k C n m mn kn

(3.18)

2. Eqs. (3.9) and (3.13) belong to the class of pseudodifferential equations. The theory of such equations was developed in the 1970s (see, e.g., [2], [3]). It follows from this theory that unique solutions of Eqs. (3.9) and (3.13) exist if the tensors Cij (x) and Bij (x) do not degenerate in the region V (det Cij (x), det Bij (x) > 0). For regions V with smooth boundaries , the solutions of these equations are analytical functions inside V if the external fields Ei0 (x) and Ji0 (x) are analytical functions. 3. Presentation of Eqs. (3.9) and (3.13) in the forms  0 (3.19) Ei (x) = Ei (x) − Kij (x − x  )Cj1k (x  )Ek (x  )V (x  )dx  ,  (3.20) Ji (x) = Ji0 (x) + Si j (x − x  )Bj1k (x  )Jk (x  )V (x  )dx  shows that the fields Ei (x) or Ji (x) in the host medium (outside the region V ) are defined in terms of these fields inside V . Thus, the fields Ei (x)V (x) or Ji (x)V (x) are the principal unknowns of the problem. 4. If the fields Ei (x), Ji (x) in V are known, the limit values of Ei (x) and Ji (x) in the host medium at the boundary  of the region V can be obtained by using the equations for the jumps of the volume potential of electrostatics indicated in Section 2.10. For instance, if the limit value of the field Ei (x) from inside the region V at point x ∈  is Ei− (x), the limit Ei+ (x) of this field from the outside of V is expressed in terms of Ei− (x) as follows:   ∗ 1 (n)Ckj (x) Ej− (x), x ∈ . (3.21) Ei+ (x) = δij + Kik Here ni = ni (x) is the external normal to the surface .

54

Heterogeneous Media

A similar equation can be written for the outside limit Ji+ (x) of the current Ji (x) on the boundary  of the region V . If Ji− (x) is this limit from the inside of V , the following equation holds:   ∗ 1 (n)Bkj (x) Jj− (x), Ji+ (x) = δij − Sik

x ∈ .

(3.22)

5. In the case of an ellipsoidal region V and a constant conductivity tensor Cij inside V , the property of polynomial conservativity for the solutions of Eqs. (3.9) and (3.13) holds [10]: A polynomial external field Ei0 (x) (Ji0 (x)) of the power m generates a polynomial field of the same power inside an ellipsoidal region. In particular, if the fields Ei0 and Ji0 are constant, the fields Ei and Ji inside an ellipsoidal region are also constant and are defined by the equations [5], [10]   −1 1 Ei = δij + Aik (a)Ckj Ej0 , Aij (a) = Kik (x)dx, V   −1 1 Ji = δij − Dik (a)Bkj Jj0 , Dij (a) = Sik (x)dx  .

(3.23) (3.24)

V

The volume integrals in these equations can be reduced to the integrals over a unit sphere Aij (a) =

1 4π

 |m|=1

Kij∗ (a −1 m)dSm ,

Dij (a) =

1 4π

 |m|=1

Sij∗ (a −1 m)dSm . (3.25)

Here, Kij∗ (k) and Sij∗ (k) are defined in Eq. (3.18) and aij is the tensor of the linear

transformation of the x-space (yi = aij−1 xj ) that converts the ellipsoid V into a sphere of a unit radius. For an isotropic medium (Cij0 = c0 δij ), calculation of the tensors Aij (a) and Dij (a) is reduced to 1D integrals. In the basis eiα (α = 1, 2, 3) of the principal axes of the ellipsoid, these tensors are presented in the forms [5], [10] Aij (a) = A1 ei1 ej1 + A2 ei2 ej2 + A3 ei3 ej3 , Dij (a) = c02 Aij (a) − c0 δij , (3.26)  dς a 1 a2 a3 ∞  , α = 1, 2, 3. Aα = 2c0 0 (aα2 + ς) (a12 + ς)(a22 + ς)(a32 + ς) (3.27) Here, a1 , a2 , a3 are the principal semiaxes of the ellipsoid.

Volume and surface integral equations for physical fields in heterogeneous media

3.2

55

Thin heterogeneities of small or large electroconductivity in homogeneous host media

Let one of the characteristic sizes of a heterogeneous region V be much smaller than other sizes. Then, if h and l are the minimal and maximal sizes of V , the ratio δ1 = h/ l is small. We introduce the second dimensionless parameter δ2 , which is the ratio of characteristic conductivities of the heterogeneity and the host medium. For thin heterogeneities with small or large values of the parameter δ2 , the fields in the medium can be approximated by the principal terms of the asymptotic expansions with respect to δ1 , δ2 (δ1 , δ2  1) or δ1 , δ2−1 (δ1  1, δ2  1). In this section, the cases of small δ1 and finite ratio δ1 /δ2 or finite product δ1 δ2 are considered. In both cases, construction of the principal terms of the asymptotic expansions of the fields Ei (x) and Ji (x) in the medium is reduced to solution of 2D integral equations on the middle surface  of the region V . It is assumed that  is a smooth surface with the normal ni (x), and h(x) is the transverse size of the region V along the normal (Fig. 3.1).

Figure 3.1 Thin inclusion and the local coordinate system on the inclusion middle surface .

3.2.1 Thin heterogeneities of small electroconductivity Let the conductivity of a thin heterogeneity be much smaller than the conductivity of the host medium. For the construction of the principal terms of the solutions of Eqs. (3.9) and (3.13) with respect to the parameters δ1 , δ2 , the method of matching of asymptotic expansions can be used. In application to electrostatic and elastic problems, this method was considered in [4], [5], where the details of the derivations can be found. For a thin heterogeneity of a small conductivity, the electric potential u(x) in the medium is presented in the forms of the potential of the double layer of electrostatics concentrated on the middle surface  of the heterogeneity. The electric field Ei (x) and current Ji (x) are expressed in terms of derivatives of u(x),  0 ∂i g(x − x  )Cik nk (x  )b(x  )d , (3.28) u(x) = u0 (x) +   Kij (x − x  )Cj0k nk (x  )b(x  )d , (3.29) Ei (x) = Ei0 (x) +   Sij (x − x  )nj (x  )b(x  )d . (3.30) Ji (x) = Ji0 (x) + 

56

Heterogeneous Media

Here, the kernels Kij (x) and Sij (x) are defined in Eqs. (3.10) and (3.14) and the density b(x) satisfies the equation  λ(x)b(x) + T (x, x  )b(x  )d = ni (x)Ji0 (x), x ∈ , (3.31) 

T (x, x  ) = −ni (x)Sij (x − x  )nj (x  ).

(3.32)

The coefficient λ(x) in this equation depends on the conductivity Cij (x) of the heterogeneity and its transverse size h(x) λ(x) =

1 ni (x)Cij (x)nj (x). h(x)

(3.33)

If the conductivity of a thin heterogeneity is equal to zero, Eq. (3.31) takes the form  T (x, x  )b(x  )d = ni (x)Ji0 (x), x ∈ . (3.34) 

Since the integral in Eq. (3.28) is the potential of the double layer of electrostatics considered in Section 2.12, the function u(x) is discontinuous on , and the jump of u(x) is equal to the density b(x). The fields Ei (x) and Ji (x) in Eqs. (3.29) and (3.30) are also discontinuous on . It follows from the results of Section 2.12 (Eq. (2.258)) that the jumps of these fields are defined by the equations   ∗ 0 ¯ [Ei (x)] = Ei+ (x) − Ei− (x) = δij − Kik (n)Ckj (3.35) ∂j b(x), [Ji (x)] = Ji+ (x) − Ji− (x) = −Sij∗ (n)∂¯j b(x), ∂¯i = ∂i − ni (x)nk (x)∂k .

(3.36) (3.37)

Here, Ei+ (x), Ji+ (x) are the limiting values of the functions from the side of the normal, E − (x), Ji− (x) are these limits from the opposite side, and Kij∗ (k) and Sij∗ (k) are the Fourier transforms of the kernels Kij (x) and Sij (x) defined in Eq. (3.18). Since ni Sij∗ (n) = 0, the normal component of the electric current Ji (x) is continuous on , so [ni (x)Ji (x)] = 0.

(3.38)

Because the kernel T (x, x  ) has a strong singularity when x → x  (T (x, x  ) ∼ |x − x  |−3 ), the integral in Eqs. (3.31) and (3.34) needs regularization. For an arbitrary unclosed surface , the regularization formula for this integral follows from Eq. (2.260) in the form       T (x, x )b(x )d = pv T (x, x  ) b(x  ) − b(x) d −  

0  0 − ni (x)Cij rotj k g(x − x )Cks τs (x  )d  b(x). (3.39)

Volume and surface integral equations for physical fields in heterogeneous media

57

Here, is the boundary contour of  and τi (x) is the tangent vector to . For a close surface , the contour integral in this equation vanishes. Let the surface  in Eqs. (3.31) and (3.34) be a finite part of the plane P defined by the equation x3 = 0 in the Cartesian coordinate system (x1 , x2 , x3 ). In this case, ni is constant, T (x, x  ) = T (x − x  ), and the integral in Eq. (3.39) can be understood in the following sense: 

T (x − x  )b(x  )d = pv 



 T (x − x  ) b(x  ) − b(x) dx,

x = (x1 , x2 ).

P

(3.40) Here, the principal value (pv) of the integral is calculated over the entire plane x3 = 0, and the function b(x) is continued by zero outside . The function T (x1 , x2 ) in this equation is generated by the functions Sij (x1 , x2 , x3 ) in the 3D space, T (x1 , x2 ) = −S33 (x1 , x2 , x3 )|x3 =0 , 0 Sij (x1 , x2 , x3 ) = −Cik ∂k ∂l g(x1 , x2 , x3 )Clj0

(3.41) − Cij0 δ(x1 , x2 , x3 ).

(3.42)

The Fourier transform T ∗ (k1 , k2 ) of the function T (x1 , x2 ) follows from Eq. (3.41) in the form  ∞ 1 S ∗ (k1 , k2 , k3 )dk3 , (3.43) T ∗ (k1 , k2 ) = − 2π −∞ 33 ∗ (k , k , k ) is the Fourier transform of the function S (x , x , x ). It folwhere S33 1 2 3 33 1 2 3 lows from Eq. (3.18) that for an isotropic host medium (Cij0 = c0 δij ), the function ∗ (k , k , k ) takes the form S33 1 2 3 ∗ (k1 , k2 , k3 ) = c0 S33

k32 k12 + k22 + k32

− c0 = −c0

k12 + k22 k12 + k22 + k32

.

(3.44)

Thus, the integral (3.43) converges absolutely and defines an even homogeneous function of the order of 1, T ∗ (k1 , k2 ) =

c0 ¯ |k|, k¯ = (k1 , k2 ). 2

(3.45)

Action of the operator T with the kernel T (x) ¯ on functions b(x) ¯ which Fourier trans¯ −2 is defined by the equation forms tend to zero at infinity faster than |k| 

T (x¯ − x¯  )b(x¯  )d x¯  =

1 (2π)2



T ∗ (k)b∗ (k)e−ik·x dk.

(3.46)

The integral on the right converges absolutely and defines regularization of the formally divergent integral on the left hand side of this equation.

58

Heterogeneous Media

3.2.2 Thin heterogeneities of large electroconductivity Let the conductivity of a thin heterogeneity V be much larger than the conductivity of the host medium. In this case, the principal terms of the electric potential u(x), the electric field Ei (x), and current Ji (x) in the medium take the forms of the following potentials concentrated on the middle surface  of the region V [4], [5]:  ∂i g(x − x  )mi (x  )d , (3.47) u(x) = u0 (x) −   Ei (x) = Ei0 (x) − Kij (x − x  )mj (x  )d , (3.48)   Sij (x − x  )Bj0k mk (x  )d . (3.49) Ji (x) = Ji0 (x) − 

Here, the kernels Kij (x) and Sij (x) are defined in Eqs. (3.10) and (3.14) and the vector mi (x) belongs to the surface  and satisfies the equations ni (x)mi (x) = 0, θij (x)mj (x) = mi (x), θij (x) = δij − ni (x)nj (x).

(3.50) (3.51)

The 2D integral equation for this vector has the form  Uij (x, x  )mj (x  )dx  = θij (x)Ej0 (x), x ∈ , μij (x)mj (x) +

(3.52)



Uij (x, x  ) = θik (x)Kkl (x − x  )θlj (x  ).

(3.53)

Here, the function μij (x) is defined by the equation μij (x) =

1 θik (x)Bkl (x)θlj (x), h(x)

Bij (x) = Cij−1 (x).

(3.54)

For a thin heterogeneity with zero resistivity (Bij = 0), Eq. (3.52) takes the form  Uij (x, x  )mj (x  )d = θij (x)Ej0 (x), x ∈ . (3.55) 

It follows from Eq. (2.231) that the potential u(x) in Eq. (3.47) is continuous on . The potential in Eq. (3.48) is discontinuous on , and its jump is defined by Eq. (2.269). We have [Ei (x)] = Ei+ (x) − Ei− (x) = ni (x)g ∗ (n)∂¯j mj (x).

(3.56)

Here, g ∗ (k) is the Fourier transform of the Green function g(x). It follows from this equation that the tangent component of this potential is continuous on , i.e., [θij (x)Ej (x)] = 0,

(3.57)

Volume and surface integral equations for physical fields in heterogeneous media

59

while the normal component has a jump that is defined by the equation [ni (x)Ei (x)] = g ∗ (n)∂¯j mj (x).

(3.58)

The kernel Kij (x − x  ) of the potential in Eq. (3.48) has strong singularity |x − x  |−3 when x → x  . Regularization of the integral operator with the kernel Uij (x, x  ) in Eqs. (3.52) and (3.55) follows from Eq. (2.269) in the form       Uij (x, x )mj (x )d = pv Uij (x, x  ) mj (x  ) − mj (x) d +  

  (3.59) + θij (x)∂j g(x − x )ek (x )d  mk (x), x ∈ .

Here, ei (x) is the normal vector to the boundary contour of  in the plane tangent to . If  is a part of a plane P , the integral in Eqs. (3.52) and (3.55) can be understood in the sense indicated in Eq. (2.271),       Uij (x, x )mj (x )dx = pv Uij (x − x  ) mj (x  ) − mj (x) dP  , x ∈ . 

P

(3.60) Here, the principal value of the integral is calculated over the entire plane P and the function mj (x) is continued by zero outside .

3.2.3 Thin ellipsoidal inclusions Let the heterogeneity be a thin ellipsoid with the semiaxes a1 , a2 , h, and h/a1 , h/a2  1. In this case,  is a planar elliptical surface with the normal ni . If Cartesian coordinates (x1 , x2 , x3 ) coincide with the principal axes of the ellipsoid, the function h(x) takes the form  2 2 x2 x1 − . (3.61) h(x1 , x2 ) = 2hz(x1 , x2 ), z(x1 , x2 ) = 1 − a1 a2 For a constant tensor of electroconductivity Cij inside the ellipsoid, a specific form of the polynomial conservativity property of the solutions of the integral equations (3.31) and (3.52) holds [9], [10]: If the external field Ji0 (x) (Ei0 (x)) is a polynomial of power m on the surface , the solutions b(x1 , x2 ) and mi (x1 , x2 ) of the integral equations (3.31) or (3.52) have the forms b(x1 , x2 ) = B(x1 , x2 )z(x1 , x2 ), x1 , x2 ∈ , mi (x1 , x2 ) = Mi (x1 , x2 )z(x1 , x2 ), x1 , x2 ∈ ,

(3.62) (3.63)

where B(x1 , x2 ) and Mi (x1 , x2 ) are polynomials of the same power m as the external field and the function z(x1 , x2 ) is defined in Eq. (3.61).

60

Heterogeneous Media

In particular, if the field Ji0 is constant, the scalar B = B0 in Eq. (3.62) is also constant and defined from Eq. (3.31) in the form B0 = (λ0 + T0 )−1 ni Ji0 , 1 λ0 = λ(0) = ni Cij nj , 2h

(3.64)

 T0 =

T (x)z(x)d.

(3.65)



Here, the integral T0 should be understood in the sense of the regularization (3.40),  T0 = T (x) [z(x) − 1] dx, x = (x1 , x2 ), (3.66) where integration is performed over the entire plane (x1 , x2 ) and the function z(x) in Eq. (3.61) is continued by zero outside . For an isotropic host medium (Cij0 = c0 δij ), the function T (x1 , x2 ) in Eq. (3.41) takes the form T (x1 , x2 ) = −

4π(x12

c0 , + x22 )3/2

(3.67)

and the scalar T0 is calculated as follows:  ∞ z(x1 , x2 ) − 1 c0 dx1 dx2 = T0 = − 2 2 3/2 4π −∞ (x1 + x2 )   2π  ∞   2π 1 z˜ (r) − 1 c0 dr dϕ dϕ c0 = , dr − a a = − a1 a2 1 2 2 2 3 3 4π 8 r r t (ϕ) t (ϕ) 0 1 0 0 (3.68) t 2 (ϕ) = a12 cos2 (ϕ) + a22 sin2 (ϕ).

(3.69)

Here, the linear transformation of the x-space ¯ (y1 = x1 /a1 , y2 =√x2 /a2 ) is performed and (r, ϕ) are the polar coordinates in the plane (y1 , y2 ), z˜ (r) = 1 − r 2 . After calculation of the last integral, we obtain

c0 E(κ), T0 = 2a2

a2 κ =1− a1

2 ,

a1 ≥ a2 ,

(3.70)

where E(κ) is the complete elliptic integral of the second kind. The fields Ei (x) and Ji (x) in the medium with a thin ellipsoidal inclusion of a small conductivity subjected to a constant external field Ji0 are presented in the forms  Kij (x − x  )c0 B0 z(x  )nj nk Jk0 d , (3.71) Ei (x) = Ei0 +   Sij (x − x  )B0 z(x  )nj nk Jk0 d , (3.72) Ji (x) = Ji0 + B0 =



−1 a2 a2 c + E(κ) . 2 c0 h c0 

(3.73)

Volume and surface integral equations for physical fields in heterogeneous media

61

In the case of a thin ellipsoidal inclusion of a large conductivity and a constant external field Ei0 , the solution of Eq. (3.52) takes the form mi (x) = Mi0 z(x), x ∈ ,

(3.74)

where Mi0 is a constant vector in the middle surface of the inclusion and the function z(x) is defined in Eq. (3.61). After substituting this expression in Eq. (3.52), we obtain θij = δij − ni nj , Mi0 = dij0 θj k Ek0 ,  −1 1 dij0 = μ0ij + Uij0 , μ0ij = θik Bkl θlj , 2h

(3.75) Bij = Cij−1 .

(3.76)

0 is the following Here, ni is the normal to the middle surface of the inclusion and Uαβ integral:

 Uij0 =

∞

−∞

θik Kkl (x1 , x2 )θlj [z(x1 , x2 ) − 1] dx1 dx2 ,

(3.77)

where the function z(x1 , x2 ) in Eq. (3.61) is continued by zero outside . For isotropic materials of the medium and the inclusion (Bij = c−1 δij , Bij0 = c0−1 δij ), we obtain μ0ij =

1 θij , 2hc

θik Kkl (x)θ ¯ lj =



x¯i x¯j − 3 θ , ij |x| 4πc0 |x| ¯3 ¯2 1

x¯ = (x1 , x2 ), (3.78)

and Uij0 is the following integral:

 ∞ x¯i x¯j 1 1 − 3 θ ¯ − 1] dx1 dx2 = [z(x) ij 4πc0 −∞ |x| |x| ¯3 ¯2

 mi (ϕ)mj (ϕ) a1 a2 2π 1 =− θij − 3 dϕ, 8c0 0 t 3 (ϕ) t 2 (ϕ) Uij0 =

m1 (ϕ) = a1 cos(ϕ),

m2 (ϕ) = a1 sin(ϕ),

(3.79)

t 2 (ϕ) = a12 cos2 (ϕ) + a22 sin2 (ϕ). (3.80)

Calculation of the last integral yields the equations for the components of the tensor Uij0 , 0 = U11 0 = U22

a2 K(κ) − E(κ) , κ 2a12 c0

0 = 0, U12

2 1 E(κ) − (1 − κ)K(κ) a2 , , κ =1− 2a2 c0 κ a1

(3.81) a1 ≥ a2 .

(3.82)

62

Heterogeneous Media

Here, K(κ) and E(k) are the complete elliptic integral of the first and second kind, i.e.,  K(κ) = 0

2π



dϕ 1 − k sin(ϕ)2

 ,

2π

E(κ) =



1 − k sin(ϕ)2 dϕ.

(3.83)

0

The fields Ei (x) and Ji (x) in the medium with a thin ellipsoidal inclusion of large conductivity take the forms  0 Ei (x) = Ei − Kij (x − x  )0j k z(x  )Ek0 d , (3.84)   Sij (x − x  )c0−1 0ik z(x  )Ek0 d , (3.85) Ji (x) = Ji0 − 0ij

 0 = θik dkl θlj ,

(3.86)

where the tensor dij0 is defined in Eq. (3.76).

3.3 Volume integral equations of static elasticity for heterogeneous media We consider an infinite homogeneous elastic medium with stiffness tensor Cij0 kl containing a heterogeneity in a finite region V . The stiffness tensor Cij kl (x) of the medium with the heterogeneity is presented in the form Cij kl (x) = Cij0 kl + Cij1 kl (x)V (x) ,

(3.87)

where Cij1 kl (x) is a finite piece-wise analytical function in V and V (x) is the characteristic function of the region V . If the medium is subjected to body forces of density qi (x), the displacement vector ui (x) satisfies the equation   (3.88) ∂j (Cj0ikl + Cij1 kl (x)V (x))∂k ul (x) = −qi (x).   Moving the term ∂j Cij1 kl (x)V (x)∂k ul (x) on the right hand side of this equation, we obtain   Cj0ikl ∂j ∂k ul (x) = −qi (x) − ∂j Cij1 kl (x)V (x)∂k ul (x) . (3.89) After applying the integral operator with the kernel gij (x), i.e., the Green function of 0 ∂ ∂ , we obtain an integro-differential equation for the vector u (x), the operator Ciklj k l i  ui (x) = u0i (x) +

  1      gij (x − x  )∂k Ckj mn (x )V (x )∂m un (x ) dx .

(3.90)

Volume and surface integral equations for physical fields in heterogeneous media

63

Here, u0i (x) is the field that would have existed in the medium without the heterogeneity and the same body forces and conditions at infinity. Using integration by parts, Eq. (3.90) is transformed as follows:  1     ui (x) = u0i (x) + ∂k gij (x − x  )Ckj (3.91) mn (x )∂m un (x )dx . V

After application of the derivative operator ∂i to this equation, we obtain the following integral equation for the strain tensor εij (x) = ∂(i uj ) (x) in the heterogeneous medium:  1 0 K ij kl (x − x  )Cklmn (x  )εmn (x  )dx  = εij (x) , (3.92) εij (x) + V

0 (x) = ∂(i u0j ) (x). Kij kl (x) = −∂i ∂k gj l (x)|(ij )(kl) , εij

(3.93)

In this equation, the kernel Kij kl (x) has singularity of the order of −3. Action of the integral operator with this kernel on finite functions is defined in Eq. (2.203). We consider the problem of thermo-elasticity, and let a homogeneous host medium with the stiffness tensor Cij0 kl and the thermo-expansion coefficient tensor αij0 contain a heterogeneous region V . The stiffness and thermo-expansion coefficient tensors of the medium are presented in the forms Cij kl (x) = Cij0 kl + Cij1 kl (x)V (x) ,

αij (x) = αij0 + αij1 (x)V (x),

(3.94)

where Cij1 kl (x) and αij1 (x) are piece-wise analytical functions. The total strain tensor

e (x) = C −1 (x)σ (x) and temperature ε T (x) = in the medium is the sum of elastic εij kl ij kl ij αij (x)T (x) strains e (x) + αij (x)T (x). εij (x) = εij

(3.95)

For the temperature field T (x) and body forces of density qi (x) applied to the medium, the stress and elastic strain tensors satisfy the system of equations (2.48), (2.49) with mij (x) = αij (x)T (x) e (x) , ∂j σij (x) = −qi (x), σij (x) = Cij kl (x)εkl e Rotij kl εkl (x) = −Rotij kl [αkl (x)T (x)] .

(3.96)

This system can be transformed into an equivalent system for the homogeneous medium with the stiffness tensor Cij0 kl , i.e.,  (x) , ∂j σij (x) = −qi (x), σij (x) = Cij0 kl εkl

 Rotij kl εkl (x) = −Rotij kl mkl (x), (3.97)

 e εij (x) = εij (x) − Bij1 kl (x)σkl (x) , mij (x) = Bij1 kl (x)σkl (x) + αij (x)T (x), (3.98)

Bij1 kl (x) = Bij kl (x) − Bij0 kl ,

Bij kl (x) = Cij−1kl (x),

Bij0 kl = (Cij0 kl )−1 . (3.99)

64

Heterogeneous Media

The system (3.97) describes the strain-stress state of the homogeneous elastic medium subjected to body forces qi (x) and containing the dislocation moments of the density mkl (x). Let the tensor σij0 (x) satisfy the equations ∂j σij0 (x) = −qi (x),

σij (x) = Cij0 kl εkl (x) , Rotij kl εkl (x) = 0

(3.100)

and given conditions at infinity. Using Eq. (2.51) for the stress tensor σij (x) in the medium with sources of internal stresses, the solution of the system (3.97) can be presented in the form  σij (x) = σij0 (x) + Sij kl (x − x  )mkl (x  )dx  , (3.101) V

0 − Cij0 kl δ(x) , Sij kl (x) = Cij0 mn Kmnpq (x)Cpqkl

(3.102)

where the function Kij kl (x) is defined in Eq. (3.93). Substituting in this equation the tensor mij (x) from Eq. (3.98), we obtain the integral equation for the stress field in the heterogeneous medium subjected to the external stress σij0 (x) and temperature T (x) fields  1 σij (x) − S ij kl (x − x  )Bklmn (x  )σmn (x  )dx  = V  0 = σij (x) + S ij kl (x − x  )αkl (x  )T (x  )dx  . (3.103) V

When T = 0, the integral on the right hand side vanishes, and we obtain  1 Sij mn (x − x  )Bmnpq (x  )σpq (x  )dx  = σij0 (x) . σij (x) −

(3.104)

V

Using Hooke’s law and Eq. (3.102) for Sij kl (x), one can prove that this equation is equivalent to Eq. (3.92). The kernels of the integral operators in Eqs. (3.92) and (3.104) have singularities of the order of −3, and the integrals in these equations should be understood in the sense of the regularizations indicated in Eqs. (2.203) and (2.204). Properties of the integral equations (3.92) and (3.104) are similar to the properties of Eqs. (3.9) and (3.13) for the fields Ei (x) and Ji (x) in a heterogeneous electroconductive medium. In particular, the following statements hold. 1. Strain εij (x) and stress σij (x) fields inside the heterogeneous region V are the principal unknowns of the problem. The fields outside V are reconstructed from Eqs. (3.92) and (3.104) if the fields εij (x) and σij (x) inside V are known, where we have  0 1 (x  )εmn (x  )V (x)dx  , (3.105) εij (x) = εij (x) − Kij kl (x − x  )Cklmn  1 (x  )σmn (x  )V (x)dx  . (3.106) σij (x) = σij0 (x) + Sij kl (x − x  )Bklmn

Volume and surface integral equations for physical fields in heterogeneous media

65

2. Unique solutions of Eqs. (3.92) and (3.104) exist if the tensors Cij kl (x) and Bij kl (x) are definitely positive. 3. If the stress and strain fields inside the region V are known, the stress and strain concentrations in the host medium at the boundary  of the region V follow from Eqs. (2.214) and (2.215) for the jumps of the potentials in Eqs. (3.105) and (3.106), i.e.,   + − 1 εij (x) = Eij1 kl + Kij∗ mn (n)Cmnkl (x) εkl (x), x ∈ , (3.107)   1 (x) σkl− (x), x ∈ . (3.108) σij+ (x) = Eij1 kl − Sij∗ mn (n)Bmnkl + (x) and σij+ (x) are the limit Here, ni = ni (x) is the external normal to the surface , εij values of the strain and stress tensors at a point x ∈  from the side of the normal, − εij (x) and σij− (x) are such limits from the opposite side, and the tensors Kij∗ mn (n) and ∗ Sij mn (n) are defined in Eqs. (2.206) and (2.207). 4. For an ellipsoidal region V and constant tensor Cij kl inside V , the property of polynomial conservativity for the solution of Eqs. (3.92) and (3.104) holds. 0 (x) (σ 0 (x)) is a polynomial of power m inside V , the strain If the external field εij ij field ε(x) (σij (x)) inside V is also a polynomial of the same power m. 0 and σ 0 , the strain and stress tensors inside In the case of constant external fields εij ij V are also constant and have the forms [10], [5] 0 εij = εij kl εkl ,

1 εij kl = (Eij1 kl + Aij mn (a)Cmnkl )−1 ,

(3.109)

σij = σij kl σkl0 ,

σij kl

1 − Dij mn (a)Bmnkl )−1 ,

(3.110)

= (Eij1 kl

where the tensors A(a) and D(a) are presented in the forms of integrals over a unit sphere S1 in the k-space 1 Aij kl (a) = 4π

 |k|=1

∗

Kij kl (a

−1

1 k)dS 1 , Dij kl (a) = 4π

 |k|=1|

Sij kl ∗ (a −1 k)dS 1 . (3.111)

Here, aij is the tensor of linear transformation of the x-space (yi = aij−1 xj ) that converts the ellipsoid in the sphere of a unit radius. Particular forms of the tensors Aij kl (a) and Dij kl (a) for ellipsoidal inclusions in anisotropic host media are presented in [10], [5].

3.4 Surface integral equations for thin inclusions in homogeneous elastic media Let one of the characteristic sizes of a heterogeneous region V be much smaller than two other sizes. The middle surface  of V is smooth and bounded by a contour .

66

Heterogeneous Media

We denote by h(x) the transverse size of V along the normal ni (x) to  and present h(x) in the form h(x) = δ1 l(x) , δ1 1.

(4.3)

The function f (x) and its quasiinterpolant f(h,H ) (x) in the interval (−1.5 < x < 1.5) are presented in Figs. 4.1–4.3 for H = 0.5, 1, 2 and h = 0.1, 0.01. Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00011-1 Copyright © 2021 Elsevier Ltd. All rights reserved.

104

Heterogeneous Media

Figure 4.1 The Gaussian quasiinterpolant of a unit impulse for H = 0.5 and the node grid steps h = 0.1 and h = 0.01.

Figure 4.2 The same as in Fig. 4.1 for H = 1.

Figure 4.3 The same as in Fig. 4.1 for H = 2.

Some important features of the approximation (4.2) can be noted from these figures. 1. For H = 0.5, the error of the approximation |f (x) − f(h,H ) (x)| is substantial and can be visually observed. The error does not vanish when increasing the number of nodes (decreasing the node grid step h). If the step h is fixed and the parameter H increases, the error decreases everywhere except the vicinities of the points of discontinuity of the function f (x). 2. More careful analysis presented in [1] reveals the principal feature of the approximation (4.2): Increasing the number of nodes in the interval of approximation does not reduce completely the error. The so-called saturation error depends on the parameter H and cannot be eliminated by decreasing h only.

Numerical calculation of volume and surface potentials

105

3. For piece-wise analytical functions, the error of approximation by the Gaussian functions can be made arbitrarily small by appropriate choice of the parameters h and H . For engineering calculations, these parameters should be taken to guarantee the error does not exceed 1 ÷ 2%. 4. For approximation of fast changing functions by the Gaussian quasiinterpolants, the number of nodes (approximating functions) should be large, and calculation of the 2D and 3D sums in Eq. (4.1) at various points x requires time consuming computations. Following [1] we consider assessment of the error of the approximation (4.2) in the 1D case for analytical functions in the interval (−∞ < x < ∞). First, we introduce the function   ∞  (x − m)2 1 exp − , H > 0. (4.4) θ (x, H ) = √ H πH m=−∞ It follows from Eq. (4.1) that θ (x, H ) is the quasiinterpolant of the unit function f (x) = 1. Because θ (x, H ) is periodic with the period T = 1, this function can be expanded in the Fourier series θ (x, H ) =

∞ 

cn e2πinx

(4.5)

n=−∞

with the coefficients cn defined by the equation  1 θ (x, H )e−2πinx dx = cn = 0



 (x − m)2 −2πinx =√ exp − dx = e H πH m=−∞ 0  2 ∞  m+1  x 1 exp − e−2πinx dx = =√ H πH m=−∞ m  2  ∞ x 1 exp − e−2πinx dx. =√ H πH −∞ 1

∞  

1

(4.6)

The last integral is calculated explicitly, and for cn we obtain the equation cn = exp(−π 2 H n2 ).

(4.7)

As a result, the Fourier series (4.5) for θ (x, H ) takes the form θ (x, H ) = 1 + 2

∞ 

exp(−π 2 H n2 ) cos(2πnx).

(4.8)

n=1

It is seen from this equation that the difference θ (x, H ) − 1 can be made arbitrarily small if the parameter H is sufficiently large. Thus, the quasiinterpolant θ (x, H ) is a good approximation of the unit function f (x) = 1 in the interval −∞ < x < ∞.

106

Heterogeneous Media

The first derivative of the function θ (x, H ) follows from Eqs. (4.4) and (4.8) in the form   ∞  2 (x − m)2 θ (x, H ) = √ (m − x) exp − = H πH 3/2 m=−∞ 

= −2π

∞ 

n exp(−π 2 H n2 ) sin(2πnx).

(4.9)

n=1

Then, we consider the quasiinterpolant (4.1) for an arbitrary analytical function f (x) 

 (x − hm)2 f (hm) exp − . f(h,H ) (x) = √ h2 H πH m=−∞ 1

∞ 

(4.10)

Taylor expansion of the function f (x) near the point x = hm has the form f (hm) = f (x) + f  (xm )(hm − x)

(4.11)

with xm between x and hm. Substituting this equation into Eq. (4.10), we obtain f(h,H ) (x) = f (x)θ

x



x , H + hf  (xm )θ  ,H . h h

(4.12)

Here, Eqs. (4.8) and (4.9) are used. As a result, we obtain for the quasiinterpolant f(h,H ) (x) the equation 

 2πnx exp(−π H n ) cos − f(h,H ) (x) = f (x) + 2f (x) h n=1   ∞  2πnx  2 2 − hf (xm )2π n exp(−π H n ) sin + O(h2 ). h ∞ 

2

2

(4.13)

n=1

  Assessment of f(h,H ) (x) − f (x) follows from this equation in the form   f(h,H ) (x) − f (x) ≤ ε1 max |f (x)| + hε2 max |f  (x)| + O(h2 ), ε1 = 2

∞  n=1

exp(−π 2 H n2 ), ε2 = −2π

∞ 

n exp(−π 2 H n2 ).

(4.14) (4.15)

n=1

For large H , the parameters ε1 and ε2 can be made arbitrarily small. But for a fixed H and decreasing h, the error of the approximation cannot be made smaller than the term ε1 max |f (x)| in Eq. (4.14). Therefore, this term defines the saturation error of the approximation.

Numerical calculation of volume and surface potentials

107

4.1.1 The “approximate approximation” concept of V. Maz’ya It is shown in [1] that the Gaussian function is an example of a class of functions that can be used for a specific approximation (“approximate approximation”) of actual physical fields. Such functions should be smooth and rapidly decaying, and satisfy the conditions   ϕ(x)dx = a, ∂i1 ∂i2 ...∂iN ϕ(x)dx = 0, ik = 1, 2, 3, k = 1, 2, ..., N (4.16) for 0 < a < ∞ and some N > 0. Let V be a finite region in d-dimensional space, and let x (n) (n = 1, 2, ..., Mn) be a homogeneous set of nodes in V . For approximation of fields in this region, the functions ϕ(x) should perform an approximate partition of unity. This means that for x ∈ V , the following assessment holds:   Mn     (n)  ϕ(x − x ) < ε, (4.17) 1 −   n=1

where ε is a small positive number and Mn is the total number of the nodes in V . In [1], several examples of functions that can be used for construction of the quasiinterpolants in the 3D space are presented: 8 2 sec h(|x|), ϕ(x) = 2 , 4 π π (1 + |x|2 )4 15 (1 − |x|2 ), |x| ≤ 1, ϕ(x) = 0, |x| > 1. ϕ(x) = 8π ϕ(x) =

(4.18) (4.19)

The quasiinterpolant of an arbitrary function f (x) can be taken in a form similar to Eq. (4.1):

Mn (n) |  |x − x f(h,H ) (x) = f (x (n) )ϕ , (4.20) h2 H n=1

where ϕ(x) is one of the functions in Eqs. (4.18) and (4.19), x (n) are the nodes of the approximation, h is an approximate distance between the neighbor nodes, and H = O(1). The quasiinterpolants (4.20) with the functions ϕ(x) in Eqs. (4.18) and (4.19) have a common property: The series f(h,H ) (x) does not converge to the function f (x) when the number of approximating nodes (approximating functions) increases. But the error of the approximation can be made small and acceptable for engineering calculations. In the book [1], various aspects of the approximation by quasiinterpolants are discussed. In particular: • Assessments of the error of the approximation in various functional spaces are presented. • Dependencies of the coefficients of the quasiinterpolants on the shape of the node grid cells are indicated.

108

Heterogeneous Media

• Improvement of accuracy of quasiinterpolants by using more sophisticated approximating functions is proposed. • Actions of many integral operators of mathematical physics on the quasiinterpolants are obtained in explicit analytical forms. • Estimations of the errors in calculation of the potentials by changing their densities with the quasiinterpolants are presented. In this chapter, the Gaussian quasiinterpolants of the densities are used for numerical calculation of the potentials introduced in Chapter 2. Advantages of this method are evident when the potential with the Gaussian density is calculated explicitly. In this case, calculation of the potentials with arbitrary densities at various points of the space is reduced to summation of known analytical functions shifted at the approximating nodes.

4.2 Fast Fourier transform algorithms for calculation of Gaussian quasiinterpolants and related sums One of the obstacles for using quasiinterpolants in practical calculations is a large number of nodes (approximating functions) that should be taken to guarantee sufficient accuracy of the approximation. Let the function f (x) be defined in a finite region V of a 3D space. For a grid of approximating nodes x (n) in the region V , the quasiinterpolant f(h,H ) (x) is the following sum: f(h,H ) (x) =

Mn 

f (x (n) )ϕ(x − x (n) ), ϕ(x) =

n=1

  1 |x|2 exp − . (4.21) (πH )3/2 h2 H

Here, x (n) are the nodes of a regular grid with the step h and Mn is the total number of nodes in V . This number depends on the size of the region V and the behavior of the function f (x) in V . For fast changing functions, Mn can be of the order of 106 and more to ensure acceptable accuracy. As a result, calculation of the sum in Eq. (4.21) at various points x becomes a time consuming computational operation. Sums similar to (4.21) but with other functions ϕ(x) appear by using quasiinterpolants for numerical calculations of volume potentials and solution of volume integral equations. Let the function A(x) be presented in the form similar to the series (4.21) A(x) =

Mn 

(x − x (n) )B(x (n) ),

(4.22)

n=1

where x (n) are the nodes of a regular grid in a region V and (x) and B(x) are given functions. We assume that V is a rectangular region (cuboid), and the function A(x) is to be calculated at the nodes x (n) . Let the region V be defined in Cartesian coordinates (x1 , x2 , x3 ) by the equations V = (l1 ≤ x1 ≤ l1 + L1 , l2 ≤ x2 ≤ l2 + L2 , l3 ≤ x3 ≤ l3 + L3 ).

(4.23)

Numerical calculation of volume and surface potentials

109

For a regular node grid, each node x (n) (n = 1, 2, ..., Mn) can be indicated by three positive integers (indices) i, j, k: (j )

x (n) = x (i,j,k) = (x1(i) , x2 , x3(k) ),

(4.24)

and the coordinates of the x (i,j,k) -node are defined by the equations (j )

(i)

(k)

x1 = l1 + h1 (i − 1), x2 = l2 + h2 (j − 1), x3 = l3 + h3 (k − 1), i = 1, 2, .., M1 , j = 1, 2, .., M2 , k = 1, 2, .., M3 .

(4.25)

Here, M1 , M2 , M3 are the numbers of nodes along the x1 -, x2 -, and x3 -axes, h1 =

L1 L2 L3 , h2 = , h3 = . M1 − 1 M2 − 1 M2 − 1

(4.26)

Thus, (l1 , l2 , l3 ) are the coordinates of the node x (1,1,1) , h1 , h2 , h3 are the node grid steps along the x1 -, x2 -, and x3 -axes, and L1 , L2 , L3 are the cuboid sizes along the axes. In the three-index numeration of the nodes, the sum in Eq. (4.22) is presented as the triple sum M3 M2  M1 

 (p) (q) (j ) (p,q,r,i,j,k) B(x1(i) , x2 , x3(k) ), A x1 , x2 , x3(r) =

(4.27)

i=1 j =1 k=1



(p) (j ) (r) (i) (q) (k) (p,q,r,i,j,k) =  x1 − x1 , x2 − x2 , x3 − x3 .

(4.28)

Here, the object (p,q,r,i,j,k) has a Toeplitz structure: It depends on the differences of the indices (p,q,r,i,j,k) =  (h1 (p − i), h2 (q − j ), h3 (r − k)) .

(4.29)

For calculation of sums similar to (4.27), the fast Fourier transform (FFT) algorithms can be used. In the next section, we consider these algorithms in detail for calculation of 1D, 2D, and 3D sums.

4.2.1 Calculation of 1D sums by the FFT algorithm Let the 1D sum in Eq. (4.22) be calculated at the nodes x (m) = l + h(m − 1) as A(x (m) ) =

Mn 

(h(m − n))B(x (n) ), m = 1, 2, ..., Mn.

(4.30)

n=1

The right  hand side of this equation can be interpreted as the product of the matrix T = tij with the elements tij = (h(i − j ))

(4.31)

110

Heterogeneous Media

and the vector b = [B(x (1) ), B(x (2) ), ..., B(x (Mn) )]T .

(4.32)

Here, [...]T is the transposition operation. Because tij depends on the difference of the indices, T is a Toeplitz matrix. Specific features of a Toeplitz matrix of n × n dimensions can be seen from its detailed form ⎤ ⎡ t−1 ... t2−n t1−n t0 ⎢ t1 t0 t−1 ... t2−n ⎥ ⎥ ⎢ ⎢ t1 t0 : ⎥ T=⎢ : (4.33) ⎥. ⎣ tn−2 t−1 ⎦ tn−1 tn−2 ... t1 t0 This matrix has 2n independent elements only, and each diagonal of T consists of the same elements. The special type of Toeplitz matrices with the additional property t−k = tn−k , 1 ≤ k ≤ n − 1

(4.34)

is called a circulant matrix, and it has n independent elements only. The product a = Tb of a circulant matrix T and a vector b = [bi ] can be performed by using the discrete Fourier transform, and the corresponding algorithm consists of the following four steps [2]: 1. 2. 3. 4.

f b = Fn b, f t = Fn t, f a = f t ∗ f b, a = Fn−1 (f a) .

(4.35)

In these equations, Fn is the discrete Fourier transform operator of the dimension n,   n 1  2πi bj exp (j − 1)(k − 1) , k = 1, 2, ..., n. (f b)k = (Fn b)k = √ n n j =1

(4.36) The vector t in step 2 of the algorithm (4.35) is composed of the independent elements of the matrix T, t = [t0 , t1 , ..., tn−1 ]T .

(4.37)

The operation (∗) in step 3 is the element per element product of two vectors: If b, c are vectors of the dimensions n, the vector d = b ∗ c has the dimension n and its elements di are di = bi ci , i = 1, 2, ..., n.

(4.38)

Numerical calculation of volume and surface potentials

111

In step 4, Fn−1 is the inverse discrete Fourier transform operator, and for an n-dimensional vector v = [vi ], we have



Fn−1 v

  n 1  2πi =√ vj exp − (j − 1)(k − 1) , k = 1, 2, ..., n. k n n

(4.39)

j =1

For calculation of the discrete direct and inverse Fourier transforms, the well-known FFT algorithm can be used [4]. If T is a noncirculant Toeplitz matrix, for application of the FFT algorithm to calculation of the matrix-vector products, T should be extended to a circulant matrix  T defined by the equation ⎤ ⎡ t2 t1 0 tn−1 ... ⎢ t1−n   0 t−1 ... t2 ⎥ ⎥ ⎢ T, T  ⎢  t1 0 : ⎥ (4.40) , T =⎢ : T= ⎥. T , T ⎣ t−2 tn−1 ⎦ t−1 t−2 ... t1−n 0 As a result, the circulant matrix  T has the dimensions 2n × 2n. If we introduce an extended vector  b of the dimension 2n associated with the vector b in Eq. (4.32),  b = [b1 , b2 , ..., bn , 0, 0, .., 0]T ,

(4.41)

the matrix-vector product  T b is      T, T b Tb  a= T b= = , T , T 0 T b

(4.42)

and the vector  a can be calculated by the FFT algorithm. The vector  t that should be used in step 2 of the FFT algorithm for calculation of the product  T b is as follows:  t = [t0 , t1 , ..., tn , 0, t1−n , t2−n , ..., t−2 , t−1 ]T .

(4.43)

The elements of the vector a = Tb coincide with the first n elements of the vector  a of dimension 2n in Eq. (4.42). As an example, we consider the Gaussian quasiinterpolant f(h,H ) (x) of the function f (x) = 1 + 0.5 sin(2πx), |x| ≤ 1, f (x) = 0, |x| > 1,

(4.44)

in the interval −1.4 ≤ x ≤ 1.4. The computational program for calculation of the sum

Mn (m) − x (n) |2  1 |x f (x (n) ) exp − , m = 1, 2, ..., Mn, f(h,H ) (x (m) ) = √ h2 H πH n=1

(4.45) x (n) = −1.4 + h(n − 1), h =

2.8 Mn − 1

(4.46)

112

Heterogeneous Media

Figure 4.4 The function f (x) in Eq. (4.44) (solid line) and its quasiinterpolant f(h,H ) (x) (dashed line) for (A) H = 2, h = 0.1, Mn = 29 and (B) H = 2, h = 0.01, Mn = 281.

is presented in Appendix 4.A.1. In this program, the FFT algorithm is adopted for calculation of the sums in (4.45). The calculation results are presented in Fig. 4.4. It is seen from these figures that for the parameters H = 2, h = 0.01, the quasiinterpolant coincides practically with the original function.

4.2.2 Calculation of 2D and 3D quasiinterpolants and related sums by FFT algorithms In the 2D case, we consider a rectangular region V with the sides L1 , L2 covered by a regular node grid. The two-index numeration x (i,j ) of the nodes in V is introduced by the equations

(j ) x (i,j ) = x1(i) , x2 , i = 1, 2, ..., M1 , j = 1, 2, ..., M2 , (4.47) (i)

(j )

x1 = l1 + h1 (i − 1), x2 = l2 + h2 (j − 1).

(4.48)

Here, (l1 , l2 ) are the coordinates of the x (1,1) -node and M1 , M2 and h1 , h2 are the numbers of nodes and grid steps along the x1 - and x2 -axes, h1 =

L1 L2 , h2 = . M1 − 1 M2 − 1

(4.49)

The one-index numeration x (M) of the nodes is introduced by the equation M[i, j ] = i + M1 (j − 1), 1 ≤ i ≤ M1 , 1 ≤ j ≤ M2 ,

(4.50)

where Mn = M1 M2 is the total number of nodes. The indices i, j of the two-index numeration are expressed in terms of the index M of the one-index numeration by the equations i(M) = M if M ≤ M1 , i(M) = Mod[M − 1, M1 (j (M) − 1)] + 1 if M > M1 ,

(4.51) (4.52)

Numerical calculation of volume and surface potentials

 j (M) = Floor

113

 M −1 + 1, M1

(4.53)

where Mod[a, b] is the remainder on division of real numbers a by b and Floor[z] is the closest integer to the number z that is less or equal to z. Thus, according

to (j (M)) Eqs. (4.51)–(4.53), the Cartesian coordinates of the node x (M) are x1(i(M)) , x2 , where M is the node number in the one-index numeration. In the 2D case, the values of the function A(x) in Eq. (4.22) at the nodes are the double sums M2 M1 

 (p) (q) (i) (j ) A x1 , x2 = (p,q,i,j ) B(x1 , x2 ), i=1 j =1

p = 1, 2, ..., M1 , q = 1, 2, ..., M2 ,

(p) (j ) (i) (q) (p,q,i,j ) =  x1 − x1 , x2 − x2 .

(4.54) (4.55)

Because the four-index object (p,q,i,j ) =  (h1 (p − i), h2 (q − j ))

(4.56)

depends on the differences of the indices p − i and q − j , it has a Toeplitz structure, and the FFT algorithm can be used for fast calculation of this sum. For this purpose, we present this sum in the form A[p, q] =

M2 M1  

T [p, q, i, j ]B[i, j ], p = 1, 2, ..., M1 , q = 1, 2, ..., M2 ,

i=1 j =1



(p) x1

(i) (q) − x1 , x2

(j ) − x2



T [p, q, i, j ] =  ,

(p) (q) (i) (j ) A[p, q] = A x1 , x2 , B[i, j ] = B(x1 , x2 )

(4.57) (4.58) (4.59)

 j ] of the dimensions 2M1 × and introduce the extended two-index object (matrix) B[i, 2M2 by the equations  j ] = B[i, j ] if i ≤ M1 and j ≤ M2 , B[i,  j ] = 0 if i > M1 or j > M2 . B[i,

(4.60)

We introduce also the objects T0 [P , Q] and T[p, q], which are expressed in terms of the object T [p, q, i, j ] as follows: T0 [P , Q] = T [i(P ), j (P ), i(Q), j (Q)], P , Q = 1, 2, ..., Mn, T[p, q] = T0 [M[p, q], 1] if p ≤ M1 and m ≤ M2 , T[1 + M1 , q] = 0, T[p, 1 + M2 ] = 0,

114

Heterogeneous Media

T[p, q] = T0 [M[1, p], M[N1 − q + 2, 1]] if p > M1 + 1 and q ≤ M2 , (4.61) T[p, q] = T0 [1, M[N1 − p + 2, N2 − q + 2]] if p > M1 + 1 and m > M2 + 1, T[p, q] = T0 [M[p, 1], M[1, N2 − q + 2]] if p ≤ M1 and q > M2 + 1. Here, N1 = 2M1 , N2 = 2M2 , M[p, q] is defined in Eq. (4.50), and i(P ), j (P ) are defined in Eqs. (4.51)–(4.53).  j ] and T[p, q] with respect to indices i, j The discrete Fourier transforms of B[i,  j ] and F T[p, q]. Then, the elements of the object F A  and p, q are denoted as F B[i, of the dimensions N1 × N2 are calculated as the element per element products of the  objects F T and F B,  j ] = F T[i, j ] ∗ F B[i,  j ], i = 1, 2, ..., N1 , j = 1, 2, ..., N2 . F A[i,

(4.62)

Application of the inverse discrete Fourier transform with respect to the indices i, j  j ]: yields the object A[i,

 [i, j ],  j ] = F −1 F A (4.63) A[i, and the sums A[i, j ] in Eq. (4.57) coincide with the following elements of the object  j ]: A[i,  j ], i = 1, 2, ..., M1 , j = 1, 2, ..., M2 . A[i, j ] = A[i,

(4.64)

Let us consider this algorithm for calculation of the quasiinterpolant of the function f (x1 , x2 ) defined by the equation  x2 x2 f (x1 , x2 ) = 1 − x12 − 2 , x12 + 2 ≤ 1, 4 4 f (x1 , x2 ) = 0, x12 +

x22 >1 4

(4.65)

in the region V : (−1.5 < x1 < 1.5, −2.5 < x2 < 2.5). The computational program for Wolfram Mathematica software is presented in Appendix 4.A.2. The results of the calculations are shown in Fig. 4.5. The quasiinterpolants are constructed for the parameters H = 2 and h = 0.1 (Mn = 1421) (Fig. 4.5A) or h = 0.01 (Mn = 135161) (Fig. 4.5B). It is seen from these figures that for h = 0.01, the quasiinterpolant coincides practically with the original function. In the 3D case, the three-index numeration of the nodes is defined by Eqs. (4.25) and (4.26). The one-index numeration is introduced by the equation M[i, j, k] = i + M1 (j − 1) + M1 M2 (k − 1), (4.66) 1 ≤ i ≤ M1 , 1 ≤ j ≤ M2 , 1 ≤ k ≤ M3 , 1 ≤ M ≤ Mn, Mn = M1 M2 M3 . Here, M1 , M2 , M3 are the numbers of nodes along the x1 -, x2 -, and x3 -axes and Mn is the total number of nodes. The indices i, j, k that correspond to the Mth node in the

Numerical calculation of volume and surface potentials

115



Figure 4.5 The function f (x1 , x2 ) = 1 − x12 − x22 /4 (solid line) and its quasiinterpolant f(h,H ) (x1 , x2 ) (dashed line) for the parameters (A) H = 2, h = 0.1, Mn = 1421 and (B) H = 2, h = 0.01, Mn = 135161.

one-index numeration are defined by the equations   M −1 k(M) = Floor + 1, (4.67) M1 M2   M − M1 M2 (k(M) − 1) − 1 + 1, (4.68) j (M) = Floor M1 i(M) = Mod[r − 1, M1 (j (M) − 1) + M1 M2 (k(M) − 1)] + 1 if M > M1 , (4.69) i(M) = M if M ≤ M1 . Thus, the Cartesian coordinates of the node x (M) are

(i(M)) (j (M)) (k(M)) x (M) = x1 , x2 , x3 .

(4.70)

In the 3D case, the sum in Eq. (4.27) is presented as the triple sum A[p, q, r] =

N3 M2  M1  

T [p, q, r, i, j, k]B[i, j, k],

(4.71)

i=1 j =1 k=1

p = 1, 2, ..., M1 , q = 1, 2, ..., M2 , r = 1, 2, ..., M3 ,

(p) (q) (r) (i) (j ) (k) A[p, q, r] = A x1 , x2 , x3 , B[i, j, k] = B(x1 , x2 , x3 ),

(p) (j ) (r) (i) (q) (k) T [p, q, r, i, j, k] =  x1 − x1 , x2 − x2 , x3 − x3 .

(4.72) (4.73)

For calculation of this sum by the FFT algorithm, we introduce the extended objects  j, k] associated with B[i, j, k]: B[i,  j, k] = B[i, j, k], i ≤ M1 and j ≤ M2 and k ≤ M3 , B[i,  j, k] = 0, j > M1 or j > M2 or k > M3 , B[i,

(4.74)

116

Heterogeneous Media

and the object T[i, j, k] of the dimensions N1 × N2 × N3 , where Ni = 2Mi (i = 1, 2, 3), T[i, j, k] = T [i, j, k, 1, 1, 1], i ≤ M1 , j ≤ M2 , k ≤ M3 , T[1 + M1 , j, k] = 0, T[i, 1 + M2 , j, k] = 0, T[i, j, 1 + M3 ] = 0, T[i, j, k] = T [1, j, k, N1 − i + 2, 1, 1], i > M1 + 1 and j ≤ M2 and k ≤ M3 , T[i, j, k] = T [i, 1, k, 1, N2 − j + 2, 1], i ≤ M1 and j > M2 + 1 and k ≤ M3 , T[i, j, k] = T [i, j, 1, 1, 1, N3 − k + 2], i ≤ M1 and j ≤ M2 and k > M3 + 1, T[i, j, k] = T [1, 1, k, N1 − i + 2, N2 − j + 2, 1], i > M1 + 1 and j > M2 + 1 and k ≤ M3 , T[i, j, k] = T [1, j, 1, N1 − i + 2, 1, N3 − k + 2],

(4.75)

i > M1 + 1 and j ≤ M2 and k > M3 + 1, T[i, j, k] = T [i, 1, 1, 1, N2 − j + 2, N3 − k + 2], i ≤ M1 and j > M2 + 1 and k > M2 + 1, T[i, j, k] = T [1, 1, 1, N1 − i + 2, N2 − j + 2, N3 − k + 2], i > M1 + 1 and j > M2 + 1 and k > M3 + 1.  j, k] and T[i, j, k] with respect to The discrete Fourier transforms of the objects B[i,  j, k] and F T[i, j, k]. Then, the elements of the the indices i, j, k are denoted as F B[i,  of the dimensions N1 × N2 × N3 are calculated as the element per element object F A  product of the objects F T and F B,  j, k] = F T[i, j, k] ∗ F B[i,  j, k], F A[i, i = 1, 2, ..., N1 , j = 1, 2, ..., N2 , k = 1, 2, ..., N3 .

(4.76)

Application of the inverse discrete Fourier transform with respect to the indices i, j, k  j, k]: yields the object A[i,

 [i, j, k],  j, k] = F −1 F A (4.77) A[i, and the sums A[p, q, r] in Eq. (4.71) coincide with the following elements of  q, r]: A[p,  q, r], 1 ≤ p ≤ M1 , 1 ≤ q ≤ M2 , 1 ≤ r ≤ M3 . A[p, q, r] = A[p,

(4.78)

As an example, we consider calculation of the quasiinterpolant of the characteristic function V (x1 , x2 , x3 ) of a ball of radius a = 1, V (x) = 1, |x| ≤ 1, V (x) = 0, |x| > 1,

(4.79)

Numerical calculation of volume and surface potentials

117

in the cube W = (|xi | ≤ 1.4, i = 1, 2, 3).

(4.80)

The program for calculation of the quasiinterpolant V(h,H ) (x) by Wolfram Mathematica software is presented in Appendix 4.A.3. The results of the calculations are shown in Fig. 4.6 for H = 1 and h = 0.1 (Mn = 24389), h = 0.05 (Mn = 185193), or h = 0.02 (Mn = 2803221).

Figure 4.6 The function V (|x|) = 1, |x| ≤ 1, V (|x|) = 0, |x| > 1 and its 3D quasiinterpolant for the

parameters H = 1 and (A) h = 0.1, Mn = 24389, (B) h = 0.05, Mn = 185193, and (C) h = 0.02, Mn = 2803221.

It is seen from these figures that for h = 0.02, the quasiinterpolant coincides practically with the original function.

4.3

Numerical calculation of volume potentials of electrostatics

We consider the volume potential of electrostatics in Eq. (2.173) with the charge density q(x) distributed in a region V (henceforth, the dielectric permittivity of the medium is taken as c = 1)  u(x) =

g(x − x  )q(x  )dx  , g(x) =

V

1 . 4π |x|

(4.81)

For calculation of this potential, we introduce a cuboid W containing the region V and cover W by a regular node grid with step h (Fig. 4.7). Then, we approximate the charge density q(x) by the Gaussian quasiinterpolant q(h,H ) (x), i.e., q(h,H ) (x) =

Mn  n=1

q (n) ϕ(x − x (n) ), ϕ(x) =

  1 |x|2 exp − . (πH )3/2 h2 H

/ V. Here, x (n) are the nods of the grid, q (n) = q(x (n) ), and q (n) = 0 if x (n) ∈

(4.82)

118

Heterogeneous Media

Figure 4.7 A cuboid W covered by a regular node grid with a heterogeneous region V inside.

After substitution of the approximation (4.82) into Eq. (4.81), we obtain the approximate equation for calculation of the potential u(x) u(x) ≈

Mn 

h2 I 0 (x − x (n) )q (n) , I 0 (x) =

n=1

1 h2



g(x − x  )ϕ(x  )dx  .

(4.83)

Here, the integral I 0 (x) is calculated over the entire 3D space. It is taken into account that each Gaussian function ϕ(x − x (n) ) in Eq. (4.82) is concentrated in a vicinity of the node x (n) , the characteristic size of which is of the order of the parameter h. Because h is much smaller than the sizes of the region V , integration over the volume V in Eq. (4.81) can be changed to integration over the entire 3D space in Eq. (4.83). For calculation of the integral I 0 (x), we introduce the variable y = x − x  and the yi xi unit vectors mi = |y| , ni = |x| , and take into account the equations |x  |2 = |x − y|2 = |x|2 − 2x · y + |y|2 ,

x · y = |x||y|(ni mi ).

Then, the function I 0 (x) is calculated as follows:    1 1 |x  |2 0 exp − 2 I (x) = dx  = |x − x  | 4πh2 (πH )3/2 h H

2      exp − h|x| 2H 1 |y|2 2x · y = exp − exp dy = |y| 4πh2 (πH )3/2 h2 H h2 H

2    ∞ exp − h|x| 2H |y|2 exp − = |y|d|y| × 4πh2 (πH )3/2 0 h2 H    2|x||y|(ni mi ) exp dSm = × h2 H |m|=1

2     H ∞ exp − h|x| 2H |y|2 2|x||y| = exp − sinh d|y| = 2|x|(πH )3/2 0 h2 H h2 H    z |x| 2 h 2 e−t dt. erf √ , erf(z) = √ = 4π |x| π 0 h H

(4.84)

(4.85)

Numerical calculation of volume and surface potentials

119

Here, erf(z) is the error function. Thus, we obtain for I 0 (x) the equation     ξ |x| 1 . , φ0 (ξ ) = erf √ I 0 (x) = φ0 h 4πξ H

(4.86)

For a cuboid W with the region V inside, the values of the sum in Eq. (4.83) at the nodes x (n) can be calculated by the FFT algorithm presented in Section 4.2. Let V be a ball of radius a with the constant charge density q = 1 inside. In this case, the integral (4.81) is calculated explicitly, and the potential u(x) = π 0 (x) takes the form (r = |x|) 1 π 0 (x) = (3a 2 − r 2 ), r ≤ a, 6

u0 (x) =

a3 , r > a. 3r

(4.87)

The computational program for numerical calculation of the potential π 0 (x) in the cube W : (|xi | ≤ 4a, i = 1, 2, 3) with the ball V inside coincides with the program for calculation of the Gaussian quasiinterpolant q(h,H ) (x) in the 3D case (Appendix 4.A.3) if the Gaussian function ϕ(x) is changed to the function φ0 (|x|/ h) in Eq. (4.86). Graphs of the potential (4.87) (solid lines) and the approximation (4.83) (dashed lines) are presented in Fig. 4.8 for a = 1. The parameters of the quasiinterpolant q(h,H ) (x) are H = 1 and h = 0.2 (Mn = 9261), h = 0.1 (Mn = 68921), or h = 0.05 (Mn = 531441).

Figure 4.8 The volume potential of electrostatics π 0 (r) in Eq. (4.87) for the charge density q = 1 inside

the ball of radius a = 1; solid lines are the exact function π 0 (r), dashed lines are the approximations by changing the density q(x) with the Gaussian quasiinterpolant q(h,H ) (x) with the parameters H = 1 and (A) h = 0.2, (B) h = 0.1, and (C) h = 0.05.

It is seen from these figures that for H = 1, h = 0.1, the approximation coincides practically with the potential π 0 (x) in Eq. (4.87). We consider the potential of electrostatics in Eq. (2.177), which is the electric field induced by dipole density Mi (x) in the region V ,  Ei (x) = Kij (x − x  )Mj (x  )dx  , Kij (x) = −∂i ∂j g(x). (4.88) V

For numerical calculation of this potential, we embed the region V in a cuboid W and cover W by a regular node grid. Then, the density Mi (x) in Eq. (4.88) is changed to

120

Heterogeneous Media

the Gaussian quasiinterpolant Mi(h,H ) (x), (h,H )

Mi

(x) =

Mn 

Mi (x (n) )ϕ(x − x (n) ) , ϕ(x) =

n=1

  1 |x|2 exp − . (4.89) (πH )3/2 h2 H

As a result, we obtain the following approximate equation for calculation of the potential Ei (x): Ei (x) ≈

Mn 

(n)

(n)

Iij (x − x (n) ) Mj , Mj = Mj (x (n) ),

n=1



Iij (x) = −∂i ∂j

g(x − x  )ϕ(x  )dx  = −h2 ∂i ∂j I 0 (x),

(4.90) (4.91)

where the function I 0 (x) is defined in Eq. (4.86). The explicit equation for Iij (x) takes the form         1 |x| |x| |x| xi xj Iij (x) = 0 + 1 δij − 31 , (4.92) 3c h h h |x|2  2 1 ξ exp − , 0 (ξ ) = 3/2 H (πH )   3H + 2ξ 2 3 ξ − erf 0 (ξ ). (4.93) 1 (ξ ) = √ 4πξ 3 2ξ 2 H For a ball V of radius a with the characteristic function V (x) (V (x) = 1, |x| ≤ a, V (x) = 0, |x| > a) the integral  0 πij (x) = Kij (x − x  )V (x  )dx  (4.94) is calculated explicitly and takes the form 1 a3 πij0 (x) = δij , |x| ≤ a, πij0 (x) = (δij − 3ni nj ), |x| > a. 3 3|x|3

(4.95)

0 (x) = π 0 (x , x , x ) of this poIn Figs. 4.9–4.12, comparisons of the component π11 11 1 2 3 tential (solid lines) and the approximation

0 π11 (x) ≈

Mn 

I11 (x − x (n) )V (x (n) )

(4.96)

n=1

(dashed lines) are presented. The region W : (|xi | ≤ 2a, i = 1, 2, 3) with the ball V of radius a = 1 inside is covered by the cubic node grid with step h. The parameters of the quasiinterpolant V(h,H ) (x) are H = 1 and h = 0.2 (Mn = 9261), h = 0.1 (Mn = 68921), h = 0.05 (Mn = 553441), or h = 0.025 (Mn = 4173281).

Numerical calculation of volume and surface potentials

121

0 (x) in Eq. (4.95) for a ball of radius a = 1 with constant potential Figure 4.9 The volume potential π11

0 (x), the dashed line is the approximation by changing density q = 1; the solid line is the exact function π11 the density to the Gaussian quasiinterpolant q(h,H ) (x) with the parameters H = 1, h = 0.2.

Figure 4.10 The same as in Fig. 4.9 for H = 1, h = 0.1.

Figure 4.11 The same as in Fig. 4.9 for H = 1, h = 0.05.

122

Heterogeneous Media

Figure 4.12 The same as in Fig. 4.9 for H = 1, h = 0.025.

It is seen from these figures that the jump of the potential on the boundary of the region V is satisfactorily described by the approximate equation (4.96) for h = 0.025. The computational program for numerical calculation of the potential (4.94) is presented in Appendix 4.B.

4.4 Far field asymptotics of static potentials and multipole expansions of the potential densities If the charge density q(x) is an integrable function inside a region V , the electric potential u(x) outside V is an infinitely differentiable function of x,  g(x − x  )q(x  )dx  . (4.97) u(x) = V

Expansion of the Green function g(x − x  ) in the Taylor series about a point x0 ∈ V has the form (x ∈ / V) g(x − x  ) =

∞        1 ∂i1 ∂i2 ...∂ik g(x − x0 ) x0 − x  i x0 − x  i ... x0 − x  i . k 1 2 k! k=0

(4.98) Substituting this expansion in Eq. (4.97), we present the potential u(x) outside the region V in the form of the series u(x) =

∞ 

∂i1 ∂i2 ...∂ik g(x − x0 )Qki1 i2 ...ik (x0 ) ,

k=0

Qki1 i2 ...ik (x0 ) =

(−1)k k!

 V

(x  − x0 )i1 (x  − x0 )i2 ...(x  − x0 )ik q(x  )dx  .

(4.99) (4.100)

Numerical calculation of volume and surface potentials

123

The same result can be obtained if the function q(x) in Eq. (4.97) is presented in the form of the following series of derivatives of Dirac’s delta function δ(x − x0 ): q(x) =

∞ 

Qm i1 i2 ...im (x0 )∂i1 ∂i2 ...∂ik δ(x − x0 ) ,

(4.101)

k=0 0 where Qm i1 i2 ...im (x0 ) is in Eq. (4.100). The first term of this expansion is Q (x0 )δ(x − x0 ), and  0 Q (x0 ) = q(x)dx (4.102) V

can be interpreted as the total charge in the region V concentrated at the point x0 . The second term in the series (4.101) Q1i ∂i δ(x − x0 ) is the dipole with the vector density  1 Qi = − (x − x0 )i q(x)dx, (4.103) V

concentrated at a point x0 . Other terms in Eq. (4.101) can be interpreted as multipoles of higher orders concentrated at x0 . The series in Eq. (4.101) is called the expansion of the charge density q(x) in the series about the multipoles concentrated at the point x0 ∈ V . The series (4.101) should be understood in the following sense: Convolution of the right and left parts of (4.101) with an arbitrary analytical function gives the same result. The kth term of the series in (4.99) has the asymptotic |x|−(k+1) when |x| → ∞. Therefore, the larger is the distance of the region V from the point x, the better is convergence of this series. In order to calculate the fields far from the region V , one can take just the first few terms of the multipole expansion (4.101). Keeping three such terms only, we present the field u(x) in the form u(x) = g(x − x0 )Q0 + ∂i g(x − x0 )Q1i + ∂i ∂j g(x − x0 )Q2ij ,

(4.104)

and g(x − x0 )Q0 is the principal term of the field u(x) when |x| → ∞. If the density q(x) is changed to its quasiinterpolant q(h,H ) (x), the integrals in Eq. (4.100) are approximated by the following sums: Q0 = h3

Mn 

q(x (n) ), Q1i = 2h3

n=1

Q2ij = h5 H

Mn  n=1

Mn  (x 0 − x (n) )i q(x (n) ),

(4.105)

n=1 Mn  (n) 3 q(x )δij − 2h (x 0 − x (n) )i (x 0 − x (n) )j q(x (n) ).

(4.106)

n=1

Thus, for calculation of the potentials in a distance from the region V , the multipole expansions can be used. Eqs. (4.83) and (4.90) are to be used for numerical calculation of the potentials in the nearest vicinity of the region V only.

124

Heterogeneous Media

4.5 Volume potential of time-harmonic acoustics The volume potential  A(x, κ) =

g(x − x  )ν(x  )dx  , g(x) =

e−iκ|x| 4π |x|

(4.107)

relates to time-harmonic acoustic pressure fields in fluids (Section 3.8). Here, κ is the wave number of the wave field. Changing the density ν(x) to the Gaussian quasiinterpolant, we obtain the following approximate equation for this potential: A(x, κ) ≈

Mn 

h2 I 0 (x − x (n) , κ)ν (n) , ν (n) = ν(x (n) ),

n=1

1 I (x, κ) = 2 h



0

(4.108)

  1 |x|2 g(x − x )ϕ(x )dx , ϕ(x) = exp − 2 . (πH )3/2 h H 





(4.109) The integral I 0 (x, κ) is calculated similar to the integral in Eq. (4.85). We have    exp(−iκ(x − x  )) 1 |x  |2 dx  = exp − 2 I (x, κ) = |x − x  | 4πh2 (πH )3/2 h H

2      −iκ|y| exp − h|x| 2H e y2 2x · y = exp − 2 exp dy = |y| 4πh2 (πH )3/2 h H h2 H

2      ∞ H exp − h|x| 2H |y|2 2|x||y| exp −iκ|y| − 2 = sinh d|y| = 2|x|(πH )3/2 0 h H h2 H      H h2 κ 2 + 4i|x|κ iH h2 κ − 2|x| h − exp − erf c = √ 8π|x| 4 2h H   iH h2 κ + 2|x| , erf c(z) = 1 − erf(z). (4.110) − e2iκ|x| erf c √ 2h H 0

As a result, the function I 0 (x, κ) depends on the dimensionless variables |x|/ h, κh, and is presented in the form  |x| , κh = I (x, κ) = 0 h       |x| |x| |x| h 2iκ|x| f+ f0+ , κh f− , κh − e , κh , = 8π|x| h h h     H q 2 ± 4iqξ iH q ± 2ξ . , f± (ξ, q) = erf c f0± (ξ, q) = exp − √ 4 2 H 

0

(4.111) (4.112)

Numerical calculation of volume and surface potentials

125

In the case of a spherical region V of radius a and the constant potential density ν = 1 inside V , the integral in Eq. (4.107) is calculated explicitly (r = |x|):  A0 (x, κ) =  =

g(x − x  )V (x  )dx  =   − κ12 1 − (1 + iaκ)e−iκa j0 (κr) , (κr) a 2 e−iκr j1κr ,

r < a, r > a.

(4.113)

Here, j0 (z), j1 (z) are the spherical Bessel functions of the first kind. Changing V (x) to the Gaussian quasiinterpolant, we obtain the approximate equation similar to Eq. (4.108) for calculation of this potential. The function A0 (x, κ) (solid lines) and the approximations (dashed lines) are presented in Figs. 4.13–4.16. The approximation is calculated inside the cube W : (|xi | ≤ 2a, i = 1, 2, 3). In dynamics, the accuracy of the approximation (4.108) depends on the wave number κ. When κ grows, the field oscillations increase, and the finer node grid is required to guarantee acceptable accuracy. It is seen from Fig. 4.13 that for κa = 1 and a node grid step of h/a = 0.1, the exact potential A0 (x, κ) and the approximation (4.108) coincide practically. For κa = 10, such a coincidence is achieved for the grid step h/a = 0.025 only (Fig. 4.16).

Figure 4.13 Real (ReA0 ) and imaginary (ImA0 ) parts of the volume potential of acoustics in Eq. (4.113) for the wave number κa = 1; solid lines correspond to the exact potential, dashed lines are the approximation (4.108) with the parameters H = 1, h/a = 0.1.

Figure 4.14 The same as in Fig. 4.13 for the wave number κa = 10 and the parameters H = 1, h = 0.1.

126

Heterogeneous Media

Figure 4.15 The same as in Fig. 4.13 for the wave number κa = 10 and the parameters H = 1, h/a = 0.05.

Figure 4.16 The same as in Fig. 4.13 for the wave number κa = 10 and the parameters H = 1, h/a = 0.025.

4.6

Surface potentials of electrostatics

It is shown in Section 3.2 that surface potentials are appropriate for presentations of fields in homogeneous host media containing thin heterogeneities. Changing the densities of the surface potentials to the Gaussian quasiinterpolants yields the approximate equations for calculation of these potentials. In contrast to the volume potentials, the surface potentials with the Gaussian densities are not calculated in explicit analytical forms but expressed in terms of standard 1D integrals. Explicit equations can be obtained for asymptotics of these integrals for large values of arguments. Thus, for numerical calculations, these integrals can be tabulated for small arguments and changed with their asymptotics for large arguments. For thin heterogeneities with planar middle surfaces, the FFT algorithms can be adopted for numerical calculation of the potentials. Let be a planar surface belonging to an infinite plane P with the normal ni in 3D space. We consider the potential of the double layer of electrostatics and its derivative related to the problem for thin heterogeneities of a small electroconductivity in a homogeneous host medium (Section 3.2)  1 u(x) = − ∂i g(x − x  )ni ν(x  )d  , g(x) = , (4.114) 4π |x|  E i (x) = Kij (x − x  )ni ν(x  )d  , Kij (x) = ∂i ∂j g(x). (4.115)

Numerical calculation of volume and surface potentials

127

Figure 4.17 A rectangular region W covered by a regular node grid with a planar surface inside.

For numerical calculation of these potentials, is embedded into a rectangular W and covered by a regular grid of nodes x (n) (Fig. 4.17). The Gaussian quasiinterpolant of the density ν(x) in Eqs. (4.114) and (4.115) has the form ν(h,H ) (x) =

Mn 

ν

(n)

ϕ(x − x

n=1

(n)

  1 |x|2 ), ϕ(x) = exp − 2 , πH h H

(4.116)

/ . Changing ν(x) to the quasiinterpolant where ν (n) = ν(x (n) ) and ν (n) = 0 if x (n) ∈ yields the following approximate equations for calculation of the surface potentials u(x) and E i (x):  Mn  u(x) ≈ I (x − x (n) ) ν (n) , I (x) = − ∂j g(x − x  )nj ϕ(x  )dP  , (4.117) P

n=1

E i (x) ≈

Mn  n=1

1 I i (x − x (n) ) ν (n) , I i (x) = h h



∂i ∂j g(x − x  )nj ϕ(x  )dP  .

P

(4.118) Here, the integrals are calculated over the entire plane P . The functions I (x) and I i (x) 0 are expressed in terms of the potential I (x), i.e.,  1 0 I (x) = g(x − x  )ϕ(x  )dP  , (4.119) h P 0

0

I (x) = −hni ∂i I (x),

I i (x) = h2 nj ∂j ∂i I (x).

(4.120)

In the Cartesian coordinate system (x1 , x2 , x3 ) with the equation x3 = 0 for the plane 0 P , the potential I (x) is presented as the triple integral over the entire x-space, i.e., 0

I (x1 , x2 , x3 ) =

1 h



∞

−∞

g(x1 − x1 , x2 − x2 , x3 − x3 )ϕ(x1 , x2 )δ(x3 )dx1 dx2 dx3 . (4.121)

128

Heterogeneous Media

Using the property of convolution, this integral is presented as the integral of the Fourier transforms of the integrand functions 0

I (x1 , x2 , x3 ) =

1 h(2π)3



∞

−∞

g ∗ (k1 , k2 , k3 )ϕ ∗ (k1 , k2 ) exp(−iki xi )dk1 dk2 dk3 . (4.122)

Here, g ∗ (k) and ϕ ∗ (k) are the Fourier transforms of the functions g(x) and ϕ(x), 2

2H k h 1 g ∗ (k1 , k2 , k3 ) = 2 , ϕ ∗ (k1 , k2 ) = h2 exp − , (4.123) 4 k   (4.124) k = k12 + k22 + k32 , k = k12 + k22 . After calculating the integral over the k3 -variable in Eq. (4.122) and then over the polar angle in the plane (k1 , k2 ), we obtain  r x

0 3 I (x1 , x2 , x3 ) = F0 (4.125) , , r = x12 + x22 , h h    ∞ 1 κ 2H exp −κ|ς| − (4.126) J0 (κρ)dκ. F0 (ρ, ς) = 4π 0 4 Here, J0 (z) is the Bessel function. The integral F0 (ρ, ς) converges absolutely and can be differentiated with respect to variables ρ and ς under the integral sign. The far field asymptotic f0 (ρ, ς) of this integral follows from Eq. (4.119) and Eq. (4.104) for the far field of a static potential. Keeping the first terms of this expansion, we obtain for f0 (ρ, ς) the equation  1 H (ρ 2 − ς 2 ) ρ2 + ς 2. , ξ = (4.127) + f0 (ρ, ς) = 4πξ 16πξ 5 The function F0 (ρ, ς) (solid lines) and its asymptotic f0 (ρ, ς) (dashed lines) are shown in Fig. 4.18 for ς = 0 and ς = 1. It is seen that F0 (ρ, ς) can be changed to its asymptotic expression if ρ > 3 or ς > 3. For ρ, ς ≤ 3, the integrals F0 (ρ, ς) can be tabulated. The function I (x) in Eq. (4.120) takes the form r x

0 3 I (x) = −hni ∂i I (x) = sign(x3 )F1 , , (4.128) h h    ∞ κ 2H 1 exp −k|ς| − (4.129) J0 (κρ)kdκ. F1 (ρ, ς) = 4π 0 4 The asymptotic expression of this integral for large values of the arguments can be obtained by derivation of the asymptotic f0 (ρ, ς) of the function F0 (ρ, ς) in Eq. (4.127). 0 For the function I i (x) = −h2 nj ∂j ∂i I (x) in Eq. (4.118), we obtain I i (x) = 1 (r, x3 )nj + 2 (r, x3 )ei , ei =

xi , i = 1, 2, e3 = 0, r

(4.130)

Numerical calculation of volume and surface potentials

129

Figure 4.18 The function F0 (ρ, ς) in Eq. (4.126) (solid lines) and its asymptotic f0 (ρ, z) in Eq. (4.127) (dashed lines) for ς = 0 and ς = 1.

1 (r, x3 ) = −F2

r x

r x

3 3 , , 2 (r, x3 ) = −sign (x3 ) F3 , . h h h h

(4.131)

Here, the term proportional to the delta function concentrated on the surface is neglected. The functions F2 (ρ, ς) and F3 (ρ, ς) of the dimensionless variables ρ = r/ h and ς = x3 / h have the form of the absolutely converging integrals 

 κ 2H exp −κ|ς| − J0 (κρ)κ 2 dκ, 4 0    ∞ 1 κ 2H exp −κ|ς| − J1 (κρ)κ 2 dκ, F3 (ρ, ς) = 4π 0 4 1 F2 (ρ, ς) = 4π



∞

(4.132) (4.133)

where Jm (z) are Bessel functions. For ς = 0 (in the surface ), the integrals F0 , F1 , F2 , F3 are calculated explicitly:   2  1 ρ ρ2 F0 (ρ, 0) = √ I0 , (4.134) exp − 2H 2H 4 πH   1 ρ2 exp − , (4.135) F1 (ρ, 0) = 2πH H     2  2   1 ρ2 ρ2 ρ ρ ρ2 1− I0 + I1 , F2 (ρ, 0) = exp − √ 2H H 2H H 2H 2H πH (4.136)   2 ρ ρ exp − F3 (ρ, 0) = . (4.137) 2 H πH Here I0 (z) and I1 (z) are the modified Bessel functions. Asymptotics fm (ρ, ς) of the integrals Fm (ρ, ς) for large values of the arguments ρ, ς have the forms f1 (ρ, ς) =

1 Hρ(3ρ 2 − 7ς 2 ) + , 4πξ 3 16πξ 9

(4.138)

130

Heterogeneous Media

ς H ς(3ς 2 − 7ρ 2 ) − , 4πξ 3 16πξ 9 2ς 2 − ρ 2 3H (3ρ 4 − 24ρ 2 ς 2 + 8ς 4 ) − . f3 (ρ, ς) = 4πξ 5 16πξ 9

f2 (ρ, ς) =

Let us consider the potential  ni Kij (x − x  )nj b(x  )d  , x ∈ , Kij (x) = −∂i ∂j g(x), π(x) =

(4.139) (4.140)

(4.141)

which is related to the electric field in a homogeneous medium with a thin inclusion of small electroconductivity (Section 3.2.1). If belongs to the plane x3 = 0, this integral takes the form  b(x1 , x2 ) 1  π(x1 , x2 ) = − (4.142)   d . 4π (x1 − x  )2 + (x1 − x  )2 3/2 1 1 Regularization of the formally diverging integral in this equation is defined in Eq. (3.40). We obtain  ∞ b(x1 , x2 ) − b(x1 , x2 ) 1   π(x1 , x2 ) = − pv (4.143)  3/2 dx1 dx2 .   2 2 4π −∞ (x1 − x ) + (x1 − x ) 1 1 Here, integration is performed over the entire plane (x1 , x2 ) and the function b(x1 , x2 ) is continued by zero outside . Changing b(x) to the Gaussian quasiinterpolant, we obtain the following approximate equation for calculation of this potential:

Mn 1 |x − x (n) | π(x) ≈ − F2 (4.144) , 0 b(x (n) ), x = (x1 , x2 ), h h n=1

where the function F2 (ρ, 0) is defined in Eq. (4.132). Let be a circle of radius a, and let b(x) be defined by the equation (r = |x|)  r2 b(x) = 1 − 2 , r ≤ a, b(x) = 0, r > a. (4.145) a In this case, the potential π(x) is calculated explicitly. Using the convolution property for the integral in Eq. (4.142), we obtain  1 ∗ K33 π(x) = (k)b∗ (k) exp(−ik · x)dk = (2π)2  a ∞ j1 (ak)J0 (rk)kdk = = 2  π0 r < a, 4a , r  (4.146) = − 1  1 + 2a arcsin a , r > a. 2 2 2 r −a

Numerical calculation of volume and surface potentials

131

Here, we have taken into account that the 2D Fourier transforms of the function b(x) in Eq. (4.145) and of the kernel K33 (x) have the forms [3] b∗ (k) = 2π

j1 (ak)

k ∗ (k) = . , K33 2 ak

(4.147)

The program for calculation of the sum in Eq. (4.144) coincides with the program for calculation of the 2D sums in Appendix

4.A.2, where the Gaussian function should |x| 1 be changed to the function − h F2 h , 0 defined in Eq. (4.136). The graphs of the function π(x) in Eq. (4.146) (solid lines) and its approximations in Eq. (4.144) (dashed lines) are presented in Figs. 4.19–4.21 for H = 2 and h = 0.1 (Mn = 1681), h/a = 0.05 (Mn = 6561), or h/a = 0.01 (Mn = 160801). In the last case, the approximation and exact function π(x) coincide practically.

Figure 4.19 The surface potential of electrostatics in Eq. (4.146) for the potential density b(r) =  1 − (r/a)2 , r < a, b(r) = 0, r > a; the solid line is the exact potential, the dashed line is the approximation in Eq. (4.144) for the parameters H = 1, h/a = 0.1.

Figure 4.20 The same as in Fig. 4.19 for the parameters H = 1, h/a = 0.05.

In the case of nonplanar surfaces, construction of the quasiinterpolants of the potential densities brings specific difficulties. First, a set of nodes on the surface with approximately equal distances between the nodes should be generated. Then, for the constructed node grid, the coefficients of the series of Gaussian functions for the quasiinterpolants should be indicated. These coefficients cannot coincide with the values of the function at the nodes. The problem of construction of Gaussian quasiinterpolants for general node grids on nonplanar surfaces (manifolds) is considered in Chapter 10 of the book [1].

132

Heterogeneous Media

Figure 4.21 The same as in Fig. 4.19 for the parameters H = 1, h/a = 0.025.

4.7

Surface potentials of time-harmonic acoustics

We consider the potential of the simple layer of acoustics (Section 3.9)    e−iκ|x| A(x, κ) = g x − x  μ(x  )d  , g(x) = , 4π |x|

(4.148)

where is a planar surface. Using the Gaussian quasiinterpolant of the density μ(x), we obtain the following approximate equation for calculation of this potential: A(x, κ) ≈

Mn 

0

hI (x − x (n) , κ)μ(n) , μ(n) = μ(x (n) ),

n=1 0

I (x, κ) =

1 h



  g x − x  ϕ(x  )dP  , P

ϕ(x) =

  1 |x|2 exp − 2 . πH h H

(4.149) (4.150)

Here, x (n) are the nodes of a regular node grid covering the surface and P is the plane which surface belongs to. If (x1 , x2 , x3 ) are Cartesian coordinates and the 0 plane P is defined by the equation x3 = 0, then we obtain for I (x, κ)    ∞ −iκ|x−x  | e h |x  |2 0  I (x, κ) = exp − δ(x3 )dx1  dx 2 dx3 =  4π 2 H h2 H −∞ |x − x |

 ∞ h3 1 |k|2 h2 H = exp −i(ki xi ) − (4.151) dk1 dk2 dk3 , 2 2 4 (2π)3 −∞ k − κ   |x| = x12 + x22 , |k| = k12 + k22 . Integrating first over the k3 -variable and then over the polar angle in the (k1 , k2 )-plane, we obtain

r x 0 3 I (x, κ) = F0 , , κh , (4.152) h h 1 ∞ τ (k, ς, q) (4.153) J0 (kρ)kdk, F0 (ρ, ς, q) = 4π 0 η(k, q)

Numerical calculation of volume and surface potentials

η(k, q) =



133

  k2H k 2 − q 2 , τ (k, ς, q) = exp −η(k, q)|ς| − . 4

(4.154)

The far field asymptotic f0 (ρ, ς, q) of the function F0 (ρ, ς, q) has the form    q 2ρ2H 1 ρ2 + ς 2. exp −iqξ − , ξ = f0 (ρ, ς, q) = 4πξ 4ξ 2

(4.155)

In Fig. 4.22, the function F0 (ρ, ς, q) and its asymptotic f0 (ρ, ς, q) for ς = 0, q = 0.1, H = 1 are presented. It is seen that for ρ > 3, the real parts of F0 and f0 coincide practically, and the imaginary parts of these functions coincide for all values of ρ. The same takes place for ς = 0 and other values of q. Thus, for ρ, ς < 3, the function F0 (ρ, ς, q) can be tabulated, and for larger values of the arguments, the asymptotic f0 (ρ, ς, q) can be used.

Figure 4.22 Real and imaginary parts of the function F0 (ρ, ζ, q) in Eq. (4.153) for ζ = 0, q = 0.1 (solid lines) and its asymptotic f0 (ρ, ζ, q) in Eq. (4.155) (dashed lines).

We consider two surface potentials related to the problem of acoustic wave scattering on a rigid screen (Section 3.9),    1 1 A (x, κ) = − ∂i g x − x  ni b(x  )d  , Ai (x, κ) =

   (4.156) = − ∂i ∂j g x − x  nj b(x  )d  .

