Table Of ContentSpringer Tracts in Natural
Philosophy
Volume 35
Edited by C. Truesdell
Springer Tracts in Natural Philosophy
Vol. 1 Gundersen: Linearized Analysis of One-Dimensional Magnetohydrodynamic Flows.
With 10 figures. X, 119 pages. 1964.
Vol. 2 Walter: Differential-und Integral-Ungleichungen und ihre Anwendung bei Abschatzungs
lind Eindeutigkeitsproblemen
Mit 18 Abbildungen. XIV, 269 Seiten, 1964.
Vol. 3 Gaier: Konstruktive Methoden der konformen Abbildung
Mit 20 Abbildungen und 28 Tabellen. XIV, 294 Seiten. 1964.
Vol. 4 Meinardus; Approximation von Funktionen und ihre numerische Behand1ung
Mit 21 Abbildungen. VIII, 180 Seiten, 1964.
Vol. 5 Coleman, Markovitz, Noll: Viscometric Flows of Non-Newtonian Fluids.
Theory and Experiment
With 37 figures. XII, 130 pages. 1966.
Vol. 6 Eckhaus: Studies in Non-Linear Stability Theory
With 12 figures. VIII, 117 pages. 1965.
Vol. 7 Leimanis: The General Problem of the Motion of Coupled Rigid Bodies About a Fixed
Point
With 66 figures, XVI, 337 pages. 1965.
Vol. 8 Roseau: Vibrations non lineaires et theorie de la stabilite
Avec 7 figures. XII, 254 pages. 1966.
Vol. 9 Brown: Magnetoelastic Interactions
With 13 figures. VIII, 155 pages. 1966.
Vol. 10 Bunge: Foundations of Physics
With 5 figures. XII, 312 pages. 1967.
Vol. II Lavrentiev: Some Improperly Posed Problems of Mathematical Physics
With 1 figure. VIII, 72 pages. 1967.
Vol. 12 Kronmuller: Nachwirkung in Ferromagnetika
Mit 92 Abbildungen. XIV, 329 Seiten. 1968.
Vol. 13 Meinardus: Approximation of Functions: Theory and Numerical Methods
With 21 figures. VIII, 198 pages. 1967.
Vol. 14 Bell: The Physics of Large Deformation of Crystalline Solids
With 166 figures. X, 253 pages. 1968.
Vol. 15 Buchholz: The Confluent Hypergeometric Function with Special Emphasis on its
Applications
XVIII, 238 pages. 1969.
Vol. 16 Slepian: Mathematical Foundations of Network Analysis
XI, 195 pages. 1968.
Vol. 17 Gavalas: Nonlinear Differential Equations of Chemically Reacting Systems
With 10 figures. IX, 107 pages. 1968.
Vol. 18 Marti: Introduction to the Theory of Bases
XII, 149 pages. 1969.
G. Capriz
Continua
with Microstructure
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Gianfranco Capriz
Dipartimento di Matematica
Universita
Pisa 56100, Italy
Mathematics Subject Classification (1980): 73B25, 73S99
Library of Congress Cataloging-in-Publication Data
Capriz, Gianfranco.
Continua with microstructure/Gianfranco Capriz.
p. cm.-(Springer tracts in natural phi}osophy; v. 35)
"Lectures given in 1986 at the Scuola Estiva di Fisica Matematica
in Ravello (ltaly),,-Pref.
Bibliography: p.
ISBN-13: 978-1-4612-8166-5
1. Continuum mechanics. 1. Title. II. Series.
QA808.2.C33 1989
531-dc19 88-8483
Printed on acid-free paper.
© 1989 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1989
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, USA), except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and re
trieval, electronic adaptation, computer software, or by similar or dissimilar method-
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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 5 4 3 2 1
ISBN-13: 978-1-4612-8166-5 e-ISBN-13: 978-1-4612-3584-2
DOl: 10.1007/978-1-4612-3584-2
To Barbara
Preface
This book proposes a new general setting for theories of bodies with
microstructure when they are described within the scheme of the con
tinuum: besides the usual fields of classical thermomechanics (dis
placement, stress, temperature, etc.) some new fields enter the picture
(order parameters, microstress, etc.). The book can be used in a
semester course for students who have already followed lectures on
the classical theory of continua and is intended as an introduction to
special topics: materials with voids, liquid crystals, meromorphic con
tinua. In fact, the content is essentially that of a series of lectures
given in 1986 at the Scuola Estiva di Fisica Matematica in Ravello
(Italy).
