Theory and numerics of higher gradient
inelastic material behavior
vom Fachbereich Maschinenbau und
Verfahrenstechnik der Universität Kaiserslautern
zur Verleihung des akademischen Grades
Doktor–Ingenieur (Dr.–Ing.)
genehmigte Dissertation
von Dipl.-Ing. Tina Liebe
aus Magdeburg
Hauptreferent: Prof. Dr.–Ing. Paul Steinmann
Korreferenten: Prof. Dr. rer. nat. B. Svendsen
Prof. Dr. A. Benallal
Vorsitzender: Prof. Dr.–Ing. H.-D. Hellmann
Dekan: Prof. Dr.–Ing. Paul Steinmann
Tag der Einreichung: 19. Februar 2003
Tag der mündlichen Prüfung: 07. April 2003
Kaiserslautern, April 2003
D 386Preface
The work presented in this thesis has been carried out during the period 1999-2003 at the Chair of
Applied Mechanics at the University of Kaiserslautern. The financial support of the DFG (Deutsche
Forschungsgemeinschaft) within the project ’Theorie und Numerik von Mono- und Polykristallplastizität
unter Berücksichtigung höherer Gradienten’ (STE-544/7-1-3) is gratefully acknowledged.
In the first place, I would like to thank Professor Paul Steinmann for his constant support and guidance,
his never ending patience, inspiring comments and the time he put into my thesis. He motivated me to
come to Kaiserslautern and take up this PhD-project and I owe him a great deal for making these last
four years a successful and enjoyable stage of my life.
My sincere thanks go to Professor Bob Svendsen and Professor Ahmed Benallal, who spontaneously
agreed to become correferees for my thesis. Many scientific discussions and valuable remarks encour-
aged and helped me getting a deeper insight into the subject of my work, and at some stages even more
enlightening were the insights they provided to the overall picture.
Special thanks go to Professor Erwin Stein who introduced me to the fascinating world of mechanics, in
particular computational mechanics, already during my study at the university in Hanover. Furthermore,
I would like to thank Professor Kaspar Willam for his hospitality and support during my stay at the
Colorado University Boulder, where I carried out my diploma thesis. The open-minded atmosphere I
encountered there raised my interest to carry on with scientific research.
I appreciated the pleasant working climate at the Chair of Applied Mechanics at the University of
Kaiserslautern, which is undoubtly due to my colleagues. In particular, I like to thank my room-mates
during the period of this work: Thomas Svedberg, Ellen Kuhl and Bernd Kleuter, who made our office-
life a great deal more enjoyable.
Last, but not least, I would like to express my gratitude to my parents and my brother, Jörg, for their
continuous encouragement and for always standing behind me. Especially, I thank Klaus Schmitt for
his support - not to forget his sometimes funny but always valuable questions and comments during the
reading and re-reading of the manuscript at various stages of its development.
Kaiserslautern, in April 2003 Tina LiebeContents
Preface i
Notation vii
Introduction 1
1 Concepts of the formulation 9
1.1 Internal degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Internal variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Continuum dislocation theory 13
2.1 Kinematics of the dislocation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Incompatibility measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Exploitation of the Positive Dissipation Principle . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Simplified gradient model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Phenomenological isotropic gradient plasticity 21
3.1 Thermodynamics of phenomenological local plasticity . . . . . . . . . . . . . . . . . . 22
3.2 Thermodynamics of gradient plasticity . . . . . . . . . . . . . . . . . 22
3.3 Isotropic local and gradient prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Well-posedness of the coupled problem . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Numerical treatment of phenomenological gradient plasticity . . . . . . . . . . . . . . . 27
3.5.1 Strong form of the coupled problem . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5.2 Weak form of the coupled . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.3 Temporal discretization of the coupled problem . . . . . . . . . . . . . . . . . . 29
3.5.4 Spatial of the coupled problem . . . . . . . . . . . . . . . . . . . 30
3.5.5 Monolithic iterative solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.5.1 Constitutive update . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.5.2 Active set search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Numerical examples of phenomenological gradient plasticity . . . . . . . . . . . . . . . 34
3.6.1 1D-model problem: bar under uniaxial tension . . . . . . . . . . . . . . . . . . 35
3.6.1.1 Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6.1.2 Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.2 Geometrically non-linear model problem: panel under tension . . . . . . . . . . 42iv Contents
4 Gradient plasticity in single and double slip 45
4.1 Kinematics of single crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Thermodynamics of single crystal gradient plasticity . . . . . . . . . . . . . . . . . . . 48
4.3 Numerical treatment of single crystal gradient plasticity . . . . . . . . . . . . . . . . . . 49
4.4 examples of single crystal . . . . . . . . . . . . . . . . . . 51
4.4.1 Single slip model problem: simple shear of a crystalline strip . . . . . . . . . . . 53
4.4.2 Double slip model simple shear of a strip . . . . . . . . . . 55
5 Phenomenological isotropic gradient damage 59
5.1 Thermodynamics of phenomenological local damage . . . . . . . . . . . . . . . . . . . 60
5.2 of gradient damage . . . . . . . . . . . . . . . . . 61
5.3 Isotropic local and gradient prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Well-posedness of the coupled problem . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5 Numerical treatment of phenomenological gradient damage . . . . . . . . . . . . . . . . 66
