335 15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour of Shear-Loaded Aluminium Panels.. The increasing necessity to obtain experimental results ofhigh quali
Trang 1Final Report of the
Collaborative Research Centre 319,
Stoffgesetze fÏr das inelastischeVerhalten metallischer
Werkstoffe – Entwicklung und technischeAnwendung
1985–1996
Edited by
Elmar Steck, Reinhold Ritter, Udo Pfeil and Alf Ziegenbein
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)
Trang 2E-Mail: (Internet RFC 822): postmaster@dfg.de
Internet: http://www.dfg.de
This book was carefully produced Nevertheless, editors, authors and publisher do not warrant the information contained therein to be free of errors Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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Trang 3Preface XV
of Face-Centred Cubic Metals, Cold-Worked and Softened
to Different States 1
Lothar Kaps, Frank Haeßner 1.1 Introduction 1
1.2 Experiments 1
1.3 Simulation 11
1.4 Summary 14
References 15
2 Material State after Uni- and Biaxial Cyclic Deformation 17
Walter Gieseke, K Roger Hillert, Gu¨nter Lange 2.1 Introduction 17
2.2 Experiments and Measurement Methods 18
2.3 Results 19
2.3.1 Cyclic stress-strain behaviour 19
2.3.2 Dislocation structures 24
2.3.3 Yield surfaces 28
2.3.3.1 Yield surfaces on AlMg3 28
2.3.3.2 Yield surfaces on copper 30
2.3.3.3 Yield surfaces on steel 30
2.4 Sequence Effects 31
2.5 Summary 34
Acknowledgements 35
References 35
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)
Trang 43.2 Experimental Details 38
3.2.1 Experimental details for room-temperature tests 38
3.2.2 Experimental details for high-temperature tests 39
3.3 Tests at Room Temperature: Description of the Deformation Behaviour 40
3.3.1 Macroscopic test results 40
3.3.2 Microstructural results and interpretation 43
3.3.3 Phenomenological description of the deformation behaviour 45
3.3.3.1 Description of cyclic hardening curve, cyclic stress-strain curve and hysteresis-loop 45
3.3.3.2 Description of various hysteresis-loops with few constants 47
3.3.4 Physically based description of deformation behaviour 47
3.3.4.1 Internal stress measurement and cyclic proportional limit 47
3.3.4.2 Description of cyclic plasticity with the models of Steck and Hatanaka 50
3.3.5 Application in the field of fatigue-fracture mechanics 51
3.4 Creep-Fatigue Interaction 53
3.4.1 A physically based model for predicting LCF-life under creep-fatigue interaction 53
3.4.1.1 The original model 53
3.4.1.2 Modifications of the model 54
3.4.1.3 Experimental verification of the physical assumptions 55
3.4.1.4 Life prediction 55
3.4.2 Computer simulation and experimental verification of cavity formation and growth during creep-fatigue 57
3.4.2.1 Stereometric metallography 57
3.4.2.2 Computer simulation 58
3.4.2.3 Results 59
3.4.3 In-situ measurement of local strain at the crack tip during creep-fatigue 61
3.4.3.1 Influence of the crack length and the strain amplitude on the local strain distribution 61
3.4.3.2 Comparison of the strain field in tension and compression 62
3.4.3.3 Influence of the hold time in tension on the strain field 63
3.5 Summary and Conclusions 64
References 65
Trang 54 Development and Application of Constitutive Models
for the Plasticity of Metals 68
Elmar Steck, Frank Thielecke, Malte Lewerenz Abstract 68
4.1 Introduction 68
4.2 Mechanisms on the Microscale 69
4.3 Simulation of the Development of Dislocation Structures 71
4.4 Stochastic Constitutive Model 73
4.5 Material-Parameter Identification 77
4.5.1 Characteristics of the inverse problem 77
4.5.2 Multiple-shooting methods 77
4.5.3 Hybrid optimization of costfunction 77
4.5.4 Statistical analysis of estimates and experimental design 79
4.5.5 Parallelization and coupling with Finite-Element analysis 79
4.5.6 Comparison of experiments and simulations 81
4.5.7 Consideration of experimental scattering 82
4.6 Finite-Element Simulation 83
4.6.1 Implementation and numerical treatment of the model equations 83
4.6.1.1 Transformation of the tensor-valued equations 84
4.6.1.2 Numerical integration of the differential equations 85
4.6.1.3 Approximation of the tangent modulus 86
4.6.2 Deformation behaviour of a notched specimen 86
4.7 Conclusions 88
References 88
5 On the Physical Parameters Governing the Flow Stress of Solid Solutions in a Wide Range of Temperatures 90
Christoph Schwink, Ansgar Nortmann Abstract 90
5.1 Introduction 90
5.2 Solid Solution Strengthening 92
5.2.1 The critical resolved shear stress,so 92
5.2.2 The hardening shear stress,sd 92
5.3 Dynamic Strain Ageing (DSA) 93
5.3.1 Basic concepts 93
5.3.2 Complete maps of stability boundaries 94
5.3.3 Analysis of the processes inducing DSA 97
5.3.4 Discussion 99
5.4 Summary and Relevance for the Collaborative Research Centre 102
References 102
Contents
Trang 66.3.1 Development of single slip bands 106
6.3.2 Development of slip band bundles and Lu¨ders band propagation 112
6.3.3 Comparison of single crystals and polycrystals 116
6.3.4 Conclusion 117
6.4 Deformation Processes at Intermediate Temperatures 118
6.4.1 Analysis of single stress serrations 118
6.4.2 Analysis of stress-time series 121
6.4.3 Conclusion 124
6.5 Deformation Processes at Elevated Temperatures 124
6.5.1 Dynamical testing and stress relaxation 124
6.5.2 Creep experiments 126
6.5.3 Conclusion 128
Acknowledgements 128
References 129
7 The Influence of Large Torsional Prestrain on the Texture Development and Yield Surfaces of Polycrystals 131
Dieter Besdo, Norbert Wellerdick-Wojtasik 7.1 Introduction 131
7.2 The Model of Microscopic Structures 131
7.2.1 The scale of observation 131
7.2.2 Basic slip mechanism in single crystals 132
7.2.3 Treatment of polycrystals 133
7.2.4 The Taylor theory in an appropriate version 133
7.3 Initial Orientation Distributions 135
7.3.1 Criteria of isotropy 135
7.3.2 Strategies for isotropic distributions 136
7.4 Numerical Calculation of Yield Surfaces 137
7.5 Experimental Investigations 140
7.5.1 Prestraining of the specimens 140
7.5.2 Yield-surface measurement 141
7.5.3 Tensile test of a prestrained specimen 142
7.5.4 Measured yield surfaces 143
7.5.5 Discussion of the results 146
7.6 Conclusion 146
References 147
Trang 78 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 Introduction 149
8.2 General Procedure 149
8.3 The Deformation Law of Inelastic Solids 150
8.4 Bending of Rectangular Beams 152
8.4.1 Principle 152
8.4.2 Experimental technique 152
8.4.3 Evaluation 155
8.4.3.1 Determination of the yield curves 155
8.4.3.2 Determination of the initial yield-locus curve 158
8.5 Bending of Notched Beams 160
8.5.1 Principle 160
8.5.2 Experimental technique 161
8.5.3 Approximation of displacement fields 163
8.6 Identification of Material Parameters 165
8.6.1 Integration of the deformation law 165
8.6.2 Objective function, sensitivity analysis and optimization 167
8.6.3 Results of parameter identification 169
8.7 Conclusions 170
Acknowledgements 172
References 173
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 Introduction 174
9.2 On Unified Models for Metallic Materials 174
9.2.1 The overstress model by Chaboche and Rousselier 175
9.2.2 Other unified models 177
9.3 Time-Integration Methods 178
9.4 Adaptation of Model Parameters to Experimental Results 181
9.5 Systematic Approach to Improve Material Models 186
9.6 Models Employing Distorted Yield Surfaces 190
9.7 Approach to Cover Stochastic Test Results 197
9.8 Structural Analyses 201
9.8.1 Consistent formulation of the coupled boundary and initial value problem 202
9.8.2 Analysis of stress-strain fields in welded joints 203
9.8.3 Thick-walled rotational vessel under inner pressure 205
Contents
Trang 810 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 Introduction 218
10.2 Material Behaviour 219
10.2.1 Material, experimental set-ups, and techniques 219
10.2.2 Material behaviour under uniaxial cyclic loading 219
10.2.2.1 Parameters 219
10.2.2.2 Results of the uniaxial experiments 220
10.2.3 Material behaviour under biaxial cyclic loading 225
10.2.3.1 Parameters 225
10.2.3.2 Relations of tensile and torsional stresses 226
10.2.3.3 Yield-surface investigations 229
10.3 Modelling of the Material Behaviour of Mild Steel Fe 510 236
10.3.1 Extended-two-surface model 236
10.3.1.1 General description 236
10.3.1.2 Loading and bounding surface 237
10.3.1.3 Strain-memory surfaces 238
10.3.1.4 Internal variables for the description on non-proportional loading 241
