Table Of ContentElasticity and Plasticity of Large Deformations
Albrecht Bertram
Elasticity and Plasticity of Large
Deformations
An Introduction
2ndEdition
123
Prof. Dr.AlbrechtBertram
Otto-von-Guericke-Univ.
Inst.Mechanik
Universita¨tsplatz2
39106Magdeburg
Germany
[email protected]
ISBN:978-3-540-69399-4 e-ISBN:978-3-540-69400-7
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(cid:2)c 2008Springer-VerlagBerlinHeidelberg
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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.
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
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
6.2 Thermo-Elasticity................................................................................179
6.3 Change of the Reference Placement....................................................180
X CONT ENTS
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......................................................................................................229
8.1 Boundary Value Problems...................................................................229
8.2 Universal Solutions.............................................................................236
9 Inelasticity...................................................................................................249
10 Plasticity......................................................................................................253
10.1 Elastic Ranges.....................................................................................254
10.2 Thermoplasticity..................................................................................284
10.3 Viscoplasticity.....................................................................................291
10.4 Plasticity Theories with Intermediate Placements...............................293
10.5 Crystal Plasticity..................................................................................303
References...........................................................................................................317
Index....................................................................................................................335
List of Frequently Used Symbols
Sets
B body (-manifold)
B domain of the body in the reference placement
0
B domain of the body in the current placement
t
E EUCLIDean point space
Ela elastic range
p
Inv set of invertible tensors (general linear group)
Lin space of linear mappings from V to V (2nd-order tensors)
Lin space of hardening variables in Chap. 10
Orth set of orthogonal 2nd-order tensors (general orthogonal group)
Psym set of symmetric and positive-definite 2nd-order tensors
R space of real numbers
Sym space of symmetric 2nd-order tensors
Skw space of antisymmetric or skew 2nd-order tensors
Unim set of 2nd-order tensors with determinant ±1
(general unimodular group)
V space of vectors (EUCLIDean shifters)
A superimposed + such as Inv+ means: with positive determinant. R + denotes
the positive reals.
Variables and Abbreviations
a = r•• ∈ V acceleration
A ∈ Unim+ symmetry transformation
B = F FT ∈ Psym left CAUCHY-GREEN tensor
b ∈ V specific body force
c ∈ V translational vector in the EUCLIDean transform.
c ∈ V lattice vector in Chap. 10.5
i
C = FT F ∈ Psym right CAUCHY-GREEN tensor
C = PT C P ∈ Psym transformed right CAUCHY-GREEN tensor in
e
Chap. 10
COOS coordinate system
XII LIST OF FREQUENT LY USED SYMBOLS
dO = ∫ rO × v dm ∈ V moment of momentum with respect to O
B
t
dα directional vector of a slip system in Chap. 10.5
D = sym(L) ∈ Sym stretching tensor, rate of deformation tensor
da element of area in current placement
da element of area in reference placement
0
da vectorial element of area in current placement
da vectorial element of area in reference placement
0
Div, div material and spatial divergence operator
dm element of mass
dv element of volume in current placement
dv element of volume in reference placement
0
dx vectorial line element in current placement
dx vectorial line element in reference placement
0
e ∈ V basis vector of ONB
i
E = sym(H) ∈ Sym linear strain tensor
Ea = ½ (I – B –1) ∈ Sym ALMANSI´s strain tensor
Eb = I – V–1 ∈ Sym spatial BIOT´s strain tensor
EB = U – I ∈ Sym material BIOT´s strain tensor
EG = ½ (C – I) ∈ Sym GREEN´s strain tensor
Egen ∈ Sym spatial generalised strain tensor
EGen ∈ Sym material generalised strain tensor
Eh = ln V ∈ Sym spatial HENCKY´s strain tensor
EH = ln U ∈ Sym material HENCKY´s strain tensor
E ∈ R internal energy
f ∈ V resulting force acting on the material body
F = Grad χ ∈ Inv+ deformation gradient
g ,g i ∈ V basis vectors of dual bases
i
g = grad θ ∈ V spatial temperature gradient
g = Grad θ ∈ V material temperature gradient
0
Grad, grad material and spatial gradient operator
G , G ik ∈ R metric coefficients
ik
H = Grad u ∈ Lin displacement gradient
I ∈ Psym unit tensor of 2nd-order
I unit tensor of 4th-order
