Elasticity and Plasticity of Large Deformations: An Introduction

339 Pages·2005·3.799 MB·English

by  Professor Dr. Albrecht Bertram (auth.)

Albrecht Bertram
Elasticity and Plasticity of Large Deformations
Albrecht Bertram
Elasticity and Plasticity
of Large Deformations
An Introduction
Springer
Professor Dr. Albrecht Bertram
Otto-von-Guericke-Universitat
Institut fiir Mechanik
Universitatsplatz 2
Gebaude N
39106 Magdeburg
Germany
[email protected] agdehurg.de
Library of Congress Control Number: 2005921096
ISBN 3-540-24033-0 Springer Berlin Heidelberg New York
ISBN 978-3-540-24033-4 Springer Berlin Heidelberg New York
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V
Preface
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 BRUGGEMANN, 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 BOHLKE and ARNOLD
KRAWIETZ, who gave countless valuable comments. The author is grateful to all of them.
VI
Contents
List of Frequently Used Symbols VIII
List of Selected Scientists XIII
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
VII
4.4 Extension of the Principles to Thermodynamics 159
5 Internal Constraints 166
5.1 Mechanical Internal Constraints 166
5.2 Thermo-Mechanical Internal Constraints 171
6 Elasticity 175
6.1 Reduced Elastic Forms 176
6.2 Thermo-Elasticity 177
6.3 Change of the Reference Placement 178
6.4 Elastic Isomorphy 180
6.5 Elastic Symmetry 183
6.6 Isotropic Elasticity 193
6.7 Incremental Elastic Laws 198
6.8 Symmetries in Thermo-Elasticity 203
7 Hyperelasticity 206
7.1 Thermodynamical Restrictions 206
7.2 Hyperelastic Materials 208
7.3 Hyperelastic Isomorphy and S3mimetry 213
7.4 Isotropic Hyperelasticity 215
8 Solutions 225
8.1 Boundary Value Problems 225
8.2 Universal Solutions 232
9 Inelasticity 244
10 Plasticity 248
10.1 Elastic Ranges 249
10.2 Thermoplasticity 275
10.3 Viscoplasticity 282
10.4 Plasticity Theories with Intermediate Placements 284
10.5 Crystal Plasticity 294
References 307
Index 322
VIII
List of Frequently Used Symbols
Sets
d§ body (-manifold)
^0 domain of the body in the reference placement
^i domain of the body in the current placement
(^ EUCLIDean point space
S'Ap elastic range
^>C^ set of invertible tensors (general linear group)
..^ space of linear mappings from Y XoY (2nd-order tensors)
^^^ space of hardening variables in Chap. 10
<^^ set of orthogonal 2nd-order tensors (general orthogonal group)
S^d^^^ set of symmetric and positive-definite 2nd-order tensors
^ space of real numbers
space of S3mimetric 2nd-order tensors
.j/^ space of antisymmetric or skew 2nd-order tensors
^^W*^ set of 2nd-order tensors with determinant ±1
(general unimodular group)
space of vectors (EUCLIDean shifters)
A superimposed + such as ^^ means: with positive determinant. ^"^ denotes
the positive reals.
Variables and Abbreviations
a = r" ^Y acceleration
A G ^^^^5^^ symmetry transformation
B = F F^ G S^6^^ left CAUCHY-GREEN tensor
b ^ Y specific body force
e G ^ translational vector in the EUCLIDean transform.
C/ G >^ lattice vector in Chap. 10.5
C = F^ F G . ^^ right CAUCHY-GREEN tensor
C^ = P^C P G ^6^^ transformed right CAUCHY-GREEN tensor in
Chap. 10
COOS coordinate system
LIST OF FREQUENTLY USED SYMBOLS IX
do = j Yo^^dm e 7^ moment of momentum with respect to O
directional vector of a slip system in Chap. 10.5
D = sym(Li) : *_>^ stretching tensor, rate of deformation tensor
da element of area in current placement
dao element of area in reference placement
da vectorial element of area in current placement
