Steck e , ritter r , et al (eds ) plasticity of metals experiments, models, computation wiley VCH (2001)

Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

Deutsche
Forschungsgemeinschaft

Plasticity of Metals:
Experiments, Models,
Computation
Final Report of the
Collaborative Research Centre 319,
Stoffgesetze fÏr das inelastischeVerhalten metallischer
Werkstoffe – Entwicklung und technische Anwendung
1985–1996
Edited by
Elmar Steck, Reinhold Ritter, Udo Pfeil and Alf Ziegenbein
Collaborative Research Centres


Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

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Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.

Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

Contents

Preface

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV

Correlation between Energy and Mechanical Quantities
of Face-Centred Cubic Metals, Cold-Worked and Softened
to Different States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lothar Kaps, Frank Haeßner
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1.1
1.2
1.3
1.4

Introduction
Experiments
Simulation .

Summary . .
References .

2

Material State after Uni- and Biaxial Cyclic Deformation . . . . . . . 17
Walter Gieseke, K. Roger Hillert, Guănter Lange

2.1
2.2
2.3
2.3.1
2.3.2
2.3.3
2.3.3.1
2.3.3.2
2.3.3.3
2.4
2.5

Introduction . . . . . . . . . . . . . . . . . . . .
Experiments and Measurement Methods
Results . . . . . . . . . . . . . . . . . . . . . . . .
Cyclic stress-strain behaviour . . . . . . . .
Dislocation structures . . . . . . . . . . . . .
Yield surfaces . . . . . . . . . . . . . . . . . . .
Yield surfaces on AlMg3 . . . . . . . . . . .
Yield surfaces on copper . . . . . . . . . . .
Yield surfaces on steel . . . . . . . . . . . . .
Sequence Effects . . . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . .

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1

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. 1
. 1
. 11
. 14
. 15

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17
18
19
19
24
28
28
30
30
31

34
35
35

V

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Contents
3

3.1
3.2
3.2.1
3.2.2
3.3
3.3.1
3.3.2
3.3.3
3.3.3.1
3.3.3.2
3.3.4
3.3.4.1
3.3.4.2
3.3.5
3.4
3.4.1
3.4.1.1
3.4.1.2

3.4.1.3
3.4.1.4
3.4.2
3.4.2.1
3.4.2.2
3.4.2.3
3.4.3
3.4.3.1
3.4.3.2
3.4.3.3
3.5

Plasticity of Metals and Life Prediction in the Range
of Low-Cycle Fatigue: Description of Deformation
Behaviour and Creep-Fatigue Interaction . . . . . . . . . . . . . . . . . . . 37
Kyong-Tschong Rie, Henrik Wittke, Juărgen Olfe
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental details for room-temperature tests . . . . . . . . . . .
Experimental details for high-temperature tests . . . . . . . . . . . .
Tests at Room Temperature: Description
of the Deformation Behaviour . . . . . . . . . . . . . . . . . . . . . . . .
Macroscopic test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Microstructural results and interpretation . . . . . . . . . . . . . . . .
Phenomenological description of the deformation behaviour . .
Description of cyclic hardening curve, cyclic stress-strain curve
and hysteresis-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Description of various hysteresis-loops with few constants . . . .
Physically based description of deformation behaviour . . . . . .

Internal stress measurement and cyclic proportional limit . . . . .
Description of cyclic plasticity with the models
of Steck and Hatanaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application in the field of fatigue-fracture mechanics . . . . . . .
Creep-Fatigue Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A physically based model for predicting LCF-life
under creep-fatigue interaction . . . . . . . . . . . . . . . . . . . . . . . .
The original model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modifications of the model . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental verification of the physical assumptions . . . . . . .
Life prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computer simulation and experimental verification
of cavity formation and growth during creep-fatigue . . . . . . . .
Stereometric metallography . . . . . . . . . . . . . . . . . . . . . . . . . .
Computer simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In-situ measurement of local strain at the crack tip
during creep-fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Influence of the crack length and the strain amplitude
on the local strain distribution . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the strain field in tension and compression . . . .
Influence of the hold time in tension on the strain field . . . . . .
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI

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37

37
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38
39

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40
40
43
45

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45
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47

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57
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61
62
63
64
65


Contents
4

4.1
4.2
4.3
4.4
4.5

4.5.1
4.5.2
4.5.3
4.5.4
4.5.5
4.5.6
4.5.7
4.6
4.6.1
4.6.1.1
4.6.1.2
4.6.1.3
4.6.2
4.7

5

5.1
5.2
5.2.1
5.2.2
5.3
5.3.1
5.3.2
5.3.3
5.3.4
5.4

Development and Application of Constitutive Models
for the Plasticity of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Elmar Steck, Frank Thielecke, Malte Lewerenz
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mechanisms on the Microscale . . . . . . . . . . . . . . . . . . . . . . . .
Simulation of the Development of Dislocation Structures . . . . . .
Stochastic Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . .
Material-Parameter Identification . . . . . . . . . . . . . . . . . . . . . . .
Characteristics of the inverse problem . . . . . . . . . . . . . . . . . . .
Multiple-shooting methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hybrid optimization of costfunction . . . . . . . . . . . . . . . . . . . . .
Statistical analysis of estimates and experimental design . . . . . .
Parallelization and coupling with Finite-Element analysis . . . . . .
Comparison of experiments and simulations . . . . . . . . . . . . . . .
Consideration of experimental scattering . . . . . . . . . . . . . . . . . .
Finite-Element Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implementation and numerical treatment of the model equations
Transformation of the tensor-valued equations . . . . . . . . . . . . . .
Numerical integration of the differential equations . . . . . . . . . . .
Approximation of the tangent modulus . . . . . . . . . . . . . . . . . . .
Deformation behaviour of a notched specimen . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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68
68
69
71
73
77
77
77
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79
79
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82
83
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84
85
86
86
88
88

On the Physical Parameters Governing the Flow Stress
of Solid Solutions in a Wide Range of Temperatures . . . . . . . . . . . 90
Christoph Schwink, Ansgar Nortmann
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solid Solution Strengthening . . . . . . . . . . . . . . . . . . . . . . . . .
The critical resolved shear stress, so . . . . . . . . . . . . . . . . . . . .
The hardening shear stress, sd . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic Strain Ageing (DSA) . . . . . . . . . . . . . . . . . . . . . . .
Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Complete maps of stability boundaries . . . . . . . . . . . . . . . . . .
Analysis of the processes inducing DSA . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary and Relevance for the Collaborative Research Centre
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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90
90
92
92
92
93
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102
102

