Continuum Physics. Mixtures and EM Field Theories

323 Pages·1976·5.387 MB·English

by  A. Eringen (Eds.)

Contributors
RAY M. BOWEN
RICHARD A. GROT
GERARD A. MAUGIN
CONTINUUM PHYSICS
Edited by A. CEMAL ERINGEN
PRINCETON UNIVERSITY
PRINCETON, NEWJERSEY
VOL UM E III-Mixtures and EM
Field Theories
@1976
ACADEMIC PRESS New York San Francisco London
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Library of Congress Cataloging in Publication Data
Main entry under title:
Continuum physics.
Includes bibliographies.
1. Field theory (Physics) I. Eringen, A. Cemal, ed.
QC174.45.C6 530 71-117094
ISBN 0-12-240803-9 (vol. 3)
PRINTED IN THE UNITED STATES OF AMERICA
List of Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
RAY M. BOWEN (1), Department of Mechanical Engineering and
Materials Science, Department of Mathematical Sciences, Rice
University, Houston, Texas
RICHARD A. GROT (129), National Bureau of Standards, Washington,
D.C.
GERARD A. MAUGIN (221), Department of Theoretical Mechanics, Uni­
versity of Paris VI, Paris, France
IX
Preface
The third volume of Continuum Physics is devoted to the discussion of field
theories for bodies composed of different substances (e.g., mixtures) and the
interaction of electromagnetic effects with the deformable bodies. The inten­
tion here is to present the mathematical foundations of nonlinear mechanical,
electrical, and magnetic phenomena that take place in mixtures as well as in
materially uniform bodies. While initial developments in these fields occurred
at the turn of the century, many questions remained open (e.g., thermody­
namics for irreversible processes, EM interactions with moving media, micro-
magnetism). As a result, most of the material presented here is the fruits of
the research that has taken place during the last decade or so.
Part I is devoted to the development of the theory of mixtures. Kinematics,
balance laws, and constitutive equations for bodies consisting of several dif­
ferent substances are discussed. The mixture may consist of N different sub­
stances consisting of solid and fluidlike species. Diffusion theory, porous media,
many-fluid theories, and chemically reacting media all fall into the domain of
the mixture theory.
Part II is concerned with a discussion of mechanics of deformable bodies
interacted by electromagnetic fields. Deformation produced by EM fields
(e.g., electrostriction, magnetostriction), EM fields resulting from the defor­
mation of bodies (e.g., piezoelectricity, pyroelectricity), and a plethora of
other physical phenomena arising from mechanical and EM interactions fall
into the subject area of this field.
The subject of micromagnetism, treated in Part III, deals with even further
considerations arising from the interaction of strong magnetic fields with the
inner structure of the body. Thus the magnetically saturated media, magnetic
domains, ferromagnetism, spin waves, etc., provide rich classes of physical
phenomena for research.
The invariance under the group of time-dependent rigid-body motions of
the spatial frame of reference (the axiom of objectivity) so essential to de­
velopment of thermomechanical theories must be replaced with the Lorentz
invariance when we come to deal with the EM fields. This basic need forces
XI
Xll PREFACE
us to reexamine the foundations of the continuum theories of deformable
bodies interacting with EM fields. Thus the approach must necessarily be
relativistic. Moreover, the nature of magnetism requires additional considera­
tions arising from the inner structure of materials. These are also discussed in
Part III.
I express my pleasure for the pleasant cooperation of contributors. Patient
revisions and friendly discussions made my task rewarding. I acknowledge
with thanks the careful and untiring efforts of the staff of Academic Press for
the high standards in production of this treatise.
Contents of Other Volumes
VOLUME I—Mathematics
Part I. Tensor Analysis
A. Cemal Eringen
Part II. Group Theory
Alfred W. Jones
Part III. Theory of Invariants
A. J. M. Spencer
Part IV. Functional Analysis
Ian N. Sneddon
Part V. Analytic Function Theory
L. C. Woods
