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The difference of the solutions of the elastic and elastoplastic
boundary value problem and
an approach to multiaxial stress-strain correction
Holger Lang
Elastic BVP solution
Elastoplastic BVP solution
Correction
Vom Fachbereich Mathematik der Universit¨at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
Gutachter
Prof. Dr. Ren´e Pinnau (Technische Universit¨at Kaiserslautern)
Prof. Dr. Martin Brokate (Technische Universit¨at Mu¨nchen)
Disputation
2. Oktober 2007
D 386For my familyContents
I INTRODUCTION 1
II THEORY 13
1 Stop and Play with respect to different scalar products 15
1.1 Definitions, Wellposedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Basic properties of stop and play . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Some generalised Lipschitz estimates . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Some transformation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 The difference of the solutions of the elastic and elastoplastic BVPs 29
2.1 The elastic model, governing relations . . . . . . . . . . . . . . . . . . . . . . 30
2.2 The elastoplastic model, governing relations . . . . . . . . . . . . . . . . . . . 30
2.3 Common definitions for both models . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 The operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.3 The elastic domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Solution of elastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Solution of elastoplastic model . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Robustness of elastoplastic model . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Difference of both models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.8 The difference of stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 The correction model for linear kinematic hardening material 53
3.1 Global correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Localisation of correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Homotopy between both models . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Two generalisations 71
4.1 Generalisation for linear kinematic plus isotropic hardening material . . . . . 71
4.2 Generalisation for the class of continuous functions with bounded variation . 82
5 The pointwise correction model for linear kinematic hardening material 85
5.1 Transfer from the global to a pointwise correction model . . . . . . . . . . . . 85
5.2 A first estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 A second estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4 A third estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
vIII PRACTICE 99
6 The pointwise correction model for Jiang material 101
6.1 The Jiang correction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
e6.1.1 Definition of functionJ . . . . . . . . . . . . . . . . . . . . . . . . . 104s
6.1.2 Definition of functionsLe andKe . . . . . . . . . . . . . . . . . . . . 107s s
6.1.3 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1.4 Some further explanations . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 The Jiang constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2.1 The stress controlled Jiang model. . . . . . . . . . . . . . . . . . . . . 111
6.2.2 The strain controlled Jiang model . . . . . . . . . . . . . . . . . . . . 113
6.3 Some theoretical discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.1 Some easy facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.2 Flux computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.3 A special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.3.4 Phases of relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 A modified correction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7 Numerical implementation of the pointwise correction model 137
7.1 The basic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.1.1 The outer loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.1.2 The inner loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1.3 The constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2.1 The butterfly test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.2 The unsmoothed noise test . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3 The modified correction model . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 Comparison with elastoplastic BVP solutions and measurements 157
8.1 Parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2 Algorithmic differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
IV APPENDIX 197
A Proof of Krejˇc´ı’s theorem 199
B Miscellaneous 203
B.1 Two useful lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
B.2 Remarks on linear operators and function spaces . . . . . . . . . . . . . . . . 205
B.3 Remarks on the D¨oring-Vormwald constitutive model. . . . . . . . . . . . . . 207
B.4 Open challenges and questions . . . . . . . . . . . . . . . . . . . . . . . . . . 211
C Notations 213Part I
INTRODUCTION
1introduction 3
Motivation
In order to give a reliable lifetime or damage prediction for an elastoplastic metallic body,
which is subjected to exterior loads, good knowledge of the local stresses and strains over
time is necessary. If measurements are not possible (experimentally or financially), numerical
analysis may be applied.
Theusualprocessoffinite element based fatigue analysis inpracticalengineeringisdepictedin
figureA.Inanycase,thestartingpointofadurabilityanalysisisafiniteelementdiscretisation
of the body.
Finite Element Analysis
Difference? Estimates?
(this thesis)
@? R ?
Elastoplastic BVPElastic BVP
(linear, quasistatic, superposition) (nonlinear, quasistatic, transient)
?
Tensorial ‘elastic’ stresses and
e estrains σ(x,t), ε(x,t)
·q ??