1

1

Here, ni is the normal to a planar surface and A (x, κ) and Ai (x, κ) are the potential of the double layer and its gradient. Using the quasiinterpolant of b(x), the approximate equations for calculation of these potentials take the forms 1

A (x, κ) ≈

Mn 

I (x − x (n) , κ)b(n) , b(n) = b(x (n) ),

n=1



I (x, κ) = − P

  0 ∂i g x − x  ni ϕ(x  )dP  = −hni ∂i I (x, κ),

(4.157) (4.158)

134

Heterogeneous Media

1

Ai (x, κ) ≈

Mn  1

I i (x − x (n) , κ)b(n) , h n=1    0 I i (x, κ) = −h ∂i ∂j g x − x  nj ϕ(x  )dP  = −h2 nj ∂j ∂i I (x, κ),

(4.159) (4.160)

P

where P is the entire plane which the surface belongs to and the scalar potential 0 I (x, κ) is defined in Eq. (4.152). Introducing the Cartesian coordinates (x1 , x2 , x3 ) with the equation x3 = 0 for the plane P , we obtain for I (x, κ) the equation 

r x 3 I (x1 , x2 , x3 , κ) = sign(x3 )F1 (4.161) , , κh , r = x12 + x22 , h h    κ 2H 1 ∞ exp −η(k, q)|ς| − (4.162) J0 (κρ)κdκ. F1 (ρ, ς, q) = 4π 0 4 For ς = 0 (in the plane P ), this integral is calculated in the explicit form   ρ2 1 exp − , F1 (ρ, 0, q) = 2πH H

(4.163)

and for large values of the variables ρ and ς, it has the following asymptotics:   q 2ρ2H ς(2ξ 2 (1 + iqξ ) − H q 2 ρ 2 ) exp −iqξ − , f1 (ρ, ς, q) = 4ξ 8πξ 5  ξ = ρ2 + ς 2. (4.164) The function I i (x, κ) in Eq. (4.160) has the form (ei = xi /r, i = 1, 2; e3 = 0)



r x r x 3 3 I i (x, κ) = −sign(x3 )F2 (4.165) , , κh ei + F3 , , κh ni , h h h h    ∞ k2H 1 exp −η(k, q)|ς| − (4.166) J1 (kρ)k 2 dk, F2 (ρ, ς, q) = 4π 0 4    ∞ 1 k2H η(k, q) exp −η(k, q)|ς| − J0 (kρ)kdk. (4.167) F3 (ρ, ς, q) = 4π 0 4 For the parameters ρ, ς < 5, this integral can be calculated numerically and tabulated. For ρ > 5 or ς > 5, the functions F2 (ρ, ς, q) and F3 (ρ, ς, q) can be changed to their asymptotics f2 (ρ, ς, q) and f3 (ρ, ς, q),   ρ|ς| q 2ρ2H 2 (3 + 3iqξ + q ξ ) exp −iqξ − f2 (ρ, ς, q) = , (4.168) 4πξ 5 4ξ 2 1  2 f3 (ρ, ς, q) = ς (2 − q 2 ς 2 + 2iqξ )− 4πξ 5    q 2ρ2H . (4.169) −2 (1 + q 2 ς 2 + iqξ ) exp −iqξ − 4ξ 2

Numerical calculation of volume and surface potentials

4.8

135

Notes

A survey of the theory of approximation by Gaussian radial and other similar functions is presented in [1]. The history of development of the FFT algorithm and its applications can be found in [4]. The method of calculation of matrix-vector products with Toeplitz matrices by the FFT algorithm is described in many books (see, e.g., [2]). Extension of this algorithm to the calculation of 2D and 3D sums of identical functions shifted at the nodes of regular grids is performed in this chapter. The equations for the volume and surface potentials of electrostatics and time-harmonic acoustics with the Gaussian densities are obtained in [1].

Appendix 4.A Computational programs for fast calculation of sums of identical functions shifted at the nodes of regular grids 4.A.1 The FFT algorithm for calculation of 1D sums In this appendix, the computational program for the calculation of the Gaussian quasiinterpolant f(h,H ) (x) of the function f (x) = 1 + 0.5 sin(2πx), |x| ≤ 1, f (x) = 0, |x| > 1

(4.A.1)

in the interval |x| ≤ 1.4 by the FFT algorithm is presented. The text of the program is written for Wolfram Mathematica software. Initial data. Mn is the number of nodes in the interval of approximation, L is the size of the interval of approximation, L h = Mn−1 is the step of the node grid, l is the coordinate of the node x (1) . (*Initial data*) Mn = 281; L = 2.8; l = -1.4; h = L/(M - 1); N1 = 2*Mn; (*The original function*) f[x_] := 1+0.5*Sin[2*Pi*x] /; Abs[x]≤ 1 f[x_] := 0 /; Abs[x] > 1 (*Coordinates of the nodes*) X = Table[l + h*(i - 1), {i, Mn}]; (*Approximating functions*) ϕ[x_] := Exp[-x ˆ 2/(h ˆ 2*H)]/Sqrt[Pi*H] H = 2; (*The basic matrix*)

136

Heterogeneous Media

[i_, j_] := ϕ[X[[i]] - X[[j]]] (*Auxiliary vector associated with the basic matrix*) V = Module[{AA}, V = Table[0, {N1}]; V[[1]] = [1, 1]; Do[AA = [i, 1]; V[[i]] = AA; V[[N1 - i + 2]] = AA, {i, 2, Mn}]; V]; (*Auxiliary vector associated with the original function*) vf[i_] := f[X[[i]]] /; i Mn Vf = Table[vf[i], {i, N1}]; (*Fourier transforms of the auxiliary vectors*) FV = Fourier[V]; FVf = Fourier[Vf]; (*Values of the Gaussian quasiinterpolant f(h,H ) (x) at the nodes*) fhH = Sqrt[N1]*InverseFourier[Table[FV[[k]]*FVf[[k]], {k, N1}]]; (*Interpolation of the vector fhH in the interval of the approximation*) IfhH = Interpolation[Table[{N[X[[k]]], fhH[[k]]}, {k, M}]] (*Plots of the original function and the quasiinterpolant*) Plot[{f[z], Re[IfhH[z]]}, {z, -1.4, 1.4}, PlotRange -> {0, 1.1}, PlotStyle -> {Black, Black}] End

4.A.2 The FFT algorithm for calculation of 2D sums In this appendix, the computational program for the calculation of the 2D Gaussian quasiinterpolant f(h,H ) (x1 , x2 ) of the function  x2 x2 x2 f (x1 , x2 ) = 1 − x12 − 2 , x12 + 2 ≤ 1, f (x1 , x2 ) = 0, x12 + 2 > 1 4 4 4 (4.A.2) in the region W = (|x1 | < 1.4, |x2 | < 2.8) is presented. The text of the program is written for Wolfram Mathematica software. Initial data. M1 , M2 are the numbers of the nodes along the x1 - and x2 -axes, Mn = M1 M2 is the total number of nodes in the area of approximation, L1 , L2 are the sizes of the region of approximation along the x1 - and x2 -axes, h = L1 /(M1 − 1) is the step of the node grid, l1 , l2 are the coordinates of the first node x (1,1) . (*Initial data*) M1 = 281; M2 = 481; L1 = 2.8; L2 = 4.8; l1 = -1.4; l2 = -2.4; h = L1/(M1 - 1);

Numerical calculation of volume and surface potentials

137

N1 = 2*M1; N2 = 2*M2; Mn = M1*M2; (*The original function*) f[z1_, z2_] := Sqrt[1 - z1 ˆ 2 - z2 ˆ 2/4] /; z1 ˆ 2 + z2 ˆ 2/4 1 (*The coordinates of the nodes*) x1 = Flatten[Table[l1 + h*(i - 1), {j, M2}, {i, M1}]]; x2 = Flatten[Table[l2 + h*(j - 1), {j, M2}, {i, M1}]]; (*Relation between the one-index and two-index numerations*) M[i_, j_] := i + M1*(j - 1) (*Transition from the one-index to two-index numeration and vice versa*) To1In[A_] := Flatten[Transpose[A]]; To2In[A_] := Transpose[Partition[A, M1]]; (*Gaussian approximating function*) H = 2; fi[r_] := Exp[-r ˆ 2/H]/(Pi*H) /; r ˆ 2/H 20 (*Distances between the nodes*) R[r_, s_] := Sqrt[(x1[[r]] - x1[[s]]) ˆ 2 + (x2[[r]] - x2[[s]]) ˆ 2] (*The basic matrix*) FI[r_, s_] := N[fi[R[r, s]/h]] TFI = Table[FI[i, 1], {i, Mn}]; (*Fourier transform of auxiliary object associated with the basic matrix*) FFI = Module[{d}, FFI = Table[0, {N1}, {N2}]; d = Module[{A}, d = Table[0, {N1}, {N2}]; d[[1, 1]] = TFI[[1]]; Do[A = TFI[[M[l, 1]]]; d[[l, 1]] = A; d[[N1 - l + 2, 1]] = A, {l, 2, M1}]; Do[A = TFI[[M[1, m]]]; d[[1, m]] = A; d[[1, N2 - m + 2]] = A, {m, 2, M2}]; Do[A = TFI[[M[l, m]]]; d[[l, m]] = A; d[[N1 - l + 2, m]] = A; d[[l, N2 - m + 2]] = A; d[[N1 - l + 2, N2 - m + 2]] = A, {l, 2, M1}, {m, 2, M2}]; d]; FFI = Fourier[d]; FFI]; (*Fourier transform of the auxiliary object associated with the original function*) Tf = Table[f[x1[[i]], x2[[i]]], {i, Mn}]; Ff = Module[{In, B}, In = To2Ind[Tf[[1 ;; Mn]]]; B = Table[0., {N1}, {N2}]; Do[B[[k, l]] = In[[k, l]], {k, M1}, {l, M2}]; Ff = Fourier[B]; Ff]; (*Calculation of the quasiinterpolant at the nodes*) fhH = Module[{AA}, fhH = Table[0, {Mn}]; AA = Sqrt[N1*N2]*InverseFourier[FFI*Ff]; fhH = To1Ind[AA[[1 ;; M1, 1 ;; M2]]]; fhH]; (*Interpolation of the quasiinterpolant fhH in the interval of approximation*)

138

Heterogeneous Media

IfhH = Interpolation[ Table[{{N[x1[[i]]], N[x2[[i]]]}, fhH[[i]]}, {i, Mn}]] (*Plots of the original function and the quasiinterpolant*) Plot[{f[z,0], IfhH[z,0]}, {z,-1.4,1.4}, PlotStyle->{Black,{Dashed, Black}}] End

4.A.3 The FFT algorithm for calculation of 3D sums In this appendix, the computational program for the calculation of the 3D Gaussian quasiinterpolant V(h,H ) (x1 , x2 , x3 ) of the function V (|x|) = 1, |x| ≤ 1, V (|x|) = 0, |x| > 1

(4.A.3)

in the cuboid W : (|xi | ≤ 1.4, i = 1, 2, 3) is presented. The text of the program is written for Wolfram Mathematica software. Initial data. M1 , M2 , M3 are the numbers of the nodes along the x1 -, x2 -, and x3 -axes, Mn = M1 M2 M3 is the total number of nodes, L1 , L2 , L3 are the sizes of the region of approximation along the x1 -, x2 -, and x3 -axes, h = L1 /(M1 − 1) is the step of the approximating grid, l1 , l2 , l3 are the coordinates of the node x (1,1,1) . (*Initial data*) M1 = 113; M2 = 113; M3 = 113; L1 = 2.8; L2 = 2.8; L3 = 2.8; l1 = -1.4; l2 = -1.4; l3 = -1.4; h = L1/(M1 - 1); N1 = 2*M1; N2 = 2*M2; N3 = 2*M3; Mn = M1*M2*M3; (*Definition of the original function*) f[z1_, z2_, z3_] := 1 /; z1 ˆ 2 + z2 ˆ 2 + z3 ˆ 2 1 (*Cartesian coordinates of the nodes*) x1 = Flatten[Table[l1 + h*(i - 1), {k, M3}, {j, M2}, {i, M1}]]; x2 = Flatten[Table[l2 + h*(j - 1), {k, M3}, {j, M2}, {i, M1}]]; x3 = Flatten[Table[l3 + h*(k - 1),{k, M3}, {j, M2}, {i, M1}]]; (*Relation between the one-index and three-index node numerations*) M[i_, j_, k_] := i + M1*(j - 1) + M1*M2*(k - 1) (*Transition from the one-index to two-index numeration and vice versa*) To1In[A_] := Flatten[Transpose[Flatten[Transpose[A], 1]]] To3In[A_] := Transpose[Partition[Transpose[Partition[A, M1*M2]], M1]] (*Approximating function*) H = 1; f0[r_] := Exp[-r ˆ 2/H]/(Pi*H) ˆ (3/2) /; r ˆ 2/H 20 (*Distances between the nodes*) R[r_, s_] := Sqrt[(x1[[r]] - x1[[s]]) ˆ 2 + (x2[[r]] - x2[[s]]) ˆ 2 + (x3[[r]] x3[[s]]) ˆ 2] (*The basic matrix*) A[r_, s_] := N[f0[R[r, s]/h]] (*Auxiliary object associated with the basic matrix*) FtMg = Module[{MG}, FtMg = Table[0., {N1}, {N2}, {N3}]; MG = Module[{AA}, MG = Table[0., {N1}, {N2}, {N3}]; MG[[1, 1, 1]] = A[1, 1]; Do[AA = A[M[l, 1, 1], 1]; MG[[l, 1, 1]] = AA; MG[[N1 - l + 2, 1, 1]] = AA, {l, 2, M1}]; Do[AA = A[M[1, m, 1], 1]; MG[[1, m, 1]] = AA; MG[[1, N2 - m + 2, 1]] = AA, {m, 2, M2}]; Do[AA = A[M[1, 1, k], 1]; MG[[1, 1, k]] = AA; MG[[1, 1, N3 - k + 2]] = AA, {k, 2, M3}]; Do[AA = A[M[l, m, 1], 1]; MG[[l, m, 1]] = AA; MG[[N1 - l + 2, N2 - m + 2, 1]] = AA; AA = A[M[l, 1, 1], M[1, m, 1]]; MG[[N1 - l + 2, m, 1]] = AA; MG[[l, N2 - m + 2, 1]] = AA, {l, 2, M1}, {m, 2, M2}]; Do[AA = A[M[1, m, k], 1]; MG[[1, m, k]] = AA; MG[[1, N2 - m + 2, N3 - k + 2]] = AA; AA = A[M[1, m, 1], M[1, 1, k]]; MG[[1, m, N3 - k + 2]] = AA; MG[[1, N2 - m + 2, k]] = AA, {m, 2, M2}, {k, 2, M3}]; Do[AA = A[M[l, 1, k], 1]; MG[[l, 1, k]] = AA; MG[[N1 - l + 2, 1, N3 - k + 2]] = AA; AA = A[M[l, 1, 1], M[1, 1, k]]; MG[[N1 - l + 2, 1, k]] = AA; MG[[l, 1, N3 - k + 2]] = AA, {l, 2, M1}, {k, 2, M3}]; Do[AA = A[M[l, m, k], 1]; MG[[l, m, k]] = AA; MG[[N1 - l + 2, N2 - m + 2, N3 - k + 2]] = AA; AA = A[M[l, 1, k], M[1, m, 1]]; MG[[N1 - l + 2, m, k]] = AA; MG[[l, N2 - m + 2, k]] = AA; AA = A[M[l, m, 1], M[1, 1, k]];

140

Heterogeneous Media

MG[[l, m, N3 - k + 2]] = AA; MG[[N1 - l + 2, N2 - m + 2, k]] = AA; AA = A[M[1, m, k], M[l, 1, 1]]; MG[[N1 - l + 2, m, k]] = AA; MG[[l, N2 - m + 2, N3 - k + 2]] = AA; AA = A[M[l, 1, 1], M[1, m, k]]; MG[[l, N2 - m + 2, N3 - k + 2]] = AA; MG[[N1 - l + 2, m, k]] = AA; AA = A[M[1, m, 1], M[l, 1, k]]; MG[[N1 - l + 2, m, N3 - k + 2]] = AA; MG[[l, N2 - m + 2, k]] = AA, {l, 2, M1}, {m, 2, M2}, {k, 2, M3}]; MG]; FtMg = Fourier[MG]; FtMg]; (*Values of the original function at the nodes*) Tf = Table[f[x1[[i]], x2[[i]], x3[[i]]], {i, Mn}]; (*Auxiliary object associated with the original function*) Ff = Module[{Inp, B}, Inp = To3In[Tf[[1;;Mn]]]; B = Table[0., {N1}, {N2}, {N3}]; Do[B[[k, l, m]] = Inp[[k, l, m]], {k, M1}, {l, M2}, {m, M3}]; Ff = Fourier[B]; Ff]; (*Values of the quasiinterpolant at the nodes*) fhH = Module[{AA}, fhH = Table[0, {Mn}]; AA = Sqrt[N1*N2*N3]*InverseFourier[FtMg*Ff]; fhH = To1In[AA[[1 ;; M1, 1 ;; M2, 1 ;; M3]]]; fhH]; (*Interpolation of the vector fhH on the interval of approximation*) IfhH = Interpolation[Table[{{N[x1[[i]]], N[x2[[i]]], N[x3[[i]]]}, Re[fhH[[i]]]}, {i, Mn}]] (*Plots of the original function and the quasiinterpolant*) Plot[{f[z,0,0], IfhH[z,0,0]}, {z,-1.4,1.4}, PlotStyle->{Black,{Dashed, Black}}] End

Appendix 4.B

The computational program for numerical calculation of a 3D potential of electrostatics

The computational program for the numerical calculation of the 3D potential of electrostatics  (4.B.1) πij (x) = Kij (x − x  )V (x  )dx  , Kij (x) = −∂i ∂k g(x), g(x) =

1 4π|x|

(4.B.2)

Numerical calculation of volume and surface potentials

141

inside the cube W : (|xi | ≤ 2, i = 1, 2, 3) is presented. Here, V (x) is the characteristic function of a ball of radius a = 1 with the center at x = 0. Initial data. M1 , M2 , M3 are the numbers of the nodes along the x1 -, x2 -, and x3 -axes, Mn = M1 M2 M3 is the total number of nodes, h is the step of the approximating grid, L1 , L2 , L3 are the sizes of the region of approximation along the x1 -, x2 -, and x3 -axes, l1 , l2 , l3 are the coordinates of the node x (1,1,1) . (*Initial data*) M1 = 81; M2 = 81; M3 = 81; h = L1/(M1 - 1); N1 = 2*M1; N2 = 2*M2; N3 = 2*M3; Mn = M1*M2*M3; L1=4;L2=4;L3=4; l1=-2;l2=-2;l3=-2; (*Cartesian coordinates of the nodes*) x1= Flatten[Table[l1 + h*(i - 1), {k, M3}, {j, M2}, {i, M1}]]; x2= Flatten[Table[l2 + h*(j - 1), {k, M3}, {j, M2}, {i, M1}]]; x3= Flatten[Table[l2 + h*(k - 1), {k, M3}, {j, M2}, {i, M1}]]; (*Relations between one-index and three-index numerations*) M[i_, j_, k_] := i + M1*(j - 1) + M1*M2*(k - 1) k[M_] := Floor[(M - 1)/(M1*M2)] + 1 j[M_] := Floor[((M - M1*M2*(k[M] - 1)) - 1)/M1] + 1 i[M_] := Mod[M - 1, NM1*(j[M] - 1) + NM1*NM2*(k[M] - 1)] + 1 /; M > NM1 i[M_] := M /; M 0

142

Heterogeneous Media

Ps1[r_, s_] := 1/(3*Pi ˆ (3/2))/; Abs[r-s]0 Ps2[r_, s_] :=0/; Abs[r-s] a. Ei (x) = Ei0 − δij − 3ni nj E 3 3c0 |x| i , E i = Ei (x) = E

(5.56) (5.57)

The exact and numerical solutions are presented in Figs. 5.2–5.4. The radius of the inclusion is a = 0.5, and the numerical solution is constructed in the cube W = (|x1 | ≤ 1, |x2 | ≤ 1, |x3 | ≤ 1). In the figures, solid lines correspond to the exact solution (5.56)–(5.57) and dashed lines to the numerical solutions. For c = 0.1, the integral equation (5.3) for the electric field Ei (x) is used, and the results are shown in the left figures 5.2–5.4. For c = 10, the integral equation (5.4) for the current Ji (x) is used (the right figures 5.2–5.4). The parameters of the Gaussian quasiinterpolants are H = 1 and h = 0.05 (Fig. 5.2), 0.25 (Fig. 5.3), or 0.0125 (Fig. 5.4). For the iterative solution of the discretized problem, the MRM is used. For inclusions with c < c0 , the numerical algorithm based on the integral equation (5.3) for the electric field converges faster and gives more accurate results, while for c > c0 , the integral equation for the current (5.4) is preferable. These specifics of Eqs. (5.3) and (5.4) can be explained as follows. The integral operator in Eq. (5.3) for the electric field is proportional to the parameter  c1 = (c − c0 ) /c0 , and if c < c0 , the 1 1 parameter | c | is less than 1. If c > c0 , then  c > 1, and this parameter can be large for a sharp contrast in the properties of the host medium and the inclusion. The integral

154

Heterogeneous Media

Figure 5.2 The components E1 (x1 , x2 , x3 ) and J1 (x1 , x2 , x3 ) of the electric field and current in the 3D medium with the conductivity c0 = 1 containing a spherical inclusion of radius a = 0.5 subjected to a constant external electric field E10 = 1 acting along the x1 -axis; solid lines are exact solutions, dashed lines are numerical solutions for H = 1, h = 0.05, Mn = 68921. (A) The electric field for the inclusion with conductivity c = 0.1. (B) The current for the inclusion with conductivity c = 10.

Figure 5.3 The same as in Fig. 5.2 for h = 0.025, Mn = 531441.

Figure 5.4 The same as in Fig. 5.2 for h = 0.0125, Mn = 4173281.

operator K in Eq. (5.3) has the symbol (the Fourier transform of the kernel) Kij∗ (k) =

ki kj , c0 k 2

(5.58)

Numerical solution of volume integral equations for static fields in heterogeneous media

155

Figure 5.5 Four spherical inclusions with the centers in the plane x3 = 0.

Figure 5.6 The component E3 (x1 , x2 , 0) of the electric field in the plane x3 = 0 for the medium with the

inclusions shown in Fig. 5.5; the conductivity of the host medium is c0 = 1 and the conductivity of the inclusions is c = 0.1. The external field E30 = 1 acts along the x3 -axis; the numerical solution is constructed for the parameters H = 1, h = 0.0125, Mn = 2099601.

which is singular because det(Kij ) = 0. Therefore, if the parameter  c1 increases, the operator K in Eq. (5.3) dominates, and the solution of this equation becomes an illposed problem. In this case, it is better to exploit Eq. (5.4) for the current. The symbol Sij∗ (k) of the integral operator in this equation, Sij∗ (k) = c0

ki kj − c0 δij , k2

(5.59)

is also singular, but this operator is proportional to the parameter  b1 = (c − c0 )/c, and 1  for c > c0 , b is less than 1. As a result, if c increases, the operator S in Eq. (5.4) does not dominate, and solution of Eq. (5.4) is a well-posed problem. In Fig. 5.5, the group of four inclusions with the centers at the points (x1 = ±0.5, x2 = ±0.5, x3 = 0) is shown. The inclusions are balls of the radii a = 0.3 with conductivity c = 0.1, c0 = 1. For the incident field with the components Ei0 = δi3 , distribution of the electric field in the plane x3 = 0 is shown in Fig. 5.6. The electric fields along the x1 - and x2 -axes that pass through the inclusion centers are shown in Fig. 5.7.

156

Heterogeneous Media

Figure 5.7 The same as in Fig. 5.6 for the electric field distribution along the x1 -axis (x2 = 0.5, x3 = 0) and the x3 -axis (x1 = x2 = 0.5).

For the numerical solution, Eq. (5.3) for the electric field is used, and the region W : (|x1 | ≤ 1, |x2 | ≤ 1, |x3 | ≤ 0.5) was covered by a regular node grid with the step h = 0.0125, H = 1, Mn = 2099601. The computational program for solution of this problem by Wolfram Mathematica software is presented in Appendix 5.A.

5.4

Numerical solutions of the volume integral equations of static elasticity for heterogeneous media

The strain tensor εij (x) in a homogeneous host medium with stiffness tensor Cij0 kl containing a heterogeneity with stiffness tensor Cij kl (x) in a region V satisfies the volume integral equation (3.89), i.e.,  1 0 (x  )εmn (x  )dx  = εij (x), (5.60) εij (x) + Kij kl (x − x  )Cklmn V   (5.61) Kij kl (x) = − ∂i ∂k gj l (x) (ij )(kl) , Cij1 kl (x) = Cij kl (x) − Cij0 kl . Here, gij (x) is the Green function of static elasticity, and for isotropic media, this function is defined in Eq. (2.46). The stress tensor σij (x) in the heterogeneous medium satisfies the integral equation (3.101), i.e.,  1 (x  )σmn (x  )dx  = σij0 (x), (5.62) σij (x) − Sij kl (x − x  )Bklmn V

0 − Cij0 kl δ(x), Sij kl (x) = Cij0 mn Kmnrs (x)Crskl



Bij1 kl (x) = Bij kl (x) − Bij0 kl , Bij kl (x) = Cij−1kl (x), Bij0 kl = Cij0 kl

−1

(5.63) . (5.64)

For numerical solution of these equations, the region V is embedded in a cuboid W and covered by a cubic node grid with step h. Then, the strain and stress tensors are

Numerical solution of volume integral equations for static fields in heterogeneous media

157

approximated by the Gaussian quasiinterpolants in W , εij (x) ≈

Mn 

(n)

εij ϕ(x − x (n) ), σij (x) ≈

n=1

Mn 

(n)

σij ϕ(x − x (n) ),

n=1

(n)

(n)

εij = εij (x (n) ), σij = σij (x (n) ), ϕ(x) =

1 (πH )d/2

(5.65)



|x|2 exp − . (5.66) H h2

Here, d is the space dimension, x (n) (n = 1, 2, ..., Mn) are the nodes of the grid, (n) (n) and εij and σij are unknown coefficients of the approximation that coincide with the values of the functions at the nodes. After substitution of the approximations in (5.65) into the integral equations of (5.60) and (5.62) and application of the collocation (n) (n) method, the systems of linear algebraic equations for the unknowns εij and σij take the forms (m)

εij +

Mn 

(m,n)

1(n)

0(m)

(n) ij kl Cklmn εmn = εij

, m = 1, 2, ..., Mn,

(5.67)

Mn  (m,n) 1(n) (n) 0(m) ij kl Bklmn σmn = σij , m = 1, 2, ..., Mn,

(5.68)

n=1 (m)

σij

−

n=1 (m,n)

(m,n)

ij kl = ij kl (x (m) − x (n) ), ij kl = ij kl (x (m) − x (n) ),   1(m) 0(m) 0 (x (m) ), Cij kl = Cij1 kl x (m) , εij = εij   1 (m) , σij0(m) = σij0 (x (m) ). Bij1(m) kl = Bij kl x

(5.69) (5.70) (5.71)

In these equations, the functions ij kl (x) and ij kl (x) are the integrals   ij kl (x) = Kij kl (x − x  )ϕ(x  )dx  , ij kl (x) = Sij kl (x − x  )ϕ(x  )dx  (5.72) calculated over the entire x-space.

5.4.1 Discretization of the volume integral equation of elasticity in the 2D case For isotropic media, the integral ij kl (x) in Eq. (5.72) is calculated explicitly, and in the 2D case, ij kl (x) takes the form

6 1  m |x| xi π Eijmkl (n), ni = , ij kl (x) = μ0 h |x|

(5.73)

m=1

π 1 = φ1 − 2κ0 φ2 , π 2 = −κ0 φ2 , π 3 = π 4 = −κ0 (φ1 − 4φ2 ) ,

(5.74)

158

Heterogeneous Media

π 5 = φ0 − 2(1 − 2κ0 )φ1 − 16κ0 φ2 , π 6 = −κ0 (φ0 − 8φ1 + 24φ2 ), (5.75) κ0 =

λ0 + μ0 , λ0 + 2μ0

(5.76)

where Eijmkl (n) are the elements of the E-basis in Eqs. (2.211) and (2.212) and three scalar functions φ0 , φ1 , φ2 of the dimensionless variable ξ = |x|/ h are defined by the equations 2

 2  1 1 ξ ξ exp − , φ1 (ξ ) = , (5.77) 1 − exp − φ0 (ξ ) = 2 πH H H 2πξ 2

  1 ξ 2 − H 1 − exp − . (5.78) ξ φ2 (ξ ) = H 2πξ 4 In the same tensor basis, the function ij kl (x) in Eq. (5.72) is defined by the equations

6  xi m |x| γ Eijmkl (n), ni = , (5.79) ij kl (x) = −4μ0 κ0 h |x| m=1

γ 1 = φ0 − 2φ1 + 2φ2 , γ 2 = φ2 , γ 3 = γ 4 = φ1 − 4φ2 ,

(5.80)

γ 5 = −2(φ0 − 4φ1 + 8φ2 ), γ 6 = φ0 − 8φ1 + 24φ2 .

(5.81)

The system (5.67) for strains can be written in the matrix form (I + B)X = F.

(5.82)

Here, I is the unit matrix of the dimensions 3Mn × 3Mn and the vectors of unknowns X and of the right hand side F are X = |X (1) , X (2) , ..., X (3M) |T , F = |F (1) , F (2) , ..., F (3M) |T , ⎧ (n) ⎪ n ≤ Mn, ⎪ ⎨ ε11 , (n−Mn) (n) X = ε22 , Mn < n ≤ 2Mn, ⎪ ⎪ ⎩ (n−2Mn) ε12 , 2Mn < n ≤ 3Mn, ⎧ 0(n) ⎪ n ≤ Mn, ⎪ ⎨ ε11 , 0(n−Mn) (n) F = , Mn < n ≤ 2Mn, ε22 ⎪ ⎪ ⎩ 0(n−2Mn) ε12 , 2Mn < n ≤ 3Mn.

(5.83)

(5.84)

The matrix B in Eq. (5.82) has the dimensions 3Mn × 3Mn and consists of nine submatrices bij of the dimensions Mn × Mn,    b11 , b12 , b13    B =  b21 , b22 , b23  , (5.85)  b31 , b32 , b33 

Numerical solution of volume integral equations for static fields in heterogeneous media

159

(m,k) 1(k) (m,k) 1(k) (m,k) 1(k) mk b11 = (m,k) = (m,k) 11ij Cij 11 , b12 11ij Cij 22 , b13 = 11ij Cij 12 , (m,k)

1(k)

(m,k)

= 22ij Cij 22 , b23

(m,k)

1(k)

(m,k)

= 12ij Cij 22 , b33

(m,k)

= 22ij Cij 11 , b22

(m,k)

= 12ij Cij 11 , b32

b21 b31

(m,k)

1(k)

(m,k)

= 22ij Cij 12 ,

(m,k)

1(k)

(m,k)

= 12ij Cij 12 .

(5.86)

(m,k)

1(k)

(5.87)

(m,k)

1(k)

(5.88)

The matrix form of the equations for stresses has a similar structure by changing εij to σij and ij kl (x) to − ij kl (x).

5.4.2 Discretization of the volume integral equation of static elasticity in the 3D case In the 3D case, the function ij kl (x) in the discretized equation for strains (5.67) takes the form

6 1  m |x| xi ij kl (x) = π Eijmkl (n), ni = , μ0 h |x|

(5.89)

m=1

π 1 = 1 − 2κ0 2 , π 2 = −κ0 2 , π 3 = π 4 = −κ0 ( 1 − 5 2 ), (5.90) π 5 = 0 − 3 1 − 4κ0 ( 1 − 5 2 ), π 6 = −κ0 ( 0 − 10 1 + 35 2 ). (5.91) Here, 0 , 1 , 2 are scalar functions of the dimensionless variable ξ = |x|/ h: 2

ξ exp − , (5.92)

0 (ξ ) = 3/2 H (πH ) 2  

√ ξ 1 ξ ξ exp −

1 (ξ ) = π erf , (5.93) − 2 √ √ H 4π 3/2 ξ 3 H H 2



 ξ H ξ ξ2 √ ξ π erf √ − 3−2

2 (ξ ) = 6 √ exp − . H H 16(π)3/2 ξ 5 H H (5.94) 1

(m)

The system of linear algebraic equations (5.67) for the coefficients εij written in matrix form as follows: (I + B)X = F,

can be

(5.95)

where I is the unit matrix of the dimensions 6Mn × 6Mn and the vectors of the unknowns X and the right hand side F are X = |X (1) , X (2) , ..., X (6Mn) |T , F = |F (1) , F (2) , ..., F (6Mn) |T ,

(5.96)

160

Heterogeneous Media

⎧ (r) ε11 , ⎪ ⎪ ⎪ ⎪ (r−Mn) ⎪ ⎪ , ε22 ⎪ ⎪ ⎪ ⎨ ε (r−2Mn) , 33 X (r) = (r−3Mn) ⎪ , ε ⎪ 12 ⎪ ⎪ ⎪ (r−4Mn) ⎪ ⎪ , ε13 ⎪ ⎪ ⎩ (r−5Mn) , ε23 ⎧ 0(r) ε11 , ⎪ ⎪ ⎪ ⎪ 0(r−Mn) ⎪ ⎪ , ε22 ⎪ ⎪ ⎪ ⎨ ε 0(r−2Mn) , 22 F (r) = 0(r−3Mn) ⎪ , ε ⎪ 12 ⎪ ⎪ ⎪ 0(r−4Mn) ⎪ ⎪ , ε13 ⎪ ⎪ ⎩ 0(r−5Mn) , ε23

r ≤ Mn, Mn < r ≤ 2Mn, 2Mn < r ≤ 3Mn, 3Mn < r ≤ 4Mn, 4Mn < r ≤ 5Mn, 5Mn < r ≤ 6Mn, r ≤ Mn, Mn < r ≤ 2Mn, 2Mn < r ≤ 3Mn,

(5.97)

3Mn < r ≤ 4Mn, 4Mn < r ≤ 5Mn, 5Mn < r ≤ 6Mn.

The matrix B in Eq. (5.95) has the dimensions 6Mn × 6Mn and consists of 36 submatrices bpq of the dimensions Mn × Mn,   (5.98) B = bpq  , p, q = 1, 2, ..., 6, 1(k) (m,k) = (m,k) bpq ppij Cij qq , p, q = 1, 2, 3, (m,k)

1(k)

(m,k)

= 13ij Cij qq , b6q

(m,k)

1(k)

(m,k)

1(k)

(m,k)

= qqij Cij 13 ,

(m,k)

1(k)

(m,k)

1(k)

(m,k)

1(k)

(m,k)

= 12ij Cij qq , b5q

(m,k)

= qqij Cij 12 , bq5

(m,k)

= qqij Cij 23 , q = 1, 2, 3,

(m,k)

= 12ij Cij 12 , b45

b4q

bq4 bq6

(5.99)

(m,k)

(m,k)

(m,k)

(m,k)

1(k)

= 23ij Cij qq ,

(5.100)

(5.101) 1(k)

(m,k)

1(k)

(5.102)

(m,k) 1(k) (m,k) 1(k) (m,k) 1(k) = (m,k) = (m,k) = (m,k) b54 13ij Cij 12 , b55 13ij Cij 13 , b56 13ij Cij 23 ,

(5.103)

(m,k)

b64

(m,k)

1(k)

(m,k)

= 23ij Cij 12 , b65

= 12ij Cij 13 , b46

(m,k)

= 12ij Cij 23 ,

b44

(m,k)

1(k)

(m,k)

= 23ij Cij 13 , b66

(m,k)

1(k)

= 23ij Cij 23 .

(5.104)

In these equations, m, k = 1, 2, ..., Mn; summation from 1 to 3 with respect to repeating indices i, j is implied. The function ij kl (x) in Eq. (5.68) for stresses has the following explicit form: ij kl (x) = −2μ0

6  m=1

γm

|x| Eijmkl (n), h

(5.105)

γ = 0 − 2 1 + 4κ0 2 , γ 2 = (1 − 2κ0 )(2 1 − 0 ) + 2κ0 2 ,

(5.106)

γ = γ = (1 − 2κ0 ) 0 − (3 − 8κ0 ) 1 − 10κ0 2 ,

(5.107)

1 3

4

Numerical solution of volume integral equations for static fields in heterogeneous media

161

γ 5 = −2 0 + 2(3 + 4κ0 ) 1 − 40κ0 2 ,

(5.108)

γ = 2κ0 ( 0 − 10 1 + 35 2 ) ,

(5.109)

6

where the functions 0 , 1 , and 2 are defined in Eqs. (5.92)–(5.94). Similar to Eq. (5.67) the discretized equation (5.68) can be presented in the matrix form (I − C)X = F.

(5.110)

In this case, the vector of unknowns X is composed from the values of the stress tensor σij at the nodes, and the vector of the right hand side F consists of the components of the external stress field σij0 (x) at the nodes similar to Eqs. (5.86) and (5.87). The matrix C consists of 36 submatrices cpq that are defined in Eqs. (5.99)–(5.104), where (r,s) 1(s) 1(s) (r,s) pqij should be changed to pqij and Cij kl to Bij kl .

5.4.3 A radially heterogeneous inclusion in a homogeneous elastic medium In this section, a spherical isotropic inclusion with radially varying elastic properties in a homogeneous elastic medium is considered. Let (x1 , x2 , x3 ) be the Cartesian coordinates with the origin at the center of the inclusion of radius a. The medium is isotropic with the Young modulus E0 and the Poisson ratio ν0 = 0.3. First, we consider an inclusion with parabolic change of the Young modulus E(x) along the radius: 2 E(x) |x| = 11 − 10 , |x| ≤ a, E0 a

E(x) = E0 , |x| > a

(5.111)

and the Poisson ratio ν(x) = ν0 = 0.3. The medium is subjected to a constant 1D external stress field σij0 = σ0 δi1 δj 1 in the direction of the x1 -axis, and σ0 is a scalar. The distributions of the stress tensor components σ11 , σ22 along the x1 - and x2 -axes are presented in Figs. 5.8 and 5.9. The functions σ11 (x1 , 0, 0) and σ22 (x1 , 0, 0) are shown on the right hand sides in these figures, the functions σ11 (0, x2 , 0) and σ22 (0, x2 , 0) on the left hand sides. The graphs in Figs. 5.8 and 5.9 correspond to the numerical solution of Eq. (5.62) for stresses inside the cube W : |xi | ≤ a, i = 1, 2, 3. The regular grids of approximating nodes with the steps h/a = 0.1 (Mn = 9261) and h/a = 0.0317 (Mn = 262144) are considered. The graphs in Fig. 5.10 show the distribution of the shear stress σ12 along the x1 - and x3 -axes for the medium subjected to the external shear stress tensor σij0 = σ0 (δi1 δj 2 + δi2 δj 1 ). The solid lines in Figs. 5.8–5.10 are the exact distributions of the components of the elastic stress tensor obtained by the method presented in [5]. It is seen that for the grid step h/a = 0.0317, the numerical solutions practically coincide with the exact ones. The parameter H = 1 is taken in the calculations. The same problem was solved using Eq. (5.60) for strains. The medium was sub0 = ε δ δ , where ε is a constant. The correjected to the external strain field εij 0 i1 j 1 0 sponding distributions of the components ε11 and ε22 of the strain tensor inside the

162

Heterogeneous Media

Figure 5.8 The component σ11 of the stress field in a spherical inclusion with parabolic distribution of the Young modulus along the radius in Eq. (5.112), the medium is subjected to a uniaxial stress field in the direction of the x1 -axis.

Figure 5.9 The same as in Fig. 5.8 for the component σ22 of the stress field.

inclusion are shown in Figs. 5.11 and 5.12. Fig. 5.13 shows the distribution of the shear 0 = ε (δ δ + δ δ ). strain ε12 by application of the external shear strain tensor εij 0 i1 j 2 i2 j 1 The solid lines in these figures are exact distributions of the corresponding components of the strain tensor. It is seen that similar to the equation for stresses, the numerical solution coincides practically with the exact one for the step of the node grid h/a = 0.0317. But the number of iterations in the MRM is almost two times more than by the use of Eq. (5.62) for stresses. This fact reflects a general situation: The iteration process based on Eq. (5.62) for stresses converges faster than the same process based on Eq. (5.60) for strains if the inclusion is stiffer than the medium. In the opposite case, when the inclusion is softer than the medium, the iteration process based on the equation for strains is more efficient. If elastic moduli of a heterogeneous medium are discontinuous, components of the stress and strain tensors have jumps on the discontinuity surfaces. In these cases, for

Numerical solution of volume integral equations for static fields in heterogeneous media

163

Figure 5.10 The same as in Fig. 5.8 for the component σ12 of the stress field; the medium is subjected to 0 . a constant shear stress σ12

Figure 5.11 The component ε11 of the strain field in a spherical inclusion with parabolic distribution of the Young modulus along the radius; the medium is subjected to a uniaxial strain field in the direction of the x1 -axis.

accurate description of the fields, a sufficiently fine grid of approximating nodes is required. In the next example, we consider elastic fields in a spherical inclusion of radius a that consists of a central kernel in the region |x|/a ≤ 0.5 with the Young modulus E/E0 = 0.2, and a layer in the region 0.5 < |x|/a ≤ 1 with the Young modulus E/E0 = 0.5. The inclusion is embedded in a homogeneous medium with the Young modulus E0 , and Poisson ratios of the medium and the inclusion are the 0 = ε δ δ , the corresame (ν = ν0 = 0.3). For a uniaxial external strain field εij 0 i1 j 1 sponding distributions of the components ε11 and ε22 of the strain tensor inside the inclusion are presented in Figs. 5.14 and 5.15. For the external shear strain field 0 = ε (δ δ + δ δ ), the component ε εij 0 i1 j 2 i2 j 1 12 (x) is shown in Fig. 5.16. For the solution, Eq. (5.60) for strains and the node grids with the steps h/a = 0.05 (Mn = 68921) and h/a = 0.01 (Mn = 8120601) were used. The solid lines in these figures are exact

164

Heterogeneous Media

Figure 5.12 The same as in Fig. 5.11 for the component ε22 of the strain tensor.

Figure 5.13 The same as in Fig. 5.11 for the component ε12 of the strain tensor; the medium is subjected to a constant shear strain ε 0 .

solutions obtained by the method presented in [5]. It is seen from these figures that for h/a = 0.01, the numerical and exact solutions coincide practically. The stress distributions in a layered spherical inclusion the Young moduli of which are more than the Young modulus of the medium are presented in Figs. 5.17–5.19. In this case, E/E0 = 10 when |x|/a ≤ 0.5 and E/E0 = 5 when 0.5 < |x|/a ≤ 1, ν = ν0 = 0.3. The graphs in Figs. 5.17 and 5.18 correspond to the uniaxial external stress field σij0 = σ0 δi1 δj 1 and the graphs in Fig. 5.19 correspond to the external shear stress field σij0 = σ0 (δi1 δj 2 + δi2 δj 1 ). For the numerical solutions, Eq. (5.62) for stresses was used with the node grid steps h/a = 0.1, h/a = 0.02, and h/a = 0.01. Figs. 5.8–5.19 demonstrate convergence of the numerical solutions to the exact distribution of the elastic fields when the step h of the node grid decreases. For finite values of the elastic constants of the heterogeneity and the host medium, the MRM in Eqs. (5.45)–(5.47) always converges. The number of iterations grows together with the contrast in the elastic properties of the host medium and the heterogeneity as well as

Numerical solution of volume integral equations for static fields in heterogeneous media

165

Figure 5.14 The component ε11 of the strain tensor inside a spherical inclusion with step-wise change of the Young modulus along the radius; the medium is subjected to a uniaxial strain ε 0 in the direction of the x1 -axis.

Figure 5.15 The same as in Fig. 5.14 for the component ε22 of the strain tensor.

Figure 5.16 The same as in Fig. 5.14 for the component ε12 of the strain tensor; the medium is subjected to a constant shear strain ε 0 .

166

Heterogeneous Media

Figure 5.17 The component σ11 of the stress tensor inside a spherical inclusion with step-wise change of the Young modulus along the radius; the medium is subjected to a uniaxial stress σ 0 in the direction of the x1 -axis.

Figure 5.18 The same as in Fig. 5.17 for the component σ22 of the stress tensor.

with the number Mn of the approximating nodes. For a moderate contrast in the elastic properties of the host medium and the heterogeneity, the CGM in Eqs. (5.49)–(5.56) converges faster than the MRM (5.45)–(5.47). But for large contrasts, the CGM can diverge, while the MRM keeps converging.

5.4.4 Several heterogeneous inclusions in a homogeneous host medium Let a homogeneous medium contain three isolated spherical inclusions of the same radius a = 0.4. We assume that the centers of the inclusions are at the vertices Y (i) of

Numerical solution of volume integral equations for static fields in heterogeneous media

167

Figure 5.19 The same as in Fig. 5.17 for the component σ12 of the stress tensor; the medium is subjected to a constant shear stress σ 0 .

Figure 5.20 A cuboid W with three spherical inclusions inside.

an equilateral triangle with the coordinates Y (1) = (0, 0.6, 0), Y (2) = (−0.579, −0.3, 0), Y (3) = (−0.579, 0.3, 0). (5.112) The numerical solution is constructed in the cuboid W : ( |x1 | ≤ 1, |x2 | ≤ 1, |x3 | ≤ 0.5) (Fig. 5.20). The Young moduli of the medium and the inclusions are E0 and E, and E/E0 = 0.001. A cubic node grid with the steps h = 0.02 (M = 1030301) and h = 0.01 (M = 8120601) that covers W was used in the calculations, and H = 1. For 0 = 1 applied along the x -axis, the distribution the external uniaxial stress field σ33 3 of the component σ33 (x1 , x2 , 0) of the stress tensor in the plane x3 = 0 is presented in Fig. 5.21. The distributions of the components σ33 , σ11 , and σ22 along the lines x2 = 0.6, x3 = 0 and x2 = −0.3, x3 = 0 that pass through the centers of the inclusions are shown in Figs. 5.22–5.24, correspondingly. For the solution, Eq. (5.62) for stresses was used.

168

Heterogeneous Media

Figure 5.21 The component σ33 of the stress field in the plane x3 = 0 inside the cuboid with three inclusions in Fig. 5.20; the medium is subjected to a uniaxial stress field in the direction of the x3 -axis.

Figure 5.22 The component σ33 of the stress field along the lines x2 = 0.6, x3 = 0 (left hand side) and x2 = −0.3, x3 = 0 (right hand side) that pass through the centers of the inclusions in Fig. 5.20; the medium is subjected to a uniaxial stress field in the direction of the x3 -axis.

Figure 5.23 The same as in Fig. 5.22 for the stress component σ11 .

Numerical solution of volume integral equations for static fields in heterogeneous media

169

Figure 5.24 The same as in Fig. 5.22 for the stress component σ22 .

5.5 Thermo-elastic deformation of heterogeneous media We consider a homogeneous elastic medium containing several heterogeneous inclusions in the regions V (k) (k = 1, 2, ..., M). The medium is subjected to an external stress field σij0 (x) and temperature field T (x). The stiffness tensor and the tensor of (k)

thermo-expansion coefficients of the host medium are Cij0 kl and αij0 , and Cij kl (x) and (k)

αij (x) are the same tensors for the heterogeneity V (k) . The tensors of the elastic stiffness and thermo-expansion coefficients of the heterogeneous medium are presented in the forms Cij kl (x) = Cij0 kl + Cij1 kl (x), αij (x) = αij0 + αij1 (x), (k)

(k)

(5.114)

V (k) .

(5.115)

Cij1 kl (x) = Cij kl (x) − Cij0 kl , αij1 (x) = αij (x) − αij0 if x ∈ V (k) , / Cij1 kl (x) = 0, αij1 (x) = 0 if x ∈

M 

(5.113)

k=1

In the case of thermo-elasticity, the stress tensor σij (x) in the heterogeneous medium satisfies Eq. (3.100), i.e.,  1 (x  )σmn (x  )dx  + σij (x) = σij0 (x) + Sij kl (x − x  )Bklmn   1 0 + Sij kl (x − x  )αkl (x)T (x  )dx  + Sij kl (x − x  )αkl T (x  )dx  .

(5.116)

Let T (x) = T be constant, and for T = 0, let the medium be free of temperature stresses. In this case, the last integral in Eq. (5.116) has the meaning of the stress field in a homogeneous medium with the elastic stiffness tensor Cij0 kl and the tensor of 0 subjected to a constant temperature field. The value thermo-expansion coefficients αkl of this integral depends on the conditions at infinity. If deformations are not restricted

170

Heterogeneous Media

at infinity, this integral is equal to zero [5]:  0 Sij kl (x − x  )αkl T dx  = 0.

(5.117)

Meanwhile, for the restricted total deformation (εij = 0), we have 

0 0 Sij kl (x − x  )αkl T dx  = −Cij0 kl αkl T.

(5.118)

The right hand side in this equation is the stress field in an infinite homogeneous host medium subjected to a constant temperature field T by the condition that the total deformation of the medium is equal to zero. In the absence of external stresses σij0 and restrictions for deformations at infinity, Eq. (5.116) takes the form   1 1 (x  )σmn (x  )dx  = Sij kl (x − x  )αkl (x  )dx  T . σij (x) − Sij kl (x − x  )Bklmn (5.119) 1 (x) by the Gaussian quasiinApproximation of the functions Bij1 kl (x  )σkl (x  ) and αkl terpolants and application of the collocation method yields the discretized form of this equation:

(m) σij

Mn  (m,n) 1(n) (n) (m) − ij kl Bklmn σmn = ij ,

m = 1, 2, ..., Mn,

(5.120)

n=1 (m)

ij =

Mn    (m,n) 1(n) 1(n) 1 ij kl αkl T , αkl = αkl x (n) .

(5.121)

n=1

This system can be presented in the matrix form similar to (5.110) the left hand sides of which coincide, but the components of the vector F on the right hand side are defined by the equation ⎧ (n) 11 , ⎪ ⎪ ⎪ ⎪ (n−M) ⎪ ⎪ , 22 ⎪ ⎪ ⎪ ⎨ (n−2M) , 22 F (n) = (n−3M) ⎪ ,  ⎪ 12 ⎪ ⎪ ⎪ (n−4M) ⎪ ⎪ , 13 ⎪ ⎪ ⎩ (n−5M) , 23

n ≤ M, M < n ≤ 2M, 2M < n ≤ 3M,

(5.122)

3M < n ≤ 4M, 4M < n ≤ 5M, 5M < n ≤ 6M. (m,n)

Because for regular node grids, the objects ij kl in Eq. (5.121) have Teoplitz properties, the FFT algorithm can be used for calculation of the vector F.

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171

Distributions of temperature stresses in the medium with three spherical inclusions considered in Section 5.4.4 are presented in Figs. 5.25–5.30. The Young modulus E0 , the Poisson ratio ν0 , and the coefficient of thermo-expansion α0 of the host medium are taken as follows: E0 = 70 GPa, ν0 = 0.35, α0 = 23 · 10−6 /◦ C, which corresponds to aluminum. The same parameter of the inclusions are E = 120 GPa, ν = 0.36, α = 9 · 10−6 /◦ C, which corresponds to titanium. The solution of Eq. (5.119) was constructed in the cuboid W : (|x1 | ≤ 1, |x2 | ≤ 1, |x3 | ≤ 0.5). The radii of the inclusions are taken as a = 0.4. Note that the stress distribution does not depend on the absolute sizes of the inclusions. The distributions of the components σ11 , σ22 , σ33 of the stress tensor along the axis x1 for x2 = −0.3, 0.6 and x3 = 0 are shown in Figs. 5.25, 5.27, and 5.29. The distributions of the same stresses in the plane x3 = 0 are shown in Figs. 5.26, 5.28, and 5.30. In the solutions, the node grid with the step h = 0.01333 and H = 1 were used.

Figure 5.25 The component σ11 of the temperature stress tensor along the axis that passes through the centers of the inclusions shown in Fig. 5.20.

Figure 5.26 The component σ11 /T (GPa/◦ C) of the thermo-stress tensor in the plane x3 = 0 for the medium with three inclusions shown in Fig. 5.20.

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Heterogeneous Media

Figure 5.27 The same as in Fig. 5.25 for the stress component σ22 .

Figure 5.28 The same as in Fig. 5.26 for the stress component σ22 .

Figure 5.29 The same as in Fig. 5.25 for the stress component σ33 .

5.6 Elasto-plastic deformation of heterogeneous media The integral equation for the stress field in an elasto-plastic heterogeneous medium T = α (x)T (x) is changed follows from Eq. (3.103) if the temperature deformation εij ij p with the plastic deformation εij (x), i.e.,

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173

Figure 5.30 The same as in Fig. 5.26 for the stress component σ33 .



1 Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  =  p = σij0 (x) + Sij kl (x − x  )εkl (x  )dx  .

σij (x) −

(5.123)

p

Because plastic deformation εkl (x) is a nonlinear function of stresses, this integral equation is nonlinear. The conventional procedure of linearization of elasto-plastic problems is well known. The process of loading is divided into small intervals, and the problem is considered as linear inside each interval. In this section, this algorithm is applied to numerical solution of the integral equation (5.123) in the 2D and 3D cases.

5.6.1 Integral equations for the stress increments by loading of elasto-plastic heterogeneous media Let an infinite homogeneous host medium with the tensor of elastic stiffness Cij0 kl contain a finite number M of isolated heterogeneous inclusions in the regions V (k) (k = 1, 2, ..., M), and let Cij(k)kl (x) be the tensor of elastic stiffness of the kth inclusion. The materials of the medium and the inclusions are elasto-plastic. The objective is to calculate stress and plastic strain distributions around the inclusions if the medium is subjected to an increasing external stress field σij0 (x). It is assumed that elastic and plastic deformations are small. M  / V, V = V (k) , the stress fields in Because in Eq. (5.123) Bij1 kl (x) = 0 if x ∈ p

k=1

the region V and in the regions V p where εij (x) = 0 are the principal unknowns of the problem. The stress field outside these regions is reconstructed from the same Eq. (5.123). Thus, the solution of Eq. (5.123) should be constructed in a region W that includes the region V and the region V p involved in plastic deformations. In particular, W can be a cuboid containing the regions V and V p . For the 2D case, the p regions W , V , and V p are shown in Fig. 5.31. Note that the plastic deformation εij (x)

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Heterogeneous Media

Figure 5.31 An inclusion V and the regions V p involved in plastic deformations included in a rectangular W covered with a regular node grid.

under the integral on the right hand side is expressed in terms of the stress tensor from the constitutive equations for the elasto-plastic medium. From Eq. (5.123) follows the equation for the increment σij (x) of the stress field caused by a small increment of the external stress tensor σij0 (x), i.e., 

1 Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  =  p 0 = σij (x) + Sij kl (x − x  )εkl (x  )dx  .

σij (x) −

(5.124)

p

Here εij is the corresponding increment of the plastic deformations. In what follows we consider an elasto-plastic host medium and elastic inclusions. This case is most interesting for applications, and its generalization for elasto-plastic inclusions is trivial. According to the theory of plasticity with isotropic hardening p [6], an infinitesimal increment εij of the plastic deformation is calculated from the equations p

εij = (J y )sij J, if J ≥ J y and J > 0, p

εij = 0,

if J < J y or J ≤ 0.

(5.125)

Here, sij is the deviator of the stress tensor and J is the stress intensity,  J=

3 sij sij , 2

1 sij = σij − σkk δij , 3

(5.126)

J y = J y (x) is the yield stress at point x that is equal to the maximal stress intensity J y y achieved at this point in the loading process if J > J0 , and J0 = σy is the initial yield stress for the host medium. The intensities of the total J ε and plastic J p deformations are defined by the equations  J = ε

2 (εij − εkk δij )(εij − εkk δij ), 3

 J = p

2 p p ε ε . 3 ij ij

(5.127)

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175

Figure 5.32 The diagram of hardening of an elasto-plastic material.

The function (J y ) in Eq. (5.125) defines the law of plastic hardening. It follows from Eq. (5.125) and Fig. 5.32 that the function (J y ) can be presented in the form

1 1 1

(J y ) = y − . (5.128) J Et (J y ) E0 Here E0 is the Young modulus of the host medium and Et (J y ) = J y /J ε is the tangent modulus of the hardening diagram J y = J y (J ε ). Let the external stress field increase from σ 0(0) to σ 0(1) , and for σ 0 = σ 0(0) , let the deformations be pure elastic. Thus, the plastic deformations appear in the process of loading. We divide this process into n small intervals (l) σ 0 (l = 1, 2, ..., n), and the external stress tensor at the kth interval is the following sum: 0(0)

σij0 = σij

+

k 

(l) σij0 .

(5.129)

l=1

For a small increment of the stress tensor, the increment of the stress intensity J is calculated as follows: J ≈

3 sij σij . 2J

(5.130)

Thus, according to Eq. (5.125), for J ≥ J y and J > 0, the increment of the plastic p deformations εij (x) is presented in the following form: p

εij (x) ≈

3 (J y ) sij skl σkl (x). 2 J

(5.131)

From this equation and Eq. (5.124) follows the equation for the stress field increment (k) σij (x) at the kth loading interval in the following form:  (k) 1 (x  )(k) σmn (x  )dx  =  σij (x) − Sij kl (x − x  )Bklmn

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Heterogeneous Media

 = (k) σij0 (x) +

Sij kl (x − x  )(k) εkl (x  )dx  . p

(5.132)

The linearized equation (5.132) yields the following equation for the increment of the p plastic deformation (k) εkl (x) at the kth interval of loading: p

(k) εij (x) =

3 (J y(k−1) ) (k−1) (k−1) (k) sij skl  σkl (x) 2 J (k−1) if J (k−1) ≥ J y(k−2) and J (k−1) > 0,

p

(k) εij (x) = 0 if J (k−1) < J y(k−1) or J (k−1) ≤ 0,

(5.133)

where the upper indices (k − 1) and (k − 2) correspond to the previous intervals, (k−1) , the stress intensity J (k−1) , and the yield stresses and the stress deviator sij J y(k−1) , J y(k−2) are supposed to be known from the solution at the (k − 1)- and (k − 2)-intervals, J (k−1) = J (k−1) − J (k−2) . From Eqs. (5.132) and (5.133) we obtain the linearized integral equation for the stress increment (k) σkl (x) at the kth interval of loading in the following form:  (k) (x  )(k) σmn (x  )dx  = (k) σij0 (x), (5.134) (k) σij (x) − Sij kl (x − x  )B klmn (k) (x) = Bij1 kl (x) + Q(k) (x), B ij kl ij kl Q(k) ij kl (x) =

(5.135)

3 (J y(k−1) (x)) (k−1) (k−1) sij (x)skl (x) 2 J (k−1) (x) if J (k−1) ≥ J y(k−2) and J (k−1) > 0,

(k)

Qij kl (x) = 0 if J (k−1) < J y(k−1) or J (k−1) ≤ 0.

(5.136)

After solution of Eq. (5.134) the stress increment (k) σ (x) is to be used for the calculation of the increment (k) ε p (x) of the plastic deformations according to Eq. (5.133). Then, we go to the next interval (k + 1) and repeat the process until the external stress field reaches its final value σ 0(1) . The total stress tensor and the total plastic deformation at the kth interval are calculated from the equations σij (x) = σij0 (x) +

k  l=1

p

p(k)

(l) σij (x), εij = εij

=

k 

p

(l) εij .

(5.137)

l=1

Here, the initial plastic deformation is assumed to be equal to zero and σij0 (x) is the stress tensor in the medium by action of the initial external stress σ 0(0) (x). Thus, the problem is reduced to the solution of the integral equation (5.134) for the increment of the stress field σij (x) in the medium caused by the increment of the external stress field σij0 (x),  klmn (x  )σmn (x  )dx  = σij0 (x). (5.138) σij (x) − Sij kl (x − x  )B

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177

ij kl (x), Here, the functions Sij kl (x) and σij0 (x) are known, as well as the function B which is defined from the solution in the previous interval of loading. In this equation, the upper subindex k that indicates the number of the loading interval is omitted.

5.6.2 Numerical solution of the integral equation (5.138) in the 2D case For numerical solution of the integral equation (5.138) in the 2D case, the increment of the stress tensor is changed to the Gaussian quasiinterpolant: σij (x) ≈

Mn 

(n) σij ϕ(x

n=1

−x

(n)



1 |x|2 ), ϕ(x) = exp − . πH H h2

(5.139)

Here, x (n) (n = 1, 2, ..., M) are the nodes of a regular grid in the rectangular W containing the regions V and the region V p involved in plastic deformations, h is the distance between the neighbor nodes, Mn is the total number of nodes in W , and (s) σij are unknown coefficients of the approximation. Substitution of Eq. (5.139) into the integral in Eq. (5.138) yields the following equation: σij (x) −

Mn  (n) (n) σmn ij kl (x − x (n) )B = σij0 (x), klmn

n=1 (n) 1(n) (n)  =B B ij kl ij kl + Qij kl ,

(5.140) (5.141)

  1(n) (n) where Bij kl = Bij1 kl x (n) , and for the kth interval of loading, the tensor Qij kl is (n)

Qij kl =

3 (J y(k−1) (x (n) )) (k−1) (n) (k−1) (n) sij (x )skl (x ) 2 J (k−1) (x (n) ) if J (k−1) ≥ J y(k−2) and J (k−1) > 0,

(k−1) < J y(k−1) or J (k−1) ≤ 0. Q(n) ij kl = 0 if J

(5.142)

Here, we take into account that klmn (x)σmn (x) ≈ B

Mn 

(n) (n) σmn ϕ(x − x (n) ). B klmn

(5.143)

n=1

The function ij kl (x) in Eq. (5.140) is defined in Eq. (5.79). The system of linear algebraic equations for the coefficients σ (n) follows from Eq. (5.140) if the latter is satisfied at all nodes. Because for a square node grid (r) σij (x (r) ) = σij , we obtain the system of linear algebraic equations for the co-

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Heterogeneous Media

efficients σ (n) in the form (m)

σij

−

Mn  (m,n) (n) 0(m) (n) ij kl B , m = 1, 2, ..., Mn, klmn σmn = σij

(5.144)

n=1 (m,n)

ij kl = ij kl (x (m) − x (n) ).

(5.145)

This system is presented in matrix form as follows: (I − B)X = F,

(5.146)

where I is the unit matrix of the dimensions 3M × 3M and the vectors of the unknowns X and the right hand side F are X = [X (1) , X (2) , ..., X (3Mn) ]T , F = [F (1) , F (2) , ..., F (3Mn) ]T , ⎧ (n) n ≤ Mn, ⎪ ⎨ σ11 , (n−Mn) (n) X = σ22 , Mn < n ≤ 2Mn, ⎪ ⎩ (n−2Mn) , 2Mn < n ≤ 3Mn, 2σ12 ⎧ 0(n) n ≤ Mn, ⎪ ⎨ σ11 , 0(n−Mn) (n) F = , Mn < n ≤ 2Mn, σ22 ⎪ ⎩ 0(n−2Mn) , 2Mn < n ≤ 3Mn. 2σ12

(5.147)

(5.148)

The matrix B in Eq. (5.146) has the dimensions 3Mn × 3Mn and consists of nine submatrices bpq of the dimensions Mn × Mn, (m,n) (n) (m,n) (m,n) (n) (m,n) = ppij B = 12ij B bpq ij qq , p, q = 1, 2, b33 ij 12 , (m,n)

b3q

(m,n) (n) (m,n) (m,n) (n) = 12ij B = qqij B ij qq , bq3 ij 12 , q = 1, 2.

(5.149) (5.150)

In these equations, m, n = 1, 2, ..., Mn; summation from 1 to 2 with respect to repeating indices i, j is implied. These equations follow from Eq. (5.144). As seen from Eqs. (5.79), (5.149), and (5.150), the elements of the matrix B have simple analytical forms and are calculated fast. Let us apply the method to the calculation of stress and plastic strain fields in an infinite elasto-plastic plane with a circular rigid inclusion of radius a. The Young modulus and Poisson ratio of the medium are E0 = 70 GPa and ν0 = 0.3, and the initial yielding stress J0s = σy = 100 MPa. For simplicity, the law of hardening is assumed to be linear with the constant tangent modulus Et . The Young modulus and Poisson ratio of the inclusion are E = 2000 GPa, ν = 0.5. The medium is subjected to a uniaxial stress along the x1 -axis, the starting external stress is σ 0(0) = 60 MPa, and the final stress is σ 0(1) = 90 MPa. For Et = 35 GPa, the contour plot of the distribution of the plastic strain intensity J p around the inclusion is shown in Fig. 5.33. The influence of the step h of the node grid and the value of the tangent modulus Et on the distributions of the stress and plastic strain intensities J and J p along the x1 - and x2 -axes are

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179

Figure 5.33 The contour plots of the intensity of plastic deformations J p around a rigid circular inclusion in the elasto-plastic plane subjected to a uniaxial stress along the x1 -axis.

Figure 5.34 Distributions of the stress intensity J (x1 , 0) along the x1 -axis in the elasto-plastic plane with a rigid circular inclusion for various values of the node grid step h (left) and various tangent moduli Et in the plastic region (right); the plane is subjected to a uniaxial stress along the x1 -axis; dashed lines are the results of the FEM calculations.

shown in Figs. 5.34–5.36. The numerical solutions were constructed inside the rectangular (3a × 3a) with the node grid steps h/a = 0.1, 0.05, 0.01. For h/a = 0.01, the total number of nodes is Mn = 80601. The parameter H in the approximation (5.139) is taken as H = 1. The value of the loading step affects smoothness of the plastic strain intensity distribution: The smaller is the step, the smoother is the function J p (x1 , x2 ), and for σ 0 < 0.5 Mpa, there is no observable change in the behavior of the functions J p (x1 , x2 ) and J (x1 , x2 ). The dashed lines in Figs. 5.34–5.36 correspond to the finite element method (FEM) applied for solution of the elasto-plastic problem in a square region of size 6a × 6a with the same elasto-plastic properties that contains an absolutely rigid inclusion of radius a in its center. For the FEM calculations, the commercial software ANSYS was used. The results of the calculations for an isotropic plane with a circular soft inclusion are presented in Figs. 5.37–5.40. The external stress field acts along the x1 -axis with

180

Heterogeneous Media

Figure 5.35 Distributions of the stress intensity J (0, x2 ) along the x2 -axis in the elasto-plastic plane with a rigid circular inclusion for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right); the plane is subjected to a uniaxial stress along the x1 -axis; dashed lines are the results of the FEM calculations.

Figure 5.36 Distributions of the plastic strain intensity J p (x1 , 0) along the x1 -axis in the elasto-plastic plane with a rigid circular inclusion for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right); the plane is subjected to a uniaxial stress along the x1 -axis; dashed lines are the results of the FEM calculations.

the initial σ 0(0) = 30 MPa and final σ 0(1) = 90 MPa values with the loading step σ 0 = 0.5 MPa. The contour plot of the intensity of the plastic deformation is presented in Fig. 5.37 for the material with E0 = 70 GPa, ν0 = 0.3, Et = 35 GPa, and σy = 100 MPa. The elastic properties of the inclusion are E = 0.001 GPa and ν = 0.3. The influence of the node grid step h and the tangent modulus Et on the distributions of the intensities of stresses and plastic deformations can be seen in Figs. 5.38–5.40. The dashed lines in these figures correspond to the FEM calculations for a square plate of size 6a × 6a with a central hole for the same elasto-plastic properties of the material and the loading regime. It is seen from Figs. 5.34–5.36 and 5.38–5.40 that the value of the tangent modulus Et does not affect the stress distributions around the inclusions but essentially changes the value of the plastic deformations. The number of iterations at each step of the

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181

Figure 5.37 The contour plots of the intensity of plastic deformations J p around a circular hole in the elasto-plastic plane subjected to a uniaxial stress along the x1 -axis.

Figure 5.38 Distributions of the stress intensity J (x1 , 0) along the x1 -axis in the elasto-plastic plane with a circular hole for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right); the plane is subjected to a uniaxial stress along the x1 -axis; dashed lines are the result of the FEM calculations.

loading depends on the value of the tangent modulus, and this number is usually small. As a rule, three or four iterations are sufficient to get a stable result. The distributions of stress and plastic strain intensities in the plane with two circular holes of the same radius a with the distance 4a between their centers are shown in Figs. 5.41 and 5.42. The centers of the holes are on the x2 -axis and the external stress acts along the x1 -axis, changing from 30 MPa to 90 MPa. The elasto-plastic properties

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Heterogeneous Media

Figure 5.39 Distributions of the stress intensity J (0, x2 ) along the x2 -axis in the elasto-plastic plane with a circular hole for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right). The plane is subjected to a uniaxial stress along the x1 -axis. The dashed line shows the results of the FEM calculations.

Figure 5.40 Distributions of the plastic strain intensity J p (0, x2 ) along the x2 -axis in the elasto-plastic plane with a circular hole for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right); the plane is subjected to a uniaxial stress along the x1 -axis; dashed lines are the results of the FEM calculations.

of the plane are the same as in the previous examples. The contour plots of the function J p (x1 , x2 ) in the region (−3a < x1 < 3a, 0 < x2 < 5a) are presented in Fig. 5.41, and the distribution of the stress and plastic strain intensities along the x2 -axis are shown in Fig. 5.42. The dashed lines in these figures are the results of the FEM calculations. It can be noted from Figs. 5.34–5.36 and 5.38–5.40 that the numerical results of the presented method and of the FEM calculation are very close in the case of a rigid inclusion and slightly deviate in the case of a hole. It can be attributed to a small Gibbs effect that takes place when the Gaussian functions are used for approximations. This effect appears at the points of jumps of approximated functions. In spite of the fact that this effect is almost not observable for elasticity problems (see Section 5.5 and [7], [8]), it can accumulate and affect final values of plastic deformations by repeating solutions of the linearized problems. In the case of a rigid inclusion, the stress con-

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183

Figure 5.41 The contour plots of the intensity of plastic deformations J p around two equal circular holes in the elasto-plastic plane subjected to a uniaxial stress along the x1 -axis; the hole centers are at the points (0, 1.5a) and (0, −1.5a), and a half-plane x2 > 0 is shown in the figure.

Figure 5.42 Distributions of the stress intensity J (0, x2 ) and the plastic strain intensity J p (0, x2 ) along the x2 -axis in the elasto-plastic plane with two equal circular holes; the calculations are performed for h/a = 0.02, Et = 35 GPa; the dashed lines are the results of the FEM calculations.

centration on the inclusion boundary is not as strong as for a hole, and the predictions of the presented method and of the FEM almost coincide.

5.6.3 Solution of the integral equation (5.138) in the 3D case In the 3D case, the discretization of Eq. (5.138) using the Gaussian approximating functions yields the following linear algebraic system for the coefficients of the ap(n) proximation σij :

(m)

σij

−

Mn  (m,n) (n) 0(m) (n) ij kl B , m = 1, 2, ..., Mn. klmn σmn = σij

(5.151)

n=1

(n) is defined in Eqs. (5.141) and (5.142), (m,n) = ij kl (x (m) − Here the tensor B klmn ij kl x (n) ), and the function ij kl (x) is defined in Eq. (5.105). The discretized Eq. (5.151)

184

Heterogeneous Media

is presented in the matrix form similar to Eq. (5.146) (I − B)X = F,

(5.152)

where I is the unit matrix of the dimensions 6Mn × 6Mn and the vectors of the unknowns X and the right hand side F are X = |X (1) , X (2) , ..., X (6Mn) |T , F = |F (1) , F (2) , ..., F (6Mn) |T , ⎧ (n) σ11 , ⎪ ⎪ ⎪ ⎪ (n−Mn) ⎪ ⎪ , σ22 ⎪ ⎪ ⎪ ⎨ σ (n−2Mn) , 33 X (n) = (n−3Mn) ⎪ , 2σ12 ⎪ ⎪ ⎪ ⎪ (n−4Mn) ⎪ ⎪ , 2σ13 ⎪ ⎪ ⎩ (n−5Mn) , 2σ23 ⎧ 0(n) σ11 , ⎪ ⎪ ⎪ ⎪ 0(n−Mn) ⎪ ⎪ , σ22 ⎪ ⎪ ⎪ ⎨ σ 0(n−2Mn) , 22 F (n) = ⎪ 2σ 0(n−3Mn) , ⎪ 12 ⎪ ⎪ ⎪ 0(n−4Mn) ⎪ ⎪ , 2σ ⎪ 13 ⎪ ⎩ 0(n−5Mn) , 2σ23

(5.153)

n ≤ Mn, Mn < n ≤ 2Mn, 2Mn < n ≤ 3Mn, 3Mn < n ≤ 4Mn, 4Mn < n ≤ 5Mn, 5Mn < n ≤ 6Mn, n ≤ Mn, Mn < n ≤ 2Mn, 2Mn < n ≤ 3Mn, 3Mn < n ≤ 4Mn,

(5.154)

4Mn < n ≤ 5Mn, 5Mn < n ≤ 6Mn.

The matrix B in Eq. (5.152) has the dimensions 6Mn × 6Mn and consists of 36 submatrices bpq of the dimensions Mn × Mn,   B = bpq  , p, q = 1, 2, ..., 6,

(5.155)

(m,n) (n) (m,n) = ppij B bpq ij qq , p, q = 1, 2, 3, (m,n)

(m,n) (n) (m,n) (m,n) 1(n) (m,n) (m,n) 1(n) = 12ij B = 13ij B = 23ij B ij qq , b5q ij qq , b6q ij qq ,

(m,n)

(m,n) 1(n) (m,n) (m,n) (n) (m,n) (m,n) 1(n) = qqij B = qqij B = qqij B ij 12 , bq5 ij 13 , bq6 ij 23 ,

b4q

bq4

q = 1, 2, 3, (m,n)

(m,n) (n) (m,n) (m,n) (n) (m,n) (m,n) (n) = 12ij B = 12ij B = 12ij B ij 12 , b45 ij 13 , b46 ij 23 ,

(m,n)

(m,n) (n) (m,n) (m,n) (n) (m,n) (m,n) (n) = 13ij B = 13ij B = 13ij B ij 12 , b55 ij 13 , b56 ij 23 ,

(m,n)

 (m,n) (n) (m,n) (m,n)B1(n) (m,n) (m,n) (n) = 23ij B = 23ij ij 13 , b66 = 23ij B ij 12 , b65 ij 23 .

b44 b54 b64

(5.156)

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185

Figure 5.43 Distributions of the stress intensity J (x1 , 0) along the x1 -axis in the elasto-plastic 3D medium with a rigid spherical inclusion for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right). The medium is subjected to a uniaxial stress along the x1 -axis.

In these equations, m, n = 1, 2, ..., Mn; summation from 1 to 3 with respect to repeating indices i, j is implied. These equations follow from Eqs. (5.151), (5.153), and (5.154). For the numerical solution of Eq. (5.151), the MRM (5.44)–(5.48) can be used. (m,n) Teoplitz properties of the object ij kl that hold for regular grids of approximating nodes allow application of the FFT algorithm for fast calculation of the matrix-vector products with the matrix B. Let us apply the method to the calculation of elasto-plastic fields in the medium with a rigid spherical inclusion of radius a centered at the origin of the Cartesian coordinate system (x1 , x2 , x3 ). The medium is isotropic with the Young modulus and Poisson ratio E0 = 70 GPa and ν0 = 0.3, and the initial yielding stress is σy = 100 MPa. The law of hardening has the form of Eqs. (5.125)–(5.128) with a constant parameter Et . The Young modulus of the inclusion E = 2000 GPa and its Poisson ratio ν = 0.5. The medium with the inclusion is subjected to a constant 1D external stress in the direction of the x1 -axis. The process of loading starts with σ 0 = σ 0(0) = 60 MPa and finishes with σ 0 = σ 0(1) = 90 MPa. The problem is axisymmetric, and the distributions of the stress and plastic strain intensities along the x1 -axis are shown in Figs. 5.43 and 5.44. Convergence of the method with decreasing the grid step h (h/a = 0.04, 0.02, 0.01) can be observed on the left hand sides of Figs. 5.43 and 5.44, and the dependence on the parameter Et can be observed on the right hand sides of Figs. 5.43 and 5.44. The case of a spherical hole in the medium with the same properties is presented in Figs. 5.45 and 5.46. The medium is subjected to an uniaxial external stress σ 0 acting along the x1 -axis and changes from 30 MPa to 90 MPa with the step σ 0 = 1 MPa. Convergence of the method with respect to the grid step h is seen in Figs. 5.45 and 5.46. It is seen from the right hand sides of Figs. 5.45 and 5.46 that the stress intensity J does practically not depend on the parameter Et , but the plastic strain intensity J p depends strongly on the Et value.

186

Heterogeneous Media

Figure 5.44 Distributions of the plastic strain intensity J p (x1 , 0) along the x1 -axis in the elasto-plastic 3D medium with a rigid spherical inclusion for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right). The medium is subjected to a uniaxial stress along the x1 -axis.

Figure 5.45 Distributions of the stress intensity J (0, x2 ) along the x2 -axis in the elasto-plastic 3D medium with a spherical cavity for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right). The medium is subjected to a uniaxial stress along the x1 -axis.

Figure 5.46 Distributions of the plastic strain intensity J p (x1 , 0) along the x1 -axis in the elasto-plastic 3D medium with a spherical cavity for various values of the grid step h (left) and various tangent moduli Et in the plastic region (right); the medium is subjected to a uniaxial stress along the x1 -axis.

Numerical solution of volume integral equations for static fields in heterogeneous media

5.7

187

Notes

The content of this chapter is based on the following publications: numerical solution of the electrostatic problem for heterogeneous media [9]; thermo-elastic deformation of heterogeneous media [7], [8], [10]; elasto-plastic deformation of a medium with heterogeneous inclusions [11].

Appendix 5.A The computational program for numerical solution of volume integral equations of electrostatics for heterogeneous media In this appendix, the computational program for calculation of the steady electric field in a conductive medium with a group of four inclusions shown in Fig. 5.5 is presented. Initial data. M1 , M2 , M3 are the numbers of the nodes along the x1 -, x2 -, and x3 -axes; Mn = M1 M2 M3 is the total number of nodes; L1 , L2 , L3 are the sizes of the region of approximation along the x1 -, x2 -, and x3 -axes; h = L1 /(M1 − 1) is the step of the approximating grid; l1 , l2 , l3 are the coordinates of the node x (1,1,1) ; c0 , c are the dielectric primitivities of the host medium and of the inclusions; SC are the coordinates of the centers of the spheres; r0 is the sphere radius. (*Initial data*) (*Dielectric primitivities of the host medium and of the inclusion*) c0=1; c=0.1; c1=c-c0; (*Inclusion radii*) r0=0.3; (*The coordinates of the centers of the inclusions*) SC={{0.5,0.5,0},{-0.5,0.5,0},{0.5,-0.5,0},{-0.5,-0.5,0}}; (*Node generation*) M1=161; M2=161; M3=81; N1=2*M1; N2=2*M2; N3=2*M3; h=2/(M1-1); Mn=M1*M2*M3; l1=-1; l2=-1; l3=-0.5; (*Cartesian coordinates of the nodes*) x1=Flatten[Table[-1+h*(i-1),{k,M3},{j,M2},{i, M1}]]; x2=Flatten[Table[-1+h*(j-1),{k,M3},{j,M2},{i, M1}]];

188

Heterogeneous Media

x3=Flatten[Table[-0.5+h*(k-1),{k,M3},{j,M2},{i, M1}]]; (*Connection between one- and three-index numeration*) M[i_,j_,k_]:=i+M1*(j-1)+M1*M2*(k-1) To1In[A_]:=Flatten[Transpose[Flatten[Transpose[A],1]]] To3In[A_]:=Transpose[Partition[Transpose[Partition[A,M1*M2]],M1]] (*Indication of the region occupied by the inclusions*) in1[k_] := Select[Table[If[Norm[X[[i]]-SC[[k]]] 0 &] Do[in[k]=in1[k],{k,4}]; IN=Join[in[1],in[2],in[3],in[4]] (*Deviation of the properties at the nodes from the properties of the host medium*) C1= Module[{},C1=Table[0,{Mn}];C1[[IN]]=c1;C1]; (*2. Distances and directions between nodes*) R[r_,s_]:=Sqrt[(x1[[r]]-x1[[s]]) ˆ 2+(x2[[r]]-x2[[s]]) ˆ 2+(x3[[r]]-x3[[s]]) ˆ 2]+ 0.001*h n1[r_,s_]:=N[(x1[[r]]-x1[[s]])/R[r,s]] n2[r_,s_]:=N[(x2[[r]]-x2[[s]])/R[r,s]] n3[r_,s_]:=N[(x3[[r]]-x3[[s]])/R[r,s]] (*3. Approximating functions*) H=1; f0[z_]:=N[1/(Pi*H) ˆ (3/2)*Exp[-z ˆ 2/H]] f1[z_]:=(3/(4*Pi*z ˆ 3))*Erf[z/Sqrt[H]]-(3*H+2*z ˆ 2)*f0[z]/(2*z ˆ 2) F00[r_,s_]:=N[f0[R[r,s]/h]] Ps1[r_,s_]:=N[f1[R[r,s]/h]] (*The right hand side of the discretized system*) F[1]= Module[{F1}, F1 = Table[0, {3*Mn}]; Do[F1[[p]] = 1, {p, Mn}]; F1] F[2]= Module[{F2}, F2 = Table[0, {3*Mn}]; Do[F2[[p+Mn]] = 1, {p, Mn}]; F2]; F[3]:= Module[{},F3 = Table[0, {3*Mn}];Do[F3[[p+2*Mn]] = 1, {p, Mn}]; F3] (*The matrix of the discretized problem*) A[r_, s_] :=Module[{ps01, ps11, n11, n21, n31, v0, AA}, p01=F00[r,s]; p02 = Ps1[r, s]; n11 = n1[r, s]; n21 = n2[r, s]; n31 = n3[r, s]; AA = Table[0.,{3},{3}]; AA[[1, 1]] = (p01+p02)/3-p02*n11 ˆ 2; AA[[1, 2]] = -p02*n11*n21; AA[[1, 3]] = -p02*n11*n31; AA[[2, 2]] = (p01+p02)/3-p02*n21 ˆ 2; AA[[2, 3]] = -p02*n21*n31; AA[[3, 3]] = (p01+p02)/3-p02*n31 ˆ 2;

Numerical solution of volume integral equations for static fields in heterogeneous media

189

AA[[2, 1]] = p01;AA]; (*Auxiliary object for the FFT algorithm of calculation of the matrix-vector products*) FtMg= Module[{MG}, FtMg = Table[0, {3}, {3}, {N1}, {N2}, {N3}]; MG = Module[{AA}, MG = Table[0., {3}, {3}, {N1}, {N2}, {N3}]; MG[[All, All, 1, 1, 1]] = A[1, 1]; Do[AA = A[M[l, 1, 1], 1]; MG[[All, All, l, 1, 1]] = AA; MG[[All, All, N1 - l + 2, 1, 1]] = AA, {l, 2, M1}]; Do[AA = A[M[1, m, 1], 1]; MG[[All, All, 1, m, 1]] = AA; MG[[All, All, 1, N2 - m + 2, 1]] = AA, {m, 2, M2}]; Do[AA = A[M[1, 1, k], 1]; MG[[All, All, 1, 1, k]] = AA; MG[[All, All, 1, 1, N3 - k + 2]] = AA, {k, 2, M3}]; Do[AA = A[M[l, m, 1], 1]; MG[[All, All, l, m, 1]] = AA; MG[[All, All, N1 - l + 2, N2 - m + 2, 1]] = AA; AA = A[M[l, 1, 1], M[1, m, 1]]; MG[[All, All, N1 - l + 2, m, 1]] = AA; MG[[All, All, l, N2 - m + 2, 1]] = AA,{l, 2, M1}, {m,2, M2}]; Do[AA = A[M[1, m, k], 1]; MG[[All, All, 1, m, k]] = AA; MG[[All, All, 1, N2 - m + 2, N3 - k + 2]] = AA; AA = A[M[1, m, 1], M[1, 1, k]]; MG[[All, All, 1, m, N3 - k + 2]] = AA; MG[[All, All, 1, N2 - m + 2, k]] = AA,{m, 2, M2}, {k, 2, M3}]; Do[AA = A[M[l, 1, k], 1]; MG[[All, All, l, 1, k]] = AA; MG[[All, All, N1 - l + 2, 1, N3 - k + 2]] = AA; AA = A[M[l, 1, 1], M[1, 1, k]]; MG[[All, All, N1 - l + 2, 1, k]] = AA; MG[[All, All, l, 1, N3 - k + 2]] = AA,{l, 2, M1}, {k, 2, M3}]; Do[AA = A[M[l, m, k], 1]; MG[[All, All, l, m, k]] = AA; MG[[All, All, N1 - l + 2, N2 - m + 2, N3 - k + 2]] = AA; AA = A[M[l, 1, k], M[1, m, 1]]; MG[[All, All, N1 - l + 2, m, k]] = AA; MG[[All, All, l, N2 - m + 2, k]] = AA; AA = A[M[l, m, 1], M[1, 1, k]]; MG[[All, All, l, m, N3 - k + 2]] = AA; MG[[All, All, N1 - l + 2, N2 - m + 2, k]] = AA; AA = A[M[1, m, k], M[l, 1, 1]]; MG[[All, All, N1 - l + 2, m, k]] = AA;

190

Heterogeneous Media

MG[[All, All, l, N2 - m + 2, N3 - k + 2]] = AA; AA = A[M[l, 1, 1], M[1, m, k]]; MG[[All, All, l, N2 - m + 2, N3 - k + 2]] = AA; MG[[All, All, N1 - l + 2, m, k]] = AA; AA = A[M[1, m, 1], M[l,1, k]]; MG[[All, All, N1 - l + 2, m, N3 - k + 2]] = AA; MG[[All, All, l, N2 - m + 2, k]] = AA, {l, 2,M1}, {m, 2, M2}, {k, 2, M3}]; MG]; Do[FtMg[[i,j]] = Fourier[MG[[i,j]]],{i,3},{j,i,3}]; FtMg]; (*Product of the matrix of the discretized problem and a vector*) ZZ[P_] := Module[{pc,RH}, PC[1] = C1*P[[1;;Mn]]; PC[2] = C1*P[[Mn+1;;2*Mn]]; PC[3] = C1*P[[2*Mn+1;;3*Mn]]; Do[pc = Table[0,{N1},{N2},{N3}];Inpc[k] = To3In[PC[k]]; pc[[1;;M1,1;;M2,1;;M3]]=Inpc[k];tpc[k]=pc,{k,3}]; Do[Ftpc[k] = Fourier[tpc[k]],{k,3}]; Do[Z0[n] = Sqrt[N1*N2*N3]*InverseFourier[Sum[FtMg[[k, n]]*Ftpc[k],{k,3}]], {n,3}]; Do[Z[n] = Z0[n][[1;;M1,1;;M2,1;;M3]],{n,3}]; RH = P + Join[To1In[Z[1]], To1In[Z[2]],To1In[Z[3]]];RH] (*The residue*) hh[P_]:=ZZ[P]-F0 (*The minimal residual method*) IT=Do[del=1;F0=F[1];P[1]=F0; While[del>0.0001, Do[hh1=hh[P[1]]; tah=ZZ[hh1]; tau=(hh1.tah)/(tah.tah); P[2]=P[1]-tau*hh1;P[1]=P[2]; del=Abs[tau*Sqrt[(hh1.hh1)/(P[1].P[1])]];Print[del]]]] (*The components of the electric field vector at the nodes*) E1[1]=P[1][[1;;Mn]]; E1[2]=P[1][[Mn+1;;2*Mn]]; E1[3]:=P[1][[2Mn+1;;3Mn]]; (*Interpolation to the entire region*) IE[k_]:= Interpolation[Table[{{x1[[i]], x2[[i]], x3[[i]]}, Re[E1[k][[i]]]}, {i, Mn}]] End

References [1] A. Peterson, S. Ray, R. Mittra, Computational Methods for Electromagnetics, IEEE Press, NY, 1997.

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[2] A. Samokhin, Integral Equations and Iterative Methods in Electromagnetic Scattering, VSP, Utrecht, Boston, Köln, Tokyo, 2001. [3] W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed., Cambridge University Press, 1992. [4] J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997. [5] S. Kanaun, V. Levin, Self-Consistent Methods for Composites, vol. 1, Static Problems, Springer, Dordrecht, 2008. [6] L. Kachanov, Fundamentals of the Theory of Plasticity, Dover Publications, New York, 2004. [7] S. Kanaun, Fast calculation of elastic fields in a homogeneous medium with isolated heterogeneous inclusions, International Journal of Multiscale Computational Engineering 7 (4) (2009) 263–276. [8] S. Kanaun, An efficient numerical method for calculation of elastic and thermo-elastic fields in a homogeneous media with several heterogeneous inclusions, World Journal of Mechanics 1/2 (2011) 31–43. [9] S. Kanaun, S. Babaii, A numerical method for the solution of thermo- and electro-static problems for a medium with isolated inclusions, Journal of Computational Physics 192 (2003) 471–493. [10] S. Kanaun, E. Pervago, Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures, International Journal of Engineering Sciences 49 (5) (2011) 420–442. [11] S. Kanaun, R. Martinez, Numerical solution of the integral equations of elasto-plasticity for a homogeneous medium with several heterogeneous inclusions, Computational Materials Science 55 (2012) 147–156.

