Plasticity of Metals: Experiments, Models, Computation 3527600116

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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)

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)

Deutsche Forschungsgemeinschaft Kennedyallee 40, D-53175 Bonn, Federal Republic of Germany Postal address: D-53175 Bonn Phone: ++49/228/885-1 Telefax: ++49/228/885-2777 E-Mail: (X.400): S = postmaster; P= dfg; A= d400; C = de E-Mail: (Internet RFC 822): [email protected] 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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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

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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 .

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Material State after Uni- and Biaxial Cyclic Deformation . . . . . . . 17 Walter Gieseke, K. Roger Hillert, Gu¨nter 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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17 18 19 19 24 28 28 30 30 31 34 35 35

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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

VI

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, Ju¨rgen 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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

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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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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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VII

Contents 6

Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Hartmut Neuha¨user

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 Lu¨ders 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . .

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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

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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

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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298 298 299 300 300 301 302 303 305 305 305 305 307 308 309 309 309 310 313 317

318 319 319 321 324 325 325

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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337 339 339 340 341 341 341 344 349 349 349 350 351 356 358 359 360

361 362 362 365 365 365 365 XIII

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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366 366 369 370 372 374 375 375 375 375 375 376 380 380 382 383

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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 XV

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 Ju¨lich, 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

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

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 XVIII

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. XIX

Preface 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 XX

Research Program 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  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. XXI

Preface Inhomogeneity and instability of plastic flow in Cu-based alloys (Neuha¨user) 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, Lu¨ders 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 XXII

Research Program 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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Preface

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 XXIV

Research Program 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, XXV

Preface • • •

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 XXVI

Research Program 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. XXVII

Preface Experimental determination of deformation- and strain fields by optical measuring methods (Ritter, Friebe) Mainly, two methods were developed and adapted for solving the mentioned problems: the object-grating method and the electronic Speckle interferometry. As known, the object-grating method leads to the local vector of each point of the considered object surface marked by an attached grating, consisting of a deterministic or stochastic grey value distribution, and recorded by the photogrammetric principle. Then, the strain follows from the difference of the displacement vectors of two neighbouring points related to two different deformation states of the object and related to their initial distance. The electronic Speckle interferometry is based on the Speckle effect. It comes into existence if an optical rough object surface is illuminated by coherent light, and the scattered waves interfere. By superposing of the interference effects of an object and reference wave related to two different object states, the difference of the arising Speckle patterns leads to correlation fringes, which describe the displacement field of the considered object. Regarding the object-grating method, grating structures and their attachment have been developed, which can be analysed automatically and which are practicable also at high temperatures up to 1000 8C, as often inelastic processes take place under this condition. Furthermore, the optical set-up, based on the photogrammetric principle, was adapted 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 interferometric principle was applied. A 3-D electronic Speckle interferometer has been developed, 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 optoelectronic components as laser diodes, piezo crystals and CCD-cameras. Furthermore, both methods are suitable for high resolution of a large change of material behaviour. Finally, the measurement can be conducted at the original and takes place without contact and interaction.

Surface deformation fields from grating pictures using image processing and photogrammetry (Andresen) The before-mentioned grating techniques are optical whole-field methods applied to derive the shape or the displacement and strain on the surface of an object. A regular grating fixed or projected on the surface is moved or deformed together with the object. 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 information 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 evaluated manually by projecting the images to large screens or by use of microscope techniques. Today, the pictures are usually digitized, yielding resolutions from 200 × 200 to 2000 × 2000 picture elements (pixels or pels) with generally 256 grey levels (8 bit). By XXVIII

Research Program suitable image-processing methods, the grating coordinates in the images are determined to a large extent automatically. The corresponding coordinates on plane objects are derived from the image coordinates by a perspective transformation. Considering spatial surfaces, first, the orientation of the cameras in space must be determined by a calibration procedure. Then, the spatial coordinates are given by intersection of the rays of adjoined grating points in the images. The sequence of the grating coordinates in different states describes displacement and strain of the considered object surface. Applying suitable interpolation gives continuous fields for the geometrical and physical quantities on the surface. These experimentally 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 shearloaded aluminium panels (Kossira, Arnst) 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 experimental and numerical methods. Beyond the up-to-now used classical theory of plasticity, the theoretical research was based on the “unified” models, which were developed and adjusted to numerical computational methods in other areas of the research project. Shear-loaded panels are in general substructures of aerospace constructions since there are always load cases during a flight mission, in which shear loads are predominant in the thin-walled structures of subsonic as well as in supersonic and hypersonic aircrafts. 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, but with a thermal protection shield, the temperatures of the load-carrying structure can be reduced to moderate values, which allow the application of aluminium alloys. Therefore, the properties of the mostly used aluminium alloy 2024-T3 are taken as a basis for the experimental and theoretical studies of the behaviour of shear-loaded panels at room temperature and at 200 8C. The primary aim of these studies is the understanding of the occurring phenomena, respectively the examination of the load-carrying behaviour of the considered structures under different load-time histories, and to provide suitable data for the design. Besides experimental investigations, which are achieved by a specially designed test set-up, the development of numerical methods, which describe the phenomena, was necessary to accomplish this intention. The used numerical model is based on a FiniteElement method, which is capable of calculating the geometric and physical non-linear – in case of visco-plastic material behaviour time-dependent – postbuckling behaviour. XXIX

Preface A substantial problem within the numerical method was the simulation of the non-linear material properties. Using a rate-independent two-surface material model and a modified visco-plastic material model of the Chaboche type, the non-linear properties of the aluminium alloy 2024-T3 are approximated with sufficient accuracy at both considered temperatures. Some results of the theoretical and experimental studies on the monotonic and cyclic postbuckling behaviour of thin-walled aluminium panels under shear load at ambient and elevated temperatures are presented. The applied loads exceed the theoretical buckling loads by factors up to 40, accompanied by the occurrence of moderate inelastic deformations. Apart from the numerical model, the monotonic loading, subsequent creep rates, the snap-through behaviour at cyclic loading, the inelastic processes during loading, and the influence of the aspect ratio are major topics in the presented discussion of the results for shear-loaded panels at room temperature and at 200 8C. Consideration of inhomogeneities in the application of deformation models, describing the inelastic behaviour of welded joints (Wohlfahrt, Brinkmann) A second practical problem was the investigation of the influence of welded joints on the mechanical behaviour of components, which is due to the high degree of “Werkstoffnutzung” in modern welded structures of high importance. Special consideration was given here to the important question of the material behaviour at cyclic loading as well from the point of view of numerical computation of these processes and the connected effects as from the point of view of the problems connected with aspects of materials sciences. The local loads and deformations in welded joints have rarely been investigated under the aspect that the mechanical behaviour is influenced by different kinds of microstructure. These different kinds of microstructure lead to multiaxial states of stresses and strains, and some investigations have shown that for the determination of the total state 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 the heat-affected zone as well as a weld metal with a lower strength than the base metal can be the reason or the starting point of a fracture in welded joints. A new investigation demonstrates that in TIG-welded joints of the high strength steel StE690, a finegrained 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 compressive range. These investigations support the approach described here that the mechanical behaviour of the different kinds of microstructure in the heat-affected zone of welded joints has to be taken into account in the deformation analysis. The influences of these inhomogeneities on the local deformation behaviour of welded joints were determined by experiments and numerical calculations over a wide range of temperature and loading. The numerical deformation analysis was performed with ttformat he method of Finite-Elements, in which recently developed deformation models simulate the mechanical behaviour of materials over the tested range of temperature and loading conditions.

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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)

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Correlation between Energetic and Mechanical Quantities of Face-Centred Cubic Metals, Cold-Worked and Softened to Different States Lothar Kaps and Frank Haeßner *

1.1

Introduction

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 parameters can in principle be determined by calorimetric methods. By combining calorimetrically determined values with characteristics measured mechanically and with microstructural data, information can be gained about the strain-hardened state and the mechanism of the softening process. This materials information can support critical assessment of the structure of material models and hence be utilized for the appropriate adjustment of constitutive models to material properties.

1.2

Experiments

One objective of the work in this particular area of research was to investigate the dependence of the softening kinetics of face-centred cubic metals on the deformation. The chosen types of deformation were torsion, tension and rolling. In the cases of torsion and tension, additional cyclic experiments with plastic amplitudes of 0.01 to 0.1 were carried 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 investigated. In the following presentation of the results, the emphasis will be on copper. To determine the mechanical data, the first step was to characterize the deformation with the aid of the crystallographic slip
, the shear stress N normalized to the * Technische Universita¨t Braunschweig, Institut fu¨r Werkstoffe, Langer Kamp 8, D-38106 Braunschweig, Germany 1

1 Correlation between Energetic and Mechanical Quantities shearing modulus G : N ˆ c =G, and the strain-hardening rate  ˆ dN =da. The conversion to crystallographic quantities was effected using calculated Taylor factors [1, 2]. This procedure permits direct comparison between different types of deformation. Figure 1.1 shows the family of curves that are obtained when copper is subjected to torsion at various temperatures. The characterization is clear because for increasing deformation temperature, a decreasing yield stress results. Figure 1.2 shows the strain-hardening rate versus the normalized shear stress of copper to extreme deformation. The strain-hardening rate can be subdivided into three regions, 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 shear stress. Region IV, as region II, is characterized by constant strain hardening. The occurrence of these different regions depends strongly on the type of deformation. Thus, for tensile deformation, in consequence of instability, only deformation to region III can be realized. Rolling permits greater deformation, but brings with it the problem of defining a specific measurement to categorize the strain-hardening regions. The temperature 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 initial III strain-hardening rate III 0 and a variable limiting stress S . This latter is affected by dynamic recovery and is therefore dependent on deformation temperature and velocity. It decreases for increasing deformation temperature and increases for higher deformation velocities. This statement is also true for the other characteristic stresses IV; V; V S : The logarithm of the characteristic stress decreases linearly with the normalized deformation temperature, TN ˆ kT=Gb3 : The normalization was proposed by Mecking et al. [5]. It

Figure 1.1: Flow curves of copper at temperatures of –20 8C to 120 8C. 2

1.2

Experiments

Figure 1.2: Strain-hardening rate of copper under torsion at room temperature versus normalized shear stress.

has been successfully applied to our own measurements. However, it may be seen that the dependence on temperature is different for the individual stresses (Figure 1.3). Careful evaluation of the experiments taking account of the effects of texture and sample shows similarities as well as differences between the two deformation types tension and torsion. Up to a slip value of a = 0.4, the flow curve shows little difference between tension and torsion. Above that value, the hardening is greater for the tension experiment (Figure 1.4). The differences are more pronounced when the hardening rate is studied rather than 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-

Figure 1.3: Characteristic stresses for the strain hardening of copper versus the normalized temperature. 3

1 Correlation between Energetic and Mechanical Quantities

Figure 1.4: Comparison of flow curves from tension and torsion experiments.

tron microscope. Other authors have described this influence of the load path on the microstructure [6–9]. The reason for this may be that different average numbers of slip systems are necessary for deformation [10]. This also affects the development of activation 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 extension rate, the material shows a higher flow stress. For the evaluation, the following ansatz was chosen for the relationship between the extension rate e_ and the flow stress r:
 DG0  Vr : 1 e_ ˆ e_ 0 exp kT The activation volume and energy are the important quantities for the constitutive equations developed in the subproject A6 [11, 12]. The comparison of the deformation types tension and torsion shows a definite difference in the development of activation volumes with N . This is manifest by the tension (strain) deformation, which exhibits a constant velocity sensitivity even for significantly smaller degrees of deformation (Figure 1.5). The activation volumes are a particularly indicative measurement for the velocity sensitivity. In region III for torsion, they show a continuous decrease, which becomes less only upon reaching region IV. For tension, on the other hand, there are also two sections with decreasing or nearly constant activation volumes. However, the transition in the curve of the activation volume versus the stress already lies in the strain-hardening region III. The activation energies DG0 for torsion were determined from the characteristic stresses for different temperatures (cf. Figure 1.3). The resultant values for the stresses IV V III S ;  and 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 determined using a calorimeter. As expected, the stored energy shows a monotonic in4

1.2

Experiments

Figure 1.5: Activation volume of copper deformed in tension and torsion at room temperature.

crease with deformation. Moreover, dynamic recovery counteracts energy storage as it does hardening. Hence, there is an unequivocal correlation between the deformation temperature and stored energy such that an increasing deformation temperature leads to less 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-

Figure 1.6: Stored energy versus shear strain for distorted copper and silver deformed at different temperatures. 5

1 Correlation between Energetic and Mechanical Quantities tial dislocations. This hinders dynamic recovery because the mechanism of cross slip is impaired. The connection between stored energy and shearing stress was studied for deformation by torsion, tension and push-pull. There is a clear tendency to store more energy with increasing deformation temperature at constant shearing stress. It would appear that energy storage by more fully condensed states is more effective. The measured values for tension and push-pull in this sequence lie above those for the greatest torsional deformation. For the same shearing stress, silver also clearly stores more energy than copper. This relationship is represented in the Figures 1.7 and 1.8. Figure 1.7 comprises torsion experiments up to extreme deformation. Figure 1.8 shows a comparison of various types of deformation. For better resolution, the abscissa here is confined to small and 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 softening kinetics to be discussed. In analogy to the strain-hardening rate, an energy storage rate HE
dES =dN has been defined. This quantity represents independent information. The development of the energy storage rate is clearly correlated with the strainhardening stages (Figure 1.9). The combination of energetic and mechanical measurements permits a statement on the change in dislocation density , to a first approximation proportional to the stored energy, with increasing flow stress. A linearly increasing energy storage rate with stress leads to a law of the type:  qES c
N 
k1 ES : qN G

2

This kind of behaviour is found only up to the middle of region III. After that, the energy storage rate increases overproportionally until region IV is reached. In region IV, it decreases slightly and then increases linearly again in region V. This time, however,

Figure 1.7: Stored energy versus normalized shear stress for copper and silver deformed at different temperatures. 6

1.2

~ Cu 235 K

Experiments

s Cu 293 K * Cu 373 K

+ Cu 293 K Tension × Cu 293 K Push-Pull

Figure 1.8: Stored energy versus normalized shear stress for copper deformed in torsion, tension and push-pull.

Figure 1.9: Stored energy (upper curve) and rate of energy storage of distorted copper versus normalized shear stress.

with a different proportionality factor
of value rather below the one pertaining to region III. The factor
may only be analytically assessed for deformation in the region of the strain-hardening stage II. For greater plastic deformation, which would then be deformation in region III of the strain-hardening stage, this factor is of a qualitative nature. The evolution of
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. 7

1 Correlation between Energetic and Mechanical Quantities Table 1.1: The constant k1 according to Equation (2) for various temperatures. The constant k2 applies to extreme deformation in the region V.

Cu 253 K Cu 293 K Cu 373 K Cu 293 K tension Ag 253 K Ag 293 K

k1

c/G

k2

6.4·10–4 5.7·10–4 5.5·10–4 5.0·10–4 6.0·10–4 5.5·10–4

1.5·10–3 1.25·10–3 1.12·10–3 1.5·10–3 1.5·10–3 1.25·10–3

4.8·10–4 4.5·10–4 4.5·10–4 – 4.5·10–4 4.4·10–4

If 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:   Q : 3 X_
f XgT
f X exp  RT Equation (3) is easily handled numerically. The activation energy of the softening Q and the form function f may be determined separately. Equation (3) offers the added advantage that, as a rate equation, it may be directly incorporated into a constitutive equation 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 the stability of the deformed state. The thermal results show that for increasing stored energy, the softening process takes place at lower temperatures. An influence of the deformation temperature becomes apparent. Higher deformation temperatures promote easier reaction for the same stored energy. Exact analysis of these facts shows that the form function makes only a negligible contribution here. The effect is induced by a reduced activation energy. Different types of deformation show a stronger influence on the reaction temperature than the deformation temperature. At lower energies, distorted samples soften faster than extended or rolled ones. At higher energies, the reverse is true: Rolled samples react faster. It is noticeable that cyclically deformed samples, for torsion as well as for push-pull, do not diverge from the unidirectionally deformed samples of the same deformation mode. This is remarkable because, particularly for tension and push-pull deformation, there are substantial differences in the activation energy. The activation energy describes the purely temperature dependence of the reaction. For small deformation and stored energies of distorted copper at a value of 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); pushpull deformed samples, on the other hand, show significantly lower activation energies (130 kJ/mol). With increasing energy, the activation energies of all deformation types fall. Figure 1.10 demonstrates these relationships. With the aid of torsional deformation, it is unequivocally proved that only upon reaching the strain-hardening stage V, one may presume constant activation energy. At 8

1.2

Experiments

Figure 1.10: Activation energy of differently deformed copper versus the stored energy.

values of 80 to 90 kJ/mol, here for all deformation temperatures, the activation energy lies in the region of grain boundary self diffusion or diffusion in dislocation cores. Tension and push-pull samples do not achieve these high stored energies; for these deformation modes, there is therefore no region of constant activation energy. Elevated deformation temperatures result in a lower softening activation energy. One may interpret this as strain hardening at higher temperature producing a microstructure that softens faster. This effect should be accounted for when setting up constitutive equations. There is a theory for the softening of deformed metals through the mechanism of primary 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 measured 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 1.2 shows the comparison in detail. For high deformation, this interpretation complies with studies according to the microstructural-path method [18]. The grain spectra of weakly deformed and recrystallized material show agreement with calculated spectra after sitesaturated cluster nucleation. Table 1.2: Effective activation energies from the JMAK theory compared with measured values for low/high deformation.

Copper Silver

Site-saturated nucleation [kJ/mol]

Continuous nucleation [kJ/mol]

Measured values [kJ/mol] low deformation

high deformation

166–120 143–115

125–86 107–86

170 ± 8 120 ± 8

85 ± 5 85 ± 5

9

1 Correlation between Energetic and Mechanical Quantities The second component of the kinetics, the pure reaction form, is described by the function f X. For all nucleation-nucleation growth reactions, this function, by way of the transformed fraction, is parabolic with zero points at the beginning and end of the reaction. A more significant picture results when this function is compared with the JMAK theory. For ideal nucleation-nucleation growth reactions, the theory demands for f X=1  X a higher order function of ln1  X with an exponent n  1=n independent of X. The Avrami exponent n takes the value 4 or 3, respectively. In 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 the transformed fraction for distorted copper. The horizontal reference line outlines the curve 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 the stored energy is taken as the comparative measure instead of the strain-hardening regions. Complementary studies using the transmission electron microscope show that the microstructural details are similar for these deformations (cf. Nix et al. [9]). The deformation types unidirectional tension and push-pull are very different from torsion. The Avrami exponents are very large for unidirectional tension. In summary, the combination of stored energy, softening temperature and activation energy as well as the softening form function is unequivocal for the material states studied here. The degree and type of deformation of a sample may thus be identified with no knowledge of its prior mechanical history.

Figure 1.11: Avrami exponent versus the transformed fraction for distorted copper with shear strains 3.4 ≤ c ≤ 7.0. 10

1.3

1.3

Simulation

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, which develop their properties through the interaction of numerous similar particles [19, 20]. They are comprehensively described by the four properties geometry, environment, states and rules of evolution. Cellular automatons were first applied to primary recrystallization for the two-dimensional case by Hesselbarth et al. [21, 22]. For the extension to three dimensions, a cubic lattice of identical cubes is defined. Each of these small cubes represents a real sample volume of about 0.6 lm3. This value is obtained by comparison with real grain sizes. The whole field is then equivalent to a mass of 0.007 mg. Compared with the mass of thermal samples at 150 mg, this is very little. The geometrically closest cells are counted as the nearest neighbours. It turns out that an alternating sequence of 7 and 19 nearest neighbours yields the best results. Stochastically changing environments influence the 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 each time-step causes all grains in odd time-steps to be identical. The resultant grain shape looks like a flattened octahedron.

Figure 1.12: Sequence of the recrystallization in the three-dimensional space. 11

1 Correlation between Energetic and Mechanical Quantities The possible states of the cells are recrystallized and non-recrystallized. For the extension to different grain boundary velocities, the non-recrystallized state was subdivided further. The fourth descriptive characteristic after the geometry, environment and possible states are the rules of evolution. These stipulate, which states the cells will adopt in the next time-step. If a cell already has a recrystallized environment, the rules predict that in the next time-step, this cell will also adopt the recrystallized state. Using this simple cellular automaton, it is possible to solve the differential equation of the JMAK theory. The quality of the solution improves with the field size. Alternatively, several calculations may be combined. The deviation of simulated from theoretical kinetics is of the order of 1%. A great advantage of cellular automatons is that boundary conditions are automatically taken into account. They do not have to be stated explicitly. This advantage should not be underestimated because the problem of collision of growing grains for arbitrary site-dependent nucleation is non-trivial. In this way, it is possible to calculate even complicated geometries not amenable to analytical solution. The objective of simulations is to support the discussion on the various possible causes for the deviation of real recrystallization kinetics from the theoretically predicted processes. In so doing, one differentiates between topological and energetic causes. Namely, the classical JMAK theory leans on two hypotheses, which strongly limit its universal applicability. The first in the assumption that all processes are statistically distributed in space; this applies to nucleation in the first instance and thus subsequent grain growth. Any kind of nucleation concentration on chosen structural inhomogeneities alters the collision course of growing nuclei and hence the correction factors of the extendedvolume model. The second restrictive assumption concerns the process rates. Nucleation and 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 stored energies, even if they are mean values, recrystallization occurs at different rates. If, therefore, we have structural components with different energies side by side in the same sample, one must be aware that a uniform process rate does not exist. Non-statistical nucleation was intensively studied for point clustering. The model postulates stochastically placed centres, which show an increased nucleation rate. The nucleation density follows a Normal distribution around the chosen centres. On a line between two concentration centres, one obtains the distribution for the nucleation rates shown 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 kinetics are very sensitive to change (Figure 1.14). It is possible to simulate the continuously decreasing Avrami exponents of the strongly deformed samples as well as the low Avrami exponents at the beginning of the transformation found for weakly deformed samples. Another structural characteristic, the contiguity, describes the cohesion of recrystallized areas. This quantity may also be calculated using the cellular automaton for various site functions of nucleation. Comparison with experimentally determined contiguity curves indicates that nucleation clustering can also be found in real materials. The evaluation of grain-size distributions also points to clustering.

12

1.3

Simulation

Figure 1.13: Model of point clustering (left); plot of the nucleation rate between two concentration centres (right).

Figure 1.14: Avrami exponent due to the restriction of nucleation to point clustering.

Introduction of site-dependent process rates is effected through an extension of possible non-recrystallized states. One differentiates between mobility and driving force. With reference to the literature [26, 27], a value around the factor 3 is taken. The result is 9 different velocities. Two degrees of recrystallization are defined, one of which refers to the energy, the other to the volume. If the kinetics of the JMAK theory are appropriately evaluated, there is hardly any difference between these two definitions. The introduction of different velocities causes the reaction rate to decline towards the end of the transformation. If the proportionality of the areas of equal velocity and the resulting grain size is changed, the kinetics may be influenced to a degree. The kinetics of strongly deformed samples may be simulated if the areas of equal velocity are larger than the resultant grain size. Smaller initial areas do not give the desired effect; the decline of the effective rates is too late and too weak. The kinetics of weakly de13

1 Correlation between Energetic and Mechanical Quantities formed samples with low Avrami exponents cannot be calculated using this ansatz. An experimental indication of rate retardation is obtained from studies on strongly rolled copper by the microstructural-path method [18]. Finally, it may be said that there are indicators for each ansatz in real recrystallization processes. Considering the experimental results, a weighted mixture of both would appear to be a realistic course, which can doubtless be applied in the model. Coupling to a constitutive equation is directly possible, for example by introducing the stored energy as a function of deformation. The cellular automaton, on the other hand, is able to calculate partially softened material structures. The strength of the composite may then be determined from this using a parallel or series network. In future, this type of model coupling will become more important in those areas, where modelling with constitutive equations on the basis of discontinuous phenomena only such as dynamic recrystallization do not produce the desired results.

1.4

Summary

Shortly summarizing this report, we can make the following basic statements: •

The diverse strain-hardening stages of face-centred cubic metals, identifiable from mechanical data, which correspond to different structures of the strain-hardened material, may also be determined from the thermally measured stored energy and from the rate of energy storage. One finds that the energy storage of more fully condensed states is particularly effective.

•

The softening kinetics investigated via the stored energy are strongly influenced by the details of the type of deformation (for example, unidirectional deformation-alternate deformation). In the case of the primary recrystallization as the cause of the softening, the process may be described well by quoting the activation energy and the Avrami exponent. Knowledge of these two parameters for a strain-hardened state allows the degree of softening to be numerically calculated for a freely chosen temperature-time programme. Qualitatively, the activation energy and the Avrami exponent are a measure of the thermal stability, that is, for the ease of reaction of the deformed material.

•

Utilizing a suitably fitted cellular automaton, it is possible to simulate the microstructural processes underlying the softening and hence to control the topological as well as the energetic model hypotheses. An important result of this simulation is the proof that the Avrami theory, which is based on stereological elements, may be applied to calorimetrically determined softening data. The kinetics in both cases are very similar.

The results presented here are the compilation of numerous data; a comprehensive publication is given in [28].

14

References

References [1] J. Gil Sevillano, P. van Houtte, E. Aernoudt: Deutung der Schertexturen mit Hilfe der Tayloranalyse. Z. f. Metallkunde 66 (1975) 367. [2] U. F. Kocks, M. G. Stout, A. D. Rollett: The influence of texture on strain hardening. In: P. O. Kettunen (Ed.): Strength of metals and alloys, Pergamon Press, Oxford, 1988. [3] J. Diehl: Zugverformung von Kupfer Einkristallen. Z. f. Metallkunde 47 (1956) 331. [4] J. Gil Sevillano: The cold-worked state. Materials Science Forum 113–115 (1993) 19. [5] H. Mecking, B. Nicklas, N. Zarubova, U. F. Kocks: A “universal” temperature scale for plastic flow. Acta metall. 34 (1986) 527. [6] M. N. Bassim, C. D. Liu: Dislocation cell structures in copper in torsion and tension. Mater. Sci. Eng. A 164 (1993) 170. [7] B. Bay, N. Hasnen, D. A. Hughes, D. Kuhlmann-Wilsdorf: Evolution of fcc deformation structures in polyslip. Acta metall. mater. 40 (1992) 205. [8] C. D. Liu, M. N. Bassim: Dislocation substructure evolution in torsion of pure copper. Metall. Trans. 24A (1993) 361. [9] W. D. Nix, J. C. Gibeling, D. A. Hughes: Time dependent deformation of metals. Metall. Trans. 16A (1985) 2215. [10] T. Unga´r, L. S. To´th, J. Illy, I. Kova´cs: Dislocation structure and work hardening in polycrystalline of hc copper rods deformed by torsion and tension. Acta metall. 34 (1986) 1257. [11] R. Gerdes: Ein stochastisches Werkstoffmodell fu¨r das inelastische Materialverhalten metallischer Werkstoffe im Hoch- und Tieftemperaturbereich. Mechanik-Zentrum der TU Braunschweig (Dissertation), Braunschweig, 1995. [12] H. Schlums, E. A. Steck: Description of cyclic deformation processes with a stochastic model for inelastic creep. Int. J. Plast. 8 (1992) 147. [13] W. A. Johnson, R. F. Mehl: Reaction kinetics in process of nucleation and growth. Trans. Am. Inst. Min. Engrs. 135 (1939) 416. [14] M. Avrami: Kinetics in phase change: I. General theory. J. Chem. Phys. 7 (1939) 1103. [15] M. Avrami: Kinetics in phase change: II. Transformation-time relations for random distribution of nuclei. J. Chem. Phys. 8 (1940) 212. [16] M. Avrami: Kinetics in phase change: III. Granulation, phase change and microstructure. J. Chem. Phys. 9 (1941) 177. [17] A. E. Kolmogorov: Zur Statistik der Kristallvorga¨nge in Metallen (russ. mit deutscher Zusammenfassung). Akad. Nauk. SSSR Ser. Mat. 1 (1937) 335. [18] R. A. Vandermeer, D. Juul Jensen: Quantifying recrystallization nucleation and growth kinetics of cold-worked copper by microstructural analysis. Metall. Mater. Trans. 26A (1995) 2227. [19] S. Wolfram: Statistical mechanics of cellular automata. Reviews of modern physics 55 (1983) 601. [20] S. Wolfram: Cellular automata as models of complexity. Nature 311 (1984) 419. [21] H. W. Hesselbarth, I. R. Go¨bel: Simulation of recrystallization by cellular automata. Acta metall. mater. 39 (1991) 2135. [22] H. W. Hesselbarth, L. Kaps, F. Haeßner: Two dimensional simulation of the recrystallization kinetics in the case of inhomogeneously stored energy. Materials Science Forum 113–115 (1993) 317. [23] J. W. Cahn: The kinetics of grain boundary nucleated reactions. Acta metall. 27 (1979) 449. [24] J. W. Cahn, W. Hagel: Decomposition of austenite by diffusional processes. In: Z. D. Zackay, H. I. Aarosons (Eds.), Interscience Publ., New York, 1960. [25] R. A. Vandermeer, R. A. Masumura: The microstructural path of grain-boundary-nucleated phase transformations. Acta metall. mater. 40 (1992) 877. 15

1 Correlation between Energetic and Mechanical Quantities [26] J. S. Kallend, Y. C. Huang: Orientation dependence of stored energy of cold work in 50% cold rolled copper. Metal Science 18 (1984) 381. [27] F. Haeßner, G. Hoschek, G. To¨lg: Stored energy and recrystallization temperature of rolled copper and silver single crystals with defined solute contents. Acta metall. 27 (1979) 1539. [28] L. Kaps: Einfluss der mechanischen Vorgeschichte auf die prima¨re Rekristallisation. Shaker Verlag, Aachen, 1997.

16

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)

2

Material State after Uni- and Biaxial Cyclic Deformation Walter Gieseke, K. Roger Hillert and Gu¨nter Lange *

2.1

Introduction

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 microstructural 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 [1–4], 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 [5, 6], it is known that the yield surfaces of small offset strains under 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. 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 inner stresses are discussed.

* Technische Universita¨t Braunschweig, Institut fu¨r Werkstoffe, Langer Kamp 8, D-38106 Braunschweig, Germany 17

2

2.2

Material State after Uni- and Biaxial Cyclic Deformation

Experiments and Measurement Methods

Copper of 99.99% purity was chosen as a material exhibiting typical wavy slip behaviour. Most of the experiments were performed using the technically important material AlMg3 of 99.88% purity. Its behaviour may be described as being somewhere intermediate between planar and wavy slip 1. Commercial austenitic steel 1.4404 (AISI 316L) was used as a material with typical planar slip behaviour. The total strain amplitude was varied from  0.25% to  0.75% for AlMg3 and steel, and between  0.05% and  0.5% for copper. All materials were previously solution annealed or recrystallized. AlMg3 and the austenitic steel were quenched in water, the copper samples cooled in the oven. After the thermal treatment, a h100i slightly fibrous texture was identified, which did not change during the subsequent cyclic deformation. The copper showed almost no texture. The yield surfaces of the initial materials were isotropic, independent of the offset used [7, 8]. Tubular samples were used in the experiments. Their outer diameter and wall thickness were 28 mm and 2 mm for AlMg3 and copper, 29 mm and 1.5 mm for steel, respectively. The measuring distance was 54 mm long for all samples. The following cyclic experiments were carried out using a servo-hydraulic Schenck testing machine, which had been augmented by a laboratory-made torsional drive [9]: uniaxial tension/ compression, alternating torsion; biaxial equal phase superposition of tension/compression and alternate torsion; a 90 8 antiphase combination of tension/compression and alternate torsion. The dislocation structures were subsequently investigated using a Philips 120 kV transmission electron microscope. For the strain-controlled experiments, a triangular nominal value signal with constant strain rate of 2 · 10–3 s–1 was chosen. The equivalent strains were calculated after v. Mises according to:
eq ˆ



1
2 ‡ 2 3

1=2

1 with
ˆ p  : 3

…1†

Two methods were applied to determine the yield surfaces. Using the definition via an offset strain of 2 · 10–4%, the load was increased in steps of 6 N/mm2 in the -direction or of 2.5 N/mm2 in the -direction until the given yield limit was reached. There was a 10 s intermission at each level. Before the next point on the curve was measured, several load cycles were run through again to set the material to the same initial state. The second measurement method was the recording of directionally dependent stress-strain diagrams. Here, a new sample was used for each point measured. It was stressed under predetermined load paths immediately following the cyclic treatment far into the plastic region. In this way, static strain ageing effects were avoided. Further, it was possible to determine yield surfaces of higher offset strain and areas of equal tangent modules. For the evaluation of the yield surfaces and the tangent module areas, besides the yield conditions after v. Mises and Tresca, a formulation developed within the scope of this project was used: 1

18

The results for copper and AlMg3 presented in this report and their interpretation are taken from the thesis by Walter Gieseke [9].

ˆ




0eq ˆ



0eq

…

…


E A † ‡ … G 2


A †2 ‡

G … E

A † A †2

2.3

Results

1=2

;

2

1=2

:

…2† …3†

The advantage of these equations lies in the fact that all equivalent stress-strain diagrams show a Young’s modulus appropriate increase in the elastic region. In the case of AlMg3, the , -hysteresis can be converted into the equivalent eq ,
eq -hysteresis, which are in almost complete agreement with the measured ,
-hysteresis values. Figure 2.1 a shows the strain paths for the measurement of a family of yield surfaces of varying offset strain and tangent modules after prior tension/compression loading. The starting point for the measurement was set here in the centre of the elastic region after load reversal in the load maximum. Figure 2.1 b shows the appropriate load paths, Figure 2.1 c the relevant equivalent stress-strain diagrams. The yield points of various offset strains were determined by parallel shift of the elastic straight line. For areas with the same tangent modules, the equivalent stress-strain curves were differentiated; for a given tangential gradient, one obtains the pertinent , -points. The yield surfaces in Figure 2.2 show that the yield conditions according to Equations (2) and (3) produce the same results as the evaluation after v. Mises or Tresca (AlMg3, tension/compression loading, starting from the stress zero crossover, offset strain  0.2% or  0.01%, respectively).

2.3

Results

2.3.1

Cyclic stress-strain behaviour

Figure 2.3 a shows a plot for AlMg3 of the stress amplitudes as a function of number of cycles for the appropriate given equivalent total strain amplitude of D
eq ˆ  0.5%. The three proportional loads are compared and that for the 90 8 anti-phase combination of tension/compression and alternating torsion. For all four load types, the saturation state is reached after about 500 cycles. The curves for proportional loading almost coincide. Larger torsional fractions cause a slight increase in the stress amplitudes. The curve for disproportional loading systematically assumes higher values. This additional hardening effect is much more pronounced at the beginning of the fatigue at about 25% than in the saturation stage, where it is only about 5%. Figure 2.3 b shows the appropriate curves for the lower total strain amplitude of D
eq ˆ  0.3%.

19

Figure 2.1: a) Strain paths for measurement of yield surfaces and areas of equal tangent modules; b) load paths for Figure 2.1 a; c) equivalent stress-strain diagrams for the stress and strain values of Figures 2.1 a and b. Calculated according to Equations (2) and (3).

2

20

Material State after Uni- and Biaxial Cyclic Deformation

2.3

Results

Figure 2.2: 0.2% and 0.01% offset saturation yield surfaces measured in the stress zero crossover. Evaluation using the v. Mises and Tresca conditions and Equations (2) and (3).

Figure 2.3 a: Cyclic strain hardening behaviour for D
eq ˆ  0.5%, material: AlMg3.

The saturation state is reached after about 900 cycles. Here too, the curves of proportional loading approximately coincide. For disproportional loading, a weak additional hardening effect appears at the beginning of the fatique stage, yet this reverses in saturation. The additional hardening effect may usually be explained by the fact that for an appropriately large plastic strain amplitude, the anti-phase loading leads to an additional hardening because more slip systems are activated than for proportional loading. This is particularly the case for the high strain amplitude of D
eq ˆ  0.5% at the beginning of the fatigue. For strain amplitudes of D
eq ˆ  0.3%, the plastic fraction of 21

2

Material State after Uni- and Biaxial Cyclic Deformation

Figure 2.3 b: Cyclic strain hardening behaviour for D
eq ˆ  0.3%, material: AlMg3.

saturation is so small that the additional hardening in consequence of anti-phase loading is not enough to compensate the overall smaller stress values of the sum of the individual components. At strain amplitudes of D
eq   0.4%, AlMg3 shows Masing behaviour for all proportional loads. Deviations occur at smaller amplitudes: The length of the elastic regions increases with decreasing strain amplitude. Similar behaviour is found for planar flowing -brass [10]. For copper, a total strain amplitude of D
eq ˆ  0.5% under phase-shifted loading produces a pronounced additional hardening effect throughout the whole fatigue region (Figure 2.4 a). As for AlMg3, the curves for proportional loading approximately coincide, though the pure torsional load yields the lowest values. The stress values of the phase-shifted

Figure 2.4 a: Cyclic strain hardening behaviour at D
eq ˆ  0.5%, material: copper. 22

2.3

Results

Figure 2.4 b: Cyclic strain hardening behaviour at D
eq ˆ  0.1%, material: copper.

loading reach saturation after about 30 cycles. Under proportional loading, on the other hand, constant stress values are only measured after about 50 cycles. For an amplitude of D
eq ˆ  0.1%, the effect occurs only at the onset of fatigue (Figure 2.4 b). For a further reduction to D
eq ˆ  0.05%, the stress values for phase-shifted loading in the saturation region lie below those for synchronous loading (Figure 2.4 c). The considerations regarding the additional hardening effect in AlMg3 are equally applicable here. The austenitic steel 1.4404 for proportional loading at D
eq ˆ  0.75% shows a relatively short strain hardening region already reaching saturation after about 20 cycles. But the 90 8 phase-shifted loading produces a strong additional hardening effect. The appropri-

Figure 2.4 c: Cyclic strain hardening behaviour at D
eq ˆ  0.05%, material: copper. 23

2

Material State after Uni- and Biaxial Cyclic Deformation

Figure 2.5 a: Cyclic strain hardening behaviour at D
eq ˆ  0.75%, material: steel 1.4404.

Figure 2.5 b: Cyclic strain hardening behaviour at D
eq ˆ  0.5%, material: steel 1.4404.

ate stress values compared with proportional loading are increased by more than 60%. The saturation plateau is only reached after about 30 cycles (Figure 2.5 a). For a strain amplitude of D
eq ˆ  0.5% too, the material reaches saturation for proportional loading after about 20 cycles. The increase of the stress amplitudes is less here, however. Phase-shifted loading (Figure 2.5 b) also yields a distinct additional hardening effect. The appropriate stress amplitudes as for D
eq ˆ  0.75% are greatly increased. The additional hardening effect may be regarded here as a consequence of the planar flow behaviour.

2.3.2

Dislocation structures

The dislocation structure of AlMg3 is characterized by walls of prismatic edge dipoles. Mobile screw dislocations lie between them. For all strain amplitudes and loading types studied, the dipolar walls lie in (111) planes at the onset of fatigue. The value and type 24

2.3

Results

of loading determine the resultant saturation structure. Using a model by Dickson et al. [11, 12], all wall orientations that differ from (111) planes can be indexed. For low strain amplitudes …D
eq   0.3%) after proportional and disproportional loading, the (110) walls and the initial (111) walls dominate. In almost all cases, formation of the (110) walls was the work of a single slip system. Hereby, the walls were compressed perpendicular to the Burgers vector. Similar structures are also found in brass with 15 at% zinc [13]. At high amplitudes …D
eq >  0.3%) after proportional loading, the (100) besides the (311), (210), (211) and (110) walls are predominant. By contrast, for phase-shifted loading, the initial orientation of the (111) walls is conserved. During proportional loading, a maximum of three slip systems are set in motion. Phase-shifted loading on the other hand, because of the rotating stress vector, usually activates more than four slip systems. Figure 2.6 a shows the typical example of a dislocation structure after proportional loading. The arrangement can be designated anisotropic since the dipolar walls in almost all grains are oriented in only one or two crystallographic directions. The anisotropy essentially results from the small number of active slip systems in the proportional loading case, expressing a certain planarity in the slip behaviour. Disproportional loading at high total strain amplitudes, however, results in generally more isotropic structures (Figure 2.6 b). Here, the dipolar wall structure is quite often destroyed along favourably oriented (111) planes (Figure 2.6 c). Parallel arrays of elongated screw dislocations are often observed in these bands, which infers high local slip activity. Depending upon loading amplitude, for copper, characteristic dislocation structures evolve, which differ much more strongly from each other than for AlMg3. In saturation, copper does not show Masing behaviour. The saturation state, depending on amplitude, is reached following various amounts of accumulated plastic strain. On the basis of the experimental results, it appears meaningful to classify into small (D
eq   0.2%), medium ( 0.2% D
eq   1%) and high (D
eq   1%) amplitudes. After Hancock and Grosskreutz [14], in the medium amplitude region (D
eq ˆ  0.375%) at the onset of fatigue, bundles of multipoles initially appear separated by dislocation poor regions. The majority of dislocations in the bundles are primary edge dislocations in parallel slip planes, which mutually interact in some sections to form dipoles and multipoles. Further, as for AlMg3, prismatic loops are formed through jog-dragging processes. Screw dislocations on the other hand are hardly found in this fatigue stage; it is assumed that they are largely annihilated through cross-slip. In the continued course of fatigue, the density of primary and particularly secondary dislocations increases in the bundles. The dipoles are divided into small pieces through cutting processes with dislocations of other slip systems. This causes additional hardening: The dipole ends now present in higher concentrations are less mobile. A similar process is also presumed for AlMg3. The bundles gradually combine to cell-like structures. Finally, elongated dislocation cells are produced, the walls of which are sharply outlined against the dislocation poor interstices. The walls comprise short dipoles of high density. In the dislocation poor regions, screw dislocations stretch from one wall to the next (Figure 2.7 a, proportional loading with D
eq ˆ  0.5%). According to Laird et al. [15], one may expect the spatial arrangement of the structure in Figure 2.7 a to yield approximately cylindrical dislocation cells, the cross-sectional areas of which are shown here. 25

2

Material State after Uni- and Biaxial Cyclic Deformation

a)

b)

c) Figure 2.6: a) Equal phase overlap of tension/compression and alternate torsion, D
eq ˆ  0.5%, saturation, Z = [100], multibeam case; b) 90 8 phase-shifted overlap of tension/compression and alternate torsion, D
eq ˆ  0.5%, saturation, Z = [01-1], g = [-111]; c) deformation band parallel to the (11-1) plane, tension/compression, D
eq ˆ  0.5%, saturation, Z = [001], g = [200].

After 90 8 phase-shifted overlap of tension/compression and alternate torsion, in copper with an equivalent strain amplitude of D
eq ˆ  0.5%, isotropic cells dominate. Their walls are composed of elongated, regularly ordered single dislocations (Figure 2.7 b) as found by Feltner and Laird for the high plastic strain amplitude D
pl ˆ  0.5% [16]. 26

2.3

Results

a) b) Figure 2.7: a) Elongated cells with dipolar walls for copper, tension/compression, D
eq ˆ  0.5%, saturation, Z = [011], g = [1-11]; b) isotropic, non-dipolar cell structure after phase-shifted loading for copper, D
eq ˆ  0.5%, saturation, Z = [011], multibeam case.

The lack of dipolar structures is explained by Feltner and Laird as being due to unhindered cross-slip. The rotating stress vector activates the slip systems required to create isotropic cell structures at even smaller stress amplitudes than in the proportional case. Annihilation of screw dislocations is facilitated, thus producing the dislocation poor inner cell regions. In addition, the enhanced cross slip ability of the screw dislocations suppresses the creation of prismatic loops. Thus copper, for proportional and disproportional loading at D
eq ˆ  0.5%, always exhibits different slip mechanisms. For proportional loading, the screw dislocations glide to and fro parallel to the walls in the dislocation poor areas. At the same time, new screw dislocations are continually being pressed out of the walls until they reach the opposite wall. In between the walls too, new screw dislocations are formed. The walls themselves take part in the slip by flip-flop movement. For disproportional loading, only slip dislocations participate in the deformation; these are pressed out of the walls and after crossing the cell interior are reincorporated into the opposite cell wall. It follows that copper shows an additional hardening effect, which is retained in saturation (cf. Figure 2.4 a). For austenitic steel 1.4404, the additional hardening effect predominates at 90 8 phase-shifted loading with equivalent total strain amplitude of D
eq ˆ  0.75% and  0.5%. Study using the transmission electron microscope shows for disproportional loading that although a large number of stacking faults are produced, there is no deformation-induced martensite. For steel 1.4306, this transformation already occurs at strain amplitudes of D
pl ˆ  0.3% under uniaxial loading [17]. Figure 2.8 a shows a typical dislocation structure after proportional loading with D
eq ˆ  0.75%. The walls of the elongated cells comprise dislocation bun27

2

a)

Material State after Uni- and Biaxial Cyclic Deformation

b) Figure 2.8: a) Dislocation structure after proportional loading, D
eq ˆ  0.75%, saturation; b) dislocation structure after 90 8 phase-shifted loading, D
eq ˆ  0.75%, saturation.

dles with a preferential orientation parallel to the {111} planes. On the other hand, the stronger tendency to multiple slip produces a labyrinthine structure after phase-shifted loading (Figure 2.8 b). The walls are sharply defined against the cell interior.

2.3.3

Yield surfaces

The discussion of yield surface measurements may be exemplified by experiments with equivalent strain amplitude of D
eq ˆ  0.5%. The materials were in the cyclic saturation state.

2.3.3.1 Yield surfaces on AlMg3 Figure 2.9 collates the dynamically measured 0.01% offset yield surfaces for AlMg3 for the four chosen loading types. The starting point each time was the reversal point of the stress hysteresis. The yield surfaces for proportional loading (in the following denoted proportional yield surfaces) are flattened in each relief direction compared with an elliptical shape. The 0.01% surfaces are in general agreement with those presented in [9]: the 2 · 10–4% offset yield surfaces determined by method 1. The yield surfaces determined after disproportional loading (hereafter denoted disproportional yield surfaces) come closest to an isotropic shape (v. Mises ellipse). The 28

2.3

Results

Figure 2.9: 0.01% offset yield surfaces measured in the load reversal points, saturation, AlMg3, D
eq ˆ  0.5%.

proportional yield surfaces, by contrast, show definitely anisotropic shapes. Compared with the axial ratio of the v. Mises ellipse, the values measured perpendicular to the loading direction (transverse yield surface values) are clearly larger than the cross sections found in the loading direction (longitudinal yield surface values). As the comparison of yield surfaces measured at the upper reversal point (Figure 2.9) and at the stress zero crossover (cf. Figure 2.2) shows, both the transverse values and the contracted longitudinal values within a cycle remain constant. The shape of the yield surface, however, changes from the flattened form at the load reversal point to an essentially symmetrical ellipse in the stress zero crossover. During the further course of the negative half-cycle, this then changes into a flattened shape once more (flattening again on the origin side). This deformation may also be observed on yield surfaces with the small offset strain of 2 · 10–4% and on tangent module areas with high tangential gradients. Figure 2.10 shows the proportional and disproportional yield surfaces measured at the load reversal points for the relatively large offset strain of 0.2%. In consequence of the high plastic fraction, during deformation, all four yield surfaces practically coincide and are almost elliptical in shape. Referring to the v. Mises condition or Equation (2), the longitudinal values are slightly less than the transverse ones. The surfaces thus show, in weaker form, the same anisotropy as those measured with small offsets. The torsional yield surface is slightly flattened in the relief direction.

29

2

Material State after Uni- and Biaxial Cyclic Deformation

Figure 2.10: 0.2% offset yield surfaces measured at the load reversal points, saturation, AlMg3, D
eq ˆ  0.5%.

2.3.3.2 Yield surfaces on copper Copper behaves in many aspects like AlMg3. In Figure 2.11, the three proportional yield surfaces measured at the load reversal points are shown in contrast with the disportional surface. The proportional surfaces are again flattened in the relief direction. The shortened axis too remains the same throughout the whole hysteresis cycle. The disproportional yield surface approaches the elliptical shape, which is significantly larger. The distinct additional hardening effect of copper thus causes an additional isotropic hardening.

2.3.3.3 Yield surfaces on steel Figure 2.12 shows 0.02% offset yield surfaces measured after equal phase superposition at the upper and lower reversal points of the saturation hysteresis. As for AlMg3 and copper, the displacement of the yield surface in the loading direction and the flattening on the origin side are clearly seen. Figure 2.13 represents the 0.02% offset yield surfaces measured after disproportional loading at the reversal points of tension and compression ( = 0). For this load path, the yield surface follows the rotating stress vector. Both yield surfaces are symmetrical to the tensile stress axis and again show the typical flattening on the origin side.

30

2.4

Sequence Effects

Figure 2.11: 0.01% offset yield surfaces measured at the load reversal points, saturation, copper, D
eq ˆ  0.5%.

Figure 2.12: 0.02% offset yield surfaces measured at the load reversal points, proportional loading (tension/compression and alternate torsion), saturation, steel 1.4404, D
eq ˆ  0.5%.

2.4

Sequence Effects

On AlMg3 and copper in the saturation state, the variation of the loading direction from tension/compression to alternating torsion, and the reverse, was investigated. The experiments were meant to show how inner stresses affect the shape of the yield surfaces. For the offset strain of 0.01%, the points of yield onset were taken from the as31

2

Material State after Uni- and Biaxial Cyclic Deformation

Figure 2.13: 0.02% offset yield surfaces measured at the load reversal points of the tension/compression hysteresis, disproportional loading, saturation, steel 1.4404, D
eq ˆ  0.5%.

cending and descending branches of the hysteresis curves and from these, the yield surfaces’ diameters (cross sections) determined. In similar fashion, the diameters of the surfaces with equal tangent modules were also determined [9, 18]. In addition, the transition of the maximum stress amplitude to the new saturation state was observed. Figure 2.14 shows the change of the 0.01% yield surface diameter of AlMg3 and copper (equivalent total strain amplitude D
eq ˆ  0.5%) for the transition from pure tension/ compression to pure alternating torsion. The broken lines show the transverse values of the tension/compression saturation yield surface, respectively (state before the change). The continuous lines represent the longitudinal values of the saturation yield surfaces, which would have appeared following pure torsion. The yield surfaces’ diameters of torsion hysteresis in the case of AlMg3 already decrease drastically in the first cycle and quickly reach a new saturation state yet without recurring to the saturation longitudinal value following pure torsion. The new loading state must therefore differ from the initial state with regard to the inner stress, or else, in consequence of isotropic hardening, the saturation yield surfaces are larger after prior tension/compression than after pure alternating torsion. The second option is confirmed by the dislocation structure. As already demonstrated, for AlMg3, an anisotropic dislocation structure evolves after proportional loading. In extensive grain areas, only few slip systems are activated; the dipolar walls generally take up only one or two crystallographic directions. Since different slip systems are involved in tension/compression loading than in alternating torsion, the dipolar walls orient themselves in different crystallographic directions. The screw dislocations move in dislocation poor channels parallel to the dipolar walls, adjacent to the respective slip systems. If a tension/compression experiment is immediately followed by one with alternating torsion, the dislocation structure is initially unfavourable for torsion. With changing loading direction, the sources of torsional slip dislocations are activated first and then later on, the dipolar walls change their orientation to one more favourable for torsional loading. As is seen from Figure 2.14, the greater fraction of the torsional slip dislocations is already activated in the first three cycles after the change of 32

2.4

Sequence Effects

Figure 2.14: AlMg3 and copper, 0.01% yield surface diameter after changing the loading direction from tension/compression to alternating torsion.

the loading direction. The result is an immediate drastic reduction of the yield surface diameter. However, further restructuring of the dislocation arrangement appears to be impeded; the diameter remains constant during subsequent cycles. The dislocation structure anticipated for pure torsional loading is clearly unable to evolve following previous tension/compression loading. The large number of activated slip systems after the change in loading direction may offer some explanation. For copper, the yield surface diameter from torsional hysteresis also seriously decreases in the first torsional half-cycle after changing the loading type. Moreover, in contrast to AlMg3, it falls continuously until the saturation longitudinal value for pure torsion is reached. The longitudinal values for saturation yield surfaces thus come about independent of previous history. The same is true for the diameters of the tangent module areas. The dislocations in copper arrange themselves in a similarly isotropic way as in AlMg3 (elongated cells, dipolar walls: see Figure 2.7 a). Yet after the change in loading direction, they reorientate themselves completely. This property characterizes materials with wavy slip behaviour [15]. In a further experiment to assess the effect of inner stress, samples of AlMg3 and copper were relieved from various points in the torsional hysteresis branch. The yield surface diameters were taken from these partial cycles and plotted in Figure 2.15 as a function of the offset strain and of the strain values (initial stress relieving points). 33

2

Material State after Uni- and Biaxial Cyclic Deformation

Figure 2.15: Yield surface longitudinal values for various offsets determined after stress relief from various points on the torsional hysteresis.

For a given offset strain, both materials showed the same yield surface longitudinal values at every point in the hysteresis. The influence of inner stress, assumed load dependent, upon the shape of the yield surfaces would therefore not appear to be significant. It seems that the dislocation structure exerts the critical influence.

2.5

Summary

We have presented yield surfaces on AlMg3, copper and austenitic steel 1.4404 (AISI 316L) after tension/compression and alternating torsional loading as well as proportional and phase-shifted superposition of both loads. The materials were first cycled to saturation with maximum deformation amplitudes of  0.75%, whereby substantial additional hardening effects occurred. The development of the appropriate dislocation structures was studied using a transmission electron microscope. 34

References Yield surfaces measured in all three materials at the reversal points of the stress deformation hysteresis, for small offset strains (0.01% or 2 · 10–4%), after proportional alternate loading, show a flattened shape in the off-load direction compared with the v. Mises ellipse. At the stress zero crossover points of the hysteresis, the yield surfaces assume a symmetrical shape. Transverse and longitudinal values of the yield surfaces remain constant independent of the starting point in the hysteresis. This behaviour and the sequence effects confirm that the anisotropy of the yield surfaces is caused by the appropriately anisotropic dislocation structure of the materials. Inner stresses obviously play a minor role. After disproportional loading, generally isotropic yield surfaces result. This may be explained quite simply by the relevant isotropic dislocation structures. Yield surfaces of higher offset strains and areas of equal tangent modules for small tangential gradients also evolve essentially isotropically since sufficient slip systems are activated during the measurement procedure and the dislocation walls participate in the slip process.

Acknowledgements The authors thank Mr. Horst Gasse for his decisive contribution to the development of the experimental apparatus, the measuring technique and the performance of the experiments.

References [1] Y. F. Dafalias, E. P. Popov: Plastic Internal Variables Formalism of Cyclic Plasticity. Journal of Applied Mechanics 63 (1976) 645–651. [2] Y. F. Dafalias: Bounding Surface Plasticity, I Mathematical Foundation and Hypoplasticity. Journal of Engineering Mechanics 12 (9) (1986). [3] D. L. McDowell: A Two Surface Model for Transient Nonproportional Cyclic Plasticity, Part 1: Development of Appropriate Equations, Part 2: Comparison of Theory with Experiments. Journal of Applied Mechanics 85 (1986) 298–308. [4] F. Ellyin: An isotropic hardening rule for elastoplastic solids based on experimental observations. Journal of Applied Mechanics 56 (1969) 499. [5] N. K. Gupta, H. A. Lauert: A study of yield surface upon reversal of loading under biaxial stress. Zeitschrift fu¨r angewandte Mathematik und Mechanik 63(10) (1983) 497–504. [6] J. F. Williams, N. L. Svensson: Effect of torsional prestrain on the yield locus of 1100-F aluminium. Journal of Strain Analysis 6(4) (1971) 263. [7] R. Hillert: Austenitische Sta¨hle bei ein- und bei zweiachsiger, plastischer Wechselbeanspruchung. Dissertation TU Braunschweig, 2000. [8] W. Gieseke, G. Lange: Vera¨nderung des Werkstoffzustandes bei mehrachsiger plastischer Wechselbeanspruchung. In SFB Nr. 319 Arbeitsbericht 1991–1993, TU Braunschweig. 35

2

Material State after Uni- and Biaxial Cyclic Deformation

[9] W. Gieseke: Fließfla¨chen und Versetzungsstrukturen metallischer Werkstoffe nach plastischer Wechselbeanspruchung. Dissertation TU Braunschweig, 1995. [10] H. J. Christ: Wechselverformung der Metalle. In: B. Ilschner (Ed.): WFT Werkstoff-Forschung und Technik 9, Springer Verlag Berlin, 1991. [11] J. I. Dickson, J. Boutin, G. L. ’Espe´rance: An explanation of labyrinth walls in fatigued f.c.c. metals. Acta Metallurgica 34(8) (1986) 1505–1514. [12] J. L. Dickson, L. Handfield, G. L. ’Espe´rance: Geometrical factors influencing the orientations of dipolar dislocation structures produced by cyclic deformation of FCC metals. Materials Science and Engineering 81 (1986) 477–492. [13] P. Luka´s, M. Klesnil: Physics Status solidi 37 (1970) 833. [14] J. R. Hancock, J. C. Grosskreutz: Mechanisms of fatigue hardening in copper single crystals. Acta Metallurgica 17 (1969) 77–97. [15] C. Laird, P. Charlsey, H. Mughrabi: Low energy dislocation structures produced by cyclic deformation. Materials Science and Engineering 81 (1986) 433–450. [16] C. E. Feltner, C. Laird: Cyclic stress-strain response of FCC metals and alloys II. Dislocation structures and mechanism. Acta Metallurgica 15 (1967) 1633–1653. [17] M. Bayerlein, H.-J. Christ, H. Mughrabi: Plasticity-induced martensitic transformation during cyclic deformation of AISI 304L stainless steel. Materials Science and Engineering A 114 (1989) L11–L16. [18] W. Gieseke, G. Lange: Yield surfaces and dislocation structures of Al-3Mg and copper after biaxial cyclic loadings. In: A. Pineau, G. Cailletaud, T. C. Lindley (Eds.): Multiaxial fatigue and design, ESIS 21, Mechanical Engineering Publications, London, 1996, pp. 61– 74.

36

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)

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue: Description of Deformation Behaviour and Creep-Fatigue Interaction Kyong-Tschong Rie, Henrik Wittke and Ju¨rgen Olfe *

Abstract Results of low-cycle fatigue tests are presented and discussed, which were performed at the Institut fu¨r Oberfla¨chentechnik und plasmatechnische Werkstoffentwicklung of the Technische Universita¨t Braunschweig, Germany. The cyclic deformation behaviour was investigated at room temperature and high temperatures. The investigated materials are copper, 2.25Cr-1Mo steel, 304L and 12%Cr-Mo-V steel. (Report of the projects A5 and B4 within the Collaborative Research Centre (SFB 319) of the Deutsche Forschungsgemeinschaft.)

3.1

Introduction

Low-cycle fatigue (LCF) and elasto-plastic cyclic behaviour of metals represent a considerable interest in the field of engineering since repeated cyclic loading with high amplitude limit the useful life of many components such as hot working tools, chemical plants, power plants and turbines. During loading in many cases after a quite small number of cycles with cyclic hardening or softening, a state of cyclic saturation is reached. This saturation state can be characterized by a closed stress-strain hysteresisloop. 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. The crack growth is correlated with parameters of fracture mechanics, which take into account informations especially about teh steady-state stress-strain hysteresis-loops. * Technische Universita¨t Braunschweig, Institut fu¨r Oberfla¨chentechnik und plasmatische Werkstoffentwicklung, Bienroder Weg 53, D-38106 Braunschweig, Germany 37

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Therefore, a more exact life prediction is possible by investigating the cyclic deformation behaviour in detail and describing the cyclic plasticity, e.g. with constitutive equations. In this paper (see Section 3.3), the investigated cyclic deformation behaviour was described by analytical relations and, moreover, by relations, which take into account physical processes as the development of dislocation structures. When components are loaded at high temperature, additional processes are superimposed on the fatigue. Besides corrosion, which is not discussed here, creep deformation and creep damage are the most important. Therefore in many cases, not one type of damage prevails, but the interaction of both fatigue and creep occurs leading to failure of components. A reliable life prediction model for creep-fatigue must consider this interaction as proposed by the authors (see Section 3.4.1). In this model, the propagating crack, which is the typical damage in the low-cycle fatigue regime, interacts with grain boundary cavities. Cavities are for many steels and some other metals the typical creep damage and also play an important role in the case of creep-fatigue. The possibility of unstable crack advance, which is the criterium for failure, is given if a critical configuration of the nucleated and grown cavities is reached. Therefore, the basis for reliable life prediction is the knowledge and description of the cavity formation and growth by means of constitutive equations. 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 pore nucleation on grain boundaries and the cyclic dependence of this process have 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 interactions between the cavities (see Section 3.4.2). Formerly, the total stress and strain have been used for the calculation of the creep-fatigue damage. However, these are macroscopic parameters, whereas the crack growth is a local phenomenon. Therefore, the local conditions near the crack tip have to be taken into consideration. The determination of the strain fields in front of cracks is an important first step for modelling (see Section 3.4.3).

3.2

Experimental Details

3.2.1

Experimental details for room-temperature tests

The materials used for the uniaxial fatigue tests at room temperature were polycrystalline copper and the steel 2.25Cr-1Mo (10 CrMo 9 10). Specimens of 2.25Cr-1Mo were investigated in as-received conditions, in the case of copper, the material was annealed at 650 8C for 1/2 h.

38

3.2 Experimental Details The tests were controlled by total strain and carried out at room temperature in air. The strain rates were
_ = 10–3 s–1 (or, for a small number of tests,
_ = 2 · 10–3 s–1) for steel, and
_ = 10–4 s–1 and
_ = 10–3 s–1 for copper. Most of the tests were single-step tests (SSTs) with a constant strain amplitude D
/2, some tests were performed as two-step tests (2STs) and other as incremental-step test (ISTs). In the case of the two-step test, the specimens had been cycled to a steady-state regime before the strain amplitude was changed in the next step. The strain amplitudes were in general in the low-cycle fatigue range and a few amplitudes in the range of high-cycle fatigue (HCF) and in the transition regime between low-cycle fatigue and extremely low-cycle fatigue (ELCF): The tests with copper were performed with strain amplitudes between 0.1 and 1.7%, the tests with steel with amplitudes between 0.185 and 1.2%. The incremental-step tests were carried out with constant strain rate and with given values for the lowest and the highest strain amplitude, (D
/2)min and (D
/2)max. The factor of subsequent amplitudes qa in the ascending part of the IST-block or, alternatively, the difference of amplitudes da is constant. For most of the tests, smooth cylindrical specimens were used. Usually, the diameter and the length of the gauge were 14 mm and 20 mm, for the tests with very high strain amplitudes (near the ELCF-regime), the diameter was 14.7 mm and the length 10 mm. For some tests, flat specimens were used with the values 8.7 × 5 mm2 for the rectangular cross-section. The steady-state microstructure of tested specimens was investigated with transmission electron microscope at the Institut fu¨r Schweißtechnik (Prof. Wohlfahrt [1]), the Institut fu¨r Metallphysik und Nukleare Festko¨rperphysik (Prof. Neuha¨user [2]) and the Institut fu¨r Werkstoffe (Prof. Lange [3]). They are all at the Technische Universita¨t Braunschweig and involved in the Collaborative Research Centre (SFB 319).

3.2.2

Experimental details for high-temperature tests

The creep-fatigue tests were carried out on 304L austenitic stainless steel and on 12% Cr-Mo-V ferritic steel. The tests were total strain-controlled low-cycle fatigue tests with a tension hold time up to 1 h at 600 8 and 650 8C for the 304L, and 550 8C for the ferritic steel. For the tests for the lifetime determination and the tests for analysing the cavity configuration, we used round and polished specimen. After low-cycle fatigue testing, the specimens were metallographically prepared for stereological analysis of the density and cavity size distribution (see Section 3.4.2.1). A furnace with a window and special optics allow high magnification observation of the specimen surface continuously during the test with a video system and a subsequent measurement of the crack growth, the crack tip opening and the crack contour on flat and polished specimens in an inert atmosphere. In-situ measurement of the strain field in front of the crack was performed by means of the grating method [4–9]. The surface of the specimen was prepared with a grating of TiO2 with a line distance of 200 lm, which was photographed at the beginning of the test and at given loads after cycling. By means of digital image analysis, the local strain at every cross of the grating was calculated by the group of Prof. Ritter [10] and Dr. Andresen [11] 39

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.1: Deformed grid and corresponding strain in the direction of the load (in-situ, 550 8C).

(TU Braunschweig, Collaborative Research Centre (SFB 319)). The following picture (Figure 3.1) shows a photograph of the grid and the digitized picture with the regions, where the local strain is higher than 4% and 5%. The position of the crack is illustrated by means of a straight line, and the line, which surrounds the 5% deformed zone at the crack tip, is shown in the figure. From the figure, the size of the 5% deformed zone in direction of its maximum expansion was taken. In the following, this distance was designed as R0.05 in analogy to Iino [12]. It has been used to describe the development of the highly deformed zone in dependence on the crack length and the tension hold time.

3.3

Tests at Room Temperature: Description of the Deformation Behaviour

3.3.1

Macroscopic test results

In single-step tests, annealed copper shows cyclic hardening in nearly the whole range of lifetime. After a quite small number of cycles, the end of a rapid hardening regime is reached. Due to the effect of secondary hardening, in some ranges of amplitudes, no saturation was observed, but, as first approximation, the effect of secondary hardening can be neglected [13]. Examples for cyclic hardening curves up to saturation are shown in Figure 3.2 a. In the case of single-step tests with 2.25Cr-1Mo, there is cyclic softening in nearly the whole range of strain amplitudes. In the first cycles, rapid hardening can be found before cyclic softening takes place. After this, a steady-state regime can be 40

3.3 Tests at Room Temperature: Description of the Deformation Behaviour

Figure 3.2: Copper; a) cyclic hardening curves,
_ = 10–4 s–1; b) cyclic stress-strain curves: amplitudes of applied and internal stress vs. amplitude of plastic strain. &: data of SSTs with
_ = 10–3 s–1; *: data of SSTs and 2STs with
_ = 10–4 s–1; n, ~: data of stress relaxation tests after SSTs with
_ = 10–3 s–1 or
_ = 10–4 s–1, respectively.

found, which continues until a failure takes place. While in the case of copper, there is a very clear effect of rapid hardening, in the case of 2.25Cr-1Mo, the effects of cyclic hardening and softening are less pronounced. For both materials, after a certain number of cycles, a state of saturation is achieved. The stress-strain behaviour is represented by a hysteresis-loop. (To avoid confusion, it may be useful to mark characteristic values of the steady-state hysteresis-loop with an index. For example, the amplitude of stress D/2 can be written in the case of saturation as (D/2)s. Nevertheless, no index is used in this paper because it is usually clear from the context whether the instantaneous or the steady-state values are referred.) In Figure 3.2 b, an example for cyclic stress-strain curves, D/2 or Di/2 vs. D
p/2, are shown, which are constructed with the aid of steady-state hysteresis-loops. The values of the plastic strain
p are given in dependence on total strain
and stress  by:
p

 =E ;

1

where E is the Young’s modulus. This equation is used to describe also the relation between the amplitudes of plastic strain D
p/2, total strain D
/2 and stress D/2. The amplitudes of the internal stress, Di/2, are found with the aid of stress relaxation tests (see [14]). Most of the experimental points shown in Figure 3.2 b were found from 24 tests with amplitudes in the range of LCF (single-step tests and two-step tests with low-high amplitude-sequences; 0:16%  D
=2  1:0%. Additionally, one test in the high-cycle fatigue (HCF) regime and three tests in the transition regime between LCF and extremely low-cycle fatigue (ELCF: compare Komotori and Shimizu [15]) are taken into consideration. In the case of copper, the 24 tests are used to study various 41

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.3: Steady-state stress-strain hysteresis-loops; a) 2.25Cr-1Mo,
_ = 10–3 s–1; b) copper, hysteresis-loops in relative coordinates.

parameters of the material, in the case of 2.25Cr-1Mo, data are used from 13 singlestep tests (12 LCF-tests, one HCF-test). Examples for steady-state hysteresis-loops are shown in the Figure 3.3 a and b. In Figure 3.3 a, stress-strain hysteresis-loops of 2.25Cr-1Mo are shown, in Figure 3.3 b, hysteresis-loops in relative coordinates, r and
r, are shown in the case of copper. The relative coordinate system is defined by an origin, which is set at the point of minimum stress and strain of the hysteresis-loop. The material exhibits Masing behaviour when the upper branches of different hysteresis-loops follow a common curve in the relative coordinate system. In contrast, copper exhibits non-Masing behaviour in single-step tests as can be seen in Figure 3.3 b. Also for 2.25Cr-1Mo, non-Masing behaviour was found. Only in a small range of the tested amplitudes, in the range of 0.185% < D
/2 < 0.4%, the steel exhibits approximately Masing behaviour. For many materials with non-Masing behaviour, it is possible to get a “master curve”, which is obtained from matching the upper branches of the hysteresis-loops through translating each loop along its linear response portion (see Jhansale and Topper [16], Lefebvre and Ellyin [17]). The construction of the master curve is possible for the tested materials in good approximation (Schubert [18], Rie et al. [19]). This behaviour is shown in Figure 3.4 for copper with the relative plastic strain
pr as the x-axis. In two-step tests with low-high amplitude-sequence, a saturation amplitude can be found, which is equal to that of an equivalent single-step test. In the range of the tested amplitudes, this behaviour can also be found in good approximation in tests with highlow amplitude-sequences. The materials are nearly history-independent (compare Feltner and Laird [20] and Hoffmann et al. [21]). Also in incremental-step tests, a state of cyclic saturation can be found. In contrast to the stress-strain behaviour in single-step tests and two-step tests, the steadystate stress-strain behaviour in incremental-step tests can be approximately expressed by Masing behaviour (see [8, 13]). 42

3.3 Tests at Room Temperature: Description of the Deformation Behaviour

Figure 3.4: Copper (_
= 10–4 s–1) shifted; hysteresis-loops and master curve.

3.3.2

Microstructural results and interpretation

For both materials, dislocation cell structures were found. For 2.25Cr-1Mo, cell structure was found in single-step tests in the range of amplitudes, in which the materials exhibit non-Masing behaviour. In the case of single-step tests with copper, the cell structure is well developed for high amplitudes, for low amplitudes, other dislocation structures are dominating as e.g. vein structure. Often, the shape of the cells is not cuboidal but elongated. With increasing strain amplitude, the cell size is decreasing (compare Feltner and Laird [22]). Schubert [18] proposed a microstructure-dependent cyclic proportional limit prop
L  2 MS G b=dm ;

2

where L is the lattice friction stress, MS is the Sachs factor, G is the shear modulus and b is the absolute value of the Burgers vector. The decrease of the mean cell size dm and the increase of prop with increasing strain amplitude is in agreement with the non-Masing behaviour of the materials [13, 14]. For 2.25Cr-1Mo, the value of dm in Equation (2) corresponds to the mean distance of precipitates for low amplitudes and to the mean cell size for high amplitudes (D
/2 > 0.4%). Therefore, in the case of low amplitudes, Masing behaviour was found [18]. A typical steady-state dislocation structure of the second step of a two-step test with an amplitude-sequence high-low is shown in Figure 3.5. A dislocation cell structure can be seen although the dominating structure of the low amplitude in the case of a single-step test is vein structure (see [18]). While in two-step tests with amplitude-sequences low-high the microstructure is history-independent, it is obviously not independent in the case of a test with an amplitude-sequence high-low (compare [21]). Nevertheless, the dependence of the macroscopic behaviour on this history-dependent microstructural behaviour is almost negligible. In incremental-step tests with sufficiently high values of (D
/2)max, dislocation cell structure can be found in cyclic saturation. The dislocation structure is assumed as 43

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.5: Copper, dislocation cell structure of a two-step test;
_ = 10–4 s–1, strain amplitude sequence 0.4–0.2%: steady-state dislocation structure of the second step.

quasi-stable: In cyclic saturation, the dislocation structure does not change within one IST-block. The quasi-stable dislocation structure correlates well with the Masing behaviour of the incremental-step test (for details: see Schubert [18]). Experimental values of the dislocation cell size or cell wall distance, respectively, are: dm
0:85 l for D
=2
0:2% ; dm
0:76 l for D
=2
0:4% ; dm
0:58 l for D
=2
0:7% ; in the case of copper and SSTs for
_
104 s1. In the case of 2.25Cr-1Mo, SSTs,
_
103 s1 , the experimental values are: dm
0:85 l for D
=2
0:6% ;

and

dm
0:65 l for D
=2
1:2% 13 : These values were used to calculate the cyclic proportional limit prop, and a good agreement with the macroscopic cyclic proportional limit defined by an offset of 0.01% was found [18]. Moreover, the values of [18] are used for further evaluation (Sections 3.3.4.1 and 3.3.4.2).

44

3.3 Tests at Room Temperature: Description of the Deformation Behaviour

3.3.3

Phenomenological description of the deformation behaviour

3.3.3.1 Description of cyclic hardening curve, cyclic stress-strain curve and hysteresis-loop It is shown by Wittke [13] that the first part of a cyclic hardening curve of a single-step test, the rapid hardening regime, can be described excellently with a stretched exponential function for stress amplitude D/2 vs. cycle number N (or more exact: N – 0.25): h  i 3 D=2
A0  As  A0  1  exp N  0:25=N0 H : The constants A0 and As are closely related to the monotonous and cyclic stress-strain curve, respectively, the constants N0 and H are found by trial and error. A simple dependence of the parameters on the steady-state value of the plastic portion of the total strain amplitude can be found [8]. Moreover, the stretched exponential function, D/2 vs. N, is applicable also for two-step tests in the case of hardening and softening in good approximation. The comparison between experimental and calculated cyclic hardening curves is given in Figure 3.2 a. It is usual to describe the cyclic stress-strain curve (css-curve) by a power law. As can be seen in Figure 3.2 b, in the case of copper, the description of the cyclic stress-strain curves by the solid line and the dotted line is quite good. The double-logarithmic cyclic stress-strain curves, D/2 vs. D
p/2, for different strain rates are nearly parallel. Also in the case of 2.25Cr-1Mo, the description of the cyclic stress-strain curve by a power law is good. In the case of 2.25Cr-1Mo, we get with

D=2
k D
p =2n

4

and by using the constants k
803 MPa and n
0:138 good agreement between experimental and calculated values _

103 s1 ; E
208 GPa. For copper, the values of the constants for the different css-curves in Figure 3.2 b are: k
554:6 MPa ; n
0:228

for
_
103 s1 ;

k
565:9 MPa ; n
0:238

for
_
104 s1 ; and

k
441:3 MPa ; n
0:220

for internal stress measurements tests :

With regard to fatigue fracture mechanics and lifetime estimation, the description of the steady-state hysteresis-loop is the most important point in this Section 3.3. In first approximation, also in the case of the hysteresis-loop, a power law between relative stress and relative plastic strain, r and
pr, can be assumed (see Morrow [23]): r
kH
bpr :

5

It should be mentioned that the parameters kH and b are dependent on the plastic strain amplitude. 45

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Although the description of the hysteresis-loop with a power law is quite rough, it may be useful to apply such a law for fracture mechanical estimation (see Rie and Wittke [24]). To get a better description of the hysteresis-loop shape, other relations are necessary. The hysteresis-loop can be described, e. g. by a two-tangents method (compare [19]), as follows:
pr
r =k0 1=b0  r =kE 1=bE :

6

For each hysteresis-loop, four constants, k0, b0, kE and bE, have to be determined. An example for the applicability of this relation in the case of the mild steel Fe510 (St-52), which was tested at the Institut fu¨r Stahlbau of the TU Braunschweig (Prof. Peil [25]), is shown in Figure 3.6. We have developed other very exact relations with only three constants. They are expressed by: r
AG 1  exp 
pr =dG G  ;

7

or alternatively by: r
Cq exp q ln 
pr =dE 2  :

8

The three constants of the stretched exponential function (Equation (7)) are G, dG and G, the constants of the exponential parabola function (Equation (8)) are Cq, jq and dE. Examples for the excellent applicability of both equations are shown by Rie and Wittke [14] and Wittke [13]. In contrast to other relations, in the case of Equations (7) and (8), a good agreement between experiment and calculation can be found even for the second derivative of the hysteresis-loop branch, d2 r/d
2r vs.
r (see Section 3.3.4.1).