da^ vectorial element of area in reference placement
Div, div material and spatial divergence operator
dm element of mass
dv element of volume in current placement
dvo element of volume in reference placement
dx vectorial line element in current placement
dxo vectorial line element in reference placement
e/ 6 r basis vector of ONB
E = sym(n) linear strain tensor
E^ = y2(i-B~^) e^ ALMANSFs strain tensor
E^ = I - \-^ G^ spatial BIOT's strain tensor
E®=U-I G^ material BIOT's strain tensor
E^ = y.(C-I) G^ GREEN'S strain tensor
jggen G^ spatial generalised strain tensor
material generalised strain tensor
•|7iGen G tJ^.
E^ = /«V e^ spatial HENCKY's strain tensor
E"=/WU G ..^^ material HENCKY's strain tensor
E internal energy
G .^
f ET resulting force acting on the material body
F = Grad % deformation gradient
G^
G r basis vectors of dual bases
g = gradO eT spatial temperature gradient
go= Grad 6 material temperature gradient
Grad, grad material and spatial gradient operator
ik, O G .W metric coefficients
H = Grad u G <=>^^^ displacement gradient
I unit tensor of 2nd-order
X
/ unit tensor of ^th-order
1' symmetriser of ^th-order
l^ antimetriser of ^th-order
J = det¥ G ^" JACOBIan, determinant of the deformation
gradient
K= V2 \ v^ dm e^ kinetic energy
K = Grad(KQ KQ'O G ^^t^ local change of the reference placement
lu = grad\ = ¥*¥~^ E:^,zi^ velocity gradient
&m
l = p-^T-L specific stress power
mo er moment with respect to O
m e^* mass
n er normal vector
0 sr zero vector
0 G ^y^^^^^ zero tensor
ONB orthonormal basis
P e^ pressure
p = j \ dm e^ linear momentum
P G oxW«^ plastic transformation in Chap. 10
PFI Principle of form invariance
PISM Principle of invariance under superimposed rigid body mo
tions
PMO Principle of material objectivity
q er spatial heat flux vector
qo = J¥-'q er material heat flux vector
Q e^ heat supply
Q e ( ^^ versor in EUCLIDean transformation
^r
r pbsition vector
r €^ specific heat source
R € ^ c^ rotation tensor
S = F-^TF-^ G tJT^^/t' material stress tensor
S^ = -CSP"^ G <^z^ plastic stress tensor in Chap. 10
T G iJT^'/n^ CAUCHY's stress tensor
T^ = ^3;m(T'^^U) G tj7^:,^/i' BIOT's stress tensor
T^^JT G fJT^i^U' KIRCHHOFF's stress tensor
Tk = F^TF G f^y^^^^ convected stress tensor
LIST OF FREQUENTLY USED SYMBOLS XI
T^ = C T^^^ G ^^ MANDEL's stress tensor
Tr = R^ T R e ^^€^ relative stress tensor
^iPK = y X F"^ G.^^ 7st PIOLA-KIRCHHOFF stress tensor
T^PK = y F -^ T F -^ G .s/^ 2nd PIOLA-KIRCHHOFF stress tensor
t = T n G ^ stress vector on a surface
t G ^ time
U = Vc G ^ay^ right stretch tensor
u G X displacement vector
V = VB G ^6f>^ left stretch tensor
F G ^^ volume in the current placement
Fo G <^^ volume in the reference placement
V = r* G ^ velocity
w G ^ spin vector (axial vector of W)
W = skw{y) G ^s/^ spin tensor
w G ^ specific elastic energy
X ^Y position vector in space
Xo ^y position vector in the reference placement
Y^ = Q* - P^ C P G t>^^ incremental plastic variable in Chap. 10
Z G ^^ hardening variables in Chap. 10
Greek Letters
X motion
5 variation, virtual
sLs^i, 4/'' s'^ KRONECKER symbols
8 G^ specific dissipation
£ G^ specific internal energy
linear strains (components of E)
^ij
m permutation or LEVI-CIVITA-symbol
V G^ specific entropy
v' spatial coordinates
<p yield criterion in Chap. 10
yield criterion for an elastic range in Chap. 10
i>P
K placement
Ko reference placement
X G ^" plastic consistency-parameter in Chap. 10
0 2 G ^" eigenvalue of C and B
A- G ^^ eigenvalue of U and V