VII

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Contents
6

Inhomogeneity and Instability of Plastic Flow
in Cu-Based Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Hartmut Neuhaăuser

6.1

6.2
6.3
6.3.1
6.3.2
6.3.3
6.3.4
6.4
6.4.1
6.4.2
6.4.3
6.5
6.5.1
6.5.2
6.5.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Some Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deformation Processes around Room Temperature . . . . . . . . . .
Development of single slip bands . . . . . . . . . . . . . . . . . . . . . . .
Development of slip band bundles and Luăders band propagation
Comparison of single crystals and polycrystals . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deformation Processes at Intermediate Temperatures . . . . . . . . .
Analysis of single stress serrations . . . . . . . . . . . . . . . . . . . . . .
Analysis of stress-time series . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deformation Processes at Elevated Temperatures . . . . . . . . . . . .
Dynamical testing and stress relaxation . . . . . . . . . . . . . . . . . . .
Creep experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

The Influence of Large Torsional Prestrain on the Texture
Development and Yield Surfaces of Polycrystals . . . . . . . . . . . . . . 131
Dieter Besdo, Norbert Wellerdick-Wojtasik

7.1
7.2
7.2.1
7.2.2
7.2.3
7.2.4
7.3
7.3.1
7.3.2
7.4
7.5
7.5.1
7.5.2
7.5.3
7.5.4
7.5.5
7.6

Introduction . . . . . . . . . . . . . . . . . . . . . . .
The Model of Microscopic Structures . . . . .
The scale of observation . . . . . . . . . . . . . .

Basic slip mechanism in single crystals . . .
Treatment of polycrystals . . . . . . . . . . . . . .
The Taylor theory in an appropriate version
Initial Orientation Distributions . . . . . . . . .
Criteria of isotropy . . . . . . . . . . . . . . . . . .
Strategies for isotropic distributions . . . . . .
Numerical Calculation of Yield Surfaces . . .
Experimental Investigations . . . . . . . . . . . .
Prestraining of the specimens . . . . . . . . . . .
Yield-surface measurement . . . . . . . . . . . .
Tensile test of a prestrained specimen . . . . .
Measured yield surfaces . . . . . . . . . . . . . .
Discussion of the results . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .

VIII

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146
146
147


Contents
8

Parameter Identification of Inelastic Deformation Laws Analysing
Inhomogeneous Stress-Strain States . . . . . . . . . . . . . . . . . . . . . . . 149
Reiner Kreißig, Jochen Naumann, Ulrich Benedix, Petra Bormann,
Gerald Grewolls, Sven Kretzschmar

8.1
8.2
8.3
8.4
8.4.1
8.4.2
8.4.3
8.4.3.1
8.4.3.2
8.5
8.5.1

8.5.2
8.5.3
8.6
8.6.1
8.6.2
8.6.3
8.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Deformation Law of Inelastic Solids . . . . . . . . . . .
Bending of Rectangular Beams . . . . . . . . . . . . . . . . . .
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental technique . . . . . . . . . . . . . . . . . . . . . . . .
Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Determination of the yield curves . . . . . . . . . . . . . . . . .
Determination of the initial yield-locus curve . . . . . . . .
Bending of Notched Beams . . . . . . . . . . . . . . . . . . . . .
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental technique . . . . . . . . . . . . . . . . . . . . . . . .
Approximation of displacement fields . . . . . . . . . . . . . .
Identification of Material Parameters . . . . . . . . . . . . . .
Integration of the deformation law . . . . . . . . . . . . . . . .
Objective function, sensitivity analysis and optimization
Results of parameter identification . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Development and Improvement of Unified Models
and Applications to Structural Analysis . . . . . . . . . . . . . . . . . . . . 174
Hermann Ahrens, Heinz Duddeck, Ursula Kowalsky,
Harald Pensky, Thomas Streilein

9.1
9.2
9.2.1
9.2.2
9.3
9.4
9.5
9.6
9.7
9.8
9.8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
On Unified Models for Metallic Materials . . . . . . . . . . .
The overstress model by Chaboche and Rousselier . . . . .
Other unified models . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-Integration Methods . . . . . . . . . . . . . . . . . . . . . . .
Adaptation of Model Parameters to Experimental Results
Systematic Approach to Improve Material Models . . . . . .
Models Employing Distorted Yield Surfaces . . . . . . . . . .
Approach to Cover Stochastic Test Results . . . . . . . . . . .
Structural Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Consistent formulation of the coupled boundary
and initial value problem . . . . . . . . . . . . . . . . . . . . . . . .

Analysis of stress-strain fields in welded joints . . . . . . . .
Thick-walled rotational vessel under inner pressure . . . . .

9.8.2
9.8.3

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149
149
150
152
152
152
155
155
158
160
160
161
163
165
165

167
169
170
172
173

174
174
175
177
178
181
186
190
197
201

. . . . . . . . . 202
. . . . . . . . . 203
. . . . . . . . . 205
IX

www.pdfgrip.com


Contents
9.8.4
9.8.5
9.8.6

Application of distorted yield functions . . . . . . . . . . . . . . . .
Application of the statistical approach of Section 9.7 . . . . . . .
Numerical analysis for a recipient of a profile extrusion press
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

On the Behaviour of Mild Steel Fe 510
under Complex Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Udo Peil, Joachim Scheer, Hans-Joachim Scheibe,
Matthias Reininghaus, Detlef Kuck, Sven Dannemeyer

10.1
10.2
10.2.1
10.2.2
10.2.2.1
10.2.2.2
10.2.3
10.2.3.1
10.2.3.2
10.2.3.3
10.3
10.3.1
10.3.1.1
10.3.1.2
10.3.1.3
10.3.1.4
10.3.1.5

10.3.1.6
10.3.1.7
10.3.1.8
10.3.1.9
10.3.1.10

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Material Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Material, experimental set-ups, and techniques . . . . . . . . . . . . . .
Material behaviour under uniaxial cyclic loading . . . . . . . . . . . . .
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results of the uniaxial experiments . . . . . . . . . . . . . . . . . . . . . .
Material behaviour under biaxial cyclic loading . . . . . . . . . . . . .
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relations of tensile and torsional stresses . . . . . . . . . . . . . . . . . .
Yield-surface investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modelling of the Material Behaviour of Mild Steel Fe 510 . . . . .
Extended-two-surface model . . . . . . . . . . . . . . . . . . . . . . . . . . .
General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Loading and bounding surface . . . . . . . . . . . . . . . . . . . . . . . . . .
Strain-memory surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Internal variables for the description on non-proportional loading .
Size of the yield surface under uniaxial cyclic plastic loading . . .
Size of the bounding surface under uniaxial cyclic plastic loading
Overshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional update of din in the case of biaxial loading . . . . . . . . .
Memory surface F' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional isotropic deformation on the loading surface
due to non-proportional loading . . . . . . . . . . . . . . . . . . . . . . . . .
Additional isotropic deformation of the bounding surface

due to non-proportional loading . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between theory and experiments . . . . . . . . . . . . . . .
Experiments on Structural Components . . . . . . . . . . . . . . . . . . .
Experimental set-ups and computational method . . . . . . . . . . . . .
Correlation between experimental and theoretical results . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.3.1.11
10.3.2
10.4
10.4.1
10.4.2
10.5

