Electrodynamics of Continua I

A. C. Eringen G.A. Maugin

Electrodynamics of Continua I Foundations and Solid Media

With 64 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong

A.C. Eringen Princeton University Princeton, N.J. 08544 U.S.A.

Library of Congress Cataloging in Publication Data Eringen, A. Cerna!.

G.A. Maugin Laboratoire de Modelisation

en Mecanique Universite Pierre et Marie

Curie et C.N.R.S. 75252 Paris 05 France

Electrodynamics of continua I A.C. Eringen, G.A. Maugin. p. cm.

Includes bibliographical references. Contents: 1. Foundations and solid media - 2. Fluids and complex media.

ISBN-13: 978-1-4612-7923-5 e-ISBN-13: 978-1-4612-3226-1 DOl: 10.1007/978-1-4612-3226-1 I. Fluid mechanics. 2. Electrodynamics. 3. Magnetohydrodynamics.

mechanics. I. Maugin, G. A. (Gerard A.), 1944-4. Continuum

II. Title. QC151.E&5 1989 537.6--dc20

Printed on acid-free paper

© 1990 by Springer-Verlag New York Inc.

Softcover reprint of the hardcover 1 st edition 1990

89-21880 CIP

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec­tion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 543 2 1

Preface

The electrodynamics of continua is a branch ofthe physical sciences concerned with the interaction of electromagnetic fields with deformable bodies. De­formable bodies are considered to be continua endowed with continuous distributions of mass and charge. The theory of electromagnetic continua is concerned with the determination of deformations, motions, stress, and elec­tromagnetic fields developed in bodies upon the applications of external loads. External loads may be of mechanical origin (e.g., forces, couples, constraints placed on the surface of the body, and initial and boundary conditions arising from thermal and other changes) and/or electromagnetic origin (e.g., electric, magnetic, and current fields). Because bodies of different constitutions respond to external stimuli in a different way, it is imperative to characterize properly the response functions relevant to a given class of continua. This is done by means of the constitutive theory. For example, an elastic dielectric responds to electromagnetic fields in a totally different way than a magnetic fluid. The present book is intended to present a unified approach to the subject matter, based on the principles of contemporary continuum physics.

In traditional studies, usually the electromagnetic theory is uncoupled from continuum mechanics, so that researchers in the mechanics community sel­dom deal with electromagnetic phenomena, while physicists and electrical engineers often ignore the mechanical effects. As a result, the extension of classical theories to strong electromagnetic fields and finite deformations has suffered long delays. Yet many interesting physical phenomena which possess important technological applications lie at the intersection of continuum mechanics and electromagnetic theory. Clearly, the electrodynamics of de­formable continua is not the sum of these two fields. It is a nonlinear composi­tion of them.

The present work (in two volumes) is written in the hope that it will fill a vacuum that exists in the library of other treatises which are devoted prin­cipally to special topics in continuum mechanics and electromagnetic theory. While there exist several works covering the subject of classical (rigid body) electromagnetism, piezoelectricity, magnetoelectricity, magnetohydrodyna­mics, etc., they generally explore a restricted class of phenomena in one or two

vi Preface

fields. By-and-Iarge, they employ the linear constitutive theory, ignoring the unified nature of the field. The division ofthe subject matter into the mechanics and electromagnetism of special bodies is responsible for drawing our atten­tion away from the study and research in the most exciting interdisciplinary areas of continuum physics. As a result, the tradition has been to place electromagnetic theory within the physics or electrical engineering curricula, and continuum mechanics within other engineering science departments. Yet recent rapid advances in engineering technology have placed severe burdens on engineers and physicists in understanding and being able to use the effects of intense fields and large deformations of bodies. Nonlinear optics, equi­librium of magnetic stars, design of space vehicles, fast computers, fusion reactors, plasma devices, and magnetically levitated vehicles are but a few examples of where a deeper understanding and more careful formulations are necessary. It is here that we must return to the foundations and then build various couplings on sound ground.

This book attempts to present a self-contained, finite deformation, and finite electromagnetic field theory from a unified viewpoint-providing ample critical illustrations by way of applications. The constitution of this work may be sketched as follows:

(i) Development of the ten basic balance laws. (ii) Establishment of the constitutive theory based on nine fundamental

axioms. (iii) Discussion of special theories and applications.

The ultimate goal for (i) is achieved at the end of Chapter 3, on macroscopic electromagnetic theory. To arrive at these results, it was necessary to discuss the kinematics of deformable continua and relevant notions. This is sum­marized in Chapter 1. In Chapter 2, we present the electromagnetic theory from a microscopic viewpoint. This provides a basis for the macroscopic theory.