10.3.1.5 Size of the yield surface under uniaxial cyclic plastic loading 242
10.3.1.6 Size of the bounding surface under uniaxial cyclic plastic loading 242
10.3.1.7 Overshooting 242
10.3.1.8 Additional update ofdinin the case of biaxial loading 243
10.3.1.9 Memory surface F ' 243
10.3.1.10 Additional isotropic deformation on the loading surface due to non-proportional loading 244
10.3.1.11 Additional isotropic deformation of the bounding surface due to non-proportional loading 244
10.3.2 Comparison between theory and experiments 248
10.4 Experiments on Structural Components 248
10.4.1 Experimental set-ups and computational method 248
10.4.2 Correlation between experimental and theoretical results 248
10.5 Summary 251
References 252
Trang 911 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 253
11.1 Introduction 253
11.1.1 General research topics 253
11.1.2 State of the art at the beginning of project B6 254
11.1.3 Aims and scope of project B6 254
11.2 Review of the 3-D Overlay Model 256
11.3 Numerical Approach to Shakedown Problems 259
11.3.1 General considerations 259
11.3.2 Perfectly plastic material 260
11.3.2.1 The special SQP-algorithm 260
11.3.2.2 A reduced basis technique 261
11.3.3 Unlimited kinematic hardening material 261
11.3.4 Limited kinematic hardening material 263
11.3.5 Numerical examples 264
11.3.5.1 Thin-walled cylindrical shell 264
11.3.5.2 Steel girder with a cope 265
11.3.5.3 Incremental computations of shakedown limits of cyclic kinematic hardening material 267
11.4 Transition to Ductile Fracture 269
11.5 Summary of the Main Results of Project B6 272
References 273
12 Parameter Identification for Inelastic Constitutive Equations Based on Uniform and Non-Uniform Stress and Strain Distributions 275
Rolf Mahnken, Erwin Stein Abstract 275
12.1 Introduction 275
12.1.1 State of the art at the beginning of project B8 275
12.1.2 Aims and scope of project B8 276
12.2 Basic Terminology for Identification Problems 277
12.2.1 The direct problem: the state equation 277
12.2.2 The inverse problem: the least-squares problem 278
12.3 Parameter Identification for the Uniform Case 280
12.3.1 Mathematical modelling of uniaxial visco-plastic problems 280
12.3.2 Numerical solution of the direct problem 282
12.3.3 Numerical solution of the inverse problem 282
12.4 Parameter Identification for the Non-Uniform Case 283
12.4.1 Kinematics 284
Contents
Trang 1013 Experimental Determination of Deformation- and Strain Fields
by Optical Measuring Methods 298
Reinhold Ritter, Harald Friebe 13.1 Introduction 298
13.2 Requirements of the Measuring Methods 298
13.3 Characteristics of the Optical Field-Measuring Methods 299
13.4 Object-Grating Method 300
13.4.1 Principle 300
13.4.2 Marking 301
13.4.3 Deformation analysis at high temperatures 302
13.4.4 Compensation of virtual deformation 303
13.4.5 3-D deformation measuring 305
13.4.6 Specifications of the object-grating method 305
13.5 Speckle Interferometry 305
13.5.1 General 305
13.5.2 Technology of the Speckle interferometry 307
13.5.3 Specifications of the developed 3-D Speckle interferometer 308
13.6 Application Examples 309
13.6.1 2-D object-grating method in the high-temperature area 309
13.6.2 3-D object-grating method in fracture mechanics 309
13.6.3 Speckle interferometry in welding 310
13.7 Summary 313
References 317
14 Surface-Deformation Fields from Grating Pictures Using Image Processing and Photogrammetry 318
Klaus Andresen 14.1 Introduction 318
14.2 Grating Coordinates 319
14.2.1 Cross-correlation method 319
14.2.2 Line-following filter 321
14.3 3-D Coordinates by Imaging Functions 324
14.4 3-D Coordinates by Close-Range Photogrammetry 325
14.4.1 Experimental set-up 325
Trang 1114.4.2 Parameters of the camera orientation 326
14.4.3 3-D object coordinates 327
14.5 Displacement and Strain from an Object Grating: Plane Deformation 328 14.6 Strain for Large Spatial Deformation 329
14.6.1 Theory 329
14.6.2 Correcting the influence of curvature 332
14.6.3 Simulation and numerical errors 333
14.7 Conclusion 335
References 335
15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour of Shear-Loaded Aluminium Panels 337
Horst Kossira, Gunnar Arnst 15.1 Introduction 337
15.2 Numerical Model 339
15.2.1 Finite-Element method 339
15.2.1.1 Ambient temperature – rate-independent problem 340
15.2.1.2 Elevated temperature – visco-plastic problem 341
15.2.2 Material models 341
15.2.2.1 Ambient temperature – rate-independent problem 341
15.2.2.2 Elevated temperature – visco-plastic problem 344
15.3 Experimental and Numerical Results 349
15.3.1 Test procedure 349
15.3.2 Computational analysis 349
15.3.2.1 Monotonic loading – ambient temperature 350
15.3.2.2 Cyclic loading – ambient temperature 351
15.3.2.3 Time-dependent behaviour 356
15.4 Conclusion 358
List of Symbols 359
References 360
16 Consideration of Inhomogeneities in the Application of Deformation Models, Describing the Inelastic Behaviour of Welded Joints 361
Helmut Wohlfahrt, Dirk Brinkmann 16.1 Introduction 361
16.2 Materials and Numerical Methods 362
16.2.1 Materials and welded joints 362
16.2.2 Deformation models and numerical methods 365
16.2.2.1 Deformation model of Gerdes 365
16.2.2.2 Fitting calculations 365
16.3 Investigations with Homogeneous Structures 365
Contents
Trang 1216.4.1.1 Experimental investigations 375
16.4.1.2 Numerical investigations 375
16.4.1.3 Finite-Element models of welded joints 375
16.4.1.4 Calculation of the deformation behaviour of welded joints 375
16.4.2 Strain distributions of welded joints with broad weld seams 376
16.4.3 Strain distributions of welded joints with small weld seams 380
16.4.4 Discussion 380
16.5 Application Possibilities and Further Investigations 382
References 383
Bibliography 384
Trang 13The Collaborative Research Centre (Sonderforschungsbereich, SFB 319), “MaterialModels for the Inelastic Behaviour of Metallic Materials – Development and TechnicalApplication”, was supported by the Deutsche Forschungsgemeinschaft (DFG) from July
1985 until the end of the year 1996 During this period of nearly 12 years, scientistsfrom the disciplines of metal physics, materials sciences, mechanics and applied engi-neering sciences cooperated with the aim to develop models for metallic materials on aphysically secured basis The cooperation has resulted in a considerable improvement
of the understanding between the different disciplines, in many new theoretical and perimental methods and results, and in technically applicable constitutive models aswell as new knowledge concerning their application to practical engineering problems.The cooperation within the SFB was supported by many contacts to scientists andengineers 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
ex-to the Deutsche Forschungsgemeinschaft for the financial support and the very structive cooperation, and to all the colleagues who have contributed by their interestand their function as reviewers and advisors to the results of our research work
con-Introduction
The development of mathematical models for the behaviour of technical materials is ofcourse directed towards their application in the practical engineering work Besides theprojects, which have the technical application as their main goal, in all projects, whichwere involved in experiments with homogeneous or inhomogeneous test specimens –where partly also the numerical methods were further investigated and the implementa-tion of the material models in the programs was performed –, experiences concerningthe application of the models for practical problems could be gained The whole-fieldmethods for measuring displacement and strain fields, which were developed in con-nection 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 hasproven to be very actual The investigations for most efficient methods for the param-eter identification will in the future still find considerable attention, where the coopera-tion of scientists from engineering as well as applied mathematics, which was started inthe SFB, will continue As is shown in a later chapter, it is of increasing importance to
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)
Trang 14the engineering practice They have however the advantage that they are based on sults of material physics and therefore can use further developments of the knowledgeabout the mechanisms of inelastic deformations on the microscale.