I would like to thank the Scientific Committee of the Gruppo di
Fisica Matematica of the Italian National Council of Research (CNR)
for the invitation to teach in the School. I also thank the Committee
for Mathematics of CNR and the National Science Foundation: they
have supported my research over many years and given me the
opportunity to study the topics presented in this book, in particular
through a USA-Italy program initiated by Professor Clifford A.
Truesdell.
My interest in the field dates back to a period of collaboration
with Paolo Podio-Guidugli and some of the basic ideas came up
during our discussions.
Successive versions of the text of the lectures, in Italian, were
circulated among friends and colleagues, who have offered welcome
viii Preface
comment and criticism; I am grateful in particular to Paolo Podio
Guidugli, Piero Villaggio an~ Epifanio Virga. My thanks are due also
to my secretary Mrs. Tao Pei Lin, who has helped me with her usual
competence and dedication.
Contents
Preface vii
§1 IntrodQction
Part I General Properties 5
§2 The Model for the Microstructure 5
§3 The Notion of Observer 8
§4 Continua with Microstructure 10
§5 Invariance Properties 13
§6 Conservation of Mass: Kinetic Energy 15
§7 Inertia 18
§8 Dynamic Equations of Balance 21
§9 Balance of Moment of Momentum 23
§1O Boundary Conditions: Change of Variables 25
§1l The Conservative Case in Statics 27
§12 Perfect Fluids with Microstructure 30
§13 Rules of Invariance and the Balance of Moment of
Momentum: Variational Principles in Dynamics 34
§14 Internal Constraints: Continua with Latent Microstructure 36
Part II Special Theories 42
§15 Continua with One-Dimensional Microstructure:
Continua with Voids 42
§16 Liquids with Bubbles 44
§17 Dilatant Granular Materials 46
x Contents
§18 The Perfect Korteweg Fluid 47
§19 Continua with Vectorial Microstructure 49
§20 Uniaxial Liquid Crystals 54
§21 Continua with Affine Microstructure 57
§22 Micromorphic Elastic Continua: Bodies with Continuous
Distribution of Dislocations 59
§23 The Continua of Cosserat 62
§24 Biaxial Nematic Liquid Crystals 64
Part III Thermodynamics 67
§25 Balance Equations 67
§26 Interpretation of the Equations of Balance 69
§27 Thermodynamics of Continua with Latent Microstructure 71
§28 Comparison with the Traditional Class of
Hyperelastic Bodies 74
Part IV Mathematical Problems Posed by the Theory 77
§29 The Influence of the Topological Properties of the
Manifold Jt 77
§30 Further Remarks on the Topological Theory of Defects 81
§31 Existence of Singular Solutions in Statics 82
§32 Phase Transitions 84
§33 Droplets of Perfect Liquids with Microstructure 87
Appendix
90
§1. Introduction
The continuum with microstructure is a refined mathematical model
for a wide class of material bodies endowed with some sort of local
microscopic order, a model that preserves the well established advan
tages accruing from the classical scheme of the continuum.
Actually, the variety of physical phenomena observed and the wealth
of specific mathematical tools invoked to represent them seem at first
to deny the possibility for a global approach, in some way similar to
that which in the classical context precedes the study of special
theories (of fluids, of hyperelastic bodies, of perfect gases, etc.) and
puts in evidence common properties. But recent work shows that such
a global approach is possible. The first part of this book describes the
proposed route and expands on general axioms and theorems. Special
properties valid in particular contexts (e.g., media with voids, liquid
crystals, Cosserat continua) are either proposed or derived in the
second part. Thermodynamic questions are discussed in the third part.
In Part IV some mathematical problems are stated which arise within
the theory.
The notation, as far as possible, is standard. IR is the set of real
numbers, tff the three-dimensional Euclidean space, "f/ the translation
space of tff (the space of three-component vectors), Lin the space of
the linear mappings of "f/ into itself (the space of second-order ten
sors), and Sym (Skw, Sph, Dev) the subspace of symmetric (skew,
