Table Of Content& SpringerWienNewYork
CISM COURSES AND LECTURES
Series Editors:
The Rectors
Manuel Garcia Velarde - Madrid
Jean Salengon - Palaiseau
Wilhelm Schneider - Wien
The Secretary General
Bemhard Schrefler - Padua
Executive Editor
Carlo Tasso - Udine
The series presents lecture notes, monographs, edited works and
proceedings in the field of Mechanics, Engineering, Computer Science
and Applied Mathematics.
Purpose of the series is to make known in the international scientific
and technical community results obtained in some of the activities
organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES - No. 453
MOVING INTERFACES IN
CRYSTALLINE SOLIDS
EDITED BY
FRANZ DIETER FISCHER
MONTANUNIVERSITAT LEOBEN, AUSTRIA
AND
ERICH SCHMID INSTITUTE FOR MATERIALS SCIENCE,
AUSTRIAN ACADEMY OF SCIENCES, LEOBEN, AUSTRIA
SpringerWien NewYork
The publication of this volume was co-sponsored and co-financed by the UNESCO Venice
Office - Regional Bureau for Science in Europe (ROSTE) and its content corresponds to a
CISM Advanced Course supported by the same UNESCO Regional Bureau.
This volume contains 126 illustrations
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.
© 2004 by CISM, Udine
Printed in Italy
SPIN 11355274
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 3-211-23899-9 SpringerWienNewYork
PREFACE
Materials with a changing microstructure are a matter of fact in many fields of
materials research as well as in the application of materials in diverse fields of
technical practice. Such a change may consist in one or more phase transformations in
the material, the growth or shrinking of grains, or the appearance of the material in a
different geometrical configuration like twinning. Both physical and thermomechanical
activities are necessary to drive or stop or, better, to control such a process. The goal
of the course "Moving Discontinuities in Crystalline Solids", on which this booklet is
based, was to bring together experts from materials physics and materials mechanics
to explain the fundamental phenomena of moving interfaces accompanied by the
change of the material on both sides of the interface. The application of both classical
and modern concepts of materials physics was demonstrated. Furthermore, the
findings of continuum mechanics and numerical methods were discussed ranging from
Eshelby's derivation of the thermodynamical force on an interface to numerical
concepts under development for multiparticle, multicomponent systems. The authors of
the various chapters, however, have tried hard not to present simply diverse concepts
but to bridge the gaps between the numerous physical and mechanical approaches to
this wide field of knowledge. Such a task is not easy and rather new. For many years
researchers from mechanics and physics have tended to take different roads which met
only by chance. Within the last fifteen years both groups felt that they had to come
nearer, and finally, the authors of this booklet have the impression that they have come
together. The inherent interdisciplinarity of the subject has been one of the strongest
motivations to perform such a course and to write the booklet at hand. Computational
methods, like the Monte Carlo Method, Ab-initio Modelling and "Enriched" Finite
Elements have contributed a lot to a better understanding of what is behind the change
of the microstructure. One may recognize this in the ever increasing number of
courses, seminars, conferences and corresponding papers on "Modelling and
Simulation". The authors also feel that the common view on the controlling
mechanisms for the microstructure will have an increasing impact on industrial
application. More effective materials that are better adapted to their respective
functions and require shorter development times are more or less a must in the world
of today's technology.
Some comments are also necessary with respect to the layout of this booklet. Since a
group of researchers got together from different fields, very often their symbols and
notations differ. A full unification would take too much time although it is a demanding
task for the future. Therefore, the authors decided to write each chapter in a self-
contained and self-explaining way with a list of notations at the beginning of the
chapter. So each chapter has a textbook character starting from the basics and stating
carefully the assumptions and limitations of the application of the theoretical
framework. The authors try to present examples that are easy to understand. The
authors have also performed a mutual reading of their chapters with the goal to bring
their own contribution in line with the related chapters.