Figure 3.6: Mild steel Fe510; hysteresis-loop in relative coordinates; D
/2 = 0.5%,
_ = 10–4 s–1; comparison between experiment and calculation; calculation according to Equation (6), k0 = 45418 MPa, b0 = 0.553, kE = 1238 MPa, bE = 0.095; E = 210 GPa. 46

3.3 Tests at Room Temperature: Description of the Deformation Behaviour 3.3.3.2 Description of various hysteresis-loops with few constants A very exact description of the shape of various hysteresis-loops with few constants can be obtained when the parameters of the power law (Equation (5)), kH and b, are given as simple functions of the plastic strain range D
p. Such functional relations are developed in [13, 26]. Furthermore, a method to calculate the parameters of the exponential parabola function (Equation (8)), Cq, q, and dE, in dependence on the plastic strain amplitude is described in [13]. As shown above, in the case of non-Masing behaviour, it is possible to get a master curve. This master curve together with the cyclic stress-strain curve can be used to construct each hysteresis-loop (see [17]). In contrast to the power-law master-curve proposed by Lefebvre and Ellyin [17], better results were achieved, e.g. by a stretched exponential function or an exponential parabola function (see [13]). In the latter case, the master curve can be described by: 
Cq exp q ln 
pr =dE 2  ;

9

where Cq , jq and dE are constants. For copper, almost independent on strain rate, the values of the parameters of the master curve (compare Figure 3.4) are: Cq
246:1 MPa ; jq
0:03576 ; dE
2:3194% : All these methods, which were used to describe various steady-state hysteresis-loops of copper with few constants, are also applicable in the case of 2.25Cr-1Mo.

3.3.4

Physically based description of deformation behaviour

3.3.4.1 Internal stress measurement and cyclic proportional limit For a physically based description of the cyclic deformation behaviour, it is necessary to take into consideration that the applied stress  can be separated into the internal and the effective stress, i and eff. The effective stress is that fraction of the total stress causing dislocations to move at a specific velocity, the internal stress can be defined as the stress needed to balance the dislocation configuration at a net zero value of the plastic strain rate (see Tsou and Quesnel [27]). At room temperature, internal stress can be easily obtained experimentally by stress relaxation tests. For this purpose, test specimens were cycled to approximated saturation in uniaxial push-pull tests in the range of LCF prior to the relaxation tests. Figure 3.7 a shows hysteresis-loops for copper with both the total stress  and the internal stress i plotted vs. the plastic strain
p. In agreement with the method of Tsou and Quesnel [27], the stress value after 30 min of relaxation is adopted as the internal stress value. Figure 3.7 b shows for a stress relaxation test performed after a monotonic strain-controlled tension test (_
 104 s1  that this is a good approximation: After 47

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.7: Copper; a) hysteresis-loops: total and internal stress vs. plastic strain; D
/2 = 0.4%,
_ = 10–3 s–1; *: experimental data of stress relaxation tests, - - - - - -: calculation analogous to Equation (7), AG = 260 MPa, dG = 0.039%, G = 0.412; b) data of strain-controlled tension test (interrupted at
= 9.3%) and relaxation test.

less than 30 min = 1800 s, the stress value is almost constant. (In Figure 3.7 b, t = 0 is defined by the start of the stress relaxation procedure.) The twelve experimental points of Figure 3.7 a are expressed by the dotted-line fit curve. The fit curve can be described by a relation, which is analogous to Equations (7) or (8), respectively. By considering different i-
p-hysteresis-loops, the cyclic stressstrain curve Di/2 vs. D
p/2 can be determined. This cyclic stress-strain curve has been shown already in Figure 3.2 b. With the same amplitude of the plastic strain, the shape of the i-
p-hysteresis-loop is assumed to be independent of the strain rate of the prior cyclic test (compare Tsou and Quesnel [27], Hatanaka and Ishimoto [28]). The proposed Equations (7) and (8) are well appropriate to fit the experimental points. Therefore, one of them (here Equation (8)) is used in the following for checking whether the i-
p-hysteresis-loops exhibit Masing or non-Masing behaviour. To investigate the Masing or non-Masing behaviour of the i-
p-hysteresis-loops, several hysteresis-loops are presented in Figure 3.8 a in relative coordinates. In this figure, ir-
pr-hysteresis-loops for a small, a medium and a quite large strain amplitude of the LCF range are shown. Non-Masing behaviour can be seen clearly. In the following, the dependence of the non-Masing behaviour on microstructure will be quantified with the model of Schubert [18]. As usual, a macroscopic cyclic proportional limit can be defined by a strain offset, e. g. 0.01%. Nevertheless, the value of the strain offset is arbitrary and has no physical meaning. Therefore, in the case of the ir-
pr-hysteresis-loops, a better way is chosen: At first, a hypothetic hysteresis-loop ir vs.
r is constructed with the given values of ir,
pr and the analogous relation to Equation (1):
r

pr  ir =E : 48

10

3.3 Tests at Room Temperature: Description of the Deformation Behaviour In the next step, the second derivative of a half branch of this hysteresis-loop, d2 ir/d
2r vs.
r, is constructed. The
r-value of the extreme point (minimum) of this second derivative is called here
r, ex. Now, we define a macroscopic cyclic yield stress: yc

r; ex E=2

11

in agreement with a statistical approach based on the distribution of elementary volumes with different yield stresses (compare Pola´k et al. [29]): yc is interpreted as the yield stress with the highest probability density within the material. By this definition, an uniquely applicable and physically better justified macroscopic cyclic proportional limit is found. With the values of dm and prop (see Equation (2)) for copper given by Schubert [18] or Rie et al. [19], respectively, the good agreement between the two cyclic proportional limits, prop and yc, can be seen in Figure 3.8 b. The effective stresses contribute also to the non-Masing behaviour of materials, but in agreement with the above mentioned model, the main reason of the non-Masing behaviour is thought to be governed by the non-Masing behaviour of the i-
p-hysteresis-loops. In the case of 2.25Cr-1Mo, the described model is also applicable. Evaluation of a stress relaxation test for another charge of the material give a value of Di/D = 0.907 for D
/2 = 0.6%. This value is quite similar to the tests with copper.

Figure 3.8: Copper; a) hysteresis-loops of relative internal stress vs. relative plastic strain (without experimental values; dotted line hysteresis-loops: calculation by Equation (8)); parameters of the former performed single-step tests: strain rate
_ = 10–4 s–1; strain amplitudes D
/2: 0.16%, 0.4% and 0.7%; b) cyclic proportional limits, prop and yc, in dependence on plastic strain range. The values of prop are calculated in dependence on experimental values of dm; the values of yc are determined with the aid of the i-
p-hysteresis-loops and described by the dotted-line fit function.

49

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

3.3.4.2 Description of cyclic plasticity with the models of Steck and Hatanaka By Schlums and Steck [30], a model was proposed, which allows to describe high-temperature cyclic deformation behaviour in terms of metal physics and thermodynamics. A modification was given by Gerdes [31] to use the model for low temperatures (e. g. room temperature). The model was applied in the case of copper. According to this model, the plastic strain rate can be calculated as follows:
_ p
CG exp



QG  RT



sinh



DVG   i  RT



:

12

The evolution equations of the internal parameters, i and DVG (internal stress and activation volume), are: _ i
HG E exp



 b G DVG i sign   i  
_ p RT

13

and DV_ G
K1 DVG2 _
p  K2 DVG _
p ;

14

where R = 8.3147 · 10–3 kJ mol–1 K–1, T = 293 K, QG = 49.0 kJ mol–1. The Young’s modulus is dependent on temperature (room temperature: E = 116 GPa), the other constants, CG, QG, HG, bG, K1, K2, and the initial value of the activation volume DVG0, have to be determined, e. g. by a parameter identification procedure. The original model is three-dimensional, but here it is used only in the uniaxial case. In cooperation with the Institut fu¨r Allgemeine Mechanik und Festigkeitslehre (Prof. Steck) at the Technische Universita¨t Braunschweig, a set of parameters was found. This set of parameters takes the results of internal stress measurements and the dependence of the deformation behaviour on strain rate into account and is given by: CG
0:3670 105 s1 ; HG
1:784 ; b G
0:3676 ; K1
47:20 MPa mol kJ1 ; K2
10:328 ; and DVG0
1:182 kJ mol1 MPa1 : With these parameters, a good description of rapid hardening and cyclic saturation is possible [13]. Results in the case of saturated hysteresis-loops are shown in Figure 3.9 a. It can be shown that the model describes the non-Masing behaviour of the material in single-step tests. Furthermore, a relatively exact description of the hysteresis-loop shape is possible. Some modifications seem to be necessary because the parameters are valid only in a limited range of amplitudes and strain rates. More modifications are needed to describe the stress-strain behaviour also in the case of incremental-step tests and two-step tests with sufficient accuracy (for details: see Wittke [13]). 50

3.3 Tests at Room Temperature: Description of the Deformation Behaviour

Figure 3.9: Application of physical based models: comparison of experiment and calculation; a) copper,
_ = 10–4 s–1, calculation with the Steck model [30]; b) 2.25Cr-1Mo,
_ = 10–3 s–1, calculation with the modified model of Hatanaka [28] (calculation: solid line; experiment: dotted line).

By Hatanaka and Ishimoto [28], another physically based model was proposed to describe cyclic plasticity. In this model, assumptions are made concerning the evolution of dislocation density and concerning the mean dislocation velocity. We have modified the original model by taking into account also the evolution of dislocation structure (for details: see [13]). It is shown that the modified model can be applied for copper and also for steady-state hysteresis-loops for 2.25Cr-1Mo [13]. An example for the latter case is shown in Figure 3.9 b.

3.3.5

Application in the field of fatigue-fracture mechanics

Usually, crack growth data are correlated with a fracture mechanical parameter such as e. g. DJ or DJeff. According to the proposals of Dowling [32] and with the results of Shih and Hutchinson [33], it is possible to estimate DJ in the case of various specimen and crack geometries. Schubert [18] measured the growth of cracks, which were approximated as half circular surface cracks in circular specimens. Crack growth of 2.25Cr-1Mo was measured with the ACPD method. In crack closure measurements, a crack closure parameter U was found, which is nearly constant: U = 0.9 (compare Rie and Schubert [34] and Schubert [18]). Crack growth was successfully correlated with DJ [18] and DJeff [34], respectively. The value U = 0.9 was used also in the case of crack growth measurements of edge cracks in flat specimens [13]. For the calculation of DJ, characteristic values of the deformation behaviour are needed. According to Dowling [32], the cyclic integral DJ is calculated, e. g. in dependence on the cyclic hardening exponent n'. As proposed by Rie and Wittke [24], n' is 51

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

replaced by the exponent b of Equation (5). By this replacement, non-Masing behaviour is taken into consideration. The values of b are calculated according to the method proposed in [26] (for details: see Wittke [13]). With this method, the effective part of DJ for edge cracks is estimated as: ! 2 4:84 DD
p 2 D a; 15  p DJeff
7:88 U 2E b 1b

where a is the crack depth. The growth of edge cracks in flat specimens for the steel 2.25Cr-1Mo was measured with optical method. Tests with three different strain amplitudes (strain rate:
_
2 103 s1 ) were performed. The relation between crack growth per cycle da/dN and DJeff is described by: da=dN
CJ DJeff J ;

16

where CJ and J are constants. With an assumed initial crack depth and with a crack depth, which defines failure, lifetime can easily predicted by integrating Equation (16) (compare Schubert [18]). With values for the characteristic parameters of hysteresisloops, D, D
p and b, the constants CJ
3:89 105  J
116 were found (with da/dN in mm and DJeff in Nmm/mm2). The correlation between da/ dN and DJeff is quite satisfactory as can be seen in Figure 3.10.

Figure 3.10: 2.25Cr-1Mo,
_ = 2 · 10–3 s–1, correlation between crack growth per cycle da/dN and DJeff.

52

3.4 Creep-Fatigue Interaction

3.4

Creep-Fatigue Interaction

3.4.1

A physically based model for predicting LCF-life under creep-fatigue interaction

In this section, the original model of the author proposed in 1985 [35] was described to illustrate in the following the modifications and the experimental verifications made successively in the last years. 3.4.1.1 The original model Unstable crack advance occurs if the crack progress per cycle, da/dN, becomes approximately equal to the spacing of the nucleated intergranular cavities [35, 36]. The crack tip opening displacement d/2 may be seen as the upper bound to crack growth [36] and the relation can be written as: da d 
  2r  dN 2

17

where  is the cavity spacing, r is the radius of the r-type cavity and  is a constant. The crack tip opening displacement may be represented in analogy to the total strain by an elastic term D
el plus a contribution due to plastic deformation D
p and by thermally activated, time-dependent processes
c [37]: d
aK1 D
el  K2 D
p  K3
c 
aCcal 

18

where K1, K2, K3 are constants [35], and a is the crack length. Under repeated loading, there will be a dependence of the number of created cavities on the number of cycles. In analogy to the Manson-Coffin relationship, we postulate a constitutive equation for the cycle-dependent cavity nucleation under cyclic creep and low-cycle fatigue condition with superimposed hold time. Assuming that only the plastic strain imposed is responsible for cavity nucleation and disregarding stress dependence, the maximum number of cavities nmax is given by: nmax
p N  D
p 

19

where D
p is the plastic strain range, N is the number of cycles, p is the cavity nucleation factor, and  is the cyclic cavity nucleation exponent. It was proposed that p was identical with the density of grain boundary precipitates. Since it was found that not every precipitation necessarily produces a cavity, experimental constant  has been used to adapt the observation in our first model.  was used as a fit factor to have best results in life prediction.

53

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Cavity nucleation under creep-fatigue condition is favoured on grain boundaries perpendicular to the load axis and the cavity spacing  can be written as: 1 
 : n

20

Nucleation of cavities is governed by a deformation of the matrix, and the cavity growth is controlled by diffusion. In the first model, the cavity growth model of Hull and Rimmer [38] was taken for describing the cavity growth rate during creep-fatigue. This was done because the Hull-Rimmer model could be integrated analytically for every cycle, and the result could be expressed in one compact equation for the life prediction. In this case, the lifetime is reached if: a K1 D
el  K2 D
p  K3
c  2 v9 8 u  Zt > > u4Xd D p < = pD
p 1 u  gb gb 1 2 t 
 p t dt N  2 > >   2kT : pN  D
p ;

21

0

By integrating, it was assumed that the kinetics of the cavity growth in tension are the same as the kinetics of cavity shrinking in compression. 3.4.1.2 Modifications of the model The empirical constant  could reduce the versatile character of the basic concept on unzipping of cavitated material as the failure criterion. Therefore, in a first step of modification, we use: d
  2r  2

22

p was taken from direct experimental observation of cavities as will be shown in the following. Therefore, it is not necessary to consider the influence of precipitation on the nucleation of cavities, and nmax in Equation (19) could be replaced by the real density of cavities on grain boundaries n. The cavities nucleated by tensile stresses can be healed during periods of compressive stress if the compressive stress is applied for a long enough time. It has been observed that the time required to heal the cavity by compressive stress is up to six times longer than the time to nucleate the cavity by tensile stress [39]. In a second step, the incomplete healing response has been modelled in dividing the rate of the radius changes dr/dt in the growth models by a factor of 6 if the stress is negative. In a third step, a numerical procedure for integrating the cavity growth models was introduced. With this, it is possible to use any model depending on the physical parameters, which may prevail. The models of Hull-Rimmer [38], Speight-Harris [40] and Riedel [41] 54

3.4 Creep-Fatigue Interaction were compared, and it could be shown that in case of the calculated lifetimes, the influence of the model on the result is negligible [42]. The model of Riedel [41] is used in the following because it is successfully checked directly by experiments (see Section 3.4.2.3). 3.4.1.3 Experimental verification of the physical assumptions Both the cavity nucleation factor p and the cavity growth were determined experimentally by means of stereometric metallography as will be shown in Section 3.4.2. These values have been used for the life prediction. A furnace with window and special optics allows high magnification observation of the specimen surface continuously during the test with a video system and a subsequent measurement of the crack growth and the crack tip opening. The value for crack tip opening was determined in a distance of 250 lm [43]. With this method, the fundamental assumption of the life prediction model about the dependence of the crack tip opening displacement on the crack length and the strain range expressed in Equation (18) could be experimentally checked. An example of the crack tip opening displacement in dependence on the crack length is shown in Figure 3.11. The slope of the straight line Cexp for the experiments ranges between 0.043 and 0.058. The calculated slope determined with Equation (18) for the same experiment is Ccal = 0.044. From that, it can be concluded that the calculation of the crack tip opening displacement in the original life prediction model leads to values, which are in the right order of magnitude. 3.4.1.4 Life prediction The fatigue life of high-temperature low-cyclic fatigue under arbitrary cyclic loading situations including wave shapes and hold time can be estimated using the unstable crack advance criterion of the critical cavity configuration expressed in Equation (22).

Figure 3.11: Crack tip opening d250 lm vs. crack length. 55

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.12: Variation of cavity growth with time, respectively number of cycles (D
p/2 = 1%).

Figure 3.13: Comparison of experimental life and predicted life.

The effective cavity spacing  can be estimated from Equations (19) and (20), the crack tip opening displacement d from Equation (18). And the cavity radius r can be obtained by integrating the history-dependent cavity growth equation expressed in Equations (24) to (27). Because the cavity spacing  influences the cavity growth rate (Equations (24) to (27)), the number of created cavities and their growth have to be calculated for every cycle separately (Figure 3.12). From the radius of every nucleated and subsequently grown cavity, we calculate the mean value rm and compare  – 2rm with d/2 (Equation (22)) to get the critical life. Figure 3.13 shows the good agreement between experimental data and predicted life using the pore growth model of Riedel [41].

56

3.4 Creep-Fatigue Interaction

3.4.2

Computer simulation and experimental verification of cavity formation and growth during creep-fatigue

The fundamental physically based assumptions in the life prediction model about the development of the cavity density (Equation (19)) and the cavity growth have been experimentally verified. The results of this gave rise to the development of a new 2-dimensional cavity growth model, which describes the complex interaction between the cavities, and thus leading to constitutive equations of the damage development, which could be directly measured.

3.4.2.1 Stereometric metallography After low-cycle fatigue testing, the specimens were metallographically prepared for stereometric analysing for the density and cavity size distribution. For this purpose, the cavities were photographed by a Scanning Electron Microscope, and the cavity density and the distribution of the radii on the polished surface were detected. For every test, nearly 100 cavities were measured. The measured values of size and density on the metallographic section are much different from the real cavity configuration in the volume. For calculating the real cavity size distribution and density on the grain boundaries, the following assumptions are made: All cavities are on boundaries oriented perpendicular to the load axis with a maximum deviation of 30 8. All grains are of identical size, which is the mean value (in this case 62 lm), and all cavities are spherically shaped. The principal procedure is divided into two steps: First, the cavity size distribution and the cavity density in the volume of the specimen were calculated. This was done from the corresponding values in the metallographic section by means of a numerical procedure. Spheres were placed in a given volume by means of the Monte Carlo method. The spheres are randomly distributed. The size distribution of the spheres was set as a logarithmic Gaussian distribution. The resulting size distribution was calculated in a section of the volume, which is designed as the imaginary metallographic surface. The determined values of this section were compared with the experiment, and this procedure was repeated by varying the density and the parameters of the Gaussian distribution. This was done until the resulting density and the size distribution were identical to the values of the metallographic section. Second, the real density on the grain boundaries ngb from the density in the volume nv was calculated by means of a formula, which was provided by Needham and Gladman [44]: ngb
nv

i : 2q

23

i is the size of the grains determined by means of the intercepted-segment method, and the constant q (q = 0.134) depends on the angle between the cavitated grain boundaries and the load axis. 57

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

3.4.2.2 Computer simulation For the simulation, the cavities were placed one after another on a given area of the grain boundary by the Monte Carlo method. After the nucleation of one additional cavity, the growth of all cavities on the grain boundary was calculated until the next was formed. The growth of every cavity in our calculations depends on the spacing to the 6 neighbouring pores in plane and is described by so extended diffusion-controlled cavity growth model. The extension is illustrated in Figure 3.14. The advantage of the proposed model in comparison to the existing cavity growth models is the inhomogeneous distribution in plane. To calculate the cavity growth, it can be assumed that the total vacancy flow to the cavity considered is the sum of the flows from all 6 segments as illustrated in Figure 3.14. We propose that the flow from every segment depends on the distance only to the nearest cavity within the segment considered. This is analogous to existing 1-dimensional cavity growth models [45–48], but overestimates the vacancy flow because the contribution of the far distant cavities within this segment is supposed to be the same as the nearest. The same considerations will be applied for other segments. A fit factor is introduced, which takes this into consideration. This factor is set to the value of 0.2 to have the best fit of the experiments. To calculate the growth rate r_ from the cavity distance  under the actual stress b, the cavity growth model proposed by Riedel [41] was chosen: r_


2XdDb b  0 1   ; k T h q r 2

2s sin ; r  2 2r
; 

0


q 
2 ln  3  1   :

Figure 3.14: Statistically distributed cavities in plane. 58

24 25 26 27

3.4 Creep-Fatigue Interaction The meaning and the values of the constants for 304L are: h (W) = 0.61 the relation between the cavity volume and the volume of a sphere with radius r, X = 1.21 · 10–29 m3 the atomic volume, dDb = 2 · 10–13 exp (–Q/RT) m3/s with Q = 167 kJ/mol the grain boundary diffusion coefficient times the grain boundary width, 2W = 70 8 the void tip angle, cs = 2 kJ/m2 the specific surface energy, k Boltzmann constant and T the temperature. This differential equation (Equation (24)) was solved numerically. A possible coalescence has been taken into account. In this case, the two cavities were replaced by a new cavity, with the volume of both at the centre of the connecting line. As is shown below, this coalescence of cavities plays an important role in the case of fatigue because the accumulated strain, which controls the cavity formation, is relatively high compared to unidirectional tests. The cavity development during creep has also been successfully simulated, but will not be the subject of this paper. In the case of low-cycle fatigue, the cavity density nGb depends on the number of cycles N and is calculated by a power law function between nGb and N (Equation (19)). This is one of the basic assumptions of our life prediction model and is verified by the experiments as will be shown below. Note that during the creep-fatigue, the stress is not constant, whereas it is constant in the case of pure creep. Therefore, in the calculation, the changing stress was taken into consideration. The cavities are formed continuously during the tension period of the cycle until the strain maximum is reached. During the hold period, stress relaxation occurs and no cavities are formed. The actual stress for calculating the cavity growth after the formation of every single pore was taken directly from the experiment. During the compression period, no further cavity formation occurs. Due to the negative stress, the cavities are shrinking. However, the influence of shrinkage is negligible for this kind of test without compressive hold time and therefore will not be further discussed in this paper. 3.4.2.3 Results In Figure 3.15, the experimentally detected cavity density on the grain boundary is plotted versus the number of cycles. The cavity density during creep-fatigue depends on the number of cycles N by a power law as suggested before. With p = 12 · 10–2 1/ lm2 and j = 0.4 in Equation (19), a good fit of the experimental data is possible (not plotted in the figure), and the fundamental idea about the cavity formation in the life prediction model is verified. With this basic assumption about the cavity density in dependence on the number of cycles, the simulation of the cavity growth proceeds as follows. After a few cycles, the distribution is cut off at the right-hand side of the curve as proposed by Riedel [41]. When cycling continues, more and more cavities coalesce, and therefore, large cavities are formed. At the end of the simulation, the distribution is nearly Gaussian. In Figure 3.16, the cumulative frequency of the cavity radii for the experiment and the simulation, which is the solid line, is given for different numbers of cycles. In the case of the experiment, the size distribution in the metallographic section is plotted. The size distribution for the simulation is transformed to the resulting distribution in the imaginary section by means of the method described in Section 3.4.2.1. 59

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.15: Experimentally detected cavity density and cavity density of the simulation vs. number of cycles.

Figure 3.16: Comparison of computer simulation with experimentally detected cavity size distribution for creep-fatigue tests (304L,
a = 1%, T = 650 8C, 1 h tension hold). 60

3.4 Creep-Fatigue Interaction The proposed model and the constitutive equations for simulating the pore configuration in the case of creep-fatigue leads to a good agreement between calculation and experimentally detected cavity size distribution, and also the cavity density as shown in Figure 3.15. From Figure 3.15, one can draw the conclusion that the cavity coalescence plays an important role in the case of creep-fatigue. The values of the given density, that is the density, which would exist without coalescence, is much higher than the resulting density on the grain boundaries by coalescence.

3.4.3

In-situ measurement of local strain at the crack tip during creep-fatigue

In the previous sections, the total strain and stress were used for calculating the damage development and predicting the fatigue life. But in the LCF-regime, failure is a local phenomenon, which takes place in front of the crack. Therefore, the strain has been measured in front of the crack for giving the basis of a local application of material laws and a local damage model. The method provided by the group of Prof. Ritter [10] is usable for long time creep-fatigue tests at high temperature. The stability of the grating is sufficient for high accuracy measurement in argon for more than two weeks [49]. 3.4.3.1 Influence of the crack length and the strain amplitude on the local strain distribution The size of the highly deformed zone in front of the crack depends on the crack length. This effect can be measured with this method. For both steels, the size of the highly deformed zone increases with the crack length, which is shown in Figure 3.17 by plotting R0.05 vs. crack length. The size of the highly deformed zone also depends on the amplitude
a of the total strain. This is also demonstrated in Figure 3.17. The increase of the plastic zone size with both the crack length and the total strain amplitude will be explained by means of the theory of Shih and Hutchinson [33] and by observations of Iino [12]. Finite-Element calculations by Shih and Hutchinson [33] showed that both the crack length a and the strain amplitude
a are directly proportional to the crack tip opening displacement d, Iino [12] observed the linear dependence of the highly deformed zone size R0.05 on the crack tip opening in the case of low-cycle fatigue: a  d ;
a  d

R0:05  d

Shih and Hutchinson Iino :

28

From both theory and experiment, the measured relationships between R0.05 and
a as well as between R0.05 and a will be expected as shown in Figure 3.17. In the case of high-cycle fatigue, these effects are well known and can be explained in terms of linear-elastic fracture mechanics [50].

61

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

Figure 3.17: Increase of the highly deformed zone size in dependence on the total strain range.

3.4.3.2 Comparison of the strain field in tension and compression The strain in front of the crack tip measured at the maximum stress in tension and compression of the same cycle is shown in Figure 3.18. It can be seen that also in compression, the local strain just in front of the crack tip is positive. This has been found in all tests and for all crack lengths in both steels. Three different explanations are possible: •

A small amount of oxygen remains in the inert gas atmosphere, which leads to oxidation of the crack surfaces. As a consequence, the crack surfaces can not return to their original position of the previous cycle [51]. Therefore, the high deformation developed at the tensile strain maximum cannot be completely reversed.

Figure 3.18: Local strain in direction of the load for the tension and compression maximum of the same cycle vs. distance from crack tip in direction of the maximum expansion of the 5% deformed zone. 62

3.4 Creep-Fatigue Interaction •

A small shifting of the crack surfaces during opening of the crack may lead to an incomplete crack closure, and therefore to positive strain at the crack tip in compression.

•

Due to the notch effect, stress and strain concentration occur at the crack tip during crack opening. However, when the crack closes, no stress concentration appears and, as a consequence, the maximum stress at the crack tip in compression is equal to the total stress. Therefore, the stress at the crack tip in tension is higher than the stress in compression. The mean value of the stress in front of the crack is positive, and the consequence is the measured positive strain.

The fact that a positive strain appears in compression supports the high importance of the local strain measurement for crack growth calculation and life prediction. For the demonstrated test, the crack advance is 4 lm per cycle. The size of the zone of positive mean strain in front of the crack is estimated at 1 mm. This means that the propagating crack advances for more than 250 cycles through a material, which has been cycled under positive mean stress. 3.4.3.3 Influence of the hold time in tension on the strain field The values of the strain in front of the crack are lower in the case of tests with tension hold times compared to tests without hold. Figure 3.19 shows the development of R0.05 (size of the 5%-deformed zone from crack tip) as a function of crack length for 304L and different hold times. The same results are given for the ferritic steel in a paper of the authors [49]. The strain field depends on the hold time of the test, but remains the same during the hold period of each cycle within the accuracy of the measurement. In-situ monitoring of the crack advance and crack path indicates that the increase of crack growth rate

Figure 3.19: Plastically deformed zone size (size of the 5%-deformed zone from crack tip R0.05) vs. crack length a in 304L. 63

3

Plasticity of Metals and Life Prediction in the Range of Low-Cycle Fatigue

with tension hold time is related to a transition from a trans- to an intercrystalline crack path. Metallographic observation of the microstructure shows that during tension, hold grain boundary cavitation and microcrack formation occur. We conclude that the microscopic changes of the material during the creep-fatigue such as the grain boundary damage lead to the change of the stress-strain behaviour in front of the crack. This phenomenon has to be emphasized particularly because the macroscopic stress-strain behaviour is not influenced by the grain boundary damage [37]. To explain the strain behaviour in front of the crack, we propose a model, which is based on the reduction of strain to rupture in front of the crack if cavities are formed [49]. Comparable results for creep-fatigue cannot be found in literature, but Hasegawa and Ilschner [52] have detected a reduction of the strains in front of cracks in the case of high temperature tension tests if cavities are formed.

3.5

Summary and Conclusions

The cyclic deformation behaviour at room temperature was investigated for copper and steel 2.25Cr-1Mo. It can be concluded that for both materials, non-Masing behaviour has to be taken into consideration. The investigation of the microstructure shows that for both copper and 2.25Cr-1Mo, dislocation cell structures were found for sufficient high strain amplitudes. The deformation behaviour can be described by analytical relations. Especially for the steady-state stress-strain hysteresis-loops, very exact relations are proposed. With the aid of stress relaxation experiments, a cyclic yield stress yc can be defined and correlated with a microstructure-dependent proportional limit prop. Calculations with the physically based models of Steck and Hatanaka, respectively, show good agreement with experimental results. The model of Hatanaka was modified by taking results concerning the dislocation structure into account. An application of test results in the field of fatigue fracture mechanics is shown by correlating da/dN and DJeff. The generalized life prediction model of the authors has the capability to predict lifetime of high temperature low-cycle fatigue under various wave shapes and hold times. Physically based constitutive equations for cavity nucleation and subsequent growth under variable loading histories are considered, and the unzipping of the cavitated grain boundary is taken as criterion for catastrophic failure. The crack tip opening displacement is seen as the upper bound to crack growth. These physically based assumptions in the model are verified by corresponding experiments. The development of intergranular cavitation in austenitic steels can be simulated by the proposed 2-dimensional cavity growth model with good agreement to the experiment. It is important that not only the cavity size distribution but also the resulting cavity density on grain boundaries are in accordance with the experiment. From this, it can be concluded that the coalescence of neighbouring voids is very important for the cavity growth during low-cycle fatigue and is the main reason for the existence of relatively large cavities. 64

References The grating method is a very useful tool for determining the local strain in front of cracks during creep-fatigue. The high accuracy of this method for measuring the plastic deformation remains even for long time tests. It can be shown that the magnitude of the local strain at the crack tip during high temperature, low-cycle fatigue testing depends on the crack length and on the total strain range. During cycling, the local strain in front of the crack tip is positive even in compression maximum. By means of the grating method, it can be shown that the high crack growth rate of creep-fatigue is associated with a relatively small size of the plastically deformed zone.

References [1] H. Wohlfahrt, D. Brinkmann: Consideration of Inhomogeneities in the Application of Deformation Models, Describing the Inelastic Behaviour of Welded Joints. This book (Chapter 16). [2] H. Neuha¨user: Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys. This book (Chapter 6). [3] W. Gieseke, K. R. Hillert, G. Lange: Material State after Uni- and Biaxial Cyclic Deformation. This book (Chapter 2). [4] D. Bergmann, K. Galanulis, R. Ritter, D. Winter: Application of Optical Field Methods in Material Testing and Quality Control. In: Proceedings of the Photome´chanique 95, Cachan/ Paris, March 1995, E´ditions Eyrolles. [5] M. Erbe, K. Galanulis, R. Ritter, E. Steck: Theoretical and Experimental Investigations of Fracture by Finite Element and Grating Methods. Engineering Fracture Mechanics 48(1) (1994) 103–118. [6] K. Andresen, B. Hu¨bner: Calculation of Strain from Object Grating on a Reseau Film by a Correlation Method. Exp. Mechanics 32 (1992) 96–101. [7] Z. Lei, K. Andresen: Subpixel grid coordinates using line following filtering. Optik 100 (1995) 125–128. [8] J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtemperatur-Low-Cycle-Fatigue. Z. Metallkde. 81 (1990) 783–789. [9] J. Olfe: Wechselwirkung zwischen Kriechscha¨digung und Low Cycle Fatigue und ihre Beru¨cksichtigung bei der Berechnung der Lebensdauer. Dissertation TU Braunschweig, Papierflieger, Clausthal-Zellerfeld, 1996. [10] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical Measuring Methods. This book (Chapter 13). [11] K. Andresen: Surface-Deformation Fields from Grating Pictures Using Image Processing and Photogrammetry. This book (Chapter 14). [12] Y. Iino: Cyclic crack tip deformation and its relation to Fatigue Crack Growth. Eng. fract. mech. 7 (1975) 205–218. [13] H. Wittke: Pha¨nomenologische und mikrostrukturell begru¨ndete Beschreibung des Verformungsverhaltens und Rißfortschritt im LCF-Bereich. Dissertation TU Braunschweig, 1996. [14] K.-T. Rie, H. Wittke: Low Cycle Fatigue and Internal Stress Measurement of Copper. In: Fatigue ’96, Proceedings of the Sixth International Fatigue Congress, Pergamon, 1996, pp. 81–86. [15] J. Komotori, M. Shimizu: Microstructural Effect Controlling Exhaustion of Ductility in Extremely Low Cycle Fatigue. In: K.-T. Rie (Ed.): Low Cycle Fatigue and Elasto-Plastic Be65

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haviour of Materials – 3, Elsevier Applied Science, London, New York, 1992, pp. 136– 141. H. R. Jhansale, T. H. Topper: Engineering Analysis of the Inelastic Stress Response of a Structural Metal under Variable Cyclic Strains. ASTM STP 519 (1973) 246–270. D. Lefebvre, F. Ellyin: Cyclic Response and Inelastic Strain Energy in LCF. Intern. Journ. Fat. 6 (1984) 9–15. R. Schubert: Verformungsverhalten und Rißwachstum bei Low Cycle Fatigue. Fortschrittsber. VDI, Reihe 18, No. 73, VDI Verlag, Du¨sseldorf, 1989. K.-T. Rie, H. Wittke, R. Schubert: The DJ-Integral and the Relation between Deformation Behaviour and Microstructure in the LCF-Range. In: K.-T. Rie (Ed.): Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials – 3, Elsevier Applied Science, London, New York, 1992, pp. 514–520. C. E. Feltner, C. Laird: Cyclic Stress-Strain Response of f.c.c. Metals and Alloys – I. Phenomenological Experiments. Acta Metallurgica 15 (1967) 1621–1632. G. Hoffmann, O. Öttinger, H.-J. Christ: The Influence of Mechanical Prehistory on the Cyclic Stress-Strain Response and Microstructure of Single-Phase Metallic Materials. In: K.-T. Rie (Ed.): Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials – 3, Elsevier Applied Science, London, New York, 1992, pp. 106–111. C. E. Feltner, C. Laird: Cyclic Stress-Strain Response of F.C.C. Metals and Alloys – II: Dislocations Structures and Mechanisms. Acta Metallurgica 15 (1967) 1633–1653. J. D. Morrow: Cyclic Plastic Strain Energy and Fatigue of Metals; Internal Friction, Damping, and Cyclic Plasticity. ASTM STP 378 (1965) 45–87. K.-T. Rie, H. Wittke: New approach for estimation of DJ and for measurement of crack growth at elevated temperature. (To be published in: Fatigue Fract. Mater. Struct. Vol. 19 (1996).) U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading. This book (Chapter 10). H. Wittke, J. Olfe, K.-T. Rie: Description of Stress-Strain Hysteresis Loops with a Simple Approach. (To be published in: Int. J. Fatigue (1996/97).) J. C. Tsou, D. J. Quesnel: Internal Stress Measurements during the Saturation Fatigue of Polycrystalline Aluminium. Mat. Sci. and Engin. 56 (1982) 289–299. K. Hatanaka, Y. Ishimoto: A Numerical Analysis of Cyclic Stress-Strain-Response in Terms of Dislocation Motion in Copper: In: H. Fujiwara, T. Abe, K. Tanaka (Eds.): Residual Stresses – III, Elsevier Applied Science, 1991, pp. 549–554. J. Pola´k, M. Klesnil, J. Heles˘ic: The Hysteresis Loop: 2. An Analysis of the Loop Shape. Fatigue of Engineering Materials and Structures 5(1) (1982) 33–44. H. Schlums, E. A. Steck: Description of Cyclic Deformation Process with a Stochastic Model for Inelastic Behaviour of Metals. Int. J. of Plasticity 8 (1992) 147–159. R. Gerdes: Ein stochastisches Werkstoffmodell fu¨r das inelastische Materialverhalten metallischer Werkstoffe im Hoch- und Tieftemperaturbereich. Braunschweiger Schriften zur Mechanik 20 (1995). N. E. Dowling: Crack Growth During Low-Cycle Fatigue of Smooth Axial Specimens; Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Growth. ASTM STP 637 (1977) 97–121. C. F. Shih, J. W. Hutchinson: Fully Plastic Solutions and Large Scale Yielding Estimates for Plane Stress Crack Problems. Journal of Engin. Mat. and Technol., Oct. 1976, Transactions of ASME, pp. 289–295. K.-T. Rie, R. Schubert: Note on the crack closure phenomenon in low-cycle fatigue. Int. Conf. Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, Munich 1987, Elsevier Applied Science, pp. 575–580. K.-T. Rie, R.-M. Schmidt, B. Ilschner, S. W. Nam: A Model for Predicting Low-Cycle Fatigue Life under Creep-Fatigue Interaction. In: H. D. Solomon, G. R. Halford, L. R. Kai-

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sand, B. N. Leis (Eds.): Low Cycle Fatigue, ASTM STP 942, American Society for Testing and Materials, Philadelphia, 1988, pp. 313–328. G. J. Lloyd: High Temperature Fatigue and Creep Fatigue Crack Propagations Mechanics, Mechanisms and Observed Behaviour in Structural Materials. In: R. P. Skelton (Ed.): Fatigue at High Temperatures, Applied Science Publishers, London, New York, 1983, pp. 187–258. R.-M. Schmidt: Lebensdauer bei Kriechermu¨dung im Low-Cycle Fatigue Bereich. Dissertation TU Braunschweig, VDI Fortschritt-Berichte Nr. 47 (1988). D. Hull, D. E. Rimmer: The Growth of Grain-Boundary Voids Under Stress. Philosophical Magazine 4 (1959) 673–687. B. K. Min, R. Raj: Hold Time Effects in High Temperature Fatigue. Acta Metall. 26 (1978) 1007–1022. H. E. Evans: Mechanisms of Creep Fracture. Elsevier Applied Science Pub. LTD., 1984, pp. 251–263. H. Riedel: Fracture at High Temperatures. Springer-Verlag, Berlin Heidelberg, 1987. K.-T. Rie, J. Olfe: A physically based model for predicting LCF life under creep fatigue interaction. In: K.-T. Rie (Ed.): Proc. 3rd Int. Conf. on Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, Elsevier Applied Science, London, New York, 1992, pp. 222–228. K. Tanaka, T. Hoshide, N. Sakai: Mechanics of Fatigue Crack-Tip plastic blunting. Engineering Fracture Mechanics 19 (1984) 805–825. N. G. Needham, T. Gladman: Nucleation and growth of creep cavities in a Type 347 steel. Mat. science 14 (1980) 64–66. S. J. Fariborz: The effect of nonperiodic void spacing upon intergranular creep cavitation. Acta metall. 33 (1985) 1–9. S. J. Fariborz: Intergranular creep cavitation with time-discrete stochastic nucleation. Acta metall. 34 (1986) 1433–1441. J. Yu, J. O. Chung: Creep rupture by diffusive growth of randomly distributed cavities – I. Instantaneous cavity nucleation. Acta metall. 38 (1990) 1423–1434. J. Yu, J. O. Chung: Creep rupture by diffusive growth of randomly distributed cavities – II. Continual cavity nucleation. Acta metall. 38 (1990) 1435–1443. K.-T. Rie, J. Olfe: In-situ measurement of local strain at the crack tip during creep-fatigue. In: Proceedings of the International Symposium on Local Strain and Temperature Measurements in Non-Uniform Fields at Elevated Temperatures, March 14–15, Berlin, 1996. K.-H. Schwalbe: Bruchmechanik metallischer Werkstoffe. Carl Hanser Verlag, Mu¨nchen, Wien, 1980. T. Ericsson: Review of oxidation effects on cyclic life at elevated temperature. Canadian metallurgical quarterly 18 (1979) 177–195. T. Hasegawa, B. Ilschner: Characteristics of crack tip deformation during high temperature straining of austenitic steels. Acta metall. 33(6) (1985) 1151–1159.

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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)

4

Development and Application of Constitutive Models for the Plasticity of Metals Elmar Steck, Frank Thielecke and Malte Lewerenz *

Abstract The macroscopic behaviour of crystalline materials under mechanical or thermal loadings is determined by processes in the microregion of the material. By a combination of models on the basis of molecular dynamics and cellular automata, it seems possible to simulate numerically the formation of internal structures during the deformation processes. The stochastical character of these mechanisms can be considered by modelling them as stochastic processes, which result in Markov chains. By a mean value formulation, this leads to a macroscopic model consisting of non-linear ordinary differential equations. The determination of the unknown material parameters is based on a Maximum-Likelihood output-error method comparing experimental data to the numerical simulations. With Finite-Element methods, it is possible to use the material models for the design of components and structures in all fields of technical application and for the numerical simulation of their behaviour under complex loading situations.

4.1

Introduction

Metallic materials show, like other crystalline substances, typical macroscopic responses on mechanical loading, which are caused by processes on the microscale. Figure 4.1 shows a typical cyclic stress-strain diagram with constant strain amplitude. Cyclic hardening can be observed as well as the Bauschinger effect, which can be recognized by the fact that plastic flow occurs after load reversal at significantly lower stresses than those, from which the load reversal was done. For the technical use of metallic materials, the description of this kind of processes in material models is of high importance. * Technische Universita¨t Braunschweig, Institut fu¨r Allgemeine Mechanik und Festigkeitslehre, Gaußstraße 14, D-38106 Braunschweig, Germany 68

4.2 Mechanisms on the Microscale

Figure 4.1: Cyclic stress-strain diagram for 304 stainless steel.

The moving of dislocations is the main microscopic mechanism responsible for the plastic deformations in metallic materials. In the following, a stochastic model is presented, which is able to consider hardening and recovery processes by means of Markov chains. During the deformation process, the dislocations arrange in a hierarchy of structures such as walls, adders or cells. This forming of structures influences the macroscopic behaviour of the materials considerably. The principle of cellular automata in combination with the method of molecular dynamics is used for the numerical simulation of these processes. For the material parameter identification, the minimization of the Maximum-Likelihood costfunction by hybrid optimization methods parallelized with PVM is considered. With a multiple shooting method, additional information about the states can be taken into account, and thus the influence of bad initial parameters will be reduced. 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 results are compared with experimental data from grating methods.

4.2

Mechanisms on the Microscale

The movement of dislocations and the connected plastic deformations caused by external loading is determined by two important activation mechanisms. The stress activation is caused by the external loads. The thermal activation supports at elevated temperatures the dislocation movements and therefore the plastic deformations. 69

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Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.2: Stress and thermal activation of dislocation motion.

Figure 4.2 shows schematically the obstacles, which resist the dislocation movements on the microscale in the form of barrier potentials U*, the possible position – determined by temperature – of the dislocations relative to these barrier potentials, and the effect of an external load and a temperature increase on the energetic situation of the dislocations. It is visible that the potential U r of the external forces by superposition changes the potentials of the actual obstacles so that the dislocation movement in the direction of the applied stress is more probable than in the opposite direction, and that the thermal activation supports this process [1–4]. The barriers, which oppose the dislocation movements, are on the one side given by the crystalline structure of the material itself, on the other hand, foreign atoms and grain boundaries can form obstacles. One of the most important reasons for the hindering of the dislocations, however, are the dislocations themselves. During plastic deformation, continuously new dislocations are produced. In the beginning, the ability of the material for deforming plastically is increased. With increasing dislocation density, a mutual influence of the lattice disturbances occurs, which results in isotropic hardening. Due to the lattice distortions connected with the plastic deformation, elastic energy is stored in the material, which also hinders the movements of the dislocations, which are generating it. This process is called kinematic hardening. The internal stresses, however, support the dislocation movements in the opposite direction and result in e.g. the Bauschinger effect. At elevated temperatures – above half of the melting temperature of the material –, thermally activated reorganization processes in the crystals occur, which reduce the mutual hindering of the dislocations and result macroscopically in recovery. Significant magnitudes for these processes are given in Figure 4.3, which shows a dislocation, which is influenced by other dislocations. The shaded area is a measure for the activation volume DV
bA, which decreases in size with increasing isotropic hardening. The Burgers vector b determines with his orientation relative to the dislocation line the character of the dislocation. qw is the density of the so-called forest dislocations, i.e. the dislocations, which hinder the movement of the others [4]. Table 4.1 shows the connection between the activation volume and the most important dislocation mechanisms for different regions of the homologous temperature T=Tm .

70

4.3

Simulation of the Development of Dislocation Structures

Figure 4.3: Activation volume and forest dislocations. Table 4.1: Activation volume depending on deformation mechanism and temperature. Mechanism

Temperature

Activation volume

Climbing Movement of dislocation jumps Cross slip

> 0.5 > 0.5

Cutting of dislocations

> 0.3

b3 remains constant during deformation 10–1000 b3, the value of the activation volume decreases during deformation 10–100 b3, the value remains approximately constant during deformation 1000 b3, the activation volume decreases due to increase of the density of forest dislocations with increasing deformation

4.3

0.2–0.4

Simulation of the Development of Dislocation Structures

For unidirectional as well as for cyclic plastic deformation, it is observed that dislocation structures are developed in the shape of e.g. adders or dislocation cells, which in a typical manner depend on the loading history and the loading magnitude (Figure 4.4). Due to the fact that this forming of dislocation patterns influences the macroscopic behaviour of the materials considerably, the simulation of these self-organization processes can result in valuable information for the choice of formulations for the modelling of processes on the microscale. The interaction of a large number of identical particles is the basic idea for the definition of cellular automata. It is an idealization of real physical systems, where space as well as time are discrete. A cellular automaton is completely characterized by the following four properties [5]: geometry of the cell arrangement, definition of a neighbourhood, definition of the possible states of a cell, and evolution rules. Each cell can during the evolution in time only assume values (states) out of a finite set. For all 71

4

Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.4: Characteristic dislocation structures.

cells, the same evolution rules are valid. The change in state of a cell depends on its own state and those of the neighbouring cells. Opposite to the usual assumptions for cellular automata, where the state of a cell only depends on the states of the next neighbours, for the simulation of dislocation movements, it has to be taken into account that the dislocations possess long-range acting stress fields. With this model, it is possible to compute the dynamics of some thousand edge- or screw-dislocations on parallel slip planes in areas of arbitrary magnitude. A basic model, for which only one slip system in horizontal direction was chosen, assumes a grid of rectangular cells, which can be occupied by edge- or screw-dislocations with positive or negative sign [5]. The transition rules are: A positive or negative occupied cell becomes an empty cell if the dislocation in the cell will move due to the acting forces to a neighbouring cell or if an annihilation with a dislocation in a neighbouring cell occurs. The step width of a dislocation is always one cell size per time step. Reachable cells are the cells left, right, up and down from the actual cell. This characterizes a so called v. Neumann neighbourhood. For the calculation of the forces acting on a dislocation, a larger neighbourhood is necessary due to the longrange acting stress of the dislocations. The balance of forces decides, if and in which direction a dislocation will move. It is computed for each time step and each dislocation for both degrees of freedom. A much more realistic simulation for the development of dislocation structures is obtained from models, which consider several glide planes [6]. Figure 4.5 shows a twodimensional projection for the glide system for a cubic face-centred lattice, and modelling of the glide processes on this system with three glide directions under angles of respectively 608. The simulation results in wall- and labyrinth-structures of the dislocations (Figure 4.6). An extension of the model with consideration of vacancies and a suitable velocity law is under progress.

72

4.4 Stochastic Constitutive Model

Figure 4.5: Cell arrangement and neighbourhood of simulation model.

Figure 4.6: Simulation of dislocation structures.

4.4

Stochastic Constitutive Model

The description of the processes responsible for plastic deformations shows that they are strongly stochastic. Figure 4.7 shows for a simplified case for processes at high temperatures, under consideration of kinematic hardening only, the used stochastic model [1, 2]. Over the state axis, which represents the value of the kinematic hardening rkin , and therefore the strength of the obstacles resisting the dislocation movements, the distribution of the “flow units” (dislocations, dislocation packages or grain boundaries) is given. The effect of the external stress is reduced by the hardening stress, therefore only the effective stress reff
r  rkin is responsible for the dislocation movements. Depending on reff , a hardening probability

73

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Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.7: Stochastic model for high temperatures.

  dDVrkin sign reff e_ ie V
c1 exp  RT

1

is formulated. This transition probability is based on the condition that thermal activation of the dislocations can be taken as an empirical Arrhenius function. R is the gas constant and c1 ; d; DV are constants, which have to be determined by experiment. It can be seen that the transition probability from a certain hardening state to the next higher decreases with increasing hardening. Hardening is opposed by a recovering process according to:      F0 jrkin j m DVrkin ; sinh E
c2 exp  RT RT r0

…2†

which is thermally activated and not dependent on the external stress. The constants c2 and m have also to be determined by experiment. The strength of the lattice distortions increases with increasing hardening. It supports the recovery process. Therefore, the transition probabilities for recovery increase with increasing hardening. The model simulates hardening and recovery by transitions of dislocations at a barrier strength rkin;i to higher barriers rkin;i‡1 and lower barriers rkin;i 1 : The probability that a flow unit remains in the actual position is given by: Bˆ1

V

E:

The transition probabilities of the model can be arranged in a stochastic matrix:

74

…3†

4.4 Stochastic Constitutive Model 

      Sˆ      

1

V1 V1

E2



B2

..

.

V2

..

.

Ei

..

.

Bi

..

.

Vi

..

.

Ek

..

.

Bk Vk

0

0

1 1 1

1

Ek Ek

      :      

…4†

The change of the structure, which is described by the state vector z, during one time step Dt is given by the Markov chain: z…t ‡ Dt† ˆ S z…t† :

…5†

For constant stress and temperature (homogeneous process), the state vector after n time steps is given by z…t0 ‡ nDt† ˆ Sn z…t0 †: The stochastic matrix given by Equation (4) can be transformed to principal axes and yields then:   1 0 0 0  0 k2 0 0    ˆ M 1 SM ˆ S …6† ;  .. 0 0 . 0 0 0 0 kn where M is the modal matrix, i.e. the matrix of the columnwise arranged eigenvectors of the matrix S. Due to the fact that the maximal principal value of stochastic matrices is 1 and all other eigenvalues have magnitudes < 1, it is visible that their magnitudes decrease with increasing time, and the eigenvalue connected with the maximal eigenvalue 1 represents a stationary state. The other principal values are responsible for transient processes [1–3]. An extension of the stochastic model, which allows for the simultaneous consideration of the development of activation volume DV and kinematic stress rkin is given in Figure 4.8. Thus, isotropic and kinematic hardening spread a state plane, which allows that with the distribution of the flow units, the state determined by both hardening types can be considered. The transition probabilities for the description of the development of the isotropic and kinematic hardening consider mutually the influences given by the other hardening process [3, 4]. By a mean value formulation, the stochastic model is transformed in a macrosocpic continuum mechanical material model, which takes a form similar to other models given in literature. This approach leads to a non-linear system of ordinary differential equations for the inelastic strain e, the kinematic back stress rkin and the activation volume DV:

75

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Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.8: Distribution function for rkin and DV= bA.



  F0 jreff j n DVreff ; …7† sinh RT RT r0



  dDVrkin F0 jrkin j m DVrkin sign reff e_ ie R exp ; ˆ H d exp sinh RT RT RT r0


e_ ie ˆ C exp r_ kin

…8† DV_ ˆ

K1 DV 2 j_eie j ‡ K2 DVj_eie j :

…9†

The material behaviour is described by a relation for the inelastic strain rate, where the actual values for isotropic and kinematic hardening occur as internal variables. This general form of the constitutive equations is also the basis for the development of a hierarchical model classification [7]. A concrete model must be chosen with respect to the intended application purpose. The values C; n; H; d; R ; m; K1 ; K2 and DV0 are material parameters, which have to be determined by comparison with experimental results. The parameter identification, which consists in integrating the non-linear, ordinary differential equations for varying parameter sets and by appropriate optimization methods to search for the optimal parameter sets, deserves special recognition in aspect of the methods [7, 8]. An additional scaling of the functions like
used mathematical

 F0 1 1 is necessary to improve the parameter identifiability and the exp R T T0 macroscopical interpretations.

76

4.5 Material-Parameter Identification

4.5

Material-Parameter Identification

4.5.1

Characteristics of the inverse problem

Under the assumption of normal distributed measurement errors with zero mean and known measurement-covariance matrix C…ti †; the costfunction is: n 1 1 L2 …x; p† ˆ jjrjj2 ˆ ‰z…ti † 2 2 iˆ1

x…ti †ŠT C 1 ‰z…ti †

x…ti †Š?min :

…10†

The minimization of this weighted least squares function yields a Maximum-Likelihood estimate of the parameters, which reproduces the observed behaviour z of the real process with maximum probability [9]. Typical features of the identification are that the constitutive model is not only highly non-linear in states x, but also in parameters p. Due to incomplete measurement information, the problem is ill-conditioned, parameters are highly correlated. Because of unbalanced parameters, the model may change its characteristics and becomes stiff or even pathological. Since replicated tests for the same laboratory conditions show a significant scattering and thus incompatibility of the data, this uncertainty must be taken into account for the development and identification of the constitutive models [7, 10].

4.5.2

Multiple-shooting methods

The measurements of the kinematic back stress, e.g. by relaxation test, yield very important informations about the deformation process and thus can be used to get more reliable parameters. In general, there are no (complete) measurements for the internal states. However, engineers have a lot of additional apriori-information, which should be used to improve the model prediction capacity. Although it is possible to formulate additional weighted least-squares terms for the Maximum-Likelihood function, a much more efficient method is to use multiple shooting (Figure 4.9) [11, 12]. The basic idea is to subdivide the integration interval by a suitable chosen grid and to treat the discretized model equations as non-linear constraints of the optimization problem. The initial state estimates at the nodes of the grid allow to make efficient use of measurement- and apriori-information about the solution [13, 14].

4.5.3

Hybrid optimization of costfunction

For the identification of the material parameters, a hybrid optimization concept is used. Starting with evolution strategies as a pre-optimization to get reliable initial parameters, the main-optimization is done with a damped Gauß-Newton method [15]. 77

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Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.9: Principle of multiple shooting.

Adaptive Evolution Strategies attempt to imitate the organic evolution process, e.g. a collective learning of a population with the principles mutation, recombination and selection [16]. The self-learning process of strategy parameters adapts this optimization procedure to the local topological requirements. Since it is possible to overcome local minima with a special destabilization method, evolution strategies even work with bad initial model parameters [8]. Damped Gauß-Newton methods are most widely used for the minimization of non-linear least-squares functions [7, 11]. Starting from initial parameters p0, improved parameters are iteratively obtained by the solution of a linear least-squares problem linearized about pk. The steplength parameter kk is chosen to enforce the convergence properties: 1 jj r…pk † ‡ J k Dpk jj2 ! min with pk‡1 ˆ pk ‡ kk D pk ; 2 solution with pseudo inverse: D pk ˆ J ‡ …pk †r…pk † :

…11†

A study of different search and gradient-based methods like the algorithms of Powell, special subspace simplex methods or sequential quadratic programming are given in [7]. The numerical sensitivity analysis is a very important and most time consuming part of the identification. Since the calculation is very closely related to the numerical integration of the differential equations and the available accuracy, the sensitivity analysis may be a critical point. Three different concepts are used to generate the sensitivity matrix. The commonly used finite difference approximation: qxi xi …pj ‡ dpj †  qpj dpj

xi …pj †

…12†

is easy to implement, but the efficiency and reliability are low. Better concepts are based on the integration of the sensitivity equations:



qx  qf qx qf ‡ : ˆ qp qx qp qp

…13†

It is obvious that the solution of the model and the sensitivity equations should be coupled. A very powerful coupling is available by Internal Numerical Differentiation 78

4.5 Material-Parameter Identification (IND) [11]. This means that the internally generated discretization scheme of the integrator is differentiated with respect to the parameters.

4.5.4

Statistical analysis of estimates and experimental design

The parameter estimates are only useful if also a statistical analysis of their reliability is computed. Using the pseudo inverse J+ at the solution of the Gauß-Newton method, the calculation of standard deviations and correlations for the parameters is quite easy. Very important for further work is to improve the calculation by better experimental designs. Based on design criteria like the minimization of det …J T J† 1 ; different methods have been considered and tested for typical growth function and a fundamental constitutive model. These studies also show that the bad identifiability of the inverse problems can be overcome with a special scaling of the states [7].

4.5.5

Parallelization and coupling with Finite-Element analysis

The separable multiexperiment structure leads to a coarse-grained parallelism of the parameter identification problem. In addition, evolution strategies and multiple shooting provide inherent parallelism on a high level. Thus, efficient parallel computation of model functions and derivatives can be easily performed on a workstation-cluster with PVM (Figure 4.10).

Figure 4.10: Parallel simulation concept. 79

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Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.11: Creep tests for aluminium.

Since the development and identification of anisotropic damage models became more and more important, a three-dimensional Finite-Element program was coupled with the estimation software by a special interface. The flexibility and modular structure of this approach may be very useful for a lot of other applications, e.g. structure optimization. For the application of the damped Gauß-Newton method, the Internal Numerical Differentiation was adapted to the Finite-Element analysis. Thus, not only the simulation results but also the sensitivities have to be transferred. The results of the simulations are compared with experimental strain fields obtained by grating methods [17].

80

4.5 Material-Parameter Identification

Figure 4.12: Cyclic tests for copper.

4.5.6

Comparison of experiments and simulations

A lot of different materials like pure aluminium, pure copper or the austenitic steels AISI 304 and AISI 316 have been extensively studied. Figure 4.11 shows some results of parameter identifications for aluminium Al 99.999. The temperature regime was between 500 8C and 700 8C. Since only monotonic tests were evaluated, a constitutive model with only one structure variable for the internal stress is used. The parameters were identified for the given stresses simultaneously so that the calculated curves were obtained by a single parameter set [7, 8, 15]. Figure 4.12 gives two examples on copper. The experimental database consists of seven strain-controlled cyclic tests at room temperature [18]. Two strain rates e_ = 10–4, 10–3, and a multitude of strain amplitudes De=2 = 0.2–0.7% are examined. For this application of the stochastic constitutive model, the special characteristics of the material and the measurements have to be considered. In the low-temperature regime, hardening is the most important phenomenon, while the recovery influence is negligible. In contrast to high temperatures, metal physical results also indicate that the 81

4

Development and Application of Constitutive Models for the Plasticity of Metals

Figure 4.13: Scattering of creep and tension-relaxation tests for AISI 316.

influence of the effective stress can be modelled only by a sinushyperbolicus function. Thus, the low-temperature model has only five parameters.

4.5.7

Consideration of experimental scattering

The experimental data to determine the parameters of constitutive equations usually consist of only one observed trajectory for each temperature and loading condition. Nevertheless, replicated tests for the same laboratory conditions show a significant scattering and thus incompatibility of the measurements (Figure 4.13). Based on a statistical analysis, this uncertainty can be taken into account for more reliable modelling and parameter identification. The modelling of the experimental uncertainties is based on the scattering of the parameters or the initial values (Figure 4.14) [10]. Based on these concepts, realistic simulations of the uncertainties in experimental data due to measurement errors and scattering are possible [7]. 82

4.6

Finite-Element Simulation

Figure 4.14: Probability density function and correlation of scattered parameters.

4.6

Finite-Element Simulation

The aim of using constitutive models is to predict the behaviour of metallic structures under mechanical and thermal loading. This requires the solution of a coupled initialboundary value problem, given by the momentum equilibrium and the constitutive equations. Since the boundary value problem is usually solved by the Finite-Element Method (FEM), the constitutive model has to be implemented in an appropriate way. Since the code ABAQUS/STANDARD is used, the theoretical aspects of the model implementation are discussed for application of the user subroutine UMAT. The developed method of implementation is described in Section 4.6.1. The main characteristics of this method are its applicability to any unified constitutive model of the class described above and to small as well as to large deformations theory. In Section 4.6.2, some numerical and experimental results are given, which show that the model presented here works well.

4.6.1

Implementation and numerical treatment of the model equations

The considered constitutive model can be mathematically classified as a coupled system of non-linear ordinary differential equations (CSNODE), which builds an initial value problem. Its solution to a time increment can be embedded in an incremental Finite-Element formulation with displacement approach, leading to the well known impli83

4

Development and Application of Constitutive Models for the Plasticity of Metals

cit FEM-problem for non-linear material equations, which has to be solved iteratively (see e.g. [19]). Since this iteration requires a repeated solution of the initial value problem, the computational cost of the FEM-simulation can be minimized by optimizing the numerical solution of the CSNODE. This can be reached by: • • •

simplifying the model equations with some appropriate transformation, the use of an efficient numerical integration scheme, and an efficient algorithm to approximate the so-called tangent modulus.

The proposals worked out to this, three aspects are summarized in the following subsections (for further details see [7, 20]).

4.6.1.1 Transformation of the tensor-valued equations Using the v. Mises hypothesis, the multiaxial formulation of the model equations takes the form: r_ ij ˆ fij …_ekl ; rkl ; rkin kl ; DV† ;

…14†

kin r_ kin ij ˆ fij …rkl ; rkl ; DV† ;

…15†

_ ˆ f …rkl ; rkin ; DV† ; DV kl

…16†

where rij is the Cauchy stress tensor, rkin ij is the back stress tensor and e_ ij is the deformation rate tensor. Each of these symmetric tensors is defined by six independent components, so that the whole CSNODE contains thirteen scalar equations. Since the v. Mises equivalent stress

3 0 0 r r rv ˆ 2 ij ij

…17†

just depends on the deviatoric stresses, the inelastic part of the tensor equations are also purely deviatoric. Therefore, the deviatoric rates r_ ij and r_ kin ij can be described in some interval ‰t0 ; t0 ‡ DtŠ as a linear combination of the three deviatoric tensors rij …t0 †; rkin ij …t0 † and e_ ij …t0 †, as long as e_ ij is constant in Dt. Using a suitable transformation, the deviatoric rates can be expressed in a subspace by only three independent components. After this transformation, the initial value problem (Equations (14) to (16)) can be written as: y_ i ˆ fi …yj † ; 84

yi …t0 † ˆ yi;0 ;

i; j ˆ 1 . . . n ;

…18†

4.6

Finite-Element Simulation

and contains only seven scalar equations. Hornberger [21] shows that the subspace dimension can be reduced to two if a special integration scheme is used. Nevertheless, this idea is neglected here in order to obtain a free integrator choice. This transformation concept can be applied to plane strain, plane stress and uniaxial states as well. Although the number of scalar equations cannot be reduced in these cases, the main advantage is that the transformed model equations in the subspace are of identical form for each of these cases. Based on this fact, the model implementation for one-, two- and three-dimensional states can be performed very easily. Using special large deformation formulations (see e.g. ABAQUS Theory Manual [22]), this form of implementation can be used with small or large deformation theory as well.

4.6.1.2 Numerical integration of the differential equations Due to its complexity and non-linearities, the CSNODE (see Section 4.6.2) has to be integrated numerically. In oder to choose an appropriate integration algorithm, the integration task is classified as follows: • • • •

medium required integration tolerances (corresponding to usual FEM-tolerances), a small integration interval (given by the incremental FEM-solution), an associated efficient method for error estimation, and a stable solution (to guarantee a stable FEM-solution).

Numerical integration methods on the other hand can be classified by their integration order p, which describes the discretization error R in dependency of the step size h by R  hp (for an overview see [23, 24]). There are: • •

methods with fixed integration order like multi-step methods, Runge-Kutta methods, and Taylor series methods, and methods with variable integration order like extrapolation methods.

Extrapolation methods are efficient only for high integration tolerances, while multi-step methods loose efficiency for small integration intervals. The use of Taylor series methods is not practicable since it requires higher derivatives of the CSNODE, which are usually not given directly. So, explicit and implicit, Runge-Kutta methods are widely used for the integration of constitutive equations in FEM-analysis (see e.g. [19, 21, 25]). Butcher [26] pointed out that the mentioned methods with fixed integration order can be combined to get new classes of integration methods. For example, so-called Rosenbrock methods result from the combination of Runge-Kutta methods and Taylor series methods based on the first derivative of the CSNODE (also called the Jacobean of the system). The main advantage of these methods is their unconditional stability – as in implicit Runge-Kutta methods – that is reached with an explicit algorithm without any iteration process. Rosenbrock methods as well as Runge-Kutta methods can be designed as embedded integration formulae, which lead directly to a method of internal error estimation without additional numerical cost. 85

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Development and Application of Constitutive Models for the Plasticity of Metals

Verner [27] proposed families of embedded explicit Runge-Kutta processes, which allow to rise integration order p without dismissing the results of an integration with a lower starting order p0. If this concept is used with Rosenbrock methods, the resulting integration process is kind of an optimal numerical integration method for constitutive rate equations in FEM-analysis because • • • •

it is an efficient algorithm especially for medium error tolerances, it is unconditionally stable, it is designed for computing the solution for the whole (but small) integration interval in one step, and an internal error estimation nearly without additional cost is possible.

For details and comparison to classical methods see [7, 20].

4.6.1.3 Approximation of the tangent modulus qDrij is used to compute the qDekl element stiffness matrix, which is the tangent operator for the applied Newton iteration method. Due to the necessity of numerical integration, the stress increment Drij is a discrete value and so, the partial derivative cannot be built analytically. Therefore, it has to be approximated numerically too. This can be done by an Internal Numerical Differentiation (IND), which was proposed by Bock [11]. Illustratively, IND means to compute the derivative of the numerical integration algorithm, which leads to the discrete stress increment. The IND computes an approximation of the partial derivative that is of similar relative exactness as the solution of the integration itself. In non-linear implicit FEM-analysis, the tangent modulus

4.6.2

Deformation behaviour of a notched specimen

Some results of the simulated relaxation behaviour of a notched flat bar are shown in Figure 4.15. Since the main advantage of the proposed method of model implementation is its easy applicability to three-dimensional as well as to plane state or even onedimensional (uniaxial) FEM-problems, the numerical results of two simulations using three-dimensional and plane stress theory are compared. Additionally, experimental results of Ritter and Friebe [17] show that the model is able to predict the material response correctly.

86

4.6

Finite-Element Simulation

Figure 4.15: Normal strain in load direction after two hours relaxation time. Comparison between experimental and numerical results. Material: SS 304 L, temperature: 923 K. ESZ means plane stress.

87

4

4.7

Development and Application of Constitutive Models for the Plasticity of Metals

Conclusions

The mechanisms on the microscale of crystalline materials can be examined on different scales of magnitude. Starting from a scale, where the processes are described by help of activation energies and activation volumes as mechanically and thermally activated, it is possible to consider their stochastical nature by stochastic processes, from which by mean value considerations, a transition to macroscopic material equations is possible. To support the formulation of these models, simulations can be useful, which consider the multi-particle properties of the processes, and use the methods of cellular automata or molecular dynamics. For the numerical simulation and the parameter identification, a variety of sophisticated methods have been considered. The results show that it is possible to use the material models for the analysis of structures even under complex loading situations.

References [1] E. Steck: A Stochastic Model for the High-Temperature Plasticity of Metals. Int. J. Plast. (1985) 243–258. [2] E. Steck: The Description of the High-Temperature Plasticity of Metals by Stochastic Processes. Res. Mechanica 25 (1990) 1–19. [3] H. Schlums: Ein stochastisches Werkstoffmodell zur Beschreibung von Kriechen und zyklischem Verhalten metallischer Werkstoffe. Dissertation TU Braunschweig, Braunschweiger Schriften zur Mechanik 5 (1992). [4] R. Gerdes: Ein stochastisches Werkstoffmodell fu¨r das inelastische Materialverhalten metallischer Werkstoffe im Hoch- und Tieftemperaturbereich. Dissertation TU Braunschweig, Braunschweiger Schriften zur Mechanik 20 (1995). [5] H. Hesselbarth: Simulation von Versetzungsstrukturbildung, Rekristallisation und Kriechscha¨digung mit dem Prinzip der zellula¨ren Automaten. Dissertation TU Braunschweig, Braunschweiger Schriften zur Mechanik 4 (1992). [6] D. Sangi: Versetzungssimulation in Metallen. Dissertation TU Braunschweig, 1996. [7] F. Thielecke: Parameteridentifizierung von Simulationsmodellen fu¨r das viskoplastische Verhalten von Metallen – Theorie, Numerik, Anwendung. Dissertation TU Braunschweig, 1997. [8] F. Thielecke: Gradientenverfahren contra stochastische Suchstrategien bei der Identifizierung von Werkstoffparametern. ZAMM Z. angew. Math. Mech. 75 (1995). [9] E. Steck, M. Lewerenz, M. Erbe, F. Thielecke: Berechnungsverfahren fu¨r metallische Bauteile bei Beanspruchungen im Hochtemperaturbereich, Arbeits- und Ergebnisbericht 1991– 1993. Subproject B1, Collaborative Research Centre (SFB 319), 1993. [10] F. Thielecke: New Concepts for Material Parameter Identification Considering the Scattering of Experimental Data. ZAMM Z. angew. Math. Mech. 76 (1996). [11] H. G. Bock: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen. Bonner Mathematische Schriften, Bonn, Vol. 183 (1985). [12] J. Schlo¨der: Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Parameteridentifizierung. Dissertation Universita¨t Bonn, 1987. 88

References [13] F. Thielecke: Ein Mehrzielansatz zur Parameteridentifizierung von viskoplastischen Werkstoffmodellen. ZAMM Z. angew. Math. Mech. 76 (1996). [14] R. Jategaonkar, F. Thielecke: Evaluation of Parameter Estimation Methods for Unstable Aircraft. AIAA Journal of Aircraft 31(3) (1994). [15] R. Gerdes, F. Thielecke: Micromechanical development and identification of a stochastic constitutive model. ZAMM Z. angew. Math. Mech. (1996). [16] I. Rechenberg: Evolutionsstrategie ’94, Werkstatt Bionik und Evolutionstechnik, Band 1. Frommann-Holzboog, Stuttgart, 1994. [17] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical Measuring Methods. This book (Chapter 13). [18] K.-T. Rie, H. Wittke: Inelastisches Stoffgesetz und zyklisches Werkstoffverhalten im LCF-Bereich, Arbeits- und Ergebnisbericht 1991–1993. Subproject B4, Collaborative Research Centre (SFB 319), 1993. [19] E. Hinton, D. R. J. Owen: Finite Elements in Plasticity: Theory and Practice. Pineridge Press, Swansea, 1980. [20] M. Lewerenz: Zur numerischen Behandlung von Werkstoffmodellen fu¨r zeitabha¨ngig plastisches Materialverhalten. Dissertation TU Braunschweig, 1996. [21] K. Hornberger: Anwendung viskoplastischer Stoffgesetze in Finite-Element-Programmen. Dissertation Universita¨t Karlsruhe, 1988. [22] Hibbitt, Karlsson, Sørensen, Inc.: ABAQUS THEORY MANUAL, Version 5.4. Pawtucket, RI, United States, 1994. [23] E. Hairer, S. P. Nørsett, G. Wanner: Solving Ordinary Differential Equations I (Nonstiff Problems). Springer, Berlin, 1987. [24] E. Hairer, G. Wanner: Solving Ordinary Differential Equations II (Stiff Problems). Springer, Berlin, 1991. [25] S. W. Sloan: Substepping Schemes for the Numerical Integration of Elastoplastic StressStrain Relations. Int. J. Numer. Meth. Eng. 24 (1987) 893–911. [26] J. C. Butcher: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd, Chichester, 1986. [27] J. H. Verner: Families of Imbedded Runge-Kutta-Methods. SIAM J. Numer. Anal. 16 (1979) 857–875.

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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)

5

On the Physical Parameters Governing the Flow Stress of Solid Solutions in a Wide Range of Temperatures Christoph Schwink and Ansgar Nortmann*

Abstract At sufficiently low temperatures, host and solute atoms remain on their lattice sites. The critical flow stress r0 is governed by a thermally activated dislocation glide (Arrhenius equation), which depends on an average activation enthalpy, DG0 , and an effective obstacle concentration, cb . The total flow stress r is composed of r0 and a hardening stress rd , which increases with the dislocation density qw in the cell walls according to rd
qw 1=2 . 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 in the Arrhenius equation. In the main, Dg depends on the activation energy Ea of the diffusing solutes and the waiting time tw of the glide dislocations arrested at obstacles. Three different diffusion processes were found for the two f.c.c.-model systems investigated, CuMn and CuAl, respectively. Under certain conditions, the solute diffusion causes 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 become freely mobile and r0 as well as Dg vanish. In any temperature region, only a small total number of physical parameters is sufficient for modelling plastic deformation processes.

5.1

Introduction

The intention of the present project was to find out the physically relevant parameters, which determine the stable flow stress r in metallic systems of model character over a given wide range of temperatures and strain rates. * Technische Universita¨t Braunschweig, Institut fu¨r Metallphysik und Nukleare Festko¨rperphysik, Mendelssohnstraße 3, D-38106 Braunschweig, Germany 90

5.1

Introduction

Any theory describing plastic deformation modes of such systems will have to make use of these – and only these – parameters. As model systems we choose single phase binary f.c.c. solid solutions. They are on the one hand simple, macroscopically homogeneous materials, on the other hand exhibit all basic processes, which occur also in more complex alloys of technical interest. To cover a wide range of different characteristics existing in various binary alloys, we studied the two systems CuMn and CuAl, which differ appreciably in some salient properties (Table 5.1). We point to the misfit parameter, the variation of the stacking fault energy with solute concentration and the tendency for short range ordering. The systems have in common a metallurgical simplicity and a large range of solubility. The samples used were rods of polycrystals, for CuMn also of single crystals oriented either for single or for multiple ([100], [111]) glide. For low enough temperatures, i.e. roughly below room temperature, host and solute atoms remain on their lattice sites in our systems. Then, the flow stress is recognized to consist of two additive parts, which are in single crystals the critical resolved shear stress s0 , and the shear stress sd produced by strain hardening. s0 is best examined on crystals oriented for single glide, while results on sd originated from studies on [100] and [111] crystals. The parameters governing s0 and sd are discussed in Section 5.2. At higher temperatures, the solute atoms become increasingly mobile and start to diffuse to sinks, e.g. dislocations. As a consequence, an additional anchoring of glide dislocations occurs, known as dynamic strain ageing (DSA), which results in an additional contribution to flow stress, DrDSA , and in a decrease of the strain rate sensitivity (SRS) with increasing deformation. If the SRS reaches a critical negative value, jerky flow sets in, the so-called Portevin-Le Chaˆtelier (PLC) effect. The mechanisms inducing DSA and the relevant parameters represent the main part of project A8 and are reported in Section 5.3. We restrict the report on the own main results. For details and further literature, the reader is referred to the publications cited. Table 5.1: Metallurgical and physical properties of CuAl and CuMn. CuAl

CuMn

Misfit d d  Da=aDc

d  0.067 (weak hardening)

d  0.11 (strong hardening)

Bulk diffusion

QD  1:86 . . . 2:01 eV D0  0:8 . . . 5:6  105 m2 s1

QD  2:03 . . . 2:12 eV 1 D0  7:4 . . . 14:2  105 m2 s

Stacking fault energy

strongly decreasing with increasing cAl

independent of cMn

Slip character

5 . . . 10 at% Al planar

0.5 . . . 5:5 at% Mn homogeneous

Short range order

marked and increasing with cAl

negligible up to  5 at%

91

5

5.2

On the Physical Parameters Governing the Flow Stress of Solid Solutions

Solid Solution Strengthening

In the frame of the project, invited overviews on “Hardening mechanisms in metals with foreign atoms” [1], “Solid solution strengthening” [2] (in collaboration with project A9), and “Flow stress dependence on cell geometry in single crystals” [3] have been published.

5.2.1

The critical resolved shear stress, s0

A detailed investigation on CuMn [4] showed that the thermally activated process governing s0 is phenomenologically completely characterized by two parameters, the average activation enthalpy, DG0 , of the effective glide barriers, and the concentration of the latter, cb . This concentration resulted about 20 times smaller than the solute concentration, cMn . For DG0, values around 1.3 eV were found. The magnitude of cb and DG0 suggest the effective glide barriers to consist of complexes of at least two solute atoms. A dislocation segment after having surpassed an effective barrier sweeps in the subsequent elementary glide step an area containing cMn =cb solute atoms on the average. Altogether, we arrive at s0  s0 DG0 ; cb ; T; e_ .