X

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209
212
214
215

218
219
219
219
219
220
225
225
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229
236
236
236
237
238
241
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242
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244
248
248
248
248
251

252


Contents
11

11.1
11.1.1
11.1.2
11.1.3
11.2
11.3
11.3.1
11.3.2
11.3.2.1
11.3.2.2
11.3.3
11.3.4
11.3.5
11.3.5.1
11.3.5.2
11.3.5.3
11.4
11.5

12

12.1
12.1.1
12.1.2

12.2
12.2.1
12.2.2
12.3
12.3.1
12.3.2
12.3.3
12.4
12.4.1

Theoretical and Computational Shakedown Analysis
of Non-Linear Kinematic Hardening Material
and Transition to Ductile Fracture . . . . . . . . . . . . . . . . . . . . . . . . 253
Erwin Stein, Genbao Zhang, Yuejun Huang,
Rolf Mahnken, Karin Wiechmann
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
General research topics . . . . . . . . . . . . . . . . .
State of the art at the beginning of project B6 .
Aims and scope of project B6 . . . . . . . . . . . .
Review of the 3-D Overlay Model . . . . . . . . .
Numerical Approach to Shakedown Problems .
General considerations . . . . . . . . . . . . . . . . .
Perfectly plastic material . . . . . . . . . . . . . . . .
The special SQP-algorithm . . . . . . . . . . . . . .
A reduced basis technique . . . . . . . . . . . . . . .
Unlimited kinematic hardening material . . . . .
Limited kinematic hardening material . . . . . . .
Numerical examples . . . . . . . . . . . . . . . . . . .
Thin-walled cylindrical shell . . . . . . . . . . . . .

Steel girder with a cope . . . . . . . . . . . . . . . . .
Incremental computations of shakedown limits
of cyclic kinematic hardening material . . . . . .
Transition to Ductile Fracture . . . . . . . . . . . .
Summary of the Main Results of Project B6 . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

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253
253
253
254

254
256
259
259
260
260
261
261
263
264
264
265

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267
269
272
273

Parameter Identification for Inelastic Constitutive Equations Based
on Uniform and Non-Uniform Stress and Strain Distributions . . . 275
Rolf Mahnken, Erwin Stein
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
State of the art at the beginning of project B8 . . . . . . . . . .
Aims and scope of project B8 . . . . . . . . . . . . . . . . . . . . .
Basic Terminology for Identification Problems . . . . . . . . .
The direct problem: the state equation . . . . . . . . . . . . . . .
The inverse problem: the least-squares problem . . . . . . . . .
Parameter Identification for the Uniform Case . . . . . . . . . .
Mathematical modelling of uniaxial visco-plastic problems
Numerical solution of the direct problem . . . . . . . . . . . . .
Numerical solution of the inverse problem . . . . . . . . . . . .
Parameter Identification for the Non-Uniform Case . . . . . .
Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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275
275
275
276
277
277
278
280

280
282
282
283
284
XI

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Contents
12.4.2
12.4.3
12.5
12.5.1
12.5.2
12.6

The direct problem: Galerkin weak form . . . . . . . . . . . . . . . . . . . .
The inverse problem: constrained least-squares optimization problem
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cyclic loading for AlMg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Axisymmetric necking problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Experimental Determination of Deformation- and Strain Fields
by Optical Measuring Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Reinhold Ritter, Harald Friebe

13.1
13.2
13.3
13.4
13.4.1
13.4.2
13.4.3
13.4.4
13.4.5
13.4.6
13.5
13.5.1
13.5.2
13.5.3
13.6
13.6.1
13.6.2
13.6.3
13.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Requirements of the Measuring Methods . . . . . . . . . . . . .
Characteristics of the Optical Field-Measuring Methods . . .
Object-Grating Method . . . . . . . . . . . . . . . . . . . . . . . . . .
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deformation analysis at high temperatures . . . . . . . . . . . .
Compensation of virtual deformation . . . . . . . . . . . . . . . .

3-D deformation measuring . . . . . . . . . . . . . . . . . . . . . . .
Specifications of the object-grating method . . . . . . . . . . . .
Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . .
General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Technology of the Speckle interferometry . . . . . . . . . . . . .
Specifications of the developed 3-D Speckle interferometer
Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-D object-grating method in the high-temperature area . . .
3-D object-grating method in fracture mechanics . . . . . . . .
Speckle interferometry in welding . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Surface-Deformation Fields from Grating Pictures
Using Image Processing and Photogrammetry . . . . . . . . . . . . . . . . 318
Klaus Andresen

14.1
14.2
14.2.1
14.2.2
14.3
14.4
14.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grating Coordinates . . . . . . . . . . . . . . . . . . . . . .
Cross-correlation method . . . . . . . . . . . . . . . . . . .

Line-following filter . . . . . . . . . . . . . . . . . . . . . .
3-D Coordinates by Imaging Functions . . . . . . . . .
3-D Coordinates by Close-Range Photogrammetry
Experimental set-up . . . . . . . . . . . . . . . . . . . . . .

XII

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Contents
14.4.2
14.4.3
14.5
14.6
14.6.1
14.6.2
14.6.3
14.7

Parameters of the camera orientation . . . . . . . . . . . . . . . . . . . . . . .
3-D object coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Displacement and Strain from an Object Grating: Plane Deformation
Strain for Large Spatial Deformation . . . . . . . . . . . . . . . . . . . . . . .
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correcting the influence of curvature . . . . . . . . . . . . . . . . . . . . . . .
Simulation and numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Experimental and Numerical Analysis of the Inelastic
Postbuckling Behaviour of Shear-Loaded Aluminium Panels . . . . . 337
Horst Kossira, Gunnar Arnst

15.1
15.2
15.2.1
15.2.1.1
15.2.1.2
15.2.2
15.2.2.1
15.2.2.2
15.3
15.3.1
15.3.2
15.3.2.1
15.3.2.2
15.3.2.3
15.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Model . . . . . . . . . . . . . . . . . . . . . . .
Finite-Element method . . . . . . . . . . . . . . . . . . .
Ambient temperature – rate-independent problem
Elevated temperature – visco-plastic problem . . .
Material models . . . . . . . . . . . . . . . . . . . . . . . .
Ambient temperature – rate-independent problem
Elevated temperature – visco-plastic problem . . .
Experimental and Numerical Results . . . . . . . . .
Test procedure . . . . . . . . . . . . . . . . . . . . . . . . .
Computational analysis . . . . . . . . . . . . . . . . . . .
Monotonic loading – ambient temperature . . . . .
Cyclic loading – ambient temperature . . . . . . . . .

