Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations
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733 Frederick Bloom
Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations
Springer-Verlag Berlin Heidelberg New York 1979
Author Frederick Bloom Department of Mathematics, Computer Science and Statistics University of South Carolina Columbia, S.C. 29208 USA
A M S Subject Classifications (1970): Primary: 7 3 S 0 5 Secondary: 53 C10 ISBN 3 - 5 4 0 - 0 9 5 2 8 - 4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 7 - 0 9 5 2 8 - 4 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data Bloom, Frederick, 1944 Modern differential geometric techniques in the theory of continuous distributions of dislocations. (Lecture notes in mathematics ; 733) Bbiliography: p. Includes index. 1. Dislocations in crystals. 2. Geometry, Differential. 3. G-structures. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 733. OD921.B56 548'.842 79-9374 ISBN 0-387-09528-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisheJ © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
FOR
HARRY
AND
MORDECHAI
Preface Among research workers
in mechanics
ematics there has been great interest, decades,
and applied math-
in the past two
in the area of continuum theories
More recently,
of dislocations.
attention has turned to the more difficult
problem connected with the motion of dislocations a continuum and its relation to various plasticity
formulations
theory for a body possessing
effort to formulate a continuous
dislocations was made by Kondo prescribing,
characterize
distribution
on the basis of certain heuristic
geometric
structures
a geometric
of the dislocation
was any serious
given to the types of constitutive
such as
which then served to
similar efforts were made by Bilby and by Kroner in none these theories
in
arguments,
on the body manifold,
certain properties
of
([ I ],[ 2 ]) and consisted
a metric and an affine connection,
however,
of
theory.
The first comprehensive
various
through
equations
distribution;
([3 ], [4 ]), consideration
which may be
associated with the body manifold. A new approach to the problem was made by Noll
[ 5 ] in
the early sixties and was later extended by Noll [ 6 ] and by Wang [ 7 ]. constitutive manifold
Here one starts with the prescription equation for particles
of a
belonging to the body
and, using the concept of a uniform reference,
VI develops
a geometric theory which in many ways
to those considered by Kondo, work differs
Bilby,
is isomorphic
and K
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