Elasticity and Plasticity of Large Deformations: An Introduction

358 Pages·2012·5.491 MB·English

by  Albrecht Bertram (auth.)

Elasticity and Plasticity of Large Deformations

Albrecht Bertram
Elasticity and Plasticity
of Large Deformations
An Introduction
Third Edition
123
Prof.Dr.-Ing.AlbrechtBertram
Otto-von-GuerickeUniversitätMagdeburg
Inst.f.Mechanik
Universitätsplatz2
39106MagdeburgSachsen-Anhalt
Germany
[email protected]
ISBN978-3-642-24614-2 e-ISBN978-3-642-24615-9
DOI10.1007/978-3-642-24615-9
SpringerHeidelbergDordrechtLondonNewYork
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PREFACE V

Preface to the First Edition

This book is based on the lecture notes of courses given by the author over the
last decade at the Otto-von-Guericke University of Magdeburg and the Technical
University of Berlin. Since the author is concerned with researching material the-
ory and, in particular, elasto-plasticity, these courses were intended to bring the
students close to the frontiers of today's knowledge in this particular field, an
opportunity now offered also to the reader.
The reader should be familiar with vectors and matrices, and with the basics of
calculus and analysis. Concerning mechanics, the book starts right from the be-
ginning without assuming much knowledge of the subject. Hence, the text should
be generally comprehensible to all engineers, physicists, mathematicians, and
others.
At the beginning of each new section, a brief Comment on the Literature con-
tains recommendations for further reading. Throughout the text we quote only the
important contributions to the subject matter. We are far from being complete or
exhaustive in our references, and we apologise to any colleagues not mentioned in
spite of their important contributions to the particular items.
It is intended to indicate any corrections to this text on our website
http://www.uni-magdeburg.de/ifme/l-festigkeit/elastoplasti.html
along with remarks from the readers, who are encouraged to send their frank criti-
cisms, comments and suggestions to
[email protected].
All the author’s royalties from this issue will be donated to charitable organisa-
tions like Terres des Hommes.

Acknowledgment. The author would like to thank his teachers RUDOLF TROSTEL,
ARNOLD KRAWIETZ, and PETER HAUPT who taught him Continuum Mechanics in the
early seventies, and since then have continued to give much helpful advice.
Many colleagues and friends also made useful comments and suggestions to improve this
book, including ENRICO BROSCHE, CARINA BRÜGGEMANN, SAMUEL FOREST,
SVEN KASSBOHM, THOMAS KLETSCHKOWSKI, JOHN KINGSTON, WOLFGANG
LENZ, GERRIT RISY, MANUELA SCHILDT, MICHAEL SCHURIG, GABRIELE
SCHUSTER, BOB SVENDSEN and, most of all, THOMAS BÖHLKE and ARNOLD
KRAWIETZ, who gave countless valuable comments. The author is grateful to all of them.

PREF ACE VII

Preface to the Second Edition

The new edition of such a textbook provides the chance to eliminate any errors
or misprints identified in the first edition, to correct any misleading expressions,
and to add new references. This has all been done here.
The valuable input of students and colleagues alike has enabled me to make
many small improvements to the first edition, without, however, changing the
main structure of the book.
I would also like to express my gratitude to everyone who helped me with this
task, in particular to Thorsten Hoffmann, Alexander Lacher, Jan Kalisch, Sven
Kaßbohm, and Uwe Herbrich, to mention just a few.
Magdeburg, May 2008

Preface to the Third Edition

Nonlinear Continuum Mechanics is a rapidly growing field of research. Since
the last edition of this book, many important results in this field have been pub-
lished. In this new edition we have done our best to refer to the most important
such results.
We have enlarged the part on hyperelastic models and anisotropic yield criteria,
included an outlook on Material Plasticity, and tried to improve the text wherever
possible.
I would like once again to express my gratitude to everyone who helped me with
this task, in particular to Rainer Glüge and Jan Kalisch. And I would also sincerely
like to thank Jim Casey for the many stimulating discussions that I enjoyed during
my stay at the UC Berkeley in spring 2010.
Magdeburg, June 2011

CONT ENTS IX

Contents

List of Frequently Used Symbols.........................................................................XI
List of Selected Scientists.................................................................................XVII
Introduction.............................................................................................................1
1 Mathematical Preparation................................................................................3
1.1 Repetitions from Vector Algebra.............................................................5
1.2 Tensor Algebra......................................................................................10
1.2.1 Tensor Product and Tensor Components.......................................17
1.2.2 The Eigenvalue Problem................................................................21
1.2.3 Special Tensors..............................................................................29
1.2.4 Tensors of Higher Order................................................................35
1.2.5 Isotropic Tensor-Functions............................................................46
1.3 Tensor Analysis.....................................................................................53
1.4 The EUCLIDean Point Space................................................................71
1.4.1 The Covariant Derivative..............................................................76
1.4.2 Integral Theorems..........................................................................87
2 Kinematics.....................................................................................................91
2.1 Placements of Bodies.............................................................................91
2.2 Time Derivatives...................................................................................94
2.3 Spatial Derivatives.................................................................................95
3 Balance Laws..............................................................................................121
3.1 Mass.....................................................................................................121
3.2 The General Balance Equation............................................................122
3.3 Observer-Dependent Laws of Motion.................................................130
3.4 Stress Analysis.....................................................................................136
3.5 The Thermodynamical Balances.........................................................149
4 The Principles of Material Theory...............................................................153
4.1 Determinism........................................................................................153
4.2 Local Action........................................................................................154
4.3 EUCLIDean Invariances......................................................................155
4.4 Extension of the Principles to Thermodynamics.................................159
5 Internal Constraints.....................................................................................167
5.1 Mechanical Internal Constraints..........................................................167
5.2 Thermo-Mechanical Internal Constraints............................................172
6 Elasticity......................................................................................................177
6.1 Reduced Elastic Forms........................................................................178
X CONT ENTS

6.2 Thermo-Elasticity................................................................................179
6.3 Change of the Reference Placement....................................................180
6.4 Elastic Isomorphy................................................................................182
6.5 Elastic Symmetry.................................................................................185
6.6 Isotropic Elasticity...............................................................................195
6.7 Incremental Elastic Laws.....................................................................200
6.8 Symmetries in Thermo-Elasticity........................................................205
7 Hyperelasticity.............................................................................................209
7.1 Thermodynamical Restrictions............................................................209
7.2 Hyperelastic Materials.........................................................................211
7.3 Hyperelastic Isomorphy and Symmetry..............................................216
7.4 Isotropic Hyperelasticity......................................................................218
8 Solutions......................................................................................................231
8.1 Boundary Value Problems...................................................................231
8.2 Universal Solutions.............................................................................238
9 Inelasticity...................................................................................................251
10 Plasticity..................................................................................................255
10.1 Elastic Ranges.....................................................................................256
10.2 Thermoplasticity..................................................................................286
10.3 Viscoplasticity.....................................................................................293
10.4 Plasticity Theories with Intermediate Placements...............................296
10.5 Crystal Plasticity..................................................................................306
10.6 Material Plasticity................................................................................319
References...........................................................................................................321
Index....................................................................................................................341