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Cracks in heterogeneous media

6

This chapter is devoted to numerical solution of the surface integral equations of the crack problem of static elasticity and quasistatic poroelasticity. For heterogeneous media, the crack problem is formulated in terms of combinations of volume and surface integral equations. Cracks with planar and curvilinear surfaces are considered. Approximation of the crack opening by Gaussian quasiinterpolants reduces the problems to solution of systems of linear algebraic equations (the discretized problems). The elements of the matrix of the discretized crack problems are expressed in terms of standard 1D integrals that can be tabulated. For planar cracks, these elements are calculated in explicit analytical forms, and for regular node grids, the matrix of the discretized problem has a Toeplitz structure. The FFT algorithm substantially accelerates the calculation of matrix-vector products by iterative solution of the discretized problem. The method is applied to analysis of crack–crack and crack–inclusion interactions. In the case of quasistatic poroelasticity, the crack problem is time-dependent, and for numerical solution, additional discretization with respect to time is required.

6.1 A planar crack of arbitrary shape in a homogeneous elastic medium Let a homogeneous elastic host medium with stiffness tensor Cij0 kl contain a crack – a cut along a smooth surface  with the normal ni (x). The medium is subjected to external stress field σij0 (x), and bi (x) is the crack opening vector. This problem is considered in Section 3.4, and the stress field in the medium with a crack is presented in the form  σij (x) = σij0 (x) + Sij kl (x − x  )nk (x  )bl (x  )d , (6.1) 

0 − Cij0 kl δ(x), Sij kl (x) = Cij0 mn Kmnpq (x)Cpqkl

Kij kl (x) = −∂i ∂k gj l (x)|(i,j )(k,l) .

(6.2)

Here, gij (x) is the Green function of the host medium. For stress-free crack sides, the boundary condition on  takes the form nj (x)σij (x)| = 0.

(6.3)

Substitution in this condition of the stress tensor from Eq. (6.1) yields the integral equation for the crack opening vector bi (x)  Tij (x, x  )bj (x  )d = ti0 (x), x ∈ , (6.4) 

Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00013-5 Copyright © 2021 Elsevier Ltd. All rights reserved.

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Heterogeneous Media

Tij (x, x  ) = −nk (x)Skij l (x − x  )nl (x  ), ti0 (x) = nj (x)σij0 (x).

(6.5)

Let  be a finite part of a plane P , and in the Cartesian coordinates (x1 , x2 , x3 ), this plane is defined by the equation x3 = 0. In this case, the normal ni is constant, and the kernel Tij (x, x  ) in the integral equation (6.4) depends on the difference of the arguments Tij (x, x  ) = Tij (x − x  ). The function Tij (x) = Tij (x1 , x2 ) in the 2D space is generated by the function Sij kl (x1 , x2 , x3 ) defined in the 3D space Tij (x1 , x2 ) = −S3ij 3 (x1 , x2 , x3 )|x3 =0 .

(6.6)

It follows from this equation that the 2D Fourier transform Tij∗ (k1 , k2 ) of the kernel Tij (x1 , x2 ) is the following integral:  ∞ 1 ∗ Tij (k1 , k2 ) = − S ∗ (k1 , k2 , k3 )dk3 . (6.7) 2π −∞ 3ij 3 For an isotropic medium, the Fourier transform Sij∗ kl (k1 , k2 , k3 ) of the function Sij kl (x1 , x2 , x3 ) is defined by Eq. (2.207), i.e.,   Sij∗ kl (k1 , k2 , k3 ) = −2μ0 Pij1 kl (m) + (2κ0 − 1)Pij2 kl (m) , mi =

ki λ0 + μ 0 , , κ0 = |k| λ0 + 2μ0

(6.8)

where λ0 , μ0 are Lame parameters of the host medium and the tensors Pij1 kl (m), ∗ Pij2 kl (m) are indicated in Eq. (2.212). After substituting S3j k3 (k1 , k2 , k3 ) in Eq. (6.7) ∗ and integrating over the k3 -coordinate, we obtain for Tij (k1 , k2 ) the equation  μ0  k δij + (2κ0 − 1)(ni nj + ei ej ) , 2  ki ei = , i = 1, 2, e3 = 0, k = k12 + k22 . k Tij∗ (k1 , k2 ) =

(6.9) (6.10)

For numerical solution of Eq. (6.4) in the case of a planar crack, the crack region  is embedded in a rectangular W (Fig. 4.17). Then, W is covered by a square node grid with the step h, and the vector bi (x) is approximated by the Gaussian quasiinterpolant in W bi (x) =

Mn 

(n)

(n)

bi ϕ(x − x (n) ), bi

= bi (x (n) ),

(6.11)

n=1



x2 1 ϕ(x) = exp − 2 , x = (x1 , x2 ). πH h H

(6.12) (n)

Here, x (n) are the nodes of the approximation, and bi = 0 if x (n) ∈ / . Substitution of Eq. (6.11) into Eq. (6.4) and application of the collocation method yields the system

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195

of linear algebraic equations for the coefficients of the approximation 1  (m,n) (n) 0(m) (m) I ij bj = ti , x (m) ∈ , bi = 0, x (m) ∈ / , m = 1, 2, ..., Mn, h Mn

n=1

(6.13) (m,n) I ij

bi(m)

= I ij (x (m) − x (n) ), = bi (x (m) ),  I ij (x) = h Tij (x − x  )ϕ(x  )dx  .

ti0(m)

= ti0 (x (m) ),

(6.14) (6.15)

Here, the integral I ij (x) is calculated over the entire plane x3 = 0. After transition to the Fourier transforms of the integrand function and integrating first over the polar angle in the (k1 , k2 )-plain and then over the radial coordinate k, we obtain  I ij (x) = h Tij (x − x  )ϕ(x  )dx  = 2

 h3 k h2 H ∗ Tij (k) exp − = − ik · x dk = 4 (2π)2 xi = 1 (|x|)δij + 2 (|x|)ei ej + 3 (|x|)ni nj , ei = , (6.16) |x| 

 μ0 |x| |x| |x| |x|

1 (|x|) = F1 − F2 + 2κ0 F1 + F2 , 2 h h h h (6.17) |x|

2 (|x|) = −μ0 (2κ0 − 1)F2 , (6.18) h   μ0 |x| |x|

3 (|x|) = (2κ0 − 1) F1 + F2 . (6.19) 2 h h In these equations, two functions F1 (ρ) and F2 (ρ) of the dimensionless variable ρ = |x|/ h have the forms

2 2  ρ2 ρ2 ρ ρ 1− I0 + I1 , H 2H H 2H (6.20)

 2 2

2  2 2 1 ρ ρ ρ ρ ρ I0 − 1+ I1 . F2 (ρ) = exp − √ 2H H 2H H 2H 2H πH (6.21)

ρ2 F1 (ρ) = exp − 2H 2H πH 1 √



Here, I0 (z) , I1 (z) are the modified Bessel functions. The system of linear algebraic equations (6.13) can be presented in the matrix form AX = F,

(6.22)

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Heterogeneous Media

where the vectors of unknowns X and of the right hand side F are defined by the equations  T T  X = X 1 , X 2 , ..., X 3Mn , F = F 1 , F 2 , ..., F 3Mn , ⎧ (m) (m) m ≤ Mn, ⎪ ⎨ κ b1 , (m−Mn) m (m−Mn) X = κ b2 , Mn < m ≤ 2Mn, ⎪ ⎩ (m−2Mn) (m−2Mn) b3 , 2Mn < m ≤ 3Mn, κ ⎧ 0(m) m ≤ Mn, ⎪ ⎨ t1 , 0(m−Mn) m F = t2 , Mn < m ≤ 2Mn, ⎪ ⎩ 0(m−2Mn) , 2Mn < m ≤ 3Mn, t3 / . κ (m) = 1, x (m) ∈ , κ (m) = 0, x (m) ∈

(6.23)

(6.24)

(6.25) (6.26)

The matrix A of the dimensions 3Mn × 3Mn consists of nine block-matrices aij (i, j = 1, 2, 3) of the dimensions Mn × Mn,    a11 , a12 , a13      (m,n) A =  a21 , a22 , a23  , aij = aij , (6.27)  a31 , a32 , a33  1 (m,n) (m,n) = I ij , i, j = 1, 2, 3, m, n = 1, 2, ..., Mn. (6.28) aij h (m,n)

Because the components of the tensor I ij

have explicit analytical forms (6.16)–

(m,n)

are calculated fast. (6.21), the elements aij The system (6.22) can be solved by the minimal residual method (MRM) or the conjugate gradient method (CGM) (Section 5.2). For the discretized system (6.22), the CGM converges faster than the MRM, and the algorithm of the CGM is indicated in Eqs. (5.45)–(5.48). The text of the computational program for numerical solution of the crack problem is presented in Appendix 6.A.

6.1.1 A planar elliptical crack under normal tension Let a tensile stress σ 0 be applied orthogonal to the plane of an elliptical crack in an isotropic elastic medium. In this case, the crack opening vector has only one nonzero component b3 (x1 , x2 ), and its analytical expression has the form  2 2 x2 2a2 (1 − ν0 )σ 0 x1 − , B0 = . (6.29) b3 (x1 , x2 ) = B0 1 − a1 a2 μ0 E(1 − a22 /a12 ) Here, the x1 - and x2 -axes are directed along the a1 - and a2 -semiaxes of the ellipse , E(k) is the elliptic integral of the second kind.

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Figure 6.1 The normalized crack opening b(x1 , x2 ) of an elliptical crack with semi-axes a1 , a2 (a1 /a2 = 2) by normal tension; solid lines are the exact solution, lines with dots and squares are numerical solutions for the parameters of the Gaussian quasi-interpolants H = 2, h/a1 = 0.02 and h/a1 = 0.01.

For the elliptical crack with a1 /a2 = 2 in the host medium with Poisson ratio ν0 = 0.3, comparison of the numerical solution of Eq. (6.4) with the exact crack opening in Eq. (6.29) is presented in Fig. 6.1. Solid lines in this figure are the normalized exact crack opening b(x1 , x2 ) =

b3 (x1 , x2 ) , B0

(6.30)

and lines with squares and triangles correspond to the normalized numerical solutions bn (x1 , x2 ) for the steps of the node grid h/a1 = 0.05 (Mn = 2911) or h/a1 = 0.01 (Mn = 17701) and H = 2. For smaller steps, the numerical solutions are practically indistinguishable from the exact one. It is seen from Fig. 6.1 that the maximal error of the numerical solution is concentrated near the crack edge.

6.1.2 A circular crack with surface contacts We consider a penny-shaped crack that has several circular surface contacts (crack side connections, Fig. 6.2). The crack is subjected to external stresses of two kinds: (a) normal tension applied orthogonal to the crack plane and (b) shear stress acting parallel to the crack plane. The integral of the crack opening vector bi (x1 , x2 ) over the crack surface  Bi (λ) =

bi (x1 , x2 )d

(6.31)



relates to the crack contribution to the effective elastic compliance of a cracked medium (see Section 9.5). Here, the parameter λ (0 < λ < 1) is defined by the equation λ = 2a/ l, where a is the radius of the contacts and l is the distance between the

198

Heterogeneous Media

Figure 6.2 A penny shape crack with N circular side contacts on the crack surface: (A) N = 7, (B) N = 19, (C) N = 37, (D) N = 61.

nearest centers of the contacts. The graphs of the dimensionless functions β1 (λ) and β3 (λ) βi (λ) =

Bi (λ) Bi0

, i = 1, 3,

(6.32)

for various numbers of the contacts on the crack surface are presented in Figs. 6.3–6.5. Here Bi0 is the integral (6.31) for a penny-shaped crack without contacts. In Fig. 6.6, the 3D plot of the crack opening for a crack with 61 contacts is shown. In the calculations, the crack is embedded into a square with the sides 2R, where R is the crack radius, the steps of the node grid are taken h/R = 0.005 (Mn = 160801) and h/R = 0.001 (Mn = 4004001), H = 1, and the host medium has a Poisson ratio of ν = 0.3.

Cracks in heterogeneous media

199

Figure 6.3 The normalized crack compliance coefficient β1 and β3 in Eq. (6.32) for a circular crack with 7 surface contacts; the parameters of the Gaussian quasi-interpolant are H = 1, h/R = 0.005 (Mn = 160801) and h/R = 0.001, (Mn = 4004001).

Figure 6.4 The same as in Fig. 6.3 for a crack with 19 surface contacts.

Figure 6.5 The same as in Fig. 6.3 for a crack with 37 surface contacts.

6.2

Cracks with curvilinear surfaces

Let  be a curvilinear smooth surface of a crack with the normal vector ni (x). If the crack sides are free from stresses, the integral equation for the crack problem

200

Heterogeneous Media

Figure 6.6 The crack opening for a crack with 61 surface contacts by normal tension; (A) the 3D-plot of the crack opening, (B) the crack opening along the x1 -axis; the dashed line is one-fourth of the opening of a penny shape crack without surface contacts.

has the form (6.4). For discretization of this equation by the Gaussian approximating functions, the crack surface should be covered by a homogeneous grid of nodes. In the following, an algorithm of generation of such a node grid is considered.

6.2.1 Generation of a node grid on the surface of a curvilinear crack Let a smooth surface  in 3D space be defined by an implicit equation (x) = (x1 , x2 , x3 ) = 0.

(6.33)

For generation of a homogeneous node grid on , an arbitrary point x ∈  is taken as the original node x (1) . The next node x = x (2) is any solution of the equations |x (1) − x| = h, (x) = 0.

(6.34)

The third node x = x (3) is one of two solutions of the equations |x (1) − x| = h, |x (2) − x| = h, (x) = 0.

(6.35)

For calculation of the coordinates of the sth node x = x (s) (s = 4, 5, ...), we consider the system of equations |x (p) − x| = h, |x (q) − x| = h, (x) = 0, p, q = 1, 2, ..., s − 1, p = q.

(6.36)

  (s) Among all the solutions x = xm (m = 1, 2, ...) of this system we choose the one x (s) that satisfies the conditions (s) (s) | ≥ h, p = 1, 2, ..., s − 1, min |xm − x (1) | = |x (s) − x (1) |. |x (p) − xm m

(6.37)

Cracks in heterogeneous media

201

Figure 6.7 A spherical surface covered by a set of small spheres with constant radii; the approximating nodes of the Gaussian quasiinterpolants are the centers of the spheres, the total number of the node is Mn = 13680.

In the case of a planar crack, this algorithm provides a regular hexagonal node grid in the crack plane. For a spherical surface, an example of the generated node grid is presented in Fig. 6.7. The centers of small spheres shown in this figure compose a set of approximating nodes.

6.2.2 Discretization of Eq. (6.4) for a curvilinear crack Suppose that  is covered by a homogeneous set of nodes x (m) (m = 1, 2, ..., Mn) with an approximate distance h between neighbor nodes. At each node, we introduce a local Cartesian basis (g1(m) , g2(m) , g3(m) ) with the vector g3(m) normal to  3(m)   g = n(m) . The local Cartesian axes (y1 , y2 , y3 ) are directed along these basis vectors. We introduce also a global Cartesian coordinate system (x1 , x2 , x3 ) with the basis (e1 , e2 , e3 ) and a global spherical coordinate system (R, ϕ, θ ) with the same origin (x = 0) and the polar axis directed along the e3 -vector (Fig. 6.8). Each normal vector n(m) at the mth node on the surface  can be parallelly transferred into   the origin of the spherical system, and the polar and azimuthal angles ϕ (m) , θ (m) of this system are associated with the vector n(m) (m = 1, 2, ..., Mn). We take the directions of the unit vectors g1(m) , g2(m) of the local basis at the mth node coinciding with (m) (m) the directions of the basis vectors eθ and eϕ of the spherical system at the point  (m) (m)  on the unit sphere. Relations between the elements of the local and global ϕ ,θ bases are given by the equations

gi(m) =

3  j =1

(m)

Qj i ej , i = 1, 2, 3, m = 1, 2, ..., Mn,

(6.38)

202

Heterogeneous Media

Figure 6.8 Crack surface  with a set of approximating nodes; gi (p) and gi (s) (i = 1, 2) are the local bases at the pth and sth nodes in the tangent planes to  at the nodes, ei (i = 1, 2, 3) is the basis of the global Cartesian system, eR (p) , eϕ (p) , eθ (p) are the basis of the global spherical coordinate system and eR (p) = n(p) . (m)

where the elements of the matrix Qj i are expressed in terms of the polar angles ϕ (m) , θ (m) of the mth node ⎡ ⎤ cos ϕ cos θ, − sin ϕ, cos ϕ sin θ ⎣ sin ϕ cos θ, cos ϕ, sin ϕ sin θ ⎦ . (6.39) Q(m) ji = − sin θ, 0, cos θ Here, ϕ = ϕ (m) and θ = θ (m) . The local bases at the mth and nth nodes are related by the equation gi(m) =

3 

(m,n) j (n)

g

Qj i

,

(6.40)

j =1 (m,n)

where the matrix Qij (m,n)

Qij

=

3 

(m)

is (n)

Qki Qkj

(6.41)

k=1 (m)

and the matrix Qj i is defined in Eq. (6.39). If the Gaussian functions are used for discretization of the integral equation (6.4), the vector bi (x) is presented in the form bj (x) =

Mn  n=1

(n)

bj ϕ(x − x (n) ),

(6.42)

Cracks in heterogeneous media

203

where ϕ(x − x (n) ) is the Gaussian function concentrated in the plane P (n) tangent to the surface  at the nth node. In the local coordinate system (y1 , y2 , y3 ) of the mth node, this function has the form

1 |y|2 ϕ(y1 , y2 ) = (6.43) exp − 2 , y = (y1 , y2 ). πH h H As a result, the stress field in the medium with a crack in Eq. (6.1) is approximated by the following sum:  σij (x) = σij0 (x) +

Iij(n)l (x) =



Sij kl (x − x  )nk (x  )bl (x  )d σij0 (x) +

Mn 

(n)

(n)

Iij l (x)bl ,

n=1

(6.44)



Sij kl (x − x  )nk ϕ(x  − x (n) )dP  . (n)

P (n)

(6.45)

(n)

The integral Iij l (x) is calculated over the entire plane P (n) tangent to the surface  at the nth node. Let us consider calculation of this integral in the local basis of the nth node. Using the convolution property, we obtain Iij(n)k (y) = Iij k (y1 , y2 , y3 ) =  ∞ Sij k3 (y1 − y1 , y2 − y2 , y3 − y3 )ϕ(y1 , y2 )δ(y3 )dy1 dy2 dy3 = = −∞  ∞   1 = Sij∗ k3 (k1 , k2 , k3 )ϕ ∗ (k1 , k2 ) exp −i(km ym ) dk1 dk2 dk3 . 3 (2π) −∞ (6.46) Here, the function Sij∗ kl (k1 , k2 , k3 ) is defined in Eq. (6.8), and

2

k h2 H ϕ (k1 , k2 ) = h exp − 4 ∗

2

2

, k = k12 + k22 .

(6.47)

Straightforward calculation of the triple integral in Eq. (6.46) yields the result Iij k (y1 , y2 , y3 ) = Iij k (r, y3 ) = −s1 e(i θj )k + s2 θij ek + s3 n(i θj )k − s4 θij nk − − s5 e(i nj ) ek + s6 ei qj nk + s7 (2e(i nj ) nk + ni nj ek ) − s8 ni nj nk − s9 ei ej ek , (6.48)  yi (6.49) (i = 1, 2), e3 = 0, θij = δij − ni nj . r = y12 + y22 , ei = r (n)

Here, ni = ni is the normal to  at the nth node and the scalar coefficients sl = sl (r, y3 ) (l = 1, 2, ..., 9) are   (6.50) s1 = 2μ0 (g1 − 4κ0 g2 ), s2 = 2μ0 (1 − 2κ0 )g1 + 2κ0 g2 ,

204

Heterogeneous Media

  s3 = 2μ0 (g3 − 4κ0 g4 ), s4 = 2μ0 (1 − 2κ0 )g3 + 2κ0 g4 ,   s5 = 2μ0 (g5 − 4κ0 g6 ) , s6 = 2μ0 (1 − 2κ0 )g5 + 2κ0 g6 ,

(6.52)

s7 = 4μ0 κ0 g7 , s8 = 4μ0 κ0 g8 , s9 = 4μ0 κ0 g9 .

(6.53)

(6.51)

The functions gm in these equations are expressed in terms of five functions Fm of the dimensionless variables ρ = r/ h and ς = y3 / h, i.e., r y3 1 F2 , g3 = (F2 − F1 ), sign(y3 )(F3 + F4 ), g2 = 2 2hr 2h 2h   1 |y3 | 1 g4 = F 1 + F2 − (F3 + F4 ) , g5 = F2 , 4h h h



1 |y3 | y3 h F2 − F4 , g7 = 2 F5 − 4 F2 , g6 = 2h h r 2h 1 |y3 | y3 (F1 + F3 ), g9 = 2 F5 . g8 = 2h h 2h g1 =

(6.54) (6.55) (6.56) (6.57)

The functions Fm = Fm (ρ, ς) (m = 1, 2, 3, 4, 5) have the forms of 1D absolutely converging integrals

 ∞ 1 k2H J0 (kρ)k 2 dk, F1 (ρ, ς) = exp −k|ς| − (6.58) 4π 0 4

 ∞ 1 k2H F2 (ρ, ς) = J2 (kρ)k 2 dk, exp −k|ς| − (6.59) 4π 0 4

 ∞ 1 k2H F3 (ρ, ς) = J0 (kρ)k 3 dk, exp −k|ς| − (6.60) 4π 0 4

 ∞ 1 k2H F4 (ρ, ς) = J2 (kρ)k 3 dk, exp −k|ς| − (6.61) 4π 0 4

 ∞ 1 k2H J3 (kρ)k 3 dk. exp −k|ς| − (6.62) F5 (ρ, ς) = 4π 0 4 In these equations, Jn (kρ) are Bessel functions of the first kind. Asymptotics fm (ρ, ς) of the functions Fm (ρ, ς) for large values of the arguments ρ and ς are presented in the following forms: 2ς 2 − ρ 2 4π(ρ 2 + ς 2 )5/2 3ρ 2 f2 (ρ, ς) = 4π(ρ 2 + ς 2 )5/2 3ς(2ς 2 − 3ρ 2 ) f3 (ρ, ς) = 4π(ρ 2 + ς 2 )7/2 15ρ 2 ς f4 (ρ, ς) = 4π(ρ 2 + ς 2 )7/2

f1 (ρ, ς) =

3H (3ρ 4 − 24ρ 2 ς 2 + 8ς 4 ) , 16π(ρ 2 + ς 2 )9/2 15Hρ 2 (ρ 2 − 6ς 2 ) + , 16π(ρ 2 + ς 2 )9/2 15H ς(15ρ 4 − 40ρ 2 ς 2 + 8ς 4 ) − , 16π(ρ 2 + ς 2 )11/2 315H ςρ 2 (ρ 2 − 2ς 2 ) + , 16π(ρ 2 + ς 2 )11/2 −

(6.63) (6.64) (6.65) (6.66)

Cracks in heterogeneous media

205

Figure 6.9 Functions F1 (ρ, z) and F2 (ρ, z) in Eqs. (6.58), (6.59) (solid lines) and their asymptotics in Eqs. (6.63), (6.64) (dashed lines) for z = 0 and z = 1.

Figure 6.10 The same as in Fig. 6.9 for the function F3 (ρ, z) and F4 (ρ, z) in Eqs. (6.60), (6.61).

f5 (ρ, ς) =

15ρ 3 15Hρ 3 (ρ 2 − 8ς 2 ) + . 4π(ρ 2 + ς 2 )7/2 16π(ρ 2 + ς 2 )11/2

(6.67)

The functions Fm (ρ, ς) and their asymptotics fm (ρ, ς) are shown in Figs. 6.9–6.11, where solid lines are the functions Fm (ρ, ς) and dashed lines are the asymptotics. It is seen from these figures that Fm (ρ, ς) can be changed to their asymptotics if the variables ρ and ς satisfy the conditions ρ > 5 or ς > 5. For ρ ≤ 5, ς ≤ 5, the function Fm (ρ, ς) can be tabulated. (m,n) In the basis of the nth node, the stress tensor σij induced at the mth node by the (n) (n)

source ni bj ϕ(x − x (n) ) at the nth node is presented in the form (n) σij(m,n) = Iij(m,n) k bk . (m,n)

Here, the tensor Iij k (m,n) ) (r (m,n) , y3

(6.68) (m,n)

= Iij k (r (m,n) , y3

) is defined in Eqs. (6.48)–(6.62), where

are the components of the vector x(m) − x(n) in the local basis of the nth node. Thus, for the calculation of the tensor Iij(m,n) k , the arguments of the functions

206

Heterogeneous Media

Figure 6.11 The same as in Fig. 6.9 for the function F5 (ρ, z) in Eq. (6.62).

Fm (ρ, ς) in Eqs. (6.58)–(6.62) should be ρ = ρ (m,n) , ς = ς (m,n) , ρ (m,n) =

1 (m) 1 |(x − x(n) ) − hς (m,n) n(n) |, ς (m,n) = (x(m) − x(n) ) · n(n) . (6.69) h h

Here, point (·) is the scalar product of two vectors. In the local basis of the nth node, (m,n) has the form the tensor σij (m,n)

σij

= Iij k

(m,n) (n) bk ,

(m,n) (m,n) (m,n) (m,n) Iij k = Qip Qj l Iplk ,

(6.70)

(m,n)

is the transfer matrix from the basis gi(n) to the basis gi(m) defined in where Qij Eq. (6.41). In the basis of the mth node, the components of the stress vector acting on the infinitesimal area with the normal n(m) are (m,n)

ti

(m) (m,n)

= nj σj i

(6.71)

.

Finally, the total stress vector at the mth node induced by the sources acting at all nodes is ti(m) =

Mn  s=1

ti(m,n) =

Mn 

(m,n) (n) n(m) j Iij k bk .

(6.72)

s=1

The equations for the vectors b(n) follow from the boundary conditions (6.3) if the latter are satisfied at all the nodes (the collocation method). As a result, we obtain the system of linear algebraic equations for the components of the vector bi(n) in the local basis of each node in the form Mn 

(m,n) (n) (m) (m) bj =  ti , x (m) ∈ , bj = 0, x (m) ∈ / , m = 1, 2, ..., Mn, Tij

s=1

(6.73) (m,n) Tij

(m) (m,n) = −nk Ikij ,

(m)  ti

(m) = nj σij0 (x (m) ).

(6.74)

Cracks in heterogeneous media

207

Here, σij0 (x (m) ) are the components of the external stress tensor at the mth node in the local basis of this node. This system presents the discretized integral equation (6.4) in the case of a curvilinear crack by using Gaussian approximating functions.

6.3 Stress intensity factors at the crack edge For the Gaussian approximating functions, the maximal error of the numerical solution is concentrated near the boundary  of the crack surface . This is due to the fact that the Gaussian functions have infinite supports, whereas the actual crack opening bi (x) is equal to zero outside . Nevertheless, accurate calculation of the asymptotics of the crack opening vector bi (x) near  is important for assessment of the stress intensity factors (SIFs) in the medium near the crack edge [1], [2]. Further, an algorithm for calculation of these asymptotics in the framework of the numerical method based on the Gaussian approximating functions is considered. In this algorithm, the analytical structure of the solution bi (x) of Eq. (6.4) near the crack edge is taken into account.

Figure 6.12 The local basis (γ , τ , n) at an edge point x 0 of a crack for calculation of stress intensity factors.

We consider an arbitrary point x 0 at the crack edge  and introduce the local Cartesian basis γ (x 0 ), τ (x 0 ), n(x 0 ) with the origin at x 0 (Fig. 6.12). Vector n is normal to the crack surface , vector τ is tangent to the crack contour , and vector γ is orthogonal to  and belongs to the plane tangent to  at the point x 0 . The axes directed along the vectors γ , τ , n are ξ1 , ξ2 , ξ3 . The asymptotic of the crack opening vector b(x) near the point x 0 has the form [3]  b(ξ1 , 0, 0) = β(x 0 ) −ξ1 + O(|ξ1 |3/2 ), (6.75) where the vector β(x 0 ) = βγ γ + βτ τ + βn n

(6.76)

defines the principal term of the crack opening vector near the crack edge. The stress intensity factors KI , KII , and KIII at the point x 0 are expressed in terms of the components of the vector β(x 0 ), and in the case of an isotropic medium, these expressions

208

Heterogeneous Media

are as follows: (λ0 + 2μ0 ) (λ0 + 2μ0 ) βn (x 0 ), KII (x 0 ) = βγ (x 0 ), 8 8 μ0 βτ (x 0 ). KIII (x 0 ) = 4 KI (x 0 ) =

(6.77)

The SIFs KI , KII , and KIII define asymptotics of the stress field in the medium near the point x 0 by the equations KI (x 0 ) KII (x 0 ) σ33 (ξ1 , 0, 0) = √ + O(1), σ31 (ξ1 , 0, 0) = √ + O(1), ξ1 ξ1 KIII (x 0 ) + O(1), σ32 (ξ1 , 0, 0) = √ ξ1

(6.78) (6.79)

where σ31 , σ32 , σ33 are the components of the stress tensor in the local basis (γ , τ , n) at the point x 0 . Thus, the crack opening vector bi (ξ1 , 0) near the edge point x 0 is presented in the form  (6.80) bi (ξ1 , 0) = −ξ1 (βi0 + βi1 ξ1 + ...), where the series in parentheses is an analytical function of ξ1 . Let  bi (ξ1 , 0) be a numerical solution of the crack problem. Then the approximate values of the coefficients βi0 and βi1 can be found from solution of the classical problem of minimization of the square functional 

i (βi0 ) =

s2 s1

   1 , 0) 2  0 (β + β 1 ξ1 ) − b(ξ  dξ1 . √ i  i −ξ 

(6.81)

1

Here, (s1 , s2 ) is the interval in the ξ1 -axis near the crack edge, where the numerical solution is reliable. Minimization of the functional in Eq. (6.81) yields the approximate asymptotics of the vector bi (ξ1 , ξ2 ) in the vicinity of the crack contour . In other words, this procedure is interpolation of the numerical solution from the region of its reliable values to the crack edge using the correct analytical asymptotics of the crack opening vector. The interval of integration (s1 , s2 ) in Eq. (6.81) depends on the node grid step h (accuracy of the numerical solution). We consider numerical calculation of the coefficient β 0 in Eq. (6.75) for the crack opening vector near the edge of a planar crack. Let the crack contour  be defined by the equation R0 (ϕ) = R0 (ϕ)er .

(6.82)

Here, R0 (ϕ) is the distance from the origin of a polar coordinate system in the crack plane to the contour  along the ray with the polar angle ϕ, and er and eϕ are the unit

Cracks in heterogeneous media

209

vectors of the polar coordinate system (Fig. 6.13). For an elliptical crack, the function R0 (ϕ) takes the form R0 (ϕ) =

cos2 ϕ sin2 ϕ + a12 a22

− 1

2

.

(6.83)

In the crack plane, the tangent τ and normal γ vectors to  are defined by the equations 

− 1

1 dR0 2 2 1 dR0 r ϕ τ (ϕ) = 1 + e +e , R0 (ϕ) dϕ R0 (ϕ) dϕ 

− 1

1 dR0 2 2 r 1 dR0 ϕ γ (ϕ) = 1 + e . e − R0 (ϕ) dϕ R0 (ϕ) dϕ

(6.84)

(6.85)

The equation for the ξ1 -axis in Fig. 6.13 that passes through the point ϕ on  and is directed along the normal vector γ (ϕ) has the form r(ϕ, ξ1 ) = R0 (ϕ) + ξ1 γ (ϕ).

(6.86)

For ξ1 < 0, the point is on the crack surface ; for ξ1 > 0, it is outside .

Figure 6.13 The local basis (γ , τ ) at an edge point of a crack and the basis (er , eϕ ) of the polar coordinate system for calculation of stress intensity factors of a planar crack.

If the crack opening vector b(x1 , x2 ) is known in the Cartesian coordinate system (x1 , x2 ) shown in Fig. 6.13, its values along the local axis ξ1 are calculated from the equation b (X1 (ϕ, ξ1 ), X2 (ϕ, ξ1 )) = bγ (ϕ, ξ1 )γ + bτ (ϕ, ξ1 )τ + bn (ϕ, ξ1 )n.

(6.87)

Here, X1 (ϕ, ξ1 ), X2 (ϕ, ξ1 ) are Cartesian coordinates of points on the ξ1 -axis ξ 1 dR0 sin(ϕ), R0 (ϕ) dϕ ξ 1 dR0 cos(ϕ), X2 (ϕ, ξ1 ) = (ξ 1 + R0 (ϕ)) sin(ϕ) − R0 (ϕ) dϕ

X1 (ϕ, ξ1 ) = (ξ 1 + R0 (ϕ)) cos(ϕ) +

(6.88) (6.89)

210

Heterogeneous Media

Figure 6.14 The normalized stress intensity factor k1 at the contour of an elliptic crack with semi-axes a = a1 , a2 , (a1 /a2 = 2) subjected to normal tension; the solid line is the exact distribution of the SIF along the contour, lines with squares, triangles, and rhombs correspond to numerical solution with the parameters H = 1, h/a = 0.01, h/a = 0.005, h/a = 0.001.





1 dR0 ξ1 = 1 + R0 (ϕ) dϕ

2 − 12 ξ1 .

(6.90)

For calculation of the SIFs at the crack edge, we consider the limits βα (ϕ) = lim

ξ1 →0

bα (ϕ, ξ1 )) , α = γ , τ, n. √ −ξ1

(6.91)

The coefficients βn (ϕ), βγ (ϕ), βτ (ϕ) relate to the SIFs at the crack edge by Eq. (6.77). The exact SIF KI for an elliptical crack under normal tension σ 0 is the following function of the polar angle ϕ: 

1/4 μ0 R0 (ϕ) sin ϕ 2 R0 (ϕ) cos ϕ 2 K1 (ϕ) = + , (6.92) B0 4(1 − ν0 ) a1 a2 where B0 and R0 (ϕ) are defined in Eqs. (6.29) and (6.83). The distribution of the normalized SIF k1 (ϕ) k1 (ϕ) =

K1 (ϕ) λ0 + 2μ0 , K1 (0) = B0 K1 (0) 8

(6.93)

along the crack edge is presented in Fig. 6.14 by the solid line (a1 /a2 = 2). The lines with white squares, triangles, and dots correspond to the numerical solutions for h/a1 = 0.01 (Mn = 17701), h = 0.005 (Mn = 62809), and h = 0.001 (Mn = 1570721), respectively. For the calculation of the approximate coefficient βn (ϕ) in Eq. (6.76) using the numerical solution  bn (ϕ, ξ1 ), the following function is introduced:  b (ϕ, ξ ) n (ϕ, ξ1 ) = n√ 1 . β −ξ1

(6.94)

For ϕ = 0 and various values of the node grid steps h, this function is shown in Fig. 6.15. In this figure, the solid line corresponds to the exact solution. In order

Cracks in heterogeneous media

211

√

Figure 6.15 The function bn (0, ξ1 )/ −ξ1 in Eq. (6.94) for a circular crack of the radius a subjected to normal tension, solid line corresponds to the exact solution, lines with squares, triangles, dots, and rhombs are numerical solutions with the parameters H = 1, h/a = 0.01, 0.005, 0.004, 0.001; the dashed line is the tangent to the numerical solution for h/a = 0.001 in the region of reliable values of the crack opening.

n (ϕ) from the numerical solution, the function to extract an approximate value of β n (ϕ, ξ1 ) is interpolated from an interval  in a distance l from the crack edge into β the origin (ξ1 = 0) by a direct line (the dashed line in Fig. 6.15). In the calculations, the  distance l was taken about (20 ÷ 30)h,  ≈ 10h, and the function  βn (ϕ, ξ1 ) was apn (ϕ, ξ1 ) ≈ βn0 (ϕ) + βn1 (ϕ)ξ1 inside the interval  proximated by a linear function β using minimization of the functional (6.81). Then, this approximation was continued to the origin ξ1 = 0. For this algorithm, the error in calculations of SIFs is about 0.8% for h/a1 = 0.001. This error decreases together with h, as observed from Fig. 6.15.

6.4 Elastic bodies containing cracks Let the surface S of a finite homogeneous elastic body V be subjected to surface forces ti (x). The stress tensor in the body can be presented in the form of the potential double layer similar to Eq. (6.1)  σij (x) =

Sij kl (x − x  )nk (x  )bl (x  )d .

(6.95)

S

Here ni (x) is the external normal to S. For the boundary condition on S ni (x)σij (x) = ti (x), x ∈ S,

(6.96)

the integral equation for the vector bi (x) in Eq. (6.95) takes the form 

Tij (x, x  )nk (x  )bl (x  )d = −ti (x), x ∈ S,

(6.97)

S

Tij (x, x  ) = −nk (x)Skij l (x − x  )nl (x  ).

(6.98)

212

Heterogeneous Media

The stress vector ti (x) in this equation should satisfy the conditions of equilibrium (the total force and moment applied to the body are equal to zero)   ti (x)dS = 0, ij k xj tk (x)dS = 0, (6.99) S

S

where ij k is the Levi-Civita symbol. Let an elastic body V contain a finite number M of cracks with surfaces (m) , m = 1, 2, ..., M. The stress tensor in V can be presented as the sum of the potentials of the double layers concentrated on the body boundary S and on the crack surfaces (m) , i.e.,  (0) (0) σij (x) = Sij kl (x − x  )nk (x  )bl (x  )dS  + S

+

M   m=1

Sij kl (x − x  )nk (x  )bl (m)

(m)

(m)

(x  )d .

(6.100)

(0)

Here, bl is the density of the potential concentrated on the body boundary S with (0) (m) (m) the normal nk (x), bl (x) is the crack opening vector of the mth crack, and nk (x) (m) is the normal to the crack surfaces  . The system of integral equations for vectors (0) (m) bi (x) and bi (x) follows from the conditions (6.96) on the body boundary S and on the crack surfaces (m) , (m)

ni (x)σij (x) = 0, x ∈ (m) , m = 1, 2, ..., M.

(6.101)

Substituting in these conditions the stress tensor from Eq. (6.100), we obtain the fol(0) (m) lowing system of integral equations for the vectors bi (x) and bi (x): 

(0,0)

S

Tij

(x, x  )bi (x  )dS  + (0)

M  

(0,n)

(n) n=1 

Tij

(x, x  )bj (x  )dS  = −ti (x), x ∈ S, (n)

(6.102) 

(m,0)

S

Tij

(x, x  )bi (x  )dS  + (0)

M   n=1

(n)

(m,n)

Tij

(x, x  )bj (x  )dS  = 0, x ∈ (m) , (n)

(6.103) m = 1, 2, ..., M. In these equations, (m,n)

Tij

(x, x  ) = −nk (x)Skij l (x − x  )nl (x  ), m, n = 0, 1, 2, ...M. (m)

(n)

(6.104)

Thus, the integral operators in Eqs. (6.102) and (6.103) have similar kernels, which is convenient for numerical solution of the system (6.102)–(6.103).

Cracks in heterogeneous media

213

6.4.1 Discretization of the integral equations (6.102) and (6.103) Discretization of the system of integral equations (6.102)–(6.103) by Gaussian approximating functions is performed similar to discretization of the integral equation (6.4) for cracks with curvilinear surfaces. In the previous notations, the discretized system takes the form N0 

(m,n) (n) bj + Tij

Mn 

(m,n) (n) (m) bj =  ti , m = 1, 2, ..., Mn, Tij

(6.105)

n=N0 +1

n=1

(m)  ti(m) = −ti (x (m) ), m = 1, 2, ..., N0 ,  ti = 0, m = N0 + 1, N0 + 2, ..., Mn. (6.106)

Here, the nodes 1, 2, ..., N0 are on the body surface S and the nodes N0 + 1, N0 + (m,n) are defined in Eqs. (6.74), 2, ..., Mn are on the crack surfaces. The tensors Tij (6.70), and (6.48). The system of linear algebraic equations (6.105) can be presented in the matrix form AX = F,

(6.107)

where vectors of unknowns X and of the right hand side F are introduced by the equations X = |X 1 , X 2 , ..., X 3Mn |T , F = |F 1 , F 2 , ..., F 3Mn |T , ⎧ (m) m ≤ Mn, ⎪ ⎨ b1 , (m−Mn) m X = , Mn < m ≤ 2Mn, b2 ⎪ ⎩ (m−2Mn) , 2Mn < m ≤ 3Mn, b3 ⎧ (m) m ≤ Mn, t1 , ⎪ ⎨ (m−Mn) m F =  , Mn < m ≤ 2Mn, t2 ⎪ ⎩ (m−2Mn)  , 2Mn < m ≤ 3Mn. t3

(6.108)

(6.109)

The matrix A of the dimensions 3Mn × 3Mn consists of nine block-matrices aij (i, j = 1, 2, 3) of the dimensions Mn × Mn,    a11 , a12 , a13    A =  a21 , a22 , a23  , (6.110)  a31 , a32 , a33  (m,n)

aij

= Iij

(m,n)

, i, j = 1, 2, 3, m, n = 1, 2, ..., Mn. (m,n)

(6.111)

of the matrix A, the tabulated functions For calculation of the elements aij Fm (ρ, ς) in Eqs. (6.58)–(6.62) can be used for small values of the arguments, and

214

Heterogeneous Media

the asymptotics fm (ρ, ς) of these functions in Eqs. (6.63)–(6.67) can be used for large values of the arguments. The matrix A in Eq. (6.107) is nonsparse and nonsymmetric. For numerical solution of the system (6.107), the iterative CGM can be used. Note that the matrix A does not have a Toeplitz structure, and by iterations, the matrix-vector products cannot be calculated by the FFT algorithm. Nevertheless, the problem is 2D, and modern personal computers allow solving the problems with acceptable accuracy.

6.4.2 The node grid refinement strategy We consider an infinite homogeneous elastic medium containing a finite number of cracks. If the medium is subjected to an external stress field σij0 (x), the crack opening vectors satisfy the system similar to (6.103) M  

(m,n)

(n) n=1 

Tij

(x, x  )bj (x  )dS  = ti (n)

(m)

(x), x ∈ (m) , m = 1, 2, ..., M, (6.112)

(m,n) (m) (x, x  ) = −nk (x)Skij l (x Tij

−x



(n) )nl (x  ),

(m) (m) ti (x) = nj (x)σij0 (x).

(6.113) For numerical solution, we have to cover each crack surface by a node grid that is sufficiently fine to guarantee acceptable accuracy. For a large number of cracks, the system of the discretized equations (6.112) has large dimensions, and the computational cost of the solution is substantial. This cost can be reduced using the following node grid refinement strategy. First, coarse node grids for all the crack surfaces in the system (6.112) are generated, and the solution of the corresponding discretized system (1) is constructed. Then, one of the cracks (1) with the surface (1) and normal ni (x) ∗(1) is separated from the rest of the cracks, and the stress field σij (x) induced on the (n)

σij (x) induced at the surface (1) by cracks 2, 3, ..., M is calculated. The stress field  point x by the source acting at the nth node (x (n) ∈ / (1) ), in the local basis of this node, has the form (n)

(n)

(n)

(n)

(n)

 σij (x) = Iij k (y1 , y2 , y3 )bk , (n)

(n)

(6.114)

(n)

where y1 , y2 , y3 are the coordinates of the point x in the local basis (g1(n) , g2(n) , g3(n) ) of the nth node. The components of this tensor in the global basis are (n)

(n)

(n)

(n)

(n)

(n)

(n)

σij (x) = Qim Qj l Imlk (y1 , y2 , y3 )bk , (n)

(6.115)

where the matrix Qim relates the elements of the local and global bases according to ∗(1) Eq. (6.39). The stress field σij (x) induced at a point x on the surface (1) is the

Cracks in heterogeneous media

215

following sum: 

∗(1)

σij (x) = σij0 (x) +

(m)

σij (x), x ∈ (1) ,

(6.116)

m(x (m) ∈ / (1) )

where summation is performed over the nodes that do not belong to the surface (1) . ∗(1) The field σij (x) can be considered as an external field applied to the crack (1) , and therefore, the equation for the crack opening vector of this crack takes the form  (1,1) (1) (1) ∗(1) Tij (x, x  )bj (x  )d = nj (x)σij (x), x ∈ (1) , (6.117) (1) Tij(1,1) (x, x  ) = −n(1) k (x)Skij l (x

 − x  )n(1) l (x ).

(6.118)

For numerical solution of this equation, the node grid on the surface (1) can be taken finer than the original coarse grid. If (1) is a planar surface, the matrix of the discretized integral equation (6.117) has a Toeplitz structure, and the FFT algorithm can be applied by the iterative solution of the discretized problem. Then, this procedure can be repeated for calculation of the crack opening vectors of all other cracks 2 , 3 , ..., M . Further, examples of application of this stratagy are presented.

A lens shape crack subjected to the hydrostatic stress field σij0 = σ 0 δij Let the crack surface  be defined by the equations r = R, 0 ≤ θ ≤ θ0 , where r and θ are the radius and azimuthal angle of a spherical coordinate system (Fig. 6.16). Numerical solution of Eq. (6.1) is constructed for the right hand side ti (x) = σ 0 ni (x), where σ 0 is a constant and ni (x) is the normal vector to the crack surface. In this case, the crack opening vector has the form b(θ ) = br (θ )er + bθ (θ )eθ ,

(6.119)

where er , eθ are the local basis vectors of the spherical coordinate system on the crack surface. The distributions of the components br (θ ) and bθ (θ ) along the meridian of the sphere are presented in Figs. 6.17 and 6.18 for different crack characteristic angles θ0 , and br (0) is the maximum value of the crack component br (θ ) at the pole. The calculations were performed for a constant number of nodes Mn = 15000 on the crack surface. For calculation of the stress intensity factors KI and KI I at the crack edge, we introduce the function b(θ ) =



R(θ0 − θ )

m 

β (n) θ 2n ,

(6.120)

n=0

where the vector coefficients β (n) provide minimum to the functional 

(β (0) , β (1) , ..., β (m) ) =

θ0

θ1

 2  b(θ ) − b(θ ) sin θ dθ.

(6.121)

216

Heterogeneous Media

Figure 6.16 A lens shape crack of the radius R and the characteristic angle θ0 .

Figure 6.17 Meridional distributions of the component br (θ) of the crack opening vector for lens shape cracks with various characteristic angles θ0 subjected to hydrostatic tension.

Figure 6.18 Meridional distributions of the component bθ (θ) of the crack opening vector for lens shape cracks with various characteristic angles θ0 subjected to hydrostatic tension.

Here,  b(θ ) is the numerical solution of Eq. (6.1). Near the edge θ = θ0 , the asymptotic of the crack opening vector has the form m  √ b(θ ) = β(θ0 ) s + O(s 3/2 ), β(θ0 ) = β (k) θ02k , s = R(θ0 − θ ). k=0

(6.122)

Cracks in heterogeneous media

217

In the local basis of the spherical coordinate system at a point x 0 on the crack edge, the vector β(θ0 ) has two components, β(θ0 ) = βr (θ0 )er + βθ (θ0 )eθ ,

(6.123)

and the stress intensity factors KI and KI I are defined by Eq. (6.77), where βn (x (0) ) = βr (θ0 ) and βγ (x (0) ) = βθ (θ0 ). For hydrostatic tension, the equations for the SIFs KI and KI I of a lens-shaped crack are obtained in [4] by another numerical method and have the forms (ν = 0.286)   (6.124) KI = 0.9896σ 0 R sin θ0 , KII = 0.2652σ 0 R sin θ0 . The dependencies of the dimensionless coefficients k1 (θ0 ) =

KI (θ0 ) KI I (θ0 ) , k2 (θ0 ) = KI (π/2) KI I (π/2)

(6.125)

on the crack angle θ0 are shown in Fig. 6.19. Solid lines in this figure correspond to the values of the SIFs in Eq. (6.124), lines with triangles are the numerical solutions for a constant number Mn = 15000 of the nodes on the crack surface, and line with squares corresponds to the constant node grid step h/R = π/150. Note that the dimensionless SIFs k1 (θ0 ) and k2 (θ0 ) almost coincide. That is why only the coefficient k1 (θ0 ) is shown in Fig. 6.19.

Figure 6.19 The dependence of the dimensionless SIF k1 (θ0 ) of the lens shape crack on the characteristic angle θ0 ; the solid line is exact values of k1 (θ0 ) presented in [4], line with squares are the numerical results for the constant node grid step h/R = π/150, and line with triangles are the results for the constant number of the nodes Mn = 15000.

An elastic sphere subjected to two concentrated forces at the poles (Fig. 6.20) For taking the stress field distribution near the concentrated forces into account correctly, the stress tensor inside the sphere is presented in the form σij (x) = σij0 (x) + σij1 (x),

(6.126)

where the tensor σij0 (x) is the superposition of the solutions of Boussinesq’s problems for concentrated forces F applied to the half-spaces z ≤ a and z ≥ −a. Here a is the

218

Heterogeneous Media

Figure 6.20 A spherical elastic body subjected to two concentrated forces F ; P + and P − are the tangent planes at the points of the force applications.

sphere radius and z is the axis of the cylindrical coordinate system (ϕ, r, z) shown in Fig. 6.20. In this coordinate system, the nonzero components of the tensor σij0 (x) have the forms [5] 0 (r, z) = sαβ (r, z + a) + sαβ (r, −z + a), α, β = ϕ, r, z, σαβ   2 F R 3r z − (1 − 2ν ) , srr (r, z) = − 0 R+z 2πR 2 R 3   F (1 − 2ν0 ) z R − , sϕϕ (r, z) = − 2πR 2 R3 R + z

szz (r, z) = −F

3z3 3rz2 , s (r, z) = −F , R 2 = r 2 + z2 . rz 2πR 5 2πR 5

(6.127) (6.128) (6.129) (6.130)

The total stress tensor σij (x) satisfies the boundary conditions on the surface S, i.e., σij (x)nj (x)|S = 0,

(6.131)

except the sphere poles, where the concentrated forces are applied. Because the field σij0 (x) takes into account the presence of the concentrated forces, the boundary conditions on S for the stress tensor σ 1 (x) in Eq. (6.126) are σij1 (x)nj (x)|S = ti0 (x), ti0 (x) = −σij0 (x)nj (x)|S .

(6.132)

In the local basis of the spherical coordinate system on S, the vector t0 (x) takes the form t0 (θ ) = −tθ0 (θ )eθ − tR0 (θ )eR , θ = 0, π, t0 (0) = t0 (π) = 0,  1 0 0 0 tθ0 (θ ) = (r, z) sin(2θ ) + cos(2θ )σrz (r, z), σrr (r, z) − σzz 2

(6.133) (6.134)

Cracks in heterogeneous media

219

0 0 0 tR0 (θ ) = sin2 (θ )σrr (r, z) + cos2 (θ )σzz (r, z) + sin(2θ )σrz (r, z),

(6.135)

r = a sin θ, z = a cos θ.

(6.136)

The solution of the second boundary value problem (6.132)–(6.136) can be found in the form (6.95)  σij1 (x) =

Sij kl (x − x  )nk (x  )bl (x  )dS  ,

(6.137)

S

where bi (x) is the solution of the integral equation (6.97) for ti (x) = ti0 (x). For numerical solution, the node grid on the surface S is generated by the algorithm presented in Section 6.2.1. This grid is shown in Fig. 6.7, where the centers of small spheres compose the set of approximating nodes. Some defects of the grid where the distances between the neighbor nodes are more than h do not affect practically the quality of the solution for sufficiently small grid steps h.

Figure 6.21 Stress distributions in the plane z = 0 for an elastic sphere subjected to two concentrated forces; solid lines are the results of the method presented in [6], lines with squares and triangles are the results of numerical calculations for the node grid steps h/R = 0.08 and h/R = 0.03.

Figure 6.22 Stress distribution along the z-axis (r = 0) in the elastic sphere subjected to two concentrated forces; solid lines are the results presented in [6], lines with squares and triangles are the results of numerical calculations for the node grid steps h/R = 0.08 and h/R = 0.03.

220

Heterogeneous Media

Figure 6.23 Dependence of the normalized SIF KI of a circular crack emanating from a spherical cavity on the crack width λ; the medium is subjected to uniaxial tension σ0 in the direction of the x3 -axis; the solid line corresponds to the solution presented in [7], lines with squares and triangles are numerical solutions for different values of the node grid step.

In Figs. 6.21 and 6.22, the components of the stress tensor σ (r, z) in the plane z = 0 and along the z-axis (r = 0) are presented. In these figures, σα (r, z) =

σαα (r, z) , α = ϕ, r, z. F a2

(6.138)

The calculations were performed for a Poisson ratio of ν0 = 0.25, and the node grid steps on the sphere surface are taken h/a = 0.08 and h/a = 0.03. Solid lines in Figs. 6.19 and 6.20 correspond to the results of another numerical method, presented in [6].

A circular crack emanating from a spherical cavity subjected to external tension (Fig. 6.23) For an external tensile stress σ 0 orthogonal to the crack plane, the crack opening vector has one nonzero component b3 only. The nonzero SIF KI is calculated by the method described in Section 6.3. The calculations were performed for a Poisson ratio of ν0 = 0.3, the step of the coarse grid is taken h/a = 0.0314 (the node number Mn varies from 15195 to 24188 for different values of the crack width λ), and the step h/(a + λ) = 0.0017 (Mn = 361201) is taken for the refined grid. In Fig. 6.23, the lines with squares and triangles are the values of KI for the coarse and refined node grids, respectively. The solid line corresponds to the results of another numerical method, presented in [7].

Elastic sphere with a peripheral edge crack subjected to hydrostatic pressure (Fig. 6.24) The crack is subjected to pressure σ 0 , the only nonzero SIF is KI , the calculations were performed for a Poisson ratio of ν0 = 0.3, the coarse grid step h/a = 0.0314 (Mn varies from 14896 to 17059 for different values of the crack width λ), and the refined grid step h/(a − λ) = 0.0017 (Mn = 361201). In Fig. 6.24, the lines with squares and triangles are the values of KI at the edge of the crack for the coarse and

Cracks in heterogeneous media

221

Figure 6.24 Dependence of the normalized SIF KI of a peripheral edge crack of an elastic sphere on the crack width λ; the crack surface is subjected to hydrostatic pressure σ0 ; solid line corresponds to KI presented in [8], lines with squares and triangles are numerical solutions for different values of the node grid steps.

Figure 6.25 Dependence of the normalized SIF K1 at the crack edge point B on the parameter λ for four penny shape cracks; lines with squares, triangles, and dots are numerical solutions for various values of the ratio l/a.

refined grids, respectively. The solid line corresponds to the results of another method, presented in [8].

6.4.3 A group of cracks in a homogeneous elastic medium We consider four parallel cracks (Fig. 6.25) under tension σ 0 orthogonal to the crack planes. The coarse grid step is taken as h/a = 0.01 (Mn = 9061). In Fig. 6.25, results for the refined grid step h/a = 0.0017 (Mn = 361201) are shown; the lines with squares, triangles, and dots are the values of the SIF KI at point B for different values of the distance parameter l/a. It can be observed that the amplifying effect of the crack interaction (SIFs increasing) has a shorter range than the shielding effect. This effect was discussed in detail in [9], [10]. If the vertical distance l is smaller than or equal to the crack radius, the amplifying effect takes place when two coplanar cracks are not stacked with the others; if the vertical distance l is larger than the crack radius, there is

222

Heterogeneous Media

Figure 6.26 Dependence of the normalized SIF K1 (θ0 ) at the central crack edge point B on the parameter λ for nine parallel penny shape cracks; lines with squares, triangles, and dots are numerical solutions for various values of the ratio l/c.

Figure 6.27 The same as in Fig. 6.26 for the central crack edge point B on the ray with the angle γ = π/4 to the x1 -axis.

no amplification until the distance c becomes much smaller than l (see similar results in [10]). Nine parallel penny-shaped cracks the centers of which are at the vertices of a cuboid (c × c × l) are shown in Fig. 6.26. The tensile stress σ 0 acts orthogonal to the crack planes. The SIF KI on the edge of the central crack was calculated for a Poisson ratio of ν = 0.3, the coarse grid step h/a = 0.0314 (Mn = 6510), and the refined grid step h/a = 0.0017 (Mn = 361201). In Fig. 6.26, the lines with squares, triangles, and dots are the values of KI at point B, which is the intersection of the x1 -axis and the edge of the central crack, for different values of the dimensionless parameters λ = 2a/c and l/c. Here, a is the crack radius, c is the horizontal size of the cuboid, and l is its vertical size. In Fig. 6.27, the SIF KI is given at the point B that is on the ray with the angle π/4 to the x1 -axis. It can be seen that at this point, the shielding effect is stronger than at the crack edge point on the x1 -axis, and the difference in the KI -values at these points become larger when the size l is smaller than c. Note that for l/c = 1, the amplifying effect is rather small.

Cracks in heterogeneous media

223

Figure 6.28 Dependencies of the integral I b = I3b in Eq. (6.139) averaged over the cracks in Fig. 6.26 on the parameter λ; lines with triangles, squares, and dots are numerical solutions for different values of the ratio l/c, I0b is the value of I b for non-interacting cracks.

The integral of the crack opening vector over the crack surface  b Ii = bi (x)d

(6.139)



characterizes the crack contribution in the average deformation of a cracked medium, and therefore, in the effective elastic compliance tensor of the medium (see Section 9.5). If the vector bi (x) is approximated by the Gaussian quasiinterpolant, this integral is calculated as follows: Iib ≈

Mn  n=1

(n)

bi

 P (n)

ϕ(x − x (n) )d = h2

Mn 

(n)

bi .

(6.140)

n=1

Here, P (n) is the plane tangent to  at the node x (n) . In Fig. 6.28, the dependencies of the mean values of the integral I b = I3b over nine penny-shaped cracks shown in Fig. 6.26 on the dimensionless parameters λ = 2a/c and l/c are presented. In this figure, I0b is the integral I3b for noninteracting cracks (λ = 0). It is seen that the values of I b are always smaller than I0b , and the difference between these integrals increases when the ratio l/c decreases. Thus, the arrangement of cracks in the centered orthorhombic lattice shown in Fig. 6.26 increases the effective elastic stiffness tensor of the cracked medium in comparison with the case of noninteracting cracks. This fact agrees with the results presented in [10].

Two intersecting penny-shaped cracks under tension The case of two penny-shaped cracks of radius a intersecting at the angle π/4 and subjected to a tensile stress σ 0 orthogonal to the plane of the first crack is presented in Fig. 6.29. The SIF KI along the crack contour are calculated for a Poisson ratio of ν = 0.3, the coarse grid step h/a = 0.01 (Mn = 9061), and the refined grid step h/a = 0.0017 (Mn = 361201). The distribution of the parameter KI /KI0 along the second crack contour is shown in Fig. 6.29 by the line with squares, where KI0 is the

224

Heterogeneous Media

Figure 6.29 Normalized SIF KI (γ )/KI0 along the edge of crack 2 in the case of two intersecting penny

shape cracks subjected to tension normal to the crack 1 surface; line with squares is the numerical solution, solid line is the approximate solution presented in [10], KI0 is the SIF for an isolated crack 2 subjected to the same external tension.

Figure 6.30 A cuboid of a heterogeneous medium covered by a regular node grid.

SIF of isolated crack 2 subjected to the same external tensile stress. The solid line corresponds to the results presented in [10].

6.5 A homogeneous elastic medium containing cracks and inclusions We consider a homogeneous elastic host medium containing finite numbers of heterogeneous inclusions v (m) (m = 1, 2, ..., M) and cracks (n) (n = 1, 2, ..., N) (Fig. 6.30). The medium is subjected to an external stress field σij0 (x). It is shown in Section 3.3 that the stress tensor in the medium with heterogeneous inclusions is presented in a form that follows from Eq. (3.104):  1 σij (x) = σij0 (x) + Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  . (6.141) V

Cracks in heterogeneous media

225

The stress tensor in the medium with cracks has the form (6.1). Combining these equations, we obtain the following integral presentation of the stress tensor σij (x) in the medium containing inclusions and cracks:  1 (x  )σmn (x  )dx  + σij (x) = σij0 (x) + Sij kl (x − x  )Bklmn V  + Sij kl (x − x  )nk (x  )bl (x  )d .

(6.142)



Here, V =

M

v (m) is the region occupied by the inclusions and  =

m=1

N

(n) is the

n=1

surface of all the cracks. If the crack surfaces are stress-free, the boundary condition (6.3) on  should be satisfied. Multiplying both sides of Eq. (6.142) with the normal vector ni (x) to  and substituting the resulting equation in Eq. (6.3), we obtain the equation  1 (x  )σmn (x  )dx  − − nj (x) Sj ikl (x − x  )Bklmn V  − nj (x) Sj ikl (x, x  )nk (x  )bl (x  )d = nj (x)σj0i (x), x ∈ .

(6.143)



The right hand side of Eq. (6.142) depends on the values of the stress tensor σij (x) in the region V and the crack opening vectors bi (x) on the crack surfaces . If the stress field in V and the vectors bi (x) on  are known, the stress tensor at an arbitrary point x of the host medium is reconstructed from the same Eq. (6.142). Eqs. (6.142) and (6.143) compose a closed system for the unknowns of the problem: the stress tensor in the region V and the crack opening vectors on the surfaces .

6.5.1 Discretization of Eqs. (6.142) and (6.143) For numerical solution, the integral equations Eqs. (6.142) and (6.143) should be discretized. Let a cuboid W containing all the inclusions and cracks be covered by a regular grid of nodes x (n) (n = 1, 2, ..., Mn) (Fig. 6.30). For simplicity, we assume that the cracks are planar and parallel to a fixed plane P . Suppose that the node grid covers the surfaces of all the cracks and the regions of the inclusions. Let in a global Cartesian coordinate system (x1 , x2 , x3 ) the plane P be defined by the equation x3 = 0. A local coordinate system (y1 , y2 , y3 ) at each node x (n) is introduced by parallel translation of the global system to this node. Using Gaussian approximating functions, the stress tensor σij (x) and the crack opening vectors bi (x) in the region W are approximated by the quasiinterpolants σij (x1 , x2 , x3 ) =

Mn  n=1

(n)

σij ϕ (n) (x1 , x2 , x3 ),

226

Heterogeneous Media



bi (x1 , x2 , x3 ) =

(l)

bi ϕ (l) (x1 , x2 , x3 ).

(6.144)

l (x (l) ∈)

Here, σij(n) = σij (x (n) ) and bi(l) = bi (x (l) ) are the values of stress tensors and crack opening vectors at the nodes. The function ϕ (n) (x) = ϕ(x − x (n) ) in the local basis y1 , y2 , y3 of the node x (n) is defined by the equation

1 |y|2 . (6.145) ϕ(y1 , y2 , y3 ) = exp − h2 H (πH )3/2 The functions ϕ (l) (x) = ϕ(x − x (l) ) are defined in the crack plane, and in the local basis of the lth node, the function ϕ(y) = ϕ(y1 , y2 ) has the form

1 |y|2 ϕ(y1 , y2 ) = (6.146) exp − 2 , y = (y1 , y2 ). πH h H Substituting approximations (6.145) and (6.146) into Eqs. (6.142) and (6.143) and satisfying the resulting equations at all the nodes, we obtain the following system of (n) (l) linear algebraic equations for the tensors σij and vectors bi : (m)

σij

−

Mn 

(m,n)

1(n)

0(m)

(n) ij kl Bklmn σmn = σij

n=1 Mn 

Mn 

(m,l) (l)

(6.147)

1(n)

(6.148)

Iij k bk , x (m) ∈ V ,

l=1

(m) (m,l) (l)

0(m)

ni Iij k bk = ti

l=1 (l) bi = 0

+

−

Mn 

(m) (m,n)

(n) ni ij kl Bklmn σmn , x (m) ∈ ,

n=1 0(m)

if x (l) ∈ / , ti (m,n)

= nj (x (m) )σj0i (x (m) ).

(6.149)

(m,l)

Here, the objects ij kl , Iij k are the stresses induced at the mth node by the sources at the nth or lth node. These tensors depend on the vector of distance between the points x (m) , x (n) and x (m) , x (l) . We have (m,n)

ij kl = ij kl (x (m) − x (n) ),

(m,l)

Iij k

= Iij k (x (m) − x (l) ).

(6.150)

The functions ij kl (x) and Iij k (x) are the following integrals:   ij kl (x) = Sij kl (x − x  )ϕ(x  )dx  , Iij k (x) = Sij kl (x − x  )nl ϕ(x  )dP  . P

(6.151) The integral ij kl (x) is calculated over the entire 3D space, and the integral Iij k (x) is spread on the plane P to which function ϕ(x) belongs. The function ij kl (x) is defined in Eqs. (5.106)–(5.110), and the function Iij k (x) is expressed in terms of standard (m,l) 1D integrals in Eqs. (6.48)–(6.67). As the result, the objects ij(m,n) are kl and Iij k calculated fast.

Cracks in heterogeneous media

227

The systems of linear algebraic equations (6.147) and (6.148) can be written in the matrix form AX = F,

(6.152)

where vectors of unknowns X and of the right hand side F are introduced by the equations  T T  X = X 1 , X 2 , ..., X 9Mn , F = F 1 , F 2 , ..., F 9Mn ,

X = s

F = s

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(s)

s ≤ Mn,

σ11 , (s−Mn)

σ22

Mn < s ≤ 2Mn,

,

(s−2Mn)

,

2Mn < s ≤ 3Mn,

(s−3Mn) , σ12 (s−4Mn) , σ13 (s−5Mn) , σ23 (s−6Mn) , b1 (s−7Mn) , b2 (s−8Mn) , b3

3Mn < s ≤ 4Mn,

σ33

(6.153)

0(s)

4Mn < s ≤ 5Mn, 5Mn < s ≤ 6Mn, 6Mn < s ≤ 7Mn, 7Mn < s ≤ 8Mn, 8Mn < s ≤ 9Mn,

σ11 ,

s ≤ Mn,

0(s−Mn) , σ22 0(s−2Mn) , σ33 0(s−3Mn) , σ12 0(s−4Mn) , σ13 0(s−5Mn) , σ23 0(s−6Mn) , t1 0(s−7Mn) , t2 0(s−8Mn) , t3

Mn < s ≤ 2Mn, 2Mn < s ≤ 3Mn, 3Mn < s ≤ 4Mn, 4Mn < s ≤ 5Mn,

(6.154)

5Mn < s ≤ 6Mn, 6Mn < s ≤ 7Mn, 7Mn < s ≤ 8Mn, 8Mn < s ≤ 9Mn.

The 9Mn × 9Mn matrix A consists of 81 block-matrices a(i,j ) (here i, j = 1, 2, . . . , 9) of dimensions Mn × Mn, i.e.,   (1,1) (1,2) (1,9)   a  (2,1) a(2,2) · · · a(2,9)    a a ··· a   A= . (6.155) , . . . .. . . ..   ..    a(9,1) a(9,2) · · · a(9,9)  (i,j )

(p,s)

1(s)

aps = δij − iikl Bkljj , i, j = 1, 2, 3, p, s = 1, 2, ..., Mn,

(6.156)

228

Heterogeneous Media (4j )

(p,s)

1(s)

(5j )

(p,s)

1(s)

(6j )

(p,s)

1(s)

(p,s)

1(s)

(5j )

(p,s)

1(s)

(6j )

(p,s)

1(s)

aps = −12kl Bkljj , aps = −13kl Bkljj , aps = −23kl Bkljj , (6.157) (i4) aps = −iikl Bkl12 , aps = −iikl Bkl13 , aps = −iikl Bkl23 , (6.158) (p,s)

1(s)

(p,s)

1(s)

(p,s)

1(s)

(p,s)

1(s)

(p,s)

1(s)

(p,s)

1(s)

(44) (45) (46) aps = 1 − 12kl Bkl12 , aps = −12kl Bkl13 , aps = −12kl Bkl23 , (6.159) (p,s)

1(s)

(p,s)

1(s)

(54) (55) (56) aps = −13kl Bkl12 , aps = 1 − 13kl Bkl13 , aps = −13kl Bkl23 , (6.160) (p,s)

1(s)

(64) (65) (66) aps = −23kl Bkl12 , aps = −23kl Bkl13 , aps = 1 − 23kl Bkl23 , (6.161) (7j )

(5j )

(8j )

(6j )

(9j )

(3j )

aps = aps , aps = aps , aps = aps , j = 1, 2, . . . , 6,

(6.162)

(1j ) aps

(r,s) = I33j , j = 1, 2, 3,

(6.163)

(r,s) = I23j ,

(6.164)

(4j ) aps

(r,s) = I11j ,

(2j ) aps

(r,s) = I22j ,

(3j ) aps

(5j ) (6j ) (r,s) (r,s) = I12j , aps = I13j , aps (i+6,j +6) (r,s) = I3ij , i, j = 1, 2, 3. aps

(6.165)

For nonregular node grids, the matrix A is nonsparse and nonsymmetric. For numerical solution of the system (6.152), the MRM can be used. The method converges for all nondegenerate stiffness tensors Cij kl (x) of the heterogeneous medium. For regular node grids, the matrix A has a Toeplitz structure, and the FFT algorithms can be used for calculation of the matrix-vector products by iterative solution of the system (6.152).

6.5.2 Interaction of inclusions and cracks We consider a homogeneous elastic medium containing a penny-shaped crack of radius r and a spherical inclusion of radius a, a/r = 2. The crack and inclusion are separated by the distance l (Fig. 6.31). The Poisson ratio of the medium and the inclusion is ν0 = ν = 0.25 and Young moduli are E0 and E. The medium is subjected to a tensile stress σ 0 orthogonal to the crack plane.

Figure 6.31 A penny shape crack and a spherical inclusion separated by distance l.

The stress intensity factor KI is calculated at the point A of the crack boundary (Fig. 6.31). For the calculations, the crack and inclusion were embedded in the cuboid W : ((2r + 2a + l) × 2a × 2a), which was covered by a cubic node grid with the step h/a = 0.005 (Mn ≈ 16 · 106 ). In Table 6.1, the values of the normalized SIF

Cracks in heterogeneous media

229

Table 6.1 The normalized SIF kI at the point A on the crack edge (Fig. 6.31) for various parameters l/a, E/E0 , and the node grid step h/a = 0.005. l/a = 0.5

l/a = 0.2

E/E0

h/a = 0.005

Shodja et al. (2003) [11]

Kushch (1998) [12]

0 0.5 2 10 1000 0 0.5 2 10 1000

1.293 1.083 0.906 0.780 0.726 1.539 1.141 0.833 0.597 0.515

1.279 1.092 0.908 0.773 0.722 1.49 1.16 0.83 0.59 0.5

1.273 1.089 0.911 0.781 0.732 1.5 1.16 0.84 0.6 0.51

Figure 6.32 A penny shape crack inside a spherical inclusion.

" ! I √ for different values of the ratio E/E0 of Young moduli of the inclusion kI = σ K 0 a and the medium and the parameter l/a are compared with semianalytical solutions presented in [11] and [12]. It is seen that the results of the proposed method are in a good agreement with calculations of other authors. Let a penny-shaped crack be inside a spherical inclusion and subjected to tensile stress σ 0 orthogonal to the crack plane (Fig. 6.32). For calculation of the SIF KI at the crack edge, the inclusion and the crack are embedded in the cube W : (2a × 2a × 2a), which was covered by a cubic node grid with the step h. A comparison of the numerical results with the solutions available in the literature is presented in Fig. 6.33. The lines with triangles and circles correspond to the numerical solutions for h/a = 0.005 (Mn ≈ 8 · 106 ) and h/a = 0.0025 (Mn ≈ 64·106 ), respectively, and solid lines are the numerical results presented in [13]. Case 1 corresponds to μ/μ0 = 0.1, ν = 0.325, and ν0 = 0.25; case 2 to μ/μ0 = 0.5, ν = 0.2, and ν0 = 0.4; and case 3 to μ/μ0 = 2, ν = 0.2, and ν0 = 0.4. Here, μ0 , ν0 are the shear modulus and Poisson ratio of the host medium, and μ, ν are the same for the inclusion. Let a crack of radius r intersect a spherical inclusion of radius a, as shown in Fig. 6.34. The medium is subjected to a tensile stress σ 0 orthogonal to the crack plane. The stress intensity factor KI at the crack edge is calculated by the method described in Section 6.3. The cuboid W containing the crack and inclusion is covered by a cubic node grid with the step h/a = 0.005 (Mn ≈ 8 · 106 ). Numerical results for a soft inclusion (μ/μ0 = 0.1) are shown in Fig. 6.35, and for a stiff inclusion

230

Heterogeneous Media

Figure 6.33 Dependencies of the SIF KI at the edge of a penny-shaped crack of radius r inside a spherical inclusion of radius a (Fig. 6.32); solid lines are numerical results presented in [13].

Figure 6.34 Intersection of a penny shape crack and a spherical inclusion.

Figure 6.35 Distribution of the SIF KI at the edge of a penny-shaped crack intersecting with a soft spherical inclusion.

(μ/μ0 = 10) in Fig. 6.36. Poisson ratios of the matrix and inclusion materials are taken as ν = ν0 = 0.25. The lines with squares, circles, triangles, and diamonds corre√ spond to the normalized SIF kI (θ ) = KI (θ )/( σ 0 r) for different values of the factor l/ (2r), where l is the length of the intersection area. Solid lines in Figs. 6.35 and 6.36 correspond to the SIF kI for an isolated crack.

Cracks in heterogeneous media

231

Figure 6.36 The same as in Fig. 6.35 for a stiff spherical inclusion.

6.6

A planar crack subjected to pressure of injected fluid in a poroelastic medium

We consider a planar crack of arbitrary shape in a homogeneous isotropic poroelastic medium subjected to the pressure of an injected fluid. In Section 3.5, this problem is reduced to a system of integral equations for the crack opening b(x, t) and the density ν(x, t) of the potential of the simple layer concentrated on the crack surface. The approximate system of integral equations for b(x, t) and ν(x, t) has the form (3.190)–(3.191)   μ(λ + μ)     4 1 (x − x )b(x , t)d + μ R(x − x  , t) ∗ ν(x  , t)d = (λ + 2μ)  α 

αμ α(λ + μ) 0 = 1− p f (x, t) + σ 0 (x, t) − p (x, t), x ∈ , (6.166) λ + 2μ (λ + 2μ)  1 2 (x − x  , t) ∗ ν(x  , t)d = p f (x, t) − p 0 (x, t), x ∈ . (6.167)  κQ2  2 (x, t), and R(x, t) of the integral operators in these equaThe kernels 1 (x),  tions are defined in Eqs. (3.153), (3.155), and (3.187). These equations serve if the pressure p f (x, t) applied to the crack surface and external stress and pressure fields σ 0 (x, t), p 0 (x, t) are slowly changing functions of time. For numerical solution, the functions b(x, t) and ν(x, t) in Eqs. (6.166) and (6.167) are approximated by Gaussian quasiinterpolants in a rectangular W that includes the crack surface  (Fig. 4.17), b(x, t) =

Mn  n=1

b

(n)

(t)ϕ(x − x

(n)

), ν(x, t) =

Mn 

ν (n) (t)ϕ(x − x (n) ),

(6.168)

n=1



x2 1 ϕ(x) = exp − 2 , x = (x1 , x2 ). πH h H

(6.169)

232

Heterogeneous Media

Here, x (n) (n = 1, 2, ..., Mn) are the nodes of a square grid covering the region W , / , and b(n) (t) = 0, ν (n) (t) = 0 if x (n) ∈ b(n) (t) = b(x (n) , t), ν (n) (t) = ν(x (n) , t).

(6.170)

After substitution of the approximations (6.168) into Eqs. (6.166) and (6.167) and satisfaction of the resulting equations at all the nodes, we obtain the following system of equations for coefficients b(n) (t) and ν (n) (t): μ(λ + μ) 

1 (x (m) − x (n) )b(n) (t)dτ = (λ + 2μ) n=1

αμ α(λ + μ) 0 (m) = 1− pf (x (m) , t) + σ 0 (x (m) , t) − p (x , t)− (λ + 2μ) (λ + 2μ) Mn  2  t

3 (x (m) − x (n) , t − τ )ν (n) (τ )dτ, x (m) ∈ , (6.171) μ − α 0 Mn

4

n=1

1 κQ2

Mn  t 

2 (x (m) − x (n) , t − τ )ν (n) (τ )dτ =

n=1 0 f (m)

= p (x b

(m)

, t) − p 0 (x (m) , t), x (m) ∈ ,

(t) = 0, ν

(m)

(t) = 0, x

(m)

∈ / , m = 1, 2, ...Mn.

(6.172) (6.173)

Here, the functions 1 (x), 2 (x, t), and 3 (x, t) are the following integrals:      2 (x − x  , t)ϕ(x  )dx  ,

1 (x) = 1 (x − x )ϕ(x )dx , 2 (x, t) =  

3 (x, t) =

(6.174) R(x − x  , t)ϕ(x  )dx  , x = (x1 , x2 ).

(6.175)

Since Gaussian functions are concentrated in small vicinities of the corresponding nodes, these integrals are calculated over the entire plane x3 = 0 instead of the crack area . The functions m (x, t) are calculated explicitly and have the forms

! 2  " ρ2 1 ρ2 ρ 2 2 exp − H − ρ I0 + ρ I1 ,

1 (x) = √ 5/2 2H 2H 2H 4h πH (6.176) Q

2 (x, t) = √ 0 (x, t), πt    (t) 8t ρ 2 − H Q (6.177)

3 (x, t) = √

0 (x, t), 1+ (t)2 h2 Q2 H 6 πt

ρ2 1 (t) = H + 4t , exp − (6.178)

0 (x, t) = , H (t) (t) h2 Q2 πH H

Cracks in heterogeneous media

Q2 =

233

α 2 + β(λ + 2μ) r , ρ= , r= κ((λ + 2μ) h



x12 + x22 .

(6.179)

Here, I0 (z) and I1 (z) are modified Bessel functions of the first kind. Calculating the coefficients ν (n) (t) from Eq. (6.172) and substituting the result in Eq. (6.171), we obtain a system of linear algebraic equations for the coefficients b(n) (t). A method for solution of this system is considered in Section 6.1. The pressure p(x, x3 , t) in the medium with a crack is defined in Eq. (3.193) and presented in the form

p(x, x3 , t) = p 0 (x, x3 , t) +

1 8κQ

 

dx 



t 0

! " |x−x  |2 +x 2 exp −Q2 4(t−τ ) 3 [π(t − τ )]3/2

ν(x  , τ )dτ. (6.180)

Approximating the density ν(x, t) by the Gaussian quasiinterpolant (6.168), we obtain for p(x, x3 , t) the equation p(x, x3 , t) = p 0 (x, x3 , t)+

! " Q2 x32 Mn  t exp −  4(t−τ ) 1 +

0 (x − x (n) , t − τ )ν (n) (τ )dτ. √ 2κQ π(t − τ ) 0 n=1 (6.181)

The equations for the coefficients ν (n) (t) follow from the boundary conditions p(x, 0, t) = p f (x, t) on the crack surface that are to be satisfied at all nodes on , 1  2κQ Mn



t

n=1 0

0 (x (m) − x (n) , t − τ ) (n) ν (τ )dτ = p f (x (m) , t) − p 0 (x (m) , t), √ π(t − τ ) (6.182)

x

(m)

∈ , ν

(n)

(t) = 0 if x

(n)

∈ / , m, n = 1, 2, ..., Mn.

Taking into account that 0 (x, t) in Eq. (6.178) is a smooth function of t, we perform time discretization of the integral in Eq. (6.182) and obtain the system Mn k−1 1  B(k, i) 0 (x − x (n) , tk − ti+1 )ν (n) (ti+1 ) = √ κQ π n=1 i=0

= p f (x, tk ) − p 0 (x, tk ),  √ B(k, i) = tk − ti − tk − ti+1 , k = 1, 2, ..., N.

(6.183) (6.184)

Here, tk are the nodes in the discretized time interval (0, t), 0 = t0 < t1 < t2 ... < tN = t. From the properties of approximation by the Gaussian functions, the following

234

Heterogeneous Media

equation holds: Mn 

0 (x (m) − x (n) , 0)ν (n) (ti ) =

n=1

Mn ! "  ϕ x (m) − x (n) ν (n) (ti ) ν(x (m) , ti ) = n=1 (m)

=ν

(ti ).

(6.185)

As a result, from Eq. (6.183), we obtain a sequence of equations for functions ν (m) (t) at discrete time points t1 , t2 , t3 , ..., tN , i.e., √  κQ π  f (m) (6.186) p (x , t1 ) − p 0 (x (m) , t1 ) , √ t1 "  √ ! 1 ν (m) (tk ) = κQ π p f (x (m) , tk ) − p 0 (x (m) , tk ) − B(k, k − 1)  k−2 Mn   (m) (n) (n) B(k, i)

0 (x − x , tk − ti+1 )ν (ti+1 ) , k = 2, 3, 4, ..., N. −

ν (m) (t1 ) =

i=0

n=1

(6.187)

6.6.1 Fluid filtration from the surface of a penny-shaped crack We consider a penny-shaped crack of radius a = 1m in the poroelastic medium with the parameters typical for geologic structures in the region of oil fields. The bulk and shear moduli of the solid skeleton are ks = 40 GPa, μs = 30 GPa, the skeleton porosity is φ = 0.2, and the medium permeability is κ = 10−14 m2 . The fluid bulk modulus is kf = 2.3 GPa, and the fluid viscosity is η = 10−3 Pa· sec. Effective bulk and shear moduli of the solid skeleton with dry pores are calculated by the equations of the effective field method [2], i.e.,

φ φ k = ks 1 − , μ = μs 1 − , (6.188) 1 − s1 (1 − φ) 1 − s2 (1 − φ) 2 3ks 1 λ = k − μ, s1 = , s2 = (3 − s1 ). (6.189) 3 3ks + 4μs 5 A pressure p f = 2 MPa is applied to the crack surface at the initial moment t = 0, and σij0 (x, t) = p 0 (x, t) = 0. In Fig. 6.37, the distributions of the density ν(r, t) along the crack radius at various time points t are presented. Pressure distributions p(0, x3 , t) in the medium along the x3 -axis at various time points are shown in Fig. 6.38. Dashed lines in Figs. 6.37 and 6.38 are the limits of the density ν(x, t) and the pressure p(0, x3 , t) at t → ∞. In the calculations, the step of the regular node grid on the crack surface is taken as h = 0.01 m (the total number of nodes on the crack surface is Mn = 31417), and H = 1. The fluid flux on the crack surface is an important characteristic of the hydraulic fracture process. It allows assessing the volume of the fluid filtrated into the medium

Cracks in heterogeneous media

235

Figure 6.37 Distributions of the density ν(r, t) along the radial coordinate r on the surface of a penny shape crack in a poroelastic medium at various time moment; the crack surface is subjected to a constant pressure at t = 0; the dashed line is the limit distribution at t → ∞.

Figure 6.38 Distribution of fluid pressure p(0, 0, x3 , t) along the x3 -axis orthogonal to the surface of a penny shape crack in a poroelastic medium at various time moment; the crack surface is subjected to a constant pressure at t = 0; the dashed line is the limit distribution at t → ∞.

from the crack surface. According to Darcy’s law, the fluid flux l(x, t) from the crack surface is defined by the equation l(x, t) = −κ lim

x3 →0

∂ p(x, x3 , t). ∂x3

(6.190)

Note that in the (x, ω)-presentation, the potential in Eqs. (3.193) and (6.180) 

 g (x − x  , x3 , ω)ν(x  , ω)d

p(x, x3 , ω) =

(6.191)



coincides with the potential of the simple layer ϑ(x) in Eq. (2.303). Therefore, p(x, x3 , t) is the potential of the simple layer, and the jump of the normal derivative of this potential on  is defined in Eq. (2.305). We have 

 1 ∂ p(x, x3 , t) = ν(x, t). ∂x3 κ x3 =0

(6.192)

236

Heterogeneous Media

Because the problem is symmetric with respect to the plane x3 = 0, the following equation holds: l(x, t) = −κ lim

x3 →0

∂ 1 p(x, x3 , t) = ν(x, t)sign(x3 ). ∂x3 2

(6.193)

As a result, the fluid flux from two sides of the crack surface and the total flux L(t) are defined by the equations  2|l(x, t)| = ν(x, t), L(t) =

(6.194)

ν(x, t)d. 

In the literature, the 1D model of filtration is usually used to assess fluid flux (leakoff) from the crack surface (the so-called Carter model). In this model, the pressure change along the x1 - and x2 -coordinates is neglected in comparison with this change along the x3 -axis. In this approximation, the equation for the pressure is accepted in the form ∂p ∂ 2p = Q2 2 ∂t ∂x3

(6.195)

with the initial and boundary conditions p(x, x3 , t)|t=0 = p 0 (x, t), lim p(x, x3 , t) = p f (x, t). x3 →0

(6.196)

For p 0 (x, t) = 0, the solution of this problem has the form

Q|x3 | p f (x, t). p(x, x3 , t) = Erf c √ 2 t

(6.197)

The corresponding fluid flux from the crack surface and the total flux are defined by the equations 2κQ 2|l(x, t)| = √ p f (x, t), πt

2κQ L(t) = √ πt

 p f (x, t)d.

(6.198)



In Fig. 6.39, time dependencies of the fluid flux at the distances r = 0, 0.85, 0.95, 0.99 m from the crack center on the surface of the penny-shaped crack of radius a = 1m are shown, and the pressure applied to the crack surface is constant, i.e., p f = 2 MPa. The dashed line in this figure corresponds to Eq. (6.198). It is seen that this equation serves in the vicinity of the crack center, but its error near the crack edge is substantial. In Fig. 6.40, time dependence of the total fluid flux L(t) from the crack surface is shown. The solid line corresponds to the numerical solution of Eqs. (6.186) and (6.187) and the dashed line to Eq. (6.198).

Cracks in heterogeneous media

237

Figure 6.39 Time-dependencies of the fluid flux l(r, t) from the surface of a penny shape crack of the radius a = 1 m at various distances r from the crack center; the crack is subjected to a constant pressure at t = 0; the dashed line is an approximate solution in Eq. (6.198).

Figure 6.40 Time-dependence of the total fluid flux L(t) from the surface of a penny shape crack; the crack is subjected to a constant pressure at t = 0; the dashed line is an approximate solution in Eq. (6.198).

6.7

Notes

Discretization of the integral equations of the boundary value problems of mathematical physics by Gaussian approximating functions was proposed in [14] (Chapter 12). There, the corresponding numerical method is called the boundary point method. Application of the method to solution of the static elasticity problem for planar cracks is performed in [15], and for curvilinear cracks in [16]. The content of this chapter is based on the works [15], [16], [17], [18], [19].