re-During the work in the different projects, a surprising number of similar problemshave been found Due to the close contacts between the working groups, they could beinvestigated with much higher quality than without this cooperation
The exchange of thought between metal physics, materials sciences, mechanicsand applied engineering sciences was very stimulating and has resulted in the fact thatthe groups oriented towards application could be supported by the projects workingtheoretically, and on the other hand, the scientists working in theoretical fields couldobserve the application of their results in practical engineering
Research Program
The main results of the activities of the SFB have been models for the tion behaviour as well as for damage development and the development of deformationanisotropies These models make it possible to use results from the investigations frommetal physics and materials sciences in the SFB in the continuum mechanics models.The research work in metal physics and materials sciences has considerably contributed
load-deforma-to a qualitative understanding of the processes, which have load-deforma-to be described by tive models The structure of the developed models and of the formulations found inliterature, which have been considered for comparisons and supplementation of ourown development, have strongly influenced the work concerning the implementation ofthe material models in numerical computing methods and the treatment of technicalproblems The models could be developed to a status, where the results of experimentalinvestigations can be used to determine the model parameters quantitatively
constitu-This has resulted in an increasing activity on the experimental side of the workand 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 Ju¨lich, KFZ Karlsruhe, E´ cole Polytechnique Lausanne) In the SFB, joint researchwas undertaken in the fields of high-temperature experiments for the investigation ofcreep, cyclic loading and non-homogeneous stress and displacement fields for technicalimportant metallic materials, and their comparison with theoretical predictions The de-veloped whole-field methods for measuring deformations have shown to be an impor-
Trang 15tant experimental method The increasing necessity to obtain experimental results ofhigh quality for testing and extending the material models has resulted in the develop-ment of experimental equipment, which also allows to investigate the material behav-iour under multiaxial loadings in the high- and low-temperature range.
The determination of model parameters and process quantities from experimentshas put the question for reliable methods for the parameter identification in the fore-ground The earlier used methods of least-squares and probabilistic methods, such asthe evolution strategy, have given satisfying results In the SFB, however, the know-ledge has developed that methods for the parameter identification, which consider thestructure of the material models and the design of optimal experiments and discriminat-ing experiments, deserve special consideration
If numerical values for the model parameters are given, the possibility exists toexamine these values concerning their physical meaning, and in cooperation with thescientists from metal physics and materials sciences to investigate the connection be-tween the knowledge about the processes on the microscale and the macroscopic con-stitutive 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,
Project area A: materials behaviour
The research in the project area A was mainly concerned with theoretical and mental investigations concerning the basis for the development of material models anddamage development from metal physics and materials sciences In the following, ashort description of the activities within the research projects is given Methods and re-sults are in detail given in later chapters
experi-Research Program
Trang 16ined 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 amaterial is cold-worked, its yield stress increases This process, denoted strain harden-ing, leads to a gain in internal energy Recovery and recrystallization act to opposestrain hardening Already upon deformation or during subsequent annealing, theseforces transform the material back into a state of lower energy Although this reciproci-
ty has been known for some time, the exact dependence of the process upon the typeand extent of deformation, upon the temperatures during deformation and softening an-neal as well as upon the chemical composition of the material is as yet only qualita-tively known Consequently, the predictability of the processes is as poor as it has al-ways been so that, even today, one is still obliged to refer to experience and explicitexperiments for help
Material state after uni- and biaxial cyclic deformation (Gieseke, Hillert, Lange)The investigations concerning the material behaviour at multiaxial plastic deformationwere performed using the material AlMg3, copper and the austenitic stainless steel AISI316L To find the connection between damage development and microstructure, the dis-locations structure at the tip of small cracks and at surface grains with differently pro-nounced slip-band development was investigated With the aim to check the main as-sumptions of the two-surface models explicitly, measurements of the development ofthe yield surface of the material from the initial to the saturation state and within a sat-uration cycle were considerably extended Consecutive yield surfaces along differentloading histories were measured The two-surface models of Ellyin and McDowellwere implemented in the computations
Technical components and structures today are increasingly being designed anddisplayed by computer-aided methods High speed computers permit the use of mathe-matical models able to numerically reconstruct material behaviour, even in the course
of complex loading procedures
In phenomenological continuum mechanics, the cyclic hardening and softeningbehaviour as well as the Bauschinger effect are described by yield-surface models If aphysical 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 tion Two-surface models, going back to a development by Dafalias and Popov, de-scribe the displacement of the elastic deformation zone in a dual axis stress area Theyield surfaces are assumed to be v Mises shaped ellipses However, from experimentswith uniaxial loading, it is known that the yield surfaces of small offset strains under
Trang 17deforma-load become characteristically deformed In the present subproject, the effect of cyclicdeformation on the shape and position of the yield surfaces is studied, and their rela-tion to the dislocation structure is determined To this end, the yield surfaces of threematerials with different slip behaviour were measured after prior uni- or biaxial defor-mation The influence of the dislocation structures produced and the effect of internalstresses are discussed.