The general constitutive theory is contained in Chapter 5. Here, by means of nine axioms, the local nonlinear theory is developed for finite deformations and electromagnetic fields. Magnetic symmetry groups and allied topics essen­tial to the development of constitutive equations for magnetic materials (ferromagnetic, ferrimagnetic, and antiferromagnetic) are given in five appen­dices. In this way, for the first time, it has been possible to express the nonlinear constitutive equations for the ninety magnetic groups relevant to magnetic crystals. Nonlinear and linear theories are presented for the elastic solids and viscous fluids that are subject to electromagnetic fields.

Out of necessity, and in consideration of the extra degrees of freedom, the constitutive theory appears again in Chapters 9, 11, 12, and 13 (volume II) for more complicated continua: elastic ferromagnets, electro hydrodynamics, ferrofluids, and memory-dependent materials.

Excluding the last two chapters in volume II, the rest of the work explores various special topics and presents solutions to several linear and nonlinear

Preface vii

problems in order to demonstrate the uses of the basic theory. Chapter 4 contains an elementary treatment of various classes of electromagnetic phe­nomena. Here the emphasis is on the physical nature of diverse electromag­netic effects. Chapter 6 discusses rigid dielectrics, including aspects of the po­tential theory, linear and nonlinear electromagnetic waves, electromagnetic shock waves, and nonlinear optics. Chapter 7 is devoted to the discussion of linear and nonlinear elastic dielectrics, piezoelectricity, surface and body waves, and the general theory of small deformati9ns and electromagnetic fields superposed on large, static deformations and fields. The latter is important to the discussion of the instability of elastic bodies that are subject to electro­magnetic fields. Photoelasticity, electro-optical effects, magneto-optical, and other allied phenomena are presented. Chapter 8 develops nonlinear and linear magnetoelasticity, including a plethora of applications to stress con­centrations, magnetoelastic waves, and the instability of plates under strong magnetic fields. Fundamental to all treatments are the basic field equations and jump conditions collected at the end of Chapter 3, as well as the constitu­tive equations given in Chapter 5.

The second volume of this treatise begins with Chapter 9 on elastic ferro­magnets. Here it is necessary to incorporate the extra degree offreedom arising from the gyroscopic nature of the spin density. Balance laws are also extended to include spin-spin interactions. Constitutive theory, based on the general axioms, is developed. The theory is then applied to various problems including waves, the Faraday effect, etc.

Chapter 10 is concerned with a discussion of magnetohydrodynamics (MHD). Bernoulli's equation, Alfven waves, incompressiple viscous flows, compressible flow and magnetohydrodynamic shock waves, magneto hydro­dynamic stability, and the equilibrium of magnetic stars are some of the problems considered.

Chapter 11 deals with electro hydrodynamics and Chapter 12 with ferro­fluids. The latter includes a discussion of extra degrees of freedom for the spin-spin interaction. Several flow problems which are solved include non­viscous and viscous ferro hydrodynamics, shear flow, stagnation-point flow, and interfacial stability.

Chapter 13 presents the theory of memory-dependent electromagnetic continua. It is well known that energy loss, dispersion, and absorption of waves cannot be explained without consideration of memory effects. This requires incorporation of past histories of strain and electromagnetic fields into constitutive equations. Thermodynamics of rate-dependent materials, and materials with continuous memory of past motions, and fields are dis-

f

cussed. Constitutive equations are obtained for memory-dependent electro-magnetic elastic solids and electromagnetic viscous fluids. Applications are made to electromagnetic waves, birefringent materials, photo viscoelasticity, and nonlinear optics.

In Chapter 14 we present an account of the recent theory of nonlocal electromagnetic elastic continua. Here the stress, polarization, magnetization,

viii Preface

and current at a reference point in the body are considered to be functionals of the strain and electromagnetic fields of all points of the body. The non­locality extends the domain of the application of the theory to microscopic and atomic scales (without the quantum effect). In this way, electromagnetic waves with short wavelengths can be treated. Point charge, anomalous skin effects, eddy currents, infrared dispersion, and lattice vibrations, and even superconductivity, fall into the domain of applicability of the nonlocal theory. This field is rather new, currently in a development stage. Consequently, a short account has been included here, in the hope that it may provide some stimuli for future researchers.

Since electrodynamics properly falls into the domain of relativity, we also include an account of this subject, to close this treatise.

All chapters on special topics and applications are supplied with relevant basic equations and jump conditions in the opening pages. This way, no need will arise for a search of the basic equations in the text.

It is hoped that this work will serve two basic needs:

(a) as a textbook for graduate studies, and (b) as a guide and reference for researchers.