Time-dependent behaviour . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Consideration of Inhomogeneities in the Application
of Deformation Models, Describing the Inelastic Behaviour
of Welded Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Helmut Wohlfahrt, Dirk Brinkmann

16.1
16.2
16.2.1
16.2.2
16.2.2.1
16.2.2.2
16.3

Introduction . . . . . . . . . . . . . . . . . . . . . . .
Materials and Numerical Methods . . . . . . .
Materials and welded joints . . . . . . . . . . . .
Deformation models and numerical methods
Deformation model of Gerdes . . . . . . . . . .
Fitting calculations . . . . . . . . . . . . . . . . . .
Investigations with Homogeneous Structures

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362
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365
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Contents
16.3.1
16.3.1.1
16.3.1.2
16.3.1.3
16.3.2
16.4
16.4.1
16.4.1.1
16.4.1.2

16.4.1.3
16.4.1.4
16.4.2
16.4.3
16.4.4
16.5

Experimental and numerical investigations . . . . . . . . . . . .
Tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creep tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cyclic tension-compression tests . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Investigations with Welded Joints . . . . . . . . . . . . . . . . . . .
Deformation behaviour of welded joints . . . . . . . . . . . . . .
Experimental investigations . . . . . . . . . . . . . . . . . . . . . . .
Numerical investigations . . . . . . . . . . . . . . . . . . . . . . . . .
Finite-Element models of welded joints . . . . . . . . . . . . . .
Calculation of the deformation behaviour of welded joints .
Strain distributions of welded joints with broad weld seams
Strain distributions of welded joints with small weld seams
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application Possibilities and Further Investigations . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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366
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383

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

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Plasticity of Metals: Experiments, Models, Computation. Collaborative Research Centres.
Edited by E. Steck, R. Ritter, U. Peil, A. Ziegenbein
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

Preface

The Collaborative Research Centre (Sonderforschungsbereich, SFB 319), “Material
Models for the Inelastic Behaviour of Metallic Materials – Development and Technical
Application”, was supported by the Deutsche Forschungsgemeinschaft (DFG) from July
1985 until the end of the year 1996. During this period of nearly 12 years, scientists
from the disciplines of metal physics, materials sciences, mechanics and applied engineering sciences cooperated with the aim to develop models for metallic materials on a
physically secured basis. The cooperation has resulted in a considerable improvement
of the understanding between the different disciplines, in many new theoretical and experimental methods and results, and in technically applicable constitutive models as
well as new knowledge concerning their application to practical engineering problems.
The cooperation within the SFB was supported by many contacts to scientists and
engineers at other universities and research institutes in Germany as well as abroad.
The authors of this report about the results of the SFB 319 wish to express their thanks
to the Deutsche Forschungsgemeinschaft for the financial support and the very constructive cooperation, and to all the colleagues who have contributed by their interest
and their function as reviewers and advisors to the results of our research work.

Introduction
The development of mathematical models for the behaviour of technical materials is of
course directed towards their application in the practical engineering work. Besides the
projects, which have the technical application as their main goal, in all projects, which
were involved in experiments with homogeneous or inhomogeneous test specimens –

where partly also the numerical methods were further investigated and the implementation of the material models in the programs was performed –, experiences concerning
the application of the models for practical problems could be gained. The whole-field
methods for measuring displacement and strain fields, which were developed in connection with these experiments, have given valuable support concerning the application
of the developed constitutive models to practical engineering.
The research concerning the identification of the parameters of the models has
proven to be very actual. The investigations for most efficient methods for the parameter identification will in the future still find considerable attention, where the cooperation of scientists from engineering as well as applied mathematics, which was started in
the SFB, will continue. As is shown in a later chapter, it is of increasing importance to
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Preface
use not only homogeneously, uniaxially loaded test specimen, but also to analyze stress
and deformation fields in complexly loaded components. In connection with these investigations, methods for the design of experiments should be developed, which can be
used for the assessment of the structure of the material models and the physical meaning of the model parameters. The results obtained up to now have shown, also by comparisons in cooperation with institutions outside the SFB, that the predictive properties
of the developed material models are of equal quality as those of other models used in
the engineering practice. They have however the advantage that they are based on results of material physics and therefore can use further developments of the knowledge
about the mechanisms of inelastic deformations on the microscale.
During the work in the different projects, a surprising number of similar problems
have been found. Due to the close contacts between the working groups, they could be
investigated with much higher quality than without this cooperation.
The exchange of thought between metal physics, materials sciences, mechanics
and applied engineering sciences was very stimulating and has resulted in the fact that
the groups oriented towards application could be supported by the projects working
theoretically, and on the other hand, the scientists working in theoretical fields could
observe the application of their results in practical engineering.

Research Program
The main results of the activities of the SFB have been models for the load-deformation behaviour as well as for damage development and the development of deformation

anisotropies. These models make it possible to use results from the investigations from
metal physics and materials sciences in the SFB in the continuum mechanics models.
The research work in metal physics and materials sciences has considerably contributed
to a qualitative understanding of the processes, which have to be described by constitutive models. The structure of the developed models and of the formulations found in
literature, which have been considered for comparisons and supplementation of our
own development, have strongly influenced the work concerning the implementation of
the material models in numerical computing methods and the treatment of technical
problems. The models could be developed to a status, where the results of experimental
investigations can be used to determine the model parameters quantitatively.
This has resulted in an increasing activity on the experimental side of the work
and also in an increase of the cooperation within the SFB and with institutions outside
of Braunschweig (BAM Berlin, TU Hamburg-Harburg, TH Darmstadt, RWTH Aachen,
KFA Juălich, KFZ Karlsruhe, E´cole Polytechnique Lausanne). In the SFB, joint research
was undertaken in the fields of high-temperature experiments for the investigation of
creep, cyclic loading and non-homogeneous stress and displacement fields for technical
important metallic materials, and their comparison with theoretical predictions. The developed whole-field methods for measuring deformations have shown to be an imporXVI

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Research Program
tant experimental method. The increasing necessity to obtain experimental results of
high quality for testing and extending the material models has resulted in the development of experimental equipment, which also allows to investigate the material behaviour under multiaxial loadings in the high- and low-temperature range.
The determination of model parameters and process quantities from experiments
has put the question for reliable methods for the parameter identification in the foreground. The earlier used methods of least-squares and probabilistic methods, such as
the evolution strategy, have given satisfying results. In the SFB, however, the knowledge has developed that methods for the parameter identification, which consider the
structure of the material models and the design of optimal experiments and discriminating experiments, deserve special consideration.
If numerical values for the model parameters are given, the possibility exists to
examine these values concerning their physical meaning, and in cooperation with the
scientists from metal physics and materials sciences to investigate the connection between the knowledge about the processes on the microscale and the macroscopic constitutive equations.