Appendix 6.A The computational program for numerical solution of the crack problem of elasticity In this appendix, the computational program for numerical solution of the elasticity problem for a planar crack in a 3D homogeneous isotropic medium is presented. In the program, the CGM and the FFT algorithm for calculation of the matrix-vector

238

Heterogeneous Media

products are adopted for iterative solution of the discretized problem. The text of the program is written for Wolfram Mathematica software. (*Initial data*) (*Young modulus E0 and Poisson ratio nu0 of the host medium*) E0=1, nu0=0.3 (*Lame parameters*) mu0=E0/(2(1+nu0)); la0=nu0*E0/((1-2*nu0)*(1+nu0)); kp0=1/(2*(1-nu0)); (*Node generation*) M1=200; M2=200; h=L1/(M1-1); N1=2*M1; N2=2*M2; Mn:=M1*M2 L1=2;L2=2; l1=-1; l2=-1; (*Cartesian coordinates of the nodes*) x1:=Flatten[Table[l1+h*(i-1),{j,M2},{i,M1}]]; x2:=Flatten[Table[l2+h*(j-1),{j,M2},{i,M1}]]; (*Connection between one- and two-index numeration of the nodes*) M[i_,j_]:=i+M1*(j-1) (*Transition from two- to one-index numeration and vice versa*) To1Ind[A_] := Flatten[Transpose[A]]; To2Ind[A_] := Transpose[Partition[A, M1]]; (*Indication of the crack area*) A1=1; A2=1; Ind :=Module[{InX}, Ind=Table[0,{Mn}]; InX=Select[Table[If[(x1[[j]]/A1) ˆ 2+(x2[[j]]/A2) ˆ 20 &]; Ind[[InX]]=1;Ind]; IND :=Join[Ind,Ind,Ind]; (*The external stress vector on the crack surface*) FF[1] := Module[{AA}, AA = Table[0, {3*Mn}]; Do[AA[[p]] = 1, {p, Mn}]; AA] FF[2] := Module[{AA}, AA = Table[0, {3*Mn}]; Do[AA[[p+Mn]] = 1, {p, Mn}]; AA] FF[3] := Module[{}, AA = Table[0, {3*Mn}]; Do[AA[[p+2*Mn]] = 1, {p, Mn}]; AA] (*Approximating function*) H=2 f0[r_]:=Exp[-r ˆ 2/H]/(Pi*H) f1[r_]:=Exp[-r ˆ 2/(2*H)]*((H-r ˆ 2)*BesselI[0,r ˆ 2/(2H)]+ r ˆ 2*BesselI[1,r ˆ 2/(2*H)])/(H ˆ (5/2)*Sqrt[Pi]) f2[r_]:=Exp[-r ˆ 2/(2*H)]*(BesselI[0,r ˆ 2/(2*H)]BesselI[1,r ˆ 2/(2*H)])/(2*H ˆ (3/2)*Sqrt[Pi]) (*Distances and directions between nodes*) R0[r_,s_]:=Sqrt[(x1[[r]]-x1[[s]]) ˆ 2+(x2[[r]]-x2[[s]]) ˆ 2] R[r_,s_]:=Sqrt[(x1[[r]]-x1[[s]]) ˆ 2+(x2[[r]]-x2[[s]]) ˆ 2]+0.0001*h

Cracks in heterogeneous media

239

nn1[r_,s_]:=(x1[[r]]-x1[[s]])/R[r,s] nn2[r_,s_]:=(x2[[r]]-x2[[s]])/R[r,s] (*Auxiliary matrices*) ps0[r_,s_]:=N[f0[R0[r,s]/h]] ps1[r_,s_]:=N[f1[R0[r,s]/h]/h] ps2[r_,s_]:=N[f2[R0[r,s]/h]/h] PS00 :=Module[{}, PS0=Table[0.,{Mn}]; Do[PS0[[i]]=Chop[ps0[i,1]], {i,Mn}]; PS0] PS10 :=Module[{}, PS1=Table[0.,{Mn}]; Do[ PS1[[i]]=Chop[ps1[i,1]],{i,Mn}]; PS1] PS20 :=Module[{}, PS2=Table[0.,{Mn}]; Do[ PS2[[i]]=Chop[ps2[i,1]], {i,Mn}]; PS2] Tn10:=Table[nn1[r,1],{r,Mn}] Tn20:=Table[nn2[r,1],{r,Mn}] (*Auxiliary object for calculation of the matrix-vector products by the FFT algorithm*) FtMg := Module[{TPs0, TPs1,TPs2, Tn1, Tn2,v1,v2, Tm11, Tm12, Tm13, Tm22, Tm23,Tm33}, TPs0 = PS00; TPs1 = PS10; TPs2 = PS20; Tn1 = Tn10; Tn2 = Tn20; v1 = mu0/2*(TPs1+(2*kp0-1)*TPs2); v2 = mu0*(2*kp0-1)/2*(TPs1-2*TPs2); Tm11 = v1+v2*Tn1*Tn1; Tm12 = v2*Tn1*Tn2; Tm13 = TPs0; Tm22 = v1+v2*Tn2*Tn2; Tm33 = mu0*kp0*TPs1; FtMg = Table[0, {3}, {3}, {N1}, {N2}]; FtMg[[1, 1]] = Module[{AA, MG11}, MG11 = Table[0., {N1}, {N2}]; MG11[[1, 1]] = Tm11[[1]]; Do[AA = Tm11[[M[l, 1]]]; MG11[[l, 1]] = AA; MG11[[N1 - l + 2, 1]] = AA, {l, 2, M1}]; Do[AA = Tm11[[M[1, m]]]; MG11[[1, m]] = AA; MG11[[1, N2 - m + 2]] = AA, {m, 2, M2}]; Do[AA = Tm11[[M[l, m]]]; MG11[[l, m]] = AA; MG11[[N1 - l + 2, m]] = AA; MG11[[l, N2 - m + 2]] = AA; MG11[[N1 - l + 2, N2 - m + 2]] = AA, {l, 2, M1}, {m, 2, M2}]; FtMg[[1, 1]] = Fourier[MG11]]; FtMg[[1, 1]]; FtMg[[1, 2]] = Module[{AA, MG12}, MG12 = Table[0., {N1}, {N2}]; MG12[[1, 1]] = Tm12[[1]]; Do[AA = Tm12[[M[l, 1]]]; MG12[[l, 1]] = AA; MG12[[N1 - l + 2, 1]] = -AA, {l, 2, M1}]; Do[AA = Tm12[[M[1, m]]]; MG12[[1, m]] = AA;

240

Heterogeneous Media

MG12[[1, N2 - m + 2]] = -AA, {m, 2, M2}]; Do[AA = Tm12[[M[l, m]]]; MG12[[l, m]] = AA; MG12[[N1 - l + 2, m]] = -AA; MG12[[l, N2 - m + 2]] = -AA; MG12[[N1 - l + 2, N2 - m + 2]] = AA, {l, 2, M1}, {m, 2, M2}]; FtMg[[1, 2]] = Fourier[MG12]; FtMg[[1, 2]]]; FtMg[[1, 3]] = Module[{AA, MG13}, MG13 = Table[0., {N1}, {N2}]; MG13[[1, 1]] = Tm13[[1]]; Do[AA = Tm13[[M[l, 1]]]; MG13[[l, 1]] = AA; MG13[[N1 - l + 2, 1]] = AA, {l, 2, M1}]; Do[AA = Tm13[[M[1, m]]]; MG13[[1, m]] = AA; MG13[[1, N2 - m + 2]] = AA, {m, 2, M2}]; Do[AA = Tm13[[M[l, m]]]; MG13[[l, m]] = AA; MG13[[N1 - l + 2, m]] = AA; MG13[[l, N2 - m + 2]] = AA; MG13[[N1 - l + 2, N2 - m + 2]] =AA, {l, 2, M1}, {m, 2, M2}]; FtMg[[1, 3]] = Fourier[MG13]; FtMg[[1, 3]]]; FtMg[[2, 2]] = Module[{AA, MG22}, MG22 = Table[0., {N1}, {N2}]; MG22[[1, 1]] = Tm22[[1]]; Do[AA = Tm22[[M[l, 1]]]; MG22[[l, 1]] = AA; MG22[[N1 - l + 2, 1]] = AA, {l, 2, M1}]; Do[AA = Tm22[[M[1, m]]]; MG22[[1, m]] = AA; MG22[[1, N2 - m + 2]] = AA, {m, 2, M2}]; Do[AA = Tm22[[M[l, m]]]; MG22[[l, m]] = AA; MG22[[N1 - l + 2, m]] = AA; MG22[[l, N2 - m + 2]] = AA; MG22[[N1 - l + 2, N2 - m + 2]] = AA, {l, 2, M1}, {m, 2, M2}]; FtMg[[2, 2]] = Fourier[MG22]; FtMg[[2, 2]]]; FtMg[[3, 3]] = Module[{AA, MG33}, MG33 = Table[0., {N1}, {N2}]; MG33[[1, 1]] = Tm33[[1]]; Do[AA = Tm33[[M[l, 1]]]; MG33[[l, 1]] = AA; MG33[[N1 - l + 2, 1]] = AA, {l, 2, M1}]; Do[AA = Tm33[[M[1, m]]]; MG33[[1, m]] = AA; MG33[[1, N2 - m + 2]] = AA, {m, 2, M2}]; Do[AA = Tm33[[M[l, m]]]; MG33[[l, m]] = AA; MG33[[N1 - l + 2, m]] = AA; MG33[[l, N2 - m + 2]] = AA; MG33[[N1 - l + 2, N2 - m + 2]] = AA, {l, 2, M1}, {m, 2, M2}]; FtMg[[3, 3]] = Fourier[MG33]; FtMg[[3, 3]]]; FtMg]; Fd:=FtMg[[1,3]] (*Auxiliary two-index objects for calculation of matrix-vector products*) tp1[P_] := Module[{AA, InP}, InP = To2Ind[P[[1;;Mn]]]; AA = Table[0., {N1}, {N2}]; Do[AA[[i, j]] = InP[[i, j]], {i, M1}, {j, M2}]; AA] tp2[P_] := Module[{AA, InP}, InP = To2Ind[P[[1+Mn;;2*Mn]]]; AA = Table[0., {N1}, {N2}]; Do[AA[[i, j]] = InP[[i, j]], {i, M1}, {j, M2}]; AA] tp3[P_] := Module[{AA, InP}, InP = To2Ind[P[[1+2*Mn;;3*Mn]]]; AA = Table[0., {N1}, {N2}]; Do[AA[[i, j]] = InP[[i, j]], {i, M1}, {j, M2}]; AA] (*Calculation of matrix-vector products by the FFT algorithm*) Z[P_] := Module[ {Ftp1, Ftp2, Ftp3, Z10, Z20, Z30, Z1, Z2, Z3},

Cracks in heterogeneous media

241

Ftp1 = Fourier[tp1[P]]; Ftp2 = Fourier[tp2[P]]; Ftp3 = Fourier[tp3[P]]; Z10 = Sqrt[N1*N2]*InverseFourier[FtMg[[1,1]]*Ftp1 + FtMg[[1,2]]*Ftp2]; Z20 = Sqrt[N1*N2]*InverseFourier[FtMg[[1,2]]*Ftp1 + FtMg[[2,2]]*Ftp2]; Z30 = Sqrt[N1*N2]*InverseFourier[FtMg[[3,3]]*Ftp3]; Y1 = Ind*To1Ind[Chop[Z10[[1;;M1,1;;M2]]]]; Y2 = Ind*To1Ind[Chop[Z20[[1;;M1,1;;M2]]]]; Y3 = Ind*To1Ind[Chop[Z30[[1;;M1,1;;M2]]]]; Join[Y1,Y2,Y3]] (*Conjugate gradient method*) IT:=Do[F0=FF[j]; S[0]=Table[0,{3*Mn}]; P[1]=IND*F0; rr[0]=P[1]; del=1; While[del>0.01, P[1]=IND*P[1]; ZZ=Z[P[1]]; nu[1]=(rr[0].rr[0])/(P[1].ZZ); S[1]=IND*(S[0]+nu[1]*P[1]); rr[1]=IND*(rr[0]-nu[1]*ZZ); mmu[2]=(rr[1].rr[1])/(rr[0].rr[0]); P[2]=(rr[1]+mmu[2]*P[1]); rr[0]=rr[1]; P[1]=P[2]; S[0]=IND*S[1]; del=Chop[rr[1].rr[1]]; Print[del]];B[j]=S[0],{j,3}] (*The FFT algorithm of summation of Gaussian functions*) Fd1:= Module[{d1, PS0}, PS0 = PS00; Fd1 = Table[0, {N1}, {N2}]; d1 = Module[{AA}, d1 = Table[0, {N1}, {N2}]; d1[[1, 1]] = PS0[[1]]; Do[AA = PS0[[M[l, 1]]]; d1[[l, 1]] = AA; d1[[N1 - l + 2, 1]] = AA, {l, 2, M1}]; Do[AA = PS0[[M[1, m]]]; d1[[1, m]] = AA; d1[[1, N2 - m + 2]] = AA, {m, 2, M2}]; Do[AA = PS0[[M[l, m]]]; d1[[l, m]] = AA; d1[[N1 - l + 2, m]] = AA; d1[[l, N2 - m + 2]] = AA; d1[[N1 - l + 2, N2 - m + 2]] = AA, {l, 2, M1}, {m, 2, M2}]; d1]; Fd1 = Fourier[d1]; Fd1]; FB:= Module[{B1,B2,B3}, Do[Module[{Inp}, Inp = To2Ind[B[j][[1 ;; Mn]]]; B1 = Table[0., {N1}, {N2}]; Do[B1[[k, l]] = Inp[[k, l]], {k, M1}, {l, M2}]; Inp = To2Ind[B[j][[1 + Mn ;; 2*Mn]]]; B2 = Table[0., {N1}, {N2}]; Do[B2[[k, l]] = Inp[[k, l]], {k, M1}, {l, M2}]; Inp = To2Ind[B[j][[1 + 2*Mn ;; 3*Mn]]]; B3 = Table[0., {N1}, {N2}]; Do[B3[[k, l]] = Inp[[k, l]], {k, M1}, {l, M2}]]; B0[j, 1] = B1; B0[j, 2] = B2; B0[j, 3] = B3, {j, 3}]; FB = Table[0, {3}, {3}, {M1}, {M2}]; Do[FB[[i,j]]=Fourier[B0[i,j]],{i,3},{j,i,3}]; FB]; (*Crack opening at the nodes*) BB:=Module[{AA},BB=Table[0,{3},{3},{Mn}];

242

Heterogeneous Media

Do[AA=Chop[Sqrt[N1*N2]*InverseFourier[Fd1*FB[[i,j]]]]; BB[[i,j]]=To1Ind[AA[[1;;M1,1;;M2]]],{i,3},{j,i,3}];BB] (*Interpolation of the crack opening on the region of solution*) IB[k_, l_] :=Interpolation[Table[{{N[x1[[i]]], N[x2[[i]]]}, BB[[k, l, i]]}, {i, Mn}]] End

References [1] I. Kunin, The Theory of Elastic Media with Microstructure II, Springer, Berlin, 1983. [2] S. Kanaun, V. Levin, Self-Consistent Methods for Composites, vol. 1, Static Problems, Springer, Dordrecht, 2008. [3] G. Eskin, Boundary-Value Problems for Elliptic Pseudo-Differential Equations, American Mathematical Society, New York, 1981. [4] M. Gosz, B. Moran, An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions, Engineering Fracture Mechanics 69 (2002) 299–319. [5] A. Mal, S. Singh, Deformation of Elastic Solids, Prentice Hall, New Jersey, 1991. [6] A. Alexandrov, Yu. Soloviev, Three Dimensional Problems of Elasticity, Nauka, Moscow, 1978 (in Russian). [7] Y. Murakami, T. Norikura, T. Yasuda, Stress intensity factors for a penny-shaped crack emanating from an ellipsoidal cavity, Transactions of the American Society of Mechanical Engineers 436 (1982) 1558–1565. [8] A. Atsumi, Y. Shindo, Axially symmetrical stress problem of an elastic sphere with peripheral edge crack, Transactions of the American Society of Mechanical Engineers 422 (1981) 1006–1011. [9] M. Kachanov, J.-P. Laures, Three-dimensional problems of strongly interacting arbitrarily located penny-shaped cracks, International Journal of Fracture 41 (1989) 289–313. [10] M. Kachanov, Elastic Solids with Many Cracks and Related Problems, Advances in Applied Mechanics, vol. 30, Academic Press, 1993, pp. 256–426. [11] H. Shodja, I. Rad, R. Soheilifard, Interacting cracks and ellipsoidal inhomogeneities by the equivalent inclusion method, Journal of the Mechanics and Physics of Solids 51 (2003) 945–960. [12] V. Kushch, Interacting cracks and inclusions in a solid by multipole expansion method, International Journal of Solids and Structures 35 (15) (1998) 1751–1762. [13] R. Kant, D. Bogy, The elastostatic axisymmetric problem of a cracked sphere embedded in a dissimilar matrix, Journal of Applied Mechanics 47 (3) (1980) 545–550. [14] V. Maz’ya, G. Schmidt, Approximate Approximation, Mathematical Surveys and Monographs, vol. 141, American Mathematical Society, Providence, 2007. [15] S. Kanaun, Fast solution of 3D-elasticity problem of a planar crack of arbitrary shape, International Journal of Fracture 148 (2007) 435–442. [16] S. Kanaun, A. Markov, S. Babaii, An efficient numerical method for the solution of the second boundary value problem of elasticity for 3D-bodies with cracks, International Journal of Fracture 183 (2013) 169–186. [17] S. Kanaun, A. Markov, Stress fields in 3D-elastic material containing multiple interacting cracks of arbitrary shapes: efficient calculation, International Journal of Engineering Science 75 (2014) 118–134.

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[18] A. Markov, S. Kanaun, Interactions of cracks and inclusions in homogeneous elastic media, International Journal of Fracture 206 (2017) 35–48. [19] S. Kanaun, Cavities and cracks subjected to pressure of injected fluid in poroelastic media, International Journal of Engineering Science 137 (2019) 73–91.

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Time-harmonic fields in heterogeneous media

7

This chapter is devoted to numerical solution of the problems of plane monochromatic wave scattering from heterogeneities in homogeneous host media. The integrodifferential equations of the 3D scattering problems of acoustics, electromagnetics, elasticity, and poroelasticity are discretized by Gaussian approximating functions. For iterative solutions of the discretized equations, the minimal residual method (MRM), the conjugate gradient method (CGM), and the FFT algorithms for the calculation of matrix-vector products are adopted. Examples of computational programs for numerical solution of basic scattering problems are presented. Numerical calculations are compared with exact solutions of the scattering problems for spherical heterogeneities.

7.1

Scattering of acoustic waves from heterogeneities in fluids

Let a homogeneous fluid with density ρ0 and bulk modulus K0 contain a heterogeneous region V . The density ρ(x) and bulk modulus K(x) in V are piece-wise analytical functions of coordinates. For time-harmonic fields, the acoustic pressure in the heterogeneous fluid is presented in the form p(x, t) = p(x)eiωt , and the integrodifferential equation for the pressure amplitude p(x) is obtained in Section 3.8 in the form 

1 (x  )∂i p(x  )dx  + p(x) = p0 (x) + ∂i g(x − x  )R V  + κ02 g(x − x  ) κ12 (x  )p(x  )dx  .

(7.1)

V

Here, p0 (x) is the amplitude of a monochromatic incident wave propagating in the host fluid, e−iκ0 |x| , 4π|x| 1 (x) = R1 (x) , R R0 2 κ (x)  κ12 (x) = 1 2 = κ0 g(x) =

ρ0 , K0 1 1 1 R1 (x) = − , R0 = , ρ(x) ρ0 ρ0 κ02 = ω2

K1 (x) , K1 (x) = K(x) − K0 . K(x)

Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00014-7 Copyright © 2021 Elsevier Ltd. All rights reserved.

(7.2) (7.3) (7.4)

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Heterogeneous Media

The integral terms in Eq. (7.1) are the field ps (x) scattered from the heterogeneity V   1 (x  )∂i p(x  )dx  + κ02 ∂i g(x − x  ) R g(x − x  ) κ12 (x  )p(x  )dx  . ps (x) = V

V

(7.5) For numerical solution, the integro-differential equation (7.1) should be discretized, and further, Gaussian radial functions are used for this purpose. In order to perform the discretization, the Gaussian quasiinterpolants of the derivatives ∂i p(x) of the acoustic pressure in Eq. (7.1) should be determined.

7.1.1 Approximation of partial derivatives ∂i p(x) by the Gaussian radial functions The Gaussian quasiinterpolant of a function p(x) has the form  p (n) ϕ(x − x (n) ), p (n) = p(x (n) ), p(x) ≈

(7.6)

n

  1 |x|2 , ϕ(x) = exp − 2 (πH )3/2 h H

(7.7)

where x (n) are the nodes of a cubic grid with the step h covering the entire 3D space. If (x1 , x2 , x3 ) are Cartesian coordinates in the space, each node can be indicated by three numbers (i, j, k) (the three-index numeration) (j )

(i)

(k)

x (i,j,k) = (x1 , x2 , x3 ), (i) x1

= ih,

(j ) x2

= j h,

(k) x3

(7.8) = kh, −∞ < i, j, k < ∞.

(7.9)

In the three-index numeration, the sum in Eq. (7.6) takes the form of a triple sum with respect to the indices i, j, k, i.e., p(x) ≈

∞ 

  p (i,j,k) ϕ x − x (i,j,k) , p (i,j,k) = p(x (i,j,k) ).

(7.10)

i,j,k=−∞

For approximation of the partial derivatives ∂i p(x) by the Gaussian functions, we take into account that the coefficients of the series (7.10) are the values of the function p(x) at the nodes. Therefore, the following finite difference equations for the derivatives ∂i p hold: p (i+1,j,k) − p (i−1,j,k) , 2h p (i,j +1,k) − p (i,j −1,k) ∂2 p(ih, j h, kh) ≈ D2 p (i,j,k) = , 2h p (i,j,k+1) − p (i,j,k−1) . ∂3 p(ih, j h, kh) ≈ D3 p (i,j,k) = 2h

∂1 p(ih, j h, kh) ≈ D1 p (i,j,k) =

(7.11) (7.12) (7.13)

Time-harmonic fields in heterogeneous media

247

As a result, the approximations of the derivatives ∂l p(x) (l = 1, 2, 3) take the forms of the series similar to (7.10) ∂p(x1 , x2 , x3 ) ≈ ∂xl

∞ 

(i)

(j )

(k)

Dl p (i,j,k) ϕ(x1 − x1 , x2 − x2 , x3 − x3 ),

(7.14)

i,j,k=−∞

where the operators Dl are defined in Eqs. (7.11)–(7.13). For numerical solution of the integro-differential equation (7.1), the heterogeneous region V is embedded in a cuboid W that is covered by a cubic node grid with the step h (Fig. 4.7). The function p(x) and its partial derivatives ∂i p(x) are approximated in W by the series p(x) ≈

Mn 

p (n) ϕ(x − x (n) ),

∂i p(x) ≈

n=1

Mn 

Di p (n) ϕ(x − x (n) ).

(7.15)

n=1

Here, x (n) are the approximating nodes, Mn is the total number of nodes, p (n) = p(x (n) ) are unknown coefficients of the approximation, and the coefficients Di p (n) are expressed in terms of the coefficients p (n) by the linear equations (7.11)–(7.13). 1 (x)∂i p(x) in Eq. (7.1) is approximated by the series The product R 1 (x)∂i p(x) ≈ R

Mn 

(n) = R 1 (x (n) ). (n) Di p (n) ϕ(x − x (n) ), R R 1 1

(7.16)

n=1

For reliable calculations of the derivatives of p(x) in the region V , the distance from V to the boundaries of the cuboid W should be not smaller than h.

7.1.2 Discretization of the integral equation (7.1) Approximating the functions p(x) and ∂i p(x) by the series (7.15) and substituting the approximations into the integrals in Eq. (7.1), we obtain p(x) −

Mn Mn   2(n) (n) Di p (n) − i (x − x (n) , κ0 )R γ (x − x (n) , κ0 ) κ1 p (n) = p0 (x), 1 n=1

n=1

(7.17) (n) R 1

1 (x =R

(n)

),

2(n)  κ1

= κ12 (x (n) ).

(7.18)

Here, i (x, κ0 ) and γ (x, κ0 ) are the integrals      2 i (x, κ0 ) = ∂i g(x − x )ϕ(x )dx , γ (x, κ0 ) = κ0 g(x − x  )ϕ(x  )dx  (7.19) calculated over the entire 3D space. The system of linear algebraic equations for unknowns p (n) in Eq. (7.15) follows from Eq. (7.17) if the latter is satisfied at all nodes

248

Heterogeneous Media

x (n) (the collocation method), p (m) −

Mn Mn   (m,n) (n) 2(n) (m) i γ (m,n) κ1 p (n) = p0 , m = 1, 2, ..., Mn, R1 Di p (n) − n=1

n=1

(7.20) (m,n) i

= i (x

(m)

−x

(n)

, κ0 ),

γ

(m,n)

= γ (x

(m)

−x

(n)

, κ0 ).

(7.21)

Since the objects Di p (n) are expressed in terms of linear combinations of the coefficients p (n) in Eqs. (7.11)–(7.13), Eq. (7.20) is a linear algebraic system for the coefficients p (n) . For the Gaussian function ϕ(x) in Eq. (7.7), the integrals i (x, κ0 ) and γ (x, κ0 ) in Eq. (7.19) are expressed in terms of the scalar function I 0 (x, κ0 ), i.e., i (x, κ0 ) = h2 ∂i I 0 (x, κ0 ), γ (x, κ0 ) = h2 κ02 I 0 (x, κ0 ),  1 e−iκ0 |x| 0 . I (x, κ0 ) = 2 g(x − x  )ϕ(x  )dx  , g(x) = 4π |x| h

(7.22) (7.23)

It is shown in Section 4.4 (Eq. (4.111)) that I 0 (x, κ0 ) is a function of the dimensionless variables ξ = |x|/ h and q = κ0 h, i.e.,   |x| , κ0 h = I 0 (x, κ0 ) = 0 h    

  |x| |x| |x| 1 (7.24) f0+ , κ0 h f− , κ0 h − e2iqξ f+ , κ0 h , = 8πξ h h h     H q 2 ± 4iqξ iH q ± 2ξ . (7.25) , f± (ξ, q0 ) = erfc f0± (ξ, q) = exp − √ 4 2 H The asymptotic φ0 (ξ, q) of the function 0 (ξ, q) for large values of ξ has the form φ0 (ξ, q) =

  q2 H q 2 + 4iξ q exp − . 4πξ 4

Thus, for the function i (x, κ0 ) and γ (x, κ0 ), we have the equations   xi |x| i (x, κ0 ) = h2 0 , κ0 h ni , ni = , h |x|   |x| , κ0 h . γ (x, κ0 ) = h2 κ02 0 h

(7.26)

(7.27) (7.28)

Here, 0 (ξ, q) is the derivative of 0 (ξ, q) with respect to ξ , and its explicit form is 0 (ξ, q) =

  d0 (ξ, q) q2 [4ξ exp −ξ 2 /H − = √ dξ 8 H π 3/2 ξ 2

Time-harmonic fields in heterogeneous media

249

√ − f0+ (ξ, q) πH ((1 + iqξ )f− (ξ, q) − exp (2iqξ ) (1 − iqξ )f+ (ξ, q))]. (7.29) The asymptotics of the functions i (x, κ0 ) and γ (x, κ0 ) for |x|/ h  1 follow from Eqs. (7.27) and (7.28) if the function 0 (ξ, q) in these equations is changed to the asymptotic φ0 (ξ, q) in Eq. (7.26). Eq. (7.20) can be presented in the matrix form (I − B)X = F,

(7.30)

where I is the unit matrix of the dimensions Mn × Mn and the vectors of unknowns X and of the right hand side F are   T T X = p (1) , p (2) , ..., p (Mn) , F = p0(1) , p0(2) , ..., p0(Mn) .

(7.31)

The matrix B in Eq. (7.30) is reconstructed from Eq. (7.20).

7.1.3 Scattering of acoustic waves from a spherical heterogeneity We consider a spherical heterogeneity of the radius a with constant density ρ and bulk modulus K in a homogeneous fluid with the parameters ρ0 , K0 . Let a plane monochromatic wave with the amplitude p0 (x) = p0 exp



−iκ0 n0i xi



, κ0 = ω

ρ0 K0

(7.32)

be scattered from the heterogeneity. Here, κ0 n0i is the wave vector of the incident pressure wave. We introduce a Cartesian coordinate system (x1 , x2 , x3 ) with the origin at the center of the heterogeneity and a spherical coordinate system (r, ϕ, θ) with the same origin and the polar axis directed along the x3 -axis (Fig. 7.1). The vector n0i is directed along the x3 -axis.

Figure 7.1 A spherical inclusion subjected to a plane monochromatic incident wave in the direction of the x3 -axis of the Cartesian coordinate system (x1 , x2 , x3 ).

250

Heterogeneous Media

In the spherical coordinates, the incident field p0 (x) is presented in the form of the series [1] p0 (x) =

∞ 

(2m + 1)(−i)m jm (κ0 r)Pm (cos(θ )).

(7.33)

m=0

Here, jm (z) are spherical Bessel functions and Pm (z) are Legendre polynomials. For constant density ρ and bulk modulus K inside the spherical heterogeneity, the pressure p(x) at r < a is presented in the form p(x) =

∞ 

αm jm (κr)Pm (cos(θ )), r < a, κ = ω

m=0

ρ . K

(7.34)

The pressure in the medium (r > a) is the sum of the incident p0 (x) and scattered ps (x) fields p(x) = p0 (x) + ps (x),

(7.35)

and ps (x) is presented in the form of the series ps (x) =

∞ 

βm hm (κ0 r)Pm (cos(θ )), r > a,

(7.36)

m=0 (2)

where hm (κ0 r) = hm (κ0 r) are spherical Hankel functions of the second kind. The conditions on the surface  of the heterogeneity (r = a) has the form of Eqs. (3.244) and (3.245) p − (x) = p0+ (x) + ps+ (x), |x| = a,  1 ∂ − 1 ∂  + p (x) = p0 (x) + ps+ (x) , |x| = a. ρ ∂r ρ0 ∂r

(7.37) (7.38)

Here, p + (x) and p − (x) are the limit values of the pressure on the surface  from the side of the external normal to  and from the opposite side. The expressions for the coefficients αm and βm in Eqs. (7.34) and (7.36) follow from the boundary conditions (7.37) and (7.38) and take the forms (−i)m−1 (2m + 1)ρ , (κ0 a 2 )[ρ0 κjm (κa)hm (κ0 a) − ρκ0 h (κ0 a)jm (κa)] ρ0 κjm (κ0 a)jm (κa) − ρκ0 jm (κ0 a)jm (κa) βm = p0 (−i)m (2m + 1) , ρκ0 jm (κa)hm (κ0 a) − ρ0 κjm (κa)hm (κ0 a) d d jm (z) = jm (z), hm (z) = hm (z). dz dz αm = p 0

(7.39) (7.40) (7.41)

Time-harmonic fields in heterogeneous media

251

Since the asymptotics of the spherical Hankel functions hm (z) for large z have the forms [1] hm (z) ≈

i m+1 −iz e , m = 0, 1, 2, ..., z

(7.42)

the scattered field ps (x) far from the heterogeneity is determined by the following asymptotic equation: ps (x) ≈ A(n)

e−ik0 r xi , ni = , r r

(7.43)

where the amplitude A(n) is the series A(n) = −

∞ 1  m−1 i βm Pm (cos θ )), cos θ = n3 . κ0

(7.44)

m=0

Here, θ is the azimuthal angle of the vector ni in the spherical coordinate system. The total scattering cross-section Q of the heterogeneity is the integral in Eq. (3.252)  Q=

|n|=1

|A(n)|2 dSn , p02

(7.45)

and the differential cross-section F (n) is the integrand function F (n) =

|A(n)|2 . p02

(7.46)

Note that the amplitude A(n) of the far scattering field is also defined in Eq. (3.251), and it is expressed in terms of the integrals over the inclusion volume V as follows: −iκ0 A(n) = 4π

 V

1 (x)ni τi (x)eiκ0 ni xi dx + R

κ02 4π

 V

 κ12 (x)p(x)eiκ0 ni xi dx. (7.47)

Here, τi (x) = ∂i p(x). For the Gaussian quasiinterpolants of the integrand functions, the approximate equation for A(n) takes the form   Mn κ02 h2 H −iκ0 3  (n) (n) (n) h + R1 ni τi exp iκ0 ni xi − A(n) ≈ 4π 4 n=1   Mn 2 h2 H κ h3 κ02  2(n) (n) (n)  κ1 (x)p (n) exp iκ0 ni xi − 0 , τi = τi (x (n) ). (7.48) + 4π 4 n=1

The exact (solid lines) and numerical (dashed lines) solutions of the scattering problem for a spherical heterogeneity are presented in Figs. 7.2–7.9 for various values of

252

Heterogeneous Media

the dimensionless wave number κ0 a of the incident wave. In the calculations, the parameters of the host fluid are ρ0 = 103 kg/m3 , K0 = 2 GPa, and the parameters of the heterogeneity are ρ = 100 kg/m3 , K = 200 MPa. The heterogeneity is embedded into the cuboid W : (|xi | ≤ a + h, i = 1, 2, 3), which is covered by a regular nod grid with the step h/a = 0.05 (Mn = 79507). Then, the discretized equation of the problem (7.20) is solved by the MRM, and the FFT algorithm is adopted for calculation of matrix-vector products. In Fig. 7.2, the distributions of the real and imaginary parts of the pressure along the x1 - and x3 -axes inside the heterogeneity are presented for the dimensionless wave number κ0 a = 0.5. The graphs of the same distributions for κ0 a = 1, 2, 4 are shown in Figs. 7.4, 7.6, and 7.8. The differential cross-sections F (n) of the inclusion in Eq. (7.46) can be presented as a 2D surface defined by the equation r = F (n), where r is the distance from the origin to the cross-section surface in the direction ni . In the case of spherical inclusion, this surface is symmetric with respect to the x3 -axis, and sections of this surface by the coordinate plane x2 = 0 are shown in Figs. 7.3, 7.5, 7.7, and 7.9 for κ0 a = 0.5, 1, 2, 4. The program for numerical solution of the integral equation (7.1) is presented in Appendix 7.A.

Figure 7.2 Distributions of real and imaginary parts of the acoustic pressure along the x1 - and x3 -axes

inside a spherical heterogeneity for the incident wave number κ0 a = 0.5; solid lines correspond to the exact solution, lines with dots to the numerical solution with the parameters H = 1, h/a = 0.05, Mn = 79507.

7.1.4 Acoustic wave scattering from a group of heterogeneities in fluid Let nine identical ellipsoidal inclusions be arranged as shown in Fig. 7.10. The inclusions have the same orientations and semiaxes a1 = 0.6, a2 = a3 = 0.3. The inclusion centers X (n) (n = 1, 2, ..., 9) are at the points

Time-harmonic fields in heterogeneous media

253

Figure 7.3 The differential cross-section of the spherical inclusion of radius a for the incident wave number κ0 a = 0.5, where the horizontal arrow indicates the direction of the incident wave vector; the solid line is the exact solution, the line with dots is the numerical solution for H = 1, h/a = 0.05, Mn = 79507.

Figure 7.4 The same as in Fig. 7.2 for the incident wave number κ0 a = 1.

Figure 7.5 The same as in Fig. 7.3 for the incident wave number κ0 a = 1.

X (1) = (−2 + a1 , −1 + a2 , −1 + a2 ), X (3) = (−2 + a1 , −1 + a2 , 1 − a2 ), X (5) = (2 − a1 , 1 − a2 , 1 − a2 ), X (7) = (2 − a1 , −1 + a2 , 1 − a2 ), X (9) = (0, 0, 0).

X (2) = (−2 + a1 , 1 − a2 , −1 + a2 ), X (4) = (−2 + a1 , 1 − a2 , 1 − a2 ), X (6) = (2 − a1 , 1 − a2 , −1 + a2 ), X (8) = (2 − a1 , −1 + a2 , −1 + a2 ), (7.49)

254

Heterogeneous Media

Figure 7.6 The same as in Fig. 7.2 for the incident wave number κ0 a = 2.

Figure 7.7 The same as in Fig. 7.3 for the incident wave number κ0 a = 2.

Figure 7.8 The same as in Fig. 7.2 for the incident wave number κ0 a = 4.

The properties of the host medium and the inclusions are the same as in the case of a spherical inclusion considered above; the wave vector of the incident wave is directed along the x3 -axis.

Time-harmonic fields in heterogeneous media

255

Figure 7.9 The same as in Fig. 7.3 for the incident wave number κ0 a = 4.

Figure 7.10 Acoustic wave scattering from a group of nine inclusions; the vertical arrow indicates the direction of the incident wave.

Figure 7.11 The acoustic scattering problem for nine inclusions shown in Fig. 7.10; the real and imaginary parts of the pressure distribution along the x3 -axis in the region of the solution for the incident wave number κ0 L = 3, H = 1, h/L = 0.05, Mn = 153467.

For numerical solution, the inclusions are embedded into the cuboid W : (4L + h, 2L + h, 2L + h), L = 1. Then, W is covered by a cubic node grid with the step h/L = 0.05 (Mn = 153467). The real and imaginary parts of the pressure distribution p(0, 0, x3 ) along the x3 -axis are presented in Fig. 7.11 for κ0 L = 3. The distribution of the real and imaginary parts of the function p(x1 , 0.7, 0.7) along the x1 -axis passing through the centers of the inclusions is shown in Fig. 7.12. The 3D image of the differential cross-section of the group of inclusions is shown in Fig. 7.13. It is seen that for κ0 L = 3, the forward scattering prevails.

256

Heterogeneous Media

Figure 7.12 The acoustic scattering problem for nine inclusions shown in Fig. 7.10; the real and imaginary parts of the pressure distribution along the axis that is parallel to the x1 -axis and passes through the centers of the two inclusions. The parameters of the numerical solution are the same as in Fig. 7.11.

Figure 7.13 The scattering cross-section of the group of nine inclusions shown in Fig. 7.10; the parameters of the solution are the same as in Fig. 7.11.

7.2

Acoustic wave scattering from a rigid screen; direct and inverse problems

We consider a planar rigid screen  in a homogeneous fluid (Fig. 3.7). Let a plane monochromatic incident pressure wave p0 (x, t) = p0 (x)eiωt , p0 (x) = p0 e−iκ0 ni xi 0

(7.50)

Time-harmonic fields in heterogeneous media

257

propagate in the medium and be scattered from the screen. The amplitude p(x) of the acoustic pressure in the medium with the screen can be presented in the form of the potential of the double layer concentrated on  (see Eq. (3.258)), i.e.,   ∂  e−iκ0 |x| g x − x  ni (x  )b(x  )d , g (x) = , (7.51) p(x) = p0 (x) − 4π |x|  ∂xi where ni (x) is the normal vector to . The boundary condition on the screen has the form indicated in Eq. (3.257)  ∂p(x)  = 0. (7.52) ∂n  The equation for the density b(x) of the potential in Eq. (7.51) follows from this boundary condition in the form  ∂p 0 (x) T (x, x  )b(x  )d = , x ∈ , (7.53) ∂n  T (x, x  ) = ni (x)∂i ∂j g(x − x  )nj (x  ) + δ(x). (7.54) Let  be a finite region in a plane P , and let this plane in the Cartesian coordinates (x1 , x2 , x3 ) be defined by the equation x3 = 0. In this case, the normal ni is constant, T (x, x  ) = T (x − x  ), and Eq. (7.53) takes the form  T (x − x  )b(x  )d = t 0 (x), x = (x1 , x2 ) ∈ , (7.55)   ∂p 0 (x, x3 )  T (x) = T (x1 , x2 ) = T (x1 , x2 , x3 )|x3 =0 , t 0 (x) = . (7.56)  ∂x3 x3 =0 The 3D Fourier transform of the function T (x1 , x2 , x3 ) follows from Eq. (7.54) and has the form T ∗ (k1 , k2 , k3 ) = −ni ki g ∗ (k)kj nj + 1,

(7.57)

where the Fourier transform g ∗ (k) of the Green function g(x) in Eq. (7.51) is g ∗ (k) =

k2

1 , k 2 = k12 + k22 + k32 . − κ02

(7.58)

As a result, for the function T ∗ (k), we have the equation T ∗ (k1 , k2 , k3 ) = −

k32 k12 + k22 + k32 − κ02

+1= ∗

k12 + k22 − κ02 k12 + k22 + k32 − κ02

.

(7.59)

The equation for the 2D Fourier transform T (k1 , k2 ) of the function T (x1 , x2 ) follows from Eqs. (7.56) and (7.59) in the form  ∞  1 1 2 ∗ T (k1 , k2 ) = T ∗ (k1 , k2 , k3 )dk3 = k + k22 − κ02 . (7.60) 2π −∞ 2 1

258

Heterogeneous Media

7.2.1 Discretization of Eq. (7.55) by the Gaussian approximating functions For numerical solution of the integral equation (7.55), the screen region  is embedded in a rectangular W and covered by a square node grid with the step h. After approximating the function b(x) by the Gaussian quasiinterpolant in W , b(x) ≈

Mn 

b(n) ϕ(x − x (n) ), b(n) = b(x (n) ),

n=1

b(n) = 0 if x (n) ∈ / , ϕ(x) =

  1 |x|2 exp − , πH H h2

(7.61) (7.62)

and substituting this series into Eq. (7.55), we obtain the equation Mn  1 n=1

h

I (x − x (n) , κ0 )b(n) = t 0 (x), x ∈ , b(n) = 0, x (n) ∈ / .

(7.63)

In this equation, I (x, κ0 ) is the integral over the entire plane P , i.e.,  I (x, κ0 ) = h

T (x − x  )ϕ(x  )dx  .

(7.64)

P

The system of linear algebraic equations for the coefficients b(n) in Eq. (7.61) follows from Eq. (7.63) by the requirement of satisfaction of these equations at the nodes x (n) belonging to . As a result, we obtain the system of linear algebraic equations for the coefficients b(m) Mn  1 n=1

h

I

(m,n) (n)

b

= t 0(m) , x (m) ∈ , b(m) = 0 if x (m) ∈ / , m = 1, 2, ..., Mn, (7.65)

I

(m,n)

= I (x (m) − x (n) , κ0 ), t 0(m) = t 0 (x (m) ).

(7.66)

The integral (7.64) is calculated by using the convolution property and transferring to the Fourier transforms of the integrand functions. Then, after integrating over the polar angle in the (k1 , k2 )-plane, we obtain I (x1 , x2 , κ0 ) =  ∞ h ∗ T (k1 , k2 )ϕ ∗ (k1 , k2 ) exp(−i(k1 x1 + k2 x2 )dk1 dk2 = = 2 (2π) −∞  2 2     h3 ∞ k h H = k 2 − κ02 exp − J0 (kr)kdk, r = x12 + x22 . 4π 0 4

(7.67)

Time-harmonic fields in heterogeneous media

259

 ∗ 2 Here, J0 (z) is the Bessel function. It is taken into account that T (k) = k − κ02 and   2 2 ϕ ∗ (k) = h2 exp − k h4 H , k = (k1 , k2 ). Therefore, the function I (x, κ0 ) is defined by the equation I (x, κ0 ) = F

r h

 , κ0 h ,

(7.68)

where the function F (ρ, q) of the dimensionless variables ρ = r/ h and q = κ0 h is the following 1D integral: F (ρ, q) =

1 4π

 0

∞

 2  k H k 2 − q 2 exp − J0 (kρ)kdk. 4

(7.69)

This absolutely converging integral can be calculated numerically and tabulated. For large values of the variable ρ, the asymptotic f (ρ, q) of the function F (ρ, q) is   1 + iqρ q 2H f (ρ, q) = − exp −iqρ − . 4 4πρ 3

(7.70)

For ρ > 5, the function F (ρ, q) can be changed to the asymptotic f (ρ, q) with sufficient accuracy. Eq. (7.65) can be written in the matrix form AX = F,

(7.71)

where the vectors of unknowns X and of the right hand side F are as follows:   T T X = X 1 , X 2 , ..., X Mn , F = F 1 , F 2 , ..., F Mn ,

(7.72)

X n = κ (n) b(n) , κ (n) = κ(x (n) ), F (n) = t 0(n) .

(7.73)

Here, κ(x) is a characteristic function of the screen region , / . κ(x) = 1 if x ∈ , κ(x) = 0 if x ∈

(7.74)

 (m,n) is a square, nonsparse matrix the dimensions Mn × In Eq. (7.71), A = h1 I Mn of which can be large if high accuracy of the solution is required. For solution of Eq. (7.71), the CGM can be used (Section 5.2). It follows from Eq. (7.66) that  (m,n) for regular node grids, the matrix I has Toeplitz properties, and therefore, the FFT algorithm can be used for calculation of the matrix-vector products by iterative solution of Eq. (7.71). The computational program for numerical solution of Eq. (7.71) is presented in Appendix 7.B.

260

Heterogeneous Media

7.2.2 The scattered field far from a rigid screen The integral term in Eq. (7.51) is the field ps (x) scattered from the screen     ∂  g x − x  ni b x  d . ps (x) = −  ∂xi

(7.75)

Far from the screen, this field is presented in the form [2] (see also Section 2.15) ps (x) ≈ A (m)

e−iκ0 |x| xi , mi = . |x| |x|

(7.76)

Here, A(m) is the amplitude of the far scattered field in the direction mi . This amplitude is expressed in terms of the integral of the function b(x) over the screen surface     iκ0 (7.77) (mi ni ) b(x) exp iκ0 mj xj d. A (m) = 4π  For the approximation (7.61) of b(x1 , x2 ), the integral in this equation takes the form    b(x) exp iκ0 mj x j d ≈    Mn   κ 2 h2 H    (n) (n) 0 2 (n) 2 2 b exp iκ0 m1 x 1 + m2 x 2 − m1 + m2 . (7.78) ≈h 4 n=1

The total scattering cross-section of the screen is the integral over the unit sphere  Q=

|A(m)|2 dS m . 2 |m|=1 |p0 |

(7.79)

The integrand function in this equation is the differential scattering cross-section F (m) of the screen F (m) =

|A(m)|2 . |p0 |2

(7.80)

In the case of a planar screen, the function F (m) is symmetric with respect to the screen plane.

7.2.3 Scattering from an elliptic screen We consider an elliptic screen the surface  of which is defined by the equations x12 a12

+

x22 a22

≤ 1, x3 = 0.

(7.81)

Time-harmonic fields in heterogeneous media

261

Let an incident plane wave p 0 (x) with the wave vector orthogonal to  be scattered from the screen. For such a wave, we have p0 (x) = p0 exp(iκ0 x3 ),

(7.82)

and Eq. (7.55) takes the form 

T (x − x  )b(x  )dx  = iκ0 p0 , x ∈ .

(7.83)



For numerical solution, the screen  is embedded into the rectangular W : (2a1 × 2a2 ), which is covered by a square node grid with the step h. In the calculations, the following parameters of the Gaussian functions are taken: H = 1, h/a1 = 0.001 (Mn = 2003001). The results of numerical solutions of Eq. (7.83) are presented in Figs. 7.14–7.16 for various values of the dimensionless wave number κ0 a1 . In these figures, the left hand sides are the functions |b(0, x2 )| /|κ0 bs (0, 0)| and the right hand sides are the functions |b(x1 , 0)| /|κ0 bs (0, 0)|. Here bs (x1 , x2 ) is the solution of Eq. (7.83) for κ0 = 0 and the right hand side equal to p0 (the static solution)   2  2  2 2a2 x1 x2 a2 bs (x1 , x2 ) = p0 1− − , k=1− . E(k) a1 a2 a1

(7.84)

Here, E(κ) is the elliptic integral of the second kind.

Figure 7.14 The normalized amplitude of the pressure jump b(x1 , x2 ) on an elliptic screen with semiaxes a1 , a2 (a1 /a2 = 2) subjected to the incident field orthogonal to the screen plane; the left hand side shows the functions |b(0, x2 )|/|κ0 bs (0, 0)| along the semiaxis a2 , the right hand side shows |b(x1 , 0)|/|κ0 bs (0, 0)| along the semiaxis a1 ; the incident wave numbers are κ0 a1 = 0.5, 1, 2, 2.5, 3; bs is the “static” limit; the parameters of the numerical solution are H = 1, h/a1 = 0.001, Mn = 2003001.

The screen differential cross-section F (m) is defined in Eq. (7.80). In Figs. 7.17 and 7.18, sections of the surface r = F (m) by the planes x1 = 0 (solid lines) and x2 = 0 (dashed lines) are shown for various values of the dimensionless wave number κ0 a1 . It is seen that for long incident waves (κ0 a1 < 2), the surface of the differential cross-section is almost symmetric with respect to the x3 -axis orthogonal to the screen.

262

Heterogeneous Media

Figure 7.15 The same as in Fig. 7.14 for the incident wave numbers κ0 a1 = 4, 5, 6, 8, 10, 12.

Figure 7.16 The same as in Fig. 7.14 for the incident wave numbers κ0 a1 = 14, 16, 20, 24.

Figure 7.17 The differential cross-sections of the elliptic screen with the semiaxes a1 , a2 (a1 /a2 = 2) subjected to the incident wave field orthogonal to the screen plane for dimensionless wave numbers κ0 a1 = 0.5, 1 (left) and κ0 a1 = 0.5, 1, 2, 2.5 (right); solid lines are sections of the function F (ϕ, θ) by the plane x2 = 0, dashed lines are the section of F (ϕ, θ) by the plane x1 = 0.

Time-harmonic fields in heterogeneous media

263

Figure 7.18 The same as in Fig. 7.17 for κ0 a1 = 4, 6, 8 (left) and κ0 a1 = 10, 14, 18 (right).

Figure 7.19 The normalized total scattering cross-section Q of an elliptic screen with semiaxes a1 , a2

(a1 /a2 = 2) subjected to the incident field orthogonal to the screen plane as the function of the dimensionless wave number κ0 a1 .

For κa1 > 2, the energy scattered in the plane x1 = 0 is less than the energy scattered in the plane x2 = 0 for vectors n0i with the same inclination to the screen plane. The dependence of the normalized total scattering cross-section Q = Q/(πa1 a2 ) on the wave number κ0 a1 is presented in Fig. 7.19. In Figs. 7.20 and 7.21, sections of the surface of the differential cross-sections by the plane x2 = 0 are presented for the wave vector n0i oblique to the screen surface for various values of the parameter κ0 a1 , n0i = (sin θ, 0, − cos θ ), θ = π/4. Here, θ is the angle between the vector n0i and the x3 -axis.

264

Heterogeneous Media

Figure 7.20 The differential cross-sections of the elliptic screen with the semiaxes a1 , a2 (a1 /a2 = 2) subjected to an incident field oblique to the screen plane by the angle θ = π/4 for dimensionless wave numbers κ0 a1 = 0.5, 1, 1.5, 2, 2.5, 3, 4; the lines are sections of the functions F (ϕ, θ) by the plane x2 = 0.

Figure 7.21 The same as in Fig. 7.20 with κ0 a1 = 4, 6, 8, 10, 14, 18.

7.2.4 The inverse problem In the numerical method used for solution of the scattering problem from a planar screen, the most time and memory consuming operation is construction of the auxiliary two-index object in Eq. (4.61) associated with the matrix A in Eq. (7.71). In the computational program presented in Appendix 7.B, this object is denoted as FtMg. The object FtMg is used in the FFT algorithm for calculation of the matrix-vector

Time-harmonic fields in heterogeneous media

265

products AX by iterative solution of the discretized problem. The object FtMg depends on the dimensions of the node grid covering the rectangular region W but not on the shape and size of the screen region  ( ∈ W ). If the object FtMg is constructed, it can be used for solution of the scattering problems for screens of various sizes and shapes. The solution process is fast and does not occupy additional computer memory. These specific features of the numerical method open the possibility for efficient solution of the inverse problem related to the rigid screen. The inverse problem is reconstruction of the screen shape and size from the amplitude of the far pressure field scattered from the screen. We focus on a specific situation when the source of the incident waves (transducer) and a system of discrete receivers are in the same plane, and this plane is parallel to the screen surface (see Fig. 7.22).

Figure 7.22 The acoustic experiment associated with the inverse problem for a rigid screen; the transducer of the incident acoustic waves is at the ship, triangles are receivers spread on an area around the transducer.

The plane of the transducer and receivers is at a distance D from the screen plane, and D >> L, where L is a characteristic screen size. The transducer is in the x3 -axis orthogonal to the screen plane, and the receivers cover a region S ∗ in the observation plane. Usually in hydro-acoustics, transducers generate a short-time (milliseconds) package of waves of basic frequency f , and receivers can measure the pressure field scattered from the screen. Spectral analysis allows calculating the amplitudes of the harmonic component of the pressure waves of the basic frequency produced by the transducer and arriving at the receivers. If the distance D is known, one can assess the amplitude of the monochromatic components of frequency f of the incident wave in the screen area. This incident wave can be considered as a plane wave for the screens the sizes of which are much smaller than the distance D. It follows from Eq. (7.76) that the amplitude of the scattered field in the plane of the receivers (x3 = D) is (c is the velocity of the acoustic waves in fluid)  2πf . (7.85) A(x1 , x2 ) = ps (x1 , x2 )|x|eiκ0 |x| , |x| = x12 + x22 + D 2 , κ0 = c Thus, the amplitude A(x1 , x2 ) can be calculated from the experimentally measured pressure ps (x1 , x2 ) at the points of the receivers.

An elliptic screen For an elliptic screen, the unknowns of the inverse problem are the values of the semiaxes a1 , a2 and the orientation of the actual screen. The experimentally measured

266

Heterogeneous Media

amplitude of the scattered field in the plane of the receivers is denoted as A∗ (x1 , x2 ). The “theoretical” amplitude in Eq. (7.85) can be constructed for screens with various semiaxes and orientations according to the algorithm presented in Section 7.2.1. For solution of the inverse problem, we consider two functions J0 (a1 , a2 ) and J (a1 , a2 ) [3]: J0 (a1 , a2 ) = |A(a1 , a2 , 0, 0) − A∗ (0, 0)|,  J (a1 , a2 ) = |A(a1 , a2 , x1 , x2 ) − A∗ (x1 , x2 )|dx1 dx2 , S∗

(7.86) (7.87)

where A(a1 , a2 , x1 , x2 ) is the theoretical amplitude of the scattered field for the screen with semiaxes a1 , a2 . The function J0 (a1 , a2 ) corresponds to the situation when the transducer and receiver are at the same point (echo-signal). For construction of the function J (a1 , a2 ), one has to interpolate the experimental values of A∗ (x1 , x2 ) obtained at the points of the receiver locations on the entire region S ∗ , and then perform integration over S ∗ . The amplitude A(a1 , a2 , x1 , x2 ) is calculated from Eqs. (7.77) and (7.78) after numerical solution of Eq. (7.83) for the function b(x). Minima of the functions J0 (a1 , a2 ) and J (a1 , a2 ) are reached for the “theoretical” screen having the same semiaxes (a1 , a2 ) and orientation as the actual screen. Let the actual screen be at the depth D = 100 m and have semiaxes a1 = 0.8 m, a2 = 0.4 m. The fluid is water with the wave velocity c = 1500 m/sec, and the transducer generates wave packages with three basic frequencies: f = 100 Hz, 1 kHz, and 10 kHz. The lengths of the corresponding incident waves are 15 m, 1.5 m, and 0.15 m. The receivers are at the nodes of a square grid with the step 10 m that covers the square S ∗ : (100 m × 100 m) with the transducer in the center.

Figure 7.23 Contour plots of the far scattered amplitude of the elliptic screen with semiaxes a1 = 0.8 m, a2 = 0.4 m for the frequency of the incident field f = 100 Hz.

In Fig. 7.23, the scattered amplitude from the actual screen for the frequency f = 100 Hz is presented. The lines of the constant values of |A∗ (x1 , x2 )| (contour plots) in this figure show that the scattered amplitude is symmetric with respect to the x3 -axis.

Time-harmonic fields in heterogeneous media

267

Figure 7.24 Contour plots of the functions J0 (a1 , a2 ) and J (a1 , a2 ) in Eqs. (7.86) and (7.87) for the frequency of the incident field f = 100 Hz.

Figure 7.25 The functions J0 (a1 , 0.32/a1 ) and J (a1 , 0.32/a1 ) for the frequency of the incident field f = 100 Hz.

The contour plots of the functions J0 (a1 , a2 ) and J (a1 , a2 ) are shown in Fig. 7.24 for 0.1 ≤ a1 , a2 ≤ 1. It is seen that minima of these functions are on the line with the equation a1 a2 = 0.32 m2 . This line corresponds to the screens of a constant area that coincides with the area of the actual screen. The functions J 0 (a1 ) = J0 (a1 , 0.32/a1) and J (a1 ) = J (a1 , 0.32/a1) are shown in Fig. 7.25. Both functions have two minima for the values of the semiaxes a1 = 0.4 m, a2 = 0.8 m and a1 = 0.8 m, a2 = 0.4 m. Thus, for a frequency of f = 100 Hz, neither the function J0 (a1 , a2 ) nor J (a1 , a2 ) can be used to determine the orientation of the screen in the plane (x1 , x2 ). Only the screen area πa1 a2 can be definitely indicated. Figs. 7.26–7.28 correspond to the frequency f = 1 kHz. It is seen from Fig. 7.26 that the contour plots of the modulus of the scattered amplitude |A∗ (x1 , x2 )| of the actual screen are not symmetric with respect to the x3 -axis, and the screen orientation can be easily found from this figure: The ellipse semiaxes are directed along the axes of symmetry of the function |A∗ (x1 , x2 )|. For f = 1 kHz, the function J0 (a1 , a2 ) has two zero minima as in the case f = 100 Hz, but the function J (a1 , a2 ) has only one

268

Heterogeneous Media

Figure 7.26 The same as in Fig. 7.23 for f = 1 kHz.

Figure 7.27 The same as in Fig. 7.24 for f = 1 kHz.

global minimum that corresponds to the correct values of the semiaxes a1 = 0.8 m, a2 = 0.4 m. It is also seen from Figs. 7.27 and 7.28 that the function J (a1 , a2 ) has three local nonzero minima. These minima result in difficulties for numerical solution of the global minimum of this function by the Newton, steepest descent, or coordinate descent method [3]. These methods do not converge to the global, but to a local minimum, depending on the initial guess. The case f = 10 kHz is shown in Figs. 7.29–7.31. From the contour plots of the amplitude of the scattered field of the actual screen in Fig. 7.29, the screen orientation can be clearly indicated. The graphs in Figs. 7.30 and 7.31 show that the function J0 (a1 , a2 ) has two zero minima at the same points as for the frequencies f = 100 Hz and f = 1 kHz, but the function J (a1 , a2 ) has only one well-pronounced zero minimum for the correct values of the semiaxes: a1 = 0.8 m, a2 = 0.4 m. Application of numerical methods for seeking the minimum of the function J (a1 , a2 ) provides correct values of the actual screen semiaxes.

Time-harmonic fields in heterogeneous media

Figure 7.28 The same as in Fig. 7.25 for f = 1 kHz.

Figure 7.29 The same as in Fig. 7.23 for f = 10 kHz.

Figure 7.30 The same as in Fig. 7.24 for f = 10 kHz.

269

270

Heterogeneous Media

Figure 7.31 The same as in Fig. 7.25 for f = 10 kHz.

Figure 7.32 The screen boundary tortuosity defined by the parameters A0 = 0.7, m = 6 in Eq. (7.88).

Screens with tortuous boundaries We consider a screen, the boundary (contour ) of which is defined by the equations R(ϕ) = r0 + r1 cos(mϕ),

(7.88)

  1 1 (7.89) 1 + 2(3A0 − 1) , r1 = 2 − 2(3A0 − 1) . 3 3 Here, R and ϕ are the radius and the polar angel of a point in the screen boundary and the parameter A0 relates to the screen area Ss : Ss = πA0 , 38 ≤ A0 ≤ 1. If A0 = 38 , then r0 = r1 = 0.5, and if A0 = 1, r0 = 1, r1 = 0. The parameter m is continuous, and 2 ≤ m ≤ 10. These equations define a two-parameter family of screens of the same characteristic size L = 2. An example of the screen with A0 = 0.7 and m = 6 is presented in Fig. 7.32. Henceforth, the units of parameters and functions are not indicated, and it is implied that lengths are measured in meters (m) and areas in m2 , and the incident wave has a unit amplitude. In the case of tortuosity of the screen contour, the inverse problem is determination of the parameters A0 and m from the measured amplitudes of the scattered field far from the actual screen. For solution of this problem, we consider the functions J0 and r0 =

Time-harmonic fields in heterogeneous media

271

J similar to (7.86), (7.87) J0 (A0 , m) = |A(A0 , m, 0, 0) − A∗ (0, 0)|,  J (A0 , m) = |A(A0 , m, x1 , x2 ) − A∗ (x1 , x2 )|dx1 dx2 . S∗

(7.90) (7.91)

Here, A(A0 , m, x1 , x2 ) is the “theoretical” amplitude for the parameters A0 and m, and A∗ (x1 , x2 ) is the actual amplitude that corresponds to the screen with the parameters A0 = 0.7, m = 6. For f = 100 Hz, the contour plots of the functions J0 (A0 , m) and J (A0 , m) are shown in Fig. 7.33. The minima of these functions are on the line A0 = 0.7, and dependencies of J0 (0.7, m) and J (0.7, m) on the parameter m are shown in Fig. 7.34. The both functions have two zero minima at m = 6 and m = 9 and one local minimum at m between 3 and 4. Thus, for this frequency, the actual values of the parameter m cannot be uniquely determined.

Figure 7.33 Contour plots of the functions J0 (A0 , m) and J (A0 , m) in Eqs. (7.90) and (7.91) for the frequency of the incident field f = 100 Hz.

The case f = 1 kHz is shown in Figs. 7.35 and 7.36. Again, the minima of J0 (A0 , m) and J (A0 , m) are in the line A0 = 0.7; the function J0 (A0 , m) has two zero minima at m = 6, m = 9, and a local minimum for m between 3 and 4. The function J (0.7, m) has one zero minimum at m = 6 and two local minima at m = 9 and 3 < m < 4. So, the global minimum of J (A0 , m) provides correct values of the parameters A0 and m. But the presence of the local minima results in difficulties in seeking the global minimum by numerical methods. In the case f = 10 kHz, shown in Figs. 7.37 and 7.38, the function J (A0 , m) has only one well-pronounced zero minimum at A0 = 0.7, m = 6, and local minima of this function do not exist. Function J0 (A0 , m) has one zero minimum and two local minima as before. So, for this frequency, the function J (A0 , m) allows reconstructing the correct tortuosity of the screen boundary.

272

Heterogeneous Media

Figure 7.34 The functions J0 (0.7, m) and J (0.7, m) in Eqs. (7.90) and (7.91) for the frequency of the incident field f = 100 Hz.

Figure 7.35 The same as in Fig. 7.33 for f = 1 kHz.

Figure 7.36 The same as in Fig. 7.34 for f = 1 kHz.

Time-harmonic fields in heterogeneous media

273

Figure 7.37 The same as in Fig. 7.33 for f = 10 kHz.

Figure 7.38 The same as in Fig. 7.34 for f = 10 kHz.

A four-parameter model of the screen boundary The two-parameter model of the screen boundary considered above can be extended to a four-parameter model if the screen boundary is defined by the equations x1 (ϕ) = a1 R(ϕ) cos ϕ, x2 (ϕ) = a2 R(ϕ) cos ϕ, 0 ≤ ϕ ≤ 2π.

(7.92)

Here, (x1 (ϕ), x2 (ϕ)) are Cartesian coordinates of the boundary point and the function R(ϕ) has form (7.88)–(7.89). In this case, ϕ does not coincide with the polar angle of the corresponding boundary point. This model describes a wide class of screen boundaries, and the parameters a1 , a2 define global sizes of the screen, while parameters A0 , m describe tortuosity of the boundary. For the solution of the inverse problem, one has to find minima of the function J (a1 , a2 , A0 , m) similar to (7.90)–(7.91). The solid line in Fig. 7.39 is the screen with the following parameter values: a1 = 0.8, a2 = 0.4, A0 = 0.7, m = 6. This screen is taken as an actual screen for solution of the inverse problem. The testing frequency was f = 10 kHz. The process of solution starts

274

Heterogeneous Media

Figure 7.39 The inverse problem for the screen with the parameters a1 = 0.8, a2 = 0.4, A0 = 0.7, m = 6. The solid line is the actual screen, and lines 1–5 are the screen boundary after the corresponding step of the coordinate descent method. The sixth iteration coincides with the actual screen boundary.

with the value of A0 = 1 (elliptic screen). Because for this A0 , r1 = 0, the parameter m can be arbitrary. The function J (a1 , a2 , 1, m) was constructed for 0.1 ≤ a1 , a2 ≤ 1. The only minimum of J is at a1 = 0.744, a2 = 0.3 (the corresponding line is indicated as 1 in Fig. 7.39). Note that the area of the ellipse with these semiaxes is very close to the area of the actual screen. Then, for the obtained values of a1 , a2 , the minimum of the function J (0.744, 0.3, A0 , m) is to be found with respect to the parameters A0 and m. The unique minimum of this function turns out to be at A0 = 0.9026, m = 6. For these values of A0 , m, the minimum of the function J with respect to a1 , a2 is to be found again. In the literature, this process is called the coordinate descent method [3]. It is necessary to perform six iterations in order to obtain the correct solution of the inverse problem. The contours that correspond to each iteration are indicated by the corresponding numbers in Fig. 7.39. The sixth iteration coincides with the actual screen boundary. At each step of the iteration process, the only minimum of the function J (a1 , a2 , A0 , m) was indicated. The presented results show that the inverse problem for a rigid screen can be successfully solved if the frequency of the incident wave f satisfies the condition 2πf L/c = O(10−1 ). For incident waves of lower frequencies, only the screen area can be definitely identified. For incident waves of higher frequencies, the scattered field is concentrated in a small vicinity of the origin, and for the considered system of the receivers, it is impossible to construct the far scattered amplitude A∗ (x1 , x2 ) with sufficient accuracy. Note that in the short wave limit (f → ∞), the function A∗ (x1 , x2 ) tends to the delta function concentrated at the origin with the coefficient proportional to the screen area. So, in this limit, the details of the screen shape cannot be revealed. In addition, attenuation of waves increases with frequency, and for a large distance D, the scattered field can be lost. If the screen is not parallel to the plane of receivers, its orientation can be found by seeking coordinates of the maximum of the modulus of the scattering amplitude |A∗ (x1 , x2 )| in the observation plane. The normal to the screen surface is directed from the screen center to the point of this maximum. If the angle between this normal to the screen and the x3 -axis is large, the point of the maximum of |A(x1 , x2 )| can be outside the observation region S ∗ , and the screen orientation cannot be found.

Time-harmonic fields in heterogeneous media

7.3

275

Electromagnetic wave scattering from a heterogeneity of arbitrary shape in dielectric media

We consider an infinite homogeneous host medium with the dielectric permittivity c0 . The medium contains a finite heterogeneous region V (inclusion) with the dielectric permittivity c0 + c1 (x); the magnetic permittivity of the host medium and the heterogeneity is μ0 = 1. The dielectric permittivity of the medium with the heterogeneity is presented in the form c(x) = c0 + c1 (x)V (x),

(7.93)

where V (x) is the characteristic function of the region V . Let an incident electric field Ei0 (x, t) Ei0 (x, t) = Ei0 (x)eiωt , Ei0 (x) = Ei0 e−iκ0 ni xi , κ02 = ω2 c0 0

(7.94)

propagate in the medium and be scattered from the region V . Here, κ0 n0i is the wave vector and Ei0 (x) is the polarization vector of the incident field. The amplitude Ei (x) of the electric field in the medium satisfies Eq. (3.271), i.e.,  c1 (x) Ei (x) = Ei0 (x) + κ02 Gij (x − x  ) c1 (x  )Ej (x  )dx  ,  c1 (x) = . (7.95) c0 V The kernel Gij (x) of the integral operator in this equation is defined in Eq. (3.272): Gij (x) = g(x)δij +

1 e−iκ0 |x| ∂ ∂ g(x), g(x) = . i j 4π |x| κ02

(7.96)

The integral term in Eq. (7.95) is the electric field Eis (x) scattered from the heterogeneity, and in the far zone, the scattered field takes the form in Eqs. (3.275) and (3.276). The deferential cross-section of the heterogeneity is defined in Eq. (3.278).

7.3.1 Discretization of the volume integral equation (7.95) Let a cuboid W containing the heterogeneous region V be covered by a cubic grid of approximating nodes at points x (n) (n = 1, 2, ..., Mn) (Fig. 4.7). For discretization of Eq. (7.95), the electric field in W is approximated by the Gaussian quasiinterpolant Ei (x) ≈

Mn  n=1

(n)

(n) Ei ϕ(x

−x

(n)

  1 |x|2 ), ϕ(x) = exp − 2 , (πH )3/2 h H (n)

where Ei are unknown coefficients of the approximation and Ei substitution of Eq. (7.97) into Eq. (7.95), we obtain Ei (x) −

= Ei (x (n) ). Upon

Mn  (n) (n) (n) ij (x − x (n) , κ0 ) c1 Ek = Ei0 (x),  c1 =  c1 (x (n) ). n=1

(7.97)

(7.98)

276

Here, ij (x, κ0 ) is the integral over the entire 3D space  ij (x, κ0 ) = κ02 Gij (x − x  )ϕ(x  )dx  .

Heterogeneous Media

(7.99)

It follows from Eq. (7.96) for Gij (x) that the function ij (x, κ0 ) is presented in the form   1 2 2 0 0 ij (x, κ0 ) = h κ0 I (x, κ0 )δij + 2 ∂i ∂j I (x, κ0 ) , (7.100) κ0 where the integral I 0 (x, κ0 ) is defined in Eq. (7.23),    1 |x| I 0 (x, κ0 ) = 2 g(x − x  )ϕ(x  )dx  = 0 , κ0 h , h h

(7.101)

and the function 0 (ξ, q) is presented in Eq. (7.24). As a result, the function ij (x, κ0 ) takes the form    

xi |x| |x| 2 2 ij (x, κ0 ) = h κ0 1 , κ0 h δij + 2 , κ0 h ni nj , ni = , h h |x| (7.102) where the functions 1 (ξ, q) and 2 (ξ, q) have the following explicit expressions:   2 1 ξ 1 (ξ, q) = − 4ξ exp − 3/2 2 3 H 8 (πH ) q ξ √  − πH (1 + qξ(i − qξ ))f0+ (ξ, q)f− (ξ, q)+  (7.103) +(1 − qξ(i + qξ ))f0− (ξ, q)f+ (ξ, q) ,   2 1 ξ (3H + 2ξ 2 )+ −4ξ exp − 2 (ξ, q) = 3/2 2 3 H 8 (πH ) q ξ √  + H πH (3 + qξ(3i − qξ ))f0+ (ξ, q)f− (ξ, q)−  (7.104) −(3 − qξ(3i + qξ ))f0− (ξ, q)f+ (ξ, q) . In these equations, the functions f0± (ξ, q) and f± (ξ, q) are defined in Eq. (7.25). Asymptotics φ1 (ξ, q), φ2 (ξ, q) of the functions 1 (ξ, q), 2 (ξ, q) for large values of the argument ξ are   1 1 2 2 φ1 (ξ, q) = exp − + 4iξ q − 1 − iqξ , (7.105) H q (qξ ) 4 4πξ 3   1 1 2 2 exp − + 4iξ q 3 + 3iqξ − H q . (7.106) φ2 (ξ, q) = (qξ ) 4 4πξ 3 The real parts of the functions 1 (ξ, q) and 2 (ξ, q) coincide practically with their asymptotics φ1 (ξ, q) and φ2 (ξ, q) when ξ > 5. The imaginary parts of these functions and their asymptotics coincide for all values of ξ .

Time-harmonic fields in heterogeneous media

277

The system of linear algebraic equations for the unknowns Ei(n) in Eq. (7.97) follows from Eq. (7.98) if the latter is satisfied at all the nodes. We have Ei(m) −

Mn 

(m,n) (n) (n)  c 1 Ej

ij

0(m)

= Ei

, m = 1, 2, ..., Mn,

(7.107)

= Ei0 (x (n) ).

(7.108)

s=1 (m,n)

ij

0(n)

= ij (x (m) − x (n) ), Ei

Here, ij (x) is defined in Eq. (7.102). The matrix form of this system is (I − B)X = F,

(7.109)

where I is the unit matrix of the dimensions 3Mn × 3Mn, the vectors of unknowns X and of the right hand side F have the dimensions 3Mn,   T T X = X 1 , X 2 , ..., X 3Mn , F = F 1 , F 2 , ..., F 3Mn , ⎧ (n) n ≤ Mn, ⎪ ⎨ E1 , (n−Mn) Xn = , Mn < n ≤ 2Mn, E2 ⎪ ⎩ (n−2Mn) , 2Mn < n ≤ 3Mn, E3 ⎧ 0(n) n ≤ Mn, ⎪ ⎨ E1 , 0(n−Mn) n F = , Mn < n ≤ 2Mn, E2 ⎪ ⎩ 0(n−2Mn) , 2Mn < n ≤ 3Mn. E3

(7.110)

(7.111)

The matrix B in Eq. (7.109) has the dimensions 3Mn × 3Mn and consists of nine submatrices bij of the dimensions Mn × Mn, ⎡

b11 , B = ⎣ b21 , b31 ,

b12 , b22 , b32 , (m,n)

with the elements bij (m,n)

bij

(7.112)

given by the equations

(m,n) (n)  c1 ,

= ij

⎤ b13 b23 ⎦ b33

i, j = 1, 2, 3, m, n = 1, 2, ..., Mn.

(7.113)

The system (7.109) can be solved by the iterative MRM. The iterations converge for any finite electric permittivity of the heterogeneity; the number of iterations depends on the contrast in the properties of the host medium and the heterogeneity and the frequency of the incident field. For regular node grids, the FFT algorithm can be adopted for calculation of the matrix-vector products in the process of iterations.

278

Heterogeneous Media

7.3.2 Electromagnetic wave scattering from a spherical heterogeneity We consider a spherical inclusion of radius a subjected to a plane monochromatic incident wave. Let the origin of the Cartesian coordinate system (x1 , x2 , x3 ) be at the center of the inclusion, and let the x1 - and x3 -axes be directed along the polarization vector Ei0 and the wave normal n0i of the incident field, respectively. The medium and the inclusion are isotropic with the dielectric permittivities c0 and c, and c/c0 = 2. Distributions of the real (Re E1 ) and imaginary (Im E1 ) parts of the component E1 (x1 , x2 , x3 ) of the electric field along the x1 - and x3 -axes inside the inclusion are presented in Figs. 7.40, 7.42, and 7.44 for the dimensionless wave numbers of the incident field κ0 a = 0.5, 1, 2. In these figures, solid lines correspond to the exact (Mie) solutions presented in [4] and dashed lines are the results of the numerical solutions for H = 1 and the node grid step h = 0.02 (Mn = 1030301). For the mentioned values of κ0 a, the graphs of the normalized differential cross-section F (φ, θ )/F (0, 0) in Eq. (3.278) are shown in Figs. 7.41, 7.43, and 7.45. Here, φ and θ are the polar and azimuthal angles of the spherical coordinate system with the polar axis x3 . These angles define the direction of the vector ni in Eq. (3.275). Because of symmetry of the problem, the differential cross-section F (φ, θ ) does not depend on the angle φ, and the diagrams in Figs. 7.41, 7.43, and 7.45 are parametrically defined by the equations x1 = F (0, θ) sin(θ ), x3 = F (0, θ) cos(θ ).

Figure 7.40 Real and imaginary parts of the component E1 of the electric field inside a spherical inclusion

of radius a for the wave number of the incident field κ0 a = 0.5 (c0 = 1, c = 2); the function E1 (x1 , 0, 0) is shown in the left figure, the function E1 (0, 0, x3 ) is shown in the right figure. Solid lines correspond to the exact (Mie) solution, and dashed lines show the numerical solution for a node grid with the step h/a = 0.02 (Mn = 103031), H = 1.

The integral relative error (h) of the numerical solution can be defined by the equation  "  h,H e (x) dx E (x) − E   i i V "  e  (h) = . (7.114)   V Ei (x) dx

Time-harmonic fields in heterogeneous media

279

Figure 7.41 The normalized differential cross-section F (ϕ, θ)/F (0, 0) of a spherical inclusion of radius a for the wave number of the incident field κ0 a = 0.5; the solid line corresponds to the exact (Mie) solution, and the line with white dots shows the numerical solution for a cubic node grid with the steps h/a = 0.02, Mn = 103031, H = 1.

Figure 7.42 The same as in Fig. 7.40 for κ0 a = 1.

Figure 7.43 The same as in Fig. 7.41 for κ0 a = 1.

280

Heterogeneous Media

Figure 7.44 The same as in Fig. 7.40 for κ0 a = 2.

Figure 7.45 The same as in Fig. 7.41 for κ0 a = 2.

Here, Eih,H (x) is the quasiinterpolant of Ei (x) with the parameters h and H , and Eie (x) is the exact solution. The function (h) decreases monotonically with decreasing h. For κ0 a of the order of 1, (h) changes from the value of about 0.1 for h = 0.1 until (h) ≈ 0.03 for h = 0.02. For larger values of the wave numbers κ0 a and for higher contrasts in the properties of the medium and the inclusion,  becomes larger, but it always decreases together with h. The local error is maximal near the boundary of the body, as seen in Figs. 7.40, 7.42, and 7.44. The numerical solution is not sensitive to the value of the parameter H in the approximation (7.97) if H is in the region 1 < H < 3. Convergence of the methods depends on the contrast in the properties of the inclusion and host medium as well as on the value of the wave number of the incident field. For |c/c0 | < 3 and κ0 a < 3, the tolerance ε = 0.001 (ε = |X(n) − X(n−1) |/|X(n) |) is achieved in five iterations. The number of iterations can increase until hundreds for larger property contrasts and/or shorter incident waves.

Time-harmonic fields in heterogeneous media

281

Figure 7.46 A cylindrical inclusion subjected to the incident electric wave with the wave normal n0 and the polarization vector E 0 .

7.3.3 Electromagnetic wave scattering from a shot cylinder Let a monochromatic incident field be scattered on a dielectric cylinder the length L of which is twice its radius a. The dielectric permittivities of the medium and the cylinder are c0 and c, and c/c0 = 2. The angle between the wave normal n0i of the incident field and the x3 -axis of the cylinder is α (Fig. 7.46), and the incident field Ei0 (x1 , x2 , x3 ) is defined by the equation Ei0 (x1 , x2 , x3 ) = (cos (α) δi1 − sin(α)δi3 ) exp [−iκ0 (sin(α)x1 + cos(α)x3 )] . (7.115) For the angle α = π/4, the numerical results are presented in Figs. 7.47–7.52 for the wave numbers of the incident field κ0 L = 0.5, 2. The distributions of the real and imaginary parts of the electric field components E1 and E3 inside the cylinder are shown in Figs. 7.47 and 7.50, and the distributions of these components along the x3 -axis are shown in Figs. 7.48 and 7.51. The diagrams of the corresponding differential cross-sections F (ϕ, θ ) are shown in Figs. 7.49 and 7.52. For x1 > 0, these graphs are parametrically defined by the equations x1 = F (0, θ) sin(θ ), x3 = F (0, θ) cos(θ ), and for x1 < 0, by the equations x1 = −F (π, θ ) sin(θ ), x3 = F (π, θ ) cos(θ ).

7.4

Scattering of elastic waves from heterogeneities of arbitrary shapes

We consider an infinite homogeneous host medium with density ρ0 and an elastic stiffness tensor Cij0 kl . The medium contains a heterogeneity or a group of heterogeneities in a finite region V . The density and elastic stiffness tensor of the medium with the heterogeneities are presented in the forms

282

Heterogeneous Media

Figure 7.47 Real and imaginary parts of the component E1 of the electric field inside a cylindrical inclusion of length L and radius a (L/a = 2) for the wave number of the incident field κ0 L = 0.5 (c0 = 1, c = 2) and the angle between the wave normal n0 and the cylinder axis α = π/4; the function E1 (x1 , 0, 0) is shown in the left figure, the function E1 (0, 0, x3 ) is shown in the right figure. Lines with white dots correspond to the numerical solution for a rectangular node grid with the step h = 0.02, H = 1.

Figure 7.48 The same as in Fig. 7.47 for the distribution of the components E1 and E3 of the electric field along the x3 -axis.

ρ(x) = ρ0 + ρ1 (x)V (x), Cij kl (x) = Cij0 kl + Cij1 kl (x)V (x),

(7.116)

where V (x) is the characteristic function of the region V . Let a monochromatic incident displacement field u0i (x, t) u0i (x, t) = u0i (x)eiωt , u0i (x) = ai e−iκ0 ni xi 0

(7.117)

propagate in the medium and be scattered from the region V . Here, κ0 is the wave number of the incident field, n0i is its wave normal, and ai is the polarization vector.

Time-harmonic fields in heterogeneous media

283

Figure 7.49 The normalized differential cross-section F (ϕ, θ)/F (0, 0) of a cylindrical inclusion of length L and radius a (L/a = 2) for the wave number of the incident field κ0 L = 0.5 and the angle of its wave normal n0 with the cylinder axis α = π/4.

Figure 7.50 The same as in Fig. 7.47 for κ0 a = 2.

For an isotropic host medium and longitudinal incident wave,

κ0 = α0 = ω

ρ0 . λ0 + 2μ0

(7.118)

Here, λ0 and μ0 are the Lame coefficients of the host medium, and vectors n0i and ai are of the same direction. For a transverse incident wave,

κ0 = β0 = ω

ρ0 , μ0

(7.119)

284

Heterogeneous Media

Figure 7.51 The same as in Fig. 7.48 for κ0 a = 2.

Figure 7.52 The same as in Fig. 7.49 for κ0 a = 2.

and the vectors n0i and ai are orthogonal. The displacement vector ui (x, t) in the medium has the form ui (x, t) = ui (x)eiωt , and the amplitude ui (x) satisfies the integro-differential equation (3.288), i.e.,  ui (x) = u0i (x) + 

+ω

2

V

      ∂k gij x − x  Cj1klm x  ∂l um x  dx  +

gij (x − x  ) ρ1 (x  ) uj (x  )dx  .

(7.120)

V

Here, gij (x) is the dynamic Green function of elasticity. For an isotropic medium, this function is defined in Eq. (2.145).

Time-harmonic fields in heterogeneous media

285

7.4.1 Discretization of Eq. (7.120) by Gaussian approximating functions For numerical solution of the integro-differential equation (7.120), the region of heterogeneity V is embedded into a cuboid W and covered by a cubic node grid with the step h. Then, the vector ui (x) and its partial derivatives ∂i uj (x) are approximated by the Gaussian quasiinterpolants similar to Eq. (7.15)

ui (x) ≈

Mn Mn   (n) (n) ui ϕ(x − x (n) ), ∂i uj (x) ≈ Di uj ϕ(x − x (n) ), n=1

(n) ui

(7.121)

n=1

= ui (x

(n)

  1 |x|2 ), ϕ(x) = exp − 2 . (πH )3/2 h H

(7.122)

(n) by linHere, the coefficients Di u(n) j are expressed in terms of the coefficients ui ear equations similar to Eqs. (7.11)–(7.13). Substituting ui (x) and ∂i uj (x) in these equations into Eq. (7.120), we obtain

ui (x) −

Mn 

1(n)

(n)

ij k (x − x (n) )Cj klm Dm ul

− ω2

n=1 0 = ui (x),

Mn 

(n) (n)

ij (x − x (n) )ρ1 uj =

n=1

(7.123)

where  ij (x) =









gij (x − x )ϕ(x )dx , ij k (x) =

∂k gij (x − x  )ϕ(x  )dx  , (7.124)

1(n)

(n)

Cj klm = Cj1klm (x (n) ), ρ1 = ρ1 (x (n) ).

(7.125)

The system of linear algebraic equations for the unknowns u(n) i in Eq. (7.121) follows (n) from Eq. (7.123) if the latter is satisfied at all nodes x (n) . Because the coefficients ui coincide with the values of the function ui (x) at the nodes, we obtain (m)

ui

−

Mn 

(m,n)

1(n)

(n)

ij k Cj klm Dm ul

n=1

− ω2

Mn  (m,n) (n) (n) 0(m) ij ρ1 uj = ui ,

(7.126)

r=1

m = 1, 2, ..., Mn, (m,n) 0(m) ij(m,n) = ij k (x (m) − x (n) ), ij = ij (x (m) − x (n) ), ui = u0i (x (m) ). k (7.127)

286

Heterogeneous Media

The integrals in Eq. (7.124) are expressed in terms of the derivatives of the scalar function I 0 (x, κ) defined in Eq. (7.23) 1 I (x, κ) = 2 h 0









g(x − x )ϕ(x )dx = 0



 e−iκ|x| |x| , κh , g(x) = . h 4π |x| (7.128)

The function 0 (ξ, q) is presented in Eq. (7.24), and its asymptotic for large ξ is given in Eq. (7.26). The integrals ij (x) and ij k (x) in Eq. (7.124) are expressed in terms of the function 0 (ξ, q) and its derivatives 0 =

d0 d 2 0 d 3 0  ,  = . , 0 = 0 dξ dξ 2 dξ 3

(7.129)

The function ij (x) has the form      h2 |x| |x| , α0 h − 2 , α, β0 h δij − 0 ij (x) = ρ0 h h   

  xi |x| |x| , α0 h, β0 h − 2 , α0 h, β0 h ni nj , ni = , − 1 h h |x| 1 1 1 (ξ, a, b) = 2 0 (ξ, a) − 2 0 (ξ, b), a b  1 1  1  2 (ξ, a, b) =  (ξ, a) − 2 0 (ξ, b) . ξ a2 0 b

(7.130) (7.131) (7.132)

The equation for ij k (x) is   1 |x|  0 , β0 h ni δj k − ij k (x) = h μ0 β02 h    

|x| |x| , α0 h, β0 h (δij nk + δik nj + δkj ni ) − 4 , α, β0 h ni nj nk , −3 h h (7.133) 1 1 3 (ξ, a, b) = 2 (ξ, a) − 2 (ξ, b), b  a 1 1   (ξ, q) = (7.134) 0 (ξ, q) − 0 (ξ, q) , ξ ξ 1 1 4 (ξ, a, b) = 2 (ξ, a) − 2 (ξ, b), (ξ, q) =  0 (ξ, q) − 3(ξ, q). a b (7.135) The asymptotics of the functions ij (x) and ij k (x) for large |x| follow from Eqs. (7.130)–(7.135) if the function 0 (ξ, q) is changed to its asymptotic expression φ0 (ξ, q) in Eq. (7.26).

Time-harmonic fields in heterogeneous media

287

Eq. (7.126) can be presented in the matrix form (I − B)X = F,

(7.136)

where I is the unit matrix of the dimensions 3Mn × 3Mn and the vectors of unknowns X and of the right hand side F have the dimensions 3Mn,   T T X = X 1 , X 2 , ..., X 3Mn , F = F 1 , F 2 , ..., F 3Mn , ⎧ (n) n ≤ Mn, ⎪ ⎨ u1 , (n−Mn) n X = , Mn < n ≤ 2Mn, u2 ⎪ ⎩ (n−2Mn) , 2Mn < n ≤ 3Mn, u3 ⎧ 0(n) n ≤ Mn, ⎪ ⎨ u1 , 0(n−Mn) n F = , Mn < n ≤ 2Mn, u2 ⎪ ⎩ 0(n−2Mn) , 2Mn < n ≤ 3Mn. u3

(7.137)

(7.138)

The matrix B in Eq. (7.136) is reconstructed from Eq. (7.126). For solution of the system (7.136), the MRM and the FFT algorithm for calculation of the matrix-vector products can be adopted.

7.4.2 Elastic wave scattering from a spherical inclusion We consider a spherical inclusion of the radius a subjected to the incident plane wave with the wave normal n0i directed along the x3 -axis of the Cartesian coordinate system (x1 , x2 , x3 ). In the calculations, we take the properties of the medium and the inclusion corresponding to aluminum (E0 = 70 GPa, ν0 = 0.3, ρ0 = 2700 kg/m3 ) and steel (E = 200 GPa, ν = 0.3, ρ = 7800 kg/m3 ). Here, E0 , E are Young moduli and ν0 , ν are the Poisson ratios of the host medium and the inclusion. The exact solution of the scattering problem of elasticity for a spherical inclusion is presented in [5].

Longitudinal incident waves Let a longitudinal wave of a unit amplitude propagate along the x3 -axis and be scattered from the inclusion. In this case, the principal component of the displacement vector is u3 (x1 , x2 , x3 ). The distributions of the modulus |u3 (x1 , x2 , x3 )| along the x1 and x3 -axes are presented in Figs. 7.53, 7.55, and 7.57 for the dimensionless wave numbers of the incident field α0 a = 0.5, 1, 5. In these figures, solid lines are the exact distributions of |u3 |, lines with white dots correspond to the numerical solution for the node grid step h/a = 0.05 (Mn = 91125), and lines with squares correspond to the numerical solution for h/a = 0.02 (Mn = 1191016). For the mentioned values of α0 a, the graphs of the differential cross-section F (φ, θ ) in Eq. (3.300) are shown in Figs. 7.54, 7.56, and 7.58. Here φ and θ are the polar and azimuthal angles of the spherical coordinate system with the polar

288

Heterogeneous Media

Figure 7.53 Distribution of the modulus of the component u3 of the elastic field along x1 - and x3 -axes

inside a spherical inclusion of radius a for the wave number of the longitudinal incident field α0 a = 0.5; solid lines correspond to the exact solutions, lines with white dots are numerical solutions for a cubic node grid with the step h/a = 0.05, H = 1.

Figure 7.54 The differential cross-section F (ϕ, θ) of a spherical inclusion of radius a for the wave number of the longitudinal incident field α0 a = 0.5; the solid line corresponds to the exact solution, and the line with white dots shows the numerical solution for the node grid with the step h/a = 0.05, H = 1.

axis x3 (Fig. 7.1). Because of symmetry of the differential cross-section with respect to the x3 -axis, the function F (φ, θ ) does not depend on the angle φ, and the graphs in Figs. 7.54, 7.56, and 7.58 are parametrically defined by the equations x1 = F (0, θ) sin(θ ), x3 = F (0, θ) cos(θ ). Solid lines in Figs. 7.53–7.58 correspond to the exact differential cross-sections of the spherical inclusion.

Transverse incident waves We consider a transverse incident wave with the wave vector in the direction of the x3 -axis and the polarization vector bi directed along the x1 -axis. In this case, u1 (x1 , x2 , x3 ) is the principal component of the displacement vector. The distributions of |u1 | along the x1 - and x3 -axes inside the inclusion are presented in Figs. 7.59, 7.61, 7.63, and 7.65 for the dimensionless wave numbers of the incident field β0 a =

Time-harmonic fields in heterogeneous media

Figure 7.55 The same as in Fig. 7.53 for α0 a = 1.

Figure 7.56 The same as in Fig. 7.54 for α0 a = 1.

Figure 7.57 The same as in Fig. 7.53 for α0 a = 5.

289

290

Heterogeneous Media

Figure 7.58 The same as in Fig. 7.54 for α0 a = 5.

Figure 7.59 Distribution of the modulus of the component u1 of the elastic field along the x1 - and x3 -axes inside a spherical inclusion of radius a subjected to a transversal incident field with the wave number β0 a = 0.5, where the unit polarization vector is directed along the x1 -axis; solid lines correspond to the exact solutions, and lines with white dots and squares show the numerical solutions for a cubic node grid with the steps h/a = 0.05 and h/a = 0.02, with H = 1.

