Electrodynamics of Continua I: Foundations and Solid Media

468 Pages·1990·9.98 MB·English

by  G. A. Maugin (auth.), A. C. Eringen

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 G.A. Maugin
Princeton University Laboratoire de Modelisation
Princeton, N.J. 08544 en Mecanique
U.S.A. Universite Pierre et Marie
Curie et C.N.R.S.
75252 Paris 05
France
Library of Congress Cataloging in Publication Data
Eringen, A. Cerna!.
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. 4. Continuum
mechanics. I. Maugin, G. A. (Gerard A.), 1944-
II. Title.
QC151.E&5 1989
89-21880
537.6--dc20
CIP
Printed on acid-free paper
© 1990 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1990
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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. De-
formable 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 elec-
tromagnetic 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 sel-
dom 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 de-
formable continua is not the sum of these two fields. It is a nonlinear composi-
tion 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 prin-
cipally to special topics in continuum mechanics and electromagnetic theory.
While there exist several works covering the subject of classical (rigid body)
electromagnetism, piezoelectricity, magnetoelectricity, magnetohydrodyna-
mics, 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 atten-
tion 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, equi-
librium 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 sum-
marized 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 essen-
tial to the development of constitutive equations for magnetic materials
(ferromagnetic, ferrimagnetic, and antiferromagnetic) are given in five appen-
dices. 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 phe-
nomena. Here the emphasis is on the physical nature of diverse electromag-
netic effects. Chapter 6 discusses rigid dielectrics, including aspects of the po-
tential 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 electro-
magnetic 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 con-
centrations, 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 constitu-
tive equations given in Chapter 5.
The second volume of this treatise begins with Chapter 9 on elastic ferro-
magnets. 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 hydro-
dynamic stability, and the equilibrium of magnetic stars are some of the
problems considered.
Chapter 11 deals with electro hydrodynamics and Chapter 12 with ferro-
fluids. The latter includes a discussion of extra degrees of freedom for the
spin-spin interaction. Several flow problems which are solved include non-
viscous 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 non-
locality 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 ferro-
fluids and elec;trohydrodynamics could take place. Alternatively, the memory-
dependent materials diseussed in Chapter 13, and the nonlocal electromag-
netic 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 indi-
cated 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 typo-
graphical, 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. v
CHAPTER 1
Kinematics of Material Continua
1.1. Scope of the Chapter.
1.2. Mass and Charge. 2
1.3. Motion and Deformation 4
1.4. Strain Measures . 6
1.5. Polar Decomposition 8
1.6. Infinitesimal Strains . 9
1.7. Volume and Area Changes 10
1.8. Compatibility Conditions 11
1.9. Deformation Rates 12
1.10. Rigid Body Motion . 14
1.11. Objectivity 15
1.12. Objective Time Rates of Tensor Fields. 16
1.13. Kinematics of Line, Surface, and Volume Integrals 19
Problems. 23
CHAPTER 2
Microscopic Electromagnetic Theory . 26
2.1. Scope of the Chapter. 26
2.2. Electric Charge 27
2.3. Electric Moments. 29
2.4. Current, Magnetic Moments 32
2.5. Microscopic Electromagnetic Fields 33
2.6. Electromagnetic Force on a Point Particle. 35
2.7. Microscopic Maxwell's Equations . 36
2.8. Electromagnetic Force on Composite Particles 40
2.9. Couple Acting on a Composite Particle 42
2.10. Power of Electromagnetic Forces 43
Problems. 45
xii Contents (Volume I)
CHAPTER 3
Macroscopic Electromagnetic Theory. 47
3.1. Scope of the Chapter. 47
3.2. Statistical Averages . 48
3.3. Maxwell's Equations. 50
3.4. Galilean Invariance of Maxwell's Equations 52
3.5. Macroscopic Densities of Mass, Force, Couple, and Power 55
A. Mass Density . 55
B. Force Density. 56
C. Couple Density 59
D. Electromagnetic Power 60
3.6. Electromagnetic Stress Tensor, Momentum, and Poynting Vector 62
3.7. Electromagnetic Force, Couple, and Power at a Discontinuity
Surface 65
3.8. Balance Laws of Continuum Physics 66
3.9. Balance Laws of Electrodynamics 72
A. Global Statement. 72
B. Local Balance Laws. 73
C. Boundary Conditions 74
3.10. Thermomechanical Balance Laws 75
3.11. Clausius-Duhem Inequality. 81
3.12. Invariance Requirements for Balance Laws 82
3.13. Principle of Virtual Power 84
3.14. Resume of Balance Laws. 85
Problems. 87
CHAPTER 4
Elementary Properties of Electromagnetic Continua 91
4.1. Scope of the Chapter. 91
4.2. Dielectric Materials . 91
4.3. Physical Theory of Dielectrics 93
A. Polarizability. 93
B. The Clausius-Mossoti Equation 96
C. Ferroelectric Crystals 96
D. Electrostriction and Piezoelectricity 99
E. Pyroelectricity 100
4.4. Magnetic Materials . 100
A. The Origin of Magnetism 100
B. Gyromagnetic Effects. 101
4.5. Different Types of Magnetism 102
A. Paramagnetism . 103
B. Ferromagnetism . 104
C. Ferrimagnetism and Antiferromagnetism 110
D. Magnons . 111
E. Magnetostriction and Piezomagnetism. 114
4.6. Electric Conduction . 114
A. Electron Conduction. 114
B. Cross Effects . 115