The SFB was during its activities organized essentially in three project areas:
A:
•
•
•

Materials behaviour
Phenomena
Material models
Parameter identification

B: Development of computational methods
• General computational methods under consideration of the developed material models
• Special computational methods (e. g. shells structures, structural optimization, shakedown)
C:
•
•
•

Experimental verification
Whole-field methods
Examination of the transfer of results
Mock-up experiments.

Project area A: materials behaviour
The research in the project area A was mainly concerned with theoretical and experimental investigations concerning the basis for the development of material models and
damage development from metal physics and materials sciences. In the following, a
short description of the activities within the research projects is given. Methods and results are in detail given in later chapters.

XVII

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Preface
Correlation between energetic and mechanical quantities of face-centred cubic
metals, cold-worked and softened to different states (Kaps, Haeßner)
One of these basic investigations is concerned with calorimetric measurements in connection with the description of recovery. After measurements based on the sheet rolling
process, final investigations were performed concerning higher deformation temperatures and more complex deformation processes. Here, torsion experiments were examined due to the fact that this process allows the investigation of very high deformations
as well as a simple reversal of the deformation direction and cyclic experiments.
Recovery and recrystallization are in direct competition with strain hardening. If a
material is cold-worked, its yield stress increases. This process, denoted strain hardening, leads to a gain in internal energy. Recovery and recrystallization act to oppose
strain hardening. Already upon deformation or during subsequent annealing, these
forces transform the material back into a state of lower energy. Although this reciprocity has been known for some time, the exact dependence of the process upon the type
and extent of deformation, upon the temperatures during deformation and softening anneal as well as upon the chemical composition of the material is as yet only qualitatively known. Consequently, the predictability of the processes is as poor as it has always been so that, even today, one is still obliged to refer to experience and explicit
experiments for help.

Material state after uni- and biaxial cyclic deformation (Gieseke, Hillert, Lange)
The investigations concerning the material behaviour at multiaxial plastic deformation
were performed using the material AlMg3, copper and the austenitic stainless steel AISI
316L. To find the connection between damage development and microstructure, the dislocations structure at the tip of small cracks and at surface grains with differently pronounced slip-band development was investigated. With the aim to check the main assumptions of the two-surface models explicitly, measurements of the development of
the yield surface of the material from the initial to the saturation state and within a saturation cycle were considerably extended. Consecutive yield surfaces along different
loading histories were measured. The two-surface models of Ellyin and McDowell
were implemented in the computations.
Technical components and structures today are increasingly being designed and
displayed by computer-aided methods. High speed computers permit the use of mathematical models able to numerically reconstruct material behaviour, even in the course
of complex loading procedures.
In phenomenological continuum mechanics, the cyclic hardening and softening
behaviour as well as the Bauschinger effect are described by yield-surface models. If a
physical formulation is chosen as a basis for these models, then it is vitally important

to have exact knowledge of the processes occurring in the metal lattice during deformation. Two-surface models, going back to a development by Dafalias and Popov, describe the displacement of the elastic deformation zone in a dual axis stress area. The
yield surfaces are assumed to be v. Mises shaped ellipses. However, from experiments
with uniaxial loading, it is known that the yield surfaces of small offset strains under
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Research Program
load become characteristically deformed. In the present subproject, the effect of cyclic
deformation on the shape and position of the yield surfaces is studied, and their relation to the dislocation structure is determined. To this end, the yield surfaces of three
materials with different slip behaviour were measured after prior uni- or biaxial deformation. The influence of the dislocation structures produced and the effect of internal
stresses are discussed.

Plasticity of metals and life prediction in the range of low-cycle fatigue: description of
deformation behaviour and creep-fatigue interaction (Rie, Wittke, Olfe)
In the field of investigations about the connection between creep and low-cycle fatigue,
the development of models for predicting the componente lifetime at creep fatigue was
the main aim of the work. Measuring the change of the physical magnitudes in the
model during an experiment results in an investigation and eventually a modification of
the model assumptions. The model was also examined for its usability for experiments
with holding-times at the maximum pressure loading during a loading cycle.
For hot working tools, chemical plants, power plants, pressure vessels and turbines, one has to consider local plastic deformation at critical locations of structural
components. Due to cyclic changes of temperature and load, the components are subjected to cyclic deformation, and the components are limited in their use by fatigue.
After a quite small number of cycles with cyclic hardening or softening, a state of cyclic saturation is reached, which can be characterized by a stress-strain hysteresis-loop.
Cyclic deformation in the regime of low-cycle fatigue (LCF) leads to the formation of
cracks, which can subsequently grow until failure of a component part takes place.
In the field of fatigue fracture mechanics, crack growth is correlated with parameters, which take into account information especially about the steady-state stress-strain
hysteresis-loops. Therefore, it can be expected that a more exact life prediction is possible by a detailed investigation of the cyclic deformation behaviour and by the description of the cyclic plasticity, e. g. with constitutive equations.
At high temperatures, creep deformation and creep damage are often superimposed on the fatigue process. Therefore, in many cases, not one type of damage prevails, but the interaction of both fatigue and creep occurs, leading to failure of components.

The typical damage in the low-cycle fatigue regime is the development and
growth of cracks. In the case of creep fatigue, grain boundary cavities may be formed,
which interact with the propagating cracks, this leading to creep-fatigue interaction. A
reliable life prediction model must consider this interaction.
The knowledge and description of the cavity formation and growth by means of
constitutive equations are the basis for reliable life prediction. In the case of diffusion-controlled cavity growth, the distance between the voids has an important influence on their
growth. This occurs especially in the case of low-cycle fatigue, where the cavity formation
plays an important role. Thus, the stochastic process of void nucleation on grain boundaries and the cyclic dependence of this process has to be taken into consideration as a
theoretical description. The experimental analysis has to detect the cavity-size distribution, which is a consequence of the complex interaction between the cavities.
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Up to now, only macroscopic parameters such as the total stress and strain have
been used for the calculation of the creep-fatigue damage. But crack growth is a local
phenomenon, and the local conditions near the crack tip have to be taken into consideration. Therefore, the determination of the strain fields in front of cracks is an important step for modelling.