0.5, 1, 5, 10. In these figures, solid lines correspond to the exact distributions of the u1 -component inside a spherical inclusion, lines with white dots are the numerical solution for H = 1 and the node grid step h/a = 0.05, and lines with squares are such solutions for h/a = 0.02. For the mentioned values of β0 a, the graphs of the differential cross-section F (φ, θ ) are shown in Figs. 7.60, 7.62, 7.64, and 7.66. In the case of transverse incident waves, the differential scattering cross-section depends on the angles φ and θ, and the lines in Figs. 7.60, 7.62, 7.64, and 7.66 correspond to two values of the polar angles φ = 0, π/2 and are parametrically defined by the equations x1 = F (φ, θ ) sin(θ ), x3 = F (φ, θ ) cos(θ ). Solid lines in Figs. 7.60, 7.62, 7.64, and 7.66 correspond to the exact differential scattering cross-sections of the spherical inclusion for the considered wave numbers of the incident field. It is seen from Figs. 7.54–7.66 that for the node grid step h/a = 0.05, the numerical solutions approximate the exact distributions of the displacement field in the inclusion

Time-harmonic fields in heterogeneous media

291

Figure 7.60 The differential cross-section F (ϕ, θ) of a spherical inclusion of radius a for the wave number of the transverse incident field β0 a = 0.5; solid lines correspond to the exact solution, and lines with white dots and squares are the numerical solutions for the node grid step h/a = 0.05 and h/a = 0.02, with H = 1.

Figure 7.61 The same as in Fig. 7.59 for β0 a = 1.

Figure 7.62 The same as in Fig. 7.60 for β0 a = 1.

with sufficient accuracy. For accurate description of the differential scattering crosssections, the node grid step should decrease to h/a = 0.02 in some cases. Convergence of the method with respect to the node grid step h as well as the behavior of the error

292

Figure 7.63 The same as in Fig. 7.59 for β0 a = 5.

Figure 7.64 The same as in Fig. 7.60 for β0 a = 5.

Figure 7.65 The same as in Fig. 7.59 for β0 a = 10.

Heterogeneous Media

Time-harmonic fields in heterogeneous media

293

Figure 7.66 The same as in Fig. 7.60 for β0 a = 10.

of the numerical solution are similar to the case of the electromagnetic wave scattering problem. The MRM converges for any finite contrast in the properties of the matrix and inclusion materials, but the number of iterations depends on the property contrast and length of the incident waves. For a contrast of about 2.5, which corresponds to the case of a steel inclusion in an aluminum matrix, the number of iterations is about 10 ÷ 20 for h/a = 0.05 and about 30 for h/a = 0.02. If the wave number α0 a or β0 a of the incident field increases, the number of nodes should also be increased for accurate description of the field oscillations inside the inclusion.

7.5

Scattering of elastic waves from a planar crack

We consider a planar crack  with the boundary contour  in an infinite homogeneous medium with density ρ and an elastic stiffness tensor Cij kl (Fig. 7.67). Let a monochromatic incident field u0i (x, t) u0i (x, t) = u0i (x)eiωt , u0i (x) = ai e−iκni xi 0

(7.139)

propagate in the medium and be scattered from the crack.

Figure 7.67 A planar crack  in an elastic medium subjected to an incident plane wave with the wave normal n0 , where W is a rectangular containing  and covered with a regular node grid for numerical solution of the scattering problem.

For time-harmonic elasticity, the vector ui (x, t) of displacements and the stress tensor σij (x, t) have the forms ui (x, t) = ui (x)eiωt , σij (x, t) = σij (x)eiωt ,

(7.140)

294

Heterogeneous Media

and for the amplitudes ui (x) and σij (x) of these fields, integral presentations in Eqs. (3.303) and (3.304) hold:      (7.141) ui (x) = u0i (x) − ∂k gij x − x  Cj klm nl bm x  d ,   Sij kl (x − x  ) nk bl (x  )d . (7.142) σij (x) = σij0 (x) + 

Here, ni is the normal to the crack surface, bi (x) is the crack opening vector, gij (x) is the Green function of time-harmonic elasticity defined in Eq. (2.145), and σij0 (x) = Cij kl ∂k u0l (x). The tensor Sij kl (x) in Eq. (7.142) is expressed in terms of the second derivatives of the Green function in Eq. (3.306). The integral term in Eq. (7.141) is the potential of the double layer of elasticity. For an arbitrary vector bi (x), this potential satisfies the equations of time-harmonic elasticity everywhere outside  and the jump of ui (x) on  is equal to the density bi (x), i.e., − [ui (x)] = u+ i (x) − u (x) = bi (x).

(7.143)

The stress tensor σij (x) in Eq. (7.142) satisfies the equation of motion ∂j σij (x) + ρω2 ui = 0 in the entire space and coincides with the tensor Cij kl ∂k ul (x) outside . The normal component ni σij (x) of this tensor is continuous on . If the crack sides are free from forces, the boundary condition on  takes the form ni σij (x)| = 0,

(7.144)

and the integral equation for the crack opening vector bi (x) follows from this condition:  Tij (x − x  )bj (x  )d = ti0 (x), x ∈ , (7.145) 

Tij (x) = −nk Skij l (x)nl , ti0 (x) = nj σij0 (x).

(7.146)

7.5.1 Discretization of Eq. (7.145) by the Gaussian approximating functions Let (x1 , x2 , x3 ) be a Cartesian coordinate system in the 3D space, and let a planar crack occupy the region  in the plane x3 = 0. For numerical solution of the integral equation (7.145), the crack region  is embedded inside a rectangular W and covered by a square node grid with the step h (Fig. 7.67). Then, the vector bi (x) is approximated by the Gaussian quasiinterpolant in W , i.e., bi (x) ≈

Mn  bi(n) ϕ(x − x (n) ), bi(n) = 0 if x (n) ∈ / ,

(7.147)

n=1

(n) bi

# $ 2 x 1 = bi (x (n) ), ϕ(x) = exp − 2 . πH h H

(7.148)

Time-harmonic fields in heterogeneous media

295

Substituting the series (7.147) into Eq. (7.145), we obtain the equation Mn 

(n)

(n)

I ij (x − x (n) )bj = ti0 (x) if x ∈ , bj = 0 if x (n) ∈ / ,

(7.149)

n=1

where  I ij (x) = I ij (x1 , x2 ) =

∞ −∞

Tij (x1 − x1 , x2 − x2 )ϕ(x1 , x2 )dx1 dx2 . (n)

The system of linear algebraic equations for the unknowns bi Eq. (7.149) if the latter is satisfied at the nodes x (n) belonging to : Mn 

(m,n) (n) bj

I ij

0(m)

(m)

= ti

if x (m) ∈ , bj

= I ij (x

(m)

(7.150)

follows from

= 0 if x (m) ∈ / , m = 1, 2, ..., Mn,

n=1

(7.151) (m,n) I ij

−x

(n)

),

0(m) ti

= ti0 (x (m) ).

(7.152)

Using the property of convolution, the integral I ij (x) in Eq. (7.150) is presented in terms of the integral of the Fourier transforms of the integrand functions I ij (x1 , x2 ) =  ∞ Tij (x1 − x1 , x2 − x2 , x3 − x3 )ϕ(x1 , x2 )δ(x3 )dx1 dx2 dx3 |x3 =0 = = −∞  ∞ 1 = Tij∗ (k1 , k2 , k3 )ϕ ∗ (k1 , k2 ) exp(−i(k1 x1 + k2 x2 ))dk1 dk2 dk3 , (2π)3 −∞ (7.153) # $ H h2 (k12 + k22 ) ϕ ∗ (k1 , k2 ) = h2 exp − . (7.154) 4 For an isotropic medium, the function Tij∗ (k1 , k2 , k3 ) is defined in Eq. (3.316). Then, from Eq. (7.153), we obtain the following equation for the function I ij (x1 , x2 ):  1 ∗ I ij (x1 , x2 ) = T ij (k1 , k2 )ϕ ∗ (k1 , k2 ) exp(−i(k1 x1 + k2 x2 ))dk1 dk2 , (2π)2 (7.155)  ∞ 1 ∗ T ij (k1 , k2 ) = T ∗ (k1 , k2 , k3 )dk3 . (7.156) 2π −∞ ij ∗

The function T ij (k1 , k2 ) is presented in Eq. (3.319), i.e., ∗

T ij (k1 , k2 ) = s1∗ (k)ni nj + s2∗ (k)mi mj + s3∗ (k)θij ,

(7.157)

296

Heterogeneous Media

∗ (k) (m = 1, 2, 3) are defined in Eqs. (3.321)–(3.323), n is where scalar coefficients sm i a unit vector of the k3 -axis, and  ki (7.158) θij = δij − ni nj , mi = , i = 1, 2, m3 = 0, k = k12 + k22 . k

For calculation of the integral (7.155), we introduce the polar coordinates in the plane (k1 , k2 ) and integrate over the polar angle. As a result, for I ij (x), we obtain the equation    r r 1  r I ij (x) = s1 , αh, βh ni nj + s2 , αh, βh ei ej + s3 , αh, βh θij , h h h h (7.159)  xi (7.160) ei = , i = 1, 2, e3 = 0, r = x12 + x22 . r Here, the functions s1 , s2 , s3 of the dimensionless variables ρ = r/ h, a = αh, b = βh are a λ2 4μ  F1 (ρ, a) + (λ + 2μ)F2 (ρ, a) − 2 F3 (ρ, q) b , λ + 2μ b (7.161) a 4μ  s2 (ρ, a, b) = μF4 (ρ, a) − 2 F5 (ρ, q) b , (7.162) b a 4μ  (7.163) s3 (ρ, a, b) = μF5 (ρ, a) − 2 F6 (ρ, q) b , b a  (7.164) Fm (ρ, q) b = Fm (ρ, a) − Fm (ρ, b). s1 (ρ, a, b) = −

In these equations, Fm (ρ, a) (m = 1, 2, 3, 4, 5, 6) are absolutely converging 1D integrals    ∞ exp − k 2 H 4 1 F1 (ρ, q) = (7.165) J0 (kρ)k 3 dk, 4π 0 η(k, q)  2   ∞ 1 k H η(k, q) exp − (7.166) J0 (kρ)kdk, F2 (ρ, q) = 4π 0 4  2   ∞ 1 k H η(k, q) exp − (7.167) J0 (kρ)k 3 dk, F3 (ρ, q) = 4π 0 4  2   ∞ exp − k H 4 1 F4 (ρ, q) = (7.168) J2 (kρ)k 3 dk, 4π 0 η(k, q)    ∞ 1 k2H η(k, q) exp − (7.169) J2 (kρ)k 3 dk, F5 (ρ, q) = 4π 0 4   1 F6 (ρ, q) = F1 (ρ, q) + F4 (ρ, q) − F2 (ρ, q), η(k, q) = k 2 − q 2 . 2 (7.170)

Time-harmonic fields in heterogeneous media

297

Here, J0 (z) and J2 (z) are Bessel functions. For ρ < 5 and q < 1, these integrals can be tabulated. For large values of the dimensionless distance ρ = r/ h, the integrals Fm (ρ, q) can be changed with their asymptotics fm (ρ, q). Details of the calculation of the integrals (7.165)–(7.169) and the equations for their asymptotics fm (ρ, q) are presented in Appendix 7.C. Eq. (7.151) can be written in the matrix form AX = F,

(7.171)

where the vectors of unknowns X and of the right hand side F have the dimensions 3Mn,   T T X = X 1 , X 2 , ..., X 3Mn , F = F 1 , F 2 , ..., F 3Mn , ⎧ (n) (n) n ≤ Mn, ⎪ ⎨ κ b1 , (n−Mn) n (n−Mn) X = κ b2 , Mn < n ≤ 2Mn, ⎪ ⎩ (n−2Mn) (n−2Mn) b3 , 2Mn < n ≤ 3Mn, κ ⎧ 0(n) n ≤ Mn, ⎪ ⎨ t1 , 0(n−Mn) n F = , Mn < n ≤ 2Mn, t2 ⎪ ⎩ 0(n−2Mn) , 2Mn < n ≤ 3Mn. t3

(7.172)

(7.173)

(7.174)

Here κ (n) = κ(x (n) ) and κ(x) is the characteristic function of the crack region , / . κ(x) = 1 if x ∈ , κ(x) = 0 if x ∈

(7.175)

The matrix A in Eq. (7.171) has the dimensions 3Mn × 3Mn and consists of nine (m,n) (i, j = block-matrices aij of the dimensions Mn × Mn with the elements aij 1, 2, 3, m, n = 1, 2, ..., Mn)    a11 a12 a13    (m,n) A =  a21 a22 a23  , aij(m,n) = I ij . (7.176)  a31 a32 a33  It follows from Eqs. (7.159) and (7.161)–(7.169) that A is a nonsparse matrix that has a Toeplitz structure for a square grid of approximating nodes. For numerical solution of the system (7.171), the iterative CGM and the FFT algorithm for calculation of matrix-vector products in the process of iterations can be adopted.

7.5.2 A penny-shaped crack subjected to longitudinal incident waves orthogonal to the crack surface In this section, we consider the scattering problem for a penny-shaped crack with the surface  : (x12 + x22 ≤ 1; x3 = 0) subjected to a longitudinal incident wave u0i (x) the

298

Heterogeneous Media

wave vector and amplitude of which are orthogonal to  (n0i = ni ), i.e., u0i (x) = ai exp(−iαx3 ), ai = −

i ni . λ + 2μ

(7.177)

For such an amplitude ai , the vector ti0 (x) = nj σij0 (x) in Eq. (7.149) coincides with the normal vector ni . In this case, the crack opening bi (x) has the form bi (x) = b(r)ni , and the scalar function b(r) depends on the distance r = |x| from the crack center, x = (x1 , x2 ). The function b(r) satisfies the equation that follows from Eq. (7.145),  T (x − x  )b(|x  |)dx  = 1, x ∈ , T (x) = ni Tij (x)nj . (7.178)  ∗

The Fourier transform T (k) of the kernel T (x) follows from Eq. (7.157) and has the form ∗

T (k) = γ0 t0 (k) + γ1 t1 (k) + γ2 t2 (k) + γ3 t3 (k),

(7.179)

t0 (k) = k,

t1 (k) = η(k, α) − k,  3  α2 1  3 3 − 1 k, t2 (k) = η(k, α) − k + 2 η (k, β) − η (k, α) − 2 β2 β   α2 2 , η(k, q) = k − q 2 , k = k12 + k22 , t3 (k) = η(k, α) μ(λ + μ) 2λμ λ2 , γ1 = , γ2 = 2μ, γ3 = − . γ0 = λ + 2μ λ + 2μ 2(λ + 2μ)

(7.180) (7.181) (7.182) (7.183)

∗

Here, T (k) is the symbol of the operator T in Eq. (7.178). When α, β → 0 (the static ∗ ∗ limit), T (k) tends to T 0 (k) = γ0 k. The operator T 0 with √ the symbol γ0 k has a remarkable property: It converts the function ψn (r) = r 2n 1 − r 2 (n = 0, 1, 2, ...), into a polynomial function of the order of 2n on . To be exact, the following equation holds:  (T0 ψn )(x) =







T0 (x − x )ψn (|x |)d = γ0 

n 

t (n, m)|x|2m ,

(7.184)

m=0

 2    32 + m  n − m − 12 , t (n, m) = − √ π (1 + m)2  (1 − m + n)

(7.185)

where (z) is the Gamma function. Therefore, there exist functions Bn (r), Bn (r) =

n   1 − r2 w(n, m)r 2m , n = 0, 1, 2, ...,

(7.186)

m=0

that are converted by the operator T0 into Chebyshev’s polynomials U2n (r), (T0 Bn ) (r) = γ0 U2n (r), n = 0, 1, 2, ...,

(7.187)

Time-harmonic fields in heterogeneous media

U2n (r) =

n 

299

v(n, m)r 2m , v(n, m) =

m=0

(−1)(n−m) 22m (n + m)! . (2m)!(n − m)!

(7.188)

The linear algebraic system for the coefficients w(n, m) in Eq. (7.186) follows from Eqs. (7.184) and (7.187) in the form n 

t (l, m)w(n, m) = v(n, l), l = 0, 1, 2, ..., n.

(7.189)

m=0

It is known [6] that the solution of the integral equation (7.178) can be presented in the form of the series b(r) =

∞ 

(7.190)

cn Bn (r),

n=0

where the functions Bn (r) are defined in Eq. (7.186). Substituting this equation into Eq. (7.178), we obtain the following system for the coefficients cn : (T b) (r) =

∞ 

  cn γ0 U2n (r) + γ1 (T1 Bn ) (r) + γ2 (T2 Bn ) (r) + γ3 (T3 Bn ) (r) =

n=0

= 1,

(7.191)

where T1 , T2 , T3 are the integral operator with the symbols t1 (k), t2 (k), t3 (k) defined in Eqs. (7.180)–(7.182). For the calculation of the √ coefficients cn , we multiply both parts of Eq. (7.191) with the functions m (r) = 1 − r 2 U2m (r) (m = 0, 1, 2, ...) and integrate over the variable √ r from 0 to 1. Using orthogonality of Chebyshev’s polynomials with the weight 1 − r 2 [7], we obtain an infinite system of linear algebraic equations for the coefficients cn : ∞  n=0

cn

π 4

γ0 δnm + γ1 τ1 (n, m) + γ2 τ2 (n, m) + γ3 τ3 (n, m) =

π δ nm , m = 0, 1, 2, ..., 4   1, m = n, 1, m = n = 0, δnm = δ nm = 0, m = n, 0, m = 0 or n = 0. =

(7.192) (7.193)

The coefficients τ1 (n, m), τ2 (n, m), τ3 (n, m) are the following absolutely converging integrals: 1 τp (n, m) = 2π



∞

tp (k)g1 (n, k) 0

π 3/2  (−1)p g(l, k) = l! 2 p! l

p=0



m 

v(m, l)g(l, k)kdk, p = 1, 2, 3, (7.194)

l=0

l p

 # $p−l−3/2 k Jp+l+3/2 (k), 2

(7.195)

300

Heterogeneous Media

g1 (n, k) =

n  m=0

d(n, m) =

# d(n, m)1 F2

2

k 1 n − m + ; 1, n − m + 2; − 2 4

$ ,

π(−1)m (2n − m)! . 4(n − m)!(n − m + 1)!

(7.196) (7.197)

  l Here, are the binomial coefficients, Jν (z) is the Bessel function, and 1 F2 (a; p b, c; z) is the generalized hypergeometric function [7]. Truncating the infinite system (7.192), we can calculate a finite number of coefficients cn in the approximate solution (7.191). The number N of the terms that are to be kept in the series (7.190) depends on the frequency of the incident field, and in the calculations, it changes from 2 (for βR = 1) to 12 (for βR = 20). The criterion of truncation is the value of the last coefficient cN in the series (7.190). If this coefficient is two orders of magnitude smaller than the maximal coefficient of the series, all other coefficients can be neglected. This criterion holds for all the considered frequencies of the incident field.

7.5.3 Numerical solutions of the scattering problem for a penny-shaped crack In this section, the results of the serial and numerical solutions are compared for a penny-shaped crack of radius R subjected to a longitudinal incident wave normal to the crack surface. Calculations of the crack opening b(r) by both methods are presented in Fig. 7.68 for√ the parameter βR = 0, 1, 2, 4, 6 and in Fig. 7.69 for βR = 8, 10, 15, 20 (β = ω ρ/μ). A medium with a Poisson ratio of ν = 0.25 is considered in the calculations. For the numerical solution, a square W with the sides 2R containing the crack was covered by a regular node grid with the step h/R = 0.01 (Mn = 40401) or h/R = 0.005 (Mn = 160801). The numerical results are shown as dashed lines in Figs. 7.68 and 7.69. Solid lines are the result of the serial solution. Convergence of the numerical method by increasing the number of nodes (decreasing the grid step h) is demonstrated in Fig. 7.70. The cases of transverse incident waves the wave vectors of which are orthogonal to the crack plane and the polarization vectors are directed along the x1 -axis are shown in Figs. 7.71 and 7.72. The dimensionless amplitude |b1 (x1 , 0)|/b0 of the component of the crack opening vector along the x1 -axis is presented in Fig. 7.71 for βR = 0, 1, 2, 4, 5, 6 and in Fig. 7.72 for βR = 8, 9, 10, 15, 20. Here, b0 = bs (0, 0) is the crack opening at the crack center by static loading (β = 0).

7.5.4 Dynamic stress intensity factors For calculation of the stress intensity factors at the crack edge in elastodynamics, the same algorithm as in the case of the static crack problem of elasticity can be used. In this algorithm, calculations of the SIFs are reduced to construct the asymptotics of the

Time-harmonic fields in heterogeneous media

301

Figure 7.68 The dimensionless amplitudes of the crack opening vector |b(r)|/b0 of a penny-shaped crack subjected to a longitudinal incident wave orthogonal to the crack plane for the frequency parameter βR = 0, 1, 2, 4, 6; b0 is the crack opening in the center by static loading (βR = 0); solid lines correspond to the serial solutions, dashed lines show the numerical solutions.

Figure 7.69 The same as in Fig. 7.68 for the frequency parameter βR = 8, 10, 15, 20.

Figure 7.70 Dependence of the opening |b(r)|/b0 of a penny-shaped crack subjected to a longitudinal incident wave orthogonal to the crack plane on the node grid step h for the frequency parameter βR = 20.

crack opening vector near the crack contour. The algorithm is described in Section 6.3, and it is transferred to the dynamic case without changing.

302

Heterogeneous Media

Figure 7.71 The dimensionless crack opening vector |b1 (x1 , 0)|/b0 of a penny-shaped crack subjected to transverse incident waves with the wave vector orthogonal to the crack plane and the polarization vector directed along the x1 -axis for the frequency parameters βR = 0, 1, 2, 4, 5, 6; b0 is the opening at the crack center by static loading (βR = 0).

Figure 7.72 The same as in Fig. 7.71 for βR = 8, 9, 10, 15, 20.

By scattering longitudinal incident waves the wave vectors of which are normal to the crack plane, the crack opening has the normal component bn (r) only, and for a penny-shaped crack, the stress intensity factor KI is constant at the crack contour . For calculation of the SIF for the crack with radius R = 1, the function βn (1 + s) =

bn (1 + s) √ −s

(7.198)

should be constructed in the vicinity of the crack contour. Here s is the coordinate along the crack radius with the origin at the point of the crack contour. Examples of behavior of real and imaginary parts of this function are shown in Fig. 7.73 for βR = 3, where solid lines correspond to the serial solution and lines with white dots to the numerical solution. In order to extract the correct value of asymptotics of bn (r) from the numerical solution, one has to construct the function βn (1 + s) inside an interval  at a distance l from the crack edge and interpolate the numerical solution into the origin (s = 0) by the tangent line to the function βn (1 + s) in this interval (the dashed line in Fig. 7.73). For a values of the distance l of about 20 ÷ 30h and  ≈ 10h, where h is the step of the node grid, the error in calculation of the SIFs

Time-harmonic fields in heterogeneous media

303

√

Figure 7.73 The real and imaginary parts of the function bn (1 + s)/ (−s) near the crack edge for a penny-shaped crack subjected to a longitudinal incident wave orthogonal to the crack plane (bn (r) is the crack opening); the solid line corresponds to the serial solution, the line with white dots shows the numerical solution, and the dashed line is tangent to the numerical solution in the distance 20h/R for h/R = 0.005 and frequency parameter βR = 3.

Figure 7.74 Dependence of the dimensionless stress intensity factor k1 for a penny-shaped crack subjected to a longitudinal incident wave orthogonal to the crack plane on the frequency parameter βR; the solid line corresponds to the serial solution, the dashed line to the numerical solution.

does not exceed 3%–4% for h/R = 0.005. This error decreases together with √ h. The dependence of the dimensionless SIF k1 on the frequency parameter β = ω ρ/μ is presented in Fig. 7.74. The coefficient k1 is defined by the equation k1 (β) =

|KI (β)| . KI (0)

(7.199)

Here KI (0) is the SIF in statics (β = 0). In Fig. 7.74, the solid line corresponds to the SIF obtained by the serial solution presented in Section 7.5.2, the dashed line shows the results of the numerical method. Note that the solid line in Fig. 7.74 coincides practically with the similar curve presented in [8]. In the case of transverse incident waves with the wave vectors normal to the crack plane and the polarization vector directed along the x1 -axis, the asymptotics of the stress field at the contour of a penny-shaped crack are defined by the stress intensity

304

Heterogeneous Media

Figure 7.75 Dependencies of the dimensionless stress intensity factors k2 and k3 at the edge of a pennyshaped crack subjected to a transverse incident wave whose wave vector is orthogonal to the crack plane and the polarization vector is directed along the x1 -axis on the frequency parameter βR; the SIF k2 is taken at the point x1 = 1, x2 = 0 (φ = 0), and the SIF k3 at x1 = 0, x2 = 1 (φ = π/2).

factors KII (φ, β) and KIII (φ, β), which are functions of the polar angle φ. Maximal values of the SIFs are achieved at φ = 0 (for KII ) and φ = π/2 (for KIII ). Dependencies of the dimensionless coefficients k2 and k3 , k2 =

|KII (0, β)| |KIII (π/2, β)| , k3 = , KII (0, 0) KII (0, 0)

(7.200)

on the frequency parameter βR are presented in Fig. 7.75. Here KII (0, 0) is the stress intensity factor in statics (β = 0).

7.5.5 Elastic wave scattering from a noncircular crack Let the crack contour  be described by the equation R0 (φ) =

1 [1 + 0.1 cos(4φ) + 0.2 sin(3φ)] . 1.3

(7.201)

The crack area  is symmetric with respect to the x2 -axis, and the contour  and auxiliary normal lines for calculating the SIFs are shown in Fig. 7.76. For a longitudinal incident field the wave vector of which is orthogonal to the crack plane, the graphs of the crack opening amplitudes |bn (x1 , x2 )|/bs (0, 0) are presented in Figs. 7.77 and 7.78 for the parameter βR = 1, 4, 6, 10, where R is the maximal distance of the crack contour from the origin (R = 1). Here, bs (0, 0) is the static limit of the crack opening in the center of a penny-shaped crack of radius R = 1 subjected to the normal stress equal to the stress amplitude of the longitudinal incident wave. Distributions of the dynamic SIFs along the crack contour for various values of the parameter βR are shown in Fig. 7.79. In the calculations, the step of the node grid was taken as h/R = 0.005 (Mn = 160801) and H = 2. In this figure, the dimensionless

Time-harmonic fields in heterogeneous media

305

Figure 7.76 The crack contour  defined in Eq. (7.201) with the lines normal to  at the points where the dynamic SIFs are calculated.

Figure 7.77 The crack opening amplitude for the crack whose contour is defined in Eq. (7.201); the crack is subjected to a longitudinal incident wave orthogonal to the crack plane for the frequency parameters βR = 1 (left) and βR = 4 (right), and bs (0, 0) is the static opening of a penny-shaped crack of radius R = 1.

306

Heterogeneous Media

Figure 7.78 The same as in Fig. 7.77 for βR = 6 and βR = 10.

SIF k1 (β, φ) is defined by the equation k1 (β, φ) =

|KI (β, φ)| , KI (0, −π/2)

(7.202)

where KI (0, −π/2) is the SIF at the contour point x = (0, −1) by static loading (β = 0).

7.5.6 The far scattered field from a planar crack The integral terms in Eq. (7.141) are the field usi (x) scattered from a crack,  usi (x) = −



    ∂k gij x − x  Cj klm nl bm x  d .

(7.203)

Time-harmonic fields in heterogeneous media

307

Figure 7.79 Distribution of the SIFs along the crack edge defined in Eq. (7.201); the crack is subjected to a longitudinal incident wave orthogonal to the crack plane with the frequency parameters βR = 0, 1, 1.5, 2, 3 (left figure) and 4, 6, 10 (right figure).

Far from the crack, this field is presented in the form [9] (see Sections 2.15 and 3.11.1) usi (x) ≈ Ai (m)

xi e−iα|x| e−iβ|x| + Bi (m) . , mi = |x| |x| |x|

(7.204)

Here, Ai (m) and Bi (m) are the vector amplitudes of longitudinal and transverse waves scattered in the direction mi . These amplitudes are expressed in terms of the integral of the crack opening vector bi (x) over the crack area ,   Ai (m) = mi mj fj (αm) , Bi (m) = δij − mi mj fj (βm), fi (qm) =

 −iq 3  λmi nj + μ(ni mj + (nk mk )δij × 2 4πρω  bj (x) exp (iqml xl ) d, (q = α, β). ×

(7.205)

(7.206)



For the approximation of bi (x) in Eq. (7.147), the integral in this equation is the following sum:  bi (x) exp (iqml xl ) d ≈ 

308

Heterogeneous Media

2 2  Mn    q h H (n) (n) (n) ≈ h2 exp − bi exp iq(m1 x1 + m2 x2 ) . m21 + m22 4 n=1

(7.207) (n)

(n)

(n)

Here, (x1 , x2 ) are coordinates of the nth node in the crack plane and bi = bi (x (n) ). The total scattering cross-section Q of a heterogeneity is defined in Eq. (3.299) and the differential scattering cross-section F (m) in Eq. (3.300) F (m) = F (ϕ, θ ) =

(λ + 2μ) α|A(m)|2 + μβ|B(m)|2 , α(λ + 2μ)|a|2 + βμ|b|2

(7.208)

where the amplitudes Ai (m) and Bi (m) are given in Eqs. (7.205) and (7.206). In the case of a planar crack, the function F (φ, θ ) is symmetric with respect to the crack plane. For a penny-shaped crack subjected to a longitudinal incident wave normal to the crack plane, this function does not depend on the φ-angle. The graphs of the function F (0, θ )/F (0, 0) for various values of the frequency parameters βR of the incident field are shown in Fig. 7.80. In this figure, solid lines correspond to the serial solution presented in Section 7.5.2, and the lines with white dots are the numerical solutions for the node grid step h/R = 0.005, H = 2.

Figure 7.80 Differential scattering cross-sections of a penny-shaped crack subjected to a longitudinal incident wave orthogonal to the crack plane for various frequency parameters βR; solid lines correspond to the serial solutions, lines with white dots to the numerical solutions.

The dimensionless differential cross-sections for the crack defined in Eq. (7.201) are shown in Figs. 7.81–7.84. The crack is subjected to a longitudinal incident field normal to the crack plane, and βR = 1, 4, 6, 10. The surfaces shown in these fig-

Time-harmonic fields in heterogeneous media

309

Figure 7.81 Differential cross-sections of the crack defined in Eq. (7.201) and subjected to a longitudinal incident wave orthogonal to the crack plane for the frequency parameter βR = 1.

Figure 7.82 The same as in Fig. 7.81 for βR = 4.

ures are the parametric graphs of the differential scattering cross-sections f (φ, θ ) = F (φ, θ )/F (0, 0) defined by the equations x1 = f (φ, θ ) sin θ cos ϕ, x2 = f (φ, θ ) sin θ sin ϕ, x3 = f (φ, θ ) cos θ, (7.209) 0 ≤ φ ≤ 2π, 0 ≤ θ ≤ π/2. The function F (φ, θ ) = F (m) is defined in Eq. (7.208), where mi is the vector on the unit sphere with spherical coordinates φ, θ.

310

Heterogeneous Media

Figure 7.83 The same as in Fig. 7.81 for βR = 6.

7.6

Scattering from heterogeneous inclusions in poroelastic media

We consider a finite region V with heterogeneous poroelastic properties in an infinite homogeneous poroelastic medium. The material parameters of the homogeneous host medium are λ0 , μ0 , ρ0 , α0 , ρf 0 , κ0 , β0 , and in the region V , these parameters are presented in the forms λ(x) = λ0 + λ1 (x),

μ(x) = μ0 + μ1 (x),

α(x) = α0 + α 1 (x),

ρf (x) = ρf 0 + ρf1 (x),

β(x) = β0 + β 1 (x),

ρt (x) = ρt0 + ρt1 (x), κ(x) = κ0 + κ 1 (x), (7.210)

where λ1 (x), μ1 (x), ρ 1 (x), α 1 (x), ρf1 (x), κ 1 (x), β 1 (x) are deviations of the parameters inside V from the parameters of the host medium. In Section 3.12, the scattering problem of poroelasticity is reduced to a system of integral equations (3.338)–(3.339) for the vector of displacements of the solid skeleton ui (x) and fluid pressure p(x) in the pore space   0    ∂j Gik (x − x )skj (x )dx + Gik (x − x  )sk (x  )dx  + ui (x) = ui (x) + V V   + ∂k i (x − x  )vk (x  )dx  − i (x − x  )s(x  )dx  , (7.211) V

V

Time-harmonic fields in heterogeneous media

311

Figure 7.84 The same as in Fig. 7.81 for βR = 10.

  p(x) = p 0 (x) + ∂k i (x − x  )ski (x  )dx  + k (x − x  )sk (x  )dx  + V V      + ∂k g(x − x )vk (x )dx − g(x − x  )s(x  )dx  . (7.212) V

V

The kernels Gik (x), i (x), and g(x) of the integral operators in these equations are the Green functions of poroelasticity presented in Eqs. (2.166)–(2.170), and the integrand functions are expressed in terms of the deviations of the poroelastic parameters inside the inclusion from the parameters of the host medium, the displacement vector of the solid skeleton ui (x), and the pressure p(x). The functions sij (x), si (x), vi (x), and s(x) are presented in Eqs. (3.340)–(3.342).

7.6.1 Discretization of the integro-differential equations (7.211) and (7.212) The integral equations (7.211) and (7.212) define displacement ui (x) and pressure p(x) in the poroelastic medium with a heterogeneity V . The principal unknowns are the fields inside the heterogeneity. Thus, the numerical solution of these equations can

312

Heterogeneous Media

be considered in any region W with the region V inside. Let W be a cuboid covered by a cubic node grid with a constant step h, and let x (n) (n = 1, 2, ..., Mn) be the nodes of the grid (Fig. 4.7). For numerical solution, the unknown functions ui (x) and p(x) and their derivatives ∂i uj (x) and ∂i p(x) are approximated by the Gaussian quasiinterpolants Mn Mn   (n) (n) (n) ui ϕ(x − x ), ∂i uj (x) ≈ Di uj ϕ(x − x (n) ), ui (x) ≈ n=1

p(x) ≈

(7.213)

n=1

Mn Mn   p (n) ϕ(x − x (n) ), ∂i p(x) ≈ Di p (n) ϕ(x − x (n) ), n=1

  1 |x|2 ϕ(x) = exp − 2 . (πH )3/2 h H

(7.214)

n=1

(7.215)

(n)

Here, ui , p (n) are unknown coefficients of the approximation and the coefficients (n) (n) Di uj and Di p (n) are expressed in terms of ui and p (n) by the linear equations (n)

(7.11)–(7.13). The coefficients ui at the nodes

and p (n) coincide with the values of the functions

u(n) = u(x (n) ), p (n) = p(x (n) ).

(7.216)

After substitution of the approximations (7.213) and (7.214) into the integral equations (7.211) and (7.212) and satisfaction of the resulting equations at all nodes x (n) (n = 1, 2, ..., Mn), we obtain the system of linear algebraic equations for the coefficients (n) (the discretized problem). For performing the discretization procedure, u(n) i and p we have to calculate the results of actions of the integral operators in Eqs. (7.211) and (7.212) on the Gaussian function ϕ(x) in Eq. (7.215). These integrals are expressed in terms of the basic integral in Eq. (7.24),    1 e−iκ|x| |x| , κh , g(x) = . I 0 (x, κ) = 2 g(x − x  )ϕ(x  )dx  = 0 h 4π |x| h (7.217) Actions of the integral operators in Eqs. (7.211) and (7.212) on the Gaussian function ϕ(x) are presented in terms of the derivatives of the function 0 (ξ, q) with respect to the variable ξ defined in Eq. (7.129). Let us introduce the functions Ii1 (x, κ), Iij2 (x, κ), Iij3 k (x, κ) by the equations    xi 1 |x| |x| , κh = 0 , κh ni , ni = , h h h |x|   |x| , κh = Iij2 (x, κ) = ∂i ∂j 0 h    

  |x| 1 h  |x| 0 , κh δij − ni nj + 0 , κh ni nj , = 2 h h h |x| 

Ii1 (x, κ) = ∂i 0

(7.218)

(7.219)

Time-harmonic fields in heterogeneous media

313

    1 |x| |x| Iij3 k (x, κ) = ∂i ∂j ∂k 0 , κh = 3  , κh ni nj nk + 0 h h h       h  |x| |x| h 0 , κh − 0 , κh δij nk + δik nj + δj k ni − + |x| h |x| h

 − 3ni nj nk .

(7.220)

The discretized integral equations (7.211) and (7.212) take the forms u(m) − i

Mn  

(n)

(n)

D2ij k (x (m) − x (n) )sj k + 2ij (x (m) − x (n) )sj



−

n=1

−

Mn  

(n) 0(s) D1ij (x (m) − x (n) )vj − 1i (x (m) − x (n) )s (n) = ui ,

(7.221)

n=1

p (m) −

Mn   (n) (n) D1ij (x (m) − x (n) )sij + 1i (x (m) − x (n) )si − n=1

−

Mn  

(n)

D0i (x (m) − x (n) )vi

− 0 (x (m) − x (n) )s (n) = p 0(m) ,

(7.222)

n=1

m = 1, 2, ..., Mn. Here, the functions 0 (x), D0i (x), 1i (x), D1ij (x), 2ij (x), and D2ij k (x) are expressed in terms of the functions I 0 (x, q), Ii1 (x, κ), Iij2 (x, κ), Iij3 k (x, κ) in Eqs. (7.217)–(7.220) as follows: 0 (x) = b1 I 0 (x, κf h) − b2 I 0 (x, κs h),

(7.223)

D0i (x) = b1 Ii1 (x, κf h) − b2 Ii1 (x, κs h),

(7.224)



1i (x) = γ Ii1 (x, κf h) − Ii1 (x, κs h) ,  D1ij (x) = γ Iij2 (x, κf h) − Iij2 (x, κs h) , 1 2 Iij (x, κt h), qt2 1 D2ij k (x) = g1 Iij3 k (x, κf h) + g2 Iij3 k (x, κs h) + 2 Iij3 k (x, κt h), qt 2ij (x) = g1 Iij2 (x, κf h) + g2 Iij2 (x, κs h) +

(7.225) (7.226) (7.227) (7.228)

where the coefficients b1 , b2 , γ , g1 , g2 are defined in Eqs. (2.166)–(2.170), κf , κs are the solutions of the dispersion equation (2.172) with positive imaginary parts, and κt (n) (n) (n) is given in Eq. (2.171). In Eqs. (7.221) and (7.222), tensors sij , vectors si , vi , and scalars s (n) are the values of the functions in Eqs. (3.340)–(3.342) at the nodes (n)

(n)

sij = sij (x (n) ), si

(n)

= si (x (n) ), vi

= vi (x (n) ),

314

Heterogeneous Media

s (n) = s(x (n) ), n = 1, 2, ..., Mn.

(7.229)

The system of Eqs. (7.221)–(7.222) can be presented in matrix form as follows: (I − B)X = F,

(7.230)

where I is a 4Mn × 4Mn unit matrix; the vectors of unknowns X and of the right hand side F have dimensions 4Mn,   T T X = X 1 , X 2 , ..., X 4Mn , F = F 1 , F 2 , ..., F 4Mn , ⎧ (n) ⎪ u1 , ⎪ ⎪ ⎪ ⎨ (n−Mn) , u2 Xn = (n−2Mn) ⎪ , u3 ⎪ ⎪ ⎪ ⎩ (n−3Mn) p , ⎧ 0(n) ⎪ u1 , ⎪ ⎪ ⎪ ⎨ 0(n−Mn) , u2 Fn = 0(n−2Mn) ⎪ , u3 ⎪ ⎪ ⎪ ⎩ 0(n−3Mn) p ,

(7.231)

n ≤ Mn, Mn < n ≤ 2Mn, 2Mn < n ≤ 3Mn, 3Mn < n ≤ 4Mn, n ≤ Mn, Mn < n ≤ 2Mn, 2Mn < n ≤ 3Mn, 3Mn < n ≤ 4Mn.

(7.232)

Matrix B in Eq. (7.230) has dimensions 4Mn × 4Mn and is reconstructed from Eqs. (7.221) and (7.222). Let us consider an example of an actual geological structure. The inclusion material consists of a solid skeleton with Young modulus Es = 30 GPa and a Poisson ratio of νs = 0.2. The skeleton porosity is φ = 0.2, the fluid in the pores has bulk modulus Kf = 1.5 GPa and viscosity η = 10−3 Pa· sec, the densities of the solid and fluid phases are ρs = 2.5 · 103 kg/m3 and ρf = 1.2 · 103 kg/m3 , and the permeability of the inclusion material is χ = 10−13 m2 . The host medium has the skeleton elastic parameters Es0 = 50 GPa, νs0 = 0.2, porosity φ0 = 0.2, fluid bulk modulus Kf 0 = 2 GPa, and viscosity η0 = 10−3 Pa· sec, the densities of the solid and fluid phases are ρs0 = 2.65 · 103 kg/m3 and ρf 0 = 103 kg/m3 , and the host medium permeability is χ0 = 10−14 m2 . These parameters correspond to typical sedimentary rocks at a depth of 1.5–2.0 km [10]. Effective bulk K0 and shear μ0 moduli of the host medium with dry pores are calculated by the equations of the effective field method [11]  K0 = Ks0 1 −

   φ0 φ0 , μ0 = μs0 1 − , (7.233) 1 − (1 − φ0 )S1 1 − (1 − φ0 )S2 3Ks0 1 , S2 = (3 − S1 ). S1 = 3Ks0 + 4μs0 5

Effective bulk and shear moduli of the inclusion are calculated from similar equations.

Time-harmonic fields in heterogeneous media

315

Figure 7.85 Distribution of the real and imaginary parts of component u3 of the displacement vector and pressure p along the x3 -axis inside a spherical inclusion; the plane longitudinal monochromatic incident wave field has dimensionless wave number q0 = 0.5 and the wave vector directed along the x3 -axis.

Figure 7.86 The same as in Fig. 7.85 for dimensionless wave number q0 = 1.

In Figs. 7.85–7.88, the numerical solutions of the system (7.230) for the displacement component u3 (0, 0, x3 ) and pressure p(0, 0, x3 ) along the x3 -axis inside a spherical inclusion are compared with the serial solution of the corresponding scattering problem presented in [13]. The wave vector of the longitudinal incident waves are directed along the x3 -axis (Fig. 7.1). The frequencies of the incident fields correspond to the values of the parameter q0 = 0.5, 1, 2, 4, where

ρto . (7.234) q0 = ωa λ0 + 2μ0 Dashed lines in these figures show the numerical solutions of Eqs. (7.221) and (7.222) for the step of the node grid h/a = 0.05 (Mn = 79507) and H = 1. Solid lines are

316

Heterogeneous Media

Figure 7.87 The same as in Fig. 7.85 for dimensionless wave number q0 = 1.

Figure 7.88 The same as in Fig. 7.85 for dimensionless wave number q0 = 1.

the results of the serial solution [13]. It is seen from these figures that the numerical solutions practically coincide with the serial solutions for displacements, but numerical values of the pressure deviate slightly from the serial solutions near the inclusion boundary. (s) The far scattered field of displacements ui (x) and pressure p (s) (x) is defined in Eqs. (3.349)–(3.355), and for longitudinal incident waves, the differential scattering cross-section F (φ, θ ) is in Eq. (3.357), where φ, θ are the angle coordinates of vector ni in the spherical coordinate system with the polar axis directed along the wave vector of the incident wave. For the considered spherical inclusion, the graphs of the function F (0, θ) for the dimensionless wave numbers q0 = 0.5, 1, 2, 4 are presented in Figs. 7.89–7.92. In these figures, dashed lines correspond to the numerical solutions and solid lines are the serial solutions.

Time-harmonic fields in heterogeneous media

317

Figure 7.89 The differential scattering cross-section of a spherical inclusion for the longitudinal wave with dimensionless wave number q0 = 0.5.

Figure 7.90 The same as in Fig. 7.89 for dimensionless wave number q0 = 1.

Figure 7.91 The same as in Fig. 7.89 for dimensionless wave number q0 = 2.

Figure 7.92 The same as in Fig. 7.89 for dimensionless wave number q0 = 4.

318

Heterogeneous Media

7.6.2 Scattering from a nonspherical heterogeneous inclusion in a poroelastic medium In this section, scattering of a plane longitudinal monochromatic wave from a semispherical heterogeneous inclusion is considered (Fig. 7.93). In the Cartesian coordinate system shown in Fig. 7.93, the inclusion occupies the region V : (x12 + x22 + x32 ≤ a 2 ; x3 ≥ 0). For numerical solution, a cuboid W with the semisphere V inside is covered by a regular node grid with the step h/a = 0.05, a = 100 m (Mn = 42527). The bulk and shear moduli of the solid phase of the host medium and the inclusion are the same, Ks = 40 GPa, μs = 30 GPa; porosity of the host medium is φ0 = 0.2, and porosity of the inclusion material is φ = 0.3. The permeabilities of the host medium and the inclusion are χ0 = 10−15 m2 and χ = 10−13 m2 .

Figure 7.93 A semispherical inclusion subjected to an incident longitudinal wave; the arrow indicates the direction of the incident wave vector.

We consider two cases. In the first case (inclusion 1), the properties of the fluid in the host medium and in the inclusion are the same, and the fluid bulk modulus is Kf = 2.25 GPa, density is ρf = 103 kg/m3 , and viscosity is 10−3 Pa· sec. In the second case (inclusion 2), the inclusion is filled with a mixture of fluid and gas bubbles. The gas bulk modulus is Kg = 25 MPa, gas density is ρg = 100 kg/m3 , and gas viscosity is η = 10−4 Pa· sec. The volume fraction φg of the gas bubbles depends linearly on the x3 -coordinate: φg (x3 ) = x3 /a (0 ≤ x3 ≤ a). The effective bulk modulus Kfg and viscosity ηfg of the fluid–gas mixture are calculated by the equation of the effective field method [11] Kf (Kg − Kf ) , Kf + (1 − φg )(Kg − Kf ) 5ηf (ηg − ηf ) . ηfg = ηf + φg 5ηf + 2(1 − φg )(ηg − ηf )

Kfg = Kf + φg

(7.235) (7.236)

Thus, the poroelastic properties of inclusion 2 are heterogeneous. In Fig. 7.94, the dependencies of the effective bulk modulus Kfg (x3 ) and viscosity ηfg (x3 ) of the fluid–gas mixture on the x3 -coordinate are shown.

Time-harmonic fields in heterogeneous media

319

Figure 7.94 Dependencies of the bulk modulus Kf g and viscosity ηf g of the fluid–gas mixture on the x3 -coordinate inside a semispherical inclusion in a poroelastic medium.

Figure 7.95 Distributions of displacement component u3 (0, 0, x3 ) in the solid phase and pressure p(0, 0, x3 ) in the fluid phase in a semispherical inclusion subjected to an incident plane longitudinal wave of frequency f = 10 Hz; inclusion 1 is filled with a homogeneous fluid, and inclusion 2 is filled with a fluid–gas mixture whose bulk modulus and viscosity depend on the x3 -coordinate (Fig. 7.94); U0 is the amplitude of the incident wave and μ is the shear modulus of the host medium with dry pores.

Distributions of real and imaginary parts of the component u3 (0, 0, x3 ) of the displacement vector in the solid phase and pressure p(0, 0, x3 ) in the fluid (inclusion 1) or in the fluid–gas mixture (inclusion 2) along the x3 -axis inside the inclusions are shown in Fig. 7.95 for a frequency of the incident field of f = 10 Hz. In Fig. 7.96, the dimensionless differential cross-sections F (ϕ, θ ) = F (0, θ) of the inclusions are presented. It is seen that the displacements u3 (0, 0, x3 ) in both inclusions coincide practically, but the pressure in inclusion 2 is much smaller than in inclusion 1. Because the contrast in the properties is larger for inclusion 2, the differential cross-section of this inclusion is larger for all the directions. For a frequency of f = 10 Hz, the back scattering prevails for both inclusions. The same dependencies for a frequency of f = 20 Hz are shown in Figs. 7.97 and 7.98. In this case, the back scattering prevails for inclusion 2, but the forward scattering dominates for inclusion 1.

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Heterogeneous Media

Figure 7.96 The differential cross-section of a semispherical inclusion subjected to an incident plane longitudinal wave of frequency f = 10 Hz (Fig. 7.93); inclusion 1 is filled with a homogeneous fluid, and inclusion 2 is filled with a fluid–gas mixture whose bulk modulus and viscosity depend on the x3 -coordinate (Fig. 7.94); the arrow indicates the direction of the wave vector of the incident field.

Figure 7.97 The same as in Fig. 7.95 for the frequency of the incident field f = 20 Hz.

Figure 7.98 The same as in Fig. 7.96 for the frequency of the incident field f = 20 Hz.

If the frequency of the incident field increases, the step of the node grid h should decrease in order to approximate the wave fields inside the inclusion with sufficient accuracy. For the poroelastic parameters of the medium and the inclusion considered in this section, the step h/a = 0.05 does not affect the accuracy of the solution for the dimensionless wave number of the incident field q0 < 5. For q0 > 5, the step h should be taken smaller, and as a result, the number of nodes Mn increases together with the computational time and the required RAM.

Time-harmonic fields in heterogeneous media

7.7

321

Notes

The content of this chapter is based on the following publications: electromagnetic wave scattering from perfectly conducting screens [14]; electromagnetic wave scattering from dielectric heterogeneities [15]; elastic wave scattering from heterogeneities of arbitrary shapes [16]; elastic wave scattering from a planar crack [17]; acoustic wave scattering from a rigid screen [18]; and the scattering problem for an isolated heterogeneity in poroelastic medium [12], [13], [19].

Appendix 7.A The computational program for solution of the problem of acoustic wave scattering from heterogeneities in fluid In this appendix, the computational program for numerical solution of the acoustic wave scattering problem for a spherical heterogeneity in fluid is presented. For heterogeneities of other shapes, the object “Ind,” which indicates the region occupied by the heterogeneity, should be changed to the object for the actual region V Ind[[i]] = 1/; x (i) ∈ V , i = 1, 2, ..., Mn. /V Ind[[i]] = 0/; x (i) ∈ In Wolfram Mathematica software, the MRM for iterative solution of the system (7.30) and the FFT algorithm for calculation of the matrix-vector products are adopted. (*Initial data*) (*Properties of the host medium*) (*Density in kg/m3 *) ro0=10 ˆ 3; (*Bulk modulus in Pa*) K0=2*10 ˆ 9 (*The square of the incident wave number in m−2 *) Q0=w ˆ 2*ro0/K0 q:=w*h*Sqrt[ro0/K0] (*Properties of the heterogeneity*) (*Density*) ro=100 (*Bulk modulus*) K=2*10 ˆ 8 Q=w ˆ 2*ro*/K (*Node generation*) M1=42; M2=42; M3=42;

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h=N[2/(M1-2)] N1=2*M1; N2=2*M2; N3=2*M3; Mn=M1*M2*M3 (*Cartesian coordinates of the nodes*) x1=Flatten[Table[(-1-h)+h*(i-1),{k,M3},{j,M2},{i,M1}]]; x2=Flatten[Table[(-1-h)+h*(j-1),{k,M3},{j,M2},{i,M1}]]; x3=Flatten[Table[(-1-h)+h*(k-1),{k,M3},{j,M2},{i,M1}]]; (*Relations between the one- and three-index node numerations*) M[i_,j_,k_]:=i+M1*(j-1)+M1*M2*(k-1) To1In[A_]:=Flatten[Transpose[Flatten[Transpose[A],1]]] To3In[A_]:=Transpose[Partition[Transpose[Partition[A,M1*M2]],M1]] (*Indication of the inclusion region*) r00=1 DR[i_]:=Norm[{x1[[i]],x2[[i]],x3[[i]]}] LI= Select[Table[If[DR[i]0 &]; Ind=Module[{},Ind=Table[0,{Mn}];Ind[[LI]]=1;Ind] (*Deviation of the properties of the inclusion from the properties of the host medium*) R01= Module[{}, R01 = Table[0,{Mn}]; R01[[LI]] = (ro-ro0)/ro; R01] Q1:=Module[{}, Q1 = Table[0,{Mn}]; Q1[[LI]] = Q-Q0;Q1] (*Approximating functions*) H=1 f0[z_]:=N[1/(Pi*H) ˆ (3/2)*Exp[-z ˆ 2/H]] fp[z_, q_] := Exp[-H*q ˆ 2/4 + I*q*z]*Erfc[(I*H*q + 2*z)/(2*Sqrt[H])] fn[z_, q_] := Exp[-H*q ˆ 2/4 - I*q*z]*Erfc[(I*H*q - 2*z)/(2*Sqrt[H])] FI0[z_,q_] := 1/(8*Pi*z)*(fn[z,q] - fp[z,q])/;z>0 FI0[z_,q_] :=(2/Sqrt[Pi*H]-q*E ˆ (-H*q ˆ 2/4)*(I+Erfi[Sqrt[H]*q/2]))/(4*Pi)/; z0 FI1[z_,q_] :=0/; z0.0001, tah=ZZ[hh1]; tau=hh1.Conjugate[tah]/(tah.Conjugate[tah]); P1=P-tau*hh1;hh2=hh1-tau*tah; P=P1;hh1=hh2; DEL=Abs[tau*Sqrt[(hh1.Conjugate[hh1])/(P.Conjugate[P])]];Print[DEL]]] (*Graphs of the pressure distribution inside the region of the solution*) Grp=Interpolation[Table[{x1[[j]],x2[[j]],x3[[j]],P[[j]]},{j,Mn}]] (*The differential scattering cross-section*) (*Integrals of the pressure over the inclusion region*) dP1 := def[P]; mt[nn_]= h ˆ 3*Exp[-q ˆ 2*H/4]*Table[Tr[Ind*tt[dP][[i]]* Exp[I*q/h*(nn[[1]]*x1+nn[[2]]*x2+nn[[3]]*x3)]],{i,3}] ms[nn_] := h ˆ 3*Exp[-q ˆ 2*H/4]*Tr[Ind*ss[P]*Exp[I*q/h*(nn[[1]]*x1+ nn[[2]]*x2+nn[[3]]*x3)]] nt[nn_]:=nn.mt[nn] (*The amplitude of the far scattered field*) a[nn_]:=(I*q/h*nt[nn]+ms[nn])/(4*Pi) Nn1[f_,t_]:=Sin[t]*Cos[f] Nn2[f_,t_]:=Sin[t]*Sin[f] Nn3[f_,t_]:=Cos[t]

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327

As[f_, t_] := Module[{n1,n2,n3,n,aa,bb}, n1 = Nn1[f, t];n2 = Nn2[f, t];n3 = Nn3[f, t]; n={n1,n2,n3};aa = a[n]; bb=aa*Conjugate[aa];bb] (*The differential scattering cross-section*) DCS[f_,t_]:=Re[As[f,t]] End

Appendix 7.B

The computational program for solution of the acoustic wave scattering problem for a rigid screen

The computational program for solution of Eq. (7.55) of the scattering problem for a planar screen consists of two parts. In the first part, the function F (ρ, q) in Eq. (7.69) is tabulated and kept in the file “Math/DinFan.” Then this file is used for construction of the element of the matrix of the discretized problem. In the program, the function F (ρ, q) is calculated with the steps ρ = 0.1 and q = 0.01 in the intervals 0 ≤ ρ ≤ 5 and 0 ≤ q ≤ 0.5. For ρ > 5, the asymptotic expression (7.70) of this function for large values of the arguments is used. (*Tabulation of the function F[r,q]*) H = 1; F[r_,q_] := 1/(4*Pi)*NIntegrate[Sqrt[k ˆ 2-q ˆ 2]*Exp[-k ˆ 2*H/4]*BesselJ[0,k*r]*k, {k,0,10}] TF = Partition[Flatten[Table[{0.1*(i-1), 0.01*(j-1), Re[F[0.1*(i-1), 0.01*(j-1)]]}, {i,51},{j,51}]],3]; IF = Interpolation[TF]; Save["Math/DinFun", IF]; End 5, q < 0.5, the integrals Fm (ρ, q) can be changed to their asymptotics fm (ρ, q) with sufficient accuracy.

References [1] Ph. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, Toronto, London, 1953. [2] A. Pierce, Acoustics, an Introduction to Its Physical Principles and Applications, McGrawHill, New York, 1981. [3] D. Colton, R. Cress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, 1998. [4] C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, Toronto, Singapore, 1983. [5] A. Eringen, E. Suhubi, Elasodynamics II, Academic, New York, 1975. [6] G. Eskin, Boundary-Value Problems for Elliptic Pseudo-Differential Equations, American Mathematical Society, New York, 1981. [7] M. Abramovitz, I. Stigan, Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55, National Bureau of Standards, 1964. [8] A. Mal, Interaction of elastic waves with a penny-shaped crack, International Journal of Engineering Science 8 (1970) 381–388. [9] J. Gubernatis, E. Domeny, J. Krumhansl, Formal aspects of the theory of the scattering of ultrasound by flaws in elastic materials, Journal of Applied Physics 48 (1977) 2804–2811. [10] G. Mavko, T. Mukerji, J. Dvorkin, The Rock Physics Handbook. Tools for Seismic Analysis in Porous Media, second edition, Cambridge University Press, 2009. [11] S. Kanaun, V. Levin, Self-Consistent Methods for Composites, vol. 1, Static Problems, Springer, Dordrecht, 2008. [12] S. Kanaun, V. Levin, M. Markov, Volume integral equations of the scattering problem of poroelasticity and their properties, Mathematical Methods in the Applied Sciences (2018) 1–17. [13] S. Kanaun, V. Levin, M. Markov, Scattering problem for a spherical inclusion in poroelastic media: application of the asymptotic expansion method, International Journal of Engineering Science 128 (2018) 187–207. [14] S. Kanaun, A numerical method for the solution of electromagnetic wave diffraction problems on perfectly conducting screens, Journal of Computational Physics 176 (2002) 170–195. [15] S. Kanaun, Scattering of monochromatic electromagnetic waves on 3D-dielectric bodies of arbitrary shapes, Progress in Electromagnetics Research B 21 (2010) 129–150. [16] S. Kanaun, V. Levin, Scattering of elastic waves on a heterogeneous inclusion of arbitrary shape: an efficient numerical method for 3D-problems, Wave Motion 50 (4) (2013) 687–707. [17] S. Kanaun, Scattering of monochromatic elastic waves on a planar crack of arbitrary shape, Wave Motion 51 (2) (2014) 360–381. [18] S. Kanaun, Scattering of acoustic waves on a planar screen of arbitrary shape: direct and inverse problems, International Journal of Engineering Science 92 (2015) 28–46. [19] S. Kanaun, V. Levin, M. Markov, Scattering of plane monochromatic waves from a heterogeneous inclusion of arbitrary shape in a poroelastic medium: an efficient numerical solution, Wave Motion 92 (2020) 102411.

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Quasistatic crack growth in heterogeneous media

8

This chapter is devoted to the problem of quasistatic crack growth in heterogeneous elastic media. The method of fast solution of the crack problem developed in Chapter 6 opens the way for efficient numerical simulation of crack growth by increasing external loading. First, the method is applied to simulation of crack growth under prescribed pressure applied to the crack surface (Sections 8.1 and 8.2). Then, crack growth caused by fluid injected inside the crack is considered. This problem relates to the so-called hydraulic fracture, which is widely used in oil and gas exploration technology. The coupled problem of solid, fluid, and fracture mechanics for simulation of hydraulic fracture is considered. For numerical solution of the governing equations of the hydraulic fracture theory, the discrete model of crack propagation is proposed (Section 8.3). An approximate three-parameter model considered in Section 8.4 substantially accelerates computer simulations of hydraulic fracture processes. Numerical examples and computational programs for the simulation of crack propagation in heterogeneous media are presented.

8.1 Crack growth under prescribed pressure applied to the crack surface Quasistatic crack growth in heterogeneous materials is an important problem of fracture mechanics. This problem relates to simulation of cleavage processes in composites and natural rocks by slowly increasing external loading. Determination of the crack shape and size in the process of crack growth is equivalent to construction of an equilibrium crack configuration. For such a configuration, the stress intensity factors at each point of the crack edge are equal to the material fracture toughness at this point. For cracks in heterogeneous media, this problem can be solved only numerically. By numerical simulations, the crack configuration has to be adjusted to the actual fracture toughness distribution in the medium. For the adjustment, numerical solution of the crack problem should be performed many times for various crack contours. It makes the computational process cumbersome and time consuming. In this section, the method of solution of the crack problem of elasticity developed in Chapter 6 is applied to simulation of a planar crack growth in layered (heterogeneous) media. It is assumed that the boundaries of the layers are parallel, and the crack plane is orthogonal to the layers. This situation is common for crack growth in deep rock formations. In this case, the layers are often parallel to the earth surface, and the geologic stresses force cracks to grow in the plane orthogonal to the layers. Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00015-9 Copyright © 2021 Elsevier Ltd. All rights reserved.

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8.1.1 The elasticity problem for planar cracks with varying contours We consider a planar crack in an isotropic homogeneous elastic medium with Lame constants λ0 , μ0 . Let in Cartesian coordinates (x1 , x2 , x3 ) the crack region  be in the plane x3 = 0. If the crack surface is subjected to pressure distribution p(x), x = (x1 , x2 ), the crack opening vector bi (x) = b(x)ni is proportional to the normal ni to , and the scalar function b(x) satisfies the equation that follows from Eq. (6.4),  T (x − x  )b(x  )dx  = p(x), x ∈ . (8.1) 

Here, the kernel T (x) is the generalized function T (x) = −

μ0 λ0 + μ0 , μ0 = μ0 . 3 λ0 + 2μ0 2π|x|

(8.2)

We consider a family of cracks the contours  of which vary from a basic contour  0 , and our objective is fast solution of Eq. (8.1) for such a crack family. We introduce the dimensionless variables ςi = xi / l (i = 1, 2), where l is the maximal size of the crack region . In these variables, Eq. (8.1) takes the form  T (ς − ς  ) b(ς  )dς  = l p(ς), ς ∈ ς . (8.3) ς

Here, ς is the crack surface in the ς-coordinates,  b(ς) = b(ςl), p (ς) = p(ςl), and it is taken into account that dx = l 2 dς and T (x) is a homogeneous function of the order of −3, T (lς) = l −3 T (ς).

(8.4)

Thus, for an arbitrary crack, the maximal size of the region ς is equal to 1. For numerical solution of Eq. (8.3), the region ς is embedded into a rectangular W and covered by a square node grid with the step h. Note that the rectangular W : (|ςi | ≤ 1, i = 1, 2) can be taken such that the crack with an arbitrary contour will be inside W . As a result, the elasticity problem for an arbitrary planar crack can be solved using the same region W covered by the node grid with a fixed step h. For cracks of various shapes and sizes subjected to an arbitrary pressure p(x), only the right hand side l p(ς) of Eq. (8.3) and the crack region ς ∈ W are changed. It accelerates substantially construction of the numerical solutions of the crack problem for the considered crack family. The Gaussian quasiinterpolant of the function  b(ς) in W has the form  b(ς) =

Mn  n=1

  1 ς2  exp − 2 b(n) ϕ(ς − ς (n) ), ϕ(ς) = , πH h H

(8.5)

Quasistatic crack growth in heterogeneous media

337

where ς (n) are the approximating nodes and  b(n) =  b(ς (n) ). After substitution of this approximation into Eq. (8.3) and satisfaction of the resulting equation at all nodes, we obtain the following linear algebraic system for the coefficients of the approximation: 1  (m,n)(n) I p (m) if ς (m) ∈ ς , b = l h Mn

n=1

 b(m) = 0 if ς (m) ∈ / ς , m = 1, 2, ..., Mn, I

(m,n)

= I (ς

(m)

−ς

(n)

), p 

(m)

=p (ς

(m)

).

(8.6) (8.7)

It follows from Eq. (6.16) that the function I (ς) has the form  |ς| , (8.8) h     2  2   1 ρ2 ρ2 ρ ρ ρ2 1− I0 + I1 , F1 (ρ) = √ exp − 2H H 2H H 2H H πH (8.9) I (ς) = μ0 F1



where I0 (z) (m,n)

and I1 (z) are the modified Bessel functions. Note that the matrix A = 1 I of the discretized problem (8.6) does not depend on the crack region ς . h The conjugate gradient method (CGM) with the FFT algorithm adopted for calculation of matrix-vector products can be used for efficient numerical solution of the system (8.6) (Chapter 6).

8.1.2 Quasistatic growth of a planar crack in the medium with varying fracture toughness Let the crack surface  be subjected to increasing with time pressure p(x, t). The process is quasistatic, and inertial effects can be neglected. The medium is elastic and isotropic with constant Lame parameters and varying in space fracture toughness KI c (x). According to the basic principle of fracture mechanics [4], at the points of the contour  of the equilibrium crack, the stress intensity factor KI (x) should be equal to the medium fracture toughness KI c (x), i.e., KI (x) = KI c (x), x ∈ .

(8.10)

This condition determines the crack contour configuration for the given pressure distribution on the crack surface. In the dimensionless variables ςi = xi / l (i = 1, 2), the discretized system (8.6) can be solved for the given pressure p(lς, t) at any time moment if the crack region ς is I (ς) at the crack edge is calculated by the method known. The stress intensity factor K presented in Section 6.3. The actual stress intensity factor at the crack contour  is

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Heterogeneous Media

I (ς) by the equation expressed in terms of K

1   x

KI (x)|x∈ = √ KI . l x∈ l

(8.11)

To satisfy the condition (8.10), we have to adjust the crack contour  to actual distribution of the fracture toughness KI c (x) in the medium. The following algorithm can be used for construction of the contour of an equilibrium crack. Let (r, ϕ) be polar coordinates in the crack plane with the origin at the crack center, and let the crack contour be defined by the equation r = R(ϕ). Here, R(ϕ) is the distance from the origin to the contour  in the direction defined by the polar angle ϕ. It is assumed that  is a star region. The initial approach for  can be taken as a circle with constant radius R (0) . Then, a set of nodes at discrete points ϕ (k) (k = 1, 2, .., M) on the crack contour is introduced, and the distances R(ϕ (k) ) are changed iteratively according to the rule R (i+1) (ϕ (k) ) = R (i) (ϕ (k) ) + α(KI (ϕ (k) ) − KI c (ϕ (k) )) if max KI (ϕ (k) ) > max KI c (ϕ (k) ) and min KI (ϕ (k) ) < min KI c (ϕ (k) ), R (i+1) (ϕ (k) ) = R (i) (ϕ (k) ) − α(KI (ϕ (k) ) − KI c (ϕ (k) )) if max KI (ϕ (k) ) < max KI c (ϕ (k) ) and min KI (ϕ (k) ) > min KI c (ϕ (k) ), R (i+1) (ϕ (k) ) = (1 − β)R (i) (ϕ (k) ) if max KI (ϕ

(k)

) > max KI c (ϕ

(i+1)

(k)

) = (1 + β)R (ϕ

R

(ϕ

(k)

(i)

) and min KI (ϕ (k)

(8.12) (k)

) > min KI c (ϕ

(k)

),

) if

max KI (ϕ (k) ) < max KI c (ϕ (k) ) and min KI (ϕ (k) ) < min KI c (ϕ (k) ), i = 1, 2, .... Here, i is the number of the iteration and α and β are empirical coefficients (α > 0, 0 < β < 1) taken for the fastest convergence of the iteration process. The iterations stop when the relative error of satisfaction of the criterion (8.10) becomes smaller than the prescribed tolerance δ, i.e.,

M (k) (k)

k=1 KI (ϕ ) − KI c (ϕ ) ≤ δ. M (k) k=1 KI c (ϕ )

(8.13)

In the calculations, δ = 0.05 is accepted. First, we consider the problem for a penny-shaped crack of radius R in an infinite homogeneous elastic medium subjected to pressure p uniformly distributed over a concentric disk of radius a on the crack surface (Fig. 8.1). The stress intensity factor KI at the crack contour is calculated by the method presented in Section 6.3. For the calculations, the crack is embedded in the square W (2R × 2R) and covered by a regular node grid with the step h/R = 0.005 (Mn ≈ 8 · 106 ), H = 1. The exact stress intensity factor KI at the crack contour is defined by the equation [1]

Quasistatic crack growth in heterogeneous media

339

Figure 8.1 A penny-shaped crack subjected to a constant pressure p applied to the concentric disk of radius a.

    a 2 p√ KI = 2R 1 − 1 − . π R

(8.14)

The numerical and exact values of KI are presented in Table 8.1 for various values √of the parameter a/R. In the table, the values of the dimensionless SIFs kI = KI /p R calculated by the numerical method are shown in the second column, and the exact values of kI are shown in the third column. Table 8.1 The numerical and exact stress intensity factors kI for the crack in Fig. 8.1. a/R 0.125 0.375 0.5 0.625 0.75 0.875 1.0

h/R = 0.005, H = 1 0.00362 0.03275 0.0598 0.0969 0.1484 0.2298 0.456

Eq. (8.14) 0.00353 0.03283 0.0602 0.0989 0.1524 0.2322 0.45

Then, we consider a crack in the elastic medium that consists of two half-spaces. For x2 < 0.5 m, the medium has Young modulus E0 = 1 Pa, a Poisson ratio of ν0 = 0.3, and a fracture toughness of KI0c = 0.412 Pa·m1/2 . For x2 > 0.5 m, the medium has the same elastic moduli and the fracture toughness KI c = 0.824 Pa·m1/2 . The initial penny-shaped crack with the center at x = 0 has radius R = 0.5 m. The crack surface is subjected to a constant pressure p applied to the disk with radius a = 0.3 m (Fig. 8.1). The results of application of the iterative scheme (8.12) for construction of the equilibrium crack contour for pressure p = 30 Pa are shown in Fig. 8.2. In this figure, the dashed line is the initial crack, and the solid lines are the contours of the crack after the indicated number of iterations. After 25 iterations the condition (8.13) is fulfilled, and the final distribution of the stress intensity factor KI (x) and the medium fracture toughness KI c (x) along the crack contour are shown in Fig. 8.3.

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Heterogeneous Media

Figure 8.2 The contour of an equilibrium crack in the medium with constant elastic moduli E = 1 Pa, ν = 0.3 and varying in space fracture toughness (KI c = 0.412 Pa·m1/2 if x2 < 0.5 m, KI c = 0.824 Pa·m1/2 if x2 > 0.5 m); the crack surface is subjected to a pressure of p = 30 Pa applied to a disk of radius a = 0.3 m with the center at the origin (Fig. 8.1). The crack contours after 5, 9, 15, and 25 iterations in the process of construction of the contour of the equilibrium crack are shown; the dashed line is the initial crack contour.

Figure 8.3 Distributions of the fracture toughness and the stress intensity factor along the contour of the equilibrium crack shown in Fig. 8.2 after 25 iterations; the solid line is the fracture toughness, the dashed line is the stress intensity factor.

The crack contours for the values of the pressure p = 6, 20, 30, ..., 90 Pa are presented in Fig. 8.4, and the corresponding crack openings b(0, x2 ) along the x2 -axis are shown in Fig. 8.5.

Quasistatic crack growth in heterogeneous media

341

Figure 8.4 The contours of equilibrium cracks subjected to increasing pressure p applied to a disk of radius a = 0.3 m on the crack surface (Fig. 8.1); the fracture toughness of the medium is KI c = 0.412 Pa·m1/2 if x2 < 0.5 m and KI c = 0.824 Pa·m1/2 if x2 > 0.5 m, and the Young modulus and Poisson ratio of the medium are constant E = 1 Pa, ν = 0.3.

The contours of the cracks are subjected to the same values of the pressure in the medium with E0 = 1 Pa, ν0 = 0.3, KI0c = 0.412 Pa·m1/2 in the layer |x2 | < 0.5 m, and the same values of the elastic moduli and the fracture toughness KI c = 0.824 Pa·m1/2 for |x2 | > 0.5 m are shown in Fig. 8.6. The corresponding crack openings are shown in Fig. 8.7. The computational program for construction of the equilibrium crack contour and crack opening is presented in Appendix 8.A.

8.1.3 A planar crack in a layered heterogeneous medium We consider a layered heterogeneous elastic medium with stiffness tensor Cij kl (x) = Cij0 kl + Cij1 kl (x). The layer boundaries are planar, and the elastic parameters and fracture toughness of each layer are constant. The medium contains a crack whose surface  is planar and orthogonal to the layers. Let Cij0 kl be the elastic stiffness tensor of one of the layers that contains the crack. The medium is subjected to an external stress field and pressure p(x) applied to the crack surface  and σij0 (x) is the stress field in the layered medium without the crack by action of the same external stress field. The stress tensor in the medium with the crack can be presented in the form σij (x) = σij∗ (x) + σij1 (x), where σij∗ (x) is the stress field in the layered medium with-

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Heterogeneous Media

Figure 8.5 Crack opening b(x1 , x2 ) along the x2 -axis for the cracks shown in Fig. 8.4 for various values of the pressure p applied to the crack surface (Fig. 8.1).

Figure 8.6 The contours of equilibrium cracks subjected to a constant pressure p applied to a disk of radius a = 0.3 m on the crack surface (Fig. 8.1); the fracture toughness of the medium is KI c = 0.412 Pa·m1/2 if |x2 | < 0.5 m and KI c = 0.824 Pa·m1/2 if |x2 | > 0.5 m, and the elastic moduli of the medium are E = 1 Pa, ν = 0.3.

out the crack and prescribed external stress field σij0 (x), and σij1 (x) is the perturbation caused by the crack presence. The field σij1 (x) satisfies the integral equation that follows from the results of Section 6.5:

Quasistatic crack growth in heterogeneous media

343

Figure 8.7 Crack opening b(x1 , x2 ) along the x1 - and x2 -axes for the cracks shown in Fig. 8.6.

 σij1 (x) − 

1 1 Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  −

Sij kl (x − x  )nk bl (x  )d = 0,

(8.15)

 −1 Bij1 kl (x) = Cij kl (x)−1 − Cij0 kl .

(8.16)

− 

Here, ni is the normal vector to . The kernel Sij kl (x) of the integral operators in Eq. (8.15) is expressed in terms of the second derivatives of the Green function gij (x) of the homogeneous medium with the stiffness tensor Cij0 mn and is defined in Eq. (6.2), i.e., 0 Sij kl (x) = −Cij0 mn ∂m ∂p gnq (x)Cpqkl − Cij0 kl δ(x).

(8.17)

If symmetric traction ti (x) is applied to the crack sides, the boundary condition on  takes the form

ni σij∗ (x) + σij1 (x) = tj (x), x ∈ . (8.18) Multiplying both sides of Eq. (8.15) with the vector ni and using the condition (8.18), we obtain  1 1 (x  )σmn (x  )dx  − − ni Sij kl (x − x  )Bklmn  ni Sij kl (x − x  )nl bk (x  )d = nj σij∗ (x) − ti (x), x ∈ . (8.19) − 

344

Heterogeneous Media

Eq. (8.15) defines the perturbation σij1 (x) of the stress tensor caused by the crack presence, and Eq. (8.19) serves in the surface  and defines the crack opening vector bi (x). Let l be the maximal size of the crack surface . Taking into account that the kernel Sij kl (x) in Eqs. (8.15) and (8.19) is a homogeneous function of the order of −3, we can rewrite Eqs. (8.15) and (8.19) in the dimensionless coordinates ςi = xi / l as follows (d = l 2 dς , dx = l 3 dς):  1 klmn l σij (ς) − l Sij kl (ς − ς  )B (ς  ) σmn (ς  )dς  −  Sij kl (ς − ς  )nk (8.20) bl (ς  )dς = 0, − ς

1 1 klmn  σij (ς) = σij1 (lς), B (ς) = Bklmn (lς),  bk (ς) = bk (lς),  1 klmn − lni Sij kl (ς − ς  )B (ς  ) σmn (ς  )dς  −  ni Sij kl (ς − ς  )nl bk (ς  )dς = lj (ς), ς ∈ ς , −

(8.22)

σij∗ (ς) −  tj (ς),  tj (ς) = ti (lς). j (ς) = nj 

(8.23)

(8.21)

ς

Here, ς is the crack surface  in the coordinates ςi and the maximal size of ς is equal to 1. For numerical solution of the system (8.20), (8.22), we introduce a cuboid W containing the crack such that outside W the stress field perturbations caused by the crack can be neglected. For discretization of Eqs. (8.20) and (8.22) by the Gaussian approximating functions, the region W and the crack surface ς should be covered by a common regular node grid with the step h. Then, the stress field  σij (ς) in W and vector  bi (ς) on ς are approximated by the Gaussian quasiinterpolants  σij (ς) =

Mn 



(n)  σij ϕ(ς − ς (n) ),  bi (ς) =

(n)  bi ϕ(ς − ς (n) ).

(8.24)

n(ς (n) ∈ς )

n=1

Here, ς = (ς1 , ς2 ), and it is assumed that the crack is in the plane ς3 = 0,     |ς| 1 |ς| 1 exp − ϕ(ς) = exp − ϕ(ς) = , . πH (πH )3/2 h2 H h2 H

(8.25)

Substituting approximations (8.24) into Eqs. (8.20) and (8.22) and satisfying the resulting equations at all the nodes, we obtain the following system of linear algebraic (n) (n) bi : equations for the coefficients  σij and  (m)

l σij

−

Mn  n=1

(m,n) 1(n) (n) lijpl B σmn − kpmn

 n(ς (n) ∈ς )

(m,n) (n) Iij k  bk = 0, ς (m) ∈ W,

(8.26)

Quasistatic crack growth in heterogeneous media

−

Mn 

(m,n) 1(n) (n) lni ij kp B σmn − kpmn

345



(m,n) (n) (m) ni Iij k  bk = lj , ς (m) ∈ ς ,

n(ς (n) ∈ς )

n=1

(8.27) 1(n) B ij kl

ij1 kl (ς (n) ), =B

(m) j

= j (ς

(m)

(8.28)

). (m,n)

(m,n)

Here, Mn is the total number of nodes and the tensors ij kl , Iij k in these equations can be interpreted as stresses induced at the mth node by the sources acting at the nth node. These tensors depend on the distance between the points ς (m) , ς (n) , i.e., (m,n)

(m,n)

ij kl = ij kl (ς (m) − ς (n) ), Iij k

= Iij k (ς (m) − ς (n) ).

(8.29)

The functions ij kl (ς) and Iij k (ς) are the following integrals:  ij kl (ς) = Sij kl (ς − ς  )ϕ(ς  )dς  ,  Sij kl (ς − ς  )nl ϕ(ς  )dP  . Iij k (ς) =

(8.30)

P

Here, the first integral is calculated over the entire 3D space and the second integral over the plane P to which the crack region ς belongs. The integral ij kl (ς) is calculated explicitly (see Eqs. (5.90)–(5.95)), while the integral Iij k (ς) is expressed in terms of five standard 1D integrals that can be tabulated (Eqs. (6.58)–(6.62)). The system (8.26)–(8.27) can be presented in the matrix form AX = F.

(8.31)

Here, the vectors X and F are defined by the equations

T

T X = X 1 , X 2 , ..., X 9Mn , F = l F 1 , F 2 , ..., F 9Mn , ⎧ (n) ⎪ n ≤ Mn, σ11 , ⎪ ⎪ ⎪ (n−Mn) ⎪ ⎪ , Mn < n ≤ 2Mn, σ22 ⎪ ⎪ ⎪ (n−2Mn) ⎪ ⎪ , 2Mn < n ≤ 3Mn, σ33 ⎪ ⎪ ⎪ ⎪ (n−3Mn) ⎪ , 3Mn < n ≤ 4Mn, σ12 ⎪ ⎨ (n−4Mn) n X = , 4Mn < n ≤ 5Mn, σ13 ⎪ ⎪ (n−5Mn) ⎪ ⎪ , 5Mn < n ≤ 6Mn, σ23 ⎪ ⎪ ⎪ (n−6Mn) ⎪ (n−6Mn) ⎪ b1 , 6Mn < n ≤ 7Mn, κ ⎪ ⎪ ⎪ ⎪ (n−7Mn) (n−7Mn) ⎪ b2 , 7Mn < n ≤ 8Mn, κ ⎪ ⎪ ⎪ ⎩ (n−8Mn) (n−8Mn) b3 , 8Mn < n ≤ 9Mn, κ

(8.32)

346

Heterogeneous Media

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨ (n−6Mn) , 1 Fn = (n−7Mn) ⎪ , 2 ⎪ ⎪ ⎪ ⎩ (n−8Mn) , 3

n ≤ 6Mn, 6Mn < n ≤ 7Mn,

(8.33)

7Mn < n ≤ 8Mn, 8Mn < n ≤ 9Mn.

Here, κ (n) = κ(ς (n) ), and κ(ς) is the characteristic function of the crack region ς . The 9Mn × 9Mn matrix A consists of 81 block-matrices a(m,n) (m, n = 1, 2, . . . , 9) of the dimensions Mn × Mn,

(1,1) (1,2) (1,9)

a

(2,1) a(2,2) · · · a(2,9)

a a ··· a

A= . (8.34)

, .. . . ..

.. . . .

a(9,1) a(9,2) · · · a(9,9)  (i,j ) (m,n) 1(n) (8.35) amn = l δij − iikl Bkljj , i, j = 1, 2, 3, m, n = 1, 2, ..., Mn, (4j )

(m,n)

1(n)

(5j )

(m,n)

1(n)

(6j )

(m,n)

1(n)

(m,n)

1(n)

(5j )

(m,n)

1(n)

(6j )

(m,n)

1(n)

(m,n)

1(n)

amn = −l12kl Bkljj , aps = −l13kl Bkljj , amn = −l23kl Bkljj , (8.36) (i4) amn = −liikl Bkl12 , aps = −liikl Bkl13 , amn = −liikl Bkl23 , (8.37)  (m,n) 1(n) (m,n) 1(n) (44) (45) = l 1 − 12kl Bkl12 , amn = −l12kl Bkl13 , amn (46) amn = −l12kl Bkl23 ,

 (m,n) 1(n) (m,n) 1(n) (54) (55) = −l13kl Bkl12 , amn = l 1 − 13kl Bkl13 , amn (m,n)

1(n)

(56) amn = −l13kl Bkl23 , (64) amn (66) amn (7j )

(8.39)

(m,n) 1(n) (65) = −l23kl Bkl12 , amn  (m,n) 1(n) = l 1 − 23kl Bkl23 , (5j )

(8j )

(8.38)

(6j )

(m,n) 1(n) = −l23kl Bkl13 ,

(8.40) (9j )

(3j )

amn = amn , amn = amn , amn = amn , j = 1, 2, . . . , 6,

(8.41)

(1j ) amn

(8.42)

(2j ) (3j ) (m,n) (m,n) = I11j , amn = I22j , amn (4j ) (5j ) (6j ) (m,n) (m,n) amn = I12j , amn = I13j , amn (i+6,j +6) (m,n) = I3ij , i, j = 1, 2, 3. amn

(m,n) = I33j , (m,n) = I23j ,

j = 1, 2, 3,

(8.43) (8.44)

Since matrix A is nonsparse and nonsymmetric, only iterative methods are efficient for solution of the system (8.31). In particular, the minimal residual method or the conjugate gradient method can be used for this purpose. For regular node grids, matrix A has Toeplitz properties and the FFT algorithms can be used for calculation of matrix-vector products in the process of iterations. Note that for numerical simulation of crack growth processes, the parameter l increases together with the crack sizes, but the node grid, in the dimensionless co-

Quasistatic crack growth in heterogeneous media

347

(m,n) ordinates ςi = xi / l, remains unchanged. As a result, the elements (m,n) ij kl and Iij k 1 (ς), the can be calculated once for all crack growth process. Only the functions B klmn region ς , and the right hand sides of Eqs. (8.20) and (8.22) should be changed. This substantially accelerates construction of the equilibrium crack configurations.

8.1.4 Quasistatic crack growth in heterogeneous layered media Let an elastic medium consist of two half-spaces with different Young moduli E, Poisson ratios ν, and fracture toughnesses KI c . The medium with the parameters E0 , ν0 , KI0c occupies the half-space x2 < a, and the medium with the parameters E, ν, KI c is in the half-space x2 > a. The initial penny-shaped crack with the center at the origin x = 0 is in the plane x3 = 0. The crack is subjected to an increasing pressure p uniformly distributed over the concentric disk of radius a = 20 m on the crack surface. For construction of the crack contour for increasing pressure, the crack is embedded in the cube W : (2L×2L×2L), where L is the maximal distance from the crack center to the crack boundary. The cube W is covered by a regular node grid with the step h such that h/L = 0.01 (Mn ≈ 8 · 106 ). The equilibrium crack contours were constructed by the iteration scheme presented in Section 8.2. Results of calculation for the host media with E0 = 15 GPa, ν0 = 0.25, and KI0c = 1 MPa·m1/2 are shown in Figs. 8.8 and 8.9. The solid lines in these figures correspond to the second medium with properties E = E0 , ν = ν0 , and KI c = 1.5KI0c , and the dashed lines correspond to the medium with E = 1.5E0 , ν = ν0 , and KI c = 1.5KI0c . The crack contours for the values of pressure p = 0.67, 1, 1.5 MPa are indicated by 1, 2, and 3 in Fig. 8.8. Crack openings b(x1 , x2 ) along the x2 -axis for both cases are shown in Fig. 8.9. For the second medium with the properties E = E0 , ν = ν0 , and KI c = 0.67KI0c (solid lines) and E = 0.67E0 , ν = ν0 , and KI c = 0.67KI0c (dashed lines), the crack contours are shown in Fig. 8.10. As before, a pressure of p = 0.67, 1, 1.5 MPa is applied to the circular disk of radius a = 20 m. Note that if the ratio of the Young moduli of the two media coincide with the ratio of their fracture toughnesses, the equilibrium crack turns out to be close to penny-shaped. Crack openings b(x1 , x2 ) along the x2 -axis for both cases are shown in Fig. 8.11.

8.2

Governing equations of the hydraulic fracture theory

In hydraulic fracture processes, crack growth is caused by the pressure of a fluid injected inside the crack at a small region on the crack surface. For simulation of hydraulic fracture crack growth, a coupled problem of solid, fluid, and fracture mechanics should be considered. Before deriving the governing equations of the hydraulic fracture theory, we consider two relevant problems of fluid mechanics: flow of viscous incompressible fluid in a slot between two rigid walls and radial flow by injection of fluid at a fixed point of the slot.

348

Heterogeneous Media

Figure 8.8 The contours of equilibrium cracks in the heterogeneous medium for various values of the pressure p applied to a disk region of radius a = 20 m (Fig. 8.1). 1 – p = 0.67 MPa, 2 – p = 1 MPa, 3 – p = 1.5 MPa. For x2 < 20 m, the medium parameters are E0 = 15 GPa, ν0 = 0.25, KI0c = 1 MPa·m1/2 , and for x2 > 20 m, E = E0 , ν = ν0 , and KI c = 1.5 MPa·m1/2 (solid lines) or E = 1.5E0 , ν = ν0 , and KI c = 1.5 MPa·m1/2 (dashed lines).

Figure 8.9 Crack openings b(0, x2 ) along the x2 -axis for the cracks in Fig. 8.8.

The linearized equations of motion of incompressible viscous fluids consist of the continuity equation [2] div υ = ∂i υi = 0

(8.45)

Quasistatic crack growth in heterogeneous media

349

Figure 8.10 The contours of the equilibrium cracks for various values of the pressure p applied to a disk of radius a = 20 m (Fig. 8.1). 1 – p = 0.67 MPa. 2 – p = 1 MPa. 3 – p = 1.5 MPa. For x2 < 20 m, the medium parameters are E0 = 15 GPa, ν0 = 0.25, KI c = 1 MPa·m1/2 , and for x2 > 20 m, E = E0 , ν = ν0 , and KI c = 0.67 MPa·m1/2 (solid lines) or E = 0.67E0 , ν = ν0 , and KI c = 0.67 MPa·m1/2 (dashed lines).

Figure 8.11 Crack openings b(0, x2 ) along the x2 -axis for the cracks in Fig. 8.10.

and the momentum (Navier–Stokes) equation dυi η 1 = − ∂i p + υi . dt ρ ρ

(8.46)

350

Heterogeneous Media

Here, υi = υi (x, t) is the velocity vector of fluid particles, ρ, η are the fluid density and viscosity, p = p(x, t) is pressure, dυi /dt is the material derivative of the velocity vector, and  is the Laplace operator.

8.2.1 Laminar flow of viscous newtonian fluid in a slot between two parallel walls We consider fluid flow between two parallel walls x3 = ±b/2 and assume that the lines of flow are parallel to the x1 -axis. If the fluid velocity does not depend the x2 -coordinate, we have υ1 = υ1 (x1 , x3 , t), υ2 = υ3 = 0.