5.2.2

The hardening shear stress, sd

Detailed mechanical and TEM-studies have been performed on CuMn-crystals oriented along [100] and [111] [5]. The hardening shear stress resulted as equal to the reduced stress, sd  s  s0 , and obeying the known relation [6]: 1=2

sd  t G b qt

:

1

Here, qt is the average total dislocation density, G the shear modulus. A surprising result was that t depends on the solute concentration, it decreases with increasing cMn . This means that for a given value of the reduced stress, the dislocation density qt is higher in an alloy than in the pure host. A further analysis showed that qt is stored nearly completely in the cell walls, which are fully developed already at small stresses and strains. The next result of relevance was the increase of the wall area fraction fw with solute concentration. Defining a mean dislocation density inside the cell walls, qw , by qw  qt =fw, we can rewrite Equation (1) as: 1=2 sd  t fw1=2 G b q1=2 w  w G b q w :

2

The prefactor w now turns out as independent of cMn and practically constant for a fully developed cell structure. w  0:25  0:03 from the experiments favourably com92

5.3 Dynamic Strain Ageing (DSA) pares with the lowest values for  calculated by theory [7] (cf. also [8]). This suggests the view that the most favourably oriented dislocation segments will cross the obstacle field and will be followed via the unzipping effect by all others at nearly the same stress, which is the lowest possible one. A TEM-investigation on Cu1.3 at% Mn crystals oriented for single glide [9], the first systematic TEM-study on a solid solution, yielded for the extended stage I a prevailing primary dislocation density, qprim , and a continuous decrease of the strain hardening rate with increasing strain. In stage II and above, the reduced flow stress was found as completely governed by the density of all secondary dislocations taken together, qsec . It is: s  s0    G b q1
2 sec ;

with   0:32  0:04 :

3

The total shear stress, s, results as a linear superposition of a “solid solution stress”, s0 , and a strain hardening stress, sd , as found also for the multiple glide crystals. It is completely described by four parameters, apart from the obvious ones, T and e_ : s  s0 DG0 ; cb ; T; e_  sd qt ; fw ; T; e_  :

4

The generalization for polycrystals adds the problem of compatibility of neighbouring grains. It is of importance mainly for small stresses and strains and introduces essentially the average grain diameter as an additional parameter in the case of a random assembly of grains (cf. [3]). At higher stresses, the relevant parameters are the same as in Equation (4).

5.3

Dynamic Strain Ageing (DSA)

5.3.1

Basic concepts

In the commonly applied models [10–12], the contribution of DSA to flow stress, DrDSA , increases proportional to the increase in the line concentration C of glide obstacles on arrested, “waiting” dislocations by DC during the waiting times, tw [11–13]. It is generally assumed that DC is a function of DTtw , where DT is the diffusion coefficient of the underlying process with the activation energy Ea. The waiting time tw is connected with the strain rate e_ via tw  X=_e, where X represents the “elementary strain” [14]. Phenomenologically, DSA can be described by an additional free activation enthalpy Dg  DgEa ; tw  entering besides DG0 the well-known Arrhenius equation [15]. Ample DSA leads to flow stress instabilities (PLC-effect). Details of the processes inducing jerky flow can be studied in the region of stable flow preceding a PLC-region by measuring with high accuracy i) stress-strain curves, re (Figure 5.1), ii) strain rate sensitivities (SRS) of flow stress, Dr=D ln e_ , along whole stress-strain curves and over a wide range of temperature. The results are presented in the following. 93

5

On the Physical Parameters Governing the Flow Stress of Solid Solutions

Figure 5.1: Schematic stress-strain curve showing the definition of the critical stresses ri  and strains ei . r0 is the critical flow stress (see [16]).

5.3.2

Complete maps of stability boundaries

We succeeded in establishing the first complete maps of boundaries of stable flow. From copper-based solid solutions, polycrystalline samples of six different Mn- and three Al-concentrations have been studied, furthermore CuMn-crystals oriented for single and multiple ([100]) glide [15–20]. Figure 5.2 shows a survey of occurrence and types of instabilities in Cu 2.1 at% Mn. The critical reduced stresses ri  r0  (gained from about 40 recurves (see Figure 5.1) running parallel to the ri  r0  axis) are plotted as function of temperature T. There are three transitional temperature intervals, labelled a, b and c, where several regions of stable and unstable deformation alternate along re-curves. Outside these intervals, the stress-strain curves are either stable or jerky throughout. In the small interval c, 290 8C 9 T 9 305 8C, an irregular sequence of bursts of type C

Figure 5.2: Mode of deformation map: dependence of reduced critical stresses ri  r0  on temperature T for Cu 2.1 at% Mn. Basic strain rate e_ 1  2:45  106 s1 . The hatched areas represent domains of unstable deformation with the predominant types of instabilities indicated (see [18]). 94

5.3 Dynamic Strain Ageing (DSA)

Figure 5.3: Reduced critical flow stresses for the beginning and end of jerky flow as functions of T. (a) polycrystals; (b) [100]-crystals, Cu 2 at% Mn (see [19]).

prevents the existence of a unique dependence of ri on T (or e_ ) as could be established for intervals a and b. For further details see [16, 18]. The strain hardening coefficient, which roughly remains constant for temperatures up to about 300 K, decreases strongly with further increasing temperature owing to recovery processes. At about T  600 8C, it becomes nearly zero [18], the critical flow stress simultaneously vanishes as well as the additional enthalpy Dg and a steady state of deformation exists across the whole deformation curve. The solutes are now moving freely through the lattice [21]. Figure 5.3 a gives stability maps for several Mn-concentrations over the intervals a and b for polycrystals, Figure 5.3 b the same for [100]-crystals of 2 at% Mn [19]. The similarity of both is closest if the boundaries for the [100]-crystal are compared with those for a polycrystal of about 1.2 at% Mn. Contrarily, the boundary map for a single glide ([sg]-) crystal of 2 at% Mn (Figure 5.4) looks quite different [19]. Only a single boundary occurs over the whole range of temperatures. However, the curve can be divided into two parts, which for good reasons are noted as intervals a and b, too (see Section 5.3.3). Finally, boundary maps for CuMn-polycrystals have been compared with those measured for CuAl [20]. Figure 5.5 presents characteristic examples. The complete correspondence of Cu 0.63 at% Mn with Cu 5 at% Al is obvious and indicates the existence of two different PLC-domains. They are labelled as domains I and III. With in95

5

On the Physical Parameters Governing the Flow Stress of Solid Solutions

Figure 5.4: Reduced critical flow stresses sxa;b for Cu 2 at% Mn single glide crystals as functions of T. In contrast with multiple glide crystals (poly-, [100]-, see Figure 5.3) at any temperature, only one boundary of stability occurs (see [19]).

Figure 5.5: Deformation-mechanism maps of CuAl (a–c) (see [20]) and CuMn (d–f) as obtained at different temperatures but constant basic strain rate e_ 1  2:5  106 s1 . The reduced critical stresses ri  r0  indicate the transition between stable and unstable deformation (stress-strain curves running parallel to the ordinate). The hatched areas indicate the PLC-regions. a) Cu 5 at% Al, b) Cu 7.5 at% Al, c) Cu 10 at% Al (values for CuAl from [20]), d) Cu 0.63 at% Mn (*) and Cu 0.95 at% Mn (n), e) Cu 1.1 at% Mn, f) Cu 2.1 at% Mn (values for CuMn from [18]).

96

5.3 Dynamic Strain Ageing (DSA) creasing concentration, a “bulge” develops on the boundary r2 T; e_ . It is clearly visible already for 0.95 at% Mn (Figure 5.5 d), and is extended to a peak for 1.1 at% Mn (Figure 5.5 e) [18]. As a consequence, an additional PLC-domain II develops bounded by the anomalous boundary r 2 at the lower temperature side (Figure 5.5 b and e). The island of stability, which appears in Cu 1.1 at% Mn and Cu 2.1 at% Mn above about 400 K is covered in CuAl by the domain II (Figure 5.5 b, c, e and f) [20].

5.3.3

Analysis of the processes inducing DSA

Precise measurements of the critical stresses ri for the onset of jerky flow [16] on the one hand, and of changes in the flow stress, Dr, after variations in strain rate [17, 18] on the other, are the basis of an analysis of DSA. Figure 5.6 shows as an example variations in shear stress measured in stages I and II of a crystal oriented for single glide [22]. Generally, one has to distinguish between the instantaneous variations, Dsi , occurring immediately after a change in e_ , and the stationary ones, Dss . (Remark: For single crystals, the flow stress r is always replaced by the resolved shear stress, s.) It is the difference, Dss  Dsi   DDsDSA ; which reflects the effect of DSA and causes a decrease of the SRS  Ds=D ln e_ T [19]. Analogously, for polycrystals is SRS  Dr=D ln e_ T [18]. The boundaries r2 , r3 of the “island of stability” in temperature interval b (see Figure 5.3) are governed by a thermally activated process as demonstrated in Figure 5.7: A decrease in T is qualitatively equivalent to an increase in e_ . We can take ri as indicating the onset of the thermally activated process and derive from ri  r0   A

B B=C C ln e_  ln e_  A0  T T

5

values for the activation energies Qm  B=C m  2; 3 [18].

Figure 5.6: Change in resolved shear stress, Ds, after a change in external strain rate of e_ 2 =_e1  2 : 1, against incremental true strain, Dc, taken in stage II at s  32:4 MPa and c  52:6%. The plot is corrected for the average strain hardening rate (see [22]). 97

5

On the Physical Parameters Governing the Flow Stress of Solid Solutions

Figure 5.7: Dependence of the reduced critical stresses on (a) 1=T at e_ 1  2:45  106 s1 , and on (b) ln e_ 1 at T  460 K; both plots for Cu 2.1 at% Mn in interval b (see [18]).

Another way of determining these energies starts from a consideration of the normalized instantaneous and stationary SRS, denoted by Si and Ss , respectively [18, 19]. Figure 5.8 gives their course with increasing stress along the stress-strain curve of a [sg]-crystal [22], Figure 5.9 shows Ss r alone for a polycrystal at various temperatures. Following the above mentioned models [11, 12], the marked dependence of SDSA  Ss  Si  on T (and also e_ ) is for short enough tw described by the relation [18]:

 DT n nEa : 6 SDSA
c
c_en exp kT e_ Here, Ea is the activation energy of pipe diffusion entering DT  D0 expEa =kT. Strain rate exponent n and Ea are best obtained from Equation (6) in regions, where SDSA varies linearly with stress yielding constant slopes M  qSDSA =qrT;e_ . One easily finds [18]:

 q ln M q ln M : 7 ; and nEa   n q ln e_ T q1=kT e_ The activation energies Qm and Ea;m found for the diffusion processes generating the PLC-domains I, II and III are compiled in Table 5.2 [20]. Where Qm and Ea;m can both 98

5.3 Dynamic Strain Ageing (DSA)

Figure 5.8: Dimensionless instantaneous Si  and stationary Ss  SRS against reduced stress s  s0 ; s0  critical resolved shear stress; T  263 K (see [22]).

Figure 5.9: The dependence of the stationary, normalized SRS on reduced stress for Cu 3.5 at% Mn at temperatures of interval a. All data points refer to states of stable deformation. The critical stresses are indicated for 73.4 8C. The Mm denote the linear slopes of the S-r  r0 curves (see [18]).

be measured for the same process, they are found equal, Qm  Ea;m , within scatter. Average values for the three DSA-processes are denoted by EI, EII , EIII. An important further quantitative result is that the strain rate exponent resulted as n  1=3 (within scatter) in all cases.

5.3.4

Discussion

The strain rate exponent n has for a long time been commonly assumed to equal 2/3 according to Cottrell and Bilby’s theory of lattice diffusion [23, 24]. Already the first experimental determination of n yielded, however, n  1=3 and has been explained by a pipe-diffusion mechanism governing the DSA-process concerned [25].

99

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On the Physical Parameters Governing the Flow Stress of Solid Solutions

Table 5.2: Activation energies of DSA-processes in CuAl and CuMn. Average value Ea1 [eV] Q1 [eV] Q2 [eV] EI [eV] a

Ea3 [eV] Q2 [eV] EII [eV] b

Ea3 [eV] b Q3 [eV] EIII [eV]

Cu x Mn 1

Cu x Al 5

10

0.63

1.3

2.1

3.5

0.74 ± 0.15 × × 0.74

0.79 ± 0.12 0.75 ± 0.10 0.76 ± 0.10 0.77

– × – –

– × 0.88 ± 0.10 0.88

0.88 ± 0.05 × 0.91 ± 0.10 0.89

0.86 ± 0.10 2 × 0.87 ± 0.10 0.86

× × ×

× 1.1 ± 0.30 1.1

× × ×

0.81 ± 0.10 × 0.81

0.86 ± 0.10 × 0.86

0.87 ± 0.10 × 0.87

1.42 ± 0.25 1.46 ± 0.30 1.44

× × ×

1.9 – 1.9

1.53 ± 0.10 1.-59 ± 0.08 1.56

1.27 ± 0.10 1.25 ± 0.05 1.26

1.15 ± 0.10 1.16 ± 0.10 1.15

1 Values for CuMn from [18]; able.

2

Cu 4.1 at% Mn; –: not measured; ×: not defined or not measur-

Shortly after, Schlipf [26] pointed out that a more general relation than Equation (6) for SDSA is conceivable, viz. SDSA
DC q , which by use of DC
DT=_er yields SDSA
cq  e_ qr (see also [27]). Now q  1=2 and r  2=3 would give an exponent n  qr  1=3 also in the case of lattice diffusion. On the other side, it became more and more clear that n  1=3 holds quite generally for any DSA-process [20, 28, 29]. To clarify the puzzling situation, a more extensive experimental analysis of SDSA has been undertaken by studying and simultaneously evaluating the dependence of SDSA on flow stress as well as on solute concentration. We found [30] that • •

the Mulford-Kocks model of DSA [12] describes the experiments clearly better than the van den Beukel model [11], and the data – taking the most reliable ones – are in favour of a simple proportionality to solute concentration, i.e. q  1.

This would exclude a lattice diffusion and is suggesting an own pipe diffusion mechanism for each DSA-process. Recent theoretical work [31] points to an even probable existence of several modes of pipe diffusion [20] along dissociated dislocations. A recently found method to measure immediately the average waiting time tw of dislocations [32] showed that the elementary strain X continuously increases with the flow stress, the total increase never exceeding a factor of only 10. In principle, X is deducible from a knowledge of the dislocation arrangement qt ; qf ; fw  and of the density of glide barriers cb  [19]. However, a general theory is still missing. Therefore, Xr and with it tw , which governs stress transients, are still to be considered as parameters. 100

Wille, Gieseke and Schwink [4]; Neuha¨user and Schwink [2]

Neuhaus and Schwink [6]; Neuhaus, Buchhagen and Schwink [33]; Heinrich, Neuhaus and Schwink [9]

Springer, Nortmann and Schwink [30] Schwarz [13]; McCormick [34]; Springer and Schwink [32] Schwink and Neuha¨user [35]; Neuha¨user [36]; Traub, Neuha¨user and Schwink [37]; Nortmann and Schwink [22]

i) average activation enthalpy, DG0 (eV) ii) effective barrier concentr. cb  f c i) total dislocation density, qt r [m–2] ii) volume fraction of disloc. walls, fw r i) activation energy Ea;m (eV), m  I, II, III, of the diffusion inducing DSA ii) tw s  X=_e  waiting time of arrested dislocations iii) strain rate exponent, n relaxation constant, B relaxation time H waiting time tw b a  1  d ln Va =d ln c_

critical flow stress, r0 (single crystal: crss, s0 )

reduced flow stress, rd rd  r  r0 ; sd  s  s0  flow stress contribution, DrDSA , or additional enthalpy, Dg  VDrDSA , Dg  DgEa ; tw n  limiting Dg-value, Dgmax 0:1DG0 transition from instantaneous to stationary flow stress, Dri  Drs active slip volume, Va

Solid solution hardening

Dislocation hardening

Dynamic strain ageing (DSA)

Exhaustion of DSA

Transitions of DSA owing to variations in e_

Variations in mobile dislocation density, owing to variations in e_

Springer and Schwink [25]; Kalk, Schwink and Springer [17]; Kalk and Schwink [18]; Nortmann and Schwink [20]

Parameter (quanitatively measured) Literature

Characteristic magnitude

Elementary process

Table 5.3: Overview of the parameters investigated quantitatively in project A8. They characterize the flow stress and its strain rate sensitivity in single phase random f.c.c. solid solutions along stress-strain curves taken over a wide range of temperatures. The DSA parameters come into play only at higher temperatures (about room temperature!). In the future, some of the parameters will prove derivable from more complete theories.

5.3 Dynamic Strain Ageing (DSA)

101

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On the Physical Parameters Governing the Flow Stress of Solid Solutions

The detailed analysis of the SRS also allows to evaluate quantitatively the variations of the additional enthalpy by varying the strain rate, DDg, along the whole stress-strain curves, and to determine the value of the quantity Dg_e itself [30]. It amounts up to about 10% of DG0  1:3 eV (see Section 5.3.1) [15, 30]. At higher flow stresses (0 100 MPa), Dg is observed to approach a limit with decreasing strain rate. The course of Dg_e can be best discussed for data on CuAl. At the time, the approach of the limit is described by a kind of relaxation parameter B. It is concluded that an exhaustion of solute atoms available for the diffusion process in question limits the increase of DrDSA , and not a saturation of glide dislocations by the diffusion-induced glide obstacles [30]. Finally, the question has been addressed, whether the SRS might be influenced besides of DSA-processes also by variations of the mobile dislocation density qm when varying the strain rate [22]. In fact, a superposition of both effects could be demonstrated. There exists a very small interval around a transition temperature, above which DSA-effects are dominating the SRS, while qm -effects dominate below.

5.4

Summary and Relevance for the Collaborative Research Centre

Shortly summarizing this report, we can say that any mechanism contributing to flow stress can be accounted for by a few measurable parameters in a model description. Whether a mechanism and parameter is relevant or negligible depends on the experimental conditions, e.g. on temperature. In any case, the total number of relevant parameters is defined and quite limited. Table 5.3 is to give a concise overview of all results obtained as far as they concern the parameters investigated. The methods developed in this project to determine these parameters (cf. Table 5.3) can be applied to any material. The parameters will enter any final constitutive material equations developed, e.g. those of project A6 of the Collaborative Research Centre (SFB). Results and experiences of our project have been also exchanged with project A1. Throughout the work, there was an intimate contact to project A9.

References [1] Ch. Schwink: Rev. Phys. Appl. 23 (1988) 395. [2] H. Neuha¨user, Ch. Schwink: In: H. Mughrabi (Ed.): Materials Science and Technology, Vol. 6. VCH Weinheim, 1993, p. 191. [3] Ch. Schwink: Scripta metall. mater. 27 (1992) 963 (Viewpoint Set No 20). [4] Th. Wille, W. Gieseke, Ch. Schwink: Acta metall. 35 (1987) 2679. 102

References [5] R. Neuhaus, Ch. Schwink: Phil. Mag. A 65 (1992) 1463. [6] For a review referring mainly to pure copper, see: S. J. Basinski, Z. S. Basinski: In: F. R. N. Nabarro (Ed.): Dislocations in Solids, Vol. 4. North-Holland, Amsterdam, 1979, p. 261. [7] W. Pu¨schl, R. Frydman, G. Scho¨ck: phys. stat. sol. (a) 74 (1982) 211. [8] G. Saada: In: G. Thomas, J. Washburn (Eds.): Electron Microscopy and Strength of Crystals. Interscience, New York, 1963, p. 651. [9] H. Heinrich, R. Neuhaus, Ch. Schwink: phys. stat. sol. (a) 131 (1992) 299. [10] For reviews see: a) Y. Estrin, L. P. Kubin: Acta metall. 34 (1986) 2455. b) Y. Estrin, L. P. Kubin: Mat. Sci. Eng. A 137 (1991) 125. c) P. G. McCormick: Trans. Indian Inst. Metals 39 (1986) 98. d) L. P. Kubin, Y. Estrin: Rev. Phys. Appl. 23 (1988) 573. e) H. Neuha¨user: In: D. Walgraef, E. M. Ghoniem (Eds.): Patterns, Defects and Materials Instabilities, Kluwer Ac. Publ., Dordrecht, 1990, p. 241. [11] A. van den Beukel: phys. stat. sol. (a) 30 (1975) 197. [12] R. A. Mulford, U. F. Kocks: Acta metall. 27 (1979) 1125. [13] R. B. Schwarz: Scripta metall. 16 (1982) 385. [14] L. P. Kubin, Y. Estrin: Acta metall. mater. 38 (1990) 697. [15] Th. Wutzke, Ch. Schwink: phys. stat. sol. (a) 137 (1993) 337. [16] A. Klak, Ch. Schwink: phys. stat. sol (b) 172 (1992) 133. [17] A. Kalk, Ch. Schwink, F. Springer: Mater. Sci. Eng. A 164 (1993) 230. [18] A. Kalk, Ch. Schwink: Phil. Mag. A 72 (1995) 315. [19] A. Kalk, A. Nortmann, Ch. Schwink: Phil. Mag. A 72 (1995) 1229. [20] A. Nortmann, Ch. Schwink: Acta metall. mater. 45 (1997) 2043-2050, 2051–2058. [21] H. Neuha¨user: This book (Chapter 6). [22] A. Nortmann, Ch. Schwink: Scripta metall. mater. 33 (1995) 369. [23] A. H. Cottrell, B. A. Bilby: Proc. Phys. Soc. Lond. A 62 (1949) 49. [24] J. Friedel: In: Dislocations, 368 Pergamon, Oxford, 1964, p. 405. [25] F. Springer, Ch. Schwink: Scripta metall. mater. 25 (1991) 2739. [26] J. Schlipf: Scripta metall. mater. 29 (1993) 287; Scripta metall. mater. 31 (1994) 909. [27] H. Flor, H. Neuha¨user: Acta metall. 28 (1980) 939. [28] C. P. Ling, P. G. McCormick: Acta metall. mater. 41 (1993) 3127. [29] S.-Y. Lee: Thesis, Aachen, 1993. [30] F. Springer, A. Nortmann, Ch. Schwink: phys. stat. sol. (a) 170 (1998) 63–81. [31] J. Huang, M. Meyer, V. Pontikis: Phil. Mag. A 63 (1991) 1149; J. Phys. III 1 (1991) 867. [32] F. Springer, Ch. Schwink: Scripta metall. mater. 32 (1995) 1771. [33] R. Neuhaus, P. Buchhagen, Ch. Schwink: Scripta metall. 23 (1989) 779. [34] P. G. McCormick: Acta metall. 36 (1988) 3061. [35] Ch. Schwink, H. Neuha¨user: phys. stat. sol. 6 (1964) 679. [36] H. Neuha¨user: In: F. R. N. Nabarro (Ed.): Dislocations in Solids, Vol. 6, North-Holland, Amsterdam, 1983, p. 319. [37] H. Traub, H. Neuha¨user, Ch. Schwink: Acta metall. 25 (1977) 437. The publications [1–5, 9, 15–20, 22, 25, 30, 32, 33] resulted from work performed in the present project of the Collaborative Research Centre (SFB).

103

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)

6

Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys Hartmut Neuha¨user *

6.1

Introduction

Slip deformation in crystal is inhomogeneous by nature as it is accomplished by the production and movement of dislocations on single crystallographic planes. Usually, only few dislocation sources are activated and produce slip on few planes, where, in particular in fcc and hcp crystals and even more pronounced in alloys with low stacking-fault energy, many dislocations move on the same plane. This is provoked in particular if the dislocations on their path through the crystal change the obstacle structure in the slip plane, e.g. in short-range ordered or in short-range segregated alloys (i.e. in nearly all alloys) so that the first dislocation feels a stronger “friction” than the succeeding ones. As the macroscopic elongation of the sample is distributed in general heterogeneously among the crystallographic planes, the quantities of resolved strain a and _ defined as: strain rate a, _ 0 l0 a ˆ l=l0 l0 and a_ ˆ l=l

1

(with l_ = macroscopic deformation rate, l0 = specimen length, l0 = Schmid orientation factor), and commonly used in the formulation of constitutive equations, cannot be directly connected with realistic dislocation behaviour. Therefore, in this work, the local strain and strain rate in slip bands, which are the active regions of the crystal [1], have been measured by a micro-cinematographic method [2]. Cu-based alloys turned out to be a convenient model system for experimental reasons: Single crystals can be grown easily in reasonable perfection and the stacking-fault energy c can be varied by changing the alloy composition. In Cu2 . . . 16 at% Al, c varies from 35 to 5 mJ/m2 with increasing Al content, while it remains (nearly) constant (c & 40 mJ/m2) for Cu-2 . . . 17 at% Mn. Thus, the effects of stacking-fault energy can be separated from those of solute hardening and short-range ordering, which are comparable for both alloy systems.

* Technische Universita¨t Braunschweig, Institut fu¨r Metallphysik und Nukleare Festko¨rperphysik, Mendelssohnstraße 3, D-38106 Braunschweig, Germany 104

6.2

Some Experimental Details

While solid solution hardening has been extensively studied and is well documented and appears well understood in the temperature range below room temperature [3–5], several open questions remain, which are particularly connected with inhomogeneity of slip above ambient temperature. In a certain range of deformation conditions, even macroscopic deformation instabilities occur like the Portevin-Le Chaˆtelier (PLC) effect. This effect appears to be a consequence of the mobility of solute atoms in the strain field of dislocations (“strain ageing”) and are extensively studied in [6]. In the following, we briefly review our local slip line observations performed during deformation and accompanied by EM and AFM (atomic force microscope) investigations of the slip line fine structure and of dislocation structure by TEM. The conclusions reached so far as well as the still open questions are summarized. According to the changes of principal mechanisms, the chapter will be divided into the ranges around room temperature, at intermediate temperatures, and at elevated temperatures.

6.2

Some Experimental Details

Observations with video records of slip line development during deformation are performed in two special set-ups with tensile deformation machines equipped with light microscopes. The slip steps are visualized in dark field illumination as bright lines, where the scattered light intensity is a measure of slip step height. The minimum step height resolved is around Smin ˆ 5 to 10 nm (depending on the quality of the electropolished crystal surface), changes of larger step heights down to dS  5 nm can be resolved. One apparatus is designed for very high resolution in time (down to 3 ls) [7, 8], using photo diodes and a storage oscilloscope with pretrigger parallel to video recording. From the rate of intensity increase and by comparison with interference microscopy of the same slip band after full development, the local rate of step height increase and thus the local shear rate can be determined. The time shift of curves of development recorded by two neighbouring photo diodes immediately yields the velocity of growth in length, corresponding to the velocity of screw dislocations if the observations are performed on the “front” surface, where the plane with Burgers vector and crystal axis cuts the crystal surface (cf. [9]). By using a second microscope and video system observing the opposite front surface of the plate-shaped crystal, the time, which slip needs to traverse the crystal thickness, can be determined. The second apparatus is designed for observations at various temperatures (up to 500 8C) [10] and with a wide field of view between 0.3 and 4.2 mm in order to check spatial correlations between activated slip bands. The video system usually records with a frame rate of 50 s–1 and can be increased up to 500 s–1. For investigation of the fine structure of slip lines, which is not resolved by light microscopy, after deformation EM replica and AFM observations are performed. In addition, in some cases, the dislocation structure developed during deformation steps has been studied by TEM. 105

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys For creep tests at elevated temperatures, a special creep set-up was designed, using the controlling system of the drive of the Instron tensile machine to keep arbitrary constant stress values and recording strain and strain rate versus time. In particular, the system permits rapid changes between deformation conditions, e.g. between testing at constant deformation rate and at constant stress. The specimen is inside a vacuum tube (p < 10–4 mbar) surrounded by a furnace reaching temperatures up to about 1000 8C (± 2 K). Load cell and extensometers (connected by rods to the grips) are situated in the cool part inside the vacuum vessel to avoid any friction effects.

6.3

Deformation Processes around Room Temperature

The macroscopic stress during deformation with various constant strain rates and during stress relaxation experiments has been measured for different alloy compositions for many single crystals oriented for single glide and for polycrystals. Crystals oriented for multiple glide have been extensively studied by Schwink and Nortmann (cf. [6]). As an example, Figure 6.1 gives the critical resolved shear stress (crss) s0 and the ef_ deterfective activation volumes V
kT=S (or strain rate sensitivities S
ds=d ln a) mined for Cu-Al single glide crystals, showing in Figure 6.1 a the typical low temperature rise indicating thermal activation as rate controlling process, the plateau region at intermediate temperatures (now interpreted as a superposition of thermal activated glide and solute mobility, cf. Section 6.4) with a range of unstable deformation ending in a maximum of the crss, and the rapidly decreasing high temperature part (cf. Section 6.5). These regions are also reflected in the strain rate sensitivity (Figure 6.1 b), which will be discussed in more detail below.

6.3.1

Development of single slip bands

The slip line observations show that for alloy concentrations c ≥ 4 at% Al and c ≥ 7 at% Mn in stage I (yield region), the deformation is constricted into small crystal volumes, which can be classified in a fractal hierarchy from slip lines on the nm-scale (e.g. for Cu-10 at% Al most frequent distances dsl = 85 nm, step heights Ssl = 25 nm), slip bands on the lm-scale (e.g. about dsb = 5 lm in slip band bundles, 80 lm at the Lu¨ders band front, Ssb = 120 nm), slip band bundles on the 100 lm-scale (e.g. average dbb = 300 lm, integrated step height Sbb up to 15 lm) and up to the Lu¨ders band (e.g. width BLB = 3.8 mm, total shear SLB = 36 lm) [11]. Direct measurements of the dislocation velocity from slip band growth in length (_xL ) [12] result in v s
x_ L  25 m=s 106

2

6.3

Deformation Processes around Room Temperature

a)

b) Figure 6.1: a) Temperature dependence of the critical resolved shear stress (crss s0 ) of Cu-2 . . . 15 at% Al single crystals oriented for single slip at a deformation rate of l_= 2 · 10–3 mm/s (a_ = 3.6 · 10–5 s–1). In the range of macroscopic slip instabilities (“PLC effect”, dotted line), the stress intervals of serrations are plotted. b) Temperature dependence of the (normalized) strain rate sensitivity S
ds=d ln a_ (determined from stationary back extrapolated stress jumps during strain rate changes) for one selected Al concentration (c = 15 at%). Interval with arrows indicates PLC effect (jerky flow). The plots a) and b) contain data from literature (` cf. refs. in [5]) in addition to our own measurements (*, • and I, indicating the interval between stress maxima and minima in serrated flow).

for the velocity of screw dislocation groups at the edge of an expanding new slip band on the front surface. For this example of Cu-15 at% Al, the velocity of edge dislocations can be estimated from the measured growth rate in height Sb if a reasonable distance de of the (edge) dislocations moving in groups is assumed. As we consider here the very first dislocation group produced by the source, we use an average distance between edge dislocations in the group as determined on TEM micrographs for single, slightly piled-up groups, i.e. de = 0.2 lm [13, 14]. Then v e
Ssb de =b  3 m=s

3 107

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys results, where Ssb is the slope of the step height versus time curve in the very first few ms. The ratio v s/v e&8 (at least 3) appears reasonable in view of the interaction strength of solute obstacles with different dislocation characters and with estimates of friction stresses on dislocations from the shape of dislocations on TEM micrographs [13]. Figure 6.2 a shows the typical slip band development recorded by video and the photo diode; in Figure 6.2 b, the local shear rate during the development is shown in a double log plot. After the very first rapid growth of step height, the rate slows down gradually when more and more slip lines are added to the slip band. While the very first dislocation group appears to move under overstress, resulting in a local slip instability with the shear rate exceeding that imposed by the deformation machine (cf. dotted line in Figure 6.2 b), the successive groups feel opposing internal stresses. This behaviour can be modelled by a (local) work hardening [11, 15]. It shows that the local strain rate a_ loc
Rm bv

4

a)

b) Figure 6.2: a) Records of slip step growth (step height Ssb versus time t) of a single slip band, evaluated from photodiode and digital storage oscilloscope (note ms time scale and high level of noise). b) Variation of the growth rate in step height Ssb (= local slopes of a)) for several slip bands in Cu-15 at% Al (compared with earlier results in Cu-30 at% Zn), plotted in double log scales versus time t. The dotted line indicates the growth rate, which would be necessary to accommodate the imposed deformation rate by one single slip band. 108

6.3

Deformation Processes around Room Temperature

(Rm = mobile dislocation density) varies with time in the activated slip zones by many orders of magnitudes and that the assumption of an average strain rate according to Orowan’s equation (Equation (4)), which is frequently used in constitutive modelling, is not realistic and somewhat arbitrary. The local strain rate a_ loc can be connected with the external deformation rate only by using the “active crystal length” la instead of the total crystal length l0 in Equation (1): _ 0 la ; a_ loc
l=l

5

where la
nab Bsb

6

(nab = number of simultaneously active slip bands, Bsb = active width of a slip band measured along the crystal axis [2]) is a function of deformation rate, stress, strain, temperature, and time in general. Instead, the nucleation rate formulation of Orowan can be used to express the average strain rate _ ; a_
NbF

7

where the rate N_ of successive source activations is required (i.e. in our case, the rate of slip band activations), and the details of slip band development do not matter because only the total area F swept by all active dislocations during the event enters. The slip instability at the onset of each slip band evolution can be detected as a slight stress drop in special experiments (using very thin, short specimens with the load cell directly connected to one crystal grip [16]) and in acoustic emission records (e.g. [17]); they are too small to be resolved in case of common specimens (y 4 mm, length 120 mm) in usual tensile machines with their large inertia. The firstly activated dislocation source of a new slip band is always on that crystal surface, which due to the bending and lattice rotation by local shear feels a slight overstress (surface “high”, see below). The average times tHL for the edge dislocation group to traverse the crystal from this front surface to the opposite one are found, for plate-shaped Cu-15 at% Al crystals of D = 170 and 220 lm thickness, to be 11.1 and 0.7 s, respectively, corresponding to average velocities of the edge dislocations of 22 and 440 lm/s (slip plane inclined by 458 to the crystal axis). The large difference to Equation (3) is due to the opposing stress gradient along this dislocation path and reflects the high strain rate sensitivity around room temperature (cf. Figure 6.1 b). An important process in the formation of dislocation groups from each activated source is the partial destruction of obstacles by the dislocations cutting across obstacles in the slip plane. In case of the present alloys Cu-Al and Cu-Mn, their well-known tendency to short-range ordering suggests that the effective obstacles in the yield region are groups of solute atoms in an at least partially ordered configuration. This will be destroyed by a cutting dislocation so that the next dislocation will be able to move at a lower stress. Although some energetically favourable solute configurations will be “repaired” by the following dislocations, the net effect is a destruction of “friction” to a lower value in the activated slip plane. This process was modelled [14] using realistic 109

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys next-neighbour pair potentials determined from diffuse X-ray scattering measurements on Cu-15 at% Al crystals [18] in Monte Carlo simulations of a model crystal with the measured short-range order, and the resulting dislocation configurations of the group compared with those observed in TEM [14] (Figure 6.3). This indicates quite high intrinsic “friction” stresses of the original alloy. The dislocation group is able to move at a distinctly lower stress because the first dislocation feels, in addition to the external stress, the internal stress from the following piled-up dislocations. The resulting fluctuations in local stress are especially pronounced in the case of Cu-Al alloys, where the dislocation groups on single slip planes are much more extended than in Cu-Mn alloys as a consequence of the low stacking-fault energy in the former case, which prevents dislocations from easy cross-slip. This tendency is clearly observed in TEM micrographs of the dislocation structure after deformation in stage I (Figure 6.4 a, b [14]) and in the slip band fine structure imaged by EM replica in Figure 6.4 c, d, and by AFM in Figure 6.4 e, f [19]. In particular, the high resolution of the

a)

b) Figure 6.3: a) Variation of the diffuse antiphase boundary energy in the slip plane by passage of a number n of dislocations crossing the slip plane and changing near neighbour short-range-ordered configurations. b) Interaction stresses between dislocations in single dislocation groups (sww dotted lines) observed by TEM for annealed and quenched Cu-10.7 at% Al crystals. Full lines sSRO give the difference between these curves (*) and the simulation result (^) from a), assuming sSRO
cSRO =b [14, 23]. 110

6.3

Deformation Processes around Room Temperature

a)

b)

c)

d)

e)

0.5 lm

f) Figure 6.4: Comparison of dislocation structures: TEM micrographs after deformation in stage I at room temperature: a) Cu-14.4 at% Al; b) Cu-12 at% Mn), and slip line structures, EM replica: c) Cu-10.7 at% Al; d) Cu-8 at% Mn; AFM micrographs: e) Cu-15 at% Al; f) Cu-17 at% Mn.

last method permits to decide that in Cu-Al, the activated slip line is indeed on a single crystallographic plane according to the measured step angle (cf. for Cu-30 at% Zn [20], for Cu-7.5 at% Al [21]). In Cu-Mn alloys, on the other hand, the high probability for cross-slip (c  cCu ) appears to be the reason for the Cu-like slip line arrangement with clusters of activated slip planes during the work-hardening stages II and III, while in Cu-Al alloys with its low c value, very strong local variations of slip behaviour occur [22]. This again indicates that the average stress and strain usually given in stress-strain 111

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys diagrams may differ considerably from the local values relevant at the active dislocation sources and for the moving dislocations.

6.3.2

Development of slip band bundles and Lu¨ders band propagation

The process of successive activation of slip lines in the slip band and of slip bands in the slip band bundle or at the front of a propagating Lu¨ders band has been investigated in detail by observations on thin flat crystals [16] and by FEM calculations of the stress around slip steps as well as calculations of the stress field resulting from excess dislocation groups below the surface necessary to shield the notch stress of the slip step (Figure 6.5). These calculations show that in the surface region, maxima of shear stress occur in characteristic distances ahead of a previously activated slip plane (irrespective of the details of dislocation arrangement in the group), i.e. in a distance of 200 nm and in a distance of 30 lm. The former corresponds to the observed distances of slip lines dse, the latter to those of slip bands dsb; the numbers depend on the positions of the front and last dislocation of the excess group. Thus, the activation of a new source occurs under a certain overstress, which explains the above-mentioned slip instability in the first stage of slip band growth. It also indicates that the externally measured crss or yield stress in stage I has to be considered with some caution, although, owing to the high strain rate sensitivity (Figure 6.1 b), the local stress will exceed the average value by only a few percent.

Figure 6.5: Resolved shear stresses in the slip planes near the upper surface of the crystal (cf. sketch below) around a slip step and from the stresses of dislocations (B) below the surface, which are necessary to shield the notch stress of about 50 MPa (A). Note the maxima of the resulting stress around distances of 200 nm and 30 lm, which are prefered locations for next source activation. Calculation for S = 100 nm, a = 200 nm, n = 50 dislocations, distance to the front dislocation= 33.5 lm. 112

6.3

Deformation Processes around Room Temperature

The above-mentioned differences between Cu-Al and Cu-Mn disappear in the mesoand macroscopic level: The appearance of slip bands, slip band bundles and the Lu¨ders band is quite similar (Figure 6.6 a–d). In observations specially designed for examining the long-range correlations of slip by applying low magnification in the light microscope, it was found [23, 24] that the neat and simple Lu¨ders band configuration (Figure 6.6 c, d) usually observed in thin flat crystals [16] can be produced also in thick cylindrical crystals (4 mm y) if the external load (i.e. applied deformation rate) is selected low enough. Such a deformation front, which is shown schematically in cross section in Figure 6.7 (left side), propagates with a certain velocity v LB from one crystal grip to the other during tensile deformation in stage I (yield region) in a nearly stable configuration (solitary wave [25]). The first source of a new slip band ahead of the front is activated at (or near) surface “high” (Figure 6.7), and slip gradually crosses the crystal towards the opposite surface “low” against a gradient of bending stress (Figure 6.5). The average plastic front is normal to the crystal axis and the propagation velocity along the crystal, as determined from the measured distances and times of front slip bands (Figure 6.8), is found to be proportional to the external deformation rate l_ if this remains below the critical value. Above that, the deformation mode changes to the formation of slip band bundles (Figure 6.6 a, b) whose trace across the crystal follows the crystallographic slip planes (Figure 6.7, right side). Now, the stress due to the increased deformation rate appears to be high enough to activate sources more or less at random along the crystal length. They grow to slip band bundles by adding neighbouring slip bands according to the mechanism shown in Figure 6.5 (cf. [26, 27]). From such a slip band bundle, the Lu¨ders band starts when the bundle has reached a certain sufficiently high integrated shear, implying enough stress concentration due to the bending moment, the thickness reduction and the lattice rota-

a)

b)

c)

d) Figure 6.6: Light micrographs of slip band structure on the crystal front surface for the two deformation modes in stage I of Cu-Al and Cu-Mn: Formation and growth of slip band bundles: a) Cu10.7 at% Al; b) Cu-17 at% Mn, and formation and propagation of a Lu¨ders band front: c) Cu15 at% Al; d) Cu-17 at% Mn. 113

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys

Figure 6.7: Schematic representation (crystal cross section along its axis) indicating the shear distribution in the Lu¨ders band front which propagates with velocity v LB, and in slip band bundles (cf. Figure 6.6 a, b). The slip bands are initiated in the Lu¨ders band at surface “high”, at the edge of the slip band bundle at its right on surface “high”, at its left on surface “low”, according to the bending stresses and the stress patterns of Figure 6.5.

a)

b) Figure 6.8: Determination of Lu¨ders band propagation rates v LB from plots of cumulated distances xF and times tF of the front slip bands of Lu¨ders bands at various external deformation rates of l_= 2, 4, and 10 lm/s (selected below the critical value) for Cu-10.7 at% Al (a) and Cu12 at% Al (b).

114

6.3

Deformation Processes around Room Temperature

a)

b) Figure 6.9: a) FEM analysis of the stress pattern around the Lu¨ders band front; b) Plot of the resolved shear stress near the surface across the Lu¨ders band front from the sheared (left) to the virgin part (right), for different radii of curvature (R) in the Lu¨ders band region (cf. [23]). The increased stress at the left, mainly due to the reduced cross section, is compensated by work hardening (kinematic stress).

tion, which accompany the local shear of the single crystal (with slip planes inclined by 458 to the crystal axis). This stress concentration then helps to propagate the Lu¨ders band constriction along the crystal. The Finite-Element Method (FEM) analysis of stresses (Figure 6.9) shows a stress maximum at the tail and a minimum at the front of the Lu¨ders band; the latter explains the large gaps between the front slip bands and indicate the necessity of local stress concentrations from neighbouring slip bands (Figure 6.5) to initiate the next new one ahead of the Lu¨ders band front. In a recent approximate treatment, Brechet et al. [28] have described such transitions between homogeneous slip, bundled slip and propagating deformation fronts in quite general terms reflecting many of the above observations. Macroscopically, the existence of stress concentrations is realized in the yield points observed during first loading of the specimen. In fact, calculating the propagation stress from the external load by using the specimen cross section at the most active part in the Lu¨ders band region, we arrive at the same stress as that is observed at the yield point calculated from load and original cross section (cf. [23, 24]). This indicates that for these alloys the initial yield point is of purely geometrical origin (cf. [29]; the yield points due to strain ageing are smaller and will be discussed below). 115

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys

6.3.3

Comparison of single crystals and polycrystals

An important further stage of the investigations concerns the possibility to transfer the single crystal results to the case of polycrystals. As a first step in thin flat specimens of Cu-5, 10 and 15 at% Al with grain sizes around 200 lm, the slip bands have been observed during several steps of tensile deformation [30] recorded by video and examined in detail after the deformation steps in the light and electron microscope. In exceedingly large grains, often fronts of slip bands propagate similar to slip band bundles or Lu¨ders bands in single crystals. In exceedingly small grains, slip activity is often retarded due to stresses from neighbouring grains. In the average sized grains, several (mostly 2 to 3) slip planes are activated, often one after the other and different ones in different parts of the grain (Figure 6.10). This reflects the local influence of compatibility stresses exerted by the neighbouring grain. It also explains why not all, but most slip systems are activated according to the magnitude of the Schmid factor. In the CuAl alloys, the plastic relaxation near grain boundaries occurs frequently, in spite of the low stacking-fault energy, by cross-slipping of primary dislocations [12, 30]. This appears to be easier than to activate new sources on secondary systems. It is important in particular that the kinetics of single slip bands in polycrystals appear to be quite similar

Figure 6.10: Video records of slip line formation in single grains of a polycrystalline thin flat Cu10 at% Al specimen shown at three stages of deformation (e = 0.5, 1.5 and 8%) at room temperature. The numbers in the scheme indicate the succession of activated slip planes. 116

6.3

Deformation Processes around Room Temperature

Table 6.1: Comparison of average times of formation of slip bands on single crystals (plate shaped, thickness 0.18 mm) and in grains of polycrystals (plate shaped, thickness 0.4 mm, grain size 0.2 mm) for the Cu-Al alloys with 5, 10 and 15 at% Al. For the single crystals, the range of observed values is given in parenthesis. tB (s)

Single Crystals (thickness 0.18 mm)

Polycrystals (grain size 0.2 mm)

Cu-5 at% Al

7.8 (3 . . . 15)

12.7

Cu-10 at% Al

0.15 (0.1 . . . 0.2)

0.73

Cu-15 at% Al

0.04 (0.02 . . . 0.06)

0.05

to that in single crystals as shown in Table 6.1 for the average total times of activity of single slip bands. Contrary to single crystals, in the investigated polycrystals, no Lu¨ders bands were observed to propagate; according to experience in the literature [31], the grain sizes for that have to be chosen much smaller. A pilot experiment was performed in cooperation with Harder [32, 33] and Bergmann [34] on a thin flat specimen of Cu-5 at% Al containing 3 grains of different known orientations. The observations of slip band activity correspond well with the measurements of local strains by the multigrid method and with the FEM calculations [35].

6.3.4

Conclusion

In single and polycrystals of the considered Cu-Al and Cu-Mn alloys, deformation proceeds by production and movement of groups of strongly correlated dislocations across slip zones. This strong correlation and the destruction of short-range order lead to localized deformation and micro-instabilities of slip. Owing to the variation of the slip kinetics during the activity of each slip band, a description of the overall kinetics by the nucleation rates of slip bands (Equation (7)) including local work hardening (i.e. kinematic stress) appears appropriate. Thus, the flow units used in [36] consist of such spatially and temporarily correlated dislocations in groups. Their local stress concentrations are important in the propagation of slip along the crystal. Details of the mechanisms and kinetics of dislocation multiplication inside the slip bands still remain to be explored. The first steps done to study the influence of surrounding grains on the activity of a considered grain in a polycrystal should be extended, in particular by combining them with FEM analyses of the local stresses.

117

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys

6.4

Deformation Processes at Intermediate Temperatures

The range of “intermediate” temperatures is characterized by an increasing mobility of the solute atoms in the alloy, in particular in the neighbourhood of dislocations. Although first atomic site changes seem to occur in the dislocation core region already at temperatures well below room temperature, as evidenced by strain ageing effects during and after stress relaxation experiments [37], well pronounced influences of solute mobility are observed at temperatures exceeding room temperature (the lower, the higher the solute concentration, cf. [6]). In a certain range of temperature and external strain rate, dynamic strain ageing leads to repeated rapid local slip events even observable as serrations in the load-time curve in ordinary deformation experiments, the well-documented Portevin-Le Chaˆtelier (PLC) effect (e.g. [31]). Supplementing the research in [6], where most investigations are performed in the range preceding this instability region, the present study concentrates on the evolution of such plastic instabilities. Their temperature region for Cu-Al single crystals oriented for single glide and deformed in stage I is indicated in Figure 6.1 a by the dotted lines. Figure 6.1 b shows that it nearly coincides with the range of negative strain rate sensitivity if this is determined from the back-extrapolated stress course during strain rate changes [38, 39]. For the more general behaviour and ranges of existence of the PLC effect during work hardening for various crystal orientations and polycrystals, cf. [6, 40].

6.4.1

Analysis of single stress serrations

Applying an especially rapid data acquisition system to record the load (or stress) simultaneously with slip line recording by video, the course of PLC load drops has been directly correlated to the formation of new slip bands at the crystal surface [38, 41]. Figure 6.11 a shows a series of several selected frames taken during the stress serration given in Figure 6.11 b. Thus, in this range of temperature, one macroscopic instability event involves the rapid formation of a whole cluster of new slip bands. Obviously, after breakaway of a first source dislocation from its solute cloud, rapid dislocation multiplication occurs, where dislocations move fast enough to develop only minor solute clouds implying high dislocation mobility. The slip transfer mechanism of Section 6.3.2 (Figure 6.5) with local stresses in the surface region rapidly produces a series of neighbouring slip bands (i.e. a slip band bundle) at a rate higher than that necessary to comply with the deformation rate imposed by the tensile machine. Therefore, the load decreases and thus the production rate and dislocation velocity, too. This in turn permits the solute cloud to grow further and to slow down the dislocation until it stops suddenly and the specimen is again elasticly reloaded up to the next breakaway event. The quantitative formulation of this behaviour [38] permits to estimate the change in the effective enthalpy dDG due to ageing: DG
DG0  dDG
DG0  Kf tw  118

8

6.4

Deformation Processes at Intermediate Temperatures

a)

b) Figure 6.11: Sequence of frames: a) with slip bands originating during a stress drop (b)), in the PLC regime (T = 500 K) of a Cu-10 at% Al crystal deformed in stage I with a rate of l_
2  103 mm/s. 119

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys in the waiting times for thermal activations tw
tw0 exp DG=kT ;

9

where K and tw0 are constants and the function f describes the ageing kinetics. We find dDG  0.16 eV during the stress drop and & 0.14 eV during reloading, i.e. & 0.3 eV in total for Cu-10 at% Al at 580 K [39], which compares quite well with the values determined from different experiments and different arguments [42, 43]. This change amounts to roughly 10% of the total effective activation enthalpy DG0 in this temperature range of stress serrations. The breakaway stress rises with temperature due to an increasing solute cloud, up to a stress maximum s0 TM 
s0M , which occurs at lower TM for higher solute concentrations c (Figure 6.1 a, in more detail Figure 6.12 a). Beyond the crss maximum, no serrations occur and slip bands can no longer be detected: Slip becomes virtually homogeneous for T > TM (cf. Section 6.5). The correlation of s0M with solute concentration (Figure 6.12 b) agrees quite well with the classical formula proposed by Friedel [44]: s0M
AWm2 c=kTM b3 

10

for the boundary between dislocation breakaway from the (unsaturated!) solute cloud (T < TM ) and continuous dislocation movement with a solute cloud (T > TM , which by rapid diffusion reforms fast enough to be “dragged along” with the moving dislocation. This relation permits to estimate the mean binding enthalpy Wm of solute atoms to the dislocation, i.e. for c = 2 . . . 15 at% Al: Wm  0.12 eV taking the structure factor A= 0.1 as determined for Cu-Mn alloys by Endo et al. [45]. These Wm values compare well with earlier results from internal friction [46] and from theoretical estimates [47]. A summary of the temperature dependence of the correlation of serrations (load fluctuations) with slip activity is given in Figure 6.13, where, on the right, the mean stress drop amplitude Ds is plotted, while, on the left, the magnitude of simultaneously active slip band bundles, naB, is given as determined from the video records according to: _ N_ b Ssb ; naB
l=

11

where l_ = external deformation rate, N_ b = formation rate of new slip bands in one recorded active slip band bundle, Ssb = average slip step height (normal to the crystal surface), which does not change noticeably with temperature from room temperature (cf. Section 6.3) up to the PLC range. It is evident that at low T, where naB is high, the fluctuations in this number average out well so that a smooth load trace results. However, when naB becomes small (1 to 10), fluctuations in the load trace are resolved, and they turn into serrations when naB formally falls below 1, i.e. when only one slip band bundle is active for a short time with intervals of elastic reloading until breakaway of the next event.

120

6.4

Deformation Processes at Intermediate Temperatures

a)

b)

Figure 6.12: a) High temperature part of the temperature dependence of the crss s0 ) (cf. Figure 6.1 a) around its maximum, measured for various Al concentrations (2 . . . 15 at%) for Cu-Al single crystals oriented for single slip, at a deformation rate of l_= 1.7 · 10–3 mm/s (crystal length l0 = 100 mm) (* Cu-15 at% Al, * Cu-10 at% Al, & Cu-7.5 at% Al, n Cu-5 at% Al, ~ Cu3.5 at% Al, s Cu-2 at% Al); b) plot of the maximum stress s0M = s0 (T = TM) at the temperatures TM of the crss maxima versus alloy concentration to check Equation (10) by Friedel [44] for the transition between dislocation breakaway from and dragging along of the solute cloud.

6.4.2

Analysis of stress-time series

The recorded time series of load (or stress) in the range of plastic instabilities were analysed by several methods with respect to deterministic chaos or randomness and under the influence of measurement noise. Different methods proposed in literature for such dynamic time series analyses have been compared [48] such as reconstruction in phase space and correlation integral [49, 50], determination of Eigenvalues [51], and determination of Lyapunov exponents and K2 entropy [52, 53]. The problems with finding optimum embedding parameters have been studied relativating first attempts to detect the existence of chaos in jerky flow [54]. More successful appears a space-time analysis 121

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys

Figure 6.13: Correlation of the temperature dependence of the number of active slip band bundles naB (Equation (11)) and the average height of stress serrations Ds for Cu-10 at% Al crystals deformed with a rate of l_= 1.7 · 10–3 mm/s. Below the shape of the load-time curves in indicated. Note the abrupt disappearance of serrations at TM.

[48], which permits to take into account temporal correlations of the correlation integral, such as in case of quasi-periodic behaviour, after checking the autocorrelation function (for determination of a proper cut-off) and the power spectrum (for detecting periodicities). In evaluations of stress-time series, special care must be taken in case of changes of the specimen structure during deformation as common in deformation due to work hardening. This is shown in the examples of Figure 6.14 [48] for polycrystals deformed in the PLC regime at different temperatures and for a single crystal oriented for single glide, both for Cu-10 at% Al. For single crystals of Cu-5 . . . 15 at% Al and for polycrystals (Cu-15 at% Al), the PLC instabilities are of statistic rather than chaotic (deterministic) nature supporting the recent theoretical treatment by Ha¨hner [55]. For polycrystals, in certain ranges of deformation conditions at least some deterministic contributions are identified, which are periodic and seem to correspond to the propagation of the various types of PLC bands. The long period “type A” serrations (at T = 100 8C in Figure 6.14) is superposed by a short period at higher temperature (“type B” at T= 150 8C in Figure 6.14), while the single crystal does not show any periodicity, but indicates a change of structure from stage I to stage II. While, according to McCormick [56], the type A serrations are associated with a continuous propagation of plastic PLC deformation bands, type B corresponds to discontinuous propagation of bands, and during type C, serrations at still higher temperature with spatially 122

6.4

Deformation Processes at Intermediate Temperatures

Figure 6.14: Sequences of stress-time series measured in the PLC region of polycrystals (T = 100 8C, T = 150 8C) and a single crystal (single glide, transition from stage I to stage II, T = 300 8C).

uncorrelated local deformation bands occur. Accordingly, for types A and B from an analysis of the time series characteristic parameters of the deformation bands (band width, local plastic shear and shear rate in the band) and of their propagation rate v B can be evaluated [48]. Figure 6.15 gives examples of the latter quantity for type A and B bands in Cu-15 at% Al polycrystrals, which show different dependences on total strain e (Figure 6.15 a) and e_ (Figure 6.15 b). The model of Jeanclaude and Fressengeas [57] would predict a decrease of v B with increasing e if spatial coupling of local deformations occurs by cross-slip, while an increase would indicate spatial coupling by internal stresses [58] (cf. Figure 6.5). The observed dependence in Figure 6.15 a then would mean a change of cross-slip transfer to internal stress transfer with increasing temperature, which does not seem quite reasonable. Further investigations appear necessary and are under way for clarification.

a)

b) Figure 6.15: a) Propagation rates v B of PLC deformation bands evaluated from time series like those in Figure 6.14 for the serrations of type A (T = 100 8C) and type B (T = 150 8C) for various total strains e at a strain rate of e_ = 1 · 10–4 s–1; b) strain dependence of type B propagation rates _ (T = 150 8C) for variations of external strain rates e_
l=l. 123

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys

6.4.3

Conclusion

The strain ageing process forming solute clouds around the dislocations leads to macroscopically pronounced plastic instabilities in a certain range of deformation conditions, which are again intimately connected with strain localization. Here, the reason is the breakaway of a dislocation from its solute cloud and subsequent rapid multiplication of less aged dislocation groups. Thus, the overall kinetics (neglecting the serrations) can be again described in the nucleation rate approach for aged dislocations, where the kinetics of ageing enters the rate equations [40, 55, 56, 58]. The evolution of each single stress instability event can be described in such an approach, too, while the details of the dislocation multiplication and in particular the role of cross-slip processes in the slip transfer from the active into the bordering region still remains to be clarified.

6.5

Deformation Processes at Elevated Temperatures

6.5.1

Dynamical testing and stress relaxation

As indicated above in connection with Figure 6.12, for T > TM, the deformation occurs in a nearly ideal homogeneous manner. This was checked by EM slip line replica and TEM: No traces of slip could be detected on the crystal surface, and TEM does not show dislocation groups, but randomly distributed heavily jogged dislocations indicating easy cross-slip of screw and climb of edge dislocations. Therefore, no slip line observations are possible. In this range, viscous glide behaviour of dislocations can be assumed, and the classical Orowan equation (Equation (4)) a_
Rm bv with a definite dislocation velocity v and mobile dislocation density Rm appears realistic. According to the analysis from Figure 6.12 b, these dislocations carry along their (unsaturated) solute cloud, which now decreases with increasing temperature for entropy reasons. Thus, the alloying effect diminishes with increasing temperature as seen in Figure 6.1 a and Figure 6.12 a. The observation of a smaller yield stress for higher alloy concentrations (for c > 5 at%) at T > TM, which looks surprising at first sight, can be explained by the wellknown increase of the diffusion constant D (c) with solute concentration c [59] in the treatment of Friedel [44]: The relation between strain rate a_ and applied stress s is a_
2Rm b=kD sinh sb2 k=kT  2Rm b3 Ds=kT ;

12

where k
bcM
bc exp Wm =kT is the distance of pinning solute atoms along the dislocation. This relation also describes well the observed strain-rate sensitivity in stage I for T > TM (Figure 6.16 c, where different c values are plotted), which agrees well if determined from either stress relaxations (Figure 6.16 a) or from strain-rate changes (Figure 6.16 b), where the initial stress jumps (constant structure) are evaluated. The course of stress relaxations in this temperature regime can be well described by a viscous dislocation velocity [60]: 124

6.5 Deformation Processes at Elevated Temperatures

a)

b)

c)

Figure 6.16: Strain-rate sensitivities (cf. Figure 6.1 b) in the range of elevated temperatures for single crystals of Cu-Al with different Al concentrations measured from stress relaxations (a) and from strain rate changes (b) taking the “initial” stress changes of the transients (i.e. without structural changes) (_e2
5_e1 ; e_ 1 = 1.7 · 10–5 s–1; symbols as in Figure 6.12 a); c) plot of the strain-rate sensitivity for T > TM (TM = temperature at the crss maxima in Figure 6.12 a) versus 1/T to check Equation (12), for initial and stationary (i.e. back extrapolated) stress changes, using all data with different alloy concentrations ≥ 5 at%. 125

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys v  s;

13

and either by a stress-dependent mobile dislocation density: Rm  sn ;

14

or by a Gaussian spectrum of free activation enthalpies. For the first approach, the observed n values correspond well with the m = n+1 values (= 3.6 . . . 3.8 ± 0.5 for Cu3.5 . . . 10 at% Al) usually found from creep experiments for this type of alloys [61]. For the latter approach, the temperature dependence of the average characteristic relaxation times changes abruptly at the temperature of the crss maximum, indicating again the change of rate-controlling mechanism, i.e. breakaway of dislocations from their solute cloud for T < TM, solute diffusion in the non-saturated solute cloud dragged along with the moving dislocation for T > TM.

6.5.2

Creep experiments

In order to check by more direct measurements and evaluations creep data for T > TM, additional creep tests have been performed in the special creep set-up described in Section 6.2. Figure 6.17 shows some typical creep curves in the plot of strain rate versus strain: a) at a fixed stress for various temperatures, and b) at a fixed temperature for various applied stresses for polycrystalline Cu-10 at% Al. After a rapid decrease, the strain_ is due to specimen rate approaches stationarity (the following rapid increase of e_
l=l constriction and should be disregarded). In Figure 6.17 b for sufficiently low stresses

a)

b) Figure 6.17: Creep tests on Cu-10 at% Al polycrystals, in plots of strain rate e_ versus strain e: a) performed at constant stress r/G (G = shear modulus) for various temperatures, and b) at constant temperature T/Tm (Tm = melting temperature) for various stresses. Note the oscillating strain rate at low stresses in b). 126

6.5 Deformation Processes at Elevated Temperatures

b)

a) Figure 6.18: a) Critical strains for the onset of dynamic recrystallization (DRX), determined from its first appearance (*) and from the distance of strain rate maxima (*); b) examples for initiating dynamic recrystallization by a change of stress to lower values during creep tests.

(or strain rates), creep occurs with an oscillating strain rate showing the characteristics known for dynamic recrystallization (e.g. [62]). For instance, the critical strain for the onset of dynamic recrystallization increases in proportion to the applied stress (Figure 6.18 a). The dynamic recrystallization can be induced by a rapid change to a lower stress in the critical range (Figure 6.18 b). This seems to be accompanied by changes of the dislocation structure, which are to be studied in more detail to obtain more information on the nature of the recovery processes in this temperature range T > TM. The stationary creep rate, approximated by the minimum rate e_ min in Figure 6.17, varies with stress and temperature (Figure 6.19) according to

a)

b) Figure 6.19: Plots of the stationary creep rate (cf. Figure 6.16) versus stress (a) and temperature (b) to determine the parameters in Equation (15). 127

6 Inhomogeneity and Instability of Plastic Flow in Cu-Based Alloys e_ min  r5 exp Q=kT

15

with Q&2 eV, which approximates the activation for diffusion of solutes in the alloy or for self diffusion. The stress exponent (m&5) is slightly higher than that quoted above from stress relaxations, which has been clarified in [63, 64]. The first rapidly decreasing part of the creep curve contains information on dislocation multiplication. At very low stresses, this part of primary creep may even show increasing strain rate for some time. Observed differences to creep tests in conventional creep machines [65] can be traced back to the different kinetics of loading. These processes will be explored further by rapid changes from strain rate to stress-controlled conditions at different levels of stress (or strain) (cf. [63, 64]).

6.5.3

Conclusion

In the temperature region T > TM, diffusion processes are dominant. The deformation kinetics can be well described by the viscous glide approach with the dislocation velocity governed by dragging of solute clouds and a stress-dependent mobile dislocation density. This is the result of dislocation multiplication and simultaneous intensive recovery processes, where dislocation climb and cross-slip are important similar to pure metals [66, 67]. The details of these processes in solid solutions have to be further clarified.

Acknowledgements This work was possible through the engagement and essential contributions of my coworkers, C. Engelke, A. Hampel, A. Nortmann, J. Plessing, in their dissertation works, and Ch. Achmus, U. Hoffmann, T. Kammler, M. Ku¨gler, H. Rehfeld, S. Riedig, M. Schu¨lke, H. Voss, G. Wenzel, A. Ziegenbein, in their diploma works. In addition, I acknowledge gratefully the continuous discussions and cooperation with Prof. Dr. Ch. Schwink, and the cooperation in SRO measurements with Prof. Dr. O. Scha¨rpf (ILL Grenoble) and Dr. R. Caudron (LLB Saclay) by neutron scattering, and with Prof. Dr. G. Kostorz and Dr. B. Scho¨nfeld (ETH Zu¨rich) by X-ray scattering (with financial support of the Volkswagenstiftung). In particular, the financial support of our work by the Deutsche Forschungsgemeinschaft in the Collaborative Research Centre (Sonderforschungsbereich, SFB 319-A9) is gratefully acknowledged.

128

References

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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)

7

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

7.1

Introduction

The simulation of forming processes applying the Finite-Element method is more and more in use today. If the results are close to reality, the simulation can save costs involved in the forming of testing tools and shorten the development stage of new products. But this aim can only be achieved if the model of the forming process is physically plausible. The treatment of contact problems and the modelling of the material behaviour, e. g., present many problems. The material properties of the anisotropy caused or at least modified by the forming process in particular are problematic. In classical continuum mechanics, the material behaviour is described by phenomenological laws; the inner structure of the material is not considered in detail. Today, the available CPU’s have reached a performance level that allows us to take the microscopic behaviour into account as in texture analysis (see Figure 7.1). Thus, it seems possible to develop constitutive laws based on an improved physical basis and to use them in Finite-Element calculations.

7.2

The Model of Microscopic Structures

7.2.1

The scale of observation

In papers on texture analysis and on theories of polycrystals, the expressions ‘microscopic’ and ‘macroscopic’are often used. It is thus necessary to define the scale of observation. The resolving power of the microscopic observer is usefully described by the

* Universita¨t Hannover, Institut fu¨r Mechanik, Appelstraße 11, D-30167 Hannover, Germany 131

7

The Influence of Large Torsional Prestrain on the Texture Development

Figure 7.1: View of material structure in continuum mechanics and in texture analysis.

following definition. The observer knows the physical phenomenon and mechanism of slipping, but he is not able to locate the area of slipping in the grain. Thus, if slipping occurs, he is forced to treat it as a homogeneous in the grain-distributed action. This also means that all points of the grain are describable by only one stress tensor or velocity gradient. The expressions ‘macroscopic’, ‘global’ or ‘polycrystal’ are not related to a deformed body of a special form; they refer to a volume of many crystals. This volume is often called a control volume or representative volume, which is large compared with the microscopic scale. Although it consists of many crystals, it is small in contrast to any deformed body. Thus, a deformed specimen consists of many representative volumes. To start calculations in the interior of the representative volume, one is forced to have some state quantities of the macroscopic scale as well as of the microscopic scale. The macroscopic information could be a velocity gradient, for example.

7.2.2

Basic slip mechanism in single crystals

The plastic deformation of a single crystal is assumed to be caused only by slipping in certain slip systems. A slip system consists of a slip direction and a slip plane. The planes and directions are determined by the structure of the crystal. In face-centred cubic crystals, e. g., the primary slip systems are formed by the {111} planes and the
110 directions. Plastic deformation by slipping of a system  is only possible if the shear stress s on the slip system exceeds a critical value sc. The deformation gradient, F and the velocity gradient L, relative to a lattice fixed frame, are then given by: F  I  c s  mT 

and L  c_  s  mT  ;

1

where s and m are the orthogonal lattice vectors of the slip direction and the slip plane. The magnitude of shear in the active slip system  is called c. The equations 132

7.2 The Model of Microscopic Structures above are only valid if single slip occurs, but generally more than one slip system will be operating simultaneously. The appropriate equations for multislip follow for the velocity gradient by superposition of single slips: X 2 c_  s  mT  : L 

Nevertheless an analogue treatment of the deformation gradient F is not valid. Furthermore, when the elastic distortion of the lattice is considered as well, the expressions become more complicated because the distorted lattice vectors must be used for an appropriate formulation (see e. g. Havner [1]).

7.2.3

Treatment of polycrystals

The main problem of modelling crystal structures is not the formulation for the single crystal. It is more difficult to find a suitable averaging method to obtain the properties of the polycrystal. The interactions of the crystals at their grain boundaries during their deformation are so complex that there are still some simplifications necessary to make the problem mathematically treatable. Several texture models have been developed to deal with this problem. Some of them ignore the grain interactions, while others try to consider them in different ways. The first and basic models are those of Sachs [2] and Taylor [3, 4]. Simulations based on the Taylor model show better results compared with textures measured in experiments. It is therefore till now often the basis of texture simulations. Generally, most methods differ from each other in terms of whether the homogeneity of deformations or the homogeneity of stresses are partly or completely satisfied. A comprehensive overview of modelling plastic deformation of polycrystals is given in [5]. All texture models require some basic data of the microscopic scale. Usually, at least the following specifications are considered: • • •

The polycrystal consists of Nj crystallites with equal volume. No restrictions about the grain form are made. The orientation of each crystal is given by the Eulerian angles u1 , y and u2 . No information about the arrangement of the crystals in the polycrystal is available. The elastic constants of the single crystals are given as well as the slip systems including their critical shear stresses. The assumption of the known critical shear stresses presents a problem in practice.

7.2.4

The Taylor theory in an appropriate version

The Taylor model, often called Full-Constrained model, is the most often used texture model. The fundamental assumption of the model is that in each crystal, five slip sys133

7

The Influence of Large Torsional Prestrain on the Texture Development

tems are activated in a way such that the microscopic velocity gradient for the incompressible flow is identical with the global one: ( ) 5 X T T grad V  L  Aj 3 c_  s  m  Aj  Xj with Xj  A_ Tj Aj : 1

|‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚} macroscopic

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚} microscopic

The expression Xj  A_ Tj Aj given by the transformation tensor A and its time derivative A_ is most important. This is the lattice spin of the crystal. To solve the problem, the equation is decomposed in symmetric and antimetric parts, D and W, respectively. Thus, if the global velocity gradient is given, the symmetric part can be solved as a set of linear equations. This leads to 384 possible solutions in the case of 12 slip systems. The correct solution is the one, which minimizes the internal power of the grain. Especially, if the critical shear stresses are equal on all slip systems, this selection criterion is not unique, and all combinations of five slip systems that lead to internal power inside a tolerance limit are supposed to be active. As more than five slip systems operate simultaneously, this is an extension of the Taylor theory. Another way to solve the ambiguous problem is to vary the initial critical shear stresses with a random generator. This solution is not a restriction of the model; in some cases, it might improve the quality of the texture models. If the magnitude of shear is known for all slip systems, the lattice spin is given by: ( ) 12 X T T 4 c_ ges Xj  W  A T  s  m  m  s  Aj ; 1

hence, A_ j can be calculated. Integration of Equation (3) leads to the new orientation of the lattice. Finally, the microscopic stresses can be calculated with the known slip systems. A macroscopic stress tensor and also a mean spin tensor are obtained by averaging the crystal data. Normally the hardening of the crystal is considered by a law of the form: ! 12 X c cl ; cb ; u_ m ; b; l 1; . . . ; 12 ; 5 s_   f l1

which must be evaluated after each step of calculation. Some examples for suitable hardening laws can be found in [6] and [1]. The calculations documented in [6] also show that the hardening law strongly effects the stress response and has hardly any influence on the texture development of the polycrystal. 134

7.3 Initial Orientation Distributions

7.3

Initial Orientation Distributions

For a practical comparison of measured and calculated textures, the initial orientations of the crystals should be measured as single orientations of single grains or as non-discretized orientation distribution function (ODF). In theoretically based works and research projects, it is quite normal to start the calculation with an isotropic state. Therefore, it is necessary to generate a distribution with initial global isotropic properties.

7.3.1

Criteria of isotropy

Before initial orientations can be used for numerical simulations, it is necessary to check whether an initial isotropy is actually guaranteed and not only orthotropy. Many criteria can be used to check this although not all of them are sufficient. In [7], e. g., the components of the average elastic stiffness tensor were regarded. But for small deviations from the isotropy configuration, there can be remarkable deviations of the elastic modulus for different directions of loading. In [6], the plastic isotropy is proved by calculating the yield surfaces of the single crystals. If all these yield surfaces are regularly distributed in the stress plane, the distribution is thought to be isotropic. This approach considers only the first possible slip system, and if multislip occurs, the isotropy might not be satisfied. It therefore seems to be best to introduce an isotropy test, which checks the elastic properties as well as the plastic properties under consideration of multislip. A suitable test of the elastic isotropy is to calculate the average elastic stiffness tensor. The method introduced by Hill [8], which leads to good estimations in the case of randomly distributed crystals, seems to be the simplest and best method of approximation. A quantity denoting the elastic anisotropy of an orientation distribution may be: AE 

Emax  Emin E
111  E
100

AE  0 elastic isotropy ;

AE  1 single crystal isotropy ;

6

where the maximum difference of the calculated average elastic modulus is related to the corresponding data of the single crystal. Thus, the value is independent of the constants of the single crystal and only the quality of the distribution is assessed. In the case of ideal isotropy, the quantity AE will vanish. A helpful visualization is to draw the elastic properties of different directions as a body of elastic moduli. Here, distributions with lower quality, concerning isotropy, show remarkable deviations from the ideal spherical form. The Taylor model is an ideal tool to check the plastic properties under consideration of multislip because at least five slip systems are active. The function

fAP e  1  AP e

with AP e 

r1 e  r2 e sc

7 135

7

The Influence of Large Torsional Prestrain on the Texture Development

can be used to judge the plastic properties. AP is the ratio of the differences of stresses orthogonal to the tension direction to a mean value of the critical shear stress. In the best case of isotropy, the function will reach fAP e  1e, and the accompanying plot will show a sphere.

7.3.2

Strategies for isotropic distributions

A special strategy is only required if the distribution should consist of as few crystals as possible. But given a later implementation of such a texture-based constitutive law in a Finite-Element program, this should always be the aim. Several authors use random distributions generated with the help of a random generator. Unfortunately, these distributions are only usuable, considering the isotropy, if they consist of many orientations (> 1000). Distributions created by a proper strategy are generally better than distributions generated randomly when the number of orientations is equal. Several strategies are based on a discretization of the Euler space. Here, the space built from the possible combinations of Euler angles is discretized. On account of the crystal symmetry, it is not necessary to consider the entire Euler space; a small portion is sufficient. For cubic crystal symmetry and orthorhombic symmetry of the specimen, the relevant Euler space was given by Pospiech [9]. Unfortunately, this field has a nonlinear boundary and therefore it is not easy to discretize it. Mu¨ller [6] and Harren [7] use a corresponding discretization and obtain distributions of 32 . . . 128 and 385 orientations, respectively. Isotropy is not satisfied in every case, but the distributions of Mu¨ller [6] are better although they consist of fewer orientations. Asaro and Needlemann [10] and Harren and Asaro [11] use a combination of specific method and random distribution. The unit triangle of the stereographic projection is used to fix one of the global axes. The attachment of the base in space is done with an angle given by a random generator. Figure 7.2 shows the elastic properties calculated with data given in [11]. The distributions show a noticeable anisotropy, which is assumed to be caused by the random generator. The method was used again to check the plastic isotropy with the result that all distributions obtained had better properties than the original ones. Another method is given by Mu¨ller [6], who discretized the surface of a sphere to obtain the positions of local basis vectors. This method leads to distributions of good quality (see e. g. Figure 7.3), but it is always combined with the problem of the spherical geometry. This problem can be avoided if one takes the area of a circle for discretization and obtains the points on the sphere by an equal area projection. A detailed description of the method is given in [12]. The quality of the distributions naturally depends on the division of the area and on the number of orientations, but for the same number of orientations, the isotropy is better or at least comparable to that of the so-called Kugel distributions. Figure 7.4 shows the isotropy test of a distribution generated with this method. Although it consists of only roughly one hundred orientations, the isotropy is nearly guaranteed. 136

7.4 Numerical Calculation of Yield Surfaces

Figure 7.2: Global elastic modulus body of some distributions given in [11].

Figure 7.3: Test of isotropy of the distribution kugel192 given in [6].

Figure 7.4: Test of isotropy of the distribution kr104 given in [12].

7.4

Numerical Calculation of Yield Surfaces

The numerical calculation of yield surfaces with data from orientation distributions can be carried out in many different ways. But with regard to a comparison with experimental data, research methods, which allow the consideration of the sequence of an experiment, should be preferred. Generally, all methods are averaging methods, but the 137

7

The Influence of Large Torsional Prestrain on the Texture Development

procedure of averaging and the basic assumptions vary. The methods can be categorized as follows: Static methods of averaging are based only on the Schmid law and no strain is considered. Methods of this type are not suitable for a comparison with experimental data, as the later are usually measured with an offset strain. In [6], a method is proposed based on averaging the single crystal yield surfaces. This method can also consider kinematic hardening when the surface lies outside the origin. Figure 7.5 shows two yield surfaces on an initial distribution. The values of the stresses rXX, rYY and sXY are related to a mean value of the critical shear stresses. This normalization is also done in the figures below. In [13], this method is combined with some offset simulations. When the offset is large, the resulting yield surfaces are similar. Another method, called MHSSS (Most Highly Stressed Slip Systems), is proposed by Toth and Kova´cs [14]. This method uses a double averaging, first in the grain and second for the polycrystal. It is shown in [12] that the calculated yield stress is the harmonic mean of the five lowest possible stresses causing yielding in different slip systems. The arithmetric or geometric mean may be used in the same way. The classical Taylor yield surfaces are based on statics as well. The yield surfaces shown in Figure 7.6 have been calculated by applying 80 loading paths, which are marked by the arrows. The best and fastest method for calculating the yield surface in this manner was introduced by Bunge [15]. Stress-controlled methods are based on a global given stress tensor. The stress is increased incrementally until the shear stress in one slip system exceeds the critical value. It is then possible to calculate the amount of shearing that is needed for a static equilibrium with the hardening law. If the critical shear stress is not reached during a step, the deformation is assumed to be purely elastic. After all deformations of the crystals have been obtained, the mean value of strain is calculated. The method continues until the offset strain is reached. Unfortunately, only the stress is given and no information about the global velocity gradient is supplied. Therefore, antimetric parts are hardly considered. But for small deformations (e. g. for a simulation of the elastic-plastic transition), this method may be suitable. Figure 7.7 shows initial yield surfaces of an initial kugel distribution. The graph is due to an ideal calculation, and the symbols correspond to a calculation under consideration of strain hardening, loading path effects and orientation alterations. The offset strain used is 0.2%, which is a standard value in material testing. One may notice that

Figure 7.5: Initial yield surfaces calculated with the radial averaging method. 138

7.4 Numerical Calculation of Yield Surfaces

Figure 7.6: Initial yield surfaces calculated with the Taylor model.