Development and application of constitutive models for the plasticity of metals (Steck,
Thielecke, Lewerenz)
The inelastic material behaviour in the low- and high-temperature ranges is caused by
slip processes in the crystal lattice, which are supported by the movement of lattice defects like dislocations and dislocation packages. The dislocation movements are opposed by internal barriers, which have to be overcome by activation. This is performed
by stresses or thermal energy. During the inelastic deformation, the dislocations interact
and arrange in a hierarchy of structures such as walls, adders and cells. This forming of
internal material structures influences strongly the macroscopic responses on mechanical and thermal loading.
A combination of models on the basis of molecular dynamics and cellular automata is used to study numerically the forming of dislocation patterns and the evolution
of internal stresses during the deformation processes. For a realistic simulation, several
glide planes are considered, and for the calculation of the forces acting on a dislocation, a special extended neighbourhood is necessary. The study of the self-organization
processes with the developed simulation tool can result in valuable information for the

choice of formulations for the modelling of processes on the microscale.
The investigations concerning the development of material models based on
mechanisms on the microscale have resulted in a unified stochastic model, which is
able to represent essential and typical features of the low- and high-temperature plasticity. For the modelling of the dislocation movements in crystalline materials and their
temperature and stress activation, a discrete Markov chain is considered. In order to describe cyclic material behaviour, the widely accepted concept is used that the dislocation-gliding processes are driven by the effective stress as the difference between the
applied stress and the internal back stress. The influence of effective stress and temperature on the inelastic deformations is considered by a metalphysically motivated
evolution equation. A mean value formulation of this stochastic model leads to a
macroscopic model consisting of non-linear ordinary differential equations. The results
show that the stochastic theory is helpful to deduce the properties of the macroscopic
constitutive equations from findings on the microscale.
Since the general form of the stochastic model must be adapted to the special
material characteristics and the considered temperature regime, the identification of the
unknown material parameters plays an important role for the application on numerical
calculations. The determination of the unknown material parameters is based on a Maximum-Likelihood output-error method comparing experimental data to the numerical simulations. For the minimization of the costfunction, a hybrid optimization concept parallelized with PVM is considered. It couples stochastic search procedures and several
Newton-type methods. A relative new approach for material parameter identification is
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the multiple shooting approach, which allows to make efficient use of additional measurement- and apriori-information about the states. This reduces the influence of bad initial
parameters. Since replicated experiments for the same laboratory conditions show a significant scattering, these uncertainties must be taken into account for the parameter identification. The reliability of the results can be tested with a statistical analysis.
Several different materials, like aluminium, copper, stainless steel AISI 304 and
AISI 316, have been studied. For the analysis of structures, like a notched flat bar, the
Finite-Element program ABAQUS is used in combination with the user material subroutine UMAT. The simulations are compared with experimental data from grating
methods.

On the physical parameters governing the flow stress of solid solutions in a wide range
of temperatures (Schwink, Nortmann)

In the area of the metal-physical foundations, investigations on poly- and single-crystalline material have been performed. The superposition of solution hardening and ordinary hardening has found special consideration. Along the stress-strain curves, the limits between stable and unstable regions of deformation were investigated, and their dependencies on temperature, strain rate and solute concentration were determined. In regions of stable deformation, a quantitative analysis of the processes of dynamic strain
ageing (“Reckalterung”) was performed. The transition between regions of stable and
unstable deformation was investigated and characterized.
At sufficiently low temperatures, host and solute atoms remain on their lattice
sites. The critical flow stress is governed by thermally activated dislocations glide (Ar 0, and an effecrhenius equation), which depends on an average activation enthalpy DG
tive obstacle concentration cb. The total flow stress is composed of the critical flow
stress and a hardening stress, which increases with the dislocation density in the cell
walls.
Detailed investigations on single crystals yielded expressions for the critical
0 ; cb ; T; e_ †, and the hardening shear stress,
resolved shear stress, s0 ˆ s0 …DG
1=2
sd ˆ w Gbqw . Here, w is a constant, w ˆ 0:25 Ỉ 0:03, G the shear modulus, and
qw the dislocation density inside the cell walls. The total shear stress results as
s ˆ s0 ‡ sd .
At higher temperatures, the solutes become mobile in the lattice and cause an additional anchoring of the glide dislocations. This is described by an additional enthalpy
Dg…tw ; Eam † in the Arrhenius equation. In the main, it depends on the activation energy
Eam of the diffusing solutes and the waiting time tw of the glide dislocations arrested at
obstacles. Three different diffusion processes characterized by EaI ; EaII; EaIII were found
for the two f.c.c.-model systems investigated, CuMn and CuAl, respectively. In both,
0 . Under certain conditions, the solute diffusion
Dg reaches values up to about 0.1 DG
causes instabilities in the flow stress, the well-known jerky flow phenomena (PortevinLe Chaˆtelier effect). Finally, above around 800 K in copper-based alloys, the solutes become freely mobile, and the critical flow stress as well as the additional enthalpy vanish. In any temperature region, only a small total number of physical parameters is sufficient for modelling plastic deformation processes.
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Inhomogeneity and instability of plastic flow in Cu-based alloys (Neuhaăuser)

In a second project, the main goal of the research is to clarify the physical mechanisms, which control the kinetics of the deformation, especially in such parameter regions, which are characterized by inhomogeneity and instability of the deformation process. It is looked for a realistic interpretation of the magnitudes, which will be used
with empirical material equations as it is necessary for a sensible application and extrapolation to extended parameter regions. Especially, reasons and effects of deformationinhomogeneities and -instabilities in the systems Cu-Al and Cu-Mn, which show tendencies to short-range order, were investigated. Determining dislocation-generation
rates and dislocation velocities in the case of gradients of the effective stress were as
well aim of the investigations as the influence of diffusion processes on the generation
(blocking, break-away) and motion (obstacle destruction and regeneration) of dislocations. Investigations were also performed concerning the use of the results for single
crystals for the description of the practically more important case of the behaviour of
polycrystals. In this case, especially the influence of the grain-boundaries on generation
and movement of dislocations or dislocation groups has to be considered.
The special technique used in this project is a microcinematographic method,
which permits to measure the local strain and strain rate in slip bands, which are the
active regions of the crystal. Cu-based alloys with several percent of Al and Mn solutes
are considered in order to separate the effects of stacking-fault energy from those of solute hardening and short-range ordering, which are comparable for both alloy systems,
while the stacking-fault energy decreases rapidly with solute concentration for CuAl
contrary to CuMn alloys. Both systems show different degrees of inhomogeneous slip
in the length scales from nm to mm (slip bands, Luăders bands), and, in a certain range
of deformation conditions, macroscopic deformation instabilities (Portevin-Le Chaˆtelier
effect). These effects have been studied in particular.