Part VI. Elements of Stochastic Processes
/. Clifton Samuels
VOLUME II—Continuum Mechanics of Single-Substance Bodies
Part I. Basic Principles
1. Deformation and Motion
A. Cemal Eringen
xiii
XIV CONTENTS OF OTHER VOLUMES
2. Balance Laws
A. Cemal Eringen
3. Thermodynamics of Continua
A. Cemal Eringen
Part II. Constitutive Equations for Simple Materials
1. General Theory
A. Cemal Eringen
2. Thermoelastic Solids
Erdogan S. Suhubi
3. Thermoviscous Fluids
Tien Sun Chang
4. Simple Materials with Fading Memory
Ellis Harold Dill
Part III. Methods of Solution
1. Exact Solutions in Fluids and Solids
Charles B. Kafadar
2. Singular Surfaces and Waves
Matthew E. McCarthy
3. Complex Function Technique
Eazil Erdogan
VOLUME IV—Polar and Nonlocal Field Theories
Part I. Polar Field Theories
A. Cemal Eringen and Charles B. Kafadar
Part II. Nonlocal Field Theories
Dominic G. B. Edelen
Part III. Nonlocal Polar Field Theories
A. Cemal Eringen
Part I Theory of Mixtures
Ray M. Bowen
DEPARTMENT OF MECHANICAL ENGINEERING
AND MATERIALS SCIENCE
DEPARTMENT OF MATHEMATICAL SCIENCES
RICE UNIVERSITY
HOUSTON, TEXAS
Introduction 2
1. Kinematics and Field Equations 2
1.1. Kinematics and Changes of Frame 2
1.2. Balance of Mass 9
1.3. Elements of Stoichiometry 12
1.4. Balance of Momentum 16
1.5. Balance of Energy 21
1.6. Second Axiom of Thermodynamics 28
1.7. Comments on the Formulation of Mixture Theories 35
2. Diffusion Theories 37
2.1. Introduction 37
2.2. Constitutive Assumptions 43
2.3. Restrictions Imposed by the Second Axiom of Thermodynamics 45
2.4. Mixtures of Elastic Materials 56
2.5. Restrictions Imposed by the Axiom of Material Frame-IndifFerence . .. 58
2.6. Restrictions Imposed by Material Symmetry 63
2.7. Mixtures of Gases. Linearized Constitutive Equations 67
2.8. Mixtures of Gases. Classical Diffusion Theories 79
2.9. Mixture of a Gas and an Isotropie Solid. Porous Media Model 93
2.10. Wave Propagation with Diffusion 105
2.11. Concluding Remarks 120
References 122
1
2 RAY M. BOWEN
Introduction
This part is concerned with certain aspects of the general continuum theory
of mixtures. For the most part, it is concerned with theories of diffusion.
Section 1 contains a discussion of the kinematics and field equations appro­
priate to the study of mixtures of general materials. In Section 2 a mixture of
a relatively general type is defined by a set of constitutive equations. These
constitutive equations are then systematically examined in order to deduce
the restrictions imposed by thermodynamics, material symmetry, and
material frame-indifference. Section 2 also contains linearized formulations
appropriate to mixtures of gases and mixtures of gases and solids. The latter
formulation yields a model of a porous medium. The porous media model
is used in Section 2.10 in order to show the effect of diffusion on the growth
and decay properties of acceleration waves.
Throughout this part a considerable effort has been made to show how the
classical diffusion theories and porous media theories can be obtained as
approximations to the general theories. Thus, it is hoped that this part will
be useful as a bridge between the purely theoretical studies of general types
of mixtures and their applications.
L Kinematics and Field Equations
In this section we formulate the kinematics and the field equations for a
mixture. In addition, in Sections 1.2, 1.5, and 1.6 we discuss certain approxi­
mations present in classical diffusion theories.
1.1. KINEMATICS AND CHANGES OF FRAME
Consider 91 continuous bodies 93^ ..., 33^, each of which is visualized by
the region it occupies in three-dimensional physical space. As in theories of
single materials, each body can be assigned a fixed but otherwise arbitrary
reference configuration and a motion
(1.1.1) χ = χ (Χ ,0
ο α
where X is the position of a particle of the ath body or constituent in its
a
reference configuration, / the time, and x the spatial position occupied at the
time / by the particle labeled X. The function χ in (1.1.1) is called the de­
a α
formation function for the ath constituent. In this work, no major effort will
be made to state every mathematical smoothness assumption. In general we