5.5.1 Strong form of the coupled problem . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5.2 Weak form of the . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5.3 Temporal discretization of the coupled problem . . . . . . . . . . . . . . . . . . 68
5.5.4 Spatial of the coupled problem . . . . . . . . . . . . . . . . . . . 69
5.5.5 Monolithic iterative solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5.5.1 Constitutive update . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6 Numerical examples of phenomenological gradient damage . . . . . . . . . . . . . . . . 72
5.6.1 1D-model problem: bar under uniaxial tension . . . . . . . . . . . . . . . . . . 72
5.6.2 2D-model panel in tension . . . . . . . . . . . . . . . . . . . . . . . . 77
5.6.3 Geometrically non-linear model problem: bar under uniaxial tension . . . . . . . 78
6 Material Force Method coupled to damage 81
6.1 Continuum format of J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2 Spatial versus material motion problem . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3 Hyperelasticity coupled to isotropic damage . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.1 Spatial motion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3.2 Material motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4 Numerical treatment incorporating the Material Force Method . . . . . . . . . . . . . . 85
6.4.1 Weak form of the coupled problem . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.1.1 Spatial motion . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.1.2 Material motion problem . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4.2 Discretization of the coupled problem . . . . . . . . . . . . . . . . . . . . . . . 87
6.4.2.1 Spatial motion problem . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4.2.2 Material motion . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4.2.3 Discretized format of J-integral: Material Force Method . . . . . . . . 90
6.5 Numerical examples of the Material Force Method coupled to damage . . . . . . . . . . 91
6.5.1 Specimen with elliptic hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5.2 with center crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.3 MBL-specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Summary and Outlook 99Contents v
A A few notes about continuum mechanics 103
A.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.1.1 Spatial motion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.1.2 Material motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.1.3 Spatial versus material motion problem . . . . . . . . . . . . . . . . . . . . . . 105
A.2 Quasi–static balance of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2.1 Spatial motion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2.2 Material motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
B Geometrically non-linear gradient plasticity 107
B.1 Discretization in time and space of the coupled problem . . . . . . . . . . . . . . . . . . 108
B.2 Prototype isotropic gradient plasticity model . . . . . . . . . . . . . . . . . . . . . . . . 109
B.3 Constitutive update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C Geometrically non-linear gradient damage 113
C.1 Discretization in time and space of the coupled problem . . . . . . . . . . . . . . . . . . 113
C.2 Prototype isotropic gradient damage model . . . . . . . . . . . . . . . . . . . . . . . . 115
C.3 Constitutive update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166
Notation
Throughout the thesis, scalars are denoted by small non-bold symbols a = a, vectors by small bold
symbols a = a e , whereas second order tensors are mainly indicated by capital bold symbols A =i i
A e
e and fourth order tensors are recognized as calligraphic symbols A =A e
e
e
e .ij i j i jijkl k l
Here, reference to the Euclidean basis e is tacitly assumed. If special distinction between materiali;:::;l
and spatial configuration is employed, capital bold symbols denote material second order tensors, while
small bold symbols define spatial second order tensors. In particular, the second order unit tensor is
denoted as 1 = e
e and the fourth order unit tensor is given by I = e
e
e
eij i j ik jl i j k l
with respect to the Kronecker delta = 1 fori = j and = 0 for i = j. Furthermore, the scalarij ij
and vector products of, e.g., vectors a;b;c are defined in standard fashion, namely a b = b a and
[a
c] b = [c b]a, respectively, whereby each indicates one contraction. The subsequent list gives a
general overview of the symbols used throughout this thesis.
sym sym 1 tf g symmetric part off g , e.g., A := [A + A ]
2
skw 1 tskwf g skewsymmetric part off g , e.g., A := [A A ]2
vol vol 1f g volumetric part off g , e.g., A := [A :1]1
3
dev dev volf g deviatoric part off g , e.g., A := A A
() variational form of ()
[] [] corresponding to slip systemII
[_] time derivative
[] temporally discretized formn+1
t[] transpose of []
’() equivalent stress
M ;K fundamental matrix and iteration matricesKL KL
A assembly of all elementse
e
B;B global solution domain and its elementwise discretizatione
B;B global and elementwise discretization node point setse
B active discretization node point setact
k;K elementwise numbering and global numbering
n ;n number of elements and number of nodal pointsen np
k k k k kN ;N ;N ;N ;N shape functions corresponding to discretization variable u;x; ; d;x u d
shear number
K;G bulk and shear modulus
c gradient parameterviii Notation
B ;TB compatible material configuration and corresponding tangent space0 0
B ;TB spatial configuration and tangent spacet t
B ;TB incompatible, isoclinic intermediate configuration and tangent spacep p
t;t reference and current time0
X;x placements of a material point at timet andt0
’ non-linear deformation map
F;f direct and inverse tangent map
ppF ;f direct and inverse inelastic tangent map
e eF ;f direct and inverse elastic tangent map
J;j Jacobian wrt F and f
tM Mandel–type second order stress tensor
t first Piola-Kirchhoff second order stress tensor
pL plastic ’velocity gradient’