As a text, this work could be used for several different courses. A first course may be based on Chapters 2, 3, and 5, with problems selected from Chapters 6 through 9. This sequence has been followed successfully by the first author during the last ten years. Another course can be composed by utilizing material selected from various chapters; for example, the topic of elastic dielectrics and magnetoelasticity may be presented, or a discussion of ferro­fluids and elec;trohydrodynamics could take place. Alternatively, the memory­dependent materials diseussed in Chapter 13, and the nonlocal electromag­netic continua presented in Chapter 14, could be offered as advanced topics.

As a guide for researchers, there is some material on foundations; e.g., constitutive theory for magnetic crystal classes, memory-dependent continua and nonlocality, formulations and solutions of problems in nonlinear optics of deformable bodies, instability of magnetized bodies, the effects of high electric fields and currents in deformable bodies and conductors, etc. Once the basic principles and the constitutive theory are well understood, extensions are possible in other directions (e.g., topics on mixtures and semiconductors, which fall beyond the scope of the present treatise).

Much of the material presented in this treatise is either new, or newly composed. Presently, there are no other treatises similar in content and composition. For example, Chapters 3 and 5, and parts of Chapters 7, 8, 13, and 14, contain original material and compositions. Some of these are indi­cated specifically in the appropriate sections.

The beginning of the fOrqlal writing of this book goes back to the Spring of 1976, when G.A. Maugin was a visiting Research Associate at Princeton University. Over the years, the project became increasingly ambitious. The mechanical and communication difficulties arising from the three thousand miles of ocean between Princeton and Paris caused many delays. In spite of

Preface ix

all our efforts, errors may have crept into the text. Whether they are typo­graphical, or otherwise, we ask the tolerance of our readers and express our deep appreciation, in advance, should they be brought to our attention.

Because of our extensive involvement with research in various sectors of this wide field, many parts of this book have, naturally, been influenced by our research and thought processes. This natural predisposition does not exclude our respect and admiration for other researchers in the field. The large bibliography stands as a testament to this. We hope that the authors of many other contributions that have been unintentionally left out will understand our limitations.

It is a pleasure to acknowledge the help received from various students and colleagues. Without the generous help provided by Dr. Erhan Kiral, Appendix B on the crystallographic magnetic groups could not have been written. A.c. Eringen and E. Kiral completed a monograph on this subject in 1977. This manuscript is now scheduled for publication by Springer-Verlag. Dr. Na~it Ari and Mr. T.M. Cheng checked parts of Chapter 7, and Mrs. (Lois) M. Nase helped in the search for some references. Eleni Maugin prepared the index. A.C.E's daughters, Meva S. Eringen and Lisa M. Eringen edited the manuscript, and we are deeply indebted to them for their invaluable help. A father's love is forever. A.C.E. expresses his thanks to the authorities of the University of Paris VI, "Laboratore de Modelisation en Mecanique", and to his host, G.A.M., for inviting him as a visiting professor in the spring of 1985. This provided an excellent opportunity to discuss several questions and check portions of Chapters 8 and 9. We acknowledge with thanks, the extreme care and attention provided in the preparation of this manuscript by our publisher Springer-Verlag.

Finally, we acknowledge the permission of the following organizations for allowing us to reprint various tables and figures:

Oxford University Press, Oxford, U.K. (Figure 4.5.3) Elsevier Science Publishers, Amsterdam, The Netherlands (Figure 9.2.1) Pergamon Press, Oxford, U.K. (Figures 9.6.1, 9.6.2, 9.6.3, 9.6.4, 9.6.5, 9.6.6,

12.10.2) and Tables Cl.1-C1.12 and C2.1-C2.12. American Institute of Physics, New York (Figures 4.5.7, 8.8.1, 8.8.2, 8.8.3,

9.8.1,9.8.3,9.8.4, 11.8.4, 12.7.2, 12.8.2, 12.8.3) Springer-Verlag, Wien, Austria (Figures 10.5.1, 10.5.2) Springer-Verlag,

Berlin, Heidelberg (Figures 13.14.1-13.14.5). John Wiley and Sons, New York (Figures 6.7.1, 7.5.2, 8.4.2, 10.6.2, 13.7.1,

13.8.1) Annual Reviews, Inc., Palo Alto, CA (Figure 11.7.1) Gauthier-Villars, Paris, France (Figures 11.8.2, 11.8.3, 12.4.1, 12.4.2, 12.4.3,

12.4.4) The MIT Press, Cambridge, MA (Figures 11.9.1, 11.9.2, 12.11.2) Academic Press, New York (Figure 1.2.1) Kluwer Academic Publishers, Dordrecht, Holland (Figures 4.5.2, 6.17.2,

6.17.3)

Contents (Volume I)

Preface.