Plasticity of metals and life prediction in the range of low-cycle fatigue: description ofdeformation 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 wasthe main aim of the work Measuring the change of the physical magnitudes in themodel during an experiment results in an investigation and eventually a modification ofthe model assumptions The model was also examined for its usability for experimentswith holding-times at the maximum pressure loading during a loading cycle
For hot working tools, chemical plants, power plants, pressure vessels and bines, one has to consider local plastic deformation at critical locations of structuralcomponents Due to cyclic changes of temperature and load, the components are sub-jected 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 cyc-lic 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 ofcracks, which can subsequently grow until failure of a component part takes place
tur-In the field of fatigue fracture mechanics, crack growth is correlated with eters, which take into account information especially about the steady-state stress-strainhysteresis-loops Therefore, it can be expected that a more exact life prediction is possi-ble by a detailed investigation of the cyclic deformation behaviour and by the descrip-tion of the cyclic plasticity, e g with constitutive equations
param-At high temperatures, creep deformation and creep damage are often posed on the fatigue process Therefore, in many cases, not one type of damage pre-vails, but the interaction of both fatigue and creep occurs, leading to failure of compo-nents
superim-The typical damage in the low-cycle fatigue regime is the development andgrowth 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 Areliable life prediction model must consider this interaction
The knowledge and description of the cavity formation and growth by means ofconstitutive equations are the basis for reliable life prediction In the case of diffusion-con-trolled cavity growth, the distance between the voids has an important influence on theirgrowth This occurs especially in the case of low-cycle fatigue, where the cavity formationplays an important role Thus, the stochastic process of void nucleation on grain bound-aries and the cyclic dependence of this process has to be taken into consideration as atheoretical description The experimental analysis has to detect the cavity-size distribu-tion, which is a consequence of the complex interaction between the cavities
Research Program
Trang 18Development 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 byslip processes in the crystal lattice, which are supported by the movement of lattice de-fects like dislocations and dislocation packages The dislocation movements are op-posed by internal barriers, which have to be overcome by activation This is performed
by stresses or thermal energy During the inelastic deformation, the dislocations interactand arrange in a hierarchy of structures such as walls, adders and cells This forming ofinternal material structures influences strongly the macroscopic responses on mechani-cal and thermal loading
A combination of models on the basis of molecular dynamics and cellular mata is used to study numerically the forming of dislocation patterns and the evolution
auto-of internal stresses during the deformation processes For a realistic simulation, severalglide planes are considered, and for the calculation of the forces acting on a disloca-tion, a special extended neighbourhood is necessary The study of the self-organizationprocesses with the developed simulation tool can result in valuable information for thechoice of formulations for the modelling of processes on the microscale
The investigations concerning the development of material models based onmechanisms on the microscale have resulted in a unified stochastic model, which isable to represent essential and typical features of the low- and high-temperature plastic-ity For the modelling of the dislocation movements in crystalline materials and theirtemperature and stress activation, a discrete Markov chain is considered In order to de-scribe cyclic material behaviour, the widely accepted concept is used that the disloca-tion-gliding processes are driven by the effective stress as the difference between theapplied stress and the internal back stress The influence of effective stress and tem-perature on the inelastic deformations is considered by a metalphysically motivatedevolution equation A mean value formulation of this stochastic model leads to amacroscopic model consisting of non-linear ordinary differential equations The resultsshow that the stochastic theory is helpful to deduce the properties of the macroscopicconstitutive equations from findings on the microscale
Since the general form of the stochastic model must be adapted to the specialmaterial characteristics and the considered temperature regime, the identification of theunknown material parameters plays an important role for the application on numericalcalculations The determination of the unknown material parameters is based on a Max-imum-Likelihood output-error method comparing experimental data to the numerical si-mulations For the minimization of the costfunction, a hybrid optimization concept par-allelized with PVM is considered It couples stochastic search procedures and severalNewton-type methods A relative new approach for material parameter identification is
Trang 19the multiple shooting approach, which allows to make efficient use of additional ment- and apriori-information about the states This reduces the influence of bad initialparameters Since replicated experiments for the same laboratory conditions show a sig-nificant scattering, these uncertainties must be taken into account for the parameter iden-tification The reliability of the results can be tested with a statistical analysis.
measure-Several different materials, like aluminium, copper, stainless steel AISI 304 andAISI 316, have been studied For the analysis of structures, like a notched flat bar, theFinite-Element program ABAQUS is used in combination with the user material sub-routine UMAT The simulations are compared with experimental data from gratingmethods
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 line material have been performed The superposition of solution hardening and ordi-nary hardening has found special consideration Along the stress-strain curves, the lim-its between stable and unstable regions of deformation were investigated, and their de-pendencies on temperature, strain rate and solute concentration were determined In re-gions of stable deformation, a quantitative analysis of the processes of dynamic strainageing (“Reckalterung”) was performed The transition between regions of stable andunstable deformation was investigated and characterized
single-crystal-At sufficiently low temperatures, host and solute atoms remain on their latticesites The critical flow stress is governed by thermally activated dislocations glide (Ar-rhenius equation), which depends on an average activation enthalpyDG0, and an effec-
tive obstacle concentration cb The total flow stress is composed of the critical flowstress and a hardening stress, which increases with the dislocation density in the cellwalls
Detailed investigations on single crystals yielded expressions for the criticalresolved shear stress, s0 s0 DG0; cb; T; _e, and the hardening shear stress,
sd wGbq1=2
w Here, w is a constant, w 0:25 0:03, G the shear modulus, and
qw the dislocation density inside the cell walls The total shear stress results as
s s0 sd
At higher temperatures, the solutes become mobile in the lattice and cause an ditional anchoring of the glide dislocations This is described by an additional enthalpy
ad-Dg tw; Ea m 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,
Dg reaches values up to about 0.1 DG0 Under certain conditions, the solute diffusioncauses instabilities in the flow stress, the well-known jerky flow phenomena (Portevin-
Le Chaˆtelier effect) Finally, above around 800 K in copper-based alloys, the solutes come freely mobile, and the critical flow stress as well as the additional enthalpy van-ish In any temperature region, only a small total number of physical parameters is suf-ficient for modelling plastic deformation processes
be-Research Program
Trang 20polation to extended parameter regions Especially, reasons and effects of inhomogeneities and -instabilities in the systems Cu-Al and Cu-Mn, which show ten-dencies to short-range order, were investigated Determining dislocation-generationrates and dislocation velocities in the case of gradients of the effective stress were aswell aim of the investigations as the influence of diffusion processes on the generation(blocking, break-away) and motion (obstacle destruction and regeneration) of disloca-tions Investigations were also performed concerning the use of the results for singlecrystals for the description of the practically more important case of the behaviour ofpolycrystals In this case, especially the influence of the grain-boundaries on generationand movement of dislocations or dislocation groups has to be considered.
deformation-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 theactive regions of the crystal Cu-based alloys with several percent of Al and Mn solutesare considered in order to separate the effects of stacking-fault energy from those of so-lute hardening and short-range ordering, which are comparable for both alloy systems,while the stacking-fault energy decreases rapidly with solute concentration for CuAlcontrary to CuMn alloys Both systems show different degrees of inhomogeneous slip
in the length scales from nm to mm (slip bands, Lu¨ders bands), and, in a certain range
of deformation conditions, macroscopic deformation instabilities (Portevin-Le Chaˆteliereffect) 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, sik)
Wellerdick-Wojta-This research project consists of a theoretical and an experimental part The topic ofthe theoretical part was the simulation of texture development and methods of calculat-ing yield surfaces The calculations started from an initially isotropic grain distribution.Therefore, it was necessary to set up such a distribution Different possibilities werecompared with an isotropy test considering the elastic and plastic properties With somefinal distributions, numerical calculations were carried out The Taylor theory in an ap-propriate 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 meth-ods or formulations, which take the elastic behaviour into account Several possibilitiesare presented, and the numerical calculations are compared with the experimental results
In order to measure yield surfaces after large torsional prestrain, thin-walled lar specimens of AlMg3 were loaded up to a shear strain of c 1:5, while torsional
Trang 21tubu-buckling was prevented by inserting a greased mandrel inside the specimens Furtherinvestigations of the prestrained specimens were done with the testing machine of theproject area B.
