Table Of ContentReinhold Kienzler .George Herrmann
MechanicsinMaterial Space
Springer-Verlag
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Reinhold Kienzler · George Herrmann
Mechanics
in Material Space
with Applications to Defect
and Fracture Mechanics
, Springer
Prof. Dr. Ing. Reinhold Kienzler
Universitlit Bremen
Postfach 330440
D-28334 Bremen
Germany
Prof. em. Dr. se. teehn. George Herrmann
Stanford University
Ortstrasse 7
CH-7270 Davos Platz
Switzerland
Ubrary of Congress Cataloging-in-Publication Data applied for
Kienzler, Reinhold
Mechanics in material space: with applications to defect and fracture mechanics / Reinhold Kienzler ;
George Herrmann. -Berlin; Heidelberg ; New York; Barcelona ; Hong Kong ; London ; Milan ; Paris ;
ISBN 978-3-642-63121-4 ISBN 978-3-642-57010-0 (eBook)
DOI 10.1007/978-3-642-57010-0
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Preface
Thetitle ofthe presenttext is somewhatunfamiliarand needs to beelucidated first,
before describing briefly the aims and scope ofthis work.
Almost half-a-century ago, the late great 1. D. Eshelby advanced the notion of a
force on a defect or a singularity in a stressed solid. This notion is radically
different from the usual Galilean or Newtonian conceptofa force, which engineers
of various disciplines encounter daily in their work, concerned perhaps with the
determinationofflight trajectories oflaunched satellites, orwith stress, deformation
and structural stability analysis of high-rise buildings. The Eshelby-type force is
always to be understood as a relative change ofthe total energy ofa given system
with respect to some quantity which alters the configuration of that system. The
latter quantity might be the displacement ofa foreign or missing atom in a lattice,
the change in location of a dislocation, the change in size or shape of a crack,
cavity or inclusion orthe change in location ofa phase boundary in a material. All
such changes ofconfiguration ofcertain objects occurwithin the material in which
they find themselves, by contrast to changes in the configuation ofa bridge under
some Newtonian loadings, which occur in what might be called the physical space
ofoursurroundings, in which thebridge finds itselfandin whichNewton'slaws are
valid.
Thus the term space, whether physical or material, is used here not in a strictly
mathematical sense, as possessingacertainmetricand possiblyotherproperties,but
in essentially a descriptive meaning.
The realization that it is helpful, in fact most desirable, to distinguish between
configurational, Eshelbian forces on one hand, and the common, usual Newtonian
forces onthe other, may lead naturally to the terminology ofmaterial forces on one
hand, and physical forces on the other. And, proceeding along this line ofthought,
itis intriguingto investigatethepossibilityofconstructingthe edificeofMechanics
in Material Space in parallel (or in analogy) with the well-established classical
mechanics, which now, for reasons ofsymmetry and aesthetics, shall bereferred to
as Mechanics in Physical Space.
Indeed, the quantities and problems commonly examined in Mechanics in Physical
Space find, to a large extent, their counterpart in Mechanics in Material Space as
detailed in the Introduction. As an example, the physical Cauchy stress tensor
corresponds to the so-called material momentum tensor, or the Eshelby tensor,
VI Preface
related to configurational forces. But there are aspects for which the parallelism is
incomplete.
For example, the quantities needed in Mechanics in Material Space can be
determined only after some related problem has been solved in Mechanics in
Physical Space. This aspect might lead to the belief that Mechanics in Material
Space is entirely superfluous. We obviously do not share this point ofview,justas
we do not think, even though it is possible, to conceive of classical mechanics
without the introduction of the concept of force, as has been tried in the past. It
might be added that most recently attempts have been made to establish the
framework ofMechanics in Material Space completely independently ofMechanics
in Physical Space, as will be mentioned in the Introduction.
The objective of this book is to present, in as complete a manner as possible, a
Mechanics in Material Space (or Configurational Mechanics) and exhibit its
usefulness inDefect andFracture Mechanics.
Even though thevarious topics discussed are essentially mathematical in nature, the
presentation is kept atalevel which should be accessible to any engineer, physicist,
material scientist or interested student with only some knowledge of calculus and
elements of partial differential equations. The mathematical foundations of the
conservation and balance laws discussed in the text rest on concepts of Lie group
theory, symmetry and invariance, but aknowledge ofthese topics is not required in
orderto derivea benefitfrom ourpresentation. Even familiarity with the calculus of
variations is notaprerequisite. In addition, all topics, with one exception in Chapter
7, are presented in a linearformulation. This restriction makes the text absorbable
by engineers, at the obvious expense of generality, which we sacrifice in favor of
readability.
The contents ofthe book may be briefly summarized as follows:
The Introduction attempts to circumscribe the essential substance of the book and
the spirit in which this substance is to be presented, drawing specific parallels
between the Mechanics in Physical Space and the Mechanics in Material Space. As
already mentioned, conservation and balance laws on both sides ofthis parallelism
form the basis for developments in subsequent chapters, and this strictly
mathematical topic is presented in Chapter I without explicit recourse to group
theory.
The elements of the linear theory of elastictiy are recalled in Chapter 2, and
conservation and balance laws in differential and integral form are established.
Chapter 3 is devoted to the discussion of the properties of the Eshelby tensor,
whereas the notions of energy-release rates, crack-extension forces and stress
intensity factors, which form the essence of linear elastic fracture mechanics, are
introduced in Chapter 4.