The influence of large torsional prestrain on the texture development and yield surface
of polycrystals – experimental and theoretical investigations (Besdo, Wellerdick-Wojtasik)
This research project consists of a theoretical and an experimental part. The topic of
the theoretical part was the simulation of texture development and methods of calculating yield surfaces. The calculations started from an initially isotropic grain distribution.
Therefore, it was necessary to set up such a distribution. Different possibilities were
compared with an isotropy test considering the elastic and plastic properties. With some
final distributions, numerical calculations were carried out. The Taylor theory in an appropriate version and a simple formulation based on the Sachs assumption were used.
Calculation of yield surfaces from texture data can be done in many different ways.
Some examples are the yield surfaces calculated with the Taylor theory, averaging methods or formulations, which take the elastic behaviour into account. Several possibilities
are presented, and the numerical calculations are compared with the experimental results.
In order to measure yield surfaces after large torsional prestrain, thin-walled tubular specimens of AlMg3 were loaded up to a shear strain of c ˆ 1:5, while torsional
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buckling was prevented by inserting a greased mandrel inside the specimens. Further
investigations of the prestrained specimens were done with the testing machine of the
project area B.
At least one yield surface, represented by 16 yield points, was measured with
each specimen. The yield point is defined by the offset-strain definition, where generally the von Mises equivalent offset strain is used. Three different loading paths were
realized with the extension-controlled testing machine. Thus, the results were yield surfaces measured with different offsets and loading paths.
The offset-strain definition is based on the elastic tensile and shear modulus.
These constants were calculated at the beginning of each loading path, and since they
strongly effect the yield surfaces, this must be done with the highest amount of care.
The isotropic specimens are insensitive to different loading paths, and the measured
yield surfaces seem to be of the von Mises type. By contrast, the prestrained specimens
are very sensitive to different loading paths. Especially the shape and the distorsion of
the measured surfaces changes as a result of the small plastic strain during the measurement. Therefore, it seems that the shape and the distortion of the yield surface were not
strongly effected by the texture of the material.

Parameter identification of inelastic deformation laws analysing inhomogeneous stressstrain states (Kreißig, Naumann, Benedix, Borman, Grewolls, Kretzschmar)
In the last years, the necessity of solutions of non-linear solid mechanics problems has
permanently increased. Although powerful hard- and software exist for such problems,
often more or less large differences between numerical and experimental results are observed. The dominant reason for these defects must be seen in the material-dependent
part of the used computer programs. Either suitable deformation laws are not implemented or the required parameters are missing.
Experiments on the material behaviour are commonly realized for homogeneous
stress-strain states, as for example the uniaxial tensile and compression test or the thinwalled tube under combined torsion, tensile and internal pressure loading. In addition
to these well-known methods, experimental studies of inhomogeneous strain and stress
fields are an interesting alternative to identify material parameters.
Two types of specimens have been investigated. Unnotched bending specimens

have been used to determine the elastic constants, the initial yield locus curve and the
uniaxial tension and compression yield curves. Notched bending specimens allow experiments on the hardening behaviour due to inhomogeneous stress-strain states.
The numerical analysis has been carried out by the integration of the deformation
law at a certain number of comparative points of the ligament with strain increments,
determined from Moire´ fringe patterns, as loads. The identification of material parameters has been performed by the minimization of a least-squares functional using deterministic gradient-type methods. As comparative quantities have been taken into account
the bending moment, the normal force and the stresses at the notch grooves.

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Project area B: development of computational methods
The essential goals of the project area B were the transfer of experimental results in
material models, which describe the essential characteristics of the complex non-linear
behaviour of metallic materials in a technically satisfactory manner. For this reason,
known formulations of material models, developments of the SFB and new formulations had to be examined with respect to their validity and the limits of their efficiency.
To be able to describe processes on the microscale of the materials, the material models
contain internal variables, which can either be purely phenomenological or be based on
microstructural considerations. In the frame of the SFB, the goal was the microstructural substantiation of these internal variables.
For the adjustment of the model parameters on the experimental results, optimization strategies are necessary, which allow judging the power of the models. The obtained results showed that this question is of high importance, also for further research.
Extensions for multiaxial loading cases have been developed and validated. For the investigated loadings of metals at high temperatures and alternating and cyclic loading
histories as well as for significantly time-dependent material behaviour, the literature
shows only a first beginning in the research concerning such extensions.
The material models had firstly to be examined concerning the materials. For the
practical application, however, their suitability for their implementation in numerical algorithms (e. g. Finite-Element methods) and the influence on the efficiency of numerical computations had to be examined.
Especially for the computation of time-dependent processes, numerically stable
and – because of the expensive numerical calculations – efficient computational algorithms had to be developed (e. g. fast converging time-integration methods for strongly

non-linear problems).
The developed (or chosen) material models and algorithms had to be applied for
larger structures, not only to test the computational models, but simultaneously also –
by reflection to the assumptions in the material models – to find out which parameters
are of essential meaning for the practical application, and which are rather unimportant
and can be neglected. This results in the necessity to perform on all levels sensitivity
investigations for the relevancy of the variants of the assumptions and their parameters.
At loading histories, which describe alternating or cyclic processes due to the alternating plastification, the question of saturation of the stress-strain histories and
shakedown are of special importance. The projects in the project area B were investigating these problems in a complementary manner. They were important, central questions conceived so that related problems were investigated to accelerate the progress of
the work and to allow mutual support and critical exchange of thought.
Development and improvement of unified models and applications to structural analysis
(Ahrens, Duddeck, Kowalsky, Pensky, Streilein)
Especially for structures of large damage potentials, the design has to simulate failure
conditions as realistic as possible. Therefore, inelastic and time-dependent behaviour
such as temperature-induced creep have to be considered. Besides adequate numerical
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methods of analyses (as non-linear Finite-Element methods), mathematically correct
models are needed for the thermal-mechanical material behaviour under complex loadings. Unified models for metallic materials cover time-independent as well as time-dependent reactions by a unified concept of elasto-viscoplasticity.
Research results are presented, which demonstrate further developments for unified models in three different aspects. The methodical approach is shown firstly on the
level of the material model. Then, verifications of their applicability are given by utilizing them in the analyses of structures. The three aspects are the following problems:
1. Discrepancies between results of experimental and numerical material behaviour
may be caused by
• insufficient or inaccurate parameters of the material model,
• inadequate material functions of the unified models,
• insufficient basic formulations for the physical properties covered by the model.