At least one yield surface, represented by 16 yield points, was measured witheach specimen The yield point is defined by the offset-strain definition, where gener-ally the von Mises equivalent offset strain is used Three different loading paths wererealized with the extension-controlled testing machine Thus, the results were yield sur-faces 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 theystrongly 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 measuredyield surfaces seem to be of the von Mises type By contrast, the prestrained specimensare very sensitive to different loading paths Especially the shape and the distorsion ofthe measured surfaces changes as a result of the small plastic strain during the measure-ment Therefore, it seems that the shape and the distortion of the yield surface were notstrongly effected by the texture of the material
Parameter identification of inelastic deformation laws analysing inhomogeneous strain states (Kreißig, Naumann, Benedix, Borman, Grewolls, Kretzschmar)
stress-In the last years, the necessity of solutions of non-linear solid mechanics problems haspermanently increased Although powerful hard- and software exist for such problems,often more or less large differences between numerical and experimental results are ob-served The dominant reason for these defects must be seen in the material-dependentpart of the used computer programs Either suitable deformation laws are not imple-mented or the required parameters are missing
Experiments on the material behaviour are commonly realized for homogeneousstress-strain states, as for example the uniaxial tensile and compression test or the thin-walled tube under combined torsion, tensile and internal pressure loading In addition
to these well-known methods, experimental studies of inhomogeneous strain and stressfields are an interesting alternative to identify material parameters
Two types of specimens have been investigated Unnotched bending specimenshave been used to determine the elastic constants, the initial yield locus curve and theuniaxial tension and compression yield curves Notched bending specimens allow ex-periments on the hardening behaviour due to inhomogeneous stress-strain states.The numerical analysis has been carried out by the integration of the deformationlaw at a certain number of comparative points of the ligament with strain increments,determined from Moire´ fringe patterns, as loads The identification of material param-eters has been performed by the minimization of a least-squares functional using deter-ministic gradient-type methods As comparative quantities have been taken into accountthe bending moment, the normal force and the stresses at the notch grooves
Research Program
Trang 22To be able to describe processes on the microscale of the materials, the material modelscontain internal variables, which can either be purely phenomenological or be based onmicrostructural considerations In the frame of the SFB, the goal was the microstructur-
al substantiation of these internal variables
For the adjustment of the model parameters on the experimental results, tion strategies are necessary, which allow judging the power of the models The ob-tained 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 in-vestigated loadings of metals at high temperatures and alternating and cyclic loadinghistories as well as for significantly time-dependent material behaviour, the literatureshows only a first beginning in the research concerning such extensions
optimiza-The material models had firstly to be examined concerning the materials For thepractical application, however, their suitability for their implementation in numerical al-gorithms (e g Finite-Element methods) and the influence on the efficiency of numeri-cal computations had to be examined
Especially for the computation of time-dependent processes, numerically stableand – because of the expensive numerical calculations – efficient computational algo-rithms had to be developed (e g fast converging time-integration methods for stronglynon-linear problems)
The developed (or chosen) material models and algorithms had to be applied forlarger structures, not only to test the computational models, but simultaneously also –
by reflection to the assumptions in the material models – to find out which parametersare of essential meaning for the practical application, and which are rather unimportantand can be neglected This results in the necessity to perform on all levels sensitivityinvestigations for the relevancy of the variants of the assumptions and their parameters
At loading histories, which describe alternating or cyclic processes due to the ternating plastification, the question of saturation of the stress-strain histories andshakedown are of special importance The projects in the project area B were investi-gating these problems in a complementary manner They were important, central ques-tions conceived so that related problems were investigated to accelerate the progress ofthe work and to allow mutual support and critical exchange of thought
al-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 failureconditions as realistic as possible Therefore, inelastic and time-dependent behavioursuch as temperature-induced creep have to be considered Besides adequate numerical
Trang 23methods of analyses (as non-linear Finite-Element methods), mathematically correctmodels are needed for the thermal-mechanical material behaviour under complex load-ings Unified models for metallic materials cover time-independent as well as time-de-pendent reactions by a unified concept of elasto-viscoplasticity.
Research results are presented, which demonstrate further developments for fied models in three different aspects The methodical approach is shown firstly on thelevel of the material model Then, verifications of their applicability are given by utiliz-ing them in the analyses of structures The three aspects are the following problems:
uni-1 Discrepancies between results of experimental and numerical material behaviourmay 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 threesources of deficits by systematic numerical investigations
2 Most of the models for metallic materials assume yield functions of the v Misestype For hardening, isotropic and/or kinematic evolutions are developed, that corre-spond to affine expansions or simple shifting of the original yield surface, whereasexperimental 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 ofdistorted yield surfaces, a hierarchical expansion of the hardening rule is proposed.The evolutionary equations of the hardening (expressed in tensors) are extended byincluding higher order terms of the tensorial expressions
3 Even very accurately repeated tests of the same charge of a metallic material show acertain scattering distribution of the experimental results The investigation of testseries (provided by other projects of the SFB) proved that a normal Gaussian distri-bution can be assumed A systematic approach is proposed to deal with such experi-mental deviations in evaluating the parameters of the material model
The concepts in all of the three items are valid in general although the overstressmodel by Chaboche and Rousselier is chosen here for convenience
In verifying the conceptual improvements, it is necessary to provide accurate andefficient procedures for time-integration processes and for the evaluation of the modelparameters via optimization In both cases, different procedures are elaborately com-pared with each other
Results of the numerical analyses of different structures are given They strate the efficiency of the proposed further developments by applying Finite-Elementmethods for non-linear stress-displacement problems This includes:
demon-• investigations of welded joints with modifications of the layers of different
micro-structures,
• thick-walled vessels in order to demonstrate the effects of different formulations of
the material model on the stress-deformation fields of larger structures,
Research Program
Trang 24Scheibe, 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 haviour of mild steel allows a load-increase well over the elastic limit To make use ofthis effect, efficient algorithms, taking account of the plastic behaviour under cyclic orrandom loads in particular, are an important prerequisite for a precise calculation of thestructure
be-The basic elements of a time-independent material model, which allows to takeinto account the biaxial or random load history for a mild steel under room tempera-ture, are presented In a first step, the material response under cyclic or random loadshas to be determined The fundamentals of an extended-two-surface model based onthe two-surface model of Dafalias and Popov are presented The adaptations have beenmade in accordance with the results of experiments under multiaxial cyclic loadings.Finally, tests on structural components are performed to verify the results obtainedfrom the calculations with the described model
Theoretical and computational shakedown analysis of non-linear kinematic hardeningmaterial and transition to ductile fracture (Stein, Zhang, Huang, Mahnken, Wiechmann)The response of an elastic-plastic system subjected to variable loadings can be verycomplicated If the applied loads are small enough, the system will remain elastic forall possible loads Whereas if the ultimate load of the system is attained, a collapsemechanism will develop and the system will fail due to infinitely growing displace-ments Besides this, there are three different steady states, that can be reached while theloading proceeds:
1 Incremental failure occurs if at some points or parts of the system, the remainingdisplacements and strains accumulate during a change of loading The system willfail due to the fact that the initial geometry is lost
2 Alternating plasticity occurs, this means that the sign of the increment of the plasticdeformation during one load cycle is changing alternately Though the remainingdisplacements are bounded, plastification will not cease, and the system fails locally
3 Elastic shakedown occurs if after initial yielding plastification subsides, and the tem behaves elastically due to the fact that a stationary residual stress field isformed, and the total dissipated energy becomes stationary Elastic shakedown (or
Trang 25sys-simply shakedown) of a system is regarded as a safe state It is important to knowwhether a system under given variable loadings shakes down or not.