Scalar ‘elastic’ stresses and Multiaxial correction
e estrains σ (x,t), ε (x,t) (this thesis)q q
? ? ?
No/Uniaxial correction Tensorial stresses and strains
(e.g. Neuber, ESED) σ˜(x,t), ε˜(x,t) resp. σ(x,t), ε(x,t)
·q
? ?
Scalar stresses and strains σ˜ (x,t), ε˜ (x,t) resp. σ (x,t), ε (x,t)q q q q
?
Uniaxial damage models, Hysteresis counting algorithms
?
DamageD(x)
-
hcf lcf
Figure A. Possible ways in fatigue analysis. Here · denote some one-dimensional scalar ‘equivalent’ magni-q
tude, e.g. a signed norm or a projection of the tensorial quantity, the tilde˜denotes the correction of a (scalar
or tensorial) elastic quantity.
For low cycle fatigue analysis (lcf), the numerically expensive transient elastoplastic bound-
ary value problem with an appropriate elastoplastic constitutive material law is solved in
order to receive the ‘exact’ elastoplastic stresses σ(x,t) and strains ε(x,t). This is the right
hand branch of figure A.
In high cycle fatigue analysis (hcf), the linear elastic boundary value problem with Hooke’s
e econstitutive material law is solved to receive the elastic stresses σ(x,t) and strains ε(x,t).
Here the linear superposition principle is essential for the low computational costs. If the
locations, where the elastic stress crosses the yield stress of the material, are small in volume4 introduction
and almost the whole body behaves elastic (the ‘effect of elastic support’), local corrections
may be applied,
e e e• after projection of the tensorial quantities σ(x,t), ε(x,t) to a scalar quantity σ (x,t),q
eε (x,t). Hereclearly multiaxiality gets lost. For uniaxialcorrections, Neuber’smethod,q
cf. Neuber [90, 91], and the equivalent strain energy density (ESED) method, cf.
Glinka/Molski [45], are well established in the meantime. This corresponds to the
very left hand branch in figure A.
• before projection on scalar quantities. Here multiaxial Neuber approaches, cf. Glinka
et al. [18, 83, 44, 104] and Chu [23], yield surface approaches, cf. Barkey et al. [3],
¨and pseudo parameter approaches have been suggested, cf. Kottgen et al. [72, 73]
and Hertel [53]. This corresponds to the centred branch in figure A.
An example for hcf. In engineering durability, the exterior load functions are typically
−1sampled with a frequency of about 400sec . Half an hour on a test track, i.e. 30min
= 1800sec, consequently yield 720000 data points. So it is – and will be in the next years –
impossible to solve the full transient nonlinear problem in an acceptable computational time
with sufficiently fine discretisation of the body. Nowadays, practicians cannot help choosing
the leftmost branch in figure A.
At theendof thefatigue analysis, whichis common for allbranches, theaccumulated damage
is computed, where it is widespread consensus to use the uniaxial Rainflow counting method,
see e.g. Dressler et al. [4, 15, 35, 36, 37], Seeger [26], Rychlik [101] or Hack [48].
Thereby, for the damage quantification, a lot of ‘damage parameters’ have been developed.
They assign a certain amount of damage to a given closed hysteresis loop. Some of the most
established ones are those of Manson, Coffin and Morrow [84, 28, 87] or Smith, Wat-
son andTopper [105]. These methods are usually available in commercial software tools for
fatigue analysis nowadays.
Independentlyof which specialkindof elastoplastic constitutive material law weconsider, the
elastoplastic BVP model (nonlinear, hysteresis operator with memory structure
between stress σ and strain ε, energy dissipation, damage accumulation, irre-
versible process!)
is much closer to nature than the
e eelastic BVP model (linear, one-to-one operator between stress σ and strain ε, no
energy dissipation, no damage, fully reversible process!).
However, for stresses, which are below the yield stress, they coincide. (Cf. lemmas 2.3.5 and
4.1.3 as well.)
Where can this thesis be situated?
We briefly summarise what is new in this thesis.
• So far, no literature exists that is concerned with the question what the difference
between the solutions of the elastic and elastoplastic boundary value problem is. No