(8.47)

It follows from Eqs. (8.45) and (8.46) that in this case, the following equations hold: ∂1 υ1 = 0, υ1 = υ1 (x3 , t), ∂2 p = ∂3 p = 0, p = p(x1 , t),

(8.48) (8.49)

η ∂ 2 υ1 1 ∂p ∂υ1 + . =− ∂t ρ ∂x1 ρ ∂x32

(8.50)

Eq. (8.50) can be rewritten in the form ∂ 2 υ1 ∂υ1 ∂p =η 2 −ρ , ∂x1 ∂t ∂x3

(8.51)

and because υ1 = υ1 (x3 , t), the function stationary flow,

∂υ1 ∂t

∂p(x1 ,t) ∂x1

depends on the t-variable only. For

= 0, and as a result, the function υ1 (x3 ) satisfies the equation

1 d 2 υ1 = ∂1 p. 2 η dx3

(8.52)

For complete sticking of fluid particles to the walls, the boundary conditions at x3 = ± b2 take the form υ1 (x3 )|x3 =± b = 0.

(8.53)

2

The solution of Eq. (8.52) with the conditions (8.53) is   1 b2 υ1 (x3 ) = x32 − ∂1 p. 2η 4

(8.54)

The total fluid flux q1 through the slot cross-section is the following integral:  q1 =

b/2 −b/2

υ1 (x3 )dx3 = −

b3 ∂1 p. 12η

(8.55)

Quasistatic crack growth in heterogeneous media

351

This equation can be extended to the case of slowly changing slot widths b(x1 , x2 , t) along the x1 - and x2 -axes and is written in the form qi (x1 , x2 , t) = −

b(x1 , x2 , t)3 ∂i p(x1 , x2 , t). 12η

(8.56)

In the literature, Eq. (8.56) is called the Poiseuille equation.

8.2.2 Radial flow of viscous newtonian fluid between two rigid walls We consider laminar flow between two parallel walls when the fluid source with injection rate Q is at the origin of the cylindrical coordinate system (r, ϕ, z). Here, the z-axis is orthogonal to the walls and the wall positions are defined by the equations z = ±b/2. Following [3] we assume that velocities in the ϕ- and z-directions are small and can be neglected in comparison with the radial velocity υr = υ. Taking into account this assumption, we can write the continuity and momentum equations (8.45) and (8.46) in the forms ∂ (rυ) = 0, ∂r dυ ∂p ∂ 2υ ρ =− +η 2 , ∂t ∂r ∂z ∂p 1 ∂p = 0, = 0. r ∂ϕ ∂z

(8.57) (8.58) (8.59)

For slow flows, the viscous term is dominant, and the inertia term in the momentum equation (8.58) can be neglected. As a result, the system (8.57)–(8.59) is simplified as follows: rυ = f (z, t), p = p(r, t),

(8.60)

∂ 2υ

∂p − η 2 = 0. ∂r ∂z

(8.61)

The boundary conditions at z = 0 and z = ± b2 take the forms

∂υ

= 0, υ|z=± b = 0. 2 ∂z z=0

(8.62)

The condition at the origin r = z = 0 is  lim 2πrq(r, t) = Q(t), q(r, t) =

r→0

b/2

−b/2

υ(r, z, t)dz.

(8.63)

352

Heterogeneous Media

Integrating Eq. (8.61) over the z-variable and taking into account the condition at z = 0, we obtain z

∂p(r, t) η ∂f (z, t) ∂p(r, t) η ∂f (z, t) − = 0 or r = = −C(t). ∂r r ∂z ∂r z ∂z

(8.64)

Here, C(t) is independent of r and z. Solution of the system (8.61) and (8.64) that satisfies conditions (8.62) takes the form   r C(t) 2 b2 p(r, t) = −p0 (t) ln + p1 (t), υ(r, z, t) = − z − , (8.65) R 2rη 4 C(t) 3 6η p0 (t) = C(t), q(r, t) = Q(t). (8.66) b , C(t) = 12rη πb3 Here, R is a characteristic size of the slot in the r-direction. This solution can be extended to the case of a slowly changing distance b = b(r, ϕ) between the walls when



∂b ∂b

,

(8.67)

∂r r∂ϕ 1. The fluid flux q(r, t) through the slot cross-section can be presented in the form similar to Eq. (8.56) q(r, t) = −

b(r, t)3 ∂p(r, t) . 12η ∂r

(8.68)

Note that the radial dependence of the pressure has the logarithm singularity at the point of injection.

8.2.3 The lubrication equation for an infinitesimal crack volume We consider a crack  with the normal ni (x) subjected to the pressure of fluid injected at a point x 0 of its surface with the rate Q(t) (Fig. 8.12). Let ω be an infinitesimal area on the crack surface with boundary contour γ that does not contain the point of injection. The crack volume Vω (t) between two sides

Figure 8.12 A crack subjected to the pressure of fluid injected at a point of the crack surface.

Quasistatic crack growth in heterogeneous media

353

of the crack surface ω is expressed in terms of the normal component bn (x, t) = bi (x, t)ni (x) of the crack opening as follows:  Vω (t) =

bn (x, t)dω.

(8.69)

ω

For incompressible fluid, balance of the crack volume Vω and fluid flux qi through the contour γ yields the equation dVω =− dt

 qi (x, t)ei (x)dγ .

(8.70)

γ

Here, ei (x) is the external normal to γ in the tangent plane to . Using Gauss’ theorem and Eq. (8.69), Eq. (8.70) can be written in the form   ω

 ∂bn (x, t) + ∂i qi (x, t) dω = 0. ∂t

(8.71)

Since ω is an arbitrary infinitesimal portion of the crack surface, the integrand function in this equation should be equal to zero, i.e., ∂bn (x, t) + ∂i qi (x, t) = 0, x = x 0 . ∂t

(8.72)

For a circular contour γε of a small radius ε with the center at the point of injection x 0 , balance of the fluid flux qi through the contour γε and the rate Q of the injected fluid takes the form  qi (x, t)ei (x)dγ . (8.73) Q(t) = lim ε→0 γε

In what follows, we assume that the fluid flux qi (x, t), crack opening bn (x, t), and pressure p(x, t) are related by the Poiseuille equation (8.56) qi (x, t) = −

b(x, t)3 ∂i p(x, t). 12η

(8.74)

After substituting this equation into Eq. (8.72), the local balance equation takes the form   bn (x, t)3 ∂bn (x, t) (8.75) = ∂i ∂i p(x, t) , x = x 0 . ∂t 12η In the literature, Eq. (8.75) is called the lubrication equation. In the case of a heterogeneous medium, the crack opening vector bi (x, t) and pressure p(x, t) on the crack surface are related by Eqs. (8.15) and (8.19). We have

354

Heterogeneous Media

 1 σij (x, t) − Sij kl (x − x  )Bklmn (x  )σmn (x  , t)dx  −  Sij kl (x − x  )nk (x  )bl (x  , t)d = 0, − (t)  1 − nj (x)Sj ikl (x − x  )Bklmn (x  )σmn (x  , t)dx  −  − nj (x)Sj ikl (x − x  )nl (x  )bk (x  , t)d = (t) = nj σj0i (x) + p(x, t)ni (x),

(8.76)

x ∈ (t).

(8.77)

Here, (t) is the crack surface at the moment t, σij0 (x) is the stress field in the heterogeneous medium without the crack subjected to the same external field sources, and σij (x, t) is the stress field perturbation caused by the crack. For a slowly changing pressure p(x, t), the fracture criterion should be satisfied at all points of the crack contour , f (KI (x), KII (x), KIII (x)) = γc (x), x ∈ .

(8.78)

Here, KI (x), KII (x), KIII (x) are the stress intensity factors at the crack edge and γc (x) is a specific characteristic of resistance of the medium to crack propagation. The function f in this equation depends of the symmetry of the medium, and for an isotropic medium, it takes the form [4] f (KI , KII , KIII ) =

1 2 (1 − ν)(KI2 + KII2 ) + KIII . 4μ

(8.79)

Condition (8.78) defines the crack shape and size at each moment of the hydraulic fracture process. Eqs. (8.75)–(8.78) compose a complete system of equations of the hydraulic fracture theory.

8.2.4 Singularities of the fluid pressure on the crack surface We consider a small circular contour γε of radius ε with the center at the point of injection x 0 = 0. From Eqs. (8.73) and (8.74) and Gauss’ theorem, we obtain the equation  Q(t) = − lim

ε→0 γε

bn (x, t)3 bn (0, t)3 ∂i p(x, t)ei (x)dγ = − lim 12η 12η ε→0

 p(t, x)dω. ωε

(8.80) Here, ωε is the area inside the contour γε and  is the 2D Laplace operator. Because the crack opening at the origin bn (0, t) is finite, the limit in the right hand side of this equation is also finite. It means that the function p(t, x) contains the 2D Dirac delta

Quasistatic crack growth in heterogeneous media

355

function concentrated at the origin p(t, x) = −

12ηQ(t) δ(x) + O(ε). bn (0, t)3

(8.81)

It follows from this equation that p(x, t) has a logarithmic asymptotic when x → 0,   |x| 6ηQ(t) ln + O(1). (8.82) p(x, t) = − 3 R∗ πbn (0, t) Here, R ∗ is a characteristic crack size. Note that singularity of pressure at x = 0 can be removed taking into account that fluid is injected inside the crack at a small but finite area on the crack surface. Let the crack be planar, and in the polar coordinate system (r, ϕ) with the origin at the point of injection, let the crack contour  be defined by the equations r = R(ϕ, t). Here, R(ϕ, t) is the distance from the origin to the crack edge in the direction defined by the polar angle ϕ. For arbitrary pressure distribution, the asymptotics of crack opening at the crack edge (r → R(ϕ, t)) has the form  1/2 r , (8.83) bn (r, ϕ, t) ∼ β(ϕ, t) 1 − R(ϕ, t) where the coefficient β(ϕ, t) depends on the crack region and details of the pressure distribution on the crack surface. It follows from this equation that the asymptotic of the left hand side of the lubrication equation (8.75) at the crack edge is  1/2 ∂bn (x, t) ∂β(ϕ, t) r + ∼ 1− ∂t ∂t R(ϕ, t) −1/2  ∂R(ϕ, t) r r . (8.84) 1 − + β(ϕ, t) R(ϕ, t) ∂t 2R(ϕ, t)2 Meanwhile, the asymptotic of the right hand side of the same equation at the crack edge is defined by the equation    1/2 b(x, t)3 β 3 (ϕ, t) r div ∇p(t, x) ∼ − 1− (xi ∂i p) + 12η 8ηrR(ϕ, t) R(ϕ, t)  3/2 r β 3 (ϕ, t) p. (8.85) 1− + 12η R(ϕ, t) Comparing the right hand sides of Eqs. (8.84) and (8.85), we can conclude that p should have asymptotics p ∼ (1 − r/R(ϕ, t))−2 at the crack edge or p(x, t) ∼ ln(1 − r/R(ϕ, t)).

(8.86)

Thus, the pressure at the edge of a moving crack has a logarithmic singularity. = 0. In this case, the principal Let the crack contour be fixed, and therefore, ∂R(ϕ,t) ∂t asymptotics of the left and right hand sides of Eqs. (8.84) and (8.85) coincide if the pressure p(x, t) and its derivative are finite at the crack edge.

356

8.3

Heterogeneous Media

Hydraulic fracture crack propagation in a homogeneous elastic medium with varying fracture toughness

We consider a homogeneous isotropic medium with a planar crack  subjected to the pressure of a fluid injected at a point x 0 ∈ . Because of symmetry, the crack remains planar in the process of hydraulic fracture, and the crack opening vector has the normal component bn (x, t) = b(x, t) only. In this case, pressure p(x, t) on the crack surface and crack opening b(x, t) are related by Eq. (8.1),  p(x, t) = T (x − x  )b(x  , t)d , x ∈ (t). (8.87) (t)

The fracture criterion yields the condition (8.10) at the crack contour , KI (x, t) = KI c (x), x ∈ (t).

(8.88)

If the fracture toughness KI c (x) is not constant, the shape of the growing crack is not circular. The system of Eqs. (8.75), (8.87), and (8.88) can be solved only numerically, and for solution of this system, the lubrication equation (8.75) should be discretized with respect to time t and presented in the form  b(x, t + t) = b(x, t) + div

 b(x, t)3 ∇p(x, t) t. 12η

(8.89)

Thus, if functions b(x, t) and p(x, t) are known at moment t, we can calculate the crack opening b(x, t + t) at moment t + t. The difficulty in constructing b(x, t + t) is that the new crack region (t + t) is not known in advance. If we assume that in the time interval t, the position of the crack contour does not change ((t + t) = (t)), Eqs. (8.87) and (8.89) determine the pressure distribution p+ (x, t + t) and the crack opening b+ (x, t + t) on the crack surface (t) at the moment t + t. For a positive fluid injection rate Q(t), the stress intensity factor KI+ (x, t + t) on the crack contour (t) will exceed KI c (x). Thus, we have to change the contour (t) in such a way that the fracture criterion (8.88) is satisfied at the points of the new contour (t + t). But in order to do this, we have to define the pressure distribution on the new crack surface (t + t) since the pressure p+ (t + t, x) changes if the crack region changes. In any numerical algorithm, the assumption about the new pressure distribution on the crack surface (t + t) should be accepted (explicitly or implicitly), but a unique natural definition of such a distribution does not exist. The second principal difficulty in numerical solution of Eqs. (8.87)–(8.89) is that the right hand side of Eq. (8.89) depends on spatial derivatives of the functions b(x, t). It is known that numerical calculation of derivatives is an ill-posed operation [5], and small errors in the definition of b(x, t) result in large errors in the calculation of ∂i b(x, t) and, as a result, in the calculation of b(x, t + t). Since Eq. (8.89) should

Quasistatic crack growth in heterogeneous media

357

be solved many times (for a number of intervals t which the total time of injection is divided into), the errors accumulate, and the reliable solution can be lost. Note that calculation of the pressure p(x, t + t) from Eq. (8.87) is also an ill-posed problem because the operator T in this equation is similar to the derivative operator (symbols of these operators are μ0 |k| and iki , respectively, and they have the same order of 1 with respect to k). The same difficulty appears if we use any conventional numerical method for constructing pressure distribution for the known crack opening b(x, t) (e.g., the finite element method or the boundary element method): Small errors in the calculation of b(x, t) result in large errors in the corresponding pressure distribution p(x, t).

8.3.1 The discrete model of hydraulic fracture crack propagation The formal time discretization (8.89) of the lubrication equation (8.75) can be interpreted as an actual physical process, shown in Fig. 8.13.

Figure 8.13 Three-stage process of crack growth during a time interval t; pi (r), Ri , and Vi are pressure,

+ crack size, and crack volume at the moment ti ; the dashed line is a new volume Vi+1 for a fixed crack size Ri after fluid injection in the time interval t (the first stage); Ri+1 is the new crack size after the crack + jump defined by the fracture criterion (the second stage); pi+1 (r), Ri+1 , Vi+1 = Vi+1 are pressure, crack size, and crack volume at the moment ti+1 = ti + t (the third stage).

Let at a time point t the crack opening b(x, t), the pressure distribution p(x, t), and the crack contour (t) be known, and let (t) be defined by the equation r = R(ϕ, t). The stress intensity factor at the crack contour satisfies the fracture criterion (8.88) KI (p(r, ϕ, t), R(ϕ, t), ϕ) = KI c (R(ϕ, t), ϕ).

(8.90)

We assume that the process of crack growth from r = R(ϕ, t) to r = R(ϕ, t + t) consists of three stages. First, in the time interval t, fluid is injected inside the crack, but the crack contour does not change. For incompressible fluid, balance of the injected fluid and increment of the crack opening (the lubrication equation (8.75)) should be satisfied, while the fracture criterion (8.90) can be violated. At the end of this stage, the crack volume increases from V (t) to V + (t + t), the pressure increases from p(x, t) to p + (x, t + t), and p + (x, t + t) can be presented in the form   r , ϕ, t + t . (8.91) p + (x, t + t) = p + R(ϕ, t)

358

Heterogeneous Media

Then, the crack jumps instantly to a new surface (t + t) with the contour (t + t) defined by the equation r = R(ϕ, t + t) (second stage). After the jump, the pressure on the crack surface changes from p + (x, t + t, x) to p(x, t + t) (third stage), and we assume that p(x, t + t) is defined by the equation     r r p , ϕ, t + t = αp+ , ϕ, t + t , (8.92) R(ϕ, t + t) R(ϕ, t + t) where the coefficient α (α < 1) is to be found from the fracture criterion (8.90). Because of an instant jump, the fluid volume inside the crack does not change, and after the pressure drop, the crack volume V (t + t) coincides with the volume V + (t + t). The crack contour r = R(ϕ, t + t) and pressure p(x, t + t) should satisfy the fracture criterion (8.90). Therefore, the new crack contour R(ϕ, t + t) and the coefficient α in Eq. (8.92) are to be found from the equations     r V (t + t) = V p , ϕ, t + t) , R(ϕ, t + t) = R(ϕ, t + t) + (8.93) = V (t + t),   r , ϕ, t + t) , R(ϕ, t + t), ϕ = R(ϕ, t + t) = KI c (R(ϕ, t + t), ϕ).

  KI p

(8.94)

The total time T of the three stages is assessed as follows: T =

V (t + t) − V (t) . Q(t)

(8.95)

Note that the lubrication equation (8.75) serves only at the first stage when the crack contour is fixed, and therefore, the fluid pressure has no singularity at the crack edge.

8.3.2 The regularized algorithm for calculation of the pressure distribution on the crack surface Following the strategy of solution of ill-posed problems [5], we approximate pressure distributions on the crack surface by a physically reasonable class of functions. For a positive fluid injection rate Q(t), the pressure should be a positive monotonic function decreasing from the point of injection x 0 to the crack edge with the logarithm asymptotics at x 0 . Let x 0 = 0, and let φm (x, t) (m = 0, 1, 2, ..., N) be a finite set of functions with the following properties. For m = 0, the function φ0 (x, t) is defined by the equation φ0 (x, t) = − ln

|x| , R ∗ (t)

(8.96)

Quasistatic crack growth in heterogeneous media

359

where R ∗ (t) is the maximal distance from the point of injection to the crack contour. For m > 0, all functions φm (x, t) are monotonic with finite derivatives on the crack surface (t). Then, we assume that at any time moment t, the pressure distribution p(x, t) can be approximated by the series p(x, t) =

N 

(8.97)

pm (t)φm (x, t)

m=0

with real positive coefficients pm (t). For such a pressure distribution, crack opening b(x, t) is presented by the series b(x, t) =

N 

(8.98)

pm (t)bm (x, t),

m=0

where functions bm (x, t) are solutions of the equations T bm (x, t) = φm (x, t) or bm (x, t) = T −1 φm (t, x), x ∈ (t), m = 0, 1, 2, ..., N. (8.99) Here, T is the operator in Eq. (8.87) of the crack problem. Note that construction of the inverse operator T −1 is a well-posed problem. For positive monotonic functions φm (x, t), the functions bm (x, t) are also positive monotonic with asymptotics √ 1 − r/R(ϕ, t) at the crack contour (t). If the set of functions bm (x, t) is constructed, we can present the pressure p + (x, t + t) and the crack opening b+ (x, t + t) at the moment t + t after the first stage of crack growth (when the crack contour does not change) in the forms p + (x, t + t) =

N 

+ pm (t + t)φm (x, t),

m=0

b+ (x, t + t) =

N 

+ pm (t + t)bm (x, t).

(8.100)

m=0

Substitution of the series (8.100) and (8.97)–(8.98) in the discretized lubrication equation (8.89) yields N 

+ pm (t + t)bm (x, t) =

m=0

=

N  m=0

pm (t)bm (x, t) +

N  m=0



 b(x, t)3 pm (t) div ∇φm (x, t) t. 12η

(8.101)

To satisfy this equation in the weak sense, we multiply both parts with the functions bk (x, t) (k = 0, 1, ..., N ) and integrate over the crack surface (t) after excluding a

360

Heterogeneous Media

small vicinity ωε ((t)\ωε ) of the injection point x 0 = 0. As a result, we obtain the + (t + t): following system of linear algebraic equations for the coefficients pm N 

+ pm (t + t)

m=0

=

N 

N 

bm (t, x)bk (x, t)d = (t)\ωε

 bm (x, t)bk (x, t)d +

pm (t) (t)\ωε

m=0

+





 div

pm (t) (t)\ωε

m=0

 b(x, t)3 ∇φm (x, t) bk (x, t)dt, k = 0, 1, 2, ...N. 12η (8.102)

Using Gauss’ theorem the integrals in the last sum are transformed as follows: 

 b(x, t)3 div ∇φm (x, t) bk (x, t)d = 12η (t)\ωε    b(x, t)3 div ∇φm (x, t)bk (x, t) d− = 12η (t)\ω  3 b(x, t) ∇φm (x, t) · ∇bk (x, t)d = − 12η (t)\ω  b(x, t)3 ei (x)∂i φm (x, t)bk (x, t)dγ − =− 12η γε  b(x, t)3 ∂i φm (x, t)∂i bk (x, t)d. − 12η (t)\ω



(8.103)

Here, ei (x) is the external normal to the boundary contour γε of the region ωε , and it is taken into account that b(x, t) = 0 when x ∈ (t). Using condition (8.80), we obtain in the limit ε → 0 −

N 

 pm (t) lim

m=0

= − lim



ε→0 γε

ε→0 γε

b(x, t)3 e(x) · ∇φm (x, t)bk (x, t)dγ = 12η

b(x, t)3 e(x) · ∇p(0, t)dγ bk (0, t) = Q(t)bk (0, t). 12η

(8.104)

It follows from Eqs. (8.103) and (8.104) that in the limit ε → 0, the system (8.102) takes the form N 

+ Mkm pm (t + t) =

m=0

k = 0, 1, 2, ...., N,

N  m=0

Mkm pm (t) + [Q(t)bk (0, t) − Fk (t)] t,

(8.105)

Quasistatic crack growth in heterogeneous media

361

 Mkm (t) =

bk (x, t)bm (x, t)d, (t)

Fk (t) =

N 



pm (t) (t)

m=0

b(x, t)3 ∂i φm (x, t)∂i bk (x, t)d. 12η

(8.106) (8.107)

Note that for the functions φm (x, t) and bm (x, t) with the mentioned properties, the integrand functions in Eq. (8.107) have weak singularities at x = 0 (m = 0) or do not have singularities (m > 0). If we introduce the vectors + X+ (t + t) = [p0+ (t + t), p1+ (t + t), ..., pN (t + t)]T ,

X(t) = [p0 (t), p1 (t), ..., pN (t)]T ,

(8.108)

Eq. (8.105) is presented in the matrix form M(t)X+ (t + t) = rhs(t), rhs(t) = M(t)X(t) + [Q(t)B(0, t) − F(t)] t.

(8.109) (8.110)

Here, the components of the matrix M(t) and vector F(t) are defined in Eqs. (8.106) and (8.107), and the components of the vector B(0, t) are bm (0, t) (m = 0, 1, 2, ..., N ). + (t + t) should be positive, we find the vector Because all the coefficients pm + X (t + t) from the equation [5] min

Ym ≥0 m=0.1,2,..N

||Y||2 =

||M(t)Y − rhs(t)|| = ||M(t)X+ (t + t) − rhs(t)||,

N 

Ym2 .

(8.111)

(8.112)

m=0

Minimum in this equation with constrains Ym ≥ 0 can be found by standard methods of dynamic programming. Natural restriction for the time interval t follows from Eqs. (8.109) and (8.110) in the form ||M(t)X+ (t + t) − rhs(t)|| ≤ δ, ||rhs(t)||

(8.113)

where 0 < δ 1 is the tolerance.

8.3.3 The numerical algorithm of solution of the hydraulic fracture problem The numerical algorithm that adopts the discrete model of crack propagation and the method of solution of the ill-posed problem of reconstruction of the pressure distribution on the crack surface consists of the following steps. 1. An initial crack contour  (1) = (t) is taken in the region of constant values of the fracture toughness KI c . The pressure distribution p (1) (x) = p(x, t) is accepted

362

Heterogeneous Media

in the form associated with Eq. (8.82) p(x, t) = −

|x| 6ηQ(0) ln + p1 . 3 R(t) πbn (0, t)

(8.114)

In this equation, the constant p1 and the crack radius R(t) are to be found from the equations V (t) = Q(0)t, KI (R(t)) = KI c , where V (t) is the crack volume  V (t) = b(x, t)d.

(8.115)

(8.116)

(t)

2. Let at the moment t = ti the contour  (i) and the crack surface (i) = (ti ) be known, as well as the series (8.97) and (8.98) for the pressure distribution p(i) (x) = p(x, ti ) and crack opening b(i) (x) = b(x, ti ). It is assumed that approximating functions φm (x, t) and bm (x, t) are also known at the time moment ti . Thus, the right hand side of the discretized lubrication equation (8.89) can be calculated. For a fixed + (t + t) in the series (8.100) for the pressure crack contour  (i) , the coefficients pm i + distribution p (ti + t, x) are to be obtained from the system (8.105). 3. From Eqs. (8.93), (8.94), the new crack contour  (i+1) and new pressure distribution p (i+1) (x) on the crack surface are to be found. For the contour  (i+1) , the condition Ri+1 (ϕ) ≥ Ri (ϕ) should be satisfied. If Ri+1 (ϕ) < Ri (ϕ) for some polar angles ϕ, the equality Ri+1 (ϕ) = Ri (ϕ) is accepted. If Ri+1 (ϕ) < Ri (ϕ) for all angles ϕ, the crack contour does not change  (i+1) =  (i) , the coefficient α in Eq. (8.92) is equal to 1, and the pressure on the crack surface p(ti + t, x) coincides with p+ (ti + t, x). 4. For the new crack contour  (i+1) , we have to define a new set of approximating functions for the pressure distribution φm (x, ti + t) and construct the corresponding functions bm (x, ti + t) in Eq. (8.99) for approximation of the crack opening b(x, ti + t). Then, we can go to the next time interval. In the next section, this algorithm is considered in detail for hydraulic fracture crack growth in a homogeneous isotropic elastic medium.

8.3.4 Hydraulic fracture crack propagation in a homogeneous isotropic elastic medium Let the medium be homogeneous and isotropic, KI c = const, and let the initial crack be circular. It follows from the symmetry of the hydraulic fracture process that the crack remains circular (penny-shaped) with increasing radius R(t). The crack opening b(r, t) and pressure distribution p(r, t) are functions of the distance r from the point of injection (r = 0). In this case, the lubrication equation (8.75) takes the form   ∂ b(ρ, t)3 ∂p(ρ, t) 1 ∂b(ρ, t) = ρ , ρ = 0, (8.117) ∂t 12η ∂ρ ρR(t)2 ∂ρ

Quasistatic crack growth in heterogeneous media

363

where the dimensionless radial coordinate ρ = r/R(t) is introduced. For radial symmetry, the pressure distribution p(ρ, t) and crack opening b(ρ, t) are related by the equations [6], R(t) b(ρ, t) = 4πμ



1



ρ



dz z2

− ρ2

 0

z

p(ς, t)ς  dς = z2 − ς 2

1

R(t) G(ρ, ς)p(ς, t)dς, 4πμ 0 μ μ = , 2(1 − ν) =

(8.118) (8.119)

where μ and ν are the shear modulus and Poisson ratio of the medium and the kernel G(ρ, ς) is an integrable function of variables (ρ, ς) with a weak (logarithmic) singularity at ρ = ς [7]. Therefore, solution of Eq. (8.118) for p(ρ, t) with a known left hand side b(ρ, t) is an ill-posed problem [5]. The stress intensity factor KI for the fracture mode I at the crack edge is the following integral [6]: √ KI (p, R) =

2R π



1 0

p(ρ)ρ  dρ, 1 − ρ2

(8.120)

and the fracture criterion (8.88) takes the form KI (p, R) = KI c .

(8.121)

For numerical solution, the pressure distribution p(r, t) is approximated by the series 

    N r r pm (t)φm + , p(r, t) = −p0 (t) ln R(t) R(t)

(8.122)

m=1

where pm (t) are positive coefficients and φm (ρ) are monotonically decreasing functions. In the calculations, the following 11 functions φm (ρ) are taken for approximation of the pressure distribution: φ0 (ρ) = − ln(ρ), φ3 (ρ) = 1 − ρ 4 , φ6 (ρ) = (1 − ρ 2 )4 , φ9 (ρ) = (1 − ρ 2 )40 ,

φ1 (ρ) = 1, φ2 (ρ) = 1 − ρ 10 , 2 φ4 (ρ) = 1 − ρ , φ5 (ρ) = (1 − ρ 2 )2 , φ7 (ρ) = (1 − ρ 2 )8 , φ8 (ρ) = (1 − ρ 2 )15 , 2 200 φ10 (ρ) = (1 − ρ ) .

(8.123)

For m > 0, these functions are polynomials with zero derivatives at ρ = 0 such that 

1 0

φm (ρ)dρ ≈ 1 − l(m − 1), l = 0.1, m = 1, 2, ..., 10.

(8.124)

364

Heterogeneous Media

Taking the step l = 0.05, one can construct 20 functions of this type. For the pressure distribution (8.122)–(8.123) the crack opening b(ρ, t) is presented in the form b(ρ, t) =

 1 N R(t)  p (t)b (ρ), b (ρ) = G(ρ, ς)φm (ς)dς. m m m πμ 0

(8.125)

m=0

Here, the function b0 (ρ) has the form  b0 (ρ) = (2 − ln 2) 1 − ρ 2 − ρ arccos ρ,

(8.126)

and the functions bm (ρ) for m = 1, 2, ...10 can be found in explicit analytical forms. The graphs of functions φm (ρ) in Eq. (8.123) and corresponding functions bm (ρ) in Eq. (8.125) are shown in Fig. 8.14. Explicit equations for the first eight functions bm (ρ) are presented in Appendix 8.B.

Figure 8.14 Approximating functions φm (ρ) for pressure p(ρ) and the corresponding functions bm (ρ) for the crack opening b(ρ) of a penny-shaped crack in the process of hydraulic fracture.

The stress intensity factor KI (t) at the crack edge follows from Eq. (8.120) in the form √  1 N 2R(t)  φm (ρ)ρ  pm (t)km , km = dρ. KI (t) = π 0 1 − ρ2 m=0

(8.127)

The coefficients km for the function φm (ρ) in Eq. (8.123) are presented in Appendix 8.B. We consider crack growth in a time interval t. According to the discrete model of crack propagation, the lubrication equation (8.75) serves at the first stage of crack growth when the crack radius does not change. The discretized form of this equation is   R(t)−2 ∂ b(ρ, t)3 ∂p(ρ, t) ρ t. (8.128) b+ (ρ, t + t) = b(ρ, t) + ρ ∂ρ 12η ∂ρ

Quasistatic crack growth in heterogeneous media

365

Approximation of the functions p(ρ, t), p + (ρ, t + t), b(ρ, t), b+ (ρ, t + t) by the series similar to (8.122) and (8.125) takes the forms p(ρ, t) =

N 

p + (ρ, t + t) =

pm (t)φm (ρ) ,

m=0

N 

p+ m (t + t)φm (ρ) ,

m=0

(8.129) b(ρ, t) =

R(t) πμ

N 

pm (t)bm (ρ),

m=0

N R(t)  + b (ρ, t + t) = pm (t + t)bm (ρ). πμ +

(8.130)

m=0

After substituting these series into Eq. (8.128), multiplying the resulting equation with bk (ρ), and integrating over the crack surface, we obtain the following system for the coefficients p+ m (t + t) that is similar to (8.105):  1 N 2R(t)3  + p (t + t) bm (ρ)bk (ρ)ρdρ = m μ 0 m=0

 1 N 2R(t)3  p (t) bm (ρ)bk (ρ)ρdρ + m μ 0 m=0    1 N  b(ρ, t)3 dφm dbk + Q(t)bk (0) − 2π pm (t) ρdρ t, k = 0, 1, ...N. 12η dρ dρ 0 =

m=0

(8.131) If we introduce the vector of the coefficients p+ m (t + t), + + T X+ (t + t) = [p+ 0 (t + t), p1 (t + t), ..., pN (t + t)] ,

(8.132)

the matrix M(t) with the elements Mmk (t), 2R(t)3 Mmk (t) = μ



1

(8.133)

bm (ρ)bk (ρ)ρdρ, 0

and the vector rhs(t) with the components rhsk (t) =

N 

Mkm (t)pm (t) +

m=0



+ Q(t)bk (0) − 2π

N  m=0



1

pm (t) 0

 b(ρ, t)3 dφm dbk ρdρ t, (8.134) 12η dρ dρ

366

Heterogeneous Media

the system (8.131) takes the form M(t)X+ (t + t) = rhs(t).

(8.135)

The vector X+ (t + t) with positive components should satisfy this equation in the sense of Eq. (8.111). The crack volume V + (t + t) at the first stage of crack growth in the interval t has the form  1 N R(t)3  + pm (t + t)vm , vm = 2π bm (ρ)ρdρ, V (t + t) = πμ 0 +

(8.136)

m=0

and after the crack jump (the second stage), the pressure and the crack volume in Eqs. (8.129) and (8.130) are N 

p(t + t, r) = α

p+ m (t

 + t)φm

m=0

r R(t + t

 ,

R(t + t)3  + α pm (t + t)vm . πμ

(8.137)

N

V (t + t) =

(8.138)

m=0

Equating the volumes V + (t + t) in Eq. (8.136) and V (t + t) yields α=

R(t)3 , R(t + t)3

(8.139)

and from the fracture criterion (8.121), we obtain √ 2/5 N 2 R(t)3  + pm (t + t)km . R(t + t) = π KI c

(8.140)

m=0

Here, Eq. (8.127) for KI (p, R) is taken into account. If Eq. (8.140) results in R(t + t) < R(t), we accept that R(t + t) = R(t) and α = 1. Thus, after the crack jump, the pressure distribution p(r, t + t) and crack opening b(r, t + t) are defined by the equations p(r, t + t) =

N 

 pm (t + t)φm

m=0

 r , R(t + t)

(8.141)

  N r R(t + t)  pm (t + t)bm , b(r, t + t) = πμ R(t + t)

(8.142)

pm (t + t) = αp+ m (t + t), m = 0, 1, 2, ..., N.

(8.143)

m=0

For this pressure distribution and crack radius, the fracture criterion (8.121) is satisfied. Then, we can go to the next time interval.

Quasistatic crack growth in heterogeneous media

367

Figure 8.15 Dependence of the crack radius R on time t and viscosity η of the fluid injected in the crack at the rate Q = 0.1 m3 /sec.

Figure 8.16 Pressure distribution p(r, t) on the crack surface for viscosity η = 0.001 Pa·sec of the injected fluid at various time points t; the injection rate is Q = 0.1 m3 /sec.

Examples of evolution of the crack radius, opening, and pressure distribution on the crack surface in the process of hydraulic fracture are shown in Figs. 8.15–8.25 for various values of the fluid viscosity η. The medium with the shear modulus μ = 6.25 GPa, the Poisson ratio ν = 0.2, and fracture toughness KI c = 1 MPa·m1/2 is considered. The fracture process is studied for the constant injection rate Q = 0.1 m3 / sec, and the initial crack radius is R0 = 2 m. Dependencies of the crack radius R on time t and fluid viscosity η are shown in Fig. 8.15. The pressure distribution p(r, t) and the crack opening b(r, t) for various values of fluid viscosity are shown in Figs. 8.16–8.23. Dependencies of the pressure and crack opening on time and fluid viscosity in the distance r = 1 m from the point of injection are shown in Figs. 8.24 and 8.25. In calculations, the time intervals t are taken to satisfy the condition (8.113) with δ = 0.001. These intervals should be small enough to avoid visual kinks and oscillations in time dependencies of the crack radius R(t), pressure distribution p(r, t), and crack opening b(r, t). In order to satisfy these requirements, t should decrease together with the fluid viscosity η. As a result,

368

Heterogeneous Media

Figure 8.17 Crack opening b(r, t) for a viscosity of η = 0.001 Pa·sec of the injected fluid at various time points; the injection rate is Q = 0.1 m3 /sec.

Figure 8.18 The same as in Fig. 8.16 for a fluid viscosity of η = 0.1 Pa·sec.

Figure 8.19 The same as in Fig. 8.17 for a fluid viscosity of η = 0.1 Pa·sec.

Quasistatic crack growth in heterogeneous media

Figure 8.20 The same as in Fig. 8.16 for a fluid viscosity of η = 1 Pa·sec.

Figure 8.21 The same as in Fig. 8.17 for a fluid viscosity of η = 1 Pa·sec.

Figure 8.22 The same as in Fig. 8.16 for a fluid viscosity of η = 10 Pa·sec.

369

370

Heterogeneous Media

Figure 8.23 The same as in Fig. 8.17 for a fluid viscosity of η = 10 Pa·sec.

Figure 8.24 Time dependencies of the pressure at a distance of r = 1 m from the point of injection for various viscosities of the injected fluid.

Figure 8.25 Time dependencies of the crack opening b(r, t) at a distance of r = 1 m from the point of injection for various viscosities of the injected fluid.

Quasistatic crack growth in heterogeneous media

371

for a fixed total injection time T that was taken as T = 2000 sec, the computational time grows when the fluid viscosity η decreases. The number of time intervals was about 1000–3000, depending on the fluid viscosity, and t started from 0.001 sec and increased proportionally to the crack radius. Increasing of the number of approximating functions φm (ρ) in series (8.122) does not change practically the graphs in Figs. 8.15–8.25 but increases the computational time.

8.4

The three-parameter model of hydraulic fracture crack growth in heterogeneous elastic media

Analysis of crack growth in homogeneous media reveals a specific feature of the hydraulic fracture process. Increasing the fluid viscosity η results in an increase in the fluid pressure near the point of injection and a faster decrease in the pressure from this point to the crack edge (see Figs. 8.16–8.23). For small fluid viscosity (η < 0.01 Pa·sec), pressure distribution turns out to be homogeneous on the crack surface except a vicinity of the point of injection (note that the viscosity of sea water widely used for hydraulic fracture in practice is about 0.001 Pa·sec). This means that in the series (8.122) for the pressure, the logarithm and constant terms dominate. Therefore, for small viscosity, only two terms in the series (8.122) can be kept, i.e.,   r p(r, t) = −p0 (t) ln (8.144) + p1 (t), R(t) and the solution depends on three parameters: the coefficients p0 (t), p1 (t) and the crack radius R(t). A similar pressure distribution is obtained in Section 8.2 by considering radial flow from an injected point between two rigid walls (Eq. (8.65)). These three parameters can be found from the fracture criterion (8.121), the total balance of the injected fluid and the crack volume V (t), and the condition (8.82) at the point of fluid injection:  t 6ηQ(t) KI (p0 , p1 , R) = KI c , V (t) = Q(τ )dτ, p0 (t) = . (8.145) πb(0, t)3 0 For a penny-shaped crack with pressure distribution in Eq. (8.144), the total crack volume V (t), the crack opening at the point of injection b(0, t), and the SIF KI (t) are expressed in terms of the coefficients p0 (t) and p1 (t) as follows:  2R(t)3  (4 − 3 ln 2)p0 (t) + 3p1 (t) ,  9μ   1 R(t) (2 − ln 2)p0 (t) + p1 (t) , b(0, t) = πμ √  2R(t)  (1 − ln 2)p0 (t) + p1 (t) . KI (t) = π V (t) =

(8.146) (8.147) (8.148)

372

Heterogeneous Media

The system of Eqs. (8.145)–(8.148) for p0 (t), p1 (t), and R(t) can be solved numerically for discrete time moments t1 , t2 , ..., tM . In the framework of the three-parameter model, the asymptotic solution for vanishing viscosity (η → 0) can be easily found. It follows from Eqs. (8.145)–(8.148) that in this limit, we have   πKI c 1 12μ V (t) 2/5 p0 (t) = 0, p1 (t) = √ . , R(t) = 2 πKI c 2R(t)

(8.149)

For p0 (t) = 0, the pressure is constant on the crack surface (Sneddon’s assumption). It was pointed out in [8] that Sneddon’s solution (8.149) corresponds to the zero limit of the fluid viscosity η. In this case, the pressure of the fluid injected in the crack rapidly reaches its hydrostatic limit, which is constant according to Pascal’s law. The pressure distribution (8.144) can be considered as a correction of Sneddon’s assumption for small but finite fluid viscosity and low rates of fluid injection. Coefficients p0 (t) and p1 (t) are expressed in terms of the crack volume V (t) and radius R(t) as follows: p0 (t) =

9μ V (t) 3πKI c 9μ V (t)(ln 2 − 1) πKI c (3 ln 2 − 4) − (t) = + , p , √ √ 1 2R(t)3 2R(t)3 2R(t) 2R(t) (8.150)

the crack opening b(r, t) takes the form      1 r r (t)b (t)b + p , p 0 0 1 1 πμ R(t) R(t)   b0 (ρ) = (2 − ln 2) 1 − ρ 2 − ρ arccos ρ, b1 (ρ) = 1 − ρ 2 .

b(r, t) =

(8.151) (8.152)

In Figs. 8.26 and 8.27, time dependencies of crack radius R(t) for the threeparameter model and the discrete model of crack growth are shown for a fluid viscosity of η = 0.01 Pa·sec, η = 0.001 Pa·sec, and various values of fracture toughness KI c (μ = 6.25 GPa, ν = 0.2, Q = 0.1 m3 / sec). In these figures, solutions of the system (8.145)–(8.148) are shown as dashed lines, and the results of the discrete model are shown as solid lines. In Fig. 8.28, the influence of the shear modulus μ of the medium on the time dependence of the crack radius in the process of crack growth is shown (ν = 0.2, KI c = 1 MPa·m1/2 , η = 0.001 Pa·sec). It is seen from these figures that the three-parameter model corresponds better to the discrete model in the case of higher values of the fracture toughness and lower values of fluid viscosity. The computational program for calculation of the crack radius, opening, and pressure distribution on the crack surface is presented in Appendix 8.C. In the program, the three-parameter model of crack growth is adopted.

Quasistatic crack growth in heterogeneous media

373

Figure 8.26 Influence of fracture toughness KI c of the medium on the time dependence of the crack

radius R(t) (μ = 6.25 GPa, ν = 0.2, η = 0.001 Pa·sec; the fluid injection rate is Q = 0.1 m3 /sec). Solid lines correspond to the discrete model and dashed lines to the three-parameter model of crack growth.

Figure 8.27 The same as in Fig. 8.26 for a fluid viscosity of η = 0.01 Pa·sec.

8.4.1 Hydraulic fracture crack propagation in a layered heterogeneous medium We consider a layered isotropic medium with constant elastic moduli and fracture toughness in the layers. Suppose that at initial moment t = 0, the crack is a circle of radius R0 subjected to fluid injected at its center x 0 . For description of the crack contour evolution, the three-parameter model of pressure distribution on the crack

374

Heterogeneous Media

Figure 8.28 Influence of the shear modulus μ of the medium on the time dependencies of the crack radius R(t) in the process of crack growth (ν = 0.2, KI c = 1 MPa·m1/2 , η = 0.001 Pa·sec); the fluid injection rate is Q = 0.1 m3 /sec). Solid lines correspond to the discrete model and dashed lines to the three-parameter model.

surface is used. Thus, the pressure is approximated by the equation   r p(x, t) = −p0 (t) ln + p1 (t), x ∈ , R∗ (t)

(8.153)

where r = |x − x 0 | is the distance from a point x ∈  to the injection point x 0 and R∗ (t) is the maximal distance of the points of the crack contour from x 0 . The condition at the point of injection x 0 yields Eq. (8.145) for the coefficient p0 (t), p0 (t) =

6ηQ(t) . πb(x 0 , t)3

(8.154)

For constructing evolution of the crack contour (t), we introduce a homogeneous reference medium with elastic constants E∗ , ν∗ and fracture toughness KI∗c and solve the problem for a given injection rate Q(t) at discrete time moments t1 , t2 , ..., tM . These solutions determine crack contours  0 (tk ) that are circles of radius R0 (tk ) and the coefficients p0 (tk ) and p1 (tk ) at the considered time moments. The radius R0 (tk ) and coefficients p0 (tk ) and p1 (tk ) are initial data for the iterative construction of the actual crack contour at the moments tk . To satisfy the fracture criterion m (8.90), N discrete points with polar coordinates {rm , ϕm } (rm = R0 (ϕm ), ϕm = 2π N , 0 m = 1, 2, ..., N) on the crack contour  (tk ) are taken. Then, the distances R(ϕm ) are changed iteratively according to Eq. (8.12). Subsequently, keeping p1 (tk ) unchanged, we correct the coefficient p0 (tk ) in order to satisfy Eq. (8.154) at the point of injection. It requires an additional iteration procedure with respect to the coefficient p0 (tk ). In these iterations, p0 (tk ) is changed to (1 − δ)p0 (tk ) if p0 (tk ) > 6ηQ/(πb(x 0 , tk )3 ) and to (1 + δ)p0 (tk ) if p0 (t1 ) < 6ηQ/(πb(x 0 , tk )3 ). Here, δ is a small parameter that should be taken for optimal convergence of the iteration process; in the calculations, δ = 0.1. Then, the crack opening b(x, tk ) and crack volume V (tk ) for the pressure

Quasistatic crack growth in heterogeneous media

375

coefficients p0 (tk ), p1 (tk ) can be calculated. The corrected time point Tk that corresponds to the pressure coefficients p0 (tk ), p1 (tk ) and crack volume V (tk ) is calculated from the equation 

Tk

Q(τ )dτ = V (tk ).

(8.155)

0

We consider first a medium that consists of two half-spaces with the boundary at x2 = 20 m. For x2 < 20 m, the Young modulus and Poisson ratio of the medium are E0 = 15 GPa, ν0 = 0.3, and the fracture toughness is KI0c = 1 MPa·m1/2 , and for x2 > 20 m, they are E = 15 GPa, ν = 0.3, KI c = 1.5 MPa·m1/2 or E = 30 GPa, ν = 0.3, KI c = 1.5 MPa·m1/2 . The initial crack is a circle with radius R = 1 m, and the fluid with viscosity η = 0.01 Pa·sec is injected at the crack center x 0 = 0 with the rate Q = 0.2 m3 / sec. The crack contours at various time points are shown in Fig. 8.29; the crack opening along the x2 -axis is shown in Fig. 8.30. In these figures, solid lines correspond to E = 15 GPa, KI c = 1.5 MPa·m1/2 and dashed lines to E = 30 GPa, KI c = 1.5 MPa·m1/2 . It can be noted that for the medium with constant Young modulus (E0 = E = 15 GPa) and varying fracture toughness, the crack shape is substantially different from a circle when the crack intersects the boundary between the two media. For the two media with the parameters E00 = 15 GPa, KI0c = 1 MPa·m1/2 and E = 30 GPa, KI c = 1.5 MPa·m1/2 , the crack shape is closer to a circle in the process of growing.

Figure 8.29 Crack contours at various time moments in a heterogeneous medium with the parameters E0 = 15 GPa, ν0 = 0.3, KI0c = 1 MPa·m1/2 for x2 < 20 m and E1 = E0 , ν1 = ν0 , KI1c = 1.5KI0c for x2 > 20 m (solid lines); and E0 = 15 GPa, ν0 = 0.3, KI0c = 1 MPa·m1/2 for x2 < 20 m and E1 = 30 GPa, ν = ν0 , KI c = 1.5KI0c for x2 > 20 m (dashed lines), with Q = 0.2 m3 /sec.

376

Heterogeneous Media

Figure 8.30 Crack opening b(0, x2 ) along the x2 -axis at various time moments for the cracks whose contours are shown in Fig. 8.29.

In the calculations, the external stresses (e.g., geologic stresses in actual rocks) and fluid filtration in the medium from the crack surface are neglected for simplicity. Accounting for these factors in the framework of the discrete or three-parameter models is straightforward.

8.5

Notes

Simulation of crack propagation in heterogeneous materials was considered in [9], [10]. In most works, the numerical algorithms are based on the finite element method (FEM) used for construction of the crack contour as well as for calculation of SIFs in the process of crack growth. For the calculation of SIFs, the technique of J-integrals is usually used. Accuracy of this technique was studied in [11], [12]. It was shown that the standard FEM does not allow constructing reliable SIF values. The accuracy can be substantially increased (up to 1%) if the so-called extended FEM (XFEM) is used (see, e.g., [13], [14]). In the XFEM, special finite elements with actual asymptotics of the stress field at the crack edge are introduced. This technique requires reconstruction of the finite element mesh at each step of the crack growth as well as in the process of adjustment of the SIFs to the actual distribution of the fracture toughness in the medium. Another way of solution of the crack problem is based on the so-called cohesive segments method proposed in [15], [16]. In these works, a crack was simulated by a set of overlapping cohesive segments that were inserted into finite elements as discontinuities of the displacement field. This method was used for simulation of crack growth processes in 2D cases only. The BEM is well suited for modeling crack propagation in homogeneous media (see, e.g., [17], [18]). However, there are few examples of application of the method to heterogeneous media. For calculation of SIFs in the framework of the BEM, the

Quasistatic crack growth in heterogeneous media

377

J-integral technique can be used (see, e.g., [19]). In [20], [21] asymptotic behavior of crack opening near the crack edge is exploited for SIF calculations. Due to its importance in gas and oil extraction technology, the process of hydraulic fracture has been the object of intense theoretical and experimental studies since the 1960s. The number of publications dedicated to analytical and numerical solutions of this problem is huge. There are several books and journal surveys where a substantial portion of these publications is indicated. Most publications before the 21st century can be found in [22], [23]. Surveys of more recent publications are presented in [24], [25]. Mathematical equations of the process of hydraulic fracture crack growth were derived at the end of the last century and presented in the works of many authors (see, e.g., [7]). Asymptotics of the fluid pressure at the boundary of a moving crack were indicated in [26]. The content of this chapter is based on the publications [27], [28], [29], [30], [31].

Appendix 8.A The computational program for construction of equilibrium crack contours in a homogeneous elastic medium with varying fracture toughness In this appendix, the computational program for construction of the contour of an equilibrium crack in the medium with varying fracture toughness is presented. A planar crack is subjected to the pressure applied to the disk of radius a = 0.3 m on the crack surface (Fig. 8.1); the Young modulus and Poisson ratio of the medium are E0 = 1 Pa, ν0 = 0.3, the fracture toughness is KI c = 0.412 Pa·m1/2 if x2 < 0.5 m and KI c = 0.824 Pa·m1/2 if x2 > 0.5 m. The contour of the equilibrium crack is constructed for pressure values of p = 6, 20, 30, ..., 90 Pa. (*Initial data*) (*Young modulus (Pa) and Poisson ratio of the medium*) E0=1; nu0=0.3; mu0=E0/(2(1+nu0)); kp0=1/(2*(1-nu0)); (*Fracture toughness (Pa·m1/2 )*) KIC1=0.412; KIC2=0.824; (*Critical asymptotics of the crack opening at the crack edge*) kic1=4*(1-nu0)/mu0*KIC1; kic2=4*(1-nu0)/mu0*KIC2; kic[y1_, y2_] := kic1/; y20.5 (*Node grid generation*) M1=401;M2=401; h=4/(M1-1);

378

Heterogeneous Media

N1=2*M1; N2=2*M2; Mn:=M1*M2; l1=-2; l2=-2; (*Cartesian coordinates of the nodes*) x1=Flatten[Table[l1+h*(i-1),{j,M2},{i,M1}]]; x2=Flatten[Table[l2+h*(j-1),{j,M2},{i,M1}]]; (*Connection between one- and two-index numeration*) M[i_,j_]:=i+M1*(j-1) (*Transition from two-index to one-index numeration and vice versa*) To1Ind[A_] := Flatten[Transpose[A]]; To2Ind[A_] := Transpose[Partition[A, M1]]; (*The initial crack definition (m)*) As[1]=0.5;as[1]=1; RR0[f_] := As[1] (*Polar angle and radius in the crack plane*) fi[x1_, x2_] := Arg[x1 + I*x2]/; x2 >= 0 fi[x1_, x2_] := 2*Pi + Arg[x1 + I*x2]/; x2 < 0 r[x1_, x2_] := Abs[x1 + I*x2] (*Nodes on the crack contour*) Nf := 400; df := 2*Pi/Nf; lr[1]:= Table[N[RR0[(i - 1)*df]], {i, Nf + 1}]; ir[1]:= Interpolation[Table[{(i - 1)*df, lr[1][[i]]}, {i, Nf + 1}]] ir[k_]:= Interpolation[Table[{(i - 1)*df, lr[k][[i]]}, {i, Nf + 1}]]/;k>1 (*Crack contour in the list-form, LR[[i]] is the distance from the origin to the ith node of the contour*) IR1 :=Interpolation[Table[{(i-1)*df, LR[[i]]},{i,Nf+1}]] (*Crack area indicator*) Ind1[k_] := Module[{AA, BB, id}, BB = Table[0,{Mn}]; Do[AA = r[x1[[i]], x2[[i]]]/IR[fi[x1[[i]], x2[[i]]]]; id = If[AA = 1 && i Nf+1 ftt[k_,i_] := FT[k][[i + Nf]]/; i < 1 FTm1[k_]:=Table[Sum[ftt[k,i + j], {j, -3, 3}]/7, {i, 1, Nf + 1}]; LRR[i_] := LR[[i]] /; i >= 1 && i Nf+1 LRR[i_] := LR[[i + Nf]] /; i < 1 (*Iterative construction of the equilibrium crack contour*) CRSs[k_] :=While[ddl > 0.05,LDR = LDR1; Ind[k] = Ind1[k]; ITs[k]; FB = FB1; BB = BB1; IB = IB1;bb[k]=IB; Print[Plot[IB[0, y2], {y2, 0, 2}, PlotRange -> All]]; Do[Dat[k, j] = Dat1[k, j], {j, Nf + 1}]; AP[k, t_] = Table[Ap1[k, j, t], {j, Nf + 1}]; Print[Table[Plot[{AN[k,m,t]/Sqrt[-t],AP[k, t][[m]]},{t,-0.2,0}],{m,{1,100}}]]; LSIF = AP[k, 0]/Sqrt[As[k]]; Print[{Max[LSIF], Min[LSIF]}]; FT[k] = FT1[k]; FTm[k] = FTm1[k]; Print[{Max[FTm[k]], Min[FTm[k]]}]; Which[Max[LSIF] > Max[FTm[k]] && Min[LSIF] < Min[FTm[k]], LR = LR-0.02*(LSIF - FTm[k]), Max[LSIF] < Max[FTm[k]] && Min[LSIF] > Min[FTm[k]], LR = LR+0.02*(LSIF - FTm[k]), Max[LSIF] > Max[FTm[k]] && Min[LSIF] > Min[FTm[k]], LR = 1.05*LR, Max[LSIF] < Max[FTm[k]] && Min[LSIF] < Min[FTm[k]], LR = 0.95*LR]; LR = Table[Sum[LRR[i + j],{j, -3, 3}]/7, {i, 1, Nf + 1}]; Print[{Max[LR], Min[LR]}]; as[k] = Max[LR];IR = IR1;ir[k]=IR; Print[ParametricPlot[{{RR0[f]*Cos[f],RR0[f]*Sin[f]}, {IR[f]*Cos[f],IR[f]*Sin[f]}},{f,0,2*Pi}]]; ddl = Tr[Abs[FTm[k]-LSIF]]/Tr[FTm[k]];Print[ddl]]; (*Crack contour and opening for the given pressure*)

382

Heterogeneous Media

CrGr = Do[LR = lr[k];IR = ir[k]; Print[{k, p[k], As[k]}]; as[k] = 1;ddl=1;CRSs[k]; As[k+1] = as[k]*As[k]; lr[k+1]=LR/as[k], {k, Dimensions[TP][[1]]}] End

Appendix 8.B

Approximating functions for simulation of hydraulic fracture crack propagation in homogeneous elastic media

The approximating functions in Eq. (8.123) for the pressure on the surface of a pennyshaped crack in the hydraulic fracture process have the forms φ0 (ρ) = − ln(ρ), φ1 (ρ) = 1, φ2 (ρ) = 1 − ρ 10 , φ3 (ρ) = 1 − ρ 4 , φ4 (ρ) = 1 − ρ 2 , φ5 (ρ) = (1 − ρ 2 )2 ,

(8.B.1)

φ6 (ρ) = (1 − ρ ) , φ7 (ρ) = (1 − ρ ) , φ8 (ρ) = (1 − ρ ) , 2 4

2 8

2 15

φ9 (ρ) = (1 − ρ 2 )40 , φ10 (ρ) = (1 − ρ 2 )200 . The corresponding functions for approximation of the crack opening in Eq. (8.125) are  z  1 dz φm (ς)ςdς   bm (ρ) = . (8.B.2) 2 2 ρ z2 − ς 2 z −ρ 0 The first eight functions bm (ρ) have the forms   2 b0 (ρ) = (2 − ln 2) 1 − ρ − ρ arccos ρ, b1 (ρ) = 1 − ρ 2 ,  b2 (ρ) = 1 − ρ 2 (0.9664 − 0.0373ρ 2 − 0.0426ρ 4 − 0.0512ρ 6 − − 0.0682ρ 8 − 0.1365ρ 10 ),  b3 (ρ) = 1 − ρ 2 (0.8933 − 0.1422ρ 2 − 0.2844ρ 4 ), b4 (ρ)  1 1 − ρ 2 (7 − 4ρ 2 ), = 9  b5 (ρ) = 1 − ρ 2 (0.66222 − 0.7467ρ 2 + 0.2844ρ 4 ), (8.B.3)  b6 (ρ) = 1 − ρ 2 (0.5350 − 1.1863ρ 2 + 1.3506ρ 4 − 0.7534ρ 6 + 0.1651ρ 8 ),  b7 (ρ) = 1 − ρ 2 (0.4139 − 1.8023ρ 2 + 4.7704ρ 4 − 7.9715ρ 6 + + 8.7296ρ 8 − 6.2899ρ 10 + 2.8842ρ 12 − 0.7654ρ 14 + 0.08972ρ 16 ).

Quasistatic crack growth in heterogeneous media

383

For the functions in Eq. (8.B.1), the integrals associated with the stress intensity factors  1 φm (ρ)ρ  dρ (8.B.4) km = 0 1 − ρ2 have the following values: k0 1 − ln 2

k1

k2

k3

k4

k5

k6

k7

k8

k9

k10

1

437 693

7 15

1 3

1 5

1 7

1 9

1 17

1 31

1 81

Appendix 8.C

.

(8.B.5)

Computer simulation of hydraulic fracture crack propagation by the three-parameter model

In this appendix, the computational program for the calculation of the hydraulic fracture crack radius R, crack opening, and pressure distribution on the crack surface in a homogeneous isotropic elastic medium is presented. The three-parameter model of the crack growth is adopted. (*Initial data*) (*Elastic constants of the medium (Pa)*) E0 = 15*10 ˆ 9; nu = 0.2; mu = E0/(2*(1 + nu)); (*Fluid viscosity (Pa·sec)*) et = 0.01; (*Fracture toughness of the medium (Pa·m1/2 )*) KIC = 1*10 ˆ 6; (*Injection rate (m3 /sec)*) Q = 0.1; (*Pressure distribution on the crack surface*) p0[R_, t_] := -3 Pi*KIC/(Sqrt[2*R] - (9*mu*Q*t)/(8*R ˆ 3*(-1 + nu)); p1[R_, t_] := -9*mu*Q*t*(-1 + Log[2])/(8*R ˆ 3*(-1 + nu)) Pi*KIC*(-4 + Log[8])/(Sqrt[2*R]); P[r_, R_, t_] := -p0[R, t]*Log[r/R] + p1[R, t] /; r R (*Crack volume V[R,t]*) V[R_, t_] := 8*(1 - nu)*R ˆ 3/(9*mu)*((4 - 3*Log[2])*p0[R, t] + 3*p1[R, t]) (*Crack opening*) b0[r_] := -(r*ArcCos[r]) - Sqrt[1 - r ˆ 2]*(-2 + Log[2]); b1[r_] := Sqrt[1 - r ˆ 2] b[r_,R_,t_] := 4*R*(1 - nu))/(Pi*mu)*(p0[R, t]*b0[r/R] + p1[R, t]*b1[r/R]) /; r R (*Crack opening at the point of injection*) b[0, R_, t_] := 4 R (1 - nu) (p1[R, t] + p0[R, t]*(2 - Log[2]))/(Pi* mu); (*Crack radius R(t) at various time moments TT*) TT = Join[Table[k, {k, 9}], Table[10*k, {k, 1, 200}]]; F[R_, t_] = p0[R, t] - 6*et*Q/(Pi*b[0, R, t] ˆ 3); S = Table[{k, Solve[F[R, k] == 0, R, Reals]}, {k, TT}]; TR = Table[{S[[i, 1]], S[[i, 2, 1, 1, 2]]}, {i, Dimensions[TT][[1]]}]; IR = Interpolation[TR]; (*Graph of the function R(t)*) Plot[{IR[t]}, {t, 1, 2000}] (*Time dependence of the pressure coefficients p0(t), p1(t)*) Plot[{p0[IR[t], t], p1[IR[t], t]}, {t, 1, 2000}] (*Crack opening at various time moments*) Plot[ Table[b[r, IR[t], t]*100, {t, {2,11,36,74,128,212,325,449,637,902,1316, 2000}}], {r, 0, IR[2000] + 3}, PlotRange -> {0,0.5}, AxesLabel -> {m,mm}] (*Pressure distribution at various time moments*) Plot[ Table[P[r, IR[t], t]/10 ˆ 6, {t, {t, {2,11,36,74,128,212,325,449,637,902,1316, 2000}}], {r, 0,IR[2000] + 10}, PlotRange -> {0, 2}, AxesLabel -> {m,MPa}] End

References [1] I. Sneddon, The distribution of stress in the neighborhood of a crack in an elastic solid, Proceedings of the Royal Society of London 187 (1946) 229–260. [2] P. Kundo, I. Kohen, D. Dowling, Fluid Mechanics, Elsevier, 2012. [3] T. Na, A. Hansen, Radial flow of viscous non-newtonian fluids between two disks, International Journal of Non-Linear Mechanics 2 (1967) 261–273. [4] G. Cherepanov, Mechanics of Brittle Fracture, McGraw-Hill, New York, London, 1979. [5] A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems, Winston & Sons, Washington, 1977. [6] I. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951. [7] A. Savitski, E. Detournay, Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions, International Journal of Solids and Structures 39 (2002) 6311–6337. [8] B. Meyer, Three-Dimensional Hydraulic Fracturing Simulation on Personal Computers: Theory and Comparison Studies, Society of Petroleum Engineers, 1989, SPE 19329. [9] S. Ramanathan, D. Erta¸s, D. Fisher, Quasistatic crack propagation in heterogeneous media, Physical Review Letters 79 (5) (1997) 873. [10] C. Duarte, O. Hamzeh, T. Liszka, W. Tworzydlo, A generalized finite element method for the simulation of three-dimensional dynamic crack propagation, Computer Methods in Applied Mechanics and Engineering 190 (15) (2001) 2227–2262.

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[11] N. Moes, J. Dolbow, T. Belytschko, Elastic crack growth in finite elements without remeshing, International Journal for Numerical Methods in Engineering 46 (1999) 131–150. [12] T. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications, International Journal for Numerical Methods in Engineering 84 (2010) 253–304. [13] N. Sukumar, N. Moes, B. Moran, T. Belytschko, Extended finite element method for threedimensional crack modelling, International Journal for Numerical Methods in Engineering 48 (2000) 1549–1570. [14] Z. Zhuang, B. Cheng, Equilibrium state of mode-I sub-interfacial crack growth in bimaterials, International Journal of Fracture 170 (1) (2011) 27–36. [15] N. Moes, T. Belytschko, Extended finite element method for cohesive crack growth, Engineering Fracture Mechanics 69 (7) (2002) 813–833. [16] J. Remmers, R. de Borst, A. Needleman, A cohesive segments method for the simulation of crack growth, Computational Mechanics 31 (1–2) (2003) 69–77. [17] Y. Mi, M. Aliabadi, Three-dimensional crack growth simulation using BEM, Computers & Structures 52 (5) (1994) 871–878. [18] B. Yang, K. Ravi-Chandar, A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks, in: Recent Advances in Fracture Mechanics, Springer, Netherlands, 1998, pp. 115–144. [19] M. Knight, L. Wrobel, J. Henshall, L. De Lacerda, A study of the interaction between a propagating crack and an uncoated/coated elastic inclusion using the BE technique, International Journal of Fracture 114 (1) (2002) 47–61. [20] R. Williams, A. Phan, H. Tippur, T. Kaplan, L. Gray, SGBEM analysis of crack–particle(s) interactions due to elastic constants mismatch, Engineering Fracture Mechanics 74 (3) (2007) 314–331. [21] J. Lei, Y. Wang, Y. Huang, Q. Yang, C. Zhang, Dynamic crack propagation in matrix involving inclusions by a time-domain BEM, Engineering Analysis with Boundary Elements 36 (5) (2012) 651–657. [22] P. Valko, M. Economides, Hydraulic Fracture Mechanics, John Wiley & Sons, Chichester, 1995. [23] M. Economides, K. Nolte (Eds.), Reservoir Stimulation, John Wiley & Sons, 2000. [24] J. Adachi, E. Siebrits, A. Peirce, J. Desroches, Computer simulation of hydraulic fractures, International Journal of Rock Mechanics and Mining Sciences 44 (2007) 739–757. [25] E. Detournay, Mechanics of hydraulic fractures, Annual Review of Fluid Mechanics 48 (3) (2016) 311–339. [26] D. Garagash, E. Detournay, The tip region of fluid-driven fracture in an elastic medium, ASME Journal of Applied Mechanics 67 (2000) 183–192. [27] A. Markov, S. Kanaun, An efficient numerical method for quasi-static crack propagation in heterogeneous media, International Journal of Fracture Mechanics 212 (1) (2018) 1–14. [28] S. Kanaun, Discrete model of hydraulic fracture crack propagation in homogeneous isotropic elastic media, International Journal of Engineering Science 110 (2017) 1–14. [29] S. Kanaun, Hydraulic fracture crack propagation in an elastic medium with varying fracture toughness, International Journal of Engineering Science 120 (2017) 15–30. [30] S. Kanaun, A. Markov, Discrete and three-parameter models of hydraulic fracture crack growth, WSEAS Transactions on Applied and Theoretical Mechanics 12 (2018) 147–156. [31] S. Kanaun, Efficient numerical solution of the hydraulic fracture problem for planar cracks, International Journal of Engineering Science 127 (2018) 114–126.

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The homogenization problem

9

When solving boundary value problems for physical fields in heterogeneous materials, it is reasonable to substitute an actual heterogeneous medium with a homogeneous effective medium with equivalent response to external loading. If such a substitution is possible, then, instead of a boundary value problem for the heterogeneous medium, a simpler problem for an effective homogeneous medium can be considered. Determination of the parameters of the homogeneous effective medium equivalent to an actual heterogeneous medium is the objective of the homogenization problem considered in this chapter. In Sections 9.1–9.4, combinations of self-consistent and numerical methods are used for the calculation of effective conductive, elastic, and elasto-plastic properties of heterogeneous materials by quasistatic loading. In Section 9.5, self-consistent methods are applied to solution of the homogenization problem for time-harmonic fields in heterogeneous media.

9.1

Effective property tensors of heterogeneous media

We consider an infinite heterogeneous medium whose property tensor C(x) is a homogeneous in space random function; C(x) can be the conductivity tensor of a conductive medium, the tensor of dielectric permittivity of a dielectric medium, the stiffness tensor of an elastic medium, etc. Let C(x) =  C be the average of C(x) over the ensemble realizations of this function. The correlation function R(x) of the field C(x) is defined by the equation     C ⊗ C(x  ) −  C . (9.1) R(x − x  ) = C(x) −  We assume that R(x) is a decaying function, and the distance l such that R(l) / R(0)  1 is called the correlation length. If the ergodic hypothesis holds, the average over the ensemble realization of C(x) coincides with the spatial average for a fixed typical realization of C(x), i.e.,  1 C(x) = lim C(x)dx. (9.2) V →∞ V V Here, V is a region that occupies the entire 3D space in the limit V → ∞. Let tensors u(x) and σ (x) satisfy the system of equations for static fields in heterogeneous media div σ (x) = −q(x), σ (x) = C(x) · u(x), Rot u(x) = −η(x).

(9.3)

Here, q(x) and η(x) are external and internal field sources and Rot is the incompatibility operator defined in Eq. (2.2) for vector fields and in Eq. (2.37) for tensor fields. If Heterogeneous Media. https://doi.org/10.1016/B978-0-12-819880-3.00016-0 Copyright © 2021 Elsevier Ltd. All rights reserved.

388

Heterogeneous Media

C(x) is a random function, the functions u(x) and σ (x) are also random, and we introduce the averages u(x) and σ (x) of these functions over the ensemble realizations of C(x). The relation between these averages can be presented in the form σ (x) = C∗ u(x) ,

(9.4)

where C∗ is the operator of effective properties. For media with microstructure, general properties of this operator are studied in [1], [2], and it is shown that C∗ can be presented in the integral form    σ (x) = C∗ (x − x  ) · u(x  ) dx  . (9.5) Here, the kernel C∗ (x) is a decaying function with the characteristic scale of the order of the correlation length l of the random function C(x). In the Fourier transform space, Eq. (9.5) takes the form σ (k) = C∗ (k) · u(k) .

(9.6)

Henceforth, the symbol ∗ is used for effective parameters, and the Fourier transforms of functions are indicated by the argument k. It is shown in [2] that if the fields in the medium satisfy Eqs. (9.3), the function C∗ (k) is an even analytical function of the vector argument k. This function can be expanded into the Taylor series in the vicinity of k = 0 C∗ (k) = C∗(0) + C∗(2) · (k ⊗ k) + C∗(4) · (k ⊗ k ⊗ k ⊗ k) + ....

(9.7)

Keeping the two first terms in this series and transferring to the x-space, we obtain C∗ (x) = C∗(0) δ(x) − C∗(2) · (∇ ⊗ ∇)δ(x),

(9.8)

σ (x) = C∗(0) · u(x) − C∗(2) · ∇ ⊗ ∇ u(x) .

(9.9)

Here, C∗(0) and C∗(2) are constant tensors and C∗(2) is proportional to the square of the correlation length l of the random field C(x). Eq. (9.5) means that the relation between the averages u(x) and σ (x) is nonlocal, and Eq. (9.9) presents the simplest nonlocal constitutive relation for the so-called medium with constrained rotations [2]. The objective of the homogenization problem is the construction of the kernel C∗ (x) in Eq. (9.5) for a known random function C(x) of an actual heterogeneous medium. For the medium with constrained rotations, this problem consists in the determination of two constant tensors C∗(0) and C∗(2) in Eq. (9.9). Note that the constitutive relations (9.5) and (9.9) result in a specific formulation of the boundary value problems. For instance, for the constitutive relation (9.9), the system of the differential equations for the field potential ϕ (u = ∇ϕ) that follows from Eq. (9.3) by η(x) = 0 has the order of 4, and not 2, as the equation for the potential ϕ of the local field u(x) in the original system (9.3). As a result, the classical Dirichlet, Neumann, or mixed boundary conditions are not sufficient for the construction of a unique solution of the

The homogenization problem

389

system for u(x) and σ (x). Formulation of the appropriate boundary conditions is a nontrivial problem (see [2]). The problem is simplified if u(x) and σ (x) are slowly changing fields the characteristic scales of which are larger than the correlation length l of the random field C(x). In this case, we can keep the first term in the series (9.7) only. As a result, the averages u(x) and σ (x) satisfy the classic system of equations for static fields in a homogeneous medium div σ (x) = −q(x), σ (x) = C∗(0) · u(x) , Rot u(x) = −η(x),

(9.10)

where C∗(0) is a constant tensor of the effective property of the heterogeneous medium. It is necessary to emphasize that the system (9.10) serves for slowly changing averages u(x) and σ (x). The solution of this system cannot be used for assessment of the fields in the regions of field concentrations (vicinities of sharp edges of macroinclusions, cavities, macrocrack tips, etc.). In these cases, the solution of the system (9.10) can be considered as the external fields acting on these regions.

9.1.1 The tensor of effective conductivity We consider an infinite heterogeneous medium with the conductivity tensor Cij (x), which is a homogeneous in space random function. We introduce a reference medium with a constant conductivity tensor Cij0 and present Cij (x) in the form Cij (x) = Cij0 + Cij1 (x).

(9.11)

Here, Cij0 is a nondegenerate tensor that can be taken equal to the ensemble average   Cij (x) of Cij (x). For a constant external field Ei0 and field flux Ji0 , the vectors Ei (x) and Ji (x) in the heterogeneous medium satisfy the integral equations (3.9) and (3.13):  (9.12) Ei (x) + Kij (x − x  )Cj1k (x  )Ek (x  )dx  = Ei0 ,  (9.13) Ji (x) − Sij (x − x  )Bj1k (x  )Jk (x  )dx  = Ji0 ,  −1 Bj1k (x) = Bij (x) − Bij0 , Bij (x) = Cij (x)−1 , Bij0 = Cij0 , Ji0 = Cij0 Ej0 . (9.14) The kernels Kij (x) and Sij (x) of the integral operators in these equations are defined in Eqs. (3.10) and (3.14). These kernels are homogeneous functions of x of the order of −3, and their Fourier transforms are the homogeneous functions of the order of zero, i.e., Kij (k) =

ki kj 0k kk Ckl l

0 , Sij (k) = Cik Kkl (k)Clj0 − Cij0 .

(9.15)

390

Heterogeneous Media

Eqs. (9.12) and (9.13) are derived in Chapter 3 for a homogeneous host medium with the conductivity tensor Cij0 containing heterogeneities in a finite region V . In this case, the functions Cij1 (x)Ej (x) and Bij1 (x)Jj (x) in Eqs. (9.12) and (9.13) have finite supports. If heterogeneities occupy the entire space, the supports of the functions Cj1k (x)Ek (x) and Bj1k (x)Jk (x) are infinite. But for such functions, the integrals in Eqs. (9.12) and (9.13) diverge formally at infinity and need regularizations. For constant external fields Ei0 and Ji0 , the functions Cij1 (x)Ej (x) and Bij1 (x)Jj (x) in Eqs. (9.12) and (9.13) are homogeneous in space random functions. Each realization of such a function can be presented as a sum of exponentials with arbitrary wave vectors (an almost periodic function) and a function with a finite support. A typical function Fi (x) of this type can be presented in the form Fi (x) = Fi0 +



(m)

Fi

exp(ik (m) · x) + fi (x),

(9.16)

m (k (m) =0) (m)

where Fi0 and Fi are constant coefficients and fi (x) is a function with a finite support. Regularizations of the integral operators with the kernels Kij (x) and Sij (x) on the functions with finite supports are defined in Eqs. (3.15) and (3.16). Actions of these operators on exponentials can be obtained by transferring to the Fourier transforms of the integrand functions  Kij (x − x  ) exp(ik (m) · x  )dx  =  (9.17) = Kij (k)δ(k (m) + k) exp(−ik · x)dk = Kij (k (m) ) exp(ik (m) · x),  Sij (x − x  ) exp(ik (m) · x  )dx  = Sij (k (m) ) exp(ik (m) · x). (9.18) Then, we have to define regularization of the integral operators with the kernels Kij (x) and Sij (x) on constants. For this purpose, we consider a model problem for a “heterogeneity” of a constant conductivity Cij = Cij0 + Cij1 that occupies the entire 3D space. If the conditions at infinity are such that the external field Ei0 is fixed, then the field Ei and field flux Ji in the medium are constant and defined by the equations  (9.19) Ei = Ei0 , Ji = Cij0 + Cij1 Ej0 . On the other hand, for constant Cij1 , Ei and Bij1 , Ji , the integral equations (9.12) and (9.13) take the forms  Ei + Kij (x − x  )dx  Cj1k Ek = Ei0 , (9.20)  (9.21) Ji − Sij (x − x  )dx  Bj1k Jk = Ji0 .

The homogenization problem

391

The solutions of these equations coincide with Ei and Ji in Eq. (9.19) if the integrals in Eqs. (9.20) and (9.21) have the values   Kij (x − x  )dx  = 0, Sij (x − x  )dx  = −Cij0 . (9.22) If the conditions at infinity are such that the external field flux Ji0 is fixed, for the fields Ei and Ji in the medium with a constant resistivity tensor Bij = Bij0 + Bij1 ,  −1 Bij0 = Cij0 , we have the equations  Ei = Bij0 + Bij1 Jj0 , Ji = Ji0 .

(9.23)

In this case, the solutions of Eqs. (9.20) and (9.21) have the forms (9.23) if the integrals in these equations are   Kij (x − x  )dx  = Bij0 , Sij (x − x  )dx  = 0. (9.24) Thus, regularization of the operators in Eqs. (9.12) and (9.13) on constants depends on the conditions at infinity that define what kind of the external field is fixed in the problem: the field Ei0 or the field flux Ji0 . Let us consider the averages of the fields Ei (x) and Ji (x) in Eqs. (9.12) and (9.13) over the ensemble realizations of the random conductivity tensor Cij (x). Because the 

averages Cij1 (x  )Ej (x  ) and Bij1 (x  )Jj (x  ) are constant, we obtain for Ei (x) and Ji (x) the equations   0 Ei (x) = Ei − Kij (x − x  )dx  Cj1k (x  )Ek (x  ) =   1 Jm , (9.25) = Ei0 + Kij (x − x  )dx  Cj0k Bkm   Ji (x) = Ji0 + Sij (x − x  )dx  Bj1k (x  )Jk (x  ) =   1 = Ji0 − Sij (x − x  )dx  Bj0k Ckl (x  )Ek (x  ) . (9.26) Here, we take into account the identity 0 1 Cij1 (x)Ej (x) = Cij1 (x)Cj−1 k (x)Jk (x) = −Cij Bj k (x)Jk (x).

(9.27)

Let conditions at infinity correspond to a fixed external field Ei0 . For this case, the integrals in Eqs. (9.25) and (9.26) are defined in Eq. (9.22), and we obtain  Ei (x) = Ei0 , Ji (x) = Ji0 + Cij1 (x)Ej (x) . (9.28)

392

Heterogeneous Media

 For linearity of the problem, the average Cij1 (x)Ej (x) is a linear function of the external field Ej0 : 

Cij1 (x)Ej (x) = Qij Ej0 .

(9.29)

Here, Qij is a constant tensor that depends on the random conductivity Cij (x) of the heterogeneous medium. Then, the average Ji (x) in Eq. (9.28) takes the form   Ji (x) = Cij0 Ej0 + Qij Ej0 = Cij∗ Ej (x) , (9.30) Cij∗ = Cij0 + Qij .

(9.31)

  In these equations, the tensor Cij∗ relates the averages Ji (x) and Ej (x) over the ensemble realizations of the function Cij (x), and therefore, Cij∗ is the tensor of effective conductivity of the heterogeneous medium. If the external field flux Ji0 is fixed in the problem, from  Eqs. (9.25), (9.26), and (9.22), we obtain the following equations for the averages Ej (x) and Ji (x):  Ei (x) = Ei0 + Bij1 (x)Jj (x) , Ji (x) = Ji0 . (9.32)  Because Bij1 (x)Jj (x) is a linear function of the external field flux Jj0 , we can write 

Bij1 (x  )Jj (x  ) = Mij Jj0 ,

Ei (x) = Bij0 Jj0 + Mkj Jj0 = Bij∗ Bij∗ = Bij0 + Mij .

(9.33) 

 Jj (x) ,

(9.34) (9.35)

Here, Bij∗ is the tensor of effective resistivity of the heterogeneous medium. Because  −1 , the tensors Qij and Mij in Eqs. (9.29) and (9.33) are related by the Cij∗ = Bij∗ equation  −1 0 0 0 Qij = −Cik Mkl Clm . (9.36) δmj + Mmn Cnj Assuming the ergodic property, the averages in Eqs. (9.30) and (9.33) for the tensors Qij and Mij can be calculated for a fixed typical realization of the random fields Cij (x)Ej (x) and Bij1 (x)Jj (x), i.e.,   1 Cij1 (x)Ej (x)dx, (9.37) Cij1 (x)Ej (x) = lim V →∞ V V   1 Bij1 (x)Jj (x) = lim Bij1 (x)Jj (x)dx. (9.38) V →∞ V V For calculation of these averages, we have to possess the solutions of Eq. (9.12) and (9.13) in infinite regions for a typical realization of the conductivity tensor Cij (x). For

The homogenization problem

393

an arbitrary random function are hardly solutions  possible to obtain. Con Cij (x), such struction of the averages Cij1 (x)Ej (x) and Bij1 (x)Jj (x) for a realization of Cij (x) in a finite region V of a heterogeneous medium is considered in the next section.

9.2

The representative volume elements of heterogeneous media

We consider a heterogeneous conductive medium whose conductivity tensor C(x) is a homogeneous in space random function. The medium is subjected to a constant external field or a constant field flux at infinity. If other field sources are absent, the field u(x) and field flux σ (x) in the medium satisfy the homogeneous system of equations (9.3): div σ (x) = 0, σ (x) = C(x) · u(x), Rot u(x) = 0.

(9.39)

In this case, the field u can be presented as the gradient of a potential ϕ(x), i.e., u(x) = ∇ϕ(x).

(9.40)

Similar to C(x), the fields u(x) and σ (x) are also homogeneous in space random functions, and the averages u(x) and σ (x) over the ensemble of realizations of the function C(x) are constant. Assuming the ergodic property, we can change ensemble averages of u(x) and σ (x) with the spatial averages for a fixed typical realization of the field C(x):   1 1 u(x) = lim u(x)dx, σ (x) = lim σ (x)dx. (9.41) V →∞ V V V →∞ V V If the averages u(x) and σ (x) are found, the effective conductivity tensor C∗ = C∗(0) can be reconstructed from the second equation of the system (9.10), and we have σ  = C∗ · u .

(9.42)

Here, C∗ is a constant tensor (henceforth, the upper index 0 is omitted). In physical and numerical experiments, the volume V cannot be taken infinite, and a finite volume VR is called the representative volume element (RVE) if the equations     1 1 1 1 lim u(x)dx = u(x)dx, lim σ (x)dx = σ (x)dx V →∞ V V V →∞ V V VR VR VR VR (9.43) are satisfied with acceptable accuracy.

394

Heterogeneous Media

Let R be the surface of the RVE VR with the external normal n(x), and let t0 (x) and ϕ 0 (x) be the boundary values of the normal component of the field flux σ (x) and the field potential ϕ(x) on R , n(x) · σ (x)|R = t0 (x),

ϕ(x)|R = ϕ 0 (x).

(9.44)

Taking into account Gauss’ theorem and the equations u(x) = ∇ϕ(x), div σ (x) = 0, we obtain    u(x)dx = ∇ϕ(x)dx = n(x) ⊗ ϕ 0 (x)d, (9.45) VR VR R    σ (x)dx = div(σ (x) ⊗ x)dx = t0 (x) ⊗ xd. (9.46) VR

VR

R

Thus, the averages u(x) and σ (x) over the RVE are expressed in terms of the boundary values of the field potential and the field flux by the equations   1 1 u(x) = n(x) ⊗ ϕ 0 (x)d, σ (x) = t0 (x) ⊗ xd. (9.47) VR R VR R These equations serve for any heterogeneous region VR . For calculation of the effective property tensor C∗ , we consider two boundary value problems for the RVE. For the first problem, the potential ϕ 0 (x) on the RVE surface R is defined by the equation ϕ 0 (x) = u0 · x, x ∈ R ,

(9.48)

where u0 is a prescribed constant. For this boundary condition, the average u(x) in Eq. (9.47) takes the form     1 1 u(x) = n(x) ⊗ u0 · x d = ∇ u0 · x dx = u0 . (9.49) VR R VR VR Thus, for the conditions (9.48), the average u(x) does not depend on the tensor C(x) in VR and is equal to u0 . Eq. (9.48) is called the condition for a fixed external field. For the second boundary value problem, the condition on R is t0 (x) = σ 0 · n(x), x ∈ R ,

(9.50)

where σ 0 is a prescribed constant. For this condition, the average of the field flux σ (x) in Eq. (9.47) is presented in the form    1 1 0 σ (x) = t (x) ⊗ xd = div σ 0 ⊗ x dx = σ 0 . (9.51) VR R VR VR Thus, for the boundary conditions (9.50), the average σ (x) does not depend on the tensor C(x) and is equal to σ 0 . Eq. (9.50) is called the condition for a fixed external field flux σ 0 .

The homogenization problem

395

In physical (numerical) experiments, we can define the potential ϕ 0 (x) on the RVE surface and measure (calculate) the vector t0 (x) on R , or vice versa, we define the normal component t0 (x) of the field flux on R and measure (calculate) the potential ϕ 0 (x) on this surface. In both cases, the averages u and σ  can be calculated from Eq. (9.47). Then, the tensor C∗ can be reconstructed from Eq. (9.42). The sizes of the RVE VR should satisfy the following conditions. • The first and second kinds of the boundary conditions on the RVE surface provide the same values of the effective property tensors. • The effective property tensors obtained for various ensemble realizations of the tensor C(x) inside VR are the same. • The shape of the RVE does not affect the values of the effective property tensors. These conditions can be satisfied with some tolerance only, which determines the accuracy of calculation of the effective properties of the heterogeneous medium for the chosen RVE. If the finite element method is used for numerical calculations of the effective parameters, the RVE is usually taken as a cube whose sides are about several correlation lengths of the random field of the material property tensors. For solution of the homogenization problem by the method of volume integral equations, a finite RVE is embedded into an infinite homogeneous reference medium and subjected to a constant external field or field flux at infinity. After numerical solutions of the volume integral equations for the fields in the RVE, the averages u(x) and σ (x) over the RVE can be calculated and the effective properties tensor C∗ can be constructed. For the method of volume integral equations, the RVE should be large enough to satisfy the following conditions. • The effective property tensor does not depend on the properties of the reference medium. • The effective property tensors obtained for various ensemble realizations of the tensor C(x) inside VR are the same. • The shape of the RVE does not affect the values of the effective property tensors.

9.2.1 Effective conductive properties of foam materials In this section, we consider application of the method of volume integral equations to the calculation of the effective conductivity of foam materials. Geometric properties of cells of foam materials have been extensively studied in a number of works (see, e.g., [3] and references therein). In these works, detailed analysis of foam cells with indication of the number of faces and edges of a typical cell is presented. For simulation of the effective conductive properties of foams, we introduce the following complex foam cell (CFC). Let an icosahedron with 12 vertices be located on the surface of a sphere of radius r = 2/3 (Fig. 9.1A). Then, these vertices and the sphere center are taken as seed points for Voronoi tessellation of the space inside the unit sphere with the same center. The edges of the Voronoi polygons inside the sphere will be the axes of the ligaments of the CFC. In Fig. 9.1B, the resulting CFC of the foam material is shown. As seen from the figure, the central cell is a dodecahedron, the 30

396

Heterogeneous Media

edges of which have the same lengths as well as the lengths of 20 external ligaments. This complex cell is embedded into the medium with the conductivity of the ligaments (solid phase). The opaque sphere in Fig. 9.1B indicates the region (RVE) where the fields will be averaged for calculation of the effective conductivities of the foams. Note that this CFC reflects important geometric properties of cells in actual foams. As indicated in [3], a large fraction of the nodes of the foams connects four ligaments, and pentagons are the most common shapes of the cell faces. The introduced CFC is isotropic in the sense that the averaged fields induced in the CFC by a constant external field applied to the medium are almost independent of the orientation of the CFC with respect to the direction of the external field.

Figure 9.1 (A) An icosahedron whose vertices and center are taken as the seed points for Voronoi tessellation of the sphere by constructing a complex foam cell (CFC). (B) The CFC with cylindrical ligaments; the opaque region is the RVE for calculation of the effective conductivity.

For simulation of the ligament shapes, the approximation proposed in [4] is used. We assume that the cross-section of a typical ligament is quasitriangular, and its area changes parabolically along the ligament axis. In the local Cartesian coordinate system (x, y, z) with the x-axis directed along the ligament, the ligament cross-section is defined by the equations 

cos(2ϕ) , (9.52) y(ϕ, x) = R(x) cos(ϕ) + a1 

sin(2ϕ) . (9.53) z(ϕ, x) = R(x) − sin(ϕ) + a1 Here 0 ≤ ϕ < 2π, and (y, z) are the coordinates in the plane of the ligament crosssection. The function R(x) is taken in the form 

 x 2  , −l < x < l. (9.54) R(x) = a2 1 − a3 1 − l This function defines the ligament cross-section area along the x-axis and reflects the fact that ligaments are thinner in the middle than at the ends x = ±l, 0 ≤ a3 < 1.

The homogenization problem

397

The parameters a1 , a2 , a3 define the ligament particular shape. The CFC with the ligament parameters a1 = 2.5, a2 = 0.3, a3 = 0.4 is shown in Fig. 9.2. If a1 → ∞, the cross-section of the ligament is circular, and an example of the CFC with such ligaments is shown in Fig. 9.3 for a2 = 0.15, a3 = 0.75.

Figure 9.2 A complex foam cell with the triangular ligaments defined in Eqs. (9.52) and (9.53) (a1 = 2.5, a2 = 0.3, a3 = 0.4).

Figure 9.3 A complex foam cell with the ligaments with circular cross-sections (a1 = ∞, a2 = 0.15,

a3 = 0.75).