The research work is based on Melan’s static shakedown theorems for perfectly plasticand linear kinematic hardening materials, and is extended to generally non-linear limitedhardening by a so-called overlay model, being the 3-D generalization of Neal’s 1-D mod-
el, for which a theorem and a corollary are derived Finite-Element method and adequateoptimization algorithms are used for numerical approach of 2-D problems A new lemmaallows for the distinction between local and global failure Some numerical examples il-lustrate the theoretical results The shakedown behaviour of a cracked ductile body is in-vestigated, where a crack is treated as a sharp notch Thresholds for no crack propagationare formulated based on shakedown theory
Parameter identification for inelastic constitutive equations based on uniform and uniform stress and strain distributions (Mahnken, Stein)
non-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 specimenwith stresses and strains assumed to be uniform within the whole volume of the sam-ple Furthermore, in order to account for possible non-uniformness of stress and straindistributions, identification is performed with the Finite-Element method, where alsothe geometrically non-linear case is taken into account In both approaches, gradient-based optimization strategies are applied, where the associated sensitivity analysis isperformed in a systematic manner Numerical examples for the uniform case are pre-sented with a material model due to Chaboche with cyclic loading For the non-uni-form case, material parameters are obtained for a multiplicative plasticity model, whereexperimental data are determined with a grating method for an axisymmetric neckingproblem In both examples, the results are discussed when different starting values areused 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 sen basic experiments One-dimensional load-displacement measurements, however, arenot providing sufficient informations to identify parameters of three-dimensionalmaterial laws For this purpose, the complete whole-field deformation respectivelystrain state of the considered object surface is needed It can be measured by opticalmethods They yield the displacement distribution in three dimensions and the straincomponents in two dimensions So, these methods make possible an extensive compari-son of the results of a related Finite-Element computation
cho-Research Program
Trang 26or stochastic grey value distribution, and recorded by the photogrammetric principle.Then, the strain follows from the difference of the displacement vectors of two neigh-bouring points related to two different deformation states of the object and related totheir initial distance.
The electronic Speckle interferometry is based on the Speckle effect It comesinto existence if an optical rough object surface is illuminated by coherent light, andthe scattered waves interfere By superposing of the interference effects of an objectand reference wave related to two different object states, the difference of the arisingSpeckle patterns leads to correlation fringes, which describe the displacement field ofthe considered object
Regarding the object-grating method, grating structures and their attachment havebeen developed, which can be analysed automatically and which are practicable also athigh temperatures up to 10008C, as often inelastic processes take place under this con-
dition Furthermore, the optical set-up, based on the photogrammetric principle, wasadapted to the short-range field with testing fields of only a few square millimeters.The object-grating method is applicable if the strain values are greater than 0.1%.For measurement of smaller strain values down to 10–5, the Speckle interfero-metric principle was applied A 3-D electronic Speckle interferometer has been devel-oped, which is so small that it can be adapted directly at a testing machine It is based
on the well-known path of rays of the Speckle interferometry including modern electronic components as laser diodes, piezo crystals and CCD-cameras
opto-Furthermore, both methods are suitable for high resolution of a large change ofmaterial behaviour Finally, the measurement can be conducted at the original and takesplace without contact and interaction
Surface deformation fields from grating pictures using image processing and grammetry (Andresen)
photo-The before-mentioned grating techniques are optical whole-field methods applied to rive the shape or the displacement and strain on the surface of an object A regulargrating fixed or projected on the surface is moved or deformed together with the ob-ject In different states, pictures are taken by film cameras or by electronic cameras.For plane surfaces parallel to the image plane, one camera supplies the necessary infor-mation for displacement and strain To get the spatial coordinates of curved surfaces,two or more stereocameras must be used In early times, the grating patterns were eval-uated manually by projecting the images to large screens or by use of microscope tech-niques Today, the pictures are usually digitized, yielding resolutions from 200×200 to
de-2000×2000 picture elements (pixels or pels) with generally 256 grey levels (8 bit) By
Trang 27suitable image-processing methods, the grating coordinates in the images are mined to a large extent automatically The corresponding coordinates on plane objectsare derived from the image coordinates by a perspective transformation Consideringspatial surfaces, first, the orientation of the cameras in space must be determined by acalibration procedure Then, the spatial coordinates are given by intersection of the rays
deter-of adjoined grating points in the images
The sequence of the grating coordinates in different states describes displacementand strain of the considered object surface Applying suitable interpolation gives contin-uous fields for the geometrical and physical quantities on the surface These experimen-tally determined fields are used for
• getting insight into two-dimensional deformation processes and effects,
• supplying experimental data to the theoretically working scientist,
• providing experimental data to be compared with Finite-Element methods,
• deriving parameters in standard constitutive laws,
• developing constitutive laws with new dependencies and parameters
Experimental and numerical analysis of the inelastic postbuckling behaviour of loaded aluminium panels (Kossira, Arnst)
shear-As a practical problem of aircraft engineering, the case of shear-loaded thin panels out
of the material AlCuMg2 under cyclic, quasistatic loading was investigated by mental and numerical methods Beyond the up-to-now used classical theory of plastici-
experi-ty, the theoretical research was based on the “unified” models, which were developedand adjusted to numerical computational methods in other areas of the research project.Shear-loaded panels are in general substructures of aerospace constructions sincethere are always load cases during a flight mission, in which shear loads are predomi-nant in the thin-walled structures of subsonic as well as in supersonic and hypersonicaircrafts The good-natured postcritical load-carrying behaviour of shear-loaded panels
at moderate plastic deformations can be exploited in emergency (fail safe) cases since
they exhibit no dramatic loss of stiffness even in the high plastic postbuckling regime.The temperature at the surface of hypersonic vehicles may reach very high values, butwith a thermal protection shield, the temperatures of the load-carrying structure can bereduced to moderate values, which allow the application of aluminium alloys There-fore, the properties of the mostly used aluminium alloy 2024-T3 are taken as a basisfor the experimental and theoretical studies of the behaviour of shear-loaded panels atroom temperature and at 2008C.
The primary aim of these studies is the understanding of the occurring ena, respectively the examination of the load-carrying behaviour of the consideredstructures under different load-time histories, and to provide suitable data for the de-sign Besides experimental investigations, which are achieved by a specially designedtest set-up, the development of numerical methods, which describe the phenomena, wasnecessary to accomplish this intention The used numerical model is based on a Finite-Element method, which is capable of calculating the geometric and physical non-linear– in case of visco-plastic material behaviour time-dependent – postbuckling behaviour
phenom-Research Program
Trang 28bient and elevated temperatures are presented The applied loads exceed the theoreticalbuckling loads by factors up to 40, accompanied by the occurrence of moderate inelas-tic deformations Apart from the numerical model, the monotonic loading, subsequentcreep rates, the snap-through behaviour at cyclic loading, the inelastic processes duringloading, and the influence of the aspect ratio are major topics in the presented discus-sion of the results for shear-loaded panels at room temperature and at 2008C.
