Table Of ContentCISM COURSES AND LECTURES
Series Editors:
The Rectors
Sandor Kaliszky - Budapest
Mahir Sayir - Zurich
Wilhelm Schneider -Wien
The Secretary General
Bernhard Schrefler - Padua
Former Secretary Gen'eral
Giovanni Bianchi - Milan
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. 427
CONFIGURATIONAL MECHANICS
OF MATERIALS
EDITED BY
REINHOLD KIENZLER
UNIVERSITY OF BREMEN
GERARD A. MAUGIN
PIERRE ET MARIE CURIE UNIVERSITY
~ Springer-Verlag Wien GmbH
This volume contains 123 illustrations
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© 2001 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 2001
SPIN 10839118
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 978-3-211-83338-4 ISBN 978-3-7091-2576-2 (eBook)
DOI 10.1007/978-3-7091-2576-2
PREFACE
A rational analysis and design of contemporary materials and mechanical systems
requires an ever finer scale of mathematical modeling and experimental
observation and measurement than has been the case in the past. It has to be
recognized that it is, in many circumstances, no longer sufficient to consider the
material at hand to be a perfect continuum on some scale of examination. One
has, rather, to descend deeper into the conditions of matter in order to make
reliable and realistic predictions regarding the possible. behaviour of the material
or structural element under a variety of environmental constitution encountered
during the lifetime of the piece. In particular, the so-called phenomenological
theories of material response, with their simplistically postulated criteria of strength
and failure, have to be replaced today by consideration of a variety of defects and
imperfections always existing, such as micro-cracks, inclusions, vacancies, voids,
dislocations etc.
One feature of these defects is that, under the influence of external sources such as,
e. g., loadings and temperature changes, they can move (by mass diffusion) or
change their shape within the material in which they find themselves. In the case of
void nucleation and growth or micro-crack advance, new surfaces are created, i.
e., the configurational of the material changes, giving rise to the encompassing
term "Configurational Mechanics". An alternative designation of this quite novel
branch of engineering science could be "Mechanics in Material Space",
emphasizing that "objects" such as voids or interfaces between some two phases of
a material may move within the material, rather then with respect to "Physical
Space", (as is the case in usual (Newtonian) mechanics), in which deformation
and motion of bodies with mass is considered. Here file term space is to be
understood in a purely descriptive rather than in some strict mathematical sense.
Still a third designation which is used for the same body of knowledge and inquiry
is the term "Eshelbian Mechanics", which honours the late J. D. Eshelby who was
most successful in originating (in a seminal paper of 1951) and developing the
concept of a force on a material defect as the change in .total energy of a given
mechanical system with respect to a possible displacement of such a defect within
the material.
In view of an ever broadening and widening of this "Configurational Mechanics of
Materials" it was deemed appropriate and even desirable to organize a one-week
course held at the International Centre for Mechanical Sciences (CISM) in Udine,
Italy, September 11 - 15, 2000, in order to present to an interested audience a
summary of some latest developments and advances in the field.
The Lecture Notes before you embody a distillate of presentations and discussions
held at CISM during the week indicated. It is obvious that the 6 lecturers could not
cover in a comprehensive fashion all the analytical, experimental and numerical
aspects of the field, yet it is believed that most of the important recent advances
were included.
In a first introductory lecture on "Conservation laws and their application in
configurational mechanics", delivered by the first author, G. Herrmann and R.
Kienzler attempted to provide, in an as simple fashion as possible, the
mathematical framework required to carry out a theoretical investigation of defects
moving in materials.
A more general and more encompassing approach was taken in the lecture on
"Elements of field theory in inhomogeneous and defective materials" by G. A.
Maugin and C. Trimarco, delivered by the first author, in which numerous topics
were discussed such as driving forces on field singularities, with application to the
propagation of shock waves, phase-transition fronts and localized nonlinear
waves, as well as computational schemes.
"Material mechanics of electromagnetic solids" by C. Trimarco and G. A. Maugin
focussed on the topic of the title, and laid the foundation for an analytical
treatment of defective electromagnetic materials, and was presented by the first
named author.
It is a surprising fact that the determination of some energy-release rates and
stressintensity factors can be carried out not only within the framework of
mechanics of extended continua, but even within theories of strength-of-materials
concerned with bars, shafts and beams. Such considerations were presented in a
paper "Configurational mechanics applied to strength-of-materials" by R. Kienzler
and G. Herrmann, and given by the first-named author.
As was shown in a paper by D. Gross entitled "Morphological equilibrium and
kinetics of 2-phase materials", configurational forces play a decisive role also in
determining the final equilibrium shape of precipitates in a variety of two-phase
materials. Analytical findings were corroborated by experimental evidence.
In a final paper by A. Chudnovsky and S. Preston on "Variational formulation of
a material ageing model", delivered by the first author, a c/assifica'tion of the
many fracture mechanisms was undertaken, followed by fatigue crack-growth
equations and life-time prediction and supported by a body of experimental
results.
For a smooth and pleasant organization of the Course as an advanced school, the
coordinators should express their gratitude, also in the name of all participants, to
the Secretary General of CISM, Professor G. Bianchi and his efficient staff, as well
as to Professor C. Tasso, editor of the CISM series of publications, for his help in
setting the standards of production of this volume.