Figure 7.7: Initial yield surfaces calculated with a stress-controlled method.

although a large offset is used, the resultant yield surfaces are smaller than the ones calculated with the Taylor model. Therefore, the later ones are only valid for comparison with yield surfaces measured when a large offset strain is used. More examples and a detailed description of this method can be found in [12]. Strain-controlled methods are based on a given deformation or velocity gradient. In a simple manner, the yield stress of the Taylor model is calculable with a strain path like an experiment. When the components of the global velocity gradient are given for the stress plane considered, e. g. in the form Y Lij  fX LX ij  fY Lij

with fX  cos b

and fY  sin b ;

8

it is possible to apply the loading path desired by choosing suitable values for the anY gle b. LX ij and Lij are the tensor components of pure loading in X and Y direction, respectively. When b changes, the equivalent strain rate changes, too; it is therefore necessary to vary the time increment of the integration to achieve a constant step of equivalent strain increment during each loading step. Thus, this method is suitable for calculating yield surfaces as shown in Figure 7.8. Exactly 80 loading paths starting with pure tension and then continuing counter clockwise round the stress plane are applied. For comparison, the ideal Taylor yield surfaces are shown too. This model shows the expected effect as the expanding of the yield surfaces caused by the loading path under consideration of hardening. The hardening law used is the isotropic PAN law with parameters proposed in [6] and the offset strain is 0.1%. 139

7

The Influence of Large Torsional Prestrain on the Texture Development

Figure 7.8: Initial yield surfaces calculated with a strain-controlled Taylor simulation.

When kinematic hardening is considered, the yield surfaces become distorted and may not be closed. The only disadvantage is that the classical Taylor theory starts with the full plastic material state. Thus, the elastic-plastic transition is not taken into consideration. A better method might be the Lin model [16], which similar to the Taylor model assumes that all crystals have the same strain. Furthermore, nearly the same hardening law can be used. This method is used in [6] for the calculation of offset-strain dependent yield surfaces. The disadvantage is the small deformation area of application. Thus, it is not useful for texture simulations. The problem is discussed further in [12], and it is shown that the numerical evaluation can be simplified without restrictions.

7.5

Experimental Investigations

The aim of the experimental investigations was to measure yield surfaces of large prestrained materials. The large deformation was achieved with a torsion-testing machine at the Institut fu¨r Mechanik of the Universita¨t Hannover. The measurement of the yield surfaces was done with a testing machine at the Institut fu¨r Stahlbau of the Technische Universita¨t Braunschweig. The material of the specimens was always the aluminium alloy AlMg3.

7.5.1

Prestraining of the specimens

The prestraining of the specimens was achieved with a torsion-testing machine, further described in [12]. In order to measure yield surfaces after the deformation, it was necessary to twist thin walled tubular specimens. The final nominal length, inside diameter and wall thickness of each specimen, were 60 mm, 24 mm and 2 mm, respectively. If the accuracy of the manufactured specimen is high (e. g. by using a CNC-controlled lathe), large deformations without buckling can be achieved. To prevent buckling and 140

7.5 Experimental Investigations to ensure that the cylindrical form of the specimens was maintained, a lubricated mandrel was inserted inside the specimens. A maximal amount of shear of ca  tan w  1:5 could be reached with that configuration, where w describes the angle of an axial direction on the surface of the specimen after the deformation. That means a twist of about 360 degrees for the specimens. The elongation of the specimen was not suppressed with the result that an elongation always occurred, which nearly depended linearly from the twisting angle in agreement with the research done by Po¨hlandt [17] with specimens of aluminium. The maximal elongation was D`  1:25 mm. Since the measurement of the yield surfaces was done in another apparatus, the specimens were fully unloaded after the torsional deformation.

7.5.2

Yield-surface measurement

Four different material states have been investigated: specimens without any prestrain and ones with ca = 0.5, ca = 1.0 and ca = 1.5 magnitudes of shear. The testing apparatus was a strain-controlled machine with the capability of combined tension-torsion loadings. The yield point of the material was detected with the offset strain definition. In all tests, one specimen was used for 16 loading paths, starting with pure tension and then continuing counter clockwise round the r-s-plane. This was done for three reasons. First, this reduced the costs of specimens. Second, it was not guaranteed that the prestrain is reproducible and last, the multipath measurement data are needed for the comparison with the theoretical models. These data are ideal to check whether the texture model including the hardening law is able to describe the material behaviour during such a loading history. The interpretation of the data measured is based on an additive decomposition of the total strain increment in an elastic and a plastic part. If the total strain increment is given by the testing machine, the plastic parts of it are given by: Depl  Deges 

Dr E

and Dcpl  Dcges 

Ds G

9

when the constants E and G are known. In determining the yield surface, the offset von Mises equivalent plastic strain was computed using the equation: DevM pl

r 1  De2pl  Dc2pl : 3

10

The yielding point was reached when the calculated plastic strain exceeded the given offset strain: X

DevM pl  eoff ;

11

where offsets between 0.0015% and 0.1% have been used. It is essential for the offset definition that the values of E and G are known with high accuracy. Otherwise, large 141

7

The Influence of Large Torsional Prestrain on the Texture Development

Figure 7.9: Calculated value of E in dependence on the number of used measuring points.

errors may be the result. If E is measured too large, the resultant yield stress will be smaller than the real one. In an extreme case, yielding is supposed although the material is still in the elastic state. On the other hand, if E is measured too small, the resultant yield stress will be larger than the real one. The shear modulus G has an appropriate influence. This possible error in determining the yield stress increases with decreasing offset. Data obtained by using very small offsets should therefore be treated with caution. The determination of the elastic constants E and G is normally done with the first measured points of a new loading path. The best way is to calculate the regression coefficients. Although the regression coefficient in the elastic range should always be constant, that is practically not the case. It always varies in a small range depending on the number of considered measuring points as shown in Figure 7.9. Thus, if another number of measuring points is selected for calculating the modulus E, the resultant value E and as a consequence also the resultant yield stress will be changed. This is an especially critical case for the modulus E; the shear modulus G shows better relations.

7.5.3

Tensile test of a prestrained specimen

This test was done to investigate the appearance of the cross-effect. A cross-effect is given when the maximum yield stress in the tensile component of stress is altered by the strain hardening in torsion and vice versa. Normally, the cross-effect and related issues are investigated by the measurement of yield surfaces when the plastic deformation at most reaches the usually small offset strain. This tensile test was realized to investigate the cross-effect on a larger scale. Two tensile test specimens DIN 50125-B 10 × 50 have therefore been produced, one of nearly isotropy material and the other of prestrained material. Hence, a cylindrical specimen was twisted up to fracture, which occurs at a shear rate of ca = 1.65. The 142

7.5 Experimental Investigations

Figure 7.10: Tensile test of pre- and unstrained material.

tensile specimen was produced from the broken rest as shown in Figure 7.10. An estimate led to an amount of shear of ca = 0.55 at the radius of the final test specimen, but as a result of the processing, the two specimens were not distinguishable. The result of the tensile test is shown as a diagram of force and elongation in Figure 7.10. Additionally, some mechanical properties are given in Figure 7.10. As expected, the prestrained material is more brittle compared with the other one. Furthermore, the mechanical strength properties are greater than those of the unstrained specimen. Values never reachable for the unstrained specimen were obtained. This shows that there is a remarkable cross-effect.

7.5.4

Measured yield surfaces

Some measured yield surfaces are presented below. Detailed discussions and further investigations about the cross-effect and the loading path are given in [12]. First, it is remarkable that 0.0015% was the smallest practicable offset-strain for the unstrained specimens, while this value was too small for the prestrained specimens. There were several runaways among the data measured and therefore the smallest offset was chosen to 0.005%. A larger one of 0.05% was also chosen for comparison. Other offsets were only used for unique specimens. Figure 7.11 shows the measured yield surfaces of unstrained specimens with increasing offset. As expected, the yield surfaces have an elliptical form and for small values, the axial ratio r/s is closely to the von Mises yield surface. This ratio, however, increases with increasing offset as well. The data obtained from the largest offset used show an expansion of the surface, which is surely caused by the specific loading path and the large offset of 0.1% inducing significant plastic deformation. 143

7

The Influence of Large Torsional Prestrain on the Texture Development

Figure 7.11: Yield surfaces of unstrained specimens.

A comparison of yield surfaces of prestrained specimens is given in Figure 7.12. There are remarkable concave areas, which seem to disappear for the more prestrained specimens. This is causually connected with the loading path because in the second and third quadrant, no such areas occur. On the other hand, this concave area is due to the first measured point and, therefore, it is the first loading differing from the torsional preloading. This might be an important fact. Furthermore, the surfaces show a hardening with increasing degree of prestrain. There is a significant distortion, a flattening of the portion of the surface opposite the loading direction, and a kinematic hardening or a so-called Bauschinger effect occurs. When the large offset is applied, the most remarkable characteristics disappear. the yield surfaces shown in Figure 7.13 are ellipses slightly shifted in the loading direction. Furthermore, a hardening with increasing degree of prestraining is noticeable. Exceptionally, the yield surface of the unstrained specimen is measured with an alternate loading path, which does not affect the shape strongly.

Figure 7.12: Yield surfaces of prestrained specimens (small offset strain). 144

7.5 Experimental Investigations

Figure 7.13: Yield surfaces of prestrained specimens (large offset strain).

Figures 7.12 and 7.13 show that it is often difficult to assign the characteristics observed. One may ask if the effects are due to the prestraining or to the parameters of the measurement. Especially, the loading path for each specimen could be the cause of some effects. In order to investigate the influence of these parameters, some specimens were applied to the measure procedure three times. First, the small offset was used; then the larger one and finally again the small offset. The resultant properties of the unstrained material are shown in Figure 7.14, where only the surfaces measured with the small offset are presented. The data measured characterized by the * is of the third measurement of this specimen. Thus, an influence of the loading path can be seen because this yield surface is slightly shifted to the direction of the last loading of the previous path with the large offset. In fact, there is an influence of the loading path, which does not seem to be too large because the form of the surface is not affected. Surprisingly, the prestrained material shows a very different behaviour. In Figure 7.15, correspondent measurement of a specimen prestrained up to ca = 0.5 is shown.

Figure 7.14: Influence of the loading path (unstrained material). 145

7

The Influence of Large Torsional Prestrain on the Texture Development

Figure 7.15: Influence of the loading path (prestrained material).

The first surface measured with the small offset shows the properties already detected in Figure 7.12. The second has nearly an elliptical form. The third surface, measured with the small offset, is a slightly shifted ellipse. Considering the previous yield surface, the third one shows properties as expected, but compared with the first one, there are hardly any common characteristics. Form, size and position have changed remarkably. Thus, the prestrained specimens are very sensitive to further deformations compared with the unstrained ones.

7.5.5

Discussion of the results

The measurement of yield surfaces is not problematic for unstrained specimens even when small offset strains are used. The data are reproducible and if the offset is small, there is only a small influence caused by the loading path. On the other hand, the prestrained specimens were very sensitive. When the form of the yield surface is not known, it is difficult to identify runaway data and to assign the effects to parameters of the measure procedure. An amplification of these effects is due to the problem of determination of the correct elastic modulus. In comparison with similar investigations (e. g. in [18–24]) agreement as well as different results can be found.

7.6

Conclusion

In several investigations, the models of polycrystals are based on the motivation that these models lead to better results in simulating the distortion of yield surfaces. The 146

References distortion and the resultant anisotropy are often assumed to be caused by the orientation distribution of the single crystals in the material. The results presented prove that especially prestrained material is very sensitive to small deformation. This means that the principle form of the yield surface is strongly sensitive to small plastic deformations. Since this small deformation hardly affects the texture of the material, it must be assumed that the texture is not the real cause for the distortion of the yield surface. Additional events and mechanisms must occur in the material during any plastic deformation. Further investigations on the numerical calculation of yield surfaces will be undertaken. Especially the question concerning, which method leads to results similar to the surfaces measured and what kind of microscopic hardening law is needed, will be considered. The fact that almost all parameters of the hardening law must be identified by the mechanical properties of the polycrystal is problematic. At least, one should be able to identify all these parameters with standard methods in material testing. Otherwise, it does not make sense to use microscopic-based material laws. The final aim is to do the calculation first and then proceed in manufacturing. The other way of doing an experiment first and then trying to reach the same results in simulation may be practicable for research projects, but this is surely not senseful for practical applications. Finally, a search for a texture model to describe small deformations as well as large deformations and all this in an acceptable calculation time will be undertaken. Then, an implementation in a Finite-Element program may be useful.

References [1] K. S. Havner: Finite Plastic Deformation of Crystalline Solids. University Press, Cambridge, 1992. [2] G. Sachs: Zur Ableitung einer Fließbedingung. Zeitschrift des Vereins deutscher Ingenieure 72 (1928) 734–736. [3] G. I. Taylor: Plastic Strain in Metals. J. Inst. Metals 62 (1938) 307–323. [4] G. I. Taylor: Analysis of Plastic Strain in Cubic Crystals. In: J. M. Lessels (Ed.): Stephen Timoshenko 60th Anniversary Volume, 1938, pp. 307–323. [5] E. Aernoudt, P. van Houtte, T. Leffers: Deformation and Textures of Metals at Large Strains. In: H. Mughrabi (Ed.): Plastic Deformation and Fracture of Materials, Vol. 6 of Materials Science and Technology: A Comprehensive Treatment (Vol.-Eds.: R. W. Cahn, P. Haasen, E. J. Kramer), VCH, Weinheim, 1993, pp. 89–136. [6] M. Mu¨ller: Plastische Anisotropie polykristalliner Materialien als Folge der Texturentwicklung. VDI Fortschrittsberichte Reihe 11: Mechanik/Bruchmechanik, VDI-Verlag, Du¨sseldorf, 1993. [7] S. V. Harren: The Finite Deformation of Rate-Dependent Polycrystals: I. A Self-Consistent Framework. J. Mech. Phys. Solids 39 (1991) 345–360. [8] R. Hill: The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc. London A 65 (1952) 349–354. [9] J. Pospiech: Symmetry Analysis in the Space of Euler Angles. In: H. J. Bunge, C. Esling (Eds.): Quantitative Texture Analysis, 1982. 147

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[10] R. J. Asaro, A. Needlemann: Texture Development and Strain Hardening in Rate Dependent Polycrystals. Acta. metall. 33 (1985) 923–953. [11] S. V. Harren, R. J. Asaro: Nonuniform Deformations in Polycrystals and the Aspects of the Validity of the Taylor Model. J. Mech. Phys. Solids 37 (1989) 191–232. [12] N. Wellerdick-Wojtasik: Theoretische und experimentelle Untersuchungen zur Fließfla¨chenentwicklung bei großen Scherdeformationen. Dissertation Universita¨t Hannover, 1997. [13] D. Besdo, M. Mu¨ller: The Influence of Texture Development on the Plastic Behaviour of Polycrystals. In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications. IUTAM Symposium Hannover/Germany 1991, Springer-Verlag, Berlin, Heidelberg, 1992, pp. 135–144. [14] L. S. Toth, I. Kova´cs: A New Method for Calculation of the Plastic Properties of Fibre Textures Materials for the Case of Simultaneous Torsion and Extension. In: J. S. Kallend, G. Gottstein (Eds.): Proc. 8th Int. Conf. on Textures of Materials ICOTOM, 1988. [15] H. J. Bunge: Texture Analysis in Materials Science. Cuvillier, Go¨ttingen, 1993. [16] T. H. Lin: Analysis of Elastic and Plastic Strains of a Face-Centered Cubic Crystal. J. Mech. Phys. Solids 5 (1957) 143–149. [17] K. Po¨hlandt: Beitrag zur Optimierung der Probengestalt und zur Auswertung des Torsionsversuches. Dissertation TU Braunschweig, 1977. [18] P. M. Nagdhi, F. Essenburg, W. Koff: An Experimental Study of Initial and Subsequent Yield Surfaces in Plasticity. J. Appl. Mech. 25 (1958) 201–209. [19] H. J. Ivey: Plastic Stress-Strain Relations and Yield Surfaces for Aluminium Alloys. J. Mech. Engng. Sci. 3 (1961) 15–31. [20] W. M. Mair, H. L. D. Pugh: Effect of Prestrain on Yield Surfaces in Copper. J. Mech. Engng. Sci. 6 (1964) 150–163. [21] J. F. Williams, N. L. Svensson: Effect of Torsional Prestrain of the Yield Locus of 1100-F Aluminium. Journal of Strain Analysis 6 (1971) 263–272. [22] A. Phillips, C. S. Liu, J. W. Justusson: An Experimental Investigation of Yield Surfaces at Elevated Temperatures. Acta Mechanica 14 (1972) 119–146. [23] P. Cayla, J. P. Cordebois: Experimental Studies of Yield Surfaces of Aluminium Alloy and Low Carbon Steel under Complex Biaxial Loadings. Preprints of MECAMAT 92, International Seminar on Multiaxial Plasticity, 1992, pp. 1–17. [24] A. S. Khan, X. Wang: An Experimental Study on Subsequent Yield Surface after Finite Shear Prestraining. Int. J. of Plasticity 9 (1993) 889–905.

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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)

8

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

8.1

Introduction

The rapid development of numerical mechanics has resulted in • •

an increased need for the identification of material parameters, new procedures, developed to solve these problems.

A common property of material parameters consists in the fact that they could not be measured directly. The classical method of the determination of material parameters is to demand a quite good agreement between measured data from properly chosen experiments and comparative data taken from numerical analysis. This will be carried out by the optimization of a least-squares functional. Furtherly, the parameter identification based on experiments with inhomogeneous stress-strain fields, the usage of global and local comparative quantities in the objective function and optimization by deterministic methods will be described.

8.2

General Procedure

In addition to classical material-testing methods, current research is done to identify material parameters of inelastic deformation laws by the experimental and theoretical analysis of inhomogeneous strain and stress fields. A new method is the parameter identification using the comparison of numerical results obtained by the Finite-Element * Technische Universita¨t Chemnitz, Institut fu¨r Mechanik, Straße der Nationen 62, D-09009 Chemnitz, Germany 149

8

Parameter Identification of Inelastic Deformation Laws

method with experimental data, for instance, with displacement fields measured by optical techniques [1–7]. The papers [1–4] were realized within the Collaborative Research Centre (Sonderforschungsbereich 319). Unlike this Finite-Element algorithm based method, in this paper, another procedure is presented to identify material parameters of inelastic deformation laws. The principle consists in the experimental determination of the strain distributions in the ligament of a notched bending specimen at several load steps and the numerical integration of the deformation law at a certain number of points along the ligament with measured strain increments as load. The actual material parameters can be found using the global equilibrium of the stresses integrated along the ligament with the known external loads. Besides also local quantities, for instance, the stresses in the grooves of the notch could be compared. A detailed scheme of this procedure is shown in Figure 8.1. Below constitutive equations, in the framework of the classical plasticity and materials as sheet metals or metal plates are studied. The elastic properties should be isotropic. Viscoplastic effects are neglected. An initial anisotropy, especially a planar orthotropy is taken into account.

8.3

The Deformation Law of Inelastic Solids

As an example, the deformation law of classical plasticity theory with small strains as used in the material subroutines of the integration algorithm (cf. Section 8.6.1) will be considered. At the yield limit holds the yield condition: F…r; h; p† ˆ 0 :

…1†

The linear elasticity law r_ ˆ E_e

…2†

is valid for loads in the elastic domain F m are determined from the condition that the approximate value of zi tj1  m should be equal to the local Taylor series for zi tj  Dt up to the potential term Dtm . Equations (11) to (14) employ formulae, which yield two approximate values, qg and q~g , of different order of error with the least possible computational effort. Thus, it is the appropriate equation system (Equations (11) to (14)) to use for the adaptive control of time steps. The iteration process is governed by the tolerance limit eact
zj1  ~zj1  < etol . In the case, the limit of tolerance is satisfied, the next time step is enlarged if not the step is taken half of the last one. The Runge-Kutta method of the 2. and 3. order is an effective integration procedure. For higher accuracy, the methods of 4. and 5. order by Fehlberg and DormandPrince should be chosen. Complete sets of formulae are given in [20].

•

Predictor-corrector method

Applying the trapezoid formula on Equation (10) yields the implicit approach: zj1
zj 

Dtj1 Ftj ; zj   Ftj1 ; zj1  : 2

15

Following Heun [20], the iteration proceeds by starting from an explicit formulation (the predictor): 0

zj1
zj  Dtj1 Ftj ; zj 

with z0
zt0  :

16

This is solved by the Euler-Cauchy method. Then, an implicit equation for corrections (corrector) is solved: v1

zj1
zj 

Dtj1 v Ftj ; zj   Ftj1 ; zj1  ; v
0; 1; . . . : 2

17

The corrections are determined in several (in most cases only one or two) iteration steps until an error limit is met. The step length may be adjusted to the result of the last steps.

179

9 •

Development and Improvement of Unified Models

Implicit method of collocation

By starting from the equivalent integral formulation of Equation (9) and by applying the principle of weighted residuals, one arrives at: tDt Z

dzT  z_  Ft; ztdt  . . .B
0 :

18

t

The errors weighted by the special functions d zi (t) should vanish in regions and at boundaries. The approximations depend on the selection of the weight functions. The Dirac function and selecting 0:5 n 1:0 for the collocation point lead to an unlimited stable iteration process. The advantages are discussed in [12, 22]. Here, it is easy to adapt the time steps: After two steps of the length Dt, the next step is taken as 2 Dt. Both results are used for determining errors. •

Efficiency of the integration methods

The different integration procedures are applied to cyclic loadings of MST+ as given in Figures 9.1 and 9.2. The set of parameters is given in [23]. In Table 9.1, the following results are presented: qg
order of error (see Equations (11) to (14)), etol
limit of errors, CPU-time
computing time (Pentium P90), r
resulting stress after 30 cycles. The comparative investigation proves that an accuracy of etol
10–2 is sufficient for this specific problem. The implicit collocation methods need much more computing time although the numbers of iteration steps are smaller. The most effective method is that of the predictor-corrector by Heun. For problems, which require higher accuracy, the Runge-Kutta method of 2. and 3. order and the one by DormandPrince are the most efficient. Braasch [9] showed for long-term creep that a combined approach of explicit and implicit methods is better than the application of just the explicit method. In each time step, it is checked whether a local stability criterion (based on the eigenvalues of the functional matrix J) is met. If not, the implicit method is used instead of the explicit procedure.

t [s]

Figure 9.1: Strain(e)-time(t)-plot (MST+). 180

9.4

Adaptation of Model Parameters to Experimental Results

Figure 9.2: Stress(r)-strain(e)-plot (MST+). Table 9.1: Efficiency of different integration procedures applied to the simulation of cyclic loadings, results after 30 cycles. Method

qg

etol [-]

CPU-time [s]

P.-C. by Heun R.-K. 2./3. order Dormand-Prince Collocation n = 0.5 Collocation n = 1.0

2 3 5 2 1

10–1 10–1 10–1 10–1 10–1

0.49 0.82 1.43 4.78 6.48

13173 12253 5402 1592 3029

304.01 303.96 303.98 303.94 304.07

P.-C. by Heun R.-K. 2./3. order Dormand-Prince Collocation n = 0.5 Collocation n = 1.0

2 3 5 2 1

10–2 10–2 10–2 10–2 10–2

0.77 0.93 2.14 7.31 14.23

21202 12987 5953 3118 9076

303.98 303.98 303.98 303.96 304.01

P.-C. by Heun R.-K. 2./3. order Dormand-Prince Collocation n = 0.5 Collocation n = 1.0

2 3 5 2 1

10–4 10–4 10–4 10–4 10–4

5.16 1.65 2.36 18.63 102.58

144204 23704 7556 13062 101738

303.98 303.98 303.98 303.98 303.98

9.4

Time steps

r [MPa]

Adaptation of Model Parameters to Experimental Results

The mathematical formulations of the material models are derived by physical understanding of the reactions of the material to various imposed loads and temperatures in time histories. The models – here of the unified type – express the physical properties by material functions (equations of evolution) and free parameters pi. These parameters have to be determined by adaptations to experimental results. Hereby, optimizing processes are applied to find the most suited set of parameters. The differences between model simulation and experimental results are expressed 181

9

Development and Improvement of Unified Models

by a target function q (p). Values of the parameters follow from the solutions of an equivalent optimization problem. Because of the manifold of parameter sets, numerical methods have to be applied, which are determining the parameter values by an iteration process. Various algorithms have been developed for this optimizing process. Stochastical methods as the strategy of selecting evolution apply the principle of random corrections. Deterministic methods apply mathematical procedures in search of minima of functionals as the methods of steepest gradients and the simplex method. Strategies of hybrid approaches combine both stochastical and deterministic methods. This section firstly describes the different methods, also by their advantages and disadvantages, and gives recently the results of an investigation, which compares these methods in their efficiency (see also [23]).

•

Target functions q (p)

They can be defined either in absolute or in relative measures of error. Absolute error values are determined by the quadratic differences between experimental and numerical values of the state variables zi: qa …p† ˆ

n X …zexp i

2 zsim i …p†† Dti :

…19†

iˆ1

Relative errors may be determined by evaluating the area of the differences between experimental and simulated curves, relating these error areas to the area underneath the experimental curve:

qr …p† ˆ

i1 nP1 tR

zexp t  zsim t; p dt

i
1 ti

n1 P tRi1

:

20

zexp t dt

i
1 ti

For simultaneous adaptations to several experiments, weighting factors wj for each of the tests may be added: qw p


m X

wj qj p :

21

j
1

Braasch proved by investigating several creep tests that the relative measure is the better one [9]. If for one and the same load-time, history tests are repeated, then as a rule, the results will deviate. In this case, the Maximum-Likelihood method may be applied, by which the adaptation is related to the mean values of the test sets, and the standard deviations at each data point are chosen for weighting and standardizing the error at each data point. The target function is defined by a relative measure of errors (see [24]). 182

9.4 •

Adaptation of Model Parameters to Experimental Results

Strategy of selecting evolution

The stochastic method, as e. g. given by Rechenberg [25] and Schwefel [26], works in analogy of the ‘natural’ mechanism of mutation, selection, heredity. It is a robust procedure with a broad path of searching minima, overcoming local relative minima, although the required computer time may be large. This method is fully developed and widely used in [9, 16, 27, 28].

•

Downhill-simplex method by Nelder and Meat [29]

The deterministic simplex method is based on geometric considerations. A simplex is a geometric hyper surface spanned between n + 1 knots within a n-dimensional space. Each of the knots is given by a parameter vector pi of n elements. For n
2 a triangle, n
3 a tetraeder, and generally, a regular polyeder of n + 1 edge points are defined. For each set of the parameters pi, the corresponding target function is evaluated and allotted to the edge point of the simplex surface. The basic idea is to replace the edge point phigh of the maximum target value qmax by a new point pnew, which is found by reflecting phigh at the centre of the other n edge points and, which as a rule, is closer to the minimum. The extension of this procedure by Nelder and Meat [29] introduces an algorithm, by which the simplex area is expanded or contracted to adjust it to a better target function q (p).

•

Methods applying gradient algorithms

In addition to the direct evaluation of the functional, for which a minimum is searched for, also the gradients of this function are taken into consideration: q…p† ˆ

qq…p† ; i ˆ 1; 2; . . . ; n : qpi

…22†

The optimum is determined by an iteration process given by: p…v1
pv  v sv
pv  v Hv qpv  :

23

H is an iteration matrix,  a step length. The different methods are applying different directions s of the search for a minimum: •

•

Gradient: sv
Iqpv  ;

24

Newton method: sv
2 qpv 1 qpv  ;

25 183

9 •

•

Development and Improvement of Unified Models

BFGS method: v s…v†
HBFGS qpv  ;

26

Conjugate gradients (CG): qpv T qpv  v1 : s sv
qpv   qpv1 T qpv1 

27

The Broyden-Fletcher-Goldfarb-Shanno method (BFGS method) proved so far to be numerically stable and of very good convergence. It determines better approximations of the inverse curvature matrix in Equation (25) only by informations of first order and by the following updating formula: v1 HBFGS




I

dv gv T gv T dv

!

Hv I 

gv dv T gv T dv

!



dv dv T gv T dv

;

28

where 0

HBFGS
I; dv
pv1  pv

and gv
qpv1   qpv  :

The efficiency is depending on the choice of suited step lengths v . To determine v , the manifold-dimensional minima-problem is reduced towards an one-dimensional by: qE v  q pv  v sv  :

29

Herein, sv is the given downwards direction. The step lengths v have to be evaluated so that the next iteration point pv1
pv  v sv is acceptable. To reduce the amount of calculation work, line search methods have been developed. Mahnken presents a comprehensive review of the one-dimensional minima-methods [30].

•

Hybrid methods

For the adaptation of the parameters in material models, the mathematical tools should provide a method, by which a set of parameters of an optimal fit to experimental results is evaluated by the least amount of computer work. The experiences of the research work in this project proved that none of the methods discussed above were satisfying enough. Therefore, a hybrid method has been developed, which combines the advantages of both the evolution and the gradient methods, where the first one avoids being trapped in a valley of a relative minimum and the second one contributes to high speed convergency. The hybrid method [23] works in three major procedures: 184

9.4

Adaptation of Model Parameters to Experimental Results

1. Sets of parameters, which cover the entire possible parameter space: By a random algorithm, na sets of parameters are chosen that are fulfilling given restrictions and are evenly spread over the entire parameter space. Thus, it is assured that the global minimum is not missed. Then, the best nb < na sets are selected as starting values for application of the evolution strategy. 2. Determination of the local fields of minima: The strategy of evolution is applied to find as many as possible surroundings of local minima. 3. Evaluation of the local minima by the BFGS method: Having passed a given limit of quality or a maximum number of generations, the best sets of parameters of each of the parent generations is taken as the starting set for the application of the BFGS method. The optimizing procedure is stopped when a local minimum is reached. The sets of parameters for all the determined local minima are then taken as starting sets for a repeated strategy of evolution, by which the global minimum is searched for. This iteration procedure is repeated as long as a given criteria for finishing is not satisfied (CPU-time or error q).

•

Efficiency of the optimizing methods

The special case investigated is the adaptation of the parameters of the material model of Chaboche and Rousselier to cyclic experimental results. The efficiency is tested by reidentification of the parameters. For the numerical simulation of the experiments, two tests are selected: one of proportional loading and another one of non-proportional loading path, for which a parameter vector popt is given. For the set of parameters popt for q (popt)
0% (see Equations (20) and (21)), two variations are investigated: a slight one (p01 for q (popt)
23%) and a stronger one (p02 for q (p02)
87%). The sets of parameters are given [23]. Five different methods for optimizing the adaptation are selected: the evolution strategy (Evo), the simplex-(Sim), the BFGS-, the CG- and the hybrid method (Hyb). For the methods using random values, the results are determined by mean values of 10 test runs. For the hybrid method, the number of starting vectors is restricted to na
1 (pstart
p01 or p02). The switch from the evolution strategy to the BFGS method is governed by the value of the relative target function of q
20%. The efficiency is evaluated by the CPU-time (Pentium P90) required for reaching the target function value of q (p)
0.02%. The results are presented in Figures 9.3 and 9.4 with respect of the CPU-time, the evolution method is in these cases requiring much more time than the other methods. This is caused by the random search that delays the precise localization of the minimum. Whereas the deterministic methods do need only about 1/6 of the computing time. The hybrid method is – taking both Figures 9.3 and 9.4 into considerations – the most efficient method, followed closely by the BFGS method. For the methods of gradients, several restarts had been necessary to localize the minimum. Possible causes are instabilities of the numerical optimizing or interim paths to only local minima. The hybrid method avoids these ‘traps’. 185

Development and Improvement of Unified Models

[s]

9

[s]

Figure 9.3: CPU-time, starting vector p01.

Figure 9.4: CPU-time, starting vector p02.

9.5

Systematic Approach to Improve Material Models

In the usual formulation of material models, the equations describing the properties of the material are expressed in closed form being valid for the entire region of the structural variables (see Equation (6)). In this formulation, adaptation of the parameters – as discussed in Section 9.4 – is the only mean for adjusting the model to experimental results. However, when the material functions do not cover the real behaviour, then adaptation of the parameters will not be very successful. For this case, an improvement is achieved when the functions are expressed (in analogy to Finite-Element methods) in discrete sections of the entire regions of the variables by shape functions. Thus, the material functions themselves can also be determined by an optimization process because the number of unknowns are increased. This method of discretization of the material functions has been developed by Braasch [9, 31, 32]. By sufficiently small discretization, even very complicated material behaviour can be covered by the simulation as accurate as desirable. If the final discrete expressions of the material functions are expressed by a complete single function – say a polynomial –, then new material functions are derived, which are closer to test results. Moreover, this procedure of discretization can also be applied to scrutinize the completeness of the model with regard to the physical assumptions. If even for very 186

9.5

Systematic Approach to Improve Material Models

fine discretizations of the given material functions and optimal adaptations of the parameters, the differences between experimental and computed results are still unacceptable, then the basic assumptions of the material model are wrong or insufficient. In this case, the material model has to be modified or extended taking into account other or additional physical effects. Hence, the approach of discretization of the material functions is also a powerful tool to check the basic assumptions of the material model.

•

Example of application

The methodical way is demonstrated here by applying the procedure to the model by Chaboche and Rousselier in its one-dimensional formulation with one variable for kinematic hardening (see Section 9.2.1). For experimental results, the tests by Styczynski [33] of pure aluminium Al550 at a temperature of 550 K are taken. The load path consists of a prior tension phase and three creep phases. For monotonous loading, isotropic and kinematic hardening cannot be distinguished. For more information on complete results, values of the parameters etc., see [9]. The optimization of the parameters for closed form, material functions may have been achieved by methods as in Section 9.4 in a first step. If there are still differences between test results and numerical simulation, it may be justified to assume firstly that the chosen functions of the material properties may be wrong. Therefore, in a second step, the material function p_ is discretized in analogy to the Finite-Element method by: _ ex † ˆ fD …rex † ˆ ai rnexi  bi for ri1 < rex ri : p…r

30

The overstress region of the model is subdivided into smaller sections. For each of the sections, the exponentials ni on the stresses are taken to be constant. The shape functions for the additional parameters ai and bi have to fulfil the conditions of continuity up to the first derivatives. For the ai and bi, the optimization algorithm for parameter adaptation is once more applied including all the other parameters. The computation for the example proves that a subdivision into more than four discrete sections does not improve the model results although the yield functions have been optimized with respect to the overstress. Figure 9.5 shows that the ‘4 sections’ discretization is covering the test results much better than the single function assumption. Nevertheless, the stress-strain curve in Figure 9.5 is still far off the test results, and the first transient creep phase is somewhat too large. These deficits prove that the model is basically insufficient. Now, in a third step, improvements of the physical properties of the material model have to be selected. They may be found by careful interpretation of the second step and by experiences gained from other material models. Here for example, the material function p_ is assumed to depend on the overstress and isotropic hardening in a form of a product: _ ex ; K
fD rex gD K : pr

31

In addition to fD rex , also the function gD K is discretized by: 187

9

Development and Improvement of Unified Models

t [s]

Figure 9.5: Computed results and test Al55007.

gD …K† ˆ cj

 lj K K dj for Kj1 < Kj : Q Q

32

The parameters cj and dj are determined by an optimization process together with all the other parameters with the restriction that the function gD(K) is continuous up to the first derivatives. Figure 9.6 shows that now the numerical results for the ‘4 sections’ are fitting much better to the experimental results. The so far achieved model is tested by applying it to a three-phase tension experiment Al550ZS, which has not been included in the adaptation of the parameters. The results are given in Figure 9.7. Both the ‘1 section’ and the ‘4 sections’ simulations do not cover the test satisfactorily. Therefore, the basic physical assumptions are further improved. The evolutionary equation for the kinematic hardening is refined by including static recovery: ! 2 qf 3 ij : 33 _ ij
C a ef  ij p_ rex   cr1 3 qsij 2 v

t [s]

Figure 9.6: Computed results and test Al55007. 188

9.5

Systematic Approach to Improve Material Models

Figure 9.7: Prediction of test Al550ZS.

t [s]

Figure 9.8: Computed results and test Al55007.

Figure 9.9: Prediction of test Al550ZS.

Herein, c and r are additional parameters. The accompanied state of stress depends now on the time rate. Additional investigations proved that a discretization of the function in Equation (33) does not lead to significant improvements compared to those already achieved by the closed form amendment of Equation (33). The improved results are shown in Figures 9.8 and 9.9 for the experiments, as which the adaptation is based as well as the Al550ZS test is not included in the adaptation procedure. 189

9

Development and Improvement of Unified Models

In comparing all the results of Figure 9.5 up to Figure 9.9, it is well demonstrated how the proposed method does have a strong capability to improve the material models. In Section 9.8.3, an example is investigated to validate the effects of this improvement approach to analyses of structures.

9.6

Models Employing Distorted Yield Surfaces

In most of the material models, the yield function is assumed to be of the v. Mises type. Isotropic hardening expands the yield surface, kinematic hardening shifts the surface translatorically. In both cases, the original geometry of the surface is not changed. Experiments especially of non-monotonous load paths, however, show that the yield surfaces may be distorted (non-proportional change of the main axis, rotation of the axis, and rotated ‘buckling’) (see [34, 35]). This section presents mathematical expansions of the material models, which cover these distortional phenomena of the yield surfaces.

•

Development of yield surfaces with different loadings

Figure 9.10 shows in the two-dimensional s-r-space yield surfaces, which have been derived by Phillips and Tang [36] from experiments on aluminium specimen (Al 1100) at normal temperature. At the start, the yield surface is a well developed v. Mises ellipse. After the load path form A to B in uniaxial direction, the double symmetry is lost, the yield surface ‘buckles’ out in direction of the loading. For the path A-B-C-D, the symmetry is lost completely and it is distorted severely. A biaxial load path from A directly to D results in quite a different yield surface although the final stress point D is the same. The main phenomena of the development of yield surface distortions are:

Figure 9.10: s-r-space yield surfaces [36]. 190

9.6

Models Employing Distorted Yield Surfaces

•

In the unloaded state, the yield surface is of elliptical form. The loading path causes a distortion in the path direction. The axis of the ellipse decreases in loading direction. Perpendicular to the loading path, the axis does not change its length. The curvature of the yield surface is larger in direction of the load path and smaller at the opposite section. The distortion of the yield surface depends very much on the load path.

•

Yield functions and hardening tensors

• • • • •

The distorted yield surfaces can be described mathematically by introducing higher order tensors. The investigations by Wegener [37] proved that these expansions can be adapted to the different distortion forms. The yield function f is defined by the stress deviator sij and internal hardening tensors h. The following equations are representing yield surfaces in a kind of hierarchy with increasing complexity and hardening properties (see Sayir [38]). They all belong to the class of polynom yield surfaces: 1

2

3

…0†  hij sij  sij hijkl skl  sij skl hijklmn smn   . . . : f …sij ; h…k† ... † ˆ h

•

34

The yield function (of 2. degree) with isotropic hardening (v. Mises) is: f0




3 sij sij 2

12

 K  k

1 with sij
rij  rkk dij : 3

35

K is a scalar hardening tensor of zero order. The yield surface expands proportional preserving its original geometry. •

The yield surface (of 2. degree) with kinematic hardening (v. Mises/Prager) is: f1




3 ef ef s s 2 ij ij

12

 K  k

1 ef ef with sef ij
rij  rkk dij 3

and ref ij
rij  ij : 36

A hardening tensor of 2. order ij is added, which results in translatoric displacement of the yield surface. ref ij are the effective stresses, which are related to the distance of the current stress point from the centre of the kinematical displaced yield surface. Equation (36) is identical with the formulation by Chaboche and Rousselier in Section 9.2.1. •

The yield surface of 2. degree with distortional hardening (Edelmann and Drucker [39]) is given by:   12 3 ef ef ef ef s s  sij b ijkl skl  K  k : f2
2 ij ij

37

191

9

Development and Improvement of Unified Models

Here, a hardening tensor of 4. order bijkl is added to Equation (36). It leads to change of the lengths of the axes and rotation of the yield surface. The double symmetry is still preserved. •

The yield surface of 3. degree with distortional hardening (Sayir [38], Rees [40], Betten [41] and Lehmann [42]) is given by: f3


  12 3 ef ef ef ef ef ef sij sij  sef b s  s s c s  K  k : ij ijkl kl ij kl ijklmn mn 2

38

A term of a hardening tensor of 6. order cijklmn is added to Equation (37). It results in the change of the curvature of the yield surface (see Figure 9.11). The symmetry of the surface form is lost completely. For the yield functions of distortional hardening (Equations (37) and (38)), numerical calculations proved that the condition of convexity is not always assured for all the possible values of the tensor. The state tensors of fourth and sixth order may lead to a numerical bursting of the convex yield surface so that some sections are no longer valid (see Figure 9.12). Special investigations proved that an always stable formulation is achieved for the yield function of 2. degree distortional hardening if a stabilized tensor of 2. order is introduced: ef ^sef ij
gijkl  b ijkl skl :

39

The constants gijkl are describing the preceding form and orientation of the yield surface in the space of principle stresses. In the case of purely isotropic hardening in the beginning, the constants follow from the v. Mises condition. In the case of anisotropy, the gijkl have to be evaluated from experimental results. The stabilized yield function is:

Figure 9.11: Variation of c121212 leads to distortion (dashed lines). 192

9.6

Models Employing Distorted Yield Surfaces

Figure 9.12: ‘Bursting’ violates the condition of convexity.

1

2 f2s ˆ …3J2 …^sef ij ††  K  k




3 ef ef ^s ^s 2 ij ij

12

 K  k :

40

The hardening properties are the same as in Equation (37). By the introduction of the 2. invariant of ^sef ij , it is assured that f has reell points of zero value. •

Material models for distortional hardening of 2. and 3. order

The hierarchy of progressively more complex hardening rules is given here for the model by Chaboche and Rousselier model (see Section 9.2.1). It starts from isotropic hardening. The additional hardening tensors are of polynomial type. Each of the following yield functions is valid for all the evolutionary equations of the actual model. •

Model with isotropic hardening, f
f0 as in Equation (35): _ ex  : K_
bK QK  K pr

•

•

Model with kinematic hardening, f
f1 as in Equation (36), additional: ! 2 qf _ ex  :  ij pr Qa _ ij
ba 3 qsef ij

41

42

For distortional hardening of 2. degree, f
f2 as in Equation (37), additional: ! _b
bb Qb qf  b _ ex  : 43 ijkl ijkl pr qb ijkl 193

9 •

Development and Improvement of Unified Models

For distortional hardening of 3. degree, f
f3 as in Equation (38), additional: ! qf _ ex † : …44† cijklmn p…r Qc c_ijklmn
bc qcijklmn

The Equations (43) and (44) are structured similarily to the known evolutionary equations as those of Equations (41) and (42). However, the distortional hardenings are not developing in direction of the inelastic strain rates – as the Prager rule in Equation (42) – but in direction of the gradients of the corresponding hardening tensors. This is in accordance to the ‘principle of the maximum hardening effect’ (Wegener [37]).

•

Numerical simulations of saturated yield surfaces

To prove the efficiency of the material models given above, a saturation process is simulated. The experimental results on mild steel Fe 510 are produced within the project B10 (Peil [35]) by Dannemeyer [43]. The material parameters of the various models (Equations (41) to (44)) are adapted to the cyclic experimental results of the saturated yield surfaces applying the optimization procedure presented in Section 9.4. The sets of parameters are determined so that each set is the best fit to all the yield surfaces of the three experiments (see Table 9.2). The parameters are given in [23]. The differences between experimental and numerical results are taken as criteria for the quality of the simulations qfl, that is the sum of all the deviations for n different points on the yield surface: qfl ˆ

n h 1 X exp rf;i 2n iˆ1

i exp sim  s  s rsim f;i : f;i f;i

45

In Table 9.2, the results are compared. It proves that the extension by distortional hardening is improving the quality of the numerical simulation considerably. Figure 9.13 presents the saturated yield surfaces for the different material models for one of the experiments: the biaxial tests as given by the stress-strain path in Figure 9.13 a. The model by Chaboche Table 9.2: Quality of the distortional material models for three different types of model. Tests

De [%]

Dc [%]

Error qfl [MPa] Chaboche and Rousselier

Tension/compression Torsion Biaxial

0.490 – 0.347

– 0.848 0.600

Mean value

194

Distortional hardening of 2. degree

3. degree

8.24 7.74 12.88

6.94 5.42 7.76

3.88 3.10 7.10

9.62

6.70

4.69

Models Employing Distorted Yield Surfaces

 s 3 [MPa]

 c= 3 [%]

9.6

a)

 s 3 [MPa]

 s 3 [MPa]

b)

c) d) Figure 9.13: Numerical simulation with different models of the biaxial test results with saturation after 5. cycle: a) load path; b) model of kinematic hardening; c) model of 2. degree distortional hardening; d) model of 3. degree distortional hardening.

and Rousselier does not cover the rotation of the axis and the distortional ‘bumbs’ of the yield surfaces. The model of 2. degree distortional hardening describes the rotation and the change in lengths of the axis correctly. Yet, only the model with 3. degree distortional hardening can reproduce the characteristic ‘bumbs’-distorsions.

•

Numerical simulation of cyclic experiments

Within project B10 (Peil [35]), specimens of hollow cylinders of mild steel Fe 510 have been tested. The results are given by Reininghaus [44]. The cyclic loading of proportional and non-proportional load paths reached saturations after 15 cycles. For the numerical simulation, all the different models as given above by Equations (41) to (44) are investigated. The parameter adaptation is considering the proportional experiments by relative weight factors to result in equivalent simulations of these tests. Hence, deficiencies are showing up only for the non-proportional load path. The sets of 195

Development and Improvement of Unified Models

 c= 3 [%]

9

a) b)

c)

d)

Figure 9.14: Saturation curves of cyclic non-proportional loading as given in a); b) model of kinematic hardening, where q = 11.60%; c) model of 2. degree distortional hardening, where q = 6.63%; d) model of 3. degree distortional hardening, where q = 5.72%.

parameters are given in [23]. As quality criterion q, the relative differences of the areas beneath the curves are chosen as explained in Section 9.4 (Equations (20) and (21)). Figure 9.14 presents some of the results after saturation. The r-e-hysteresis-loops for non-proportional loading as given by Figure 9.14 a are shown. The model by Chaboche and Rousselier (Equation (42)) is underestimating the maximum stress. Whereas the models of distortional hardening do fit much better (see Figure 9.14 c and d). The relative error q is reduced to half the value of model b (Figure 9.14 b).

•

Adaptation results for one- or multiaxial test results

The adaptation is usually applied to uniaxial tests. Hence, it is questionable whether the set of parameters, which is adapted to these simpler tests, is also covering multiaxial load paths. Within the project A2 [34], experiments have been carried out for uniaxial as well as for biaxial tests of AlMg3. The results are given by Gieseke [45]. These results are 196

9.7

Approach to Cover Stochastic Test Results

taken for a numerical investigation applying both models, that by Chaboche and Rousselier and that of the distortional hardening (Equation (43)). The adaptation is applied alternatively to the experimental data of uniaxial and of biaxial test results. More details are given in [23, 46]. Some of the results of these investigations are the following: • •

As a rule, the non-proportional tests are not sufficiently covered by models adapted to uniaxial tests. Uniaxial test data do not include the distortional phenomena.

Hence, the adaptation has to be applied to multiaxial load paths if models should be derived, which can safely be used for a broader application of more complex load paths as in actual structures.

9.7

Approach to Cover Stochastic Test Results

For the investigation of the effects of distributed material properties on the results of structural analyses, a special kind of steel is selected (Warmarbeitsstahl 1.2344 DIN X40CrMoV5 1). This steel is especially suited for high temperature-loaded steel tools (see also Section 9.8.6). The temperature may be as high as 700 K. The experimental data are provided by the project A5 (Rie et al. [47]). To limit the experimental effort, here only one type of test is selected. 27 specimens (of the same production charge) are tested by a two-step tension load. The creep curves are

X40CrMoV51 T
733 K tension phases: e_ = 5·10–5s–1

t [s] Figure 9.15: Experimental creep data of 27 specimen. 197

9

Development and Improvement of Unified Models

shown in Figure 9.15. They are distinctively scattered around a mean value curve. There may be several sources for this stochastic behaviour. However, for the following approach, it is assumed that such scattering curves are given by experimental tests as material phenomena.

•

Statistical interpretation of the data

From Figure 9.15, the supposition may be justified that the results are clustered around some mean value for each specific time. Therefore, the data of strains are selected as stochastic variables. Arguments for satisfying the conditions of a normal Gaussian distribution are: The mean values e of the strains (see Figure 9.15) are a smooth function well in the centre of the distributed test results. • The curves for the standard deviation se of these tests (see Figure 9.16) are also smooth and follow the creep phases very well. • The coefficient of variance v is almost constant and has a value of 0.05 in the first creeping phase and inclines to 0.07 in the second phase. • Each test value is compared with the probabilistically expected one. The cumulative distribution functions turned out to be almost equal. • The coefficients of variance have values below the limits given by Sachs [48]. Therefore, a logarithmic normal distribution is not required, a Gaussian distribution can be assumed. • The distribution satisfies further criteria as those for adaptation to normal distribution given by Kolmogoroff-Smirnoff.

•

t [s] Figure 9.16: Expected mean values e for strains of the experimental results of 27 tests and standard deviation se. 198

9.7

Approach to Cover Stochastic Test Results

In Figure 9.16, the characteristic statistical values of means and standard deviations are shown for the test data. In the small table, the same characteristics are given if only a group of nine specimens (91, 92, 93) would have been investigated. The differences are small. •

Concept for the statistical approach

It should be possible to express stochastically distributed test results by correspondingly distributed sets of parameters in the functional formulations. Some considerations have already been given by Braasch [24]. As it is shown before, the experimental data obey a normal Gaussian distribution. The estimated mean values and the variances are determined from the test results …e ) le ; s2e ) r2e †. Then, for the assumed Gaussian distribution, an infinite number of ‘artificial’ tests is defined, which possesses the same characteristic statistical values. From this infinite number of tests, discrete strain-time curves can be simulated. In Figure 9.17, such curves are presented for quantils of equal distances of 5%. This results in 19 ‘artificial tests’. It proved to be necessary to follow this concept to find the parameter distribution. •

Evaluation of stochastic parameters

The ideal curves of Figure 9.17 are the basis for the adaptation of the parameters of the model by Chaboche and Rousselier. The adaptation of the parameters follows the procedures described in Section 9.4. The example of Figure 9.17 is of special nature because the adaptation process proved that assuming only isotropic hardening is sufficient. The parameters C and a corresponding to kinematic hardening (see Equation (8)) can be dropped. Furthermore, the initial yield limit k can be neglected (see Equation (4)).

t [s] Figure 9.17: Idealized creep curves derived from infinite Gaussian distribution. 199

9

Development and Improvement of Unified Models

Hence, only 5 parameters E, n, D, b, Q of Equations (2) to (7) have to be evaluated. It is obvious also to assume only one value for the Young’s modulus E. Figure 9.18 a shows the results of the parameter evaluations for each of the 19 curves in Figure 9.17. The four parameters are presented along the axis representing the cumulative distribution values. The functions are largely antimetrical to the centre of F(x)
0.5. The parameter D is almost constant. Therefore, in Figure 9.18 c, the same functional distribution is presented for only three free parameters. If the same evaluation of the parameters is applied – not to the ideal distribution, but directly to the curves of the test results –, then the functional distributions of the parameters are not at all smooth curves (see Figure 9.18 b and the corresponding Figure 9.18 a). Better parameters for the test results are achieved when only three parameters are free for adaptations (see Figure 19.8 d compared to Figure 9.18 b). The results are: • •

E
1.80 · 105 MPa, D
8.88 · 102 MPa s1/n, three functional distributions given in Figure 9.18 c for the parameters n, b, Q of Equations (6) and (7). fixed values for

In the next step of the approach, the distribution functions of Figure 9.18 c should be presented by fit polynomials. Thus, all the stochastical parameters are described as functions of the cumulative distributions, the quantils.

a)

c)

cumulative distribution function F…x†

empirical cumulative distribution function S…x†

cumulative distribution function F…x†

empirical cumulative distribution function S…x†

b)

d)

Figure 9.18: Results of the parameter evaluation for the idealized creep curves of Figure 9.17: a) ideal distribution, 4 free parameters; b) test results, 4 free parameters; c) ideal distribution, 3 free parameters; d) test results, 3 free parameters. 200

9.8 •

Structural Analyses

Application to analyses of structures

Having arrived at material models, where – at least some, if not all – the parameters are given in form of Gaussian distributions, random selections of material properties may be produced and allocated to physical points of the structure, say to the Gaussian points of Finite-Element discretizations. Hereby, it should be considered that the material model is already adapted to test results, which are integrals of the behaviour of the entire test specimen. Therefore, the pointwise distribution over a structure should have distances not smaller than the dimensions of the specimen. If the areas of identical material properties are too small, then the stochastic approach of the material model is suppressed because now structural analysis is driving towards the mean values. For the practical application, it is proposed to assign random values of stochastic material properties (the parameters of the model) to somewhat larger areas. An example is presented in Section 9.8.5.

•

Generalization

So far, the proposed statistical approach has been applied and verified only for a single type of experimental tests. This procedure offers best possibilities to take into account simultaneously different types of tests with complex loadings, which is necessary to describe the distributed material behaviour of real structures. Therefore, the sets of parameters have to be adapted simultaneously to test results of different loading types, e.g. tension tests with different strain rates, creep or relaxation in different levels of stresses or of strains, cyclic loadings. Before the simultaneous adaptation for parameter optimization is applied, all the different types of test results have to be replaced by idealized distributions like those in Figure 9.17. For weighting the different test results, the standard deviations of each of them may be taken as weight measure. The procedure follows the same steps as given for the single type example.

9.8

Structural Analyses

The stress-strain developments in structures exposed to high thermo-mechanical outer actions are in general geometrically as well as physically non-linear and also time-dependent. For the following examples of applications, it is assumed that strains and deformations are sufficiently small for applying a geometrical linear theory and that the states of temperatures may be stationary. For the Finite-Element method, the formulation in displacements is chosen (see [9, 10, 12, 15]). For a mixed Finite-Element method formulation, algorithms and analysis are presented in [19].

201

9

9.8.1

Development and Improvement of Unified Models

Consistent formulation of the coupled boundary and initial value problem

The problem is described by the following basic equations and by additional boundary and initial value conditions: •

equilibrium rij;j  pi
0 in X ;

•

46

kinematic relations 1 eij  ui;j  uj;i 
0 in X ; 2

•

47

material model 1 r_ kl  e_in e_ij  Eijkl ij
0 in X ;

_ ij ; qij  e_in ij  pr

48

qf
0 in X ; qsef ij

49

q_ ij  hrij ; qij   rrij ; qij 
0 in X :

50

The boundary value problem is solved by Finite-Element method in a weak formulation of the equilibrium conditions [49, 50]. The integration of the initial value problem is discussed in Section 9.3. The collocation method is applied, assuming linear approximations within time steps in implicit formulation, for all examples except in Section 9.8.4, where higher order tensors for hardening are investigated. The non-linear equations are solved by Newton-Raphson iteration. For the structural discretization, isoparametric elements with quadratic approximations are chosen. The analysis proceeds along the following equations: 1. Stiffness matrix of the total structure: Z ~ H dX ; K
D HT ED

51

X

D operational matrix

202

H

element functions

E~




 ij A

partial derivatives

E

1

q e_in  1   A23 A33 A32  nDt qr

1

ü ï ï ï ý , ï ï ï ï þ

(52)

9.8

Structural Analyses

2. Residuum: Z Z Z Dr
HT DpdX  HT Df dC r  D HT DrdX X



Z X

Cr

X

  _ D HT E~ D HDv  E1 Dr  e_in Dt  A23 A1 dX : 33 Dq  qDt

53

3. Global iteration: K  DDm
Dr :

54

4. Total increments of displacements: Dm
Dm  DDm : 5. Local iteration:   _ Dq  qDt ; DDr
E~ D HDm  E1 Dr  e_in Dt  A23 A1 33

9.8.2

55

56

_  A32 DDr ; DDq
A1 33 Dq  qDt

57

Dr
Dr  DDr ; Dq
Dq  DDq :

58

Analysis of stress-strain fields in welded joints

In project C4, Wohlfahrt [51] investigated experimentally welded joints for steel StE 460 NA. Uniaxial tension tests under room temperature have been performed for specimens of the base metal, of the weld metal and for four different modifications of the microstructure within the heat affected zone (HAZ). The results have been used for adaptions of the parameters of the model by Chaboche and Rousselier applying the methods discussed in Section 9.4. In Figure 9.19, the stress-strain curves are shown for the different materials as given by the numerical simulation after parameter adaptation. The experimental results are not shown because of only small differences to the numerical evaluations. It proved that the original model by Chaboche and Rousselier is covering the tests sufficiently. For the numerical analysis of the 100 mm long welded joint, symmetry is assumed. The numerically simulated welded joint is elongated by 4.0 mm within 900 s. This would be equivalent to a medium strain of em
40:0‰ and a strain rate of e_m
4:4  105 s1 within a homogeneous material. The structural model (see Figures 203

9

Development and Improvement of Unified Models

e_ = 5·10–5s–1

Figure 9.19: Tension tests for the different materials of a welded joint.

9.20 and 9.21) is composed of the three zones, where the HAZ of a total thickness of approximate 2.1 mm is divided into four different kinds of microstructures. The structural model is two-dimensional. Further details are given in the contribution to this book by the project C4 [51]. Figures 9.20 and 9.21 show the results of the analysis: the distribution of the longitudinal strains caused by the material inhomogeneity for mean strains of em
2:8‰ (t
63 s) and for em
8:0‰ (t
180 s).

Figure 9.20: Strain ex [–] for em = 2.8‰, t = 63 s. 204

9.8

Structural Analyses

Figure 9.21: Strain ex [–] for em = 8.0‰, t = 180 s.

From Figure 9.20, it is proved that high inelastic strains (and, hence, also higher strain rates) are developing within the weld metal and at the edges of the HAZ. These strain rates are larger as those of the tests in Figure 9.19 for the parameter adaptation. This is caused by the stiffer layers C and M within the HAZ being also restrained in the direction along the welded joint (see Figure 9.19). For higher loads, as in Figure 9.21 for mean em
8:0‰, the strains within the weld are only slightly larger than 8.0‰, whereas the strains in the base metal are approaching 8‰. For greater em, the relative results do not change remarkable. The analysis of the welded joint proved in general what has been measured also in the experiments of the project C4 [51]: The heat affected zone is responsible for non-uniform distributions of strains and stresses. The present investigations do not cover, however, the three-dimensional inhomogeneity within the HAZ.

9.8.3

Thick-walled rotational vessel under inner pressure

The small vessel of Figure 9.22 under pressure is taken as a structural example to investigate the effects of the different material models of Section 9.5 on the stress-strain fields in the walls [9]. Compared are the following variants (see Section 9.5, Equations (30) to (33)): A

Discretization of material function

_ ex † ˆ fD …rex † p…r

B

Discretization of both functions

_ ex ; K† ˆ fD …rex †gD …K† p…r

C

Additional term of static recovery (see Equation (33)).

The structure of Figure 9.22 is discretized by 8-nodes elements (reduced integration in 2 × 2 Gaussian points). The load history of inner pressures is chosen so that for the vari205

9

Development and Improvement of Unified Models

Figure 9.22: Geometry of a thick-walled vessel.

Figure 9.23: Stationary stresses along section II-II of Figure 9.22.

ant C, the most critical Gaussian point is stressed and strained equal to the material curves, which are used in Section 9.5 to determine the parameters. These conditions yield the following load path: Firstly, the inner pressure is raised to pi
1:4 MPa within t
100 s. Then, the increase of pressure is slowed down so that for t
480 s, pi
4:82 MPa is reached. Figure 9.23 shows the differences in the stress fields of one section by applying the three variants A, B, C. Different assumptions for material models produce also distinctly different stress fields in structures. The same structure as in Figure 9.22 has also been taken as an example for the investigations with the models by Hart, Chaboche and Rousselier, and by Miller (see [52]).

9.8.4

Application of distorted yield functions

This section reports on the applicability of the concept given in Section 9.6 of higher order hardening tensors for considering also distorted yield surfaces. Also the effects of 206

9.8

Structural Analyses

t [s] a) b) c) Figure 9.24: Structural example: a) case I; b) case II; c) imposed cycles for v and u.

the extension of the model by Chaboche and Rousselier given in Section 9.6 are evaluated. The structural example is a strip with a central hole (see Figure 9.24). Two different cases are investigated for cyclic patterns of imposed displacements u and v. In the first case, the strip is strained only in v direction. In the second case, imposed v and u displacements of the edges are following in cycles of 908 phase shifting as shown in the time-dependent diagram in Figure 9.24 c. Two models are applied for both cases I and II. In the first one, the original model and the kinematic hardening rule (Equation (42)) as given in Section 9.6 are employed. The second model is that of distortional hardening of 3. degree (Equations (43) and (44)). The structural analysis by the Finite-Element method yields results given in Figures 9.25 and 9.26. They show the v. Mises stresses rv at the time of the third cycle peak. For the uniaxial imposed displacements v, Figure 9.25 shows no significant differences in the stress fields of both applied models. In the case of two-dimensional imposed displacements v and u, in Figure 9.26, the model of distortional hardening (b) yields a very different v. Mises stress field compared to the simple one of kinematic hardening (a), especially in the region above the hole. More refined investigations, not shown here, revealed that the differences between a and b in Figure 9.26 are caused mainly by the additional term for distortional hardening of 2. degree (Equation (43)), whereas the 3. degree refinement of Equation (44) is – at least in this example – of minor effect. However, other applications of non-proportional stress or strain paths may prove that also the 3. degree hardening term is significant. For the same example of a structure (Figure 9.24), the case I is chosen to investigate the efficiency of the different time integration methods discussed in Section 9.3. For the material model, that of Equation (42) with kinematic hardening is chosen. The integration process is stopped when the accuracy of etol
10–3 is reached. In Table 9.3, the different integration methods are compared with regard to their efficiency when applied to a structure (see also Section 9.3), where this was done at the level of the model. qg is the order of error (Equations (11) to (14)). The required computing time is related to the employment of a Pentium P90 computer. The v. Mises stress rv is taken from a point of maximum value. On the level of structural analyses, the Runge-Kutta methods proved to be more efficient than in comparison on the level of models (see Ta207

9

Development and Improvement of Unified Models

a) b) Figure 9.25: v. Mises rv stress fields for cyclic imposed displacements v as in Figure 9.24 a after 3. cycle, t = 90 s: a) model of kinematic hardening; b) model of 3. degree distortional hardening.

a) b) Figure 9.26: v. Mises rv stress fields for cyclic imposed displacements v and u as in Figure 9.24 b after 3. cycle, t = 90 s: a) model of kinematic hardening; b) model of 3. degree distortional hardening.

ble 9.1). The lowest time required is that of the Runge-Kutta version by Dormand-Prince. This is caused by the additional efforts needed for structures by expressing the load vector and for determining the stresses etc. This is proportional to the time steps needed. The predictor-corrector method by Heun requires more than twice the number of time steps as the two best methods although the higher order of error, qg
5, is connected with a 208

9.8

Structural Analyses

Table 9.3: Efficiency of different time integration methods (Section 9.3). Method

qg

etol [–]

CPU-time [s]

Time steps

rv,max [MPa]

P.-C. by Heun R.-K. 2./3. order R.-K. Fehlberg Dormand-Prince

2 3 5 5

10–3 10–3 10–3 10–3

2126 1652 1663 1571

28597 21446 16065 13090

431.6 431.6 431.6 431.4

larger amount of efforts within the evaluation of the material model. However, on the level of structural analyses, these additional efforts are relatively small. If the model of distortional hardening is applied in the comparative investigation then, the Runge-Kutta method by Dormand-Prince needs the following computer times: • •

model of distortional hardening of 2. degree (Equation (43)): 8201 s, model of distortional hardening of 3. degree (Equation (44)): 38421 s.

This shows that the distortional hardening of higher degree requires considerably more computer time because of much more unknowns in each Gaussian point to be evaluated.

9.8.5

Application of the statistical approach of Section 9.7

For the investigation of the applicability of material models covering the statistical distribution of test results, the strip with two circular holes and two half circle cuts in the centre line from the project C2 (Ritter and Friebe [53]) is taken (see Figure 9.27). The metal strip is 40 mm long, 13 mm broad and 3 mm thick. The material is that of Section 9.7. The structure is loaded by a longitudinal tension force of 37 kN under a temperature of 733 K. The Finite-Element method analyses took advantage of the double symmetry (see Figure 9.27) although this may not be allowed for a more consistent statistical scattering of deviations in the material properties. The investigated net consists of 168 isoparametric 8-nodes elements. For the statistical distribution, the results of Section 9.7 are applied, where the free parameters of the material model are expressed in a normal Gaussian distribution. In a first step, the structure is analysed by assuming homogeneous material all over the plate, where, for the material parameters, the values of the mean expectation le are selected as well as those of the band of distribution le ± re . The results of this homogeneous approach are needed to evaluate the differences to results when the material differs over the plate. In the second step, the material properties are inhomogeneously scattered over the plate. Because of experimental restrictions, the metal strip has the same extensions as the uniaxial specimen. Therefore, the subdivision in rectangular sections is not conve209

9

Development and Improvement of Unified Models

t [s] Figure 9.27: Displacement uj over time for different material distributions.

nient. Nevertheless, as an example, one quarter of the strip is divided into 10 × 3 rectangular sections (see Figure 9.28). To each of these sections, then a value of the cumulative distribution function corresponding to Figure 9.18 c is allotted by a random process. One of these distributions is presented in Figure 9.28 c. This random process is applied 25 times, which yields 25 different distributions of material properties over the plate. In Figure 9.27, the results are shown: the development of the edge displacement uj with time for different distributions of the material parameters. The uj for the 25 random distributions are rather close to the homogeneous case of mean values le as should be expected. The homogeneous cases for the le ± re values are, of course, upper and lower bounds relatively far away from the random cases. The coefficient of variance is 0.038 for homogeneous material and 0.010 for the 25 cases of different material in the 3 × 10 sectional rectangles. In order to demonstrate the local effects of statistical distributions, the computed strains ex are shown in Figure 9.28 for two cases: a) homogeneous material (le), b) scattered distribution corresponding to the cumulative distribution values in c). The distribution of ex along the plate does not differ very much between case a) and case b). In spite of large differences in c) (Figure 9.28 c), the same results can be drawn from Figure 9.29, where the strains ex along the section I-I (see Figure 9.27) are plotted for the different cases of material distribution. The conclusion of these investigations is that statistical scattering of the test results even of larger deviations in the material models is levelled out if this scattering is distributed over the entire structure. This is especially true for mechanical quantities, which follow from integrals like displacements. For local quantities as stresses, the statistical scattering may be more of relevance.

210

9.8

Structural Analyses

a) b) c) Figure 9.28: Strains ex [–] for t = 15 000 s: a) homogeneous (le); b) 3/10 areas of cumulative distribution values as in c).

Figure 9.29: Strains ex along section I-I (see Figure 9.27).

211

9

9.8.6

Development and Improvement of Unified Models

Numerical analysis for a recipient of a profile extrusion press

With this structural example, the applicability is proved of the material model based on the microphysical approach by Estrin [14] (see also Section 9.2.2) to structural analyses. Detailed results are given in [15, 16]. Moreover, the capacity and efficiency in structural applications are investigated for the following material models: microphysical model by Estrin, extended microphysical model by Kowalsky, model by Chaboche and Rousselier with an additional term for static recovery, Burgers model.

• • • •

Figure 9.30 shows the main geometrical parts of the extrusion press taken here as an example for application. The working temperature is 733 K. For the material, the ‘Warmarbeitsstahl’ X40CrMoV5 1 is chosen (see also Section 9.7). Recipients of rectangular cross-sections for receiving the hot aluminium billets are more prone to failure than those with circular cross sections. Therefore, alternative solutions are investigated, where, e. g., the inner part is composed of some independent segments. The structural discretization and the load-time sequences of the working processes are given in more details in [16], where also experimental results from project C2 [53] are documented, on which the application is based. The recipient is loaded by temperatures and by pressures. Some of the thermal actions are already imposed in the course of assembling the recipient. The intermediate ring is heated and then placed over the internal segments (see Figure 9.30). Cooling imposes prestressing. Then, this is repeated also for the outer cover ring. In the manufacturing periods, the entire recipient is loaded by the temperature of the billet and by the press cycles (see Figure 9.31).

Figure 9.30: Extrusion press for aluminium profiles. 212

9.8

Structural Analyses

Figure 9.31: Loading history by internal pressure.

Some of the results are presented in Figures 9.32 to 9.34. The radial displacements of the inner surface at the vertical axis of symmetry are plotted over time in Figure 9.32. The different models applied do not result in significant different displacements within the time of the assemblage. They are shrinking deformations caused by cooling. However, larger differences are calculated for the working cycles. Since the original model by Estrin covers only isotropic hardening, there is no incremental increase of the deformations during load reversal of the cycles. The Burgers model does not cover hardening. Therefore, the deformations increase linearly with each cycle. The extended microphysical model and the model by Chaboche and Rousselier lead to saturation of the deformations with increased numbers of cycles. In Figure 9.33, the accumulated inelastic strains after finishing the assemblage are shown for each of the models in one of the quadrants. They are especially large for the analyses assuming the model by Burgers. The other differences are small because the stresses are rather small in this load situation. The inner working pressure reverses the loading of the rings. Hence, different assumptions for hardening rules result in different stresses. In Figure 9.34, the maximum v. Mises rv-stresses are plotted during the fourth loading cycle. The maximum values are 996 MPa for the model by Chaboche and Rousselier and 1490 MPa for the model by Estrin. The latter one does not simulate inelastic stress release because the model does not include kinematic hardening.

s

Figure 9.32: Radial displacements uy. 213

9

Development and Improvement of Unified Models

Figure 9.33: Accumulated inelastic strains after assemblage for four models.

Figure 9.34: v. Mises stresses during the fourth loading cycle for four models.

Acknowledgements This final report draws conclusions and results from many individual papers and coworkers at the institute (see [9–12, 15–19, 22–24, 28, 31, 32, 46, 52]). The authors are indebted to them for these contributions. 214

References

References [1] J. L. Chaboche, G. Rousselier: On the Plastic and Viscoplastic Constitutive Equations. J. Pressure Vessel Technology (ASME) 105 (1983) 153–158, 159–164. [2] A. K. Miller: An Inelastic Constitutive Model for Monotonic, Cyclic and Creep Deformation. J. Engineering Materials Technology (ASME) 98 (1976) 97–105, 106–112. [3] E. W. Hart: Constitutive Relations for Nonelastic Deformation of Metals. J. of Engineering Materials and Technology (ASME) 98 (1976) 193–202. [4] Y. Estrin: Stoffgesetze der plastischen Verformung und Instabilita¨ten des plastischen Fließens. VDI-Verlag, Heft 642, Hamburg, 1987. [5] E. Steck, F. Thielecke, M. Lewerenz: Development and Application of Constitutive Models for the Plasticity of Metals. This book (Chapter 4). [6] O. T. Bruhns, P. S. White, J. L. Chaboche, J. V. D. Eikhoff: Constitutive Modelling in the Range of Inelastic Deformations. EUR 17999, 1988. [7] D. Nouhailhas: A Viscoplastic Modelling Applied to Stainless Steel Behavior. Proceedings of the Second International Conference on Constitutive Laws for Engineering Materials: Theory and Application, Tucson, Arizona, USA, 1987. [8] D. N. Robinson, P. A. Bartolotta: Viscoplastic Relationships with Dependence on Thermomechanical History. NASA CR-174836, 1985. [9] H. Braasch: Ein Konzept zur Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle. Bericht Nr. 92-71, 1992. [10] G. Kracht: Erschließung viskoplastischer Stoffmodelle fu¨r thermomechanische Strukturanalysen. Bericht Nr. 93-69, 1993. [11] M. Schwesig, H. Ahrens, H. Duddeck: Erfahrungen aus der Anwendung des inelastischen Stoffgesetzes nach Hart. Festschrift Schardt, Schriftenreihe Wissenschaft und Technik 51, TH Darmstadt, 1990, pp. 423–438. [12] M. Schwesig: Inelastisches Verhalten metallischer Werkstoffe bei ho¨heren Temperaturen – Numerik und Anwendung. Bericht Nr. 89-57, 1989. [13] Y. Estrin, L. P. Kubin: Local Strain Hardening and Nonuniformity of Plastic Deformation. Acta metallurgica 34(12) (1986) 2455–2464. [14] Y. Estrin, H. Mecking: Microstructural Aspects of Constitutive Modelling of Plastic Deformation. Elastic-Plastic Failure Modelling of Structures with Applications – PVP – Vol. 141, ASME, 1988. [15] U. Kowalsky: Mikrophysikalisch begru¨ndetes Werkstoffmodell zur Berechnung thermomechanisch beanspruchter Konstruktionen. Bericht Nr. 94-78, 1994. [16] U. Kowalsky, H. Ahrens: FE-Analysis of the Recipient of an Extrusion Press Applying Microstructure-Related Material Model. Computers & Structures 64 (1–4) (1997) 655–665. [17] U. Eggers: Verification of a Microstructure-Related Constitutive Model by Optimized Identification of Material Parameters. Low Cycle Fatigue and Elasto-Plastic Behavior of Matiersl – 3, Berlin, 1992, pp. 405–410. [18] E.-R. Tirpitz, M. Schwesig: A Unified Model Approach Combining Rate-Dependent and Rate-Independent Plasticity. Low Cycle Fatigue and Elasto-Plastic Behavior of Materials – 3, Berlin 1992, pp. 411–417. [19] E.-R. Tirpitz: Elastoplastische Erweiterung von viskoplastischen Stoffmodellen fu¨r Metalle – Theorie, Numerik und Anwendung. Bericht Nr. 92-70, 1992. [20] G. Engeln-Mu¨llges, F. Reuter: Numerische Mathematik fu¨r Ingenieure. B.I.-Wissenschaftsverlag, Mannheim, 1987. [21] P. Rentrop, K. Strehmel, R. Weiner: Ein Überblick u¨ber Einschrittverfahren zur numerischen Integration in der technischen Simulation. GAMM-Mitteilungen, Heft 1, 1996, pp. 9–43. [22] D. Dinkler, M. Schwesig: Numerische Lo¨sung von Anfangswertproblemen in der Statik und Dynamik. Festschrift Heinz Duddeck, Springer-Prod.-Ges. 1988, pp. 99–116. 215

9

Development and Improvement of Unified Models

[23] T. Streilein: Erfassung formativer Verfestigung in viskoplastischen Stoffmodellen. Bericht Nr. 97-83, 1997. [24] H. Braasch: Erfassung streuenden Materialverhaltens in Stoffmodellen. Bericht Nr. 93-75, 1993, pp. 1–14. [25] I. Rechenberg: Evolutionsstrategie – Optimierung technischer Systeme nach den Prinzipien der biologischen Evolution. Friedrich Frommann Verlag, Stuttgart – Bad Cannstatt, 1972. [26] H.-P. Schwefel: Evolutionsstrategie und numerische Optimierung. Dissertation, TU Berlin, 1975. [27] D. Mu¨ller, G. Hartmann: Identification of Materials Parameters for Inelastic Constitutive Models Using Principles of Biologic Evolution. J. of Engineering Materials and Technology (ASME) 111 (1989) 299–305. [28] M. Bergmann: Lo¨sung von Optimierungsproblemen mit parallelisierten Evolutionsalgorithmen. Bericht Nr. 93-75, 1993, pp. 15–28. [29] J. A. Nelder, R. Meat: A Simplex Method for Function Minimization. Computer Journal 7 (1965) 308–313. [30] R. Mahnken: Duale Methoden fu¨r nichtlineare Optimierungsprobleme in der Strukturmechanik. Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universita¨t Hannover, F 92/3, 1992. [31] H. Braasch, H. Duddeck, H. Ahrens: A New Approach to Improve Material Models. J. Engng. Mat. Technol. (ASME) 117 (1995) 14–19. [32] H. Braasch: Concept to Improve the Approximation of Material Functions in Unified Models. Low Cycle Fatigue and Elasto-Plastic Behavior of Materials – 3, Berlin, 1992, pp. 405–410. [33] A. Styczynski: Experiments on pure Aluminium. Unpublished data. [34] W. Gieseke, K. R. Hillert, G. Lange: Material State after Uni- and Biaxial Cyclic Deformation. This book (Chapter 2). [35] U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading. This book (Chapter 10). [36] A. Phillips, J. L. Tang: The Effect of Loading Path on the Yield Surface at Elevated Temperatures. Int. J. Solids Struct. 8 (1972) 463–474. [37] K. Wegener: Zur Berechnung großer plastischer Deformationen mit einem Stoffgesetz vom Überspannungstyp. Braunschweiger Schriften zur Mechanik, Nr. 2-1991. [38] M. Sayir: Zur Fließbedingung der Plastizita¨tstheorie. Ingenieur-Archiv 39 (1970) 414–432. [39] F. Edelmann, D. C. Drucker: Some Extensions of Elementary Plasticity Theory. J. Franklin Institute 251 (1951) 581–605. [40] D. W. Rees: Yield Functions that Account for the Effects of Initial and Subsequent Plastic Anisotropy. Acta Mechanica 43 (1982) 223–241. [41] J. Betten: Plastische Anisotropie und Bauschinger-Effekt; allgemeine Formulierung und Vergleich mit experimentell ermittelten Fließortkurven. Acta Mechanica 25 (1994) 79–94. [42] T. Lehmann: Anisotrope plastische Forma¨nderungen. Rheol. Acta, Darmstadt 3 (1964) 281–285. [43] S. Dannemeyer: Zum Einfluß der Prozeßparameter bei der experimentellen Ermittlung von Fließfla¨chen. Dissertation aus dem Institut fu¨r Stahlbau der TU Braunschweig, 1995. [44] M. Reininghaus: Baustahl St52 unter zweiachsiger plastischer Wechselbeanspruchung. Dissertation aus dem Institut fu¨r Stahlbau der TU Braunschweig, 1994. [45] W. Gieseke: Fließfla¨chen und Versetzungsstrukturen metallischer Werkstoffe nach plastischer Wechselbeanspruchung. Dissertation aus dem Institut fu¨r Werkstoffe der TU Braunschweig, 1995. [46] T. Streilein: Viskoplastische Werkstoffmodelle – formatives Verfestigungsverhalten bei einund mehraxialer Beanspruchung. Numerische Methoden der Plastomechanik, Tagungsband, Institut fu¨r Mechanik der Universita¨t Hannover, 1995, pp. 307–321.