Consideration of inhomogeneities in the application of deformation models, describingthe inelastic behaviour of welded joints (Wohlfahrt, Brinkmann)
A second practical problem was the investigation of the influence of welded joints onthe mechanical behaviour of components, which is due to the high degree of “Werk-stoffnutzung” in modern welded structures of high importance Special considerationwas given here to the important question of the material behaviour at cyclic loading aswell from the point of view of numerical computation of these processes and the con-nected effects as from the point of view of the problems connected with aspects ofmaterials sciences
The local loads and deformations in welded joints have rarely been investigatedunder the aspect that the mechanical behaviour is influenced by different kinds of mi-crostructure These different kinds of microstructure lead to multiaxial states of stressesand strains, and some investigations have shown that for the determination of the totalstate of deformation of a welded joint, the locally different deformation behaviour has
to be taken into account It is also published that different mechanical properties in theheat-affected zone as well as a weld metal with a lower strength than the base metalcan be the reason or the starting point of a fracture in welded joints A new investiga-tion demonstrates that in TIG-welded joints of the high strength steel StE690, a fine-grained area in the heat-affected zone with a lower strength than that of the base metal
is exclusively the starting zone of fracture under cyclic loading in the fully compressiverange These investigations support the approach described here that the mechanical be-haviour of the different kinds of microstructure in the heat-affected zone of weldedjoints has to be taken into account in the deformation analysis The influences of theseinhomogeneities on the local deformation behaviour of welded joints were determined
by experiments and numerical calculations over a wide range of temperature and ing The numerical deformation analysis was performed with ttformat
load-he method of Finite-Elements, in which recently developed deformation modelssimulate the mechanical behaviour of materials over the tested range of temperatureand loading conditions
Trang 291 Correlation between Energetic
and Mechanical Quantities of Face-Centred Cubic Metals, Cold-Worked and Softened to Different States
Lothar Kaps and Frank Haeßner *
Cold-worked metals soften at higher temperatures The details of this process depend
on the material as well as on the type and degree of deformation The kinetic ters can in principle be determined by calorimetric methods By combining calorimetri-cally determined values with characteristics measured mechanically and with micro-structural data, information can be gained about the strain-hardened state and the mech-anism of the softening process
parame-This materials information can support critical assessment of the structure ofmaterial models and hence be utilized for the appropriate adjustment of constitutivemodels to material properties
One objective of the work in this particular area of research was to investigate the pendence of the softening kinetics of face-centred cubic metals on the deformation Thechosen types of deformation were torsion, tension and rolling In the cases of torsionand tension, additional cyclic experiments with plastic amplitudes of 0.01 to 0.1 werecarried out The materials studied were aluminium, lead, nickel, copper and silver.Thus, in this order, metals of very high to very low stacking fault energy were investi-gated In the following presentation of the results, the emphasis will be on copper
de-To determine the mechanical data, the first step was to characterize the tion with the aid of the crystallographic slip a, the shear stress sN normalized to the
deforma-* Technische Universita¨t Braunschweig, Institut fu¨r Werkstoffe, Langer Kamp 8,
D-38106 Braunschweig, Germany
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)
Trang 30copper to extreme deformation The strain-hardening rate can be subdivided into threeregions, which, following the literature, may be denoted strain-hardening regions III to
V [3] Regions III and V show a linearly decreasing strain-hardening rate with shearstress Region IV, as region II, is characterized by constant strain hardening
The occurrence of these different regions depends strongly on the type of tion Thus, for tensile deformation, in consequence of instability, only deformation to re-gion III can be realized Rolling permits greater deformation, but brings with it the problem
deforma-of defining a specific measurement to categorize the strain-hardening regions The ture effect of the deformation fits well into the scheme proposed by Gil Sevillano [4].According to this scheme, all flow curves in region III may be described by a fixed initialstrain-hardening rateHIII
tempera-0 and a variable limiting stresssIII
S This latter is affected by namic recovery and is therefore dependent on deformation temperature and velocity Itdecreases for increasing deformation temperature and increases for higher deformationvelocities
dy-This statement is also true for the other characteristic stressessIV; sV; sV
S: The
loga-rithm of the characteristic stress decreases linearly with the normalized deformation
temperature, TN kT=Gb3: The normalization was proposed by Mecking et al [5] It
Figure 1.1: Flow curves of copper at temperatures of –208C to 1208C.
Trang 31has been successfully applied to our own measurements However, it may be seen thatthe dependence on temperature is different for the individual stresses (Figure 1.3).Careful evaluation of the experiments taking account of the effects of texture andsample shows similarities as well as differences between the two deformation types ten-
sion and torsion Up to a slip value of a = 0.4, the flow curve shows little difference
be-tween tension and torsion Above that value, the hardening is greater for the tension periment (Figure 1.4)
ex-The differences are more pronounced when the hardening rate is studied ratherthan the flow curve From the start, the former lies higher for tension than for torsion.The different procedures may be followed microstructurally using a transmission elec-
Trang 32tem-tron microscope Other authors have described this influence of the load path on themicrostructure [6–9] The reason for this may be that different average numbers of slipsystems are necessary for deformation [10] This also affects the development of activa-tion energies DG0 and activation volumes V To determine these quantities, velocities
are varied in tension and torsion experiments, i.e during a unidirectional experiment,the extension rate is momentarily increased In those sections with an increased exten-sion rate, the material shows a higher flow stress For the evaluation, the following an-satz was chosen for the relationship between the extension rate _e and the flow stress r:_e _e0 exp DG0 Vr
equa-The activation energies DG0 for torsion were determined from the characteristicstresses for different temperatures (cf Figure 1.3) The resultant values for the stresses
sIII
S; sIVandsV
S are 3.15, 2.79 and 2.79 eV/atom, respectively
To obtain the energy data, the stored energy ES of the plastic deformation was termined using a calorimeter As expected, the stored energy shows a monotonic in-
de-Figure 1.4: Comparison of flow curves from tension and torsion experiments.
Trang 33crease with deformation Moreover, dynamic recovery counteracts energy storage as itdoes hardening Hence, there is an unequivocal correlation between the deformationtemperature and stored energy such that an increasing deformation temperature leads toless stored energy (Figure 1.6).
Figure 1.6 demonstrates the great influence of the stacking fault energy The value
of the reduced stacking fault energy for silver lies at 2.4·10–3 compared with the value
of 4.7·10–3 for copper Lower stacking fault energies lead to a greater separation of
par-1.2 Experiments
Figure 1.5: Activation volume of copper deformed in tension and torsion at room temperature.
Figure 1.6: Stored energy versus shear strain for distorted copper and silver deformed at different temperatures.
Trang 34torsional deformation For the same shearing stress, silver also clearly stores more ergy than copper.
en-This relationship is represented in the Figures 1.7 and 1.8 Figure 1.7 comprisestorsion experiments up to extreme deformation Figure 1.8 shows a comparison of vari-ous types of deformation For better resolution, the abscissa here is confined to smalland intermediate values of stress The variable behaviour of the materials and the effect
of the types of deformation may also be demonstrated in measurements of the ing kinetics to be discussed In analogy to the strain-hardening rate, an energy storagerateHE dES=dsNhas been defined This quantity represents independent information.The development of the energy storage rate is clearly correlated with the strain-hardening stages (Figure 1.9) The combination of energetic and mechanical measure-ments permits a statement on the change in dislocation densityq, to a first approxima-
soften-tion proporsoften-tional to the stored energy, with increasing flow stress A linearly increasingenergy storage rate with stress leads to a law of the type:
en-Figure 1.7: Stored energy versus normalized shear stress for copper and silver deformed at ent temperatures.
