The Nonlocal Continuum Theory of Lattice Defects
Crystal imperfections, like those of human beings, come in seemingly endless variety. However, not all are equally interesting or easy to deal with. In these lectures we restrict ourselves to the most important kinds such as point defects and dislocations
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NONLOCAL THEORY OF
MATERIAL MEDIA EDITED BY D. ROGULA POLISH ACADEMY OF SCIENCES
SPRINGER-VERLAG WIEN GMBH
This volume contains 27 figures.
This work is subject to copyright. AU 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.
© 1982 by Springer-Verlag Wien Originally published by Springer-Verlag Wien New York in 1982
ISBN 978-3-211-81632-5
DOI 10.1007/978-3-7091-2890-9
ISBN 978-3-7091-2890-9 (eBook)
PREFACE
Although tbe idea of interaction distance appears as old as human knowledge, tbe Nonlocal Tbeory of Material Media is a relatively recent discipline. Tbe fact tbat tbis tbeory bas not been developed until recently can perhaps be explained by the great success both of tbe classical continuum theory witb contact interactions and, on the other band, of the statistical molecular mecbanics. However, the growing need for understanding tbe phenomena intermediate between these two extremities promoted the development of tbe theory of material media, based on nonlocal concepts. This tbeory bridges tbe continuum mecbanics and the molecular pbysics by a general representation of interaction force and kinematical properties of condensed matter. The classical models of solids, i.e. the c/assical continuum witb its contact forces and a discrete lattice with interaction of molecules distant from each other, may be regarded as the extreme instances of nonlocal material media. In spite of the fact tbat at present tbe tbeory is far from being complete, a number of ideas bave developed wbicb can be of broad interest, even for the non-specialist. Tbe present t•olume contains essentially tbe lectures on Nonlocal Tbeory of Material Media given at tbe Centre International des Seiences Mecaniques. The aim of the volume is to sketch tbe pbysical and mathematical foundations of the nonlocal tbeory of material media, its generat results, applications, connexions with related domains of mechanics, and many questions open for future research. Special attention is paid to the problems of structural defects and boundaries of solids. lt is a pleasure to acknowledge the Centre International des Seiences Mecaniques for its helpfulness. Our gratitude is also due to tbe autbors who contributed to this volume.
Dominik Rogula Warsaw, june 1982
CONTENTS
Page
Defects in Crystalline Media by A.G. Cracker Perfeet and lmperfect Crystals
Geometrical Theories of Defects Continuum Theory of Defects . Discrete Theories of Defects . . Relationships between Theories of Defects References . . . . . . . . . . . . . . . . .. The Nonlocal Continuum Theory of Lattice Defects by B.K.D. Gairola I. Introduction . . . . . . . . . . . . II. Basic Theory . . . . . . . . . . . 1. Nonlocal Theory of Elasticity
1 9 21
26 • 43'
.44
52 54 54
2. The Relationship between the Force Constants and the Elastic Constants 66 3. Equilibrium Equation and its Solution in
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