216

References [47] K.-T. Rie, H. Wittke, J. Olfe: Plasticity of Metals and Life Prediction in the Range of LowCycle Fatigue: Description of Deformation Behaviour and Creep-Fatigue Interaction. This book (Chapter 3). [48] L. Sachs: Angewandte Statistik. Springer-Verlag, Berlin, 1992. [49] O. C. Zienkiewicz: The Finite Element Method. 3rd. edn., McGraw-Hill Book Company, London, 1977. [50] H. Ahrens, D. Dinkler: Finite-Element-Methoden. Teil I. Bericht Nr. 88-50, 1994, Teil II. Bericht Nr. 88-51, 1996. [51] H. Wohlfahrt, D. Brinkmann: Consideration of Inhomogeneities in the Application of Deformation Models, Describing the Inelastic Behaviour of Welded Joints. This book (Chapter 16). [52] M. Schwesig, H. Braasch, G. Kracht, H. Duddeck, H. Ahrens: Erfahrungen aus der Anwendung inelastischer Stoffgesetze bei ho¨heren Temperaturen. In: D. Besdo (Ed.): Numerische Methoden der Plastomechanik, Tagungsband, Hannover, 1989. [53] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical Measuring Methods. This book (Chapter 13). The publications [9–12, 15–19, 22–24, 28, 31, 32, 46, 50, 52] resulted from the Institut fu¨r Statik, TU Braunschweig.

217

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)

10

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

10.1

Introduction

The aim of this project is to develop a material model for the prediction of the material behaviour of mild steel Fe 510 under multiaxial cyclic plastic loading. First of all, detailed information about the material response under cyclic plastic loading are necessary. Therefore, extensive experimental investigations are made including uniaxial single- and multiple-step tests and biaxial tension-torsion tests with various prestrains, increasing or decreasing strain amplitudes, and several proportional and non-proportional biaxial loading paths, respectively. Cyclic hardening or softening in the uniaxial case, or additional hardening under non-proportional loading are some of the observed effects. To describe the material peculiarities, the two-surface model of Dafalias-Popov has been modified. The improvements result in a rate-independent, isothermal two-surface model, which is presented here. Experimental data from the uniaxial and biaxial tests, and, in addition, from tests on structural components are compared with corresponding calculations made with the new model. The results demonstrate the capabilities of the extended-two-surface model to predict the behaviour of mild steel and steel constructions under multiaxial cyclic plastic loading.

* Technische Universita¨t Braunschweig, Institut fu¨r Stahlbau, Beethovenstraße 51, D-38106 Braunschweig, Germany 218

10.2

Material Behaviour

10.2

Material Behaviour

10.2.1

Material, experimental set-ups, and techniques

The investigated material was mild steel Fe 510. The chemical and mechanical characteristics of the different heats used in the investigations can be found in the corresponding papers (Scheibe [1], Reininghaus [2]). Two types of specimens were used in the investigations. Figures 10.1 and 10.2 show sketches of both the cylindrical specimens used in the experiments with uniaxial loading, and the tubular specimens for the biaxial investigations. Different servohydraulic testing machines were used in the uniaxial investigations. The biaxial investigations were performed on a 160 kN-tension-compression “SCHENK” universal testing machine, extended with a 1000 Nm-torsion drive and an extensometer.

10.2.2

Material behaviour under uniaxial cyclic loading

10.2.2.1

Parameters

In the strain-controlled experiments, the strain rate
_ was chosen for values between 3.5 and 24‰/min. For the force-controlled experiments, the loading rate was 3 kN/s. All

Figure 10.1: Cylindrical specimen for uniaxial experiments.

Figure 10.2: Tubular-type specimen for biaxial experiments. 219

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.3: Varied parameters of the uniaxial experiments.

tests were performed at room temperature. The varied parameters in the uniaxial loaded tests were the strain amplitude
a (2, 3, 5, 8, and 12‰), the mean strain
m (–40, –25, 0, 10, 25, and 40‰), and the strain history (see Figure 10.3 for explanation). The uniaxial tests were performed paying particular attention to effects of the sequence of loadings, the evolution of cyclic hardening and softening, the relaxation of mean stresses, and the size of the elastic region. To investigate the ratchetting effects, stress-controlled experiments were performed varying the initial strain
m, the stress amplitude ra, the mean stress rm, and the stress ratio R
ru/ro. 10.2.2.2

Results of the uniaxial experiments

Strain-controlled experiments Typical results of the different multiple-step tests without mean strain are shown in Figures 10.4 and 10.5. In these figures, the courses of the half-range of stresses vs. the number of halfcycles are shown. In addition, the strain vs. time is plotted for explanation in the upper left part of the diagram. The resulting stress vs. strain is given in the upper right part.

Figure 10.4: Multiple-step test (MST), ea
12-8-5-3-2‰. 220

10.2

Material Behaviour

Figure 10.5: Multiple-step test (RFE), ea ˆ 2-12-2‰.

After passing the linear-elastic region, the distinct yield point and the yield plateau, mild steel Fe 510 shows the well known Bauschinger effect if the specimen is unloaded. During the cyclic loading, the maximum stresses of a hysteresis-loop are not constant: For amplitudes ea smaller than 5‰, the maximum stresses decrease from cycle to cycle, and a saturated state is reached after 500 or more cycles (cyclic softening). Amplitudes ea higher than 5‰ cause an increase of maximum stresses during the first 40 cycles (cyclic hardening). Note that cyclic softening or hardening is understood as a decrease or an increase of the stress amplitude in comparison with the stress level of the monotonic stress-strain curve at this strain level. For ea  12‰, the stress level of the monotonic stress-strain curve is almost constant (with r ˆ rF). Therefore, cyclic hardening or softening can be observed easily. Stress levels Dr/2 (for symmetric amplitudes) greater than rF are seen by definition as cyclic hardening, and stress levels Dr/2 lower than rF as cyclic softening. A cyclic stress-strain curve is achieved by plotting the corresponding stress ranges Dr/2 at the stabilized states vs. the corresponding strain amplitudes ea. Figure 10.6 shows a typical monotonic stress-strain curve along with two cyclic stress-strain curves. The intersection of the curves at a strain of 5‰ characterizes the border between cyclic softening and cyclic hardening.

Figure 10.6: Cyclic stress-strain curves. 221

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading The cyclic stress-strain curves obtained after 40 cycles and the one after 500 cycles differ if smaller amplitudes are used. This shows that a saturated state is not reached after 40 cycles at all. With higher amplitudes, the two curves tend to become identical: The steady state is reached during the first 40 cycles. However, after prestraining with a higher amplitude (e.g. path MST, Figure 10.4), the stabilized state is already reached after a few cycles. Figure 10.7 shows a multiple-step test (MST) with a mean strain of em
+ 40‰. The maximum stress at 52‰ (or the maximum stress level Dr/2) decreases within the first cycles. Note that this transient behaviour is not a cyclic softening according to the above definition because, during the first loops of the amplitude ea
12‰, only the mean stress caused by the mean strain returns to zero. Parallel to this so-called mean-stress relaxation, a cyclic hardening of the ea
12‰ amplitude takes place, but due to the dominance of the mean-stress relaxation, only a softening of the stress level Dr/2 can be observed. A comparison of the stabilized loops for amplitudes ea
12‰ with different mean strains shows that the different mean strains do not significantly influence the shape or the maximal stress amplitudes of the hysteresis-loops. Figure 10.8 shows the cyclic stress-strain curves for different mean strains. It is seen that the maximum stress amplitude Dr/2 depends only on the strain amplitude ea. This result is the basis for the above definition of cyclic hardening or cyclic softening under a given strain amplitude. Another important point besides the effect of cyclic hardening is the influence of the cyclic loading on the size of the elastic region. The elastic regions were determined using an offset-proof-strain method (see Scheibe [1] for further explanation) with a proof strain of 0.03‰ and a constant elastic modulus E0 of 206 000 MPa. Figure 10.9 deals with the evolution of the elastic domain during uniaxial cyclic loading. The diagram shows that the size of the elastic domain k is a function of the maximum strain amplitude emax. If the former is greater than 5‰, the size of the elastic region is reduced to the value ks&100 MPa. For strain amplitudes lower than 5‰, the elastic region decreases very slowly (ea
2‰ in Figure 10.9). The dashed line of the amplitude ea
2‰ shows that the size of the elastic region tends to reach the same value as that of the greater amplitudes within some hundred cycles.

Figure 10.7: Multiple-step test (MST) 12-8-5-3-2‰, with a mean strain of 40‰. 222

10.2

Material Behaviour

Figure 10.8: Cyclic stress-strain curve for different mean strains.

Figure 10.9: The evolution of the elastic region under cyclic loading.

Stress-controlled experiments As a result of the stress-controlled experiments concerning the ratchetting effect (Kuck [3]), different variations of the ratchetting effect were found (Figure 10.10): a) b) c) d)

quick saturation, especially with small ra and rm, positive increase of strain without saturation, negative increase of strain with gradual saturation (at a large number of cycles), increase of strain with gradual saturation, reversal, and then constant increase of strain.

In addition, the influence of the mentioned cross-section of the specimen and with it the kind of stress (initial cross-section: technical stress, and actual cross-section: effective stress) on the number of cycles needed to reach the saturated state, was investigated. The tests show an increase of ratchetting with decreasing minimum stress at constant maximum stress (Figure 10.11), and, in addition, a distinct mean stress (34.5 MPa), where no ratchetting is found (Figure 10.12). The amount of the occurred ratchetting depends on the difference between the actual mean stress and the mean stress, which leads to zero ratchetting (Kuck [3]). 223

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.10: Different appearances of the ratchetting effect.

Figure 10.11: Influence of R
ru =ro :

224

10.2

Material Behaviour

Figure 10.12: Influence of the mean stress.

10.2.3

Material behaviour under biaxial cyclic loading

10.2.3.1

Parameters

For the biaxial tests, the loading path (Figure 10.13), the sequence of different loading paths, the loading intensity eB (defined by Equation (1)), and the ratio D between tensile and torsional loads (defined by Equation (2)) were varied. Note that eB is not a mechanical derived equation, it is used only to compare biaxial and uniaxial loads here. For the experimental investigations, different loading intensities eB between 1.90‰ and 7.10‰ were chosen. 
















1 eB
e2a ‡ c2a ; 3 p


ea Dˆ 3 : ca

…1† …2†

The achieved stress curves and the calculated uniaxial equivalent stresses (based on the theory of v. Mises) show the influences of various sequences of loading paths or the additional-hardening effect. Moreover, the yield-surface investigations give additional insight into the evolution of the elastic region under complex loadings. Here, the influence of the intensity eB and the loading path on the location, shape, and size of the measured yield surface were investigated. Additional experiments were made to determine the influence of several parameters coming out of the process of yield point probing itself, i.e. technique and sequence of yield point probing, choice of the starting point, or fixing of the elastic modulus.

225

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.13: Biaxial loading paths.

10.2.3.2

Relations of tensile and torsional stresses

Figure 10.14 gives some characteristic results of the evolution of the tensile and torsional stress in the transient state of biaxial cyclic-loaded mild steel. The evolution of the stresses of the biaxial proportional and non-proportional tests differs with increasing intensities. For a proportional loading, only the maximum stresses increase and the stress curve grows without changing its shape (Figure 10.14 a).

" Figure 10.14 a)–h): Tensile and torsional stresses under biaxial proportional and non-proportional loadings. 226

10.2

Material Behaviour

a) Path 03; D
1:0

b) Path 07; D
1:0

c) Path 10; D
1:0

d) Path 09; D
1:0

e) Path 09; D
1:0

f) Path 09; D
2:4

g) Path 08; D
1:0

h) Path 08; D
1:0

227

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

i) Path 08; D
1:0

j) Step 1: Path 03

k) Step 2: Path 07

l) Step 3: Path 03

Figure 10.14 i)–l): Tensile and torsional stresses under biaxial proportional and non-proportional loadings.

On the other hand, for some of the non-proportional loadings, a significant change in the shape of the stress course in addition to the increase of the maximum stresses can be found (Figure 10.14 b, g–i). Note that the loading path and the relation D between ea and ca does not change. The results of a varying D can be seen in Figure 10.14 d–f. In Figure 10.15, the uniaxial equivalent stresses rv are plotted vs. the maximum equivalent strain or intensity eB. In these experiments, the cyclic hardening in the uniaxial cyclic stress-strain curve starts at about 3‰. The difference between the cyclic stress-strain curve of the uniaxial experiments described in Section 10.2, and the curve seen in Figure 10.15, depends on the different heats of Fe 510 used in the two investigations. Comparing the cyclic stress-strain curve with the equivalent stresses of the proportional loadings, no significant difference in the maximum stresses between the uniaxial and the biaxial proportional loadings can be found. A significant difference between the cyclic stress-strain curve and the equivalent stresses is found however for non-proportional loading paths. This effect is named additional hardening here. The additional hardening appears to be strongly dependent on the type of nonproportional loading. In general, the additional hardening increases by an increasing phase angle u (Figure 10.13). For higehr values of u (between 60 and 90 degrees, path 06 and 07), the additional hardening is almost constant. The amount of additional hardening of path 10 corresponds to that of path 07. The highest amount of additional hard228

10.2

Material Behaviour

Figure 10.15: Uniaxial equivalent stresses of biaxial loading paths.

ening is found in experiments with the strain path 09 (“Butterfly”). Here, the equivalent stress in the saturated state reaches the level of the uniaxial tensile strength. The additional hardening fades when the non-proportionality of the loading has decreased. Figure 10.14 j–l shows the results of an experiment, where a specimen was first uniaxially loaded (ea
6‰) up to a saturated state, then underwent a non-proportional loading path 07 (eB of 8‰), and finally was uniaxially loaded again. The cyclic hardening during the first loading stage is followed by an additional hardening in the second stage. During the third loading, the additional hardening fades so that the saturated stress-strain curves of the first and the third loading level are nearly identical.

10.2.3.3

Yield-surface investigations

Yield surfaces were investigated in the transient and in the saturated state of the material behaviour. To reduce time-consumptional manual controlling, a computer program was developed, which allows yield surfaces at any point of any complex loadings to be established automatically. This program and the experimental set-up is described comprehensively by Dannemeyer [4]. An offset-proof-strain method was used to define the onset of plastic flow. For an increasing offset-proof strain, the yield surfaces of mild steel Fe 510 in saturation show the same effects as described for other materials (e.g. Michno and Findley [5]). The distinct corner in preloading direction and the flattening opposite to it tend to blur, and the size of the surface increases, mainly against the preloading direction (Figure 10.16). The Bauschinger effect is responsible for this uneven expansion when the offset value increases. An offset-proof strain of 0.05‰ was used for the yield-surface investigations in the saturated state. To determine yield surfaces in the transient state of the material behaviour, the offset-proof strain was reduced to 0.03‰. A further decrease of the offsetproof strain was restricted by the precision of the technical set-up. In this investigation, several yield-surface determinations were performed on a single specimen. 16 yield probings with different ratios of sizes of tensile and torsional strain increments are combined to build up a yield surface. To investigate the influence 229

p


3 s

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.16: Effect of various offset-proof strains on the yield surface of mild steel Fe 510.

of the sequence of the different yield-probing directions, yield surfaces were determined using two different sequences (Figure 10.17). For materials, which reach a cyclically stable state during cyclic loading, the influence of the probing sequence can be excluded if the specimen is loaded with several cycles between two yield probings. The small hardening effects drawn by the previous yield probing disappear completely, and a similar loading state can be reached at the beginning of every new yield probing (Figure 10.18). This allows yield surfaces to be investigated with negligible dependence on the probing path (Figure 10.19). (The turnover points and the first yield points are marked with arrows here. The loading path for each of the yield surfaces is given by symbols.) This method is called the single-point technique (SPT) here, in contrast to the multiple-point technique (MPT), where several yield points are investigated one after another, each time unloading to a starting point located somewhere within the yield surface.

Figure 10.17: Sequences of yield point probings. 230

10.2

Material Behaviour

p


3 s

Figure 10.18: Single-point technique.

Figure 10.19: Yield surfaces determined with the single-point technique and current elastic moduli under proportional and non-proportional loading paths.

231

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

p


3 s

The influence of the probing sequence on a yield surface determined with the multiple-probing technique is systematic by nature (Figure 10.20). So, the measured yield surface can be corrected, at least in quality, if a surface investigated with the single-probing technique is used as a reference. Another effect, which influences a yield-point determination on mild steel Fe 510, should be described here. In Figure 10.21, the stress-strain curve of the initial loading and the elastic region after the turnover of the 20th cycle of a specimen under uniaxial tension-compression load is plotted. In order to compare the two gradients, the turnover point of the 20th cycle is moved to the origin. It is seen that the slope of the initial loading curve is nearly constant over the plotted range of 1.3‰. After being loaded 20 times into the plastic range, a distinct proportional region is missing. This non-linearity of the stress-strain curve under preloading affects the yield-surface determination in different ways. In general, for the determination of a yield locus, it is necessary to start the yield-point determinations somewhere in the elastic region, if possible in the centre of the elastic region to secure an almost rectangular touch of the elastic-plastic border. Due to the non-linear area after a turnover point, the unloading strain es becomes a parameter of the size of the established yield surface. The results of an experiment, shown in Figure 10.22, demonstrate the effect of various unloading strains on the yield surface of a uniaxial cyclic tension-compressionloaded specimen. It is obviously that the expansion of the yield surface against the preloading direction depends on the amount of the unloading strain. The diameter of the yield surface rectangular to the preloading direction is not affected, however. After preloading, the lack of a distinct area of proportionality influences the yieldsurface determination not only in the setting of a starting point but also in the calculation of plastic strains. Using an offset-proof-strain definition in combination with an automa-

Figure 10.20: Yield surfaces determined with the multiple-point technique and current elastic moduli under proportional and non-proportional loading paths. 232

10.2

Material Behaviour

p


3 s

Figure 10.21: Sections of the stress-strain curve of mild steel Fe 510 under uniaxial cyclic tension-compression loading.

Figure 10.22: The effect of various unloading strains es on yield loci under cyclic uniaxial tension-compression loading.

tized experimental procedure, a continuous separation of elastic and plastic strains has to be carried out on-line during a yield probing. The elastic moduli (E and G) can be set either to constant values for the whole experiment, or they can be determined from the gradient of a defined number of data considered to be linear-elastic. If the current elastic moduli change during a cyclic loading and constant moduli are considered in the on-line calculation of plastic strains, the resulting yield point does not correspond to the existing yield point. Figure 10.23 shows the results of yield-surface determinations with fixed and current elastic characteristics. In comparison to the Figures 10.19 and 10.20, it can be seen that the size of the yield surface depends strongly on how the elastic moduli are given. 233

p


3 s

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.23: Yield surfaces investigated with constant elastic moduli under proportional and non-proportional loading paths.

The SPT-loci with constant elastic moduli show a corner and a distinct flattening, whereas the MPT-loci of the same type are smoother and have increased in size. If non-proportional loading paths are used, the difference in size of the SPT- and MPTloci is at its greatest (Figure 10.23, upper left yield surface). In the transient state of material behaviour, yield-surface investigations are more sensitive than in the saturated state. Even in the area of the yield plateau, a yield-point determination with its incursion into the plastic region influences strongly the later yield-point probes. In general, the intensive hardening during the first few cycles excludes the use of the single-point technique in the transient state. The multiple-point technique and a further decrease of the offset-proof strain to smallest values seem to help getting realistic results for mild steel at this stage of a cyclic loading. When a new specimen is first loaded, the yield surface contracts and starts to move in the stress space immediately after p


the elastic limit is passed. In addition, a distortion of the initial round yield surface (r- 3s-space) occurs. The pronounced isotropic softening is strongest during the first cycle, and in the case of a proportional load, it is usually completed after a few cycles depending on the load intensity (see also Figure 10.9). If the load is of a non-proportional type, the shrinkage of the yield surface is almost completed at the beginning of the second cycle (Figure 10.24). This isotropic softening is always connected with a kinematic hardening. In further states of the cyclic loading, the yield surface continues to move to higher stresses after the shrinkage is already completed. In Figure 10.24, the results of an experiment in the transient state are presented. During the first cycle, the yield surface decreases at constant uniaxial equivalent stresses rv, and from the second to the fifth cycle, a distinct increase of the maximum stresses obtained by a movement of the yield surface takes place. It can be stated that the additional hardening of a non-proportional load is also a type of kinematic hardening. 234

Material Behaviour

p


3 s

10.2

Figure 10.24: Yield surfaces of the first, second and fifth cycle of a non-proportional cyclic loaded specimen in the transient state.

This distortion of the yield surface (formative hardening), including the deviation of the initial round shape by forming a corner in preloading direction and a flat side at the opposite, is the third mechanism of hardening besides the isotropic and kinematic hardening found on cyclic loaded mild steel Fe 510. In general, the shape of a preloaded yield surface depends on the type of load. All investigated proportional loading paths of the same intensity show yield surfaces, which are nearly identical in size and shape. They show a distinct corner in preloading direction and a flattening opposite to it. In addition, the two diameters of the yield surface change differently during a proportional load. In the saturated state, the diameter in direction of the preloading (d1, Figure 10.25) has shrunk to 40% of the initial value, whereas the second diameter rectangular to the preloading direction (d2) decreases only to 70% as seen in Figure 10.19, for example. A non-proportional load forms a more

Figure 10.25: Definition of diameters of a yield surface. 235

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading rounded and smoother shape, and the diameters decrease more homogeneously because of the changes in the direction of the load increments. An approximate alignment of the yield surface with the corner and the flattening towards the direction of the stress increment can be found only at proportional loading paths. At some of the non-proportional loading paths, the direction of the stress increment at the turnover point differs distinctly from the direction of the suggested axis of symmetry of the yield surface (see Figure 10.19). Additional yield-surface investigations were performed for the subproject A10 (Prof. Besdo [6]) on torsional preloaded AlMg3.

10.3

Modelling of the Material Behaviour of Mild Steel Fe 510

10.3.1

Extended-two-surface model

10.3.1.1

General description

Extensive investigations were made into the suitability of several models to describe the characteristical effects of the material behaviour of cyclic loaded mild steel Fe 510 (Heuer [7], Scheibe [1]). Based on these investigations, it can be concluded that the two-surface model of Dafalias and Popov [8] represents a suitable basis for further developments. The extensions of the original two-surface model are presented here. The new extended-two-surface model (ETS-model) is described completely by Scheibe [1], Reininghaus [2], Scheer et al. [9], Peil and Kuck [10], Peil and Reininghaus [11], and Reininghaus [12]. Here, only the fundamental description of the model is given. The fundamental parts of this model are: • • • •

three memory surfaces in the strain space, the consideration of the additional-hardening effect based on experimental findings to describe non-proportional loadings and the consideration of a softening of the loading surface, and an isotropic hardening of the bounding surface.

One important aspect of the presented model is that the material or model parameters, which are determined once for the mild steel Fe 510, are fixed for this material. All calculations shown in this paper were carried out with the same set of parameters. There was no extra fitting necessary for any special kind of loading path or calculation.

236

10.3 10.3.1.2

Modelling of the Material Behaviour of Mild Steel Fe 510

Loading and bounding surface

The original two-surface model of Dafalias and Popov assumes that the yield surface or loading surface and the memory or bounding surface (Figure 10.26) harden kinematically and isotropically, while the two surfaces are in contact. If there is no contact between the two surfaces, the memory surface remains unchanged, while the loading surface moves according to the Mro´z rule [13]. This rule secures a tangential contact without any intersection if the two surfaces come into contact. Both the loading surface and the bounding surface are assumed as hyperspheres in the deviatoric stress space and are represented as: 1 F
…r D 2 ~

 †…rD ~ ~

† ~

1 2 k ˆ 0; 3

…3†

1 F^ ˆ …rD 2 ~

^†…rD ~ ~

^† ~

1 ^2 k ˆ0 3

…4†

with k radius of the yield or loading surface, ^k radius of the bounding surface,  centre of the loading surface (kinematic hardening), ~^ centre of the bounding surface, ~rD deviatoric stress tensor. ~ In this basic formulation, the model has some disadvantages: • • •

the yield plateau of mild steel cannot be predicted, there is no possibility to distinguish between monotonic and cyclic loadings, and an update problem of the variable din (overshooting problem) occurs.

In the ETS-model, which is described here, the loading surface remains unchanged: The surface can contract or expand, move, but not be distorted. In contrast to the origi-

Figure 10.26: Loading and bounding surface. 237

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading nal model of Dafalias and Popov, the bounding surface in the ETS-model is only able to harden or soften isotropically. Furthermore, two bounding surfaces instead of one are implemented. The inner one corresponds to the original bounding surface in the model of Dafalias and Popov. The outer one is used as a control surface and is activated only under non-proportional loadings. The plastic modulus in the ETS-model is calculated from Equation (5): P
P^0 ‡ h

d din

…5†

d

with d din P^0 h

distance between the actual stress point and the bounding surface (see Figure 10.26), distance between the yield point and the bounding surface (see Figure 10.26), plastic modulus of the strain hardening region, shape parameter.

The plastic modulus P is split up into a modulus for the kinematic hardening Pa and the isotropic hardening Pk : P ˆ Pa ‡ Pk :

…6†

Pk is calculated from the formulation of the isotropic hardening or softening of the yield surface. Pk is a function of the size of the new implemented memory surface in the strain space. The plastic modulus Pa for the kinematic hardening is determined from the difference between the total plastic modulus P and the plastic modulus for the isotropic hardening Pk . During the development of the model, different rules for the kinematic hardening of the surface were tested (Reininghaus [2]). The best results are obtained with the original kinematic hardening rule of Mro´z [13], therefore all results presented here are calculated with this rule.

10.3.1.3

Strain-memory surfaces

The strain-memory surfaces in the strain space allow the monotonic and cyclic material behaviour to be taken into account.

•

Strain memory Mm for monotonic behaviour

The size qm of this memory surface can be understood as the maximum plastic strain amplitude during the whole cyclic loading: 1 Mm ˆ …ep 2 ~ 238

b m †…ep ~ ~

bm† ~

3 2 q ˆ 0; 4 m

…7†

10.3

qm




Modelling of the Material Behaviour of Mild Steel Fe 510

…8†

dqm ;

1 dqm ˆ Hm nn mdevp ; 2 ~~  bm ˆ dbm ; ~ ~

…9† …10†

1 d b m ˆ Hm dep 2 ~ ~

…11†

with

n ~n m ~ep ~b m ~devp qm

•

Hm ˆ 0

for



Hm ˆ 1

for

Mm ˆ 0

Mm  0 Mm ˆ 0

and and

nn m < 0; ~~ nn m  0; ~~

…12† …13†

normal to the loading surface, normal to the monotonic strain-memory surface, plastic strain tensor, centre of the monotonic strain-memory surface, equivalent plastic strain increment, size of the monotonic strain-memory surface.

Strain memory Ms for cyclic behaviour

The increase of size qs of the memory surface Ms during a cycle only depends on an additional factor (cs in Equations (15) and (16)). The change in the size of this memory surface describes cyclic hardening or softening: 1 Ms ˆ …ep 2 ~ dqs ˆ …1

b s †…ep ~ ~

bs ˆ ~

dbs ~

3 2 q ˆ 0; 4 s

Hm †Hs cs nns devp ; ~~

d b s ˆ ‰…1 Hm †Hs …1 ~  qs ˆ dqs ; 

bs† ~

cs † ‡ Hm Šdep ; ~

…14† …15† …16† …17† …18† 239

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading with (

Ms  0 Ms ˆ 0

Hs
0

for

Hs ˆ 1

for Ms ˆ 0

and

nns < 0 ; ~~ nns  0 ; ~~

and

ns ~b s ~qs cs

normal to the saturated strain-memory surface, centre of the saturated strain-memory surface, size of the saturated strain-memory surface, factor for the isotropic hardening per cycle.

•

Strain memory Ma for the actual loading direction

…19† …20†

The memory surface Ma for the actual loading direction will decrease to zero when the angle between the normal vector of the loading surface and the normal vector of the memory surface Ma exceed 908: 1 Ma ˆ …ep 2 ~ dqa ˆ

b a †…ep ~ ~

ba† ~

3 2 q ˆ 0; 4 a

1  nna devp ; 2 ~~

…22†

1 d b a ˆ dep ; 2 ~ ~  qa ˆ dqa ;

…23† …24†

Ha † e p ‡ ~

b a ˆ …1 ~

…21†



…25†

dba ~

with Ha ˆ



1

for

0

for

n na < 0 ~~ n na  0 ~~

;

na normal to the actual strain-memory surface, ~b centre of the actual strain-memory surface, a ~qa size of the actual strain-memory surface.

240

…26†

10.3 10.3.1.4

Modelling of the Material Behaviour of Mild Steel Fe 510

Internal variables for the description of non-proportional loading

The variable Z is used to distinguish between a proportional and a non-proportional loading within the current loading increment. Z is defined as: drD Z
1  ~D dr ~ Zˆ0

rD ~ r D

for

~

drD  0 ; ~

…27†

for jdrD j ˆ 0 : ~

…28†

The internal variables FS and FL are functions of Z and devp . In the case of proportional loading, FS and FL are assigned to zero. During non-proportional loading, both variables rise to the value one. During the process, FL has a temporal delay to FS . Effects, which occur immediately with the set-in of a non-proportional loading, are controlled by the variable FS , and those processes, which occur slowly during a non-proportional loading, are controlled by FL. If the non-proportional loading is followed by a proportional one, both internal variables decrease to zero again to simulate the erasure of the additional hardening found in the experiments (see Section 10.2.3.2):  …29† FS ˆ dFS ; dFS ˆ W2 …Z with





Z  FS ; Z < FS :

…31†

dFL ;

dFL ˆ W3 …Z with

…30†

W2 ˆ 0:1 tanh…qm =q† for for W2 ˆ 0:01

FL ˆ



FS †devp

W3 ˆ …1 W3 ˆ FL

…1 FL †0:1

…32†

cos 308††devp

for for

Z1 Z dlim, this additional hardening of the bounding surface decreases. The plastic modulus of the additional isotropic deformation of the bounding surface is given by:  …FB;max FB † 0:2 ^ PZ ˆ P FS FB;max P^Z ˆ

P

FB …0:001 ‡ FS † FB;max

for

d  dlim ;

for d > dlim

…48† …49†

with FB;max ˆ …D^k1 FS ‡ FD 2:0†FL ;

…50†

Figure 10.27: Multiple-step test. 245

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.28: Path 03, eB
7.1‰.

Figure 10.29: Path 07, eB
2.8‰.

Figure 10.30: Path 07, eB
4.9‰.

246

10.3

Modelling of the Material Behaviour of Mild Steel Fe 510

Figure 10.31: Path 08, eB
4.9‰.

Figure 10.32: Path 09, eB
2.8‰.

Figure 10.33: Path 09, eB
4.9‰.

247

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

FB




P^Z devp ;

dlim ˆ 1:0 …D^k1 FS0:2 ‡ FD 2:0† ; P^Z D^k FB

…51† …52†

plastic modulus for the additional isotropic hardening of the bounding surface due to non-proportional loading, material parameter, additional isotropic hardening of the bounding surface due to non-proportional loading.

The additional hardening resulting from a non-proportional loading FB (Equation (51)) is added to the size of the bounding surface ^k (Equations (34) to (36)) regardless of the amount of qm and qs .

10.3.2

Comparison between theory and experiments

Figures 10.27 to 10.33 show the experimental results in the left column and the results of the calculations with the ETS-model in the right column. Note that all calculations (uniaxial, proportional and non-proportional) are made with the same set of parameters. It can be seen from these figures that the response of mild steel Fe 510 under uniaxial, proportional and non-proportional loading histories is well predicted by the model.

10.4

Experiments on Structural Components

10.4.1

Experimental set-ups and computational method

Calculations using several models were made on typical components of steel constructions like necked girders, girders with holes, or plates with holes. Figures 10.34 to 10.36 show specimens used in these investigations. All experiments were performed force-controlled. The longitudinal strains in the interesting sections were measured with strain gauges. For the description of the structural behaviour, the Finite-Element method was used. Precise informations concerning details of the computational methods can be found in the corresponding papers [1, 2, 9, 14].

10.4.2

Correlation between experimental and theoretical results

First, the results of an experiment with a necked girder and the corresponding calculations are presented. The cyclically loaded girder (Figure 10.35) shows repeating plastic deformations in the area of the neck. 248

10.4

Experiments on Structural Components

Figure 10.34: Plate with a hole.

Figure 10.35: Necked girder. 249

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading

Figure 10.36: Girder with holes.

Figure 10.37: Force-strain diagrams of a necked girder (experiment and calculation).

In the right diagram of Figure 10.37, the force-strain relation measured directly at the edge of the neck is presented. The force-strain relation predicted by the ETS-model is plotted in the left diagram of this figure. The calculated force-strain relation shows a good correspondence with the results of the experiment. The amount of the increase of the plastic strains per cycle is well predicted for both amplitudes. The strains of the upper turnover points are almost 10% smaller than the strains in the experiment. The higher difference of the measured and calculated strains in the area of the lower turnover points is caused by the more intensive bulge of the calculated hysteresis-loops. 250

Summary

[½]

10.5

Figure 10.38: Plate with a hole, LK2, cycle 1–10, experiment and calculations.

As a second example, Figure 10.38 shows a comparison between measured and calculated strains exx of a plate with a hole (X
0:0 in Figure 10.34) for the first 10 load steps. It is seen that the differences between the results of the calculations with the models of Reininghaus (biaxial) [2] and Scheibe (uniaxial, Z
0) [1] are small. As an additional result, Figure 10.38 shows that the “biaxial” extensions of the ETS-model to include nonproportional effects do not influence the results of the calculations of the uniaxial loaded plate with a hole. If a non-proportional load occurs, distinct differences are obtained merely because this kind of load is not mentioned in the model of Scheibe.

10.5

Summary

Both the exact knowledge of the material behaviour and a model to simulate this behaviour are necessary for a precise calculation of the response of structures under plastic cyclic loads. Extensive investigations were carried out into the material behaviour of structural mild steel Fe 510 under uniaxial, load- and strain-controlled loads as well as under different biaxial proportional and non-proportional loads combined with yield-surface investigations. 251

10 On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading The two-surface model of Dafalias-Popov was extended to fit the individual characteristics of mild steel behaviour under cyclic loads. This extended-two-surface model is able to simulate precisely the behaviour of Fe 510 under various proportional and non-proportional multiaxial cyclic loads. Linking up with a Finite-Element program, structures and structural elements undergoing cyclic or random plastic deformations are calculated.

References [1] H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Beru¨cksichtigung in Konstruktionsberechnungen. Tech. Univ. Braunschweig, Institut fu¨r Stahlbau, Bericht Nr. 6314, 1990. [2] M. Reininghaus: Baustahl Fe 510 unter zweiachsiger Wechselbeanspruchung. Tech. Univ. Braunschweig, Institut fu¨r Stahlbau, Bericht Nr. 6326, 1994. [3] D. Kuck: Experimentelle Untersuchungen zum Ratchetting-Verhalten bei Baustahl ST52-3. Dissertation TU Braunschweig, 1996. [4] S. Dannemeyer: Zur Vera¨nderung der Fließfla¨che von Baustahl bei mehrachsiger plastischer Wechselbeanspruchung. Dissertation TU Braunschweig, 1999. [5] M. J. Michno, W. N. Findley: A Historical Perspective of Yield Surface Investigations for Metals. Int. J. Non-Linear Mechanics 11 (1976) 59–82. [6] D. Besdo, N. Wellerdick-Wojtasik: The Influence of Large Torsional Prestrain on the Texture Development and Yield Surfaces of Polycrystals. This book (Chapter 7). [7] H. Heuer: Untersuchung zur Anwendbarkeit des Einfließfla¨chen-Modells auf das zyklische Materialverhalten von Baustahl. Diplomarbeit, Tech. Univ. Braunschweig, 1988. [8] Y. F. Dafalias, E. P. Popov: A Model of Nonlinearly Hardening Materials for Complex Loadings. Acta Mechanica 21 (1975) 173–192. [9] J. Scheer, H.-J. Scheibe, D. Kuck, M. Reininghaus: Stahlkonstruktionen unter zyklischer Belastung. Arbeits- und Ergebnisbericht 1987–1990. Subproject B5, Collaborative Research Centre (SFB 319): Stoffgesetze fu¨r das inelastische Verhalten metallischer Werkstoffe – Entwicklung und Technische Anwendung, Tech. Univ. Braunschweig, Institut fu¨r Stahlbau, 1990. [10] U. Peil, D. Kuck: Stahlkonstruktionen unter zyklischer Belastung. Arbeits- und Ergebnisbericht 1991–1993. Subproject B5, Collaborative Research Centre (SFB 319): Stoffgesetze fu¨r das inelastische Verhalten metallischer Werkstoffe – Entwicklung und Technische Anwendung, Tech. Univ. Braunschweig, Institut fu¨r Stahlbau, 1993. [11] U. Peil, M. Reininghaus: Baustahl unter mehrachsiger zyklischer Belastung. Arbeits- und Ergebnisbericht 1991–1993. Subproject B5, Collaborative Research Centre (SFB 319): Stoffgesetze fu¨r das inelastische Verhalten metallischer Werkstoffe – Entwicklung und Technische Anwendung, Tech. Univ. Braunschweig, Institut fu¨r Stahlbau, 1993. [12] M. Reininghaus: Baustahl ST52 unter plastischer Wechselbeanspruchung. Dissertation TU Braunschweig, 1994. [13] Z. Mro´z: An Attempt to Describe the Behavior of Metals under Cyclic Loads Using a More General Workhardening Model. Acta Mechanica 7 (1968) 199–212. [14] J. Scheer, H.-J. Scheibe, D. Kuck: Zum Verhalten ausgeklinkter Tra¨ger unter zyklischer Beanspruchung. Bauingenieur 65 (1990) 463–468. 252

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)

11

Theoretical and Computational Shakedown Analysis of Non-Linear Kinematic Hardening Material and Transition to Ductile Fracture Erwin Stein, Genbao Zhang, Yuejun Huang, Rolf Mahnken and Karin Wiechmann *

Abstract The research work of this project is based on Melan’s static shakedown theorems for perfectly plastic and linear kinematic hardening materials. Using a 3-D generalization of Neal’s 1-D model for limited hardening, a so-called overlay model, a new theorem and a corollary are derived for general non-linear kinematic hardening materials. For the numerical treatment of the structural analysis of 2-D problems, the Finite-Element method (FEM) is used, while enhanced optimization algorithms are used to perform the shakedown analysis effectively. This will be demonstrated with some numerical examples. Treating a crack as a sharp notch, the shakedown behaviour of a cracked ductile body is investigated and thresholds for no crack propagation are formulated.

11.1

Introduction

11.1.1

General research topics

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, * Universita¨t Hannover, Institut fu¨r Baumechanik und Numerische Mechanik, Appelstraße 9 a, D-30167 Hannover, Germany 253

11

Theoretical and Computational Shakedown Analysis

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 simply shakedown) of a system is regarded as a safe state. It is important to know if a system under given variable loadings shakes down or not.

11.1.2

State of the art at the beginning of project B6

In 1932, Bleich [1] was the first to formulate a shakedown theorem for simple hyperstatic systems consisting of elastic, perfectly plastic materials. This theorem was then generalized by Melan [2, 3] in 1938 to continua with elastic, perfectly plastic and linear unlimited kinematic hardening behaviour. Koiter [4] introduced a kinematic shakedown theorem for an elastic, perfectly plastic material in 1956, that was dual to Melan’s static shakedown theorem. Since then, extensions of these theorems for applications of thermoloadings, dynamic loadings, geometrically non-linear effects and internal variables have been carried out by different authors (Corradi and Maier [5], Ko¨nig [6], Maier [7], Prager [8], Weichert [9], Polizzotto et al. [10]). However, little progress has been made in the formulation of a corresponding shakedown theorem for materials with non-linear kinematic hardening. The only attempt was made by Neal [11], who formulated a static shakedown theorem for materials with non-linear kinematic hardening in a 1-D stress state by using the Masing overlay model [12]. Several papers were published concerning especially 2-D and 3-D problems (Gokhfeld and Cherniavsky [13], Ko¨nig [14], Sawczuk [15, 16], Leckie [17]). The shakedown investigation of those problems leads to grave mathematical problems. Thus, in most of these papers, approximate solutions based on the kinematic shakedown theorem of Koiter [4] or on the assumption of a special failure form were derived. But these solutions often lost their bounding character due to the fact that simplifying flow rules or wrong failure forms were estimated. Until the beginning of project B6, only a few papers were published, in which the Finite-Element method was used for the numerical treatment of shakedown problems (Belytschko [18], Corradi and Zavelani [19], Gross-Weege [20], Nguyen Dang and Morelle [21], Shen [22]).

11.1.3

Aims and scope of project B6

In the framework of the geometrically linearized theory, the shakedown behaviour of linear elastic, perfectly plastic, of linear elastic, linear unlimited kinematic hardening and of linear elastic, non-linear limited kinematic hardening materials were taken under 254

1.1

Introduction

consideration. The theoretical and numerical treatment of the shakedown behaviour of these material models was one major scope of project B6. Based on static shakedown theorems, the numerical treatment of 2-D and 3-D field problems for arbitrary non-linear kinematic hardening materials by Finite-Element method should be realized. One special task was the formulation and the proof of a static shakedown theorem for limited non-linear kinematic hardening materials. In Section 11.2, a 3-D overlay model is presented, that was developed to describe non-linear, limited kinematic hardening material behaviour. This model is an extension of the 1-D overlay model of Neal [11]. A static shakedown theorem and a corollary, that were formulated and proved for the proposed material model, are extensions of Melan’s static shakedown theorems for perfectly plastic and linear kinematic hardening materials [2]. While analytical solutions of shakedown problems can only be derived for very simple systems, Finite-Element methods based on displacement methods should be used for the numerical treatment and solution of 2-D and 3-D shakedown problems. After discretizing the system and accounting for the shakedown conditions, usually a non-linear mathematical optimization problem is derived, that is very large scaled. Solving optimization problems like these is normally very difficult. Thus, effective optimization algorithms should be formulated and implemented, that were designed especially to take account of the special structure of the problems. Section 11.3 is concerned with the numerical approach based on static shakedown theorems. The discretized optimization problems for the proposed material models are discussed briefly. In Section 11.3.5, numerical examples show the effectivity of the proposed formulation. Solutions for perfectly plastic and kinematic hardening materials are compared. One important scope of project B6 was the examination of hardening and softening materials. While classic shakedown theorems imply implicitly that a material under cyclic loading behaves stable after only one or two loading cycles, experimental investigations show that stable cycles can be reached only after several loading cycles and sometimes only asymptotically. Thus, the influence of cyclic hardening and softening on the shakedown behaviour of materials had to be taken under consideration. The examination of cyclic hardening material with the Chaboche constitutive equation [23], will be considered in Section 11.3.5.3. An incremental-failure analysis for this material is carried out, and the results are compared with those of the 3-D overlay model described in Section 11.2. Stress singularities occur if macroscopic cracks develop in a solid material. Under these circumstances, classical shakedown theorems cannot be used. Thus, one major aim of project B6 was to apply shakedown theory directly to fatigue fracture to include stress singularities into shakedown investigations. In Section 11.4, we will apply shakedown theory to fatigue fracture to derive thresholds for no crack propagation. Classic shakedown theorems predict a zero shakedown limit load for a cracked body because of singular stresses at the crack tip. But experiments for ductile materials show that limits exist, for which no crack propagation occurs. We will consider the crack as a sharp notch, the notch root of it being a material constant at threshold level (Neuber [24]). The threshold of a fatigue crack follows then from the stationarity of the plastic energy dissipated in the cracked body.

255

11

11.2

Theoretical and Computational Shakedown Analysis

Review of the 3-D Overlay Model

There exist many mathematical models to describe the kinematic hardening behaviour of materials, for example the Prager linear kinematic hardening model [25], its modification by Ziegler [26], Mro´z multisurface model [27], Dafalias and Popov’s two-surface model [28] and so on. In Stein et al. [29], a so-called 3-D overlay model was developed to describe the non-linear kinematic hardening material behaviour. We will give here a brief review of the proposed model. A macroscopic material point x 2 X  IR3 is assumed to be composed of a spectrum of microscopic elements (or microelements). Each microelement is numbered with a scalar variable n 2 ‰0; 1Š. Stresses and strains are separately defined for the macroscopic material point (macrostress and macrostrain) and the microelements (microstresses and microstrains). They are denoted by r…x†; e…x† and w…x; n†; g…x; n†, respectively. The macrostress r has to fulfil the equilibrium condition: div r…x† ˆ b…x†; 8 x 2 X :

…1†

In the framework of geometrically linear continuum mechanics, the kinematic relation 1 e…x† ˆ …ru…x† ‡ …ru…x††T †; 8 x 2 X ; 2

…2†

holds between the displacement u and the strain e. By assuming that the stress r of a macroscopic material point (macrostress) is the weighted sum of the stresses w of all microelements (microstresses), the macroscopic material point and the corresponding microelements deform in the same way, and we get the following static and kinematic relations:

r…x† ˆ


1

w…x; n†dn ;

…3†

0

g…x; n† ˆ e…x†; 8 n 2 ‰0; 1Š :

…4†

Furthermore, we suppose that all microelements are linear elastic, perfectly plastic and have the same temperature, the same elastic moduli, but different yield stresses k…n†. For convenience, k…n† can be considered as a monotone growing function of n. Additionally, the validity of the additive decomposition of the microstrain g in an elastic and a plastic part is supposed. Thus, the following relations can be derived for the microelements: g…n† ˆ gE …n† ‡ gP …n† ; 256

…5†

11.2

Review of the 3-D Overlay Model

gE n† ˆ E 1 w…n† ;

…6†

U…w†  k2 …n† ;

…7†

qU _ _ ; k…n†  0 ; g_ P …n† ˆ k…n† qw

…8†

_ k…n†‰U…w†

…9†

k2 …n†Š ˆ 0 ;

where E stands for the symmetrical elasticity tensor and U…
† is the yield function. For simplicity, the argument x of the fields will be omitted partly. Assume that at a certain macrostress, the microelement number n begins to yield, the function k…x; n† is then uniquely determined by the macroscopic r; e-function in the 1-D case:

r…n† ˆ


n

k… n†d n ‡ …1

n†k…n† :

…10†

0

It is easy to show that, similar to k…n†; r is also a monotone growing function of the variable n (see Figure 11.1). For n ˆ 0, we have r ˆ r…n ˆ 0† ˆ k…n ˆ 0† ˆ k0 . The maximum value of r (denoted by rY or K) is derived by setting n ˆ 1 in Equation (10):

rY ˆ r…n ˆ 1† ˆ K ˆ


1

k…n†dn :

…11†

0

Thus, k0 and K can be identified with the initial yield stress ro and the ultimate stress of the macroelement rY , respectively. For the 3-D case, there is an analogous relation between r and k…n† as Equation (10), namely:

Figure 11.1: Kinematic hardening of a macroscopic material point and yield stresses of the microelements in 1-D case. 257

11

Ur† ˆ

Theoretical and Computational Shakedown Analysis


n

k… n†d n ‡ …1

n†k…n† :

…12†

 nˆ0

Now, we define the difference between the microstress w…n† and the macrostress r as the residual microstress and denote it by p…n†: p…n† ˆ w…n†

…13†

r:

For the residual microstress defined in this way, we have: p…n† ˆ 0; n 2 ‰0; 1Š;

for krk  ro ;

…14†

p…n† 6ˆ 0; n 2 ‰0; 1Š;

for krk > ro :

…15†

and

It is necessary to notice that Equation (15) holds for all n 2 ‰0; 1Š even if U…r† is smaller than k…n†. Integration of Equation (13) yields:
1

p…x; n†dn ˆ 0; 8 x 2 X ;

…16†

0

where Equation (3) has been used. From Equation (13) can be concluded that the resultant of p…n† does not contribute to the macrostress r. All microstress fields, which fulfil Equation (16), can be regarded as residual microstress fields. For a kinematic hardening material, it is possible to represent the residual microstress p…x; n† by a backstress  …x†, which describes the translation of the initial yield surface: p…x; n† ˆ

U…r† U…r†

k…n†  …x† : k…0†

…17†

It is easy to show that the microstress in Equation (17) satisfies Equation (16). For the 3-D overlay model described above, the following static shakedown theorem has been formulated and proved (e.g. Stein et al. [30], Zhang [31]): Theorem 1: If there exist a time-independent residual macrostress field q…x† and a  …x; n†, satisfying time-independent residual microstress field p U‰m p…x; 0†Š  ‰K…x†

k0 …x†Š2 ; 8 x 2 X ;

such that for all possible loads within the load domain, the condition 258

…18†

11.3

Numerical Approach to Shakedown Problems

 …x; 0†Šg  k02 …x† Um‰rE …x; t† ‡  q…x† ‡ p

…19†

is fulfilled 8 x 2 X and 8 t > 0, where m > 1 is a safety factor against inadaptation, then the total plastic energy dissipated within an arbitrary load path contained within the load domain is bounded, and the system consisting of the proposed material will shake down. The static shakedown theorem 1 is formulated by using the residual microstress p. Considering the relation between the backstress  and the residual microstress p in Equation (17), it is also possible to formulate the following corollary directly in terms of the backstress  , i.e.: Corollary 1: If there exist a time-independent residual macrostress field q…x† and a time-independent field  …x†, satisfying U‰m  …x†Š  ‰K…x†

k0 …x†Š2 ; 8 x 2 X ;

…20†

such that for all possible loads within the load domain, the condition Ufm‰rE …x; t† ‡  q…x†

 …x†Šg  k02 …x†

…21†

is fulfilled 8 x 2 X and 8 t > 0, then the system will shake down. For the formulation of the static shakedown theorem 1 and the corresponding corollary 1, only the values of k0 and K have been used. That means that the shakedown limits for systems of the considered material do not depend on the particular shape of the function k…n†, and therefore do not depend on the particular r; e-curve but, solely, on the magnitudes of k0 and K. For K…x† ˆ k0 …x† (an elastic, perfectly plastic material), we have  ˆ 0 due to Equation (20), and theorem 1 reduces to Melan’s theorem [2, 3] for an elastic, perfectly plastic material. For K…x† ! ‡1 (materials with unlimited kinematic hardening), the constraint (Equation (20)) imposed on the backstresses  …x†, can never become active and therefore may be dropped. In this case, we get the static shakedown theorem of Melan for a linear, unlimited kinematic hardening material.

11.3

Numerical Approach to Shakedown Problems

11.3.1

General considerations

For the numerical approach of shakedown problems, both the static and the kinematic theorems can be employed. However, the use of the static theorems has the advantage that the discretized optimization problem is regular. Therefore, only static theorems were used for the following considerations. Furthermore, for the Finite-Element discre259

11

Theoretical and Computational Shakedown Analysis

tization, only elements based upon the displacement method were used. In the following sections, we will limit our attention to plane-stress problems (for more details and other stress situations, see e.g. Zhang [31], Stein et al. [32], Mahnken [33]).

11.3.2

Perfectly plastic material

We are interested in finding the maximal possible enlargement of the load domain allowing still for shakedown. Thus, for systems consisting of linear elastic, perfectly plastic material, we get the following optimization problem in matrix notation: b  max ;

22†

NG 

…23†

iˆ1

Ci qi ˆ Cq ˆ 0 ;

U‰brEi …j† ‡ qi Š  r2o ; 8…i; j† 2 I  J ;

…24†

for load domains of the form of a convex polyhedron with M load vertices. Here, b is the maximal possible enlargement of the load domain, NG is the number of Gaussian points. I ˆ ‰1; . . . ; NGŠ and J ˆ ‰1; . . . ; MŠ are sets containing all Gaussian points and load vertices, respectively. C is a system-dependent matrix and rEi …j† is the elastic stress vector at the i-th Gaussian point, which corresponds to the j-th load vertex P…j† of the load domain. The Equations (23) and (24) represent the discretized static equilibrium conditions for the residual stresses and the shakedown conditions controlled at the Gaussian points, respectively. qi is the residual stress vector in the i-th Gaussian point. In general, the discretized shakedown problem (Equations (22) to (24)) is a large scaled optimization problem. Direct use of standard optimization algorithms such as the sequential quadratic programming method (SQP-method) is not effective. Thus, two optimization algorithms were formulated and implemented to take account of the special structure of the problem (Equations (22) to (24)).

11.3.2.1 The special SQP-algorithm Dual methods do not attack the original constrained problem directly, but instead attack an alternative problem, the dual problem, whose unknowns are the Lagrange multipliers of the primal problem. In order to solve the dual problem, a projection method is used. Each subproblem is then solved iteratively. The iteration matrix needed to do so was implemented due to Bertsekas’s algorithm. To make the algorithm even more effective for large sized problems, a Quasi-Newton method was used and a BFGS-update formula was implemented (for details of the proposed methodology, see Stein et al. [32], Mahnken [33]). 260

11.3

Numerical Approach to Shakedown Problems

11.3.2.2 A reduced basis technique The main idea is to solve Equations (22) to (24) in a sequence of reduced residual stress spaces of very low dimensions. Starting from the known state b k 1† and q…k 1† , a few …r† basis vectors bp;k ; p ˆ 1; . . . r, are selected from the residual stress space Bd . They form a subspace Br;k (or reduced residual stress space) of Bd . The improved state …k† is determined by solving the reduced optimization problem. Due to its low dimension, the reduced problem can be solved very efficiently by using a SQP- or a penalty method. The k-th state is obtained by an update for qk and Db k . Selecting new reduced basis vectors, the process is repeated until a convergence criterion for Db k is fulfilled. One way for generating the basis vectors is to carry out an equilibrium iteration. The intermediate stresses during the iteration are in equilibrium with the same external load, and their differences are thus residual stresses. Details can be found in [29–31].

11.3.3

Unlimited kinematic hardening material

For the investigation of the shakedown behaviour of systems consisting of unlimited kinematic hardening material, we restrict the load domain in the same way as in Section 11.3.1, and the following optimization problem can be formulated [31]: b ! max ;

…25†

NG 

…26†

iˆ1

Ci qi ˆ 0 ;

U‰brEi …j† ‡ qi

 i Š  r2o ; 8…i; j† 2 I  J ;

…27†

where  i is the affine backstress vector at the i-th Gaussian point. Defining now vectors yi in such a way that yi ˆ qi

i ;

…28†

where the yi are not restricted. Thus, the equality constraints (Equation (26)) of Equations (25) to (27) can be dropped: b ! max ;

…29†

U‰brEi …j† ‡ yi Š  r2o ; 8…i; j† 2 I  J :

…30†

A modified optimization problem is derived, that has a very simple structure. Zhang [31] formulated and proved a lemma to solve this problem:

261

11

Theoretical and Computational Shakedown Analysis

Lemma 1: The maximal shakedown load factor b s of Equations (29) and (30) can be derived through  b s ˆ b;

with b ˆ min bi ; i2I

…31†

where b i , …i 2 I †, is the solution of the subproblem b i ! max ;

…32†

U‰b i rEi …j† ‡ yi Š  r2o :

…33†

The dimension of Equations (32) and (33) is very low. Thus, it can be solved effectively with a usual SQP-method. Relationship between perfectly plastic and kinematic hardening material The shakedown load of a system consisting of unlimited kinematic hardening material is determined by that point xip , where the maximum possible enlargement of the elastic stress domain S Eip is the least in comparison to other points. Thus, the failure is of local character. This reflects the fact that a system, that consists of unlimited kinematic hardening material and is subjected to cyclic loading, can fail only locally in form of alternating plasticity [34]. Incremental collapse cannot occur since it is connected with a non-trivial, kinematic compatible plastic strain field, which has global character. The shakedown load of a system consisting of perfectly plastic material cannot be larger than the shakedown load of the same system consisting of unlimited kinematic hardening material with the same initial stress ro . However, it is possible that the shakedown loads for perfectly plastic and kinematic hardening material are identical. This is the case only if alternating plasticity is dominant in both cases. It is easy to show that qip ˆ yip

…34†

holds. Concluding, the following conclusions can be drawn from lemma 1: 1.

A system consisting of unlimited kinematic hardening material and subjected to variable loading can fail only locally in form of alternating plasticity.

2.

The kinematic hardening does not influence the shakedown load if the same system with perfectly plastic material, subjected to the same loading, fails in form of alternating plasticity in such a manner that there exists at least one point xip , for which the enlarged elastic stress domain b s S Eip is just contained in the yield surface shifted to qip . A further shift of the yield surface at this point would cause that a portion of the enlarged elastic domain b sS Eip leaves the yield surface. In the sequel, the alternating plasticity with the special character mentioned before will be denoted by APSC. In all other cases, an increase of the shakedown load due to kinematic hardening is expected.

262

11.3

Numerical Approach to Shakedown Problems

3.

If the shakedown loads for perfectly plastic and unlimited hardening material are identical, then alternating plasticity is the dominant failure form in both cases.

4.

The shakedown load determined for perfectly plastic material is exact if it is identical with that determined for unlimited kinematic hardening material, provided the latter is determined exactly.

11.3.4

Limited kinematic hardening material

As mentioned in Section 11.2, the shakedown limit of a system consisting of limited kinematic hardening material depends only on the values k0 and K. For this reason, the given function kn† may be replaced by a step function of n 2 ‰0; 1Š. The step function has to be chosen such that its minimum is equal to k0 and its area is equal to K. That is, Kˆ

m  lˆ1

Dnl kl ; m  2 ;

…35†

where m is the number of the intervals of the step function and kl is the value of the step function of the l-th interval with the length Dnl . For plane stress problems, the microelements may be incorporated in a natural way. The thickness t of the structure is divided into several (m  2) layers with the thicknesses tl , l ˆ 1; 2; . . . ; m. Each layer behaves elastic, perfectly plastic and has a corresponding yield stress kl (one value of the step function). The thicknesses of the layers have to be chosen such that tl ˆ Dnl ; 8 l 2 ‰1; . . . ; mŠ : t

…36†

By doing so, a unique relation between the layers of the structure and the intervals of the step function is established. The microelements of the l-th interval are replaced by the l-th layer of the structure. The parallel connection of the microelements is realized by discretizing all layers in the same way. That is, elements that lie on top of each other have the same nodes. Thus, it is guaranteed that elements of different layers have the same kinematics. Dividing the structure into Ne elements with NG Gaussian points and m layers, we get the discretized shakedown problem (Stein et al. [32]): b ! max ; NG  iˆ1

Ci

…37†

m  ts sˆ1

t

qi;s ˆ

NG  iˆ1

Ci qi ˆ Cq ˆ 0 ;

U…brEi …j† ‡ qi;1 †  k12 ˆ k02 ; 8 …i; j† 2 I  J ;

…38† …39† 263

11

Theoretical and Computational Shakedown Analysis

Upi;1 †  …K

k1 †2 ˆ …K

k0 †2 ; 8 i 2 I :

…40†

The Equations (38), (39) and (40) represent the discretized static equilibrium conditions for the residual macrostresses and the shakedown conditions controlled at the Gaussian points, respectively. qi;s and pi;s are the residual stress vector and the residual microstress vector in the s-th layer of the i-th Gaussian point. Between qi;s ; pi;s and the residual macrostress vector qi, the following relation holds: qi;s ˆ qi ‡ pi;s :

…41†

For m ˆ 1 and K ˆ k1 , Equations (37) to (40) reduce to the discretized shakedown problem for a perfectly plastic material. In this case, we have pi;1 ˆ 0; qi ˆ qi;1 and the constraint (Equation (40)) can be dropped. We come to the other extreme case by assuming K ! 1, which corresponds to an unlimited kinematic hardening material. Due to K ! 1, the constraint (Equation (38)) can never become active, and thus pi;1 is not constrained. Correspondingly, the constraint (Equation (40)) may be dropped as well. To solve the optimization problem (Equations (37) to (40)) effectively, the reduced basis technique presented in Section 11.3.2.2 was extended (for details, see e.g. Zhang [31], Stein et al. [32]).

11.3.5

Numerical examples

In this section, numerical examples of different structures are considered. The influence of kinematic hardening on the shakedown limit is demonstrated.

11.3.5.1 Thin-walled cylindrical shell A cylindrical shell with wall thickness d and radius R is subjected to an internal pressure p and an internal temperature Ti (Figure 11.2). The external temperature Te is equal to zero for all times t. For Ti ˆ Te and p ˆ 0, the system is assumed to be stressfree. The pressure and the temperature can vary between zero and their maximum values p and Ti . The corresponding load domain is defined by: 0  p  bc1 po ˆ p ; 0  c1  1 ; 0  Ti  bc2 Tio ˆ Ti ; 0  c2  1 :

…42†

To visualize the influence of kinematic hardening on the shakedown limits, the following three constitutive laws are considered: 1.

Elastic, perfectly plastic material with initial yield stress ko (curve 1 in Figure 11.2 b).

2.

Limited kinematic hardening material with K ˆ 1:35 ko (curve 2 in Figure 11.2 b).

3. 264

Unlimited kinematic hardening with K ˆ ‡1 (curve 3 in Figure 11.2 b).

11.3

Numerical Approach to Shakedown Problems

Figure 11.2: Thin-walled cylindrical shell: a) system and loads; b) shakedown diagram.

Curves 1, 2 and 3 in Figure 11.2 b show that the kinematic hardening does not always increase shakedown limits. The common part of the curves represents load domains, which lead exclusively to APSC (Section 11.3.3.1) in all three cases, whereas both alternating plasticity and incremental failure can occur for the remaining load domains.

11.3.5.2 Steel girder with a cope A steel girder with a length of 4 m will be investigated. It consists of an IPB-500 profile with a cope on either side. The girder is simply supported and, in the middle, it is subjected to a concentrated single load P. Both corners of the copes are provided with a round drill hole of radius r ˆ 8 mm (see Figure 11.3 a). The material is St 52-3 with an initial yield stress ro ˆ 37:5 kN/cm2 and a maximal hardening rY ˆ 52:0 kN/cm2. The hardening can be regarded as kinematic. Experimental investigations At the Institute for Steel Construction of the University of Braunschweig, the girder was investigated experimentally [35]. First of all, the behaviour of the system subjected to cyclic loading was of interest. Additionally, for comparison, the ultimate load was determined experimentally as well. Firstly, the girder was subjected to different cyclic load programs. The load program of the first 15 cycles is shown in Table 11.1. Then, the girder was subjected to a load program, where the load varied between 0 and 600 kN with a velocity of 600 kN/ min. The number of load cycles, that led to a crack (with a length of 1 mm) at a drill hole, was 145. The number of load cycles, that led to a collapse of the girder, was 372. After collapse due to cyclic loading, the girder was shortened on either side by 50 cm, and it was recoped as before. Then, for this system, the ultimate load was determined as 887 kN. Note that this value can also be regarded as the ultimate load of the initial system since only those cross-sections near the cope are responsible for failure of the system.

265

11

Theoretical and Computational Shakedown Analysis

Figure 11.3: a) Discretization of the steel girder; b) load-strain diagram at one of both drill holes of the girder. Table 11.1: Loading program of the first 15 cycles for a steel girder. Cycles 1–5

Cycles 6–10

Cycles 11–15

P: 0 > 540 kN

P: 0 > 320 kN

P: 0 > 540 kN

Numerical investigations Due to the symmetry of the system and the loading, only one quarter of the system was considered for numerical investigation. For the Finite-Element discretization, two different types of elements were employed. The web of the girder was discretized by using 8128 isoparametric elements each with 4 nodes (see Figure 11.3 a). The upper and the lower flanges were discretized with 768 and 640 DKT (discrete Kirchhoff triangle)-elements, respectively. Apart from bending forces, the DKT-elements can also be stressed in plane in order to take account of the fact that the flanges are not subjected to pure bending. For the numerical investigation, both the ultimate load and the shakedown load were calculated. The solutions were obtained by the reduced basis technique. In order to demonstrate the influence of kinematic hardening on the ultimate load and on the shakedown load, respectively, the calculations were performed both for a perfectly plastic and a kinematic hardening material. The results are shown in Table 11.2. Note that the value for the ultimate load determined numerically (877.7 kN) was 1% lower than the value determined by experiment (887 kN). From Table 11.2, it can be seen that, while the ultimate load increases by a factor of rY =ro due to kinematic hardening, the shakedown load remains unaltered. In this case, the girder fails due to alternating plasticity near the drill holes (see Section 11.3.2.1). 266

11.3

Numerical Approach to Shakedown Problems

Table 11.2: Numerical ultimate and shakedown load for a steel girder. Material type

Ultimate load in kN

Shakedown load in kN

1. Perfectly plastic 2. Kinematic hardening

633.2 877.7

164.2 164.2

It should be pointed out that, originally, the experiment was not intended to analyse shakedown behaviour. During the experiment, the amplitudes of the cyclic loads were higher than the theoretical shakedown load. Thus, no shakedown behaviour could be observed. However, some valuable information can be drawn from the load-strain diagram for the first 5 load cycles shown in Figure 11.3 b. The strains were measured directly at one of the drill holes. In Figure 11.3 b, a region can be observed, where the load P is linearly proportional to the strains, i.e., where the system behaves purely elastic. The amplitude of the region is between 155.2 kN and 179.5 kN. A comparison shows that the numerical results are in good agreement with the experiment.

11.3.5.3 Incremental computations of shakedown limits of cyclic kinematic hardening material To describe the cyclic kinematic hardening behaviour of materials, many models have been developed. Mro´z’s multisurface model [27], the two-surface model of Dafalias and Popov [28] and Chaboche’s model [23] are three of the best known examples. Here, we use Chaboche’s model as an example and investigate its shakedown behaviour. Due to the fact that no shakedown theorems for cyclic hardening materials have been formulated yet, an incremental method will be used to calculate the shakedown limit. Examples and comparison As our first example, we consider the square plate with circular hole illustrated in Figure 11.4. The length of the plate is L and the ratio between the diameter D of the hole and the length of the plate is 0.01. The thickness of the plate is t ˆ 1 cm. The system is subjected to the biaxial loading p1 and p2 . Both can vary independently between p2 . The corresponding load domain is defined zero and their maximum values p1 and  by: p 1 ; 0  c1  1 ; 0  p1  bc1 ro ˆ 

…43†

p 2 ; 0  c2  1 : 0  p2  bc2 ro ˆ 

…44†

The results are shown in Table 11.3, where b pih denotes the shakedown limit for the path-independent hardening material and b cyh the shakedown limit for the cyclic kinematic hardening material. 267

11

Theoretical and Computational Shakedown Analysis

Figure 11.4.: Geometry and loading conditions of a square plate with a centric circular hole. Table 11.3: Shakedown limits for a plate with centric hole.
2
1 =

b pih

b cyh



0.0/1.0 0.2/1.0 0.4/1.0 0.6/1.0 0.8/1.0 1.0/1.0

0.69633 0.65458 0.61758 0.58433 0.55471 0.52775

0.69629 0.65417 0.61750 0.58417 0.55417 0.52707

300 20 20 20 20 20

It is shown that the results from optimization methods with cyclic independent hardening properties are upper bounds for those from incremental computation with cyclic kinematic hardening materials. It turned out that for about 20 cycles, the values b cyh approach b pih with an error less than 1%. It should be pointed out that the computation efforts for the cyclic processes are much higher in comparison with optimization methods. The second example to be considered is a CT-specimen with a notch as shown in Figure 11.5. It is subjected to a uniaxial loading p, which may vary between 0 and p. Five different values of notch root radius r are used to obtain a wide range of shakedown limits. Table 11.4 shows the results of the incremental computations. For this example too, the shakedown limits are the same as those of numerical optimization apart from the fact that the number of load paths, which we have taken for the incremental hardening, has no influence on the shakedown limit of the system. Remark 1: Elastic shakedown does not implement damage and creep phenomena. Engineers are interested in the admissible number of cycles for low-cycle fatigue, which cannot be derived from classical shakedown theorems. An approximated reduced load factor b  < b for scalar-valued damage can be achieved in a postprocess assuming conservatively that the loads always alternate between their maximum and minimum values in all load cycles. 268

11.4

Transition to Ductile Fracture

Figure 11.5: Geometry and loading conditions of a compact tension specimen. Table 11.4: Shakedown limits for a CT-specimen.

 ˆ 0:1  ˆ 0:2  ˆ 0:3  ˆ 0:4  ˆ 0:5

b pih

b cyh



0.069325 0.085133 0.092504 0.101117 0.107392

0.069292 0.085083 0.092083 0.101042 0.107292

20 20 20 20 20

Remark 2: There is an important connection between shakedown theory and structural optimization admitting inelastic deformations, which is a demanding task in the frame of the design under modern safety considerations related to the failure of a structure as given in the new EURO-CODES.

11.4

Transition to Ductile Fracture

To solve the optimization problem (Equations (37) to (40)) for a system with complicated geometry and load domain, the numerical methods like Finite-Element method should be usually used (see [32, 36]). For a problem with no more than two loading parameters (load domain with four vertices), Stein and Huang [37] developed an analytical method for determination of the shakedown-load factor b. The advantage of this method is that only the maximum effective stress in the system must be calculated. The shakedown-load factor b follows directly from a closed form. For a load domain with only one parameter (load domain with two vertices), the result for b is especially simple, it reads: 269

11

Theoretical and Computational Shakedown Analysis

Figure 11.6: a) A compact tension specimen under cyclic loading; b) modified notch; c) modified crack.

bˆ

2ro ; reff

where reff is von can be concluded elastic limit. Making use load of a notched

…45† Mises effective stress and ro is the elastic limit. From Equation (45) that the shakedown limit load of the system is as large as twice as its of this result, Huang and Stein [38] calculated the shakedown limit body under variable tension r as illustrated in Figure 11.6 a. It reads:

p 2ro ro pr ˆ p ; reff K 1 m ‡ m2

…46†

p ro pr Ksh ˆ bK ˆ p : 1 ‡ m m2

…47†

bˆ

where K is the stress-intensity factor (SIF), m is Poisson’s ratio. Thus, the maximal stress-intensity factor Ksh, under which the system will still shake down, reads:

If the applied SIF K does not exceed the shakedown limit SIF Ksh, the notched body shakes down. Otherwise, alternating plasticity occurs at the notch root. Shakedown limit SIF for a cracked body In [38], Huang and Stein applied Neuber’s material block concept [24] to the shakedown investigation of a cracked body. Accordingly, the continuum ahead of a sharp notch is considered as a material block with finite linear dimension e (Figure 11.6 b). Across this block, no stress gradient can develop. The original notch should be replaced by an effective notch with radius r …> r†. The stress-concentration factor is re270

11.4

Transition to Ductile Fracture

duced due to the enlarged notch radius. The classic strength theory is then still usable. The effective notch radius r is obtained in such a way that the average stress over the block e of the original notch is equal to the maximum stress of the modified notch, i.e.: 1 rymax r ˆ e


e

ry jr dr1 :

…48†

0

Using Craeger’s relation for the stress distribution [39] at the notch root, the following relation can be derived: 2K 1 p ˆ  e pr


e 0

  K r 2K  1 ‡ dr1 ˆ  : † 2…r=2 ‡ r 2p…r=2 ‡ r1 † p…r ‡ 2e† 1

…49†

For r, one gets: r ˆ r ‡ 2e :

…50†

Thus, the effective notch radius is equal to the original one plus two times Neuber’s material block size e. In the case of a crack (Figure 11.6 c), the effective crack-tip radius, denoted by q, can be obtained immediately by setting r  ) r ‡ 2e and r ˆ 0 in Equation (47), yielding: q ˆ 2e :

…51†

Equation (51) indicates implicitly that a crack can be treated as a notch with radius q. In fact, experiments with different materials done by Frost [40], Jack and Price [41], and Swanson et al. [42] show that a cracked body under cyclic loadings behaves in an identical manner as a notched body does if the latter has the same geometry as the cracked body and the notch root radius r is small enough (see also [43, 44]). The shakedown limit stress-intensity factor reads: p ro pq Ksh ˆ p : 1 m ‡ m2

…52†

Thus, the shakedown limit stress-intensity factor of a cracked body is proportional to the initial yield stress ro times the square root of the effective crack-tip radius q. For a material, ro is usually given. The problem now is to establish the effective crack-tip radius q. A direct measurement of this parameter is difficult. Using an indirect method, Kuhn and Hardrath [45] calculated the effective crack-tip radii for metallic materials and proposed a relationship between q and the ultimate strength rY of a material depicted in a diagram. For a given material, the initial yield stress ro and ultimate stress rY can be measured by a simple tension experiment. The effective crack-tip radius q is taken directly from the diagram given in [45]. Knowing these quantities, the shakedown limit SIF Ksh 271

11

Theoretical and Computational Shakedown Analysis

Table 11.5: Shakedown limit SIFs sh and fatigue thresholds th for various materials. Material

ro [MPa]

rY [MPa]

2 1/4 Cr-1Mo SA 387-2.22 SA 387-2.22 SA 387-2.22 Docol 350 SS 141147 HP Steel HP Steel HP Steel

345 390 340 290 260 185 210 160 120

528 550 520 500 360 322 304 279 242

 1/2 qm

13.549 · 10–3 12.433 · 10–3 13.708 · 10–3 14.346 · 10–3 19.128 · 10–3 22.316 · 10–3 25.504 · 10–3 28.692 · 10–3 36.662 · 10–3

sh [Mnm–3/2] th [Mnm–3/2] Ref. 8.25 8.6 8.14 7.3 8.9 7.6 9.5 8.1 7.8

8.3 8.3 8.6 9.6 5.4 6.0 6.2 6.7 8.2

[45] [45] [45] [45] [46] [46] [46] [46] [46]

follows immediately from Equation (52). In Table 11.5, the shakedown limit SIFs Ksh for some materials are selected. At the same time, fatigue thresholds for the same materials are listed there. They were obtained by other authors with experimental methods of other theoretical approaches [46, 47]. It can be seen that the shakedown limit SIFs Ksh of cracked bodies agree quite well with their fatigue thresholds Kth . This agreement indicates that the reason for crack arrest in these materials is the shakedown of the cracked bodies. In these cases, the fatigue threshold of a cracked body can be predicted by using shakedown theory.