It is shown that more consistent formulations can be achieved for all these three
sources of deficits by systematic numerical investigations.
2. Most of the models for metallic materials assume yield functions of the v. Mises
type. For hardening, isotropic and/or kinematic evolutions are developed, that correspond to affine expansions or simple shifting of the original yield surface, whereas
experimental results show a distinctive change of the shape of the yield surfaces
(rotated or dented) depending on the load path. To cover this material behaviour of
distorted yield surfaces, a hierarchical expansion of the hardening rule is proposed.
The evolutionary equations of the hardening (expressed in tensors) are extended by
including higher order terms of the tensorial expressions.
3. Even very accurately repeated tests of the same charge of a metallic material show a
certain scattering distribution of the experimental results. The investigation of test
series (provided by other projects of the SFB) proved that a normal Gaussian distribution can be assumed. A systematic approach is proposed to deal with such experimental deviations in evaluating the parameters of the material model.
The concepts in all of the three items are valid in general although the overstress
model by Chaboche and Rousselier is chosen here for convenience.
In verifying the conceptual improvements, it is necessary to provide accurate and
efficient procedures for time-integration processes and for the evaluation of the model
parameters via optimization. In both cases, different procedures are elaborately compared with each other.
Results of the numerical analyses of different structures are given. They demonstrate the efficiency of the proposed further developments by applying Finite-Element
methods for non-linear stress-displacement problems. This includes:
•
•

investigations of welded joints with modifications of the layers of different microstructures,
thick-walled vessels in order to demonstrate the effects of different formulations of
the material model on the stress-deformation fields of larger structures,
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•
•
•

distorted yield functions to a plate with an opening,
effects of stochastic distribution of material behaviour to a plate with openings,
the application of material models based on microphysical mechanisms to a larger
vessel, the recipient of hot aluminium blocks for a profile extrusion press.

On the behaviour of mild steel Fe 510 under complex cyclic loading (Peil, Scheer,
Scheibe, Reininghaus, Kuck, Dannemeyer)
The employment of the plastic bearing capacity of structures has been recently allowed
in both national and international steel constructions standards. The ductile material behaviour of mild steel allows a load-increase well over the elastic limit. To make use of
this effect, efficient algorithms, taking account of the plastic behaviour under cyclic or
random loads in particular, are an important prerequisite for a precise calculation of the
structure.
The basic elements of a time-independent material model, which allows to take
into account the biaxial or random load history for a mild steel under room temperature, are presented. In a first step, the material response under cyclic or random loads
has to be determined. The fundamentals of an extended-two-surface model based on
the two-surface model of Dafalias and Popov are presented. The adaptations have been
made in accordance with the results of experiments under multiaxial cyclic loadings.
Finally, tests on structural components are performed to verify the results obtained
from the calculations with the described model.

Theoretical and computational shakedown analysis of non-linear kinematic hardening
material and transition to ductile fracture (Stein, Zhang, Huang, Mahnken, Wiechmann)
The response of an elastic-plastic system subjected to variable loadings can be very
complicated. If the applied loads are small enough, the system will remain elastic for
all possible loads. Whereas if the ultimate load of the system is attained, a collapse

mechanism will develop and the system will fail due to infinitely growing displacements. Besides this, there are three different steady states, that can be reached while the
loading proceeds:
1. Incremental failure occurs if at some points or parts of the system, the remaining
displacements and strains accumulate during a change of loading. The system will
fail due to the fact that the initial geometry is lost.
2. Alternating plasticity occurs, this means that the sign of the increment of the plastic
deformation during one load cycle is changing alternately. Though the remaining
displacements are bounded, plastification will not cease, and the system fails locally.
3. Elastic shakedown occurs if after initial yielding plastification subsides, and the system behaves elastically due to the fact that a stationary residual stress field is
formed, and the total dissipated energy becomes stationary. Elastic shakedown (or
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simply shakedown) of a system is regarded as a safe state. It is important to know
whether a system under given variable loadings shakes down or not.
The research work is based on Melan’s static shakedown theorems for perfectly plastic
and linear kinematic hardening materials, and is extended to generally non-linear limited
hardening by a so-called overlay model, being the 3-D generalization of Neal’s 1-D model, for which a theorem and a corollary are derived. Finite-Element method and adequate
optimization algorithms are used for numerical approach of 2-D problems. A new lemma
allows for the distinction between local and global failure. Some numerical examples illustrate the theoretical results. The shakedown behaviour of a cracked ductile body is investigated, where a crack is treated as a sharp notch. Thresholds for no crack propagation
are formulated based on shakedown theory.

Parameter identification for inelastic constitutive equations based on uniform and nonuniform stress and strain distributions (Mahnken, Stein)
In this project, various aspects for identification of parameters are discussed. Firstly, as
in classical strategies, a least-squares functional is minimized using data of specimen
with stresses and strains assumed to be uniform within the whole volume of the sample. Furthermore, in order to account for possible non-uniformness of stress and strain
distributions, identification is performed with the Finite-Element method, where also

the geometrically non-linear case is taken into account. In both approaches, gradientbased optimization strategies are applied, where the associated sensitivity analysis is
performed in a systematic manner. Numerical examples for the uniform case are presented with a material model due to Chaboche with cyclic loading. For the non-uniform case, material parameters are obtained for a multiplicative plasticity model, where
experimental data are determined with a grating method for an axisymmetric necking
problem. In both examples, the results are discussed when different starting values are
used and stochastic perturbations of the experimental data are applied.

Project area C: experimental verification
Material parameters, which describe the inelastic behaviour of metallic materials, can
be determined experimentally from the deformation of a test specimen by suitable chosen basic experiments. One-dimensional load-displacement measurements, however, are
not providing sufficient informations to identify parameters of three-dimensional
material laws. For this purpose, the complete whole-field deformation respectively
strain state of the considered object surface is needed. It can be measured by optical
methods. They yield the displacement distribution in three dimensions and the strain
components in two dimensions. So, these methods make possible an extensive comparison of the results of a related Finite-Element computation.
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