Reinhold Kienzler
Gerard A. Maugin
Participants in the CISM Advanced School "Configurational Mechanics of Materials"
CONTENTS
Page
Preface
Photo of the participants to the school
Conservation Laws and their Application iil Configurational Mechanics
by G. Herrmann and R. Kienzler. .............................................. :. ............................................................ l
Elements of Field Theory in Inhomogeneous and Defective Materials
by GA. Maugin and C. Trimarco ........................................................................................................ 55
Material Mechanics of Electromagnetic Solids
by C. Trimarco and GA. Maugin ..................................................................................................... l29
Configurational Mechanics Applied to Strength -of -Materials
by R. Kienzler and G. Herrmann ....................................................................................................... l73
Morphological Equilibrium and Kinetics of Two-Phase Materials
by D. Gross ........................................................................................................................................................ 22l
Variational Formulation of a Material Ageing Model
by A. Chudnovsky and S. Preston ................................................................................................... 273
Conservation Laws and Their
Application in Configurational Mechanics
G. Herrmann', and R. Kienzler
1Stanford University, Stanford, CA, USA
2University of Bremen, Bremen, Germany
Abstract: Conservation laws play a leading role in establishing the necessary
mathematical apparatus for the analysis of various problems in configurational mechanics.
Two essentially different, yet related methodologies are offered for the establishment of
such laws and those methologies are then applied to several simple one-dimensional
problems, as well as to plane elastostatics and elastodynamics.
Introduction
Conservation (or balance) laws constitute the mathematical foundation for most basic
principles in various branches of physics and mechanics. The one important exception is
perhaps the 2"d law of thermodynamics which is expressed as an inequality. Thus it is essential
to be concerned with such laws also in the special area of Configurational Mechanics (CM). As
discussed more fully in this text, CM is concerned with that class of mechanical continua,
which undergo processes in which not only the deformation of a body takes place, but also its
configuration changes. As prime examples of such processes one might mention crack
advance, void nucleation and growth, change in geometry of a free boundary, as well as motion
of an interface surface between two phases of a material.
In the present contribution, an attempt has been made to discuss not only the possible
mathematical methodologies for establishment of conservation laws, but also to include some
applications, particularly to one-dimensional elastic bars and to plane elastostatics and
elastodynamics. Applications to coupled fields, such as piezoelectricity and thermoelasticity
should be discussed in other parts of this text and are also considered in the monograph by
Kienzler and Herrmann (2000).
In order to begin on a firm foundation, the first Section here is concerned with the
definition of conservation (or balance) laws, both in differential (strong) and integral (weak)
form.
2 G. Herrmann and R. Kienzler
The possible methodologies of establishing conservation laws are presented in the
following two sections. If a Lagrangian function can somehow be postulated for a system
under consideration, then Noether's method, together with the important extension by Bessel
Hagen, represents a suitable and powerful tool for the establishment of conservation laws.
The method is based on Noether's theorem on invariant variational problems. This
theorem associates every conservation law of a system with an underlying symmetry property
that results from an invariance of (Hamilton's) action integral under a continuous group (in
Lie's sense) of transformations. Bessel-Hagen showed that the strict invariance of the action
integral might be weakend by inclusion of the divergence of a function defined on the space of
independent and dependent variables and their derivatives, (jet bundle space). For a detailed
treatment of this material reference should be made to the books by Olver ( 1993), as well as
Bluman and Kumei (1989).
If a Langrangian function for the system under consideration is not available (e.g.,
because dissipation is present) and the system is described merely by a set of differential
equations, then a recently developed systematic procedure, called the Neutral Action method,
has become available, (cf. T. Honein et al., 1991, Chien, 1992, Chien et al., 1993 a). This
method amounts essentially to replacing the so-called characteristic of a conservation law,
completely determined in Noether's method in terms of the transformation selected, by a
function to be cietermined in such a way that its (inner) product with the equations of motion
represents formally a null Lagrangian, and whose action therefore, as is known, does not
change variationally, i. e., the action behaves neutrally under its variation. Thus the term
"Neutral Action" has been adopted for this methodology.
It can be shown that if the Neutral Action method is applied to the Euler-Lagrange
equation of a variational problem, i. e., a Lagrangian function exists, then the same
conservation laws ensue as if the Noether method had been applied, but together with the
Bessel-Hagen extension.
Applications of these two different methodologies to establish conservation laws, partly
already given as illustrative examples in the respective Sections 2 and 3, are given in the
following 2 sections. Section 4 is concerned with conservation laws for homogeneous and
inhomogeneous plane elastostatics. Here the by now classical J, L and M integrals are
rederived for a homogeneous linearly elastic body and then Rice's J integral is extended to a
class of inhomogeneous elastic materials. Relations to stress-intensity factors of fracture
mechanics are also investigated for this class and several specific examples are worked out.
Section 5 is devoted to elastodynamics. Here only one possible methodology of inclusion of
the additional independent variable time is investigated, and illustrated with a simple example
of an elastic bar. The section concludes with a brief discussion of the complete energy
momentum tensor appropriate for linear elastodynamics. The final Section 6 concludes with
some general remarks.