11.5

Summary of the Main Results of Project B6

One major issue of project B6 was the formulation of a 3-D overlay model for non-linear hardening materials. For this class of materials, a static shakedown theorem and a corresponding corollary were formulated and proved, which are generalizations of Melan’s static shakedown theorems for perfectly plastic and linear kinematic hardening materials. A systematic investigation of the numerical treatment of shakedown problems using Finite-Element method was carried out. The findings were used to employ efficient optimization strategies and algorithms developed to take advantage of the special structure of the arising optimization problems. As an important result of these investigations, explicit conclusions about the failure forms of cyclically loaded systems could be drawn, i.e. incremental collapse or alternating plasticity. The influence of cyclic hardening and softening on the shakedown behaviour of structures was studied incrementally. The results were compared with those derived for the 3-D overlay model. A new methodology was proposed to include stress singularities into shakedown investigations allowing for the prediction of fatigue thresholds of ductile cracked bodies. Thus, a transition from shakedown theory to cyclic fracture mechanics was achieved. 272

References

References [1] H. Bleich: Über die Bemessung statisch unbestimmter Stabwerke unter der Beru¨cksichtigung des elastisch-plastischen Verhaltens der Baustoffe. Bauingenieur 13 (1932) 261–267. [2] E. Melan: Der Spannungszustand eines Mises-Henckyschen Kontinuums bei vera¨nderlicher Belastung. Sitzber. Akad. Wiss. Wien IIa 147 (1938) 73–78. [3] E. Melan: Zur Plastizita¨t des ra¨umlichen Kontinuums. Ing.-Arch. 8 (1938) 116–126. [4] W. T. Koiter: A New General Theorem on Shakedown of Elastic-Plastic Structures. Proc. Koninkl. Acad. Wet. B 59 (1956) 24–34. [5] L. Corradi, G. Maier: Inadaptation Theorems in the Dynamics of Elastic-Workhardening Structures. Ing.-Arch. 43 (1973) 44–57. [6] J. A. Ko¨nig: A Shakedown Theorem for Temperature Dependent Elastic Moduli. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 17 (1969) 161–165. [7] G. Maier: A Shakedown Matrix Theory Allowing for Workhardening and Second-Order Geometric Effects. In: Proc. Symp. Foundations of Plasticity, Warsaw, 1972. [8] W. Prager: Shakedown in Elastic-Plastic Media Subjected to Cycles of Load and Temperature. In: Proc. Symp. Plasticita nella Scienza delle Costruzioni, Bologna, 1956. [9] D. Weichert: On the Influence of Geometrical Nonlinearities on the Shakedown of ElasticPlastic Structures. Int. J. Plasticity 2 (1986) 135–148. [10] C. Polizzotto, G. Borino, S. Caddemi, P. Fuschi: Shakedown Problems for Material Models with Internal Variables. Eur. J. Mech. A/Solids 10 (1991) 787–801. [11] B. G. Neal: Plastic Collapse and Shakedown Theorems for Structures of Strain-Hardening Material. J. Aero Sci. 17 (1950) 297–306. [12] G. Masing: Zur Heyn’schen Theorie der Verfestigung der Metalle durch verborgen elastische Spannungen. Technical Report 3, Wissenschaftliche Vero¨ffentlichungen aus dem Siemens Konzern, 1924. [13] D. A. Gokhfeld, O. F. Cherniavsky: Limit Analysis of Structures at Thermal Cycling. Sijthoff & Noordhoff, 1980. [14] J. A. Ko¨nig: Theory of Shakedown of Elastic-Plastic Structures. Arch. Mech. Stos. 18 (1966) 227–238. [15] A. Sawczuk: Evaluation of Upper Bounds to Shakedown Loads of Shells. J. Mech. Phys. Solids 17 (1969) 291–301. [16] A. Sawczuk: On Incremental Collapse of Shells under Cyclic Loading. In: Second IUTAM Symp. on Theory of Thin Shells, Kopenhagen, Springer Verlag, Berlin, 1969. [17] F. A. Leckie: Shakedown Pressure for Flush Cylinder-Sphere Shell Interaction. J. Mech. Eng. Sci. 7 (1965) 367–371. [18] T. Belytschko: Plane Stress Shakedown Analysis by Finite Elements. Int. J. Mech. Sci. 14 (1972) 619–625. [19] L. Corradi, I. Zavelani: A Linear Programming Approach to Shakedown Analysis of Structures. Comp. Math. Appl. Mech. Eng. 3 (1974) 37–53. [20] J. Gross-Weege: Zum Einspielverhalten von Fla¨chentragwerken. PhD Thesis, Inst. fu¨r Mech., Ruhr-Universita¨t Bochum, 1988. [21] H. Nguyen Dang, P. Morelle: Numerical Shakedown Analysis of Plates and Shells of Revolution. In: Proceedings of 3rd World Congress and Exhibition on FEMs, Beverley Hills, 1981. [22] W. P. Shen: Traglast- und Anpassungsanalyse von Konstruktionen aus elastisch, ideal plastischem Material. PhD thesis, Inst. fu¨r Computeranwendungen, Universita¨t Stuttgart, 1986. [23] J. L. Chaboche: Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity. Int. J. Plast. 3 (1989) 247–302. [24] H. Neuber: Kerbspannungslehre. Springer Verlag, 1958.

273

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Theoretical and Computational Shakedown Analysis

[25] W. Prager: A New Method of Analyzing Stresses and Strains in Workhardening Plastic Solids. J. Appl. Mech., 1956, pp. 493–496. [26] H. Ziegler: A Modification of Prager’s Hardening Rule. Quart. Appl. Math. 17 (1955) 55– 65. [27] Z. Mro´z: On the Description of Anisotropic Workhardening. J. Mech. Phys. Solids 15 (1967) 163–175. [28] Y. F. Dafalias, E. P. Popov: A Model of Nonlinearly Hardening Materials for Complex Loading. Acta Mechanica 43 (1975) 173–192. [29] E. Stein, G. Zhang, R. Mahnken, J. A. Ko¨nig: Micromechanical Modeling and Computation of Shakedown with Nonlinear Kinematic Hardening Including Examples for 2-D Problems. In: Proc. CSME Mechanical Engineering Forum, Toronto, 1990, pp. 425–430. [30] E. Stein, G. Zhang, J. A. Ko¨nig: Shakedown with Nonlinear Hardening Including Structural Computation Using Finite Element Method. Int. J. Plasticity 8 (1992) 1–31. [31] G. Zhang: Einspielen und dessen numerische Anwendung von Fla¨chentragwerken aus ideal plastischem bzw. kinematisch verfestigendem Material. PhD thesis, Institut fu¨r Baumechanik und Numerische Mechanik, Universita¨t Hannover, 1992. [32] E. Stein, G. Zhang, R. Mahnken: Shakedown Analysis for Perfectly Plastic and Kinematic Hardening Materials. In: Progress in Computational Analysis of Inelastic Structures, Springer Verlag, 1993, pp. 175–244. [33] R. Mahnken: Duale Methoden in der Strukturmechanik fu¨r nichtlineare Optimierungsprobleme. PhD thesis, Institut fu¨r Baumechanik und Numerische Mechanik, Universita¨t Hannover, 1992. [34] J. A. Ko¨nig: Shakedown of Elastic-Plastic Structures. PWN-Polish Scientific Publishers, 1987. [35] J. Scheer, H. J. Scheibe, D. Kuck: Untersuchung von Tra¨gerschwa¨chungen unter wiederholter Belastung bis in den plastischen Bereich. Bericht Nr. 6099, Institut fu¨r Stahlbau, TU Braunschweig, 1990. [36] E. Stein, G. Zhang, Y. Huang: Modeling and Computation of Shakedown Problems for Nonlinear Hardening Materials. Computer Methods in Mechanics and Engineering 321 (1993) 247–272. [37] E. Stein, Y. Huang: An Analytical Method to Solve Shakedown Problems with Linear Kinematic Hardening Materials. Int. J. of Solids and Structures 18 (1994) 2433–2444. [38] Y. Huang, E. Stein: Shakedown of a Cracked Body Consisting of Kinematic Hardening Material. Engineering Fracture Mechanics 54 (1996) 107–112. [39] M. Craeger: Master Thesis, Lehigh University, 1966. [40] N. E. Frost: Notch Effects and the Critical Alternating Stress Required to Propagate a Crack in an Aluminium Alloy Subject to Fatigue Loading. J. Mech. Engng. Sci. 2 (1960) 109–119. [41] A. R. Jack, A. T. Price: The Initiation of Fatigue Cracks from Notches in Mild Steel Plates. International Journal of Fracture Mechanics 6 (1970) 401–409. [42] R. E. Swanson, A. W. Thompson, I. M. Bernstein: Effect of Notch Root Radius on Stress Intensity in Mode I and Mode III Loading. Metallurgical Transactions A 17A (1986) 1633– 1637. [43] N. E. Dowling: Fatigue at Notches and the Local Strain and Fracture Mechanics Approaches. In: Fracture Mechanics, 1979. [44] D. Taylor: Fatigue Thresholds. Butterworth, 1989. [45] P. Kuhn, H. F. Hardrath: An Engineering Method for Estimating Notch-Size Effect in Fatigue Test on Steel. Technical report, NACA technical note, 1952. [46] R. O. Ritchie: Near-Threshold Fatigue Crack Growth in 2 1/4 Cr-1Mo Pressure Vessel Steel in Air and Hydrogen. J. of Eng. Materials and Technology 102 (1980) 293–299. [47] J. Wasen, K. Hamberg, B. Karlsson: The Influence of Grain Size and Fracture Surface Geometry on the Near-Threshold Fatigue Crack Growth in Ferritic Steels. Mat. Sci. Engng. 102 (1998) 217–226. 274

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)

12

Parameter Identification for Inelastic Constitutive Equations Based on Uniform and Non-Uniform Stress and Strain Distributions Rolf Mahnken and Erwin Stein *

Abstract In this contribution, various aspects for identification of material parameters are discussed. The underlying experimental data are obtained from specimen, where stresses and strains can be either uniform or non-uniform within the volume. In the second case, the associated simulated data are obtained from Finite-Element calculations. A gradient-based optimization strategy is applied for minimization of a least-squares functional, where the corresponding sensitivity analysis if 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 effect of different starting values and stochastic perturbations of the experimental data are discussed.

12.1

Introduction

12.1.1

State of the art at the beginning of project B8

The project B8 of the Collaborative Research Centre (SFB 319) has been started in 1991 with the intention to identify material parameters of constitutive models for inelastic material behaviour. Though the interest for reliable modelling has always been very high in the engineering community, up to that time, the concepts for parameter identification concerning experimental and numerical issues were fairly limited. In particular the state of the art was as follows: * Universita¨t Hannover, Institut fu¨r Baumechanik und Numerische Mechanik, Appelstraße 9 a, D-30167 Hannover, Germany 275

12

Parameter Identification for Inelastic Constitutive Equations

•

The experiments producing the experimental data were mostly conducted in simple tension, compression or torsion, respectively. In this way, the sample, e.g. a cylindrical hollow specimen, is subjected to an axial load (force or displacement), which produces strains and stresses assumed to be uniform within the whole volume of the specimen.

•

The identification process for the underlying material models was performed in the framework of a geometric linear theory.

•

For optimization of the resulting least-squares functional (at least within the SFB 319) evolutionary strategies were preferred.

•

In some situations it may occur that more than one set of parameters can give reasonable fits. This issue of instability (or even non-uniqueness in the case of identical fits) has not been considered.

12.1.2

Aims and scope of project B8

As a main consequence of the first item in the above overview it was observed that parameters derived from an optimal least-squares fit of uniaxial experiments do not necessarily predict non-uniform deformations. This is due to the facts that (i) a uniaxial experiment does not provide enough information to obtain an accurate simulation of the non-uniform case, and (ii) the ideal test conditions of uniformness often cannot be realized in the laboratory. In nearly all mechanical tests deformations eventually cease to be uniform due to localization, fracture and other failure mechanisms. E.g., non-uniformness is unavoidable in the case of necking of the sample in tension tests or barreling due to friction of the sample in compression tests. Therefore a main object of project B8 was to develop a more general approach, which accounts for this inhomogeneity by performing parameter identification using Finite-Element simulations. The next issue is concerned with the geometric setting. It is quite obvious that material parameters obtained from a fit within a geometric linear setting in general do not carry over to the finite-deformation regime. This in particular holds if extreme loads are subjected to the specimen thus yielding large deformations. To this end parameter identification within a geometric non-linear theory has been performed. In a common – classical – approach, parameter identification is formulated as an optimization problem, where a least-squares functional is minimized in order to provide the best agreement between experimental data and simulated data in a specific norm. Algorithms for solution of this problem, basically, may be classified into two classes, i.e. methods, which only need the value of the least-squares function (zero-order methods) and descent methods, which require also the gradient of the least-squares function (first-order methods). Very often an evolutionary method is preferred in practice because of its versatility (see e.g. Mu¨ller and Hartmann [1] and Kublik and Steck [2]). However, in general, these methods are very time-consuming due to many function evaluations (several hundred thousand). Thus for reasons of efficiency an optimization strategy based on gradient evaluations has been developed. For determination of the gradient of the associated least-squares functional, basically two variants are known from the literature: (1) The finite-difference method: This tech276

12.2

Basic Terminology for Identification Problems

nique, though conceptionally very simple in general, is regarded as inefficient due to many function evaluations and accuracy problems. (2) The sensitivity analysis: In this concept the gradient is determined analytically consistent with the formulation of the underlying direct problem. As part of the work of project B8 the latter concept has been developed firstly to the uniform case, and then it was carried over to the non-uniform case. Another object of project B8 was to discuss and investigate the stability of the results for the identification process since instability is a typical feature of inverse problems (see Baumeister [3], Banks and Kunisch [4]). To this end two indicators are investigated: We examine the eigenvalues of the Hessian of the least-squares function, and we study the effect of perturbations of the experimental data on the parameters. Furthermore, for the case of numerically instable results, we introduce a regularization due to Tikhonov, which can be interpreted as an enhancement of the basic least-squares functional by adequate model information. Parameter identification essentially relies on experimental data obtained in the laboratory. In this respect it is obvious that for the non-uniform case spatially distributed data give more information as data evaluated only at certain points, e.g. using strain gauges. Therefore optical methods turned out to be the ideal tools in order to obtain the underlying data sets, and in our examples the experimental data were obtained with a grating method in collaboration with the projects C1 (Dr. Andresen [5]), C2 (Prof. Ritter [6]) and B5 (Prof. Peil [7]). In this contribution we will describe our approach for parameter identification, firstly, to the conventional uniform case, and secondly, to the non-uniform case, where the Finite-Element method is applied. To specify, this work is structured as follows: In the next Section 12.2 the basic terminology for identification problems pertaining to the direct problem and the inverse problem is introduced. In Section 12.3 a systematic concept for parameter identification is briefly described for the uniform case, and in Section 12.4, it is extended to the non-uniform case. In Section 12.5 two examples for parameter identification based on experimental data obtained within the Collaborative Research Centre (SFB 319) are presented. In the first example the Chaboche model [8] is considered with a sample in cyclic loading, and in the second example we investigate an axisymmetric necking problem of mild steel. Section 12.6 gives a summary of the main results of project B8, and furthermore, we discuss issues of future research work.

12.2

Basic Terminology for Identification Problems

12.2.1

The direct problem: the state equation

In the sequel, we denote by K a (vector-)space with elements j of admissible material parameters, and g…j; u^…j†† is the state equation, which may represent e.g. the (non-linear) state of the discretized form of an initial value problem or the variational form of an initial boundary value problem. The state equation g may be dependent on the parameters j, both explicitly and implicitly, where the implicit dependency is defined via 277

12

Parameter Identification for Inelastic Constitutive Equations

the state variable u^j† 2 U, and where U is a (function-)space of admissible state variables u^…j†. With this notation, we formulate the direct problem: Find u…j† 2 U

such that g…j; u…j†† ˆ 0

for given j 2 K :

…1†

In what follows, we assume existence of the solution u…j†:ˆ Arg fg…j; u^…j†† ˆ 0g for all j 2 K.

12.2.2

The inverse problem: the least-squares problem

~ denote an observation space, and let d~ 2 D ~ denote given data from experiments. Let D In general experimental data are not complete, e.g. for cyclic loading tests very often they are available only for a part of the cycles. To account for this possible incompleteness, we introduce an observation operator M mapping the trajectory u…j† to points M u…j† in the observation space P (Banks and Kunisch [4], p. 54). With this notation we formulate the inverse problem: Find j 2 K

such that M u…j† ˆ d~ for given

~: d~ 2 D

…2†

An identification process based on experimental data is typically influenced by two types of errors: Using the notations u^ for the true state and j for the correct parameter vector, then the following situations may arise (Banks and Kunisch [4]): • •

M u^ 6ˆ d~ due to measurement errors, u^ 6ˆ u…j† due to model errors.

In general, the first error type is taken into account by statistical investigations of the data. The second type is reduced by increasing the complexity of the model, which in general is accompanied by an increase of material parameters np . Referring to the classical definitions of Hadamard [9], a problem is well-posed if the conditions of (i) existence, (ii) uniqueness and (iii) continuous dependence on the data for its solution are satisfied simultaneously. If one of these conditions is violated, then the problem is termed ill-posed. Since in practice the number of experimental data is larger than the number of unknown parameters, problem (2) in general is overdetermined, thus excluding the existence of a solution due to measurement and/or model errors. The classical strategy therefore uses an optimal approach of simulated data u…j† and experimental ~ thus replacing problem (2) by the least-squares optimization problem: data d, ~ : f …j† :ˆ 1 kM u…j† Find j 2 K such that for given d~ 2 D 2 •

~ D2~ ! min : dk

…3†

j2K

Remarks

1. In practice, experimental data are given at discrete time- or load-steps. Therefore, ~  IRndat for the observation space, for the following discussions it is natural to set D 278

12.2

Basic Terminology for Identification Problems

with ndat as the number of experimental data. Furthermore, very often parameters are independent of each other such that K  IRnp can be separated, where np is the number of material parameters, Next, we use the short hand notation M u…j† ˆ: d…j† 2 IRndat , thus indicating the transformation of the simulated data to the observation space (e.g. by an interpolation procedure). Then, the resulting least-squares problem reads: 1 f …j† ˆ kd…j† 2

~ 22 ! min ; K ˆ dk j2K

np Y

Ki ; Ki :ˆ fai 
i  bi g :

…4†

iˆ1

Here, ai ; bi are lower and upper bounds for the material parameters, respectively. 2. In many situations, the problems (3) or (4), though well-posed, may lead to numerically instable solutions, i.e. small variations of d~ then lead to large variations of the parameters j. These difficulties are caused if (a) the material model has (too many) parameters, which yield (almost) linearly dependencies within the model, or if (b) the experiment is inadequate in the sense that some effects intended by the model are not properly “activated”. It has already been mentioned that a typical step for reducing the model error is to increase the complexity of the model, which generally is accompanied by an increase of the material parameters. A typical example is the modification of the standard J2-flow theory with the linear Prager rule in order to account for anisotropic hardening effects. A further extension is possible with the non-linear Chaboche model [8] (see also Equation (8) in the forthcoming Section 12.3) in order to account for non-linear kinematic hardening effects. In doing so, it should be realized that the introduction of additional material parameters may also result into the aforementioned numerical instability for the identification process if appropriate steps are not performed when planning the experiment. To summarize, the contradictory requirements for numerical stable results and reducing the model error have to be carefully balanced. 3. In some cases, even non-uniqueness for the parameter set may occur: This was observed by Mahnken and Stein [10, 11] for two material models under certain loading conditions. The main consequences are that at least cyclic loading becomes necessary in case of the Chaboche model [8], and for identification of the Steck model [12] experiments have to be performed at different temperatures. 4. As a consequence of the above Remark 2, it is strongly recommended to study the effect of perturbations of the experimental data on the parameters. This may indicate possible instabilities of the identification process. Furthermore, the eigenvalue structure of the Hessian of f …j† gives further information about the stability. However, a systematic strategy to detect possible instabilities so far is not available. 5. A mathematical tool, suitable to overcome possible numerical instabilities, is a regularization of the functional in Equation (4), and this leads to the more general problem:

1 W  …d…j† f …j† :ˆ 2

2



~ W  …j d† ‡ 2 2

2

~ † min np : j

! j2KIR

…5†

2

279

12

Parameter Identification for Inelastic Constitutive Equations

Here, the matrices W  2 IRndat  IRndat and W  2 IRnp  IRnp , the scalar  2 IR‡ and the  2 IRm are regularization parameters (see Baumeister [3]). Note a priori parameters j that the first part of the functional in Equation (5) is also obtained when considering parameter identification based on statistical investigations in the context of a Maximum-Likelihood method in order to account for measurement errors. It is noteworthy that the r.h.s. of the functional is also related to the Bayesian estimation (see Bard [13], Pugachew [14]). The above functional provides the opportunity to include physical interpretation of some parameters, obtained e.g. by “hand fitting”, into the optimization process if numerical instabilities occur. However, a systematic concept for determination of the regularization parameters in the context of parameter identification for viscoplastic material models so far is not available. 6. Problems of the above kind like Equations (4) or (5) with separable constraints may be solved with the projection algorithm due to Bertsekas [15]: …j†

 rf …j…j† †g ; …Pfjg† :ˆ min…bi ; …max…
i ; ai †† ; j…j‡1† ˆ Pfj…j† …j† H i i ˆ 1; . . . ; np :

…6†

Note that the above iteration scheme requires the gradient of the associated leastsquares functional. This task is generally performed in the sensitivity analysis, where the gradient is determined consistent with the formulation of the underlying direct prob has to be “diagonalized” in order to insure lem. Note also that the iteration matrix H descent properties of the search directions for algorithm (see Bertsekas [15] and Mahnken [16] for an explanation of this terminology and further details).

12.3

Parameter Identification for the Uniform Case

In this section, we briefly describe a systematic strategy for parameter identification for the uniform case within a geometric linear setting. A detailed description is given in Mahnken and Stein [11, 17].

12.3.1

Mathematical modelling of uniaxial visco-plastic problems

Let I ˆ ‰0; TŠ be the time interval of interest. The uniaxial stress is designated by  ˆ 11 : I ! IR, while el and in : I ! IR, are the elastic and inelastic parts of the el small strain tensor components in ij and ij , respectively. The model equations representing one-dimensional visco-plasticity with small strains are summarized as follows:  ˆ el ‡ in el ˆ

280

1  E

additive split of total strains ;

…7 a†

elastic strains ;

…7 b†

12.3

Parameter Identification for the Uniform Case

_ in ˆ ^_ in …; q; ; in ; . . . ; j†

evolution for inelastic strains ;

…7 c†

q_ ˆ q^_ …; q; ; in ; . . . ; j†

evolution for internal variables .

…7 d†

Here, additionally, we defined the temperature , the elastic modulus E, and j 2 IRnp is a vector of np material parameters characterizing the inelastic material behaviour. There exists a great variety of constitutive relations in the literature according to the above skeletal structure (Equations (7 a) to (7 d)) (see e.g. Miller [18], Lemaitre and Chaboche [19] and references therein). Many approaches intend to provide for a number of different characteristic effects such as strain rate-dependent plastic flow, creep or stress relaxation. In doing so, a yield criterion with the inherent specification of loading and unloading conditions as in time-independent classical plasticity is not needed. The resulting equations are currently referred to as “unified models”. Concerning the internal variables, in principle they are argued for macroscopic or microscopic reasons depending on the basic conception. Three representative examples for the evolution equations _ in and q_ in Equation (7) were treated by Mahnken and Stein in [11, 17] within project B8, i.e. the models of Chaboche [8], Bodner and Partom [20] and Steck [12] (see also Kublik and Steck [2]). In this contribution only the Chaboche model with the evolution equations 8  0 n > < F sign … † if F > 0 in …8 a† _ ˆ K0 > : 0 else R_ ˆ b…q

R†_in sign …_in †

isotropic hardening

…8 b†

_ ˆ c…

 sign …_in ††_in

kinematic hardening

…8 c†

F ˆ …

† sign …

†

R

k0

overstress

…8 d†

shall be considered, where j :ˆ ‰n0 ; K 0 ; k0 ; b; q; c; ŠT is the vector of material parameters related to the inelastic material behaviour. In addition to the Equations (7 a) to (7 d), we assume that initial conditions …t ˆ 0† ˆ 0 ; in …t ˆ 0† ˆ in 0 ; q…t ˆ 0† ˆ q0

…9†

are given, which complete the formulation of the initial value problem. The representation above – and in the forthcoming two subsections – is based on stress-controlled experiments. Of course, analogous arguments hold for the complementary tests, i.e. strain-controlled experiments, where experimental data are given for a ~…t†; t 2 I . stress distribution 

281

12

12.3.2

Parameter Identification for Inelastic Constitutive Equations

Numerical solution of the direct problem

We define N as the number of time steps Dtk ˆ tk tk 1 ; k ˆ 1; . . . ; N; t0 ˆ 0; tN ˆ T. Using the second order midpoint-rule at each time step, from Equations (7 a) to (7 d), we obtain the update relations k ˆ  k

1

qk ˆ qk

1

‡ Dtk _ in k

‡ Del k

1=2

‡ Dtk q_ k

1=2

1

…10†

;

…11†

;

where we applied the notation: _ in k

1=2

ˆ ^_ in …1=2…k

q_ k

1=2

ˆ q^_ …1=2…k

Del k

1

ˆ

1 …k E

1

1

‡ k † ; 1=2…qk

‡ k † ; 1=2…qk

1

1

‡ qk †; . . . ; j† ;

‡ qk †; . . . ; j† ;

k 1 † :

…12† …13† …14†

Since the state variables k and qk are not known in advance, the following non-linear system of equations has to be solved at each time step: gk;1 …k ; qk † :ˆ k

k

gk;2 …k ; qk † :ˆ qk

qk

1

1

Dtk _ in k Dtk q_ k

1=2 1=2

Del k

1

ˆ0;

ˆ0:

…15† …16†

Defining Gk :ˆ ‰gk;1 ; gTk;2 ŠT and a vector of state variables Y k :ˆ ‰k ; qTk ŠT , Equations (15) and (16) may be summarized as: Gk …Y k † ˆ 0 :

…17†

Referring to the notation of Section 12.2.1, Equation (17) will be termed as the state equation, describing the state of the variables Y k :ˆ ‰k ; qTk ŠT at the k-th time step. Furthermore, using the notation of Section 12.2.1, we have the direct problem: Find Y k …j† such that Gk …Y k …j†† ˆ 0; k ˆ 1; . . . ; N for given j 2 K :

…18†

The iterative solution of Equation (17) is obtained with a Newton method. Details of this strategy with applications to the material models of Bodner-Partom, Chaboche and Steck are described in Mahnken and Stein [11, 17].

12.3.3

Numerical solution of the inverse problem

For reasons of simplicity, in the sequel we will assume that the discrete values for time dat integration ftk gNkˆ1  I and for the experimental data ftkexp gnkˆ1  I do coincide for both 282

12.4

Parameter Identification for the Non-Uniform Case

Figure 12.1: Schematic flow chart of the optimization strategy for the uniform case with outer and inner iteration loops.

the numerical values k j† ˆ …tk ; j† and the observations ~k ˆ ~…tk †; k ˆ 1; . . . ; N. The following considerations can be extended to more complex situations in a straightforward manner. The resultinginverse problem then becomes the least-squares optimization problem: ~: Find j 2 K such that for given d~ 2 D f …j† ˆ Kˆ

N 1 …k …j† 2 kˆ1

np Y

~k †2 !

min

j2KIRnp

;

Ki ; Ki :ˆ fai 
i  bi g :

…19†

iˆ1

As before, ai ; bi are lower and upper bounds for the material parameters. A schematic flow chart for solution of problem (19) with a simplified description is shown in Figure 12.1. It can be seen that basically an outer loop for iteration of the material parameters and an inner loop for iteration of the state variables Y^k …j† are performed. In the outer loop the Bertsekas algorithm (Equation (6)) is applied, where the gradient is determined in a sensitivity analysis. For details pertaining to this strategy with applications to the material models of Bodner-Partom, Chaboche and Steck, we also refer to Mahnken and Stein [11, 17].

12.4

Parameter Identification for the Non-Uniform Case

As already mentioned in Section 12.1.2, very often the assumption of uniform stress and strain distributions during the experiment cannot be guaranteed due to the experi283

12

Parameter Identification for Inelastic Constitutive Equations

mental conditions or failure mechanisms. Therefore, we will consider parameter identification with Finite-Element simulations in order to take into account inhomogeneities within the sample. In what follows, we give a brief review for a geometrically non-linear continuum-based formulation of the direct problem as a variational problem and furthermore of the associate least-squares problem. For solution of these problems, a standard linearization procedure is applied for the direct problem and a sensitivity analysis is performed for the inverse problem. In the forthcoming sections, we will comment on the similarities of these associated concepts. More details of our approach are documented in Mahnken and Stein [10, 21] for the geometric linear case, and in Mahnken and Stein [22], this concept has been extended to the geometric non-linear case.

12.4.1

Kinematics

Let   IRndim be the reference configuration of a continuous body  with smooth boundary @, and let X 2 B  IRndim be the position vector in the Euclidian space IRndim with spatial dimension ndim ˆ 1; 2; 3. We shall denote by @ B and @ B those parts of  and boundary tractions are the boundary @B, where configurations are prescribed as u prescribed as t, respectively. As usual, we assume @ B [ @ B ˆ @B and @ B \ @ B ˆ ;. In addition, b denotes the body force per unit volume. As before, we define I:ˆ ‰t0 ; TŠ 2 IR‡ as a time interval of interest, and K  IRnp designates the (vector-)space of material parameters. Following Barthold [23] the fundamental mapping for describing the current configuration of the body for varying time t 2 I and varying parameter j 2 K is given as:

B  I  K ! IRndim , ^ :ˆ …20† u ^…X; t; j† : …X; t; j† 7! u ˆ u As usual, we restrict ourselves to configurations u satisfying J :ˆ det…F† > 0 and ^  on @ B, where we use the shorthand notation F :ˆ F…X; uˆu t; j† :ˆ @X u for the deformation gradient at …X; t; j†. The exposition that follows crucially depends on the basic assumption: ^…X; t ˆ t0 ; j†8 …X; j† 2 …B  K† ; Xˆu

…21†

i.e. the initial configuration B at time t ˆ t0 is independent of the parameter set j. It is noteworthy that this restriction, e.g., does not hold for more complex situations in shape optimization, and would necessitate the introduction of a reference configuration invariant of the design variables j (Barthold [23], Haber [24]). Thus, we will regard the set …X; t; j† 2 B  I  K as the independent variables in the ensuing considerations. An illustration of the mapping (Equation (20)) for the body B at fixed time t for two different parameter sets j1 ; j2 2 K is shown in Figure 12.2.

284

12.4

Parameter Identification for the Non-Uniform Case

Figure 12.2: Illustration of two-parameter-dependent configurations at fixed time.

12.4.2

The direct problem: Galerkin weak form

Let j 2 K be given, and let us assume a partition of the time interval I ˆ

N 

‰tk 1 ; tk Š

kˆ1

into N subintervals (for problems of elasticity and plasticity tk refers to the load step). ^…; tk ; j† the configuration at time tk for the parameter set j, the Denoting by uk ˆ u balance equation of linear momentum and the set of Dirichlet and Neumann boundary conditions at the (k)-th step read: divrk ‡ qb ˆ 0 k uk ˆ u rk n ˆ tk

in B ; on @ B ; on @ B :

…22†

Here, rk designates the Cauchy stress tensor. Using the notation h : i for the L2 dual pairing on B of functions, vectors or tensor fields, an equivalent formulation is the classical weak form (principal of virtual work) of the momentum equations at time tk . In a spatial description this results into the direct problem: Find uk such that g…uk † ˆ hs : d ijtˆtk

gjtk ˆ 0 8 du and for given j 2 K ;

…23†

where the Kirchhoff stress tensor s ˆ Jr is introduced. Furthermore, a spatial rate of deformation tensor induced by the virtual displacement du is defined as d :ˆ sym…qu du†, and g :ˆ hb
dui ‡ ht
dui@ B designates the external part of the weak form. For the case of inelastic problems the above set of equations has to be supplemented by initial conditions Z…X; t0 ; j† ˆ Z0 , where Z denotes a set of history variables. The iterative solution of the non-linear problem (Equation (23)) is based on a standard Newton method, in which a sequence of linearizations of the weak form (Equation (23)) is performed. To this end the Gaˆteaux derivative qD g…uk † of problem (Equation (23)) as shown in Table 12.1 is determined. Here, lD :ˆ q Du and dD :ˆ sym…lD † are a velocity gradient and a spatial rate of deformation tensor, respectively, induced by the linearization increment Du. Additionally, c is the fourth order spatial material operator. 285

12

Parameter Identification for Inelastic Constitutive Equations

Table 12.1 : Weak form, Gaˆteaux derivative for linearization and linear equation for parameter sensitivity in a spatial formulation. • Weak form (principle of virtual work)
…u † ˆ hs : d ijˆ
j ˆ 0 • Gaˆteaux derivative for linearization qD
…u † ˆ h…c : dD † : d ‡ lD s : d ijˆ • Linear equation for parameter sensitivity qj
…u † ˆ h…c : dj † : d ‡ lj s : d ‡ qpj s : d ijˆ ˆ 0

12.4.3

The inverse problem: constrained least-squares optimization problem

As in Section 12.3, we assume identical time (load) steps ftk gNkˆ1 and observation ntdat ~ are available, and where D ~ denotes the , where experimental data d~j 2 D states ftj gjˆ1 ~ observation space. In particular, dj may contain stresses, strains, displacements, reaction force fields etc. Since in general only incomplete data are available from the experiment, we introduce an observation operator M mapping the configuration trajectory ~ Note that this defini^…; tk ; j†; k ˆ 1; . . . ; N to points of the observation space D. uk ˆ u tion of M also accounts for quantities such as stresses, strains, reaction force fields etc. since these quantities can be written in terms of the basic dependent variable uk . Then we consider the least-squares optimization problem: N ~ : f …j† :ˆ 1 kMuk Find j 2 K such that for given d~ 2 D 2 kˆ1

d~k jjD2~ ! min ;

…24†

j2K

where uk satisfies the weak form of the direct problem (Equation (23)). The solution procedure for problem (Equation (24)) is schematically illustrated in Figure 12.3. As for the uniform case in Figure 12.1, an outer loop is performed with the Bertsekas algorithm (Equation (6)). Then determination of Muk , needed for evaluation of the least-squares functional (Equation (24)), is performed only at the converged state of the direct problem, i.e. when problem (Equation (23)) is satisfied at the k-th time (load) step. Next we will briefly resort to the parameter sensitivity qj uk. To this end firstly, the parameter sensitivity qj g…uk † of the weak form (Equation (23)) is determined, at equilibrium, analogously as in the linearization procedure. The resulting expression for qj g…uk † is given in Table 12.1 in a spatial setting, where now a spatial rate of deformation tensor induced by the parameter sensitivity qj uk ˆ vj is defined as dj :ˆ sym …q v j †. Note that qj g…uk † also requires the spatial material operator c of the linearization procedure. Furthermore we defined the sensitivity-load term hqpj s : d ijtˆtk , which excludes dependencies of s via the configuration u. The determination of this term becomes a major task of the sensitivity analysis, and we refer to Barthold [23] and Mahnken and Stein [22] for further details. 286

12.5

Examples

Figure 12.3: Schematic flow chart of the identification process for the non-uniform case with outer and inner iteration loops.

In the practical implementation, firstly, the sensitivity load term is determined in a preprocessing procedure consistent with the underlying integration algorithm. Having obtained qj uk by solution of the associate linear equation, the total derivative qj wk u of any quantity wuk  is performed in a postprocessing procedure. Further details of the procedure are described in Mahnken and Stein [22]. We close this section with the remark that the above results can be easily extended to the enhanced element formulation described by Simo and Armero [25].

12.5

Examples

12.5.1

Cyclic loading for AlMg

In this example parameters for the Chaboche model (Equations (7) and (8)) are determined in the case of an aluminium/magnesium alloy. The underlying experimental data were obtained from project A2 (Prof. Lange [26]). The experiments were performed at room temperature for a cylindrical hollow specimen with an outer radius of 28 mm and a thickness of 2 mm. The specimen were subjected to a periodic strain of an amplitude of max ˆ 0:3% at a strain rate of _ ˆ 0:2% s–1. 110 cycles were generated during the test, however, experimental data are available only for 20 cycles out of these. Youngs modulus has been predetermined as E ˆ 1:09
105 MPa. As an objective function for the inverse problem the simple least-squares function 287

12

Parameter Identification for Inelastic Constitutive Equations

1 f j† ˆ kr…j† 2

~ k22 r

…25†

is minimized, which is the analogue of Equation (19) for the case of strain-controlled ex~ contains data only for 20 periments. Note that due to the incompleteness of the data set r cycles out of the total of 110. The minimization was performed with an evolutionary strategy as described in Schwefel [27] and with the Bertsekas algorithm (Equation (6)). For the first method we used three “parents” and “twenty descendants”, whilst for the latter the BFGS-matrix was used as an iteration matrix and a Gauss-Newton matrix for preconditioning. The computer runs were performed on an IBM-250T. The starting vector and the solution vectors are given in Table 12.2. Concerning the Bertsekas algorithm, three different runs were made. Run 1 and Run 2 were started with the vector in the second column, however, for Run 2, a regularization was performed using the extended functional Equation (5). Here, for B and for B, the unity matrix is chosen, and we set  ˆ 10 5 . It can be seen that no convergence was attained for Run 1 after 2000 iteration steps, whilst minimization with the regularized functional attained convergence after 201 steps. The corresponding minimal eigenvalue of the Hessian at the solution point is 7.62 · 10–2, thus indicating stable results. This also is confirmed by Run 3, where each data was perturbed stochasticly with a maximal value of 10%, and where the effect of this perturbation is negligible. In the last two columns of Table 12.2 results for the evolutionary strategy are shown. After 897 iterations the results are still poor (Run 4), and after 10 256 iterations and 168 h, the value for the objective function is still above that obtained by the Bertsekas algorithm (Run 2) in 24 min. Table 12.2: Cyclic loading for AlMg: starting and obtained values for the material parameters of the Chaboche model for AlMg in case of different optimization strategies and least-squares functions. Concerning Run 3 see Section 12.5.1. ITE and NFUNC denote the number of iterations and function evaluations, respectively. Bertsekas algorithm Start

Run 1

Evolutionary strategy Run 2

Run 3

Run 4

Run 5

0 [–]  0 [MPa] 0 [–]  [MPa]  [–]  [MPa] 0 [MPa]

5.0 · 100 1.0 · 102 1.0 · 102 1.0 · 102 1.0 · 102 1.0 · 102 1.0 · 101

4.582 · 100 2.344 · 102 5.195 · 100 6.233 · 101 0.206 · 10–1 9.840 · 104 0.000 · 100

1.360 · 101 4.242 · 101 4.824 · 100 6.827 · 101 1.542 · 103 4.719 · 101 0.000 · 100

1.360 · 101 4.242 · 101 4.824 · 100 6.827 · 101 1.542 · 103 4.719 · 101 0.000 · 100

4.971 · 100 1.972 · 102 5.115 · 100 6.488 · 101 1.173 · 102 1.792 · 102 8.543 · 10–1

1.250 · 101 3.896 · 101 5.068 · 100 6.697 · 101 1.546 · 103 4.736 · 101 2.238 · 100

…j†  …j† CPU [min] ITE NFUNC Remark

3.491 · 105 – – –

9.620 · 103 – 192 2000 2072 no convergence

2.135 · 103 3.582 · 10–5 24 201 250 regularized

3.335 · 103 – – – – perturbed

8.936 · 103 – 1440 897 17752 –

2.135 · 103 – 605 · 103 10 256 20 493 –

288

–

12.5

Examples

t [s]

Figure 12.4: Cyclic loading for AlMg. Up: Stress versus time for the solution parameter set for the first 18 out of 110 cycles. Note the incompleteness of the experimental data set. Down: Stresses versus strains for three different cycles. The numbers 1, 30, 110 correspond to the specific cycles.

289

12

Parameter Identification for Inelastic Constitutive Equations

Figure 12.4 depicts the stresses versus strains for three different cycles for the solution vector of the material parameters. It can be seen that very substantial agreement of experimental and simulated data is obtained, except for the first cycle, where the model is not able to simulate the horizontal plateau. This explains the relatively high model error for the values of the objective function at the solution point in Table 12.2.

12.5.2

Axisymmetric necking problem

In this section numerical results for the necking of a circular bar are presented. The material of the specimen is a mild steel, Baustahl St52, due to the german industrial codes for construction steel. The experimental data were obtained with a grating method. For this purpose, firstly, grid marks were positioned on the surface of the sample, and these were recorded by a digital CCD-camera at different observation states ti ; i ˆ 1; . . . ntdat in the displacement-controlled experiment. In Figure 12.5, the sample with the grating is shown at four observation states 5, 7, 10, 13 as introduced in Figure 12.7. More details concerning the grating method are given in the contribution of project C2 (Prof. Ritter [6]). The next step concerns the image processing by use of numerical methods in order to obtain the final data for the identification process. This task is described in more detail in the contribution of project C1 (Dr. Andresen [5]). The elastic constants are E ˆ 20 600 kN/cm2 for Youngs modulus and m ˆ 0:3 for Poissons ratio. The material is assumed to be elasto-plastic, modelled by large strain multiplicative von Mises elasto-plasticity with non-linear isotropic hardening summarized in Table 12.3 (see Simo and Miehe [28] for further details). Solution of the direct problem is done with a product-formula algorithm according to Simo [29]. The solution of the inverse problem is based on the general setting described in the previous section. Details

Figure 12.5: Axisymmetric necking problem: photographs with a CCD-camera of the sample and the grating at four different observation states NLST as introduced in Figure 12.7. 290

12.5

Examples

Table 12.3: Large strain multiplicative von Mises elasto-plasticity. s ˆ dev…ln bel † ‡  ln g 1

2 0 …s; ; j† ˆ kdev…s†k …; j† 3

Kirchhoff stress

0 …; j† ˆ 0 ‡ …1

flow stress

1 L …bel †bel 2 2 _ ˆ  3

1

yield function

exp… ††

dev…s† ˆ kdev…s†k

flow rule variable evolution

  0;   0;  ˆ 0

loading and unloading conditions

j :ˆ ‰0 ; ; ŠT ; p ˆ dim…j† ˆ 3

vector of material parameters

Figure 12.6: Axisymmetric necking problem: different levels for spatial discretization.

Table 12.4: Axisymmetric necking problem: starting and obtained values for the material parameters of a mild steel, Baustahl St52, for three different optimization runs. ite denotes the number of iterations. Run 1 (Q1/E4)

Run 2 (Q1/E4)

Run 3 (Q1/E4)

Starting

Solution

Starting

Solution

Starting

Solution

0 [MPa]  [–]  [MPa]

300.0 10.0 800.0

360.26 3.949 436.72

400.0 20.0 2000.0

360.26 3.949 436.72

400.0 20.0 2000.0

346.09 3.951 419.73

…j† [–] Perturbed ite

1.738 · 104

4.210 · 102

3.012 · 104

4.210 · 102

2.685 · 104

no 34

no 34

3.245 · 102 yes 39

291

12

Parameter Identification for Inelastic Constitutive Equations

Figure 12.7: Axisymmetric necking problem: comparison of simulation and experiment. Up: Load versus elongation. Down: Necking displacement versus total elongation.

concerning the sensitivity analysis consistent with the product-formula algorithm of Simo [29] for determination of the load term qpj sk can be found in Mahnken and Stein [22]. An axisymmetric enhanced strain element (Q1/E4) described by Simo and Armero [25] is used in the element formulation, and only a quarter of the bar is considered for discretization using the appropriate symmetry boundary conditions. The object is to identify the 3 parameters b; q; 0 of Table 12.3, which characterize the inelastic behaviour of the material. To this end the following least-squares functional is minimized 292

12.5

Examples

Figure 12.8: Axisymmetric necking problem: comparison of experiment and Finite-Element simulation for the configurations at four different observation states NLST as introduced in Figure 12.7. The circles represent the experimental data.

f j† ˆ

ntdat 1 iˆ1

2

kuj …j†

~ j k2 ‡ u

ntdat 1 iˆ1

2

…w…Fi …j†

F~i ††2 ;

…26†

~j and F~i …j†, F~i , i ˆ 1; . . . ; ntdat denote data for configurations and the total loads, where u respectively. The number of load steps is N ˆ 40, the number of observation states is ntdat ˆ 9, and we have nmp ˆ 12 for the number of observation points. A multilevel strategy is applied in order to accelerate the optimization process by using solutions on coarser grids as starting values on finer grid. In this example the total number of levels is five (Figure 12.6). In Table 12.4 results for the parameters j of three different runs are listed. Whilst Run 1 and Run 2 differ in their starting vectors in order to take into account possible local minima, the purpose of Run 3 is to investigate the effect of a perturbation of the experimental data on the final results [11]. The perturbation was performed stochasticly, whereby each data was varied by a maximum value of 5%. It can be observed that Run 1 and Run 2 give identical results, thus indicating no further local minima. The results for Run 3 differ only slightly from the results of Run 1 and Run 2, thus indicating a stable solution with respect to measurement errors. In Figure 12.7 the results for the total load versus total elongation and maximal necking displacement versus total elongation are compared for simulation and experiment. Figure 12.8 depicts a 2-D illustration of the final Finite-Element grid at different observation states and the corresponding experimental data along the side of the sample. It can be observed that for both quantities of different type, displacements and forces, excellent agreement is obtained after optimization.

293

12

12.6

Parameter Identification for Inelastic Constitutive Equations

Summary and Concluding Remarks

In this contribution the main issues of project B8 of the research period 1991–1996 for identification of parameters for inelastic constitutive equations were presented. The main purpose of the project was to develop a general concept of gradient-based optimization strategies for the uniform case and to extend this strategy to the non-uniform, geometrically non-linear case in order to take into account inhomogeneities of stresses and strains within the sample during the experiment. The main results of the project and the cooperation with other projects from the Collaborative Research Centre (SFB 319) are listed as follows:

Theoretical results •

A gradient-based optimization algorithm has been developed for minimization of a least-squares functional. In particular a projection algorithm due to Bertsekas is applied, which accounts for possible upper and lower bounds for the parameters. Quasi-Newton methods with Gauss-Newton preconditioning are used as iteration matrices.

•

A unified strategy for an analytical sensitivity analysis in the case of uniform stress and strain distributions is obtained, valid for a certain class of constitutive equations with internal variables. The resulting scheme is consistent with the corresponding time integration scheme, and as a main result a recursion formula is obtained. It has been applied to the material models of Chaboche, Steck and Bodner-Partom.

•

A unified strategy for an analytical sensitivity analysis of the variational (Galerkin) form in the case of non-uniform stress and finite-strain distributions is obtained. As for the uniform case it is consistent with the time-integration scheme.

•

Investigations of uniqueness (or stability, respectively) for the inverse problem for certain material models were performed (see Remark 3 of Section 12.2.2). The main consequences are that at least cyclic loading becomes necessary in case of the Chaboche model [11], and for identification of the Steck model experiments have to be performed at different temperatures [10].

•

A regularization technique for stabilization of the least-squares problem based on a priori information has been introduced. Applications were done for the Bodner-Partom model [11].

Numerical results •

294

Parameter identification for the methods of Chaboche, Steck and Bodner-Partom based on experimental data for experiment with uniform stresses and strains was performed.

12.6

Summary and Concluding Remarks

•

Comparative results with the evolutionary strategy showed great advantage of gradient-based schemes with respect to execution time.

•

Parameter identification with the Finite-Element method was performed in the frame of non-linear multiplicative plasticity using experimental data obtained with a grating method.

Co-work with other projects of the Collaborative Research Centre (SFB 319) •

Experimental work was done for a compact tension specimen and a necking problem in cooperation with the projects C1 (Dr. Andresen [5]), C2 (Prof. Ritter [6]), B5 (Prof Peil, Prof. Scheer [7]).

•

Results for the compact tension specimen were published in a joint publication in Der Bauingenieur (see Andresen et al. [30]).

•

For parameter identification for an aluminium/magnesium alloy subjected to cyclic loads, data of project A2 (Prof. Lange [26]) were used.

Concluding remarks The concepts proposed in this paper provide a flexible approach for identification of inelastic material models. This opens the possibility to obtain more insight into the nonuniformness of the samples during the experiment and on the reliability of the numerical results. However, some open questions remain to be considered for future work: •

Further development in this area should take into account phenomena such as damage, localization and temperature-dependent effects, which very often are highly non-uniform during the experiment.

•

In practice we know that measurement techniques possess limited accuracy. A probabilistic investigation of these dispersion phenomena can be done with the Maximum-Likelihood method (see Bard [13], Pugachew [14] and project B1 (Prof. Steck [31]). Furthermore we know that repetition of the same experiment with different samples in general yield different values. Reasons for this scattering of the data may be due to inhomogeneities, load uncertainty, production, nature of the phenomenon. Therefore this randomness also necessitates a probabilistic approach, where e.g. the Bayesian estimation is a common strategy (see Bard [13] and Pugachew [14]). The consideration of the above uncertainties seems to be a major task when performing parameter identification in the future.

•

In the work done so far the effect of discretization errors both in space and in time is not considered, especially in the frame of a proper error control. Therefore it would be of interest to take into account adaptive strategies in the context of the optimization process. 295

12

Parameter Identification for Inelastic Constitutive Equations

•

The shape of the least-squares functional, which is the objective function of the resulting optimization problem, may not be convex, and thus different local minima may occur. In this respect, a more systematic approach could be a hybrid method, that is to combine a stochastic method with our deterministic strategy.

•

The material model at hand might be insufficient for specific physical effects such that an error-controlled adaptive modelling might be necessary.

References

[1] D. Mu¨ller, G. Hartmann: Identification of material parameters for inelastic constitutive models using principles of biologic evolution. J. Eng. Mat. Tech. (ASME) 111 (1989) 299– 305. [2] F. Kublik, E. A. Steck: Comparison of Two Constitutive Models with One- and Multiaxial Experiments. In: D. Besdo, E. Stein (Eds.): IUTAM Symposium Hannover 1991, Finite Inelastic Deformations – Theory and Applications, Springer-Verlag, Berlin, 1992. [3] J. Baumeister: Stable solution of inverse problems. Vieweg, Braunschweig, 1987. [4] H. T. Banks, K. Kunisch: Estimation Techniques for Distributed Parameter Systems. Birkha¨user, Boston, 1989. [5] K. Andresen: Surface-Deformation Fields from Grating Pictures Using Image Processing and Photogrammetry. This book (Chapter 14). [6] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical Measuring Methods. This book (Chapter 13). [7] U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behaviour of Mild Steel Fe 510 under Complex Cyclic Loading. This book (Chapter 10). [8] J.-L. Chaboche: Viscoplastic Constitutive Equations for the Description of Cyclic and Anisotropic Behavior of Metals. Bull. Acad. Pol. Sci. Ser. Sci. Tech. 25 (1977) 33. [9] J. Hadamard: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven, 1923. [10] R. Mahnken, E. Stein: The Parameter-Identification for Visco-Plastic Models via Finite-Element-Methods and Gradient-Methods. Modelling Simul. Mater. Sci. Eng. 2 (1994) 597– 616. [11] R. Mahnken, E. Stein: Parameter Identification for Viscoplastic Models Based on Analytical Derivatives of a Least-Squares Functional and Stability Investigations. Int. J. Plast. 12(4) (1996) 451–479. [12] E. A. Steck: A Stochastic Model for the High-Temperature Plasticity of Metals. Int. J. of Plast. 1 (1985) 243–258. [13] Y. Bard: Nonlinear Parameter Estimation. Academic Press, New York, 1974. [14] V. S. Pugachew: Probability Theory and Mathematical Statistics for Engineers. Pergamon Press, Oxford, New York, 1984. [15] D. P. Bertsekas: Projected Newton methods for optimization problems with simple constraints. SIAM J. Con. Opt. 20(2) (1982) 221–246. [16] R. Mahnken: Duale Verfahren fu¨r nichtlineare Optimierungsprobleme in der Strukturmechanik. Dissertation, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universita¨t Hannover, F 92/3, 1992. 296

References [17] R. Mahnken, E. Stein: Gradient-Based Methods for Parameter Identification of Viscoplastic Materials. In: H. D. Bui, M. Tanaka (Eds.): Inverse Problems in Engineering Mechanics, A. A. Balkama, Rotterdam, 1994. [18] A. K. Miller: Unified Constitutive Equations for Creep and Plasticity. Elsevier Applied Science, London New York, 1987. [19] J. Lemaitre, J. L. Chaboche: Mechanics of solid Materials. Cambridge University Press, Cambridge, 1990. [20] S. R. Bodner, Y. Partom: Constitutive equations for elastic-viscoplastic strain-hardening materials. Trans. ASME, J. Appl. Mech. 42 (1975) 385–389. [21] R. Mahnken, E. Stein: A Unified Approach for Parameter Identification of Inelastic Material Models in the Frame of the Finite Element Method. Comp. Meths. Appl. Mech. Eng. 136 (1996) 225–258. [22] R. Mahnken, E. Stein: Parameter Identification for Finite Deformation Elasto-Plasticity in Prinicpal Directions. Comp. Meths. Appl. Mech. Eng. 147 (1997) 17–39. [23] F. J. B. Barthold: Theorie und Numerik zur Berechnung und Optimierung von Strukturen aus isotropen, hyperelastischen Materialien. Dissertation, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universita¨t Hannover, F 93/2, 1993. [24] R. B. Haber: Application of the Eulerian Lagrangian Kinematic Description to Structural Shape Optimization. Proc. of NATO Advanced Study Institute on Computer-Aided Optimal Design, 1986, pp. 297–307. [25] J. C. Simo, F. Armero: Geometrically Nonlinear Enhanced Strain Mixed Method and the Method of Compatible Modes. Int. J. Num. Meth. Eng. 33 (1992) 1413–1449. [26] W. Gieseke, K. R. Hillert, G. Lange: Material State after Uni- and Biaxial Cyclic Deformation. This book (Chapter 2). [27] K. P. Schwefel: Numerische Optimierung von Computer-Modellen mittels der Evolutionsstrategie. Birkha¨user Verlag, Basel, 1977. [28] J. C. Simo, C. Miehe: Associative Coupled Thermoplasticity at Finite Strains: Formulation, Numerical Analysis and Implementation. Comp. Meths. Appl. Mech. Eng. 98 (1992) 41– 104. [29] J. C. Simo: Algorithms for Static and Dynamic Multiplicative Plasticity that Preserve the Classical Return Mapping Schemes of the Infinitesimal Theory. Comp. Meths. Appl. Mech. Eng. 99 (1992) 61–112. [30] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifikation fu¨r ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Der Bauingenieur 71 (1996) 21–31. [31] E. Steck, F. Thielecke, M. Lewerenz: Development and Application of Constitutive Models for the Plasticity of Metals. This book (Chapter 4).

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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)

13

Experimental Determination of Deformationand Strain Fields by Optical Measuring Methods Reinhold Ritter and Harald Friebe *

13.1 Introduction The experiment is an essential basis for the development of material laws, which describe the inelastic behaviour of metallic materials. It is needed first of all to observe such a behaviour in order to get knowledge of the process of the corresponding methods. Furthermore, the parameters for such material laws must be measured. They can be achieved from experimentally determined, multi-dimensional load-deformation distributions. The experiment is finally necessary for the comparison with the calculation in order to verify the implanted material laws. The requirements of the measuring methods result from the tasks of the experiment.

13.2 Requirements of the Measuring Methods Since the Finite-Element programs, set up with the developed material laws, lead to two- or three-dimensional distributions of the searched values, also such measuring methods are needed, which allow larger object areas to be analysed connected and twodimensional. As one must furthermore plan on a large local change of the material behaviour, a high local resolution of the measuring method is also required. In addition, measuring systems must be developed, with which deformation- and strain fields can be measured even in the transitional areas from inelastic to elastic material behaviour. The measurements should preferably be able to be executed on the original because the model laws for the transfer of results from the model to the original are in general very complicated, especially for inelastic material behaviour. Inelastic material behaviour is often observed at high temperatures up to approx. 1000 8C. The measuring methods should also be usable in such temperature areas. A measurement directly on the testing machine in the testing field during an ongoing test is practical. * Technische Universita¨t Braunschweig, Institut fu¨r Messtechnik und Experimentelle Mechanik, Schleinitzstraße 20, D-38106 Braunschweig, Germany 298

13.3

Characteristics of the Optical Field-Measuring Methods

Finally, measuring methods without contact and interaction are advantageous. All of these requirements lead to the development and the use of the optical field-measuring techniques. The corresponding methods are distinguished by the following characteristics.

13.3 Characteristics of the Optical Field-Measuring Methods First, two-dimensional structured patterns are generated in the form of intensity distributions, which are related to the searched values of the considered object surface. The Figures 13.1 and 13.2 show two such patterns. In Figure 13.1, there are two groups of lines, which consist of straight lines of constant width. They include an angle of 90 8 and form the so-called cross grating. If this is e. g. firmly attached to the considered surface, then both will be deformed in the same way when by a loading. The pattern in Figure 13.2 is produced e. g. if laser light illuminates a rough surface. The remitted beams interfere. As a result of the distribution of different amplitudes and phases, a granular-like intensity distribution comes into existence, which is called the Speckle effect [1].

Figure 13.1: Cross grid.

Figure 13.2: Speckle pattern. 299

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Experimental Determination of Deformation- and Strain Fields

In a second step, such patterns are recorded in the image plane by recording cameras, and their image points are determined from the digital image processing. From this, the searched object values can be determined by a calibrated test set-up [2, 3]. This process is described in the following by the example of the object-grating method, which is suited mostly for the deformation analysis of inelastic material behaviour.

13.4 Object-Grating Method 13.4.1 Principle The precondition is a grating structure, which is firmly attached to the considered object surface. This is recorded from two or more different positions and orientations in reference to the object by cameras [4] (Figure 13.3). By retransforming the digitally determined image coordinates, e. g. the grating intersection points, into the object space, the local vectors of the corresponding object points can be determined. The difference of the local vectors of an object point as a result of a deformation of the object leads to the deformation vector. The change of the distance between two neighbouring object points related to their original distance describes the strain [5]. The stereophotographic set-up of Figure 13.3 represents the simplest arrangement according to the photogrammetric principle [6]. There in general, only the local vector of an object point is of interest. In case of plane deformation of the object, only one recording camera of the optical set-up is needed. With regard to the previously described 3-D object-grating method, then this is called a 2-D object-grating method. The following results have been achieved for the development of this measuring method for the deformation analysis of objects with inelastic material behaviour.

Figure 13.3: Principle of the object-grating method.

300

13.4

Object-Grating Method

13.4.2 Marking First, the technology for marking the object had to be improved as marks are needed, which not only remain attached to the object caused by greater deformations of the object surface, but which also can be recognized at high temperatures. The well-known screen print principle was the basis for the further development [7]. If a mixture of TiO2 -particles of approx. 0.3 lm diameter and ethanol is sprayed through a screen-like mask on the polished object surface, then a grating-like structure comes into existence after the ethanol is vaporized and the mask is removed. This is composed of the blank polished object surface and local limited fields, which each consist of a great number of such TiO2 -particles. Figure 13.4 shows a REM-picture of such a grating structure. By dark field illumination, the object surface appears dark, and each grating field is light due to the diffuse remission of the individual TiO2 -particles. Figure 13.5 is one example for this type of illumination and also the recognizability of the grating at high temperatures [8].

Figure 13.4: TiO2-grating in REM.

Figure 13.5: Cross grating at 850 8C. 301

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Experimental Determination of Deformation- and Strain Fields

a) b) Figure 13.6: Cross grating on a curved object surface; a) not deformed; b) strongly deformed.

This type of grating structure can also follow a very large deformation of the object surface without being destroyed (Figure 13.6).

13.4.3 Deformation analysis at high temperatures The object-grating method is also suitable for deformation analyses at high temperatures. The specimen with the attached TiO2 -grating is enclosed by a radiation heater in the testing machine. It is heated up by infra-red radiation. The visible radiation part serves as a dark field illumination of the test surface with the grating. This is recorded through a glass window built in the wall of the heater by a camera (Figure 13.7). So far, the method has been developed for only in-plane deformation analysis. In the case of the 3-D object-grating method, the heater would have to be furnished with a second window. However, for this, another heater or illumination concept would be

Figure 13.7: Principle of the optical deformation analysis at high temperatures (top view). 302

13.4

Object-Grating Method

necessary in order to produce the necessary dark field illumination of the grating for both cameras also at high temperatures. In addition, the test set-up cannot be calibrated at high temperatures in the existing heater.

13.4.4 Compensation of virtual deformation As a result of complicated initial conditions, the recorded grating images for determining the searched deformation are perhaps superimposed by unwanted rigid body movements of the considered specimen. These can lead to large errors in determining the deformation according to the 2-D grating method. In Figure 13.8, the possible grating distortions are listed. With the aid of an eight parameter pseudo-affine transformation [9] by Equations (1) and (2), it is possible by knowing the parameters a1 to a8 to transform the image coordinates distorted by rigid body movements to undistorted ones:

Figure 13.8: Grating distortions as a result of translatory and rotatory rigid body movements in relation to the camera coordinate system. 303

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Experimental Determination of Deformation- and Strain Fields

xt
a1  a2 x  a3 y  a4 xy ;

1

yt
a5  a6 x  a7 y  a8 xy :

2

The following steps are necessary in order to compensate possible virtual deformations with an otherwise in-plane deformation of the object: A reference object with an attached grating is fastened to the specimen so that the rigid body movements of both are equal, but such that the reference object is not deformed. After orienting the plane specimen surface and reference object parallel to the image plane of the recording camera, the simultaneous recording of the specimen and reference grating takes place. From the coordinates of the reference grating referring to the non-deformed and deformed specimen state, the parameters for the retransformation i.e. its virtual deformation can be determined from Equations (1) and (2). With the aid of the now known retransformation instruction, the true deformation of the specimen can be obtained from the image coordinates referring to both loading states observed in the specimen. Figure 13.9 shows two fields of lines of the same strain, one with (a) and the other without (b) virtual strains.

Figure 13.9: Lines of the same strain a) with and b) without virtual strains. 304

13.5

Speckle Interferometry

13.4.5 3-D deformation measuring A strategy has been developed for the exact calibration of the system for a 3-D deformation measurement [10, 11]. For this purpose, formulations and algorithms have been developed, which are based on the photogrammetric principle. Afterwards, the inner and outer orientation of the cameras used are determined with an appropriate calibration object and the well-known bundle adjustment.

13.4.6 Specifications of the object-grating method In Table 13.1, the exemplary data for the accuracy of the object-grating method when using a camera with 1024 × 1024 pixels has been put together. These refer in one case to the camera and as an example to an object measuring area of 20 × 20 mm2 . A camera with a higher resolution leads either to a higher resolution of the measuring area or at the same resolution to recording a larger object area. This method can currently be used for temperatures up to 1000 8C and measuring areas from 0.1 × 0.1 mm2 to any size. It primarily provides the field of the local vectors of the observed object points and the displacement- and strain fields derived from them. Table 13.1: Example for the accuracy of the object-grating method.

Measuring area: Number of measuring points approx.: Accuracy of the displacement approx.: Reference length of the strain: Accuracy of the strain approx.:

Camera

Object

1024 × 1024 Pixel 75 × 75 0.02 Pixel 13 Pixel 0.2%

20 × 20 mm2 75 × 75 0.4 lm 0.25 mm 0.2%

13.5 Speckle Interferometry 13.5.1 General When developing material laws for the inelastic behaviour of metallic materials, the transition to elastic procedures must be included. Therefore, optical field-measuring methods are needed, with which both areas can be analysed. Since the object-grating method despite of all of its advantages is not sensitive enough for determining strains, which are smaller than 0.3%, a supplementary measuring method had to be developed, with which lower scales are also attainable. 305

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Experimental Determination of Deformation- and Strain Fields

Figure 13.10: Correlation fringes.

The measuring method developed for this is based on well-known optical paths of rays of the Speckle interferometry for measuring the in- and out-of-plane displacement of an object surface [12, 13]. With this, one can e. g. make each one of the both in-plane components of the displacement visible directly in the form of correlation fringes without the out-of-plane component being included (Figure 13.10). The visibility and thus also the possibility of obtaining qualitative values on-line can be done e. g. on a video monitor. This offers the possibility of selectively controlling the deformation processes. The quantitative evaluation of the Speckle interferometric measuring is carried out according to the well-known phase-shift principle [14]. The primary results consist in the phase differences (Figure 13.11), from which the field distributions of the single displacement components can be derived (Figure 13.12). The strain distributions are obtained from the displacement field by numerical differentiation. An optical differentiation can be realized by the shearographic principle. Due to the relative shift of the interfering paths of rays, only a small number of correlation fringes can be obtained in comparison to the previously mentioned displacement mea-

Figure 13.11: Phase pictures of x-, y- and z-displacement. 306

13.5

Speckle Interferometry

Figure 13.12: Lines of constant displacement for x-, y- and z-direction.

suring. This strain information is overlapped by large geometric influences and slope influences. Therefore, no on-line observation of the strain is possible. Studies have shown that shearography is better suited for a qualitative proof of deformations.

13.5.2 Technology of the Speckle interferometry The use of the electronic Speckle interferometry (ESPI) in the material- and construction testing requires a compact and transportable measuring head, which can be directly adapted on a testing machine. The developed and practically tested measuring instrument is based on the application of modern optoelectronic elements such as laser diodes as a light source, piezo crystals for a nanometer exact phase shift of the light and a CCD-camera [15]. For every displacement component, a path of rays of illumination with a laser diode, a phase shift device and a shutter is built in the measuring head. The three displacement directions are recorded nearly simultaneously by rapidly switching between the illumination directions. Switching to the individual sensitivity directions takes only a few milliseconds. In this way, it is possible to record slow running deformation processes in 3-D. The development of the measuring head includes the construction of a control device as well as the programming of the software to control and evaluate the measuring data saved in an adapted computer. Figure 13.13 shows the measuring head. It has the dimensions 250 × 250 × 350 mm3 and is adapted on a testing machine (Figure 13.14). For a quantitative evaluation of correlation fringes by the phase-shift principle, an initial value for the phase order is needed. This is given in the easiest case manually by on-line observation. In principle, the heterodyne method can be used for the automation of the order determination. It is based on using two light sources of different wave307

13

Experimental Determination of Deformation- and Strain Fields

Figure 13.13: 3-D-ESPI measuring head.

Figure 13.14: 3-D-ESPI on a testing machine.

lengths. For this technique, the measuring head has been expanded to two illumination sources for each path of rays. However, this procedure requires, in addition to a very high degree of accuracy for the phase determination, also a stronger protection against disturbing surrounding influences than with the measuring set-up introduced here.

13.5.3 Specifications of the developed 3-D Speckle interferometer The essential specifications of the developed 3-D-ESPI are represented in Table 13.2. This method primarily leads to the field of displacements and by numeric differentiation the strains of the observed object surface.

308

13.6 Application Examples Table 13.2: Specifications of the 3-D-ESPI. Measuring surface: Measuring area Accuracy Strain resolution: Local resolution: Object distance: Measuring head dimensions: Displacement measurements

10 × 7 mm2 to 600 × 450 mm2 out-of-plane: 0.4 . . . 20 lm in-plane: 1 . . . 50 lm out-of-plane: 0.04 lm in-plane: 0.1 lm ca. 10–6 768 × 580 Pixel or 1024 × 1024 Pixel 100 mm to 2000 mm 250 × 250 × 350 mm3 qualitative: on-line quantitative: 3-D by phase evaluation

13.6 Application Examples The application possibilities of the object-grating method and the electronic Speckle interferometry are introduced by three examples. They are related to their use in deformation- and strain analysis at high temperatures, in fracture mechanics as well as welding.

13.6.1 2-D object-grating method in the high-temperature area To examine the inelastic material behaviour at high temperatures, a tensile test was done with a notched tensile specimen (Figure 13.15) according to the set-up in Figure 13.7 at 650 8C. The task was to determine its plane deformation by using the 2-D objectgrating method. To correct possible virtual deformations, the specimen was furnished with a reference object (Section 13.4.4). The essential test data are listed in Table 13.3. In Figure 13.16, a grating section of the non-deformed and the deformed state of the specimen are shown. Figure 13.17 shows the determined strain fields for the in-plane directions.

13.6.2 3-D object-grating method in fracture mechanics This example refers to the deformation- and strain analysis in the area of the crack tip of a fracture mechanic CT-specimen (Figure 13.18). Since in addition to the in-plane strain, also the out-of-plane displacement was searched, the 3-D object-grating method was used.

309

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Experimental Determination of Deformation- and Strain Fields

Figure 13.15: Tensile specimen; a) incl. reference object; b) geometry. Table 13.3: Data from the tensile test at high temperature. Temperature: Material: Measuring method: Dimensions: Testing field: Material behaviour: Displacement measurement:

650 8C Steel: X2CrNi18 9 2-D object-grating method (p = 0.2 mm) H = 100 mm, W= 13 mm 19 mm × 12.2 mm elastic/inelastic in-plane

Figure 13.19 shows the calibrated testing set-up with two CCD-cameras directed at the specimen. In Table 13.4, the essential testing data are listed. From the recorded local vectors describing the different form states, the displacement- and strain fields were derived. Figure 13.20 shows the searched out-of-plane displacement and Figure 13.21 the strain distribution in the tensile direction.

13.6.3 Speckle interferometry in welding For the experimental testing of the elastic and inelastic behaviour of a cold pressure butt welding Copper-Aluminium specimen (Figure 13.22), the electronic Speckle interferometry was applied. The ESPI shown in Figure 13.14, which was adapted to the tensile machine, was used.

310

13.6 Application Examples

Figure 13.16: Section of the tensile specimen; a) non-deformed; b) deformed.

Since the measurement of all three displacement directions is possible for small load intervals, the deformation of the elastic state could be observed on-line and quantitatively recorded as well as the change between two purely inelastic states. In Figure 13.23, the phase images of the in-plane displacements (corresponding to lines of constant displacement) of an elastic, and in Figure 13.24 of an inelastic deformation are shown. Finally, the displacement fields were determined by evaluating the phase images according to the mentioned phase-shift principle, and the 2-D strain distributions were

Figure 13.17: Lines of constant strain a) in x- and b) in y-direction. 311

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Experimental Determination of Deformation- and Strain Fields

Figure 13.18: Geometry of the CT-specimen.

Figure 13.19: Testing arrangement with two CCD-cameras.

Table 13.4: Data of the fracture mechanic test. Material: Measuring method: Dimensions: Test field: Material behaviour: Displacement measurement:

312

AlMg3 3-D object-grating method (p = 77 lm) W= 20 mm, B = 2 mm 6 × 4.5 mm2 inelastic in-plane, out-of-plane

13.7

Summary

Figure 13.20: Out-of-plane displacement field.

derived from these. Figure 13.25 shows the isolinear representation of these strains from the inelastic deformation.

13.7 Summary With the optical field-measuring methods, object values can be determined two- or three-dimensionally. Therefore, they are used especially when contours, deformations and strains of a larger area of the observed object surface should be measured together. Although these methods are based on optical principles, which have been wellknown for a long time, they were first able to be used when compact lasers for the generation of coherent light and efficient PCs including adapted software for digital image processing of a large quantity of optical measuring data were developed.

313

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Experimental Determination of Deformation- and Strain Fields

Figure 13.21: ex-strain field.

Figure 13.22: Geometry of the Cu-Al specimen.

314

13.7

Summary

a) b) Figure 13.23: Phase image in the elastic area a) in x- and b) in y-direction.

a) b) Figure 13.24: Phase image in the inelastic area a) in x- and b) in y-direction.

Optical field-measuring methods are primarily used today in the manufacturingand quality control as well as in the material- and construction testing. In the manufacturing- and quality control, they are used among other things for the automatic recording of form dimensions and to recognize global or locally limited defects. In the material- and construction testing, these methods are preferably used for determining displacement- and strain fields when testing objects with complicated structures with respect to their dimensioning. In connection with the development of material laws for the description of the inelastic behaviour of metallic materials, especially the object-grating method and the electronic Speckle interferometry have been further developed. Here, the main goal was their adaptation for the solution of three essential tasks: Firstly, they should be used for the observation of inelastic processes in order to get knowledge for the development of such material laws. Secondly, parameters had to be measured for these laws. Finally, experimentally determined displacement- and strain fields were required as a comparison to the corresponding achieved data obtained by Finite-Element calculations in order to verify the material laws included in them. 315

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Experimental Determination of Deformation- and Strain Fields

Figure 13.25: Lines of constant strain a) in x- and b) in y-direction from the inelastic deformation.

The essential result of the further development of the object-grating method and the Speckle interferometry for material- and construction testing consists in realizing compact measuring instruments, which can be adapted directly on a testing machine thereby making measurements directly in the testing field possible. This was achieved by applying modern optoelectronic elements such as fibre optics, laser diodes and CCD-cameras. In addition, both elastic and inelastic processes can be recorded from room temperature up to high temperatures (approx. 1000 8C). The measurement is made without contact and interaction. The methods yield primarily the field of the 3-D local vectors (object-grating method) or the field of the 3-D displacement vectors (Speckle interferometry). During the further development of the object-grating method, principles of the near-field photogrammetry were used; the Speckle interferometric measuring method, which is now available, is based on well-known interferometric paths of rays, which have been integrated in a compact 3-D system here. The further developed field-measuring methods have, in the meantime, been used many times in various ways in material testing with respect to their reliability. The results obtained and experiences made have encouraged further tests. It should be tested in this way whether these field methods are also suitable for an online measurement with the goal of process control. Furthermore, it is planned to modify the methods so that larger object surfaces can be recorded in order to e. g. carry out an automated construction or building supervision. 316

References

References [1] R. K. Erf: Speckle Metrology. Academic Press, INC., London, 1978. [2] R. Ritter: Messung von Weg und Dehnung mit Feldmeßmethoden. Materialpru¨fung 36(4) (1994) 130–133. [3] R. Ritter: Optische Feldmeßmethoden. In: W. Schwarz (Ed.): Vermessungsverfahren im Maschinen- und Anlagenbau. Verlag Konrad Wittwer GmbH, Stuttgart, 1995, pp. 217–234. [4] R. Ritter: Moireverfahren. In: C. Rohrbach (Ed.): Handbuch fu¨r experimentelle Spannungsanalyse. VDI-Verlag, Du¨sseldorf, 1989, pp. 299–322. [5] M. Erbe, K. Galanulis, R. Ritter, E. Steck: Theoretical and experimental investigations of fracture by finite element and grating methods. Engineering Fracture Mechanics 48(1) (1994) 103–118. [6] K. Kraus: Photogrammetrie, Band 1: Grundlagen und Standardverfahren. Du¨mmler-Verlag, Bonn, 1986. [7] V. Cornelius, C. Forno, J. Hilbig, R. Ritter, W. Wilke: Zur Formanalyse mit Hilfe hochtemperaturbesta¨ndiger Raster. VDI-Berichte Nr. 731, 1989, pp. 285–302. [8] J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messung von Dehnungsfeldern bei Hochtemperatur-Low-Cycle-Fatigue. Zeitschrift fu¨r Metallkunde 81(11) (1990) 783–789. [9] D. Winter: Optische Verschiebungsmessung nach dem Objektrasterprinzip mit Hilfe eines fla¨chenorientierten Ansatzes. Dissertation TU Braunschweig, 1993. [10] D. Bergmann, R. Ritter: 3D Deformation Measurement in Small Areas Based on Grating Method and Photogrammetry. SPIE’s Proceedings Vol. 2782, Besanc¸on, 1996, pp. 212–223. [11] J. Thesing: Entwicklung eines Versuchsstandes und eine Auswertestrategie zur dreidimensionalen Verformungsmessung nach dem Objektrasterprinzip. Studienarbeit am Institut fu¨r Technische Mechanik, Abteilung Experimentelle Mechanik, TU Braunschweig 1995 (unpublished). [12] A. Felske: Speckle-Verfahren. In: C. Rohrbach (Ed.): Handbuch fu¨r experimentelle Spannungsanalyse. VDI-Verlag, Du¨sseldorf, 1989, pp. 372–397. [13] R. Jones, C. Wykes: Holographic and Speckle Interferometry. Cambridge University Press, 1983. [14] D. Bergmann, B.-W. Lu¨hrig, R. Ritter, D. Winter: Evaluation of ESPI-phase-images with regional discontinuities: Area based unwrapping, SPIE’s Proceedings Vol. 2003, Interferometry VI, San Diego, 1993, pp. 301–311. [15] J. Hilbig, K: Galanulis, R. Ritter: Zur 3D-Verformungsmessung mit einem Elektronischen Speckle Pattern Interferometer (ESPI). VDI-Berichte Nr. 882, 1991, pp. 233–242.