Trang 35differ-with a different proportionality factora of value rather below the one pertaining to
re-gion III The factora may only be analytically assessed for deformation in the region
of the strain-hardening stage II For greater plastic deformation, which would then bedeformation in region III of the strain-hardening stage, this factor is of a qualitative na-ture The evolution of a for various materials, deformation temperatures and types of
deformation is collated in Table 1.1 The stress in the second column indicates the end
of the linear storage rate in the strain-hardening region III
Trang 36If X denotes the softened fraction of the material, one may attempt to describe the
softening kinetics _X by a product of functions, which combines the thermal activation
and the nature of the reaction in one appropriate multiplier:
vantage that, as a rate equation, it may be directly incorporated into a constitutive
equa-tion if the quantities Q and f X are known The simpler analysis considers the product
and in its place the reaction temperature This temperature is a direct measure of thestability of the deformed state
The thermal results show that for increasing stored energy, the softening processtakes place at lower temperatures An influence of the deformation temperature be-comes apparent Higher deformation temperatures promote easier reaction for the samestored energy Exact analysis of these facts shows that the form function makes only anegligible contribution here The effect is induced by a reduced activation energy.Different types of deformation show a stronger influence on the reaction tempera-ture than the deformation temperature At lower energies, distorted samples soften fast-
er than extended or rolled ones At higher energies, the reverse is true: Rolled samplesreact faster It is noticeable that cyclically deformed samples, for torsion as well as forpush-pull, do not diverge from the unidirectionally deformed samples of the same de-formation mode This is remarkable because, particularly for tension and push-pull de-formation, there are substantial differences in the activation energy
The activation energy describes the purely temperature dependence of the tion For small deformation and stored energies of distorted copper at a value of
reac-170 kJ/mol, it lies below the activation energy of volume self diffusion (200 kJ/mol).Unidirectionally extended samples show a higher activation energy (190 kJ/mol); push-pull deformed samples, on the other hand, show significantly lower activation energies(130 kJ/mol) With increasing energy, the activation energies of all deformation typesfall Figure 1.10 demonstrates these relationships
With the aid of torsional deformation, it is unequivocally proved that only uponreaching the strain-hardening stage V, one may presume constant activation energy At
Trang 37values of 80 to 90 kJ/mol, here for all deformation temperatures, the activation energylies in the region of grain boundary self diffusion or diffusion in dislocation cores Ten-sion and push-pull samples do not achieve these high stored energies; for these defor-mation modes, there is therefore no region of constant activation energy Elevated de-formation temperatures result in a lower softening activation energy One may interpretthis as strain hardening at higher temperature producing a microstructure that softensfaster This effect should be accounted for when setting up constitutive equations.There is a theory for the softening of deformed metals through the mechanism ofprimary recrystallization by Johnson and Mehl [13], Avrami [14–16] and Kolmogorov[17] In the following, this will be denoted the JMAK theory Comparison of the mea-sured activation energies with those predicted by the JMAK theory allow conclusions
to be drawn regarding the basic mechanisms of primary recrystallization
Accordingly, for high deformation continuous nucleation must be assumed,whereas for low deformation site, saturated nucleation is more probable Table 1.2shows the comparison in detail For high deformation, this interpretation complies withstudies according to the microstructural-path method [18] The grain spectra of weaklydeformed and recrystallized material show agreement with calculated spectra after site-saturated cluster nucleation
1.2 Experiments
Figure 1.10: Activation energy of differently deformed copper versus the stored energy.
Table 1.2: Effective activation energies from the JMAK theory compared with measured values for low/high deformation.
Site-saturated Continuous Measured values [kJ/mol]
nucleation nucleation
[kJ/mol] [kJ/mol] low deformation high deformation
Trang 38In reality, however, independent of the measurement method, one finds Avrami
exponents that decrease with X The thermal data show this particularly clearly As an
example, Figure 1.11 shows the curve of the Avrami exponent as a function of thetransformed fraction for distorted copper The horizontal reference line outlines thecurve for low degrees of deformation c 0:8 or 1:4, the central reference line applies
to intermediate degrees of deformation c 2:4 or 3:0:
Rolling and cyclic torsion act in the same way as unidirectional torsion if thestored energy is taken as the comparative measure instead of the strain-hardening re-gions Complementary studies using the transmission electron microscope show that themicrostructural details are similar for these deformations (cf Nix et al [9]) The defor-mation types unidirectional tension and push-pull are very different from torsion TheAvrami exponents are very large for unidirectional tension
In summary, the combination of stored energy, softening temperature and tion energy as well as the softening form function is unequivocal for the material statesstudied here The degree and type of deformation of a sample may thus be identifiedwith no knowledge of its prior mechanical history
activa-Figure 1.11: Avrami exponent versus the transformed fraction for distorted copper with shear strains 3.4 ≤c≤7.0.
Trang 391.3 Simulation
Primary recrystallization as one of the main processes of thermal softening was simulated
by a cellular automaton (CA) These latter are networks of computational units, whichdevelop their properties through the interaction of numerous similar particles [19, 20].They are comprehensively described by the four properties geometry, environment, statesand rules of evolution Cellular automatons were first applied to primary recrystallizationfor the two-dimensional case by Hesselbarth et al [21, 22] For the extension to threedimensions, a cubic lattice of identical cubes is defined Each of these small cubes repre-sents a real sample volume of about 0.6lm3
This value is obtained by comparison withreal grain sizes The whole field is then equivalent to a mass of 0.007 mg Compared withthe mass of thermal samples at 150 mg, this is very little The geometrically closest cellsare counted as the nearest neighbours It turns out that an alternating sequence of 7 and 19nearest neighbours yields the best results Stochastically changing environments influencethe kinetics in consequence of the resultant rough surface of the growing grains.Figure 1.12 shows the 7 nearest neighbours on the left and the 19 on the right,starting with a nucleus in the second time-step The change of environment with eachtime-step causes all grains in odd time-steps to be identical The resultant grain shapelooks like a flattened octahedron
1.3 Simulation
Figure 1.12: Sequence of the recrystallization in the three-dimensional space.
Trang 40JMAK theory The quality of the solution improves with the field size.
Alternatively, several calculations may be combined The deviation of simulated fromtheoretical kinetics is of the order of 1% A great advantage of cellular automatons is thatboundary conditions are automatically taken into account They do not have to be statedexplicitly This advantage should not be underestimated because the problem of collision ofgrowing grains for arbitrary site-dependent nucleation is non-trivial In this way, it is pos-sible to calculate even complicated geometries not amenable to analytical solution.The objective of simulations is to support the discussion on the various possiblecauses for the deviation of real recrystallization kinetics from the theoretically predictedprocesses In so doing, one differentiates between topological and energetic causes.Namely, the classical JMAK theory leans on two hypotheses, which strongly limit its uni-versal applicability The first in the assumption that all processes are statistically distrib-uted in space; this applies to nucleation in the first instance and thus subsequent graingrowth Any kind of nucleation concentration on chosen structural inhomogeneities altersthe collision course of growing nuclei and hence the correction factors of the extended-volume model The second restrictive assumption concerns the process rates Nucleationand nuclear growth are assumed to be site-independent and constant in time However,comparison of various strongly deformed samples shows at once that for different storedenergies, even if they are mean values, recrystallization occurs at different rates If, there-fore, we have structural components with different energies side by side in the same sam-ple, one must be aware that a uniform process rate does not exist
Non-statistical nucleation was intensively studied for point clustering The modelpostulates stochastically placed centres, which show an increased nucleation rate Thenucleation density follows a Normal distribution around the chosen centres On a linebetween two concentration centres, one obtains the distribution for the nucleation ratesshown in Figure 1.13
This yields two boundary cases, which are also being discussed in the literature[23–25] First, we have very broad scatter of nuclei and, secondly, a high concentration
on the chosen sites In a narrow parameter range between these boundary cases, the netics are very sensitive to change (Figure 1.14)
ki-It is possible to simulate the continuously decreasing Avrami exponents of thestrongly deformed samples as well as the low Avrami exponents at the beginning ofthe transformation found for weakly deformed samples
Another structural characteristic, the contiguity, describes the cohesion of tallized areas This quantity may also be calculated using the cellular automaton forvarious site functions of nucleation Comparison with experimentally determined conti-guity curves indicates that nucleation clustering can also be found in real materials.The evaluation of grain-size distributions also points to clustering