We consider an infinite homogeneous reference medium with the conductivity tensor Cij0 of the ligament phase. The medium contains a region VR with the conductivity Cij (x). The tensor Cij (x) coincides with Cij0 in the region of ligaments and is equal to the conductivity of the filler Cij = Cij0 + Cij1 outside the ligaments. The field Ei (x)

398

Heterogeneous Media

in VR satisfies the integral equation (9.20)  Ei (x) + Kij (x − x  )Cj1k (x  )Ek (x  )dx  = Ei0 , x ∈ VR .

(9.55)

VR

Here, Ei0 is an external field applied to the medium. The effective conductivity tensor Cij∗ is defined by the equation   Ji  = Cij∗ Ej ,

(9.56)

where Ei  and Ji  are the averages over the region VR of the CFC   1 1 Ei  = Ei (x)dx, Ji  = Ji (x)dx. VR VR VR VR

(9.57)

For the numerical solution of Eq. (9.55), we consider a cube W : (2 × 2 × 2) with the spherical CFC inside. The cube is covered by a regular (cubic) grid of nodes with the step h, and the function Ei (x) is approximated by the Gaussian quasiinterpolant inside W , Ei (x) ≈

Mn 

(n) Ei ϕ x − x (n) , ϕ(x) = n=1

1 (πH )3/2

 |x|2 exp − . H h2

(9.58)

Here, x (n) are the nodes of the approximation, Mn is the total number of the nodes, (n) Ei = Ei (x (n) ), and H is a dimensionless parameter of the order of 1. After substituting Eq. (9.58) into Eq. (9.55) and satisfying the resulting equation at the nodes (the collocation method), we obtain the following system of linear algebraic equations for the coefficients of the approximation: (m)

Ei

+

Mn

(m,n) 1(n) (n) ij Cj k Ek = Ei0 , m = 1, 2, ..., Mn, n=1

(m,n) = ij (x (m) − x (n) ), ij (x) = ij 1(n)

Cij

= Cij1 (x (n) ).



(9.59)

Kij (x − x  )ϕ(x  )dx  , (9.60)

Here, the function ij (x) is defined in Eq. (5.34). Note that the well-known Maxwell Garnett equation for the effective conductivity of the medium with spherical particles can be presented in the form c∗ = c 0 +

3(1 − p)(c − c0 )c0 . 3c0 + p(c − c0 )

(9.61)

Here, c0 is the conductivity of the matrix phase, p is the volume fraction of this phase, and c is the conductivity of the spherical particles (both materials are isotropic). If

The homogenization problem

399

c0 >> c and p 0.03. In Fig. 9.5, the solid lines correspond to results of the calculation of the effective conductivities of foams with cylindrical ligaments (a1 = ∞, a3 = 0) and parabolic ligaments (a1 = ∞, a3 = 0.75). For the calculations, the conductivities of the ligaments c0 and of the filler c were taken such that c/c0 = 0.001. The averages of the field and field flux are calculated over the region inside the opaque sphere in Fig. 9.1B. It is seen that the model is in good agreement with the experimental data.

400

Heterogeneous Media

Figure 9.5 Theoretical predictions and experimental data for effective conductivity of aluminum foams. Black dots and squares correspond to experimental data presented in [6] (DW) and [7] (KHN); the complex foam cells with parameters a3 = 0, a3 = 0.75 (a1 = ∞), and c/c0 = 0.001 were used in the calculations (solid lines).

Figure 9.6 Dependencies of the effective conductivity of open-cell foams on the parameter a3 (a1 = ∞) in Eqs. (9.52) and (9.53) for the ligament shapes and on the volume fraction p of the highly conductive phase, c/c0 = 0.001.

The dependence of the relative effective conductivity on the parameter a3 , which governs the distribution of the material along the ligament, is presented in Fig. 9.6. It is seen that the influence of this parameter is negligible if a3 < 0.5. The influence of the parameter a1 , which controls the shapes of the ligament cross-sections, on the effective conductivity is also negligible, as also noted in [4] and [8].

Conductivity of closed-cell foams For simulation of the conductivity of closed-cell foams, we introduce the following CFC. Let the center and 12 vertices of the icosahedron in Fig. 9.1A be the centers of spherical pores of the same radius r0 . When r0 < 1/3, the CFC corresponds to a closed-cell foam. When r0 > 1/3, there appear holes in the walls of the cells inside the CFC, and finally, we come to the open-cell foam with specific ligament shapes,

The homogenization problem

401

which are the regions between the intersected spherical pores. The region of the CFC, where the averages of the fields and the volume fraction p of the solid phase are calculated, is the sphere of radius 2/3 that passes through the icosahedron vertices. The numerical method of calculation of the effective conductivity of the closed- and open-cell foams is the same. Results of calculations are presented in Fig. 9.7 by solid lines for c/c0 = 0.001. The line with white dots corresponds to the Maxwell Garnett equation (9.61). It is seen that the effective conductivity changes from the values that correspond to open-cell foams to the values predicted by the Maxwell Garnett equation (9.61). For p > 0.35, the numerical solution practically coincides with the Maxwell Garnett equation, which is in agreement with experimental data for porous materials presented in [9].

Figure 9.7 Theoretical predictions and experimental data for effective conductivity of aluminum foams; p is the volume fraction of the solid phase. Black dots and squares correspond to experimental data presented in [6] (DW) and [7] (KHW); line 1 corresponds to the complex foam cell with cylindrical ligaments (a1 = ∞, a3 = 0); line 2 corresponds to the medium with spherical pores; and line 3 corresponds to the Maxwell Garnett equation (9.61).

The proposed method of calculation of the effective conductivities of foam materials predicts results that are in a good agreement with experimental data for the conductivity of open- and closed-cell foams. Although, strictly speaking, the RVE should contain not one but several cells, the experimental data can be described with sufficient accuracy by the RVE of smaller sizes considered in this section.

9.3

The effective field method

If the volume integral equation method is used for solution of the homogenization problem, a finite RVE VR is embedded into a homogeneous reference medium and subjected to a constant external calculation of the  field or field flux at infinity. After 1 1 fields inside VR , the averages Cij (x)Ej (x) or Bij (x)Jj (x) over the RVE volume can be constructed and the tensor of effective conductivity Cij∗ or resistivity Bij∗ can be

402

Heterogeneous Media

found from Eq. (9.31) (Eq. (9.35)). For calculation of the effective conductivity tensors with acceptable accuracy, the RVE sizes should be of the order of several correlation lengths of the random function Cij (x). If the RVE is not large enough, the error in the calculation of Cij∗ is the result of neglecting the fields induced in VR by heterogeneities outside this volume in the actual heterogeneous medium. From here on, we consider the effective field method (EFM), which takes into account this additional field by solution of the homogenization problem. Consider an infinite heterogeneous medium with the resistivity tensor Bij (x) = Bij0 + Bij1 (x), where Bij0 is the resistivity tensor of the reference medium. Let a finite region VR of the heterogeneous medium be taken as an RVE. The equation for the field flux Ji (x) in the RVE can be presented in the following form, which follows from Eq. (9.13):  Ji (x) − Sij (x − x  )Bj1k (x  )Jk (x  )dx  = Ji∗ (x), x ∈ VR , (9.64) VR  Ji∗ (x) = Ji0 + Sij (x − x  )Bj1k (x  )Jk (x  )dx  . (9.65) R\VR

Here, R\VR is the complement of the region VR to the entire 3D space R. The function Ji∗ (x) in these equations can be interpreted as the external field flux acting on the RVE.   The function Ji∗ (x) does not coincide with Ji0 , and the average Ji∗ (x) of Ji∗ (x) over the ensemble realizations  of the random function Cij (x) is called the effective field flux. The equation for Ji∗ (x) follows from Eq. (9.65) in the form  Ji∗ (x) = Ji0 +



 R\VR

 Sij (x − x  )dx  Bj1k (x)Jk (x) .

(9.66)

 Here, it is taken into account that the average Bj1k (x)Jk (x) is constant. For conditions at infinity that correspond to a fixed external field flux Ji0 , the integral in this equation is transformed in the integral over the region VR  Dij (x) = − Sij (x − x  )dx  = R\VR   Sij (x − x  )dx  = = − Sij (x − x  )dx  + R VR  = Sij (x − x  )dx  . (9.67) VR

Here, Eq. (9.24) is taken into account. For a spherical region VR , the integral Dij (x) is constant inside VR (see Section 3.1), and if the reference medium is isotropic, it takes the form 2 Dij = − c0 δij , 3

(9.68)

The homogenization problem

403

where c0 is the conductivity  of the reference medium. Changing the external field Ji∗ (x) to the average field Ji∗ (x) in Eq. (9.64), we obtain the following integral equation for the field flux Ji (x) inside the RVE:   Ji (x) − Sij (x − x  )Bj1k (x  )Jk (x  )dx  + Dij (x) Bj1k (x)Jk (x) = Ji0 , VR

x ∈ VR .

(9.69) The condition of self-consistency is the assumption that the average Bj1k (x)Jk (x) in this equation can be changed to the following volume average of the function Bj1k (x)Jk (x) over the RVE: 



 1 B 1 (x)Jk (x)dx. Bj1k (x)Jk (x) = VR VR j k

(9.70)

This assumption makes Eq. (9.69) to be closed with respect to the field flux Ji (x) in VR , i.e.,    1 Ji (x) − Dij (x) Bj1k (x  )Jk (x  )dx  = Ji0 , x ∈ VR . Sij (x − x  ) − VR VR (9.71) The EFM can be applied to Eq. (9.12) for the field Ei (x) in the RVE. This equation can be written in the form similar to Eq. (9.64)  Kij (x − x  )Cj1k (x  )Ek (x  )dx  = Ei∗ (x), x ∈ VR , (9.72) Ei (x) + VR  Ei∗ (x) = Ei0 − Kij (x − x  )Cj1k (x  )Ek (x  )dx  , (9.73) R\VR

where Ei∗ (x) is the external field acting on the RVE. Averaging the field Ei∗ (x) over the ensemble realization of the conductivity tensor Cij (x) yields the following equation for the effective external field:    ∗  Ei (x) = Ei0 − Kij (x − x  )dx  Cj1k Ek , x ∈ VR . (9.74) R\VR

If the conditions at infinity correspond to a fixed external field Ei0 , the integral in this equation is calculated as follows:  Kij (x − x  )dx  = Aij (x) = − R\VR   Kij (x − x  )dx  = = − Kij (x − x  )dx  + R VR  = Kij (x − x  )dx  . (9.75) VR

404

Heterogeneous Media

Here, Eq. (9.22) is taken into account. For a spherical RVE and an isotropic reference medium, the integral Aij is constant inside the RVE and has the form Aij =

1 δij . 3c0

  As a result, we obtain the following equation for the effective field Ei∗ (x) :   ∗  Ei (x) = Ei0 + Aij (x) Cj1k (x)Ek (x) , x ∈ VR .

(9.76)

(9.77)

 The assumption of self-consistency that the average Cj1k (x)Ek (x) can be calculated over the RVE VR yields the equation   1 1 C 1 (x)Ej (x)dx. (9.78) Cij Ej = VR VR ij Changing Ei∗ (x) in Eq. (9.72) to the effective field E ∗ (x) and taking into account Eqs. (9.77) and (9.78), we obtain the equation for the field Ei (x) inside the RVE in the form    1  Aij (x) Cj1k (x  )Ek (x  )dx  = Ei0 , x ∈ VR . Kij (x − x ) − Ei (x) + VR VR (9.79) For computation of the tensor Qij in Eqs. (9.29)–(9.31) for the effective conductivity Cij∗ , Eq. (9.79) should be solved three times with the right hand sides 0(k)

= δik , k = 1, 2, 3.

(9.80)  (k) (k) If the solutions of these problems are Ei (x), then the averages Cij1 (x)Ej (x) over the RVE in Eq. (9.78) determine the tensor Qij in Eq. (9.33) by the equation  (k) (9.81) Qik = Cij1 (x)Ej (x) , k = 1, 2, 3. Ei

Discretization of the equations of the effective field method Using the Gaussian approximating functions, the integral equations of the EFM can be discretized similar to the case of a homogeneous medium with a finite number of heterogeneous inclusions considered in Chapter 5. For this purpose, the RVE is embedded in a cuboid W , which is covered by a regular grid of approximated nodes x (m) (see Fig. 9.8). Then, the field Ei (x) (Ji (x)) in W is approximated by the Gaussian quasiinterpolant and substituted in Eq. (9.79) (Eq. (9.71)). The collocation method provides the discretized forms of these equations, i.e., (m)

Ei

+

Mn 

(m,n) 1(n) (m) 1(n) (n) ij Cj k − τ Aij Cj k Ek = Ei0 , m = 1, 2, ..., Mn, (9.82) n=1

The homogenization problem

405

Figure 9.8 The representative volume element VR and the region W of the heterogeneous medium for numerical solution of the integral equations (9.71) and (9.79). 1(m)

(m)

= Ei (x (m) ), Cj k

(m)

−

Ei Ji

(m)

= Cj1k (x (m) ), Aij = Aij (x (m) ),

(9.83)

Mn 

(m,n) 1(n) (m) 1(n) (n) ij Bj k − τ Dij Bj k Jk = Ji0 , m = 1, 2, ..., Mn, (9.84) n=1

(m) Ji

1(m)

= Ji (x (m) ), Bj k

(m)

= Bj1k (x (m) ), Dij = Dij (x (m) ).

(r,s)

In the 2D case, the objects ij

(r,s)

and ij

(9.85)

are defined in Eqs. (5.13)–(5.15) and in

τ = h2 /VR . = h3 /VR .

(r,s)

(r,s)

In the 3D case, ij and ij are (5.23) and (5.24), respectively, and defined in Eqs. (5.34)–(5.36), and τ Eqs. (9.82) and (9.84) can be presented in the matrix forms similar to Eqs. (5.16) and (5.25). Iterative methods of solution of these equations are discussed in Section 5.2.

9.3.1 Effective conductivity of matrix composites We consider an isotropic homogeneous host medium with the conductivity c0 containing a set of isotropic spherical inclusions of the conductivity c and take the reference medium coinciding with the host medium. For a spherical RVE, the tensors Dij and Aij in Eqs. (9.67) and (9.75) take the forms 2 1 Dij = − c0 δij , Aij = δij . 3 3c0

(9.86)

Let the RVE VR be a sphere of radius R = a/p 1/3 containing only one spherical inclusion v of radius a, and let VR and v have the same center. Here, p is the volume fraction of the inclusions in the composite. In this case, Eqs. (9.71) and (9.79) have exact solutions that are constant inside the inclusion region v:  −1 1 , x ∈ v, Ji = ij Jj0 , ij = δij − (1 − p)Dik Bkj  −1 1 , x ∈ v. Ei = ϒij Ej0 , ϒij = δij + (1 − p)Aik Ckj

(9.87) (9.88)

406

Heterogeneous Media

  Thus, for the averages Bij1 Jj and Cij1 Ej in Eqs. (9.33) and (9.29), we have  1 Bij1 Jj = Mij Jj0 , Mij = pBik kj ,  1 ϒkj . Cij1 Ej = Qij Ej0 , Qij = pCik

(9.89) (9.90)

As a result, the tensors of the effective resistivity Bij∗ and conductivity Cij∗ of the composite in Eqs. (9.31) and (9.35) are  −1 1 , δkj − (1 − p)Dkl Blj1 Bij∗ = Bij0 + pBik  −1 1 Cij∗ = Cij0 + pCik . δkj + (1 − p)Akl Clj1

(9.91) (9.92)

It is easy to verify that (B ∗ )−1 = C ∗ , and for isotropic matrix and inclusions, these equations coincide with the Maxwell Garnett formula for the effective conductivity c∗ of the 3D composite with spherical inclusions   c − c0 c − c0 −1 c∗ =1+p . 1 + (1 − p) c0 c0 3c0

(9.93)

Cubic lattice of spherical inclusions Let a cube W of the size (2 × 2 × 2) contain the RVE with 27 identical spherical inclusions whose centers compose a simple cubic lattice (Fig. 9.9A). The inclusions and the host medium are isotropic with the conductivities c and c0 . The integral equation (9.71) is used for calculation of the field flux vector Ji (x) inside W in the case of the contrast c/c0 = 1000. The results of the calculation of the relative effective conductivity c∗ /c0 are shown in Fig. 9.10 for the grid step h = 0.01 (Mn = 8120601). The crosses in Fig. 9.10 indicate exact values of the effective conductivities presented in [10], [11]. The effective conductivity can be also calculated from the equations J1 (x)V , E1 (x)V   1 1 J1 (x)V = J1 (x)dx, E1 (x)V = E1 (x)dx. V V V V

c∗ =

(9.94) (9.95)

Here, the external field flux is assumed to be Ji0 = δi1 , V = V0 or V = V1 , and V0 , V1 are the regions indicated in Fig. 9.9. In Fig. 9.10, the values of c∗ obtained from Eq. (9.94) by the averaging over the region V0 are indicated by triangles and over the region V1 by squares. The dashed line in Fig. 9.10 shows the results of the Maxwell Garnett equation (9.93). The results of calculations of the effective conductivity of the composite with the inclusion-matrix conductivity contrast c/c0 = 0.01 are shown in Fig. 9.11. Crosses in this figure are the exact values of the effective conductivity presented in [10], [11].

The homogenization problem

407

Figure 9.9 The RVE of the composite with a cubic lattice of spherical inclusions; V0 and V1 are the averaging regions for calculation of the effective conductivity from Eqs. (9.94) and (9.95). (A) Original material before the percolation threshold (dark regions correspond to the inclusion phase). (B) The material after the percolation threshold (dark regions correspond to the matrix phase).

Figure 9.10 Effective conductivity of the composite with a cubic lattice of spherical inclusions (c/c0 = 1000). The line with white dots corresponds to the EFM based on Eq. (9.71) for the field flux; the line with triangles corresponds to Eq. (9.94) by averaging the fields Ji (x) and Ei (x) over the region V0 in Fig. 9.9A; the line with squares corresponds to averaging Ji (x) and Ei (x) over the region V1 ; dashed lines correspond to the Maxwell Garnett equation (9.93); crosses show the exact values of c* presented in [10], [11]; and the vertical dashed line indicates the percolation threshold pc = 0.526.

In this case, numerical solution of Eq. (9.79) for the field (line with white dots) gives results that are closer to the exact solution than the solution of Eq. (9.71) for the flux (line with black dots) for the same value of the grid step h = 0.01. The dependence of the relative effective conductivity c∗ /c0 of the composite on the volume fraction p of the inclusions in the entire region of the volume fractions (0 < p < 1) for c/c0 = 1000 is presented in Fig. 9.12. In the calculations, Eq. (9.71) for the flux was used when p was less than the percolation threshold pc = 0.526, and the model of the material with phase inversion (c/c0 = 0.001) and Eq. (9.79) for the field was used when p > pc . The RVE of the composite after phase inversion is

408

Heterogeneous Media

Figure 9.11 Effective conductivity of the composite with a cubic lattice of spherical inclusions (c/c0 = 0.01). The line with white dots corresponds to the EFM predictions based on Eq. (9.79) for the field; the line with black dots corresponds to the EFM based on Eq. (9.71) for the field flux; the dashed line corresponds to the Maxwell Garnett equation (9.93); crosses show the data from [10], [11]; and the vertical dashed line corresponds to the percolation threshold pc = 0.526.

Figure 9.12 Effective conductivity of the composite with a cubic lattice of spherical inclusions (c/c0 = 1000). The line with white dots corresponds to the EFM predictions based on Eq. (9.71) for the field flux; the line with triangles corresponds to Eq. (9.94) by averaging the fields J and E over the region V0 in Fig. 9.9A; the line with squares corresponds to averaging J and E over the region V1 ; dashed lines correspond to the Maxwell Garnett equation (9.93) for the original material (lower line) and for the material with phase inversion (upper line); the solid line corresponds to the exact solution presented in [10], [11]; and the dashed vertical line indicates the percolation threshold pc = 0.526. After the percolation threshold, the medium with phase inversion was considered (c/c0 = 0.001), and Eq. (9.79) for the field was used.

shown in Fig. 9.9B, where the dark regions correspond to the matrix phase. The EFM and Eq. (9.94) by the average over small (V0 ) or large (V1 ) cells give close results. The solid line in Fig. 9.12 corresponds to exact values of the effective conductivities presented in [10], [11] for p < pc , and dashed lines correspond to the Maxwell Garnett equation (9.93): The low line is for the original composite material, and the upper line is for the material with the phase inversion.

The homogenization problem

409

Thus, the solution of the homogenization problem for matrix composite materials can be obtained on the basis of the EFM and the RVE containing an inclusion and its nearest neighbors.

9.3.2 The one-particle version of the effective field method Let a heterogeneous medium consist of a set of nonoverlapping isolated heterogeneous inclusions (particles) distributed in a homogeneous host medium. In this case, the one-particle version of the EFM can be used for solution of the homogenization problem. In this version, each inclusion is considered as an isolated one in the homogeneous host medium, and the influence of the surrounding heterogeneities is taken into account by the local external field acting on this inclusion. The local external field Ei∗ (x) acting on the inclusion v is the sum of the field Ei0 applied to the medium and the fields induced in the region v by the surrounding inclusions. In the simplest version of the method, the local external field (the effective field) is assumed to be constant and the same for all the inclusions. Let v m (m = 1, 2, ..) be a homogeneous in space random set of heterogeneous inclusions in a homogeneous host medium. It follows from Eq. (9.12) that the integral equation for the field inside the mth inclusion can be presented in the form  Ei (x) + Kij (x − x  )Cj1k (x)Ek (x  )dx  = Ei∗ (x), x ∈ v m , (9.96) vm 

Ei∗ (x) = Ei0 (x) − Kij (x − x  )Cj1k (x  )Ek (x  )dx  . (9.97) n =m vn

Eq. (9.96) shows that the inclusion v m can be considered as an isolated one in the host medium with conductivity Cij0 by action of the local external field Ei∗ (x). For a constant field Ei∗ , the solution of Eq. (9.96) is presented in the form Ei (x) = ij (x)Ej∗ , x ∈ v m , (m)

(9.98)

(m)

where the function ij (x) determines the field inside an isolated inclusion v m subjected to a constant external field Ei∗ . This function depends on the conductivity tensor Cij (x) and the shape of the mth inclusion. As a result, the function Cj1k (x)Ek (x) in Eqs. (9.96) and (9.97) takes the form  vm. (9.99) Cj1k (x)Ek (x) = Cij1 (x) j k (x)Ek∗ , x ∈ V = m (m)

Here, the function ij (x) coincides with ij (x) inside the mth inclusion. Let us introduce the functions V (x) and V (x; x  ) by the equations



v n (x), V (x; x  ) = v n (x  ) if x ∈ v m . V (x) = n

n =m

(9.100)

410

Heterogeneous Media

Here, v n (x) is the characteristic function of the region v n occupied by the nth inclusion. Using the function V (x; x  ), the local external field (9.97) at a point x ∈ V is presented in the form  ∗ 0 Ei (x) = Ei − Kij (x − x  )Cj1k (x  ) kl (x  )V (x; x  )dx  El∗ , x ∈ V . (9.101) Averaging this equation under the condition that the point x belongs to the region V , we obtain the equation     ∗ Ei (x)|x = Ei0 − Kij (x − x  ) Cj1k (x  ) kl (x  )V (x; x  )|x dx  El∗ . (9.102) Here ·|x is the average over the realizations of the random set of inclusions by the condition that x ∈ V . The conditional mean f (x)|x of a random function f (x) is defined by the equation f (x)|x =

f (x)V (x) , V (x)

(9.103)

and this mean is constant for  a homogeneous in space random functions f (x) and  V (x). If the mean Ei∗ (x)|x is identified with the effective field Ei∗ ,  ∗  Ei (x)|x = Ei∗ , (9.104) Eq. (9.102) is a closed equation with respect to Ei∗ ,   ∗ 0 Ei = Ei − Kij (x − x  ) Cj1k (x  ) kl (x  )V (x; x  )|x dx  El∗ .

(9.105)

For statistical independence of inclusion properties on their spatial locations, the conditional mean under the integral in this equation is presented in the form     Cij1 (x  ) j k (x  )V (x; x  )|x = Cij1 (x  ) j k (x  ) V (x; x  )|x =    V (x; x  )|x 1     = Cij (x ) j k (x ) V (x) (x, x ), (x, x ) = . (9.106) V (x) If for the random function Cij1 (x) j k (x) the ergodic property holds, we can write 



Cij1 (x) j k (x)

1 = lim W →∞ W = lim

W →∞

1 W

 Cij1 (x) j k (x)dx = V N 

(n) Cij1 (x) j k (x)dx, n=1 vn

(9.107)

where N is the number of inclusions in the region W , which occupies the entire space in the limit W → ∞. After averaging both sides of this equation once more over the

The homogenization problem

411

ensemble realizations of the random set of inclusions, we obtain N v Pik = pPik , Cij1 (x) j k (x) = lim W →∞ W   1 Cij1 (x) j k (x)dx , p = V (x) , Pik = v v 

(9.108) (9.109)

where v is the mean volume of the inclusion regions v n and p is the inclusion volume fraction. The integral in Eq. (9.109) is calculated over the volume of inclusions and then, it is averaged over the inclusion sizes and properties. Accounting for Eqs. (9.104), (9.106), and (9.108), Eq. (9.105) takes the form  ∗ 0 Ei = Ei − p Kij (x − x  )(x − x  )dx  Pj k Ek∗ . (9.110) Here, it is taken into account that for homogeneous in space random sets of inclusions, (x, x  ) = (x − x  ). Let us consider this function in detail. From the definition of the conditional mean (9.103), we obtain     V (x; x  |x V (x; x  )V (x)  (x − x ) = . (9.111) = V (x) V (x)2   The numerator in this equation relates to the covariance V (x  + x)V (x) . If the ergodic property holds for the random function V (x), we have   V (x  + x)V (x) = ⎤ ⎡  

1 ⎢

⎥ = lim v n (x  + x)v n (x)dx + v n (x  + x)v m (x)dx ⎦ . ⎣ W →∞ W W W m,n n (m =n)

(9.112) 

Note that the integral v n (x  + x)v n (x)dx in this equation is the volume of the intersection of two regions v n moved in a vector x  with respect to each other.  identical   The average V (x; x )V (x) in Eq. (9.111) coincides with the average (9.112) if the realizations with the points x and x  inside the same inclusion are omitted. The latter follows from the definition (9.100) of the function V (x; x  ). Therefore, taking away the first sum in Eq. (9.112), we present the function (x) in the form  1

1 (x) = 2 lim v n (x + x  )v m (x  )dx  , p = V (x) . (9.113) p W →∞ W m,n W (m =n)

It follows from this equation that (x) is a continuous function with the properties  1 (x)dx = 1, (0) = 0. (9.114) lim W →∞ W W

412

Heterogeneous Media

The second equation holds because the regions v n (n = 1, 2, 3, ...) do not overlap. For a homogeneous in space random set of inclusions, the function (x) is a sum of a constant (equal to 1), an oscillating function with the mean value equal to zero, and a finite function. Such a function is sketched in Fig. 9.13.

Figure 9.13 Correlation function (r) of a homogeneous in space random set of nonintersecting spherical inclusions of radius a.

For a statistically isotropic set of inclusions, (x) = (|x|). An example of a such a set is presented in Fig. 9.14A. Deviation of the random set of inclusions from statistical isotropy results in anisotropy of the macroproperties of the composite material (texture) in spite of isotropy of the material properties of the matrix and inclusion phases (Fig. 9.14B).

Figure 9.14 (A) An isotropic distribution of spherical inclusions in space. (B) The distribution that results in anisotropy (texture) of the effective medium.

The mentioned properties of the function (x) and the regularization (9.22) allow calculating the integral in Eq. (9.110) in the explicit form 







Kij (x − x )(x − x )dx = −



 Kij (x) [1 − (x)] dx +

Kij (x)dx =

The homogenization problem

413

 =−

Kij (x) (x)dx = −A ij , (x) = 1 − (x).

(9.115)

Regularization of the last integral is defined in Eq. (2.186). Substituting Eq. (9.115) into Eq. (9.110), we find the following equation for the effective field Ei∗ :  −1 ∗ij = δij − pA . (9.116) ik Pkj  The equation for the average Cij1 (x)Ej (x) in Eq. (9.108) follows from Eq. (9.116) in the form   1 Cij1 (x)Ej (x) = Cik (x) kj (x)) Ej∗ = Qij Ej0 , Qij = pPik ∗kj , (9.117) Ei∗ = ∗ij Ej0 ,

where the tensor Pik is defined in Eq. (9.109). Finally, Eq. (9.117) yields the equation for the tensor of the effective conductivity Cij∗ of the composite that follows from Eq. (9.31)  −1 . Cij∗ = Cij0 + Qij = Cij0 + pPik δkj − pA km Pmj

(9.118)

In this equation, statistical properties of the random set of inclusions determine the tensor A ij , and the tensor Pij depends on the shapes and conductivities of the inclusions.

9.3.3 The one-particle problem of the effective field method The tensor Pij in Eq. (9.109) is to be found from the solution of the one-particle problem of the EFM. This problem is the determination of the field Ei (x) inside an isolated inclusion v embedded in the host medium and subjected to a constant external field Ei∗ . The field Ei (x) satisfies the integral equation  Ei (x) + Kij (x − x  )Cj1k (x  )Ek (x  )dx  = Ei∗ , x ∈ v, (9.119) v

and is presented in the form Ei (x) = ij (x)Ej∗ , x ∈ v.

(9.120)

The numerical method for solution of this equation is discussed in Chapter 5, and for construction of the tensor ij (x), this equation should be solved three times for the right hand sides ∗(k)

Ei

= δik , k = 1, 2, 3.

(9.121) (k)

If the solutions of these problems are Ei (x), then the components of the tensor ij (x) are defined by the equation (k)

ik (x) = Ei (x), x ∈ v, k = 1, 2, 3.

(9.122)

414

Heterogeneous Media

Then, after calculating the integral  1 ij = Cik (x) kj (x)dx

(9.123)

v

and averaging the result over ensemble realizations of the inclusion shapes, orientations, and properties, we obtain the tensor Pij in Eq. (9.109) in the form Pij =

1   ij . v

(9.124)

9.3.4 Matrix composites with ellipsoidal inclusions In the case of ellipsoidal inclusions with constant conductivities, the one-particle problems have exact solutions. Let an isotropic matrix (Cij0 = c0 δij ) contain isotropic ellipsoidal inclusions with the conductivity c. The ellipsoids with semiaxes a1 , a2 , a3 are homogeneously distributed over the orientations. In this case, the function (x) in Eq. (9.115) is spherically symmetric, and the tensor A ij takes the form A ij =

1 δij . 3c0

(9.125)

As a result, the composite is isotropic (Cij∗ = c∗ δij ), and its effective conductivity c∗ is defined by the equation

 p −1 c∗ = c0 + p 1 − , 3c0 3  −1

1 , 1 + (c − c0 )A(k) = (c − c0 ) 3

(9.126) (9.127)

k=1

where the coefficients A(k) are the 1D integrals  dσ a 1 a2 a3 ∞ (k)  A = , k = 1, 2, 3. 2c0 0 (ak2 + σ ) (a12 + σ )(a22 + σ )(a32 + σ ) (9.128) For spherical inclusions, the expression (9.126) for c∗ is simplified and takes the form   c − c0 −1 , c∗ = c0 + p(c − c0 ) 1 + (1 − p) 3c0

(9.129)

which coincides with the Maxwell Garnett equation. Let the inclusions in the composite be spheroids of the same orientation with semiaxes a1 = a2 = a and a3 , and let the function (x) be spherically symmetric. In this

The homogenization problem

415

case, the tensors Pik and A ij in Eq. (9.118) take the forms 1 δij , θik = δik − mi mk , 3c0

−1 −1 1 1 P1 = + A1 , P2 = + A3 , c − c0 c − c0 1 1 a A1 = f0 (γ ), A3 = (1 − 2f0 (γ )), γ = , c0 c0 a3

Pik = P1 θik + P2 mi mk ,

A ij =

(9.130) (9.131) (9.132)

where mi is the unit vector of the x3 -axis and the function f0 (γ ) is defined by the equations  1 − g(γ ) γ2 arctan γ 2 − 1, γ > 1, (9.133) f0 (γ ) = , g(γ ) = 2(1 − γ ) γ2 −1 g(γ ) =

γ2 1−γ2

lg

1+

1−γ2

1−

1−γ2

, γ < 1.

(9.134)

The considered composite material is transverse isotropic, and its tensor of the effective conductivity Cij∗ has the form ∗ ∗ θij + C33 mi mj , (9.135) Cij∗ = C11

−1

−1 p p ∗ ∗ C11 = c0 + pP1 1 − P1 , C33 = c0 + pP2 1 − P2 . 3c0 3c0 (9.136)

If the symmetry of the function (x) coincides with the symmetry of a spheroid with the semiaxes α1 = α2 = α, α3 (ξ = α/α3 ) coaxial to the inclusion, the tensor A ij is presented in the form A ij = A1 θij + A2 mi mj ,

A 1 =

1 f0 (ξ ), c0

A 2 =

1 (1 − 2f0 (ξ )), c0

(9.137) (9.138)

where the function f0 (ξ ) is defined in Eq. (9.133). The composite material is also transverse isotropic with the following principal components of the effective conductivity tensor:  −1 ∗ = c0 + pP1 1 − pA , (9.139) C11 1 P1   −1 ∗ C33 = c0 + pP2 1 − pA . (9.140) 2 P2 If the aspect ratio ξ coincides with the inclusion aspect ratio, A 1 = A1 , A2 = A3 , and Eq. (9.136) are transformed in the equations  −1 1 ∗ C11 = c0 + p + (1 − p)A1 , (9.141) c − c0

416

Heterogeneous Media

∗ C33



1 = c0 + p + (1 − p)A3 c − c0

−1 .

(9.142)

In the limit γ → 0, we obtain a material containing long parallel fibers aligned in the x3 -direction, i.e.,   2p(c − c0 ) ∗ , (9.143) C11 = c0 1 + 2c0 + (1 − p)(c − c0 ) ∗ = c0 + p(c − c0 ). (9.144) C33 The limit γ >> 1 corresponds to the inclusions in the form of flat spheroids (disks). In this case, 

π 1 π A1 ≈ , A3 ≈ . (9.145) 1− 4c0 γ c0 2γ For highly conducting disks (c  c0 ), we have   c0 4pγ ∗ ∗ C11 = c0 1 + = , C33 , π(1 − p) 1−p and for nonconducting thin inclusions (c = 0),

 2pγ −1 ∗ ∗ = (1 − p)c0 , C33 = c0 (1 − p) 1 + . C11 π

(9.146)

(9.147)

Note that for dilute inclusion concentrations, interactions between inclusions can be neglected, and each inclusion can be considered as an isolated one in the homogeneous host medium subjected to the external field Ei0 applied to the medium. Thus, the effective field Ei∗ coincides with Ei0 , Ei∗ = Ei0 ,

(9.148)

and in this case, the tensor Cij∗ is a linear function of the volume fraction p, Cij∗ = Cij0 + pPij .

(9.149)

The equations for the effective conductivity tensor obtained by the one-particle version of the EFM serve for the volume fractions of the inclusions in the region p < 0.3–0.4 [16].

9.4 The homogenization problem for elastic heterogeneous media Let the stiffness tensor Cij kl (x) of a heterogeneous elastic medium be a homogeneous in space random function. After introducing the reference medium with a constant

The homogenization problem

417

stiffness tensor Cij0 kl , the function Cij kl (x) is presented in the form Cij kl (x) = Cij0 kl + Cij1 kl (x).

(9.150)

0 or stress σ 0 fields applied to the medium, the strain For constant external strain εij ij εij (x) and stress σij (x) tensors in the medium satisfy the integral equations (3.92) and (3.104):  0 1 (x  )εmn (x  )dx  , (9.151) εij (x) = εij − Kij kl (x − x  )Cklmn  1 (x  )σmn (x  )dx  . (9.152) σij (x) = σij0 + Sij kl (x − x  )Bklmn

Here,  −1 . (9.153) Bij1 kl (x) = Bij kl (x) − Bij0 kl , Bij kl (x) = Cij−1kl (x), Bij0 kl = Cij0 kl The kernels Kij kl (x) and Sij kl (x) of the integral operators in these equations are expressed in terms of the second derivatives of the Green function gij (x) of the homogeneous reference medium Cij0 kl ,   0 − Cij0 kl δ(x). Kij kl (x) = − ∂i ∂k gj l (x) (ij )(kl) , Sij kl (x) = Cij0 mn Kmnrs (x)Crskl (9.154) 0 and σ 0 , solutions ε (x) and σ (x) of Eqs. (9.151) and For constant fields εij ij ij ij (9.152) are homogeneous in space random functions. Because the random functions 1 (x)σmn (x) are also homogeneous in space, the averages of Cij1 kl (x)εkl (x) and Bklmn these functions over the ensemble realization of the field Cij kl (x) are constants. Therefore, for the averages of the strain and stress tensors in Eqs. (9.151) and (9.152), we obtain the equations     0 1 − Kij kl (x − x  )dx  Cklmn (x)εmn (x) = εij (x) = εij   0 0 1 + Kij kl (x − x  )dx  Cklmn (x)σpq (x) , (9.155) Bmnpq = εij     1 (x)σmn (x) = σij (x) = σij0 + Sij kl (x − x  )dx  Bklmn   0 1 (x)εpq (x) . (9.156) Cmnpq = σij0 − Sij kl (x − x  )dx  Bklmn

Here, the equality −1 1 Cij1 kl (x)εij (x) = Cij1 kl (x)Cklmn (x)σmn (x) = −Cij0 kl Bklmn (x)σmn (x)

is taken into account.

(9.157)

418

Heterogeneous Media

As in the case of the conductivity problem, regularization of the operators with the kernels Kij kl (x) and Sij kl (x) on constants depends on conditions at infinity. If these 0 , the integrals in Eqs. (9.155) conditions correspond to a fixed external strain field εij and (9.156) are similar to the integrals in Eq. (9.22) [16], i.e.,   Kij kl (x − x  )dx  = 0, Sij kl (x − x  )dx  = −Cij0 kl . (9.158)     As a result, the averages εij (x) and σij (x) in Eqs. (9.155) and (9.156) take the forms     0 , σij (x) = σij0 + Cij1 kl (x)εkl (x) . εij (x) = εij



(9.159)

 Because of linearity of the problem, the average Cij1 kl (x)εkl (x) is a linear function 0: of the external field εij



0 . Cij1 kl (x  )εkl (x  ) = Qij kl εkl

(9.160)

Here, Qij kl is a constant tensor that depends   on the microstructure of the heterogeneous medium. Then, for the average σij (x) in Eq. (9.159), we have 

 0 0 σij (x) = Cij0 kl εkl + Qij kl εkl = Cij∗ kl εkl (x) ,

Cij∗ kl

= Cij0 kl

+ Qij kl ,

(9.161) (9.162)

where Cij∗ kl is the tensor of effective elastic stiffness of the heterogeneous medium. If the conditions at infinity correspond to a fixed external stress field σij0 , the integrals in Eqs. (9.155) and (9.156) are [16]   Kij kl (x − x  )dx  = Bij0 kl , Sij kl (x − x  )dx  = 0. (9.163)     As a result, for the averages εij (x) and σij (x) in Eqs. (9.155) and (9.156), we obtain     0 + Bij1 kl (x)σkl (x) , σij (x) = σij0 . εij (x) = εij



(9.164)

 Because the average Bij1 kl (x)σkl (x) is a linear function of the external field σij0 , the following equations hold: 

Bij1 kl (x)σkl (x) = Mij kl σkl0 ,   εij (x) = Bij0 kl σkl0 + Mij kl σkl0 = Bij∗ kl σkl (x) ,

(9.166)

Bij∗ kl

(9.167)

= Bij0 kl

+ Mij kl .

(9.165)

The homogenization problem

419

Here, Bij∗ kl is the effective compliance tensor of the heterogeneous medium. Because  −1 Cij∗ kl = Bij∗ kl , the tensors Qij kl and Mij kl in Eqs. (9.29) and (9.33) are related by the equation  −1 0 1 0 Qij kl = −Cij0 mn Mmnpq Cpqrs + Mrstv Ctvkl . Erskl

(9.168)

Here, Eij1 kl is the unit four rank tensor defined in Eq. (2.210).

9.4.1 The effective field method for static elasticity Let a finite region VR in an infinite heterogeneous elastic medium be considered as an RVE. The stress field in the region VR satisfies the equation, which follows from Eq. (9.152):  1 σij (x) − Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  = σij∗ (x), x ∈ VR , (9.169) VR  1 σij∗ (x) = σij0 + Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  . (9.170) R\VR

Here, R\VR is the complement of VR to the entire space R of the dimension d (d = 2, 3). The stress field σij∗ (x) on the right hand side of Eq. (9.169) is the external stress field acting on the region VR . The average of the field σij∗ (x) over the ensemble realizations of the random function Cij kl (x) is the effective external stress  field σij∗ (x) . On the basis of this average of Eq. (9.170), we obtain the equation for this field in the form    1 σij∗ (x) = σij0 + Sij kl (x − x  )dx  Bklmn (x)σmn (x) . (9.171) R\VR

Here, it is taken into account that for a constant external field σij0 , the average  Bij1 kl (x)σkl (x) is constant. The integral in Eq. (9.171) is calculated as follows: 

Sij kl (x − x  )dx  =   Sij kl (x − x  )dx  = = − Sij kl (x − x  )dx  + R VR  = Sij kl (x − x  )dx  .

Dij (x) = −

R\VR

(9.172)

VR

Here, it is implied that the conditions at infinity correspond to a fixed external stress  0 ∗ field σij , and therefore, Eq. (9.163) holds. As a result, the effective field σij (x) in

420

Heterogeneous Media

Eq. (9.171) takes the form   1 (x)σmn (x) , x ∈ VR . σij∗ (x) = σij0 − Dij kl (x) Bklmn

(9.173)

The second term on the right hand side of this equation takes into account the stress field induced in VR by the heterogeneities in the region R\VR .  After changing the external field σij∗ (x) in Eq. (9.169) to the effective field σij∗ in Eq. (9.173), we obtain the following equation for the stress field σij (x) in VR :  1 Sij kl (x − x  )Bklmn (x  )σmn (x  )dx  + σij (x) − VR  1 + Dij kl (x) Bklmn (x)σmn (x) = σij0 , x ∈ VR . (9.174)   1 (x)σmn (x) can be calculated over If the RVE is sufficiently large, the average Bklmn the RVE volume VR instead of the entire space, i.e.,   1 B 1 (x)σkl (x)dx. (9.175) Bij1 kl (x)σkl (x) ≈ VR VR ij kl This is the condition of self-consistency of the considered version of the EFM. Finally, Eq. (9.174) for the stress field inside the RVE takes the form    1 1 σij (x) − Dij kl (x) Bklmn (x  )σmn (x  )dx  = σij0 , Sij kl (x − x  ) − V R VR x ∈ VR .

(9.176)

Thus, the homogenization problem is reduced to solution of Eq. (9.176) for the stress field σij (x) inside the RVE and calculation of the average of the field Bij1 kl (x)σkl (x) over the RVE in Eq. (9.175). Then, the tensor Mij kl and the tensor of the effective compliance Bij∗ kl can be constructed from Eqs. (9.165) and (9.167). The EFM can be developed on the basis of Eq. (9.151) for the strain tensor εij (x). This equation can be presented in the form  1 ∗ εij (x) + Kij kl (x − x  )Cklmn (x  )εmn (x  )dx  = εij (x), x ∈ VR , (9.177) VR  ∗ 0 1 εij (x) = εij − Kij kl (x − x  )Cklmn (x  )εmn (x  )dx  , (9.178) R\VR

∗ (x) is an external strain field acting on the region V . Averaging Eq. (9.178) where εij R over the ensemble realization  ∗ of the  function Cij kl (x) yields the following equation for the effective strain field εij (x) :    ∗ 0 1 Kij kl (x − x  )dx  Cklmn (x)εmn (x) = εij (x) = εij − R\V  R 0 1 (x)εmn (x) , x ∈ VR , (9.179) = εij + Aij kl (x) Cklmn

The homogenization problem

 Aij kl (x) = −

421

Kij kl (x − x  ) +

R



Kij kl (x − x  )dx  = VR



Kij kl (x − x  )dx  .

VR

(9.180) 0 is fixed in the problem, By deriving this equation, we accept that the external field εij  and therefore, Eq. (9.158) holds. The assumption that the average Cij1 kl (x)εkl (x) can be calculated over the region VR yields the following equation for the strain field εij (x) in the RVE:



 εij (x) +

Kij kl (x − x  ) −

VR

 1 1 0 Aij kl (x) Cklmn (x  )εmn (x  )dx  = εij , x ∈ VR . VR (9.181)

This equation  is reciprocal to Eq. (9.176). After solution of this equation, the av1 erage Cij kl εkl over the RVE can be calculated and the tensors Qij kl and Cij∗ kl in Eqs. (9.160) and (9.162) can be constructed.

9.4.2 Discretization of the integral equations of the effective field method For numerical solution of Eqs. (9.176) and (9.181), the RVE VR is embedded in a cuboid W covered by a regular grid of approximating nodes x (n) . Then, the stress and strain fields in Eqs. (9.176) and (9.181) are approximated in W by the Gaussian quasiinterpolants σij (x) ≈

Mn Mn



(n) (n) σij ϕ(x − x (n) ), εij (x) ≈ εij ϕ(x − x (n) ), n=1

ϕ(x) =

1 (πH )d/2

(9.182)

s=1

 |x|2 exp − . H h2

(9.183)

Here, Mn is the total number of nodes x (n) in W , h is the node grid step, H is a dimen(n) sionless parameter of the order of 1, d is the space dimension, and σij = σij (x (n) ) and (n)

εij = εij (x (n) ) are unknown coefficients of the approximation. After substitution of Eq. (9.182) into Eqs. (9.176) and (9.181) of the EFM and application of the collocation method, we obtain the discretized equations for the coefficients of the approximations (m)

σij

−

Mn 

(m,n) 1(n) (m) 1(n) (n) = σij0 , m = 1, 2, ..., Mn, ij kl Bklmn − τ Dij kl Bklmn σmn n=1

(9.184) 1(n) Bij kl

= Bij1 kl (x (n) ),

(m) Dij kl

= Dij kl (x

(m)

),

(9.185)

422

Heterogeneous Media

(m)

εij +

Mn 

(m,n) 1(n) (m) 1(n) (n) 0 = εij , m = 1, 2, ..., Mn, ij kl Cklmn − τ Aij kl Cklmn εmn

s=1

(9.186) Cij1(n) kl

= Cij1 kl (x (n) ),

A(m) ij kl

= Aij kl (x

(m)

).

(9.187)

(m,n) Here, τ = hd /VR , and the objects (m,n) ij kl and ij kl are (m) (m) (m,n) − x (n) ), ij(m,n) − x (n) ), ij kl = ij kl (x kl = ij kl (x

(9.188)

where the functions ij kl (x) and ij kl (x) are the integrals over the entire x-space      ij kl (x) = Kij kl (x − x )ϕ(x )dx , ij kl (x) = Sij kl (x − x  )ϕ(x  )dx  . (9.189) Explicit expressions for ij kl (x) and ij kl (x) are presented for the 2D case in Eqs. (5.74)–(5.79) and for the 3D case in Eqs. (5.90) and (5.106). The discretized equations (9.184) and (9.186) can be solved by the iteration methods considered in Chapter 5, and for regular node grids, the FFT algorithms can be adopted for calculation of the matrix-vector products in the process of iterations.

9.4.3 Homogenization of matrix composites In this section, we consider an isotropic homogeneous 3D medium containing a set of isotropic spherical inclusions (Fig. 9.8). The RVE is assumed to be a sphere with a finite number of inclusions inside. For such an RVE, the tensors Aij kl and Dij kl in Eqs. (9.180) and (9.172) are constant and have the forms

 1−κ 0 2 5 − 2κ 0 1 Eij kl + Eij1 kl − Eij2 kl , (9.190) Aij kl = 9μ0 15μ0 3

 4μ0 (1 − 4κ0 ) 2 2μ0 (5 + 4κ0 ) 1 Dij kl = Eij kl − Eij1 kl − Eij2 kl , 9 15 3 λ0 + μ 0 . (9.191) κ0 = λ0 + 2μ0 Here, Eij1 kl , Eij2 kl are the elements of the tensor basis in Eq. (2.210) and λ0 , μ0 are Lame parameters of the host medium. Let the RVE VR contain only one spherical inclusion v, the spheres VR and v have the same centers, and v/VR = p, where p is the volume fraction of inclusions. In this case, Eqs. (9.176) and (9.181) have exact solutions that are constant inside the inclusion v and have the forms  −1 0 1 , ϒij kl = Eij1 kl + (1 − p)Aij mn Cmnkl , x ∈ v, (9.192) εij = ϒij kl εkl

The homogenization problem

 −1 1 σij = ij kl σkl0 , ij kl = Eij1 kl − (1 − p)Dij mn Bmnkl , x ∈ v.

423

(9.193)

  Thus, for the averages Bij1 kl σkl and Cij1 kl εkl in Eqs. (9.160) and (9.165), we have 

Bij1 kl σkl = Mij kl σkl0 , Mij kl = pBij1 mn mnkl ,  0 , Qij kl = pCij1 mn ϒmnkl . Cij1 kl εkl = Qij kl εkl

(9.194) (9.195)

As a result, the tensors of the effective compliance Bij∗ kl and effective stiffness Cij∗ kl of the composite in Eqs. (9.167) and (9.162) are  −1 1 1 − (1 − p)Dmnpq Bpqkl , Bij∗ kl = Bij0 kl + pBij1 mn Emnkl  −1 1 1 + (1 − p)Amnpq Cpqkl . Cij∗ kl = Cij0 kl + pCij1 mn Emnkl

(9.196) (9.197)

−1  = Cij∗ kl . It is possible to verify that Bij∗ kl If the materials of the host medium and the inclusions are isotropic, then the equations for the effective bulk and shear moduli K∗ , μ∗ of the composite take the forms K0 (K − K0 ) 3K0 , , s1 = K + (1 − p) s1 (K − K0 ) 3K0 + 4μ0 μ0 (μ − μ0 ) 6 (K0 + 2μ0 ) , s2 = . μ∗ = μ0 + p μ0 + (1 − p) s2 (μ − μ0 ) 5 (3K0 + 4μ0 )

K∗ = K0 + p

(9.198) (9.199)

Here, K0 , μ0 and K, μ are the bulk and shear moduli of the matrix and inclusion materials.

Simple cubic lattice of spherical inclusions If identical spherical inclusions compose a simple cubic lattice in the matrix material, the tensor of the effective elastic stiffness has cubic symmetry. A convenient basis for presentation of such tensors consists of the following three linearly independent rank four tensors: Hij1 kl = δ1i δ1j δ1k δ1l , Hij2 kl = Eij2 kl − Hij1 kl , Hij3 kl = 2(Eij1 kl − Hij1 kl ). (9.200) In this basis, the tensors Bij0 kl and Mij kl in Eq. (9.167) for the effective compliance tensor Bij∗ kl are presented in the form Bij0 kl = b10 Hij1 kl + b20 Hij2 kl + b30 Hij3 kl , Mij kl = m1 Hij1 kl + m2 Hij2 kl + m3 Hij3 kl , (9.201)

424

Heterogeneous Media

where the scalar coefficients b10 , b20 , b30 are b10 =

μ0 + 3K0 2μ0 − 3K0 1 , b20 = , b30 = . 9K0 μ0 18K0 μ0 4μ0

(9.202)

As a result, the tensor Bij∗ kl takes the form Bij∗ kl = Bij0 kl + Mij kl = b1∗ Hij1 kl + b2∗ Hij2 kl + b3∗ Hij3 kl , b1∗

= b10

+ pm1 ,

b2∗

= b20

+ pm2 ,

b3∗

= b30

(9.203)

+ pm3 .

(9.204)

The coefficients m1 , m2 , m3 are to be calculated from Eq. (9.165):  0 δ1i δ1j + Bij1 kl σkl = (m1 − 2m3 )σ11     0 0 0 δ2i δ2j + δ3i δ3j + (σ22 + σ33 )δij + 2m3 σij0 . + m2 σ11

(9.205)

In order to find the values of m1 , m2 , m3 , Eq. (9.176) should be solved two times for the external stress fields σij0 in the forms σij0(1) = δ1i δ1j , σij0(2) = 2δ1(i δ2j ) .

(9.206)

The products of these tensors with the tensor Mij kl are 0(1)

Mij kl σkl

= m1 δ1i δ1j + m2 (δ2i δ2j + δ3i δ3j ),

0(2) Mij kl σkl

= m3 (δ1i δ2j + δ2i δ1j ).

(9.207) 0(1)

If the solutions of Eq. (9.165) for the right hand sides σij

0(2)

and σij

(1)

are σij (x) and

σij(2) (x), then the coefficients m1 , m2 , m3 are expressed in terms of the components of   (1) (2) the tensors Bij1 kl σkl and Bij1 kl σij kl as follows:    (1) (1) (2) 1 1 1 σkl , m2 = B22kl σkl , m3 = B12kl σkl . m1 = B11kl

(9.208)

The tensor of effective elastic stiffness Cij∗ kl is presented in the form similar to (9.203) Cij∗ kl = c1∗ Hij1 kl + c2∗ Hij2 kl + c3∗ Hij3 kl ,

(9.209)

where scalar coefficients c1∗ , c2∗ , c3∗ are expressed in terms of the coefficients b1∗ , b2∗ , b3∗ in Eq. (9.204), i.e., c1∗ =

b∗ 1 ∗ 1 (b1 + b2∗ ), c2 = − 2 , c3∗ = ∗ ,  = (b1∗ − b2∗ )(b1∗ + 2b2∗ ). (9.210)   4b3

The homogenization problem

425

Another form of presentation of the tensor Cij∗ kl is

  1 Cij∗ kl = K∗ Eij2 kl + 2μ∗ Eij1 kl − Hij1 kl + 2M∗ Hij1 kl − Eij2 kl , 3

(9.211)

where the effective bulk modulus K∗ and shear moduli μ∗ and M∗ are 1 1 K∗ = (c1∗ + 2c2∗ ), μ∗ = c3∗ , M∗ = (c1∗ − c2∗ ). 3 2

(9.212)

Let the cube W : (2 × 2 × 2) contain 27 identical spherical inclusions, the centers of which compose a simple cubic lattice. In the case of stiff inclusions with E/E0 = 1000 and ν = ν0 = 0.3, the integral equation (9.176) is used for calculation of the stress tensor σij (x) inside W . Here, E, ν and E0 , ν0 are Young moduli and Poisson ratios of the inclusion and matrix materials. The results of the calculations of the relative effective bulk modulus K∗ /K0 and shear moduli μ∗ /μ0 , M∗ /μ0 are presented in Fig. 9.15 for the grid steps h = 0.1, 0.04, 0.02, 0.01 and H = 1. The crosses in Fig. 9.15 are the values of the effective elastic constants of the composite with a simple cubic lattice of absolutely rigid spheres presented in [12] (NK). The results of calculation for the RVE with 8 and 27 inclusions are shown in Fig. 9.16. The effective elastic constants of the composite with a cubic lattice of soft inclusions (E/E0 = 0.001, ν = ν0 = 0.3) are shown in Fig. 9.17. Crosses in this figure are the values of the effective elastic moduli of the composites with a simple cubic lattice of spherical pores presented in [13] (IN) and [14] (Ku). In this case, the numerical scheme based on Eq. (9.181) for strains turns out to be more efficient than the scheme based on Eq. (9.176) for stresses.

Random set of spherical inclusions Let identical spherical inclusions be randomly distributed in the cube W of the sizes (2 × 2 × 2). The following statistical model of the inclusion set is accepted. First, we take a regular lattice of identical spherical inclusions of radius R 0 centered at the points X (i) = (x1(i) , x2(i) , x3(i) ), i = 1, 2, ..., N . Then, the center of the ith inclusion is (i) (i) (i) moved to the point x1 + r1 , x2 + r2 , x2 + r2 , where r1 , r2 , r3 are independent random variables homogeneously distributed in the interval {−(R p − R 0 ), (R p − R 0 )} and R p is the radius of the inclusions corresponding to the percolation threshold of the initial regular lattice of inclusions embedded in the cube W . The inclusion overlap and their translations outside the region W are not allowed. Spatial orientation of a cell with a fixed realization of the inclusions is also random, and the corresponding distribution over orientations is homogeneous. An example of the randomized FCC cell is shown in Fig. 9.8 for the volume fraction of the inclusions p = 0.2. In order to calculate the effective elastic constants of the composite, we take a fixed realization of a random set of inclusions in the cube W and solve Eq. (9.176) for stresses six times for the following external stress fields σ 0(k) :

426

Heterogeneous Media

Figure 9.15 Dependencies of the relative bulk modulus K∗/K0 and shear moduli μ∗/μ0 and M∗/μ0 of the composite with a simple cubic lattice of stiff spherical inclusions (E/E0 = 1000, ν = ν0 = 0.3) on their volume fractions p. The RVE contains 27 inclusions; h is the step of the cubic node grid; + are the data for the composite with a cubic lattice of absolutely rigid spheres presented in [12] (NK). 0(k)

σij

0(5) σij

0(4)

= δki δkj , k = 1, 2, 3, σij = 2δ1(i δ3j ) ,

0(6) σij

= 2δ1(i δ2j ) ,

(9.213)

= 2δ2(i δ3j ) .

(9.214) (k)

We denote the corresponding solutions as σij (x), k = 1, 2, ..., 6. In order to calculate the tensor Mij kl in Eq. (9.165) for Bij∗ kl , we multiply the solutions σij (x) by the tensor Bij1 kl (x) and average the results over the region VR . Finally, we obtain six  (k) (k) constant tensors Tij = Bij1 lm σlm (k = 1, 2, ..., 6) whose components are related to the components of the tensor Mij kl in Eq. (9.165) by the equations (k)

(k)

(4)

Mij kk = Tij , k = 1, 2, 3, Mij 12 = Mij 21 = Tij ,

(9.215)

The homogenization problem

427

Figure 9.16 Dependencies of the relative bulk modulus K∗/K0 and shear moduli μ∗/μ and M∗/μ0 of

the composite with a cubic lattice of stiff spherical inclusions (E/E0 = 1000, ν = ν0 = 0.3) on their volume fractions p; the RVE contains 8 or 27 inclusions; h = 0.02; + are the data for the composite with a cubic lattice of absolutely rigid spheres presented in [12] (NK).

(5)

(6)

Mij 13 = Mij 31 = Tij , Mij 23 = Mij 32 = Tij .

(9.216)

of the cell with Then, the tensor Mij kl should be averaged over the spatial   orientations the fixed realization of the inclusions. As a result, for Mij kl , we obtain the equation (see Appendix 9.A)  1 Mij kl = m∗1 Eij2 kl + m∗2 (Eij1 kl − Eij2 kl ), 3 1 (1) 1  (1) ∗ (2) ∗ m1 = (M + M ), m2 = 2M − M (2) + 3M (3) , 9 15 M (1) = M1111 + M2222 + M3333 , M (2) = M1122 + M2211 + M1133 + 

+ M3311 + M2233 + M3322 , M (3) = M1212 + M1313 + M2323 .

(9.217) (9.218)

(9.219)

428

Heterogeneous Media

Figure 9.17 Dependencies of the relative bulk modulus K∗ /K0 and shear moduli μ∗/μ0 and M∗/μ0

of the composite with a cubic lattice of soft spherical inclusions (E/E0 = 0.001, ν = ν0 = 0.3) on their volume fractions p; the RVE contains 27 inclusions; h is the step of the cubic node grid; + are the data for the medium with a simple cubic lattice of spherical pores presented in [13] (IN); × are the data presented in [14] (Ku).

In the next step, the coefficients m∗1 , m∗2 should be averaged over several     realizations of the inclusions in VR . The corresponding averages are denoted as m∗1 and m∗2 . Finally, the tensor of effective elastic stiffness of the composite Cij∗ kl is isotropic and takes the form

 1 Cij∗ kl = K∗ Eij2 kl + 2μ∗ Eij1 kl − Eij2 kl , (9.220) 3 K0 μ0  ∗  , μ∗ =  . (9.221) K∗ = 1 + 9pK0 m1 1 + 2pμ0 m∗2 The dependencies of the relative bulk K∗ /K0 and shear μ∗ /μ0 moduli of the composite with stiff spherical inclusions are presented in Fig. 9.18 (E/E0 = 1000,

The homogenization problem

429

Figure 9.18 Dependencies of the relative bulk modulus K∗/K0 and shear modulus μ∗/μ0 of the compos-

ite with a random set of stiff spherical inclusions (E/E0 = 1000, ν = ν0 = 0.25) on their volume fractions p; the RVE contains 14 inclusions; h is the step of the cubic node grid; + are the data for the composite with a random set of absolutely rigid spheres presented in [15] (SL).

Figure 9.19 Dependencies of the relative bulk modulus K∗/K0 and shear modulus μ∗/μ0 of the compos-

ite with a random set of stiff spherical inclusions (E/E0 = 1000, ν = ν0 = 0.25) on their volume fraction p; the RVE contains 1, 8, or 14 inclusions; h = 0.02; + are the data for the composite with a random set of absolutely rigid spheres presented in [15] (SL).

ν = 0.49, ν0 = 0.25). The calculations are performed for the RVE with 14 inclusions for the steps of the node grid h = 0.1, 0.04, 0.02. Crosses in these figures correspond to the results of the calculations of the effective elastic constants based on the finite element method for the RVE containing 30 identical absolutely rigid balls with periodical boundary conditions on the RVE sides [15] (SL). In Fig. 9.19, the dashed line correspond to the RVE that contains 1 inclusion (Eqs. (9.198) and (9.199)), lines with squares to the RVE with 8 inclusions, and lines with circles to 14 inclusions for the initial FCC latice, h = 0.02. Crosses correspond to the data presented in [15] (SL).

430

Heterogeneous Media

The bulk and shear moduli of the composite with a random set of soft inclusions (E/E 0 = 0.001E0 , ν = ν0 = 0.25) are compared with the numerical solution presented in [15] for a medium with spherical pores in Fig. 9.20. In this case, Eq. (9.181) of the EFM for strains was used, and it is seen that calculations based on the RVE with eight inclusions predict the values of the effective elastic constants of the composite that practically coincide with the results presented in [15]. For construction of the tensor Qij kl in Eqs. (9.160), Eq. (9.181) for the strain tensor is solved six times with the right hand side similar to Eqs. (9.213) and (9.214) for each fixed realization of a random set of inclusions inside the RVE. Then, the tensor Qij kl is averaged over the orientations similar to tensor Mij kl (Appendix 9.A). In both cases of stiff and soft inclusions, the deviations of the effective elastic constants in different realizations were small and do not exceed 3%.

Figure 9.20 Dependencies of the relative bulk modulus K∗ /K0 and shear modulus μ∗/μ0 of the composite with a homogeneous random set of soft spherical inclusions (E/E0 = 0.001, ν = ν0 = 0.25) on their volume fractions p; the RVE contains 1, 8, or 14 inclusions; h = 0.02; + are the data for the porous medium presented in [15] (SL).

9.4.4 The one-particle version of the effective field method for matrix composites The one-particle version of the EFM can be used for approximate solution of the homogenization problem in the case of matrix-inclusion composites. Let v n (n = 1, 2, ...) (n) be the regions occupied by the inclusions with the stiffness tensor Cij kl (x) in a homogeneous host medium (matrix) with the stiffness tensor Cij0 kl . The hypotheses of the method are formulated in Section 9.3.2 and allow reducing the homogenization problem to the one-particle problem. The solution of this problem defines the field inside the nth inclusion embedded in the homogeneous host medium and subjected to a con∗, stant external effective field εij ∗ εij (x) = ij kl (x)εkl , x ∈ v n , n = 1, 2, .... (n)

(9.222)

The homogenization problem

431

Here, the function (n) ij kl (x) depends on the elastic properties and the shape of the nth ∗ inclusion. Repeating derivations of Section 9.3.2, we obtain that the effective field εij acting on each inclusion has the form  −1 ∗ 0 εij = Eij1 kl − pA εkl , ij mn Pmnkl

(9.223)

  1 Cij1 mn (x) mnkl (x)dx , v v  A Kij kl (x) (x)dx. ij kl = Pij kl =

(9.224) (9.225) (n)

Here, the function ij kl (x) coincides with ij kl (x) in the region v n and (x) is a specific correlation function of the random field of inclusions defined in Eqs. (9.113) 1 and (9.115). The equation for the average Cij kl (x)εkl (x) follows from Eqs. (9.222) and (9.223) in the form   −1 1 0 Cij1 kl (x)εkl (x) = pPij mn Emnkl − pA εkl . (9.226) mnpq Ppqkl Then, Eqs. (9.161) and (9.162) yield the following equation for the effective elastic stiffness tensor Cij∗ kl :  −1 1 Cij∗ kl = Cij0 kl + pPij mn Emnkl − pA . mnpq Ppqkl

(9.227)

In this equation, the tensor A ij kl is determined by statistical properties of the random set of inclusions, and inclusion elastic properties and shapes define the tensor Pij kl . In the cases of ellipsoidal inclusions, the tensor ij kl in Eq. (9.222) is constant and has the form   −1 1 ij kl = Eij1 kl + Aij mn (a)Cmnkl , Aij kl (a) = Kij kl (x)dx, (9.228) v

where aij = ai δij (no sum on i!) and a1 , a2 , a3 are the ellipsoid semiaxes. As a result, the tensor Pij kl in Eq. (9.224) takes the form    −1 1 1 1 + Amnrs (a)Crskl v(a) , (9.229) Pij kl = Cij1 mn Emnkl v(a) 4 v(a) = πa1 a2 a3 , (9.230) 3 where averaging is performed over the ensemble realizations of the ellipsoid semiaxes, the orientations, and the elastic properties of the inclusions. Let the inclusions be spheres of a random radius a homogeneously distributed in space. In this case, the function (x) = (|x|) has spherical symmetry, and A ij kl = Aij kl ,

(9.231)

432

Heterogeneous Media

where the tensor Aij kl has the form (9.190) and aij = δij . As a result, Eq. (9.227) for the tensor Cij∗ kl takes the form  −1 1 1 Cij∗ kl = Cij0 kl + pCij1 mn Emnkl + (1 − p)Amnrs Crskl .

(9.232)

Particular forms of Eq. (9.232) for the media with spheroidal inclusions, thin ellipsoidal discs, and long cylinders are presented in [16]. For the composite with spherical isotropic inclusions, Eq. (9.232) yields Eqs. (9.198) and (9.199) for the effective bulk and shear moduli. The approximation (9.227) was compared with experimental data and exact and numerical solutions for matrix composites with various types of inclusions in [16], and for moderate volume fractions of inclusions (p < 0.3–0.4), it provides reliable values of the effective elastic constants.

9.5 Homogenization of elastic media with multiple cracks In this section, we consider an infinite homogeneous elastic medium containing a homogeneous in space random set of cracks with surfaces (q) (q = 1, 2, ...). The medium is subjected to a constant external stress field σij0 . Similar to Eq. (6.1), the stress tensor σij (x) in the medium is presented as the sum of integrals over the crack surfaces

 (q) (q) σij (x) = σij0 + Sij kl (x − x  )nk (x  )bl (x  )d . (9.233) q

(q)

(q)

(q)

Here, ni (x) is the normal to the surface (q) , bi (x) is the opening vector of the qth crack, and the kernel Sij kl (x) of the integral operator in this equation is defined in Eq. (6.2). If the crack sides are free of forces, the following boundary conditions should be satisfied on the crack surfaces: (q)

nj (x)σj i (x)|(q) = 0, q = 1, 2, ....

(9.234) (q)

The system of integral equations for the vectors bi (x) follows from the boundary conditions in the form

 (p,q) (q) (p) Tij (x, x  )bj (x  )d = nj (x)σj0i (x), x ∈ (p) , (9.235) q

(q)

(p,q)

Tij

(x, x  ) = −nk (x)Skij l (x − x  )nl (x  ), p, q = 1, 2, ... . (p)

(q)

The strain tensor εij (x) in the cracked medium is defined in Eq. (3.115):

 (q) (q) 0 εij (x) = εij + Kij kl (x − x  )Cklmn nm (x  )bn (x  )d , q

(q)

(9.236)

(9.237)

The homogenization problem

433

−1  0 εij = Bij kl σkl0 , Bij kl = Cij kl .

(9.238)

Here, the kernel Kij kl (x) is in Eq. (6.2) and Cij kl is the stiffness tensor of the homogeneous host medium.

9.5.1 The effective compliance tensor of a cracked medium For homogeneous in space random crack sets and a constant external stress field σij0 , the averages of the stress and strain tensors over the ensemble realizations of the crack sets are constant and are related by the Hooke law 

 εij (x) = Bij∗ kl σkl (x) .

(9.239)

Here, Bij∗ kl is the effective compliance tensor of the cracked medium. If the boundary   condition at infinity corresponds to a fixed external stress field, then σij (x) = σij0 .   The following equation for the average strain tensor εij (x) follows from Eq. (9.237):  0 + εij (x) = εij





Kij kl (x − x  )dx  Cklmn nm (x)bn (x)(x) ,

ni (x)bj (x)(x) =



(q)

(q)

ni (x)bj (x)(q) (x).

(9.240) (9.241)

q

Here, (q) (x) is the delta function concentrated on the crack surface (q) . For homogeneous in space random crack sets, the average ni (x)bj (x)(x) in Eq. (9.240) is constant, and assuming the ergodic property, it is presented in the form   1 ni (x)bj (x)(x)dx = ni (x)bj (x)(x) = lim W →∞ W W ⎤ ⎡

 M 

  1 M (q) (q) ni (x)bj (x)d⎦ = m0 ni (x)bj (x) , (9.242) = lim ⎣ W →∞ W M (q)



q=1

M   M  1

(q) (q) m0 = lim ni (x)bj (x)d. , ni (x)bj (x) = lim (q) W →∞ W W →∞ M  q=1

(9.243) Here, W is a region that occupies the entire space in the limit W → ∞ and M is  the number of cracks in W . For linearity of the problem, the average ni (x)bj (x) is a linear function of the tensor σij0 , i.e., 

 ni (x)bj (x) = ij kl σkl0 .

Here, the tensor ij kl is constant and depends on the random set of cracks.

(9.244)

434

Heterogeneous Media

  The equation for the average strain tensor εij (x) follows from Eqs. (9.240)– (9.244):    0 0 + m0 Kij kl (x − x  )dx  Cklmn mnrs σrs . (9.245) εij (x) = εij Because the external stress field σij0 is assumed to be fixed, the integral in this equation is defined in Eq. (9.163), i.e.,  −1  Kij kl (x − x  )dx = Bij kl = Cij kl , (9.246) 0 =B 0 and because εij ij kl σkl , we obtain the following equation for the average strain tensor of the cracked medium:     0 + m0 ij kl σkl0 = Bij kl + m0 ij kl σkl  . εij (x) = εij (9.247)

Comparing this equation with Eq. (9.239), we obtain the equation for the effective compliance tensor of the cracked medium in the form Bij∗ kl = Bij kl + m0 ij kl .

(9.248)

9.5.2 The representative volume elements of cracked media The RVE VR of the cracked medium is defined by the equations similar to Eq. (9.43)     1 1 εij (x)dx ≈ εij (x)dx, (9.249) εij (x) = lim V →∞ V V VR VR     1 1 σij (x) = lim σij (x)dx ≈ σij (x)dx. (9.250) V →∞ V V VR VR Thus, the RVE VR should be as large as the average of the stress and strain tensors over VR coinciding with the averages over the entire x-space. The equation for the stress tensor inside the RVE follows from Eqs. (9.233) and (9.241) in the form  ∗ Sij kl (x − x  )nk (x  )bl (x  )(x  )dx  , x ∈ VR , (9.251) σij (x) = σij (x) + VR  σij∗ (x) = σij0 + Sij kl (x − x  )nk (x  )bl (x  )(x  )dx  . (9.252) R\VR

Here, R\VR is the complement of the RVE VR to the entire space R and the tensor σij∗ (x) is the external stress field acting on the RVE. The integral term in Eq. (9.252) is the stress field induced in the region VR by the cracks in the region R\VR . Averaging Eq. (9.252) over the ensemble realizations of the crack set yields the following  ∗ equation for the effective field σij (x) acting on VR : 

 σij∗ (x) = σij0 +

R\VR

Sij kl (x − x  )dx  nk (x)bl (x)(x) =

(9.253)

The homogenization problem

435

 = σij0 + m0

Sij kl (x − x  )dx  nk (x)bl (x) .

(9.254)

R\VR

Here, Eqs. (9.242) and (9.243) are taken into account. In the spirit of the EFM, we  assume that the average ni (x)bj (x) can be calculated over the cracks inside the RVE if VR contains a sufficiently large number of cracks M0 , i.e., M0      1

(q) (q) ni (x)bj (x) ≈ ni (x)bj (x) V = ni (x)bj (x)d. R (q) M0 

(9.255)

q=1

 As a result, the effective field σij∗ (x) in Eq. (9.253) takes the form 

 σij∗ (x) = σij0 + m0

R\VR

Sij kl (x − x  )dx  nk (x)bl (x)VR , m0 =

M0 . VR (9.256)

The integral in this equation is defined in Eq. (9.172),   Sij kl (x − x  )dx  = Sij kl (x − x  )dx  . Dij kl (x) = − R\VR

(9.257)

VR

It is taken into account that for a fixed external stress σij0 , Eq. (9.163) holds.

Figure 9.21 An ellipsoidal RVE with a number of cracks inside.

If VR is an ellipsoidal region (Fig. 9.21), then the integral Dij kl in Eq. (9.257) is constant for x ∈ VR . Let VR be a spheroid with the semiaxes R1 = R2 = R and R3 . In the Cartesian basis coinciding with the spheroid semiaxes, the tensor Dij kl takes the form

   1 Dij kl = d1 Pij2 kl + d2 Pij1 kl − Pij2 kl + d3 Pij3 kl + Pij4 kl + d5 Pij5 kl + d6 Pij6 kl , 2 (9.258) λ+μ d1 = −μ0 (4κ − 1 − 2 (3κ − 1) f0 − 2f1 ) , κ = , (9.259) λ + 2μ d2 = −2μ (1 − (2 − κ) f0 − f1 ) , d3 = −2μ ((2κ − 1) f0 + 2f1 ) , (9.260) d5 = −4μ (f0 + 4f1 ) , d6 = −8μ (κf0 − f1 ) , (9.261)

436

Heterogeneous Media

where λ and μ are Lame constants of the host medium, and  1−g κ 2 2 , f = 2 + γ g − 3γ , 1 2(1 − γ 2 ) 4(1 − γ 2 )2

  R γ2 arctan γ2 −1 , γ = > 1. g= R3 γ2 −1

f0 =

(9.262) (9.263)

Tensors Pijk kl = Pijk kl (m) (k = 1, 2, ..., 6) in Eq. (9.258) are defined in Eqs. (2.212) and (2.213), where mi is the unit vector of the x3 -axis. If R3 → 0 (γ → ∞), then f0 , f1 → 0 and

 1 (9.264) Dij kl = −μ (4κ − 1) Pij2 kl − 2μ Pij1 kl − Pij2 kl . 2 Note that the size of the RVE does not affect the value of the tensor Dij kl : It depends only  on the aspect ratios of the ellipsoid VR . Thus, the equation for the effective field σij∗ follows from Eqs. (9.256)–(9.258) in the form 

M0 

m0 (q) (q) Dij kl nk (x)bl (x)d. σij∗ = σij0 − M0 q

(9.265)

q=1

 Replacing in Eq. (9.251) the field σij∗ (x) with the effective field σij∗ , we obtain the following equation for the stress field σij (x) in the RVE: σij (x) = σij0 +

  m0 (q) (q) Dij kl nk (x  )bl (x  )d , Sij kl (x − x  ) − M0 (q)

M0 

q=1

x ∈ VR .

(9.266)

The boundary conditions (9.234) on the crack surfaces yield the following system of (q) integral equations for the crack opening vectors bi (x) inside the RVE: M0 

(q) q=1 



(p,q) Tij (x, x  ) +

 m0 (p) (q)  (q) (p) nk (x)Diklj nl (x ) bj (x  )d = ti (x), M0

x ∈ (p) , (p,q) (p) Tij (x, x  ) = −nk (x)Skij l (x

(9.267) −x



(q) )nl (x  ),

(p) (p) ti (x) = nj (x)σij0 ,

(9.268)

p, q = 1, 2, ..., M0 . Unlike the infinite system (9.235), only cracks inside the RVE are involved in the system (9.267). The term proportional to the tensor Dij kl in Eq. (9.267) takes into account the presence of cracks outside VR .

The homogenization problem

437

9.5.3 The one-particle version of the effective field method for a medium with elliptical cracks Let the crack surfaces be planar ellipses with random semiaxes and orientations. If the RVE VR contains only one crack with normal ni to its surface  (M0 = 1), then Eq. (9.267) takes the form    Tij (x, x  ) + m0 nl Dlikj nk bj (x  )d = nj σj0i , x ∈ , (9.269) 

Tij (x, x  ) = Tij (x − x  ) = −nk Skij l (x − x  )nl .

(9.270)

The solution of this equation follows from the property of polynomial conservativity for elliptical cracks (Section 3.4) in the form bi (x1 , x2 ) = Bi z(x1 , x2 ), x1 , x2 ∈ ,  z(x1 , x2 ) = 1 − (x1 /a1 )2 − (x2 /a2 )2 ,

(9.271) (9.272)

where x1 , x2 are the Cartesian axes directed along the semiaxes a1 , a2 (a1 ≥ a2 ) of the ellipse , Bi is a constant vector, and from Eq. (9.269) and regularization (3.122), we obtain  −1 τ 2 0 0 nk Dkij l nl nm σmj , τ = π(a1 a2 )3/2 m0 , (9.273) Bi = Tij + √ 3 a1 a2  ∞ Tij (x1 , x2 )[z(x1 , x2 ) − 1]dx1 dx2 . (9.274) Tij0 = −∞

In the case of an isotropic medium, explicit expression for the tensor Tij0 is Tij0 = 0, i = j, μa2 0 T11 = 2 (c1 + ν(c2 − 2c1 )) , 2a1 (1 − ν) μa2 a2 μc1 0 0 T22 , = 2 = 2 (c1 + ν(c3 − 2c1 )) , T33 2a1 (1 − ν) 2a1 (1 − ν) c1 =

(9.275) (9.276) (9.277)

2 E (k) E(k) − K(k) a2 , , c2 = c 1 − , c3 = 3c1 − c2 , k = 1 − 1−k k a1 (9.278)

where K(k) and E(k) are the complete elliptic integrals of the first and second kinds and μ and ν are the shear modulus and Poisson ratio of the medium. For bi (x1 , x2 ) in  the form (9.271), the average ni (x)bj (x)(x) in Eq. (9.243) takes the form   ni (x)bj (x)(x) = !  −1 " τ τ ni Tj0k + √ nn Dnj km nm nl σlk0 . (9.279) = √ a1 a2 a1 a2

438

Heterogeneous Media

Here, the averaging is performed over the crack semiaxes (a1 , a2 ) and orientations ni . Comparison with Eqs. (9.242), (9.244), and (9.248) yields the equation for the effective compliance tensor Bij∗ kl of the cracked medium in the form Bij∗ kl = Bij0 kl + m0 ij kl , !  −1 " τ τ 0 ni Tj k + √ nn Dnj km nm nl m0 ij kl = √ a1 a2 a1 a2

(9.280) .

(9.281)

(ij )(kl)

In the one-particle version of the EFM, the semiaxes R1 , R2 , R3 of the ellipsoidal VR reflect statistical properties of the random crack set. If the spatial positions of cracks are uncorrelated (Poisson’s crack set), the orientations of the ellipsoid VR and of the crack  coincide, and the semiaxis R3 of the RVE VR tends to zero. As a result, the tensor Dij kl takes the form (9.264) with mi = ni . For such a tensor, the product (nn Dnj km nm ) in Eq. (9.280) vanishes. The equation for the effective compliance tensor Bij∗ kl is simplified as follows:   −1  τ Bij∗ kl = Bij0 kl + √ ni Tj0k nl . a1 a2 (ij )(kl)

(9.282)

The same equation for the effective compliance tensor can be obtained if the second term on the right hand side of Eq. (9.265) for the effective field σij∗ is absent. It means that each crack is subjected to the external field σij0 applied to the medium, and crack interactions are neglected. Thus, Eq. (9.282) is the tensor of the effective compliance of a cracked medium in the approximation of noninteracting cracks.

9.5.4 Numerical solution of the system (9.267) If the RVE contains a finite number of cracks, the system (9.267) can be solved only numerically. For discretization of this system, we cover the crack surfaces by a set of uniformly distributed nodes x (n) (n = 1, 2, ..., Mn) and approximate the crack opening vectors bi (x) by the equation bj (x) =

Mn

n=1

(n) bj ϕ(x

−x

(n)

 1 |x|2 ), ϕ(x) = exp − 2 . πH h H

(9.283)

Here, h is the distance between the neighboring nodes, H is a dimensionless parameter (n) of the order of 1, bi are constant vectors, and ϕ(x − x (n) ) is the Gaussian function concentrated in the plane P (n) tangent to the crack surface at the node x (n) . In the local Cartesian coordinates (y1 , y2 , y3 ) with the origin at the nth node and the y3 -axis directed along the normal ni (x (n) ) to , the function ϕ(y1 , y2 ) has the form # $ y12 + y22 1 ϕ(y1 , y2 ) = exp − 2 . (9.284) πH h H

The homogenization problem

439

As a result, the sums in Eq. (9.266) for the stress tensor in the RVE are approximated as follows: M0 

(q) q=1 

Sij lk (x − x  )nl (x  )bk (x  )d ≈ (q)

M0 

q=1

(q)

q

Mn

(n)

(n)

Iij k (x − x (n) )bk ,

(9.285)

n=1



(n)

Iij k (x − x (n) ) =

(q)

Sij kl (x − x  )ϕ(x  − x (n) )dP  nl , (n)

P (n) q

ni (x)bj (x)d ≈ h2

Mn

(9.286)

(n) (n)

(9.287)

ni bj .

n=1 (n)

In the local coordinates (y1 , y2 , y3 ) at the nth node, the integral Iij k (x − x (n) ) = Iij k (y1 , y2 , y3 ) is defined in Eq. (6.48). The stress tensor σij(m) = σij (x (m) ) at the mth node takes the form (m)

σij

= σij0 +

Mn

−

n=1 (m,n)

(n) (n) m0 2 h Dij kl nk bl . M0 Mn

(m,n) (n) bk

Iij k

(9.288)

n=1

(n)

(n)

Here, Iij k = Iij l (x (m) − x (n) ). The system of equations for vectors bi follows from the boundary conditions (9.234) satisfied at all nodes (the collocation method) Mn 

 h2 (m) (m,n) (n) (n) 0(m)  + m0 nj Dj ikl nk bl = ti , x (m) ∈ , Til M n=1  (n) / ,  = (q) , bj = 0 if x (n) ∈

(9.289) (9.290)

q (m,n) (m) (m,n) 0(m) (m) = −nk Ikij , ti = nj σj0i . Tij

(9.291)

This system can be written in the matrix form AX = F,

(9.292)

where the vectors of unknowns X and the right hand side F are defined by the equations X = [X 1 , X 2 , ..., X 3Mn ]T , F = [F 1 , F 2 , ..., F 3Mn ]T , ⎧ (n) ⎪ ⎨ b1 , (n−Mn) n X = , b2 ⎪ ⎩ (n−2Mn) , b3 ⎧ 0(n) ⎪ ⎨ t1 , 0(n−Mn) n F = , t2 ⎪ ⎩ 0(n−2Mn) , t3

(9.293)

n ≤ Mn, Mn < n ≤ 2Mn, 2Mn < n ≤ 3Mn, n ≤ Mn, Mn < s ≤ 2Mn, 2Mn < s ≤ 3Mn.

(9.294)

440

Heterogeneous Media

The matrix A of the dimensions 3Mn × 3Mn is reconstructed from Eq. (9.289). This matrix is nonsparse and nonsymmetric. For numerical solution of the system (9.292), the iterative methods discussed in Chapter 5 can be used.

A periodic system of penny-shaped cracks of the same orientation We consider a periodic system of penny-shaped cracks of radius a1 and the same orientations, the centers of which are at the nodes of a regular cuboid lattice with the sides α1 = α2 = α, α3 . The host medium is isotropic with a Poisson ratio of ν = 0.3. Let (x1 , x2 , x3 ) be Cartesian coordinates directed along the sides of the cuboid, and let the x3 -axis be orthogonal to the crack planes. In this case, the tensor ij kl in Eq. (9.244) has two nonzero components 1313 and 3333 , and the effective compliance tensor Bij∗ kl in Eq. (9.248) takes the form Bij∗ kl = Bij0 kl + m0 1313 Pij5 kl + m0 3333 Pij6 kl .

(9.295)

Here, the tensors Pijk kl = Pijk kl (n) (k = 1, 2, ..., 6) are defined in Eqs. (2.211) and (2.212) and ni is the unit vector of the x3 -axis.

Figure 9.22 (A) A cuboid with 27 elliptical cracks inside. (B) An elementary cell with an elliptical crack.

We consider first a cubic cell (α1 = α2 = α3 ) and take the RVE as a sphere containing 27 cracks (a central crack and its nearest neighbors, Fig. 9.22). For calculation of the components 1313 and 3333 of the tensor ij kl in Eq. (9.295), the system (9.289) was solved two times for the external stress fields σij0(1) = δi3 δj 3 and σij0(2) = 2δ1(i δ3j ) . (q)

according to the algoThe vectors bi (x) of the cracks in the RVE were  calculated  rithm presented in Chapter 6. Then, the average ni bj V was found from Eq. (9.244). R ∗ and shear modulus μ∗ on the paDependencies of the effective Young modulus E33 13 rameter α1 /a1 are the lines with squares in Figs. 9.23 and 9.24 (α1 /α3 = 1),

1 + m0 3333 E0 μ0 (3λ0 + 2μ0 ) E0 = . λ0 + μ0

∗ = E33

−1

, μ∗13 =

1 + m0 1313 μ0

−1 , (9.296)

The homogenization problem

441

∗ of an isotropic medium with Figure 9.23 Dependencies of the normalized effective Young modulus E33

a cuboid lattice of penny-shaped cracks on the parameter α1 /a1 for various aspects of the cuboid α1 /α3 ; lines with squares correspond to numerical solutions for the ellipsoidal RVE with 27 cracks; lines with dots show the results of the one-particle EFM; solid lines correspond to the results presented in [18].

Figure 9.24 The same as in Fig. 9.23 for the normalized effective shear modulus μ∗13 .

Solid lines in these figures represent the results presented in [18] and obtained by another numerical method, and the lines with dots are predictions of the one-particle EFM in Eqs. (9.280) and (9.281). Let the aspect ratio α1 /α3 of the cuboid cell be larger than 1. If we take a spherical RVE (as in the case of a cubic cell), the number of cracks inside the RVE will increase as the aspect ratio α1 /α3 increases, and the RVE will contain more than 27 cracks (a 2D analogue of this situation is shown in Fig. 9.25A,B). For instance, for α1 /α3 = 2, a spherical RVE will contain 47 cracks, and the volume of calculations increases. We can reduce the volume of calculations by choosing an ellipsoidal (spheroidal) RVE with the aspect ratio equal to the aspect ratio α1 /α3 of the elementary cell (Fig. 9.25C). In this case, the RVE contains the same number of cracks (27) as the spherical RVE for a cubic cell. The results are presented in Figs. 9.23 and 9.24 for α1 /α3 = 1, 2, 8. For the spherical RVE and 47 cracks inside (α1 /α3 = 2), the results of calculation do not differ from the case of a spheroidal RVE with aspect ratio 2 and 27 cracks inside. As seen from Figs. 9.23 and 9.24, the results of the presented method and the method used in [18] coincide practically. Note that predictions of the one-particle EFM become to differ substantially from the results of [18] with increasing aspect ratio of the cell α1 /α3 . In the one-particle version of the EFM, the tensor Dij kl in Eq. (9.281) was

442

Heterogeneous Media

Figure 9.25 2D analogy of spherical and ellipsoidal RVEs. (A) A spherical RVE for a cubic lattice of cracks with the parameters α1 = α2 = α3 . (B) A spherical RVE for a cuboid lattice of cracks with the parameters α1 = α2 , α1 /α3 = 2. (C) A spheroidal RVE with the aspect ratio a1 /a3 = 2 for the cuboid lattice of cracks with the parameters α1 = α2 , α1 /α3 = 2.

calculated for the aspect ratio of the RVE that corresponds to the aspect ratio of the actual crack cell. All calculations were performed for the hexagonal node grid on the crack surfaces with the step h/a1 = 0.0556 (the number of nodes on each crack surface is 1161), and H = 1.