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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)

14

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

14.1

Introduction

Grating methods provide a well-known technique for deriving the shape, the displacement or the deformation of the surface of an object [1]. A regular periodic grating may be projected or fixed on the surface of an observed object. If the surface is flat, a single image is sufficient to derive the physical grating coordinates on the object. For a curved surface, two or more images taken from different locations are needed to calculate spatial coordinates by photogrammetric methods. In the images, the coordinates of suitable marks will be determined. Depending on the experimental set-up, the physical coordinates of a plane surface or the 3-D coordinates of a curved surface will be derived from these data [2]. According to the application, a different type of grating is applied, e.g. a line grating, a point grating, a cross grating or a circle grating. Here, cross gratings generated by two mainly orthogonal bands of lines will be investigated because the related image-processing methods proved to be most stable when analysing largely deformed grating patterns by line-following algorithms. Moreover, the cross point coordinates could be determined in most cases automatically with subpixel accuracy [3, 4]. Projected gratings provide a simple and cheap means to deliver cross points of the surface if only the shape of the object is asked for. However, if the deformation is needed, only a fixed grating is applicable because the displacement of material points from an undeformed and a deformed state must be given. The experimental set-ups and different techniques of fixing cross gratings on a surface will be explained in the next section. The accuracy of the grating coordinates in the images of roughly 0.1 pixel limits the possible applications. Depending on the resolution of the digitized image and on the scale between image and object, one has approximately a relative error of “0.1/ number of pixels”. When a CCD-video camera with 512 pixel is used, an accuracy of 20 lm is obtainable in an object region of 100  100 mm2. The local accuracy of the * Technische Universita¨t Braunschweig, Rechenanlage des Mechanik-Zentrums, Schleinitzstraße 20, D-38106 Braunschweig, Germany 318

14.2

Grating Coordinates

strain within a mesh, however, is almost independent of the scale and is about  0:002 . . . 0:004 if the pitch of the grating lines is about 15–20 pixels and the line width is about 5–7 pixel, which is an optimal assumption. This means that only inelastic deformation of metals in a range larger than 0.01 or 1 percent strain delivers a suitable accuracy. Whole-field methods for elastic strains are based on interference or Speckle techniques. The related optical patterns need quite different image-processing methods as e.g. Moire´ or phase-shift algorithms [5], which will not be treated in this text.

14.2

Grating Coordinates

For deformation analysis, initially periodic gratings will be applied in practice. Their basic patterns are points given by filled circles, crosses given by two intersecting bands of lines or overlapping circles for very large deformation of sheet metal. Each pattern may be distorted to a certain extent during the deformation process. The coordinates are principally defined as the centre of the circle or by the intersection of the two arms of a cross. The geometrical structure of the grating points is assumed to be matrix-like, i.e. a single point is characterized by its row index i and its column index j. Also the digital image of a grating taken by a CCD-camera is stored in a rectangular matrix of e.g. 512 rows (index x) and columns (index y), respectively. Each element contains a grey or intensity value (0 . . . 256; 8 bit), which is proportional to the light intensity of a related small region of the object surface. The grating coordinates in the images are expressed in pixel. By suitable filtering techniques, subpixel accuracy is reachable even in noisy images with low contrast. For a simulated image in Figure 14.1 with a relatively large deformation of the grating, the frequency distribution of the errors [pixel] is given in Figure 14.2. The results are derived with a line-following filter as described in the next section. Obviously, about 85% of the deviations from the theoretical coordinates are less than 0.1 pixel. The larger deviations will be observed in regions with a big curvature.

14.2.1

Cross-correlation method

For less deformed cross pattern, a correlation-filter method has proved to supply cross coordinates with subpixel accuracy [4]. Such a filter will be constructed according to an idealized intensity distribution of a cross, where the filter constants cij are proportional to that distribution. Then, a filtered value f~kl is calculated by a convolution sum:

319

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Surface-Deformation Fields from Grating Pictures Using Image Processing

Figure 14.1: Simulated cross grating.

Figure 14.2: Frequency distribution of deviation [pixel].

f~kl 

K=2 X

L=2 X

cij fk

i;l j

:

iˆ K=2 jˆ L=2

…1†

If this filter is applied to a picture, a smooth correlation function is resulting, which mainly amplifies the cross region and which shows its maximum values in the centre of each cross. The pixel indices (xm ; ym ) of the maximum points could be taken as cross coordinates, however, with a limited accuracy of one pixel. This can be improved if the maximum …~ xmax ; y~max † of a local 2-D polynomial of the form f …~ x; y~† ˆ a0 ‡ a1 x~ ‡ a2 y~ ‡ a3 x~2 ‡ a4 x~y~ ‡ a5 y~2

320

…2†

14.2

Grating Coordinates

is calculated, which approximates the grey values of the central maximum point and its 8 neighbouring points using local coordinates ~ x  x xm ; y~ ˆ y ym †. This technique generally provides an accuracy of  0:1 pixel, and it was applied to different deformation processes showing relatively small changes of the original rectangular cross grating structure.

14.2.2

Line-following filter

For highly deformed cross gratings, the above described technique failed because of the big change of the width and the inclination of the crosses and its arms. For these images, a line-following filter was developed [6]. It is used to determine both bands of cross lines separately with subpixel accuracy. The intersection of these two bands then delivers the wanted cross coordinates with high accuracy. This technique proved to be stable even for strongly deformed and curved grating lines. For initializing the line-search procedure, first, one point on the line and the related line direction must be given. This is performed manually by two click points perpendicular to the line or automatically with a rotational invariant filter [7], which determines the centre of the line and its direction. But this filter is not stable when passing through cross points. Hence, a new elliptic correlation filter of the following form was developed: G…x; y† ˆ Gx …x†Gy …y†Wy …y† ˆ cos

p 3p p x cos y cos y : 2A 2B 2B

…3†

A and B in Equation (3) may be regarded as the principal semi-axes of an ellipse and hence as half the filter length and the filter width, respectively (Figure 14.3). This filter is rotated locally into grating-line direction (Figure 14.4). The hat-like filter form Gx in x-direction amplifies all values on the line and the relatively large extension in that direction guarantees a stable line-following quality. The cosine-like filter function Gy with two negative side lobes is known to detect lines with an intensity distribution similar to the central lobe. The negative side lobes provide a zero filter response if applied to a constant grey level region in the image. The weight function Wy decreases the function Gx steady to zero at y ˆ  B, which provides a smooth filter response.

Figure 14.3: Filter function of an elliptical filter. 321

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Surface-Deformation Fields from Grating Pictures Using Image Processing

Figure 14.4: The rotatable elliptic line searching filter.

This filter is moved perpendicular through a line in 5


7 discrete steps. The resulting filter response takes on a maximum value in the centre of the line. To provide subpixel accuracy, the filter responses are approximated by a second order polynomial in the maximum point. The maximum value of the polynomial then delivers coordinates with subpixel accuracy. A similar method determines the centroid of an area between the filter responses and a suitable threshold to define the centre coordinates of a line. In practice, the filter is moved in column or line direction in the image according to which direction is closer to the perpendicular direction because a shift in an arbitrary angle through a grating line needs complex interpolation procedures. For optimal results, the filter width 2B should be about 2


2:5 times the gratingline width W and a filter ratio A=B ˆ 2


2:5 with the larger values for noisy data guarantees a stable and robust line-following characteristic even through cross points and small gaps in the line. The needed line direction for the filter rotation is derived from the foregoing points by extrapolation. To demonstrate the power of the described technique, a low quality image of the surface of a metal block, deformed by forging, is evaluated (Figure 14.5). The originally rectangular grating pattern of equal line width, etched into the material, becomes strongly curved in some regions. Also, the line width and the intensity distribution of the lines are quite different in horizontal and vertical direction according to the compression and extension of the material. Hence, different filters must be used for each line. Figure 14.6 directly shows the grey values of a small subsection marked in Figure 14.5 by a box, and Figure 14.7 supplies the same information in a 3-D representation. In both Figures, the resulting lines of the filter process are drawn into the images. Obviously, the centre lines are smooth even when the image is noisy and when the line pattern has a very low contrast.

322

14.2

Grating Coordinates

Figure 14.5: Deformed grating on the surface of a metal block deformed by forging.

Figure 14.6: Grey distribution of a cross.

Figure 14.7: 3-D intensity distribution. 323

14

14.3

Surface-Deformation Fields from Grating Pictures Using Image Processing

3-D Coordinates by Imaging Functions

When looking at the 3-D displacement of small flat deformation fields – e.g. crack tips in a volume of 10  10  2 mm3 –, a simplified numerical method can be used instead of stereo photogrammetry to derive the spatial grating coordinates [8]. A calibrated grating on a glass plate is moved exactly parallel and perpendicular to its plane in precise 3 . . . 5 steps DZ in Z-direction. The coordinates …Xijk ; Yijk ; Zijk † form a dense rectangular grid in space, and they are known exactly. Since from each step, an image is recorded, also the related image coordinates are known. Hence, a pair of 3-D polynomial-imaging functions n ˆ f …X; Y; Z†; g ˆ g…X; Y; Z† for each camera can be calculated, which transforms each grid point …X; Y; Z† into an image point …n; g†, i.e. it approximates the real stereo-imaging function. Suitable approximating functions are polynomials with free parameters since that provides a linear-equation system when using a Gaussian least-squares fit. Hence one has: X X bijk X i Y j Z k ; …4† n…a† ˆ aijk X i Y j Z k ; g…b† ˆ where the vectors a ˆ …aijk †; b ˆ …bijk † describe free parameters. Each vector is determined separately by minimization. For a, one has: X …5† …nijk;measured nijk …a††2 ˆ min :

A similar equation holds for g…b†. For each camera, this supplies a set of two functions, which transform any point within the grid volume into an image point. Vice versa, also a point in space …XP ; YP ; ZP † can be calculated if its image coordinates …n1m ; g1m †, …n2m ; g2m † are known in at least 2 images. Then, one has: n1 …XP ; YP ; ZP † ˆ n1m ; n2 …XP ; YP ; ZP † ˆ n2m ;

…6†

g1 …XP ; YP ; ZP † ˆ g1m ; g2 …XP ; YP ; ZP † ˆ g2m :

…7†

These are 4 equations for the 3 unknown spatial coordinates …XP ; YP ; ZP †, which easily are solved by numerical iteration. This technique also works if the image plane in the camera is tilted (Scheimpflug condition), which provides a larger, well-focussed area in space when using stereo cameras. Moreover, the method is easy to program and there proved to be no convergence problems. It was applied to the propagation of a crack tip [9, 10].

324

14.4

14.4

3-D Coordinates by Close-Range Photogrammetry

3-D Coordinates by Close-Range Photogrammetry

14.4.1 Experimental set-up A general method for measuring spatial coordinates of an object grating is adopted from close-range photogrammetry. A measuring device was developed consisting of 2 or 3 stereo cameras in a stiff framework. A movable support, which first holds a calibrating glass plate later on holds the considered objects [11, 12]. Before any measurement can take place, the exterior and the intrinsic orientation of the cameras must be known. The related calibration procedure is based on a high quality cross grating, which is fixed on a plane glass plate. The orthogonal grating lines define a global coordinate system X; Y; Z†; …X; Y† in the plane and …Z† perpendicular to it. With respect to this system, the exterior orientation – the projection centre …X0 ; Y0 ; Z0 † and a rotation matrix R describing the rotation of the local camera system …x; y; z† into the global system – must be determined. The constants of the intrinsic orientation are the focal length c, called camera constant, and lens distortion factors A; B; R0 , described later on. The glass plate is moved in 3 parallel steps in DZ 0 -direction, which might be inclined by small angles x  y against the Z-direction; in Figure 14.8, only a plane configuration is shown. Given the pitch DX DY of the cross grating and an arbitrary shift of the origin …X0  Y0 † in that plane, the coordinates of the spatial grid coordinates are: Xijk ˆ XS ‡ iDX ‡ ax Zk 

…8†

Yijk ˆ YS ‡ jDY ‡ ay Zk 

…9†

Zijk ˆ Zk

…10†

Figure 14.8: Set-up for camera calibration, plane configuration. 325

14

Surface-Deformation Fields from Grating Pictures Using Image Processing

with i  1; . . . ; M; j  1; . . . ; N in the plane and k  1; . . . ; L in shift direction, where q ax  tan x , ay  tan y and Zk  Zk0 = 1 ‡ a2x ‡ a2y ; which is the related distance on

the Z-axis due to a parallel shift of Zk0 . XS ; YS ; DX; DY are given parameters of the cross grating, while x  y , and Zk …k ˆ 1 . . .  L 1† are unknown quantities, which will be determined together with the parameters of the camera orientation in a modified bundle-block adjustment. Instead of moving the cross grating, it is also possible to shift the cameras and fasten the grating if this results in a simpler set-up. In each shifted position, the cross grating is recorded. This means that each camera takes the images of a spatial grid, which approximately coincides with the measuring volume of the device.

14.4.2

Parameters of the camera orientation

Here, only a short summary will be given for the theory of the bundle-block adjustment because it is well-known in the relevant publications [2]. The intrinsic and exterior orientation of a camera in space is described by its projection centre …X0  Y0  Z0 †, the focal length c, the rotation matrix R ˆ …rij † usually given by 3 Euler angles, and the distance …n0  g0 † of the origin in the image plane to the optical axis. Then the transformation from the space coordinates …X Y Z† to the image coordinates …n g† is for each point (i j k) within the grid: n ˆ n0 ‡ c

…X …X

X0 †r11 ‡ …Y X0 †r13 ‡ …Y

Y0 †r21 ‡ …Z Y0 †r23 ‡ …Z

Z0 †r31 ‡ u…n g†  Z0 †r33

…11†

g ˆ g0 ‡ c

…X …X

X0 †r12 ‡ …Y X0 †r13 ‡ …Y

Y0 †r22 ‡ …Z Y0 †r23 ‡ …Z

Z0 †r32 ‡ v…n g† : Z0 †r33

…12†

u…n; g†; v…n; g† describe the lens distortion, which are assumed to be radial symmetric: u…n; g† ˆ …A1 …R20

R2 † ‡ A2 …R40

R4 ††…n

n0 † ;

…13†

v…n; g† ˆ …A1 …R20

R2 † ‡ A2 …R40

R4 ††…g

g0 † ;

…14†

where the radius R is given by: q R ˆ …n n0 †2 ‡ …g g0 †2 :

…15†

R0 is a constant of the objective, usually about 70% of the maximum width of the image plane, and A1 ; A2 are the required distortion parameters. Now p will be defined as the vector of the unknown parameters:

326

14.4

3-D Coordinates by Close-Range Photogrammetry

p  fX0 ; Y0 ; Z0 ; n0 ; g0 ; rij ; c; A1 ; A2 gl ; ax ; ay ; Z1 ; . . . ; ZL 1 ; fXijk ; Yijk ; Zijk gm † ;

…16†

where subscript l means that this set of parameters is repeated for each camera. Further on, a certain number m of spatial coordinates are not used directly, but they are taken to be unknown parameters. This provides a higher accuracy because it couples the parameters of the cameras in a global least-squares fit. …n0ijkl; g0ijkl† are measured image coordinates of the camera l. By a bundle-block adjustment, the unknown parameters in p are determined altogether by minimizing the sum of the squared differences between the measured and the calculated image coordinates. Here, the expression for n is given, a similar expression holds for g: F…p† ˆ

grid camera X X l

ijk

…nijkl…p†

n0ijkl†2 ˆ min :

…17†

To solve Equation (17), it is linearized with respect to an initial parameter vector p0 : F…p0 † ‡

qF…p0 † Dp ˆ 0 qp

for every …i; j; k; l†

…18†

yielding an overdetermined system of linear equations for an increment D p, which is solved by the least-squares method. Starting from suitable initial values p0 , a global iteration is performed until D p is less than a given threshold. p and its standard deviation are the final result. Numerical experience has proved that about half the number of grating points should be dealt with as unknown points to get an optimal convergence and accuracy of the non-linear iteration process. Programming of the bundle-block adjustment and also its application requires a lot of experience, especially choosing suitable initial values becomes a very sensitive task. Meanwhile, commercial products are available [13].

14.4.3

3-D object coordinates

If the parameters of the orientation are known, there are well-known algorithms [2] to determine spatial object points by ray intersection, provided the coordinates of the adjoined points in the stereo images are given. This is generally true for grating images. However, it is also possible to determine whole elements in space without knowing adjoined points if the elements are to be described analytically by a set of free parameters [14, 15]. Practical examples are circles, straight lines and curved lines. Also cylinders and spheres in space can be treated if a sequence of contour points in the images are detected.

327

14

14.5

Surface-Deformation Fields from Grating Pictures Using Image Processing

Displacement and Strain from an Object Grating: Plane Deformation

The strain tensor of a local grating point in a plane (x; y) can be calculated from the displacement of the vectors dr1i i  1; . . . ; 4† in the deformed state and dr0i in the undeformed state, where dri means the connection to the neighbouring 4 grating points. Generally, the displacement is due to a rigid body motion and a deformation of the object. For the calculation of strain in a grating point, first, the centre of the deformed element will be shifted parallel into the related undeformed centre. Then, the variation of the four dr-vectors describes a rotation and the desired plastic deformation or strain. The rotation will be separated from the strain in a theory of large deformation as follows. Assuming, the vectors …dr01 ; dr02 † will be deformed into …dr11 ; dr12 †, then a deformation gradient F is calculated from the linear relations: dr11 ˆ Fdr01 ; dr12 ˆ Fdr02 :

…19†

F will be split into the left rotation tensor R and a right deformation tensor U: F ˆ RU :

…20†

From the right Cauchy-Green tensor G: G ˆ FT F ˆ UT RT RU ˆ UT U ;

…21†

a deformation tensor [16] is given: Uˆ

p G ˆ c0 1 ‡ c1 G

…22†

with 1 c1 ˆ p p ; c0 ˆ c1 det G : trG ‡ 2 det G

…23†

trG ˆ g11 ‡ g22 is the trace of G and det G is the related determinant. The elements of the deformation tensor U     1 ‡ ex exy U11 U12 ˆ …24† exy 1 ‡ ey U21 U22 include the well-known plane-strain components ex ; ey ; exy . In a cross point, 4 strain values according to the four meshes surrounding the point are calculated. The average of these values is taken to define the local strain tensor in the central point. A similar technique can be applied to spatial surfaces if the curvature is relatively small within the considered area. Then, 4 pairs of vectors as in the plane case are taken 328

14.6

Strain for Large Spatial Deformation

to calculate the strain in its centre point. After moving the deformed centre point and the related vectors into the undeformed state, each of the pairs of spatial vectors, forming a triangle, are rotated into an arbitrary reference plane, in which a plain-strain tensor can be calculated as described in the foregoing section.

14.6

Strain for Large Spatial Deformation

14.6.1 Theory The described procedure for calculating the spatial deformation fails if the curvature is large. Then, virtual strains arise already from the rotation of an element into a reference plane. Geometrically based methods for evaluating large strain are published in [17, 18]. Here, an improved method based on a deformation function is proposed. It delivers a deformation gradient for the central point using the 8 neighbouring points in the grating. To derive the deformation function, 4 meshes of a plane grating are considered with basic coordinates x; y; z† as shown in Figure 14.9. The coordinates of the undeformed grating are …xij ; yij ; zij ˆ 0† with indices (i; j ˆ 1; 0; 1) or written as a vector xij ˆ …xij ; yij ; 0†T . The coordinates of the deformed element are xij ˆ …xij ;yij ;zij †T .

Figure 14.9: Undeformed grating element and spatially deformed one. 329

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Surface-Deformation Fields from Grating Pictures Using Image Processing

The total displacement of each point again is given by a rigid body translation, a rotation and a plastic deformation of the element. To eliminate the rigid body motion, first, the central vector  x00 will be subtracted: ~ xij xij  

…25†

 x00 :

Then, a rotation matrix will be determined, which moves the normal vector n of the surface element into the z-axis. Approximately, the normal vector is derived from the cross product of the difference vectors: x10 dx ˆ ~

~ x

10

; dy ˆ ~ x01

~ x

0 1

;

n ˆ dx  dy :

…26† …27†

The unit vector n0 delivers the 3. row of the rotation matrix R. The first row is taken from the unit vector d0x , which means that the deformed x-direction nearly coincides with the related undeformed direction after rotating the element. The 2. row is given by the cross product n0  d0x . Now, the rotated coordinates are: xij x^ij ˆ R

…i; j ˆ

1; 0; 1† ;

…28†

and the related displacement vectors: qij ˆ x^ij

xij :

…29†

Regarding these vectors, a deformation function for each coordinate can be assumed, which describes the displacement from the undeformed to the deformed state: x^ ˆ x ‡ f …x; y† ; y^ ˆ y ‡ g…x; y† ; ^z ˆ h…x; y† :

…30†

The functions f ; g will be approximated by polynomials of second order, e.g.: f …x; y† ˆ f0 ‡ f1 x ‡ f2 y ‡ f3 x2 ‡ f4 xy ‡ f5 y2

…31†

with f0 ˆ 0, since ^x00 ˆ 0. For g, a similar function with coefficients gk is taken. The deformation function h for ^z is less relevant since only the deformation in the tangential plane is considered. From these functions, the deformation gradients in the central point x00 are derived. Hence, one has: x=qy ˆ f2 ; q^ x=qx ˆ 1 ‡ f1 ; q^

…32†

y=qy ˆ 1 ‡ g2 : q^ y=qx ˆ g1 ; q^

…33†

To determine the coefficients fk …k ˆ 1 . . . 5†, a least-squares method applied to the xcomponent of displacement vector qxij requires 330

14.6

X ij

fqxij

Strain for Large Spatial Deformation

! f …xij ; yij †g2 ˆ min :

…34†

In the case of rectangular meshes with pitches Dx; Dy in the basic plane, the matrix of the normal equations can be calculated analytically. With respect to symmetry, the coefficients f1 ; f2 are totally uncorrelated and one has: X X X X y2ij ; …35† yij qxij = xij2 ; f2 ˆ xij qxij = f1 ˆ ij

ij

ij

ij

and similar equations for g1 ; g2 : X X X X y2ij : yij qyij = xij2 ; g2 ˆ xij qyij = g1 ˆ

…36†

ij

ij

ij

ij

Generally, weight coefficients wij may be introduced P according P to the significance of xij qxij wij = wij xij2 . If the totally 8 the displacement values qij , leading e.g. to f1 ˆ ij

ij

edge points are existing, the following simple expressions are resulting (wij ˆ 1): f1 ˆ f…qx11 ‡ qx10 ‡ qx1 1 †

…qx

11

‡ qx

10

‡ qx

1 1 †g=…6Dx†

;

…37†

f2 ˆ f…qx11 ‡ qx01 ‡ qx

…qx1

1

‡ qx0

1

‡ qx

1 1 †g=…6Dy†

:

…38†

11 †

Similar equations hold for g1 ; g2 with qxij replaced by qyij. Taking weight coefficients wij ˆ 0 in the four corner points, the simple central differences, e.g. f1 ˆ …qx10 qx 10 †=2Dx, are resulting. This proved to yield the least errors for noise-free grating coordinates as shown in the next section. Now, the deformation gradient F   f2 1 ‡ f1 …39† Fˆ g1 1 ‡ g2 is given in point (0,0), and the strain can be calculated according to Equations (19) to (24). Generally, already the undeformed element may be spatially curved and no longer rectangular. Then, both elements, undeformed 1 and deformed 2, will be shifted into the origin. There, they are rotated as described in the foregoing section, meaning that the normal vector n1 ; n2 coincide with the z-axis and that the deformed x-directions show into the basis x-direction. Then, a displacement vector is calculated from the differences of the rotated coordinates: qij ˆ x^2ij

x^1ij ;

…40†

and according to Equations (30) to (34), a system of normal equations can be determined. In this case, Equation (34) must be built up and solved numerically because no symmetry of the meshes is provided. 331

14

Surface-Deformation Fields from Grating Pictures Using Image Processing

Figure 14.10: Correcting the displacement vector by arc length.

Figure 14.11: Hat-like deformed metal sheet.

Figure 14.12: Principal strain eI .

14.6.2

Correcting the influence of curvature

Numerical simulation, as described in the next section, shows a systematic error for the strain. In the average, it is always too small because only the tangential projection of the curved surface is used. A noticeable improvement is derived when the arc length of 332

14.6

Strain for Large Spatial Deformation

Figure 14.13: Difference of eIth

eI without compensation of curvature.

Figure 14.14: Difference of eIth

eI with 70% compensation of curvature.

deformed surface is considered in the x-z- or y-z-plane, respectively. This can be performed approximately by calculating a second-order polynomial of displacement in zdirection through the deformed but translated and rotated points x^ 10 , x^00 , x^10 in Figure 14.10. Using numerical integration, the arc length of this curve is determined. Then, it proved to give optimal deviations with almost zero average when adding 50% to 70% of the difference between the length of the curve and its projection to the displacement in x-direction. A similar procedure holds for the y-direction. The errors of the strain could be decreased by 20% to 50% depending on the pitch of the grating.

14.6.3

Simulation and numerical errors

The spatial strain procedure was tested on a hat-like deformed metal sheet. The deformation functions u…r† and w…r† in z-direction were assumed to be radial symmetric: u…r† ˆ cu sin …2pr=R0 † ; w…r† ˆ cw cos …pr=R0 † ;

…41† 333

14

Surface-Deformation Fields from Grating Pictures Using Image Processing

Figure 14.15: Relative maximum error of strain eIth

r

eI with respect to grating pitch.

p x2 ‡ y2 † ;  ˆ arctan …y=x† :

…42†

Then, the deformed hat is given by: x^ ˆ x ‡ u…r† cos …† ; y^ ˆ y ‡ u…r† sin …† ; ^z ˆ w…r† :

…43†

The exact principal strains in r-direction are: eIth

q ˆ …1 ‡ du=dr†2 ‡ …dw=dr†2

1 ; eIIth ˆ u=r :

…44†

Regarding the following figures, the parameters were cu ˆ 0:5, cw ˆ 4; R0 ˆ 5; Dx ˆ Dy ˆ 0:33. In Figure 14.11, the hat-like deformed sheet is given for 30  30 lines. Figure 14.12 shows the principal strain eI calculated according to Sections 14.5 and 14.6.1. Figure 14.13 demonstrates the related error eIth eI if no compensation of the curvature is taken into account. Obviously, the difference is always positive with an average value of 0.006. Figure 14.14 shows the same deviation with 70% curvature compensation due to Section 14.6.2. The average value is reduced to 0.002 and the maximum error from 0.011 to 0.007. Hence, in this example, the maximum deviation from the theoretical values is always less than 0.7% of the maximum strain eI ˆ 0:96. A 100% compensation does not reduce the maximum values but increases the minimum values to –0.007. Figure 14.15 shows the influence of the grating pitch on the relative maximum error for the above chosen example. The error increases with the pitch in the interesting 334

References interval approximately like a second-order polynomial. For a 10  10 grating, the related maximum error becomes already 6%.

14.7

Conclusion

The grating techniques, as described in this chapter, were applied to a large variety of deformation processes, e.g. crack-tip propagation, low-cycle fatigue, high-temperature creep, cold-welded zones of Cu-Al-specimen, necking of cylindrical tension specimen, holes in thin-sheet metal, crystal deformation and many examples more. This was performed mainly for research purposes to get insight into deformation processes, to test Finite-Element results or to determine the parameters in new constitutive laws for inelastic deformation [19, 20]. With some modifications, the photogrammetric equipment and the software also were applied to measure the dimension of industrial parts [8] to determine the radius of cutting tools and to derive the contour of a fossil fish. A new field of practical applications comes from metal forming. Especially for sheet metal forming of the body works of cars, the grating methods are used for failure analysis. Often, it must be decided whether the characteristics of the material or the shape of the pressing tool cause the defects on the surface. Also, the influence of oil films on the flow of the material and on the friction between tool and sheet metal shall be investigated to optimize the forming process and the mechanical equipment. For these future applications, the image-processing hardware and software must be improved further with respect to the following topics: • • • •

developing fully automatic programs for incrementally deformed pattern series, improving the portability of the software as to support PC’s and workstations, looking for low-cost hardware equipment for technology transfer into industry, and implementing state of the art graphical user interfaces and plotting software.

These goals seem to be in the reach within the next years because the computing power still increases every year, thus allowing more sophisticated algorithms for automatic image processing.

References [1] P. J. Sevenhuijsen, J. S. Sirkis, F. Bremand: Current trends in obtaining data from grids. Exper. Techn. 27 (1993) 22–26. [2] K. Kraus: Photogrammetrie, Vol. 2. Du¨mmler, Bonn, 1984. [3] J. S. Sirkis, T. J. Lim: Displacement and strain measurement with automated grid methods. Exp. Mech. 31 (1991) 382–388. 335

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[4] K. Andresen, B. Hu¨bner: Calculation of Strain from an Object Grating on a Reseau Film by a Correlation Method. Exp. Mech. 32 (1992) 96–101. [5] K. Andresen, Q. Yu: Robust phase unwrapping by spin filtering combined with a phase direction map. Optik 94 (1993) 145–149. [6] Z. Lei, K. Andresen: Subpixel grid coordinates using line following filtering. Optik 100 (1995) 125–128. [7] P.-E. Danielsson, Q.-Z. Ye: A new procedure for line enhancement applied to fingerprints. Report of Linko¨ping University, Dept. of Electrical Engineering, Linko¨pin, Sweden, 1983, p. 581. [8] K. Andresen: 3D-Vermessungen im Nahbereich mit Abbildungsfunktionen. Mustererkennung 92, 14. DAGM Symposion, Dresden, 1992, pp. 304–309. [9] K. Andresen, B. Kamp, R. Ritter: 3D-Contour of Crack Tips Using a Grating Method. Second International Conference on Photomechanics and Speckle Metrology, San Diego 1991. SPIE Proceedings Vol. 1554A (1991) 93–100. [10] K. Andresen, B. Kamp, R. Ritter: Three-dimensional surface deformation measurement by a grating method applied to crack tips. Opt. Eng. 31 (1992) 1499–1504. [11] K. Andresen, Z. Lei, K. Hentrich: Close range dimensional measurement using grating techniques and natural edges. SPIE 2248 (1994) 460–467. [12] K. Andresen, K. Hentrich, B. Hu¨bner: Camera Orientation and 3D-Deformation Measurement by Use of Cross Gratings. Optics and Lasers in Engineering 22 (1995) 215–226. [13] CAP – Combined Adjustment Program, Users Manual. Fa. Rollei Braunschweig, FRG, 1989. [14] K. Andresen: Ermittlung von Raumelementen aus Kanten im Bild. Zeitschrift fu¨r Photogrammetrie und Fernerkundung 59 (1991) 212–220. [15] F. Neugebauer: Calculation of curved lines in space from non-homologues edgepoints. Optical 3-D Measurement Techniques III, Eds. Gruen/Kahmen, Vienna 1995, pp. 506–515. [16] J. Stickforth: The Square Root of a Three-Dimensional Positive Tensor. Acta Mechanica 67 (1987) 233–235. [17] F. Bredendick: Methoden der Deformationsermittlung an verzerrten Gittern. Wiss. Zeitschrift der Techn. Univ. Dresden 18 (1969) 531–538. [18] L. Eberlein, P. Feldmann, R. V. Thi: Visioplastische Deformations- und Spannungsanalyse beim Fliesspressen. Umformtechnik 26 (1992) 113–118. [19] K. Andresen, R. Ritter, E. Steck: Theoretical and Experimental Investigations of Fracture by FEM and Grating Methods. Defect Assessment in Components – Fundamentals and Applications. Mechanical Engineering Publications, London, 1991, pp. 345–361. [20] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifikation fu¨r ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Bauing. 71 (1996) 21–31.

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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)

15

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

15.1

Introduction

The engineering problem of the presented research project is based on the design and loading of the aerospace structure shown in Figure 15.1. Although the example in Figure 15.1 depicts a possible structure of a hypersonic vehicle, the construction is typical for supersonic and common subsonic transport aircrafts. To examine the basic phenomena of the load-carrying behaviour in the postbuckling range, the analysis of such structures can be reduced to a simplified mechanical model of an initially flat, shear-loaded panel. For practical reasons, our investigations were limited to aluminium (Al2024-T3) panels at ambient temperature and 200 8C. Within the high postbuckling regime or at high load levels and elevated temperatures in supersonic vehicles, moderate inelastic strains occur and the behaviour of the considered structure becomes geometric and physically non-linear. Due to the high complexity of the problem, analytical investigations must be accompanied by tests in order to validate the numerical model. It is based on the FiniteElement method and therefore can be easily applied to different geometries and boundary conditions. The main problem of the numerical model is the choice of suitable material models since no universal material model for the description of the inelastic behaviour of arbitrary metallic materials exists. To simplify the adaption of the presented numerical model to different materials, the used solution algorithm is designed to allow a very easy implementation of different material models. In case of the considered aluminium alloy, the performance of several material models is examined for the rate-independent plasticity at ambient temperature and visco-plastic behaviour at elevated temperature. The identification of their parameters from suited material test results is demonstrated. All shear tests, including quasistatic monotonic, cyclic and creep- and relaxation tests at elevated temperatures, are conducted with a specially designed test set-up * Technische Universita¨t Braunschweig, Institut fu¨r Flugzeugbau und Leichtbau, Hermann Blenk Straße 35, D-38108 Braunschweig, Germany 337

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.1: Typical structure.

(“PApS”, Figure 15.2) generating pure shear load with clamped boundary conditions. The rigid edges of the picture frame are pin-jointed, where the pins are located exactly at the corners of the square specimen. The pins are parted so that the tested area of the specimen forms a square field with no cutouts with a dimension of 500 × 500 mm2. Therefore, the comparison of numerical and theoretical results achieved with the mechanical model shown in Figure 15.3 are not influenced by uncertainties in the assumption of the geometry or the boundary conditions. The test set-up is equipped with nine infrared radiators in front of the shear panel and eight heating elements, which are placed directly on the edges of the shear frame. In combination with five separate temperature controllers, a nearly constant temperature distribution can be achieved in the tested area of the panel up to 200 8C. Until now, 78 monotonic and cyclic tests at ambient and elevated temperature and different panel thicknesses have been performed on this test set-up.

Figure 15.2: Test set-up PApS. 338

15.2

Numerical Model

Figure 15.3: Mechanical model.

15.2

Numerical Model

15.2.1

Finite-Element method

Two partly different formulations of the fundamental equations and affiliated solution methods are used. The behaviour of the considered material at room temperature can be described by means of rate-independent material models for spontaneous plasticity. This type of non-linear material models are implemented in the framework of the geometrically non-linear static equations of the plate theory, and the problem is solved by incremental-iterative methods. At the other hand, the visco-plastic problem introduces real time-dependency with higher demands regarding the time-integration accuracy and stability. Therefore, a more closed formulation of the fundamental equations of the continuum theory and the constitutive equations is used to apply a suitable time-integration method. Both methods base on the following assumptions of the continuum theory: Adopting a total Lagrange formulation with an additive decomposition of the strain tensor, large deformations but only small strains are admissible. The mixed variational principle, which is used to derive the finite elements, bases on the Kirchhoff-Love plate theory. This theory yields reasonably good results since the considered panels are sufficiently slender. All material models are implemented in the numerical model by means of the normality rule of the classical theory of plasticity, applying a v. Mises type of inelastic potential. A mixed principle is chosen since such a formulation produces displacements and stresses with the same degree of approximation as primary unknowns, and therefore provides some advantages concerning the computational effort in the treatment of the non-linearities. All primary unknowns of the described variational formulations are approximated with bilinear polynoms, yielding a four-noded plate element.

339

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Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

15.2.1.1 Ambient temperature – rate-independent problem The used variational functional is derived from the principle of virtual displacements with the strain-displacement relation as a restriction and reads for a certain state t: t

dA
d


 F

t 

rc

 1 t 1 r C r dF 2

Z

dv t p dF ˆ 0 :

…1†

F

This principle is transformed into an incremental form, yielding the linear stiffness matrix, the tangent and the secant matrix. The solution for each increment is achieved by an arc length-controlled modified Newton-Raphson iteration. In this form, the material law in the incremental form De ˆ DDr is comprised in the tangent matrix, and the non-linear constitutive equations of the rate-independent material models can be easily included. The in-plane, coupling and bending stiffnesses of the plate material are determined by integration of the actual elastic-plastic tangent moduli D given by the used material model over the plate thickness. This integration assumes discrete layers with constant properties in thickness direction, and the description of the material behaviour is reduced to the plain stress constitutive equation in each layer. The elastic-plastic tangent modulus is updated only once for each load increment using the results of the previous load step, yielding an Euler-Cauchy type of integration of the non-linear constitutive equations. This method considerably reduces the numerical effort. Furthermore, no differentiation of the constitutive equations is needed as it would be the case in implicit integration methods, and therefore, the material model can be changed very easily. Stability problems in the integration of the used material models were never observed, and it can be shown that the error in the solution for the used constitutive equations remains sufficiently small since the magnitude of the load increments, which is restricted by the geometric non-linearities, is small enough. Details can be found in [1] and [2]. The described solution method is in principle capable of calculating snap-through effects. However, due to the quasistatic basis of the method, a so-called instable equilibrium path connects the starting- and the end point of the snap-through. This path represents a fair approximation only for moderate snap-throughs. Simulations of severe snap-throughs, which occur in the range of unloading during cyclic shear tests after very high load amplitudes, lead – in connection with the development of plastic strains during the snap-through – to obviously wrong solutions and numerical problems. To improve the capabilities of the used method, the described Finite-Element formulation was extended to calculate dynamic effects by solving the complete equation of movement. For simplicity, the damping matrix is formed by a linear combination of the used consistent mass matrix and the system-stiffness matrix. The magnitude of damping is fitted to experimental results. The accelerations and the velocities are derived from the displacements using the Newmark scheme. To reduce the numerical effort, the dynamic method is only used if the quasistatic method detects a limit point in the loading path and a snap-through is starting. In this point of loading, the determinant of the system matrix changes its sign. When after the snap-through, the velocities of the structure are small enough, the quasistatic method is used again. 340

15.2

Numerical Model

15.2.1.2 Elevated temperature – visco-plastic problem The temporal derivative of the basic variational principle (Equation (1)) is used to derive the Finite-Element formulation. To achieve a closed formulation of the problem, the equations related to the material model are included in the principle by means of Lagrange multipliers. In case of the Chaboche model, those equations are: the consistency condition, the overstress function and the evolution equations for isotropic and kinematic hardening. The corresponding Lagrange multipliers can be determined by the requirement that each term of the variational principle has to form an energy. Like in the rate-independent problem, a layered model is adopted. However, the plate stiffness is formulated directly in terms of the primary unknowns. In the sense of a “rate approach” [3], the primary unknowns are the velocities, the temporal derivatives of the stresses, respectively the membrane forces, and the bending moments plus the temporal derivative of the visco-plastic potential and the effective strain rate of each layer. The complete visco-plastic problem is solved by a predictor-corrector method similar to a midpoint-type time-integration algorithm. One predictor and one corrector step is used for solving the equations in one time-increment. A comprehensive discussion of this method is given in [4]. This time-integration scheme was chosen in order to avoid the analytical or numerical determination of the tangent-stiffness matrix needed in the framework of implicit time-integration schemes to establish the Newton-Raphson iteration. Therefore, this method reduces the difficulties of the implementation of new constitutive equations. An automatic time-step control is established by limiting the magnitude of the increment of equivalent total strain within each time-step. The critical value for this increment of equivalent total strain is determined by numerical experiments.

15.2.2

Material models

All investigations are made for the aluminium alloy 2024-T3 (AlCuMg2/3.1354T3), where T3 denotes the rolling and prestraining pretreatment. The results of tension tests at room temperature and elevated temperature are given in Figure 15.4. There is a distinct plastic orthotropy at room temperature, which must be taken into consideration in the material model. This orthotropy vanishes at higher temperatures. The results of the tests at 200 8C with different loading rates give an impression of the changed mechanical properties of the material at elevated temperatures.

15.2.2.1 Ambient temperature – rate-independent problem The cyclic elasto-plastic behaviour of the considered aluminium alloy 2024-T3 at ambient temperature can be described by two or more surface rate-independent material models based on the classical theory of plasticity. In the conducted investigations, the performance of twelve different material models, respectively combinations of their basic characteristics, were examined. 341

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.4: Tensile tests at room temperature and at 200 8C.

Those basic characteristics are: • • • •

the number (Mroz) and shape of the yield surfaces, i. e. v. Mises, Hill, Rees, Doong-Socie, the rule for the translation of the yield surface, respectively kinematic hardening, i. e. Mroz, Phillips-Weng, Tseng-Lee, the rule for the isotropic hardening, i. e. Ellyin, and the way, the plastic tangent modulus is determined, i. e. Dafalias or McDowell.

A more detailed description of the used material models is given in [1, 5, 6]. The influence of the translational rule on the performance material models shall be discussed in more detail. All of the described models use v. Mises-type yield surfaces. Deformable yield surfaces shall not be discussed here in more detail. The Mroz model for the material used is sketched in Figure 15.5. Four surfaces are located in the stress space. The inner one surrounds the elastic region and is of the Hill-type. This type of yield surface is based on the v. Mises surface, but can be deformed by adjusting additional shape parameters. All surfaces can move in the stress space in the sense of kinematic hardening expressed in terms of the backstress tensor. This effect is substantial for the representation of the Bauschinger effect occurring in cyclic loading of metallic materials. Furthermore, the description of the plastic anisotropy is made possible by adjusting the starting values of the backstress tensor. Therefore, kinematic hardening is necessary even for the simulation of tests with monotonic loading. Plastic loading takes place when the stress point is located on the yield surface and moves in an outward direction. Stress states beyond the yield surface are not admissible. This is controlled in all types of material models of this category by the socalled consistency condition. Therefore, the movement of the yield surface during plastic loading with kinematic hardening is restricted by this condition. The direction of the movement of the yield surface has to be established separately. For the Mroz model, 342

15.2

Numerical Model

Figure 15.5: Mroz model.

the translation of the yield surfaces is chosen in a way that subsequent yield surfaces get into contact in points with equal directed normals. In other words, the surfaces approach each other tangentially. An isotropic hardening during plastic loading, which means an expansion of the surfaces, is possible. This type of hardening describes the stabilization of the materials hysteresis and is controlled by means of the accumulated effective plastic strain. Its contribution to the entire hardening is small since the considered alloy shows a rapid stabilization under cyclic loading. Each surface of the Mroz model is connected with a constant plastic tangent modulus. This causes a piecewise linear approximation of the inelastic stress-strain relation, which is a central advantage of this model since the initial position of the yield surfaces, their diameter, the shape parameters of the yield surfaces and the plastic tangent moduli can easily be determined as the graphic in Figure 15.5 demonstrates. Unfortunately, the storage capacity needed for this model in numerical analyses is comparatively high since three backstress tensors have to be stored. The second described material model is the Tseng-Lee model. It consists of only one yield surface and a so-called memory surface in the stress space. The yield surface shows kinematic and isotropic hardening, whereas the memory surface can only expand in the sense of isotropic hardening. The translational rule for the kinematic hardening is chosen in an approximation of the test results of Phillips [7]. These results indicate that the translation on the yield surface is directed along the actual stress increment. To avoid an intersection of the yield- and the memory surface, Tseng and Lee [8] formulated their translational rule as a combination of the “Phillips direction” and a direction of movement, which is determined by the distance measure between the actual stress point on the yield surface and a point on the memory surface with the same outer nor343

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

mal. Unfortunately, the Mroz direction predominates the kinematic hardening in some constellations of the position of the yield surface and loading direction. To improve this behaviour, an own modified formulation for a combined translational rule was developed, which enforces the movement along the “Phillips direction”. In both models, the tangent modulus is determined as a function of a distance measure between the actual stress point and a point on the memory surface. Considering non-proportional hardening effects, the distance measure used in the equation of Dafalias for the tangent modulus is directed along the stress increment. If the yield surface is in contact with the memory surface and the distance measure becomes zero, the tangent modulus is determined by a simple Ramberg-Osgood power law, which is fitted to the properties of the material under monotonic loading. The usage of two different equations for the tangent modulus – for repeated loading and for monotonic loading – is affirmed by tests conducted by Phillips [7]. The fourth model in this comparison is formed by a two-surface model with a pure Mroz-type translational rule and the Dafalias equation for the tangent modulus. All parameter identifications for the considered models are performed by means of stochastic, respectively evolutional optimizing methods, using a least-square formulation of the object function. To use the material models for the calculation of the behaviour of the shear-loaded panels, their parameters are identified simultaneously by the results of four uniaxial tension-compression tests with specimen made of the considered aluminium alloy in the typical pretreatment state. The results of the tension-compression tests are fairly good approximated by all four described material models. An example is given for the Tseng-Lee model in Figure 15.4, where only the first tensile loadings are depicted. Since wide areas of cyclic shear-loaded and buckled panels exhibit non-proportional paths for one component of normal strain and the shear strain, special emphasis is laid on the capabilities of the material models in reproducing non-proportional hardening effects. Results of strain-controlled tension-torsion test (provided by G. Lange et al. [9], Institute of Material Science, Techn. Univ. Braunschweig) are used to examine the performance of the material models considering the effect of non-proportional hardening. The results of the simulation of a typical non-proportional strain path are shown in Figure 15.6. The first loading of the tubular specimen leads to pure shear. Then, the specimen is unloaded and a combined tension-torsion loading starts. After total unloading, this load cycle starts again. Obviously, the models using a pure Mroz-type translational rule show a poorer correlation with the test data, especially in the development of the tensile stress. In contrast, the models using the combined translational rules are able to give a good simulation of the behaviour even up to high cycle numbers. With one indicated exception, all results given in this paper, concerning the behaviour of shear-loaded panels at ambient temperature, are calculated with the described Tseng-Lee material model.

15.2.2.2 Elevated temperature – visco-plastic problem The visco-plastic problem is treated with unified, respectively overstress material models. The models of Steck [10] (in an isothermal formulation, see Equations (2) and (3)) 344

15.2

Numerical Model

Figure 15.6: Results of tension-torsion tests compared to different material models.

and Chaboche [11] with several modifications are in examination. Several creep, stressand strain-controlled uniaxial tests were performed (supported by K.-T. Rie et al. [12], Institute of Surface Engineering and Plasmatechnology, Techn. Univ. Braunschweig) to provide results for the process of parameter identification. This is necessary because of the strongly underdetermined character of such a problem, and since it is known that a parameter identification with only one distinct type of test result usually cannot represent the properties of the material sufficiently. •

Steck model:

e_ in
A1 e

V2 riso

‰sinh …V1 reff †ŠN sign …reff † with r_iso ˆ h1 e iso

r_ kin ˆ h2 e…V1 r •

e_

A3 sinh …V6 rkin †

je_ in j

iso

A2 eV4 r ;

and reff ˆ r

rkin :

…2† …3†

Chaboche model:

 N f ; for e_ ˆ K in

V5 rkin sign …reff †† in

V3 riso

f >0

with f_ ˆ D…T

and R_ ˆ c …Q

R†e_ in :

X†

 2 _ R ; X ˆ C aein 3

X e_ in

 …4†

Figure 15.7 depicts the results for the simulation of three creep tests with the Steck model and the basic Chaboche model (Equation (4)). The parameters of both models are simul345

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.7: Simulation of creep tests with the Steck and the basic Chaboche material model.

taneously identified from the three creep tests. The creep rates of the uniaxial tests are of the same magnitude as the typical rates occurring in certain spots of the corresponding shear-panel tests. However, the results of Finite-Element calculations performed with these sets of material parameters are not in good correlation with the shear tests. It is assumed that better results – regarding the calculation of the behaviour of the shear panels – can be achieved if a fair approximation of a set of creep tests, tension tests at higher loading rates, and a special transient, stress-controlled test are achieved. The simultaneous identification of the parameters of the material models from those tests is performed by optimizing methods using a combination of gradient and stochastic algorithms. Unfortunately, parameter identifications with both models are not successful. Numerical experiments show that the main problem is the reproduction of the tension test at high loading rates (the result of a test at 1 MPa/s is depicted in Figure 15.4). To avoid this problem, the overstress function of the Chaboche model is modified by adding a term accounting for additional, rate-independent inelastic strains. This “overlaying method” is among others described in [13] and [14]. Within the engineering approach, the interaction between both parts of inelastic strains is neglected. Good results are achieved with a very simple approximation of those rate-independent strains by a type of Ramberg-Osgood power law for isotropic hardening. It is the same approximation as it is assumed for the monotonic hardening regime within the rate-independent Tseng-Lee model described above. With this additional term, two new parameters are introduced into the material model. A very good starting value (regarding 346

15.2

Numerical Model

Figure 15.8: Simulation of creep tests with the modified Chaboche model (combined identification).

further parameter identification of the complete material model) for those parameters can be found manually within a few iterations if it is assumed that the major part of the inelastic strain, occurring in a fast tension test (see Figure 15.4), is described only by this Ramberg-Osgood term. The approach without kinematic hardening is possible here since no inelastic orthotropy is present. As a matter of fact, no cyclic effects concerning this part of inelastic strains can be simulated. The results of a simultaneous identification of the parameters of this model are given in Figure 15.4 for the tension tests, in Figure 15.8 for the creep tests, and in Figure 15.9 for a transient test. Additional tests with notched specimen (measurement and processing of strain distribution was provided by R. Ritter and H. Friebe [15], Institute of Measurement Techniques and Experimental Mechanics, Techn. Univ. Braunschweig) were conducted to characterize the multiaxial behaviour of the considered material and to examine the accuracy of the material models in the multiaxial case. Figure 15.10 shows the measured strain distribution (left-hand side) and the numerical results (right-hand side) achieved with the modified Chaboche model for a creep test after 12 hours. The specimen was loaded to a nominal value of tensile stress of 180 MPa in the smallest cross section. Considering the resolution of the optical measuring method of 0.1–0.2% strain, the numerical results are in very good correlation with the test results. All results given in this paper concerning the behaviour of shear-loaded panels at elevated temperature are calculated with the described modified Chaboche material model. 347

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Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.9: Simulation of transient tests with the modified Chaboche model (combined identification).

Figure 15.10: Test and numerical results for a creep test with an inhomogeneous specimen. 348

15.3

Experimental and Numerical Results

15.3

Experimental and Numerical Results

15.3.1

Test procedure

All tests conducted on the test set-up PApS at ambient temperatures are incrementalstep tests. The temporal course of loading of the specimen at elevated temperature is controlled by a computer. In case of tests at elevated temperatures, the specimen is mounted loosely in the shear frame at ambient temperature first. Then, the shear frame and the panel are heated. After that, the 100 screws, which clamp the specimen between the halves of the shear frame, are tightened. This procedure avoids a preloading of the panel and the occurrence of significant thermal buckles due to the different thermal strains of the aluminium panel and the steel shear frame. Nevertheless, the imperfections of the panel at the beginning of the mechanical loading are slightly higher than in tests at room temperature. The angle of shear (see Figure 15.3) and the central deflection of the buckled panel are determined by inductive displacement transducers. Additionally, strain gauge rosettes are positioned at different points of the shear panels, and in some cases, the entire displacement field has been measured by means of engineering photogrammetry.

15.3.2

Computational analysis

The shear panels are discretized with regular meshes as it is shown in Figure 15.11. Simulations of shear-angle-controlled tests are conducted with prescribed deformations of the edges of the panels. The resulting load is obtained by integrating the stress resultants along the edges. In case of load-controlled tests like creep tests, the rigid clamping of the panels is simulated by introducing constraints for the nodal deformations on the edges into the Finite-Element equation system. The load is then applied as a single force on one corner of the panel. An examination of the convergency of the spatial discretization of the panels showed that regular meshes with 20 × 20 elements are sufficient since a further refinement gives no significant improvement of the results for the central deflection (Figure 15.11) or the effective shear load. Since the computational effort – especially for the solution of the visco-plastic problem – is very high even 16 × 16 element meshes are used. Meshes with refinements near the clamped boundaries of the panel improve the solution only when the whole mesh is very coarse. It has been shown by comparative analyses that the idealization of the panel with ten layers in thickness direction is sufficient for the rate-independent problem. Typical creep analyses are conducted with only seven layers without a significant loss of accuracy in the global and local results since the distribution of the inelastic strains along the cross section of the panels is more smooth than in the spontaneous plasticity problem. The determination of the critical time-step sizes related to accuracy and stability shall not be discussed here in detail. Numerical experiments show that the creep behaviour of the shear panels is reproduced within acceptable accuracy for the current param349

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.11: First loading of shear panels with a thickness of 1.4 mm.

eters of the material model if a time-step of about 10 s is used at the very beginning of the creep process. This comparatively small time-step can be increased very rapidly. The critical time-step for stability can be determined also by numerical experiments and is larger than 160 s.

15.3.2.1 Monotonic loading – ambient temperature Figure 15.11 shows some experimental and numerical results for the first loading of panels with 1.4 mm thickness at ambient temperature. The angle of shear is plotted versus the central deflection. For an undamaged plate, the first symmetric buckling mode always corresponds to the lowest eigenvalue. For this reason, the plate will buckle symmetrically. Within the pre- and the lower postbuckling range, the behaviour of the panels is strongly influenced by initial geometric imperfections. The influence of the geometric imperfections vanishes at least when the angle of shear reaches values of about 0.12 8. As the first plastic deformation occurs at an angle of shear of about 0.2 8, the geometric imperfections do not affect the plastic deformation. Numerous numerical analyses show that the angle of shear at first yielding is approximately a constant for panel thicknesses between 1.2 mm and 3.0 mm. The first yielding takes place at a spot on the edges, where the main buckle is constrained by the clamping. Further load increase leads to a propagation of plastic regions at this spot and along the tension diagonal on the concave sides of the buckle. 350

15.3

Experimental and Numerical Results

The load-angle-of-shear diagram shows no direct influence of the first plastic deformations on the overall stiffness of the panels until – depending on the panel thickness – the angle of shear reaches values of 0.5 8 or more. Figure 15.12 a depicts theoretical results for the deformation state and current distribution of the tangent modulus in three planes of the plate for a 1.6 mm panel at different loading states. The tangent modulus is a measure for the local stiffness and the development of plastic strains since it represents the slope of the uniaxial reference-stress-strain curve of the material. Therefore, in the regime of elastic straining, its value is equal to the elastic modulus and decreases with increasing plastic loading. The distribution of the tangent modulus at an angle of shear of 0.35 8 (superceding the value of the critical buckling load by a factor of 20), which is shown on the left-hand side of Figure 15.12 a, shows still large areas of nearly elastic states. Higher loads lead to a distribution of the tangent modulus, which is shown on the right-hand side of Figure 15.12 a. In this case – at an angle of shear of 0.6 8 –, the plastic regions cover the whole plate, and in the tension field, the plastic or tangent modulus decreases considerably due to large plastic deformations. From a comparison of the deformation states follows that the magnitude of the buckling deflections are not very much increased from the lower to the higher load as the load-carrying mechanism of the panel is shifted from bending to membrane tension.

15.3.2.2 Cyclic loading – ambient temperature The load reversal represents the most critical point of the behaviour of the cyclically loaded panel since remaining deformations from prior plastic loading act like geometri-

Figure 15.12 a: Deformation states and distributions of the plastic tangent modulus at maximum load. 351

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.12 b: Deformation states and distributions of the effective plastic strain before snapthrough after different maximum loads.

cal imperfections each time when zero load is passed. The computed distribution of the deflection and the corresponding distribution of the effective plastic strain at zero load are shown in Figure 15.12 b. As a matter of fact, the larger remaining deformations after higher loads lead to more intense snap-throughs from the current buckling form to the buckling form after load reversal with buckles perpendicular to the former ones. This can be seen in Figure 15.13, where the results of the first cycle of shear tests (1.4 mm panel thickness) at increasing amplitudes of the applied load, respectively of the angle of shear, are illustrated. Due to inevitable small disturbances in the tests, the direction of the central deflection after passing zero load is not predictable. Therefore, there are two possible paths of the central deflection after each snap-through. With increasing load amplitude, the snap-through becomes more complicated since it can run through different even unsymmetric buckling forms as intermediate states. In cases when the central deflection has the same sign in both load extrema, the load reversal can lead to a “double-snap-through” with an intermediate state with opposite central deflection. For the theoretical computations, a change of the deformation path after reaching the bifurcation point at load reversal – the so-called branch switching – is obtained by using small geometric imperfections or disturbances corresponding to the desired buckling mode. In cases of a more intense and complicated snap-through behaviour, the described dynamic method is used. In this case, the branch switching is managed by applying a suitable distribution of accelerations, which disturbs the system and induces the dynamic snap-through procedure. This is illustrated in Figure 15.14, where the subsequent deformation states and the distribution of the velocity normal to the plate in a moment corresponding to the depicted intermediate deformation state are shown. 352

15.3

Experimental and Numerical Results

Figure 15.13: Angle of shear vs. central deflection, results for 1.4 mm panels at different load amplitudes.

Figure 15.14: Dynamic snap-through, deformations and velocity distribution.

353

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.15: Angle of shear vs. central deflection, results for 1.6 mm panels at very high load amplitude (left) and at higher numbers of cycles (right).

The snap-through in Figure 15.14 was computed in the first cycle of a cyclic shear test (panel thickness 1.6 mm) at the very high amplitude of the angle of shear of 0.7 8. The results for the central deflection of this test are given in Figure 15.15. Additionally to the numerical results achieved with the Tseng-Lee model, the starting- and ending points of the first and second snap-through computed with the Mroz model are shown in this figure. As can be easily seen, the accuracy of the used material model has a large influence on the reproduction of the snap-through behaviour since it depends strongly on the remaining plastic deformations at the bifurcation points. In most cases, the repeated buckling behaviour remains the same after the second load cycle. In this test, the cyclic buckling behaviour did not change. Only in one of 22 cyclic tests, a random behaviour in changing the sign of the central deflection was observed. Furthermore, Figure 15.15 shows that the global cyclic load-deformation course is obviously stabilized after the second cycle, and the results for the 10th and 50th cycle are nearly identical. Since the correlations between numerical results and test results are fairly good, the numerical model is applied to different aspect ratios (a/b, see Figure 15.3). In Figure 15.16, the development of the central deflection and the buckling pattern of two panels with different aspect ratios are shown. Typical results, which can be used in the design of shear panels for defining the distance of stringers and frames, are shown in Figure 15.17. 354

15.3

Experimental and Numerical Results

Figure 15.16: Buckling modes and central deflection, cyclic loading, aspect ratios a/b = 1.5 and 2.0.

Figure 15.17: Load at angle of shear of 0.3 8 and 0.5 8 vs. thickness, different aspect ratios. 355

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

15.3.2.3 Time-dependent behaviour The general buckling behaviour and the buckling mode of panels loaded at 200 8C are very similar to those at room temperature. Results for the monotonic first loading of panels at different temperatures are shown in Figure 15.18. The loading rate for the tests at elevated temperature is 0.17 kN/s. It is known from the parameter identification of the material models, that the tangent modulus at the beginning of loading at 200 8C is only 10% smaller than at room temperature. Therefore, the “global stiffness”, which means the ratio between the global angle of shear and the load, is nearly the same for all examined temperatures in the prebuckling regime. When the panel starts to buckle, the occurring bending stresses together with the shear stress lead to early inelastic strains, which yield a stronger decrease of the global stiffness than at room temperature. Further analyses show that in the range of panel thicknesses between 1.2 and 1.8 mm, an increase of the panel thickness of approximately 0.2 mm covers the loss of global stiffness in the postbuckling regime due to the increase of the temperature from 20 8C to 200 8C. The development of the central deflection during monotonic loading of the shear panels at 200 8C is almost equal to the situation at room temperature if the postbuckling regime is concerned. The theoretical buckling loads at 200 8C tend to be slightly higher, but due to the undetermined imperfections, a proper measurement of this effect is impossible. The influence of different loading rates on the monotonic be-

Figure 15.18: Load vs. angle of shear for 1.6 mm panels at different temperatures. 356

15.3

Experimental and Numerical Results

Figure 15.19: Angle of shear vs. time for different creep tests, thickness 1.6 mm.

haviour of the panels is up to now examined between 0.17 kN/s and 0.01 kN/s. All loading rates lead to no significant changes in the monotonic behaviour regarding buckling load and postbuckling stiffness within the normal scatter of the test data. Figure 15.19 depicts results of creep tests. The theoretical creep rates in the global stationary creep regime are in good correlation with the measurements. The creep rates in the primary phase are underestimated by the numerical model, especially at lower load levels since there is no smooth transition between loading and creep phase. This is not only a problem of the description of the primary and secondary creep behaviour of the material since there is a second more geometric non-linear effect. In this first creep phase, a more rapid relaxation of the bending stresses due to the buckling must take place. In the case of spontaneous plasticity, the increase of the load reduces the share of bending within the whole load-carrying mechanism and the tension component along the diagonal increases. During a creep process, this load-carrying state is reached due to the permanent generation of inelastic strains. The used numerical model gives better results for the first creep phase if this tension load-carrying state is already reached during the loading phase by applying higher creep loads (Figure 15.19). The development of the central deflection indicates only small changes in the deformation of the panel during creep. The creep test at a load of 65 kN shown in Figure 15.19 yields to an increase of the central deflection of 0.7 mm within the first 360 min. Tests at lower creep rates yield even lower increases of the central deflection. 357

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

Figure 15.20: Increase of the angle of shear after 360 min.

Figure 15.20 illustrates the influence of creep load and panel thickness on the creep rates. In this figure, the creep-shear angle is defined as the difference between the shear angle at the start of the creep process and the value after 360 min giving an integral measure for the occurring creep rates. Finally, Figure 15.21 shows the computed development of the effective inelastic strain for a creep test at 35 kN. The distributions at the start of the creep process on the left-hand side are scaled by factor 104, and those at the end of the creep process on the right-hand side of Figure 15.21 by factor 103. In general, the distributions are again similar to those of the rate-independent problem. During the creep process, the highest increases of the inelastic strain can be found at the corners on the lower side and along the edges on the upper side in the vicinity of the tension diagonal.

15.4

Conclusion

The combination of numerous experimental results and a well-established numerical model give a good insight of the behaviour of shear-buckled aluminium panels as far as the behaviour at monotonic, cyclic and creep loading is concerned. In case of the rate-independent problem, the Tseng-Lee material model is best suited to simulate the inelastic behaviour of the material under consideration since this model describes the effect of non-pro358

List of Symbols

Figure 15.21: Effective inelastic strain distributions for a creep test at 35 kN; left-hand-side: factor 104, t = 5 min; right-hand side: factor 103, t = 360 min.

portional loading very accurately. The visco-plastic problem is treated with the Chaboche material model. To describe the material behaviour at higher load levels, an additional rateindependent term is added to the Chaboche model. Very high amplitudes of the external shear loads in cyclic tests require a Finite-Element method with an algorithm accounting for dynamic effects to describe the complex snap-through behaviour. The simulation of the behaviour of panels with different geometries was performed successfully. The accuracy of the developed method is proved by the fact that very good results are achieved for durability analyses using the output of the numerical simulations as input.

List of Symbols C D F p v r c*

matrix of in-plane, coupling and bending stiffness yield tensor area of the plate midsurface discrete force vector vector of the midplane displacements vector of membrane forces and bending moments, properties of the 2. Piola-Kirchhoff stress tensor vector of the in-plane and bending strain with Green-Lagrange properties 359

15

Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

a, Ai, Vi, N, parameters of the material models K, Q, C, c uniaxial inelastic strain ein r uniaxial stress state t [. . .]t

References [1] K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus kohlenstoffaserversta¨rktem Kunststoff. Inst. f. Flugzeugbau u. Leichtbau, Technische Universita¨t Braunschweig, 1989. [2] P. Horst: Zum Beulverhalten du¨nner, bis in den plastischen Bereich zyklisch durch Schub belasteter Aluminiumplatten. ZLR-Forschungsbericht 91-01, ISBN 3-9802073-5-8, Inst. f. Flugzeugbau u. Leichtbau, Technische Universita¨t Braunschweig, 1991. [3] J. L. Chaboche: A Review of Computational Methods for Cyclic Plasticity and Viscoplasticity. Proc. of Int. Conference: Computational Plasticity – Models, Software and Applications, Barcelona, 1987, pp. 379–411. [4] J. Knippers: Eine gemischt-hybride FE Methode fu¨r viskoplastische Fla¨chentragwerke unter dynamischen Einwirkungen. Berichte aus dem Konstruktiven Ingenieurbau, Heft 18, ISBN 3-79831548-5, Technische Universita¨t Berlin, 1993. [5] H. Kossira, P. Horst: Cyclic Shear Loading of Aluminium Panels with Regard to Buckling and Plasticity. Thin-Walled Structures 11 (1991) 65–84. [6] P. Horst, H. Kossira, G. Arnst: On the Performance of Different Elasto-Plastic Material Models Applied to Cyclic Shear-Buckling. Proc. of the Int. ECCS-Colloquim: On the Buckling of Shell Structures on Land, in the Sea and in the Air, Lyon, France, 1991. [7] A. Phillips: A Review of Quasistatic Experimental Plasticity and Viscoplasticity. Int. J. Plasticity 2 (1986) 315–328. [8] N. T. Tseng, G. C. Lee: Simple Plasticity Model of Two-Surface-Type. J. Engg. Mech. 109 (1983) 795–810. [9] W. Gieseke, K. R. Hillert, G. Lange: Material State after Uni- and Biaxial Cyclic Deformation. This book (Chapter 2). [10] H. Schlums, E. Steck: Description of Cyclic Deformation Processes with a Stochastic Model for Inelastic Behaviour of Metals. Int. Jour. Plasticity 8 (1992) 147. [11] J. L. Chaboche, G. Rousselier: On the Plastic and Viscoplastic Constitutive Equations – Part I: Rules Developed with Internal Variable Concept. J. Pressure Vessel Technology (ASME) 105 (1983) 153–158. [12] K.-T. Rie, H. Wittke, J. Olfe: Plasticity of Metals and Life Prediction in the Range of LowCycle Fatigue: Description of Deformation Behaviour and Creep-Fatigue Interaction. This book (Chapter 3). [13] E.- R. Tirpitz, M. Schwesig: A Unified Model Approach Combining Rate-Dependent and Rate-Independent Plasticity. Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials – 3, Berlin, 1992, pp. 411–417. [14] E.-R. Tirpitz: Elastoplastische Erweiterung von viskoplastischen Stoffmodellen fu¨r Metalle – Theorie, Numerik und Anwendung. Report 92-70, Inst. of Structural Mechanics, Techn. Univ. Braunschweig, 1992. [15] R. Ritter, H. Friebe: Experimental Determination of Deformation- and Strain Fields by Optical Measuring Methods. This book (Chapter 13). 360

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)

16

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

16.1

Introduction

The local loads and deformations in welded joints have rarely been investigated under the aspect that the mechanical behaviour is influenced by different kinds of microstructure [1]. These different kinds of microstructure lead to multiaxial states of stresses and strains, and some investigations [2–4] have shown that for the determination of the total state 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 the heat-affected zone (HAZ) [5] as well as a weld metal with a lower strength as the base metal [6] can be the reason or the starting point of a fracture in welded joints. A new investigation demonstrates [7] that in TIG-welded joints of the high strength steel StE 690, 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 compressive range. These investigations support the approach described here that the mechanical behaviour of the different kinds of microstructure in the heat-affected zone of welded joints has to be taken into account in the deformation analysis. The influences of these inhomogeneities on the local deformation behaviour of welded joints were determined by experiments and numerical calculations over a wide range of temperature and loading. The numerical deformation analysis was performed with the method of Finite Elements, in which recently developed deformation models simulate the mechanical behaviour of materials over the tested range of temperature and loading conditions. The starting point of these investigations was the question if such deformation models are able to describe the deformation behaviour of welded joints sufficiently.

* Technische Universita¨t Braunschweig, Institut fu¨r Schweißtechnik und Werkstofftechnologie, Langer Kamp 8, D-38106 Braunschweig, Germany ** Volkswagen AG, D-38436 Wolfsburg, Germany 361

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16.2

Materials and Numerical Methods

16.2.1

Materials and welded joints

The investigations were carried out with the microalloyed steel StE 460, of which the microstructure in the normalized state consists of ferrite and minor amounts of bainite and pearlite. The hardness has a value of 220 HV. The chemical composition is listed in Table 16.1. For the deformation analysis, manual arc weldings were manufactured with two different widths of the weld seam (24 mm and 16 mm) using the same welding parameters – with the exception of the number of layers – and the same welding electrodes. The different joints were welded by varying the distance between the two welded plates. The chemical composition of the electrodes is also given in Table 16.1 and the welding parameters (Uw = 22.5 V, Iw = 170 A, vw = 0.2 cm/s) lead to a heat input per unit length of 20 kJ/cm, which is on the upper limit for the use of these electrodes. Microsections and hardness distributions of the welded joints show clearly the three different sections of the joints base metal, heat-affected zone and weld metal (Figures 16.1 and 16.2). The microsections and the hardness distributions were not only used to identify these different zones, but also to establish Finite-Element models for the calculations. Detailed experimental and numerical investigations indicated that the heat-affected zone must also be divided into zones because the mechanical properties are not constant over its width. On the basis of the microsections, four significantly different kinds of microstructure could be identified. The differences between these kinds of microstructure are caused by the peak temperature and the number of weld cycles. To gain the mechanical properties of each microstructure, it must be identified and then prepared in specimens with a large diameter and a large measurement length by using the so-called weld simulation. During the weld simulation, various specimens of the base metal were conductively heated up to different peak temperatures and then cooled under nitrogen with different t8,5-cooling times. The simulation parameters for each structure can be determined first of all numerically by using the thermal conduction equation and subsequently optimized experimentally by comparison with microsections. The specimens with homogeneously simulated microstructures over a measurement length of 25 up to 30 mm were used in tensile tests, creep tests and tension-compression tests. Micrographs of all four kinds of microstructure are shown in Figure 16.3. The various microstructures are listed in Table 16.2 together with their hardness values.

Table 16.1: Chemical composition of the base metal StE 460 and the weld electrodes. C

Si

Mn

P

N

Cu

Ni

StE 460

0.14%

0.45%

1.62%

0.012%

0.006%

0.021%

0.56%

Tenacito 70

0.06%

0.5%

1.6%

–

–

–

0.9%

362

Nb, V, S

16.2

Materials and Numerical Methods

Figure 16.1: Microsections of manual arc-welded joints of the steel StE 460; up: width of the weld metal: 24 mm; down: width of the weld metal: 16 mm.

Figure 16.2: Areas of the welded joints with hardness values below 230 HV (grey: base metal, weld metal) and above 230 HV (black: HAZ).

In addition to the investigations with the four kinds of microstructure of the heat-affected zone, the same mechanical tests were carried out with the base metal and the weld metal. The specimens containing the weld metal were taken from welded joints vertical to the weld seam. They were machined in that way that in the mechanical tests, the deformation is concentrated in the weld metal.

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Figure 16.3: Microstructure of the base metal and microstructures N, F, C and M in the heat-affected zone (from left to right).

Table 16.2: Kinds of investigated microstructures and Vickers hardness. State of material

Microstructure

Hardness

Base metal Microstructure N Microstructure F Microstructure C Microstructure M Weld metal

Ferrite (bainite, pearlite) Ferrite (bainite, pearlite) Ferrite (bainite, pearlite) Bainite, martensite Martensite (bainite) –

220 230 270 280 375 220

364

HV HV HV HV HV HV

Notice

fine-grained as base metal fine-grained as base metal

16.3

16.2.2

Investigations with Homogeneous Structures

Deformation models and numerical methods

16.2.2.1 Deformation model of Gerdes In these investigations, the high-temperature formulation of the deformation model of Gerdes [8] was used:      Fh jr rkin j n1 bAh …r rkin † ; sinh RT r0 RT   1 bAh rkin sign …reff † e_ in ˆ H1 E exp RT      Fh bAh rkin sinh ; R1 exp RT RT

e_ in
C1 exp

r_ kin



A_ h ˆ dV1 je_ in j ‡ dV2 jrj ‡ dV3 :

…1†

…2†

…3†

The model parameter r0, which is used for the stress standardization, was substituted by the Young’s modulus. The time-dependency of the activation volume is described by a three-parametric function and is only used for the simulation of cyclic tests.

16.2.2.2 Fitting calculations The fitting calculations were carried out in cooperation with project number B1 with an evolution algorithm to gain the model parameters for the calculations. The parameter calculations were performed here only phenomenologically for each temperature and each kind of microstructure. Additionally, the different types of tests were simulated separately because the qualities of the model parameters became much better then, and the fitting calculations needed even less time than the parameter estimations made for common tensile and creep tests.

16.3

Investigations with Homogeneous Structures

All investigations were carried out to prove whether deformation models are able to describe the mechanical behaviour of the steel StE 460, of four significant kinds of microstructure of this steel and of the weld metal of manual arc weldings. Tensile, creep and tension-compression tests were performed over a wide range of temperatures (room temperature up to 700 8C) and loading conditions in order to characterize the mechanical behaviour of each state of the base metal. 365

16

16.3.1

Consideration of Inhomogeneities in the Application of Deformation Models

Experimental and numerical investigations

16.3.1.1 Tensile tests The results of the tensile tests made at room temperature are registered in Figure 16.4. The base metal and the weld metal show a clearly visible yield strength, which is highly pronounced in the base-metal deformation, whereas a proof strength has to be attributed to the various kinds of microstructure produced through weld simulation. The arrangement of the stress-strain curves in Figure 16.4 corresponds with the hardness values. The microstructure M with the highest hardness values has the highest flow stresses. For the parameter estimation, the stress-strain curve of the base metal has to be filtered so that the yield strength is transformed into a proof strength. The fitting calculations with the deformation model of Gerdes indicate that the mechanical behaviour of the various kinds of microstructure and the weld metal can be described sufficiently well, but differences arise in the simulation of the stress-strain curve of the base metal and its yield strength cannot be simulated by the deformation model. The largest differences occur in the yieldstrength range, where the stress values are underestimated. At large strains (≥ 3%), the experimental and the calculated values differ less. The calculated stress-strain curve of the high strength microstructure M shows a stress state of saturation, whereas a steadystrain hardening is observed in the experiments with this kind of microstructure. The tensile tests carried out at 300 8C show very similar results as the tests at room temperature (Figure 16.5). The arrangement of the stress-strain curves has the same order and corresponds also to hardness values. The mechanical behaviour of the base metal differs from that at room temperature because the yield strength of the base

Figure 16.4: Stress-strain curves of the investigated microstructures at room temperature; symbols: experimental curve; lines: fitted curve. 366

16.3

Investigations with Homogeneous Structures

Figure 16.5: Stress-strain curves of the investigated microstructures at 300 8C; symbols: experimental curve; lines: fitted curve.

metal is not as pronounced as at room temperature. Comparing Figure 16.4 and Figure 16.5, one sees that at strains above 3%, the strength values of all microstructures are higher at 300 8C than at room temperature. The Young’s modulus of all microstructures decreased to a value of 170 000 MPa. The mechanical behaviour of all kinds of microstructure can be described again sufficiently by the fitting calculations and here, the yield strength of the base metal is also adequately simulated. Only the calculated stress-strain curve of the microstructure M includes a stress state of saturation at large strains although the real stress-strain curve exhibits a strain-hardening behaviour. The stress-strain curves at 500 8C (Figure 16.6) are again arranged in the same order as at room temperature. At this temperature, the base metal shows no yield-strength effects and its stress-strain curve is nearly the same as the curve of the microstructure N. The stress-strain curve of the microstructure M reveals a softening behaviour at large strains, and the differences between the curves of the structures M and C decreased. The fitting calculations carried out by using a Young’s modulus of 150 000 MPa simulate the mechanical behaviour of all kinds of microstructure very well, only the softening of the microstructure M is unsuitably modelled with a stress state of saturation. It can be seen in Figure 16.7 that the stress-strain curves taken at 700 8C are not arranged in the same order as at room temperature. At 700 8C, the base metal has a higher strength than the microstructures N and F because the especially low grain size of these two microstructures favours plastic deformation at high temperatures. The two high strength microstructures M and C show a softening behaviour caused by a transformation of the microstructure. The fitting calculations (Young’s modulus = 130 000 MPa) correspond relatively well with the experimental results although the softening behaviour is unsuitably modelled by horizontal lines. 367

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Consideration of Inhomogeneities in the Application of Deformation Models

Figure 16.6: Stress-strain curves of the investigated microstructures at 500 8C; symbols: experimental curve; lines: fitted curve.

Figure 16.7: Stress-strain curves of the investigated microstructures at 700 8C; symbols: experimental curve; lines: fitted curve.