CHAPTER 1

Kinematics of Material Continua

1.1. Scope of the Chapter. 1.2. Mass and Charge. 1.3. Motion and Deformation 1.4. Strain Measures . 1.5. Polar Decomposition 1.6. Infinitesimal Strains . 1.7. Volume and Area Changes 1.8. Compatibility Conditions 1.9. Deformation Rates 1.10. Rigid Body Motion . 1.11. Objectivity 1.12. Objective Time Rates of Tensor Fields. 1.13. Kinematics of Line, Surface, and Volume Integrals

Problems.

CHAPTER 2 Microscopic Electromagnetic Theory .

2.1. Scope of the Chapter. 2.2. Electric Charge 2.3. Electric Moments. 2.4. Current, Magnetic Moments 2.5. Microscopic Electromagnetic Fields 2.6. Electromagnetic Force on a Point Particle. 2.7. Microscopic Maxwell's Equations . 2.8. Electromagnetic Force on Composite Particles 2.9. Couple Acting on a Composite Particle 2.10. Power of Electromagnetic Forces

Problems.

v

2 4 6 8 9

10 11 12 14 15 16 19 23

26

26 27 29 32 33 35 36 40 42 43 45

xii Contents (Volume I)

CHAPTER 3 Macroscopic Electromagnetic Theory.

3.1. Scope of the Chapter. 3.2. Statistical Averages . 3.3. Maxwell's Equations. 3.4. Galilean Invariance of Maxwell's Equations 3.5. Macroscopic Densities of Mass, Force, Couple, and Power

A. Mass Density . B. Force Density. C. Couple Density D. Electromagnetic Power

3.6. Electromagnetic Stress Tensor, Momentum, and Poynting Vector 3.7. Electromagnetic Force, Couple, and Power at a Discontinuity

Surface 3.8. Balance Laws of Continuum Physics 3.9. Balance Laws of Electrodynamics

A. Global Statement. B. Local Balance Laws. C. Boundary Conditions

3.10. Thermomechanical Balance Laws 3.11. Clausius-Duhem Inequality. 3.12. Invariance Requirements for Balance Laws 3.13. Principle of Virtual Power 3.14. Resume of Balance Laws.

Problems.

CHAPTER 4 Elementary Properties of Electromagnetic Continua

4.1. Scope of the Chapter. 4.2. Dielectric Materials . 4.3. Physical Theory of Dielectrics

A. Polarizability. B. The Clausius-Mossoti Equation C. Ferroelectric Crystals D. Electrostriction and Piezoelectricity E. Pyroelectricity

4.4. Magnetic Materials . A. The Origin of Magnetism B. Gyromagnetic Effects.

4.5. Different Types of Magnetism A. Paramagnetism . B. Ferromagnetism . C. Ferrimagnetism and Antiferromagnetism D. Magnons . E. Magnetostriction and Piezomagnetism.

4.6. Electric Conduction . A. Electron Conduction. B. Cross Effects .

47

47 48 50 52 55 55 56 59 60 62

65 66 72 72 73 74 75 81 82 84 85 87

91

91 91 93 93 96 96 99

100 100 100 101 102 103 104 110 111 114 114 114 115

Contents (Volume I) xiii

4.7. Interaction Between Light and Electromagnetic Continua A. Electromagnetic Optics . B. Inducement of Optical Anisotropy Problems.

CHAPTER 5

Constitutive Equations

5.1. Scope of the Chapter. 5.2. Resume of Balance Laws. 5.3. Raison d'~tre . 5.4. Axioms of Constitutive Theory .

A. Axiom of Causality B. Axiom of Determinism C. Axiom of Equipresence D. Axiom of Objectivity. E. Axiom of Time Reversal. F. Axiom of Material Invariance G. Axiom of Neighborhood. H. Axiom of Memory I. Axiom of Admissibility

5.5. Integrity Basis Under Crystallographic Point Groups 5.6. Integrity Basis Under Magnetic Point Groups. 5.7. Integrity Basis of Vectors and Tensors for Isotropic Materials 5.8. Constitutive Equations of Electromagnetic Elastic Solids. 5.9. Isotropic Electromagnetic Elastic Solids 5.10. Anisotropic Electromagnetic Elastic Solids 5.11. Linear Constitutive Equations 5.12. Electromagnetic Fluids 5.13. Resume of Constitutive Equations

Problems.