Periodic system of elliptical cracks of the same orientation Let the centers of identical elliptical cracks be at the nodes of a cuboid lattice. The cracks have the aspect ratio a1 /a2 = 2 and the same orientations, the crack semiaxes are directed along the axes x1 , x2 , and the x3 -axis is orthogonal to the crack planes. Let a spheroidal RVE contain 27 cracks and have the same aspect ratio and orientation as the elementary cell of the crack lattice. In this case, the tensor ij kl in Eqs. (9.244) and (9.248) has three nonzero components: 3333 , 1313 , and 2323 . For calculation of 0(1) these components, three types of external stress field are applied to the RVE: σij = 0(2)

0(3)

2δ(i1 δj )3 , σij = 2δ(i2 δj )3 , and σij = δi3 δj 3 . Each crack surface is covered by a hexagonal node grid with the step h/a1 = 0.077 (1227 nodes on each crack), H = 1. The host medium is isotropic with a Poisson ratio of ν = 0.3. ∗ and shear moduli μ∗ and The dependencies of the effective Young modulus E33 13 μ∗23 on the parameter α1 /a1 for the cell aspect ratio α1 /α3 = 1, 2, 4, 8 are shown in ∗ and μ∗ are given in Eq. (9.296), and μ∗ Figs. 9.26–9.28. The effective moduli E33 13 23 is

−1 1 + m0 2323 . (9.297) μ∗23 = μ0

Random set of parallel penny-shaped cracks in a homogeneous isotropic host medium An experimental study of elastic wave propagation in a medium containing a random set of parallel penny-shaped cracks of the same radius a was performed in [19]. In these experiments, the host medium was isotropic with Young modulus E0 = 8.969 GPa, a Poisson ratio of ν0 = 0.263, and density ρ = 1712 kg/m3 . For

The homogenization problem

443

∗ /E of an isotropic medium conFigure 9.26 Dependencies of the relative effective Young modulus E33 0

taining a cuboid lattice of elliptical cracks with the aspect ratio a1 /a2 = 2 on the parameter α1 /a1 ; lines with squares, triangles, dots, and diamonds correspond to numerical solutions for different values of the aspect ratio α1 /α3 .

Figure 9.27 The same as in Fig. 9.26 for the relative effective shear modulus μ∗13 /μ0 .

Figure 9.28 The same as in Fig. 9.27 for the normalized effective shear modulus μ∗23 /μ0 .

numerical calculation of the effective elastic stiffness of the cracked medium, a spherical RVE with 21 cracks was considered (an example of a crack set realization is shown in Fig. 9.29). The cracks were randomly distributed in nine parallel planes, and crack intersections were prohibited. The calculations were performed for the parameter l/a = 0.758, where l is the distance between planes, and for the crack density parameter τ = 23 πa 3 m0 = 0.2094. The surface of each crack is covered by a hexago-

444

Heterogeneous Media

Figure 9.29 A spherical RVE containing 21 randomly distributed penny-shaped cracks.

nal node grid with the step h/a = 0.0476. For a fixed crack set realization, the problem 0(1) was solved three times by the following external stress tensors: σij = 2δ(i1 δj )3 , 0(2)

0(3)

σij = 2δ(i2 δj )3 , and σij = δi3 δj 3 (the x3 -axis is orthogonal to the crack planes).   The average of the function ni (x)bj (x) over 21 cracks was calculated, and nonzero components of the tensor ij kl in Eqs. (9.244) and (9.248) were found. Five realizations of the random crack set were generated, and the values of the effective elastic constants are averages over these realizations. The final effective compliance tensor Bij∗ kl is transversely isotropic and has the form in Eq. (9.295). In the experiments performed in [19], the lengths of propagating waves were much larger than the crack sizes. Hence, for calculation of wave velocities, the static effective moduli of the cracked material can be used. The phase velocities of quasilongitudinal and quasishear waves measured in [19] are expressed in terms of the effective elastic moduli and the angle θ between the x3 -axis and the wave vector of the propagating wave by the following equations (VP is quasilongitudinal wave velocity, VSV is quasishear wave velocity, and VSH is pure shear wave velocity) [20]: ) √ ∗ ∗ ∗ C1111 sin2 θ + C3333 cos2 θ + C2323 + M∗ , (9.298) VP (θ ) = 2ρ ) √ ∗ ∗ ∗ C1111 sin2 θ + C3333 cos2 θ + C2323 − M∗ , (9.299) VSV (θ ) = 2ρ ) ∗ ∗ C1212 sin2 θ + C2323 cos2 θ VSH (θ ) = , (9.300) ρ   2  2 2  ∗ ∗ ∗ ∗ sin θ − C3333 cos θ + − C2323 − C2323 (9.301) M ∗ = C1111   ∗ 2 ∗ sin2 (2θ ) . + C1133 + C2323 (9.302) Dependencies of quasilongitudinal and pure shear phase velocities on the angle θ are shown in Fig. 9.30. The lines with triangles show the results of the calculations, and crosses correspond to experimental data presented in [19]. Lines with squares

The homogenization problem

445

Figure 9.30 Quasilongitudinal VP and pure shear VSH phase velocities of elastic waves for different directions (θ) of the wave vector in the medium with a random distribution of penny-shaped cracks of the same orientation. Crosses (×) show the experimental data presented in [19]; lines with triangles represent the numerical solutions for a spherical RVE; lines with squares correspond to the noninteracting crack approximation; and lines with dots correspond to the one-particle EFM.

correspond to the approximation of noninteracting cracks given in Eq. (9.282). Predictions of the one-particle EFM in Eqs. (9.280) and (9.281) are shown as lines with dots. The tensor Dij kl in Eq. (9.258) is taken for a spherical RVE (R1 = R2 = R3 ). As seen from Fig. 9.30, the numerical solution and the one-particle EFM yield results that are in close agreement with the experimental data. The approximation of noninteracting cracks gives a noticeable error even for a rather small crack density considered in [19]. The velocities of SV -waves are predicted with the same accuracy as the velocities of P - and SH -waves.

A random set of penny-shaped cracks homogeneously distributed over the orientations We consider an isotropic medium with a Poisson ratio of ν = 0.3 containing a random set of penny-shaped cracks of the same radius a with homogeneous distribution of crack planes over orientations. Spatial positions of crack centers are independent but crack intersections are prohibited. This statistical model of a crack set is considered in [21]. In this work, the velocities of longitudinal and shear waves in the cracked medium were calculated by another numerical method. In this case, tensor ij kl in Eq. (9.280) is isotropic and has two essential components, i.e., ij kl = (1) Eij1 kl + (2) Eij2 kl , Eij1 kl = δi(k δj )l , Eij2 kl = δij δkl .

(9.303)

The effective compliance tensor Bij∗ kl in Eq. (9.281) is also isotropic and has the form Bij∗ kl

=

  1 λ (1) 1 (2) + m0 − m0 Eij kl − Eij2 kl , 2μ 2μ(3λ + 2μ)

(9.304)

and the effective stiffness tensor Cij∗ kl = (Bij∗ kl )−1 is Cij∗ kl = 2μ∗ Eij1 kl + λ∗ Eij2 kl ,

(9.305)

446

Heterogeneous Media

μ , 1 + 2μm0 (1) λ − 2μ(3λ + 2μ)m0 (2) . λ∗ = (1 + 2μm0 (1) )[1 + (3λ + 2μ)m0 ( (1) + 3 (2) )] μ∗ =

(9.306)

The phase velocities of longitudinal VP and shear VS waves in the cracked medium are defined by the equations ) * λ∗ + 2μ∗ μ∗ , VS = . (9.307) VP = ρ ρ For numerical calculations of the coefficients (1) and (2) in Eq. (9.303), a spherical RVE with 21 cracks inside is considered; the surface of each crack was covered by a hexagonal node grid, with the step h/a = 0.0476. Five realizations of the random crack set are generated, and the crack opening vectors in the RVE are calculated for two types of external stress fields: axial tension σij0(1) = δi3 δj 3 and pure shear 0(2)

σij = 2δ1(i δ2j ) . Then, the coefficients (1) and (2) are calculated from Eqs. (9.280) and (9.303). The resulting phase velocities are averaged over the realizations of the crack set. It turns out that the range of the result dispersion does not exceed 2% of the average. The dependencies of longitudinal and shear phase velocities on the crack density parameter τ are shown in Figs. 9.31 and 9.32, where VP0 , VS0 are the velocities of longitudinal and shear waves in the host medium. The lines with triangles show results of the numerical calculations, and crosses correspond to the numerical results presented in [21]. Lines with squares and dots are the results of the approximation of noninteracting cracks and the one-particle EFM. In the last case, the tensor Dij kl in Eq. (9.258) corresponds to a spherical RVE VR . As seen from Figs. 9.31 and 9.32, the numerical results are close to the results presented in [21]. The one-particle EFM also shows relatively good agreement with these results (the difference is about 3%

Figure 9.31 Dependencies of the normalized velocity VP of longitudinal waves in an isotropic medium with a random set of penny-shaped cracks homogeneously distributed over orientations. The line with triangles shows the numerical solution for a spherical RVE with 21 cracks; the line with dots corresponds to the one-particle EFM; the line with squares shows the approximation of noninteracting cracks; and crosses (×) show the numerical results of another method presented in [21].

The homogenization problem

447

Figure 9.32 The same as in Fig. 9.31 for velocities VS of shear waves.

for τ = 0.3). The approximation of noninteracting cracks diverges noticeably from the numerical solutions if τ > 0.15.

9.6

Homogenization of elasto-plastic composites

We consider an infinite homogeneous elasto-plastic host medium with the tensor of elastic stiffness Cij0 kl containing isolated heterogeneous inclusions in regions v n with (n)

the elastic stiffness tensors Cij kl (x) (n = 1, 2, ...). If V (x) is the characteristic function of the region occupied by the inclusions, the tensor of the elastic stiffness Cij kl (x) of the heterogeneous medium is presented in the form (n)

Cij kl (x) = Cij0 kl + Cij1 kl (x)V (x), Cij1 kl (x) = Cij kl (x) − C 0 if x ∈ v n . (9.308) The objectives of the homogenization problem are calculation of the average stress and strain tensors in the medium and construction of the stress–strain relation for an increasing external stress field σij0 applied to the medium. Let the constant external stress field increase from σij0(0) , and for σij0 = σij0(0) , let the deformations be purely elastic. Thus, the plastic deformations appear in the process of loading. If the loading process is divided into n small intervals (l) σ 0 (l = 1, 2, ..., n), the external stress at the end of the kth interval is σij0(k) = σij0(0) +

k

(l) σij0 .

(9.309)

l=1

The linearized integral equation for the stress increment (k) σkl (x) at the kth interval of loading has the form in Eq. (5.140),  (k) (k) (x  )(k) σmn (x  )dx  = (k) σij0 ,  σij (x) − Sij kl (x − x  )B (9.310) klmn (k) (x) = Bij1 kl (x) + Q(k) (x), B ij kl ij kl

(9.311)

448

Heterogeneous Media

where the tensor Q(k) ij kl (x) is defined in Eq. (5.138). The total stress tensor and the total plastic deformation at the end of the kth interval are the following sums: (k)

σij (x) = σij0 (x) +

k

p(k)

(l) σij (x), εij (x) =

l=1

k

p

(l) εij (x).

(9.312)

l=1

Here, the initial plastic deformation is assumed to be equal to zero, σij0 (x) is the 0(0)

stress tensor in the medium by action of the initial external stress σij , and the increp ment of the plastic deformation (l) εij (x) is defined in terms of the stress increment (l) σij (x) in Eq. (5.135). Thus, the problem is reduced to solution of the integral equation (9.310) for the stress field increment σij (x) in the heterogeneous medium caused by the increment of the external stress field σij0 . In this equation, the functions Sij kl (x) and σij0 ij kl (x), which is defined from the solution in the are known as well as the function B previous interval of loading.

9.6.1 The effective field method for elasto-plastic heterogeneous media We consider a spherical RVE VR in an infinite heterogeneous elasto-plastic medium. The stress field increment σij (x) in the region VR caused by the increment of the external stress field σij0 satisfies the equation that follows from Eq. (9.310):  σij (x) − VR

klmn (x  )σmn (x  )dx  = σij∗ (x), x ∈ VR , Sij kl (x − x  )B

σij∗ (x) = σij0 +

(9.313)



klmn (x  )σmn (x  )dx  . Sij kl (x − x  )B

(9.314)

R\VR

Here, R\VR is the complement of VR to the entire space R. The stress field σij∗ (x) in the right hand side of Eq. (9.313) is the increment of the external stress field acting on the region VR . Further, we assume that the elasto-plastic properties of the medium are homogeneous in space random functions. The average of the field σij∗ (x) over the ensemble realizations of the random properties is the effective increment of the  ∗ external stress field σij (x) acting on the region VR . After averaging Eq. (9.314), we obtain     klmn (x)σmn (x) . σij∗ (x) = σij0 + Sij kl (x − x  )dx  B (9.315) R\VR

Here, it is taken into account that for a constant external stress field increment σ 0ij ,   klmn (x)σmn (x) is constant, and the integral in Eq. (9.314) is calcuthe average B

The homogenization problem

449

lated as follows:    Sij kl (x − x  )dx  = Sij kl (x − x  )dx  − Sij kl (x − x  )dx  = R\VR R VR  =− Sij kl (x − x  )dx  = −Dij kl (x), x ∈ VR . (9.316) VR

Here, we assume that the conditions at infinity correspond to a fixed external stress field σij0 . For an isotropic host medium and a spherical RVE, the tensor Dij kl in Eq. (9.316) is constant and defined in Eq. (9.191). As a result, the effective external  ∗ field increment σij in Eq. (9.315) is constant in the region VR and takes the form 

  klmn σmn , x ∈ VR . σij∗ = σij0 − Dij kl B

(9.317)

The second term on the right is the average stress field increment induced in VR by the inclusions and plastic deformations in the region R\VR . After changing  the ex∗ ternal field increment σij (x) in Eq. (9.313) for the effective increment σij∗ in Eq. (9.317), we obtain the following equation for the stress field increment in the RVE VR :  σij (x) −

  klmn (x  )σmn (x  )dx  + Dij kl B klmn σmn = Sij kl (x − x  )B

VR

= σij0 ,

x ∈ VR .

(9.318)

  klmn σmn can be calculated over the If the RVE is sufficiently large, the average B region VR (the condition of self-consistency), 

   ij kl (x)σkl (x) ≈ B ij kl (x)σkl (x) = 1 B VR VR



ij kl (x)σkl (x)dx. B VR

(9.319) After substituting Eq. (9.319) into Eq. (9.318), we obtain the equation for the stress field increment in the RVE in the form   1  klmn (x  )σmn (x  )dx  = σij (x) − Dij kl B Sij kl (x − x ) − VR VR 

= σij0 , x ∈ VR .

(9.320)

Thus, the problem is reduced to calculation of the stress field increment σij (x) inside the RVE only. A numerical method of solution of this problem is considered in Section 5.6.

450

Heterogeneous Media

Discretization of the integral equations of the effective field method For numerical solution of Eq. (9.320), σij (x) is approximated by the Gaussian quasiinterpolant σij (x) ≈

Mn

(n)

σij ϕ(x − x (n) ), ϕ(x) =

n=1

1 (πH )3/2

 |x|2 exp − . H h2

(9.321)

Here, x (n) (n = 1, 2, ..., Mn) is a set of approximating nodes that cover a cuboid W , (n) and VR ⊂ W , σij are the coefficients of the approximation, h is the step of the (n)

node grid, and H = O(1). For a cubic node grid, the coefficients σij coincide with the values of the functions σij (x) at the nodes σ (n) = σ (x (n) ). Substitution of the approximations (9.321) into the integral equation (9.320) and satisfaction of the resulting equations at all nodes yields a linear algebraic system for the coefficients of the approximations: (m)

σij

−

Mn 

(m,n) (n) (n) σmn = σij0 , n = 1, 2, ..., Mn. ij kl − τ Dij kl B klmn

n=1

(9.322) (n) Here, τ = h3 /VR , and ij(m,n) kl and Bij kl are defined by the equations  (m,n) (m) (n) ij kl = ij kl (x − x ), ij kl (x) = Sij kl (x − x  )ϕ(x  )dx  , ij kl (x (n) ). (n) = B B ij kl

(9.323)

The explicit expression for the function ij kl (x) is presented in Eqs. (5.106)–(5.110). The system (9.322) can be presented in the matrix form similar to the system of Eqs. (5.152)–(5.156) for a finite number of inclusions in an elasto-plastic host medium AX = F,

(9.324)

where the vector X of unknowns is expressed in terms of the components of the tensor σij at the nodes and the vector F in terms of the components of the increment of the external field σij0 , whereas the matrix A is reconstructed from Eq. (9.322).

9.6.2 Average stress–strain relations for elasto-plastic composites The system (9.322) should be solved at each step of the loading process. The stress and strain fields σ (k) (x) and ε (k) (x) at the kth step of loading are calculated as follows: (k)

σij (x) = σij0 (x) +

k

l=1

(l) σij (x),

(9.325)

The homogenization problem

(k)

451

0 εij (x) = εij (x) +

k 

p e (x) + (l) εij (x) , (l) εij

(9.326)

l=1 e (x) = Cij−1kl (x)(l) σkl (x). (l) εij

(9.327)

    The average intensities of stresses J s(k) and strains J ε(k) at the kth step of loading are calculated over the region VR , i.e.,     1 1 J s(k) (x)dx, J ε(k) = J ε(k) (x)dx. (9.328) J s(k) = VR VR VR VR Here, the intensities J s = J and J ε are defined in Eqs. (5.129) and (5.130). The relation between the average stress and strain intensities in the process of loading is the objective of the homogenization problem 

   J s = F ( J ε ).

(9.329)

We consider a homogeneous elasto-plastic host medium with a random set of identical spherical inclusions. The host medium is isotropic with the shear and bulk moduli μ0 = 26.32 GPa and K0 = 68.36 GPa. The law of plastic deformations of the host medium is described by the equation  n Jy = B Jp

(9.330)

with B = 400 MPa, and the plastic strain hardening exponent n is in the range 0–0.4. The function (J y ) in the constitutive equation (5.128) is taken the form (J y ) =

3 2nB 2

Jy B

( 1 −2) n

.

(9.331)

The inclusions are purely elastic, with the shear and bulk moduli μ = 166.67 GPa and K = 222.26 GPa. The volume fraction p of the inclusions is equal to 0.25. For simulation of elasto-plastic behavior of the composite, the following statistical model of the random set of inclusions is accepted. Starting with a regular arrangement of the inclusions, their centers are moved by independent random vectors with restrictions that the inclusions do not intersect. We consider the RVE with 9 inclusions firstly arranged in the cubic centered lattice and 14 inclusions arranged in the face centered lattice. Then, the positions of these inclusions are randomized inside a spherical RVE. Examples of the RVE realizations are shown in Fig. 9.33. The uniaxial external tensile stress σij0 = σ 0 δ1i δ1j is applied to the composite starting from 1 MPa and increasing to 400 MPa with the step 1 MPa. In the calculations, the region W is a cube W : (2 × 2 × 2) covered by the regular cubic node grid with the step h = 0.1. The region VR is a sphere with radius R0 = 0.93, and the radii of the inclusions r0 are taken to have the prescribed inclusion volume fraction p inside VR .

452

Heterogeneous Media

Figure 9.33 The RVE VR with 1, 9, and 14 inclusions in the cube W for numerical solution of the homogenization problem for an elasto-plastic composite with random sets of spherical inclusions.

Figure 9.34 The stress–strain curves for the composites with elasto-plastic matrix and a random set of stiff elastic inclusions. The hardening law is defined by Eq. (9.330); lines with white dots correspond to results of the finite element calculations presented in [22].

The predicted stress–strain curves for the hardening exponents n = 0.05, 0.15, 0.25, 0.4 in Eq. (9.330) are given in Fig. 9.34. The curves for 9 and 14 inclusions in the RVE correspond to the averages over five realizations of the inclusion sets. The lines with white dots correspond to the finite element method applied to the solution of the same problem in [22]. In this work, a cubic RVE with 30 inclusions and periodic boundary conditions on the RVE boundaries was considered. As seen from Fig. 9.34, the predictions of the presented method for the RVE with 9 and 14 inclusions practically coincide and are very close to the results of the finite element calculations. If the region VR contains only one inclusion, there is an observable difference in the predicted stress–strain curves in comparison with the results for the RVE with 9 or 14 inclusions. Increasing the number Mn of the nodes does not affect practically the stress–strain curve behavior. The number of iterations and, as a result, the time of calculations increase with decreasing values of the hardening exponent n in the plastic deformation law (9.330).

The homogenization problem

453

9.7 The homogenization problem for time-harmonic fields in heterogeneous media This section is devoted to the homogenization problem for time-harmonic fields in heterogeneous media. In order to indicate specific features of this problem, we consider an example of a heterogeneous elastic medium subjected to dynamic loading. Let the elastic stiffness tensor Cij kl (x) and the density ρ(x) of the medium be homogeneous in space random functions. The displacement field ui (x, t) in the medium satisfies the equation ∂j Cij kl (x)∂k ul (x, t) − ρ(x)

∂2 ui (x, t) = −qi (x, t). ∂t 2

(9.332)

The objective of the homogenization problem is the determination of material parameters of a homogeneous medium (effective medium) with equivalent response to external loading. Let ui (x, t) be the average of the field ui (x, t) over the ensemble realization of the random functions Cij kl (x) and ρ(x). For elastic medium with microstructure, the equation for this average takes the form similar to Eq. (9.332) [2] ∂j Cij∗ kl ∂k ul (x, t) − ρ∗

∂2 ui (x, t) = −qi (x, t), ∂t 2

(9.333)

where Cij∗ kl and ρ∗ are some operators. Because space and time dispersions are the attributes of heterogeneous media, these operators are nonlocal with respect to x- and t-variables and can be presented in the forms    Cij∗ kl ∂k ul (x, t) = Cij∗ kl (x − x  , t) ∗ ∂k ul (x  , t) dx  , (9.334)   ∂2 ∂2  ρ∗ 2 ui (x, t) = ρ∗ (x − x  , t) ∗ 2 ul (x  , t) dx  . (9.335) ∂t ∂t Here, (∗) is the operator of convolution with respect to the time variable t  f (t) ∗ ϕ(t) =

∞

−∞

f (t − t  )ϕ(t  )dt  .

(9.336)

The homogenization problem consists in constructing the kernels Cij∗ kl (x, t) and ρ∗ (x, t) of the integral operators in Eqs. (9.334) and (9.335) for the known random functions Cij kl (x) and ρ(x) of the heterogeneous medium. A more specific but important problem is homogenization in the case of timeharmonic fields. If the dependence on time is defined by the multiplier eiωt , the equation for the amplitude ui (x) of the wave field follows from Eq. (9.332) in the form (qi (x, t) = 0) ∂j Cij kl (x)∂k ul (x) + ω2 ρ(x)ui (x) = 0.

(9.337)

454

Heterogeneous Media

The average ui (x) over the ensemble realizations of the random functions Cij kl (x) and ρ(x) satisfies the equation that follows from Eq. (9.333) ∂j Cij∗ kl ∂k ul (x) + ω2 ρ∗ ui (x) = 0.

(9.338)

Here, Cij∗ kl and ρ∗ are integral operators over the x-variable. Let ui (x) be a plane monochromatic wave in the effective homogeneous medium with the wave vector κ∗ mi , ui (x) = Ui e−iκ∗ (m·x) . It follows from Eq. (9.338) that this function satisfies the equation    ∗ κ∗2 mk Ciklj (κ∗ , ω)ml − ω2 ρ∗ (k∗ , ω)δij uj (x) = 0.

(9.339)

(9.340)

Here, Cij∗ kl (k, ω) are ρ∗ (k, ω) are the Fourier transforms of the kernels Cij∗ kl (x, t) and ρ∗ (x, t) in Eqs. (9.334) and (9.335) with respect to spatial and time variables. Thus, the effective wave number κ∗ is the solution of the dispersion equation that is a consequence of Eq. (9.340),   ∗ (κ∗ , ω)ml − ω2 ρ∗ (k∗ , ω)δij = 0. (9.341) det κ∗2 mk Ciklj The real part of the wave number κ∗ determines the phase velocities υ∗ of the corresponding wave, and the imaginary part of κ∗ is the attenuation factors γ∗ of this wave, and we have ω (9.342) , γ∗ = − Im(κ∗ ). υ∗ = Re(κ∗ ) Thus, for calculation of the velocities υ∗ and attenuation factors γ∗ of the averaged monochromatic waves that can propagate in a heterogeneous elastic medium, the functions Cij∗ kl (κ∗ , ω) and ρ∗ (k∗ , ω) in the dispersion equation (9.341) should be constructed. In the literature, the field (9.339) is called the coherent part of the timeharmonic random field ui (x, t) propagating in a heterogeneous medium.

9.7.1 Specific features of time-harmonic fields in heterogeneous media The dispersion equation (9.341) defines the wave numbers of all types of monochromatic waves that can propagate in a homogeneous effective medium equivalent to a given heterogeneous medium. If the vectors Ui and mi in Eq. (9.339) have the same direction, such waves are longitudinal. If the vectors Ui and mi are orthogonal, the waves are transversal. For an anisotropic effective medium, purely longitudinal and transverse waves can exist for specific directions of the wave normal mi only. For other directions, the vectors Ui and mi are neither parallel nor orthogonal, and the propagating waves are called quasilongitudinal or quasitransversal [23].

The homogenization problem

455

Space and time dispersion and attenuation of monochromatic waves are not the only attributes that differ between the effective medium and a classical homogeneous elastic medium. Let the functions Cij kl (x) and ρ(x) in Eq. (9.337) be nonrandom, periodic functions of coordinates. It is known that in this case, there exist intervals of frequencies ω, where time-harmonic waves propagate through the medium without attenuation (pass bands) and the regions of ω with exponential attenuation of the waves (stop bands). The wave numbers of these waves correspond to various branches of solution of the dispersion equation (9.341). The branch that starts from the origin (ω = 0, κ∗ = 0) of the (ω, κ∗ )-plane is called the acoustical one, and other branches are called optical. Optical branches correspond to internal modes of oscillations of the heterogeneous medium that do not exist in classical homogeneous elastic media. By randomization of a periodic heterogeneous medium, all the propagating waves attenuate, but different branches of the dispersion equation do not disappear and can be called quasiacousical and quasioptical ones. Application of the concept of the RVE to solution of the homogenization problem for time-harmonic fields encounters principal and technical difficulties. First, for the RVE of finite sizes, the conditions on the RVE boundary should simulate wave propagation in an infinite heterogeneous medium. These conditions are not trivial, and their formulation for a wide range of frequencies is questionable. Second, even if these conditions are formulated, construction of detailed wave fields in the RVE is a time and memory consuming computational problem. The difficulty increases for the waves whose lengths are much shorter than the RVE sizes. For the boundary conditions that generate plane waves in a homogeneous RVE, the time-harmonic field ui (x)eiωt in the heterogeneous RVE is not a plane wave. Extraction of the coherent part ui (x, t) = Ui ei(ωt−κ∗ mi xi ) from a finite number of realizations of wave fields in the RVE of a random medium is a nontrivial problem. Note that for ensemble averages of time-harmonic fields, the ergodic hypothesis does not serve. Spatial averaging results in losing the oscillating (coherent) part of the random field ui (x), which is the principal unknown of the problem. If volume integral equations are used for solution of the homogenization problem, the RVE of a heterogeneous medium is embedded into a homogeneous reference medium and subjected to an incident plane monochromatic wave. In this case, the wave field in the RVE depends not only on the heterogeneities inside the RVE but also on the ratio of the RVE sizes and the incident wave length as well as on the RVE shape. Because the effective properties should not depend on the RVE specifics, the possibility of extraction of the reliable coherent part of the wave field from a number of numerical solutions for realizations of the random medium in a finite RVE is questionable. These difficulties can be overcome by using self-consistent methods in application to solution of the homogenization problem for time-harmonic fields in heterogeneous media. These methods serve for an important class of heterogeneous materials that consist of a homogeneous host medium and a set of isolated heterogeneous inclusions. The methods are based on physically reasonable hypotheses that, however, cannot be proved strictly. These hypotheses allow extracting the coherent parts of the random wave fields and reducing construction of the dispersion equation for the effective wave

456

Heterogeneous Media

numbers of the coherent waves to solution of one-particle problems. Further, examples of the applications of the methods to homogenization of a model heterogeneous medium are considered.

9.7.2 The effective medium method We consider a model time-harmonic problem with a scalar potential u(x) that satisfies the equation ∂i C(x)∂i u(x) + ω2 ρ(x)u(x) = 0.

(9.343)

This equation is an analogy of Eq. (9.337) for an elastic heterogeneous medium. The scalar function C(x) is similar to the elastic stiffness tensor of the medium and ρ(x) is the medium density. For simplicity, we assume that C(x) = C is constant, and the heterogeneities are spherical inclusions with constant densities ρ distributed in a homogeneous host medium with the density ρ0 . Thus, the function ρ(x) can be presented in the form ρ(x) = ρ0 + ρ1 V (x), ρ1 = ρ − ρ0 ,

(9.344)

where V (x) is the characteristic function of the region occupied by the inclusions. Eq. (9.343) for the function u(x) can be written in the form C∂i ∂i u(x) + ω2 ρ0 u(x) = −ω2 ρ1 V (x)u(x),

(9.345)

and after application of the inverse operator with respect to the operator L0 = C∂i ∂i + ω2 ρ0 to this equation, we obtain the integral equation for the field u(x):  u(x) = u0 (x) + ω2 ρ1 g(x − x  )u(x  )V (x  )dx  . (9.346) Here, u0 (x) is the incident wave field that would have existed in the medium without inclusions and the same conditions at infinity. It is assumed that u0 (x) = exp(−iκ0 mi xi ) is a plane wave with the wave vector κ0 mi (|m| = 1) propagating in the homogeneous host medium. The kernel g(x) in Eq. (9.346) is the Green function of the operator L0 , * exp(−iκ0 |x|) ρ0 g(x) = , κ0 = ω . (9.347) 4πC|x| C For this specific heterogeneous medium, the objective of the homogenization problem is the determination of the effective wave number κ∗ of the average (coherent) field u(x) that propagates in the heterogeneous medium. For an isotropic and homogeneous in space random function V (x), the field u(x) and u0 (x) have the same wave normal mi , and u(x) is presented in the form * ρ∗ u(x) = exp(−iκ∗ mi xi ), κ∗ = ω . (9.348) C

The homogenization problem

457

Here, ρ∗ is the effective density of the heterogeneous medium. The effective medium method (EMM) is based on the following hypothesis. The field inside each inclusion coincides with the field in an isolated inclusion embedded into a homogeneous medium with the effective density ρ∗ and subjected to the wave field u∗ (x) = u(x) = e−ik∗ m·x , which is the averaged (coherent) wave field in the heterogeneous medium. Using this hypothesis, the field inside each inclusion is presented in the form  (9.349) u(x) = ( u∗ ) (x) = e−ik∗ m·x , where the operator is to be found from the solution of the one-particle problem of the EMM: scattering a plane monochromatic wave propagating in the homogeneous effective medium with the density ρ∗ from an isolated spherical inclusion of the density ρ. Let the center of an inclusion be at a point x 0 . Because of linearity of the oneparticle problem, the field inside the inclusion can be presented in the form  0 0 u(x) = ( u∗ ) (x) = e−ik∗ m·(x−x ) e−ik∗ m·x =  0 0 (9.350) = e−ik∗ m·(x−x ) eik∗ m·(x−x ) e−ik∗ m·x = λ(x − x 0 )u∗ (x),  (9.351) λ(y) = e−ik∗ m·y eik∗ m·y . If the centers of the inclusions are at the points x (n) (n = 1, 2, ...), the integrand function in Eq. (9.346) is the following sum

λ(x − x (n) )v n (x)u∗ (x), (9.352) u(x)V (x) = n

where v n (x) is the characteristic function of the region occupied by the nth inclusion, and in each inclusion, the function λ(x − x (n) ) coincides with the function in Eq. (9.351). Thus, the hypothesis of the EMM allows extracting the coherent component from the random field u(x)V (x), and the average of this field takes the form " !

(n) n u(x)V (x) = λ(x − x )v (x) u∗ (x). (9.353) n

+

Because n λ(x − x (n) )v n (x) is a stationary function that does not contain the coherent component of the field u(x), the average in Eq. (9.353) can be calculated + using the ergodic property, and for a fixed typical realization of the field V (x) = n v n (x), we can write " ! 

1 (n) n λ(x − x )v (x) = lim λ(x − x (n) )v n (x)dx = W →∞ W W n n = pHρ (κ∗ , κ),

(9.354)

458

Heterogeneous Media

1 W →∞ W

p = lim



W

v n (x)dx, Hρ =

n

1 v

*

 λ(x)dx, κ = ω v

ρ . C

(9.355)

The equation for the field u∗ (x) = u(x) follows from Eqs. (9.346), (9.353), and (9.354) in the form    u∗ (x) = u0 (x) + ω2 ρ1 g(x − x  ) u(x  )V (x  ) dx  =  2 (9.356) = u0 (x) + ω pρ1 Hρ (κ∗ , κ) g(x − x  )u∗ (x)dx  . Applying the operator L0 = C∂i ∂i + ω2 ρ0 to this equation and taking into account that L0 u0 = 0, we obtain C∂i ∂i u∗ + ω2 ρ∗ (κ∗ , κ)u∗ = 0, ρ∗ (κ∗ , κ) = ρ0 + pρ1 Hρ (κ∗ , κ).

(9.357) (9.358)

Substituting in this equation u∗ = exp(−iκ∗ m · x), we obtain the dispersion equation for the effective wave number κ∗ Cκ∗2 − ω2 ρ∗ (κ∗ , κ) = 0.

(9.359)

9.7.3 The one-particle problem of the EMM The one-particle problem of the EMM is the scattering problem for an inclusion v with the density ρ embedded in the host medium with the density ρ∗ and subjected to the external field e−iκ∗ (x·m) . This problem is reduced to solution of the following integral equation:  −iκ∗ (x·m) 2 +ω g∗ (x − x  )ρ1∗ (x  )u(x  )dx  , (9.360) u(x) = e v

e−ik∗ r , ρ1∗ (x) = ρ − ρ∗ , r = |x|. g∗ (x) = 4πCr

(9.361)

For a spherical inclusion, this integral equation is equivalent to the following system of differential equations: C∂i ∂i u(x) + ω2 ρu(x) = 0, r < a, C∂i ∂i u(x) + ω2 ρ∗ u(x) = 0, r > a, with the conditions at the inclusion boundary r = a , , ∂u+ ,, ∂u− ,, + − u (a) = u (a), = . ∂r ,r=a ∂r ,r=a

(9.362)

(9.363)

Here, u+ (a), u− (a) are the limit values of the function u(x) on the inclusion boundary  from the side of the external normal to  and from the opposite side. The field u(x)

The homogenization problem

459

at infinity should satisfy the asymptotic equation

u(x) = exp(−iκ∗ mi xi ) + O

e−iκ∗ r r

 , r → ∞.

(9.364)

The solution of this problem has the form u(x) = u(x) =

∞

m=0 ∞

αm jm (κr)Pm (cos θ ),

(9.365)

r < a,

βm hm (κ∗ r)Pm (cos θ ) + exp(−iκ∗ mi xi ),

r > a,

(9.366)

m=0

hm (z) = jm (z) − iym (z).

(9.367)

Here, jm (z) and ym (z) are spherical Bessel functions of the first and second kind, Pm (z) are Legendre polynomials, and θ is the azimuthal angle of the spherical coordinate system with the origin at the inclusion center and the polar axis directed along the mi -vector. The coefficients αm and βm are to be found from the boundary conditions (9.363), and for αm , we obtain (−i)m+1 (2m + 1) , a 2 κ∗ [κ∗ jm (κa)hm (κ∗ a) − κjm (κa)hm (κ∗ a)] djm  dhm jm (z) = , hm (z) = . dz dz

αm =

(9.368) (9.369)

It follows from Eqs. (9.351) and (9.355) that the function Hρ (κ∗ , κ) takes the form Hρ (κ∗ , κ) =

1 v

 u(x)eiκ∗ m·x dx = v

∞

i m αm gm ,

(9.370)

m=0

  3 κjm+1 (κa)jm (κ∗ a) − κ∗ jm+1 (κ∗ a)jm (κa) . gm = 2 2 a(κ − κ∗ )

(9.371)

The far scattering field us (x) from the inclusion follows from Eq. (9.360) and is presented in the form e−iκ∗ |x| xi , ni = , |x| |x|   κ 2 ρ1∗ ω2 ρ1∗ u(x)eiκ∗ n·x dx = ∗ u(x)eiκ∗ n·x dx. A(n) = 4πC v 4π ρ∗ v

us (x) ≈ A(n)

(9.372) (9.373)

Here, A(n) is the amplitude of the scattered field in the direction ni , and if ni = mi , then A(m) is the forward scattering amplitude.

460

Heterogeneous Media

The total normalized scattering cross-section of the inclusion Q is expressed in terms of the forward scattering amplitude as follows [24]:

 4π A(m) Q=− Im . (9.374) S κ∗ Here, S is the maximal area of the intersection of the inclusion by the plane orthogonal to the wave normal mi . For a spherical inclusion, we have   ρ1∗ 4 Hρ (κ∗ , κ) . (9.375) Q = − Im aκ∗ 3 ρ∗ It is known that in the short wave limit (ω → ∞), Q → 2, and this limit does not depend on the properties of the medium and the inclusion (the paradox of extinction [24]). A consequence of this paradox is that in the short wave limit, the following asymptotic equation for the function Hρ (κ∗ , κ) in Eq. (9.370) holds: Hρ (κ∗ , κ) =

∞

i m αm gm ≈ −

m=0

3iρ∗ . 2aκ∗ ρ1∗

(9.376)

Long and short wave asymptotics of the solution of the dispersion equation of the EMM The principal terms of the function Hρ (κ∗ , κ) for small values of frequency ω (wave number κ0 ) follow from Eq. (9.370) in the forms Hρ (κ∗ , κ) = 1 − i

ρ1∗ (κ∗ a)3 + O(κ∗4 ). 3ρ∗

(9.377)

As a result, the principal terms of the effective wave number κ∗ for small ω are defined by the equation $ * # * 2 ρ∗s ρ∗s 3 p(1 − p)ρ1 3 ω =ω (9.378) 1 − ia − iγ∗ , κ∗ ≈ ω s 3/2 C C 6 ρ∗ C ρ∗s = ρ0 + pρ1 , γ∗ = a 3 p(1 − p)

ρ12 4 ω . 6C 2

(9.379)

Because γ∗ is proportional to ω4 , this factor corresponds to the so-called Rayleigh attenuation of waves in the composite. It follows from Eq. (9.376) that the function Hρ (κ∗ , κ) tends to zero when ω → ∞. Therefore, for the effective density ρ∗ and effective wave number κ∗ in the short wave limit, we have the equations 3pρ1 ρ∗ , ρ∗ ≈ ρ0 − i 2aκ∗ ρ1∗ * ρ∗ (κ∗ , κ) 3p κ∗ = ω ≈ κ0 − i . C 4a

(9.380) (9.381)

The homogenization problem

461

As the result, in the short wave limit, the velocity υ∗ and the attenuation factor γ∗ of the average wave field in the composite are υ∗ =

ω 3 ω ≈ = υ0 , γ∗ a = − Im κ∗ a ≈ p. Re(κ∗ ) Re(κ0 ) 4

(9.382)

Thus, υ∗ coincides with the wave velocity in the host medium υ0 , and the attenuation factor γ∗ does not depend on the inclusion and host medium densities and is proportional to the volume fraction of the inclusions p only.

Acoustic and optical branches of the solution of the dispersion equation of the EMM We consider numerical solutions of the dispersion equation (9.359) in the region 0 < κ0 a < 6 of the wave numbers of the incident wave field. The graphs of the real Re(κ∗ a) and imaginary γ∗ a = − Im(κ∗ a) parts of the effective wave number for the composite medium with the parameters C = 1, ρ0 = 1, ρ = 0.1 (light inclusions), and p = 0.1, 0.3 are presented in Fig. 9.35. In this region, there exists one branch of the roots of the dispersion equation (9.359) for each volume fraction p.

Figure 9.35 Real and imaginary parts of the quasiacoustic branches of the solution of the dispersion equation (9.359) of the EMM for the composite with the parameters of the host medium C0 = 1, ρ0 = 1 and of the inclusions C = 1, ρ = 0.1 for inclusion volume fractions p = 0.1 and p = 0.3.

The situation is changed dramatically for the inclusions with the density ρ = 10 (heavy inclusions). In the mentioned interval of the incident wave numbers κ0 a and p = 0.1, there exist two different branches of the roots of the dispersion equation (9.359), which are shown in Fig. 9.36. Branch 1 is quasiacoustical, and branch 2 is quasioptical. Attenuation factors γ∗ a along each branch are shown in the right side of Fig. 9.36. In the left side of Fig. 9.36, solid parts of the branches indicate the regions of frequencies with γ∗ a < 1, and the dashed parts correspond to γ∗ a > 1. In the regions with γ∗ a > 1, the amplitude of the corresponding wave decreases in 10 times in the distance about the inclusion diameters. Thus, only waves that correspond to the solid parts of the curves in Fig. 9.36 can be observed in the distances about several inclusion diameters from the wave sources.

462

Heterogeneous Media

Figure 9.36 Real and imaginary parts of the (1) quasiacoustical and (2) quasioptical branches of the solution of the dispersion equation (9.359) of the EMM for the composite with the parameters of the host medium C0 = 1, ρ0 = 1 and of the inclusions C = 1, ρ = 10; the inclusion volume fraction is p = 0.1.

The dispersion curves for heavy inclusions with the density ρ = 10 and the volume fraction p = 0.3 are shown in Fig. 9.37. In the region 0 < κ0 a < 6, there exist four branches of the solution of the dispersion equation (9.359). Branch 1 is quasiacoustical, and branches 2–4 are quasioptical. Attenuation factors γ∗ a along the branches are shown in the right side of Fig. 9.37.

Figure 9.37 Real and imaginary parts of the (1) quasiacoustical and (2–4) quasioptical branches of the solutions of the dispersion equation (9.359) of the EMM for the parameters of the composite medium C0 = 1, ρ0 = 1, and C = 1, ρ = 10, p = 0.3. Dashed parts of the curves correspond to the attenuation factors γ a > 1.

9.7.4 The effective field method We consider a homogeneous host medium with the parameters C, ρ0 containing a random set of isolated spherical inclusions with the parameters C, ρ. Thus, the density

The homogenization problem

463

ρ(x) is defined by Eq. (9.344), and the integral equation for the field u(x) has the form (9.346)  u(x) = u0 (x) + ω2 ρ1 g(x − x  )u(x  )V (x  )dx  . (9.383) Here, u0 (x) = exp(−iκ0 mi xi ) is a plane incident wave propagating in the host medium. The hypothesis of the EFM is formulated as follows. The field inside each inclusion in the composite coincides with the field in an isolated inclusion embedded into the homogeneous host medium with the density ρ0 and subjected to an external wave field u∗ (x) = U∗ e−ik∗ m·x that does not coincide with the incident wave field u0 (x). This field is a plane wave that takes into account the waves scattered from other inclusions. Note that the field u∗ (x) neither coincides with the averaged wave field u(x) in the composite. We consider a local external field u∗ (x) acting on an arbitrary inclusion in the composite. Let us introduce the functions V (x) =

n

v n (x), V (x, x  ) =



v n (x  ) if x ∈ v m .

(9.384)

n =m

Here, v n (x) is the characteristic function of the region occupied by the nth inclusion. Similar functions are considered in Section 9.3.2, Eq. (9.100). In these notations, the field u(x) inside the mth inclusion is presented in the form that follows from Eq. (9.383)  u(x) = u∗ (x) + ω2 ρ1 g(x − x  )u(x  )dx  , x ∈ v m , (9.385) m v  u∗ (x) = u0 (x) + ω2 ρ1 g(x − x  )u(x  )V (x, x  )dx  . (9.386) Here, u∗ (x) is the local external field that acts on the inclusion v m . The average u∗ (x)|x of the field u∗ (x) by the condition that x ∈ V is the effective field acting on the inclusions in the composite. According to the hypothesis of the EFM, it is a plane wave with the wave number κ∗ . For a homogeneous and isotropic random function ρ(x), the wave normals mi of the effective and incident waves coincide. Thus, the effective field u∗ (x)|x is presented in the form u∗ (x)|x = u∗ (x) = U∗ e−iκ∗ m·x ,    u∗ (x) = u0 (x) + ω2 ρ1 g(x − x  ) V (x, x  )u(x  )|x dx  .

(9.387) (9.388)

Here, ·|x is the average by the condition that x ∈ V . Let x n be the center of the inclusion v n . The hypothesis of the EFM allows presenting the field inside the region v n in the form

464

Heterogeneous Media

u(x) = (U∗ e−iκ∗ m·x ) = = (e−iκ∗ m·(x−x ) )eiκ∗ m·(x−x ) U∗ e−iκ∗ m·x = λ(x − x n )u∗ (x), n

λ(y) = (e

−iκ∗ m·y

n

)e

iκ∗ m·y

.

(9.389) (9.390)

Here, is a linear operator of the solution of the one-particle problem of the EFM. From Eq. (9.388), we obtain a closed equation for the effective field u∗ (x),     λ(x )V (x, x  )|x u∗ (x  )dx  . u∗ (x) = u0 (x) + ω2 g(x − x  )ρ1  (9.391) Here, the function  λ(x) coincides with λ(x − x n ) inside the nth inclusion. Changing  λ(x) to the mean value λ of this function over the inclusion volumes, we obtain     λ(x )V (x, x  )|x = pλ(x − x  ), (9.392)  N  1 1 1 λ = lim λ(x − x n )dx, (x − x  ) = V (x, x  )|x . (9.393) n N→∞ N v vn p n=1

Here, (x) is a specific correlation function considered in Section 9.3.2 and p is the inclusion volume fraction. As a result, Eq. (9.391) takes the form  2 u∗ (x) = u0 (x) + ω pρ1 λ g(x − x  )(x − x  )u∗ (x  )dx  . (9.394) The average u(x) of the wave field in the composite follows from Eq. (9.383) and the hypothesis of the EFM, i.e.,    2 u(x) = u0 (x) + ω ρ1 g(x − x  ) u(x  )V (x  ) dx  =     λ(x )V (x  ) u∗ (x  )dx  = = u0 (x) + ω2 ρ1 g(x − x  )   2 (9.395) = u0 (x) + ω pρ1 λ g(x − x  )u∗ (x  )dx  .   Here, it is taken into account that  λ(x)V (x) = λ V (x) = pλ. The difference of Eqs. (9.394) and (9.395) yields  u∗ (x) = u(x) − ω2 pρ1 λ g(x − x  ) (x − x  )u∗ (x  )dx  , (9.396) (x) = 1 − (x).

(9.397)

For a homogeneous and isotropic random set of nonintersecting inclusions, the function (x) has the properties (x) = (r), r = |x|, (0) = 1, lim (r) = 0. r→∞

(9.398) (9.399)

The homogenization problem

465

Figure 9.38 Correlation functions (ξ ) for a random set of identical spherical inclusions of the volume fraction p = 0.1, 0.3, 0.5 that correspond to Percus–Yevick correlation functions of the centers of nonpenetrating identical spheres.

Examples of such functions for various volume fractions of inclusions p are shown in Fig. 9.38. Applying the Fourier transform operator to Eq. (9.396), we obtain for the Fourier transform u∗ (k) of the effective field the equation  −1 u∗ (k) = (k) u(k) , (k) = 1 + ω2 pρ1 λg (k) ,  g (k) = g(x) (x)eik·x dx, λ = λ(κ0 , κ, κ∗ ).

(9.400) (9.401)

Action of the operator L0 = C∂i ∂i + ω2 ρ0 on Eq. (9.395) yields L0 u(x) + ω2 pρ1 λu∗ (x) = 0.

(9.402)

Applying the Fourier transform to this equation and taking into account Eq. (9.400), we obtain 

 −Ck 2 + ω2 (ρ0 + pρ1 λ(k)) u(k) = 0.

(9.403)

The dispersion equation for the effective wave numbers κ∗ of the coherent parts of monochromatic waves propagating in the heterogeneous medium follows from this equation in the form Ck∗2 − ω2 ρ∗ (κ0 , κ, κ∗ ) = 0,

(9.404)

ρ∗ (κ0 , κ, κ∗ ) = ρ0 + pρ1 λ(κ0 , κ, κ∗ )(κ∗ ).

(9.405)

466

Heterogeneous Media

9.7.5 The one-particle problem of the effective field method According to the hypothesis of the EFM, the integral equation of the one-particle problem of the method has the form  −iκ∗ (m·x) 2 u(x) = e + ω ρ1 g(x − x  )u(x  )dx  , (9.406) v

e−iκ0 r , r = |x|. g(x) = 4πCr

(9.407)

For a spherical inclusion of radius a, this integral equation is equivalent to the system of differential equations u(x) + κ 2 u(x) = (κ02 − κ∗2 )e−iκ∗ m·x ,

r < a,

u(x) + κ02 u(x) = (κ02 − κ∗2 )e−iκ∗ m·x ,

r > a,

with the following conditions on the inclusion boundary r = a: , , ∂u+ ,, ∂u− ,, + − = . u (a) = u (a), ∂r ,r=a ∂r ,r=a

(9.408)

(9.409)

The solution of this system is presented in the form of the series u(r, θ ) = u(r, θ ) =

∞

m=0 ∞

αm jm (κr)Pm (cos θ ) +

κ02 − κ∗2 −iκ∗ r cos θ e , r < a, κ 2 − κ∗2

βm hm (κ0 r)Pm (cos θ ) + e−iκ∗ r cos θ , r > a.

(9.410)

(9.411)

m=0

Here, jm (z) and hm (z) are spherical Bessel and Hankel functions, Pm (z) are Legendre polynomials, and (r, θ ) are the radial and azimuthal coordinates of the spherical system with the origin at the center of the inclusion and the polar axis directed along the wave normal mi of the incident wave. The constants αm and βm are to be found from the boundary conditions (9.409), and for αm , we have  αm = (−i) (2m + 1)  m

κ 2 − κ02 κ 2 − κ∗2

 

 κ∗ jm (κ∗ a)hm (κ0 a) − κ0 jm (κ∗ a)hm (κ0 a) . κjm (κa)hm (κ0 a) − κ0 jm (κa)hm (κ0 a) (9.412)

The parameter λ in Eq. (9.393) is calculated as follows:  1 λ= u(x)eiκ∗ m·.x dx = v v  1  π ∞ 3

κ 2 − κ∗2 = αm jm (κr)r 2 dr Pm (cos θ )eiκ∗ r cos θ sin(θ )dθ + = 2 κ 2 − κ02 0 0 m=0

The homogenization problem

=

∞

i m αm gm +

m=0

gm =

467

κ02 − κ∗2 , κ 2 − κ∗2

(9.413)

  3 κjm+1 (κa)jm (κ∗ a) − κ∗ jm+1 (κ∗ a)jm (κa) . a(κ 2 − κ∗2 )

(9.414)

Long and short wave asymptotics of the solution of the dispersion equation of the EFM The principal term of the asymptotic of the function ρ∗ (κ0 , κ, κ∗ ) in the dispersion equation (9.404) for small values of the frequency ω has the form ρ∗ (κ0 , κ, κ∗ ) = ρ0 + pρ1 λ(κ0 , κ, κ∗ )(κ∗ ) ≈ i √ ≈ ρs − a 3 p(1 − 3pJ )ρ12 ρ0 ω3 , 3/2 3C  ∞

ρs = ρ0 + pρ1 , J = 0

r (aξ )ξ 2 dξ, ξ = . a

(9.415) (9.416) (9.417)

Here, the principal terms of the order ω0 in the real part of ρ∗ and of the order of ω3 in the imaginary part are kept. As a result, in the long wave limit, the principal term of the effective wave number κ∗ takes the form κ∗ ≈ κ∗s − iγ∗ , * a 3 ρ0 ρ 2 ρs , γ∗ = p(1 − 3pJ ) 2 √ 1 ω4 . κ∗s = ω C 6C ρs ρ0

(9.418) (9.419)

In the short wave limit (ω → ∞), we find the effective wave number κ∗ in the form similar to Eq. (9.381) κ∗ ≈ κ0 − iγ∗ ,

(9.420)

where the attenuation factor γ∗ does not depend on the frequency. In this limit, the principal term of the coefficient λ in Eq. (9.413) takes the form 

3iρ0 2iγ∗ ρ0 2iρ0 3 λ≈− + =− (9.421) − γ∗ a . 2aκ0 ρ1 κ0 ρ1 aκ0 ρ1 4 The integral g (κ∗ ) in Eq. (9.401) for large ω is assessed as follows: 

e−iκ0 r 1 (r)eiκ∗ m·x dx = 4πCr C



∞

e−iκ0 r (r)j0 (κ∗ r)rdr = g (κ∗ ) = 0  ∞ ia 1 e−iκ0 r (r) sin(κ∗ r)dr ≈ − I (γ∗ a), (9.422) = Cκ∗ 0 2Cκ0  ∞ r exp (γ∗ aξ ) (aξ )dξ, ξ = . (9.423) I (γ∗ a) = a 0

468

Heterogeneous Media

As a result, for large ω, the factor (κ∗ ) in Eq. (9.400) has the following asymptotic expression:

−1  −1  3 ≈ 1−p . (9.424) − γ∗ a (κ∗ ) ≈ 1 + ω2 pρ1 λg (κ∗ ) 4 Because λ tends to zero if ω → ∞, we have * ρ1 ρ1 κ0 λ ≈ κ∗ = κ0 1 + p λ ≈ κ0 + p ρ0 2ρ0 

 −1

p 3 3 ≈ κ0 − i . − γ∗ a 1 − p − γ∗ a I (γ∗ a) a 4 4

(9.425)

Comparison of this equation with Eq. (9.420) for κ∗ yields the equations



 −1 ω 3 3 υ∗ ≈ = υ 0 , γ∗ a = p . − γ∗ a 1 − p − γ∗ a I (γ∗ a) Re κ0 4 4 (9.426) Thus, in the short wave limit, the velocity of propagating monochromatic waves coincides with their velocity in the host medium. The attenuation factor γ∗ of these waves does not depend on properties of the inclusions and the host medium, and it is determined by the inclusion volume fraction p and the correlation function (r). This function presents in Eq. (9.426) via the integral I (γ∗ a) in Eq. (9.423). It follows from Eq. (9.426) that for small p, the equation for the attenuation factor γ∗ a takes the form 3 γ∗ a = p + O(p 2 ). 4

(9.427)

Thus, for small volume fraction of inclusion, γ∗ coincides with the short wave asymptotic solution of the dispersion equation (9.359) of the EMM. If p is not small, then γ∗ a < 34 p, and this is the result of spatial correlations in the positions of the inclusions.

Numerical solution of the dispersion equation of the EFM For numerical solution of the dispersion equation (9.404) of the EFM, the correlation function (r) of the random set of inclusions should be determined. A reliable correlation function of a homogeneous set of nonoverlapping spheres can be constructed by using the so-called Percus–Yevick pair correlation function of the centers of nonintersecting spheres. Details of the construction of the corresponding function (r) are presented in [16]. The function (r) depends on the volume fractions p of inclusions, and the graphs of (r) for p = 0.1, 0.3, 0.5 are shown in Fig. 9.38. The real and imaginary parts of the solution of the dispersion equations (9.404) of the EFM for the parameters C = 1, ρ0 = 1, ρ = 0.1 and the inclusion volume fractions p = 0.1 and p = 0.3 are presented in Fig. 9.39. In these cases, the dispersion equation of the EFM (as well as the dispersion equation of the EMM) has one branch of roots for each volume fraction p.

The homogenization problem

469

Figure 9.39 The quasiacoustic branches of the solution of the dispersion equation (9.404) of the EFM for the parameters of the composite with the parameters of the host medium C0 = 1, ρ0 = 1, and of the inclusions C = 1, ρ = 0.1, p = 0.1, 0.3.

For heavy inclusions (C = 1, ρ0 = 1, ρ = 10), and p = 0.1, the dispersion equation (9.405) has quasiacoustical (1) and quasioptical (2) branches, shown in Fig. 9.40. The dashed part of branch 1 in Fig. 9.40 (left) corresponds to the attenuation factor γ∗ a > 1, and the attenuation factors along the branches are shown in Fig. 9.40 (right).

Figure 9.40 Real and imaginary parts of the (1) quasiacoustical and (2) quasioptical branches of the solution of the dispersion equation (9.404) of the EFM for the parameters of the composite medium C0 = 1, ρ0 = 1, and C = 1, ρ = 10, p = 0.1. The dashed part of line 1 corresponds to the attenuation factor γ∗ a > 1.

For heavy inclusions and p = 0.3, the dispersion equation (9.405) has a quasiacoustical branch (1) and three quasioptical branches (2–4) shown in Fig. 9.41. The real parts of the effective wave numbers are shown in Fig. 9.41 (left), and the attenuation factors along the branches are shown in Fig. 9.41 (right).

470

Heterogeneous Media

Figure 9.41 The real and imaginary parts of the (1) quasiacoustical and (2–4) quasioptical branches of the solutions of the dispersion equation (9.404) of the EFM for the parameters of the composite medium C0 = 1, ρ0 = 1, and C = 1, ρ = 10, p = 0.3.

9.7.6 Conclusion In application to the homogenization problem for time-harmonic fields in heterogeneous media, the EFM and EMM predict qualitatively similar results. The dispersion equations of both methods have one branch of the solutions for the composites with light inclusions and several branches for the composites with heavy inclusions. If the only parts of the acoustical and optical branches with the minimal attenuation factors are taken into account by construction of the dispersion curves, the dependence κ∗ = κ∗ (κ0 ) will consist of discrete intervals belonging to different branches, and the effective wave numbers κ∗ in various intervals of κ0 correspond to different modes of internal oscillations contributing to the wave fields propagating in the composite. The self-consistent methods can be extended to solution of the homogenization problem for the composites with inclusions of arbitrary shapes. In this case, the integral equations (9.360) and (9.406) of the one-particle problems should be solved numerically by the methods presented in Chapter 7. By seeking the roots of the dispersion equations for a fixed incident wave number κ0 (frequency ω), the one-particle problems should be solved several times for various values of the effective wave number κ∗ . Note that the one-particle problem of the EFM requires less volume of calculations than such a problem of the EMM. The integral operator in the integral equation (9.406) does not depend on the effective wave number κ∗ . As a result, for a fixed κ0 , the matrix of the corresponding discretized problem is also fixed, and only the right hand side of the discretized system depends on κ∗ . This accelerates construction of the branches of the roots of the dispersion equation (9.405) in comparison with the dispersion equation (9.359) of the EMM. In the last case, the matrix of the discretized problem depends on κ∗ and should be reconstructed for various values of κ∗ in the process of solution of the dispersion equation of the EMM.

The homogenization problem

9.8

471

Notes

The one-particle version of the EFM in application to the homogenization problem for static fields in heterogeneous media was proposed in [25]. In [26], this version of the EFM was used for construction of nonlocal constitutive relations for averaged stress and strain fields in elastic composites. Combination of the EFM and numerical methods for solution of the homogenization problems for elastic composites is proposed in [29]. Historical surveys of application of self-consistent methods to solution of the homogenization problems are presented in [16], [17], [27]. In [17], the EMM and EFM are applied to solution of the homogenization problems for time-harmonic electromagnetic and elastic fields in matrix composite materials. This chapter is based on the publications [28], [29], [30], [31], [32], [33], [34].

Appendix 9.A Averaging of rank four tensors over orientations Let three mutually orthogonal vectors e1 , e2 , e3 compose a Cartesian basis in 3D space. In this basis, the tensor M in Eq. (9.165) is presented in the form 3

M=

Mij kl ei ⊗ ej ⊗ ek ⊗ el .

(9.A.1)

i,j,k,l=1

The rotated basis qi is defined by the equation q = i

3

Qij (ϕ, θ, γ )ej , i = 1, 2, 3,

(9.A.2)

j =1

0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ γ ≤ 2π, where Qij (ϕ, θ, γ ) is an orthogonal matrix and (ϕ, θ, γ ) are Euler angles that define the orientation of the new basis qi . If a fixed configuration of inclusions is rotated by the Euler angles (ϕ, θ, γ ), the tensor M that corresponds to the new configuration takes the form 3

M =

Mij kl qi ⊗ qj ⊗ qk ⊗ ql ,

(9.A.3)

i,j,k=1

where Mij kl are the components of the tensor M in the original basis ei . The average of the tensor M over the orientations (ϕ, θ, γ ) is calculated as follows: M =

3

i,j,k=1

 Mij kl qi ⊗ qj ⊗ qk ⊗ ql ,

(9.A.4)

472

Heterogeneous Media



 1 qi ⊗ qj ⊗ qk ⊗ ql dμ, dμ = sin(θ )dϕdθ dγ , q i ⊗ q j ⊗ qk ⊗ q l = 8π 2 (9.A.5)

where the integration is spread over the values of the Euler angles: 0 ≤ ϕ, γ ≤ 2π, 0 ≤ θ ≤ π. It is assumed that the distribution over the orientations is homogeneous. The tensor M is isotropic, and for derivation of its explicit form, the following equations should be taken into account:  1  2 q i ⊗ qi ⊗ q i ⊗ q i = E + 2E1 , i = 1, 2, 3, (9.A.6) 15  1  2 q i ⊗ q i ⊗ q j ⊗ qj = 2E − E1 , i = j, i, j = 1, 2, 3, 15  1  1 q i ⊗ q j ⊗ qj ⊗ q i = 3E − E2 , i = j, i, j = 1, 2, 3, 15  i j i j (9.A.7) q ⊗ q ⊗ q ⊗ q = 0, i = j, i, j = 1, 2, 3,   q i ⊗ q i ⊗ q j ⊗ q k = q i ⊗ q j ⊗ qk ⊗ q i =  = qj ⊗ qk ⊗ qi ⊗ qi = 0, i = j, i = k, j = k, i, j, k = 1, 2, 3,   q i ⊗ q i ⊗ qi ⊗ qj = q i ⊗ q i ⊗ q j ⊗ qi =   = qi ⊗ qj ⊗ qi ⊗ qi = qj ⊗ qi ⊗ qi ⊗ qi = 0, i = j, i, j = 1, 2, 3. As a result, the averaged tensor M takes the form 1 M = m1 E2 + m2 (E1 − E2 ), 3 1 m1 = (M1111 + M2222 + M3333 + M1122 + M2211 + M1133 + 9 + M3311 + M2233 + M3322 ), 1 m2 = [2(M1111 + M2222 + M3333 )− 15 − (M1122 + M2211 + M1133 + M3311 + M2233 + M3322 )+

(9.A.8)

(9.A.9)

+ 3(M1212 + M1313 + M2323 )]. Here, the rank four isotropic tensors Eij1 kl and Eij2 kl are defined in Eq. (2.210).

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The homogenization problem

473

[2] I. Kunin, The Theory of Elastic Media with Microstructure II. Three-Dimensional Models, Springer, Berlin, 1983. [3] A. Kraynik, D. Reinelt, F. Van Swol, Structure of random monodisperse foam, Physical Review E 67 (2003) 1. [4] S. Kanaun, O. Tkachenko, Effective conductive properties of open-cell foams, International Journal of Engineering Science 46 (6) (2008) 551–571. [5] R. Lemlich, Theory of limiting conductivity of polyhedral foam at low density, Journal of Colloid and Interface Science 64 (1978) 107–110. [6] K. Dharamasena, H. Wadley, Electrical conductivity of open-cell metal foams, Journal of Materials Research 17 (2002) 625–631. [7] A. Kim, M. Hasan, S. Nahm, A nondestructive electrical method for evaluation mechanical properties of AL-foam, in: 16th WCNDT-2004-World Conference on NDT, Aug. 30– Sep. 3, 2004, Montreal, Canada, 2004, pp. 504–511. [8] E. Sadeghi, N. Djilali, M. Bahrami, Thermal conductivity and thermal contact resistance of metal foams, in: Proceedings of HT2009, 2009 ASME Summer Heat Transfer Conference, July 19-23, 2009, San Francisco, California, USA, 2009, pp. 1–11. [9] K. Feitosa, S. Marze, A. Saint-Jalmes, D. Duriam, Electrical conductivity of dispersions: from dry foams to dilute suspensions, Journal of Physics. Condensed Matter 17 (2005) 6301–6305. [10] R. McPhedran, D. McKenzie, The conductivity of lattice of spheres, I. The simple cubic lattice, Proceedings of the Royal Society of London. Series A 359 (1978) 45–63. [11] V. Kushch, Conductivity of a periodic particle composite with transversely isotropic phases, Proceedings of the Royal Society of London. Series A 453 (1997) 65–76. [12] K. Nunan, J. Keller, Effective elasticity tensor of periodic composites, Journal of the Mechanics and Physics of Solids 45 (1984) 1421–1448. [13] T. Iwakuma, S. Nemat-Nasser, Composites with periodic microstructure, Computers & Structures 16 (1983) 13–19. [14] V. Kushch, Microstresses and effective elastic moduli of a solid reinforced by periodically distributed spherical particles, International Journal of Solids and Structures 34 (1997) 1353–1366. [15] J. Segurado, J. Llorca, A numerical approximation to the elastic properties of spherereinforced composites, Journal of the Mechanics and Physics of Solids 50 (2002) 2107–2121. [16] S. Kanaun, V. Levin, Self-Consistent Methods for Composites, vol. 1, Static Problems, Springer, Dordrecht, 2008. [17] S. Kanaun, V. Levin, Self-Consistent Methods for Composites, vol. 2, Wave Propagation in Heterogeneous Materials, Springer, Dordrecht, 2008. [18] V. Kushch, A. Sangani, Stress intensity factor and effective stiffness of a solid containing aligned penny-shaped cracks, International Journal of Solids and Structures 37 (2000) 6555–6570. [19] J. Rathore, E. Fjaer, R. Holt, L. Renlie, P- and S-wave anisotropy of a synthetic sandstone with controlled crack geometry, Geophysical Prospecting 43 (1995) 711–728. [20] G. Mavko, T. Mukerji, J. Dvorkin, The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media, 2nd edition, Cambridge University Press, Cambridge, 2009. [21] E. Saenger, O. Kruger, S. Shapiro, Effective elastic properties of randomly fractured soils: 3D numerical experiments, Geophysical Prospecting 52 (2004) 183–195. [22] C. Gonzalez, J. Segurado, J. Llorca, Numerical simulation of elasto-plastic deformation of composites: evolution of stress microfields and implications for homogenization models, Journal of the Mechanics and Physics of Solids 52 (2004) 1573–1593.

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[23] F. Fedorov, Theory of Elastic Waves in Crystals, Springer, New York, 1968. [24] C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, Toronto, Singapore, 1983. [25] S. Kanaun, The effective field method in linear problems of statics of composite media, Applied Mathematics and Mechanics (PMM) 46 (1982) 655–665. [26] S. Kanaun, V. Levin, Microstresses in composite materials in a region of sharply varying external fields, Mechanics of Composite Materials 3 (1984) 625–629. [27] S. Kanaun, V. Levin, Effective field method in the theory of heterogeneous media, Chapter 3, in: M. Kachanov, I. Sevostianov (Eds.), Effective Properties of Heterogeneous Materials, in: Solid Mechanics and Applications, vol. 93, Springer, 2013. [28] S. Kanaun, S. Bababii, Effective conductive and dielectric properties of matrix composites with inclusions of arbitrary shapes, International Journal of Engineering Science 46 (2008) 147–163. [29] S. Kanaun, On the effective elastic properties of matrix composites: combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities, International Journal of Engineering Science 48 (2010) 160–173. [30] S. Kanaun, Calculation of electro and thermo static fields in matrix composite materials of regular or random microstructures, International Journal of Engineering Science 49 (2011) 41–60. [31] S. Kanaun, E. Pervago, Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures, International Journal of Engineering Science 49 (2011) 420–442. [32] S. Kanaun, En efficient homogenization method for composites with elasto-plastic components, International Journal of Engineering Science 57 (2011) 36–49. [33] A. Markov, S. Kanaun, An efficient homogenization method for elastic media with multiple cracks, International Journal of Engineering Science 82 (2014) 205–211. [34] S. Kanaun, Efficient homogenization techniques for elastic composites: Maxwell scheme vs. effective field method, International Journal of Engineering Science 103 (2016) 19–34.

Index

A Acceptable accuracy, 2, 108, 125, 151, 214, 331, 393, 402 Acoustic pressure, 20, 51, 82, 245, 246, 257 Anisotropic medium, 11, 12, 15, 16, 22 Approximating functions, 2, 3, 105, 107, 108, 135, 141, 151, 188, 322, 328, 362, 371, 378, 382 Gaussian, 3, 4, 103, 151, 183, 200, 207, 213, 225, 237, 245, 258, 285, 294, 344, 404 Approximating nodes, 3, 103, 107, 108, 151, 152, 163, 166, 185, 201, 219, 247, 297, 337, 450 Asymptotics, 85, 91, 126, 134, 205, 207, 208, 214, 216, 249, 251, 276, 286, 297, 300, 302, 303, 331, 333, 355, 376, 377, 381 crack opening, 355 B Boundary contour, 36, 40, 43, 57, 59, 89, 293, 352, 360 Boundary contour crack, 92 C Cartesian coordinates, 11, 36, 59, 70, 113, 115, 132, 134, 138, 141, 161, 187, 194, 209, 238, 246, 273, 322, 328, 378, 438, 440 Cavity surface, 75, 76, 78, 79, 81, 82 Characteristic function, 28, 29, 51, 62, 83, 87, 90, 116, 120, 141, 259, 275, 282, 297, 346, 410, 447, 456, 457, 463 Circulant matrix, 110 Circular surface contacts, 197 Close surface, 57 Coefficients pressure, 384 Collocation method, 2, 147, 148, 151, 157, 170, 194, 206, 248, 398, 404, 421, 439

Complete elliptic integral, 60, 62, 437 Complex foam cell (CFC), 395 Compliance tensor, 419, 433, 434, 438 Computational program, 111, 119, 122, 135, 136, 138, 140, 156, 187, 196, 237, 245, 259, 264, 321, 327, 335, 341, 372, 377, 383 Conjugate gradient method (CGM), 152, 196, 245, 337 Contour integral, 39, 41, 43, 57 Coplanar cracks, 221 Correlation function, 387, 468 Crack, 4, 67–69, 92, 193, 197–199, 293, 300, 302, 306–308, 335–338, 432–435 boundary, 228, 347 contour, 207, 208, 223, 301, 302, 304, 335, 337–339, 374 density, 443, 445, 446 edge, 197, 207, 208, 210, 211, 215, 217, 222, 229, 236, 300, 302, 335, 337, 354, 355, 358, 363, 364, 371, 376, 377 growth, 4, 335, 346, 347, 357, 359, 364, 366, 371, 372, 376, 383 in heterogeneous media, 335 interaction, 221 opening, 70, 73, 92, 193, 196–198, 207, 208, 212, 215, 216, 220, 223, 231, 241, 242, 294, 298, 300–302, 304, 330, 340, 341, 347, 353 vectors, 4, 214, 215, 225, 226, 446 problem, 4, 69, 92, 99, 193, 196, 199, 208, 335, 336, 359, 376, 378, 379 propagation, 335, 354, 361, 364, 376 semiaxes, 438, 442 set, 443, 444, 446 shape, 335, 354, 375 surface, 4, 72, 75, 92, 95, 98–101, 198, 200, 207, 209, 212, 231, 294, 297, 300, 335–339, 432, 433, 436–438, 442

476

volume, 352, 353, 358, 362, 366, 371, 372, 383 Cracked material, 444 medium, 197, 223, 432–434, 438, 443, 445, 446 Cuboid, 108, 119, 138, 149, 156, 167, 171, 173, 222, 228, 247, 252, 255, 285, 318, 404, 440, 450 Curvilinear crack, 200, 201, 207, 237 Curvilinear surfaces, 193, 199, 213 D Delta function, 100, 123 Density, 11, 14, 22, 31, 35, 36, 56, 62, 64, 66, 67, 103, 123, 124, 132, 212, 231, 233, 234, 245, 257, 281, 293, 294, 318, 453, 456–458, 462, 463 crack, 445 Gaussian, 108 potential, 143 Derivative operator, 52, 63 Dielectric heterogeneities, 321 Dielectric heterogeneous medium, 87 Dipole density, 35, 119 Discrete Fourier transforms, 110, 114, 116 Discretization, 3, 4, 145, 146, 149, 151, 157, 159, 183, 200–202, 213, 225, 237, 246, 247, 258, 275, 285, 294, 311, 344, 421, 438 procedure, 312 Discretized integral equations, 313 problem, 2–4, 145, 153, 188, 190, 193, 215, 238, 265, 312, 327, 379, 470 system, 188, 196, 213, 214, 323, 328, 329, 470 Dislocation density, 13 Displacement vector, 12, 25, 31, 41, 62, 67, 98, 287, 288, 311, 319 Double layer, 42, 47, 55, 66, 86, 93, 99, 133, 212, 257, 294 electrostatics, 35, 42, 56, 126 potential, 46, 211 static elasticity, 42 Dry pores, 16, 25, 46, 234, 314

Index

E Effective field method (EFM), 402, 437, 448, 462, 463, 466 Effective medium method (EMM), 456, 457 Electric field, 7–9, 21, 27, 33, 36, 39–41, 51, 52, 55, 58, 88, 89, 119, 130, 145, 146, 149, 150, 153, 155, 156, 187, 275, 278, 281 field vector, 190 potential, 7, 10, 27, 33, 35, 51, 55, 58, 122 Electrostatics, 10, 11, 27, 33, 42, 45, 117, 146, 153 double layer, 35, 42, 56, 126 fields, 4 for heterogeneous media, 145 potential, 27, 117, 119 surface potentials, 33, 40, 126, 135 Ellipse semiaxes, 267 Ellipsoid semiaxes, 431 Ellipsoidal inclusions, 59, 65, 101, 252, 414, 431 Ellipsoidal region, 54, 65, 435 Elliptic integral, 261 Elliptical crack, 101, 196, 197, 209, 210, 437, 442 Ensemble realizations, 387, 388, 391, 392, 395, 402, 411, 414, 419, 431, 433, 434, 448, 454 Equilibrium crack, 347 F Fast Fourier transform (FFT), 3, 103, 109 algorithm, 3–5, 103, 109, 111, 112, 145, 153, 170, 185, 189, 193, 214, 215, 228, 237, 239, 241, 245, 252, 259, 264, 277, 287, 297, 346, 379, 422 Fields, 1, 2, 4, 7, 9, 10, 21, 51, 53, 54, 107, 123, 124, 126, 145, 151, 163, 178, 250, 294, 311, 320, 389, 391, 393, 395 flux, 389–391, 393–395, 399, 401–403, 406 in heterogeneous media, 2, 4, 5, 151 in homogeneous media, 3, 27 pressure, 96, 99

Index

quasistatic, 4, 16 random, 392 static, 4, 389 temperature, 14 Finite element method (FEM), 179 Fourier series, 105 Fourier transform, 10–48, 53, 56–58, 71, 72, 92, 93, 111, 114, 116, 128, 131, 136, 137, 154, 194, 195, 257, 258, 295, 298, 388–390, 454, 465 G Gas density, 318 Gaussian density, 108 function, 3, 4, 103, 105, 118, 119, 131, 145, 151, 182, 183, 200, 202, 207, 213, 245, 246, 248, 258, 261, 285, 294, 344, 380, 404 quasiinterpolant, 5, 103, 105, 108, 111, 146, 149, 153, 157, 170, 177, 194, 223, 231, 233, 246, 251, 258, 275, 285, 294, 312, 336, 344, 398, 404, 421, 450 radial functions, 2, 5, 103, 145, 246 Grid step, 125, 161, 185, 219, 300, 406, 407, 425 H Heterogeneity, 4, 55, 56, 58, 151, 156, 164, 166, 169, 245, 246, 249, 250, 390, 402, 420, 455, 456 in poroelastic medium, 97, 321 Heterogeneous inclusions, 4, 166, 169, 173, 187, 224, 310, 404, 409, 447, 455 medium, 2, 4, 5, 51, 87, 89, 95, 145, 169, 172, 187, 193, 376, 387, 393, 416, 453–455, 470, 471 poroelastic medium, 95 region, 2, 52, 55, 63–65, 84, 88, 89, 95, 96, 98, 145, 146, 149, 245, 247, 275, 394 Homogeneous elastic medium, 11, 13, 22, 31, 41, 62, 64, 68, 161, 169, 193, 214, 221, 224, 228, 338, 356, 432, 455

477

host medium, 2, 52, 63, 87, 89, 126, 145, 156, 166, 170, 173, 275, 281, 310, 390, 409, 416, 430, 433, 455, 456, 462, 463 medium, 4, 13, 14, 41, 51, 63, 67, 92, 130, 163, 166, 169, 237, 293, 310, 343, 356, 387, 389, 404, 453, 454, 457 Homogenization problem, 4, 5, 387, 388, 395, 401, 402, 409, 416, 420, 430, 447, 451, 453, 455, 456, 470, 471 Hydraulic fracture, 4, 234, 335, 347, 354, 356, 361, 362, 367, 371, 377, 382 crack growth, 347, 362, 371, 377 propagation, 356, 357, 362, 373 I Inclusions, 4, 153, 155, 166, 167, 171, 173, 180, 187, 188, 224, 225, 228, 252, 254, 255, 319, 405–407, 409–411 heterogeneous, 4, 166, 169, 173, 187, 224, 310, 409, 447, 455 Inhomogeneous temperature fields, 13 Integral equation for acoustic pressure, 83 equations for heterogeneities, 51 static fields, 145 form, 8, 12, 13, 15, 17, 20, 22, 25, 72, 92, 96, 99, 388 operator, 1, 3, 8, 27, 44, 48, 52, 53, 59, 62–64, 67, 70, 108, 146, 153–155, 212, 231, 275, 299, 311, 312, 343, 389, 390, 417, 432, 453, 454 presentations, 7, 294 surface, 33, 35 Integrand function, 27, 31, 33, 34, 88, 96, 128, 195, 251, 258, 260, 295, 331, 353, 361, 390, 457 Inverse Fourier transform, 10, 15, 18, 21, 22, 26, 111 Inverse operator, 1, 359 Isotropic homogeneous medium, 16, 24 medium, 12, 24, 32, 54, 57, 60, 90, 92, 93, 147, 149, 153, 194, 207, 283, 284, 295, 354, 373, 437, 445, 449 spherical inclusions, 405, 422 tensors, 472

478

K Kernels, 9, 19, 27, 44–48, 53, 56, 58, 64, 69, 70, 73–75, 146, 212, 231, 389, 390, 417, 418, 453, 454 L Layered heterogeneous medium, 341, 373 Layered spherical inclusion, 164 Linear algebraic equations, 2, 5, 145, 147, 148, 151, 152, 157, 159, 177, 193, 195, 206, 226, 227, 233, 247, 258, 277, 285, 295, 299, 312, 344, 360, 398 Local field, 388 Logarithm asymptotics, 358 M Macrocrack tips, 389 Macroinclusions, 389 Magnetic field, 21, 22, 87 Magnetostatic fields in homogeneous media, 7 Minimal residual method (MRM), 151, 196, 245 Monotonic functions, 359 Multiple cracks, 432 N Noncircular crack, 304 Nonconducting thin inclusions, 416 Nondegenerate tensor, 389 Noninteracting cracks, 223, 438, 445–447 Nonintersecting inclusions, 464 Nonplanar surfaces, 68, 131 Nonsingular square matrix, 152 Numerical method, 3, 4, 103, 145, 207, 217, 220, 237, 264, 265, 268, 271, 300, 303, 339, 357, 387, 401, 413, 441, 445, 449, 471 solution, 2, 4, 145, 151, 153, 193, 194, 197, 245–247, 252, 255, 258, 335, 336, 344, 356, 395, 398, 401, 407, 421, 430 O Object FtMg, 264, 265

Index

P Parallel cracks, 221 Plane crack, 197, 201, 208, 209, 220–222, 226, 228, 229, 300, 302–304, 308, 335, 338, 378, 440, 442, 444, 445 elliptic surface, 68 longitudinal, 318 monochromatic, 85, 88, 90, 245, 249, 278 orthogonal, 335 rectangular region, 146 screen, 260, 263, 265 tangent, 59, 207, 223 wave, 84, 265 Plastic deformations, 13, 173–177, 180, 182, 447, 449, 451 Poisson ratio, 161, 163, 171, 178, 185, 197, 198, 220, 222, 223, 228, 287, 300, 314, 339, 347, 363, 367, 375, 377, 425, 437, 440, 442, 445 Polarization vector, 278, 288, 300 Poroelastic medium, 4, 26, 51, 68, 69, 75, 79, 81, 95, 98, 100, 231, 234, 310, 311, 318 Poroelasticity, 2, 7, 16–18, 43, 45, 69, 76, 81, 96, 193, 245, 310 Poroelasticity crack problem, 73, 74 Potential, 7, 29–31, 52, 53, 55, 56, 58, 108, 117–119, 212, 231, 235, 257, 294 density, 93, 143 double layer, 211 electric, 7, 10, 27, 33, 35, 51, 55, 58, 122 electrostatics, 27, 117, 119 static, 128 vector, 12 Pressure, 4, 18, 25, 69, 73, 75, 78, 220, 231, 233, 234, 236, 250, 252, 311, 316, 336, 337, 341, 347, 350, 352 amplitude, 20, 82, 245 coefficients, 375, 384 constant, 73, 79, 339 distribution, 234, 255, 326, 337, 355–359, 361–364, 366, 367, 371–373, 383, 384 fields, 96, 99, 231 gradient, 84, 325 surface, 68 waves, 265

Index

Principal semiaxes, 54 term, 48, 85, 123, 207, 467 unknowns, 173 Property tensors, 1, 2, 387, 395 Q Quasiinterpolant, 103, 105–108, 112, 114, 116, 119, 120, 123, 127, 131, 133, 136–138, 140, 225, 280 Gaussian, 5, 103, 105, 108, 111, 146, 149, 153, 157, 170, 177, 194, 223, 231, 233, 246, 251, 258, 275, 285, 294, 312, 336, 344, 398, 404, 421, 450 Quasistatic crack growth, 4, 335, 347 fields, 4, 16 poroelasticity, 16, 18, 43, 45, 69, 193 poroelasticity crack problem, 73 poroelasticity surface potentials, 43 R Radius crack, 198, 221, 222, 234, 302, 364, 366, 367, 371, 372, 384 Random crack, 433, 438, 444, 446 fields, 392, 455 function, 393 Rectangular region, 108, 112, 146, 265 Region, 2, 10, 13, 27, 29, 51, 53–55, 103, 107, 108, 151, 156, 163, 208, 225, 232, 234, 242, 247, 257, 265, 275, 280, 281, 336, 344, 347, 360, 387, 390, 393, 396, 397 crack, 70, 297, 336, 337, 345, 346, 355, 356 heterogeneity, 285 heterogeneous, 84, 96, 98 spherical, 402 Regularization, 7, 27, 29, 31, 33, 35, 44, 56, 57, 59, 60, 67, 68, 86, 89, 130, 390, 391, 412, 413, 418, 437 Regularized integral, 41 Representative volume element (RVE), 393 surface, 394 Resistivity tensor, 391, 402 RVE, see Representative volume element

479

S Scattered field, 85, 88, 91, 95, 97, 251, 260, 265, 266, 268, 270, 274, 275 Screen plane, 260, 263, 265 region, 258, 259, 265 semiaxes, 268 surface, 86, 260, 263, 265, 274, 327 Semiaxes, 59, 252, 265, 266, 268, 274, 414, 435, 437 crack, 438, 442 principal, 54 screen, 268 Series, 46, 108, 122, 123, 208, 246, 247, 250, 251, 258, 295, 299, 300, 359, 362, 363, 365, 371, 388, 389, 466 SIF, see Stress intensity factors Singly connected regions, finite number, 51 Singly connected surfaces, 67 Sphere surface, 220 Spherical coordinate system, 28, 201, 215, 217, 218, 249, 251, 278, 287, 316, 459 Hankel functions, 251 heterogeneities, 245 inclusion, 153, 166, 171, 228, 229, 252, 254, 287, 288, 290, 315, 316, 405, 406, 414, 422, 423, 425, 428, 451, 456–458, 460, 462, 466 isotropic inclusions, 432 region, 125, 402 RVE, 404, 405, 441, 443, 445, 446, 448, 451 surface, 91, 201 Spheroid semiaxes, 435 Spheroidal inclusions, 432 Star region, 338 Static crack problem, 92, 300 elasticity, 31, 41, 156, 419 crack problem, 193 double layer, 42 for heterogeneous media, 62, 156 integral equation, 67, 159 problem, 237 surface potentials, 41 fields, 4, 389 fields in heterogeneous media, 4, 387 potential, 122, 128

480

Stiff inclusions, 425 Stiffness tensor, 11, 46, 62, 66, 156, 169, 193, 228, 387, 433 Stress, 13, 78, 79, 82, 100, 159–162, 164, 199, 425, 451 field, 31, 43, 64, 169, 170, 172–175, 193, 203, 208, 214, 217, 225, 303, 341, 344, 354, 376, 419, 420, 448, 449 tensor, 4, 12, 13, 16, 31, 42, 64, 65, 69, 70, 72, 77, 156, 161, 167, 169, 171, 174, 193, 205, 207, 208, 211, 212, 341, 344, 417, 425, 432, 434, 439, 448 vector, 43, 212 Stress intensity factors (SIF), 207, 208, 210, 211, 217, 221, 300, 302, 304, 376 dimensionless, 217 Subregions, 2, 3, 151 Surface, 4, 7, 22, 28, 33, 53, 57–59, 65, 66, 129, 131, 132, 200, 203, 211, 212, 214, 250, 252, 260, 341, 344, 352, 358, 382, 394, 395, 432 contacts, 197 crack, 4, 72, 75, 92, 95, 98, 99, 198, 200, 207, 209, 212, 294, 297, 300, 335–338, 432, 433, 436–438, 442 current, 89 forces, 211 integral, 33, 35 integral equations, 2, 4, 51, 65, 87, 93, 101, 193 potentials, 2, 3, 7, 43, 45, 49, 66, 88, 103, 126, 127, 132, 133 pressure, 68 screen, 86, 260, 263, 265, 274, 327 spherical, 91, 201 Symmetric tensor, 24

Index

T Tangent plane, 40, 67, 89, 353 Taylor series, 122, 388 Temperature fields, 14, 63, 64, 169 stresses, 169 Tensor, 1, 7, 8, 10–12, 52–54, 59, 61, 145, 146, 169, 173, 194, 205, 214, 217, 226, 294, 313, 345, 387, 389, 392 basis, 32, 422 fields, 387 space, 32 stress, 4, 12, 13, 16, 31, 42, 64, 65, 69, 70, 72, 77, 156, 161, 167, 169, 171, 174, 193, 205, 207, 208, 211, 212, 341, 344, 417, 425, 432, 434, 439, 448 Toeplitz structure, 3, 109, 113, 193, 214, 215, 228, 297 Traction vector, 90, 99 Triple integral, 127, 203 U Unclosed surface, 43, 56, 67 Uniaxial external stress field, 163, 164 V Vector, 87, 90, 103, 110, 147, 159, 178, 184, 283, 284, 361, 379, 471 fields, 387 potential, 12 W Wolfram Mathematica software, 114, 117, 135, 136, 138, 156, 238, 321

9780128198803 | 152x229mm Paperback | Spine: 24.384 mm

HETEROGENEOUS MEDIA LOCAL FIELDS, EFFECTIVE PROPERTIES, AND WAVE PROPAGATION SERGEY KANAUN This book outlines new computational methods for solving volume integral equation problems in heterogeneous media. It starts by surveying the various numerical methods of analysis of static and dynamic fields in heterogeneous media, listing their strengths and weaknesses, before moving on to an introduction of static and dynamic Green functions for homogeneous media. Volume and surface integral equations for fields in heterogeneous media are discussed next, followed by an overview of explicit formulas for numerical calculations of volume and surface potentials. The book then covers Gaussian functions for the discretization of volume integral equations for fields in heterogeneous media, static problems for a homogeneous host medium with heterogeneous inclusions, and volume integral equations for scattering problems, and concludes with a chapter outlining solutions to homogenization problems and calculations of effective properties of heterogeneous media. The book also features multiple appendices detailing the code of basic programs for solving volume integral equations, written in Mathematica.

HETEROGENEOUS MEDIA

ELSEVIER SERIES IN MECHANICS OF ADVANCED MATERIALS

ELSEVIER SERIES IN MECHANICS OF ADVANCED MATERIALS

HETEROGENEOUS MEDIA LOCAL FIELDS, EFFECTIVE PROPERTIES, AND WAVE PROPAGATION

About the author

KANAUN

Dr. Sergey Kanaun is a Professor of Mechanical Engineering at the Technological Institute of Higher Education of Monterrey, State Mexico Campus, Mexico. His core areas of research are continuum mechanics, mechanics of composites, micromechanics, elasticity, plasticity, and fracture mechanics. Prior to his current teaching post, he was a Professor at the Technical University of Novosibirsk in Russia and also Chief Researcher at the Institute of Engineering Problems of the Russian Academy of Sciences, Saint Petersburg, also in Russia. He has published over 140 articles in peerreviewed journals and two books.

ISBN 978-0-12-819880-3

9 780128 198803

SERGEY KANAUN