Elastostaticsin material spaceisextended to inhomogeneouselastostaticsin Chapter
5 and to elastodynamics in Chapter 6.
Preface VII
The treatment of dissipative systems of various kinds is taken up in Chapter 7,
while Chapter 8 deals with interacting fields, where the elastic field is coupled in
one instance to the electric field (piezoelectricity, dielectrics) and in another
instanceto thethermal field (thermoelasticity). Thelatter, basedonananalogy, may
also be interpreted as a fluid- saturated porous medium.
It is a remarkable circumstance that some elements offracture mechanics, such as
energy-releaserates andstress-intensityfactors, mightbedevelopednoton thebasis
ofcontinuum theories, but on the basis ofthe much older and simpler theories of
strength-of-materials. Barswith cracksintension-compression, shafts intorsionand
beams and cylinders (pipes) inbending aredealt withinChapter9, whileplates and
shells are considered in the final Chapter 10.
The principal parts ofthis book were written during the authors' several stays of
several weeks each at the Mathematisches Forschungsinstitut Oberwolfach,
Germany, under aprogram called "Research in Pairs" invited bythe Directorofthe
Institute, Professor Dr. Matthias Kreck. The authors are greatly indebted to Prof.
Kreck and to the Institute for having provided such generous hospitality.
The authors are also grateful for the partial support provided both by the U. S. Air
Force Office ofScientific Research through a grant to Stanford University and by
the University ofBremen.
The book could have never taken the form it has without the benefit ofdiscussions
andcollaborationwithnumerousformergraduatestudentsfrom StanfordUniversity,
in particular Dr. N. Chien, Dr. 1. Eischen, Dr. Y. Pak and Dr. H. Sosa. Especially
Dr. T. Honein contributed many novel ideas which have found their entry into the
book.
Greatly appreciated were also the fruitful comments offeredbymany colleagues, in
particularProf. D. Barnett(Stanford University),Prof. H. Gao(Stanford University),
Prof. D. Gross (Technical University ofDarmstadt) and Prof. J. Willis (University
ofCambridge).
The preparation of the manuscript was skillfully and faithfully carried out by
B. Neumeister-Schulze and the responsibility for drawing the Figures rested with
M. Grabendorffand S. Stiihrmann. Theiraccurate work is gratefully acknowledged.
Finally, we would like to express our thanks to the publisher, and in particular to
Dr. Merkle, for excellent cooperation.
Contents
Preface v
Introduction 1
1 Mathematical Preliminaries 13
1.1 General Remarks 13
1.2 What is a Conservation Law? 14
1.3 Trivial Conservation Laws 17
1.4 System with a Lagrangian; Noether's Method 19
1.5 System without a Lagrangian; Neutral-Action Method 42
1.6 Discussion 48
2 Linear Theory ofElasticity 51
2.1 General Remarks 51
2.2 Elements ofLinear Elasticity 51
2.3 Conservation Laws ofLinear Elastostatics 62
2.4 Alternative Derivations of Conservation Laws 72
3 Properties ofthe Eshelby Tensor 81
3.1 General Remarks 81
3.2 Physical Interpretation ofthe Components ofthe Eshelby Tensor 82
3.3 Invariants, Principal Values, Principal Directions and
Extremal Values of the Eshelby Tensor 86
4 Linear Elasticity with Defects 95
4.1 General Remarks 95
4.2 Path-Independent Integrals and Energy-Release Rates 96
4.3 Example: Hole-Dislocation Interaction 100
4.4 Path-Independent Integrals ofFracture Mechanics 107
x
Contents
5 Inhomogeneous Elastostatics 121
5.1 General Remarks 121
5.2 Symmetry Transfonnations 122
5.3 The Homogeneous Case 124
5.4 The Inhomogeneous Case 124
5.5 Relation to Stress-Intensity Factors 127
5.6 Examples 128
6 Elastodynamics 133
6.1 General Remarks 133
6.2 Time t as an Additional Independent Variable 134
6.3 Convolution in Time 143
6.4 Domain-Independent Integrals 146
6.5 Energy-Release Rates 151
6.6 Wave Motion 156
7 Dissipative Systems 161
7.1 General Remarks 161
7.2 Diffusion Equation 161
7.3 Non-Linear Wave Equation 163
7.4 Viscoelasticity 165
8 Coupled Fields 173
8.1 General Remarks 173
8.2 Piezoelectricity 174
8.3 Thennoelasticity 179
8.4 Mechanics ofa Porous Medium 192
9 Bars, Shafts and Beams 195
9.1 General Remarks 195
9.2 Elements ofStrength-of-Materials 196
9.3 Balance and Conservation Laws for Bars and Shafts 200
9.4 Balance and Conservation Laws for Beams 203
9.5 Energy-Release Rates and Stress-Intensity Factors 205
9.6 Examples 211
Contents XI
10 Plates and Shells 219
IO.I General Remarks 2I9
10.2 Plate Theories 220
10.3 Conservation Laws for Elastostatics ofMindlin Plates 224
10.4 Reduction to the Classical Theory 229
10.5 Conservation Laws for Shells 233
Appendix A 239
Conservation Laws for Inhomogeneous Bars under
Arbitrary Axial Loading 239
Appendix B 245
B.I Elastodynamics ofInhomogeneous Bernoulli-Euler Beams 245
B.2 Reduction to Statics 252
Appendix C 259
C.I Elastodynamics ofInhomogeneous Mindlin Plates 259
C.2 Reduction to Statics 269
References 273
Symbol Index 281
Author Index 287
Subject Index 291