CHAPTER 6 Itigid I>ielectrics

6.1. Scope of the Chapter. 6.2. Fundamental Equations. 6.3. Potential Theory .

A. Green's Theorem. B. Uniqueness Theorem. C. Representation Theorem. D. Green's Function. E. Green's Function for Dirichlet's Problem in Spherical Regions and

in Half-Space . F. Eigenfunction Expansions

6.4. Charged Cylinder. 6.5. Potential in Half-Plane 6.6. Charged Disk. 6.7. Plane Electromagnetic Waves in Isotropic Bodies.

117 117 121 125

128

128 129 131 133 134 135 136 136 138 138 141 143 144 145 150 154 156 159 163 165 171 174 178

180

180 181 183 184 184 185 186

187 190 192 193 196 198

xiv Contents (Volume I)

6.8. Spherical Waves . 6.9. Nonlinear Theory of Rigid Dielectrics 6.10. One-Dimensional Problem . 6.11. Two-Dimensional Nonlinear Problem. 6.12. Solutions of Two-Dimensional Problems for Special Dielectrics. 6.13. Nonlinear Electromagnetic Waves. 6.14. Electromagnetic Shock Waves in Rigid Stationary Dielectrics 6.15. Moving Rigid Dielectrics

A. Dragging of Light by a Moving Dielectric . B. The Rotation of a Rigid Dielectric in a Magnetic Field C. Unipolar Induction

6.16. Nonlinear Optics. 6.17. Solitary Waves

Problems.

CHAPTER 7 Elastic Dielectrics.

201 203 205 207 209 213 217 224 224 226 229 230 233 236

239

7.1. Scope of the Chapter. 239 7.2. Resume of Basic Equations 240 7.3. Uniqueness Theorem. 243 7.4. Piezoelectric Moduli. 246 7.5. Piezoelectrically Excited Thickness Vibrations of Plates 253 7.6. Extensional Vibrations of Piezoelectric Rods 257 7.7. Surface Waves 259 7.8. Radially Symmetric Vibrations. 261 7.9. Spherically Symmetric Vibrations of Thin Ceramic Shells 263 7.10. Piezoelectrically Generated Electric Field . 263 7.11. Elastic Dielectrics Subject to Finite Deformations and Fields 265 7.12. Cylindrically Symmetric Deformation of a Tube Subject to a Radial

Field . 269 7.13. Axisymmetric Oscillations of a Tube 273 7.14. Small Deformations and Fields Superimposed on Large Static

Deformations and Fields. 278 7.15. Photoelastic Effect 287 7.16. Electro-Optical Effect 290 7.17. Magneto-Optical Effects. 297 7.18. Elastic Ionic Crystals, Ferroelectrics, and Other Polarizable Solids. 304

Problems . 306

CHAPTER 8

Magnetoelasticity .

8.1. Scope of the Chapter. 8.2. Resume of Basic Equations 8.3. Static Magnetoelastic Fields. 8.4. Uniformly Magnetized Sphere in Nonpermeable Medium 8.5. Two-Dimensional Magnetoelasticity

307

307 308 314 316 319

Contents (Volume I) xv

8.6. Stress Concentration at a Circular Hole 321 8.7. Radial Motions of a Cylinder Undet an Axial Magnetic Field 326 8.8. Propagation of Plane Waves 329 8.9. Waves Induced by a Thermal Shock 338 8.10. Homogeneous Strain in a Magnet 343 8.11. Simple Shear . 347 8.12. Torsion of a Cylindrical Magnet 349 8.13. Electric Conduction in a Twisted Cylinder. 354 8.14. Theory of Magnetoelastic Plates 359 8.15. Plate under Transverse Magnetic Field. 363 8.16. Magnetoelastic Buckling of a Circular Plate 367 8.17. Other Works: Dynamic Buckling and Nonlinear Waves. 370

Problems . 371

APPENDICES

A. Crystallographic Point Groups 373 B. Crystallographic Magnetic Groups 379 C. Integrity Bases of Crystallographic Groups. 384 D. Some Theorems on Symmetric Polynomial Functions 399 E. Representations ofIsotropic, Scalar, Vector, and Tensor Functions 402 F. Maxwell's Equations in Various Systems of Units. 405

References 407

Index 11

Contents (Volume II)

Preface to Volume II

CHAPTER 9 Elastic FerromQgnets

CHAPTER 10

Magnetohydrodynamics .

CHAPTER 11

Electrohydrodynamics

CHAPTER 12

Ferrofluids .

CHAPTER 13 Memory-Dependent Electromagnetic Continua.

CHAPTER 14 Nonlocal Electrodynamics of Elastic Solids

CHAPTER 15

Relativistic Electrodynamics of Continua.

References

Index

v

437

502

551

574

611

675

716

753

11