368

16.3

Investigations with Homogeneous Structures

16.3.1.2 Creep tests The creep tests were carried out at 500 8C and 700 8C. Already at 500 8C (loading: 275 MPa), all kinds of microstructure show a remarkable creep (Figure 16.8). The order of all creep curves agrees within the range of reproducibility with the results of the tensile tests. The differences between the base metal and the microstructure N and the differences between the microstructures M and C are very small. The fitting calculations show a very good applicability of the deformation model of Gerdes to the creep behaviour of all microstructures. Differences between the experiments and the calculations are not noticeable in Figure 16.8. The analysis of the creep behaviour at 700 8C (loading: 50 MPa) reveals results analogous to those of the tensile tests. The highest creep strains occur in the microstructures F and N (Figure 16.9), whereas the smallest creep strains occur in the microstructures C and M. The base metal takes a middle position of all creep curves. The behaviour of the microstructures F and N is highly influenced by the fine-grained microstructure, which favours the plastic deformation. The fitting calculations simulate the creep behaviour sufficiently well. The softening behaviour of the two microstructures M and C caused by a transformation of the microstructure is modelled as steady-state creep. It must be stated that it cannot be anticipated to find the consequences of the transformation in the results because this behaviour has not been implemented in the model equations.

Figure 16.8: Creep curves of the investigated microstructures at 500 8C, loading: 275 MPa; symbols: experimental curve; lines: fitted curve.

369

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Consideration of Inhomogeneities in the Application of Deformation Models

Figure 16.9: Creep curves of the investigated microstructures at 700 8C, loading: 50 MPa; symbols: experimental curve; lines: fitted curve.

16.3.1.3 Cyclic tension-compression tests Strain-controlled cyclic tension-compression tests were performed with varying strain amplitudes at room temperature (± 0.6%) and at 500 8C (± 0.4%). The strain rates of the tests lie between 1 · 10–4 1/s and 5 · 10–4 1/s. The fitting calculations were made in two steps because of the hardening or softening behaviour found in the different kinds of microstructure. In the first step, the first five cycles of the tension-compression tests were used to estimate the first model parameters. If necessary, these model parameters were utilized in the second step as the starting set of parameters for new parameter estimations to simulate the behaviour during the whole tension-compression tests. Figure 16.10 shows the results of the first five cycles of all microstructures at room temperature (strain rate: e_
5
10 4 l/s). The order of all curves is the same as that of the stress-strain curves achieved from tensile tests. The base metal exhibits a pronounced yield strength, which has to be filtered for the fitting calculations. The mathematical simulations describe the mechanical behaviour of all microstructures sufficiently although the yield strength cannot be modelled by the deformation model. The cyclic hardening of all microstructures can be described by the model. Figure 16.11 contains the stress-time curves of all microstructures over the complete testing time. The printed symbols are the points of return in the fully tension range. It can be seen that all microstructures with the exception of the base metal soften after the first hardening cycles. The hardening period lasts about 5 cycles and the softening leads very fast (10 to 20 cycles) to a state of cyclic saturation. Only the base metal already softens from the beginning of the test, and the cyclic state of saturation is reached after less cycles. 370

16.3

Investigations with Homogeneous Structures

Figure 16.10: Stress-time curves of the investigated microstructures at room temperature, first five cycles, strain amplitude: ± 0.6%; symbols: experimental curve; lines: fitted curve.

Figure 16.11: Stress-time curves of the investigated microstructures at room temperature, 200 cycles, strain amplitude: ± 0.6%; symbols: experimental curve; lines: fitted curve.

371

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Consideration of Inhomogeneities in the Application of Deformation Models

In Figure 16.12, a photo taken with the transmission electron microscope shows the saturated state of the base metal. The state of saturation can be identified through the array of dislocations in a cell structure. The fitting calculations made in the second step indicate that the deformation model of Gerdes is not able to simulate the cyclic softening behaviour of all microstructures. Figure 16.11 demonstrates that the stresses in the initial hardening range were underestimated and in the following softening range overestimated. The model describes the state of saturation by a horizontal line for all microstructures. The results of cyclic tension-compression tests carried out at 500 8C are represented in the following figures (strain rate: e_
5
10 4 l/s). Figure 16.13 contains the first five cycles of the tension-compression tests of all microstructures. The curves are arranged in the same order as the curves of the tensile tests. The cyclic behaviour agrees with the results found at room temperature. It can be seen that all microstructures with exception of the base metal firstly harden and then soften during the cyclic loading (Figure 16.14). The amounts of hardening and softening are not as high as at room temperature. The fitting calculations simulated the first five cycles of the stress-strain behaviour sufficiently (Figure 16.13). Because of the relatively weak softening behaviour, the model parameters achieved from the first five cycles could also be used to describe the behaviour during all following cycles. A second fitting improves the quality of the parameters only insignificantly. It can be seen from Figure 16.14 that all curves lead after few cycles to states of cyclic saturation. The cyclic stress-time curves of all microstructures can be simulated sufficiently well by the deformation model. But, as to be seen, the cyclic softening after the cyclic hardening cannot be described by the model. The stresses are underestimated in the range of hardening and overestimated in the range of softening, but the differences between the experiments and calculations remain relatively small.

16.3.2

Discussion

All results indicate that recently developed deformation models can be successfully applied to describe the mechanical behaviour of a microalloyed steel and of the different microstructures identified in the heat-affected zone of its weldings. But, in view of accuracy and calculation time, it is useful for all fitting calculations to estimate the parameters separately for each microstructure, each temperature and each testing sequence. The fitting calculations are carried out phenomenologically as it was not possible to find relations between model parameters and microstructural parameters. Some special aspects of the mechanical behaviour of the microalloyed steel and its various kinds of microstructure cannot be modelled by the deformation model of Gerdes. Firstly, the simulation of the yield strength is not successful because the stresses in this range are underestimated and the model calculates only a proof strength. Secondly, the softening behaviour of some kinds of microstructure at high temperatures cannot be described with the deformation model. The model simulates the softening behaviour through a steady-state behaviour. This behaviour is also detectable in the simulation of cyclic tension-compression tests. In this case, the hardening and softening behaviour is 372

16.3

Investigations with Homogeneous Structures

Figure 16.12: TEM-photo of the state of saturation due to cyclic loading, base metal at room temperature after 200 cycles.

Figure 16.13: Stress-time curves of the investigated microstructures at 500 8C, first five cycles, strain amplitude: ± 0.4%; symbols: experimental curve; lines: fitted curve.

373

16

Consideration of Inhomogeneities in the Application of Deformation Models

Figure 16.14: Stress-time curves of the investigated microstructures at 500 8C, 200 cycles, strain amplitude: ± 0.4%; symbols: experimental curve; lines: fitted curve.

also modelled by a steady-state or saturation behaviour. The experimentally observed saturation behaviour of all microstructures is modelled sufficiently well. From these results, a simplification for the Finite-Element calculations of the deformation behaviour of welded joints can be derived. In order to lower the calculation time, the number of zones in the heat-affected zone can be reduced if the mechanical behaviour of the microstructures is nearly the same.

16.4

Investigations with Welded Joints

The deformation behaviour of welded joints was investigated with the two kinds of welded specimens, which differ in the width of the weld seam (see Section 16.2.1). Tensile tests at room temperature were made to find the strain distributions during loading, and calculations were performed with the Finite-Element code ABAQUS with the same control of the strain rate as in the experiments.

374

16.4

16.4.1

Investigations with Welded Joints

Deformation behaviour of welded joints

16.4.1.1 Experimental investigations The experiments to gain the strain distributions were performed in cooperation with the project C2. In these experiments, flat specimens taken from the welded joints vertical to the weld seam were tested in tensile tests with the same strain control as in the tests with the homogeneous structures. The strain distributions were determined with the grating method [9]. For reasons of symmetry and of the grating size, only less than one half of the welded specimens was observed during the tests. The inspected regions were the heat-affected zone, the weld metal and small parts of the base metal. The welded specimens show relatively large rigid body motions during the tests so that reference objects had to be fixed to the specimens. These reference objects are subject to the same rigid body motions as the welded specimens, but they are not deformed during the tests. With these object motions, the fictitious strains can be detected and thus, the real strains can be determined.

16.4.1.2 Numerical investigations For the numerical investigations, the Finite-Element code ABAQUS has been used in cooperation with the project B1.

16.4.1.3 Finite-Element models of welded joints The Finite-Element meshes of the welded joints have been derived from hardness distributions and microsections. It turned out during the investigations that only those points had to be determined describing the transition from the base metal to the heat-affected zone and from the heat-affected zone to the weld seam (see Figure 16.2). Then, the mesh for the heat-affected zone was modelled with four equidistant zones containing the information of the mechanical behaviour of the affiliated microstructures. The Finite-Element calculations were performed with the model parameters gained from the room-temperature fittings. After the first experiments, it could be observed that the deformations are concentrated in the weld metal so that it should be possible to model the heat-affected zone with one microstructure only. In some calculations, the heat-affected zone was modelled only with the microstructure C. This procedure reduces the calculation times for the tensile test simulation with ABAQUS. Therefore, the following sections include calculations for 6-material models (weld metal, four regions of the HAZ, base metal) and 3-material models (weld metal, HAZ, base metal).

16.4.1.4 Calculation of the deformation behaviour of welded joints For the Finite-Element calculations, the same control of the strain rate as in the real tensile tests was used. The load was attached as a boundary condition on one side of 375

16

Consideration of Inhomogeneities in the Application of Deformation Models

the Finite-Element mesh. The first calculations revealed that the calculation times are very high in comparison to the real tests. The cpu-time to calculate a tensile test of 900 seconds was about 20 000 cpu-seconds with the 6-material model (2300 elements) and about 4000 cpu-seconds with the 3-material model (900 elements). A serious problem for the Finite-Element calculations is the time-stepping during the first part of the simulation (0 to 40 seconds). During this period, the time-steps for the numerical integration of the model equations are reduced from 5 seconds to 0.05 seconds because the differences between the material properties of the weld seam and the microstructures M or C are too large for greater time-steps in order to calculate the equilibrium state.

16.4.2

Strain distributions of welded joints with broad weld seams

The first investigations with welded structures were carried out with tensile specimens with a weld metal length of about 24 mm. The analysis of the measured strain distributions during loading shows the following results. The first remarkable strains occur in the weld metal possessing the lowest flow stresses of all microstructures. The deformations in the heat-affected zone and in the base metal are smaller. Figure 16.15 illustrates the distributions of the longitudinal strains measured with the grating method at a stage of 2.2% medium strain of the whole specimen. Only the strains in the weld-metal zone and in the heat-affected zone are visible because the grating was fixed only on these zones. The longitudinal strains have maximum values of about 4.8% in the weldmetal and less than 1% in the heat-affected zone. The curvature of the strain isolines indicates that the soft weld metal is backed up by the harder heat-affected zone. The hindered vertical deformations in the transition zone between weld-metal and heat-affected zone influence not only the vertical strains (Figure 16.15 right) but also the distribution of the longitudinal strains (Figure 16.15 left). The strain calculations made with a 6-material model show nearly the same results (Figure 16.16) as the experiments, but there are also some differences. The first remarkable strains occur in the heat-affected zone in the region of the microstructures N and F because they have a proof strength lower than the yield strength of the base metal. But at a medium strain stage of 2.2%, the longitudinal strains in the weld metal are much higher than the strains in the heat-affected zone. The calculated values reach 3.4% in the weld-metal and less than 1% in the heat-affected zone. The strains in the base metal are already higher than in the heat-affected zone but smaller than in the weld metal. The backing up of the soft weld metal by the harder heat-affected zone is also determined. At the points, where the contact faces between weld-metal and heat-affected zone break through the free surfaces, singularities appear in the calculated stress and strain distributions. These singularities are caused by the sudden change of the material properties between two neighbouring elements of the weld-metal and the heat-affected zone. These numerical effects have been stated before by other authors [1, 5] and must not be taken into account in the comparison between numerical and experimental results. Only the strain distributions in a wider range around the singularities can be compared with experimental results. 376

16.4

Investigations with Welded Joints

Figure 16.15: Strain distributions of a welded specimen with a long weld-metal zone measured with the grating method, medium strain: 2.2%; left: longitudinal strains; right: vertical strains.

Figure 16.16: Strain distributions of a welded specimen with a long weld-metal zone calculated with the 6-material model, medium strain: 2.2%; left: longitudinal strains; right: vertical strains.

377

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Consideration of Inhomogeneities in the Application of Deformation Models

The singularities also occur in the analysis of the strains calculated with a 3material model. Here, the first remarkable strains do not occur in the heat-affected zone but in the weld metal as observed in the experiments. The strain values at a medium strain stage of 2.2% are about 3.5% in the weld-metal and less than 1% in the heat-affected zone (Figure 16.17). The backing up of the soft weld metal is also determined in these investigations. The next examined load step at a medium strain of 4.3% was at the end of the tensile test. Figure 16.18 illustrates the strain distributions of this state. The curvatures of the strains are the same as in the load step discussed before. They also demonstrate the backing up of the soft zone by a harder zone. Figure 16.18 shows the longitudinal strains, which have values of more than 8% in the weld-metal and less than 2% in the heat-affected zone. The vertical strain distributions illustrate also the backing up clearly. The strains along a line parallel to the transition area are higher in the middle of the specimen than at the free surface. The calculations for a load step of 4.5% show qualitatively the same results as the experiments. The strains calculated with the 6-material model and the 3-material model are represented in Figures 16.19 and 16.20. In both figures, the calculated strains in the weld metal are lower than the measured strains. Both maximum values of the longitudinal strains in the weld metal are slightly lower than 7%. The arrangement of the strain isolines in the figures corresponds with the experimental results and confirms the backing up of the soft weld metal by the harder microstructure of the heat-affected zone. The figures representing the vertical strain distributions calculated with the 6-material model (Figure 16.19 right) and the 3-material model (Figure 16.20 right) verify also the experimental results.

Figure 16.17: Strain distributions of a welded specimen with a long weld-metal zone calculated with the 3-material model, medium strain: 2.2%; left: longitudinal strains; right: vertical strains. 378

16.4

Investigations with Welded Joints

Figure 16.18: Strain distributions of a welded specimen with a long weld-metal zone measured by the grating method, medium strain: 4.3%; left: longitudinal strains; right: vertical strains.

Figure 16.19: Strain distributions of a welded specimen with a long weld-metal zone calculated with the 6-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.

379

16

Consideration of Inhomogeneities in the Application of Deformation Models

Figure 16.20: Strain distributions of a welded specimen with a long weld-metal zone calculated with the 3-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.

16.4.3

Strain distributions of welded joints with small weld seams

The strain distributions measured after the loading of welded joints with a narrow zone of weld metal are shown in Figure 16.21. At the end of the tensile tests (medium strain 4.3%), the longitudinal strains have values of about 9% in the weld-metal and lower than 2% in the heat-affected zone (Figure 16.21). The strains in the base metal are somewhat higher than 2%. The vertical strains in Figure 16.21 indicate the backing up of the soft weld metal and base metal through the hard heat-affected zone. The curvature of the strain isolines demonstrates this effect clearly. It can be seen that the strains measured in specimens of a welded joint with a narrow weld seam are slightly higher than the strains measured in specimens of a welded joint with a wider weld-metal zone. The numerical results achieved with the 6-material model (Figure 16.22) and the 3-material model (Figure 16.23) correspond with the experimental results. The backing up of the weld metal is also confirmed by the longitudinal and vertical strain distributions. But the calculated strain values in the weld metal are lower than the measured ones. The maximum strains in the weld metal have values of 7%, and the strains in the heat-affected zone are about 1% and are lower than the measured ones.

16.4.4

Discussion

The results of all calculated strain distributions show that modern deformation models are able to describe the mechanical behaviour of welded joints sufficiently well. Problems arise in the numerical investigations with the automatic time-stepping in the Fi380

16.4

Investigations with Welded Joints

Figure 16.21: Strain distributions of a welded specimen with a short weld-metal zone measured with the grating method, medium strain: 4.3%; left: longitudinal strains; right: vertical strains.

Figure 16.22: Strain distributions of a welded specimen with a short weld-metal zone calculated with the 6-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.

381

16

Consideration of Inhomogeneities in the Application of Deformation Models

Figure 16.23: Strain distributions of a welded specimen with a short weld-metal zone calculated with the 3-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.

nite-Element calculations, which increases the needed cpu-times of the calculations. The small time-steps are caused by the great differences in the properties of the neighbouring materials in the transition range between the weld metal and the hard microstructure in the heat-affected zone. Also, the calculated strain and stress singularities are not found in the experiments, and they are too caused by the sudden change of the material properties. The 3-material model and the 6-material model lead to nearly the same calculated strains so that the number of different regions in the heat-affected zone of a welded joint can be reduced if the locally highest strains do not occur there.

16.5

Application Possibilities and Further Investigations

All results show that modern deformation models can be applied successfully to welded joints. The mechanical behaviour of the base metal and of various kinds of microstructure is in a reasonable good accordance with the experimental results. But the analysis demonstrates also that some special aspects of the mechanical behaviour of microalloyed steels cannot be simulated by the deformation model. This lack of simulation includes yield-strength effects and the softening behaviour, which occurs in the high temperature and cyclic range. New deformation models have to describe these effects if they should be applied in further investigations. A big problem in the calculations was the handling of the equations in Finite-Element calculations and in 382

References the parameter estimation. Both numerical methods need very high calculation times so that numerical investigations can only be made if the hardware is big enough. Further investigations may solve this problem if numerical methods or deformation equations are developed shortening the calculation times by faster algorithms or if model equations are established, which are easier to handle in modern calculation methods. The efforts of modelling welded joints can also be reduced according to these investigations. If the highest strains or fractures do not occur in the heat-affected zone, it must not be modelled with more than one microstructure. In this case, the Finite-Element meshes can directly be derived from hardness distributions of welded joints. In other cases, it can be recommended from these investigations that only two different microstructures have to be taken into account for the calculation of the mechanical behaviour of welded joints.

References [1] U. H. Clormann: Örtliche Beanspruchungen von Schweißverbindungen als Grundlage des Schwingfestigkeitsnachweises. Dissertation, Technische Hochschule Darmstadt, 1986. [2] W. Schieblich: Rechnerische und experimentelle Ermittlung des Zeitstandverhaltens einer austenitischen Schweißverbindung. Dissertation, Technische Hochschule Darmstadt, 1992. [3] W. Eckert: Experimentelle und numerische Untersuchungen zum Zeitstandverhalten von Schweißverbindungen der Werkstoffe X20 CrMoV 12 1 und GS-17 CrMoV 5 11. Dissertation, Staatliche Materialpru¨fanstalt (MPA) Stuttgart, 1992. [4] M. Kaffka: Beitrag zum Zeitstandverhalten artgleicher Schweißverbindungen einer Nickelbasislegierung unter besonderer Beru¨cksichtigung des lokalen Verformungsverhaltens. Dissertation, Technische Hochschule Aachen, 1985. [5] U. Pa¨tzold: Verformungsanalyse von Schweißverbindungen. Dissertation, Technische Universita¨t Braunschweig, 1992. [6] H. Hickel: Eigenspannungen und Festigkeitsverhalten von Schweißverbindungen. Dissertation, Universita¨t Karlsruhe, 1973. [7] J. Pucelik, Th. Nitschke-Pagel, H. Wohlfahrt: Relationship of tensile residual stresses and fatigue crack propagation under cyclic loading in the fully compressive range. Poster at the Fourth European Conference on Residual Stresses, 4.–6. 06. 1996, Cluny (F), to be published. [8] R. Gerdes: Ein stochastisches Werkstoffmodell fu¨r das inelastische Materialverhalten metallischer Werkstoffe im Hoch- und Tieftemperaturbereich. Dissertation, Technische Universita¨t Braunschweig, 1995. [9] D. Winter: Optische Verschiebungsmessung nach dem Objektrasterprinzip mit Hilfe eines fla¨chenorientierten Ansatzes. Dissertation, Technische Universita¨t Braunschweig, 1993.

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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)

Bibliography

Theses Resulted from the Collaborative Research Centre 1985–1987 Subproject A2 D. Rode: Ermu¨dungsverhalten anisotroper Aluminiumlegierungen unter mehrachsigen Beanspruchungen im Zeitfestigkeitsbereich. Subproject A3 C. Schulze: Optische Verformungsmessungen an Schalen nach dem Rasterprinzip. Subproject A6 I. Go¨bel: Modellbildung fu¨r die Hochtemperaturplastizita¨t mit Hilfe metallphysikalischer Ergebnisse. T. Lo¨sche: Entwicklung eines Stoffgesetzes fu¨r die Hochtemperaturplastizita¨t auf der Grundlage von Markov-Prozessen. R. Schettler-Ko¨hler: Entwicklung eines makroskopischen Stoffgesetzes fu¨r Metalle aus einem stochastischen Zwischenmodell. G. Wilhelms: Ein pha¨nomenologisches Werkstoffgesetz zur Beschreibung von Plastizita¨t und Kriechen metallischer Werkstoffe. Subproject B4 W. Kohler: Beitrag zur Wasserstoffverspro¨dung metallischer Werkstoffe im LCF-Bereich. Subproject D1 R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Methoden.

1988–1990 Subproject A1 U. Meyer: Einfluss verschiedener Verformungsparameter auf die Rekristallisationskinetik von Kupfer. J. Schmidt: Untersuchung der Erholungs- und Rekristallisationsvorga¨nge in tieftemperaturverformten Metallen mit Hilfe kalorischer Messungen. Subproject A5/B4 R. Schubert: Verformungsverhalten und Risswachstum bei Low-Cycle-Fatigue. Subproject B2 M. Schwesig: Inelastisches Verhalten metallischer Werkstoffe bei ho¨heren Temperaturen – Numerik und Anwendung. 384

Theses Resulted from the Collaborative Research Centre Subproject B5 H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Beru¨cksichtigung in Konstruktionsberechnungen. E. Beißner: Zum Tragverhalten sta¨hlerner Augensta¨be im elastisch-plastischen Zustand. Subproject C3 K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus kohlenstoffaserversta¨rktem Kunststoff.

1991–1993 Subproject A5/B4 H. Klingelho¨ffer: Rissfortschritt und Lebensdauer bei Hochtemperatur Low-Cycle-Fatigue in korrosiven Gasen. Subproject A6/B1 H. Hesselbarth: Simulation von Versetzungsstrukturbildung, Rekristallisation und Kriechscha¨digung mit dem Prinzip der zellula¨ren Automaten. F. Kublik: Vergleich zweier Werkstoffmodelle bei ein- und mehrachsiger Versuchsfu¨hrung im Hochtemperaturbereich. H. Schlums: Ein stochastisches Werkstoffmodell zur Beschreibung von Kriechen und zyklischem Verhalten metallischer Werkstoffe. H. Gro¨hlich: Finite-Element-Formulierung fu¨r vereinheitlichte inelastische Werkstoffmodelle ohne explizite Fließfla¨chenformulierung. Subproject A8 R. Neuhaus: Entwicklung der Versetzungsstruktur und Ha¨rtung bei [100] und [111] orientierten Kupfer und Kupfer-Mangan Kristallen im einachsigen Zugversuch. A. Kalk: Dynamische Reckalterung und die Grenzen der Stabilita¨t plastischer Verformung. Systematische Untersuchungen an Viel- und Einkristallen aus CuMn. Subproject A9 A. Hampel: Struktur und Kinetik der lokalisierten Verformungen in kubisch fla¨chenzentrierten Legierungseinkristallen – Experimente und Modellrechnungen. Subproject A10 M. Mu¨ller: Plastische Anisotropie polykristalliner Materialien als Folge der Texturentwicklung. Subproject B2 G. Kracht: Erschließung viskoplastischer Stoffmodelle fu¨r thermomechanische Strukturanalyse. E.-R. Tirpitz: Elasto-plastische Erweiterung von viskoplastischen Stoffmodellen fu¨r Metalle. H. Braasch: Ein Konzept zur Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle. Subproject B5/B7 M. M. El-Ghandour: Low-Cycle Fatigue Damage Accumulation of Structural Steel St52. Subproject B6 G. Zhang: Einspielen und dessen numerische Behandlung von Fla¨chentragwerken aus ideal plastischem bzw. kinematisch verfestigendem Material. Subproject B8 R. Mahnken: Duale Methoden fu¨r nichtlineare Optimierungsprobleme. Subproject C2 W. Wilke: Photogrammetrische Verformungsmessung durch Überlagerungseffekte frequenzmodulierter periodischer Bildstrukturen. 385

Bibliography J. O. Hilbig: Zur Beeinflussung der Speckleauswertung durch eine Kombination der Speckle- und der Objektrastermethode. D. Winter: Optische Verschiebungsmessung nach dem Objektprinzip mit Hilfe eines fla¨chenorientierten Ansatzes. Subproject C3 P. Horst: Zum Beulverhalten du¨nner, bis in den plastischen Bereich zyklisch durch Schub belasteter Aluminiumplatten. D. Hachenberg: Untersuchungen zum Nachbeulverhalten stringerversta¨rkter schubbeanspruchter Platten aus kohlenstoffaserversta¨rktem Kunststoff. Subproject C4 U. Pa¨tzold: Verformungsanalyse von Schweißverbindungen.

Publications Resulted from the Collaborative Research Centre 1985–1987 Subproject A1 W. Witzel, F. Haeßner: Zur Vergleichbarkeit von Werkstoffzusta¨nden nach Dehnen, Stauchen und Tordieren. Z. Metallkunde 78 (1987) 316. F. Haeßner, H. Mu¨ller: Einfluss von Korngro¨ße und Beanspruchungsrichtung auf die Fließspannung texturbehafteter Zink-Bleche. Z. Metallkunde 78 (1987) Juli-Heft. Subproject A2 G. Lange, D. Rode: Ermu¨dungsverhalten anisotroper Aluminium-Legierungen unter mehrachsiger Beanspruchung im Zeitfestigkeitsbereich. Hauptversammlung der Deutschen Gesellschaft fu¨r Metallkunde, 20.–23. Mai 1986, Vortrag Nr. EBO7. D. Rode: Ermu¨dungsverhalten anisotroper Aluminium-Legierungen unter mehrachsiger Beanspruchung im Zeitfestigkeitsbereich. Dissertation TU Braunschweig, 1987. Subproject A3 K. Andresen, R. Ritter: The phase shift method applied to reflection Moire´ pattern. Proc. of the VIII. Int. Conf. on Exp. Stress Analysis, Amsterdam, 1986, pp. 351–358. K. Andresen, R. Ritter: Dehnungs- und Kru¨mmungsermittlung mit Hilfe des Phasenshiftprinzips. Technisches Messen 6 (1987). S. Angerer, J. P. Lo¨wenau, B. Morche, W. Wilke: 3D-Verformungsmessung an einem gummiartigen Da¨mpfungselement mit Hilfe der Stereophotographie und der digitalen Bildverarbeitung. Fachkolloquium Experimentelle Mechanik, Stuttgart, 1986, pp. 33–40. H. C. Go¨tting, R. Schu¨tze, R. Ritter, W. Wilke: Dehnungsmessungen an Faserverbundwerkstoffen mit Hilfe des Beugungsprinzips. VDI-Berichte 631, VDI Verlag Du¨sseldorf, 1987, pp. 275–285. J. Hilbig, R. Ritter: Zur Bestimmung von Neigungsa¨nderungen schalenfo¨rmiger Objekte mit Hilfe der Laser-Speckle-Photographie. Contribution to Joint Conf. on DGaO, NOC, OG-IoP, SFO “Optics 86”, Scheveningen, 1986. R. Ritter, M. Hahne: Interpretation of Moire´-effect for curvature measurement of shells. Proc. of the VIII. Int. Conf. on Exp. Stress Analysis, Amsterdam, 1986, pp. 331–340. R. Ritter: Raster- und Moire´-Methoden. In: C. Rohrbock, N. Czaika (Eds.): Handbuch der experimentellen Spannungsanalyse, Du¨sseldorf, 1987. R. Ritter, W. Wilke: Dehnungsmessung nach dem Gitterprinzip. Lecture at the Colloquy of the Collaborative Research Centre (SFB 319), Goslar, 4.–5. Dezember 1986. 386

Publications Resulted from the Collaborative Research Centre C. Schulze: Optische Verformungsmessung an Schalen nach dem Rasterprinzip. Dissertation an der Fakulta¨t fu¨r Maschinenbau und Elektrotechnik der TU Braunschweig, 1986. Subproject A5 K.-T. Rie, R.-M. Schmidt, B. Ilschner, S. W. Nam: A Model for Predicting Low-Cycle Fatigue Life under Creep-Fatigue Interaction. In: H. D. Solomon, G. R. Halford, L. R. Kaisand, B. N. Leis (Eds.): Low Cycle Fatigue, ASTM STP 942, American Society for Testing and Materials, Philadelphia, 1988, pp. 313–328. K.-T. Rie, R.-M. Schmidt: Life Prediction for Low-Cycle Fatigue under Creep-Fatigue Interaction. Fifth International Conference on Mechanical Behaviour of Materials, Peking, 1987. K.-T. Rie, R.-M. Schmidt: Lifetime Prediction under Creep-Fatigue Conditions. Proceedings of the Second International Conference on Low-Cycle Fatigue and Elasto-Plastic Behaviour of Materials, September 1987, Mu¨nchen. R.-M. Schmidt: Low-Cycle Fatigue under Special Aspects of Quality Control. Braunschweig Kolloquium 1985, DVS, pp. 107–119. K.-T. Rie, R.-M. Schmidt: High Temperature Low-Cycle Fatigue of Austenitic and Ferritic Weldments. ECF 6, Amsterdam, 1986, pp. 1096–1113. K.-T. Rie, R.-M. Schmidt: Ermu¨dung von Schweißverbindungen im Bereich geringer Schwingspielzahlen. Schweißen und Schneiden 38 (1986) 509–514. K.-T. Rie, R.-M. Schmidt: Low-Cycle Fatigue of Welded Joints. Welding and Cutting 38 (1986) E172–E175. Subproject A6, B1 E. Steck: Entwicklung von Stoffgesetzen fu¨r die Hochtemperaturplastizita¨t. Grundlagen der Umformtechnik. Internationales Symposium, Stuttgart 1983, Springer Verlag, 1984, pp. 83–113. E. Steck: A stochastic model for the high-temperature plasticity of metals. Intern. Journal of Plasticity 1 (1985) 243–258. E. Steck: A stochastic model for the high-temperature plasticity of metals. Trans. 8th Intern. SMiRT-Conf., North-Holland, 1985. E. Steck: Ein stochastisches Modell fu¨r die Wechselwirkung von Plastizita¨t und Kriechen. Workshop Werkstoff und Umformung, Stuttgart 1986, Springer Verlag, 1986. Subproject A8 Th. Wille, W. Gieseke, Ch. Schwink: Quantitative analysis of solution hardening in selected copper alloys. Acta Met. 35 (1987) 2679–2693. Th. Steffens, C.-P. Reip, Ch. Schwink: Anomalous dislocation densities in fcc solid solutions. Scripta Met. 21 (1987) 335–339. Subproject A9 H. Neuha¨user, O. B. Arkan, H. H. Potthoff: Dislocation multipoles and estimation of frictional stress in fcc copper alloys. Mat. Sci. Eng. 81 (1986) 201–209. H. Neuha¨user: Physical manifestation of instabilities in plastic flow. In: V. Balakrishnan, C. E. Bottani (Eds.): Mechanical Properties and Behaviour of Solids: Plastic Instabilities World Scientific, Singapore, 1986, pp. 209–252. O. B. Arkan, H. Neuha¨user: Dislocation velocities in Cu-Ni alloys determined by the stress pulseetch pit technique and by slip line cinematography. phys. stat. sol. (a) 99 (1987) 385–397. H. Neuha¨user, O. B. Arkan: Dislocation motion and multiplication in Cu-Ni single crystals. phys. stat. sol. (a) 100 (1987). Subproject B2 H. K. Nipp: Temperatureinflu¨sse auf rheologische Spannungszusta¨nde im Salzgebirge. Report Nr. 82-36 from the Institut fu¨r Statik der TU Braunschweig. A. Schmidt: Berechnung rheologischer Zusta¨nde im Salzgebirge mit vertikalen Abbauen in Anlehnung an In-situ-Messungen. Report Nr. 84-43 from the Institut fu¨r Statik der TU Braunschweig. 387

Bibliography Subproject B4 K.-T. Rie, R. Schubert: Einfluss einer Druckwasserstoffumgebung auf das Ermu¨dungs- und Risswachstumsverhalten bei Low-Cycle Fatigue; Wasserstoff in Metallen. Results of the Schwerpunktprogramm, Deutsche Forschungsgemeinschaft, Bonn, 1986. K.-T. Rie, R. Schubert: Note on the Crack Closure Phenomenon in LCF. 2. Int. Conf. on LCF and Elasto-Plastic Behaviour of Materials, Mu¨nchen 1987, Elsevier Applied Science, pp. 575–580. Subproject C1 K. Andresen, R. Ritter, R. Schuetze: Application of grating methods for testing of material and quality control including digital image processing. SPIE-Potics in Engineering Measurement, Cannes, 1985. K. Andresen: Auswertung von Rasterbildern mit der digitalen Bildverarbeitung. Seminar of the Collaborative Research Centre (SFB 319), Braunschweig. K. Andresen, F. Hecker: Das Phasenshiftverfahren zur digitalen Auswertung von Moire´-Mustern und spannungsoptischen Bildern. Nieders. Mechanik-Kolloquium, Braunschweig. K. Andresen, R. Ritter: The Phase Shift Method Applied to Reflection Moire´ Pattern. Int. Conf. Exp. Stress Anal., Amsterdam, 1986. S. Angerer, J. P. Lo¨wenau, B. Morche, W. Wilke: 3D-Verformungsmessung an einem gummiartigen Da¨mpfungselement mit Hilfe der Stereophotographie und der digitalen Bildverarbeitung. Fachkolloquium Experimentelle Mechanik, Stuttgart, 1986, pp. 33–40. K. Andresen: Das Phasenshiftverfahren zur Rasterbildauswertung. Colloquy of the Collaborative Research Centre (SFB 319), Goslar. K. Andresen: Verformungsmessung mit Rasterverfahren und der digitalen Bildverarbeitung. Oberseminar Mechanik, Universita¨t Hannover. K. Andresen: Konturermittlung und Verformungsmessung mit der digitalen Bildverarbeitung. Mechanik/Colloquy of the Collaborative Research Centre (SFB), TU Berlin. Subproject D1 J. Ruge, R. Linnemann: Festigkeits- und Verformungsverhalten von Bau-, Beton- und Spannsta¨hlen bei hohen Temperaturen. Arbeitsbericht 1984–1986. Subproject B4, Collaborative Research Centre (SFB 148), TU Braunschweig. R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Methoden. Dissertation TU Braunschweig, 1987. Subproject D2 G. Bahr: Kommunizierende Versuchstechnik und simultane theoretische und experimentelle Tragwerksuntersuchung. Dissertation TU Braunschweig, 1984. W. Maier, M. Klahold: Das Konzept der kommunizierenden Versuchstechnik fu¨r Stabtragwerke. Nachdruck eines Vortrages auf dem Fach-Kolloquium Experimentelle Mechanik 1986, Eigenverlag, Ingenieurwiss. Zentrum, Inst. fu¨r Modellstatik, Uni Stuttgart.

1988–1990 Subproject A1 F. Haeßner, K. Sztwiertnia: The Misorientation Distributions Associated with the Texture of Polycrystalline Aggregates of Cubic Crystals. In: J. S. Callend, G. Gottstein (Eds.): ICOTOM 8, The Metallurgical Society, 1988, pp. 163–168. F. Haeßner, J. Schmidt: Recovery and recrystallization of different grades of high purity aluminium determined with a low temperature calorimeter. Scripta Met. 22 (1988) 1917. U. Meyer: Einfluss verschiedener Verformungsparameter auf die Rekristallisationskinetik von Kupfer. Dissertation TU Braunschweig, 1989. 388

Publications Resulted from the Collaborative Research Centre J. Schmidt: Untersuchung der Erholungs- und Rekristallisationsvorga¨nge in tieftemperaturverformten Metallen mit Hilfe kalorischer Messungen. Dissertation TU Braunschweig, 1990. J. Schmidt, F. Haeßner: Stage III-Recovery of Cold Worked High-Purity Aluminium Determined with a Low-Temperature Calorimeter. Z. f. Phys. B-Condensed Matter 81 (1990) 215. F. Haeßner: The Study of Recrystallization by Calorimetric Methods. In: T. Chandra (Ed.): Recrystallization ’90, The Minerals, Metals and Materials Society, 1990, pp. 511. Subproject A2 D. Rode, G. Lange: Beitrag zum Ermu¨dungsverhalten mehrachsig beanspruchter Aluminiumlegierungen. Metall 42 (1988) 582. Subproject A5/B4 R. Schubert, K.-T. Rie: Verfestigung, Fließfla¨che und Versetzungsstruktur bei Low-Cycle-Fatigue. Mat.-wiss. und Werkstofftech. 19 (1988) 376–383. K.-T. Rie, R. Schubert, J. Olfe: Untersuchungen zur Oberfla¨chenausbildung, Mikrostruktur und Ermu¨dungsscha¨digung im LCF-Bereich. DFG-Kolloquium: Schadensfru¨herkennung und Schadensablauf bei metallischen Bauteilen, Darmstadt, Sept. 1989, Berichtsband DVM, Berlin, 1989, pp. 55–62. R. Schubert: Verformungsverhalten und Risswachstum bei Low-Cycle-Fatigue. Dissertation TU Braunschweig, 1989. J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtemperatur-LCF. Zeitschrift fu¨r Metallkunde 81(11) (1990) 783–789. Subproject A6/B1 E. Steck: Constitutive Laws for Strain-, Strainrate- and Temperature Sensitive Materials. Keynote, Proceedings 2. Intern. Conf. on Technology of Plasticity (ICTP), Stuttgart, Springer Verlag, Berlin, 1987. E. Steck: Development of Constitutive Equations for Metals at High Temperatures. Proceedings 2. Intern. Conf. on Technology of Plasticity (ICTP), Stuttgart, Springer Verlag, Berlin, 1987. E. Steck: On the Development of Material Laws for Metals. Festschrift Heinz Duddeck zu seinem 60. Geburtstag, Institut fu¨r Statik, TU Braunschweig, 1988. E. Steck: A Stochastic Model for the Interaction of Plasticity and Creep in Metals. In: F. Zeigler, G. I. Schueller (Eds.): Nonlinear Stochastic Dynamic Engineering Systems, IUTAM Symposium Igls, 1987, Springer Verlag, Berlin, 1988. E. Steck: Marcov-Chains as Models for the Inelastic Behaviour of Metals. In: D. R. Axelrad, W. Muschik (Eds.): Constitutive Laws and Microstructure, Institute of Advanced Study, Berlin, Springer Verlag, Berlin, 1988. E. Steck: A Stochastic Model for the Interaction of Plasticity and Creep in Metals. Nuclear Engineering and Design 114 (1989) 285–294. E. Steck: The Description of the High-Temperature Plasticity of Metals by Stochastic Processes. Res. Mechanica 25 (1990) 1–19. E. Steck, H. Schlums: Discrete Models on the Microscale for the Plastic Behaviour of Metals. Proceedings of Plasticity ’89, Second Intern. Symp. on Plasticity and its Current Applications, Tsu, Japan, 1989, pp. 581–584. Subproject A8 Ch. Schwink: Hardening mechanisms in metals with foreign atoms. Revue Phys. Appl. 23 (1988) 395–404. R. Neuhaus, P. Buchhagen, Ch. Schwink: Dislocation densities as determined by TEM in
100 and
111 CuMn crystals. Scripta Met. 23 (1989) 779–784. L. Diehl, F. Springer, Ch. Schwink: Studies of hardening mechanisms of symmetrically oriented single crystals of fcc solid solutions. In: P. O. Kettunen et al. (Eds.): Strength of Metals and Alloys 1, 1988, pp. 313–319. 389

Bibliography Subproject A9 A. Hampel, H. Neuha¨user: Investigation of slip line growth in fcc Cu alloys with high resolution in time. phys. stat. sol. (a) 104 (1987) 171–181. H. Neuha¨user: The dynamics of slip band formation in single crystals. Res. Mechanica 23 (1988) 113–135. H. Neuha¨user: On some problems in plastic instabilities and strain localization. Rev. Phys. Appl. 23 (1988) 571–572. A. Hampel, O. B. Arkan, H. Neuha¨user: Local shear rate in slip bands of CuZn and CuNi single crystals. Rev. Phys. Appl. 23 (1988) 695. A. Hampel, H. Neuha¨user: Recording of slip line development with high resolution. In: O. Y. Chiem, H.-D. Kunze, L. W. Meyer (Eds.): Proc. Int. Conf. on Impact Loading and Dynamic Behaviour of Materials, DGM Verlag, 1988, pp. 845–851. H. Neuha¨user: Slip propagation and fine structure. In: L. P. Kubin, O. Martin (Eds.): Proc. Coll. Int. CNRS on Nonlinear Phenomena in Materials Science, Trans. Tech. Publ., Aedermannsdorf, 1988, pp. 407–415. A. Hampel, M. Schu¨lke, H. Neuha¨user: Dynamic studies of slip line formation on single crystals of fcc solid solutions. In: P. O. Kettunen, T. K. Lepisto¨, M. E. Lehtonen (Eds.): Proc. 8. Int. Conf. on Strength of Metals and Alloys, Pergamon Press, Oxford, 1988, pp. 349–354. J. Olfe, H. Neuha¨user: Dislocation groups, multipoles, and friction stresses in -CuZn alloys. phys. stat. sol. (a) 109 (1988) 149–160. H. Neuha¨user: Plastic instabilities and the deformation of metals. In: D. Walgraef, N. M. Ghoniem (Eds.): Proc. NATO Adv. Study Inst. on Patterns, Defects and Materials Instabilities, Kluwer Acad. Publ., Dordrecht, 1990, pp. 241–276. H. Neuha¨user, J. Plessing, M. Schu¨lke: Portevin-LeChaˆtelier effect and observation of slip band growth in CuAl single crystals. J. Mech. Beh. Metals 2 (1990) 231–254. Subproject A10 D. Besdo: Finite-Element-Analyses of Strain-Space-Represented Elastic-Plastic Media Using Simplified Stiffness Matrices. Proc. of the Intern. Conf. on Applied Mechanics, Beijing, China, Pergamon Press, 1989, pp. 1403–1408. D. Besdo, E. Doege, H.-W. Lange, M. Seydel: Zur numerischen Simulation des Tiefziehens. Lecture at the 13. Umformtechnischen Kolloquium Hannover 14./15. Ma¨rz 1990, HFF-Bericht Nr. 11 (Ed.: E. Doege), HFF Hannoversches Forschungsinstitut fu¨r Fertigungsfragen E.V. D. Besdo, L. Ostrowski: On the Creep-Ratchetting of AlMgSiO.5 at Elevated Temperature – Experimental Investigations. Proc. of the Fourth IUTAM Symposium on Creep in Structures, Krakow, 10.–14. Sept. 1990. Subproject B2 M. Schwesig: Inelastisches Verhalten metallischer Werkstoffe bei ho¨heren Temperaturen – Numerik und Anwendung. Report Nr. 89-57 from the Institut fu¨r Statik der TU Braunschweig, 1989. M. Schwesig, H. Ahrens, H. Duddeck: Erfahrungen aus der Anwendung des inelastischen Stoffgesetzes nach Hart. Festschrift Richard Schardt, THD Schriftenreihe Wissenschaft und Technik S1, Darmstadt, 1990. M. Schwesig, H. Braasch, G. Kracht, H. Duddeck, H. Ahrens: Erfahrungen aus der Anwendung inelastischer Stoffgesetze bei ho¨heren Temperaturen. In: D. Besdo (Ed.): Numerische Methoden der Plastomechanik, Tagungsband; Hannover, 1989. D. Dinkler, M. Schwesig: Numerische Lo¨sung von Anfangswertproblemen in der Statik und Dynamik. Festschrift Heinz Duddeck, Braunschweig, 1988. H. Duddeck, B. Kro¨plin, D. Dinkler, J. Hillmann, W. Wagenhuber: Berechnung des nichtlinearen Tragverhaltens du¨nner Schalen im Vor- und Nachbeulbereich. Nichtlineare Berechnungen im Konstruktiven Ingenieurbau, Hannover, DFG-Gz.: Du 25/28-7, 1989. D. Dinkler: Stabilita¨t elastischer Tragwerke mit nichtlinearem Verformungsverhalten bei instationa¨ren Einwirkungen. Ingenieur-Archiv 60 (1989). 390

Publications Resulted from the Collaborative Research Centre D. Dinkler: Stabilita¨t du¨nner Fla¨chentragwerke bei zeitabha¨ngigen Einwirkungen. Report Nr. 8852 from the Institut fu¨r Statik der TU Braunschweig, 1988. H. Duddeck, D. Winselmann, F. T. Ko¨nig: Constitutive laws including kinematic hardening for clay with pore water pressure and for sand. Numerical Methods in Geomechanics, Innsbruck, 1988. L. Pisarsky, H. Ahrens, H. Duddeck: FEM-Analysis for time-depending cyclic pore water cohesive soil problems. Eurodyn 90, European Conference on Structural Dynamics, Bochum, DFGGz.: Du 25/34-1-2, 1990. R. Meyer, H. Ahrens: An elastoplastic model for concrete. Conference Proceedings of the Second World Congress on Computational Mechanics, Stuttgart, 1990. Subproject B5 H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Beru¨cksichtigung in Konstruktionsberechnungen. Dissertation TU Braunschweig, 1990. J. Scheer, H.-J. Scheibe, M. Reininghaus: Wirtschaftliche Bemessung von Schraubenanschlu¨ssen bei Ausnutzung des duktilen Verhaltens von Stahl. Report Nr. 6018, Institut fu¨r Stahlbau, TU Braunschweig, 1989. J. Scheer, H.-J. Scheibe, D. Kuck: Untersuchungen von Tra¨gerschwa¨chungen unter wiederholter Belastung bis in den plastischen Bereich. Report Nr. 6099, Institut fu¨r Stahlbau, TU Braunschweig, 1989. E. Beißner: Zum Tragverhalten sta¨hlerner Augensta¨be im elastisch-plastischen Zustand. Dissertation TU Braunschweig, 1989. H.-J. Scheibe: Zur Berechnung zyklisch beanspruchter Stahlkonstruktionen im plastischen Bereich. Lecture 8. Stahlbau-Seminar, Steinfurt, 1989. Subproject B6 R. Mahnken, E. Stein, D. Bischoff: A globally convergence criterion for first order approximation strategies in structural optimations. Int. J. Meths. Eng. 31 (1990). E. Stein, G. Zhang, R. Mahnken, J. A. Ko¨nig: Micromechanical modelling and computation of shake down with nonlinear kinematic hardening inducing examples for 2-D problems. Proc. of CSME Mechanic Engineering Forum, Toronto, 1990. E. Stein, G. Zhang, J. A. Ko¨nig: Micromechanic modelling of shake down with nonlinear kinematic hardening inducing examples for 2-D problems. In: Axelrad, Muschik (Eds.): Recent Developments in Micromechanics, Springer Verlag, 1990. Subproject C1 K. Andresen, R. Helsch: Automatische Rasterkoordinatenermittlung mit Hilfe digitaler Filter. Informatik Fachberichte 149, Mustererkennung, 1987, pp. 228. K. Andresen, R. Helsch: Calculation of Grating Coordinates Using a Correlation Filter Technique. Optik 80 (1988) 76–79. K. Andresen, B. Kamp, R. Ritter: Verformungsmessungen an Rissspitzen nach dem Objekt-RasterVerfahren. VDI-Berichte 679 (1988) 393–403. K. Andresen, B. Morche: Die Ermittlung von Rasterkoordinaten und deren Genauigkeit. Mustererkennung 198, pp. 277–283. 10. DAGM-Symposium Zu¨rich, Berlin, New York, Tokio, 1988. K. Andresen, H. Horstmann: Ermittlung der Verformungen und Spannungen in einer gelochten Gummimembran mit Hilfe von Rasterverfahren. Forsch. Ing.-Wes. 55 (1989) 33–36. K. Andresen, K. Hentrich: Vergleich von Frequenz- und Ortsfilterverfahren zur Moire´-Bildauswertung. Optik 83 (1989) 113–121. K. Andresen, P. Feng, W. Holst: Fringe Detection Using Mean and n-Rank Filters. Fringe 89, Berlin, Physical Research 10 (1989) 45–49. K. Andresen, R. Ritter: Digitale Bildverarbeitung in der Werkstoffpru¨fung und Qualita¨tskontrolle. Tagungsband: Bildverarbeitung: Forschen, Entwickeln, Anwenden, Techn. Akad. Esslingen, 1989.

391

Bibliography K. Andresen, R. Helsch: Calculation of Analytical Elements in Space Using a Contour Algorithm. ISPRS-Commission V. Symp. on Close Range Photogrammetry and Machine Vision, Zu¨rich. SPIE 1395 (1990) 863–869. K. Andresen: Evaluation of Moire´ Fringes Using Space Filtering. Proc. 9th Int. Conf. Exp. Mechanics, Copenhagen, 1990, pp. 1650–1659. Subproject C2 J. Hilbig, R. Ritter, W. Wilke: Hochtemperaturdehnungsmessung nach dem Rasterprinzip am Beispiel des LCF-Versuchs. Akademie der Wissenschaften der DDR/Institut fu¨r Mechanik, Report Nr. 24, 8, Chemnitz, 1989, pp. 255–258. R. Ritter, W. Wilke: Optische in-situ-Dehnungsfeldmessung unter Hochtemperatureinfluss mit dem Rasterverfahren am Beispiel des LCF-Versuchs. Vortrags- und Diskussionstagung „Werkstoffpru¨fung 1990: Aussagefa¨higkeit von Pru¨fungsergebnissen fu¨r das Verarbeitungs- und Bauteilverhalten“ 6./7. 12. 1990 Bad Nauheim, veranstaltet von DVM im Auftrag der Arbeitsgemeinschaft Werkstoffe. J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtemperatur LCF. Zeitschrift fu¨r Metallkunde 81(11) (1990) 783–789. R. Ritter, J. Strusch, W. Wilke: Formanalyse mit Hilfe des Reflexions-Rasterverfahrens und des photogrammetrischen Prinzips. Österreich. Ingenieur- und Architektenzeitung 135(7/8) (1990) 346–348. K. Galanulis, J. Hilbig, R. Ritter: Strain Measurement by the Diffraction Principle. Österreich. Ingenieur- und Architektenzeitung 134(7/8) (1989) 392–394. W. Cornelius, J. Hilbig, R. Ritter, W. Wilke, C. Forno: Zur Formanalyse mit Hilfe hochtemperaturbesta¨ndiger Raster. VDI-Bericht 731, 5, Du¨sseldorf, 1989, pp. 295–302. K. Galanulis, J. Hilbig, B. W. Lu¨hrig, R. Ritter: Strain Measurement by the Diffraction Principle for Curved Surfaces. Proceedings of the 9th Int. Conference on Experimental Mechanics, Technical University of Denmark, Lyngby/Da¨nemark, 20.–24. Aug. 1990. Subproject C3 P. Horst, H. Kossira: Zum Beulverhalten du¨nner Aluminiumplatten bei wechselnder Schubbelastung. In: Proceedings der Jahrestagung der Deutschen Gesellschaft fu¨r Luft- und Raumfahrt (DGLR), Jahrbuch 1988 I der DGLR, Bonn, 1988. K. Wolf: FIPPS – Ein Programm-Paket zur numerischen Analyse des linearen und nichtlinearen Tragverhaltens von Leichtbaustrukturen. In: H. Kossira (Ed.): 50 Jahre IFL, ZLR-Bericht 8901, ISBN 3-9802073-0-7, Braunschweig, 1989. K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus kohlenstoffaserversta¨rktem Kunststoff. Dissertation, Inst. f. Flugzeugbau und Leichtbau, Technische Universita¨t Braunschweig, 1989. P. Horst: Plastisches Beulen du¨nner Aluminiumplatten. In: H. Kossira (Ed.): 50 Jahre IFL, ZLRForschungsbericht 89-01, ISBN 3-9802073-0-7, Braunschweig, 1989. P. Horst, H. Kossira: Theoretical and experimental investigation of thin-walled aluminium-panels under cyclic shearload. In: Proceedings of the International Conference on Spacecraft Structures and Mechanical Testing of ESA, CNES and DFVLR, ESA-Special Report SP-289, Noordwijk, 1989. P. Horst, H. Kossira: Cyclic Shear Buckling of Thin-Walled Aluminium Panels. Proceedings of the 17th Congress of the International Council of the Aeronautical Sciences (ICAS) (Paper Nr. 904.4.1), Stockholm, 1990.

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Publications Resulted from the Collaborative Research Centre Subproject D1 (C4) J. Ruge, C. X. Hou, U. Pa¨tzold: Bestimmung von Gefu¨geinhomogenita¨ten in der Wa¨rmeeinflusszone von Schweißverbindungen. Schweißen und Schneiden 41(3) (1989) 134–137. R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Methoden. Fortschr.-Ber. VPI-Reihe 18, Nr. 55, VPI-Verlag, Du¨sseldorf, 1988. J. Ruge, S. Zhang, U. Pa¨tzold: Spannungsberechnungen in Schweißnahtmodellen mit Hilfe neuer Werkstoffgesetze. Mat.-wiss. und Werkstofftech., Heft 9, 1990.

1991–1993 Subproject A1 M. Zehetbauer, J. Schmidt, F. Haeßner: Calorimetric study of defect annihilation in low temperature deformed pure Zn. Scripta Metallurgica et Materialia 25 (1991) 559. F. Haeßner, K. Sztwiertnia, P. J. Wilbrandt: Quantitative analysis of the misorientation distribution after the recrystallization of tensile deformed copper single crystals. Textures and Microstructures 13 (1991) 213. J. Schmidt, F. Haeßner: Recovery and recrystallization of high purity lead determined within a low temperature calorimeter. Scripta Metallurgica et Materialia 25 (1991) 969. E. Woldt, F. Haeßner: Aspekte des Entfestigungsverhaltens von Kupfer. Z. f. Metallkunde 82 (1991) 329. F. Haeßner: Calorimetric investigation of recovery and recrystallization phenomena in metals. In: R. D. Shull, A. Joshi (Eds.): Thermal Analysis in Metallurgy, The Minerals, Metals and Materials Society, 1992, pp. 233–257. F. Haeßner, K. Sztwiertnia: Some microstructural aspects of the initial stage of recrystallization of highly rolled pure copper. Scripta Metallurgica et Materialia 27 (1992) 2933. P. Kru¨ger, E. Woldt: The use of an activation energy distribution for the analysis of the recrystallization kinetics of copper. Acta Metallurgica et Materialia 40 (1992) 2933. E. Woldt: The relationship between isothermal and non-isothermal description of Johnson-MehlAvrami-Kolmogorov kinetics. J. Phys. Chem. Solids 53 (1992) 521. F. Haeßner, J. Schmidt: Investigation of the recrystallization of low temperature deformed highly pure types of aluminium. Acta Metallurgica et Materialia 41 (1993) 1739. K. Sztwiertnia, F. Haeßner: Orientation characteristics of the microstructure of highly pure copper and phosphorus copper. Textures and Microstructures 20 (1993) 87. H. W. Hesselbarth, L. Kaps, F. Haeßner: Two dimensional simulation of the recrystallization kinetics in the case of inhomogeneously stored energy. Materials Science Forum 113–115 (1993) 317. Subproject A2 G. Lange, W. Gieseke: Vera¨nderung des Werkstoffzustandes von Aluminium-Legierungen durch mehrachsige plastische Wechselbeanspruchungen. Festschrift zur Vollendung des 65. Lebensjahres von Prof. Dr. rer. nat. Dr.-Ing. E. h. Eckhard Macherauch, DGM-Verlag Sept. 1991, pp. 49. M. Heiser, G. Lange: Scherbruch in Aluminium-Legierungen infolge lokaler plastischer Instabilita¨t. Z. f. Metallkunde 83 (1992) 115. G. Lange: Schadensfa¨lle durch Schwingbru¨che. Ingenieur-Werkstoffe 4(10) (1992) 62. G. Lange: Schwingbru¨che durch Steifigkeitsspru¨nge. 15. Vortragsveranstaltung des Arbeitskreises „Rastermikroskopie in der Materialpru¨fung“, Deutscher Verband fu¨r Materialforschung und -pru¨fung, Berlin, 1992, pp. 285. G. Lange: Schwingbru¨che durch konstruktiv bedingte Spannungsspitzen (Teil 1: Sprunghafte Querschnittsa¨nderungen und Steifigkeitsspru¨nge an Schweißverbindungen). Ingenieur-Werkstoffe 5(3) (1993) 58. 393

Bibliography G. Lange: Schwingbru¨che durch konstruktiv bedingte Spannungsspitzen (Teil 2: Verbindungs- und Befestigungselemente). Ingenieur-Werkstoffe 5(4) (1993) 74. G. Lange: Fractures in Aircraft Components. In: H. P. Rossmanith, K. J. Miller (Eds.): MixedMode Fatigue and Fracture, Mechanical Engineering Publications Limited, London, 1993, pp. 23. Subproject A5/B4 K.-T. Rie, R. Schubert, H. Wittke: Cyclic Deformation Behaviour and Crack Growth in Low-Cycle Fatigue Range. Mechanical Behaviour of Materials – VI, The Sixth International Conference, Kyoto, Japan, Preprints of Additional Papers and Extended Abstracts, 1991, pp. 243– 244. K.-T. Rie, H. Wittke, R. Schubert: The DJ-Integral and the Relation Between Deformation Behaviour and Microstructure in the LCF-Range. In: K.-T. Rie (Ed.): Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials – 3, Elsevier Applied Science, London/New York, 1992, pp. 514–520. K.-T. Rie, J. Olfe: Lokale Werkstoffbeanspruchung bei Hochtemperatur Low-Cycle Fatigue. IX. Symposium Verformung und Bruch, Teil 1, August 1991, pp. 150–154. K.-T. Rie, J. Olfe: Crack Growth and Crack Tip Deformation under Creep-Fatigue Conditions. In: M. Jono, T. Inoue (Eds.): Mechanical Behaviour of Materials – VI, Volume 4, Pergamon Press, 1991, pp. 367–372. K.-T. Rie, J. Olfe: A Physically Based Model for Predicting LCF Life under Creep Fatigue Interaction. In: K.-T. Rie (Ed.): Proc. 3rd Int. Conf. on Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials, Elsevier Applied Science, London/New York, 1992, pp. 222–228. K.-T. Rie, J. Olfe: Dehnungsfelder vor Riss-Spitzen bei Kriechermu¨dung. Z. Metallkde. 84 (1993). Subproject A6/B1 E. Steck: Stochastic Models for the Plasticity of Metals. In: O. Bru¨ller, V. Mannl, J. Najar (Eds.): Advances in Continuum Mechanics, Springer Verlag, 1991, pp. 77–87. K. Rohwer, G. Malki, E. Steck: Influence of Bending-Twisting Coupling on the Buckling Loads of Symmetrically Layered Curved Panels. Proceedings Intern. Colloquium on Buckling of Shell Structures, on Land, in the Sea and in the Air, Lyon, France, Sept. 1991. E. Steck, F. Kublik: Application of Constitutive Models for the Prediction of Multiaxial Inelastic Behaviour. SMIRT 11, Transactions Vol. 1, Tokyo, Japan, 1991, pp. 557–567. E. Steck, H. W. Hesselbarth: Simulation of Disclocation Pattern Formation by Cellular Automata. In: J.-P. Boehler, A. S. Kahn (Eds.): Anisotropy and Localisation of Plastic Deformation, Proceedings of Plasticity ’91, Elsevier, London, 1991, pp. 175–178. E. Steck: Stochastic Modelling of Cyclic Deformation Process in Metals (Reprint). In: S. I. Andersen et al. (Eds.): Proceedings of the 13th Riso International Symposium on Materials Science: Modelling of Plastic Deformation and its Engineering Applications, Riso National Laboratory, Roskilde, Denmark, 1992. F. Kublik, E. Steck: Comparison of Two Constitutive Models with One- and Multiaxial Experiments. In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications, IUTAM Symposium Hannover, Germany 1991, Springer-Verlag, Berlin, Heidelberg, 1992. H. Schlums, E. Steck: Description of Cyclic Deformation Process with Stochastic Models for Inelastic Behaviour of Metals. Int. J. Plasticity 8 (1992) 147–169. Subproject A8 F. Springer, Ch. Schwink: Quantitative investigations on dynamic strain ageing in polycrystalline CuMn alloys. Scripta Metall. Mater. 25 (1991) 2739–2745. R. Neuhaus, Ch. Schwink: The flow stress of [100]- and [111]-oriented Cu-Mn single crystals: a transmission electron microscopy study. Phil. Mag. A65 (1992) 1463–1484. H. Heinrich, R. Neuhaus, Ch. Schwink: Dislocation structure and densities in tensile deformed CuMn crystals oriented for single glide. phys. stat. sol. (a) 131 (1992) 299–309. 394

Publications Resulted from the Collaborative Research Centre A. Kalk, Ch. Schwink: On sequences of alternate stable and unstable regions along tensile deformation curves. phys. stat. sol. (b) 172 (1992) 133–145. Ch. Schwink: Flow stress dependence on cell geometry in single crystals. Scripta Metal. Mater. 27 (1992) 963–969 (Viewpoint Set No. 20). H. Neuha¨user, Ch. Schwink: Solid solution strengthening. In: R. W. Cahn, P. Haasen, E. J. Kramer (Eds.): Materials Science and Technology, Vol. 6 (Vol.-Ed.: H. Mughrabi), VCH, Weinheim, 1993, pp. 191–251. A. Kalk, Ch. Schwink, F. Springer: On sequences of stable and unstable regions of flow along stress-strain curves of solid solutions – experiments on Cu-Mn polycrystals. Mater. Sci. Eng. A 164 (1993) 230–234. Th. Wutzke, Ch. Schwink: Strain rate sensitivities and dynamic strain ageing in CuMn crystals oriented for single glide. phys. stat. sol. (a) 137 (1993) 337–350. Subproject A9 H. Neuha¨user: Collective Dislocation Behaviour and Plastic Instabilities – Micro and Macro Aspects. In: J.-P. Boehler, A. S. Kahn (Eds.): Anisotropy and Localization of Plastic Deformation, Elsevier Appl. Sci., London, 1991, pp. 77–80. C. Engelke, P. Kru¨ger, H. Neuha¨user: Stress Relaxation in Cu-Al Single Crystals at High Temperatures. Scripta Metal. Mater. 27 (1992) 371–376. J. Vergnol, F. Tranchant, A. Hampel, H. Neuha¨user: Mesoscopic Observations Related to Twinning Instabilities in -CuAl Crystals. In: O. Martin, L. Kubin (Eds.): Non-Linear Phenomena in Materials Science 11, Trans. Tech. Publ., Zu¨rich, 1992, pp. 303–316. H. Neuha¨user, Ch. Schwink: Solid Solution Hardening. In: R. W. Cahn, P. Hansen, E. J. Kramer (Eds.): Materials Science and Technology – A Comprehensive Treatment, Vol. 6: Plastic Deformation and Fracture of Materials (Vol.-Ed.: H. Mughrabi), VCH Verlagsgemeinschaft, Weinheim, 1993, pp. 191–250. A. Hampel, T. Kammler, H. Neuha¨user: Structure and Kinetics of Lu¨ders Band Slip in Cu-5 to 15at%Al Single Crystals. phys. stat. sol. (a) 135 (1993) 405–416. H. Neuha¨user: Collective Micro Shear Processes and Plastic Instabilities in Crystalline and Amorphous Structures. Int. J. Plasticity 9 (1993) 421–435. C. Engelke, J. Plessing, H. Neuha¨user: Plastic Deformation of Single Glide Oriented Cu-2 to 15at%Al Crystals at Elevated Temperatures. Mater. Sci. Eng. A 164 (1993) 235–239. H. Neuha¨user: Problems in Solid Solution Hardening of Alloys. Physica Scripta T 49 (1993) 412– 419. H. Neuha¨user, A. Hampel: Observation of Lu¨ders Bands in Single Crystals. Scripta Metal. Mater. 29 (1993) 1151–1157. Subproject A10 D. Besdo: Eine Erweiterung der Taylor-Theorie zur Erfassung der kinematischen Verfestigung. ZAMM-Z. Angew. Math. Mech. 71 (1991) T264–T265. D. Besdo, M. Mu¨ller: The Influence of Texture Development on the Plastic Behaviour of Polycrystals. In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications, IUTAM Symposium Hannover, Germany 1991, Springer Verlag, Berlin, Heidelberg, 1992. M. Mu¨ller, D. Besdo: Simulation globaler Anisotropie mit Hilfe eines Vielkristallmodells. ZAMMZ. Angew. Math. Mech. 73 (1993) T658. Subproject B2 G. Kracht: Erschließung viskoplastischer Stoffmodelle fu¨r thermomechanische Strukturanalyse. Report Nr. 93-69 from the Institut fu¨r Statik der TU Braunschweig, 1993. E.-R. Tirpitz: Elasto-plastische Erweiterung von viskoplastischen Stoffmodellen fu¨r Metalle. Report Nr. 92-70 from the Institut fu¨r Statik der TU Braunschweig, 1992. H. Braasch: Ein Konzept fu¨r Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle. Report Nr. 92-71 from the Institut fu¨r Statik der TU Braunschweig, 1992.

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Bibliography L. Pisarsky: Zur Berechnung nichtmonoton beanspruchter wassergesa¨ttigter Tonbo¨den. Report Nr. 91-65 from the Institut fu¨r Statik der TU Braunschweig, 1991. Z. Huang: Beanspruchungen des Tunnelausbaus bei zeitabha¨ngigem Materialverhalten von Beton und Gebirge. Report Nr. 91-68 from the Institut fu¨r Statik der TU Braunschweig, 1991. B. Hu: Berechnung des geometrisch und physikalisch nichtlinearen Verhaltens von Fla¨chentragwerken aus Stahl unter hohen Temperaturen. Report Nr. 93-72 from the Institut fu¨r Statik der TU Braunschweig, 1993. H. Braasch, H. Duddeck, H. Ahrens: A New Approach to Improve and Derive Materials Models. J. Eng. Mat. Tech. (ASME) 117 (1995) 14–19. E.-R. Tirpitz, M. Schwesig: A Unified Model Approach Combining Rate-Dependent and Rate-Independent Plasticity. Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials-3, Berlin, 1992, pp. 411–417. M. Schwesig, U. Kowalsky: Zur Formulierung und Anwendung eines elasto-plastischen Modells fu¨r reibungsbehafteten Kontakt. Report Nr. 93-75 from the Institut fu¨r Statik der TU Braunschweig, 1993, pp. 75–94. H. Braasch: Concept to Improve the Approximation of Material Functions in Unified Models. Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials – 3, Berlin, 1992, pp. 405–410. U. Kowalski: Verification of a Microstructure-Related Constitutive Model by Optimized Identification of Material Parameters. Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials – 3, Berlin, 1992, pp. 405–410. T. Streilein: Anwendung eines Überspannungsmodells zur Beschreibung ein- und mehraxialer zyklischer Versuche. Report Nr. 93-75 from the Institut fu¨r Statik der TU Braunschweig, 1993, pp. 29–46. H. Braasch: Erfassung streuenden Materialverhaltens in Werkstoffmodellen. Report Nr. 93-75 from the Institut fu¨r Statik der TU Braunschweig, 1993, pp. 1–14. Subproject B5/B7 J. Scheer, H.-J. Scheibe: Einachsige Zug-Druck-Versuche an Baustahl St52-3. Institut fu¨r Stahlbau, TU Braunschweig, unpublished internal report. J. Scheer, H.-J. Scheibe: Untersuchungen von zyklisch beanspruchten Lochscheiben aus Baustahl St52-3. Institut fu¨r Stahlbau, TU Braunschweig, unpublished internal report. S. Dannemeyer: Verhalten von thermomechanisch behandelten Bausta¨hlen unter zyklischer Beanspruchung im elastisch-plastischen Bereich. Experimentelle Studienarbeit, Institut fu¨r Stahlbau, TU Braunschweig, 1992. M. M. El-Ghandour: Low-Cycle Fatigue Damage Accumulation of Structural Steel St52. Dissertation TU Braunschweig, 1992. L. Reifenstein: Verhalten modifizierter CT-Proben aus Baustahl St52-3 unter zyklischer Belastung im elastisch-plastischen Bereich. Experimentelle Studienarbeit, Institut fu¨r Stahlbau, TU Braunschweig. U. Peil: Dynamisches Verhalten abgespannter Maste. VDI Bericht Nr. 924, 1992. Subproject B6/B8 E. Stein, G. Zhang, J. A. Ko¨nig: Shakedown with non-linear hardening including structural computation using finite element methods. Int. J. Plasticity 8 (1992) 1–31. E. Stein, G. Zhang: Theoretical and numerical shakedown analysis for kinematic hardening materials. In: Proc. 3rd Conf. on Computational Plasticity, Barcelona, 1992, pp. 1–25. E. Stein, G. Zhang, Y. Huang: Modelling and computation of shakedown problems for non-linear hardening materials. Computer Methods in Mechanics and Engineering 103(1/2) (1993) 247– 272. E. Stein, G. Zhang, R. Mahnken: Shake-down analysis for perfectly plastic and kinematic hardening material. In: E. Stein (Ed.): Progress in computational analysis of inelastic structures, CISM courses and lecture No. 321, Springer Verlag, Wien, New York, 1993, pp. 175–244. E. Stein, Y. Huang: An analytical method to solve shakedown problems for materials with linear kinematic hardening materials. Int. J. of Solids and Structures 18 (1994) 2433–2444. 396

Publications Resulted from the Collaborative Research Centre G. Zhang: Einspielen und dessen numerische Behandlung von Fla¨chentragwerken aus ideal plastischem bzw. kinematisch verfestigendem Material. Ph. D. Thesis, Institut fu¨r Baumechanik und Numerische Mechanik, Universita¨t Hannover, 1992. R. Mahnken: Duale Verfahren fu¨r nichtlineare Optimierungsprobleme in der Strukturmechanik. Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universita¨t Hannover, F92/3, 1992. R. Mahnken, E. Stein: Parameter-Identification for Visco-Plastic Models via Finite-Element-Methods and Gradient Methods. IUTAM-Symposium on Computational Mechanics of Materials, Brown University, 1993. E. Stein, R. Mahnken: On a solution strategy for parameter identification of visco-plastic models in the context of finite elements methods. Proc. of Plasticity: Fourth Int. Symposium on Plasticity and its Current Applications, 1993. Subproject C1 K. Andresen, B. Hu¨bner: Calculation of Strain from an Object Grating on a Reseau Film by a Correlation Method. Exp. Mechanics 32 (1992) 96–101. Q. Yu, K. Andresen, D. Zhang: Digital pure shear-strain moire´ patterns. Applied Optics 31 (1992) 1813–1817. K. Andresen, B. Kamp, R. Ritter: 3D-Contour of Crack Tips Using a Grating Method. Second International Conference on Photomechanics and Speckle Metrology, San Diego 1991, SPIE Proceedings 1554 A (1991) 93–100. K. Andresen, B. Kamp, R. Ritter: Three-dimensional surface deformation measurement by a grating method applied to crack tips. Optical Engineering 31 (1992) 1499–1504. K. Andresen: 3D-Vermessungen im Nahbereich mit Abbildungsfunktionen. Mustererkennung 92, 14. DAGM Symposion, Dresden, 1992, pp. 304–309. K. Andresen, Q. Yu: Robust phase unwrapping by spin filtering combined with a phase direction map. Optik 94 (1993) 145–149. K. Andresen, Q. Yu: Robust Phase Unwrapping by Spin Filtering Using a Phase Direction Map. Fringe 93-Bremen. K. Andresen: Ermittlung von Raumelementen aus Kanten im Bild. Zeitschrift fu¨r Photogrammetrie und Fernerkundung 59 (1991) 212–220. K. Andresen, R. Ritter, E. Steck: Theoretical and experimental investigations of crack extension by FEM- and grating methods. Defect assessment in components. Fundamentals and application. ESIS/EGF9, Mechanical Engineering Publication, London, 1991, pp. 345–361. Subproject C2 R. Ritter, W. Wilke: Gliederung der Moire´verfahren. Österreich. Ingenieur- und Architekten-Zeitung 136 (1991) 218–222. R. Ritter, W. Wilke: Slope and Contour Measurement by the Reflection Grating Method and the Photogrammetric Principle. Optics and Lasers in Engineering 15 (1991) 103–113. K. Galanulis, J. O. Hilbig, R. Ritter: Zur 3D-Verformungsmessung mit einem Elektronik SpecklePattern Interferometer (ESPI). VDI-Berichte Nr. 882, 1991, pp. 233–242. J. O. Hilbig, R. Ritter: Speckle measurement for 3D surface movement. Proceedings of the Second International Conference on Photomechanics and Speckle Metrology, San Diego 1991, SPIE Proceedings 1554 A (1991) 588–592. J. O. Hilbig, K. Galanulis, R. Ritter: Zur 3D-Verformungsmessung mit einem Elektronischen Speckle-Interferometer. DVM-Tagungsband „Werkstoffpru¨fung 1991“, Bad Nauheim, 1991, pp. 103–110. D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, D. Winter, H. Wohlfahrt: Zur Anwendung der Speckle-Meßtechnik bei der Verformungsmessung in der Verbindungszone von KaltpressSchweißverbindungen verschiedener Werkstoffe. DVM-Tagungsband „Werkstoffpru¨fung 1992“, Bad Nauheim, 1992, pp. 261–271.

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Bibliography D. Brinkmann, K. Galanulis, M. Kassner, Ritter, D. Winter, H. Wohlfahrt: Deformation analysis in the joining zone of cold pressure butt welding of different materials. International Symposium on Mis-Matching of Welds, GKSS-Research Centre, Geesthacht, 1993. D. Bergmann, B.-W. Lu¨hrig, R. Ritter, D. Winter: Evaluation of ESPI-phase-images with regional discontinuities. SPIE Proceedings 2003, 1993. K. Galanulis, R. Ritter: Speckle interferometry in material testing and dimensioning of structures. SPIE Proceedings 2004, 1993. R. Ritter, H. Sadewasser, D. Winter: Evaluation of wrapped phase images with regional discontinuities. International Workshop “Fringe”, Akademie Verlag, Berlin, 1993. Subproject C3 H. Kossira, P. Horst: Cyclic Shear Loading of Aluminium Panels With Regard to Buckling and Plasticity. Thin-walled Structures 11 (1991) 65–84. P. Horst: Zum Beulverhalten du¨nner, bis in den plastischen Bereich zyklisch durch Schub belasteter Aluminiumplatten. Dissertation, ZLR Forschungsbericht 91-01, ISBN 3-9802073-5-8, Inst. f. Flugzeugbau und Leichtbau, Technische Universita¨t Braunschweig, 1991. P. Horst, H. Kossira, G. Arnst: On the Performance of Different Elastic-Plastic Material Models Applied to Cyclic Shear Buckling. Proc. of the Int. ECCS-Colloquium: On the Buckling of Shell Structures on Land, in the Sea and in the Air, Lyon, France, 1991. H. Kossira, M. Haupt: Buckling of Laminated Plates and Cylindrical Shells Subjected to Combined Thermal and Mechanical Loads. Proc. of the Int. ECCS-Colloqium: On the Buckling of Shell Structures on Land, in the Sea and in the Air, Lyon, France, 1991. M. Haupt, H. Kossira, M. Kracht, J. Pleitner: A Very Efficient Tool for the Structural Analysis of Hypersonic Vehicles under High Temperature Aspects. Proc. of the 18th ICAS-Congress, Peking, China, 1992. K. Wolf, H. Kossira: An efficient test method for the experimental investigation of the post buckling behaviour of curved composite shear panels. Proceedings of the European Conferences on Composite Materials (ECCM), Amsterdam, 1992. M. Haupt, H. Kossira: Integrated Thermal and Mechanical Structural Analysis of Hypersonic Vehicles by Using Adaptive Finite Element Methods. Proc. of the Third Aerospace Symposium 1991, Braunschweig. In: Orbital Transport – Technical, Metereological and Chemical Aspects, Springer, 1993, pp. 165–178. Subproject C4 J. Ruge, U. Pa¨tzold: Einsatz einer vollautomatischen Ha¨rtepru¨fstation zur Pru¨fung von Schweißverbindungen und Optimierung von Schweißverfahren. Tagungsband der DVM-Tagung Werkstoffpru¨fung, 1991. U. Pa¨tzold: Verformungsanalyse von Schweißverbindungen. Dissertation, Institut fu¨r Schweißtechnik, Technische Universita¨t Braunschweig, 1992. D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, H. Wohlfahrt: Zur Anwendung der SpeckleMeßtechnik bei der Verformungsmessung in der Verbindungszone von Kaltpress-Schweißverbindungen verschiedenartiger Werkstoffe. Tagungsband der DVM-Tagung Werkstoffpru¨fung, 1992, pp. 261–271. D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, D. Winter, H. Wohlfahrt: Deformation Analysis in the Joining Zone of Cold Pressure Butt Welds of Different Materials. Proceedings of the Conference on Mis-Matching of Welds, MEP, London, 1993.

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