The authors hope that the booklet will provide a sufficient basis for the understanding
of this interdisciplinary field of materials mechanics and materials physics and will
further mutual understanding. Both researchers and industrial developers should
profit from this rather unique presentation of the motion of interfaces in solids.
The authors express their thanks to the Director of CISM, Prof M. G. Velarde, for
supporting the course with the staff of CISM and for the strong encouragement they
received to write this booklet. Finally, the authors are grateful to Prof. C. Tasso for
accepting this booklet for publication and his help in the editing.
Franz Dieter Fischer
CONTENTS
Preface
Application of Configurational Mechanics to Elastic Solids with Defects and
Cracks
by R. Kienzler and G. Herrmann 1
Phase Separation in Binary Alloys-Modelling Approaches
by P. Fratzl andR. Weinkamer 57
Utilization of the Thermodynamic Extremal Principle for Modelling in Material
Science
by J. Svoboda 117
Thermodynamics and Kinetics of Phase and Twin Boundaries
by F. D. Fischer and N. K. Simha 169
Moving Grain Boundaries During Hot Deformation of Metals: Dynamic
Recrystallization
by F. Montheillet 203
Application of Configurational Mechanics
to Elastic Solids with Defects and Cracks
R. Kienzler^ and G. Herrmann^
^University of Bremen, Bremen, Germany
^Stanford University, Stanford, CA, USA
Abstract. Classical mechanics, i. e., in physical space is concerned with forces, stresses and
strains and attempts to describe the motion and/or deformation of bodies with mass. In this
context, the notions of tractions, trajectories, balance and conservation laws, stability of
equilibrium etc. are well established. Mechanics in material space (or configurational
mechanics) describes the behaviour of defects (e. g., voids, dislocations, cracks) as they move
relatively to the material, in which they find themselves. Concerning this change of
configuration, similar notions, as given above, are introduced in material space. After providing
the elements of configurational mechanics the method is applied to elastic solids with defects
and cracks. In particular, the hole-dislocation interaction problem is discussed and the use of
path-independent integrals and local failure criteria in fracture mechanics are demonstrated.
List of Notations
a abbreviation
a crack length
Gi material properties (not further specified)
A area, mostly used as differential area element dA
A cross-sectional area mostly in connection with compressional stiffness EA
A action integral
b, bi, b Burgers vector, its components and its magnitude
b^j components of Eshelby tensor, material momentum
B material force in one-dimensional problem
B body, domain of integration
Cijj^i components of tensor of elasticity
C compliance
d thickness of bar
E Young's modulus
E modified Young's modulus
EA compressional or axial stiffness
E Euler operator
R. Kienzler and G. Herrmann
fi characteristics (Neutral Action method)
/, / vector of body forces and its components
fij ? // 5 //^ dimensionless geometry functions correlated with crack-opening modes /, //
and///
F,Fj,F (physical) force vector, its components and its magnitude
g gravitational acceleration
g() function
G shear modulus
G, Gj material traction vector acting across a surface with normal vector n and its
n n
components
^ energy release rate
^ Energy release rate of strength-of-material theories
h height of bar
^ /i first invariant of stress tensor
J material force (J integral)
J, JI vector of material forces and its components (components of J integral)
Kj, Kjj, Kjjj stress-intensity factors correlated with crack-opening modes, /, // and ///
i length of bar
/ Lagrangian
L null Lagrangian
L,L^,L vector of material momentums (L integral) its components and 1 = 1^ in
plane problems
m mass
M material virial (M integral)
n, rij unit outward normal vector and its components
A^ axial force in one-dimensional problem
0(S") terms of (vanishing) order n and higher
P,Pi,P i -component vector (current), its components and current in one-dimensional
problem
pr^"^"* n-th prolongation
Q^ characteristics (Noether's formalism)
r radial coordinate in polar coordinate system
r^ radius of hole
s arc length, mostly used as differential arc-length element ds
s abbreviation
S surface, domain of integration
Sf surface of body 5 with prescribed tractions
S^ surface of body 5 with prescribed displacements
